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Robert E. Gross CoUeftion
A Memorial to the Founder of the
Business Administration Library Los Angeles
0
ly
^%^^^j^ ^-X"*^.* »r-*- '^
:*■*•
V
OBSERVATIONS
O N
REVERSIONARY PAYMENTS;
O N
Schemes for providing Annuities for Widows, and for Perfons in Old Age ;
O N
The Method of Calculating the Values of Assurances on Lives ;
AND ON
The national DEBT.
To which are added,
FOUR ESSAYS
On different Subjects in the Do6trine of Life Annuities and Political Arithmetick.
ALSO,
AN APPENDIX,
Containing a complete Set of Tables; particularly. Four New Tables, fnewing the Probabilities of Life in London, Norwich, and Northampton; and the Values of joint Lives.
By RICHARD PRICE, D.D. F. R. S.
L 0 N D O N:
Printed for T. C A D E l l, in the Strand,
M.DCC.LXXI.
cr
ill
CONTENTS.
/.
NTRODUCTION. - - Page vli
CHAP. I..
^lejiions relating to Schemes for granting Re^
'verJiGiiary Annuities y and the Values of Af-
furances on Lives. - - - p. i
CHAP. II.
Sect. I. Of the London Annuity, and Laudable Societies for the Beneft of Widows.
p. 64
Sect. II. Of the Ajj'ociation among the Lon- don Clergy and the Minifters in Scotland, for providing Annuities for their Widows,
p. 84
Sect. III. Of the beji Schemes for providing Annuities for Widows, - - P* 97
Sect. IV. Of Schemes for providing Life- Annuities^ which are not to commence till particular Ages ; andy particularly y cf the Societies lately ejlablijhed in London for the Scfiejit of Old Age. - - - p. 108 A 2 Sect,
iv CONTENTS.
Sect. V. Of the Amicable Corporation yi?r
a perpetual A ffurance -Office .• and the Society
for Equitable AfTurances on Lives and Sur-
vivorffiips. - - - - p. 122
CHAP. IIL
Of Puhlic Credit, and the National Debt,
ESSAY I.
Ohfervations on the Expeciations of Lives ; the. Increafe of Mankind ; the Nimiber of Inha- bit ajits in London j and the Influence of great fl'oivns on Health and Population. In a Let- ter to Benjarnin Franklin, Efq-y L L.T>, and F. R' S. p. 167. To which is added, a Pojicripty containing Obfervations on Edin- burgh, Paris, and Berlin. - p. 209
ESSAY IL
On Mr, De Moivre'j Rides for calculating tke Values of]omx. Lives. - - p. 222
ESSAY IIL
On the Method of calculating the Values of P^cverfions dependifig on Survroorfloips. p. 22 8
ESSAY IV.
On the proper Method of conftrudling Tables for determining the Rate of human Mortality^
the
CONTENTS. V
the Number of Inhabitants y ^and the Values of Lives in any Town or DiJiriB,from Bills of Mortality in winch are given the Numbers dying annually at all Ages, - - p. 235
APPENDIX.
Containing Algebraical Bemonjlrations -, Tables i and Rules for co?nputi?jg the Increafe of Money bearing Compound Liter eft, ^' '2.JJ
A3 IN-
Puhlijljcd by the fame Author ,
I. A Review of the principal Q,ueftions and £\. Difficulties in Morals. Particularly, thole relating to the Original of our Ideas of Vir- tue, its Nature, Foundation, Reference to the Deity, Obligation, Subject-matter, and Sanctions, The Second Edition corrected.
11. Four Dissertations. 1. On Provi- dence. -II. On Ph AVER.- III. On the Rea-
fons for expefting that virtuous Men fliall meet
after Death in a State of Happincfs. IV. On
the Importance of Chrifcianity, the Nature of Hi- {lorical Evidence, and Miracles. The Second Edition, with Additions.
Printed for T. Cad ell, in the Strand..
INTRODUCTION
EFORE the Reader enters upon this Work, it will not be improper to give him the following information cbricerniiig it»
A few years ago, many gentlemen..; ot the firfl eminence in the law, formed themfelvcs into a Society, for providing annuities for the tvidows of all fuch perfons in judicial ofHces, barrifters, civilians, and folHcitors, as fhould chufe to become members. A plan was agreed tipon and printed ; but, fome doubts happen- ing to arife with refped: to it, the diredors refolved to afl<. the opinion arid advice of three gentlemen, well known for their fkill in cal- culation. This occafioned a further reference ■^o me 5 and the iiHie was, that the plan be- ing found to be infufficient, the whole defign: Was laid alide.
About the fame time, feverat othef focie-
ties were formed with the fame views ; but
^ll on plans alike improper and infufncient.
A 4 Findin-g
vili INTRODUCTION.
Finding, therefore, that the public wanted information on this fubjecfl, I was led to un- dertake this work ; imagining, that it might be foon finiilied, and that all I could fay might be brought into a very narrow com- pafs. But in this I have been much miflaken. A defign, which I at iirfl thought v/ould give little trouble, has carried me far into a very wide field of enquiry ; and engaged me in many calculations that have taken up much time and labour, I (hall, however, be fuffi- ciently rewarded for my b.bour, flaould it prove the means of preventing any part of that diftrefs, which is likely to be hereafter produced by the focieties now fubfifting for
the benefit of widows. 1 have proved the
inadequatenefs of their plans, by undeniable i?.cls and mathematical demonftration {a). — 1 have, further, given an account of fome of the bed plans, that are confident with a fuf- ficient probability of permanency and fuc- ccia. Should, therelore, any of tbeie fo-
(a) In the note p. 68, I take notice of the five guineas J7r/e required, by ioir.c ot thcle (bcieties at admilfion ; and mention lome reafons for not making any particulai al- lov^aiice for if. Uur I have fmce learot, that it is indeed no more, than the iirft of the yearly payments, which I have always fuppofed to bef;in immediately. No al- lowance, therefore, was neceflary to be made for it.
4 cicties
INTRODUCTION. ix
cieties determine to reform themTelves; or fhould any inlHtutions of the fame kind be hereafter eftablifhed, they will here find di- rcdion and affiflance ^/;).
la
{o] I have lately learnt, that Mr. Caciell, the p':b;i-r.-r of this work, and alio Mr. Beckety Bookfeller in the Strand, are commiflioned to deliver in London^ printed accounts of the fcheme of a fociety, efiabliihed five years ago at Amjierdnm, for granting annuities on furvi-
vorfliip. 1 cannot faiisfy my own mind vvithout intro^
ducing here, though an improper place, the folloviring remarks on this fcheme.
From the folution of Qiieftions I and IV. in the Firft Chapter of rhe following Work, ic may be gathered, that, (reckoning intcreft at ■^l- per cent, and the probiibilitiesof life as they are in I'ables III. IV. and V. in the y^- pendix) the value of ?n annuity of i/. for life, to be en- joyed by a perfon aged 20, provided he furvives another perfon aged 60, is b'/. lbs. bd. in one prcfent payment; and 185. 6d. in annual payments, during the two joint lives : the firft payment to be made immediately. A Jingle payment^ therefore, of j^O Jlorins^ entitles to an annuity ot i^Jlorlm ; and an atinual payment of no florins, to an annuity of i ig florins ; and both together, to an annuity of i7^fiorins. if the annual payments are to be made, not during xhz joint lives, but during the whole continuance of the o\dt[k flugle life, they will, together with the fingle payment, entitle to an annuity of \^^ florins. Butthis fo- ciety promifes, for thefe payments, an annuity of lOO florins, if the oldefi: life fails in the fi.ft year after admif- fion j 100 fliOrins, if it fails in the 2d year ; "jpo ftorins, if it fails in the third ; ^00 florins, if it fails in the 4th ; and ^00 flori'is, if it fails in ihc flfth year, or at any time af- terwards. It is, therefore, evident that the fcheme of this fociety is, in this inflance, grofsly defedfive. There are other inltances in which it is even more defedlive j and the ivhek of it, like the fchemcs of moft of the Lon-
din
X INTRODUCTION.
In Queflion VI. Chap. I. a general me- thod is defcribed of finding the values, in Jingle and annual payments, of all life-annui- ties which are to begin after a given term of years -, and, in the 4th Section of the 2d Chapter, the plans of the focieties for grant- ing fuch annuities are particularly confidered,
and proved to be extremely deficient.- ^'
Indeed, the general difpofition which has lately fhewn itfelf to encourage thefe focieties, is a matter of the moft ferious concern j and ought, I think, to be taken under the no- tice of the Legifi.ature. The leadiiig perfons among the prefent members, will be tli^jirji
don focieties, appears to have been contrived by perfons who had no principles to go upon. And yet it has been much encouraged. Many have entered themfelves into it from different parts of Europe ; and the printed plan acquaints us, that it is now in poffeiTion of an annual in- come of 100.000 florins. What difappointment then muft
it in time producer it is provided by its rules, that the
terms of admiffion Ihall become lefs andlefs advantageous, the longer it has fubfifled ; juft as if the value of the an- nuities it promifes depended, not on the probabilities of life, and the improvement to be made of money, but on
the ae;e of the fociety. 1 have taken notice of a fimilar
abfuidity in the rules of our own focieties. But it is eafy to fee v/hat is meant by it.
Mr, Cadell can procure from his correfpondents in Holland^ any information for thofe who may want to know more of this fociety. But indeed I fiiould be ferry to find it much enquired after in LoNii'ON.
annul-
INTRODUCTION. xi
annuitants ; and they^ are fure of being gainers : and the more infufficient the fcheme is, on which a fociety is formed, the greater will be the gains of the firft annuitants. The fame principle, therefore, that has pro- duced and kept up other buhbJesy has a ten- dency to preferve and promote thefe; and, for this reafon, it is to be feared, that, in the prefent cafe, no arguments will be attended with any effe6t (c). The coniideration, that ** the gain made by fome in thefe focieties, ** will be fo much plunder taken from ** otliers," ought immediately to engage all to withdraw from them, who have any regard to juftice and humanity; but experience proves, that this argument, when oppofed to
(^) This apprehenfion has been verified by fa£l. — At the beginning; of laft winter a letter was published to the Provident Society^ containing a clear proof of the infuf- ficiency of the plans of all thefe focieties. It was at leaft to be expected, that fuch a publication (vould prevent the rife of new focieties, formed on 7mre inadequate plans. But this was fo far from being the efrect, that, foon af- terwards, a fociety fprung up which calls itfelf the Ra- tional Annuity Society-', and which, though it does not take half the values of the annuities it promifes, has had the fhamelefihefs to ojjiire the public, that it is formed on a plan inconte/lctbJy durahlc. The Conj'didatcd^ the Public An- nuitant^ and the JVef^mirJicr Union Societies, are yet worfe jnllitutions, which have been fince formed; and there may, for ought 1 know, be many more : for, indeed, all LoNDOw fecnis to b; now ent'^rinji into aflociations of this kind.
^ private
xii INTRODUCTION.
private interefl:, is apt to be too feeble in its influence.
It cannot be faid with precilion, how long thefe focieties may continue their payments to annuitants, after beginning them. A con- tinued increafe, and a great proportion of young members, may fupport them for a longer time than I can forefee. But the longer they are fupported by fuch means, the more mifchief they mufl occafion. — So, a tradefman, who fells cheaper than he buys, may be kept up many years by increafing bufinefs and credit; but he will be all the while accumulatifig diftrcfs i and the longer he goes on, the more extenfive ruin he will produce at laft.
In the latter end of the firO: chapter, I have ilated very particularly, the method of computing the values of ajfurances on lives and furvivorfhips, in all cafes where no more than two lives are concerned : and, in the 3d EiTay, I have pointed out a confiderable error, into which there is danger of falling in computing fome of thefe values. The fo- cieties and offices for tranfadling bulinefs in this way, are very ufeful ; and it is necelTary that they (hould go upon the befl principles,
and
INTRODUCTION. xlU
and poflefs all the information that can be given them.
But there is no part of this work in which the public is fo much concerned, as the -^d Chapter. It will be there proved, that had the fums raifed for public fervices lince the Revolution, been much greater than they have been, the increafe of the public debts to their prefent flate might have been pre- vented in the eafiefl manner, and at a triflin^>- expence. A method, likewife, of reducinp- within due bounds thefe debts, heavy as they
now are, will be propofed. All competent
judges will, I believe, fee, that this method, being founded on the moil perfed: improve- ment that can be made of money, is the mod: expeditious and effectual that the natures of things admit of. N^r, in my opinion, if the nation is not yet too near the limit of its re- fources, can there be any |;-5Ci:/ reafon againft
carrying it into execution. It is well
known, to vyhat prodigious fums, money, improved for fome time at compound inter eft ^ will increafe [a). A flate, if there is no mif-
{a\ A penny, put out to 5 per cent, coumpnund intereft ^l our Saviour's birth, would, by this time, have i;icrcarctl to more money than would be contained in 150 millions of globes, each t.qual to the earth in magnitude, and all iplid geld.
application
XIV INTRODUCTION.
application of money, muft neceffarily make this improvement of any favings, which can be applied to the payment of its debts. It need never, therefore, be under any difficulties ; for, with the fmallejl favings, it may, in as little time as its interefl: can require, pay off the largejl debts.- — -Several of the obferva- tions I have made on this fubjedt, have not, perhaps, been duly confidered. Could they engage the attention of the managers of our public affairs, they might, I think, be of ibme fervice. Bat this, I am fenfible, I can- not exped. I have, however, in fome de- gree, fatisfied rny ov^n mind ; and I (hall always refle<5t v;ith pleafure, that, in this part of the follovv^ing treatife, I have endeavoured to convey to the public, an information which is of particular importance to it.
In ihtjirjl Ellay I have made many obfer- fervations on the expectations of lives, the pernicious influence of great towns on health, and manners, and population ; the increafe of mankind ; and other fubjedts in the dodrine
of Annuities and Political Arithmetick.-
In the Lafl Effay I have flated carefully the proper method of forming tables of the pro- babilities of human life, from given obferva-
tions :
INTRODUCTION. xv
tions : And, in the Appendix, befides feveral new Tables, I have thought it necefTary to give Mr. Shnpfons Tables of the values and expectations of London lives ^ and all the other Tables which can be wanted in the
perufal of this work.' 1 have alfo, in the
Appendbcy given the Demonftrations of the Anfwers to the 9^eJlions in Chap. I. Thefe Demonftrations I have chofen to keep out of fight in the body of the work, in order to avoid difcouraging fuch readers as may be unacquainted with mathematics.
Upon the whole. A great part of this work is, I believe, new ; and I am in hopes alfo, that it will be found to contain fome improvements in thofe branches of philofo- phical enquiry, which are the fubjeds of it.
CHAP.
The Reader is defired to corre6l the follow-^ ing Error.
In page 148, line 11. tnjlead of 11, read 9 years pur- chafe ; and in the ne,xi line, in/lead of b-l, read i^l year?; purchafe.
CHAP. I.
ihieflions relaiijtg to Sche^nes for grant- ing Reverfionary An72uities^ a7td the . Values of AJftirances on Lives.
Q^ E S T I O N i.
it*i'>^^<"^ Set of married men enter into a X <c A i *' focicty for fecuring annuities to § ^ *' their widows. What fuin of
£!}'53^i^<(^ " money, in a lingle prefent pay- " ment, ought every member to ** contribute, in order to entitle his widovV" *' to an annuity of 30/. per annum for her iife^ ** efiimating interell at 4 per cent V*
A isr s w E R .
it Is evident, that the value of fuch an ex- pectation is different, according to the diffe- rent ages of the purchafers, and the propor- tion of the age of the wife to that of the hufband. Let us then fuppofe, that every perfon in fuch a fociety is of the fame age with his wife, and that one with another all the members when they enter may be reck- B oned
2 ^lejiions concerning
oned 40 years of age, as many entering above this age as below it. It has been demonlbat- ed by Mr. Tie Moivre and Mr, Simp/on, that ** the value of an annuity on the joint con- *' t'lJiuance of any two lives, fubtrad:ed from ' ** the value of an annuity on the life in ex- ** peculation," gives the true prefent value of an annuity on what may happen to remain of the latter of the two lives after the other.
In the prefent cafe, the value of an annuity to be enjoyed during the joiJit continuance of two lives, each [a) 40, {b) is 9.826, accord- ing
\a) See Table VII. Appendix.
(b) The values o^ joint lives and reverfions, as deduced from the Brejlaw obfervations, are not given in any part of this work from Mr. De Moivre % rules in his treatife on annuities on lives. For thefe rules are approximations, which give refults fo far from the truth, as to be, not only ufelefs, but dangerous. In the fecond efTay in the Ap- pendix, a particular account of this will be given, and alfo of the method in which thefe values have been cal- culated.
Mr. De Moivre has calculated the values oifmgle lives, on the fuppofition of an equal decrement of life thro' all its ftages till the age of 86, which he confidered as the ut- mod probable extent of life. Thus ; let there be 56 per- fons ^Jive at 30 years of age. It is fuppofed that one will die every year till, in 56 years, they will be all dead. The fame will happen to 46 at 40, in 46 years. To 36 at 50, in 36 years, and fo on for all other ages. The number of years which a given life wants of 86, he calls the complement of that life. Fifty-fix, therefore, is the com- ^)lene72t of 50 ; 46 of 40, and 36 of 50.
This hypoiheas eafcs very much the labour of calcula- ting the values of lives ; and it is fo conformable to Dr. Hollef^ table of obfervations, that there is little or no rea-
fon
Rcverjionary Annuities, &c. \
irig to the probabilities of life in the table of oblervations formed by Dr. Hailey, irom the bills of mortality oi hrejlaw in Stlejia, The value of a fingle life 40 years of age, as given by Mr. De MoivrCy agreeably to tiie fame ta- ble, is 13.20 (^) ; and the former fubtraded from the latter, leaves 3.37, or the true num- ber of years purchafe, which ought to be paid for any given annuity, to be enjoyed by a
fon for diftinguifhing between the values of lives as de- duced from this Table, and the fame values deduced from the hypothefis.
In order to avoid putting the reader to trouble, T have given this table at the end of this vv^oric. And I have alfo given two other tables which I have formed from the bills of mortality at Northamptcn and Norwich. Thefe laft ta- bles anfwer more nearly to Mr, De Moivri^ hypothefis than even Dr. Halleys table; and the difference between the values oi fingle and joint lives by the hypothefis.^ and the fame values computed ftri£tly from the tables, is gene- rally lefs in thefe tables than in Dr. HalleySy as will be (hewn in the laft Eflay. When, therefore, in the courfe of this work the values o'i f ingle zn6 joint lives are men- tioned, as given agreeably to Dr. Halli-ys table, it mufl be Underftood, that they are taken from Tables VI. and VII. in the Appendix, and given in ftrict agreement only to the hypothefis -y and that for this reafon, they are in reality ftill more conformable to the Northampton and Norwich tables.
The inhabitants of London, as is well known, not liv- ing fo long as the reft of mankind, the values offngie and joint lives there, are confiderably lefs than in ::ny other place where obfervations have been kept. Whenever, therefore, I have had London lives in view, I have given particular notice of it, and taken their values from Mr. Simpfony who has calculated them with much accuracy from the London tables of obfervation. See Tables X. and XI.
(a) See Table VI. Appendix.
B 2 perfon
4 ^ejlions concerning
perfoti 40 years of age, provided he furvlves another perfon of the fame age, intereft being reckoned at 4 per cent, per annum. The an- nuity, therefore, propofed in this queftion be- ing 30/. the prefent value of it is 30 multi- pUed by 3.37, or 101/. 2s.
By calculating from Mr. Simpfoiis tables [a), formed from the bills of mortality of London, this value comes out 102/.
The difference in the value of the reverfion will be inconfiderable, whether the common age is taken a few years more or lefs than 40. Thus married men of 30 ought not, accord- ing to Dr. Halleys table, to give two fifths of a year's purchafe more, for any given re- verfionary annuity for their wives, than mar- ried men of 50, provided they are of the fame ages with their wiv^s -, and one quarter more, according to Mr. SimpfoJis table. If the wives are younger (as is generally the cafe) there will indeed be a confiderable difference; for the value now determined would be 120/. according to the Brejlaw obfervations, fup- pofing the two lives to be 40 and 33, or that wives are one with another feven years younger than their hufbands j and 118/. \os, according to the London obfervations.
(a) See Table X. and XI. Appendix.
QuES-
C(
Reverjionary Annuities, &c. 5
Qjj E s T I 0 N II.
" Suppofing fuch a fociety as that defcrib- ed in the preceding queftion, to be limit- ** ted to a certain number of members, and ** conflantly kept up to that number, by the ** admifiion of new members as old ones arc ** loll, in confequence of their own deaths, ** and the deaths of their wives : What is the ** number of annuitants which, in fome time ** after its eflabhfhment, will come to be *^ conftantly upon it ?" ^
Answer.
Since every marriage produces either a widow or widower -, and fince all marriages taken together would produce as many wid- ows as widowers, were every man and his wife of the fame age, and the chance equal which (liali die firll: ; it is evident, that the nimiber of widows that have ever exifted in the world would, in this cafe, be equal to half t\\& number of marriages. And what would take place in the world muft alfo, or^ the fame fuppofitions, take place in this fo- ciety. ——In other words ; every other per- fon in fuch a fociety leaving a widow, there muft arife from it a number of widows equal to half its own number. — But this does not determine what number, all living at one and '^he fam^ time, the fociety may expcdl will B \ come
6 ^.ejlions concerning
come to be conftantly upon it. For if every widow lived no more than a year, the fociety would never have more annuitants upon it than came on in a year. And on the cour trary, if none ever died, the number of an- nuitants would go on increafing for ever. — : *Tis, therefore, neceffary, in order to anfwer the prefent enquiry, to determine how Jong the duration oi fiirvivorJlDtp between perfons of equal ages will be, compared with the du- ration of marriage. And the truth is, that, fuppofing the probabilities of life to decreafc uniformly [a), the former is equal to the latter; and confequently, that the number oifurvivors, or (which is the fame, fuppo- fmg no fecond marriages) of widows and wid- owers alive together, which will arife from any given fet of fuch marriages conftantly kept up, will be equal to the whole num- ber of ii^arriages ; or half of them (the num- ber of widows in particular) equal to half,
(a) 7'hat is, fLiopofing that out of any given number alive at any age the fame number will die every year 'till all are dead. See the preceding note. That on this hy- pothefis, the duration of furvivorfhip is equal to the du- ration of marriage, when the ages are equal ; or, in other words, that the expe^ation of two joint lives, the ages being equal, is the fame with the expe£iation of furvivor- ibip, may be learnt from the i8th and 20th problems of Mr. De Afoivre's treatife on annuities ; and a demonftra- tion of it, together with a particular explanation of this lubje£^, may be found at the beginning of the firft Eflay to which I muft beg the reader to turn, if he is at any lofs about the full meaning of what is here faid,
the
Reverjionary Annuities ^ 6cc. y
the number of marriages. — Now, it appears that the decreafe in the probabiUties of hfe, is in fadl nearly uniform. According to the BreJlaWy the Northatnpton and Norwich ta- bles of obfervation, almoft the fame num- bers die every year from 20 years of age to yj {a). After this, indeed, fewer die, and the rate of decreafe in the probabilities of life is retarded. But this deviation from the hypothelis is inconfiderable ; and its effed:, in the prefent cafe, is to render the duration of furvivorfliip longer than it would other- wife be. According to the London table of obfervations, the numbers dying every year begin to grow lefs at 50 years of age j and from hence to extreme old age, there is a conftant retardation in the decreafe of the probabilities of life [b). Upon the whole, therefore, it appears in anfwer to the pre- fent queftion, that " according to the three ** former tables of obfervations, and fuppo- ** fing no widows to marry, the number ** enquired after is fotnewhat greater than ** half the number of the fociety ; but, ac- " cording to the hondon table, a good deal ** greater.''
It muft be carefully remembered, that thi* has been determined on the fuppofition, that
(a) See Tables III. IV. and V. Appendix.
[h) The reafon of this difference between iheXomicu and other tables, will be given at the end of the fourth Effay.
B 4 hufbands
8^ ^teJiio?js concerning
hufbands and their wives are of equal ages, and that in this cafe it becomes an equal chance which fliall die lirft. In reality neir ther of thefe fuppofitions is juil:. Hufbands in general are older than their wives j and in equal ages the mortality of males has been found to be greater than the mortality of fe- males. For both thefe reafons, it is much more than an equal chance that the hufband will die before his wife, or that the wojnan fhall be the furvivor of a marriage, and not the man. This will increafe conliderably the duration of furvivorfhip on the part of the woman, and confequently the number enquired after in this queflion. The mar- riage of widows will alfo diminifh this num- ber, and the operation of thefe caufes will be different in different fituations. But it is by no means to be expeded (in the fituation of the focieties I have in view) that the di- minution from the latter caufe will be con- fiderable enough, to overbalance the opera- tion of all the other caufes which have been mentioned, and reduce the number under confideration fo low, as half the number of marriages {a).
Scholium.
In London it appears, that there is a re- tardation of the decreafe in the probabilities
(a) It will be obferved hereafter, that this obfervaiion ^as been found to be true m fa(5lc
^ pf
l^eijerJiQnary Annuities^ Sec, g
pf life, which renders the duration of furvl- vorfhip between two lives of equal ages, con- fiderably longer than their joint continuance. It fcems worth obferving, that this is the reafon why, though the probabilities of life, and therefore the values of fingle and joint lives, are lefs in London than in other places, yet the values of reverfions depending on fur- vivorfhips, are in fonie cafes greater there. It is proper to add, that this likewife is the reafon why, in calculating the values of joint lives and reverfions, the prefent value of an annuity payable yearly to the furvivor of two equal lives, may come out equal to, or even greater than, the prefent value of a like an- nuity for the joint lives. As an annuity, du- ring fuch furvivoriliip, will probably not be- come payable for fome years, and therefore the money given for it will have time to ac- cumulate, it is manifeft, that the value of it could never be equal to the value of an an- nuity on the joint lives, the payment of which begins immediately, were not the ob- fervation now made true.
QjJ E S T 1 O N III.
" Such a fociety as that defcribed in the *' preceding queftions being fuppofed, in ." what time will the number of anriuitants f* upon it come to a viaxhriumV
Ans-
iO ^ ^lefJons concerning
Answer.
In order to be more clear in anfwering this queftion, I will firft fuppofe the fociety to comprehend in it from its firft eftablifhrnent, all the, married perfons of all ages in any town or country, where the number of people con- tinue conftantly the fame. In this cafe, the whole colledive body of members will be at their greateft age, at the time of the eftabliOi- ment of the fociety -, and the number of members, together with the number of wi- dows left every year, will, taking one year with another, admit of no increafe or dimi- nution. The number of widows in life to- gether, derived from any given number com- ing on a fociety every year, will increafe con- tinually, till as many die off as are added e- very year; that is, till they come to die off as faft as poffible. But they cannot die off as faft as poffible, till the whole colledive body of widows are at their greateft age j or, till there is among them the greateft number pofTible of the oldeft widows ; and, therefore, not till there has been time for an accefhon to the oldeft widows, from the youngeft part of the widows that come on annually.
Let us, for the fake of greater precifion, divide the whole medium of widows that come on every year, into different claffes ac- cording to their different ages, and fuppofe fome to be left at 56 years of age, fome at 46,
fome
ReverJio7tary An^tultieSy &c. 'n.
fome at 36, and fome at 26. The widows, conftantly in life together, derived from the iirfl clafs, will come to their greateft age, and to a maximum^ in 30 years, fuppoling with Mr. De Moivrc, 86 to be the utmoft extent of life. The fame will happen to the fecond clafs in 40 years, and to the third in 50 years (^). But the whole body, compof- cd of thefe clafTes, will not come to a maxi- mum, till the fame happens to the fourth or youngefl clafs ; that is, not till the end of 60 years. After this, the affairs of the fociety will become Jiationary, and the number of annuitants upon it of all ages will keep al- ways nearly the fame.
Such is the anfwer to this queftion, fup- poling a fociety to begin with its complete number of members, confifling of married perfons of all ages, in the fame proportions to one another, with the proportions in which
they exift in the world. If it begins with
its complete number of members, but at the" fame time admits none above a particular age : If, for inftance, it begins with 200 members all under 50, and afterwards limits itfelf to this number, and keeps it up by ad- mitting every year, at all ages between 26 and 50, new members as old ones drop off;
(a) In theJppendix, note (a), a rule is given, by which the numbers alive at the end of any particular number of y.ears may be very eafily determined.
in
12 ^ejlions concerning
m this cafe, the period neceffary to bring on the maximum of annuitants will be juft dou- bled. For, in the firft place, the v/hole col- le(5tive body of members will be 60 years in getting to their greatefl age, as may eafily appear from what has been juft faid. The annual medium of widows, therefore, that will come on the fociety will increale con- tinually for 60 years, it being evident, that the older any fet of married men are, taken one with another, the fafter thev will leave widows. And after this annual medium is increafed to a maximimi, 60 years more will be neceflary to bring to a ?naximiim the num- ber in life together, derived from fuch 2ijixed
annual medium conftantly coming on. -If
fuch a fociety is any number of years in gain- ing its maximmn of members, the time ne- cellary to bring on the maximum of annuitants will be ftill further prolonged, and will be equal to twice 60 years with that number of years added. — Moft of the focieties for grant- ing annuities to widows are of this kind ; and, therefore, fuppofmg them to gain their complete number of members in ten years, and for ever afterwards to preferve it, the number of annuitants upon them will go on increafing for 130 years. — It is proper, how- ever, to be remembered, that the increafe will be quicker at firft, and afterwards flower; and that, within 20 or 30 years of the end
5 of
Reverjionary Annuities , 6cc. £3
of this term. It will be fo flow as fcarcely to be fenfible, though ftill real.
All who will befl-ow due attention on this fubjed: muft fee thefe decifions to be juft ; ' and a demonftration of them might be given, in a form more flridUy mathematical, were it neceflary.
Qjj E s T I o N IV.
*' Suppofe the members of fuch a fociety " as that defcribed in the preceding. quefli- ** ons,to chufe making annual pay tnents during *' the continuance of marriagCy in lieu of the ** fum which the reverfionary annuity for ** their widows is worth In prefent money : ** What ought thefe annual payments to be, \* cftimating intereH: at /^ per centV
Answer.
This will be eaiily determined, by finding what annual payments, during two joint lives of given ages, are equivalent to the value of the reverfionary annuity \n prefent inoney, — Suppofe, as in Qaefi:Ion I. the two joint lives to be each 40, and the reverfionary annuity 30/. per annum. An annual payment during the continuance of two fuch lives is worth, according to Dr. Halleys table of obfervati- ons, 9.82 {a) years purchafe. The annual
. {a) See Table VII.
payment
14 ^ejlions concerning
payment then ought to be fuch as being mul-» tiplied by 9.82, will produce (c) /. loi.i, the prefent value of the annuity in one payment by Queftion I. Divide then /. joi.i by 9.82, and the quotient , or /. 10.3 will be the anfwer.— This is very nearly the an- nual payment of all the members at an ave- rage, fuppoling equal numbers to offer them- felves for admifiion of every age between 30 and 50. As much as fome give lefs, others ought to give more, according to their ex- cefs of age. Thus, the annual payment of a married perfon, 30 years of age, ought to be /. 9.39 ; and of a perfon 50 years of age
/. 11.33- If the values of joint lives and
the reverfionary annuity are taken agreeably to the London table of obfervations, thefe an- nual payments will be, for 30 years of age [b\ I. 10.9, — for 40, /. 12.5, — for 50, /. 14.5.
If
{a) Particular notice fhould be taken of the method of notation here ufed, becaufe it will be carried through
the whole of this work. The figures on the right
hand of the full-point, fignify the decimal parts of i /. Thusi /. loi.i, is /. lOi and the loth of 1 /. or /, 10 i
and IS. ^•9>39» 's /.g, and 39 hundredths of i/. or
l.c^: li,: jod. /. 11.33, ^^ l-iiy and 33 hundredths
of i/. or /. 1 1 : 6 J. : 7 d. In general ; it fhould be
remembered, that 2 fhillings allowed for every unit in the firft place of decimals, and two-pence half-penny for every unit in the fecond place of decimals, will give, nearly enough, the value of the decimal part of every fuch expreffion.
(b) The value of two joint lives of 3c, taken from Ta- ble XI. is 9.6. This fubtravSted from the value of the life in expectation, or from 13. i, by Table X. gives 3.5,
the
Reverjionary AnnuitJes, Sec, 15
If either the rate of intereil is fuppofed lower, or wives are fuppofed younger than their hufbands, the annual payments will be increafed. But there is no occafion for point- ing out particularly the difference. It may be ealily found in any cafes by the directions now given. There is, however, one obfer- vation which ought to be here carefully at- tended to. — This method of calculation fup- pofes, that the firft annual payment is not to be made 'till the end of a year. If it is to be made immediately, the value of the joint lives will be increafed one year's purchafe; and, therefore, in order to find in this cafe the annual payments required, the value in prefent money found by Quefl:. I. muft be divided by the value of the joint lives in- creafed by unity, and, in this way, the pre- ceding values at /\. per cent, according to the Brejlaw obfervations, will be found to be 7.8.62 — /•9'35 — 7. 10.07.— According ^^ ^^^ London obfervations, /. 10, — /. 1 1.2, — L\2.j,
the number of years purchafe which an annuity for a life of 30 years of age, after another life of the fame age, is worth. This remainder, multiplied by 30, gives 105/. the value in a fingle payment, fuppofing the rcverfionary annuity to be 30/. And 105/. divided by 9.6, gives /. 10.9, the value of the fame annuity in annual payments, during the joint continuance of the two lives, according to the London obfervations. — By fimilar operations all thff other values above given have been found.
QUES-
l6 ^ejliem concerning
QjJ E S T I O N V.
*' A foclety may chufe to make abate- ** ments in thefe annual payments, and to re- ** quire the remainder of the value of the ** reverfionary annuity to be given, in fines *' or premiums at the time of admiffion -, it ** may, for inftance, chufe to fix the annual '* payments of all the members to 5 guineas. ** What, in this cafe, would be the premium ** due at admiffion, the annuity being fup- " pofed 30/. per annitm, and interefi: being " at 4 per cent F "
Answer.
From the whole prefent value of the an- nuity in one payment, fubtra(ft the value of 5 guineas per annum^ during the joint lives i and the remainder will be the anfwer.
Suppofing the joint lives both 40, the whole prefent value of the annuity in one payment is, according to the Brejlaw obfer- vations, /. loi.i, by Queft. I. — The value of 5 guineas per annutiiy or oi I. ^.1^ per anmun^ during two fuch joint lives, is 5.25, multi- plied by the value of the joint lives ; that is, 5.25, multiplied by 9.82, or /.51.55; and this fubtraded from /. loi.i, gives l.\c).^y the anfwer required for two lives at the age of 40. — The anfwer found in the fame way for two lives whofe common age is 30, is /.46.5, — and for two lives at 50, 50/.
Accord-
Reve'rfiondry Anrimties, Sec. ij
According to the Londojz obfervationSjthefe values are, for two lives at 30, /.54.6. — At 40, /.59.4.-— x\t 50, 7.63. 3.
If the lirft of the annual payments is to be made immediately, the true arifwer will, in every inflance, be the values found in the manner now diredled, diminiihed by the an- nual payment s or, in the prefent cafe, ^ guineas lefs than the values fpecified.
The values, in premiumi and annual pay- me?itSi ot any other reverfionary annuity, will be as much greater or lefs than thefe, as the annuity itfelf is greater or lefs.
Qjj E s T I o N VI„
*' A perfon 3 9 years of age wants to buy ** an annuity, for what may happen to re- '^ main of his life after _^o years of age. ** What is the value of fuch an annuity in ** ready moneyj> and alfo in annual payments, *' till he attains to the faid age ; that is, in *' annual payments for 1 5 years, fubjed: in ** the mean time to failure, ihould his life **faiir'
Answer.
The prefent value of fuch an annuity Is the prefent value of a lite at 50, in money to be received 15 years hence, and the payment of which depends on the contingency ol the continuance of the given lite i 5 years. That isj it is equal to the value of a life at 50, C multi-
J 8 ^lejllons cmcer'ning
multiplied by the prelent value of i /. to be received at the end of 15 years, and alfo by the probability that the given life will con- tinue fo long. — A life at 50, according to Mr. Dc Molvres, valuation of lives, and reck- oning interefl at ^ per cent, is worth 1 1, 34 year's purchafe. The prefent value of i /. to be received at the end of 1 5 years, is, by Table I, 0.5553. And the probability that a life at 35, will continue 15 years, is, ac- cording to the Brejlaw obfervations -144 (^). And thefe three values, multiplied by one another, give /. 4.44, or the number of years purchafe that ought to be given for the an- nuity.— The annuity then being fuppofed ^o/. its value in prefent money is 222/.
[a) The probability that a given life fhall continue dfty number of years, or attain to a given age^ is (as is well known) the fraction, whofe numerator is the number of the living in any table of obfervations oppofitc to the given age, and denominator, the number oppofite to the prefent age of the given life. — Thus, in the prefent inftance ; 346 is the number in Dr. Halleys table oppo- fite to 50, and 490 the number oppofite to 35. — |^^, (or the odds of 17 to 7) is, therefore, the probability that a perfon whofe age is 35 (hall attain to 50, or live 15 years. in the fame manner it will appear, that, according to the fame table, the probability that a perfon at this age {hall live 25 years, is -^^J ; or nearly an even chance.
At Nortl)ampton and Noriuieh a perfon at the fame age, has an even chance of living 26 years; but in London, fcarcely 20 years. See Tables III, IV, V^ and VIIL Appendix. I will add, though foreign to my prefent purpofe, that a perfon at the fame age has in thefe towns a better chance of living one year, than in London^ in th« proportion of 3 to 2.
In
Reverjionary Annuities ^ &:c. 19.
In order to find this value in atinual pay- tncntSy while the given life is attaining to 50, it is neceflary to find the value of an annuity for 15 years, fubje6l to failure on the ex- tindlion of the given hfe. And the value of fuch an annuity is, evidently, the laft value fubtraded from the value of the given life ; or, in the prefent inftance, /. 4.44, fubtra<5led from /. 13.97. (See Table VI, Appendix) that is, /. 9.53, — -222/. then, being the prefent va- lue of an annuity of 50/. for the remainder of a life now 35, after attaining to 50; and 9.53 being the number of years purchafe, which ought to be given for an annual pay- iiient to lall: 15 years, if a life now 35 lafts folong, it follows, that the value of the fame annuity in annual payments till this life attains to 50, is 222/. divided by 9.53 ; or
^•23.3.
This calculation fuppofes, that the fu-fl of the annual payments is not to be made till the end of a year. If the firi^ payment is made immediately, the value will be, the Jingle pay ?nent divided by the value of the life for the given term increafed by unity; that '\i, in the prefent cafe, 222/. divided by 10.53 ' or /. 21.080
If the value of the annuity is required in a fingle payment^ over and above any given annual payment -, dedu6l the value of the an- nual payment from the whole value in a fin- gle prefent pavmentj and the remainder will C 2 be
20 ^lejiions concerning
be the anfvver. — Thus ; let 5 guineas, in the prefent inflance, be the given annual pay- riient for the afligned term; and let the en- quiry be, how much more in prefent money the fuppofcd annuity is worth. By what has been juil: faid, 9.53, multiplied by 5 guineas, that is, 50/. is the value of the annual pay- ment; and this fum deduced from 222/. leaves 172/. the anfwer.
If the annual payment begins immediately, its value is 10.53, niultiplied by 5 guineas, and the anfwer comes out /. 166.75.
In this way may be found the value, in fmgle and annual payments, of any other an- nuity, payable to an afligned life, after a given term of years, taking any valuation of lives or interell: of money. But care muft be ta- ken to remember, that it is the title to the annuity that will commence at the end of the given term^ and that the firfl payment is not to be made 'till a year afterwards ; that is, in the cafe here fpecihed, not 'till the end of 16 years.
Scholium.
The value of the remainder of two joint lives, after a given term of years, is likewife the value of r /. due at the end of the given term, multiplied by the value of two joint lives, each older by the given term than the given lives j and this produd:, multiplied by the probability, that the given joint lives fl:iall
not
"Reverfionary Annuities^ 6cc. 21
not fail in the given term; or (which is the fame) by the producft of the two probabiH- ties, that the fingle hves (hall each continue the given term. And the value of an an- Huity, on any given joint lives for a term of years beginning now, is this lafl value, fub- traifted from the whole prefent value of the jointlives. Thus; the value of two joint lives, one 40 years of age, and the other 50, (fee Table VII.) is 8.9 i; which, multiplied by 0.6755, ^^ value of i/. due 10 years hence, and by 44r, (the probability that a life at 30 fhall continue 10 years) and alfo by Iff-, (the probability that a life at 40 fliall continue 10 years) gives 3.92, the prefent value of the remainder of two joint lives, aged 30 and 40, after 10 years; and this value, lubtracfted from 10.43 (^^^^ value in Table VII. of two joint lives, aged 30 and 40) leaves 6.51, their value for 10 years.
As the value of the longefl of two lives is always the value of ih^ joint lives, fubtraded from the fum of the values of the two Jingle lives; their value alfo for any given term, is the value of lh.c joint lives for the given term, fubtrad:ed from the fum of the values of the Jingle lives for the given term.
The truth of thefe rules may eafily appedr without particular proof. I have, however, pointed out the method of demonflrating them in a note [a) at the end of this work.
(a) See note (B) in the Appendix.
C 3/ By
22 ^lejlions concerning
By {imilar operations, may be found the values of 3 or morcjowt lives, or the longeft of thre or more lives, for a given term of years, or of what fliall remain of them after a given term of years,
Qu E s T I o N VIL
** The prefent value is required of an an- '^' nuity to be enjoyed by one life, for what ^* may happen to remain of it beyond ano- '' ther life, after a' given term; that is, pro- ** vided ht& lives continue, from the pre- *5 fent time, to the end of a given term of
years
Answer.
Find the value of the annuity for two lives greater, by the given term of years, than the given lives. Difcount this value for the gi- ven term ; and then, multiply by the proba- bility, that the two given lives fhall l^ot^ continue the given term; and the produ(5t will be the anfwer.
Example.
Let the two lives be each 30. The term feven years. The annuity, 10/. Intereft, 4
percent. rThe given lives, increafed by
7 years, become each 37. The value of two joint lives each 37, is (by Table VII) 10.25.
The
' Reverjionary Annuities , tec, 23
The value of a fing'e life at 37, is (by Ta-- bleVI) 13.67. The former, fubtradted from the latter, is 3.42, or the value of an annuity for the life of a perfon 37 years of age, after another of the fame age, by QucH:. I.^^3. 43 difcounted for 7 years, (that is, multiplied by 0.76, the value of i /. due at the end of {tvtvi years, by Table I.) is 2.6. — The probability that a {ingle life at 30 fliall continue 7 years, is (by the hypothefis explained page 2.) X5- (^). The probability, therefore, that two
fuch
[a) In tiiis-eafe, it is on fome accounts beft, as well as eafieft, to take the probabilities of life from the hypo- thefi; , rather than immediately from the Tables. — Fitty- fix perfons being fuppofcd a'ive at 30, one will die every year, according to the hypothefis. At the end of feven years then, the number of the living will be 49, and ^, or the odds of 7 to i, is, by note p. 18. the probability, that a life, aged 30, will continue 7 years ; and this fraction, multiplied by itfelf, is the probability, that two lives of this age, (hall both continue 7 years. In general, it muft be remembered, that the probability, that any two or more events fliall all happen, is the product arifing from multiplying by one another, the probabilities of all the events taken feparately. The probability, therefore, that any number of perfons will all live any given time, is rightly found by multiplying into one another the proba- bilities that each of them will live that time. — It may further jbe of ufe to fome, that I (hould obferve here, that the difrerence between unity and the fra£tion exprefling the probability, that an event will happen, gives the probability that it will tiot happen. Thus ; the probabj- iity, that a perfon 40 years of age will live 1 1 years, is by the BreJJatv Table -|-^. The probability, there- fore, that he will not live 11 years, iS'^^-|> ^"btradted C 4 from
24 ^lejtions concerning
fuch lives (liall both continue 7 years, is jT7r, or, in decimals 0.765. And 2.6 mul- tiplied by 0.765, is 1.989, the number of years purchale which ought to be given for an annuity, to be enjoyed by a life now 30 years of age, after a life of the fame age, pro- vided both continue 7 years. The annuity then being 10/. its prefent value is /. 19.89.
By fimilar operations, it may be found, that fuppofing the term one year, and the ages and the rate of interefl the fame, ths prefent value of the fame reverfionary an- nuity is /. 32.4; and that if the term is 15 years, the value is /, 9.7,
For two lives each 40, thefe values arc /. 30.33.-— /. 17.44. — /. 7-3. the term being 1, 7, or 15 years.
For two lives each ^jo, the fame values for the fame tcrn:i5, are /. 28.2, — /. 13.86,— 7.4.34 (^).
Thefe Vci.lues, according to the London ob- fervations and Mr. Sf/nffons Tables of the values of iingle and joint lives, are,
from unity or ~ °. — In like manner: The probability that two perfons aged 30, fliall ioth live 7 vears, being 0.765, the probability that they will not both live fo long, or that cfie or other ui them will die in 7 years, is 0.765 fub- lra<Sled from unity, or .235.
If any reader is unwilling to take thefe aflertions for granted, he fhould confult the beginning of Mr. De Aloivre's, or Mr. Si/npfons Treatifes on the Doctrine of Chances, where he will find them demonftrated.
(<?) See Nots (C) Appendix.
IR^everfionary jinnuitte^, &c. 25
For z lives at 30 — /.32.05— r/. 18.62— /.7.66„ at40— /.30.7 —A 15.6 — /.5.45. at5o— /.29.36— /.i2.33-r-/.3.24,
Q^ E s T I o N yilL
*' Let the fcheme of a foclety for granting *' annuities to widows, be, that, if a member ^' lives a year after admiflion, his widow (hall ^* be entitled to a life annuity of 20/. If *' /even years, to 10/. more, or 30/. in the f' whole. \^ fifteen years ; to another addi- ^« tional 10/. or 40/. in the whole. What ^* ought to be the annual payments of the ^* members for the ages of ^^^o^ 40, and 50, ^* fuppofing them of the fame ages with their ^* wives, and allowing compound intereft at ** 4 per cent ? "
Answer.
According to the hypothefis^ explained p. 2j and, therefore, ; very nearly, according to the Tables of obfervation for Brejlaw, Nor-* wichy and Northampton^
/. 8.44 — /,'8.69 — l. 9.05.
According to the London obfervations^ /. 9.41 — /. 10.17 — /. 10.92.
Thefc
i^6 ^.efiions concerning
Thefe values are eafily deduced from ths values in the !aft queftion. For example, The value of lo/. per annum for u.e to 40 after 40, provided the joint lives do not fail in one year, is, according to the hypothe/is, I. 30.33. The value of 20/. per annum^ in the fame circumflances, is, tiierefore, /. 60.66. -— In like manner, the value of 10/. 2Ji\.t\' Jeven years, is /. 17.44. And of 10/. after 15 years L']»1* — Thefe values together make /.85.4, or the value of the exped:ation, defcribed in this queftion, in 2. Jingle pre] ent payment \ which, divided by 9.82, (the value by Table VII. of two joint lives at 40) gives /. 8.69, the value of the fame expedlation in annual payment s^ during the joint lives. — In the fame manner may be found the anfwer in all cafes to any queftions of this kind.
Thefe calculations fuppofe, that the an- nual payments do not begin till the end of a year. If they are to begin immediately ^ the true annual payments will be, as was before obferved, ihtfmgle payments, divided by the value of the joint lives increafed by unity ^ and in the prefent cafe they will be, by the hypothejisy
^' 7-7 S—^' 7-9— ^- ^-o?-
By the London obfcrvations,
/. 8.52 — /. 9.06 — /.9.51.
By
Reverfw7iary Annuities, 6cc. 2j
6y the method of calculation now ex- plained, may be eafily found in all cafes, fuppofing the annual payments preyioufly fettled, what the reverfionary annuities are correfponding to them in value. — Thus, the annuities being the fame with thole menti- oned in this queftion, the mea?i annual pay- ments for all ages between 30 and 50, are nearly 8/. according to tlie higheji probabi- lities of life; 9/. according to the lowe/li and 8 guineas the mediim {a) ; interefl being at 4 per cent, and the tirfl payment to be made immediately.
If the mean annual payments, beginning immediately, are fixed to five guineas, the correfponding life annuities will be nearly (by the bypothejis) 12/. if the contributor lives a year, and 24/. it he lives feven years ; or (by th^ London obfervations) 12/. if he lives a year, and 20/. if he lives i^vtn years [b).
It
(«) The value of this expstftation, fuppofing married men 40 years of age, and their wives 30, is, in zjhigle payment, 113/. In annual payments beginning immedi- ately /. 9.88, by the hypotbefis. And 107/. — and /. 10.93, by the London obfervations.
[b) If the annuities in expectation are 14/. provided a member lives a year, and 20/. provided he lives feven years, the proper mean fingU payments for all :)ges, taken one with another, under 50 or 52, is 50 guineas nearly, according to all the Tables of obfervation, fuppofing equality of age between men and their wives. And the addition which ought to be made, on account of e.xcefs of age on the man's fide is, taking the neareft and the
cafieft
2^ §upJlions concernhtg
It is obfervable, that the difference in the X'alues of the annuities, arifing from diffe- rence of ages, and the difference in the pro- babilities of life, is lefs in this queftion than in qiieftion 4th \ and that, confequently, the plan propofed in it, is the fafeft, as well as the moil equitable and encouraging, that a fociety can adopt.
It is neceffary to remark here further, that yearly payments which begin immediately, are more advantageous than half-yearly pay- ments which begin immediately. Mr. Simpfon (in his Treatife on 'The DoBrtne of Annuities and Reverfons, pag. j?)y and alfo i?2 his SeleB Exercfesj p. 283) has fliewn, that, in the cafe of life annuities, half-yearly payments, which begin at the end of half a year, are ^ of a year's purchafe better than yearly payments, which begin at the end of a year. And it is manifefl:, that half-yearly payments, which begin immediately, are no
eafiefl: round fums, about a guinea and 4 l^or every year as far as 17 years; or, in the annual payments, (fup- pofcd 5 guineas) •'- a guinea />^r annum for five years ex- cels, and 4 a guinea more for every four years excefs be- yond five years, till the excefs comes to be 17 years. And, I believe, that 60 guineas \w f.ngk payments^ and fix guineas in flH«z^(?/ ^^^rj'w^w/i beginning immediately, may very well be flated as the lowejl common payments proper to be required, fuppofing all married men under 52, ta- ken into a fociety, without enquiring into the difference of age bctv/ccn them atid their wives, the annuities be- ing all along fuppofcd to be ///> annuities, and intereft reckoned at \-per cent,
c more
Keverjionary Annuiticsy Sec. '2;.^
more than half a year's purchafe better than thofe which begin at the end of half a year. But yearly payments, which begin immedi- ately, are a ivhokycars purchafe better than the fame payments to begin at the end of a year. The difference of value, therefore, between yearly and half-yearly payments, fuppofmg both to begin immediately, is a quarter of a year's purchafe in favour of the
q
former.
QjJ E 8 T I O N IX.
*' The value is required of an annuity to *' be enjoyed for what may happen to re- *' main of one life after another, provided ** the life in cxpecftation continues a given '* time?"
Answer.
Find by Qiieflion VI. the prefcnt value of the annuity tor the remainder of the life in expectation, after the given time, and multi- ply this value by the probability, that the other life fliall fail within that time. Find alfo, by Queftion VII. the value of the revcr- fion, provided both lives continue the given time. Add thefe values to one another, and thQ fiun will be the anfwer in a fingle pre- fent payment. /
Exam-
^^- ^ejiions concerning
Example.
An annuity of lo/. for the life of a perfofi how 30, is to commence at the end of 1 \ years {a\ if another perfon now 40, fhould be then dead ; or, if this lliould not happen^ at the end of any year beyond 1 1 years in which the former fliall happen to furvive the latter. What is the prefent value of fuch an annuity, reckoning intereft at 4 per cent, and taking the probabilities of life as they are in Dr. Halley^ Table ?
The value of 10/. fer annum, for the re- iiiainder of the life of a perfon now 30, after 1 1 years, found by Queft. VI; is /. 69.43. — ^ The probability that a perfon 40 years of age fhall live 1 1 years, is, by Dr. Halley\ Table^ ^\\. The probability, therefore, that he will die in 1 1 years, is \\\ fubtraded from unity (^), or ^^t; which multiplied by 7.69.43,' gives /. 17.16. — The value of the reverfion, provided both live 1 1 years, found by Queft, VII. is 17/. And this value added to the
{a) That is, the title to the anhuity is to commence at the end of ii years, and the firft payment to be made a year afterwards, in cafe the life in expectation fhould continue fo long, and the other fail. But if both liveS fhould continue the given term, the firft payment is al- ways to be made at the end of the year, in which the former life fhall happen to furvive the latter. See Queft,
VI.
{b) See the Note, p. 23,
former.
keverjionary Annuities ^ Sec, jt
former, makes /. 34. 16, the value required in 2, Jingle prefent payment 'i which payment divided by /. 1 1 .43, (the value by Table VJL of two joint lives, aged 30 and 40, with unity added) gives 3/j [a) or the value re- quired in annual payments during the joint lives, the firfl payment to be made immedi- ately. If, every thing elfe being the fame^
the affigned term' is 15 years, the value re- quired will be 29/. in 2, fmgle payjnent, and /. 2.55 in annual payments.
O^u E s T I o N X.
** What money in hand, and alfo in an- ** nual payments during life, ought a perfon ** of an affigned age to give for a fum of mo-*- ** ney, payable at his death to his heirs [b) ? — ** In other words, what money in hand, and " in annual payments during life, ought a ** perfon of a given age to pay for an affu- ** ranee of any given fum on his life ?"
Answer.
Subtra(ft the value of the life from the perpetuity. Multiply the remainder by the
[a) See the demonftration of this rule in Note (D) Appendix.
{b) This queftion is the fame with Problem i6th, in. Mr. De Moivre's Treatife on Annuities, and Problem; 26th, in Mr. Simpfon's Seletl Exercifes ; but the anfwera there given are right only when applied to reverfionary elates, and therefore mull be materially wrong, when applied to reverfionary Jurm, as will appear from the Scho- lium to this Queftion, and from note (E) in the Appendix.
product
2 2 '^eftmis concernmg
prodadt of the given fum into the interefl of lOo/. for a year: and this laft product, di- vided by I go/, increafed by its interefl for a year, will give the anfwer in z. Jingle prefent payment. i\nd this payment, divided by the Value of the life, will give the anfwer in an^ nual payments, during the continuance of the lite.
Example. Let the life be 30. The furn loo/. The rate of interefl: /^per ce?2t. And the valuation of lives, that in Table VI. The perpetuity, therefore [a)y is .?^. The inte- refl; of 100/. for a year, is 4/. 100 i. increaf- ed by its interefl: for a year is 104/. And the value of the life 14.68. — -The valiK-^of the life, fubtraded from the perpetuity, gives 10.32, which, multiplied by the produd: of 100/. into 4, or by 400, gives 4128. And this, divided by 104, gives/. ^9.7, the va- lue of 100/. payable at the death of a perfon aged 30, in a fingle prefent payment.— And this payment, divided by 14,68, is /. 2.7, the fame value in annual payments during the continuance of the life.
Thefe values found in the fame way, agree- ably to the valuation of lives for Lo/ido?!, in Table X, are /. 45.76, and /. 3.49. — If the life is 36, and interefl: /^per cent, thefe values are 43A and /. 3.1, by Table VI, and L/a^g.t,
[a) That is; the value of the fee ftmple of an eftate found by dividing 100/. by the rate of interefl:.
and
Reve7'Jio?jary Annul ties, &c. ^^
and /. 4.1, by Table X. — If interefl is reck- oned at 3 per cent, the fame values are» by Table VI, for 30 years of age, /. 48.14. — 2.86. — For 36 years of age, /. 51.43, and /.3.28.
It appears here, that difference of interefl makes no confiderable difference in the anf- wers to queffions of this kind, except when the values are required in a lingle payment.
If the firft of the annu-il payments is to be made immediately, the fmgle payment is to be divided by the value of the life, with uni- ty added to it, agreeably to w'hat has beeii already obferved ; and the annual payments in this cafe (interefl fuppofed at ^ per cent.) will be by Table VI, for a life at 30, /. 2.53— At 36, /. 2.9.
If the payments are half yearly payments beginning immediately, the fingle payment mufl be divided by the value of the life in- creafed by I, or .y^, (fee Quefl. VIII.) And the half yearly payments, ior the age of 36, will be half 2.9, or 1.45. And half 1.45, or .725, is likewife nearly the proper quar- terly payments.
Again ; if an annual payment, beginning immediately, of 7.2.9, o'^E^^ (reckoning in- terefl at /\. per cent.) to purchafe 100/. pay- able at the failure of a life now 36 ; 5/. by the rule of proportion, ought to purchafe 172/. And in like manner, it mav be found, that the fame annual contribution, in half^
D yearly
34 ^ejtlom concerning
yearly or quarterly payments, beginning im- mediately, ought to purchafe 170/. — Thefe Turns, according to the London obfervations, are 132/. and 130/. nearly.
The reafon of mentioning thefe particu- lars will be ittw in the next chapter.
Scholium.
If the reverfion is not Tifuniy but an an- nuity for ever, or an efiate mfeejimpk, to be entered upon after a given life, its prefent value, in a Jingle payment, will be ** the value ** of the life fubtraded from the perpetuity, ** and the ^^remainder multiplied by the an- ** nuity, or the annual rent of the eftate." — • And the value, in annual payments, will be, as before, the fingle payment divided by the value of the life. — Univerfally. It ought to be remembered, that a reverfionary ejiate, after any given life or lives, is worth as much more than a correfponding reverfionary yzif;;/^ as 100/. increafed by its intereft for a year, is greater than lOo/. — Thus, the prefent va- lues, in fingle and annual payments, of 4/. per annum for ever, and of 100/. in money after any afligned life, are to one another, (intereft being at \ per cent.) as 104 to 100, or 1.04 to I. — The reafon of this difference is, that the calculations fuppofe, that the re- verfion aryyz^/?;, and the firfl: yearly rent of the ejlate^ or firfl payment of the annuity,.
are
Reverjionaty An?mitiesy Sec. ^5
are to be received at the fame time, after the extindtioii of the lives in poffefTion. It is eafy to fee^ that this is a circumftance which muft make the latter of moil value. But to prevent any doubts about it, I ihall explain it more particularly in a note in the Appen- dix {a).
QjJ E s T I o N XL
" A perfon of a given age, having an year- ** ly income which will fail with his life, ** wants to make provifion for another per- *' fon of a given age, in cafe the latter fhould " happen to furvive. What ought the for- ** mer to give in a fingle payment, and alfo " in annual payments during their joint lives, ** for a given fum, payable at his death to «* the latter?"
It is manifell, that the value of the given fum in this cafe, mufl: be lefs than in the cafe Hated in the lad Queftion ; becaufe, here the payment of it is fui'pended on the contingen- cy, that one life fiiall furvive another, where- as in the other cafe, it is certainly to be paid at the failure of a given life.
Answer.
Find, by the folution of problem 3 2d, p. 297, Mr. Simpfo7i\ Sele(5t Exercifes, the
{a) Vid. Appendix note (E).
D 2 value
2 6 ^ejlions concerning
value of an eftate, correfponding to the givefi fum, and depending on the given furvivor- lliip. Divide this value by i /. increafed by its intereft for a year, and the quotient will be the value of the given fum in a fmgle pre- fent payment. And the fmglc payment, di- vided by the value of the given joint iives-, will be the anfwer in annual payments during the joint lives.
The folution I have referred to is as fol- lows.
** Find the value of an annuity on two e- *' qual joint lives, whereof the common age " is equal to the age of the older of the two ** propofed lives ; which value, fubtradt from *' the perpetuity, and take half the rcmain- ** der. Then fay, as the cxpcclation oi thie *' duration of the younger ot the two lives rs " to that of the elder, fo is the faid half re- " mainder to a 4th proportional, which will ** be the number of years purchafe to be gi- '* ven for the eflate when the life in expec- *' tation is the oldefl: of the two. But if this ^^ life is the youngeft, then add the number " of years purchafe juft found to the value *' of the joint lives, and let the fum be fub- ** trac'led from the' perpetuity, and you will ** alfo have the anfwer in this cafe (^/)".
Let
((?) Mr. Slmpfon has given the following examples of
this folution, adapted to London lives. Example I.
" Suppofe the age of the expciiant to be 40 ; of the pof- ** fcilbr 30. The rate of interefl:4p^r cmt, and the
" given
'Reverftonary Annuities^ Szc. 37
Let the life in expecftation be 30 ; and the other life 40 : The fum, 100/. Intereft, 4 J>er cent. The valuation of lives, that in Ta- ble VI.
The expcBatioji of the firft life, is 28 ; of the fecond life 23, by Mr. De Moivres hy- potbejis. The value of the joint lives is 10.43,
" given legacy 5000/. or 200/. -per annum. Then the ■" value of two equal joint lives of 40, being 8.1, by *' Table XI, and the perpetuity 25, the remainder o-r ** difference will be here 16,9; whereof the half is 8.45. *' Therefore, it will be as 23.6 to 19.6, fo 8.45 to 7.02 ^' years purchafe, or /. 1404, the required value."
Example II. " Let the age of the expe^ant be 30, of *' the poffejjor 40, and the relt as in the preceding exam- *' pie. Here the value of the joint lives 30 and 40, will *' be 8.8; which added to 7.02, (found above) the funi " will be 15.82; whence the anfwer, in this cafe, is *' 9.18 years purchafe, or 1836."
1 have Ihewn, that the values of reverfionary ejlates, and reverfionary _/«/;«, are not the fame as is here fup- pofed. — The rule gives the true value when applied to -the former; but, when applied to the latter, the values ^iven by it muft be divided by \l. increaCed by its inte- reft for a year, as above diredled. — The fame obfcrvatiou is to be applied to Mr. S'nnpfons next Problem, or the
In thefe Examples, 23.6 and 19.6, are the expectations, in Table IX, of 30 and 40, according to the London Ta- bles of Obfervation ; and the method of finding them for any age, and from any Tables of obfervation, is ex- =plained at the beginning of the firft EiTay.
In Mr. Dc Ahivrcs hypothrf:s^ the expedation of a life, is always half the complement. See note p. 2. — Some- times the compknie7it of a life is mentioned without any jview to Mr. De Moivris hypothefis, and it then means .double the expetlation of the life whatever that may be according to any Table of obfervations.
D 3 by
^8 ^cfilons concernhig
hy Table VII. The value of two joint lives, both 40, is 9.82, by the fame Table. The eftate correfponding to ipo/. is /\.l, per an7i. and the prelent value of fuch an eftate to be entered upon by a perfon 30 years of age, pro- vided he furvives a perlon 40 years of age is, by the rule jufl quoted, 7.33.32. And this value, divided by i /. increafed by its intereft for a year, or by 1.04, is /. 32.03. the value in 2^ Jingle prefent payment of the fum of 100/. dependent on the given furvivorfnip. And this fingle payment, divided by 10.43, is /. 3,07, the required value in annual payment s, during the joint lives, if the firll: payment is not to be made till the end of a year. But if the firft payment is to be made immediately, the required value in annual payments will be /. 32.03, divided by i 1.43, or /. 2.8. — Thefe values, according to the London obiervations, or Mr. Simpfon^ Tables founded upon them, are /. 35.30, in 2, Jingle payment^ and 7.3.6, in Mnniial payments y beginning immediately.
Mr. Simpjon, in the Problems following that here quoted, has given folutions of moft other Qaellions, concerning the values of re- verfions depending on furvivorfliips, where the whole duration of two or three lives is concerned. And I am acquainted with no, other folutions of thefe Queftions, which are applicable to all Tables of obfervations, and which at the fame time (proper regard being I paid
Reverjionafj Annuities^ &c. 39
paid to the correcftion explained in the laft Queftion) may be confidered as fufficiently corred {a),
Q^ E s T I o N XII.
** Suppofe an inftitation for the relief of
*' widows to extend its afliftance likewife *' to the families of married men, provided " they leave no widows. Suppofe, for in- ** fiance, that in this cafe children are to be •* entitled to 100/. What is fuch an expec- " tation worth, in prefent payment, accord- " ing to Dr. Halkys Table, intereft being at ^' /^per centf"
Answer.
If 40 is the mean age at which members are admitted on fuch an inftitution, and 32 the mean age of their wives, the anfwer (fuppofing no fubfequent marriages) is, by the 33d Problem in Mr. Simp/on s Selecfl Ex- erciles, p. 298, and the corredion already explained, /. 13.80 {b).
But
[a) See the third Eflay.
[b) This Problem and its folution are given by Mr. Simpfon in the following words. *' A and his heirs are *' entitled to an eftate of a given value, upon the deceafe " of B, provided B furvives A j to find the value of ** their expeftation in /);-^K/ money." — Solution. "Find *' the value of an annuity on the longeft of two equal
D 4 " lives.
40 ^.ejlions concerning
But there is a reducftion neceffliry, on ac- count of the chance there is, that a widower may marry again. Suppofe, therefore, one half of all widowers to marry a fecond and third time, and tliat two fifths of fuch wid- owers furvive thefe fubfequent marriages. In this cafe, 4- added to ^ of 4, or tV of all who become widowers, will die without leaving widows, and therefore A of/. 13.8, or 7.9.66, will be the anfwer. If only ox\t fourth of all who become widowers marry again, and two fifths of thefe furvive, the anfwer v^ill be /. 1 1.73.
" lives, whereof the common age is that of the older of " the lives A and B; which value fubtracSt from the ^' perpetuity, and take half the remainder; then it will *' be as the expectation of duration of the younger of *' the lives .A and B, is to that of the older, fo is the f' faid half remainder to the number of years purchafe f required, when the life B /i the older of the two. But " if B l>e the younger ; then to the number thus found, " add the value of an annuity on the longeft of the lives " A and B, and fubtra6t the fum from the perpetuity, for ?' the anfwer irj this cafe."
If the eftate is 4/. per annum., the age of B 40, and of A 32, intercft ^ per cent, the anfwer by this rule comes out /. 14.35, which divided (as in the preceding queftion) by J04, gives /. 13.80, the value, as above, of too/, in money. If B is 30 and A 40, the fame value is 20/.
N. B. The value of the longeft of two lives is always the difference between the value of i\\e joint lives, and the fum of the values of the two given y?K^/^ lives. Thus; the value of a life at 40, is, by Table VI, 13.2. The J'ufn of the Vclues of two fuch lives, is 26.4. The value 'of tvv'o joint lives, whofe common age is 40, is, by Ta- ble VI], g.82 ; and tlie dift'erence is 16.58, or the value of the Iof;^e/i of two lives at 40.
This
Keverfionary Annuities^ Sec. 41
This calculation fuppofes all marriages to leave children who furvive their parents. If this is conlidered as uncertain, the values now determined mufh be diminiflied in the pro- portion of this uncertainty. — Thus; if one •marriage in (even fails of leaving children (a) that furvive their parents ; thefe values will be reduced a feventh part, or to /. 8.28, if halj\ and /. 10.05, if a quarter of all widow- ers marry.
In this way may any other queftions of the fame kind be anlwered on any fuppoiitions that may be thought moft reafonable,
QjJ E s T I o N XIII.
** Let an efl:ablifl:iment be fuppofed which '* takes in at once all the marriages in a *' country, or all marriages among perfons ** of a particular profeffion within a given '' diftrid, and fubjeds them for perpetuity ** to a certain equal and common tax, or an- " nual payments, in order to provide life an- ** nuities for fuch widows as fhall refult from ** thefe marriages. What ought the tax to ** be, fuppofing the annuity 20/. and calcu- ** lating at ^pcr cent, from Mr. Ds Moivre's ** valuation of lives ; or, which is nearly the •' fame, from the probabilities of life in Dr. ** Halleys Table of obfervations ?"
[a) This for many years has been nearly the facl a- mong the minifters und profefibrs in Scotland,
Answer,
42 ^ejlions concerning
Answer.
Since at the commencement of fuch an eftablifliment, all the oldeft, as welt as the voungeft marriages, are to be entitled equally to the propofed benefit, a much greater num- ber of annuitants will come immediately up- on it, than would come upon any fimilar eftablilliment, which limited itfelf in the admiffion of members to perfons not exceed- ing a given age. This will check that ac- cumulation of money, which fl:iould take place at firfl:, in order to produce an income equal to the dilburfements at the time when the number of annuitants comes to a maxi" 7nuniy and, therefore, will be a particular bur- den upon the eilabliniment in its Infancy. For this, fome compenfation muft be provided \ and the equitable method of providing it, is, by levyingj^^z/L^j- at the beginnmg of the efla- blifliment, on every member exceeding a g.i- ven age, proportioned to the number of years which he has lived beyond that age. But in theprefent queftion, It Is fuppofed, that fuch fines cannot be convenient! v levied, or that every paym.ent mud be equal and common, whatever difoarity there may be In the value of the expedations of different members. The fines, therefore, mull: be reduced to one common one, anfwering as nearly as poiTible to the difadvantage I have mentioned, and
payable
Reverfionary Annuities^ 5cc. 43
payable by every member at the time when the eftablilLment begins. After this, the eftabhfhment will be the fame with one that takes upon it all at the time they marry; and the. tax or annual payment of every mem- ber adequate to its fupport, will be the annu- al payment during marriage, due from perfons who marry at the mean age at which, upon an average, all marriages may be confidered as commencing. — There are then two points to be here determined. Tho, fines necellary to be paid at firfl, according to the account I have jufl given ; and the conjiant annual payment^ neceilary to be made by every member, as an equivalent for the expedation provided by the eftablifhment. — T\\q fines to be paid at firft are, for every particular member, the fame with the difference betv/een the value of the expedation to him at his prefent age, and what would have been its value to him had the fcheme beo-un at the time he marri-
o
ed ? Or, they are, for the whole body of members, the difference between the value of the common expedation, to perfons at the mean age of all married perfons t:.ken toge- ther as they exill in the world, and to per- fons at that ape, which is to be deemed their mean age when they marry.
Thus; let 33 for the man, and 25 for the woman, be the mean ages of all that marry annually. Let alfo 48 be the mean age of all the married men in the world, and 40 of
married
44 ^cjiions concerning
married women {^a). — Now, he that will calculate for thefe ages, in the manner di- i^eded in Quefl. IV. will find, that the value in ammal payments during marriage, and be- ginning immediately, of the expediation of an annuity of 20/. per amiwn^ by a perfon 25 years of age, after a life whofe age is 33, is /. 6.64. — And that /. 8.04, is the value of the fame expedation, the ages being 4S and 40.
The former, therefore, is the payment for perpetuity from every member of the efta- blilhment J and the value of the difference be- tween it and the latter, or of /. i.^per ann. payable during two joint lives, whofe ages are 40 and 48, that is, /. 14.2, is the fine n^- ceilary to be levied on every married mem- ber at the beginning of the eilabliihment [b).
It would be eafy to extend the benefit of fuch an eflabliihment, fo far as to provide jog/, for the children of members, provided
{a) I mift beg leave to refer to note (F) in the Ap- pendix, for an explanation of what 1 mean by the mean ages of married men and women, and alfo for a confir- mation of the anfwer I have given to this queftion.
[h) An annuity for ever, the firfl: payment of which is to be made immediately, is worth 26 years purchafe, intereft being at \per cent. L \6f.l therefore, is equiva- lent in value to c.55/. or \is. per annum^ for ever. Add 'this to /. 6.64, and it v.'ill appear, that l.'j.i'^per anniim^ lieginniiig immediately, is the anfwer to this queftio,), I'uppolint/ the value of the yW to be provided for in the perpetual annua! payments.
they
Reverjionary Annuities y Sec. 45
they leave no widows ; and the neceflary ad- dition on this account to the perpetual an- nual payments, can fcarcely, in the circurn- . ftances this queflion fuppofes, be much more than about 15^. payable during life, and ex- cluding from all benefit fuch as happen to be widowers at the commencement of the eftablifhment, and do not afterwards marry.
If, in fuch an eftablifhment, all perfons of a particular denomination, whether mar- ried men, widowers, or batchelors, are fub- jed:ed alike to the taxes and fines ; they ought to be as much k/s, as the whole num- ber of perfons fubjedted to them, is greater than the number of marriages conflar.tlv exifting.
In carrying thefe fchemes into execution, there cannot be a more eafy, or equitable way of raifing the neceffary fines, than by providing, that none fhall be entitled to any expectation for a few of the^firll years. Thus ;. an eftablifhment, entitling widows to 20/. per annum for life, and conhfting of 667 married members, and 344 unmarried, al- ways kept up at an average, ought to begin with a capital of /. 14.2 multiplied by 667, or 9471/. beiides one pryment in hand of the conftant annual payments. That is, (the proper annual payment of every member be- ing in this cafe toVt, multiplied by /.6.64V or /.4.38) it ought to begi;i with a capital
of
46 Slueflions concerning
of 13,899/. over and above the payment of /.4.38, at the end oi every year for ever af- terwards [a). — The exclufion of all the lirft members from any benefit, unlefs they fur- vive the firfl: two years, or live to make three payments, would raife this capital nearly. And fuch an exclufion for three or four ycsirs, would be an advantage lb confiderable, that it would probably give fecurity and ftability to the fcheme for all fubfcquent time<
In thefe obfervations, I have had in vieWj feveral fchemes of the kind defcribed in ity which are now adually eftablifhed in this kingdom ; but more particularly, one begun among the London and Middlcfex clergy, and another which is eftablifhed by ad: of par- liament, among the clergy in Scotland, of both which, I (hall have occafion in the next chapter to take further notice.
I have chofen to calculate here only from Dr. Halleys Table, or Mr. De Molvres hy- pothefts grounded upon it, becaule the London Table is, by no means, adapted to the cafes in view.
The difference of eight years betvv'een the ages of men and their wives, as here taken> is probably too little ; and for this reafon^-
{a) Or, fiippofing the value or947i /. (the fine) provid- ed for in the annual payments, it ought to receive every year, at the beginniyig of the ytar, a contribution from each member of /. 4-74.
and
ReverfwJtary AnnuiUeSt &c. 47
and alfo on account of the greater mortality of males, the values I have given fliould be confidered as the lowefl: that any fcheme ought to provide.
It fhould be further remembered, that when the mean ages, at which marriages commence, are fuppofed to be 33 and 25, all fecond and third marriages are included; and that it is to be expeded, that almoft all thefe marriages will begin after thefe ages ; and likewife, that a confiderable proportion of the firft marriages will begin a much lon- ger time after tlicle mean ages, than any of the other firft marriages will begin before them. — Probably, therefore, thefe mean ages fliould not be taken younger. One or two years, however, more or lefs, in every fup- pofition I have made, will make no diffe- rence of any confequence.
Qjj E s T I o N XiV.
** A perfon of a given age has an eftate de- '* pending on the continuance of his life for ** a given term. What ought he to give for ** having it ajjiired to him for that term ?"
Answer.
From the value of an annuity certain for the given term, found by Table II, fuh- tra(^ the value of the life for the given term,
found
48 ^cjlions concerning
found by Queft. VI. and referve the remain- der.— Multiply the value of i /. due at the end of the given term, (found by Table I.) by )^^ perpetuity y and alio by \.\\q probability, that the given life fhall fail in the given term. The producl added to the referred remainder, and xh&Jhn multiplied by the given annuity, will be the required value of the alTurance in one prefent payment [a).
Example.
An eftate or annuity of 10 L for ever, will be loft to the heirs of a perfon now -^4, fhould his life fail in 1 1 years. What ought he to give for the ajjurance of it for this term ? — That is; v/hat is the prefent value of fuch an annuity to be entered upon at the failure of fuch a life, fl:iould that happen in 1 1 years ?
The value of the life of a perfon whofe ■iig^ is 34 for 1 1 years, is, by Quell:. VI. (reckoning Interefl: at 4 per cent, and calcu- lating from Dr. Halley's Table of obferva- tions) 7.76 ', which, fubtraded from 8.760, (the value of an annuity certain for 1 1 years) leaves i /. the remainder to be referved.
The value of i /. to be received at the end of 1 1 years, is, 0.6496, by Table I. The pro- bability that the life of a perfon, aged 34,
(a) See the demonftration in note (G) Appendix.
fhall
Rc'uerfio?iary Annuities ^ Sec. 49
lliall fiiil in 1 1 years* is, by Dr. Hal/ey'sTd.- ble, 4tv; and tiie perpetuity is 25. Thefe numbers, multiplied by one another, and i added to the product, make 4.34, which, multiplied by 10, (the given annuity) gives /. 43.4, the required value in a fingle prel'ent payment.
7.43.4, divided by 1.04, gives /. 41.7, the true value, by Scholium to Queft. X. of the aiTurance of an equivalent fum, or of 250/. for 1 1 years on the given life.
Again. 41.7, divided by 8.76, (the value of the given life for the given time with unity added to it) gives 4.76, the fame value in an- nual payments beginning immediately, for 1 1 years (a), fubjedt to failure fhould the life faiL
Scholium.
In a fimilar way may the price of afTuran- ces on any two joint lives, or the lo?igeJl of two lives for any given terms, be calculated ; the rule being as follows :
** From the value of an annuity certain ** for the given term, fubtrad: the value of ** the joint lives, or the longefl: of the two *' lives for the o"/'L'(?;z /^r;;^, lound by Scho- " Hum to Queft. VI. and referve the remain- ** der. — Multiply the value of i /. to be re-
(fl) The laft payment to be made at the end of the nth year J or 12 payments in all.^
E , ♦' ceived
5<> StueJIions concerning
^^ ceived at the end of the given term by the ** perpetuity, and alfo by the probabihty •' that the joint lives, or the hngeji of the two *' liveSi fhall fail within the given term. This *' product added to the referved remainder, *' and the fum multipHed by the annuity to be ** allured, will be the value of the aflurancc *' in a fingle prefent payment."
Example.
** What is the value of lo/. per annum, to ** be entered upon, fliould either of two '* perfons, one 40 and the other 30 years of *' age, die in ten years, reckoning intereft ** at /^ per cent, and calculating from Dr. «' Halleys Table."
The value of two joint lives at thefe ages, for 10 years, (found by Scholium toQueft. VI.) is, 6.51 i which, fubtracfted from 8.1 1 1, (the value of an annuity certain for 10 years, at /^per cent.) leaves 1.60, the remainder to be referved.
The value of i /. to be received at the end of 10 years, is, .6755, by Table I.
The probabihty, that the lives of one or other of two perfons, aged 30 and 40, ihall fail in 10 years, is, 44f, by Table III i^a).
{a) The probability taken from the Table, that a per- fon aged 30, (hall live 10 years, is, ^-^r- That a perfon, aged 40, Ihall live 10 years, is, ij-^. That they fhall both live 10 years, is, \±^%^ multiplied by .\\\.^ or 4*4' That they fhall not both live 10 years, or that one or other of them (hall die in this time, is, A*4, fubtraded from unity, or, i^-J. See note p. 23.
And
Reverjionary AnmdtieSj Sec. 51
And the perpetuity 25. Thefe numbers, multiplied by one another, and 1.60 added to the produdt, make 7.48, which, multiplied by 10, (the given annuity) gives 7.74.8, the anfvver in a lingle prefent, payment.
/. 74.S, divided by 1.04, gives /. 71.92, the value of the aiTurance of an eqiihmJent Jmi ; or of 250/ — 7.7 1. 92, divided by 7.51, (the value of the two joint lives for 10 years with unity added) gives 9.57, the value of the fame fum in annual payments beginning im- mediately, for 10 years, fubject to failure Ihould the joint lives fail.
Example II.
** What is the value of i oLper amium^ to be *' entered upon, fliould two perfons one 30, ** and the other 40, both die ; that is, iliould *' the longeji of the two lives fail in 10 years, *' reckoning intereft at 4 per cent, and cal- *' culating from Dr. Halleys Table?"
The value of the longeji of the two lives for 10 years, (that is, the value of the joint lives for 10 years, fubtra(5ted from the fum of the {a) values of the lingle lives for 10 years) is, 7.9 ij which, fubtraded from 8.1 1 1, the value of an annuity certain for 10 years, leaves .20 the remainder to be referved.— The value of i /. to be received at the end
{a) See Scholium to Queft. VI,
E 2 of
52 ^efilons concerning
of 10 years, is, '6y^^. The probability that the Hves of two perfons, aged -^o and 40, lliall fail in 10 years, is, by Table III, tVt, multiplied by -m, or T-|-5-44-r ; and the per- petuity 25. Thefe numbers, multiplied by one another, and .20 added to the product, make .740, which, multiplied by 10, (the given annuity) gives 7.4, the anfwer in a fin- gle payment.
7.4, divided by 1.04, gives i i, the va- lue of the affurance of 250/.
Remark I.
The values of fingle lives for given terms, when thefe terms are lefs than 10 years, muft, in anfwering thefe Queftions, and alfo in anfwering the following Queftions, be found true to at leaft 2- or 3 places of deci- mals. But they cannot be found to this ex- aftnefs by any Tables that are extant ; and, therefore, thev muft be calculated in the fol- lowing manner :
** Multiply the probability, taken out of ** the Table of obfervations, that the life *' fhall exift i, 2, 3, &c. years, by the value ** of i/. due at the end of i, 2, 3, &c. years ; ** and the funi of the products will be the ** value of the life for i, 2, 3, &c. years."
For Example. The probability, that a perfon whofe age is 34, fliall live a year, is,
by
'Reverjionary Annuities^ &c. 53
by Dr. Hallefs Table, -IIt. The probabili- ty, at the fame age, of living 2 years, is, ^-Ai 3 years, tw- — ^14 multiplied by .9615, (the value, by Table I, of i /. due at the end of a year, interefl being at ^ per ce?jt.) is, .942; or the value of the life for o?2e year. — *|4» multiplied by .9245, (the value of i /. due at the end of 2 years) is, .891. And this added to the former produ6l, gives 1.833; or the value of the life for 2 years. — 4It, multiplied by .8890, (the value of i/. due at the end of 3 years) is, .841; and this produdt, added to 1.833, makes 2.674, or the value of the given life for 3 years.
When the term exceeds 10 years, the rule in Queft. VI. will give thefe values with fuf- £cient exadtnefs -, and it would do the fame in all cafes, were the values of lives given true to 3 or 4 places of decimals, and in flrid agreement to the Tables of obfervation ufed.
The remark now made is to be extended to the values o^ joint lives for given terms. For thefe values, like thofe oi Jingle lives, cannot be found in folving thefe Queftions with fufficient accuracy, when the terms are fmall, by any method, except the tedious one, of multiplying the probability that the 2 lives fliall both continue i, 2, 3, &c. years, by the value of i/. due at the end of i, 2, 3, &c. years, and taking the fum of the pro- duds in the manner jult defcribed.
E 3 Remark
54 ^lejllons concerning
Remark II.
If the annuity is to he entered upon, In cafe of the failure within a given time of any life or lives, at the end of that time', and not at the end of the year in which the failure may happen ; its prefent value will be the pro- dud: ariling from the continual multiplication by one another of the perpetuity increafed by unity ^ the value of i /. due at the end of the given time; the annuity; and the proba- bility that the life, or lives, fliall fail within the given time. And care (hould be taken not to confound thefe two forts of Quefliong with one another. — Thus ; the value in one payment of lo/. per ami. to be entered upon eleven years hence, in cafe a perfon aged 34 fhould not live fo long, is 26, (the perpetuity increafed by unity, interefl being at ^per cent.) multiplied by .6496, and by 10/. and alfo by 4?i; or 34.8. — This value, divided by 1. 04, is, 3V5> ^^^^ value of an equivalent fum, or of 250/. to be obtained on the fame conditions.
The value of the afjurance of any annuity on the whole continuance of any lingle life is, by Queft. X. the excefs of the perpetuity above the value of the life, multiplied by the annuity. And in like manner; the value of the ajj'urance of any annuity on the whole contmuance of any two joint lives, or the longeji of two lives, is the excefs of the per^
petiiit^
Reverjionary Annuities, Sec. 55
petuity above the value of the joint lives, or of the longeft of two lives, multiplied by t'he annuity. This is very obvious; but no ge- neral method has been yet explained of find- ing the values q{ a[]urances on lives and fur- vivorfhips for terms of years lefs than the whole continuance of the lives. For this rea- fon, I have been here more explicit than I fhould otherwife have been ; and, as fuch af- furances are now much pradtifed, and may be very ufeful if their values are rightly de- termined, I have thought proper to add the two following Queftions, which, when join- ed to Queftion XI. and Mr. Sitnpfons 33d Problem given in the note p. 39, will, I be- lieve, exhaufi: this fubjedt as far as two lives can be concerned.
Qjj E s T 1 o N XV.
** B, expectant, will lofe a given fum, ** (hould he furvive A, within a given time. ** What ought he to pay for the ajjiirance of ** it ? — In other words : " What ought he to *^ pay for a given fum to be received at the " death of A, fliould he happen to furvive ^* him within a given time ?"
Answer.
Divide thtfwn of the decrements of life
In the Table of obfervations from the age
of A, for the given time, by the given time;
and, by the quotient, divide the number of
E 4 the
56 ^ejiions concerning
the living In the Table at the age of A ; and again, by thhfecond quotient (a)^ divide the given fum referving the third quotient.
Find the value of an annuity on the life of B, for the given time. To this value add the quotient, that will arife from dividing the value of an annuity certain, for the given time, by twice the complement of the life of Bj and xhQfu?n, multiplied by the referved quotient^ will be the required value in a fingle prel'ent [b) payment.
Example.
Let the Table of obfervations be Mr. Simp- Jon $ for London, or Table VIII. Let the rate of interefl be 3 per cent. A, feven years of age. B, 30. The given time 14 years. The given lum 100/. — The fum of the de- crements, in Table VIII. for 14 years from the age of feven, is 73, which, divided by 14, gives 5.2. The number of the living at fe- ven is 430, which, divided by 5.2, an^ 100/. divided by the quotient, gives /. 1.21, the quotient to be referved.
(a) When the age of A is under 60, and the term fo large as to exceed the difi'erence betv/een it and 70, it will be beft, when the London Table is ufed, to divide the given fum, not by the fecond quotient here menti- oned, but by the complement of the life of A, taken oqt of Table IX.
{h) See the demonftration of this rule, and alfo of the jule that will be given for folving the next Queftion, in the Appendix, note (H),
The
Keverjionary Anmiittes, Sec, 57
The value of an annuity for 14 years on the life of B, is, by Quell. VI. 9.5.-— The value of an annuity certain for 14 years, is, by Table II. 11.296, which, divided by 94.4, (twice the complement of the life of B, by Table IX) (^), gives . 1 2, which, added to 9.5, gives 9.62 ; and this again multiplied by 1.2 1, the referved quotient, gives 11.64, t\\Q prefent value in one payment of 100/. pay- able at the death of A aged 37, to B aged 30, (hould A die and leave B the furvivor within 14 years.
The prefent value for 14 years of two joint lives, one 7 and the other 30 years of age, may be found, by the help of Table XI, and the rule in the Scholium to Queft. VI. to be nearly 9 years purchafe s and, /. 11.64 ^^~ vided by this value with unity added, or by 10, gives 1. 1 64, the foregoing value in ^7;/- nual payments during the joint lives for 14 years, the Jirji payment to be made imme- diately, and the la/l payment at the end of 14 years, fhould the joint lives not fail.
Scholium.
It deferves particularly to be remembered, that in this method likewife may be calcu- lated, what fums ought to be paid on any furvivonliip, within a given time, of one life
(fl) This Table gives the expeSiations only, but it fiiould be remembered, that twice the expeSfation is always the ^ompUment of a lif»;. See note, p. 37.
beyond
58 ^lejfions concerning
beyond unother, in confideration of any gi- ven fum now advanced. — The following Ex- ample of this is a cafe which has offered it- felf in pradice.
** A perfon, aged 30, has in expe(ftation ** an eftate which is to come to him, pro- ** vided he furvives a tji'mor, aged 7, before ** he is out of his minority ; that is, pro-. ** vided he fliould be himTelf living at the •* time of the minor's death, ihould that hap- ** pen before he is 21. — In thefe circum- «* ftances, he wants to borrow looo/. on his ** expeSiation. What reverfion out of the ** eltate depending on fuch a furvivorfliip, is '* a proper equivalent for this fum now ad- ^* vanced, interelt being reckoned at 3 per ** cejit, and the probabilities of life being ^* fuppofed the fame with thofe in Mr. Simp- **/on^ Table oi London obfervations ?"
Answer.
It appears from what has been jufl: deter- mined, that for /. 1 1.64 now advanced, the proper equivalent in fuch circumftances, is, 100/. to be paid, in cafe the furvivorfliip ihould take place s or, by the correSiion in page 34, as much of the eftate as 100/, will buy at 3 per cent, fuppofing the iiril rent to be received immediately; (that is, fup- pofing the eltate worth 34.33 years purchafe.) or /. 2.9 1 2 per annum. — By the rule of pro- portion, therefore, for looo/. the proper
equi-
Reverjionary AmiuitieSt &c, 59
equivalent will be 8591 /. in money, or 250 A fer annum out of the cltate.
Qjj E s T I o N XVI,
" 100/. will be loft to B's heirs, fhould he ^* happen to die after A, within a given time^ *' What is the price of the ajj'urance of it ?— ; ^* That is : What is the prefent value of ^* 100/. payable at the death of B, provided ^' his death ihould happen after A's death, ^^ within a given time f "
Answer,
Divide the fum of the decrements of life in the Table of obfervations from tlie age of B, for the given time, by the given time; and by the quotient divide the number of the living at the age of B; and again, by this Jecond quotient [a)y divide the given fum, re- ferving the third quotient.
Find the value of an annuity on the life A for a number of years, lefs by one year than the given time, which fubtrad: from the var |ue of an annuity certain for the fame num- ber of years. Multiply the reinainder by the referved quotient:, and divide the produSi by the amount of 1 /. for one year, and let this be 2ifecond referved quotient.
{a) Or rather, if the London Table is ufed, by the com- plement of<the life of B, when his age is under 60, and ;ije term exceeds the diffcrcncs betwesa it and 70.
Again.
6o ^ejlions concerning
Again. Multiply into one another tht Jirji refervcd quotient, and the value of an an- nuity certain for the given time; and divide the produd: by twice the compk7nent of A's life. This lajl quotient, added to thtfecond referved quotient, will be the anfwer in a pre* fent lingle payment.
Example.
Let the age of B be 40. Of A 30. The fum 100/. Rate of intereft \per cent. The given time 20 years. The Table of obfer- vations, Mr. SimpfaiSy or Table VIII. — The fum of the decrements of life, in this Table, from the age of 40 for 20 years, is 1 27, which, divided by 20, (the given time) gives 6.38. — The number of the living at 40 is 229, which, divided by 6.38, gives 35.8; and 100/. (the given fum) divided by 35.8, gives 2.79, the Jirft quotient to be referved.
The value of an annuity for 19 years on a life at 30 years of age, is 10.3; which, fubtradled from 13.134, (the value of an an- nuity certain for 19 years, by Table II) and the remainder multiplied by 2.79, gives 7.89. This product divided by 1.04, (the amount of I /. in one year) gives 7.60 , the Jecond referved quotient.
2.79 multiplied by 13.59, (the value of an annuity certain for 20 years) gives 37.9165 and this product divided by 94.4, (twice the
com-
Reverjionary Ajznuities, &c. 6i
complement of A's life by Table IX.) gives .401, which, added to 7.60, gives 8/. the Anfwer y or, the value of 100/. payable at the death of B, on the contingency of his furviv- ifig A aged 30, and both dying in 20 years.
It is plain, that this is likewife the fum that ought to be lent to B now, on the ex- peftation of 100/. at his death, provided it fliould happen after A's death in 20 years.
This rule gives the jufl: folution in all cafes, except when B, the expedtant, is \\\q youngejl of the two lives, and at the fame time the term of years greater than the complement of A's life. In this particular cafe the fol- lowing rule muft be ufed.
Find, by the preceding rule, the value of the alTurance of the given fum for a term of years, equal to the complement of A's life, and let this value be referved. Multiply by one another the gwtn fw7i i the value of iL to be received at the end of a number of years equal to the complement of A's life; and the value of an animity certain for as many years as the given term exceeds this complement. And the produB^ divided by the complement of B's life, and the quotient added to the value referved, will be the true value fought.
Example.
Let the age of B be 30; of A 40. The
term 47 years ; and every thing elfe as in
2 the
6^ ^ejlions tonctrniitg
the laft Example. The complement of A's life, is, by Table IX. 39.2. The value of 100/. to be received at the death of B, if he furvives A within 39 years, may be found by the preceding rule to be /. 16.15; the va- lue to be referved. — The value of i/. to be received at the end of 39 years is, by Table II, .2166. The value of an annuity certain for 8 yearSj (thu -.xcei's oi the given term above the complement of the life of B by Table IX.) is, 6.733.
And thefe two values multiplied by one another, and by 100/. give 145.835 which^ divided by 47.2, (the complement of the life ofB) and 16.15, added to the quotient, make A 19.23, the value fought.
Remark.
As after finding the prefent value of an cftate, or annuity, it is neceffary to divide that value by the amount of 1 /. in one year, in order to find the prefent value of 2,fum equivalent to the amiidty ; fo, after finding the value of a fum, it is neceffary to multiply that value by the faid amount, in order to find from it the value of an equivalent annuity.
In the firit Example, therefore, the valiie of an eflate of 4/. per annum, would be 7.8.32, In the fecond Example, 20/. And this is< as it ought to be, the value for the whole duration of the lives, agreeably to the Pro- blem in the note pngc 37.
2 In
Re^'oerfionary Atuiuities, 6cc. 63
In folving this Queftion, care alfo mufl be taken not to forget the fijl Remark under the foregoing Queftion.
In the fame way with that in which the rules in the three laft Queftions have been difcovered, it is poffible to find rules for cal- culating the values of ajfurances, for given termSi on lives and furvivorihips, where three or more lives are concerned. But this is of lefs importance ; and I chufe to leave to others the further profecution of this fubje^ft.
CHAP,
[ 64 ]
CHAP. II.
Containing an Applicatio7t of the ^uef-^ tions in the foregoing Chapter to the Schemes of the Societies in Great Britain, for making Affiirances 07t Lives a7id Survivor fJoips^ a?id for graJtting An7iuities to Widows^ and to Perfons in old Age,
SECT. L
Of the London Annuity, and the Laudable
Societies for the Beiicfit of Widows.
THE fcheme mentioned in Queft. VIII. is nearly that of the hondon Annuity Society. The Laudable Society is alfo formed on a fimilar plan. In both, the annual con^ tribution of every member is five guineas, payable half-yearly ; and for this a title is given to an annuity of 20/. to every widow during widowhood, if the hufband, after ad- miiHon, lives one year according to thejirfi fcheme ', or three years according to the {a)
(a) In this fociety a member who lives but one year, is entitled to no more than an annuity of 10/. for his tvrdow ; if he lives two years, 15/. If he lives three years, 20/. four years, 25/. feven years, 30/. ten years, 35/, thirteen years, 40/.
Jecond'i
Of the London Annuity, ^c. 6^
fccond'y of 30/. if thehuiband livesy^^v« years, according to both fchcmesj and 40/. accord- ing to ihtf.rjl fcheme, if he lives 15 years, or 13 years, according to the fecond. — In both fchemes alfo, there Is no other pre- mium or fine required, than five guineas ex- traordinary, at admiflion, from every mem- ber whofe age does not exceed 45. The Laudable Society admits none above 45, and the London Annidty Society obliges every per- fon between 45 and 55 to pay, at admilfion, five guineas extraordinary, for every year that he is turned of 45.
Thefe are the main particulars in thefe fchemes; and, therefore, both of them, were the annuities to be enjoyed for life, would receive (fuppofing the members all under 46 at admilTion, and of the fame ages vfith their wives, and money at ^ per cent.) but little more than three fifths of the true value of the annuities ; or about one half, fuppo- fing wives, one with another, 10 years young- er than their bulb and s 3 as appears from Queftion Vill.
It appears further in that Queflion,, that, fuppoli ng the annuities to be life annuities, and men and their wives of equal ages, the expetftation to which an annual payment of five guineas beginning immediately, entitles, is nearly 14/. if the contributor lives a year, and 20/. ii he lives icwtn years (<^), taking
[a) The fame annual payment will, on the fame fup- pofitions, entitle to 14/. if a member lives a year, and i8/. jf he lives three years.
F thf:.
66 Of the London Annuity
the medium between the London and tlje other Tables of Oblervation.
It is Hkely, that many perfons will be very unwilHng to beHeve, that thefe Ichemes are fo deficient as they have been now reprelent- ed. I will, therefore, endeavour to prove this in a way which, tho' lefs ftridt, is fuffi- ciently decilive, and may be more Hkely to be intelligible to perfons unikilled in mathe- matical calculation. — ^I ihall here confine my- felf to the fcheme of the London Annuity So- ciety. The differences between it and the fcheme of the Laudable Society are inconfi- derable, and what ihall be faid of the one will be fully applicable to the other.
According to this fcheme, as it has beeft jud defcribcd, all that live 15 years in the. fociety will be entitled to annuities of 40/, ■per ajinuni for their widows. Suppofe the whole fociety, at admiffion, to be men of 40 years of age, taken one with another. A perfon of this age has an even chance of liv- ing 23 years ; and he has an even chance of continuing with a wife of the fame age, (that is, of continuing in the fociety) 13 years, and ~ [a). Not much lefs, therefore, than
half
• (a) This is the exacl truth according to Mr. De Mol- vre's Hypothefis, and the Norzvicb Table. • But accord- ing to Dr. Halley's and the Northcmipion Table, a man 40 years of age has an even chance of living no more t'fian 22 year?, and of joint conup-,ua'j;ce wiih a wife of the
fame
and Laudable Societies ^ &c. 67
half the members will continue in the focie- ty 15 years; and, confequentiy, not much lefs than half the widows that will come upon the fociety will be annuitants of 40/. per an- num. Thefe widows, however, being older than the refl when they commence annui- tants, will continue on the fociety a fiiorter time -J and, therefore, the number conftantly in life together, to which they will in a courfe of years incrcafe, will be proportion- ably fmaller. Putting every thing as favour- ably as poi'iihle, let us fuppofe, that out of 20 annuitants conftantly on the fociety, ^i?^ will be annuitants of ^ol. fix of 30/. and nine of 20/. To 20 annuitants then the fo- ciety will pay ^6qL per amiwn, or the 20th part of this lum, that is 28/. to e'-oery annui- tant at an average. But fuch an annuity for a life at 40, after another equal life, provided both furvive one year, is worth (by Queft. VII. p. 24.) in a lingle prefent payment, 85/. nearly, according to the Lo?idon, and all tlie Tables of Obfervations, intereft being all along fuppofed at 4 per cent.
It cannot appear improbable to any one, that this fhould be the true value of fuch a reverfion. It is not credible, that there is
fame ajvf*, 13 3-e?.rs. — Forty mud- be more th?n the mean age of the members cf the'focicty at admiffion, and ca this account the number of annuitants of 4c/. muil he proportionably g-eater. The mean age, therefore, has been t^skcn very moderately.
F 2 any
68 Of the London Annuity
any {ituation in which' the decrements of h'fc are fuch as can make it a tenth part more or lefs. — 85/. in prefent payment is the fame with 3/. 8 J. per annum for ever. — But is an annual payment of five guineas, which muft ceale as foon as either of two lives each 40, fails, equal in value to fuch a perpetuity } Every one mull fee, that there is a great dif- ference.— A fet of marriages between per- fons all 40, will, according to the probabi- lities of life in Dr. Halleys Table, laft, one with another, 15 years \a) -, and an annual payment beginning immediately, during the joint continuance of two perfons of this age, is worth 10 years purchale (/>). The com- parison then, in the prefent cafe, is between 3/. ^s. per annum for ever, and five guineas per annwnfor i ^ years ; or between an an- nuity of 3/. Ss. worth 25 years purchafe, and an annuity of five guineas worth only 10 years purchafe.
But to throw this fubjccft into another light.
(^) See the beginning of Efiay I.
[b) The value of fuch an annual' pavmcnt, by Table XI, or the London Obfervations, is 9. i ; and 10,8, by Mr. Dc Moivre's Hypothefis. — I have not taken into this ac- count the five guineas yf«^ paid at admiflion, bccaufe it is obvioufly of too little confequence to make any confide- rable diffeience. The allov/ances I have made in favour of thefe fchemes are more than equivalent to it. In par- ticnlar; it fhould be remembererl, that the payments re- quired by thcfe fchemes, are Ijalf-yearly paymenls begin- ning immediately; and that thtfe, by Q^icflr. Vill. are lefs advantageous than the payments I all alotig fuppofc them to require, or, '•'■ vuirly payments b/gimiing imme- " diately."
J. Left
and Laudable Societiesj Sec. 69
Let the number to which the fociety is kept up be luppofed to be 200. It has been dcmonftrated in Qneft. II, that at leaft half this number of widows will in time come to be conf^antly on the fociety ; and it has alfo been juil now (liewn, that the medium of an- nuities, payable to them, will be at leaft 28/. After a courfe of years, then, the fociety will have a conftant expence to bear of 2800/. per annum. — But what will be its income ? — In order to determine this, we muft confider, that there are two fources from whence its income will be derived. Firft, the annual payments of the members. And, fecondly, the money accumulated, or the capital raifcd during the time the number of annuitants is coming to a maximum. — The firft of thefe fources affords 1000 guineas, or 1050/. per annum. This wants 1750/. of the annual expence juft mentioned; and, therefore, in order to have the income of the fociety equ^l to the burden upon it, when the annuitants come to a )naximu?n, there muft be a fund raifed in the mean time equal to 43,750/. or to an eftate in perpetuity of 1750/. per annum. — But 1050/. per annum beginning immediately, and forborn 24 years, and im- proved, without lofs or delay, all that time at 4 per cent, compound intereft, will but juft raife fuch a capital i^a). There is, there- fore,
(rtj Every Queftion of this kind may be cafily folvcd
in the following manner. In Table I, find the value
F 3 of
^o Of the London Annuity
fore, the fuUeft proof, that the khernc I arn confiderin'^ is extremely delicient. The truth is, that fcarcely a third of fuch a capi- tal could be raifed, as will uppear from the following obfervations.
Out of 200 perfons, all 40 years of age, viore than five, according to the Lendon Ta- ble of Obfervations, and not fo many by Dr. Hallefs Table, may be expected to die in a year. Suppofe then five to be the real num- ber of members that will die the firfi; year of the iociety. In fubiequent years the col- ledive body of members will be continually growing older; and, therefore, the propor- tion of them that will die every year, will be continually increafing, till it gets to a maximum. I will, however, fuppofe, that during the firft 20 years no more than the
of I /. payable at the end of any number of years; and any given annuity divided by that value, will be the an- nuity to which the given annuity v/ill in that number of years increafe. — Thus ; the piefent value of 1/. pay- able at the end of 25 years, is .3751, reckoning intereft •&1 ^ per cent, and ic^cl. per annwn divided by .3751, gives 2,800/. per anuian^ the increafed annuity arifing irom 1050/. per ann. In the fame manner ; it may be found, that the fiime annuity, forborn 1 1 years, will in- creafe to 1610/. per annum. This fuppofes the firfl pay- ment of the annuity to be made a year hence. If the £ril payment is to be made immediately, which is the prefcnt cafe, the annuity v/ill incre.n'e to the fame fums in one year icfs time. — But a more paiticular account of this wi 1 be given in the rules annexed to the Tables at the end of this work.
number
^nd Laudable Societies, &c. yi
number juft fpecificd will die every year; and that, conicquently, no more than //bf loidows will come every year on the. fociety. The ages of all thefe widows, when they com- mence widowhood, will, it is evident, be between 40 and 60. One with another then, they maybe conlidered as having commenced widowhood at 50 years of age. Now, live widows left every year at this age, will, in 10 years, increafe to 43 conflantly in life together, according to the expectations of life in Tables III, IV, and V 5 and, in 20 years, to 70 [a). Suppofe the true number alive together at the end of 20 years to be only 62. the greater part of thefe will be annuitants of 30/. and 40/. per arm. and the reft 20L Were the former only equal to the latter, the medium of annuities payable to them would exceed 25/. Suppofe then this medium to be no more than 26/. and it will
[a] Every calculation of this kind is cafily made by the rule in note (A) in the Appendix. — I have put the number living together at the end of 20 years at 62, not only that the reader may be better fatisfied that I have kept low enough, but alio to make an allovv-ance for fuch widows as will be left by thofe members who die within a year after admiHion, and who, therefore, according to thefe fchemes, will be entitled to no annuities. This allowance is too large : For, after the firft year of the fcheme, it will not happen above once in 4 or 5 years, that the death of a member will be fo circumftanced, fuppofmg the probability that a man at 40 will live a year, to be, as ?A\ but the London Tables make it, 50 to 1.
V A. folio V/
72 Of the London Annuity
foliOvv, that, at the end of 20 years, the {o^ ciety will have an anniKil rent to pay of 26/. multiplied by 62 or 1612/. and, if then able to bear fach an expence, it muft, in the in- termediate time, have acquired an incrcafe of inconie equal to the difference between 1050/, and \612l. per ann. That is j it mud, with its favings, have accumulated a ftock equal to c^i)il. per ami. and worth 14,050/. But, as during this time, there will be a number of annuitants conftantly increafing, to whom yearly payments mull: be made, the favings of the fociety cannot certainly be one half of what they would have been had it been all the time free from all burdens. Suppofe then the ftock produced by thefe favings, to be equal to the flock that would arife from an income of 10 ^oi. per arm. beginning imme- diately, and improved perfectly at 4 per cent, compound intereft, for half the time 1 have mentioned, or for 10 years, without being fubje^l to any checks or deductions. Such ar^ income thus improved, would, in 10 years, produce an additional income of 560/. per (inmwi, or a capital of 14,000/,— According to thefe obfervations, therefore, the annuaj income of the fociety at the end of 30 years, and before a third part of the higheft annui- tants could come upon it, would begin to fall riiort of its expences. About that time then it Vv'culd neceffarily run aground ; and \on^ before the number of annuitants could
rife
a?id Laudable Societies j &c. 73
rife to a 100, it would fpcnd Its whole ftock, and iind itlelf under a neceflity of either doubling the annual payments of its mem- bers, or of reducing the annuities one half.
All I have now faid is meant on the fup- poiition, that the fociety begins with 200 members at 40 years of age, and is afterwards limited to that number, by admitting no more new members than will juH: fupply the vacan- cies cccalioncd by the lols of old members. If it is allowed to increafe, it may continue a longer time. And, for this reafon, a fociety that wants half the income neceilary to ren- der it permanent, may very well fubfift, and even profper for 30 or 40 years. — Thus, the Laudable Society^ was it to keep to its prefent number of members, might poflibly feel no deficiencies for 20 or 30 years to come ; but if it fliould continue to increafe at the rate of 70 or 80 every year, it would, at the end of that time, pollcfs a balance fo m.uch in its fa- vour, as might enable it to fupport itfelf for 20 or 30 years more [a). But bankruptcy would
cgme
{a) What has been before deinonflrated In Queft. III. ihould be here recolledlec!, that the number of annuitants on fuch a fociety as this, mull go on to increafe for more than ICO years, after acquIrinL; its greatefl number of members.
The Laudable Society, I am informed, took its rife from a calculation contained in a pamphlet entitled. The Pojfibility and Probability of a ScHEME intended for the Be- neft cf Widows being able to fupport itfelf. The fchem^ here iefgrr«d to, is the farne with that which this Society has
ftnce
74 Of '^'^^ London Annuity
come at laft, and with the more terrible weight the longer it had been deferred.
The rule in the London Annuity Society, which obliges every perfon between the ages of 45 and 55, to pay at admifTion ^ guineas extraordinary, for every year that he exceeds 45, is an advantage to it, but it is a very inadequate, and aifo a very unequitable ad- vanta^'-e. For at the fame time, that it ob- liges a perfon ^^ years of age, to give more than the value of his expectation, it takes above two fifths lefs than the value from a perfon who is 45 years of age.
Should any perfons remain (IIH doubtful about what 1 have faid, I would beg them to attend to one further argument.
It mud be reckoned upon that every other member of thefe focieties, fuppoftng them to confiti of perfons all of the fame ages with their wives, will leave widows to v.^hom one with another, (as already Ihewn) at leaft 28/.
fince fo'lowcu ; and I ^m afraid I fiiall not be credited, V.'hen I fay, that the calculation to prove its capacity of fiippoiting itfeU, is founded on the iuppofition, that a hundred married men whofe common age is 36, will leave but one widow every year, tho' at the fame time it is fuppofcd that two of them will die every year.
This miftake has made the whole calculation one half wrong. — Nothing can be plainer than that, if the death of a married man does not leave a widow at the end of the year, the rcafon muft be, tl\at both himielf and his wife h.ave happened to die in the year, But it is always very improbable thii fiiuuld happen.
1 . .pr
and Laudable SGcicties, &c. 7^
per a?itu miift be allowed, for as many years as there have been payments from each mem- ber. For every 10 guineas then received they mull Tome time or other hereafter pay 28/. But let it be well confidered what can enable them to do this. Did money bear no intercll:, for any given fum now received, they could not afford at any time hereafter to pay more than an equal i\xm. That is; (fince the duration o\ furvivorJJjip is in the prefent cafe, by Queft. J I, equal to the du- ration of marriage) the proper confideration for any given reverfionary annuity, to be al- lowed to ^<//the furvivors of a fet of marriages, Vv'ould be, fuppofmg no interefl of money, an equal annuity payable by each marriage during its exiftcnce; and juft balj ihc rever- fionary annuity, if it is to be allowed only to half the furyivors, or to widows exclufive of widowers. The annual payment then oifive guineas, during marriage, can entitle widows to no more than an annuity of ten guineas, fuppofmg money to bear no intereif. But if it does bear interefl, the fame payment will entitle them to more, in proportion to the degree in which it is capable of being im- proved, during the time between that iji which the annual payments begin, and the commencement of widowhood. Now, it is eafy to fee, that unlefs money bears very high intereft, this improvement cannot be likely in any circumflances to produce a capital, the
interefl
76 Of the London Annuity
intercil of which fliall be equal to the annual payment itfelf. Any given annual payment perfec^Hy improved at 4 per cent, compound interefl, requires 17 years to double itfelf, iuppohng the firft payment made immediate- ly j or, near 18 years (^), if the firft payment is not made till the end of a year. But no marriages are likely to laft fo long as this, except thofe among perfons who are very young. A marriage between two perfons, both 40, will not probably laft longer than
13 years, according to the probabilities of life in Dr. Halkys Table. A marriage be- tween two perfons, both 50, will not pro- bably, by the fame Table, laR longer than eleven years j nor a marritige between two perlons, both 30, longer than 16 years. Sugh marriages, it is true, may poffibly lafl 30 or 40 years. But this circumftance is more than balanced by the fa6t, that no lefs poiTibly they may not lad one year. The an- nual payments, then, being incapable of fuch an improvement as iliall produce an additi- onal income equal to themfelves ; it is obvi- ous, that no fociety ought to go fo far as to allow to widows annuities twice as great as thofe which might be allowed, fuppoiing no
(o^ At -ipn' cevt. the period of doubling money by compound inteicit, is nearly 23 years. At 5 per cenU
14 years.
ifiterefl
and Laudable SjcietieSf &C4 77
intereft of money {a) ; To far, for inAance, as to allow, iiiflead of 10 guineas, 20 gui- neas for an annual payment of five guineas. In the circumflances of moft of thefe focie- ties three fifths addition may be the full al- lowance. That is ; fuppofing the annual payment of each member to be five guineas, time may be expeded for gaining from hence a capital of 75 guineas, or that fliall produce three guineas per annum intereft ; and the proper reverfionary annuity will be 16 gui- neas i or fix guineas more than the proper reverfionary annuity, did money admit of no improvement.
The preceding obfervations have gone 011 the fuppofition, that the reverfionary annui- ties are to h^ for life. What difijrence in favour of thefe focieties arifes from the cir- cumftance, that the annuities are to be paid ovX'j for widowhood^ cannot be exactly deter- mined. Some judgment, however, may be formed of it from what has been faid at the conclufion of Queft. II. Were even one half
(a) The money accumulated will not be exatflly the fame with that to which the annual payment would in- creal'e, if improved at compound interell for a number of years, equal to that which the joint lives have an equal chance o^ tx\^\v\z- Miicli lefs will the incrcafc be ths fame with that which would arife.from the annual pay- ment foiborn, and improved for a number of years equal to the expeflatiOHS of the joint lives. It v^iU be lefs thaa either of thefe, for a re«fon explained in note (L) Ap- pendix.
of
78 Of the London Annuity
of the widows to marry, ftill the fchemes I have been coniidering would probably be in- liifficient. But in the circumftances of thefe ibcieties it cannot be expedted, that above one in 10, or perhaps one in 20, will marry. The perfons moll: likely to enter into them, are fuch as have not tlie profpect or abi- lity of making competent provifions for their widows in other wavs. The widows left, therefore, will in general be unprovided for, and being alfo left v/ith families of chil- dren, it is quite unreafonable to exped, that any confiderable proportion fliould marry. This is true of fuch as may happen to be left young ; but when a fociety has fubfiiled fome time, the greater part will not be young when left, and thefe, at the fame time that no ad- vantage can be expedled from their marrying, will be in general the /6/_g-/6^ annuitants, and, therefore, the heaviejl burdens. — Moreover, the profpeet of the lofs of their annuities will have a particular tendency to check marriage am.ono; them. — For all thefe reafons it feems to me likely, that the benefit, which thefe focieties will derive from marriage among their annuitants, will not be very confiderable; or at leafl not fo confiderable as to be equal to the advantages I have allowed them, by calculating on the fuppolitions, that the mo- ney they receive will be always improved per- JeSllyy without lofs or delay y at the rate of 4 per cent, compound inter ejly that the probabilities
of
and Laudable Societies^ &c. 79
of life among males and females are the fame, and all hufbands likewife of the fiime ages with their wives, and that confequently the maximum of widows on fuch focieties can a- mount to no more than half the number of marriages [a), — With refpe(ft to the lafl: of thefe fuppofitions, it deferves to be particu- larly obferved, that by an enquiry made fome years ago in Scotlandy it was found, that the widows of 7nin'iflers and prqfeJJ'ors there, [b) notwithftanding the diminution occafioned by their marrying, did exceed half the number of marriages. And certainly it would be un- reafonable in thefe focieties not to reckon that the fame will happen among them. — Indeed it feems certain that, notwithli:andin<y
[a] Care fhould be taken in thefe focieties, not to judge of the proportion of widows that will marry, from the proportion that may happen to marry during th^ir firf!: years. For moil of the widows that will be left at iwH. will be young; whereas the greater part will not be young when they commence widowhood, after a fociety has rubfiftcd 30 or 40 years ; and, therefore, tho' one in 3 or 4 fhould marry at firft, it will not be realbnabie to ex- pe£t, that half fo many fhould marry after the affairs of the fociety become ftationary.
[b) 364 widows, all living at one time, were counted; and the number of married minifters and prcfefibrs for many years paft has been, at an average, 667. — Twenty widows likewife are left one year with another; and, for 10 years, ending in the year 1767, but nine of thefe had married. — Of the annuitants likewife (about 160 in num- ber) on the fund tftablillied among the Diffenters in London^ for relieving tiie v/idowj of indigent minift'jrs, it is found that fev/ ever marry. See the latter end of the 4th Eifay; and nitc [k) in the Appendix.
the
So Of ike London Annuity
the hazards that attend child-bearing, the probability, that the woman fliall furvive in marriage, and not the man, is much greater [a] than ib commonly imagined. It vvih be £hewn in the laft Eilay, that it is not lets than the odds of 1^ to 2 > and had I calculated agree- ably to this fa6t, the values of annuities for widov/s, would have been given near a quar- ter greater than they have been given on the fuppofitlcn, that the chance of furvivorfhip is equal between men and their wives. — It muft be added, that I have made no account of any expences attending the execution and management of the fchemes of thefe focieties* Some fuch expences there muft be, and fome advantages fliould be always provided in or- der to compcnfate them.
There are in this kingdom feveral inftitu- tions for the benefit of widows, befides the two on which I have now remarked; and in p^eneral, as far as 1 have had any information concerning them, they are founded on plans
(<?) Parths as obfei ved in page 8, on account of the greater mortality of males, but chiefly on account of the cxcefs of age on the man's fide. — According to the print- ed articles of agreement, the Laudable Society pays no re- gard to this excels of age ; and the allowance required on this account by the London Annuity Society is fo tri- fling that it defervcs no notice.
In March 1770, thirty-two hufbands had died in the Londahh Society.^ and 27 wives. They feem, therefore, to be already beginning to experience, that the chances of turvlvorfnip iii m;:rriage arc in favour, of the wife.
equally
dnd Laudable Societies, Sec 8i
equally inadequate. The motives which in- fluence the contrivers of thele inftitutions are, without doubt, laudable; but they ought, I think, to have informed themfelves better. This appears fufficiently from what has been faidj but I will juft mention one further proof of it.
The London Annuity Society promifes that, if in 21 years j and the Laudable Society that, if in 25 years, it fhall appear that there has been all along an annual furplus in favour of the focieties, it ihall be employed in either raifino: the annuities, or in fmkin? the annual payments. Now, they may be afTured, that, if at the end of thefe periods, they ihould not be polTefled of a confiderable furplus, the true reafon will be, their having granted much higher annuities than the annual contribu- tions are able permanently to fupport : For it has been demonftrated, that the number of annuitants, and confequently the amount ai the annual expences, will go increaling for a long courfe of years beyond thefe periods. The efFed, therefore, of carrying into exe- cution this regulation will be, precipitating that bankruptcy which would have come too foon had there been no fuch regulation.
It has been faid in defence of thefe Socie- ties, that the deficiencies in their plans cannot be of much confequence, becaufe their rules oblige them to preferve a conftant equality between their income and expences, by re- ducing the annuities as there fliali be occafion.
G And
82 Of the London Annuity
And from hence it is inferred, that they can never be in any danger of a bankruptcy. — Iti anfvver to this, it has appeared, that the time when they will begin to feel deficiencies is fo diiiant, that it will be too late to remedy paft errors, without finking the annuities fo much, as to render them inconfiderable and trifling. All that is given too much to prefent an- nuitants is fo much taken away horn f tit ure annuitants. And if a fcheme is very defici- ent, the firft annuitants may, for 30 or 40 years, receive fo much more than they ought to receive, as to leave little or nothing for any who come after them. Deficient fchemes, therefore, are attended with particular in- juftice; and this injuftice will be the fame, if, inftead o'i rediicmg the annuities, the annual payments fl-jould be increafed; for all the dif- ference this can make will be, to caufe the injuilice to fall ovi future contributors, inflead Qi future annuitants.
But what requires mofi: to be confidered here is, that, after either the annuities have been for fome time in a ftate of reduction, or the contributions in a ilate of increafe, it v/ill be feen that thefe Societies have gone upon wrong plans, and, therefore, they will be deferted and avoided ; the confequence of which udll prove rtili greater deficiencies in their annual income, and a more rapid de- fertion and decline, 'till a total diflolution and bankruptcy take place. — This will be the 5 death
and Laudable Societiesy &c. S?
death of Pxiofl: of the prefent focietles for pro- viding for widows, if they continue to be en- couraged, and do not foon alter their plans : And at that period the nun^h^r of anniikants will be greater than ever , whofe annuities, having no other fupport than the poor re- mains of a ftock ahvays infufficient, will be foon left, without the poilibility of relief, to lament that ignorance and credulity which gave rife to thefe focieties, and which had fo long fupported them.
In the Loiidon Annuity Society, there is an encouri!gement to batchehrs and widowei's to join them, ariling from the additional an- nuities to which they will be immediately entitled, when they marry, in confequence of having made their payments a greater number of years ; and it is imagined, that particular advantages will be derived from fuch members. But even thefe will in ge- neral pay much lefs than the value of their expectations. — A perfon who begins an an- nual contribution of five guineas at the age of 24 will, fhould he live 1 1 years, and mar- ry a woman of the fame age at the end of that time, entitle her immediately to 35/. per ann. during furvivorfliip, and to j\.\l. per annum fhould he live four years after marry- ing, (interefl being at ii^per cent.) (^). In this
par-
(-7) The value of five guineas /)£r anniim\^'i^ payment
made immediately) for 11 years, iubjecl to failure fliould
•■i, life now 24 fail \ and, after ii years, for the jointlives
G 2 ' of
84 Of the Ajfociation among
particular cafe, therefore, a perfon will pay nearly the true value of his expedation. But all at all ages who marry ; and mojl of thofe who die^ in lefs time than 1 1 years after ad- miiTion, will pay lefs than the value of their expedtations.
SECT. IL
Of the Afociatlon among the London Clergy ^ and the Minifiers in ^coi\?indy J cr providing Annuities for their Widows,-
N April, 1765, the clergy within the bills of mortality, and the county oi Middlefex, at a general meeting in Sion-College, agreed to form themfelves into a fociety for the fup- port of their widows and orphans. Many in this refpedtable body may be capable of do- ing, in a better manner, what I have attempt- ed in this Treatife ; and they are, perhaps, already fenfible of the deficiencies in the plan which they have eftablillied. I fliall not, however, I hope, do wrong, in taking the liberty to recite briefly this plan, in order to introduce a few obfervations upon it.
of two perfoJis both 35, is, by the Table of London Ob- fervatioiis, 1. 69.3. — By Dr. Halleys Table /. 76.44. — The- prefcnt value of 35/. per anninn for life to the widow of a perfon now 24, (iiould he live 11 years, and marry a wo- man of the fame age with himfelf at the end of that time ; and alfo of 6/. more, or 41 /. per anmmi in all, fhoiild he live after marriage four years ; is, by the Table of London Obfervations, ^69.36, — By Dr. Halley% Table, /. 76.03. 2 Accord-
the London Clergy t &c. 85
According to the printed articles, every clergyman poffeiTed of any benefice, lediire- fhlp, or licenfed curacy, within the bills of mortality, and the county of Middle/ex, who fubfcribes annually one guinea, or two gui- neas, or more, fhall entitle his widow to an annuity ; or, if he leaves no widow, he fhall entitle any fuch children as he (liall leave, to the fame annuity for iQvtn years as his wi- dow would have had. And, in cafe a widow poflefled of an annuity, fliould either die or marry before the lapfe of 10 years, from the commencement of her annuity, fuch children of her former hulband, as fliall be then alive, are to be entitled to as many of the ten years payments of the annuities as ihe fhall not have received. — The annultv is fixed to no particular fum, but inftead of this, it is or- dered, that a fourth part of the annual fub- fcriptions and intereft fliall be divided the firft three years after the eftablifliment of the fociety ; half only the next four years -, and 3-4ths the next i; years; provided, however, that in no one of thefe 1 2 years the dividend fliall exceed 20/. to the widows and orphans of the clergy fubfcribing two guineas or more; and 10/. to the widows and orphans of the Ibbfcribers of one guinea. And, after the ex- piration of 12 years, the whole amount of the fubfcriptions, and of the Intereft of the capi- tal Itock, is to be divided proportlonably for ever. — It is further provided, that every cler- gyman, who iliail be married, or have chll- G 3 dren.
86 Of the AJj'ociatlon among
dren, at the time of his fubfcription, fliall paj^ a fine of two guineas towards a capital flock, if a fubfcriber of two guineas or more, and 40 years of age or upwards. If 50 years of age or upwards, he fhall pay a fine of three gui- neas ; if 60 or upwards, five guineas. But, if not married art the time of his fubfcribing, and fhall afterwards marry, he fliall pay a fine ac- cording to the age he fliall be of at the time of his marrying. The obligation laid upon all, whether married or unmarried, to become fubfcribers,is, an incapacity of being admitted members without the confent of a general court, unlefs, within two years after becom- ing pofTeiTed of any ecckfiadical employment, they fubfcribe.
Every one who has attended to the obfer- vations in this and the preceding chapter, muft know what judgment to form of thefe regulations.
Let us fuppofe, that all the clergy in Lon- don and Middle/ex came into this afibciation from the firft; and that one with another they are fubfcribers of tv^o guineas annually; and that there are among them as many un- married perfons as married.
In this cafe, it may be learnt from Queft. XIII, that the annuity to which widows fhouid be entitled, (fuppofing no allowance to the children of any that die) ought not to exceed 10 or 11 guineas at moft, and that, befides the annual fubfcriptions, there ought
to
the London Clergy) &c. 87
to have been a fine paid at the commencement of the fcheme, by every married perfon, of fix guineas at leaft, or, by the whole number of fubfcribers, three guineas. If the number of married members is double the unmarried, the annuity ought not to exceed eight gui- neas ; and the fine from every member {ht)uld be about four guineas. — The order, that only a fourth part of the annual fub- fcriptions and intereft fliall be divided the firft three years, half the next four yedrs, and three quarters the next five, is vs^ith- out reafon j becaufe the number of clai- mants, for the firit 12 years of the fcheme, will be fo i&w,' that it will not be polTibie, during that time, that there fliould be occa- fion for dividing any proportions fo large of the annual fubfcriptions and interefl, unlefs they are indeed beyond all bounds too little/ — After 1 2 years, the number of annuitants will go on increafing for near 50 years, as appears from Queft. III. The confequencc, therefore, of dividing, after this time, the whole amount of the annual fubfcriptions and intereft, will be a conitant yearly diminution in the dividends for near 50. years ; and mak- ing the payments to the firil claimants mucli more confiderable than they ought to be, at the expence of all fubfequent claimants. — For thefe reafons ; it appears to me out of all doubt, that this fcheme is by no m.eans iikeiy to anfvver the good ends propofed by it ; and thatj therefore, it will be beft to Ir.y it afide. G 4 At
85 Of the Eftablifliment among
At the time it was fettled it was, I find, fur-» ther agreed, that the annual fubfcriptions o£ the laityy together with the intereft of their benefactions, unlefs otherwife directed by the donors -, and the annual fubfcriptions of fuch of the clergy as Ihall fo dired, fhall make a charitable J imd to be applied to the relief of the diftreffed widows or children of all the clergy within the limits I have mentioned, whether fubfcribers or not, provided that ia no one year of the iirft twelve more than 20/. be given out of the fund to any one family. — ? This is an excellent delign j and if the mo- ney arifing from all the fubfcriptions is thrown into this fund, an important means of relief may be provided, for fuch of the more indi- gent widows and families as will accept the help of charity.
There is one more affociation of particular confequence, which it is neceiTary I Ihould take notice of. I mean, the ajjhciation among the minifters and profefTors in Scotlafid^ for making provifions for their widows and orphans. The lafl-mentioned alTociation, and alfo feveral others of the fame kind {a) in this kingdom, have been formed on the
[a) There is one among the DifTenting Minifters in
the counties oi Chejier and Lancajhire^ and another among the Diflenting Minifters in Cumberland^ Northunihcrland, }VeJimor eland ^ and Durham. — Even the London Annuity Society, tho' its plan is totally different, pioiedes to forni itfelf on the principles of the Scotch eftabliOiment, and to deri'/e encouragement from ic.
model
the Mlnijlers in Scotland, 6cc. 89
model of this eftablifliment ; and the fuc-' cefs with which it has been hitherto attend- ed, has been the principal caufe to which they owe their rife. — I am afraid of being too tedious, and therefore I will not attempt to recite all the particulars contained in the plan of this eflabliiliment. It may be fuf- ficient to obferve, that for " an annual ^^ payment, which begun immediately, of ** five guineas from loii contributors, 667 ^' of whom are married perfons ; befides a ** tax on weddings producing about 142/. ** per afimwi', it entitles every widow to an " annuity of 20/. during widowhood; and al- *' fo, every family of children that fiiall be *' left by fuch members as die v/ithout leav- ** ing widows, to 200/." Now, by particu- lar enquiry at the commencement of the fcheme, it had been found, that there was reafon to believe that, for many years back, 20 widows had been left annually by the whole body of minifiers and profefTors -, and that, alfo, fix had died annually and left chil- dren without widows \ and thefe facSts have been fince confirmed by the experience of 25 years. — Subfi:ra(5l, therefore, from 5,450/. (the whole annual income) 1,200/. payable every year to fix families of children ; and 4,250/. per arm. (the firfl: payment of which was made (^) immediately) or/.4.2 per
annum
(a) Thetruth is, that a double payment wa,s madeatths l^eginning of the fecond year. This is of lefs value than
two
go Of the Eflabllrtiment among
annum from each member, will remain as the ftanding provilion for bearing the burden of the annuitants, — This provifion, according to the calculation in Queft. XIII, and note (F) Appendix, ought to be at leaft 4,745/. ^^r anmim^ from each member; from whence it feems to follow, that this eftablilliment has not a fufiicient income to afford it a perma- nent fupport.-- 'But I do not by any means defis.n to alTert this. The difference between the real and calculated incomes, as it has been now Hated, is not confiderable enough to give fufficient reafon for fuch an affertion. This eftablifliment may have fome advan- tages that I know nothing of, and that are not mentioned in the printed accounts ; or, in confequence of the increafe of luxury, and the higher price of all the means of fublift- ence, marriage may decline among the mini- fters; or, poffibly, the probabilities of life among them, tho' much higher 'till towards 50, than is common among mankind in towns, may yet afterwards decreafe much fafter. The income, therefore, of this efta- blifhment, properly improved, may continue to be adequate to all its expences and bur- dens.— One obfervation, however, ought to be carefully attended to. The fuccefs it has hitherto met with, is no good reafon for en- two payments, one of v/hich is made Immediately, and the other a year hence ; but, the difference not being confiderable, 1 have reckoned them the fame.
tertaining
the Minijicrs in Scotland, &c. pt
tertaining this expe(ftation, with any degree of conlidence. It appears from Queft. Ill, that the number of annuitants, on fuch an .eflablidiment, muft go on increafing for 60 years, from the time of its commencement; land it is obvious, that the continuance of a fiiccefs which has not lafted half xKis period, cannot be abfolutely depended on. I know, indeed, that, according to the calculations v/hich were made when this eflabliiliment begun {a)y the number of widows upon it will not increafe fenfibly after the year 1780, or for more than 10 years to come; and, were this true, all diffidence about it would, perhaps, be unreafonable. But thefe calcu- lations cannot, in this inftance, be right; for they imply, that none are left widows under 52 years of age. 'Tis certain, on the con- trary, that many are left widows under 32; and that, confequently, the whole body in life together mud go on to increafe for 25 years, beyond the period alTigned in thefe cal- culations; or till the year 1805 [b). It is ne-
ceiTary
[a) See Table III, in a book printed at Edinburgh in 1748, entitled, Calculations, with the Principles and Data on which they are inftituted, relative to a late acl of parliament, entitled. An aSt for ra'ifing aiul ejiallijhing a fund for a pro'Sificn for the ividows and children of the mi- tiijiers of the churchy and of the heads., principals and majiers of the univerfities (j/"Scotland ; fiiewing the rile and pro- grefs of the fund.
{h) This is faid on the fuppofition, that all the mini- fters and profefTors acceded to the fchcnie from tlie firfr.
92 Of the Eftablifhment among
cefTary I fhould add, that the whole num- ber alive, when the increale flops, will, mod probably, be greater than the number provid- ed for in thele calculations. They are made on the fuppofition, that 52 is the mean age at which women commence widows. If this fuppofition is right, it is impoflible that, according to the probabilities of life in Dr. Halleys Table, the number of widows living together at one time, derived from 20 left annually, fliould increafe to more than 323, if none marry; .or about 300, if one marries every year [a) : Nor does it appear likely, that
this
135 did not accede ; but all their fucceflbrs have been obliged to accede. This circumftance muft add 30 or 35 years to the period of increafe which T have men- tioned ; that is, as many years as are necefiary to caufe all the non-contributors to die off. See Qtieft. 111.
[a) The expe£iation of a perfon at the age of 52 is, v/hen taken exactly from Dr. i/a/^y's Table, 16.16; and this number multiplied by 20, gives the maximum^ to which 20 widows left annually, at 52, will increafe in 34 years, fuppofing 86 the utmoft limit of life. Vid, Effay I. — In the calculation to which I have referred, there is no account taken of thofe that die \n the year in which they are left widows ; and, for this reafon, they are made to increafe to a greater number than is confiftent with the fuppofition, that 52 is their mean age when they commence widowhood.- — It fhould be remembered here, that fuppofing this mean age, as explained in note (F) Appendix, rightly taken, the maximum of widows will be rightly found in the manner juft fpecified. But the period, in which they would attain to that maximum^ would be as many years greater than the difference between the mean age and the utmofl 'limit of life, as the mean age is greater
than
th2 Mimjlers hi Scotland, &c. 91
this eftabllfliment will be able to bear the expence of above 30 or 40 more than the
lafl-
than the Icafl age at which widowhood ever commences. The calculation, therefore, which I have in view muft be very wrong. It fuppofes not only, that all the v/i- dows left at all ages, will increafe to a tiiaximum in the fame time with thofe left at the mean age; but that all left in the courfe of every year will certainly live to the beginning of the next year. It fuppofes, likewife, that no widows will marry ; and thofe concerned will under- hand mc when I add, that it fuppofes further, that ac- ceders, tho' moftly young minifters unmarried, will leave widows as faft as the noncontributors whom they fucceed. In confequence of thefe omiflions it has all along given the numbers in life much higher than they ought to have been given ; but yet the event has been, that thefe num- bers have in fa6l correfponded nearly to the calculation : From whence it follows, that the widows in life have hitherto increafed at a much greater rate than they could have done, according to Dr. Haliey^s Table, had their mean age, when left, been 52. Either, then, their mean age has been confiderably lefs than 52 ; or, their pro- babilities of life muft be confiderably greater than thofe in Dr. //<3//^j;'s Table ; and, it ought, therefore, to be expe£led, that they will at laft increafe to higher num- bers than thofe alTigned in this calculation.
I cannot help here mentioning one more reafon for entertaining this expedlation. — Were the decrements of life uniform, and the chances of furvivorfhip between men and their wives equal, the number of widows and widow- ers in the world, if none married, would alfo be equal ; and both together equal to the whole number of mar- riages. See Queft. II, and note (F) Appendix.
If the chances for the furvivorfhip of the wife are great- eft, the number of widows in the world will be alfo greatcft, and the whole number of widows and widowers greater than the whole number of marriages. In the prefent cafe experience proves, that the chances of fur- vivorfhip in marriage are as 5 to 3 at leaft, in favour of
the
94 Of the Eflablifhment among
laft-mentioned niim'oe*'. But it was founds by enquiry, that the number of widows liv-
ijiJ^
the wife ; or that there are 20 widows left to 12 widow- ers. (See note (F) Appendix, and the end of the laftEllayj. The number of widows and widowers then would cer- tainly, if none married, exceed 667, the whole number of marriages. Suppofe, however, that they would be only equal ; which is the fame with fuppofing, that zvicbtvs would not increafe to more in life together, in proportion to the number left, than widowers , or that one with ano- ther they are of equal ages. 667 then being the number of marriages, this will likewife be the number of widows and widowers, 4.16 of whom mufl: be widows and 250 widowers ; that is, 3 to 3. Now as zvidozvs arc certainly, one with another^ younger than widowers; and likewife, very probably, more long-lived at equal ages j and as alfo, in the prefent cafe, but one marries of the Hventy left annually ; it follows, with demonftrative evidence, that if the annuitants on this eftablilhnient fhould not increafe to 400, the reafon muft be, that the decreafe in the pro- babilities of life, inftead of continuing always uniform, is flow in the firft ftages of mature life, and accelerated af- terwards, to a degree of which there is no example in any Tables. And this, poffibly, may be indeed the cafe ; for it is uncertain whether, in this particular fituation, life may not wafte according to a law not yet obferved. This uncertainty it is in the power of the conduciors of this fchcme to remove, by keeping an account of the ages at which all the minifters and profeflbrs enter upon their offices and die ; and alfo of the ages of their wives when they marry and die. From fuch an account kept for a courfe of years. Tables of Obfervation, adapted to the befl fort of lives of both fexes, might be formed, which would contribute much to the improvement of this part of knowledge.
I have faid nothing above of the advantage which this eilabliflunent derives from the njarriases amono; widows. This advantage, it has appeared, were it enjoyed without abatement, would not be confiderable ; but it is in (ome mcafure given up by the order whicji makes a part of this
efta-
the Minifters in Scotland, &c. 9^
ing in 1748, was at k aft 364; (^) and it may* he learnt from notes (A and F) in the Appen- dix, that, according to Dr. Kalky^ Table, 2vO widows left annually, muft in time increafe to near 400, tho' one of the yoiingeft marri- ed every year. And it may be farther learnt from note (A) Appendix, that the widows on this ertablifiiment have not hitherto increaf- ed more ilowly, than is confident with their adually increaiing to 400 : Nor, indeed, (as the probabilities of life in this cafe are higher than thofe given by Dr. Halleys Table) fhould I much doubt of their increafing to mo?'e than this number, were it not that 364 has been given as the number found upon enquiry. With refped; to this, however, it ought to be mentioned, that another account had been taken which made the number of widows only 321. As, therefore, a more care- ful enquiry difcovered 43 new widows ; per- haps, an enquiry yet more careful would have made yet further difcoveries. In ta- king fuch accounts, none can well be added ;
eftablifhrnent, that fuch children of an annuitant as arc under i6, fliall be entitled, if flie marries, to as much as (hall happen to be then unpaid of ten years purchafe of her annuity. The fame provifion is made for the chil- dren of annuitants that die. There are olhcr burdens on this eft.Tblifhment, and it has alfo ndvaniages of vvhicli I have taken no notice; but, as far as 1 am acquainted with them, they are of no particular confequence, and they alfo nearly balance one another.
[a) See Calculations, v/ith their Principles and Data, &c. introduction.
but
^6 Of the Eftablifhinent, ^c.
but In the llriden: fearch it can hardly btf poffible to avoid omitting fome.
The refult of the laft enquiry, in particu- lar, is laid to have been^ not that it was found that there were no more than 364 wi- dows, but that they did exceed this number. {a) However, let 364 be the true number living in 1748. Before that time^ there had probably been more marriages among them, than there have been fmce ; and this may have rendered their number iefs than it would otherwife have been, and Iefs than it will be hereafter : For it feems to me, that this eftablifliment, at the fame time that it encourages marriage among the minifters, has a tendency to check it among their widows, by making the confequence of marrying to be the lofs of their annuities.
I hope the venerable minifters and profef- fors concerned in thefe remarks, and at pre- fent fo eminent in all the departments of fcience, will excufe what has been now faid. It may, perhaps, be of fervice, if not to them, yet at leaft to fome in this part of the unit- ed kingdoms, by fliewing them, that this eftablilhment has been copied in it much too rafhly ; and that, however fuccefsful it may in the end prove, it is yet too foon to make it a model and an authority for fimilar eflablifliments.
{a) See Calculations with their Principles, &c. p. 44.
SECT.
tjf the heft Schemes for ^ &c. 97
SECT. III.
Of the hejl Schemes for providing Annuities for Widows,
INflitutlons for providing widows with aii- . nulties would, without doubt, be ex- tremely ufeful, could fuch be contrived as would be durable^ and at the fame time eafy and encouraging. The natures of things do not admit of this In the degree that is com- monly imagined. The calculations and rules, ifi the preceding chapter, will enable any one to determine in all cafes to what reveriion- ary annuities any given payments entitle, ac- cording to any given valuation of lives or rate of intereft. From Queft. VII and VIII,- in particular, it may be inferred that (intereft being at 4 per cent, and the probabilities of life as in Mr. De Moivre's Flypothefis, or the Breflaw^ Norwicht and Northampton Tables) for an annual payment beginning immedi- ately o{ four guineas during marriage; and alfo for a guinea and half in hand, on ac- count of each year that the age of the huf- band exceeds the age of the wife, every mar- ried man, under 40, might be entitled to aii annuityj during life, for his widow of 5/. if he lives a year, 10/. if he lives three years, and 20/. if he Ywc^feven years. Money can fcarcely now in this kingdom be improved H at
9^ hf the heji Schemes for
at fo high a rate as 4 per cent. But, per- haps, it might be realbiiably expected, that an advantage, fufficient to compenfate this difad\^antage, would be derived, from chang- ing the annuities I have mentioned into an- nuities during widowhood. One may, at leaft, venture to pronounce, that nothing much worie could befall a fociety that went on this plan, than the neceffity of fome time or other adding half a guinea to the annual payments.
If fuch a fociety chufesj that thofe who Ihall happen to continue members the long- eft time> {hall be intitled to flill greater an- nuities, fix c;uineas, additional to all the o- ther payments at admilhon, would be the full payment for an annuity of 25/. and 12 guineas for an annuity ot 30/. if a mem- ber ihould live 1 1; years.
All batchelors and widowers might be en-^ couraged to join fuch a fociety, by admitting them on the following terms. — Four guineas to be paid on admiflion, and three guineas every year afterwards, during celibacy j and* on marriage, the fame payments with thofc made by perfons admitted after marriage; in coijfideration of which, i/. per annum, for every iingle payment before marriage, might be added to the annuities, to which fuch members would have been otherwife enti- tled.
For
providing Annuities for Widows. 99
For example. If they have been members four years, or made five payments before marriage, inftead of being entitled to life-an- nuities for their widows of only 5/. 10/. 20/. 25/. and 30/. on the conditions I have fpecified, they might be entitled to annuities of 10/. I ;/. 25/. 30/. and 35/. Or, if they have been members nine years, and made 10 payments, they might, inftead of the fame annuities, be entitled to annuities of 15/. 20/. 30/. 35/. and 40 A — In this cafe, the contributions of fuch members as fliould hap- pen to defert, or die in celibacy, would be lb much profit to the fociety, tending to give it more ftrength and fecurlty.
This is one of the beft fchemes that I ani able to think of, or would chufe to recom- mend. There are, however, others no lefjj fafe and encouraging which fome may pre- fer, and which therefore, I will juft prO" pofe*
Let the probabilities of life be the famef with thofe in the Tables juft mentioned. Let money be fuppofed to be improved at no higher interefl; than 3 per cent. Let the re- verfionary annuities promlfcd to widows be 10/. for lifey if a member lives five years after ad mlfTion, and 15/, more, or 25/. in alj^ if he lives 1 1 years. The proper payments for fuch an expedation, from married men not exceeding 50 years of age, will, in the H 2 neafelif
JQ6 Of the befi Schemes fof
neareft and mod convenient round fums. Be four guineas in annual payments beginning immediately, and two guineas in hand for every year that his age exceeds his wife's, not admitting any greater excefs than 15 years : Or, if the whole value is given in one prefent payment, 40/. added to a guinea, for every year that his age falls fhort of 50, befides the pay- ment juft mentioned on account of difparity of age. — ^For example. Four guineas in an- nual payments, belides 10 or 20 guineas in? hand, according as the age of the hufband exceeds the wife's 5 or 10 years. Or, if the whole value of the expedation is given in one payment, 10 guineas added to 40/. (that is 50/. 10 J.) from a man whofe age is 40; and, in like manner, 20 guineas added to 40/, (that is 61 /.) from a man whofe age is 30 ; befides the payment juil mentioned on ac- count of difparity of age.
If money is improved at 4 per cent, or, on- account of any advantages attending a fcheme, may be ju^ftly eonfidered as fo improved, the full payments for the expedation I have mentioned will be about one eighth, or half a guinea, lefs in the afinual payments during marriage y and a quarter lefs in all the other payments. That is : A married man,^^ or tinder 50, would, befides three guineas and halfin annual payments during marriage, be bound to add a guinea and half for every year he is older than his wife ; Or, if he I chufes
prcrolding Annuities for Widows, loi
chufes to give the value of his expectation in one payment ; befides the common contribu- tion of 30/, and a guinea and half for every year his age exceeds his v/ife's; he w^ould be bound to pay three quarters of a guinea, for ■every year he is lefs than 50 years of age ; that is, 53/. 1 2 J". 6^. in all, fuppofing him 40 years of age, and 10 years older than his wife. — All thefe payments doubled would entitle to double annuities.
There is one particular advantage which focieties formed on a plan of this kind would enjoy [a] — Perfons who know themfelves fubjed; to diforders, which are likely to ren- der them fliort-lived, will have no great temp- tations to endeavour to gain admiffion into fuch focieties ; and, if admitted, the danger from them will be lefs than on any other plan. Were it not for this danger, one might recommend the following plan, as one of the moft inviting.
In the plans hitherto mentioned it is im- plied, that, if either a member or his wife dies within any of the periods fpecified, the additional annuities, that would otherwife have become due, will be loft. But it would be much more agreeable to a purchafer, that they (liould be made certain to his wife, pro- vided fhe lives to the end of thefe periods,
{a) See another advantage mentioned under Qiiefl, VIII, p. 2fc'.
H \ tho"
102 Of the left Schemes for
tho' in the mean time his own life fhoiil4 fail. The value of fuch annuities may be computed by the rule in Queft. IX.
Suppofe, for in fiance, the fcheme to be " that a wife (liall be entitled certainly to a ^' life-annuity of 20/. the firfl payment of *' which (liali be made at the. end of i 2 years, *^ provided flie (liould be then alive, and her *' huiband dead; or at the end of any year ^* beyond this term in which Oie may hap- *• pen to be left a \yidow." Suppofe it alfo ilipulated, ** that flie Oiall be entitled to ?' 10/. more, or ':^o/. in all, on the fame *' terms, provided fhe fliould live 16 years.'* — The value of fuch an expe<ftation (interell: being at 3 per cent, and the probabilities of life as in Mr. De Mohre's Hypothefis) will be, in i\\Q mofl convenient round fums, fup- pofmg none admitted above 50 years of age, feven guineas in annual payments to be con- tinued during marriage, and to begin imme- diately'3 belides four guineas in prefent mo- ney for every year, as far as 1 5 years, that the huiband's age exceeds the wife's, if he is between 40 and ^o, and three guineas on the fame account if he is under 40 : Or, if the wliole value of the expedation is given in one prefent paynient, 70/. added to a guinea and half, for every year that the huiband's age fiiils fliort of 50, befides the payment juft inentioned on account of difparity of age,
J If
frGvidi?ig Annuities for Widows, 103
If the annuities are made to be annuities during widowhoods and not during lije^ and the advantage arifing from hence, is fuppof- pd equivalent to the difference between the improvement of money at ^per ce?it. and its real improvement ; the value of the expecfta- tion jult mentioned, (that is, its value at 4 per cent.) will be fix guineas in annual pay- ments; befides three guineas in prefent mo- ney, for every year that the huiband's age exceeds the wife's, if he is between 40 and 50 j and 2 guineas, if he is under 40 : Or, if the whple value of the cxped:ation is given in one prefent payment, 56/. added to 1/. 5^. for every year that his age falls fliort of ^o, be- lides the payment laft mentioned on account pf inequality of age. (^a)
{a) Suppofing 16 years the only term, tlie annuity 20/. and intercft at 4. per cent, the proper payments wiil be nearly, in the cafe of equal ages zx\A fingle payments, 46/. — 40/. — 29/. as the age of the man is 30, 40, or 50. Or, in (7«;/«.7/ payments, /,3.8o. — /.3.66. — /.3-i3' — Sup- pofing the woman's age lo years lefs than the man's, the fame values v/ili be, in finglc payments, l.^'i.()2. — /.56.56. — /.53.66. — In ^;7«Kj/ payments /.4.63. — /•5. — /.5.41. — It appears, therefore, that a fociety, fuppofmg money im- proved at the rate of 4 per cent, might entitle all married men Ind'ijcrminately^ who are under 50 years of age, to fuch 2n expectation as this for their wives, for either 60/. ;n c/.Y payment, or five guineas in «««;/,7/ payments. — But equity requires, that ditferent payments (hould be made, according to the different comparative ages of men and their wives ; and Tables might be formed for fhewing, at one view, what theie different payments aught to be in all cafes. If fuch Tables are wanting, recourie muft be had JO fome fuch eafy rules as thofe I have ftated above.
H 4 . He
104 Of the beji Schemes for
He that will give himfelf the trouble to calculate, agreeably to the directions in the Queftions to which I have referred, will find that, taking all particular cafes together, the rules now given come as near the truth as there is reafon to dsfire in an affair of this nature, the defers in fome cafes being nearly compenfated by the exccfes in others.
I have calculated here, as well as in moft other places, from Mr. T>e Moivres Hypo^ thejis, becaufe its conformity to the three Tables which I have fo often mentioned, convinces me, that it gives a proper medium between the different values of tow?i and country lives. In the country the probabili- ties of life are much higherj but in hondon^ and probably in all great towns and fome fmaller ones, they are much lower.
It is proper to add, that, according to the values of lives and furvivorfliips deduced both from the Lojidon and Dr. Halley^ Table, and taking intereft as low as 3 per cent, all wo- men whofe hufbands are under 50 years of age, might be entitled to an annuity of 24/. dur- ing Ufe (the firfl payment to be made at the end of the year in which they fliall be left widows) for the fum of 100/. fuppofing 3/, additional given on account of every year that they are younger than their hufbands. — ? At 4 per cent, an annuity of 30/. might be granted on the fame terms.
providing Annuities for Widows* loj
In the year 1690, the company of Mercers in London, adopted fuch a fcheme as that laft mentioned. For 100/. in one prefent pay- ment, they entitled every fubfcriber to a life' annuity for his widow of 30/.; and this, at that time, (when money bore 6 per cent, in- tercft) was confiderably lefs than the value of the money advanced, fuppofing men and their wives of equal ages. As the intereft of money funk, they funk alfo the annuity, firft to 25/. and then to 20/. and 15/. But, at laft, after carrying on the fcheme for above 50 years, finding the burden of the annui- tants too heavy, and likely to go on increa-? fing, they were obliged to drop the fcheme and to flop payment. In a little time, how- ever, by a parliamentary aid of 3000/. perann. which they are now enjoying, they were re- ftored to a capacity of making good all their engagements, and of paying their arrears. — » Their failure, is, indeed, much to be lamented ; for, in confequence of it, the public has lofl the benefit of an inftitution, that for many years promifed the happieft effeds, by en- couraging marriage, and affording relief to indigence. The rapid fall of the interefl of money; their admitting purchafers at too ad- vanced ages j and, particularly, their paying no regard to the difference of age between huf- bands and their wives, mufl: have contribut- 'ed much to hurt them. Some of the prin- cipal caufes, therefore, which have rendered
then^
Io6 Of the hejl Schemes for
them unfuccefsfiil, may be now avoidecli and for this reafon 1 fhould be glad to lee fome fimilar Icheme, providing, as this did, Annuities for life^ and not for widowhood, un- dertaken. If well planned it would, I think, be a proper objedl: of parliamentary encou^
ragement.
It muft, however, be remembered, that the iflue of the beft fchemes of this kind mufi: be in fome degree uncertain. For want of proper obfervations, it is not poffible to determine what allowances ought to be made, on account of the higher probabilities of life among females than males. No prudence can prevent all lofTes in the improvement of money; nor can any care guard againfl the inconveniencies to fuch fchemes, which mufi arile from thofe perfons being mofl ready to fly to them who, by reafon of con-r cealed diforders, feel themfelves moft likely to want the benefit of them.
The focieties, therefore, on which I have remarked in the firft Sedlion of this chapter, would have reafon to take warning from what has happened to tlie Mercer s Company, were the fchemes on which they are formed per- fediy unexceptionable. But I have demon- ftrated that thefe fchemes are very defedive; and that the longer they are carried en, the more mifchief they mull produce. 'Tis vain (as appears from (^efl. III.) to form fuch eftablifhments with the expedljition of fee-
ing
providing Annuites for Widows, 1 07
ing their fate determined foon by experience. If not more extravagant than any ignorance can well make them, they will go on prof- peroufly for 20 or 30 years; and, if at all tol- erable, they may llipport themfelves for 50 or 60 years ; and at laft end in diftrefs and ruin. No experiments, therefore, of this fort fhould be tried haftily. An unfuccefsful ex- periment mud be produ(5tIve of very perni- cious efleds. All inadequate fchemes lay the foundation ol prefent relief on Jutmr cala- mity, and afford alfiilance to ^Jew by difap- polnting and opprefiing multitudes.
As the perfons v^^ho conduct thefe fchemes can mean nothing but the advantage of the public, they ought to liflen to thefe obferva- tions. At prefent their plans are capable of being reformed ; but they cannot continue fo always ; for the greater number of exor- bitant paym.ents they now make to annui- tants, the more they confume the property of future annuitants, and the lefs pradi- cable a retreat is rendered to a rational and equitable and permanent plan {a). They fliould, thcreiore, immediately [b) either reduce
{a) See p. 82, 83. Sea. I.
[b) Thus; was the Zcw^aftfw Annuity Society to make their loweft annuity lo/. the next 20/. and the higheft 30/. they would probably be fafe. But, after procseding on their preltnt plan fome years longer, fuch a redudion ■would by no means be luflicient.
their
JoS Of Schemes for providing
their fchemes, or change them into one of thofe which I have propofed. But, I am afraid, this IS not to be expeded. The negled with which they have received fome remonftrances that have been already made to them gives reafon to fear, that what has been noWfaid Will be in vain; and that thofe who are to come after them, muft be left to rue the con. Sequences of their miftakes.
SECT. IV.
Of Schemes for providing Annuities for Old
Age.
A General difpofition has lately /hewn it- lelf, to encourage fchemes for granting- annuities to perfons in the latter flages of life^ and this has occafioned the 6th Queftion in t\\Q former Chapter; and, as a further and more particular diredion in cafes of this kind, J have thought it necelTary here to give the i:oJIowin<? Table.
Q
Valuo
AnniiiUes for Old Age,
109
VaJoMof i/./fr artr.. for life, af- ter 50, to per- fons whofe ages are
10
15
20
25
40
Values in me 1 |
Intereft |
prefent payment, 1 intereA 4 /i«r |
3 per cint. |
cent. |
|
1-235 1.583 2.028 |
2.015 2.444 2.989 |
2.594 3-369 4.446 |
3-644 4.508 5,667 |
5-953 |
7.232 |
Values in aitnti- |
Intereft |
al payments, till 50, to begin at the end of a year, intereft 4 pr ct. |
3 per cenTt |
.0789 .106 |
•113 .146 |
.146 |
•193 |
.203 .297 .466 |
.259 .366 '559 |
.822 |
.950 |
Values of the fame annuity af- ter 60, to ages
35 40
45 50
1.667 2.234 3-043 4-255
2.290 2.923 3.81 1 5.061
•135 .203
-327 .600
Values of the |
||||
fame annuity, |
||||
after 5 5,to ages |
||||
30 |
2. 114 |
2.937 |
.167 |
.21 1 |
35 |
2.722 |
3-632 |
.241 |
.297 |
40 |
3-732 |
4.708 |
•394 |
.464 |
45 |
5.088 |
6.1 15 |
•703 |
.803 |
,i6S
'245 .384
.679
The numbers In the 2d and 3d columns of this Table, multiplied by any annuity, will give the value of that annuity in ajmgk
pay.
110 Of Schemes for providing
payment, to be enjoyed for life, by the ages correfponding to thofe numbers in the firfl column, after the age mentioned at the head of that column. — And in the fame manner; the numbers in the 4th and 5th columns will give the values in annual payments. — Thus. The value of 44/. per annum^ to be enjoyed for life, after 50, by a perfon now 40, (inter- eft at A^ per cent.) is 5.95, multiplied by 44, or/. 261.9, in 2. fngle payment; and .822,' multiplied by 44, or /. 36.16, in annual pay- ments till 50, the firft payment to be made at the end of a year.
In order to find the fame values, partly in annual pay?nents, and partly in any given en- trance or adtnifjion-inojiey % fay; " As the va- *' lue of the groeti annuity in ^fngie payment, *' (found in the way juft mentioned) is to the ** given entrance-money ', fo is its value in an- ** nual payments, to a fourth proportional ; *' which, fubtra(fled from the value in annual " payments, the refnainder will be the annual " payment due, over and above the given en* ** trance-money."
E X A M P L E.i
Suppofe a perfon now 40, to be willing to pay 200/. entrance-money, befides fuch an annual payment for 10 years as iliall, toge- ther with his entrance-money, be fufficient
Afmukies for Old Age, 1 1 1
to entitle him to a life annuity of 44/. after ^o. What ought the annual payment to be?
Answer.
Z/.8.55. — For, /. 261.9, is to 200/. as /. 36. 1 6, to /. 27.6 1 3 which, fubtraded from /. 36.16, the remainder is /. 8*55.
This Table has been calculated from the .probabilities and values of lives in Tables III. and VI. The probabilities of life among the inhabitants oi London^ are (as I have of- ten had occafion to obferve) much lower than among the generality of mankind ; and the values in the preceding Table, had they been given agreeably to the LoTidon Obferva- tions, would have been lefs. But, certainly, an office or fociety, that means to be a per- manent advantage to the public, ought al- ways to take higher rather than lower va- lues, for the fake of rendering itfelf more fecure, and gaining (ovnQ profits to balance lojjes and expences.
There have lately been eflablifl:ied, 111 London, feveral focieties for granting fuch annuities as thofe now mentioned i and he that will compare their true values, as they may be learnt from the preceding Table, with the terms of admiffion into thefe focie- ties, as given in their ^x'miQ^ Abfra^s and
Tables,
1 1 2 Of Schemes for providing
Tables t muft be furprifed and fliocked. They are all impofitions on the pubHc, proceeding from ignorance, and encouraged by creduUty and folly.
It has been {hewn ; that the proper pay- ment, (allowing compound intereft at 4^ per cent.) for an annuity of 44/. to be enjoyed by a perfon now 40, for what may happen to remain of his life after 50, is 200/. in admif- fion-money-y befides 7.8.5 5, or 8/. i ^ -^m i^^ ^"* nual payments till he attains to 50, the firft of thefe payments to be made at the end of a year. — The conditions of obtaining this annuity, according to the Tables of the Lau^ dahle Society of Annuitants for the benefit of age y are 76/. 17J. in admiffion-money, and 6/. 14^. in annual payments, — According to the Ta- bles of the fociety of Lo?idon Annuitants for the benefit of age, the conditions of obtaining the fame annuity are 30/. in admifjion-mojieyy and 10/. in annual pay?nents.-—T\'\Q Equitable Society of Annuitants requires for the fame annuity 38/. ioj-. m admijjion- money ^ and 13/. in annual payments. The true value is, over and above the admifiion- money juft mentioned, an annual payment of 30/. \']s. (interelt reck- oned at 4 per cent.) or an annual payment of 36/. 1 5 J. intereft reckoned at 3 per cent. — •■ The London Union Society for the comfortable fupport of aged members promifes an annuity of no lefs than 50 guineas for life, after 50,
to
Annuities for Old Jge, 113
to a perfon now 40 for 40/. \os. in admif- fion-money, and 7/. in annual payments.
The Amicable Society of AnniiitiVits for the benefit of age y promifes an annuity of 26/. />£"/' annuWy for life, to a perfon now 40, after at- taining to 50j for 28/. 165. in admifwn-rno- neyy and 6/. in annual payments. — Tiie true value of this annuity is 28/. i6j-. in admif fion-money, and 17/. 8/. in annual payments., (interefl fuppofed -aX. a^ per cent.) -, or the fame fum in adfnijjion- money y and 20/. 185. in <^/Z- nualpayme7tts, intereft fuppofed at 3 per cent.
The Provident Society for the ben ft of age promifes an annuity of 25/. to a perfon now 40, after attaining to 50, for 34 guineas in admiffion-money, and eight guineas in annu- al payments. The true value is, 34 guineas in admifion- money, and 15/. \2s. in annual payments, interefl: at 4 per cent. -, or, the lame fum in admijjion- money , and j 9/. in aniiual payments, interefl: being at ^ per cent [a).
But I will not tire the reader, by going, in this manner, thro' the fchemes of all thele io- cieties. The contrivers of them, it is certain, can know nothing of the principles on which the rule in Quefl:. VI, and the demonftration of it in the Appendix is founded ; and, there- fore, if unwilling to be guided by the autho-
(a) The acccunt here given of the terms on v/hich i perfon whofe age is 40, is admitted into thefe focieties, I have taken from their printed Tables as they flood at the end of the year 1770. — In the younger agej the defici- encies are gre:ucr.
I rity
1 1 4 Of Schemes for providing
rity of mathematicians, it may not be poflr* ble to convince them of their miftakes. I will, however, offer to them the following demonflration, which will be underftood, without difficulty, by every one who knows how to compute (^) the increafe of money at compound intereft.
The value of a life at 50, (intereil being at 4 per ce}2t.) is ii4 years purchafe by Table VI. For an -annuity, therefore, of 44/. per anniun for life, to be enjoyed by a perfon at this age, 498/. ought to be given. T^wo in three of a number of perfons at the age of 32 will, (by Tables III, IV, and V,) live to 50; and therefore, in order to be able to pay an annuity to them of 44/. for life, after 50, the money now advanced by every three^, ought to be fuch as Vv'ill, in confequence of being laid up to be improved, increafe in 18 years to double 498/. or to 996/. — From the pre- ceding Table it may be learnt, that the mo- ney which ought to be advanced by every fingle perfon is 165/. or by three perfons 495/. and this, in 18 years, will double it- felf, or increafe to ju(l the fum that will then be the value of the annuities to be paid. — But the money required in this cafe by the Laudable Society, is 14/. 11^. 9^. from each member at admiffion, befides an amiiial pay- ment of 4/. The admiffion-money, there- fore, of two members, being 29/. '^s, 6d.
{a) The eafieft method of doing this, is taught in the rules annexed to the Tables iii the Appendix.
may
Annuities fir Old Agd i j ^
may be increafed to twice this fum, or to 58/. ys. An annual payment of 4/. for iS years will, ifperfeiflly improved at \ per cent. compound interefi:, increafe to 102/3 iind two fuch annual payments will increafe to 204/.
The whole pay, therefore, of two mem- bers will produce at the end of 18 years 262/. yj. -^A third part, 1 have faid, will die without attaining to 50, and thefe will live one with another 10 years. An annuity of 4/., tor this time, will produce a capital of 48/. and this capital improved for eight years more will increafe to 66/. The whole profit, therefore, from the member who will die is, his admiffion-money doubled and ad- ded to 66/. or 95/. 3J-. bd* And this fum added to 262/. 7J. makes 357/. loj-. td. the whole money with which the fociety can be provided, at the end of 1 8 years, to bear the expence of /wolife-annities, worth toge- ther 996/.
By a fimilar computation it may be found, that the improvement of money at only ^ per cent. v^'iiXJink i\\Q former fum to 329/. at the fame time that the value of the annuities will be raifed to 1 100/.
The deficiencies in the fchemes of all the other focieties, except the Provident Society, are no lefs confiderable (^/). — What eonfu-
lion
[a) Some of thefe focieties tell us, that the payments
on admiffion fliall increafe, as the number of nVembers
increafes i and they have praitifed on this rule juft as if
I ?. ihQ
1 1 6 Of Schemes for providing
fion then mufl they produce Tome time or other ? How barbarous is it thus to draw mo- ney
the value of an annuity was nothing determinate in itfelf, but depended on the number of perfons who have been purchafeis. But the true defign may perhaps be, to quicken the public in their applications.
Should any of thefe focieties, fenfible of their miftakes, refolvc to reform themfeivcs, they ought toconfider, that this cannot he done by only obliging //v/ar^ members to pay the juft values of the annuities promifed them. All the prefcnt members mufl: likewife, befides raifing their payments, make compenfation for what they have hi- therto paid too little ; and this compenfation is to be cal- culated in the following manner. — "Find the whole *' amount to the prefent time of the payments which have '•' been made. Subtradl this from the whole amount of " the payments which Jhould have been made; and the *' remainder will be the compenfation required."
Example. In the Laudable Society of Armziitants, the condition of a titld to 44/. per annum for life, after 50, to a perfon at the age of 40, was, 4 years ago, 34/. lyj. in admiffion-money, befides an annual payment of 6/. 141. 'till he attained to 50. — The admilHon- money will, (reckoning compound intereft zf^perceiit.) amount in four years to 39/. 45. and the annual payment to 28/. The whole aniount, therefore, of the payments of a member admitted 4 years ago, is 67/. 4^. — But the value of the annuity was 37/, 45. in annual payments, befides 34/. I'js. in admilTion-money ; and thefe payments, during the 4 years, would have amounted to 195/. The diffe- rence, therefore, betv/een thefe two amounts, or 127/. 16 i. is the cojifpefifation vi^hich fuch member ought to pay ; and if he continues a member without paying it, (be- fides raifing his annual contribution to 37/. 45.) he mufl: either lofe his annuity, or owe it to injultice.
I have taken intereft here at '^pcr cent, becaufe I think thefe focieties cannot reafonably depend on always im- proving the money they receive at a higher rate.
Since I Writ the above, I have found, that the admif- fioii-moncy required by this fucicty has lately received
another
Annuities for Old Age, 1 1 7
neyfrom the public by promifes of advantages that cannot be obtained ? Have we not al- ready fuffered too much by hubbies -y and, if nothing elfe can check the frenzy that en- courages them, ought not the legiflature to interpofe its authority ?
I do not, however, mean to condemn all inftitutions of this kind. They may be very ufeful, if the full values are taken, and pro- per care is ufed in the improvement of money. Intereft, in thefe cafes, ought not to be reck- oned higher than 3 per cent, and, fuppoiing money improved at this rate, a perfon, for a fingle payment of 50/. before he is 40, might be entitled to a life-annuity of 10 gui- neas after ^^-y or, if he chufes it, to a life-an- nuity of 17/. after 60. But if he pays the fame fum before he is 34, he might be entitled to a life annuity of 14/. after 55, or 22/. after 60. 25/. might purchafe for him half \\it{c annuities; and 100/. double.
A fociety or office that would go on this plan, might do great fervice. Perions in
another advance. At the age of 40, in particular, it is advanced to ic8/. 7^. — when they have further either ad- vanced the admiffion-money to douhlc this fum, or tripled the annual payments, they will be nlmiJi right v/ith refpe6t to this particular age, provided the cofnpe}ifalion-77:oncy,]^^^ mentioned, has been paid.
I'hefe focieties, tho' their plans are fo infufficient, may, after beginning their payments to annuitants, continue them 15, or, perhaps, 20 years j but it will be by rob- bing all the younger members.
I 3 the
1 1 8 Of Schemes foj- providing
the lower flations of life might be brought to a habit of induftry, in the beginning of life, by driving to get 25/. or 50/. before- hand in order to purchafe fuch annuities, and thus to make provifions for themfelvesin the more advanced parts of life, when they will be incapable of labour.
There are now eftabliflied \n Hollcmd (omt inilitutions of this kind. — Any poor perfons there, I am informed, who can, betore they attain to a particular age, lay up 50/. may make ufe of it in buying for themfelves a right to be admitted, when 50, or at any time afterwards, to houfes prepared on pur- pofe, for providmg them with all the con- veniencies of lodoinp and board. This is an excellent infiiitution ; and I wiili there was fome imitation of it in this kingdom.
Confiderable profits would, in this, cafe be received, from the payments o^ fome who would chufe to delay going into fuch houfes ; and of others who would grow rich enough to be above them.
It is proper to obferve here, that inftitutir ons of this kind would furnidi one of the fafef W'^ys of providing for widows. — A mar- ried man might, by paying 100/. before his wife attained to 40, entitle her, after ^^, or 60, to a life annuity of 21/. or 34/. Or, by paying the fame fum before (he attained to, 3|, he might entitle her;, after the fame ages^
■ to.
Annuities for Old Age, 1 1 9
to a life annuity of 28/. or 44/. (^) ; and in this cafe he would have a chance of (haring himfelf in the beneiit of the annuity.
I have called this ihtjafeji way of provid- ing for widows, becaufe attended with none of the dangers arifing from difproportion of age between men and their wives, and from the admilTion of perfons labouring under con- cealed diflempers.
I cannot conclude this Sedion, without mentioning the following plan of a provilion for Old Age.
Let 1 3 guineas be given as entrance-money ; and let befides i /. 2/. 3/. 4/. &c. be given at the beginning of the ifl, 2d, 3d, 4th, &c. years, as the payments for thefe years refpec- tively ,• and let the lail: payment be 16/. at the beginning of the i6th year. All thefe payments put together will, according to the probabilities of life in the 3d, 4th and 5th Tables, (intereft being at4^6T cent.) entitle a perfon, whofe age was 40 when he begun them, to an annuity, after 1 5 years, begin- ning with 15/. and increaling at the rate of i/. every year, 'till, at the end of 15 years more, or (^) v/hen he has attained to 70, it
[a) The fame payment before 30, would entitle to an annuity of 22/. after 50.
[b) According to the probabilities of life in the London Table, this annuity (hould be greater. — A Theorcin for finding what the annuity ought to be in thefe cafes, is given in the Appendix, Note (1).
I 4 becomes
I20 Of Schemes for providing^ &c.
becomes a ftanding annuity of 30/. for the remainder of his life.
If the addition of three guineas is made to the entrance-money, iox every year that any life betvveen 30 and 40 falls iliort of 40, the va- lue will be obtained nearly, of the fame an- nuity to be enjoyed by that life, after the iame number of years, and increafing in the lame manner, 'till, in 30 years, it becomes ftationary and double .-—'X\\\^ plan is particu- larly inviting, as it makes the Icirgeji payments become due, when the near approach of the annuity renders -the encouragement to them grcatcji ; and as, likev/ife, the annuity is to increafe continually with age, till it comes to be highell [a], when life is moil in the de- cline;^
{flj The lower part of mankind are objccls of parti- cular compaiTion, when rendered Incapable, hy accident, ficknels, or age, of earning their fubliftence. This has given rile to many very uieful focitties among them, for granting relief to one another, out of little funds fupplicd by weekly contributions. A fociety of this kind, formed on the following plan, would probably thrive, and might, on fome accounts, be even more ufeful than the inftitu- tions in Holland^ mentioned in p. 118.
Let the fociety, at its firfl: eftablifliment, confift of lOO perfons, all between 30 and 40 ; and whofe mean age may therefore be reckoned 36; and let it be fuppofed to be always kept up to this number, by the admiffion of new members, between the ages of 30 and 40, as old members die off. Let the contribution of each member be four-pence ^rr week, making, from the whole body, an annual contribution of 85/. 17^. — Let it be further luppofcd, that feven of them will fall every year into dif- ordcrs, that fhall incapacitate them for ievcn weeks.-r—
Of the Amicahk Corporation^ Sec. 1 2 1
cline, and when therefore it will be moft ufeful. — -It is further a recommendation of this plan, that lefs depends in it on the im- frovement of money than in moft other plans. — But I mull leave thefe hints to be purfued by others.
SECT. V.
Of the Amicable Society for a perpetual Affu-^ ranee Ojice : And the Society J or equitable Afjiirances on Lives and Survivorfiips,
THE loth Problem has been given, with a particular view to the corporation of the Amicable Society, for a perpetual Aflu-
30/. I2X. of the annnal contribution will be juft fuffici- ent, to enable the fociety to grant to each of thefe \zs, per week, during their illnelfes. And the remaining 55/. per annum, laid up and carefully improved, at 3^ per cent, will incrcafe to a capital that fhall be fufficient, accord- ing to the chances of life in Tables III, IV, and V, to enable the fociety to pay to every member, after attaining to 67 years of age, or upon entering his 68th year, an annuity, beginning with 5/. and increafing at the rate of I /. every year for fcven years, 'till, at the age of 75, it came to be a Handing annuity of 12/. for the remain- der of life.
Were fuch a fociety to make its contribution j^iv?;- pencc per v/cek, an allowance of 155. might be made, on the fame fuppofitions, to every member during ficknefs ; bsfidcs the payment of an annuity beginning with 5/. when a member entered his 64th year, and increafing for 15 years, 'till, at 79, it became fixed for the remainder of life at 20/.
\i the probabilities of life are lower among the labour- ing poor, than among the generality of mankind, this plan will be fo much the more fure of fucceeding.
ranee-
122 Of the Amicable Coj^poration
rance-ofiice on fingle lives, kept In Serjeant' s- Inn. This fociety was eftabliflied in 1706, and is the only one I am acquainted with, which has flood any confiderable trial from time and experience. The annual payment of each member ufed to be 6/. \s. payable quarterly ^ but it has been lately reduced to 5/. The whole annual income, hence arifing, is equally divided among the nominees^ or heirs of fuch members as die every year; and this renders the dividends among the nominees in different years, more or lefs, according to the number of members who have happened to die in thofe years. But the fociety now en- gages, that the dividends fliall not be lejs than 150/. to each claimant, tho' they may be more. — None are admitted whole ages are greater than 45, or lefs than 12 ; nor is there any difference of contribution allowed on ac- count of difference of age.
This fociety has, I doubt not, been very ufeful to the public; and its plan is fuch, that it cannot well fail to continue to be fo. It might, however, certainly have been much more ufeful, had it gone from the firfl: on a different plan. It is obvious, that regulating the dividends among the nominees by the number of members who die every year, is not equitable ; becaufe it makes the benefit Vv'hich a member is to receive to depend, not on the value of his contribution, bu£ on a contingency y that is, the number of members that iball happen to die the fame
yeav
for affiiring Lives. 123
year with him. This regulation muft allb have been diiadvantageous to the focietyj as will appear from the following account of the natural progrefs of the affairs of fuch a fociety, when eftablillied on a right plan.
Suppofe a thoufand perfons, whofe com- mon age is 36, to form themfelves into a fo- ciety for the purpofe of aJJ'uring a particular fum at their deaths, to fuch perfons as they iliall name, in confideration of a particular annual-contribution to be continued during their lives. Suppofe the annual contribution to be 5/. and the firft payment [a) to be made immediately. Suppofe, likewife, the original number of the fociety to be con- flantly kept up by the admiffion of new meniibers, at 36 years of age, in the room of fuch as die. — In Queft. X. p. 33, it appears, that an annual payment, beginning immedi- ately, of 5/. during a life at the age of 36,{hould entitle, at the failure of fuch a life, to 172/. reckoning intered at 4 per cent, and taking Mr. De Moivre?, valuation of lives. — A /'/60//- 7^;^^ perfons, all 36 years of age, will die off at the rate of 20 every year. The difburfe- ments, therefore, of fuch a fociety will be, the firft year, 20 times 172/. or 3,440/. and its income will be 5000/. It will, there- fore, at the end of the year, have a furplus
{a) Such payments, it has been fliewn, Queft. VIII. p. 28, are better than any half yezx\y or quarterly payments, ^nd at the fame time they lave fome trouble.
of
1 24 Of the Amicable Corpoi'atiQU
of 1560/. to put to intereft. — In confe- quence of the yearly acceffions to fupply va- cancies, the number dying annually will be always increafnig after the firft year. In 50 years it will attain to a maximum ; and then, the affairs of the fociety will become Jiatio- nary, and the number dying annually will be 40, and its annual expence will be 6,880/. exceeding the annual contribution 1,880/. But, in the mean time, by improving its fur- plus moneys, it will have raifed a capital equal to this excefs, and, confequently, its af- fairs will be fixed on a firm bafis for all fub- fequent times.
Suppofe now, that fuch a fociety, at its eftablilhment, fliould refolve to divide its whole yearly income among the nominees of deceafed members. The effed of this would be, that no capital could be raifed; that the dividends payable to nominees would diminifh continually, till, at the time that the great- eft number of members came to die annuaUy^ or at the end of 50 years, they would be re- duced to half; and all claimants, after this period, receive too little, becaufe the firft clai- mants had received too much {a).
At
[a) The reverfe of this will take place, if fuch a fo- ciety begins with admitting all at all ages, and afterwards changes its plan, and limiti the age of admiirion. In this cafe, the number o{ yearly deaths will he great ejl at firft, and the dividends ftnallcji. In confequence of altering its plan, \.hc yearly deaths will leffen gradually, and the divi- dends rife; but in time both would return a^ain to their original ftate. ' ■ ' The,
for ajjuring Lives* 125
At the time of the inflitution of the Ami- cable Corporation^ the intereil: of money was at 6 per cent, and, as they admit all between 12 and 45, the mean age of admiffion can- not probably be fo great as 36. It appears, therefore, that had they avoided the error now mentioned, and gone from the firft on the plan I have defcribed -, they might have all along paid to each 72ominee 172/. befides railing a capital much greater, in proportion to the number of members, than that I have ipecified; by the help of the excefs of their annual payments above 5/. and fome other advantages which they have enjoyed [a). Indeed, I cannot doubt but that, with thefe advantages, they might, before this time, have found themfelves able to pay at leaft 200/. to each nominee , and at the fame time
The following fails incline me to fufpecl, that this remark may be applicable to the Amicable Corporation,
Firft. In their original charter, as it is given in their printed abftraiSls, there is no limitation of age menti- oned ; but 31 years afcerwards, I find a bye-law made againft admitting any perfon who fliould be above the age of 45, or under 12. — iJecondly. In their printed advertifements in 1770, it is faid, that in 59 years they had paid, among 3643 claimants, 378,184/. from whence it follows, that tho' the average of their divi- dends, for the laft 17 years, has been 154/. the fame ave- rage, for 59 years, is only 104/.
(a) A furplus from a thoufand members of only Jive /hillings per annum, duly improved, at ^percent, would, in 41 years, produce a capital of 25,000/.
reftrided
126 Of the Amicable Corporation
reftridled themfelves, as they now do, to' an annual payment of 5/. {a).
I have already mentioned one inftance in Vv'hich the plan of this fociety is not equita- ble. Another inftance of this is, their re- quiring the fame payments from all per- fons under 45, without regarding the diffe- rences of their ages ; whereas, the annual payments of a perfon admitted at 45, Ought to be double the annual payment of a perfon" admitted at 12.
Further. The plan of this fociety is fo narrow, as to confine its ufefulnefs too much. It can be of no fervice to any perfon whofe age exceeds 45. It is, like wife, far from be- ing properly adapted to the circumftances of perfons, who want to make alTurances on theii" lives, for only fhort terms of years. — Thus ; the true value of the affurance of 150/. for 10 years, on the life of a perfon whofe age is 30, is, by Queft. XIV, (intereft being at 3 per cent.) it. 13^. in annual payments for ro yeats, to begin at the end of the firfc year ; and fubjed to failure when the life fails. But fuch an affurance could not be made, in this fociety, without an annual payment of 5/. — Neither is the plan of this fociety at all
((j) It (liould be remembereJ, that all this is faid or» tlie fuppofition, that proper care has been taken to Iccep out unhealthy perfons; and that the probabilities of life among the meir.bers of this fociety, are the fame with? thofe in the 3d. 4th, and 5t:h Tables, in the Jppendix.
4 adapted
for ajjurhig Lives. 127
adapted to the circumftances of perfons, who want to make aflurances on particular lur- vivorfliips. — For example. A perfon pof- iefled of an eftate, or falary, which muft be loft with his life, has a perfon dependent Upon him, for whom he delires to iecure a fum of money, payable at his death. But, he deiires this only as a provilion againft the danger of his dying^r/?, and leaving a wife, or a parent, without fupport. In thefe cir- cumftances, he enters himfelf into this fo- ciety ; and by an annual payment of 5/. en- titles his nominee to 150/. in a few years, perhaps, his nominee happens to die ; and, having then loft the benefit he had in view, he determines to forfeit his former payments, and to withdraw from the foclety. In this way, probably, this fociety muft have gain- ed fome advantages. But the right method would have been, to have taken from fuch a perfon the true value of the fum aflured, ** on the fuppofttion of non-payment, pro- ** vided he fhould furvive." \w this way he would have chofen to contract Vv''ith the fo- ciety ; and had he done this, he would have paid for the ajjurance, (fuppofmg intereft at 3 per cent, his age 30, the age of his no- minee 30, and the probabilities of life as in the 3d, 4th, and 5th Tables) 3/. %s. (a) in annual payments, to begin immediately, and
to
{a) The value of 150/. payable at the death of a per- ibfi, aged 30, provided he lurviveii another perfon of the
fame
128 Of the AJfiirances for
to be continued during the joint continuance of his own life, and the Hfe of his 7iominee,
All thefe obje6lions are removed by the plan of the fociety kept in Nicholas-Lane^ Lcmbard-Street, which has juftly filled itfelf the Society for Equitable Ajjurajices on Lives and Stirvivorjhips, This Society, if due care is taken, may prove a very great public be- nefit. It v^as founded, in confequence of propofals which had been made, and lec- tures, recommending fuchadefigUjwhich had been read by Mr. Dodfon^ the author ot the Mathcfnatical Repojitory, It aflures any fums^ or reverfionary annuities, on any liveSj for any number of years, as well as for the whole continuance of the lives, at rates fettled by particular calculation, and in any manner that may be beft adapted to the views of the perfons affured. That is j either by making the afiured fums payable certainly at the fai- lure ot any given lives -, or on co?iditio7t of fur- vivorfhip ; and alfo, either by taking the price of the affurance in one prefent payment ; or in annual payments, during any lingle or
fame age, is, by Queft. XI. Chap. I. 7.45.65; and this value divided by 13-43, (the value increafed by unity, of two joint lives both 30) gives /. 3.4, or 3/. gj. — The value of the fame reverlion, according to the pro- babilities of life in London^ is, A 49,19, in <j«t payment ;■ and 4.16, in amwal payments, during the joint lives, the firft payment to be made immediately.
join^
^qtiitdhk Affuranccs on Lives, iig
joint lives, or any terms lefs than the whole continuance of the lives. — In fhort ^ the plan of this fociety is fo extenlive, and fo impor- tant, that I cannot fatisfy my own mind, without offering to the gentlemen concerned in the direction of it, the following obfer-^ vations, hoping they will not think, them impertinent or improper.
Firft. They fliould confider what diftrefs would arife from the failure of fuch a fcheme in any future time ; and v/hat dangers there are, which ought to be carefully guarded againft in order to fecure fuccefs. I have al- ready more than once obferved, that thofe perfons will be moft for flying to thefe efta- bliiliments, who have feeble confcitutions, or are fubje(5t to diftempers, which they know render their lives particularly precarious i and it is to be feared, that no caution will be fufficient to prevent all danger from hence.
Again. In matters of chance, it is impof^ iibie to fay, that an unfavourable run of events will not come, which may hurt the beft contrived fcheme. The calculations on- ly determine probabilities; and, agreeably to thefe, it may be depended on, that events will happen on the whole. But at particu- lar periods, and in particular inflances, great, deviations will often happen ; and thefe de- viations, at the commencement of a fcheme, muft prove either very favourable, or very unfavourable.
K But
^3*^ ^f ^^^ Society fiy
But further. The calculations fuppofe, that all the monies received are put out im- mediately to accumulate at compound inte- reft. They make no allowance for lofTes, or for any of the expences attending manage- ment. On thefe accounts, the payments to a fociety of this kind, ought to be fomewhat more than the calculations will warrant. The interefl of money ought to be reckoned low; and fuch Tables of Obfervations ufed as give the higheft values. Mr. Dodfon^ I find, has paid due attention to all this, by reckoning interefl, in his calculations for this fociety, at 3 per cent, and taking the lowed of all the known probabilities of life, or thofe deduced from the London bills of mortality [a). There is, befides, a liberty provided of making a call on all the members, in cafe of any par* ticular emergency. It is, therefore, highly probable, that this fociety is fecure. The laft expedient, however, would be a very dif- agreeable one, fliould there be ever any oc-
[a] It ought, however, to be remembered here, that in. felling life-annuities to commence either immediately. Or after given terms ; and alfo in fome other cafes, the values come out kfs in confequence of /(jzc^rr probabilities of life. Would it, m fuch inlhmces, be taking an unfair advantage, to eftimate the values by the 3d, 4th, or 5th Table in the Appendix, rather than the Z(57?<'/(5« Table ?— Thus ; was the fociety to fell 20/. per annum, for life, to a perfon nov/ 30, after attaining to 50, the value, ac- cording to Dr. Halley\ Table, would, reckoning interefl zx iper cent, be 90/. in a fingle payment ; but, accord- ing to the £o«fl'c« 'liable, the value would be only 70/.
cafion
Equitable Ajfurances on Lives, 131
Cafion for having recourfe to it ; and, in order to guard ftill more effedually againft danger, it would not, I think, be amils to charge a profit of 3 or d^per cent, on all the payments. —Should the confequence of this prove, that in fome future period the fociety fhall find itfelf pofTefTed of too large a capital, the harm will be trifling, and future members will reap the advantage. But this leads me to repeat an obfervation of particular confe- quence.
As this fociety is guided in every inftance, by ftri(ft calculation, it is not to be exped:ed that it can meet with any difficulties for many years j becaufe, not till the end of many years after it has acquired its maximum of members, will the maximum of yearly clai- mants and annuitants come upon it ? Should it, therefore, thro' inattention to this remark, and the encouragement arifing from the pof- feffion of a large furplus, be led to check or flop the increafe of its flock by enlarging its dividends too foon, the confequences might prove pernicious.
Again ; I would obferve, that it is of great importance to the fafety of fuch a fociety,
that its affairs fhould be under the infoediion
i.
of able mathematicians. Melancholy expe- rience fhews, that none but mathematicians are qualified for forming and conducing fchemes of this kind. — In (bort; dangerous miftakes may fometimes be committed if the K 2 affairs
132 Of the Society for
affairs of fuch a fociety are not managed care- fully and prudently. One inilance of this I cannot avoid mentioning.
A perfon, who delires to affure a particu- lar fum, to be paid at the failure of his life, on condition of the furvivorfhip of another life, may chufe to pay the value in annual contributions during the continuance of his own lingle life, rather than during the con- tinuance of the joint lives, becaufe the an- nual contributions, in this cafe, ought to be much lefs. But a fociety that would prac- tife fuch a method of afj'uraiice would hurt itfelf; for, as foon as the life, on whofe fur- vivorfhip the aifurance depends, is extindl, the perfon affiired, if then living, would have no longer any benefit in view , and, there- fore, would make his payments with reluc- tance, and, in time, perhaps, entirely with- draw them; the confequence of which would be, that the fociety would fuffer a lofs by being deprived of the juft value of the ex- pedation it had granted. The plan of a fo- ciety ought always to be fuch, as that the lolTes arifing from difcontinuance of payment, Ihould fall on the purchafer, and never on the fociety.
I mud not forget to add, that it is necef- fary, that fuch a fociety fhould be furnifhed with as complete a fet of Tables as poffible. This will render the bufinefs of the fociety much more eafy, and alfo much more capa- ble of being condu(5ted by perfoxTS unfkilled
in
Equitable Ajurances on Lives. 133
in mathematics. It will alfo contribute much to lis fafefy. For in all cafes to which Ta- bles can be extended, there would be no oc- cafion for employing any calculators ; and, confequently, a danger would be prevented to which, tho' it is not now, it may hereaf- ter be expofed ; I mean, the danger of hap- pening to truft unfkilful, or carelefs calcula- tors.— Mr. Dodfon, I find, has furnifhed this fociety with fome important Tables ; and his fkill was fuch, that there is no reafon to doubt, but they may be depended on. They have alfo others which, I believe, are fafe and accurate. But there are fome ftill want- ing which fhould be fupplied ; and all fliould be fubjeded to the examination of the beil judges, and afterwards publiflied^ together with a minute account of the principles af- fumed, and the method taken in compofing them. Such a publication would be a valu- able addition to this part of fcience ; and it would alfo be the means of increafing and eftablifliing the credit of the fociety.
In Queftions 4th, 6th, icth, iith, 14th, 15th, and 1 6th, I have, with a particular view to this fociety, given rules, by which may be formed every Table it can want, for {hew- ing the values of affurances on the whole du- ration, or any terms, of any one or two lives, in all poffible cafes ; and nothing but care and attention can be neceflary to enable any good arithmetician to calculate from them. K 3 Perhaps,
134 Of the Society for, &c.
Perhaps, this may be as much bufmefs as any one foclety {hould undertake. Rules, how^ ever, for finding the values of aJJ'urancesy in moft cafes, where the whole duration of any three lives is concerned, may be found in Mr. Stmpfons, Select Exercifes, from pag. 299 to p. 307 'y and it is not poffible they ihould follow a better guide.
CHAP,
[ ^35 ]
CHAP. III.
CyPuBLic Credit, and the National Debt.
THE National Debt is a fubjedl in which the public is deeply interefted. Some obfervations have occurred to me upon it, which I think important ; and for this rea- fon, though foreign to my chief purpofe in this work, I cannot help here begging leave to offer them to the reader's attention.
The practice of ralfing the neceffary fup- plies for every public fervice, by borrowing money on intereft, to be continued 'till the principal is difcharged, muft be in the high- efl degree detrimental to a kingdom, unlefs a plan is fettled, for putting its debts into a regular and certain courfe of payment. When this is not done, a kingdom, by fuch a prac- tice, obliges itfelf to return for every fum it borrows infinitely greater fumsj and, for the fake of a prefent advantage, fubjeds itfelf to a burden which muft be always growing heavier and heavier, 'till it becomes infup- portable.
This feems to be now the very ftate of
this nation. At the Revolution, an asra
K 4 in
1^6 Of Public Credit,
in other refpeds truly glorious, the pradlice I have mentioned begun. Ever fince, the public debt has been increafing fall, and every new war has added much more to it, than was taken from it, during the preced-* ing period of peace. In the year 1700, it was 16 millions. In 1715} it was ^^ mil- lions. A peace, which continued 'till 1740, funk it to 47 millions ; but the fucceeding war increafed it to 78 millions ; and the jiext peace funk it no lower than 72 mil^ lions. In the laji war it rofe to 148 mil- lions ', and, at a few millions lefs than this fum it now Hands, and probably will {land, 'till another v/ar raifes it perhaps to 200 mil- lions.— One cannot refleft on this without terror.-— No refources can be fufficient to fup- port a kingdom long in fuch a courfe. 'Tis obvious, that the confequence of accumulat- ing debts fo rapidly j and of mortgaging pof- terity, and funding for eternity, in order to pay the intereft of them ; muft in the end prove deftrudive. Rather than go on in this way, it is abfolutely neceifary, that no mo- ney fhould be borrowed, except on annuities, which are to terminate within a given pe- riod. Were this pradiifed, there would be a LIMIT beyond which the national debts pould not increafe -, and time would do that fiecejj'arily for the public, which, if trufted to the oeconomy of the condu6^ors of its affairs, jii.ight poffibly never be done.
3 This,
and the National Debt. 137
This, therefore, is one of the propofals to which, on this occafion, I wifh I could en- gage attention. — I am fenfible, indeed, that the frefent burdens of the ilate would, in this cafe, be increafed, in confequence of the greater prefent intereft, that would be necef- fary to be given for money. But I do not confider this as an objection of any weight. For let the annuity be an annuity for a 100 years. Such an annuity is, to the prefent views of men, nearly the fame with an annui- ty for ever j and it is alfo nearly the fame in. calculation, its value at 4 per cent, being 244 years purchafe, and therefore only half a year's purchafe lefs than the value of a per- petidty. Suppofing, therefore, the public able to borrow money at 4 per cent, on annuities for ever, it ought not to give above is. jd. per cent, more for money borrowed on annui- ties for 100 years : But Ihould it be obliged to give a quarter, or even an half per cent, more {a), the additional burdens derived from hence,
{a) Thefe annuities might be kept r8 years without being much diminifhed in value; for, fuppofing intereft at \ per cent, an annuity for 82 years, is within a 49th part, or 2/. in 98/. worth as much as an annuity for a JOO years.
Perhaps, in this way of raifing money, it might be beft to offer a higher intereft at firft, which fliould fall to a lower, at the end of given intervals. Thus, tho' 4I for 100 years is equal in value to ^ per cent, for 17 years, and after that ^percent, for 83 years, yet the latter might jippear more inviting.
would
138 Of Public Credit,
■would not be fuch as could be very fenfibly felt 'y and the advantages, arifing from the necefTary annihilation of the pubHc debts by time, would abundantly overbalance them.
Thefe advantages would be, indeed, un- fpeakably great. By fuch a method of raifing money, the expence of one war would, in time, come to be always difcharged, before a new war commenced; and it would be im- poflible, that a ftate fliould ever have upon it, at any one time, the expence of many wars; or any larger debts than could be con- tracted, within the limited period of the an- nuities : and, confequently, it would enjoy the invaluable privilege of being rendered, in fome degree, independent of the manage- tnent of its finances by ignorant or unfaithful fervants.
I muft add, that it is by no means necef- fary, that the limited period of the annuities fhould be fo long as I have mentioned, or 100 years : And that, at any time before the expiration of this period, the public might employ any furplus monies, in extinguKhing part of the annuities, by purchafing them for itfelf at the market price i and thus it might aid the operations of time, and keep its debts within any bounds, that its interell rendered necefTary. — Our government has, I know, in fome inftances adopted the plan now propof- ed ; but it is to be wiflied that, inftead of
retrading
and the National Debt. 139
retrading [a) it, as was once done, it had been carried much further.
I am, however, far from intending to re- commend this plan as the beft a ftate can. purfue. There is another method of gaining the fame end, which is, on many accounts, preferable to it. I mean, " by providing an ♦' annual faving, to be applied invariably, ** together with the interefl of all the fums *' redeemed by it, to the purpofe of difcharg- ** ing the public debts : Or, in other words, ** by the eflablilhment of a permanent sink-
** ING FUND."
It is well known, that this plan has been alfo adopted by our government; but, tho' capable of producing the greateji efFedls in the eqfieji and fureji manner, it has never been carried into execution. It will abundant- ly appear from what follows that this obfer- vation is jufl,
Suppofe the annual faving to be 100,000/. This fum, applied now to difcharge an equal debt, bearing intereft at \per cent, will tranf- fer to the public, from its creditors, an an- nuity of 4,000/. At the end of a year, then, there would be a faving of 104,000/. which would transfer to the public another annuity of 4, 160/. and make the faving, at
{a) In the year 1720, the nation was put to the ex- pence of three millions, in order to reduce feveral long and fhort annuities then fubfifting, to redeemable perpe- \uitieit
the
140 Of Public Credit^
the end of two years, to be 108,160/.— Thus, the original fund would go on increaf- ing, at the fame rate with money improved at 4 per cent, compound intereft. — At the end of three years it would be 1 12,486/. At the end of 18 years, 202,587/. Of 36 years, 410,393/. and of 95 years (^), 4,151,138/. — At the end of 93 years, then, the nation might be eafed of above 4 milHons per annum in taxes; and above 100 millions of its debts would be difcharged, gradually and infenfibly, at no greater expence than 100,000/. per an- num-, and, without interfering with any of the refources of government; or making any other difference, than q2sx{\x\^ funds to be en- gaged for a courfe of time to the public^ that would have been otherwife neceifarily engag- ed to its creditors, and which, therefore, muft have been entirely ufelefs to it.
It is an obfervation that deferves particular attention here, that, on this plan, it is of lit- tle importance what intereft a ftate is obliged to give for money : For the higher the in- tereft, the fooner will fuch a fund pay oft" the principal. Thus; a 100 millions borrow- ed at 8 per cent, and bearing an annual in- tereft of eight millions, v/ould be paid off" by a fund, producing annually 100,000/. in 56 years; that is, in -^9 years lefs time, than if the fame money had been borrowed at d^per cent.
(a) See the Qiieftions annexed to the Tables in ih« Appendix.
It
ajid the National Debt, 141
It follov/s from hence, that redudlons of intereft would, on this plan, be no great ad- vantage to a flate. They would, indeed, lighten its prefent burdens 5 but this advan- tage would be balanced, by the addition that would be made to its future burdens, in con- fequence of the longer time, during which it would be neceffary to bear them. — I mean this on the fuppofition, that the favings pro- duced by redudions of intereft, are imme- diately applied to the relief of the ftate, by annihilating taxes equivalent to them. But if this is not the cafe ; and if, likewife, there is either no plan eftablifhed for putting the public debts into a certain courfe of payment, or it is not faithfully carried into execution ; in thefe circumftanccs, reductions of intereft may prove hurtful. For, firft. They would only furniih v^^ith more money for fupplying the deficiencies arifing from bad manage- ment. And, fecondly. As, in fuch circum- ftances, they would only retard, and not pre^ vent the increafe of the burdens occafioned by the public debts, a period would come when the affairs of the flate would get to a cri/is', and, at fuch a period, its danger would be increafed, in proportion to the redudlions of intereft that had been made.
In order to underftand this ; let us fuppofe, that a debt, bearing an annual intereft of five millions, is the whole debt, which a ftate can bear without being £0 much oppreft as
to
j^2 Of Fiibllc Credit^
to be near finking. Let it, however, be fup-" pofed to have flill fome laft refources left, which may enable it to bear, for 23 years to come, this load, together with every addi- tional load, which, during this time, may be neceffary to be thrown upon it. — Let it fur- ther be fuppofed, that at this time, the ftate, urged by the fear of an approaching bank- ruptcy, refolves upon entering into fome ef- fectual meafures for preferving itfelf. — Cer- tain it is, that in this cafe, no meafureyS ef- fedlual can be purfued, as the eftablifhment of 2l finking fund t and fuch a faithful applica- tion of it as I have explained. Let this then be the meafure entered upon ; and let the ftate be fuppofed capable of providing a fund, producing a million annually. If all the debts bear intereft at 6 per cent, this fund would pay off three fifths of them, within the time I have mentioned; or, in 23 years; and the ilate might be faved. But if, in confequence of redudtions, they bear intereft at no more than -^per cent, the fame fund would not give the fame relief, in lefs than double that time; and, therefore, a bankruptcy might prove unavoidable [ci).
{a) In fome other kingdoms a fpunge might be ap- plied in luch circumftances, or the funds reduced one half by an a£t of defpotifm, without occafioning any con- vulfions \ but this is not pofTible in this free country ; and, it is to be hoped, never will be pofiible.
and the National Debt, i^j
I wifh I could think, that there Is nothing in this reprefentation, that can be applied to the prefent ftate of this nation. The intereft of the public debts has been reduced, at dif- ferent periods, from 6 to 5, and from 5 to 4^ and 3 per cent.', but flill they have grown with rapidity; and we now fee ourfelves over-* loaded, and in no way of gaining relief. Had there been no redudionsof intereft, we {hould, indeed, have been in the fame condition foonerj but, we might have been relieved alfo fooner, and with lefs difficulty and dan-
What I have now faid implies, that a ftatff always difcharges its debts, whatever intereft they bear, by paying the original fum bor- rowed. It may, perhaps, be imagined, that when a loan is under /i^r, it may be dlfcharg- cd at a lefs cxpence. But this is by no means fo pratlicable as it may feem; for it fhould be confidered, that a public loan, now under favy would not long keep fo, after being put into acourfe of payment : And, for this rea- fon, as a ftate can never be obliged, in re- deeming its debts, to pay t}iore than the ori- ginal fum borrowed, fo neither ought it to cxpedl, in general, to be able to redeem them by paying lefs. I have faid, in general-, for I am fenlible, that at the beginning of the operations of a fund, when its produce is fjivall j and alfo, in a time of war, a flate might derive great advantages from the low
price
i44 ^f P^i^^^<^ Credit,
price of its debts. And I am fenfible alfo^ that confiderable advantages might be de- rived from lotteries [ci), in paying the public debts : But lotteries do great mifchief in a flate, by foftering the deftrudive fpirit of gaming. It is wretched policy to make them familiar, by recurring to them in the ordinary courfe of government. There are great oc- cafions on which they may be necelTary, and for fuch occalions they {hould be referved.
But to return to the fubjed I have princi- pally in view.
The advantages of putting the public debts into fuch a courie of payment, as I have defcribed, are fcarcely to be imagined. It would give a vigour to public credit, which would enable a ftate always to borrow money eafily, and on the bed: terms. And the en- couragement to lenders might be always im- proved, without any inconvenience, by mak- ing every loan irredeemable, during the firft 20 or 30 years ; for, there could feldom be any occalion, for beginning to difcharge any one loan fooner.
It might be eafily iliewn, that the faithful application, from the beginning of the year
{d) Thus; 800,000/. of the 'i^pcr cenU. at 87 ; of 3,000,000, at 70, might be redeemed with half a million pf money, confiiling of 50,000 lottery tickets at 10/. each, real value ; but capable of being fold at 14/., as was done in feme of the laft lotteries.
1700,
and the National Dek, 14^
1700,.' of only 200,000/. annually, would long before this time, notwithftanding the redudions of interefl, and every wafce that has been made of the public money, have caufed above half the public funds to revert to the public, and paid off above 80 millions of its debts. The nation might therefore, fome years ago, have been eafed of the great- eft part of the taxes Vv^ith v/hich it is load- ed. The moft important relief might have been given to its trade and manufactures; and it might now have been in much better circumftances, than at the beginning of the laft war ; its credit firm 5 refpeded by fo- reign nations; dreaded by its enemies; and ready to punifh any infult that could be of- fered to it. The near view, likewife, of fuch a period, during the courfe of the laft war, would have given higher fpirits to the na- tion, and encouraged it to bear the expence occafioned by the war with more chearful- nefs, and to continue it with vir^our for two or three years longer; the confcqucnce of which would, probably, have been, gaining a full indemnification from our enemies, and weakening them to fuch a degree, as would have given us efi^e<ftual fecurity againft them for many years to come. — A new account might alfo now have been begun ; and an- other fund, not much more confiderable, ap- plied in the fame way,would, in 60 or 70 years more, have paid, not only all that would
L have
146 Of Public Credit,
have been now unpaid, but alfo, probablyj, a great proportion of fuch further debts as muft be contradled within this time [a). And thus, without any expence that could be (Qn-- libly felt, its debts, as foon as they began to grow heavy, might have been conftantly re- duced to a half] or a third , and not only all danger^ but all confiderable inconvenience from them prevented.
All I have now faid, fuppofes 2. Jingle fund with a general appropriation to the payment of the public debts. The faitie ends might be anfwered by particular funds, with fmall furpluffes, appropriated to particular debts. In the wars of King William and Queen Anne, 6 per cent, intereft was given for all loans. It would have been eafy to have annexed to each loan a fund producing a furplus of i /. per cent, after paying the intereft ; and fuch a furplus would have been fufficient to annihi- late the principal of every loan in 33 years. Had this plan been followed, the difengage- ment of the public funds, and the relief at- tending it, would have begun 50 years ago; and the debts contraded, during the reigniJ
{a) One of the propereft objeiSts of taxation in a ftate is celibacy. I doubt not, tut that hy a fund fupplied only from hence, the end 1 have in view might have been cafily accomplifhed ; and, confequently, the very means of paying off the debts of the nation, rendered at the fame time the means of increafmg its chief ftrengrh, by promoting population in it.
of
and the National Debt, 147
bf Ring William and Queen AnnCy would have been all cancelled near 20 years ago, without any of that trouble, tumult and diftrefs, which have been occafioned by reductions of intereft, and by the various fchemes which have been tried for lelTening the debts (^z). — A fund, yielding i /. per cent, furplus, annexed to a loan at 5 per cent., would difcharge the prin- cipal in 37 years {b). At j^ per cent.) in 41 years. At 3 per cent., in 47 years.
Thefe obfervations relate only to what might have been the ftate of the nation with relpedl to its debts, had a right plan been purfued from the firft. But it Vv'ill be aiked. What can be done with them as they are? — I wifh I was able to give a more fatisfadtory anfwer to this enquiry. Every one muft fee Gur profped: to be difcouraging, and our ftate hazardous. Some have thought, that a good
[a) The fums to be laid out would, in this cafe, be fo fmall at firft, that it would be proper to employ them in purchafing part of the loan to be annihilated at the prices m the public market; and this, as far as it can be car- ried, is the moll eafy and quiet and filent way pofTiblc of extinguifliing the public debts.
[b) \ have all along fuppofed the produce of the pub- lic funds to come in yearly. The truth is, that it comes in /'^//^-yearly ; but this gives no advantage in the pay- ment of the public debts worth taking into account, i /. per annum^ together with its growing intereft, at 4 fer cent. ti^i^r\ yearly out of loo/. will reduce it to nothing in 41 years; if taken balf-yQ?i:\y , it will annihilate th« fame capital only four months and 12 days fooner. See the queftions annexed to the Tables in the appendix.
L 2 method
148 Of Public Credit y
method might be found out of difchargmg the national debt, by life annuities. The fol- lowing obfervations will l]iew how vain an imagination this is.
Let us fuppofe, that 33,3-^3,000/. is to be paid off, by offering to the public cre- ditors life-annuities, in lieu of their 3 per cents. A life at 60, fuppofing intereft at 3.I per cent., and the probabihties of life as in the BreJIaw, Norwich, and Northampton Ta- bles of Obfervation, is worth 1 1 years pur- chafe. A life at 30 is worth 61 years pur- chafe. Certainly, therefore, no fcheme of this kind would be fufficiently inviting, which did not offer 8 per cent, at an average, to all fubfcribers. Let us, however, fuppofe, that no more than 7I is given; and that there are 3:^333 fubfcribers, at 1000/. ftock each, for which a life-annuity is to be granted of 7^;/. or, for the whole flock fubfcribcd, two mil- lions and a half. A million and a half ex- traordinary, therefore, muft be provided to- wards paying thcfc annuities.
Let us further fuppofe, that the fubfcri- bers are perfons between the ages of 30 and ■60; and that the numbers of them, at all the intermediate ages, are in the fame pro- portions to one another, with the proportions of the living at thefe ages, as they exift in the world, or, as they are given in T^ahks of 0/frrvdtion. Let us again ilippofe, that as thefe annuitnnts die o!F, ti\ey arc immediately 3 replaced
and the National Debt. I/J9
replaced by others, who arc continually of- fering tlicmfclves at the fame ages, and in the fame proportional numbers at thefe ages, with thoi'e of the original fubfcribers at the time they fubfcribed ; in confcqucnce of which, the whole number of annuitants will be kept always the fame. In thefe circum- ftances, it will be 30 years, at lead, before a number will die off, [a) equal to the whole number; that is, before 33 millions of debts will be annihilated. But had tlie extraordinary million and half provided for paying thefe annuities, been employed during this time, in paying off fo much of the debt at par every year, extinguifhing at the fame time every year an equivalent tax, 45 milHons would have been paid. But had the favings, alfo, inftead of being funk as they arofe, been employed in the fame manner, 71 millions would have been paid.
The nation, therefore, mufl, without doubt, lofe greatly by all fchemes of tliis kind; and yet they have been often much talked of; and, indeed, I ffiall not wonder, fhould I hereafter fee an attempt made to pay off the national debt in this way.
I mufl: beg leave to detain the reader here fome time longer. A more particular ex- planation of this fubje(51:, will lead to fome
{a) A (Icmonflration of this will be given in the Ap- pendix, note (K).
L -; obfcr-
T^o Q/* PMc Credit,
obfervations on the beft methods of raifing money which, I think, deferve to be carer fully confidered.
When any fum is faid to be the value of a life-annuity, the meaning is, that, in confe- quence of being improved at intereft, and allowing for the chances of mortality, it will bear the whole expence of the annuity. If, therefore, inftead of being laid up for im- provement, it is either immediately applied to particular ufes, or has been long fmce fpent; there will be a lofs, equal to the fum which would have been added to the purchafe-mo- ney, had it h^tn improve d.-^Hihis is the rea- fon of the lofs which, I have fliewn, the pub- lic would fufFer by offering life -annuities, in lieu o^ Jlock, in order to extinguifli its debts. And for the fame reafon, it muft always lofe confiderably by raifing money on life-annui- ties.
Suppofe a million raifed hy annuities on a fet oi lives, all at 30 years of age. Perlons at this age have, (according to Tables III, IV, and V,) an expccfation of 28 years. That is; the duration of their lives, taking them one with another, will be 28 years j (fee the beginning of the firfl Effay) and they will he entitled, fuppoUng intereft at /\. per cent, to y L per an?ium, for every 100/. advanced. For a million then, the public would make 28 pay- ments of 70,000/. — -Let us fuppofe next, that
' a fun4
and the National Debt. 151
a fund producing this fum annually, inftead of being engaged to pay thefe life-annuities, is engaged for 28 years, to pay the principal and intereft of a million, borrowed on f'edeetn- able ^^x^^X.mtx^^y'iX \per cent . There will, at the end of the firftyear, be a furplus of 30, 000/.— In confequence of applying this to the ex- tin(ftion of the principal, it will be reduced to 970,000/. on which, at the end of the fe- ■cond year, the intereft due will be 38,800/. There will, therefore, be a faving of 1200/. Inftead of employing this faving in further linking t\i& principal, which would caufe the fund to accumulate in the fame manner with money at compound intereft, let it be taken and employed in any other way: And let the fame be done with all the fubfequent favings, referving only 30,000/. annually, for thepur- pofe of finking the principal. At the end of the fecond year, the principal will be 940,000/. ; and the faving of intereft upon it, at the end of the third year, 2400/ At the end of the 28th year, the principal will be reduced to 1 60,000/. The faving of intereft that year will be, 1 200/. multiplied by 27, or 32,400; and the fum of all the favings will be 453, 600/, — Dedudtfrom hence i6o,oooA remaining then undifcliarg- ed of the principal; and 293,600/. will be the lofs the public would fuftain, in the cir- cumftances I have fuppofed, by raifmg mo- ney on life-annuities. But if we fuppofe the favings, as they arife, as well as the con^ L 4 ftant
152 Of Public Credit,
ftant fum of 30,000/. to be applied to the difcharge of the principal, inftead of being fpent on current fervices -, the v/hole million will be annihilated in 21 years and half; and the lofs to the public by life-annuities, will be 6^ years purchafe of the annuities ; or 455,000/. — By fimilar dedudions it may be eafily found, that the lofs, in younger lives, is greater; in (^/^^/fcT lives lefs 5 but never incon- fiderable, except in the oldeji lives.
It appears, therefore, that, in confcquence of fuch a way of railing money, the public muft always pay much more in intereft than there is any occafion for; and ivajle a fum nearly equal to half the principal borrowed {a).
This,
{a) It is obvious, that the obfervations here made, may be applied to the common methods of raifing money, on life-annuities, for building churches, paving ftreets, making navigations, &c. &c. And, in general, to all cafes v/here the money received, is not laid up to be improved. -—For, to view this fubjeiSl in another light, let us fup- pofe 10,000/. borrowed for any public v/oric, on perpe- tuities, at 4 ^rr cent. And, if that will afford more en- couragement, let them be made irredeemable for any nnmber of years lefs than fevcnteen. Let us further fup- pofe, fuch rates, or tolls, cftablifhed for the payment of the intereft and principal, as fhall produce double the in- tereft of the fum borrowed ; or 8oo/. fer annwn^ inftead cf 400/. per amuvn. Let the furplus^ as it comes in kalf- ycnrly^ be laid up to accumulate in the public funds. \\\ J 7 years and half, reckoning intereft at 4 /><??' cent, a ca- pital will be raifed, equal to the whole fum borrowed ; and, therefore, at the end of that time, the whole debt may be difcharged, and the whole tranfa6lion finiihed. — But if the fame fum had been borrowed on annuities, fo;-
hi
and the National Debt. 153
This, however, tho' fo wafleful, Is a more frugal way of procuring money than by bor- rowing on perpetuities, without putting them into a courfe of redemption ; for in this cafe, (if a fpunge is not appHed) the lofs mufl be infinite.
I mufl add, that thefe obfervations are par- ticularly applicable to all the ways of raifing money by the fale of reverfions. — The pub- lic, for inftance, might procure a million, by offering for it a fund, that will be difengag- ed at the end of 1 8 years ; and then produce 80,000/. per annum for ever. This, fuppo- fmg interefl at /^percent., would be the very fame with offering tivo millions, 18 years hence, for one million now : And a private man, or an office for the fale of reverlions, might gain by fuch a tranfaftion ; becaufe, the money advanced, in co'nfequence of being improved, might, in 18 years, be more than
the lives of a fet of perfons 50 years of age, at 8 fer cent. which is I /-. per cent, lefs than the true value of fuch an- nuities : Had this, I fay, been done, half the annui- tants would have been alive at the end of the term I have mentioned; (fee Tables III, IV, and V,) and the whole tranfa6lion, together with the expences and trouble at- tending the management of it, could not have been finally clofed 'till the extinction of all the lives ; that is, not in lefs time, moft probably, than 35, or, perhaps, 40 years. ■ — It is a neceflary obfervation here, that, if public credit maintains its ground, much will not depend, in the plan now propofed, on the rife and fall of Stocks. If a ivar finks them, the money laid out, while the v/ar lafls, will accumulate fafter. If a peace raifesthem, the money that had been pre-vioufly laid out will be proportionably in- <;rcafcd.
doubled.
154 0/ Public Credit f
doubled. But, as the pubUc always borrows for immediate fervices, and never lays up money., it would necefTarily lofe a fum equal to the whole fum borrowed : And the fame money might have been borrowed on a fund, producing 50,000/. per annum', which would not only pay the intereft, but difcharge the whole principal in 41 years {a).
By railing money on life-annuities, the frefent me??ibers of a ftate take a heavier load on themfeives, in order to exempt pojlerity -, and there would be a laudable generofity in this, were it not for tht folly of it ; the fame exemption being equally pradlicable at half the expence. — On the other hand. By bor- rowing on reverfionary grants, the prefent members of a flate exempt themfeives entire- ly^ by throwing the load doubled on pofterity ; and there is a cruelty and injuftice in this that nothing can excufe.
fi is well known, that both thefe methods of railing money have been pradtifed among us. This, however, is, by no means, the worftthat has been done. It has been common to borrov/ money to pay the intereft of mo^ ney borrowed, and thus to give compound in- tcrcjl for money ; and our parliaments have, fometimes, exprefsly provided, that this fliall be done for a fucceflion of years.
(a) The fmallnefs of the fums, which I have here and clfe where fometimes fuppofed to be employed in dif- charging the public debts, can create no difficulties, be- caufe there is no fum which may not be applied to this yfe by rurchafmg flock.
But
and the National Debt, j^^
But to return to the main point I had in yiew.
The enquiry which has occafioned this digreflion, mufl be highly interefting to every perfon who wifhes well to his country. — All fchemes for difcharging the public debts, by life-annuities, have been fhewn to be abfurd and extravagant. — In general ; it may be obr- ferved, that it is far from probable, that any money which the nation can fpare, if ap- plied fo as to bear only Ji??ipk intereft, can be capable of reducing its debts within due bounds; or of doing us, in ourprefent circum- ftances, any effentjal fervice. A fund, pro- ducing a furplus of even two millions annu- ally, would, when thus applied, pay no more than 40 millions in 20 years; and, in that time, a war might probably come, which would interrupt the application of it; and increafe our debts much more than fuch a fund had leffened them.
Certain it is, therefore, that if our affairs are to be retrieved, it mufl be by 2ifund in- creafing itfelf in the manner I have explain- ed. The fmallefl fund of this kind is, in- deed, omnipotent, if it is allowed time to ope- rate. But we are, I fear, got fo near to the limits of the refources of the nation, that it cannot be allowed much time : And, in or- ^er to make amends for this, it is neceffary |hat it flioul4 be larger — Let us then fuppofe, 3 that
156 Of Public Credit,
that the nation Is ftill ftrong enough to ena- ble it to provide a fund, that (hall yield a million and halfanmiallyy for 20 years to come : And alfo, that, together with all its prefent burdens, it is capable of bearing every addi- tional burden that 20 years more can bring upon it. If this is not true, we have, I think, nothing to do but to wait the iffue, and tremble.
A fund, producing annually a million and a half, would increafe to three millions per Gnn, in 20 years (<2.) At the end of this term, the nation might be eafed of the moft op- preffive taxes, to the amount of a million and a half; and the confequence would prove, that, if there fliould have been a war, either the whole, or much the greateft part of the addition occaiioned by it to the public bur- dens, would be taken off, and the nation reinftated nearly in its prefent circumllances. But, if there fhould have been no war, the national debt and the taxes charged with it, would be reduced a third below the fums at which they now flandj and the nation would be fo much relieved as to be prepared for a war. — The remaining million and half would,
(a) It (hould be remembered, that in the year 1779* 1/. per cent, on the confolidated 4 per cents^ will be anni- hilated, and that I fuppofe thefavings derived from hence to be taken at that time as a part of the fund. — Methods might be eafily contrived for gettin2; this faving imme- diately, which would be fome advantage.
and the National Debt. 1^7
in 23 years, increafe again to three millions per aiiniim ; and then, fo much more of the public taxes would be fet free; 50 millions more, or 93 millions in all, of the public debts would be difcharged, and the difficul- ties of the nation would be, in a great mea- fure, conquered. — During this whole courfe of time, there may poffibly be but one war^ and fliould that happen, the appropriation at the end of it, of about 400,000/. per annunif might be enough to anfwer all purpofes.
In thefe obfervations, I fuppofe the 3 per cents to be paid off ^.tpar -, and no advantage taken at any time of their low price. By taking this advantage, and with the help of a little management, a fund, producing an- nually a million and half, might be made to increafe to another million and half, in lefs time than I have affigned. Should there be a war in a few years, the 3 per cents, would probably fall below j^ ; and then the pro- prietors of them muft be glad to part with them at this price; thejconfequence of which, fuppofing the war to laft eight years, would be, that the fund would double itfelf, and the nation be relieved in the manner I have mentioned, in 18, inflead of 20 years.— The advantage will be the fame, fuppofing the government at fuch a time to go on in paying off the 3 per cents 2Xpar. For the effecft of this would be, that money might be bor- rowed for the public fervice on proportion- ably
158 Of Public Credit,
ably better terms. Suppofe, for inftaricCjj that four millions muft be borrowed for the fervice of the year ; and let the produce of the fund be then increafed to two millions; and the interefl: of money in th^Jlocks, above 4 per cent. In thefe circumflances, it would be the interefl of the lenders of money, to take 3^- per cent, for the fums they advanced, in conlideration of having their 3 per cents paid off at par, to the amount of half thefe fums.— War, therefore, would accelerate the redemption of the public debts ; and it would do this the more, the longer it lafted, and the higher it raifed the intereft of money. Or if, in confequence of paying always at par, this could not happen ; an equivalent effecft would be produced in the way jufl; mentioned. The flocks would be always kept up by the operations of the fund ; and, in proportion to the fums yielded by it, the public would be able to borrow money more advantageoufly, and Icfs would be added to its burdens. — This feems to me an obferva- tion of particular confequence. It demon- itrates, that the invariable application, in ivar as well as peace, of the oroduce of the fund I am fuppofing, to the payment of the national debts, rather than to any cur- rent fervices, would. Independently of its ef-^ fe«5t in {a) redeeming theic debts, be attend- ed
{a) So true is this, that a war, were we now engagect ir it; would only render the ptefent time lb much tine
and the Natio?ial Debt, i^(^
cd with great advantages to the public. But this is a fubjecft on which I fhall have occa- lion to fay more prefently.
fiiore proper for entering into meafures for paying the public debts. And the following obfervations will put this out of doubt.
As it is now become the pra£lice to have recourfe to lot- teries \n peace, we may be fure, that no year will pafs with- out them in war. I would, therefore, propofe, that, inftead of making ufe of them in raifing the annual fupplies in war, they fliould be then applied as an aid in difcharging the public debts. — Suppofe the war to laft lo years, and the 3 per cents at 70. Suppofe alfo, each lottery to confift of 750,000/. in tickets, which, when difpofed of to fub- fcribers, will bring in 1,050,000/. On thefe fuppofitions, the whole lofs to the public, from applying the lotteries to the payment of the public debts, rather than to the cur- rent fupplies, will be 1,050,000/. annually, or 10 millions and .\ in all. — The gain will be as follows. 750,000/. of the produce of the finking fund, formed into tickets, will be the fame with 1,050,000/. j and this fum will pay ofFa million and a half of the "^per ce?its, every year, or 15 millions in all ; and the growing favings arifmg from thefe payments will, at the end of 10 years, have paid, at lealt, two millions more. The nation, therefore, hav- ing paid ofF 17 millions of its debts, and added to them only 10 millions and ', will gain fix millions and ■^. But this will be the fmalleft part of its gains. All the produce of the finking fund, over and above 750,000/. might be charged with the payment of the interertoffuch new debts as would be neceflary to be contracted during the war; and, at the end of it, the nation, with the help of 200,000/. to be difengaged in 1779, by the redudion of the ^per cents, would find itl'elt pofTefl'ed of a fund, producing 1,450,000/. annually ; which, faithfully employed, might probably be fufficient to extricate it from all its difficul- ties.—Befides this ; iuch a fchcme would not only pre- ferve, but raife and ejiablijl) the credit of the public : And he only can be duly fenfible of the importance of this, who will confider, what danger there would be in ano- ther
l6o Of Public Credit,
'X\it finking fund, in its prefent flate, and, after fupplying the deficiencies of the peace cftablilhment, yields, I fuppofe, a confidera- ble part of the milHon and a hah^I have men- tioned. An annual lottery might eafily raife 200,000/. more. But this is a meafure which I cannot wifh to fee carried into execution, unlefs abfolutely necefTary. Were the ma- nagers of our affairs fufiiciently in earneft in this bufinefs, I cannot doubt but that fuch favings might be made in the colledlion and expenditure of the national revenue, as would caufe the finking fund to yield, for 18 or 20 years to come, the "whole of this fum, vv^ith- out impofmg any new burdens on the pub- lic. But, were there, indeed, no way of providing any part of it, but by creating new funds, or impofing new taxes; it ought to be done, becauie it 7mfl be done, or the nation be ruined.
The evils and dangers, attending an exov" hitant public debt in this country, are fo great, that they cannot be exaggerated. — Without repeating, what has been fo often faid, of its increafing the dependence on the crown, by jobs and places without number; occafioning
ther war, fliould it coiitinue long, of either cverwhehnlng public credit; or ot being terrified, by the apprehenfion of fuch a calamity, into an ignoniinious and fatal peace: The cftablilhment, therefore, of fome fuch plan as that now propofed, would, at the beginning of a war, be the moft important of all works,
execrable
and the National Debt, i6i
execrable pradiccs of the Alley ; rendering us tributary to foreigners; and railing the price of provifions and labour; and, confequently, checking population, and loading our trade and manufadtures ; I will only take notice of the two follow! no; evils which attend it.
In the firil place. It mud check the ex- ertions of the fpirit of liberty in the kingdom. The tendency of every government is to de- fpotifm ', and in this it mufi end, if the peo- ple are not conftantly jealous and watchful. Oppofition, therefore, and refiftance are often neceffary. But they may throw things Into confufion, and occafion the ruin of the public funds. The apprehenfion of this mud in- fluence all who have their intereft conneded with the prefervation of the funds, and in- cline them always to acquiefcence and fer- vility.
But further. It expofes us to particular danger hom foreign as well as dcmeji'ic ene- mies, by making us fearful of war, and in- capable of engaging in it, hov/ever neeefTary, without the hazard of bringing on terrible convulfions by overwhelming public credit.
All thefe are evils which mud is^creafe with every increafe of the national debt; and there is a point at which, when they arrive,' the ccnfequences mud be fatal i^ci). — I am
{a) " Either the nation (Mr. Himie fays, EfTdys Vol. II. p. 145,) muft deftroy public credit ; or puhlic " credit will deflroy the nr^tron." — A dreadful ahenia- tjve ! furcly.
M new'
i62 Of Public Credit,
now writing under a convidion, that I am doing the little in my power to preferve m.j country from this danger. I have fhewn, that an annual fupplyof a million and a half for i8, or at moil 20 years, may be made the means of reftoring and faving us. This, therefore, is our remedy ; aud it ought to be applied immediately, lead it fhould not be applied time enough.
But to proceed to fome further obferva- tions.
What has been faid, has all along fup- pofed a facred 2.nd ininolahle application of the fund I have defcribed, and of all its earnings, to the piirpofe of finking the national debt. The whole etfedl: of it depends on its being allowed to operate, without interrup- tion, a proper time. But it may be alTced, how this can be fecured ? Or, by what me- thod an objetfi:, that mull: be continually growing more and more tempting, can be defended againft invalion and rapine? — I might here mention the fuperintendency and care of the reprefentatives of the kingdom, the faithful guardians of the flate, to whom miniflers are refponlible for the ufe they make o\ tlie public money. But experience has lliewii, that we cannot rely on this fe- curity. — The ditiiculty, therefore, now men- tioned, is the very greatefl difficulty the na- tion
and the Natmial Debt, 1 63
tion has to ftruggle with in the payment of its debts.
The finking fund was eftabhfhed in the year 17 16, or foon after the acceffion of the prefent family, at a time when the public debts, tho' not much more than a third of what they are now, were thought to be fo confiderable as to be alarmmg and danger- ous. It was intended as a sacred depo- sit never to be touched; the law which eftabHdied it declaring, that it was to be ap- plied to the payment of the principal and in- tereft of fuch national debts and incumbran- ces, as had been incurred before the 25th of December 1716; and to ?io other uje, intent or purpoj'e whact'ver. — The faith of parlimnent^ therefore, as we ll as the fecurity of the king- dom, feetned to require, that it fhould be prefervc^d careiuUy and rigoroufly from alie- nation. But, notwithflanding this, it has been generally alienated -, and the produce of it employed, in helping to defray fuch current expences as the exigencies of the flate ren- dered necv llary.
In order to juftify this, it has been ufual to plead, that when money is wanted, it makes no difference, whether it is taken from hence, or procured by making a new loan. There cannot be a worfe fophifm than this. The difference between thefe two methods of procuring money is no lefs than infinite. — For, let us fuppofe, a million want- ed for any public lervice. If it is borrowed at M 2 4 fsr
r^4 Q/* P^i^^^^(^ Credit,
/j^per cent, the public will lofe by the payment of intereft 40,000/. the firft year, and the fame the fecond year, and the fame for ever afterwards. But if it is taken out of the Jinking fund, the public will lofe 40,000/. they/r// year; 4160/. theyt^i^W year; 80,000/. the 1 8th year; a million the 85th year: For thefe are the fums that would, at thefe times, have otherwife neceffarily reverted to the public. It lofes, therefore, the advantage of paying in 85 years, with money of which otherwife no ufe could have been vn^^d^, twen- ty-Jive millions of debt. — In other words; by employing the sinking fund, in bearing current expences, rather than borrowing new money ; the ftate, in order to avoid giving Jimple intereji for money, is made to alienate money, that miijl have otherwife been im- proved at compound intereji ', and that, in time, would have neceffarily Increafed \.oa?2y ium[d). — Had a faithful ufe been made from the firft, of only one third of the produce of this fund, near three Jour ths of our prefent debts might nov/ have been difcharged ; and, in a few years more, the whole of them might have been difcharged [b). — Can it be pofTible then
(«) The principal obfervations in this Chnpter, I have given juft as they occurred to my thoughts, without knovv- )ng that anv of them had been made by other writers. Some propoHiis of afimiiar nature, but very differently re- prefented, I have fmce found in Mr. PoJIleihivayt's Dictio- nary, under the articles Public Credit, Debts, Fundi, Sec.
(A) See a particular explanation and proof of this iji the Qj^ieftions following the Tables in the Jppendlx.
to
and the National Debt, 165
to think, without regret and indignation, of that mifapplication of this fund, which, with the confent of parliaments always complying, our minlfters have praclifed ? — I find it dif- ficult here to fpeak with calmnefs. — But I muft reftrain myfelf. Calculation, and not cenfure, is my bufinefs In this vv'ork. — I muft believe, that the grievance i have mentioned, has proceeded more from inattention and miflake, than from any defign to injure the public.
M 3 ESSAY
[ i67 ]
ESSAY I.
*
Conta'inhig Obfervations on the Ex- pedations of Lives ; the Increafe of Ma?ikmdj the Number of Inhabit tants in London ; a7td the Influ- e7ice of great Towns ^ on Health and Population.
In a Letter to Benjamin Franklin, Ef(i'y LL. D. and F. R. S.
D EAR Sir,
Beg leave to fubmit to your perufal the following obfervations. If you think them of any importance, I fliall be obliged to you for communicating them to the Royal So- ciety. You will find, that the chief fubjedt of them is the prefent ftatc of the city of London^ with refped: to healthful nefs and number of inhabitants, as far as it can be colleded from the bills of mortality. This is a fubjed: that has been confidered by others j but the proper method of calculating
* This EfTay was read to the Royal Society, April 27th, 1769, and has been publifhcd in the Philo- fophical Tranfaclions, Vol. 59. It is here republiflicd with fcveral additions; particularly, the Pnfifcript.
M 4 from
naar
168 Of the Expediation of Lives ,
from the bills has not, I think, been fuffici- ently explained.
No competent judgment can be formed of the following observations, without a clear notion of what the writers on LiJ'e-Ammities and Reverfions have called the ExpeBation of Lfe, Perhaps this is not in common pro- perly underftood; and Mr. De Mohre\ man- ner of expreinng himfelf about it is very lia- ble to be millaken.
The mofl obvious fenfe of the expeBation of a given life is, ** That particular number ** of years which a life of a given age has an ** equal chance of enjoying." This is pro- perly, the time that a perfon may reafonably expeB to live -, for the chances againfi his liv- ing longer are greater than thofe/or it ; and, therefore, he cannot entertain an expeBation of living longer, conlidently with probability. This period does not coincide with what the writers on Annuities call the expeBationof Ife, except on the fuppofition of an uniform de- creafe in the probabilities of life, as Mr. Simpfo?i has oblerved in his SeleB Exercfes, p. 273.— It is necelTary to add, that, even on this fuppofition, it does not coincide with what is called the expeBation of life, in any cafe of joint lives. Thus, two lives of 40 have an even chance, according to Mr. De Mohres hypothefis [a), of continuing to- gether only 13^ years. But the cxpeciation
[a) See the Notes in pa^;. 2, and 23.
the State o/'London, Populatiofiy Sec, 1 69
of two equal joint lives being (according to the fame hypothecs) always a third of the commoji complement \ it is in this cafe 1 5 j years. It is neceffary, therefore, to obferve, that there is another fenfe of this phrafe, which ought to be carefully diftinguifhed from that now mentioned. It may fignify, ** The ** mean continuance of any ^vitvi Jingle, joint, ** ov furviving lives, according to any given ** Table of obfervations :" that is, the num- ber of years which, taking them one with another, they adually enjoy, and may be confidered as fure of enjoying; thofe who live or furvive beyond that period, enjoying as much more time in proportion to their num- ber, as thofe who fall 0:ort of it enjoy lefs^ Thus; Suppofing 46 perfons alive, all 40 years of age; and that, according to Mr. t)e Moivres hypothejis, one will die every year *till they are all dead in 46 years; half 46, or 23, will be their expectation of life : That is; The number of years enjoyed by them all, will be juil the fame as if every one of them had lived 23 years, and then died; fo that, fuppoiing no intereft of money, there would be no difference in value between annuities payable for life to every Ungle perfon in fuch a fet, and equal annuities payable to another equal fet of perfons of the fame common age, fuppofed to be all furc of living jud 23 years and no more.
In
170 Of the "Expeclatmi of Lives-,
In like manner;, the third oi 46 years, or 1 5 years and 4 months (^), is the expeclation of two joint lives both 40 \ and this is alfo the expeuiation of the furvivor. That is ; fuppofing a fet of marriages between perfons all 40, they will, one with another, laft juft this time; and the furvivors will laft the fame time. And annuities payable during the con- tinuance of fuch marriages would, fuppofing no intereft of money, be of exactly the fame value with annuities to begin at the extinc- tion of fuch marriages, and to be paid, during life, to the furvivors. In adding together the years which any great number of fuch marriages, and their furvivorfhips have lafted, the fums would be found to be equal.
One is naturally led to underftand the expeBation of life in the firft of the fenfes now explained, when, by Mr. Smpfon and Mr. De Mororc, it is called, the number of years which, upon an equality of chance, a per- fon may expeci to enjoy ; or, the time which a pcrfon of a given age may jujlly expcB to continue in being', and, in the lafi: fenfe, when it is called, the fare of Ife due to a perfn. But, as in reality it is always ufed in the lad of thefe fenfes, the former language fhould not be appplied to it: And it is in this lall: lenfe, that it coincides with th^ funis of the prefent probabilities, that any given fmgle ■or joint lives lliall attain to the end of the
[a] Sec Note (L) Appejidi.v.
3 ' ' " ift.
the State ^London, Population, &c. 17 j
J ft, 2d, 3d, &c. momejits, from this time to the end of their poffible exiftence ; or, (in the cafe of furviorfliips) with the fum of the probabiHties, that there fliall be afurvivorat the end of the ill, 2d, 3d, &c. moments, from the prefent time to the end of the poffible exiftence of furvivorfliip. This coin- cidence every one converfant in thefe fubjedts muft fee, upon reflecting, that both thefe fenfes give the true prefent value of a life- annuity, fecured by land, vi^ithout intereil of money {a).
This period in joint lives, I have obferv- ed is never the fame with the period which they have an equal chance of enjoying ; and in fingle lives, I have obferved, they are the fame only on the fuppofition of an uniform decreafe in the probabilities of hfe. If this dccreafe, inftead of being always uniform, is accelerated in the laft ftages of life ; the for- mer period, in fmgle lives, will be lefs than the latter ^ if retarded^ it will be greater.
It is neceflary to add, that the number expreffing the former period, multiplied by the number of fingle or joint lives whofe ex- pedation it is, added annually to a fociety or town, gives the whole number living toge- ther, to which fuch an annual addition would in time grow. Thus; lince 19, or the third of 57, is the expeBation of two
{a) See Note (L) in the Appendix.
joint
172 Of the ExpeBatiojz of Lives',
joint lives whofe common age is 29, or com- mon coviplement ^j -, twenty marriages every year between perfons of this age would, in K^j years, grow to 20 times 19, or 380 mar- riages always exifting together. The num- ber o^ ftirvivors alfo ariling from thefe mar- riages, and always living together, would, in twice ^j years, increafe to the fame number. And, fince the expeBation of a fingle life is always half its complejnent -, in ^'j years like- wife, 20 lingle perfons aged 29, added annu- ally to a town, would increafe to 20 times 28.5 or 570 ; and, when arrived at this num- ber, the deaths every year will juft equal the acceflions, and no further increafe be poffi- ble.
It appears from hence, that the particu- lar proportion that becomes extindt every year, out of the whole number conftantly exifting together of fingle or joint lives, muft, wherever this number undergoes no variation, be exadlly the fame with the ex~ peBation of thofe lives, at the time when their exigence commenced. Thus; was it found that a 19th part of all the marriages among any body of men, whofe numbers do not vary, are diffolved every year by the deaths of either the hufband or wife, it would appear that 19 was, at the time they were contracted, the expeBation of thele marriages. In like manner; was it found in a fociety, limited to a fixed number of
members^
the State c/' London , Popniatw?:, &c . 175
members, that a 28th part dies annually out of the whole number of members, it would appear that 28 was their common expeda- tion of life at the time they entered. So likewife ; were it found in any town or di- flridl, where the number of births and burials are equal, that a 20th or 30th part of the inhabitants die annually, it would appear, that 20 or 30 was the expeBation of a child raft born in that town or diftridt. Thefe ex~ peclations, therefore, for all Jingle lives, are eafily found by a T^ahle ofOhfervations, fliew- ing the number that die annually at all ages, out of a given number alive at thofe ages j and the general rule for this purpofe, is ** to divide ** the fum of all the living in the Table, at ** the age whofe expectation is required, and ** at all greater ages, by the fum of all that *' die annually at that age, and above it j or, ** which is the fame, by the number in the *' Table of the living at that age y and half ** unity fubtraded from the quotient will be " the required expeBation [a)" Thus, in Dr. Halley\ Table, the fum of all the living at 20 and upwards is, 20,724. The number living- at that age is 598 ; and the former number divided by the latter, and half unity
(^) This rule, and alfo rules for finding in all cafes the expeilations of joint lives and furvivorfliips, may be deduced with great eafe, by having recourfe to the doc- trine of fluxions. In this method, Mr. De Mo'ivre fays, he difcovered them. See Appendix, Note (L), where an account will be give.a of thefe deductions, omitted by Mr. De Mo'ivre.
A fub-
1 74 Q/^ ^'^^ Expediation of Lives ;
(a) fubtraded from the quotient, gives 34.1^ for the expectation of 20. The expecSlation of the fame hfe by Mr. Si?npfons Table, formed from the bills of mortality of Lon- don, is 28.9 [b).
Thefe
[a) If we conceive the recruit necefiary to fupply the vjajle of every year to be made always at the end of the year, the dividend ought to be the inediimi between the numbers living at the beginning and the end of the year. That is, it ought to be taken Icfs than the funi of the liv- ino- in the Table at and above the given age, by Jjulfihe number that die in the year; the eiFeft of which di;ni?JU' iion will be the fame with \.\\t fubWaSiion here directed. — The reafon of this fubtracStion will be further explained, in the beginning of the laft Efiay.
[b) It appears in p. 169 and 170, that the expe£iations of finale and '-joint lives are the fame with the values o{ annuities on thefe lives, fuppofing no intereft or improvement of money. — In confidering this fubje6l, it will, probably, occur tofome, that, allowing intereft for money, the values of lives muft be the fame with the values of annuities certain for a number of years equal to the expeSiatiom Cii the lives. But care muft be taken not to fall into this miftake. The latter values are always greater than the former: And the reafon is, that, tho' a nuniber of fvigle or joirit lives of given ages v/ill, among them, enjoy z gi- ven number of years, yet forr.eofthem will enjoy a much vreoter^ and fome a much Ufs number of years. Thus ; 100 marriages among perfons, all 29, would, as I have faid, one with another, exift 19 years; and an office bound to pay annuities to fuch marriages during their continuance, might reckon upon making 19 payments for each marriage. But then, many of thefe payments would not be made 'till the end of 30, and fome not 'till the end of 40 years. And it is apparent, that on account of the o;reatfr value of quick than late payments, when nionev bean intereft, 19 payments fo made cannot be woah as much, as the fame number of payments made
rejularlv
the State of London, Fopiilation, Sec. i j^
Thefe obfervations bring me to the prin- cipal point which I have had all along in view. They fuggefh to us an eafy method of finding the number of inhabitants in a place, from a Table of Obfervatioiis, or the bills of mortality for that place, fuppofing the yearly births and burials equal. ** Find by *' the Table, in the way juil defcribed, the '' expe5latio?2 oi ^n infant jufi: born, and this, ** multiplied by the number of yearly births, *' will be the number of inhabitants." At Brejlaw, according to Dr. Halleys Table, though half die under 16, and therefore an infant juft born has an equal chance oi liv- ing only 1 6 years ; yet his expeBation, found by the rule I have given, is near 28 years ; and this, multiplied by 1238 the number born annually, gives 34,664, the number of inhabitants. In like manner, it appears from
regularly at the end of every year, 'till in 19 years they are all made.
This obfervation might be employed, to dcmonftrate further, the error of thofe who have maintained, that the value of a given life is the fame, with the value of an annu- ity certain, for as many years as the life has an equal chance of exiftina;. Were this true, an annuity on a life, fuppofed to be expofed to fuch danger in a particular year, as to create an equal chance, whether it will not fail that year, would, at the beginning of the year, be worth nothing., though fuppofed to be fure of continuing for ever, if it efcaped that danger.-- But there can be no occafion for tak- ing notice of an opinion, which has been embraced only by perfons ignorant of mathematics, and plainly unac- quainted with the genuine principles of calculation on this fubjecir. — See a Pamphlet o-n Life- Annuities by /i^- man Lce^ Efq; of the Inner TcmpU.
Mr.
iy6 Of foe Explication of Lives*,
Mr. Simpfons Table, that, though an infant juft born in hondon has not an equal chance of living 3 years, his expeBatio?i is 20 years ; and this number, multiplied by the yearly births, would give the number of inhabitants in Londo?2, wQi'Q the births and burials equal. — The medium of the yearly births, for 10 years, from 17 1^9 to 1768, was 15,710. This number multiplied by 20, is 314,2005 which is the number of inhabitants that there v/ould be in LQndo?2j according to the bills, were the yearly burials no more than equal to the births : that is, v/ere it to fupport itfelf in its number of inhabitants, without any fupply from the country. But for the period I have mentioned, the burials were, at an average, 22,956, and exceeded the chriftenings 7,246. This is, therefore, at prefent, the yearly ad- dition of people to London from other parts of the kingdom, by whom it is kept up. Suppofe them to be all, one with another^ perfons who have, when they remove to Z/o;z- do7if an expeclation of life equal to 30 years. That is \ fuppofe them to be all of the age of 18 or 20, a fuppofition certainly far be- yond'the truth. From hence will arife, ac- cording to what has been before obfcrved, an addition of ^^o multiplied by 7.246; that is, 217,380 inhabitajits. This number, added to the former, makes 53IJ580; and this, I think, at moft, would be the number of in- b.abitants in London were the bills perfed.
But
ihe State ^London, Vopiilatio7i, &c. 177
But it is certain, that they give the number of births and burials too Httle. There are many burying-places that are never brought into the bills. Many alfo emigrate to the navy and army and country ; and thefe ought to be added to the number of deaths. What the deficiencies arifmg from hence are, can- not be determined. Suppofe them equiva- lent to 6000 every year in the births, and 6000 in the burials. This would make an addition of 20 times 6000, or 1 20,000, to the laft number ; and the whole number of in- habitants would be 651,580. If the burials are deficient only two-thirds of this number, or 4000 J and the births, the whole of it; 20 multiplied by 6000, mufl: be added to 314,290, on account of the defeds in the births : And, fmce the excefs of the burials above the births will then be only 5,246; 30 multiplied by 5,246 or 157,380, will be the number to be added on this account; and the fum, or number of inhabitants, will be 59 1 ,580. — But if, on the contrary, the bu- rials are deficient 6ooOj and the births only 4000; 80,000 mull be added to 314,290, on account of the deficiencies in the births ; and 30 multiplied by 9,246, or 277,380, on account oF the excefs of trie burials above the births ; and the v/hole number of inhabitants will be 671,580.
Every fuppofition in thefe calculations
is too high. Emigrants from London are, in
N par^
27S On the 'Expedldtion of Lives-,
particular, allowed the fame expeciation of continuance in London with thoie who are born in it, or who come to it in the firmed part of life, and never afterwards leave it; whereas it is not credible that the former expeSlation fliould be fo much as half the lat- ter. But I have a further reafon for think- ing that this calculation gives too high num- bers, which has with me irrefidible weight. It has been feen, that the number of inha- bitants comes out lefs on the fuppofition, that the defeats in the chriftenings are greater than thofe in the burials. Now it feems evi- dent that this is really the cafe ; and, as it is a fad: not attended to, I will here endeavour to explain diflincflly the reafon which proves it.
The proportion of the number of births in London^ to the number who live to be 10 years of age, is, by the bills, 16 to q. Any one may hnd this to be true, by fubtradting the annual mcdiiun of thofe who have died under 10, for fome years paft, from the an- nual medimn of births for the fame number
of years. Now, tho', without doubt, Lon~
don is very fatal to children, yet it feems in- credible that it (hould be fo fatal as this im- plies. The bills, therefore, probably, give the number of thofe who die under 10 too great in proportion to the number of births; and there can be no other caufe of this, than a greater deficiency in the births than in the q burials^
^'
the State c/' London, Population y &cc, 1 79
Min'als. Were the deficiencies in both equal; that is, were the btiriahy in proportion to their number, jufl as deficient as the births are in proportion to their number, the pro- portion of thofe who reach 10 years of age to the number born, would be right in the hills, let the deficiencies themfelves be ever fo confiderable. On the contrary; were the deficiencies in the burials greater than in the births, this proportion would be given too great; and it is only when the former are leaft, that this proportion can be given too
little. Thus ; let the number of annual
burials be 23,000; of births 15,700; and the number dying annually under 10, J 0,800. Then 4,900 will reach 10, of 15,700 born annually; that is, 5 out of 16. -—Were there no deficiencies in the burials, and were it fad: that only half the number born die under 10 ; it would follow, that there was an annual deficiency equal to 4,900 fubtraded from 10,800, or '5,900, in the births. — Were the births a third part too lit- tle, and the burials alfo a third part too little, the true number of births, burials, and o'i chil^ dr en dying under 10, would be 20,9-^3 — 30,666, and 14,400; and, therefore, the number that would live to 10 years of age, would be 6,5:^3 out of 20,933, or 5 of 16 as before. — Were the birthi a third part, and the burials fo much as two-fifths wrong, the number of births, burials, and children dying under 10 would N 2 be
i8o On the Expedfation of Lives ;
be 20,933 — 32,200 — and 15,120. And,- therefore, the number that would live to i o would be £;,8i3 out of 20,933, or five out of 18. — Were the births a 3d part wrong;, and the burials but a 6th, the foregoing num- bers would be 20,933 — 26,833 — i2,6oo> and, therefore, the number that would live to 10 would be 8,333 out of 20,933, or 5 out of 12.56 : And this proportion feems as low as is confident with probability. It is fomew^hat lefs than the proportion in Mr. Simpfons Table of London Obfervatio?ts ; and much lefs than the proportion in the Table of Obfervations for Brejlaw. The de- ficiencies, therefore, in the births muft be greater than thofe in the burials [a); and the leaft number I have given, or 591,580 is neareft to the true number of inhabitants. However,, (bould any one, after all, think that it is not improbable that only 5 of 16 fliould live in London to be 10 years of age; or that above t'lvo-thirds die under this aire : the con- fequence will dill be, that the foregoing cal-
[a) One obvious reafon of this fa£t is, that none of the h'lrths among Jeivs^ j^akers, Papljis^ and the three deno- mlnaiioni of Dijfentcrs are included in the bills, whereas ?nany of their burials are. It is further to be attended to, that the abortive and ftill-born, amounting to about 600 annually, are included in the burials, but never in the births. If we add thefe to the chriftenings, preferving the burials the fame, the proportion of the born, accord- ing to the bills, who have reached ten for the laft fixteen years, will be very nearly one third inftead of five fix - t, enths,
culatlon
the State o/" London, Population t Sec, iSi
dilation has been carried too high. For it will from hence follow, that the expediatlon of a child juft born in London cannot be fo much as I have taken it. This expediation is 20, on the fuppofition that half die under 3 years of age, and that 5 of 16 live to be 29 years of age, agreeably to Mr. Simpfo?is Table. But if it is indeed true, that half^ die under 2 years of age, and 5 of 16 under 10, agree- ably to the bills, this expe6lation cannot be fo much as 17 [a) ; and all the numbers before given will be confiderably reduced.
Upon the whole : I am forced to con- clude from thefe obfervations, that the fe- cond number I have given, or 651,580, though fliort of the number of inhabitants commonly fuppofed in London, is, very pro- bably, much greater, but cannot be lejs, than the true number. Indeed, it is in ge- neral evident, that in cafes of this kind num- bers are very much over-rated. The inge- nious Dr, Brakenridge, 14 years ago, when the bills were lower than they are now, from the number of houfes, and allowing fix to a houfe, made the num.ber of inhabi- tants 751,800. But his method of deter- jnining the {b) number of houfes is too pre- carious;
[a) This may be deduced from the obfervations in the laft EfTay; and it will be there proved, that, in reality, this expectation does not exceed 18.
{b) Vid. Phil. Tranfaaions, Vol. XLVIIT, p. 788.
Jii a paper fubfequent to this, Dr. Brakenridge tells us,
N 3 that
182 On the Expc Station of Lives 5
carious , and, belides, iix to a houfe is too large an allowance. — ?vlany families now have two houfes to live in. — The ma<rill:rates of Norwichy in 1752, took an exad account of both the number of houfes and indivi- duals in that city, [a) The number of
houfes
that in a late furvey it appeared, that in all M'uldlcfex^ London^ Wejlm'injier^ and Southwark^ there were 87,614 houfes, of which 19,324 were cottages, and 4810 empty. And he acknowledges, that this, if right, proves London to be much lefs populous than he had made it. See Phil. Tranf. Vol. 50, p. 471. He does not mention how this furvey was taken ; but moft probably it muft have been incorrect. — Mr. Maitland gives two accounts of the number of houfes within the bills. One carefully taken from the books of all the parifhes and precin(5ls belong- ing to London ; and another taken from a particular fur- vey in 1737, made by himfclf with incredible pains. The firfl: account makes the number of houfes 85,805. The fecond account makes it 95,968. And the reafon of the difference he obferves, is, that many landlords of fmall places, paying all taxes, they are in the parifli books reckoned as fo many fuigle houfes, tho' each of them contain feveral houfes. See Mr. Maitland'^ Hiftory of London^ 2d Book at the end. — Thii-, periiaps, may be alfo the reafon of the deficiencies which, I fuppofe, there muft be in the furvey, mentioned by Dr. Brakenr'idge. — \t will be obferved prefently, that the number of inhabitants in London m 1737, was eonfidcrably greater th.m it is now.
(rt) Vid. Gentleman's Magazine for 1752, and Dr. Shorf s Comparative Hjjlory of the Increafe of Mankind^ P* 38. In page 58 of this laft Vv'ork the author fays, that, in or- der to be fully fatisfied about the number of perfons to be allowed to a family, he procured the true number of famihes and individuals in 14 market towns, fome of them confiilerable for trade and populoufnefs ; and that in them were 20,371 families, and 97,611 individuals,
ox
the State o/'London, PopuIatlo?t, See. 183
houfes was y,i29* ^"^ of individuals 36,169, which gives nearly 5 to a houfe. Ano-
the
or but little more than 4-| to a family. He adds, that, ia order to find the difference in this refpedl between towns of trade and country parifhes, he procured, from divers parts of the kingdom, the exa<Sl number of families and individuals^ in 65 country parifhes. The number of fa- milies wa.s 17,208; individuals 76,284; or not quite 4! to a family. — in the place I have juft referred to, in the Gentleman's Magazine, there is an account of the num- ber of /;'oz^j- and i«/;^^zVflK/5 in Oxford^ exclufive of the colleges ; and in IP'olverbatnpton^Coventry zn^ Birmingham^ for 1750. The number of pcrfons to a houfe was, by this account, 4i in the two former towns, and 54 in the two latter. — Dr.Davenant, from M.r. King's Obfervations, gives 4^, as the number of perfons to z family for the whole kingdom. See Jn Efjay on the probable method of making a people gainers by the balance of trade. From an account with which a friend at Shrewjbury has favoured me, it appears, that in that town, in 1750, the number of inhabitants to a houfe was 4^. — Very exaiSl accounts, of which I {hall take further notice, prove, that in the parifli of Holy-Crofs, one of the fuburbs of Shrew/bury^ and at 'Northampton^ the fame proportion is 4 1 to a houfe in the former; and 4! in the latter. — It feems, therefore, that five perfons to a houfe is an allowance large enough for London^ and too large for England in general. FVom whence it will follow, that Dr. Braken- ridge has likewife over-rated the number of people in England. In a letter to George Lewis Scott, tfq; publifh- cd in 1756, in the Phil. Tranf. Vol. 49, p. 877 ; he fays, that he had been certainly informed, that the number of houfes rated to the window-tax was 690,000. The number of cottages not rated, he adds, was not accu- rately known ; but from the accounts given in, it appear- ed, that they could not exceed 200,000 ; and from thefe data, in confequence of allowing fix to a houfe, he makes the number of people in England to be 5,340,000. Per- haps the number of houfes in this account is too little. N 4 Suppofe
1 §4 On the ExpeBatiojt of Lives ;
ther method which Dr. Brakcnndge took to determine the number of inhabitants in Lon- don was from the annual number of burials, adding 2000 to the bills for omiffions, and fuppofmg a 30th part to die every year. \w order to prove this to be a moderate fuppo- fition he obferves that, according to Dr. Halleys Obfcrvations, a 34th part die every year at Brejlaw. But this obfervation was made too inadvertently. The number of annual burials there, according to Dr. Hal- Iljs account, was iij^, and the number of inhabitants, as deduced by him from his Table, was 34,000 ; and therefore a 29th part died every year. Befides ; any one may find, that in reality the Table is con- firudcd on the fuppofition, that the wholp number born, or 123B, die every year; from whence it will follow that a 28th part died every year, [a) Dr. Bi'akenridge, therefore,
^ had
Suppofc It a million ; and let five be allowec! to a houfe ; and the number of people in England will be five milli- ons : Which, fince five to a houfe is too large an allow- ance, ought to be confidcred as, probably, more than the true number. — The number of people in Scotland and Ireland, Dr. Brakenridge eftimatcs at three rnilliqns. See Phil. Tranf. Vol. 5c, p. 473.
{a) C31C {hould he taken, in confidering Dr. Halley's Table, not to take the firft number in it, or looo, for (o many juft born. 1238, he tells us, was the annual me- dium of births, and 1000 is the number he fuppofes all jiving at one vear and under. It was inattention to thjs tnat hd Dr. Brakenrids:e to his miftake.
I?
the State o/'London, Fopiihtion^ kc. i8^
liad he attended to this, would have ftated a 24th part as the proportion that dies in London every year, and this would have taken off I CO, coo from the number he has given. But even this muft be lefs than the jufl pro- portion. For let three-fourths of all who either die in London or migrate from it, be fuch as have been born in London; and let the reft be perfons who have removed to London from the country, or from foreign nations. The expeSlation of the former, it has been lliewn, cannot exceed 20 years; and 30 years have been allowed to the latter. One with another, then, they will have an expeBation oi 22t years. That is; one of 224. will die every year, [a) And, confe-
quently,
Ijt will be fhewn in the 4th EfTay, that the number of the living, under 20, is given too high in this Table ; and from hence it will follow, that more than a 28th part of the inhabitants die at Brejlaw annually.
(a) The whole number of inhabitants in ^i?w/? in 1743, was 147,476, and the annual medium of burials for three years, from 1 741 to 1743, was 6338. A 23d part, there- fore, died every year. See a Treatife in German^ on the different de2;rees of human mortality in different fitua- tif>ns, by SufmiUh, firft counfellor of his P.uJJian Majefty's Confiftory, and member of the Royal Academy of fciences at Berlin^ p. 15.
In 1761, the whole number of inhabitants in the fame town, was 157,452. The annual medium of births, for three years, from 1759 to 1761, was 5167 ; and of bu- rials 7153- One in 22, therefore, died annually. See Dr. Shorfs Comparative Hijiory of the Increaje and Decreafc f)f Mankind in England and favercil Countries abraad, p.|59,
60.
I S 6 On the Expe5iatlo?i of Lives ;
quently, fuppofing the annual recruit from the country to be 7000, the number of
bh'ths
60. — In 1752, the accurate and diligent Mr. Struyk^ took particular pains to determine the number of inhabitants in Amjlerdain^ and the refult of his enquiry was, that very probably it did not amount to 200,000. The an- nual medium of burials for fix years, from 1747 to 1752, was 8247. One in 24, therefore, died annually. See Suftnilch ibid. — At Anijlerdam^ there is a great number of Jews, and their burials are not included in the bills. There muft, I fuppofe, be other deficiencies, and an al- lowance for thefe would, I doubt not, incrcafe the pro- portion of inhabitants, who die annually, to one in 21 or 22. — At Dublin^ in the year 1695, the number of in- habitants was found, by an exa£l furvey, to be 40,508, (fee Philof. Tranfadions, N° 261). I find no account of the annual burials juil at that time; but from 1661 to 1681, the medium had been 1613; and from 1715 to 1728 it was 2123. There can, therefore, be no mate- rial error in fuppofing that, in 1695, it ^^^ 1800; and this makes i in 22 to die annually. — In 1745 the number of familici in the fame city appeared, by an exaiSl account Jaid before the Lord Mayor, to be 9,214. It is highly probable this number of families did not confift of fo many as 45,000 individuals. Suppofe them, however, 50,000; and, as at this time the medium of annual burials appears to have been 2,360, i in 21 died annually: fee Dr. Short's Co?nparative Hojhry^ p. 15, and New Obfcrvations, p. 228. — Thefe facls prove that I have been very mode- rate in making only i in 224-, including emigrants, to die in London annually. — In 1631 the number of people in the city and liberties oi London was taken, by order of the Privy Council, and found to be 130,178. — This ac- count was taken five years after a plague that had fwept oft' near a qu;!rter of the inhabitants ; and when, there- fore, the town being full of recruits in the vigour of life, the medium of annual burials muft have been lower than ufual, and the births higher. Could, therefore, the me- dium
the State of London, Population^ &c. 187
■births 3 times 7000 or 21,000, and the bu- rials and migrations 28,000 (which are all high fuppoiitions), the number of inhabi- tants will be, 224 multiphed by 28,000, or 630,000.
I will juft mention here one other in- ftance of exaggeration on the prefent fub- je(ft.
dium of annual burials at that time, within the walls, and in the 16 parifhes without the walls, be fettled, ex- clufive of thofe who died in fuch parts of the 16 pariflies without the walls, as are not in the liberties, the propor- tion dying annually obtained from hence might be de- pended on, as lefs than the common and juft proportion. But this medium cannot be difcovered with any accuracy. Grai4nt eftimates that two-thirds of thefe 16 pariflies are within the liberties i and, if this is right, the medium of annual burials in the city and liberties in 1631, was 5,500, and I in 23^ died annually; or, making a Imall allow- ance for deficiencies in the bills, i in 22. — Mr. Mait- land, in his Hiftory of London, Vol, II. p. 744, by a la- borious, but too unfatisfacSlory, inveftigation, reduces this proportion to i in 24^; and on the fuppofitions, that this is the true proportion dying annually, at all times, in. London, and that the deficiencies in the burials (includ- ing the burials in Marybone and Pancrafs pariflies) amount to 3,038 annually ; he determines, that the number of in- habitants within the bills was 725,903, in the year 1737.
The number of burials not brought to account in the bills is, probably, now much greater than either Dr. Brakenridge or Mr. Maitland fuppofe it. I have reck- oned it fo high as 6cco, in order to include emigrants, and alfo to be more fure of not falling below the truth.
It will appear in the laft Efl'ay, with an evidence little fliorc of demonftration, that, at leafl:, i in 20-|^ die an- nually in London, and that, confequently, the number of inhabitants, if the omilfions in the burials are 6000, cannot exceed 601,750.
Mr.
iS8 On the ExpeSiation of Lives ',
Mr. Corbyn Morris, in his ufeful Ohferva^ tions on the pq/i growth and prefentjtate of the city of London, publiihed in 175 i, luppofes that no more than a 60th part of the inhabi- tants of Z/O/zJi?;?, who are above 20, die every year, and from hence he concludes that the number of inhabitants was near a milhon. In this fuppofition there was an error of at leail one half. According to Dr. Haileys Table, it has been fliewn, that a 34th part of all at 20 and upwards, die every year at Brejlaw, In London, a 29th part, according to Mr. Bimpfon^ Table, and alfo according to all other Tables of London Obfervations. And in Scotland it has been found for many years, that, of 974 minifters and profellbrs whofe ages are 27 and upwards, a 33d part have died every year. Had, therefore, Mr. Morris Hated a 30th part of all above 20 as dying annually in London, he would have gone be- yond the truth, and his conclufion would have been 400,000 lels than it is.
Dr. Brakenridge obferved, that the num- ber of inhabitants, at the time he calculat- ed, was 127,000 lefs than it had been. The bills have lately advanced a little, but flill they are much below what they were from J717 to 1743. The medium of the annual births, for 20 years, from 171 6 to 1736^ was 18,000, and o^ burials 26,529; and, by cal- culating from hence on all the fame fuppo- fitions with thofe which made 651,580 to
be
. the State ^London, Population^ &c. 189
be the prefent number of inhabitants in Loti- don, it will be found that the number then was 735,840, or 84,260 greater than the number at prefent. London, therefore, for the laft 30 years, has been decreafmg; and though now it is increafmg again, yet there is reafon to thuik that the additions lately made to the number of buildings round it, are owing, chiefly to the increafe of luxury, and the inhabitants requiring more room to live upon [a).
It fhould be remembered, that the num- ber of inhabitants in hondon is now fo much lefs as I have made it, than it was 40 years
{a) The medium of annual burials in the 97 parifhes within the walls was,
From 1655 to 1664, 3264
From 1680 to 1690, 3^39
From 1730 to 1740, 2316
From 1758 to 1768, 1620
This account proves, that though, fnice 1655, London has doubled its inhabitants, yet, tv'ith'in the tvalh^ they have decreafed ; and fo rapidly for the laft 30 years as to be now reduced to one half. — The like may be obferved of the 17 pariflies immediately without the walls. Since 1730, thefe parifhes have been decreafmg fo faft, that the annual bttrials in them have funk from 8,672 to 5,432, and are now lovi'er than they were before the year 1660, In JVcJlminJler^ on the contrary, and the 23 out-parifhes in Middlesex and Surrey^ the annual bur lah have fmce 1660,
advanced from about 4000 to 16,000. Thefe fa6ts
prove, that the inhabitants o^ London are now much lefs crowded together than they were. It appears, in parti- cular, that zvithin the walls the inhabitants take as much room to live upon as double their number did formerly. — The very fame conclufions mav be drawn from an exa- mination of the chr!jkni7i^s^
ago.
190 On the FiXpeUtatlon of lLwes%
ago, on the fuppodtion, that the proportion of the omiffions in the births to thofe in the burials, was the fame then that it is now. But it appears that this is not the fadt. — From 1728, (the year when the ages of the dead were firft given in the bills^ to 1742, near five- fixths of thofe who were born died under 10, according to the bills. From 1742 to 1752 three quarters: And ever lince 1752, this proportion has flood nearly as it is now, or at fomewhat more than two-thirds. The omiffions in the births, therefore, compared with thofe in the burials, \NQrQ greater former- ly ', and this muft render the difference be- tween the number of inhabitants how and formerly lefs confiderable than it may feem to be irom the face of the bills. One rea- fon, why the proportion of the amounts of the births and burials in the bills, comes now nearer than it did, to the true proportion, may, perhaps, be, that the number of Diffen- ters is leffcned. The Foundling Hofpital alfo may have contributed a little to this event, by leirening the number given in the bills as having died under 10, without tak- ing off any from the births; for all that die in thib hofpital are buried at Pancrafs church, which is not within the bills. See the preface to a coUecfticn of the yearly bills of mortality from 1657 to 1758 inciufive,
I will
the State of London, Population^ &c. 1 9 1
I will add, that it is probable that London is now become lefs fatal to children than it was; and that this is a further circumftance which muft reduce the difference I have mentioned; and which is likewife neceffary to be joined to the greater deficiencies in the births, in order to account for the very fmall proportion of children who furvived 10 years of age, during the two firfl: of the periods I have fpecified. — Since 1752, London has been thrown more open. The cuflom of keep- ing country-houfes, and of fending children to be nurfed in the country, has prevailed more. But, particularly, the deflruvflive ufe fpirituous liquors among the poor has been checked.
I have flicwn that in London, even in its prefent ftate, and according to the moft mo- derate computation, half the number born die under three years of age. But it appears from Graunt's (a) accurate account of the births, weddings, and burials in three coun- try pariilies for 90 years; and alfo, from Dr. Short's colledlionof obfervations in his Com^ parative Hi/lory, and his treatife entitled. New Ohfervations on Town and Country Bills
[a) See Natural and Political Ohfervations on the Bills of Mortality, by Capt. John Graunt, f . R. S.— See alfo Mr. Derhani's Phifico-Theology, p. 174, where it appears, that in the parifh of Jynho in Northamptonjhire^ tho' the births had been, for 118 years, to the marriages as 6 to I j yet the burials had been to the marriages only as 3|. to i.
of
192 On the ExpeBation of Lives-,
of Mortality, that in country villages and pariilies, the major part live to mature age, and even to marry.
In the parifli of Holy-Crofs [a], in Salop, it appears from a curious regifler, which has been kept by the Rev. Mr. Gcrfucb, the vi^ car, that of 655 who have died there at all ages for the laft 20 years, 321, or near one half, have lived to 30 years of age : And, by forming a Table ot Obfervations from this
(a) This parifli contains in it a villnge which is a parfc of the fuburbs of Shreivjlmry. It confifts of 140O acres of arable and pafture land ; befides 300 acres taken up by houfes and gardens. Jtis fix miles in circumference; half of which lies along the banks of the river Severn. — ^ I mention thefe particulars to fliew, that it may be reck- oned a country pariih ; tho', perhaps, not perfectly fo, ort account of its nearnefs to Shrewfoury, — The chriftenings in it exceed the burials a little; and the number of in- habitants (mofily labouring people) has, for the laft 20 years, kept nearly to JC50, without any confiderable in- creafe. — The regifter of this parifli, from 1750 to 1760, has been publifhcd in the 52d volume of the Philofoplncal "TranjaLlions^ Part I. Art. 25. And a continuation of it from 1760 to J770, has been lately communicated and read to the Royal Society. It is to be wiihed, that more fuch accounts, fpecifying, as this does, the males and' females dying at all ages, v/ere kept in different fituations in the country. This is the only one that I have ever heard of. It is kept with particular care and accuracy by Mr. Gorfnch ; and furnifhes very ufeful data for deter- niming the difference in value between town and coun- try lives. — It defcrvcs to be mentioned particularly, that ivo foreigners or Jiravgers^ who happen to die in this parifh, (T who may be brought into it to be buried, are entered into the regifter : Nor are any of the fixed inhabitants omitted, tho' earned out to be buried.
3 regifbr,
the State ofLondon, Pvpidationi &c. 193
regifter, in the manner which will be de- Icribed in the lail EiTay, I find that a child juH: born in this pariHi has an expedation of 3-:^ years; and that in general, under the age nt ^o, the expeclatiofis of lives here exceed thole in London, in the proportion of about 4 to 3. — So great is the difference, eipecially to children, between livinsr in rrcat towns and in the country. — Hut nothing can place thi^ obfervation in a more ftriking light, than the account given by Dr. 'Thomas lieberden, and publilhed in the Philofophical Tranf- adions (vol. LVII. p. 461), of the increafe and niortality of the inhabitants of the if -and of Madeira, \x\ this ifland, it feems, the wed- dings have been to the births, for 8 years, from 1759 to 1766, as 10 to 46.8 j and to the burials, as 10 to27.v Double thefe proportions, therefore, or the proportion of 20 to 46.8, and of 20 to 27. c, are the pro- portions of the number marrying annually, to the number born and the iiumber dying, L.et I marriage in 10 be a 2d or 3d mar- riage on the lide of either the man or the woman; and 10 marriages \\\\\ imply 19 individuals who have grown up to maturity, snd lived to marry once or oftener ; and the proportion of the number marrying aniiually the firli time, to the number dying annually, will be 19 to 27.5, or near 3 to 4. It may Jeem to follow from hence, that in this iiland nrar iliree-iourths of thoi'e who die have
O been
1 9-4 Of the Expccfiithn of Lives j
been mcirried ; and, confequcntly, that not many more than a quarter of the inhabi- tants die in childhood and ceUbacy ; and this would be a jull: conelufion were there no increafe, or had the births and bu- rials been equal. But it mud be remem- bered, that the general eftecft oi an increafe while it is going on in a country, is to ren- der the proportio-n of perfons marrying an- nually, to the annual deaths, greater^ and to .the annual births lefs^ than the true propor- tion marrying, out of any given number born. This proportion generally lies be- tween the other two proportions, but always jieared to the firil [ii] ; and, in the prefent
cafe,
((?) In a country where there is no increafe or de- creafe of the inhabitants, , and where alfo life, ia its firft periods, is fo ftable, and marriage fo much encouraged, ns that half of all who arc horn live to be married, the,?;;- 7/?/<7/ births and burials muil be equal, and alio quadruple the number of weddings, after allowing for 2d and -^d marriages. Suppofe in thcfe circumlianccs (every thing clfe remaining the fame) t\^Q probabUhics of life, during its firft ftages, to be improved. In this cafe, more than half the born will live to be married, and an increal'e will take pi. ice. The births will exceed the burials, and bo-th fall below qiuulruple the weddings j or, which is the fame, below double the number annually married. — Suppofe next i^iht probabilities of life and the encouragement to mar- riage remaining the fame) the proHficknefs orily of the marriages to be improved. In this cafe it is plain, that an increafe alfo will take [■)lacc ; but i\\'^ tmnualh\r\.\\s and burials, inflead of being lefs, will now both rife above quadjuplc ilie weddings i and therefore the proportion of the borii to that part of the born who marry (being by hippbiUioa Uvo to one) will be lefs than the proporuon
of
the State of London, Population, &c. 19^
cefe, it is fufficiently evident that it cannot be much leis than two-thirds.
In
of either the annual births or the annual burials, to the number marrying <7«««(7//^-. — Suppole again {\.\\<z encourage- ment to marriage remaining the fame) that t\\t probahilities of life and l\\e prolif.chncfs of marriages are both improved* In this cafe, a more rapid increafe will taice place, or a greater cxcefs of the births above the burials ; but at the fame time they will keep nearer to quadruple the wed- dings, than if the latter caufe only had operated, and produced the fame incresjfe.-^l {hould be too minute and tedious, were I to explain thefe obfervations at large. ]t: follows from them, that, in every country or fituation where, for a courfe of years, the burials have been either equal to or lefs than the births, and both under quadruple the marriages ; and alfo that, wherever the burials are Lfs than quadruple the annual marriages, and at the fame time the births greater, there the major p:^-rt of all that are born live to marry. In the inftance which I have confidered above, and which occafions this note, the an- nual births arc fo much greater than quadruple the marri- ages, and at the fame time the annual burials fo much lfs, that the proportion that live to marry of thofe who ire born, can fcarcely be much lefs than I have faid j or two-thirds.
I have fhewn how the allowance is to be made for 2d and 3d marriages ; but it is not fo confulerable as to be of any particular confequence; and, befides, it is, in part, compenfated by the natural children which are in- cluded in the births, and which raife the proportion of the biiths to the weddings higher than it ought to be, and therefore bring it nearer to the true proportion of the number born annually, to ihofe who marry annually, af« ter dedufling thofe who marry a 2d or 3d time.
in drawing conclufions from the proportion o^ annual
birrhs and burials, in different fituations, fome writer?
on the increafe of mankind, have not given due attention.
to the difference in thefe proporciens, arifing from the
O 2 different
196 Of the ExpcBation of Lives ;
In LondoHy then, half die under three years of age; and in Madeira about two- thirds of all who are born live to be married. Agreeable to this, it appears alfo from the account I have referred to, that the expedla- tion of a child juft born in Madeira is about 39 years ; or double the expedation of a child juil born in London. For the number of inhabitants was found, by a furvey made in the beginning of the year 1767, to be 64,614. The annual medium of burials bad been, for eight years, 1293; o{ births 2201. The number of inhabitants, divided by the annual medium of burials^ gives 49.89; or i\\t expe&ation nearly of a child jufl born, fuppofing the births had been 1293, and confiantly equal to the bnrialsy the number of inhabitants remaining the fame. And the fame number, divided by the annual medium of birthsy gives 29.35; or the expectation of a child juft born, fup- pofing the burials 2201, the number of births
different circumftances of Increafe or decreafe among a people. One inftance of this I have now mentioned ; and one further inftance of it is necefTary to be mentioned. 'I'he proportion of annual births to weddings has been confidered as giving the true number of children derived from each marriage, taking all marriages one with ano- ther. But this is true only when, for many years, the births and burials have kept nearly equal. Where there is an excefs of the births occafioning an increafe, the proportion oi onnual births to weddings muft be lefs than the proportion of children derived from each mar- riage ; and the contrary rauft take place where there is a decreafe.
and
the State ^/London, Population t &c. 1 97
and of inhabitants remaining the fame. And the true expeBation of life mud be fome- where near the mean between 49.^9 and
Again : A 50th part of the inhabitants of Madeira, it appears, die annually. In Lon- dofi, I have (hewn, that above twice this proportion dies annually. In fmaller towns a fmaller proportion dies; and the births alfo come nearer to the burials. In general; there feems reafon to thing that in towns (allow- ing /or particular advantages of (ituation, trade, police, cleanlinefs, and openncfs, which fome towns may have,) the excefs of the burials above the births, and the annual deaths are more or lefs as the towns are greater or fmaller. In Londo?2 itfelf, about 160 years ago, when it was fcarcely a fourth of its prefent bulk, the births were much nearer to the burials, than they are now. But in country parifhes and villages the births almoll always exceed the burials; and I be- lieve it feldom happens that fo many as a 30th, or more than a 40th part of the in- habitants, die annually {a). In the four
provinces
[a] In the year I7;^3, a furvcy was taken of the in- habitants of the parifli of Stcke-Damerel in DcvonJJnre^ and the number of men, women, and children was found to be 3361. — The chrijhnhigi for the year were 122 — the xveddings 28 — burials 62. — No more, therefore, than the 54th part of the inl'.abitants died in the year, — In part of this year an epidemical fever prevailed in the parifh. See Mart} h'k Ahrukment of the Phihf. Tranfaftkns, vol. O 3 IX,
ig% Of the BxpcBntion of Lives t,
provinces of New-Ejigiand there is a vcvf rapid increaie of the inhabitants ; but, not- vvithftanding this, at Bofton, the capital, the inhabitants would decreafe were there no fupply from the country : for, if the account I have {ctn is juft, from 1731 to 1762, the burials all along exceeded the births [a). So remarkably do towns, in confequence of their unfavourablenefs to health, and the
IX. p. 32^ — According to G'r(7?m^'s account of a parifh in Hampjhlre^ not reckoned, he fays, remarkably health- ful, a 50th part of the inhabitants had died annually for 90 years. Natural and political Obfcrvatlom^ &c. Chap, xii.
In icg8 country parifhes, mentioned by Sufmikh^ the annual aver.ige of deaths, tor fix years, ending in 1749, was 5255. The number of inhabitants was 225,357. One, therefore, in 43 died annually. — in ic6 other pari flies, mentioned by him, this proportion was i in 50. • — In the Dukedom of JVuricmherg^ the inhabitants, he fays, are numbered every year; and from the average of five yeais, ending in 1754, it appeared that, taking the f^iwns and country together, i in 32 died annually. — In another province, which he mentions, confifting of 635,990 inhabitants, i in 33 died annually. From thefe fads he concludes, that, taking a whole country \n grofs, including ail cities and villages, manjcind enjoy among them ab-ut 32 or 33 years each of exiftcnce. And this, very probably, may not be far from the truth in the pre- fent {tate of moit of the kingdoms oi Europe. And it will follow, that a child born in a country parifli or village, has, at Icali", an expedfation of 36 or 37 years ; fuppofing the proportion of country to town inhabitants to be as 3[- to 1 5 which, I tiiink, this ingenious writer's obfervationa prove to be nearly the cafe in Pomerania, Brandenburgh^ and fome other kingdoms.
[a) See a particular account of the births and burials in this town from 1731 to 1752 ia the Gentleman i Ma- gazi-iifioi: 1/53' P- 4' 3-
} luxury
the S'tate ^London, Popn!atkny Sec. icjg
luxury which generally prevails in them, check the increale of countries.
Healthfulnefs and Prolificknefs are, pro- bably, caufes of increale feldoni feparated. In conformity to this obfervation, it appears from comparing the births and weddings, in countries and towns where regiftcrs of them have l>£en kept, that in the former, marriages, one with another, fcldom produce lefs than four children each j generally be- tween four and five, and fomctimes above five. But in towns feldom above four ; ge- nerally between three and four; and fome- times under three (a).
I have fometimes heard the great num- ber of old people in London mentioned, to prove its favourablenefs to health and long life. But no obfervation can be more erroneous. There ought, in reality, to be more old people in London^ in proportion to the number of inhabitants, than in any fmailer towns ; becaufe at lead one quarter of its inhabitants are perfons who come in-
{a) Any one mny fee what evidence there is for thi'^, by confiilting Dr. S/jcrfs two books alreiidy qiioLed, and the Abyidment of the Philofiphical Trunjaftkns^ vol. Vll. partiv. p. 46, and Gr(7.v/7/'s account already quoted, of the births, weddings and burials in three country paiilhes (or 90 years; compared with fnnilar accounts in towns. In confidcring thefe accounts, it fl^ould not be forgotten that allowances mull be made for the different circum- ftances of uicri-afe or decrcafe in a place, agreeably to theobfervaiion at ihe end f)f tl^e note in pai2;e 104.
O 4 ' ' to
^,o.o of ibe ExpeBation of Lives ,
to it, from tb.c country, in the mod: robuft part oi life, and with a much greater proba- bility of attaining to old age, than if they had come into it in the vveaknefs of infancy. But, notvvithltanding this advantnge, there are much fewer perfons who attain to great ages in Loudon, than in moft other places where obfervations have been made. — At Vienna; of 22,704 who died in the four years 1717, 1718, 1724, 1725 (^), 109 reached 90 year?i that is, 48 in 10,000. But in London^ for 30 years before 1769 only 35 of the fame number have reached this age. — At Brefuiw it appears, by Dr. Halleys Tabic, that 41 of i238 born, or a 30th part live to be Ho years of age. — In the pnriHi of jili-JaintSy in Northampton^ an account has been kept ever fmce 1 733 of the ages at which the inhabitants die; and I find that a 22d part die there turned of H'o. At Nor-^ njDich iX\V,^ account his been kept; anditap-* pears, that for the laft 30 years, a 27th part of the inhabitants have died, turned of the fairjs age. — According to Mr. /itr//c'/^(5^w's Table of Obfervations, published at the end of the laft edition of xMr. De Moivre% Treatife on the Doctrine of Chances, a 1 A^tb part of all that are born live to 80. — And in the parifh of Holy- Crofs, already mentioned p. 183 and p. 192,
[a) ViJ, Abridgment of the Philorophic:^! Tranfac- tions, vol. V'li. part iv. p. 46. — It appears allb that more than ihree-hfihs of all who died in thtTe years at Fienna were boys and {Tii lii, by whom, I fuppofe, arc meant per-
I'jns under 16.
the
(be State c/^London , PoptilatwH, Szc. 201
tlie e/eventh part of the inhabitants live to 80 (a). — Bat in Lom-fo?!, for 30 years, ending nt the year 1768, only 25 of every 1000, who have died, or a 40tli part, have lived to this age ; which may be eafily difcovered, by dividing the fum of all who have died during tliefe years at all aii^es, by the fum of GJl wlio have died above 80.
Among the peculiar evils to which great towns are fubjed, 1 might further mention the Plague. Before the year 1666, this dreadful calamitv laid Loncion almoft walte once in every i ^ or 20 years ; and there is no reafon to think., that it was not generally bred within itfelf. A moft happy alteration has taken place ; which, perhaps, in part is Dv/ing to the greater advantages of clean- linefs and opennefs, which London has en- joyed fmce it was rebuilt; and which lately have been very wifely improved.
The fads I have now taken notice of are fo important that, 1 think, they defervc more attention than has been hitherto beitowed
(^) This, however, v/ill appear itfelf inconfuierable, whe/j compared with the following account: " In ij6i *' the burii.]s in the diilrict of Ckr'ijTianna^ in Norway., *' amounted to 6,929^, and the chriucnings to 11,024. *' Among; thofe who died, 394, or i in 18, had lived to *' the a;^e of 90 ; 63 to the age of 100, and {^wcn to the *' age ot^" lOi. — In the diocefe of Bergen., the perfons *' who dhd amounted only to 2,580, ot whom 18 lived ** to the acre of ico ; o:;c woman to the ace of lOJ., and *' another woman to the age of 108/'
ice the Ar.uual Rigijur for 1761, p. 191.
upon
20 2 Of the Expecfutton of Lhes ;
upon them. Every one knovv-s that tne itreneth of a ftate confifts in the number of people. The encouragement of population, therefore, ought to be one of ths iirfh ob- jedls of pohcy in every ftate ; and fome of the word enemies of population are the lux- ury, the licentioufnefs, and debility produced and propagated by great towns.
I have obferved that Lovidon is now [a) increafing. But it appears tliat, in truth, this is an event more to be dreaded than de- iired. The more London increafes, the more the reft of the kingdom mud be deferted ; the fewer hands muft be left for agriculture j and, confequently, the lefs mufl be the plen- ty and the higher the price ot all the means of fubfiftence. — Moderate towns, being feats of refinement, eiDulation, and art<, may be public advantages. But g7-cat towns, long before they grow to half the bulk of London^ become checks on population of too hurtful a nature, nurferies of debauchery and volup-- tuoufnefs; and, in many refpedts, greater evils than can be compenfated by any advan- tages (/;).
Dr.
(a) This increafe is greater than the bills f!ievv, on account of the oiriifTiOn in them of the two parifhes v/hicll have been mod increafed by new buildings ; I mean Mary-,- hme ami Pancrafs parifnes. The former of thefe parifhes is, I fuppofe, now one of the largeft in London.
(/>) The r.'xcs.n :iv.nu:.\ births^ zveddinzs^ and burials ia the following towns, for ibme of the bit years, have been nearly,
At
the State o/London, Fopiihtioriy 6cc. 203
Dr. Heberden obferves that, in Madeira, the inhabitants double their ov/n number in 84 years. But this (as you. Sir, well know) is a very flow increafe, compared with that which takes place among cur colonies ia America. In the back lettlements, where the inhabitants apply themfelves entirely to agriculture, and luxury is not known, they double their own number in i 5 years ; and all thro' the northern colonies, in 25 years [^a). This is an inflance of increafe fo rapid, as to have fcarcely any parallel. The births in thefe countries mud exceed the burials much more than in Madeira y and a greater proportion of the born muft reach maturi- ty.— -In 1738, the number of inhabitants in New Jt'i'Jty was taken by order of the go- vernment, and found to be 47,369. Seven years afterwards, the number of inhabitants ivas again taken; and found to be increafed,- by procreation only, above 14,000; and very- pear one half oi the inhabitants were found
Births. Weddings. Burials,
At Paris, — 19,100 — 4,400 — 19,400
Vienna, — 5,600 — — 6,8co
Amllcrdam, — 4,600 — 2,400 — 8, coo
Copenhagen, -- 2,700 — bb6 — 3;3<^o Berlin, for 5
^rlin, for 5 1 years, end- > 7, ingat 1751 )
years, end- ^ 3,634 — 936 — 4,092
ingat 1751 )
[a) See a difcourfe on Chrijlian Union, bv Dr. Styles, Bc/iotu 1761, p. 103. lOQ, &c. — See alfo 77?^ Inter r^ of Great Britain confukred zviti? regard to her Colonies^ together wit}} Ohfervotions eouLerning the increoje of mankind^ peopling ofcountriiij is't. p. 35. 2d edit. Ljiuicn, ijbi.
to
2 04 Of the Expe^atwn of'Lkesj
to be under {a) i6 years of age. In 22 years, therefore, they nuift have doubled their own number, and the births muft have exceeded the burials 2000 annually. As the increafe here is much quicker than in Madeira^ we may be fure that a fmaller proportion of the inhabitants mull die annually. Let us, however, fuppofe it the fame, or a 50th part. This will make the annual burials to have been, during thefe feven years, looo; and the annual births 3000; or an 18th part of the inhabitants. — Similar obfervations may be made on the much quicker increafe in Rhode Ijlandy as related in the prefiice to the Collec- tion of the London Bills of Mortality \ and alfo in the valuable pamphlet, lait quoted, on the Inter eft of Great Britain 'with regard to her Colonies, p. 36. — What a prodigious differ- ence mull there be, between the vigour and the happinefs of human life in fuch lituati- ons, and in fuch a place as London? — The original number ol perfons who, in 1643, had fettled in New-England, was 21,200. Ever lince, it is reckoned, that more have left them than have gone to them(^). In the year 1760, they were increafed to half a million. They have, therefore, all along
[a) According to Dr. Hallcfs Table, the number of the living under 16, is but a third of all the living at aU
ages.
{b) See Dr. Siyleis pamphlet juft quoted, p. iro. Sec.
doubled
the State o/'London, Population, Sec. 205
doubled their own number in 25 years. And if they continue to increafe at the fame rate, they will, 70 years hence, in New-England alone, be four millions ; and in all North America, above twice the number of inhabit- ants in Great-Britain [a). — But I am wan- dering
[a) The rate of increafe, fuppofing the procreative powers the fame, depends on two caufes ; The " encou- ** ragenient to marriage ;" and the *' expeSiation of a child juft born." When one of thefe is given, the increafe wiU be always in proportion to the other. That is ; As much greater or lefs as the ratio is of the numbers who reach maturity, and of thofe who marry, to the number born, fo much quicker or Jlovjer will be the increafe. — Let us fuppofe the operation of thefccaufcs fuch,as to produce an annual excels of the births above the burials yt<\wdX to a 36ih part of the whole number of inhabitants, it may fecm «o follow from hence, that the inhabitants would double their own number in 36 years; and thus fome have cal- culated. But the truth is, that they would double their own number in n^uch lefs time. Every addition to ths number of inhabitants from the births, produces a pro- portionably greater number of births, and a greater ex- cefs of thefe above the burials ; and if we fuppofe the excefs to increafe annually at the fame rate with the in- habitants, or fo as to preferve the ratio of it to the num- ber of inhabitants always the fame, and call this Viitlo
--, the period of doubling will be, the quotient prodyced
by dividing the logarithm of 2, by the difference between the logarithms of r -}- i and r \ as might be eafily demon- flrated. In the prefent cafe, r being 36, and ; -f- i be- ing 37, the period of doubling comes cut 25 years. If r is taken equal to 22, the period of doubling will be 15 years. — But it is certain that this ratio may, in many lituations, be greater than ^\; and, inftead of remaining the fame, or becoming lefs, it may increafe^ the confe- quence of which will be, that the period of doubling will he fliortcr than this rule give? it. — According to Dr. Hallefi 1 able, the number of perfons betv/een 20 and
4?,
2o6 Of fhe ExpeElatlon of Lhes;
dering frotii my purpoie In this letter. Tlie point 1 had chietly in view was, the pre lent
Hate
4.2 years of age is a third part of the whole number living at all ages. The prolific part, therefore, of a country may very well be a 4th of the whole number of inhabi- tants ; and fuppofing four of thefe, or every other mar- riage between perfons all under 42, to produce one birih every year, the annual number of births will be a i6th part of the v^'hole number of people. And, therefore^ fuppofing the burials to be a 48th part, the annual excefs of the births above the burials will be a 24th part, and the period of doubling 17 years. — The number of inhabi- tants in Neiv-Enghmd was, as I have faid from Dr. Styles's pamphlet, half a million in 1760. If they have gone oii increafing at the fame rate ever fmce, they muft be now 640,000 5 and it feems to appear that in facSl they are now more than this number. For, fince writing the above obfervations, I have feen a particular account, grounded chiefly on furveys lately taken with a view to taxation and for other purpofes, of the number of males, between J 6 and 60, in the four provinces. According to this ac- count, the number of fuch males is 218,000, The whole number of people, therefore, between 16 and 60, muft be nearly 4365000. In order to be more fure of a- voiding exccfs, 1 will call them only 400,000. In Dr. Ualley's Table the proportion of all the living under 16 and above 60, to the reft of the living, is 13.33 ^^ ^° 5 and this will make the number of people now living in the four provinces of New-Engiand to be 666,000. But on account of the rapid increafe, this proportion muft be ronliderably greater m hhiv- England^ than that given by Dr. Hallcys Table. In Neiu-Jerfey, I have faid the num- ber of people under 16, was found to be almoft equal to the number above 16. Suppofe, however, that in Neiv- England^ where the increafe is flower, the proportion i have mentioned is only 16 to 20 j and then the whole number of people will be 720,000.
I eannot conclude this note vv'iihout adding a remark to remove «u objecStion which may occur to fome in reading
l>r.
the State ^^London, Population ^ 6cc. 207
ftate of London as to healthfulnefs, number ot inhabitants, and its influence on popula- tion. The obfervations I have made may, perhaps, help to Ihew, how the moft is to be made of the lights afforded by the London bills ; and ferve as a fpecimcn of the proper method of calculating from them. It is in- deed extremely to be vviOied, that they were lefs imperfed: than they are, and extended further. More pariflies round London might be taken into them ; and, by an eafy im- provement in the parifli regifters now kept, they might be extended through all the pa- ri/hes and towns in the kingdom. The ad- vantages arifing from hence would be very confiderable. It would give the precife law according to which human life waftes in its different ftages ; and thus fupply the neceffary data for computing accurately the values of all life- annuities and reverfions. It would, likewife, (hew the different degrees of health-
Dr. Heberdcns account of Madeira, to which I have re- ferred. In that account 5945 is given as the number of children under feven in the ifland, at the beginning of the year 1767. The medium of annual births, for eight years, had been 2201 ; of burials 1293. 1'^ ^^'^ years, therefore, 13,206 muft have been born*, and if, at the end of fix years, no more th.m 5945 of thefe were alive, 1210 muft have died every year. That is; almoft all the burials in the ifland, for fix years, muft have been buiialb of children under {tt\c\\ years of age. This is plainly incredible; and, tlierefore, it feems certain, that the number of children under feven yeisrs of age muft, through fome miftake, be given, in that account, 3-300 or 4000 too little.
fulnels
••oS Of the Expecfdiio?i of Lives ;
lulncfs of different lituations, mark the pro- grefs of population from year to year, keep always in view the number of people in the kingdom, and, in many other refped^s, fur- Dilh indruLtion of the ^reatelt imoortance to tfie flate. Mr. De Moivrc, at the end of his book on the dodirine of chances, has rccom- iDended a general regulation of this kind ; and obferved, particularly, that at leail it is to be widied, that an account was taken, at proper intervals, of all the living in the king- dom, with their ages and occupations ; which would, in feme degree, anfwer mod of the purpofes 1 have mentioned. — But, dear Sir, i am feniiblc it is high time to finifli thefe remarks. I liave been carried in thei«n far beyond the litnits I at firfl intend- ed. I ahvays think with pleafure and gratis tude of your friendfhip. The world owes to you many important difcoveries ; and your n.une muft live as long as there is any know- ledge of philofophy among mankind. That your happinefs in this, and every other re- fpedl, may continually increafe, is the finccre Willi of, Si r.
Your much obliged,
and very humble fervant,
NcwinG;tnn-Grccn, -r» t-,
Aprils, 1709. Richard riuci:.
POST-
the State g/'London, PopuIailoUy &C. io^
A
POSTSCRIPT.
T Edmhurgb, bills of mortality, of the f^me kind with thofe in London^ have been kept for many years. I have, fince the foregoing letter was written, examined thefe bills, and formed a Table of Obferva- tions from them, as I found them for a period of 20 years, beginning in 17393 and ending in 17^8. — As this is a tov/n of moderate bulk, and feems to have a particular advantage of lituation ; 1 expedted to find the probabili- ties of life in it, nearly the fame with thofe at BreJlaWy Northampton and Norvjich j but I have been furprized to obferve, that this is not the cafe. During the period I have men- tioned, only one in 42 of all who died at Edinburgh, reached 80 years of age j which is a fmaller proportion than attains to the fame age m London. See p. 201. — In general ^ it appears, that the probabilities of life in this town are much the fame, thro' all the ilages of life, with thofe in London, the" chief diffe- rence being, that after 30, they are rather lower at Edinburgh. — It is not didicult to ac- count for this. It affords, 1 think, a ftrik- ing proof of the pernicious cffcdts arifing from uncleanlinefs, and croudin^!; t09;ether on one fpot too many inhabitants. At Edinburgh, Mr. Maitland fays, " the build- ** ings, elfewhere called houjh^ are denomi- P ' " nated
2 1 o On the ExpeSfation ef Lives ;
** nated lands \ and the apart?nents, in other " places mxn^AJiories^ here called houfesy are ** fo many freeholds inhabited by different "families; whereby the houfes are fo ex- *' ceffively crouded with people, that the " inhabitants of this city may be juftly pre- *^ fumed to be more numerous than thole of ** fome towns oi triple its dimenfions." See Maitlanis Hijlory of Edinburgh, p. 140.
In the year 1748, the whole number of apartments or families in the city and liberties of Edinburgh, was 9064. This Mr. Mait- land mentions as the refult of particular exa- mination, and undoubtedly right, lb, p. 2 17, 218. — In 1743, an accurate account was taken, by the defire of this writer, of the number oi families and inhabitants in the pa- n^hoi^t.Ciithhert. lb. p. 171. The number oi families was 2370, and o^ inhabitants at all ages, 9731. The proportion, therefore, of inhabitants to Jamilies, v/as 4t'o- to i ; and, fuppofing this the true proportion for the whole town, the number of inhabitants will be 4^-0 multiplied by 9064, or 37,162. — The yearly medium of deaths in the town and liberties for eight years, from 1741 to 1748, was 1783. lb. p. 220 and 222. And, confequently, one in 20* died annually.
Mr. Maitland, tho' poirefTed of the data from which thefe conclufions neceffarily fol- low, lias made the number of inhabitants -50,120, in confequence of a difpofition to 3 exag-
the State 0/ London, Fopulatmiy 5cc. 211
exaggerate in thefe matters, and of afiumlng, without any reafon, a 28th part of the inha- bitants as dying annually.
\\\ page 220, he expreffvis much furprize at finding, that the number of males in this town was lefs than the number of females, in the proportion of 3 to 4. Bul this is by no means peculiar to Edinburgh.
All I have been faying mull be underllood of the flate of Edinburgh, before the year 2758. The bills, for the lafl 12 years, have been fo irregular, and fo different from the fame bills tor the preceding years, and from all other bills, that I cannot give tl^em any credit. Either fome particular incorredneis has crept into the method of keeping them; or there has been fome change in the ftate of the town which renders them of no ufe. Probably the former is the truth.
From the note in p. 203, It appears, that the chriftenings and burials at Paris, come very near to equality. This once led me to fufped:, that there muft be fome particular lingularity in the ftate of Paris, v/hich ren- dered it much lefs prejudicial to health and population than great towns commonly are. But better information has lately obliged me to entertain very different fentiments. — The difference between the births and burials at Paris, is much greater than the bills (iievv. P 2 " Children
212 On the ExpeBation of Lives ;
*' Children here are baptized the inflant ** they are born ; and, in a day or two af- " terwards, it is the cuftom to fend them to ** the adjiicent viiliges to be nurfed. A *' great number therefore, of the infants born ** at Paris, die in the country, and thefe " appear only in the regifter of chriften- ** ings." See a book entitled the Police of France, page 127. And Buffon% Natural Hiftory, Tom. II. at the end. — '* All the " children alfo received into the FGundling- *' Hofpital, sre immediately fent to be nurf- *' ed in the country, at a diftance from Paris, '« where they remain 5 or 6 years j at the end ** of which time they are brought again to ** Paris, the boys to be placed in the fuburbs ** of St. Antoine, and the girls at SalpetrierCy " to be further maintained till they arrive at " the age of twelve years." Police of France, p. 81. — The following pafTage in the fame writer, containing a further account of this Hofpital, is important; and, therefore, tho* long, I cannot help tranfcribing it — ** Let *' us fuppofe, that out of 4000 children an- *' nually carried into the country, two thirds " rnay die, during the five years they are de- *' iliined to remain at nurfe ; fo that only ** 133-^ would conftantly be the annual " number fent b.ick to Paris-, who, being *' kept at the tw^o Hofpitals St. Antoine and *' *SVz^6'/r/('r£'juf]: mentioned, 'till they are 12, *' and fucceeded by ^ like number each year,
" the
the State o/' London, Population, &c. 2 1 3
** tlie total number compofcd of all brouglu " in the ruccefTive years, would make the " condant reding- flock to amount to 9331. " But of thefe we will fuppofe a 5th part ** to die every year. Yet even then the •* conflant reftins; ftock of children ought to ** be 7465. How greatly then mud: we be ** furprized to find, by the authentic account ** taken from their own books, only 640 ** boys in the college of St. Antoiney and not *' more tham 600 girls at the Salpetriere ; " fo that the refting ftock of returned found- '* lings appears to be no more than 1240, *' which being deducted from 7465,willmake *' the difference in the deficiencies 6225. " What then becomes of thefe ? — Are they " reclaimed by their parents ? — Or do they *' periih for want of care? — In anfwer to " which queftions it was explained to me; ** that as many of the lower clafs of people " were induced to marry, in order to be ex- ** cufed from ferving in the militia ; fo when ** thefe have children, which they are una- ** ble to maintain, they ufually fend them to *' this hofpital ; which, therefore, mud be ** looked upon, as not only a charity for the ** care of expofed and deferted children whofe ** parents are unknov^n, but alfo as a public *' nurfcry for the fuftcnance of the children *' of poor people, who, tho' regiftered at tiie ** office, are often reclaimed from their coun- " try nurfes by their parents. This accounts, P 3 ' *' in
214 0?i the ExpeBation of Lives ;
" in foine meafure, for the fmall flock of ** children brought back to the hofpital at " Pan's. — The further difference is fufpefted " to be owing to the infufficient nourishment ** they receive i as this particular charity, as *' well as the General Hofpital, adopts that " prcpofterous method of taking in an un- " limited number, while there is only a li- " mited income for their fubfiftence." 16. page 83.
Thele h&s prove, that, at the fame time that the reriftcr of chrijlenings at Faris muft be iul], the regifter of burials muft be very deficient. Let tlie deficiencies be reckoned at 4600 ; and, confequently, the annual burials at 24,000. The annual average of weddings, given in p. 203, is 4300; and, therefore, the number of perlbns who marry annually mufl be 8600. The difference be- tween the chriflenings and burials is 5000; which, therefore, is the number of annual recruits from the country. Thefe, in gene- ral, mull: be perfons in mature life. Suppofe 3600 of them to marry after fettling at Paris. Then, 8600 lefTened by 3600 or 5000, will be the number of perfons born at Paris who marry annually; and 14,100, or near three- fourths of all who are born at Paris ^ will be the number dying annually in childhood and celibacy.
The fuppofitions on which I have made this computation feem moderate; but if any
one
the State o/' London, Population, &c. 215
one thinks otherwife, he may make the fame calculation 011 any other fuppofitions.
The births at Paris arc above four times the weddings ; and it may feem, therefore, that here, as well as in the mod healthy country lituations, every wedding produces above four children. I have obferved no- thing like this in any other great town. Many children born in the country are, I fuppofe [a), brought to the Foundling-Hof- pital, and there chrlftened. This Hofpital may likewife occafion a more than common number of illegitimate births. And, befides, fome who leave the country to fettle at Paris, may come thither already married. Thefe are circumflances that will fwell the regifter of births, without having any effect on the weddings. I do not, however, know that any of them take place at Paris ; and, per- haps, it muft be granted, that it is diftin- guifhed in this refped; from moft other towns. Nor can I wonder at this, if it be indeed true, not only, that all married men in France are excufed ferving in the militia from whence draughts are made for the army, but alfo, that "Sijifth of all the children born at Paris are fent to \\\^Foundling-Hofpital {b). Thefe
are
[a] " If the parents of a child brought to this Hofpital '' are known, the regifter of its baptifm muft be pro- " duced. If the parents are unknown, the child muft ** be baptifed after being received." Police of France, pag. 82.
[b) See Poh'ce of France, &c. p. 83. — This writer adds, that a third of all that die at Paris die in Hofpitnls.
P 4 "In
2 1 6 On the ExpeSlation of Lives ;
are encouragements to marriage that no other city enjoys. It has been fecn thatthe Found- ling-Hojpital, tho' attended with this efFeift, is, probably, in the higheft degree pernici- cus; but it is to be vvillied, that fome poHcy oi the fame kind wtTih that firft mentioned, was piirfiied in this kingdom. — hi the end of the 2d vol. of Monfieur De Buffhns Natural Hijiory^ there are Tables formed from the Obfervations of M. Du Pre de S. Mour, of the French Academyy containing an account of the ages as: v/hich 13,1 89 perfons died in three pariiiies Aiparis], and alio, of the ages at which 10,805 perfons died in 1 2 country pariflies and villages near Paris. — According to thefe Ta- bles, many ??tore die in the beginning of life, and much /'kyjr in the latter part of life, in the country than in Paris. But the circum- ll:anccs of Paris, and the country round it, are fuch, that no argument can be drawn
*' In \n(t Hotel Dien (a great Kofpital fiturted in the l^' micJJIe of the city) we may, he fays, beliold a hoiri4 *' icene of nufery j for, the becis being too few for the " numbers admitted, it is common to fee 4, or 6, or *' even 8 in a bed together, lying 4 at one end, and 4 <"• at the other, ill of various diflempers in feveral de- i^' grccs ; feme bad, others worfe ; fome dying, others *' dead. — Above ^ fifth o^ all admitted to this flolpital *' die; the annual numbers admitted being 21,823. -^^^ f medium of deaths for three years from 1751 to 1753, *' 4&50. — The medium of deaths for the lame years in '' ^//7 the Kofpitals was 6/8i." lb. p. 85. — In our tv/o great city Hofpitals, St. TLsivms^s and St. Bartholonieius^ about 600 die annually ^ or one in 13 of all admitted as jn pati^'irvS,
3 from
the State of London , Fopulation^ &c . 217
from hence in favour o{ Paris. Many of the children dying in the country, are children fent thither from Pans to be nurfed; and, on the other hand, many, perhaps mqfty of thole who die in old age at Paris ^ are per- fons who have removed thither from the country, fome to Plofpitals, and fonie to places and fettlemeiits. It is evident, there- iore, that thefe Tables give a reprelentation of the probabilities of life at Paris, which, when compared with thofe in the adjacent country [a), is jull, the reverfe of the truth. Were the children born at Paris, who die in the country, to be transferred to the town fegifter; and, on the contrary, the adults born in the country, who die at Paris, to be transferred to the country regifter, there is no reafon to doubt, but that the proba- bilities of. life at Paris, would be found as low, in comoarifon with thofe in ihe coun- try, as the probabilities of life in London are; or, perhaps, much lower. — This ob- fervation is applicable, in fome degree, to mOil other great towns ; and, in general, on account of the migrations from the coun- try to towns, navies and armies, we may be fatisfied, that we err on the iide of defeB,
[a) It is for this reafon that thefe Tables, when com- bined, exhibit juftly the mean probabilities of life for town and country taken together ; and that the Table of the dccrcncnis of life deduced from them by M. Buffon and M. Du Pre, agrees nearly with Dr. Hallcy-i Table.
whenever
2i8 On the Expedlation of Lives 'j
whenever v/e judge of the probabilities of life in the country y from the numbers dy- ing in the feveral ll:ages of life -y and, on the fide of excefsy whenever, in the fame way, we judge of the probabilities of life in towns. And this, it is obvious, has a tendency to confirm all that has been faid in the preced- ing Eflay, concerning the pernicious effed;s of great towns on human life.
There are feveral ordonances and arrets of council which fix the boundaries of Paris, and prohibit all new buildings beyond thofe boundaries. — The reafons of this regulation, as fet forth in one of thefe arrets, are re- markable I and it will not be improper to recite them. — ** By the exceilive aggrandiz- " ing of the city, it is faid, the air would be ** rendered unwholefome, and the cleaning ** the flreets more difficult." — " Augment- *' ing the number of inhabitants would aug- ** ment the price of provifions, labour and *' manufadures." — ** Tiiat ground would be ** covered with buildings, which ought to be " cultivated in raifing the necellary fublift- ** ence for the inhabitants j and thereby ha- ** zard a fcarcity." — *' The people in the *' neighbouring towns and villages would be *' tempted to come and fix their refidence in *' tlie capital, and defert the country." — *' And laflly; the difficulty of governing fo ** great a number of people would occafion *' a diforder in the Police, and give an oppor-
*' tunity
the St0te of "LonAony Population, &c. 219
*' tiinitv to rofTues to commit robberies and
J ■J
** murders {n)."
No one can think overgrown cities greater evils than I do. But, yet, I can by no means approve of tliis policy. The efFecft of it mufl be, crouding together too many people with- in the prefcribed boundaries, and rendering a town more the feat of uncleanlinefs, infec- tion and difeafe. — The number of houfes in P^j- is reckoned to be 28,000 [b), or 30,000; but the number of inhabitants, fuppofing a 20th part to die annually, cannot be much lefs than 480,000, or 16 times the number of houfes.
It is happy for London, that there have been no laws to reftrain its increafe. In con- fequence of being allowed to extend itfelf on all lides into the country, the inhabitants now take near twice the room to live upon that they did; and it is become lefs the means of fliortening human life. See p. 189, 191, and 201.
In page 203, I have given the annual me- dium of births, weddings and burials at Ber- lin, from 1747 to 175 1. — In 1747, an ac- count was taken with the utmoft care, by the order of the King of Prussia, of the number of inhabitants in this town ; and, it was found to be 107,224. — In order to be more certain, 2.Jkond account was taken the
{a) Vid. Police of France, p. 130. [b] Ibid. — There are other accounts which makes this number above 50,000,
fame
220 On the ExpeSiation of Lives -,
fame year; and the number found the fame within 200. — In 1749, the inhabitants were increafed to 1 10,933. Their number, there- fore, compared with the annual burials, was as 27 to I. — This is a higher proportion than could be expeded in a town lb confiderable ; and alfo fo much crouded, as to have, at an average, 16 inhabitants in every houfe. But there is a plain reafon to be given for this fad:. — Berlin, for many years, had been increaling very faft, by a conflux of people from the furrounding country and provinces. About the year 1700, the medium of annual burials was no more than 1000. In 50 years, tb.erefore, it quadrupled itfelf.— In a city in- jcreafing with fuch rapidit3^ the ratio of in- habitants to the annual deaths, mu(l be great- ly below the juft (landard. — Were there now, fuch acceffions to London of deferters from the country, in the beginning of mature life, as would caufe the number of inhabitants to increafe at the rate of 10,000 every year, it would in 60 years be doubled ; and the pro- portion of inhabitants to deaths would rife gradually, 'till it came to be about one third greater. Berlin, we have feen, has, in fad, increafed at fmre than double this rate ; and, therefore, the proportion of inhabitants dy- ing annually in it is in reality very low.
The ingenious Sufmilch, to whofe curious book, already quoted, I owe my information concerning Berlin, makes the proportion
of
the State of London, Population^ &c. > 2 2 1
of people who die annually in great towns, to be from ^ + to -rV ; in moderate towns, from -V to tV j and, in the country from Vo- to -5^. — The obfervations and fads in this EiTay, joined to thofe which will be found in the 4th EfTay, prove, I think, that thefe propor- tions may be more truly ftated as follows.—- Great towns, from aV to Vt or Vt- Moderate tovv'ns, from Vt to aV. The country, from tV or ^, to -s^ or -sV. — This, however, muH: be underflood with exceptions. There may be moderate towns fo ill fituated, or whofe inhabitants may be fo crouded together, as to render the proportion of deaths in them greater than in the largeft towns : And, of this, Edinburgh, if it is not now, was 20 years ago, an example. — There may be alfo great tov/ns in which, from a fudden increafe, this proportion may be lefs than in fmall towns : And of this, I have juft given an example in Berlin.
ESSAY
[ 222 ]
ESSAY II.
On Mr. De Mo IV re's Rules for cal- culaWtg the Values of Joint Lives,
THE calculation of the values o^ Ji?Tgle 2Lnd joint lives, from given Tables of Obfervation, being tedious and troublefome; Mr. De Moivre has had recourfe to two Hy- pothefesy which give eafy rules for this pur- pofe ; and which, he thought, correfponded with fufficient exadnefs to Obfervations. — The firft of thefe Hypothefes is, that the pro- babilities of life decreafe, as we advance from childhood to old age, in an arithmetical pro- grej/ion-, or in fuch a manner, that the diffe- rence is always the fame, between the num- ber of perfons living at the beginning of any one year, and the number living at the be- ginning of the next following year. — The other Hypothejis is, that the probabilities of life decreafe in a geometrical progreffion j or in fuch a manner, that the proportion is al- ways the fame, between the number of per- fons living at the beginning of any one year, and the number living at the beginning of the next following year. — All the Tables of Obfervation Hiew, that the real law, ac- cording to which human life waftes. comes
much
Of the Method of calculating^ Sec. 223
much nearer to the former llypothefiSy than the latter. — In Tables III, IV, and V, in the Appendix t it is fo near the former HypotheJiSt that the ditference is fcarcely worth regard- ing. According to this Hypothejis^ therefore, (accommodated to the Brejlaw Table, in the manner mentioned in the note, page 2. ) Mr. De Moivre calculated the values oi Jingle lives; and the rules founded upon it for this purpofe are fo eafy, that an operation which would otherwife take up much time, may be performed almoft immediately.
By proceeding on the fame principles, the values o^ joint lives might have been calcu- lated ; but the rules for this purpofe derived from thefe principles, are far from being equally eafy in pradife. Here, therefore, Mr. De Moivre quitted hhfrji Hypothefis ; and finding, that the y?^<9;?<^ Hypothefis af- forded, in the cafe of joint lives, rules that were as eafy, as the rules given by the other Hypothefis were in the cafe o^ fmgle lives, he chofe to adopt this Mypcthefis ; believing at the fame time, that the values of joint lives, obtained by rules derived from it, would not deviate much from rhe truth. But in this he was greatly mifl:ak.en. The values of two joint lives obtained by thefe rules are fo wrong, that in finding the prefent value, in 2i Jingle payment, of one life after another, they generally give refults which are near ^ of the true value too great; and about tivo-
fifths
224 Of the Method of calculating
fifths too great, when the value is fought in annual payments during the joint lives. — * Thefe are errors To confiderable, that I think it is of particidar importance that the public fhould be informed of them, in order to pre- vent the inconveniencies and perplexities they may occafion.
Mr. Simpfofi (in the Appendix to his Trea- tife on the DoBrine of Annuities and Rever^
fions) has obferved, that ?vlr. De Moivres rules for finding the values of joint lives are wrong. But I don't knov/, that it has been ever at- tended to, that they ^ivcfo wrong as I have found them. Mr. Si?npJon% remarks point out chiefly the errors in thefe rules, when the values of three or more joint lives are calcu- lated by them ; but, 'till 1 was forced to a particular examination of this fubjefl by fome difficulties into which I found myfelf brought by following Mr. De Moivre too implicitly, I did not at ail fufpe6:, that any fuch errors as I have mentioned, could arife from thefe rules, when the values of only two ]o\nt lives are calculated by them. Mr. De Moivre, in coni'equence of other remarks con- tained in Mr. Simpfon?, Appendix, altered in the 4th edition of his Treatife fome of his rules. It is furprizing he did not fee reafon at the fame time to alter thefe.
That there may be no doubt about the truth of thefe obfervations, I will juH: men- tion a few examples of the difference between
the
the Fakes of Joint Lhes. 225
the values of a given reverfionary annuity, according to the rules to which 1 have ob- jeded, and the true values, according to the cxa6t method of deducing them from Mr*. De Mohresfuji Hypot be/is.
Let the propofed annuity be 30/., to be enjoyed for what fliall happen to remain of the life of a perfon now 40 years of age, af- ter the life of another perfon of the fame age* The value of the joint lives (intereft being at ji^ per cent.) is, by problem 2d of Mr. De Moivre's Treatife on Life-annuities, 8.964; which fubtracted from 13.196, (the value by Table VI, of a fingle life at 40) gives 4. 23 ; v/hich remainder, multiplied by 30, gives /. 126.9, or the value of the reverfion in a fingle prefent payment. And 126.9, di- vided by the foregoing value of the joint lives, is A 14. 16 ; or, the value of the rever- lion in annual payments during the joint lives.— But the true values are /. loz.i in a Jingle payment, by Queft. L chap. L; and /. 10.3, in aimual payments, by Queft. IV.—-- The former values, therefore, are near a qiiarter of the true value too great in ih^Jin- gle payment j and near fivo-fjths too great in the ^/;;? ?/<:?/ payments.
The true value of the fame annuity for a life at 66, after another life of the fame age, is, (reckoning interefl: as before, at ^per cent.) 68/. in ayZ/z^^ payment ; and 13-5, in an- ?iual payments.-— But tliefe values, according
2 26 Of the Method of cakiilating
to the Problem jud quoted, are 91/., and 21/. one of which is near a thirdy and the other above ha ftho, true value too great.
In unequal lives tlicfe errors may be no lefs conliderable. — Thusj if the value of the pro- pofed annuity be required for a life at jo^ after a life at 30 years of age ^ it will, by the fame Problem, be Lib.^^ in 2. f ingle pay- ment; and /. 5.1, in annual payments during the joint lives. But the /r//f values are 17/. and /. 3.05.
Where 3 or more lives are concerned the errors will be flill greater.
The true values of the joint lives, men- tioned in thefe Examples, have been calcu- lated by a rule in pag. 16, of Mr. Simpfons Treatife on the Dodrine of Annuities and Reverfons, and explained in note (M) Ap- pendix.— To fave, however, a great deal of trouble hereafter, I have thought proper to calculate Table VII, which gives the exad: values, according to Mr. De Moivre^ fojl Hypothefis, of two joint lives, for every five years of human life from, 10 to 70.
This Hypotbejis, I have obferved, does not differ much from the Tables of Obfervation in the Appendix^ for Bre/laiv, Northampton and Norwich. Between the ages of '^o and 40, it gives the values oifingle lives almoft the lame with the Brejlaw Table. Under 30, it gives them fomewhat hfs ; and above 40, lumewhat greater. But it ought to be re- membered.
the Vdlucs of 'Joint Lives. 227
membcred, that wherever it docs tills, it sives, at the lame a<?cs, the values of the joint lives alio too little or too great; and that, conlequently, the refults from it, in calcula- ting the values oiReverfionSy and of the longefi of given lives, conie fo much nearer to exact- nels.
The rules to which I have objeded are the only ones given by Mr. T): Moivre, in all the editions of his Treatife on Life-Annuities. But it feems, this great mathematician be- came at lall: fenfibie, that they were too in- correct ; and, therefore, at the end of the lall edition of his Treatife on the DoBrhie of Chances y pag. 320, (a work which gets into comparatively few hands) he has given other rules which come nearer the truth. But even thefe rules produce errors fo great in many caf^s, (particularly when combined with the errors of the Hypothefis) that it will be beft never to ufe them.
<i^2 ESSAY
[ 22^ 3
ESSAY IIL
Of the Method of calculating the Values of Reverfons depe?jdi72g o?i ' Survivotfjips.
ALL Qaeftlons relating to the values of lives and revcrfions, are at pi'efent of par- ticular importance in this kingdom. Much bufinefs is continually ti'anfad:ed in this way; and any confiderable errors in the methods of folving fuch queftions, nuifl: in time pro- duce very bad confequences. — The defign of the following obfervations is to point out a particular error, into which there is danger of falling, in finding the values of fuch re- vcrfions as depend on furvivorfliips. In do- ing this, I fhall, in order to be as plain as poffible, take the following cafe. ** A, aged ** 40, expe(5ls to come to the poirellion of " an eilate, ihould he furvive B, aged like- " wife 40. In thefe circumftances he ofters, " in order to raife a prefent fum, to give fe- *' curity for 40/. per anmuiiy out of the edate '* at his death, provided he (liuukl get into
" polief-
Of the Values of Reverfions^ 6cc. 229
^■* pofledion ; that is, provided he rtiould fur- ** vive B. What is the Turn that ought now ** to be advanced to him, in conlideration of *' luch fecurity, reckoning compound intereft ** at 4 pi'r centV*
Mr. T)e Mohre'% directions in his Trea- tife on Annuities, Problems 17th and 20th, lead us to feek the required fum in this cafe, by the following procefs.
Find firft, the prefent fum A iliould re- ceive, for the reverfion of 40/. per anmun for ever after his dsath ; fuppofing it not depen- dent on his furviving B. The prefent value of fuch a reverfion is *' the (.7) value of the lite '* fubtradted from the pei'petuity, and the re- ** mainder multiplied by the annual rent." — The value of the life is, by Table VI, 13.196. This fubtraded from 25, the perpetuity ^ leaves I 1.80; which, multiplied by 40, gives 7.472 ; the value of the fuppofed eflate, after the life of A. But, as Mr. T>e Moivre obferves, the lender having a chance to lofe his money, a compenfation ought to be made to him for the rifk he runs, which is founded on the poifibility, that a man of 40 years of age may not furvive another perfon of the fame age. This chance is an equal chance ; and, there- fore, half the preceding fum, or 236/. is the money which fliould be advanced now on the expectation mentioned.
[a) By Scholhnr.y p. 34, and Problem 26th, p. 293, cf Mr. Simpfon's Selccl Kxercifcs.
Q..3 TiiU
230 Oj the Values cf Reverjions
This iolation carries a plaullble appear- ance ; and moft perfons will, probably, be ready to pronounce it right; nor will this be at all wonderful, as lb great a mafter of thefe fubje*fls as Mr. De Moivre appears to have been mifled by it. — Nothing more is necelTary to prove it to be fallacious, than proceeding in the fame way to folve the fol- lowing limilar Queftion.
*' A, aged 40, offers to give fecurity for ** 40/. per anmimy to be entered upon at his *' death, provided it fhould happen before ths ** death of B, aged likewife 40. V/uat fum ** fhould now be advanced to him for fuch a ** reverlion, interefl being reckoned at ^fer '* ce7it V
In folving this Problem, agreeably to the method juft defcribed, we are to find the va- lue of j^oL per annum, to be entered upon certainly at the death of A ; and then to mul- tiply this value by the chance that A fliail not furvive B, or by 4- ; and in this way the anfwer comes out the fame with that already
given,
Now it may be eafily feen, that this muft be wrong. The value of a reverfion, to be received when a perfon of a given age dies, cannot be the fame, whether the condition of obtaining it is, that he fliall die before^ or that he (hall die after another perfon. That is, whether it is provided, that a purchafer, if he fucceeds, Ihall get into pofTelTionySi?;?^;- or 1 later.
depending on Zurmivorjlnps. 2 3 1
later. The reverfion in the latter cafe muH:, without doubt, be of lefs value than in the former.
The firfl Queftion here propofed, refolves itfelf into the following general Q^ieftion.
** What is the prefent value of a given re- " vcrfionary eftate, to be entered upon after " the failure of two lives, provided one in '^ particidar of them fliould be the longeji ''lifer-
Now, the prefent value of an eftate to be enjoyed for ever, after the failure of the long- eji of two lives, is *' the value of the longeJl ** of the two lives, fubtraifled from the per- *^ petuity 'y and the remainder multiplied by '* the annual rent of the eftate." — The value of the /ongeft of two lives is, (as is well-known) the value of the two Joint lives, fubtrad:ed from tht/wn of the [a) values of the two Jingle lives. In the prefent cafe, therefore, it is 9.82, (the value of two joint lives at the age of 40 by Table VII,) fubtraded from twice 1 3. 196 ; (the value of 2i Jingle life at the fame age by Table VI,) that is, 16.57 year's pur- chafe. And this fubtraded from 25, (the perpetuity) gives 8.43 ; which, multiplied by 40, gives /•337.2, the value of the given eftate were it certainly to be enjoyed, after the ex-
[oi) See Mr. De Motvre on Annuities, Problem IV; or Mr. Simpfon's Doiirine of Annuitiei and Rever/ions,Pioh.
0^4 tindtion
o ^ '7
,32 Of the Values of Reverfioiis
tinflion of the longed: of two lives both 40; that is, whether one or other of them failed laft. But that A's life in particular fhould jRiil lad, rather than B's, is an even chance. The true value of the reverfion, therefore, is half the lail value, or /. 16H.6.
In like manner. The fecond Queftion is the fame with the Queftion, *' What is the pre- ** fent value of 40/. per ann. for ever, to be en- " tered upon after the extinction oi' iv^o jomt ** lives both 40; that is, whenever either of *' them (iiall fail ; provided the firft that fails ** fhould happen to be A's life in particular r" — Arid the anfwer is found Uy fubtratling the prefent value of the two joint lives from the perpetuity, and multiplying the remainder by ', or by the chance that A in particular iliall die firft : And this will give the re- quired value, /. 303.4 [a).
In (liort. It appears in both thefe cafes, that, according to the firft method of folu- tion, we are to fubtrad; from the perpetuity the value of one of the fingle lives ; when, in xX^z former cafe, the value of the hngeji of the two lives J and, in the latter c^^Q^ the value of x\\<z\x joint continuance, ought, in reality, to be fubtraded. I need not fay what prodigi- ous errors may often arife from hence; aiid how unfit fuch a method of folution is fer practice,
{a) 1 have, thvo' fcarcelv necefiary, given a demonftr?-
::on, of laeTe Solutijiis in t!ic Appendix, note (N).
Mr.
depend! fig on Survi'vorjhips. 233
Mr. Simpfon^ in p. 322, of his Sele<ft Ex- ercifcvS, ipeaks on this fubjedt in the follow- ing manner. — ** I have been very particular ** on thefe kinds of Problems; and the more *' fo, as there has been no method before pub- ** li(hed, that I know of, by which they can ** be rightly determined. "Tis true, the man- ** ner of proceeding, by firft finding the pro- ** bability of furvivoriliip, (which method is ** ufed in my former work, and which a cele- *' brated author has largely infiftcd on in three *' fucceliive editions) may be applied to good " advantage, when the given ages are nearly ** equal ; but then it is certain, that this is *' not a genuine way of going to work, and ** that the conclufions hence derived are at ** bed but near approximations."
This excellent mathematician has here ex- prefled himfelf much too favourably of the method of folution on which I have remark- ed. — In both the cafes I have fpecificd the ages are equal; and yet, in one of them the error is a good deal above a third of the true value, and in the other, 2<Jifth: And, it is obvious, that in cafes where three equal lives are taken, the errors will be much greater. — Mr. Si?7ipfo?is Obfervations in this pafTage are true only, when applied to a different me- thod ufcd by himfelf, in the 28th and fol- lowing Problems of his Treatife on the Dec- trim of Annuities and Revtfjions. This me- thod is exa^ when the lives are equal , hut,
it
2 34 Of the Values of Rcverfionsy 6lc.
it gives reiults which are too far from the I truth, when there is any coniiderable inequa- lity between the lives..
It is with reluc^lance I have made fome of thefe remarks. Mr. De Mohre has made very important improvements in this branch of fcience; and the higheil refpedl is due to his name and authority. This, however, only renders thefe remarks more iieceiTary.
In the firfl: Chapter (Quellions loth, i ith, I2th, i4.th,&c.) I have given a minute ac- count of the method of finding, in all cafes, the values of the reverlions which have been the fubjedt of this Elfay.
Ohfer-
f 235 ]
Obje7'vat'i07is on the p7'oper Method of conftruBing tables jcr deter 77it7ting the Rate of /ju77tan Mortality ^ the Number of I7ihabita7its^ a7id the Values of Lives JTt a7ty T'oizm or Diff7'i&y fro77t Bills of Mortality 171 "which are give7t^ the Nu7nbers dyi7tg a?2nually at all Ages.
N every place that jufl: Aipports itfelf in the number of its inhabitants, without any recruits from other places; or where, for a courfe of years, there has been no increafe or clecreafe, the number of perfons dying every year at any particular age, and above it, muft be caual to the number of thelivins; at that age.— The number, for example, dy- ing every year, at all ages, from the beginning to the utmoft extremity of life, muft, in fuch a fituation, be jufl equal to the whole num- ber born every year. And for the fame rea- fon, the number dying every ye^Tr at C/^f year of age and upwards ; at two years of age and upwards ; at three and upwards, and fo on ; muft: be equal to the numbers that attain to thofe ages every year 3 or, which is the fame,
to
2j6 Of the Method of forming
to the numbers of the living at thofe ag«. It is obvious, that iinlefs this happens, the number of inhabitants cannot remain the fame. If the former number is greater than the latter, the inhabitants vcw^^ decreafs y if lefs, they mull mcreafc, — From this obfer- vation it tollows, that in a town or country, where there is no increale or decreafe, bills of mortality which give the ages at w^hich all die, will fliew the exadl number of inha- bitants; and alfo the exa6t lav/, according to which human life waftes in that town or country.
In order to find the number of inhabitants; the mean numbers dying annually, at every particular age and upwards, muft be taken as given by the bills, and placed under one another in the order of the lecond column of the 1 2th Table in the Appendix. Thefe numbers will, it has appeared, be the numbers of the living at i, 2, 3, &c. years of age; and, confequently, the fum, diminiihed by half tiie number born annually (<^), will be the
whole
(rt) This fubtraclion is necelTary for the following rea- fon. — In a Table formed in the manner here direiled, it is fuppored, that the numbers in the fecond column are all Irving together at the beginning of every year. Thus; the number in the fecond column oppofite to 0 in the pjl column, the Table fuppofes to be all juft born to- gether, on the firft day of the year. The number, like- wife, opjjofite 10 I, it fuppofes to attain to one year of
a^9
Tdl>/t's of Ohftr-vaUons, &c. 2\j
whole number of Inhabitants. — In fucli a fe- ritrs of numbers, the excefs of each number above that whicli immediately follows it, will be the number dying every year, out of the particular number alive at the beginning of the year; and thefe cxceffes fct down regu- larly as in the third column of the Table to which I have referred, will (hew the diffe- rent rates at which human life wafi:es thro* all its different periods, and the different pro- babilities of life at all particular ages.
It muft be remembered, that Vv^hat has been now faid goes on the fuppofition, that the place whofe bills of mortality are given, fupports itfelf, by procreation only, in the number of its inhabitants. In towns this verv feldom happens, on account of the luxury and debauchery which generally prevail in them. They are, therefore, commonly kept up by a conflant acceffion of flrangers or fettlersy
age juft at the fame time that the former number is born. And the like is true of every number in the fccond co- lumn.— During the courfe of the year, as many will die at all ages as were born at the beginning of the year; and, confequently, there will be an excefs of the number alive at the beginning of the year, above the number alive at the end of the year, equal to the whole number of the annual births ; and the true number conftantly alive to- gether, is the arithmetical mean between theie two num- bers; or, agreeably to the rule I have given, thcfum cf the numbers in the fecond column of the Table, lefrenei by half iht number of annual births. See Eir:ty 1, pa^e 174.
who
238 Of the Method of jormlng
who remove to them from country parlflies and viUases. lii thefe circua;iflanccs, in or- der to find the true number of inhabitants, and probabihties of life, from bills of morta- lity containing an account of the ages at which all die ; it is neceiTary, that the pro- portion of the annual births to the annual lettlers lliould he known; and alio the period of life at which the latter remove. — Both thefe particulars may be difcovered in the following method.
If for a courfe of years there has been no fenfible increale or decreafe in a place, the number of annual fettlers will be equal to the excefs of the annual burials above the annual births. If there is an increafey it will be greater than this excefs. it there is a de- er eafe^ it will be hfs.
The period of life at which thefe fettlers remove, will appear in the bills by an in- creafe in the number of deaths at that pe- riod and beyond it. Thus; in "Cri.^^ London bills, the nuir.ber of deaths, between 20 and 30, is generally above double; and between 30 and 40, near triple the number of deaths between io and 20 : And the true account oi this is, that Irom the age of 18 or 20, to 35 or 40, thiCre is an afflux of people every year to London from the country, which oc- cafions a great increafe in the number of in- habitants at thefe ages; and, confequently, raifes the deaths ioi ail ages above 20, con-
fiderably
ll'ahks of Ohfcrvaiionsy 6cc. 239
fiderably above their due proportion, when, compared with the number of deaths before 20. — This is oblervable in all the bills of mortality for towns with which I am ac- quainted, not excepting even the Brejlaw bills. Dr. Hallcy takes notice, that thefe bills save the number of deaths, between 10 and 20, too fmall. This he confidered as an irregularity in them, owing to chance ; and, therefore, in forming his Table of Obferva- tions, he took the liberty fo far to corretft it, as to render the proportion of thofe who die to the living in this divifion of life, nearly the fame with the proportion which, he fays, he had been injorincd [a) die annually of the young lads in Chrijl-Church Hofpitai. But the truth is, that this irregularity in the bills was derived from the caufe I have jufl aflign- ed. — During the five years for which the Brejla-w bills are given by Dr. Hallcy, the births did, indeed, a little exceed the burials; but, it appears, that this was the effed: of fome peculiar caufes that happened to ope- rate luft at that time; for, during a com- plete century from 1633 to ijl^, the annual medium oi births was 1089 (^), and of bu-
[a) See Lowthorp's Abridgment of the Philofophicni Tranfaclions, vol. III. p. 670. — Dr. HaU:y's information in this inftance was not right, as will appear prefently ; and, therefore, he has by no means fufnciently correclcJ the irregularity I have mentioned.
[b) See Dr. .9/,-;'7's Comparative I-Iincrv, p. 63.
rials
240 Of the Method of forming
rials 1256 {a)> This town, therefore, mud have been all along kept up by a number of yearly recruits from other places, equal to about zfevcnth part of the yearly births.
What has been now obferved, concerning the period of life at which people remove from the country to fettle in towns, would appear fufficiently probable, were there no fuch evidence for it as I have mentioned ; for it might be well reckoned, that thefe people in general, mud be lingle perfons in the beginning of mature life, who, not hav- ing yet obtained fettlements in the places where they were born, migrate to towns in que ft of employments.
Having premifed thefe Obfervatlons, I diall next endeavour to explain di(lin<ft ly, the effect which thefe acceflions to towns mufl: have, on Tables of Obfervation formed from their bills of mortality. This is a fubje(fi: proper to be infilled on, becaufe miftakes have been, committed about it ; and becaufe alfo, the difcuffion of it is neceflary tofliew, how near to truth the values of lives come as deduced from fuch Tables.
The following general rule may be given on this fubjed.
[a) It appears from the account in the Phikfophicctl Tran/a.^ffons^^Ahudgmemy vol. ViI,No. 380, p. 46, &c.} that from 17 17 to 1725, the annual medium of births an Brejlavo was 1252, of burials 1 507 ; and alfo, that much the greaicft part of Lhc_ births died under 10 years of
If
Tables of Ohfervations, &c. 241
If a place has, for a courfe of years, been inaintained in a (late nearly ftatlonary, as to number of inhabitants, by recruits coming in every year, to prevent the decreafe thatwould arife from the excefs of the burials above the births ; a Table formed on the principle, *' that the number dying annually, after every ** particular age, is equal to the number liv- ** ing at that age," will give the number of inhabitants, and the probabilities of life, too greats for all ages preceding that at which the recruits ccafe ; and after this, it will give them right. — If the acceffions are fo great as to caufe an increafe in the place, fuch a Ta- ble will give the number of inhabitants, and the probabilities of life, too little, after the age at which the accelTions ceafe {a) ; and too great, if there is a decreafe. Before that age it will in both cafes give them too great ; but mod conlidcrably io in the former cafe, or when there is an increafe.
[a) Agreeably to thefe Obfervations ; if a place in- creafes, not in confequence of acceflions from other places, but of a conftant excefs of the births above the deaths ; a Table, conili uflcd on the principle I have mentioned, will give the probabilities of life too lovv through the luhole extent of life ; bccaufe, in fuch cir- cumfijnces, jthe number of deaths in xhc firj} ilages of life muft be too great, in comparifon of tlie number of deaths in the latter ilages j and inore or Icio {o., as the increafe is more or lefs rapiJ. — The contrary, in all re- fpecls, takes place where there is a decreafe, ariling from the excels of the deaths above the births.
R For
542 C^f the Method of forming
For example. Let us fuppofe, that 244 of thofe born in a town, attain annually to 20 years of age; and that 250 more, all like- wife 20 years of age, come into it annually from other places ; in confequence of which, it has, for a courfe of years, been juft main- tained in the number of its inhabitants, with- out any fenfible increafe or decreafe. In thefe circumfiances, the num.ber of the living in the town of the age of 20, will be always 244 ?7atives and 7.^0 fcttlers, or 494 in all ^ and, fince thefe are fuppofed all to die in the town, and no more recruits are fuppofed to come in -, 494 will be likewife the number dying annually at 20 and upvv^ards. — In the fame manner; it will appear on thefe fup- pofitions, that the number of the living, at every age, flibfequent to 20, will be equal to the number dying annually at that age and above it; and, confeqnently, that the num- ber of inhabitants and the decrements of life, for every fuch age, will be given exactly by the Table I have fuppofed. But for all ages bfore 20, they will be given much too great. For let 280 of all born in the town, reach 10. In this cafe, 280 will be the true num- ber of the livins; in the town at the ao;e of ID; and the recruits not coming in 'till 20, the number given by the bills, as dying be- tvv^een 10 and 20, will be the true number dyino; annually of the livinoj in this divifion of life. Let this number be 36 ; and it will 3 follow.
tables of OhfcrvatlonSy &c. 243
follow, that the Table ought to make the numbers of the livins; at the a<{es between 10 and 20, a feries of decreafing means between 280 and (280 diminiflied by 36, or) 244. But in forming the Table on the principle I have mentioned, 250 (the number above 20 dying annually in the town who were not born in it) will be added to each number in this feries \ and, therefore, the Table will give the numbers of the living, and the pro- babilities of life in this divifion of life, almofl twice as great as they really are. — This ob- fervation, it is maniieil:, may be applied to all the ages under 20.
It is neceilary to add, that fuch a Table will give the number of inhabitants, and the probabilities of life, equally wrong before 20, whether the recruits all come in at 20, agree- ably to the fuppofition juffc made, or only begin then to come in. In this laft cafe, the Table will give the number of inhabitants, and probabilities of life, too great throughout the whole extent of life, if the recruits come in at all ages above 20. But if they <z<:i\\iQ 2lX. any p.irticular age, it will give them right only from that age; and before, it will err all along on the fide of excefs ; but Icfs confi- derably between 20 and that age, than be- fore 20. For example. If, of X^^q 250
I have fuppofed to come in at 20, only 1 50 then come in, and the reft at 30; the num- bers oi the living will be given loo too high, R 2 at
244 Q/' ^'^^^ Method of forming
at every age between 20 and 30 ; but, as jufl ihewn, they will be given 250 too high at every age before 20. — In general, therefore, the number of the living at any particular age, muft be given by the fuppofed Table, as many too great as there are annual fettlers after that age -, and, if thefe fettlers come in at all ages indifcriminately, during any cer- tain interval of life ; the number of inhabi- tants and the probabilities of life will be continually growing lefs and lefs wrong, the nearer any age is to the end of that interval. - — Thefe obfervations prove, that Tables of Obfervation formed in the common way,from bills of mortality for places, where there is an excefs of the burials above the births, muft be erroneous, for a great part of the du- ration of life, in proportion to the degree of that excefs. They ihew likewife, at what parts of life the errors in fuch Tables are mod conliderable, and how they may be in a great meafure correded.
All this I fliall beg leave to exemplify and illuftrate a little further, in the particular cafe
of London,
The number of deaths, betwe<^n the ages of 10 and 20, is always fo fmall in the Lon^ don bills, that it feems certain few recruits come io London under 20 j or at ieaft, not fo many as before this age are fent out for edu- cation to fchools and univerlities. After 20, great numbers come in 'till 30, and fome,
perhaps.
Tables of Ohfcrvationsy See. 245
perhaps, 'till 40 or 50. — But, at every age after 50, it is probable, that more retire froai Loftdon than come to it. — The hondon Tables of Oblervation, therefore, being formed on the principle I have mentioned, cannot give the probabilities of life right 'till 40. Be- tween 30 and 40 they rnuft be a little too high J but more fo between 20 and 30 ; and molf of all 'io before 20. — It follows alio, that thel'e Tables mud give the number of inha^ bitants in London much too great.
Table Xll, in the Appendix, is aTable form- ed in the manner I have explained, from the London bills for 10 years, from 1759 to 1768; and adapted to a 1000 born as a radix. The fum of the numbers in the fecond column, diminidiedbyhalf the number born, 1525,757. According to this Table then, for every 1000 deaths in London^ there are 25+- as many in- hibitants ; or, in other words, the expectation of a child juft: born is 2;i; and the inhabi- tants are to the annual burials, as 25^ to i. — But it has appeared, that the numbers in the fecond column being given on the fup- pofition, that all who die in London were born there, muft be too great; and v/e have from hence a demonstration, that the probabi- lities of life are given in the common Tables ^{London Obfervations, too high, for, at leaft, the firft 30 years of life ; and alfo, that the number of inhabitants in London muft be Icfs^ than 25^^, multiplied by the annual burials. !PrrThe common Tables, therefore, oi London R 3 Ubfer-
246 Of the Method of forming
Obfervations, undoubtedly want to be cor- reded [a) ; and the way of doing this, and in general, the right method of forming ge- nuine Tables of Obfervation for towns, may be learnt from the folio v/ing rule.
'* From the fum of all that die annually, ** after any given age, fubtradt the number '* of annual fettlers after that age j and the ** remainder will be the number of the liv- ** ing at the given age."
This rule can want no exphcation or proof, after what has been ah-eady faid.
If, therefore, the number of annual fettlers in a town at every age could be afcertained; a perfedc Table of Obfervations might be formed for that town, from bills of mortality, containing an account of the ages at which all die in it. But no more can be learnt in this inftance, from any bills, than the whole number of annual fettlers, and the general divifion of life in which they enter. This, however, may be fufficient to enable us to form Tables that lliall be tolerably exact.— For inflance. Suppofe the annual deaths in a town which has not increafed or decreafed,
(i/) The ingenious and accurate Mr. ShnpfGU faw that it was necefiary 10 correal the London Tables, and he has done it with great judgment; but, I think, too imper- fectly, and without going upon any fixt principles, or fliewing particularly, how Tables of Obfervation ought to be formed, and how far in different circumftances, and tt differeiit a^es, they are to ^e depended on.
to
t
Tables of Ohfcrvations, &c. 247
to have been for many years, In the propor- tion of 4 to 3, to the annual births. It will hence follow, that -I- of the perfons who die in fuch a town ^{'cfeukrs, or emigrants from other places j and not natives: And the fud- den increafe in the deaths after 20, will alfo fliew, agreeably to what was before obferved, that they enter after this age. In forming, therefore, a Table for fuch a town, a quar- ter of all that die at all arres throup;hont the whole extent of life, mull be deduced from the fum of all that die after every given age before 20; and the remainder will be the true number living at that given age. And if, at 20, and every age above it, this de- dudlion is omitted, or the number of the living at every fuch age is taken the fame with the fum of all that die after it, the re- fult will be (fuppofmg mojl of the fettlers to come in before 30, and all before 40) a Ta- ble exadt 'till 20 ; too hii^h between 20 and 30 -y but nearly right for fome years before 40 ; and after 40 exad: again. — Such a Table, it is evident, will be the fame with the Ta- ble laft defcrlbed at all. ages aboVe 20; and different from it only under 20. — It is evi- dent alfo that, on account of its giving the probabilities of life too great for fome years after 20, the number of inhabitants deduced from it may be depended on as fomewhat greater than the truth; and m.ore orlefsfo,as the annual recruits enter iu general later or fooner after 20.
R 4 Let
24^ Of the Method of forming
Let us now confider, what the refiilt of thefe remarks will be, when applied particu-? larly. to the hondon bills.
It muft be here firO: obferved, that, at leaft, one quarter of all that die in London •^xtfettlers from the country, and not natives, — The me- dium of annual burials for lo years, from 1759 to 1768, was 22,956; of births 15,710. The excefs is 7246; or near a third of the burials. — The fame excefs, during 10 years, before 1750, was 10,500; or, near /6^^ the burials. London was then decreafing. For the lail 1 2 or 1 5 years it has been increafmg. This excefs, therefore, agreeably to the foregoing obfervations, was then greater than the num- ber of annual fettlers; and it is now lefs. I have chofen, however, to fuppofe the num- ber of annual fettlers to be now, no more than a quarter of the annual burials, in order to al- low for more omiflions in the births than the burials j and alfo, in order to be more fure of obcaining refults that fliall not exceed the truth.
Of every thoufand then who die in Lon- don, only 750 are natives, and 250 ^xt fet- tlers, who come to it after 18 or 20 years of age : And, confequently, in order to obtain from the hills a more correal Table than the 1 2th in the Appendix, 250 mud be fub- traded from every one of the numbers in the fecond column 'till 20; and the numbers in the third column muft be kept the fame, the bills always giving thefe right.— After 20, -i the
'Tfihles of Olife'rvatlons, &c. 249
the Table is to be continued unaltered ; and the refult will be, a Table which will give the numbers of the living at all ages in Lon- don much nearer the truth, but 11:111 fome- what too high. — Such is the 13th Table in the Appendix. — The Turn of all the numbers in the fecond column of this Table, dimi- niflied by 500, is 20,750. For every 1000 deaths, therefore, in London^ there are, ac- cording to this Table, 20,750 living perfons in it; or for every fingle death, 2o|: inhabi- tants. It was before lliewn, that the num- ber of inhabitants in London could not be io great as 25 times i the deaths. It now ap- pears, (fince the numbers in the fecond co- lumn of this Table are too high) that the number of inhabitants in Lojidon cannot be fo great as even 20 times i the deaths. And this is a conclufion which, I believe, every one who will beflow due attention on what has been faid, will find himfelf forced to re- ceive. It will not be amifs, however, to confirm it by the following fa(fl, the know- ledge of which I owe to the particular en- quiry, and kind information of Mr. Harris^ the ingenious mafter of the Royal Mathema- tical School in Chrift-Church Hofpital. — The average of lads in this fchool has, for 30 years part, been 831. They are admitted at all ages between leven and eleven ; and few flay beyond 16. They are, therefore, in ge- neral lads between the ages of eight and 16. They have better accommodations than it
can
250 Of the Method of forming
can be fuppofed children commonly have ; and about 300 of them have the particular advantage of being educated in the country. In fuch circumflances it may be well reck- oned, that the proportion of children dying annually, muft be lefs than the general pro- portion of children dying annually at the fame ages in London. — The fad: is, that, for the laft 30 years, i \t have died annually; or one-
in 70T.
According to Table XIII, one in 73 dies between 10 and 20, and one in 70 between eight and 16. That Table, therefore, pro- bably gives the decrements of life in hondon, at thefe ages, too little, and the numbers of the living too great : And, if this is true of thefe ages, it mud be true of all other ages under 20 \ and it follows demonilrably, in conformity to what was before (liewn, that more people fettle in London after 20, than the ~ I have fuppofed -, and that from 20 to at lead 30 or 35, the numbers of the living are given too great, in proportion to the de- crements of life.
In this Table the numbers in the fecond column are doubled at 20, agreeably to what really happens in London-^ and the fum ot the numbers in this column diminiflied by half the whole number of deaths, gives the expetlation of life, not of a child juft born, as in other Tables, but of all the inhabitants of London at the time they enter it, whether that be at birth, or at 20 years of age. The
ex-peihj->
^ahles of OhfcrvatlonSibiQ. 251
expeBatwis, therefore, and tht values o^ Lo?i- don lives under 20, cannot be calculated from this Table. But it may be very eafily fitted for this purpofe, by finding the number of births which, according to the given decre- ments of lite, will leave 494 alive at 20; and then adapting the intermediate numbers in fuch a manner to this radix, as to preferve all along the number of the living, in the fame proportion to the numbers of the dead. This is done in the 14th Table in the Ap- pendix; and this Table may, I fancy, be re- commended as better adapted to the prefent ftate of London than any other Table. The values of lives, however, deduced from it, are in general nearly the fame with thofe de- duced by Ml'.. SimpJoj2, from the London bills as they flood 40 years ago. The main dif- ference is, that after 52, and in old age, this Table gives them fomewhat lower than Mr. Simp/on^ Table.
It has fuiiiciently appeared, what judgment we are to form of the values of lives thus deduced. During the greateft part of the interval of life, in which the annual recruits that keep up London come to it, thefe values err on the fide of excefs ; and after that in- terval, they err, perhaps, a little on the fide of defeB [a), on account of retirements from London in the lafi; fiiages of life.
The
{a) I have not taken into account the effect of migra- i\OT\$^rom towns^ on T;ib!es formed in the manner I have
explained \
2^1 Of the Method of forming
The number of inhabitants in London may alfo be learnt from what has been offered,
more
explained ; becaufe, towns in general being kept up by recruits from the country, the migrations y;-5;.v thein are of little confequence, compared with the migrations to them. — Thus; in London, it appears, from the much greater number of deaths between 40 and 50, than in any other equal interval of life after 10, that more peo- ple come to it than leave it, at every age between 20 and 50. After 50, it is probable, that the contrary happens. But, it fhould be confidered, that emigrants from Lon- don after 50, are chiefly pcrfons who, having got for- tunes in bufinefs, chufe to leave off, and to fpcnd the latter part of their lives in country retirements. But how few are thefe compared with the multitudes who, tho' poffeffed of good fortunes, never retire ; and with the bulk of the inhabitants in lower (lations, who never can he able, without the greateft inconveniencics, to quit the fettlements by which they are fupported .•* It is, how- ever, likely, that retirements from London arc now niore numerous than they ever were ; and that they have fame effeit on the bills of mortality, and onTahlcs forme4 from them; by caufing thefe Tables to give the number of the living too little, in comparifon with the decrements of life, at every age, from that at which the migrations to and from London become equal, to the age at which the latter ceafe. — To explain this ; let us fuppofe, that none fettle in London after 50 ; but that, between 35 and 50, as many come to it as retire from it at all ages after 35; and that thefe retirements ceafe at 70. In this cafe, the Tables will give the proportion of the living to the decrements of life too high '//// 35. At 35, this proportion v/ill be given right. Jfter 35, it will begin to be given too low; and this error will increafe 'till 50 } from which age it will decreafe gradually 'till it vanilhes at 70: And atter 70, the Tables will be exadtly right again. — This is the exa6t flate of the effe£l of retire- ments from London, on the London Table of Obfervations. ^jut this eftedi appears, indeed, to be inconfiderable ; for,
afte^
Tables of OhfcrvationSi See, 253
more nearly than by any method which has been hitherto taken. It cannot, it has been (hewn, exceed 20 times -^ the number of an- nual deaths. Could, therefore, the annual deaths be afcertained, we fliould know the number of inhabitants within pretty narrow limits. But the omiflions in the bills are fuch, that it is not poffible to afcertain, with cxadtnefs, the annual deaths. Dr. Braken^ ridge fuppofed thefe omiflions to amount to 2000 annually. The refult of a very minute enquiry by Mr. Maitland is, that in the year 1729, they amounted to 3038. But they arc probably now much more conliderable, than either of thefe writers have reckoned them {a). Let them be 6000 -, and the num- ber of inhabitants will be 601,750 at mofl.
All the preceding Obfervations are, it is plain, applicable to bills of mortality for towns in general -, and point out the way of deducing from them genuine Tables of Ob- after 50, the values of lives by the London Table, are continually approaching nearer and nearer to the fame values by other Tables ; which could not happen were retirements attended with any great effect. — It is proper to add, that in fumming up, as above-explained, the numbers of the living, in order to find the number of inhabitants in Lomkn, the circumftance that thefe num- bers may be too fmall for fome years after 40 or 50, in confequence of retirements, is, undoubtedly, much more than balanced by their being given too high between 20 and 40.
(a) Vid. Preface to a Colledlion of the Bills of Mor- tality from 1657 to 17585 p. 4, &:c.
ferv.itions.
254 Of the Method of forming
lervations, which fliall give the true proba- bilities and Vcilues or lives, and the true num- ber of inhabitants, in the town whofe bills are s^iven. — I fliall be^: leave to confirm and illuitrate this, in the particular cafe of the town of Northampton.
In this town, containing four pari{hes, namely. All- Saint Sy St. Sepulchre's, St. Giles ^ and St. Peter s, an account has been kept ever fince the year 1741, of the number of males and females that have been chriflened and buried (Diffenters included) in the whole town. And in the parifli oi All- Saints, con- taining the greateft part of the town, an ac- count has been kept ever fince 1735, of the ages at w^hich all have died there.
In 1746, an account was taken of the num- ber of hoifcSi and of inhabitants in the town. The number of i6(9?^j- was found to be 1083; and the number of inhahita-nts 5136. — In the pari flies oi All- Saints and St. Giles, the num- ber o^ j/iale ^nd female heads of families, ^r- lants, lodgers, and children, were particularly diflinguifhed.— The heads of families wert,yoj malcs:; and 846 females. — Children, males 624; females 759. — Servants, males 203; females 280. — Lodgers, males 137; females 287. — In St. Peter s, males 99 ; females 129. — In St. Sepulchre s, adults 638 ; children 427. In this panfh the fexes were not diftinguifli- ed.
The
Tables of Ohfervations, &c, 255
The Chrifiejiings and Burials in the ivhole to'-^n for 28 years, from 1741 to 1770, have been as follows.
Chriftened < p ^ o\?,?, \ 4^49 — Annual medium 155
ri . J f Males 2860 ? „ ^ A 1 J-
Buried > p.^^^ ^g^g J 5747 — Annual medium 191
In the parifh oi All- Saints, from 1735 to 1770, or 36 years,
Chriftened \ p^ ^ ? 3242 — Annual medium 90
^ . , 5 Males 1856? ^ A 1 J-
curieJ I Fern i8?<is3 ^^ — Annual medium 102^
Of thefe died, Under 2 years of age — 1206
Between |
2 |
and |
5 |
— |
276 |
Between |
5 |
and |
10 |
"^SS |
|
Between |
10 |
and |
20 |
— |
"^SS |
Between |
20 |
and |
30 |
297 |
|
Between |
30 |
and |
40 |
— |
257 |
Between |
40 |
and |
50 |
297 |
|
Between |
50 |
and |
60 |
-— |
300 |
Between |
60 |
and |
70 |
— |
293 |
Between |
70 |
and |
80 |
285 |
|
Between |
80 |
and |
90 |
— |
^ss |
Between |
90 |
and |
100 |
— |
14 |
Total |
3690 |
||||
/ |
ATa- |
2^6 Of the Method of forming
A Table formed from thefe data in the manner of Table XII ; or, on the fuppofition, that all who die in Northampton were born there, would give the exped:ation of a child jufl born 28.83 years j or, the proportion of the inhabitants to the annual deaths, as 28.83 to I. It has been Ihewn, that this propor- tion, in a place where the burials exceed the births, mud hQ greater than the true propor- tion of the number of inhabitants to the an- nual deaths : And this appears to be the real cafe. For the bills rtiew, that, from 1741 to 1750, or for 10 years, about the time when the number of inhabitants was 5136, the annual medium of burials was 197.5; which, multiplied by 28.83, gives 56933 or a 9th part more than the true number.
A Table formed in the manner of Table XIII, would give the proportion of inhabi- tants to the annual deaths, as 26.41 to i ; and this makes the inhabitants 5216; or very near the true number.
The IVth Table, in the Appendix, is form- ed in the fame manner with Table XIV, for London: And this is the genuine Table of Obfervations for Northampton, from which may be calculated the true probabilities and values of lives, at all ages, in that town.
At Norwich, bills of mortality, of the fame kind with thofe in London and North- ampton, have been kept for many years. I
have
Talks df Ohfervationsi ^c. 257
have been favoured with a copy of thefe bills for 30 years, from 1740 to 1769. The an- nual medium of chrijienings, during this pe- riod, has been 1057 {a), of burials 1206. And from hence, together with the account of the numbers dying in the feveral decads of life, after ic, I have formed Table V, which fliews the true probabilities of life in this town.
The following particulars feem to defervd notice here.
Firft. Had thefe Tables been formed from the Northampton and Norwich bills, for no longer time than any 10 years taken together, of the periods 1 have men- tioned; they would have given the probabi- lities and values of lives nearly the fame. Thefe Tables, therefore, are founded on a fufficient number of Obfervations ; and it appears, that there is an invariable law which
(a) In this regifler all that die before baptifm, and alfo all that are born and die among ^uahrs, ynvSy &:c. are omitted. There are alfo fome other omiflio'ns; and the true annual medium of births and burials mull ht greater than they are given in the bills. But this will have no efFcdl: on a Tabic of Obfervations, fuppofing the proportions of the births to the burials, and of the numbers dying in the different (tages of life, given right. — It is proper I fliould mention further here, that theftj bills give only the whole number of children dying un- der IC, without fpecifying the numbers d\ing under two years of age, between 2 and 5, and between 5 and 10, as in other bills. I have, therefore, in forming the Ta- ble for Norwich, fuppofed the proportions of rhefc num- bers the fame that they are at North ampton.
S governs
258 Of the Method of forming
governs the wafte of human life in thefe towns. — The. fame remark might be made concerning London {a). See p. 251.
Secondly. An account was taken at Shrewsbury, in 17^0, of the whole num- ber of inhabitants^ diftinguifliing, particular- ly, the number at the age of 21 and up- wards,-— The former number was 8141; and the latter, 5187. — According to a Ta- ble formed for Northampton, in the fame manner with Table XIII, for London, the whole number of the living is to the num- ber of the living at 21 and upwards, as 26,41 i to 16,586 ; that is, as 8 141 to 51 13. — -Ac- cording to a like Table for Norwich, thefe numbers are to one another, as 24,500 to
(«) Some have entertained a very v/rong nption of the imperfections in the Lontdon bills. They do, indeed, give the tvhole number of births and deaths much too lit- tle ; but the conclufions with rcfpedt to the probabilities of life in London, and the proportion of inhabitants dying annually, depend only (agreeably to the obferva- tion in the laft note) on the ■proportions of the numbers dying in the feveral divihons of life ; and thefe are given right in the London bills. — For firft. There feems no- thing in this cafe, that can be likely to caufe the defici- encies in the bills to fall in one divilion of life more than in another : But vi^hat decides this point is, that thefe proportions, as given by the bills {or any ten, or even any five years, come out nearly the fame with one another; and always very different from the proportions given by other bills. — There are no other variations, than fuch as muft arife from the fiuttuations of London, as to increafe and decre?fe; and alfo from fome improvements in its ftate, which have lately taken place. SccEliayl. p. 190, 191, 201.
15,680;
Tables of Obfervationsy Sec. 259
i«:,68o; that is, as 8 141 105210. — Thcfd Tables, therefore, give the proportion of the w/joie number of inhabitants, to the number of the living at 21 and upwards, almoft ex- adly the fame with the true proportion, as it is at Shrewsbury (a) : And this affords a kind of demonllration ot the reditude of the principles on which thefe Tables have been formed.
In the parilli ofHoLY-CRoss nearSHREws- BURY, an account was taken, in 1760 and 1770, of the i:;ho?e number of inhabitants; diftinguin)ing, bot/j times, the number at the age of 70 and upwards ; and the /afi time, the number at 10 and upwards : And, 1 find, that a Table formed from the Regijier of this parifh, mentioned p. 192, gives, likewife, thefe numbers as nearly the fame as could poflibly be expedcd.
But further.— The number of inhabitants, not reckoning children, in the parilhes o^ St*
(«) The annual medium of births at Shrewsbury, for 7 years, from 1762 to 1768, was 301 ; of burials, 329. It appears, therefore, that one in 24^ of the inha- bitants die annually. But it Ihould be rer.lembered, that in 1766, the fmall-pox and meafles increafed very much the mortality in this town j and I find alfo, that, fince 1750, a nurfery iox foundlings from London, was eftab-. lilhed here; and that in 1768 this nurfery contained 660 children and fcrvants. It fcems, therefore, prubable, that the ixut mediuiit of burials about the year 175O, mult have been lefs than 329 ; and that the proportion of inhabitants dying annually, may not be much (greater than it is at Northampton ; or i in 26,41.
S 2 Giles
26 o Of the Method of forming
Giles and All-Faints, Northampton, was, ..J in 1746, 2460 ; and the 'Z£'/6o/t' number of in- «
habitants in thefe two parifies was 3843. i
See p. 254. — In the account 1 have received, the particular age at which the Hmit of child- hood was fixed in taking this furvey, is not mentioned ; but there is fufficient reafon to believe, that it was 2 1 : And, taking this for granted, the number of inhabitants, not chil- dren, will come out, (by fuch a Table for Northampton, as Table XIII for Lon- don) 2414; or, nearly the fame with the iwimhtv really found in thefe pariflies. — Had this number been computed, from a Table forn^edforNoRTHAMPTON,in the manner of Table XII, Appendix, it would have come out only 2 176. This remark is applicable to the Table for Brefauj, formed by Dr. Halley, compared with the fame Table, corrected for all the ages under 20 (^), by the rule, p. 246.
[a) I have given Dr. Halleys Table in the Appendix juft as he framed it. A correction of it might be made from the proportion of births to buridls, mentioned p. 239. And it would then appear, that a 25th part of the inha- bitants at Brrjlaiu die annually ; and that half the num- ber born die there under fix, as well as at Norwich. This Table, as we now have it, makes half live to 16; but the account mentioned in the note, p"ge 240, (hews this not to be the truth. It liicewife makes the number of inhabitants at Shrew^sbury, above the age of 21, to be 4730 J and in the parishes of All Saints and St. Giles in ' Northampton, 2230. It gives, therefore, thefe num- bers wrong ; whereas, as oblerved above, a correiled Ta- ble would give them true.
The
Tables of Obfervations, &c. 261
The necefilty, therefore, of this correftion is verified by fads -, and it appears, abundantly, that the Tables I have given for North- ampton and Norwich may be depended on.
But, thirdly. In comparing thefe two Ta- bles, it may be obferved, tfiat there is a dif- ference between them in favour of Noi^rn- AM^T on, fewer dying there in childhood, and i?2ore in old age. The fame would be found to be true, were the Northampton Table to be compared with a corrcded Bres- LAW Table. It appears, therefore, agree- ably to what might have been expeifled, that Northampton, being a fmall town com- pared with Breslaw and Norwich, is lefs unfavourable to health and longevity. The difference, however, i^ not confiderable. Af- ter the age of 20, there is a firiking confor- mity between all the three Tables, Vvhich gives them great weight and authority.
Further. It ought to be particularly no- ted, that thefe Tables prove, the decrements of life in moderate towns, to be nearly equal thro' moft of its ftages. At Northamp- ton it appears that, of a given number of perfons alive at 20, the lame number die every year 'till 78, without any interruption worth notice, except between the ages of 30 and 40. — A Hke uniform decreafc in the probabilities of life appear in the Breslaw
262 of the Method of forming
and Norwich Tables; but not fo remark- ably. It was this clrcumftance in the Bres- LAW Table, that led Mr. De Moivre to the HypothefiSj defcribed in p. 2, and fo often mentioned in this work.— The values of lives, I have faid, deduced from this Hypothejis, agree fo nearly with the fame values deduced immediately from theTablcs, that it is fcarce- ]y worth while to diftinguifh them. But that every one may be able to judge of this for himfelF, I have calculated {a) the following Table.
Value of a life at the age |
By Brcjlaiv Table. |
By Norivkb Table. |
By Northamp- ton Table. |
By Mr. De Mot 'vre's Hypotbtfii. |
12 |
17.617 |
17.48 |
17.20 |
16.69 |
RecKon- |
16.49 |
16.41 |
15-93 |
15.89 |
iDg in- IjO tereft at \ 4 per 40 |
12.90 |
14.85 13.10 |
14.68 i3-'9 |
|
-- 50 |
10.87 |
II. 13 |
11.25 |
1^-34 |
60 |
8.58 |
8.54 |
9.02 |
9.01 |
70 |
5-59 |
5-99 |
6.26 |
6.06 |
75 |
4.21 |
4.86 |
479 |
4.29 |
It may be obferved in this Table, that the values, by the Iijpothefsy come nearer to the true values by the Northampton and Norwich Tables, than by the Breslaw Table ; and alfo, that, before the age of 60, they are all much higher than the values for
(fl) Every calculation of this kind may be made with- out much bbour, by a rule explained in note (O) Ap' pendlx,
the
Tables of Obfervationsy &c. 263
the {\m^ ages in London by Table X; the inhabitants oi London y (as Mr. De Mohre ob- ferves) being ** for caufes {ci) too well known, " more fliort-lived than the reft of mankind." — The HypotbefiSy therefore, is by no means applicable to London lives. It is proper to add, that neither can it be applied to the va- luation of Country lives. — It appears, from the regifter of the parifli of Ho/y-Crofs [b)y that the expeBations of lives there are much greater than the expe6lations by the Hypotbejis, — 'The expcSiation there of a life [c] At 20 is 38 V>^Hypoth.ii \nLorJ.2%.c^ 27 33.9 29.5 25.1
30 32 28 23.6
40 25.7 23 19.6
50 20 18 16
60 14.5 13 12.4.
70 10 8 8.8
From
{a) Do6lrine of Chances, p. 347.
{})) Sec Eflay I. p. 191. — I have not given the Tabic? of Obfervations from whence thefe condufions are de- duced, becaufe it is poilible, feme may think 20 years not a period long enough, toafrord data in this cafe of I'ulnci- ent authority. I have in p. 257, mentioned a fa£l which feems to prove the contrary. It is, however, certain, that the fame regifter continued 10 or 20 years longer, will afford data more to be depended on.
(^) The expe£tation of a child juft born in this parifh, is 33. At Northampton, 25I. At Norwich, 23I. In London, 18. — In this parifh, i in 11 dies at 80, and upwards. In NoRTHAiMPTON ; 1 in 22. InKoRWiCH; 1 in 27. In London ; i in 40. See Eflay I. p. 200.
S 4 1 will
264. Of the Method of forming
From this comparifon it appears, that th? Ilypothefis , from 20 to 60, gives nearly the
medium
I will aJd, that the probabilities of life here, appear to be much the fame, with the probabilities of life among the minifters and profefTors inScoTLAND. — This is a fa6t of fomc confequencc ; and, therefore, 1 fhall beg leave ^o give a brief account of it.
The mean age at which the minifters and profeflbrs enter into benefices and profelTorfhips in Scotland, is reck- oned to be 27. Their number is 974. The eftablifh- ment among them for providing for their widows, begun on the 25th oi A'larcb 1744J from which time to Novem- ber 22, 1768, 721 have died: That is, 29.23 annually; or I in 33-5-. The expcilation, therefore, of a life among them, at the age of 27, is 33^; which is neaily the fame with the expectation^ as given above, of a life of the fame age in the parifh of Holy-Crofs; and y^ years more, thati the expe^ation of the fame age by Tables III, IV and V. — Now, the expcSlation at a given age, being compofed of all the probabilities of life from that age to the extre- mity of life; there arifes from hence reafon for conclud- ing, that the probabilities of life among the minifters in Scotlmul, cannot differ much in any part of life, from thofe in this parifh.-^ But there is another fatl that con- firms this obfervation.
The annual average of weddings among the minifters 2nd profeflbrs in Sco rLAND, for the laft 24 years, has been at moft 32. The average of married perfons among them, for 17 years ending in 1767, had been 667. This number, divided by 32.. gives 20.84, ^^e expectation of mar- riage among them ; which is 1~ years more than the ex- fetation of marriage would be, by Dr. Halley's Table, on the fuppofition, that all ift, 2d and 3d marriages may be juftly confidercd as commencing, one v^ith another, fo early as the age of 30. — The expc£iation of two equal joint lives is to the expeSiation of zfi7igle life of the fame age, 3S 2 to 3, by note (L) Appendix, it follows, therefore, thac among the minifters in Scotland^ the expe^ation of zfingU
life
Tables of Ohfcrvationsy &c. 265
medium between the expeBations of London and Country lives ; and for tiiis reafon it is excellently adapted to general ufe. — After 60, the cxpeBatlons and values oflives in London approach nearer and nearer to the expectations and values of lives in Northampton, Norwich and Bre/la-zu ; 'till, at 70, they come to be almofl the fame. This is a circumftance which, I believe, has not been attended to; And it is the more furprizing, as there is no caufe known, which can produce any error in the values of lives after 60, deduced from the London Taole, except migrations from London ; and the effe6t of thefe mull be to diminiPd thefe values.
The following obfcrvatlons will, perhaps, account for this.
It has been proved, that at leaft half ih^ inhabitants of London, turned of 20 years of age, are emigrants to London from the country. So great a change as that, from the count?')' air and modes of life, tp the air and modes of life in Londcn, muft be parti- life at 30 cannot be lefs than 31.26. Moft probably it is more; on account of the later commencefnent of mar- riage in the fituation of the Scotch miniftcrs. — \ reckon alfo, that 27 muft be lefs th<>n the mean a'^^e at which they enter their benefices and profefioidiips ; meaning by it, not the age on each fide of which equal numbers enter j but the age at which, the exccfs of the interval of time taken to enter on one fide, is juft fuch as to com- penfatc the greater numbers who enter on the other fide. See the conclufion of note (F) Appendix.
cularly
266 Of the Method of forming
cularly hurtful to thefe perfons; and, there- fore, (except infants) it is in them, probably, that the pernicious influence o^ London on its inhabitants chiefly appears. They come in at every age 'till near 50 ; and this is the rea- ibn why the deaths continually increafe in London 'till that age ; but, after that age, the inhabitants confiding chiefly of perfons, who (like men ifed to drink) have h^^n feafoned to London^ or with v/hom it does not happen particularly to difagree; the number of deaths becomes lefs, and the values of lives begin to approach nearer to the common flandard in other towns.
There is one more facft which I {hall here take notice of j and which deferves more at- tention than has been hitherto befl:owed upon it. I mean ; " the difference between the ** probabilities of life among males and fe- " males, in favour of the latter."
From the account in p. 254, it appears, that at Northampton, tho' nioxQ ?nales are born l\\?iX\ females, and nearly the fame number die; yet the number of livingy^Z/'M/cj- is greater than the number of males, in the proportion of 2301 to 1770, or 39 to 30. This cannot be accounted for, without fuppoling, that males are more fhort-lived thany^;72^/^j. — One obvious rcafon of this fadl is, that males are more fubje(5l to untimely deaths by accidents of various kinds ; and alfo, in general, more addided to the excelTes and irregularities c which
Tables of Obfcrvatlcns, &c. 267
which (liorten life. But this is by no means the 07ily reafon. For it ihould be obferved, that at Northampton the number of/t'- male children was, in 1746, greater than the number of male children, in the proportion of 759 to 624. — The greater mortahty of males, therefore, takes place among children. — But this, together with the greater mortality in general of males at all ages, will more parti- cularly appear from the following recital of fads.
In the pariOi o^ Holy-C?'ofsy Salop, the Inge- nious Vicar, Mr. Gorfuch, in 1760, and again in 1770, took the number of male and fe- male inhabitants turned of 70. In 1760, the number of females turned of this age, was 35; of males, 8. In 1770, thefe numbers were, females, 35; males, 26. And for the Jaft 10 years 11, cut of 365 have died be- tween the ages of "^^^ and 102; and they were all females.
At Berlin, it appeared, from the accu- rate account which was taken of the inhabi- tants in 1747, and which lias been mentioned in p. 219, that the number oi jejnale citizens exceeded the number oi male citizens, in the proportion of 459 to 391 : And yet, out of this fmaller number of males, more had died, for 20 years preceding 1751, in the propor- tion of 19 to 17 (^).
{a) Vid. SiifiiiUch, p. 8, and p. 32, &c. where a mi- nute account is given of the number of males and females at Berlin in 1747; and alfo, of the numbers of each fex that had died from 1722 to 1750.
At
263 Of the Method of forming
At EdIxNBUrgh, in 1743, the number of females was to the number oi males y as 4 to 3 j (See Eflay 1. p. 2 1 1 ) but the ferhales that died finnually, from 1749 to 1758, were to the males, in no higher proportion than 34- to 3. Before 1749, the bills give the totals of bu- rials, without diftinguifliing them into the totals of males and females dying every year.
Mr. Kefeboom, in his Effay on the num- bers of people in Holland, informs us, that from the Tables of allignable Annuities for lives in Holland, which had been kept there for 125 years, wherein the ages of the perfons dying are truly entered; it appears, that females have, in all accidents of age, lived about 3 or 4 years longer than the fame number of males. See Pbilofophical Tran- faSlions abridged. Vol. IX, p. 326.
In Volume the 7th of the Pbilofophical ^ranfaBions abridged. Part IV, p. 46, &c. there is an account of the numbers of male and female ftill-born children and chryfoms, and of boys and girls under 10, of married men and married women, and of widows and widowers, who died for a courfe of years at Viennay BrefaWy Drefden, LeipfiCy Ratifbon, and fome other towns in Germany.
He that will take the pains to examint? thefe accounts will find that, though in thefe towns the proportion of males and females bqrn is no higher than 19 to 18, yet the
pro^
Tables of Obfervations^ &c. 269
proportion of boys and girls (^) that die i§ 8 to 7 ; and that, in particular, \\\tJiili-born and chryfom males, are to the ftill-born and chryfomyt';;W<?j-, as 3 to 2.
In thefe accounts it appears alfo, that of 7270 7narried perfons who had died in thefe towns {U), 43'?6 were married men, and but 2934. married 'Z£j(5;;;^«j that is, t/jree mzTntdmeft died to two married women. — The fcheme for making provifion for the widows and orphans of the minifters in Scotland, has obliged them to keep an account of tlie number of weddings among them, and the number of widows left annually ; and it appears, from the reports of the trujlees for carrying this fcheme into execution, that the annual me- dium of weddings [c), is (as obferved in the note, p. 264) at mofi: 32. And the annual
[a] In the accounts from Brejlaw it is particularly mentioned, that by boys and girh are meant children to 10 years of age, of whom, for 8 years fron^ 17^7 ^o ^725, y^z/^n males died x.o fix females, exclufivcly of ihcjilll-born and chryfoms.
{b) In Brejlaw alone, for the eight years mentioned in the laft note, 1891 married men died, to Iig6 married women ; that is 5 to 3, — in Drefden alone, for five years, thefe numbers were 1080 and 849.
[c) The annual medium of weddings, among the mi- nifters admitted to benefices, has been, for 24. years from the commencement of the fcheme, 27. Befides thefe I find there have been 4 weddings annually among them, before admiffion to benefices. The whole annual medium, therefore, is no more than 31. But 1 have fuppofed ic 32, in order to go upon more Xure grounds.
medium
270 Of the Method of forming
medium of widows, who have come upon the fcheme for 24 years, is 20. Of 32 marriages then contracted annually, 20 become extincft by the deaths of /^E/y^^^Wj; and but 12 by the deaths oi wives. That isj among the mini- flers and pro^efTors in Scotland, 20 mar- ried men die to 1 2 married women ; or 5 to 3. It appears, therefore, that there is the chance of 3 to 2, and in fome circumftances even a greater chance, that the ivoman fhall be the furvivor of a marriage, and not the j]2a?i. In order to account for this by the difference of age between men and their wives, this dif- ference ought to be at leaft 12 years [a). That is J fuppoling the mean age at which women marry to be 23, the mean age at which men marry ought to be 35. But this feems to exceed the bounds of credibility; and, therefore, very probably, the greater mortality of males muft operate in this cafe.
It is further obfervable in the accounts from Germanyy to which I have referred, that the number of widows dying annually, is four times the number of widowers {b) ; and, as
widows
(a) The chance of furvlvorfhip between two perfons aged 21 and 34, is nearly 3 lo 2 in favour of the former. 1 here is the fame chance of furvivorlhip between 25 and 37; and 20 and 39. This may be learnt from Pro- blem XVI, in Mr. De Mohr/s Treatife on Llfe-Jnnui- ties.
(b) In DrefJe?j alone, the number oi widoivs who died, in four years, was 584. The number of widowers^ 149- That is i 4 to i. — It appears from note (^} in the laft
Rubles of Ohfervaiions, &c. 271
widows are certainly, one with another, feve- ral years younger than widowers ; it may be concluded from hence, that the number of the former in life together could not be lefs than five times the latter. — This fad: is like- wife confirmed, by the obfervations which have been made among the miniflers in jScot^ land. At the commencement of the fcheme which I have fo often had occafion to men- tion, an account was taken of the number of the widows of miniflers in the whole coun- try. 364 were counted ; and, probably, the true number was greater. See p. 95 and 96. On the contrary ; the number of widowers among the miniflers has, one year with an- other, been fcarcely 90 -, that is, not fo much as a quarter of the number oi widows, — It may be eafily feen, and it would not be dif- ficult to denionftrate, that neither the greater number of perfons left widows, nor any pro- bable fuppofition concerning the greater fre- quency of marriages among widowers, can completely account for this, without admit- ting the greater- mortality of males. — This, therefore, appears, on the whole to be a fadt
page, that the chance of furvlvorfhip in this city in fa- vour of the vjife^ is Icfs than among the miniiters ia Scotland. — Does not this fail afford a reafon, additional to that mentioned above, and in p. 92, 93, 94, he. and notes (A) and TFj, for believing^ or at [c<\\k fufpeuting that the number ot Vindows on the Scotch cflablifhment, if marriage does not decline among the minirierb, v/ill not at lad be found to be fo little as 400 ?
well
272 Of the Method of forming
well cflablidied : And it follows from it, that in order to calculate the values of Lfe- An- nuities and Reverfoiis with exadtnefs, there ought to be diftindt Tables of the probabili- ties of life for males -^v^^ females. But there are no fuch Tables extant ; nor, indeed, has it been fufpeded, that there is fo much oc- cafion for them as the fads I have mention- ed feem to (hew. All that is neceifary to obtain the proper data for forming fuch Ta- bles is, that the fexes as well as the ages of the dead (hould be fpecified in the bills ; and this is an improvement of bills [a) of mor- tality which would give little trouble, and which, therefore, I hope, will be fome time or other made.
It has been obferved, that the author of nature has provided, that more viales {hould- be born than females, on account of the par- ticular wafte of males, occafioned by wars and other caufes. Perhaps it might have been added, that this provifion had alfo in view, that particular weaknefs or delicacy in the conftitution of males, which makes them more fubjed to mortality -, and which, con-
{a) T'his improvement would be rendered more com- plete, by diftiiiguifhing the waks that die, under the de- nominations ot married 7ncn, widowers, and batchelors ; and the females, under the denominations of 7;Zv'7r;7V(:/ Wij- men, widows, and virgifis. — The ufe I have made oFfome accounts of this kind which have been kept in Germany, {hews that this would be of confiderable lervice.
fequently.
T^ables of Ohfer'-oatlonSi Scci^ 273
fequently, renders it necefTary, that more of them fhould be produced, in order to pre- ferve in the world a due proportion between the two fexes.
In the courfe of this Eftay^ it has often appeared, that I have been particularly in- debted to an information which I have re- ceived from Northampton. — I {hould be inexcufable, did I not mention, that I owe this information to Mr. Lawton, an ingeni- ous gentleman in that town, who has pre- ferved the bills of mortality there with much care, and been very obliging in communica- ting them to me. — It is much to be defired, that like accounts were kept in every town and parifli. It would be extremely agree- able to learn from them the different rates of human mortality in different places, and the number of people and progrefs of popu- lation in the kingdom. The trouble of keep- ing them would be trifling j but the inflruc- tion derived from them (^), would be very important. — I have already propofed one im- provement of fuch accounts. I will add, that they would be flill more uleful, did they give the ages of the dead after 10, within periods ofjivt'y inftead of fen years. — During every period, fo fliort as Jive years, the decrements of life may, in conltrudiing Tables, be fafely
(rt) See EfTay I. p. 207, 2c8.
T taken
274 Q/' ^^^ Method of forming
taken to be uniform. But this cannot be e- qually depended on, in periods fo long as ten years.
There is yet another improvement of thefe accounts, which I will take this opportunity to mention. They fhould contain not only a lift of the diftempers of which all die, like that in the London bills; but they (liould fpeci- fy particularly the numbers dying of thefe dif- tempers, in the feveral divilions of life. Ac- curate registers of mortality kept in this man- ner, in all parts of the kingdom j and com- pared with records of the feafons, and of the weather, and with the particular circum- ftances which difcriminate different iituations, might contribute, more than can be eafily imagined, to the increafe of />^jy^r<j/ know- ledge. — But to proceed no farther in thefe Obfervations ; I (hall now beg leave to fhut up this v/hole work with the following ge- neral reflexion.
I have reprefented particularly, the great difference between the probabilities of hu- man life in towns and in country pariOies ; and from the fadts I have recited, and the obfervations I have made, it appears, that the further we go from the artificial and ir- regular modes of living in great towns, the fewer of mankind die in the frjl flages of life, and the more in its kf lUges. The
lower
Tables of Obfervations, &c. 275
lower animals (except fuch {a) as have been taken under human management) feem in general to enjoy the full period of exiftence allotted them, and to die chiefly of old age : And were any obfervations to be made among favjgcsy perhaps the lame would be found to be true ot them. — -Death is an evil to which the order of providence has fubjeded every inhabitant of this earth ; but to man it has been rendered unfpeakably more art evil than it was defigned to be. The greateft part of that bhck catalogue of difeafes which ravage human life, is the off-fpring of the tender- nefs, the luxury, and the corruptions intro- duced by the vices and falfe refinements of civil fociety (^). That delicacy which is in^
jured
[a) Calves are the only animals taken under our pe- culiar care immediately afier birth ; and, in confequcnce of then adminiltring to them the fame fort of phyiic that is given to infants^ and treating them in other rei'peds in the fame manner, it is probable, that more of them die foon after being born, than of ull the other fpecies of animals, which we fee in the fame circumftances. See the Comparative Vieiv cf the State and Faculties of Man ivifh thofe cf the JniTr.al IFcrld^ p. 23. — It is, indeed, melan- choly to think of the havock made among the humarl fpecies by the unnatural cujhms as well as the vices^ which prevail in polifbed focieties. I have no doubt, but that the cuftom, in particular, of committing infants, as foon as born, to the care oi fojler -mothers^ deftroys more lives than the fword, famine and peftilence put together.
(h') The ingenious and excellent writer quoted in the laft note, obferves, that the whole clafs of dileafes which arife from catching cold, are found only among the ci- vilized part of mankind, p. 51. — And, concerning that T a lofs
276 Of the Method of forming T'ahles, 8cc.
jured by every breath of air, and that rot- tennefs of conftitution which is the effedt of intemperance and debauchery, were never in- tended by the author of nature; and it is impoflible, that they fhould not lay the foun- dation of numberlefs fufferings, and termi- nate in premature and miferable deaths. — Let us then value more the fimplicity and innocence of a life agreeable to nature ; and learn to confider nothing as favagenefs but malevolence, ignorance and wickednefs. The order of nature is wife and kind. In a conformity to it confifts health and long life 'y grace, honour, virtue and joy. But nature turned out of its way will always pu- nifh. 'The wicked JJd all not live out half their days. Criminal exceffes embitter and cut fhort our prefent exijlence , and the highefl: authority has taught us to expecft, that they will not only kill the body, but xh^fouly and deprive of an everlasting existence.
lofs of all our higher powers which often attends the de- cline of life, and which is fo humiliating to human pride ; he ohfcrvcs, that it exhibits a fcene fingular in nature, and that there is the greateft reafon to believe, that it proceeds from adventitious caufes, and would not take place among us if we led natural lives, p. 62.
A P P E N-
APPENDIX.
Note (A). See Queftlon III. Page ii.
T ET E be any given expedation of life; and
•■"^ — ^ Xpx will be the number of perfons
alive at the end of x years, arifing from p perfons left annually as widows, (or added annually to a town or fociety) at the age whofe expectation is E. The maximum^ therefore, is always ^E — . In Mr. D<? Moivre's Hypothefis^ E is always -^ the difference between the given age and 86. See the note page 2, and the latter end of the note in page t^']. See likewife the beginning of the Firft EfTay, and note (L) in this Appendix, where the invelligation of this rule will be given.
It will not be amifs to o-lve the followino- ex- ample ot the application ot this rule.
At the time of the commencement of the fcheme, among the minifters and profefTors in Scotland, for making provifion for their widows, it was neceffary, that a calculation fhould be made of the number of widows that would be upon the fcheme at the end of every year, till they came to a maximum^ on the fuppofition that, (agreeably to what particular enquiry had lliewn to have hap- pened for many preceding years,) 20 new widows would be left every year. In order to make this calculation, let 4 of the 20 widows be fuppofed to T 3 be
e;^ [APPENDIX.
be under 32 years of age when left-, and let 2S be Rippofed their mean age. Let the fime num- ber be left between 32 and 39, and let 35 be their mean age-, between 39 and 47, and 43 their mean age-, between 47 and sy-, and 52 their mean age; between 57 and the extremity of life, and 6^ their mean age. The number in life together, to which, in ID years, 4 widows left annually at the age of 28 will grow, is, by the rule, (E being 29)
7 — X 40, or 36.55 The number alive at
the end of 20 years, will be - — — — X 80, or 66.2, ^ 1 16
At the end of 20 years, the number alive will be 89 i of 40 years, 104.82; of 58 years 1 16' Thefe numbers, found in the fame way, for the 2d clafs, (E being 25.5), at the end of 10, 20, 30, 40, and 5 1 years, will be 36. 7 — 64.31 — 84.7 — 97-25—
102 for the 3d clais, (E being 21.5) at the end
of 10, 20, 30, 40, and 43 years, 35.34 — 61.4 — '
78.13 — 05.6 — 86 For the 4th clafs, (E being
17) at the end of 10,20, 30, and 34 years, 34.11
■ — 56.47 — 67 — 68 For the 5th clafs, (E being
31.5) at the end of 10, 20, and 23 years, 31.3—
45.2 — 46 The whole number, therefore, con-
lilling of all the claiTes, will come to a maximum nearly in 58 years -, and the totals in life, at the end of JO, 20, 30, 40, 50, and 58 years, will be 173-37— 293.58— 364.83— 401.67— 41 8.
Thefe determinations fuppofe none to marry. In 10 years, from 1757 to 1767, I have been in- formed, that but 9 widov;s married. Let us then fuppofe, that one widow of the firft clafs marries every year j and let all that marry, be fuppofed to continue, one with another, 5 years in widow- ]io,od before they marry. On thefe fuppofitions, -5 the
APPENDIX, 279
the foregoing totals will, at the end of the fame periods of years, be 169.23 — 282 — 347-5 — 380.47—394.
Thcfe calculations are made from Mr. De Moivre^s Hypothefis. Had they been made ex- adly from Dr. Halicy*s table, or any other of the Tables I have given at the end of this work, ex- cept the London one, the refults would have been very nearly the fame. — It appears, that the pro- babilities of life, among thefe widows, are greater than thofe given by thefe tables. See the Lad Effay, pages 263, 264 266, 270, (kc. The effedt of this muft be, to raife the maximum with- out fenfibly increafing the numbers in life for the firll 20 or 25 years: and its effecl may be, to raife the maximum^ and at the fame time even to diminilh thefe numbers.
Twenty-five years have now elapfcd fmce the commencement of this fcheme ; and the number of widows living every year have, in faA, corref- ponded to the lalf numbers I have given, as nearly as could poflibly be expedcd in an aifair of this nature.
T 4 Note
28o APPENDIX.
Note (B). Queflion VI. Page 21.
LET r fignify the fum of i/. and its interefl, for one year. The value of a life, whofc complement is », being (by Mr. Dz Moivre on Annuities, 4th edition, page 14. and p. 100.)
1 — - A ^ H tj &c. the preient va-
lue of the remainder of it after two years muft be
— ^ -^ §^c. which IS equal to - X X
=^ 4- =^ -t =^, &C. 7z — 2r K — 2r* « — 2r^
Now - is the prefent value of il. due at the
}? — 2
end of tv/o years. is the probability that a
life, whofe complement is », fhall continue two
?z — 3 ;* — 4 ;? — 5 years, and -f i:^ -f- — , &c. is the va-
K— 2r 7; — 2r^ « — 2r^
lue of a life two years older than the life whofe complement is n. And, therefore, (fince any num- ber of years lefs than n may be fubftituted for two years) the firfl: rule given in this queftion is right.
The fame procefs, applied to joint lives, will demonftrate what is faid in the Scholium.
Note
L
APPENDIX. 281
Note (C). See Queftion VII. Page 22.
ET the complements of any two affi^ned -, lives be n and m. The prelent value ot the fidt poffible payment of an annuity to be enjoyed by the life whole complement is //, provided ^./£r lives continue 7 years, and the life whofe comple- ment is ., furvives the other .//.. that term is the probability, that the lite ^^^ .^^^^.^P^^^"^, ^^^.^ continue 8 years, and the other life 7 years and
then fail in the 8 th year, multiplied by ^, or by
1/ difcounted for 8 years.— The probability that the life of the tx^t^ant fhall continue 8 years is
""— , The probability that the olher life lliall con-
n tinue 7 years is ^ . The probability that it llaall continue 7 years! and fail in the 8th year, is ffl— 7 __ "'—^ — I , The probability, there-
fore that the life of the expe^ant fhall continue 8 years, and the other life 7 and fail in the 8th year, is
!i:5 X -; and the prefent value of the firft pofli- ble payment of the annuity fuppofed, h -^ X -
X -.
See rhe Bo^rine of Annuities by Mr to#?/,
^ 6—1';, or his Sele^ Exercifes, p. 315. &:c- •
in like manner, the prefent value ot the 2d pay- p^ent, at the end of the 9th year, may be found
sSa APPENDIX.
n — n m — 7 m — 9 n — Q 2
to be — - X X I -, or — f X -.
and the prefent value of all the polTible payments,
I A — b 1 n — Q 2 « — 10 3 „
r^ nr m^ nr'- m^ nr^ m
,./-.. 1 I ^^ — 7 '" — 7 But this leries is equal to — X X X
. ■ * • . ' . . ' ' . .
« — 8 T n — 9 2 n — 10
3 « — 8 I n — 9 2
, &c. Now==r X 1- =~~ X ,
m — -7 n — yr ^ 7 n — yr~ ^ 7
&:c. is the value of an annuity for a life feven
years older vthan the expeftant, after another life
ieven years olcier than the life whofe complement
w — 7 m — 7
IS m, X is the probability that both
n m '■
the afTigned lives fhall continue 7 years. And -
is the value of i/. due at the end of 7 years. The rule, therefore, given for folving this queftion, is right.
This demonilration, as well as that in the laft: note, is, for the fake of more eafe and clearnefs, applied to the hypothefis of an equal decrement of life. It does not, however, depend upon it, but may be applied to any table of obfervations.
Note
APPENDIX, 283
Note (D). Quenion IX. Page 29.
LET the complement of any two affigned lives be n and m, and the given term be feven years, as in note (C). The probability that^ the former life (fuppofed to be the life in expcclation) fhall laft 8 years, is, by Mr. De Moivre's Hypo-
thefis, -^ — i and the probability that the latter
life fnall fail in 8 years, is - •, and the firft pay- ment of the annuity mentioned in this queftion, depends on the happening of both thefe events,
the probability of which is X - .
The prefent value, therefore, of the firft pofiible
payment of the annuity is — ^ X — . In like
manner ; the prefent value of the [econd poffible payment is ^^^ x - •, and of all the payments.
n — 8 8 n — Q q , n — lO lO © t»
—T- X --, ~ X - + —7^ X — , &c. But
n—S 8 K — 8 I n—S 7 , «— 9 ,
— r X - == — r X - -f — r X -; and — ^ X
- := — ^ X — V — ~ X - . The forefroing fe-
ries, therefore, is equal to the two feries's - X
n — 8 1 , n — Q 2 , 71 — 10 3 o J
' X - 4- — r X - + — 7- X - , &c. and
Jir 771 717-^ 771 nr^ m
284 APPENDIX.
I n — 8 7 , 7i— 9 7 , «— 10 7 ^
-7 X X - + — r X - 4- — r- X - , &c. or
I « — 7 w — 7 n — 8 I n — Q
to - X X X =- X h =^ X
r n m j^ — ^^ ;„ — ^ ^ — ^^z
2 7z — 10 3 « ,17 « — 7
+ =^ X -i-, &C. + p7 X £ X — ^ X
7« — 7 n — 7r^ »2 — 7 ' '-^ "
7? — 8 n — 9 n — 10 1 • 1 .
+ + — , &c. which is the very
n — 7r « — 7r* ;z — jr^
rule given for folving this quellion, as will appear from notes (B) and (C).
Note
APPENDIX. 285
Note (E). See the Scholium to Queft. X.
ACCORDING to the calculations, the time in which the firft yearly payment of a rever- fionary annuity becomes due, is the end of the year in which the event happens that entitles to it, how- ever little or much of the year may then happen to be unelapfed. And this, likewife, is the time when a reverfionary fum becomes due. Thofe who know how the calculations of the values of rever- fions are inftituted, mud know this. But an an- nuity, the firft payment of which is to be made at the fame time with another payment of a fum in hand, fufficient to buy an equal annuity, is worth one year's purchafe more than the ilim. For inftance. Reckoning intereft at 4 per cent, and r being i/. increafed by its intereft for a year, or 1.04,
- + — + -^ , &c. 1:1 25/. is the prefent value of
an eftate of 1/. per annum for ever. That is, it is the value of it, fuppofing the firft rent of it is to
be paid a year hence,- If the firft rent is to be
received immediately, or at the fame time with another payment of 25/. it is worth one year's purchafe more, or equivalent to 26/. — —I have not found, that any of the writers on annuities and reverfions, have attended to this obfervation. It fuggefts a correction necefl^ary to be applied to the common folutions of feveral important problems : particularly to the 2 i ft and 22d in Mr. Smpfon'^s "Trea' tife on Annuities^ and the 26th, 27th, 3 2d, 33d, and 40th problems in his Sele^ Exercifes ; and to all other problems of the fame kind in other writers. There 5 can
2§6 APPENDIX.
can be no great occafion for being more explicit. Ic will not, however, be amifs to add the following
demonftration : ^ is the prefent probability
that a life, whofe complement is n, will fail in any
one afTiofnable year of its duration. S X -{ r
-f. — , &c. («), or the prefent value of il. per an-
S num for n years, multiplied by- , is the prefent
value of the fum or legacy denoted by S, payable at the failure of the given life. Therefore, {n be- ing 56 ; the life 30-, intercft 4 per cent. m:i.04; the fum 25/.) the value of the expedtation, by Mr. De Moivre's hypothefis, is 9.919.
Further. The value of il. to be received at the end of a year, provided the life whofe complement is n fails, is the probability of the failure of the life multiplied by i/. difcountcd for a year, or
X -. In like manner J the value of lA
il r
to be received at the end of 2 years, if the fame
life fails in 2 years, is i X -r* And, there-
fore, the value of all the -pojjihle payments of an eftate or annuity of 1/. for ever, to be entered
upon after the p-iven life, is i ■ X - + i —
cj n r
-2 I n — 2 I o , s I
I
APPENDIX. 287
T 9 I I I 5 n — I «— 2 ,
, &c. that is, the value of the life fubtraded
from the perpetuity -, or, in this example, 14/. 684, (the value of a life at 30) fubtrafted from 25 ; that is, 10/. 316. But 10.316 is to 9.919, in the fame ratio with 104 to 100, or 26 to 25, agreeably to the rule in the Scholium,
Note
2S8 APPENDIX.
Note (F.) Quefllon XIII. Page 44.
WFIEN I here call 48 the mean age of all mar- ried men, and 40 the mean age of married women, I do not intend to fuppofe, that there are as many married perfons who exceed thele ages, as there are w ho fall (hort of them. It is likely that the latter are moft numerous ; and it is neceffary that this fhould be the cafe, to render the fuppofition I
make juft If all marriages commenced at 33 for
the man, and 25 for the woman, one half of them would be diflblved by the time the men were 50, and the women 42 •, for (by the Hypothejis, and alfo nearly by the Brejlaw^ Norwich^ and 'Northamp- ton tables) there is an equal chance for the joint continuance of two lives, whole ages are 25 and 33, fcventeen years. Forty-two and fifty then would be properly the mean ages at which widow- hood would commence •, meaning by thefe, " the *' ages on each fide of which equal numbers are
*' left v^idows and widowers." But, tho' in this
cafe half the marriages of every year would be dif- folved in 17 years, they would not be all diflblved in twice that time. So far would this be from happening, that about a 7th part would continue beyond twice 17 years-, nor would it be certain, that they would be all diflblved till near the ex- tremity of the poflible extent of life. Tho*, there- fore, an equal number of marriages would be dif- jolved, or an equal number of widows and wi- dowers left kfore 50 and 42, and afierzvards, yet the aores of the latter would, one with another, much more exceed 50 and 42, than the ages of
the
APPENDIX. 289
the former (that is, of the widows and widowers left before 50 and 42) would fall fhort of them. And the number of marriages alfo in the world, am.ong per- fons of greater ages than thefe, v/ould be much fewer than among perfons of lefier ages— — In other words : the period, at which the m.arriages that have been contrafted are half diflulved, is not the period at which the number of marriages conftantly cxifting is equally divided, but this period falls feme years fooner ; and the period I have in view, falls in that part of the interval between thefe two periods, Vv'hcre the greater ages of the marriages on one fide, are iuft enough to compenfate (in fuch a calculation as that I have given) their deficien- cies in number, compared with the number of marriages on the other tide.
In Ihort. Suppofe 35 marriages every year, between perfons 33 and 25. In 12 years there would be half as many in the world, as could poffibly arife from fuch a number of yearly wed- dings. In 17 years, half every fet would be ex- imtt. The expe5!ation of every marriage would be 19 years, by prob, 21 of Mr. De Moivre^s Treatife en Ammities^ or by the note p. 299 : that is, taking them all together, they would exill juft as long as an equal number oifingk perfons, fuppofed to be fure of living juft 19 years, and no more; or, as long as an equal number of lingle perfons, all 48 years of age, fuppofed to be fubjedt to the common laws of mortality. One with another, then, they will be all extinft in 19 years: the marriages which continue beyond this term, tho' fewer in number, enjoying among them juft as niuch more duration, as thofe that fall fliort of it enjoy lefs. Widows, then, at a medium, will com- mence widowhood at 44 (that is, 25 increafcd by 19)
U year^
290 APPENDIX.
years of age, and ividowers at 52. The values, therer fore, of the lives of the former, when they commence widowhood, will, one with another, be the fame with the value of a life at 44; or, (reckoning intereft at 4 per cent.) 12.5 years purchafe, in prefent pay- ment, (the annuity to begin at the end of a year); and their expectations of life will be 21 years, or half the difference between 44 and 86. The value of the lives of the latter will be 10.92, and their £x^_ pe^aiion 17 years — The whole number of marriages conftantly exifting, which would refult from 35 fuppofed to commence annually, would be it)X355 or 665 -, and 53 years (the difference be- tween 33 and 86) would be the time in which they
would increafe to this number- The chance of
furvivorfhip would be 6g to p,2-> ^y prob. i8th, Mr. Be Moivre on Annuities ; that is, in p,^ years, 0^^ relicts of thefe marriages would be left every year, and the number of widows would be to the number Oi widowers ^ as 69 to 53 •, or 19.S widows would be left annually, and 15.2 widowers. The tiiaximum of widows in life together, if none mar- ried, would be 21 X 19.8, or 416-, and they would increafe to this number in 114 years (or 61 years after the number of marriages had attained
to' a rnaxiriiura) -The rnaxiinum of widozvers would
be 15.2X17, or 258; and they would increafe to this nudiber in 106 years.
An eafy method may be hence deduced of folv- ing the queifion v/hich occafions this note If the number of the members of the cftablifhrnent I have fuppofed, is 66^, and the mean ages at which marriage m.ay be deemed to commence are 25 and -^^^ 19.8 widows will (ithasjuft appeared) be left every year-, and the values of their lives, when they commence widowhood, will be, one
4 "y^^^!?^
APPENDIX. 291
■with another, i24r years piirchafe. An annuity of 20/. will, thereiorc, be worth, to each widow, 250/. and 19.8 fuch annuities muft be worth 4950/. •which, confequently, is the annual income necel- fary for the Tupport of the eftablifhment, the firfl: payment to be received immediately : or /. 7.44 Ironi each of the 66§ members ; which anfwers nearly to the determination in page 44.
In the Lad Eflay, p. 296, it has been flicwn, that obfervations determine the chance of furvivorihip in favour of the wife in marriage, to be really lb great as 3 to 2 i and in fome circumitances greater. I have alio there obferved, that in order to account for this, from the difference of age between men and their wives, this difference mule be at leaft 12 years, and the mean ages of ail who marry annually, muft be fuppofed to be about 23 and 35. In this cafe, 19, as before, will nearly be the expe<^cUion of all mar- riaD;es. The mean age at which widows and wi- dowers will commence fuch will be 42 and 54. The number of annual marriages ncceflary to keep up 66^ marriages conftantly exilling, will be ^^, The number of widows left annually, by fuch a number of marriages, will be 2 i •, and the values of their lives, at the time they commence widow- hood, will be 12.85 years purchafe by Table VI: and therefore, the whole annual income neceiiary for the fupport of the fuppofed cilablifhment, will be 5397/. or anjinnual paym.ent, beginning immediately, of /. 8.1 1 from each member — The number of widows Qn fuch an eftabliflimeni; v/ill, in 6^ years, grow, if none marry, to 462 ; and the number of widowers
to 224. It may be depended on, that all this
would happen as far as Dr. Halky\ table, or the tables for Norwich and Northampton^ exhibit the Uue ftate of human mortality.
-92 APPENDIX.
Arnon<^ the iTiinifters and profeflbrs in Scot- land, the number of married men being 667, or jieaiiy that here mentioned, the number of annual weddings has, for manyyears, been at an average 32, and tlie number of widows left annually 20-, and, therefore, the chance of furvivorfnip in favour of the wife, as 20 to 12, or 5 to 3. See Elfay IV. p. 269. This is not more different, from the refults I have given, than might hive been expected; and the chief reafon of the difference is, that the esp£oI<xlwns oi fuigle and joi'it lives among the mi- riillers and their wives in Scotland, are greater than thofe given by Dr. HcUey^?,, and the oiher ta- bles of obfcrvacion '1 hele tables give the ex- pectations of lives as they are among the bulk of mankind in moderate tov/ns. The expeftations of lives among the berter fore of men, living mofdy in country villages and parillies, are much greater. The fad is, that among the minifters in ScGthmd^ the expedation of a fingle life at the age of 27, is three years and an half greater; and, confequently, of joint lives, above two years greater, than the fame expedlations by Dr, Halley\ 'rable. Ibid, page 264.
I cannot help juR' mentioning another remark
here. It may be obferved, that fuppofing no
kcond marriages, and, at the fame time, that the odds for the woman's furviving in marriage is 3 to 2, the number of imdoisos in the world would be double the number of iridcissers. But it has been found, in fadt, that the number of widows is five times the number of vvidowers. How this is to be accounted for, I have iliewn in the Eflay juft re-* fcrred to, page 270, 271.
Kote
APPENDIX. 293
Note (G). QuePcion XIV. Page 48.
LET r be i/. increafcd by its intercfl: for one year; /the given time or number of years tor which the affurance is to be made ; rt, b^ c^Scc, the probabilities taken out of a table of obferva- tions, that the perlbn whofe age is given fiiall live I, 2, 3, &c. years ; and P the probability that he
fhall live / years. Then ~ -f — ; — \ — , &:c«
(t — i) 4- • 4 ■ A &c. — - 4- — f-
&c. (/)—- + - + _, &c. ,(/ - 1) + - -f X - + — + - , &c. vi'iil be the exadl value
r I— P
of an annuity to be entered upon at the failure of the given life, provided it happens in / years. And the rule is nothing but this value exprefied in words. In a fimilar manner may be demon- ftrared the other rule for findins; the values of aflurances, tor a given time, on two joint liveSj or the long-ell of two lives.
IT cj Kote
294 APPENDIX.
Note (H). Queftion XV. page ^6,
ET r fignify as before-, S the given fum to be afllired ; / the given time; N and n the number of the living in the table of obfervations, at the age of A and B refpedtively •, A, B, C, &c. and a, h^ c^ &c. the number of the living in the table, at the end of i, 2, 3, &c. years from the ages of A and B ; D, D, D, D, &c. and d^ d, d, d, &c.
I II ni I n III
the decrements of life in the table, at the end of I, 2, 3, &:c. years from the fame ages. Then, by reafoning in the fame manner with Mr. Simp/on, in p. 3 1 6, &c, Sek^ Exercifes, it will appear that S X
+ |
ax a I |
Cx |
d " , &c. |
co- s - X 71 |
f s |
X . |
Dd iNnr |
-h |
||
&c. S |
• 3 (0- |
|||||||||
Dd I T |
.+ |
Dd II II 2Nw |
Ad Nr |
+ |
hd |
+ |
||||
C'i |
hd |
Dd |
_J1., &c. (/) -i X \-2-L, &c. (/). This
Nr^ 2N nr nr'^
is the exact anfv/er to Queflion XV. and the rule is as near an aproximation to it as there is reafon to defire.
In the fame manner, retaining all the fame fym- bols, it may be found, that the anfwer to Qiieition XVI. is
SX
iy,l Dd D-^Uxd D-i-iv-j-ux^
+ ^^-^ +
iHur Nw- N/jr^ Nw
■ Dd Dd Dd
(/), &c. + S X -^-i- + "': + -iiiiii- , &c.
2iN«r* 2Hnri 2Nw*
APPENDIX.
195
(/— i) = — X _L + I iL ^-
Nr Nr- Nr^
S D^/ Di D^
&c. r/— 0+^ X — 4-J4- + iif, &c. (-/;.
D D+D D+D+D
But ^ 4- .^„ 4- • / " , &c. (/— i) is the
fame with the excefs of the value of an annuity certain for a number of years lefs by one year than the given term, above the value of an annuity on the life of A, for the fame number of years ; from whence the reafon of the rule for folving this queftion, may be eafily difcovered.
U 4 .iioiii
296 A P t^ E N D I X.
Note (I). Page 119, Sec.
LET / be any given term of years; p the va- lue of i/. due at the end of the given term ; A the value of an annuity certain for the fame term -, ;z the complement of a given life -, G the va^ lue for the given term, of two joint lives, both equal to the given life-, (to be found by Queft.VI.) P the perpetuity -, r, i/. increafed by its interelt for one year.
Then A — G X «-+/X/'XP — AxPxr will be the prefent value of i/. 2/. 3/. &c. (/) payable at the end of i, 2, 3, &c. (/) years; but fubjed to failure when the given life fails.
If fuch a courfe of payment is to begin im- mediately, and to be made at the beginning of every year, till t-\-i payments are made in / years ; add to the preceding value, the value increafed by unity of an annuity on the given life for / years, found by Qiieflion VI. and the fiim will be the va- lue fought. And this value, divided by the pre- fent value of what may happen to remain of the given life after t years, found by Quellion VI. will give the Jlanding annuity to which fuch a feries of increafing annual payments, beginning immedi- ately, will entitle, for the remainder of the given life after t years.
With the ailiiLance of this theorem, all that is faid in page 119, &c. may be inveftigated. It would be too tedious to enter into a more minute account.
Note
APPENDIX. 297
Note (K). Page 149.
LET ^ fignify the difference between the compli- ments of the youngefl and the oldeft life in the body of Annuitants, here defcribed, at the •time they enter ; let S fignify the fum of thefe tomfkments ; n any given nun^.ber of years not
C J
greater than - — •, and .v the ratio of the whole ° 22
number of Annuitants to . Then
2
A'x^ will be the number that will die the ift year ; .vx^-{- -n- the number that will die the 2d yearj
.vx^-f-^ 4-|i, 3d year;
■vX^-r J + ^. + s^ » 4tn year j
d ,^ id .
and X X nd-\- ?/ — n X "^ -f- ''^ — 2 + n — 2 r X -r; + « — 3
+ « — 3^'X -■ + ^ — 4 + « — 4I* X -^,&c. («)
will be the whole number dying in n years. When
n is greater than , this feries is greater than
t3 22 °
the whole number dying in n years ; but in all other cafes it gives this number exadlly, fuppofing the probabilities of life to decreafe uniformly.'
I In
298 APPENDIX.
In the prefent inftance, the youngeft life being 30, and the oldeft 60^ the two complements are §6 and
26. S — 82. ^ — 30.— = 1230. And therefore a: ~
- '^ 1Z27.1. Take » = 30 years, and the fore- going feries will be 27.1 X 900 + 318.2 + 7-2424-
.164Z: 33.214, which is a little greater than the whole number dying in 30 years, but at the fame time lefs thaa the whole number of Annuitants.
Note
APPENDIX. 399
Note (L). See EiTay I. Page 169, 170, 172.
THE fum of the probabilities that any given lives will attain to the end of the ift, 2d, 3d, Sic. years from the prefent time to the utmolt extremity of life (for inftance, ^4- + ^t + ^i, &c. to -:fV — 224- for lives of 40, by the hypothefis) may be called their expe^aiion^ or the number of pay- ments due to them, zs, yearly annuitants. The fum of the probabilities that they will attain to the end of the ift, 2d, 3d, &c. half years (or, in the parti- cular cafe fpecified, -|4 + || -i- 4| + I?., &c. zi V ^^/^ years, or 12^ years) is their expedlation as half yearly annuitants. And the fums jufl: men- tioned of the probabilities of their attaining to the end of the ift, 2d, 3d, &c. moments (equal in the fame particular cafe to 23 years) is properly their expectation of life^ or their expectation as annuitants fecured by land.
Mr. De Moivre has omitted the demonftCations of the rules he has given for finding the expeCla- tio7is of lives, and only intimated, in general, that he difcovered them by a calculation deduced from the method of fluxions. See his Treatife on Anmd- ties.^ page 6(>. It will, perhaps, be agreeable to fome to fee how eafily they are deduced in this method, upon the hypothefis of an equal decre- ment of life.
Let X ftand far a moment of time, and n the
complement of any affigned life. Then , ,
n — 3;r
, &c. v/ili be the prefent probabilities of its
n
con-
3G0 APPENDIX.
continuing to the end of the ift, id, 3d, &c. mo-* ments ; and -— the probability of its continuing
to the end of .v time. — - X -v will therefore be
71
the fluxion of the fum of the probabilities, or of :in area reprefenting this fum, whofe ordinates are
J and axis a\— The fluent of this exprefiion, or
W X
n
X , is the fum itfelf for the time x ; and this.
when ;c'=:;?, becomes In, and gives the expe5lation of the afligned life, or the fum of all the probabilities juft mentioned for its whole poffible duration.— — In
like manner: fince — p- is the probability that
71 A'
two equal joint lives will continue x time, — r^ ><; v will be the fiHxion of the fum of the probabilities.
n
The fluent is ;; — ' — \- ■ — , vv'hich, when n — x, is -,
or the expedation of two equal joint lives.
Agam : fince ■— — X — is the probability that there will be a furvivor of two equal joint lives at the end of x time, -^ — - X ^ x i will be
fl 71
the fluxion of the fum of the probabilities ;
2Lnd.\h.t fluent, or 7-7 is (when xznn) -^n, or
the expeSlation of furvivorfhip between two equAl lives i which, therefore^ appears to be equal to the
APPENDIX. 301
expeBaticn of their joint continnance. The expefta- tion of two unequal joint lives, found in the fame
way, is - — 7-, m beinp; the compknient of the
oldefl life, and n the complement of the youngefV.
The whole expe<5bation of furvivorfhip is h
r— . The expectation of furvivordiip, on the part of the oldefl is — -, and the expedlation, on the
part of the youngeH:, is ~ '^"fT' ^^ ^^ ^^^^ ^°
apply this inveftigation to any number of joint lives, and to all cales of furvivorfhip.
It may be obferved, concerning the firfl of thp fluents here given, that it exprelfes not only the expeflation of a given life for the time «■, and therefore its whole expeflation when .v=«, but likewife, the number of perfons alive, to which one perfon added annually to a fociety, at a given age,
will increafe in x time. Thus : Suppofe one
annuitant, v/hofe age is 28, (and whole complement of life, therefore, is 58, or expe^alion of life 2g) to come upon a fociety every year •, the number of annuitants alive, deduced from hence, vi'ill, in x
, X' A X 20 — ,r^ J ,
years, be :i^ , or x^v; and, there-
^ 4^29' 4 + 29
fore, the number of annuitants alive, deduced in the fame time from p annuitants left annually at
the fame age, will be -^^^ — ^ xp^' In like
° 4x29-^
manner, the 2d fluent, or — - f- a\ gives the
number
3oa APPENDIX.
number of marriages in being together, that will, in X years, grow out of one yearly marriage, be- tween perfons of equal ages, v/hofe complement of life is n. If they are of unequal ages, and the com- plement of the oldeft life is m, and of the youngeft
», this number will be |-x And
' rimn 171! It
if the number of years is required, in which any given number of yearly marriages, betv;een men and women at given ages, will increafe fo far as to be in any given proportion to the greatell num- ber that can poiTibly grow out of fuch marriages, tihis expreiTion muft be made equal to the ex'pe^la- 4ton of the joint lives, or of each marriage, multi- plied by the fradion expreffing the gi-ven propor- tion •, and the root of the equation will be the an- fwcr. Thus : it may be found, that one marriage every year, between perfons o^'i and 25 years of agCj v/ould in 10 years increafe to 8.35 •, in 15 years, to 11.38 -, and in c^^ years, to 19, or their greateft poffible number : and, confequently, that 35 fuch yearly marriages would, in 10 years, in^ creafe to 292 -, in 15 years, to 398 •, and in 53 years, to 665.— —And if it is enquired in what number of years 35 fuch yearly marriages v;Duld increafe to half the number in being together, pofiible to be derived from them, the value of .v, in the cubic
A"^ n-\-mxx^ , m m ^ n.
equation -j- ^si' — r~ X a? mult
•* 3«OT 2nm 2 d;z
be found ; which, in the prefent inftance, is nearly 12.
I have, in fome parts of this work, had occafion to m.ake fuch deduftions as thefe. See note (A>, p. 277 i and note (F), p. 288 ; and Quefuons III, and XIII.
APPENDIX. -.O'
Note (M). EiTay II. Page 226.
ET r fignify i/, increafed by its intercH for one year.
V the PERPETUITY.
■n the difference between the age of the youngefl life, nnd 86; or lis complement .
m the complement of the oldeft life.
P the value (in Table II.) of an annuity certain for m years.
And the exad value of any two given joint lives, according to the hypothefis of an equal decrement
V+r ~ ?
of life, v;ill be V — x^ — ^ — 21; — ix —
-\- 2v. Example :
Let the ages be 27 and 38 ; and the rate of in- tereit ^per cent. Then n — 59. m — 48. V;;:^ 25. P — 21.195. n — m — 2v — 1 = — 40. n — m —
- P
2v — I X - ^'-21; = 50 — 17.660 = 32.340. And
V X n — m — 2V — I X ~ -{■2Vzz2k — j-t
X 32.340= 10.748, the value of two joint lives \yhofe ages are 27 and ^^.
Note
-04 APPENDIX.
Note (N). ElTay III. Page 232.
IT is plain, that the purchafer of A's right, as ftated in the firil of the queftioris, to which this note refers, cannot get into pofTefnon, till the year when A and B fhall be both dead ; nor then, un- lefs A happens to die lajl. Nov/, fuppofing the common complement of life n -, the probability that A and B iliall be hlb dead at the end of the
frjl year, and A die laft, is i — x i
n )i
X - = ; h —.---. -In hlce
manner, the probability that they fhall be hoih dead at the end of the 2d, 3d, &c. years, and A
{. . -I n-. — 2 n-—i n — 2I ' i n—T. lurvive, IS \-
2 in in 171' 2 in
-^r-^ -i 4:-, &c. The prefent value, therefore,
in in- ^ ■^
of the ift, 2d, 3d, &c. rents of the reverfionary
„ . I n — r n — i , aT^'- i n — 2
eitate is V ,
2/" mr inr 2?/r 11- irir-
n — 2 , n — 21' I n — -;> 7z— r? n — -iV- -
— r ^ —, -T T -r — ~r , &C.
ini" i>rr- ir^ Tnr^ iny iwr^
Suppofing r to fignify 1/. increafed by its intereft for a year •, and the eitate to be i/. ■per annum. And thefum of thefe terms continued in infinitum^ is the
value required. But \ — ■, -1 : , &c, is half
-' IT ir- lr^
the
^o^
^ — 2 , « — 2 2»r* 2«r-
APPENDIX.
the perpetuity. And \- _ — — +
2«T^ 2/.'r-> 2«r^ 2H^r^ is half tht value of the joifit lives, fubtracled from half ihc fum of the values of the two Jifi^le lives ;' that is, /?\'7// the value of the ioiigefl: of the two. lives.
A fimilar demon (Iration may be. applied to the, ether qucilion.
Note
^,o6 APPENDIX.
Note (O). E% IV. Page 262.
LET r be i/. increafed by its intereft for one year.
Let S reprefent any given interval of time, or number of years, during which the decrements of lite in a table of obfervacions continue equal.
a the number of the living in the table at the beginning of the firfl: year of that interval.
b the number of the living in the table at the beginning of the year immediately following the lame interval.
P the value of an annuity certain for S years.
p the valucjin Table I. of i/. due at the end of S yars.
Q^the value, in Table Vl. of an annuity for the life of a perlbn vvhofe age wants S years of 86.
N the value, in ftri6b agreement with the given table of obfervarions, of an annuity on the life of a pcrfon whofe ao;e is S years grreater than the age at which the interval of equal decrements begins. Then
Q_"l- - X P- — -Q^vvill be the value, according
to the table of cbfervations, of an annuity for S years, on a life of the fame age with that at which the interval of equal decrements begins. And
h ' —
Q_+ - X P — Q^+ /'N will be the value of at!
annuity on the v/hole duration of that life.
When S reprefents one year, Q^vaniflies, and the
lafi: exprefllon becomes — x 1 + K ; which is the
ar
3 rule
APPENDIX* 307
rule for finding, from the value given of any life, the value of a life one year younger.
Thefe Tlieorems fave much labour In calculat- ing the values of life-annuities from tables of ob- fervations.
The firft of them, with its Inveftigation, may be found in page 341, 3d edition, of Mr. D: Moivrc\ Treat! fe on the Doclrine cf Chances. But i& is neceflary to obferve, that the direction Mr, Be Moivre has given for finding the value of Q is wrong. In confequence of calculating agreeably to this diredion, he gives the value of a lite at the age of 42, by Dt. HclUy'^ table, greater than the value of the fame life by his own hypothefis ; whereas, it is evident, that the probabilities of living after 42, being all along lefs in Dr. Halleyh tabic, tlian in the hypothefis, the z^ahio of the life muft be aho lefs.
The mathematical reader may cafily fatisfy him- felf, that the value of Q^ought to be taken from Table VI. as I have diredtcd.
An eafy and accurate method of finding the values of fingles lives, agreeably to any given table of obfervations, is given by Mr. Dodfon in his Mathematical Repcfitory^ vol. II. page 161.
There is alio in Mr. Simpfoji's Sek5i Extrcifes^ page 27^, a very eafy rule for approximating to the values cf fingle lives, according to Dr. HalUy^^ table. But this rule mud not be depended on j for I have found it 4- a year's purchafe, and fome- times three-quarters of a year's purchafe wrong.
To prevent the danger of miftaking the Theo- rem I have given, 1 have thought proper to fubjoin the following exam.ple.
Let the table oFoblervations be iht Brrjla-'jj Ta- ble, or Table III. The value of a life at 78, by this X 2 Table,
3o8 APPENDIX.
Table, is ^^+^ + P7^> &c. to the end cf
life. The number of terms in this feries being Imall, it may be eafily found to be 3.514, fuppof-
ing intereft at j^per cent, and - , — , -j, &c. be- ing the values, in Table I. of i/. at the end of I, 2, 3, &c. years. ^ From 78 to 74 the decre- ments of life Gontinue equal ; and therefore 8=4. a n: 98. b — 58, P — 3.6298, by Table II •, pzz .8548, by Table li Qji: 1.406, by Table VI ;
N zz 3.514. P — Q^+Z'N = 5.227 -^ and Qj-|
^^ — Q^-i-^N = 4,500, or the value of a life at
74-
From 74 to 70 there is another interval of equ^l
decrements ; and, by a like eafy operation, ths
taiue of a iix^e at 70 will be found to be 5-595.
T A E L £
APPENDIX.
3^9
TABLE I.
The prefent Value of i/. to be received at the end of any number of years, not exceeding loo; difcounting at the rates of 3, 3^, 4, 4^, 5 and 6 per cent, compound interelt.
1 |
3 per Ct. |
3! per Ct. |
4 per Ct. |
4ipcrCt |
5 per Ct. |
6 per Ct. .943396 |
,970874 |
,966184 |
,961538 |
,956938 |
952381 |
||
2 |
,942596 |
.935511 |
,924556 |
■ .91573c |
,907029 |
,^^89996 |
3 |
,915142 |
,90,1943 |
,888996 |
,876297 |
,863838 |
,839619 |
4 |
,888487 |
,871442 |
,854804 |
,838561 |
,822702 |
,792094 |
s |
.862609 |
.841973 |
,821927 |
,802451 |
.783526 |
,747258 |
6 |
,837484 |
,813501 |
'790315 |
,767896 |
,746215 |
,704961 |
7 |
,813092 |
.7^^599' |
,759918 |
,734828 |
.710681 |
,665057 |
8 |
,789409 |
.759412 |
,730690 |
,703185 |
.676839 |
,627412 |
9 |
,766417 |
•73373J |
.7025^7 |
,672904 |
,644609 |
,59189s |
10 1 1 |
,744094 |
,7o^9'9 |
,675564 |
,■643928 |
,613913 |
'55^395 .526788 |
,722421 |
,684946 |
,649581 |
,616199 |
,584679 |
||
12 |
,701380 |
,661783 |
,624597 |
,589664 |
.556837 |
,496969 |
'3 |
,680951 |
,639404 |
,600^74 |
,564272 |
,530321 |
,468839 |
H |
,661118 |
,617782 |
.577475 |
.559973 |
,505068 |
,442501 |
'5 |
,641862 |
,596891 |
.555265 |
,516720 |
,481017 |
,4:7265 |
lb |
,623167 |
,576706 |
,533908^ |
.494469 |
,45^1 12 |
.393646 |
'7 |
,605016 |
,557204 |
-513373 |
'473 '76 |
.436297 |
•371364 |
ly |
<5«7^^95 |
.53^36' |
,493628' |
,452800 |
,415521 |
■353344 |
'9 |
,570280 |
,520156 |
,474642 |
.433302 |
'^95734 |
•330513 |
zo 21 |
,553676: |
,502566 |
,456387 |
414643 |
.376889 |
,311805 |
.537549 |
.485571 |
,438834 |
,396787 |
.358942 |
.294155 |
|
22 |
,521893 |
.469151 |
,421955 |
.379701 |
341850 |
■277505 |
23 |
,506692 |
,453286 |
,405726 |
.363350 |
•32557' |
,261797 |
2+ |
'491934 |
.457957 |
,390121 |
.3477P3 |
,310068 |
,246979 |
^5 |
) 47 7 606 |
,423147 |
•375i'7 |
.33273^ |
.295303 |
232999 |
26 |
,463695 |
,408838 |
,360689 |
,318402 |
.281241 |
219810 |
27 |
,450-189 |
,395012 |
.3468^7 |
,30469: |
.26784!! |
207368 |
28 |
.437077 |
,381654 |
■333477 |
,291571 |
255004 |
195630 |
29 |
,424346 |
.368748 |
,320651 |
.=790'5 |
242946 |
'%57 |
30 |
,41 19S7 |
.356278 |
,308319 |
,267000 |
231377 |
174110 |
,390987 |
,344230 |
296460 |
255502 |
220359 |
164255 |
|
32 |
.3883^7 ,332590! |
28505S ,244500! |
209866 |
'54957 |
X3
^10 APPENDIX.
TABLE I. Continued,
3 ptr Ct.
:> ■» 34 35
i^> 37 5^
39 40
41
42
43 44
45 46
47 48 49
377026 ,366045
,345032
.334983 ,^25226
.315754 »3o6557
3iperCt.l 4perCt. U^perCt
,321343 ,310476
.299977 ,289853 ,280032 ,270562 ,261413 ,252572
,297628 ,288959
'^'■'-543 ,272372 ,264439
.-56737 ,249259 ,241999
J'234950 I50 ,228107
,274094 ,263552
.2534' 5 ,243669
,234297
,225285
,216621
,208289
,233971 ,223896 ,214254 ,205028 ,196199 ,18775c ,179665 171929
5 per Ct. I 6 per Ct. I
51
52
53 54 5 5 56
57 5S
59
60
,244031
.235779 ,227800!
,220102'
,212659
,205468
,198520
,191806
,185320
.179053
,200278
.192575 ,185168 ,178046 ,171198 ,164614 ,158283 ,152195 ,146341 ,140713
,221463
.215C13
,208750
,202670!
,196767
,191036
,185472
,180070
,174825
.169733
64
,1 047^9 ,159990 .15533c
,! 50806 051-146413 661,142149
67 1,138009
; 68 1,133989
I ••9 j,i iCoS6 I -o j,i 2629T
,17299^ ,167148
,161496
.156035
.150758
,14566c
140734
135975
131377 126934
.155301 ,130057 ,125093 ,120282 ,115656 ,1 1 1207 ,106930 ,102817 ,098963 ,095060
,164525 ,157440 ,1 50663
.144173 ,137964
.132023
,126338
,120898
,115692
,1 10710
,199873
.190355 ,181290
,172657 ,164436 ,156605 ,149148 , 1 42046
,105942 ,ioi3i;o ,097014
,092837
,0!:
^■•39
122642 118495
,114487 , 1 1 06 1 6 ,106875 .103261 ,099760 ,096395 .093136 .019986
,085013 ,081353 ,077849
.074497 ,071289
,135282 ,128840 ,122704 ,116864 ,111297 ,105997 ,100949 ,096142 ,ogi 564 ,087204
,146186 ,137912 ,130105 ,122741
.115793 ' ,109239 ,103056 ,097222
,091404 ,087889 ,084508 ,081258 ,078133 ,075128 ,072238 .069460 ,066788 ,064219
,083051 ,079096 ,075330
.071743 ,068326 ,065073 ,061974 ,059023 ,056212 ,053536
,091719 ,086527 ,081630 ,077009 ,072650 ,068538 ,064658 ,060998!
.057546 ,054288]
,051215
,048316 ,045582 ,043001 ,040567 ,038271 ,036105 ,034061
.032133 ,030310
,068219 ,065281 ,062470 ,059780 ,057206 ,054742 ,052385 ,050129 ,047971 ,045905
,050986 ,048558 ,046246 ,044044 ,041946
.039949 ,038047 ,036235
.034509 ,032866
,028598 ,026989 ,025453 ,02401: ,027653 ,021 37c ,020161 ,019020 ,017943 , 016927
APPENDIX. TABLE I. Continued.
3n
71 |
3 per Ct. |
3^ per Ct. |
4 per Ct. |
n\ per Ct. |
5 par Ct. |
6 per Ct. |
,i?26i9 |
,086943 |
.061749 |
,043928 |
,031301 |
,015969 |
|
72 |
.119047 |
,084005 |
.059374 |
,042037 |
,029811 |
.015065 |
73 |
,115580 |
,081 162 |
,057091 |
,040226 |
,028391 |
,014212 |
74 |
,112214 |
,078418 |
,054895 |
,038491 |
,027039 |
,oi34Cii |
75 |
.108945 |
,075766 |
,052784 |
,036836 |
,025752 |
,012649 |
76 |
,105772 |
,073204 |
,050754 |
,035250 |
,024525 |
.01193^ |
77 |
,102691 |
,070728 |
,048801 |
.033732 |
,023357 |
,01 1258 |
78 |
,099700 |
,068336 |
,046924 |
,032280 |
,022245 |
,010620 |
79 |
,096796 |
,066026 |
,045120 |
,030890 |
,021 186 |
,010019 |
80 81 |
.093977 |
,063793 |
,943384 |
,0:9559 |
,020177 |
,009452 |
,091240 |
,061636 |
.041716 |
,028287 |
,019216 |
,008917 |
|
82 |
,088582 |
,059551 |
,0401 1 1 |
,027068 |
,018301 |
,008412 |
83 |
,086002 |
•057538 |
,038569 |
,025903 |
,017430 |
,007936 |
84 |
,083497 |
,055592 |
,037085 |
,024787 |
,016600 |
.007487 |
85 |
,081065 |
,053712 |
,035659 |
,023720 |
,015809 |
,007063 |
86 |
.078704 |
,051896 |
,0342^7 |
,022699 |
,015056 |
,006663 |
87 |
,076412 |
,050141 |
,032968 |
,021721 |
,014339 |
,006286 |
88 |
,074186 |
.04^445 |
,031700 |
,020786 |
,013657 |
,005930 |
89 |
,072027 |
,046807 |
,030481 |
,019891 |
,013006 |
,005595 |
90 91 |
,069928 |
,045224 |
,029309 |
,019034 |
,012387 |
,005278 ,004979 |
,067891 |
,043695 |
,028182 |
,018215 |
,011797 |
||
92 |
,065914 |
,042217 |
,027098 |
,017430 |
,011235 |
,004697 |
93 |
,063994 |
,040789 |
,026055 |
,016680 |
,010700 |
,004432 |
' 94 |
,062130 |
,039410 |
,025053 |
,015961 |
,010191 |
,004181 |
95 |
,060320 |
,038077 |
,024090 |
,015274 |
.009705 |
.003944 |
96 |
,058563 |
,036790 |
,023163 |
,0146:6 |
,009243 |
,003721 |
97 |
,05685s |
,035546 |
,022272 |
,013987 |
, 008003 |
,0035 10 |
98 |
,055202 |
.034344 |
,021416 |
.013385 |
,008384 |
.003312 |
99 |
.053594 |
,033182 |
,020592 |
,012808 |
,0079^ 5 |
.CO3124 |
ICO |
,052053 |
,032060 |
,019800 |
,oi2v;s7 |
.oo-6od |
,O070s'7 |
X4
TABLE
3U
APPENDIX.
TABLE II.
The prefent Value of an Annuity of One Poundj for any Number of Years not exceeding 100, at the feveral Rates of 3, 3|-, 4, 5, and 6/. per Cent»
Ye. I |
I3 perCt. |
3|perCc. |
4 per Ct. |
5 per Ct. |
6 per Ct. |
.970S |
.9662 |
.9615 |
'95^3 |
•9433 |
|
2 |
^•9133 |
1.8997 |
1.8860 |
1.8594 |
^'^333 |
3 |
2.8286 |
2.8016 |
2.7750 |
2.7232 |
2.6730 |
4 |
Z'1^1^ |
3-^73^ |
3.6298 |
3-5^59 |
3-4651 |
5 6 |
A-5797 |
4-5151 |
4.4518 |
4.3294 |
4.2123 |
5-4971 |
5.3286 |
5' 242 I |
5-0756 |
4-9173' |
|
7 |
6.2302 |
6.1 145 |
t).0020 |
5-7'-63 |
5-5823 |
8 |
7.0196 |
6:8740 |
6.7S2y |
6.4632 |
6.2097 |
9 |
7.7861 |
7.6077 |
7-4-353 |
7.1078 |
6.8016 |
10 1 1 |
S.5302 |
8.3166 |
8.1108 |
7.7212 |
y.^6oQ |
9.2526 |
9.0015 |
8.7604 |
8.3064 |
7.8868 |
|
12 |
9.9540 |
g.6633 |
9.3850 |
8.8632 |
8.3838 |
13 |
10.6349 |
10.3027 |
9-9^5^ |
9-3935 |
8.8526 |
14 |
11.2960 |
;io.92o5 |
10.5631 |
9.8986 |
9.2949 |
15 16 |
11.^379 |
11.5174 |
11.1183 11.6522 |
10.3796 |
9.7122 |
12.561 1 |
12.0941 |
10.8377 |
10.1058 |
||
17 |
1^.1661 |
12.6513 |
12.1656 |
11.2740 |
10.4772 |
.i8 |
^3-7535 |
13.1897 |
12.6592 |
11.6895 |
10.8276 |
19 |
14.5238 |
13.7098 |
13-1339 |
12.0853 |
11.1581 |
20 21 |
14.8774 |
14.2124 |
^3-5903 |
12.4622 12.S211 |
1 1.4699 |
15-4150 |
14.6980 |
14.0291 |
1 1.7640 |
||
22 |
i5-93^'9 |
15.1671 |
14.4A11 |
13.1D30 |
12.0415 |
•231 |
16.4436 |
15.6204 |
14.8568 |
13.4885 |
12.3033 |
24 |
1^-9355 |
16.0584 |
15.2469 |
13.7986 |
12.5503 |
25 |
1*7.4131 |
16.4815 |
15.6220 |
14.0939 |
12.7833 |
APPENDIX. TABLE ir. Continued.
Tf«!
-'Ye: 26 |
3 per Ct. |
3^; per k„c. |
4pcr Cc. |
5 per CC. |
0 per U;. |
17.876S |
16.8904 |
15.9827 |
^'^■375^ |
13.0031 |
|
•'27 |
^8.3270 |
17.2854 |
16.3295 |
146430 |
13.2105 |
28 |
18.7641 |
17.6670 |
16.6630 |
14.8981 |
13.4061 |
29 |
19.1884 |
18.0358 |
16.9837 |
15.1410 |
^3-59^7 |
30 31 |
19.6004 20.0004 |
18.3920 |
17.2920 |
15-3724 |
13.7648 |
1 8. 7363 |
17.5884 |
15.5928 |
13.9290 |
||
3-2 |
20.3887 |
19.06S9 |
^7-'^735 |
15.8026 |
14.0S40 |
33 |
20.7657 |
19.3902 |
18.1476 |
16.0025 |
14.2302 |
34 |
2 I . I 3 1 8 |
19.7007 |
18.41 II |
16.1929 |
14.3681 |
35 36 |
21.4872 |
20.0G07 |
18.6646 |
16.3741 |
14.4982 |
21.8322 |
20.2905 |
18.9082 |
16.5468 |
14.6209 |
|
37 |
22.1672' |
20.5705 |
19.1425 |
16.71 12 |
H-7367 |
38 |
22.4924 |
20.841 1 |
19.3678 |
16.8678 |
14.8460 |
39 |
22.8082 |
21.1025 |
19.5844 |
17.0170 |
14.9490 |
40 41 |
23.1147 |
21-3551 21.5991 |
19.7927 |
17.1590 |
15.0462 |
23.4124 |
19.9930 |
17.2943 |
15.1380 |
||
42 |
23.7013 |
21.8349 |
20.1856 |
17.4232 |
15-2245 |
43 |
23.9819 |
22.0627 |
20.3707 |
^7-5fj9 |
15.3061 |
44 |
24.2542 |
22.2828 |
20.548S |
17.6627 |
^5-3'^3'^ |
45 46 |
24.5187 24-7754 |
22.4955 |
20.7200 |
17.7740 |
^5-455!^ |
22.7009 |
20.8846 |
17.8800 |
15-5243 |
||
47 |
25.0247 |
22.8994 |
21.0429 |
17.9810 |
15.5S90 |
4H |
25.2667 |
23.0912 |
21.1951 |
18.0771 |
J 5.6500 |
49 |
25.5016 |
23.2766 |
21.3414 |
18.1687 |
^5-7^75 |
50 5i) |
25.7297 |
23-4556 |
21.4821 |
18.2559 |
15.7618 |
25.9512 |
23.6286 |
21.6174 |
18.33S9 |
15.8130 |
|
52 |
26.1662 |
23-'795^ |
21.7475 |
i8.^.i8o |
15.8613 |
53 |
26.3749 |
23-9^^^73 |
21.8726 |
18.4.9^4 |
15.9069 |
54 |
26.5776 |
24-1133 |
2 1.9929 |
1S.5651 |
15-9499 |
55 |
26.^744 |
24.2641 |
22.IGi^6 |
18.6334115.990,:! |
APPENDIX.
TABLE If. Continued.
yi;.
56
51 58
3 P
er Ct.
26.9654
OQ
27-i5^y 27.3310
c,(^ 27.5058 60 '2.'].6']^S
61 27.(5403
62 28.0003
67, 28.1556
64 28.3064 6^ 28.4528
66
67 68
72
73 74 75
76
77 78
79 80
81
82
84
28.5950
28.7330
28.8670
28.9971
29.1234
29.2460
29.3650
29.4806
29.5928
29.7018
3ipfrLt.
4 per Ct.
24.4097
24.5504 24.6864 24.8178 24.9447
25.0674 25.1859 25.3004 25.41 10 25.5178
29.8076 29.9102 30.0099 30.1067 30.2007
30.2920 30.3805 30.4665 30.5500 85 30.6311
25.6211 25.7209 25.8173
25.9104 26.0004
26.0873 26.1713 26.2525
22.2198 22.3267 22.4295 22.5284 22.6234
22.7148 22.S027 22.8872 22.9085 23.0466
23.1218 23.1940 23-2635 23.3302
23.4562 23-5^56 23-5727
26.3309123.6276 26.4067 23.6804
26.4799 23.7311 26.5506 23.7799 26.6190 23.826S 26.6S5O 23.872c 26.7488 23.9153
26.8104 23.9571 26.8700 23.9972 26.9275 24.0357 26.983il24.072i: 27.0368 24.1085
•; per Ct.
18.6985 18.7605 18.8757 18.9292 |
13.9802 19.0288 19.0750 19.1191 19.1610 |
19.2010 19.2390 Lg,Q.jS3 19.3098 19.3426 |
^9-3739 19.4037 19.4321 19.4592 19.4849 19.5094 19.5328 ^9-5550 i^.5y6i 19.5964. |
19.6156 ^9-^339 19.6514 19.6680 19.68:^8 |
b per (Jt.
6.0288 6.0649 6.0989 6.I3U 6.1614
6.1900 6.2170 6.2424 6.2664 6.2891
6.3104
6.^S^6 6.3496 6.3676 6.3845
6.4005
6.4155 6.4297
6.4431
6.4558
6.4677 6.4790 6.4896 6.4996 6.5091
6.518Q
6.5264
6-53^3 6.541b
6.5489
APPENDIX. TABLE IL Continued.
^^S
Ve. 86 87 88 89 80 91 92 93 94 95 96 97 98 99 100 |
3 per Cc. |
3ipcrCt. |
Vf-"'" ^• |
5 per Ct. |
6 per Ci. |
30.7098 30.7862 30.8604 30.9324 31.CO24 31.0703 31.1362 31.2002 31.2623 31.3226 |
27.0887 27.1388 27.1873 27.2341 27.2793 27-3230 27.3652 27.4060 27-4454 27-4835 |
24.1428 24-1757 24.2074 24.2379 24.2672 |
19.6988 19.7132 19.7268 19.7398 19.7522 |
16.5556 16.5618 16.567S 16.5734 16.5786 |
|
24.2954 24.3225 24.3486 24-3736 24-3977 |
19.7640 19.7752 19.7859 19.7961 19.8058 |
16.5836 16.5883 16.5928 16.5969 16.6009 |
|||
31.3812 31.4380 31-4932 31.5468 3I-59S9 |
27-5203 27-555S 27.5902 27.6234 27-6554 |
24.4209 24.4431 24.4646 24.4852 24.5050 |
19.815I 19.8239 19.8323 19.8403 19.8479 |
16.6046 16.6081 16.6114 16.6145 16.6175 |
3i6 appendix;
TABLE III.
Shewing the Probabilities of the Duration of Life, as deduced by Dr. Hallcy from Obferva-r tions on the Bills of Mortality of Breslaw.
Perfons |
Deer, i |
- |
I'erfons |
Deer. |
Perfons |
Deer. |
||
Ages I |
Jiving. 1000 |
of Life. 145 1 |
Ages. 31 |
livine. 523 |
of L'fi. |
Ages |
living. |
of Lif<^ 10 |
8 |
61 |
232 |
||||||
2 |
855 |
57 |
32 |
515 |
8 |
62 |
222 |
10 |
3 |
798 |
38 |
33 |
507 |
8 |
63 |
212 |
10 |
4 |
760 |
28 |
34 |
499 |
9 |
64 |
202 |
10 |
5 |
732 |
22 |
35 |
490 |
9 |
65 |
192 |
10 |
,6 |
710 |
18 |
36 |
481 |
9 |
66 |
182 |
10 |
7 |
692 |
12 |
37 |
472 |
9 |
6y |
172 |
10 |
8 |
680 |
10 |
38 |
463 |
9 |
6S |
162 |
ID |
9 |
670 |
9 |
39 |
454 |
9 |
69 |
152 |
10 |
lO |
661 |
8 |
40 |
445 |
9 |
70 |
142 |
II |
II |
653 |
7 |
41 |
436 |
9 |
71 |
13^ |
u |
12 |
646 |
6 |
42 |
427 |
10 |
72 |
120 |
II |
i3 |
640 |
6 |
43 |
417 |
10 |
73 |
109 |
II |
14 |
634 |
6 |
44 |
407 |
10 |
74 |
98 |
ro |
15 |
628 |
6 |
45 |
397 |
10 |
75 |
88 |
10 |
i6 |
622 |
6 |
46 |
3^7 |
10 |
76 |
78 |
10 |
17 |
616 |
6 |
47 |
377 |
10 |
77 |
68 |
10 |
i8 |
610 |
6 |
48 |
3^7 |
10 |
78 |
58 |
9 |
19 |
604 |
6 |
49 |
357 |
1 1 |
79 |
49 |
8 |
20 |
598 |
6 |
50 |
346 |
II |
80 |
41 |
7 |
21 |
592 |
6 |
51 |
335 |
1 1 |
81 |
34 |
6 |
22 |
586 |
7 |
52 |
324 |
1 1 |
82 |
28 |
5 |
23 |
579 |
6 |
53 |
313 |
'' |
83 |
23 |
4 |
24 |
573 |
6 |
54 |
302 |
10 |
84 |
19 |
4 |
25 |
.5^7 |
7 |
55 |
292 |
10 |
85 |
15 |
4 |
26 |
560 |
7 |
5'o |
282 |
10 |
86 |
1 1 |
3 |
27 |
553 |
7 |
57 |
272 |
10 |
87 |
8 |
3 |
28 |
546 |
/ |
5S |
262 |
10 |
88 |
5 |
2 |
29 |
539 |
8 |
59 |
252 |
10 |
89 |
3 |
2 |
30 |
53^ |
8 |
60 |
242 — _ — |
10 |
90 |
I |
I |
APPENDIX, 317
TABLE IV.
Shewins the Probabilities of Life at Nortii- See page 255, 256.
aMpton.
Ferlens |
Deer. |
Pcrlbns |
Deer, f , . |
Ferlons |
Deer. |
|||
Ages. |
living. |
of Life. 300 |
Ages. 31 |
living. |
of Life. |
/\ges |
1 living. |
of Life. |
0 |
1149 |
428 |
7 |
62 |
187 |
8 |
||
I |
849 |
127 |
32 |
421 |
7 |
63 |
179 |
• 8 |
2 |
722 |
50 |
33 |
414 |
7 |
64 |
171 |
8 |
3 |
672 |
26 |
34 |
407 |
7 |
65 |
163 |
8 |
4 |
646 |
21 |
35 |
400 |
7 |
66 |
^55 |
^ |
5 |
625 |
16 |
36 |
393 |
7 |
67 |
147 |
|
6 |
609 |
13 |
37 |
386 |
7 |
68 |
139 |
8 , |
7 |
59^ |
10 |
38 |
379 |
7 |
% |
^3' |
8 |
8 |
586 |
9 |
39 |
372 |
7 |
70 |
123 |
8 |
9 |
577 |
7 |
40 |
3% |
8 |
71 |
115 |
8 |
10 |
570 |
6 |
41 |
357 |
8 |
72 |
107 |
8 |
II |
564 |
6 |
42 |
34-9 |
8 |
73 |
99 |
8 |
12 |
558 |
5 |
43 |
341 |
8 |
74 |
91 |
8 ;• |
^3 |
553 |
5 |
44 |
333 |
8 |
75 |
83 |
8 |
14 |
548 |
5 |
45 |
3^5 |
8 |
76 |
75 |
8 |
15 |
543 |
5 |
46 |
3^7 |
8 |
77 |
67 |
7 |
16 |
538 |
5 |
47 |
3^9 |
8 |
78 |
60 |
7 |
17 |
533 |
5 |
48 |
301 |
8 |
79 |
53 |
7 |
18 |
528 |
6 |
49 |
293 |
9 |
80 |
46 |
7 |
19 |
522 |
7 |
50 |
284 |
9 |
81 |
39 |
7 |
20 |
515 |
8 |
5i |
275 |
8 |
82 |
32 |
6 |
21 |
507 |
8 |
52 |
267 |
8 |
83 |
26 |
5 |
22 |
499 |
8 |
53 |
259 |
8 |
84 |
21 |
4 |
23 |
491 |
8 |
54 |
251 |
8 |
85 |
17 |
4 |
24 |
483 |
8 |
55 |
243 |
8 |
86 |
13 |
3 |
25 |
475 |
8 |
56 |
235 |
8 |
^7 |
10 |
2 |
26 |
467 |
8 |
57 |
227 |
8 |
88 |
8 |
2 |
27 |
459 |
8 |
58 |
219 |
8 |
89 |
6 |
2 |
28 |
451 |
8 |
59 |
211 |
8 |
00 |
4 |
2 |
29 |
443 |
8 |
60 |
203 |
8 |
i?' |
2 |
i |
30 |
435 |
7 |
61 |
J 95 |
8 |
I92 |
I |
I |
^iS
A p p E N D I ii:.
TABLE V.
Shewing the Probabiliiies of Life at NoRwiCH-i See page 256, 257.
177 |
Perl'ons |
Deer. |
Ages 32 |
Perfohs |
Deer. |
Ages 63 |
i'erfons |
j Deer. |
Ages 0 |
living. |
of Lite. |
living. |
of Lite. |
living. ^74 |
[of Lifi, |
||
1185 |
320 |
392 |
6 |
9 |
||||
I |
865 |
160 |
33 |
386 |
6 |
64 |
165 |
9 |
2 |
705 |
60 |
34 |
380 |
6 |
65 |
156 |
9 |
3 |
^45 |
32 |
35 |
374 |
6 |
66 |
H7 |
9 |
4 |
613 |
23 |
36 |
368 |
6 |
67 |
^38 |
9 |
5 |
590 |
20 |
37 |
362 |
6 |
6S |
129 |
9 |
6 |
570 |
16 |
38 |
35^ |
6 |
69 |
120 |
9 |
7 |
554 |
13 |
39 |
350 |
7 |
70 |
II I |
Q |
8 |
541 |
II |
40 |
3^3 |
6 |
,7^ |
102 |
8 |
9 |
53^ |
9 |
41 |
337 |
6 |
72 |
94 |
8 |
10 |
521 |
7 |
42 |
33^ |
6 |
'73 |
S6 |
8 |
II |
5H |
6 |
43 |
325 |
7 |
74 |
78 |
8 |
12 |
508 |
6 |
44 |
318 |
7 |
'75 |
70 |
8 |
13 |
5C2 |
5 |
45 |
3if |
7 |
J76 |
62 |
7 |
14 |
497 |
5 |
46 |
304 |
7 |
77 |
55 |
7 |
i^ |
492 |
5 |
47 |
297 |
7 |
78 |
48 |
6 |
16 |
487 |
5 |
48 |
290 |
7 |
79 |
42 |
5 |
17 |
482 |
5 |
49 |
28? |
7 |
So |
37 |
5 |
18 |
477 |
5 |
50 |
276 |
7 |
^^ |
32 |
4 |
19 |
472 |
5 |
51 |
269 |
7 |
82 |
28 |
4 |
20 |
467 |
6 |
52 |
262 |
7 |
83 |
24 |
4 |
21 |
461 |
6 |
53 |
255 |
8 |
84 |
20 |
3- |
22 |
4 1- r |
6 |
54 |
247 |
8 |
S5 |
17 |
J |
23 |
449 |
6 |
55 |
239 |
8 |
86 |
14 |
3 |
24 |
443 |
6 |
56 |
23 f |
8 |
87 |
II |
rf |
25 |
43; |
6 |
57 |
223 |
8 |
88 |
9 |
2 |
26 |
43 i |
7 |
ss |
215 |
8 |
89 |
7 |
2 |
27 |
424 |
7 |
59 |
207 |
8 i |
90 |
5 |
2 |
28 |
417 |
7 |
60 |
199 |
8 ' |
91 |
3 |
2 |
29 |
410 |
6 |
6l |
191 |
8 i |
02 |
I |
1 |
^0 |
/[.04 |
6 |
62 |
^^3 |
9 |
93 |
I |
I |
^^ f |
39^' |
6 1 |
1 |
1 |
APPENDIX. TABLE VI. {a).
\9
Shewing the prefent Values of an Annuity of i/. on a fingle life, according to Mr. De Mohreh hypothefis ; and, therefore, nearly, according to the probabilities of life at Breslaw, Norv/ich, and Northampton. See p. 2, and p. 262.
Age. 8 9 10 II 12 «3 H 15 16 >7 18 '9 20 21 22 23 24 ^5 26 27 28 29 30 31 32 33 |
3 per Ct. |
3iperCt. |
4 per Ct. 16,791 16,882 16.882 |
4-1 per Ct. |
5 per Ct. 14.544 14,607 14.607 '4.544 14,480 14,412 14,342 14,271 14.197 14,123 14,047 13,970 13,891 13,810 '3.727 13,642 '3.555 13,466 13.375 13,282 13,186 13,088 12,988 12,855 12,780 12,673 |
6 per Ct. 12,790 12,839 12,859 |
19,736 19,868 19,868 |
18,160 18,269 18,269 |
'5-595 15,672 15,672 |
||||
19,736 19,604 19.469 19-331 19.192 19,050 18,905 18,759 18,610 18,458 |
18,160 18,049 17.937 17,823 17,707 17.588 17.467 17,344 17,220 17.093 |
16,791 16,698 16,604 16,508 16,410 16.311 16,209 16,105 '5.999 15.891 |
15.595 15.517 15.437 '5.356 '5'273 15,189 15,102 15,015 '4-923 14,831 |
12,790 12,741 12,691 12,639 12,586 12,532 12,476 12,419 12,361 12,301 12,239, 12,177 12,112 12,045 11,978 11,908 11,837 11.763 11,683 ii,6io |
||
18.305 18,148 17,990 17.827 17,664 '7.497 17.327 17.154 16,979 16,800 |
16,963 16,830 16,696 '6,559 16,419 16,277 16,133 15.9JJ5 15.^35 15,682 |
15,781 15,669 '5.554 '5-437 15,318 15. '97 '5.073 14,946 14,816 14.684 14,549 14,411 14.270 |
'4.737 14,641 14.543 14,442 14,340 14*235 14,128 14,018 '3.905 13.79' 13,673 '3.553 '3.43^' |
|||
16,620 16,436 16,248 |
15,526 '5.367 15.204 |
11,530 11,449 11.365 |
||||
the V |
This Ts alues of |
ible is the iingle li\ |
: fnme w es, publ |
th Mr. D lihed in . |
1 Moi'vre lis Tread |
J Table of /e on Life |
Anmdtiei, and carried aS far as the age of 79 to three place's 'of decimals by Mr. Dodjon in his Mcithc^natical Rsp'nlton, vol. ii, p. 169.
gzo
APPENDIX. TABLE VI. Continued.
Age 34 |
j 3 P" Cc. |
3f perCt. |
4 per Ct. |
4-1 per Ct. |
5perCt. |
6 per Ct- |
16,057 |
15,039 |
14,126 |
13^304 |
12,562 |
11,278 |
|
3S |
15.^^ |
14.871 |
13.979 |
13.175 |
12,449 |
1 1,189 |
56 |
15,666 |
14,699 |
13.829 |
13,044 |
12.353 |
1 1,098 |
57 |
15.465 |
14,524 |
13,676 |
12,909 |
12,214 |
11,003 |
3« |
15,260 |
H'345 |
i3.5'9 |
12,771 |
12,091 |
10,907 |
39 |
J5.053 |
14,163 |
13.359 |
12,630 |
1 1,966 |
10,807 |
40 41 |
. I4'842 |
13,978 |
13,196 |
12.-485 |
11,837 |
10,704 |
14,626 |
i3>789 |
13,028 |
12,337 |
11,705 |
10,599 |
|
42 |
14.407 |
'3.596 |
12,858 |
12,18^ |
11,570 |
10,49c |
: 43 |
14-185 |
•3'399 |
12,683 |
12,029 |
11,431 |
10,378 |
44 |
'3'958 |
13,199 |
12,504 |
11,870 |
11,288 |
10,263 |
45 |
13.728 |
12.993 |
12,322 |
I 1,707 |
I 1,142 |
10,144 |
46 |
13.493 |
i2,7«4 |
12,135 |
I 1.540 |
10,992 |
10,021 |
47 |
13.254 |
12,571 |
11,944 |
11,368 |
10,837 |
9.895 |
4d |
13,012 |
12354 |
11,748 |
1 1,192 |
10,679 |
9,765 |
49 |
12,764 |
1 2, 1 3 1 |
11,548 |
1 1,012 |
10,515 |
9,630 |
50
5' |
12,51 1 |
1 1,904 |
11.344 |
10,827 |
50,348 |
9,492 |
12,255 |
11,673 |
11,135 |
10,638 |
10,176 |
9.3.49 |
|
3- |
11,994 |
J 1.437 |
10,921 |
10,443 |
9.999 |
9,201 |
53 |
1.1,729 |
11,195 |
10,702 |
10,243 |
9.S17 |
9,049 |
54 |
11.457 |
10,950 |
10,478 |
10,039 |
9,630 |
8,891 |
55 |
11,183 |
10,698 |
10,248 |
9,829 |
9.437 |
8,729 |
5^ |
10,902 |
10,443 |
10,014 |
9,614 |
9.239 |
8,561 |
57 |
10,616 |
1 0, 1 8 1 |
9.775' |
9.393 |
9.036 |
8.387 |
5« |
10325 |
9.913 |
9.527 |
9,166 |
8,826 |
8,208 |
S9 |
10,029 |
9,640 |
9.275 |
8,953 |
8,6x1 |
8,023 |
bo 61 |
9.727 |
9,361 |
9.017 |
8,694 |
8,389 |
7.831 |
9.4'9 |
9,076 |
8.753 |
8.449 |
8,161 |
7.63:3 |
|
62 |
9,107 |
8,786 |
8,482 |
8,197 |
7.926 |
7.428 |
.^^ |
- b',787 |
8,488 |
8,205 |
7.938 |
7.684 |
7,216 |
64 |
8,462 |
8.185 |
7,921 |
7.672 |
7.435. |
6.997 |
^5 |
8,132 |
7.875 |
7,631 |
7.399 |
7.179 |
6,770 |
66 |
7-794 |
7.558 |
7.333 |
7.119 |
6,915 |
6.535 |
67 |
7,450 |
7.234 |
7.027 |
6.831 |
6,643 |
fy,z<^- |
68 |
7.099 |
6.902 |
6,714 |
6,534 |
6,362 |
6,040 |
6q |
6,743 |
6,565 |
6,394 |
6,2^0 |
6,073 |
5.779 |
7c |
6.rS |
6,2 IQ |
6.065 |
5.918! |
5-775 |
5,508 |
A P I^ E N D I X. TABLE VI. Continued.
32t
Age. |
3perCt. |
ji per Ct. |
4 per Ct. |
4i P" Ct |
S per Ct. |
6 per Ct. |
7' |
6,008 |
5,865 |
5,728 |
5.596 |
5,468 |
5,228 |
72 |
5.631 |
5.505 |
5.383 |
5.265 |
5.^2 |
4,937 |
73 |
5,246 |
5.m6 |
5,029 |
4.926 |
4,826 |
4,636 |
74 |
4.8H |
4.759 |
4,666 |
4.576 |
4,489 |
4.324 |
7S |
4.453 |
4.373 |
4,293 |
4,217 |
4. '43 |
4,000 |
76 |
4,046 |
3.978 |
3,912 |
3.847 |
3.734 |
3.664 |
77 |
3.632 |
3.J75 |
3,520 |
3.467 |
3,415 |
3.3'5 |
78 |
3,207 |
3.'63 |
3,111 |
3076 |
3.03 + |
2,953 |
79 |
2.776 |
2,741 |
2,707 |
2,673 |
2,641 |
2.578 |
«o |
2.334 |
2,309 1,867 |
2,284 |
2,259 1,832 |
2,235 1,816 |
2,188 |
81 |
1,886 |
1,850 |
1.783 |
|||
82 |
1.429 |
1,411 |
1,406 |
1.394 |
1,384 |
1,362 |
«3 |
0,961 |
0.955 |
0,950 |
0.943 |
0,937 |
0.925 |
84 |
0,484 |
0,483 |
0,481 |
0.479 |
0,476 |
0,472 |
«5 |
o.coo |
0,000 |
0,000 |
0,000 |
OjOOO |
0,000 |
TABLE
322
APPENDIX. TABLE Vir.
I
Shewing the Value of an Annuity on the joint con- tinuance of two lives, according to Mr. De Moi-vre's Hypoihefis ; and, therefore, nearly ac- cording to the probabilities of life at Breslaw, Norwich, and Northampton. See ElTay 11^ and p. 2, 3, 226, 262.
• u . |
U |
to , |
■*^ |
^o . |
•5« ° c |
0 .g |
JO |
||
> ^ |
> t |
^g. |
||
10 |
15.206 |
13-342 |
11.855 |
|
^5 |
14.878 |
13-093 |
ii.66i |
|
20 |
14-503 |
12.808 |
11.430 |
|
25 |
14.074 |
12.480 |
11. 182 |
|
10 |
30 |
13-585 |
I2.I02 |
10.884 |
■ |
^5 |
13.025 |
i 1.665 |
10.537 |
40 |
12.381 |
II. 156 |
10.128 |
|
45 |
11.644 |
10.564 |
9.646 |
|
50 |
10.796 |
9.871 |
9.074 |
|
■ |
55 |
9.822 |
9.059 |
8.391 |
60 |
8.704 |
8.105 |
7-57^ |
|
65 |
7-417 |
6.980 |
6.5^5 |
|
|
70 |
5-936 |
5-652 |
5-391 |
15 |
14-574 |
12. i^6o |
11.478 |
|
20 |
14.225 |
12.593 |
11.266 |
|
25 |
13. 822 |
12.281 |
11.022 |
|
30 |
13-359 |
II. 921 |
10.736 |
|
35 |
12.824 |
II. 501 |
10.402 |
|
15 |
40 |
12.207 |
II.0I3 |
10.008 |
45 |
11.496 |
10.440 |
9-541 |
|
50 |
10.675 |
9-7^7 |
8.985 |
|
55 |
9.727 |
^■975 |
8.318 |
|
bo |
8.632 |
8.041 |
7-515 |
|
65 |
7-377 |
6-934 |
6.544 |
|
70 |
5-932 |
5.623 |
5-364 |
A |
P P E N D I X. ; |
|||
TAB |
L E Vir. Cor |
itinued. |
||
u |
u |
ro , |
■* . |
<o . |
-5« |
-c |
«5 |
to 0 |
" s |
o ^ |
°-s |
JO |
?e U |
"1 ^ |
fijD O |
0 ~ |
(M 4> |
% ^ |
|
< >-• |
< 20 |
> ^ |
> ^ |
1 1.067 |
13-904 |
12.341 |
|||
25 |
13-531 |
12.051 |
10.840 |
|
50 |
13.098 |
11.71 I |
10.565 |
|
35 |
12.594 |
11.314 |
10.278 |
|
40 |
12.C08 |
10.847 |
9.870 |
|
20 |
45 |
11-325 |
10.297 |
. 9.420 |
50 |
10.536 |
9.648 |
8.880 |
|
55 |
9.617 |
S.Syg |
8-233 |
|
60 |
8.549 |
y.g6y |
7-44« |
|
65 |
7.308 |
6.S82 |
6-495 |
|
70 |
5.868 |
5-590 |
5-333 . |
|
25 |
13.192 |
11.786 |
10.621 |
|
30 |
12.794 |
11.468 |
10.367 |
|
35 |
12.333 |
11.095 |
10.067 |
|
40 |
11.776 |
10.655 |
9.708 |
|
25 |
45 |
II. 130 |
10.131 |
9.278 |
50 |
10.374 |
9-509 |
8.761 |
|
55 |
9.488 |
S.yb6 |
8.134 |
|
60 |
8.452 |
7.8S0 |
7'37i |
|
65 |
7.241 |
6.826 |
6.440 |
|
70 30 |
5.826 |
5-551 |
5.294 |
|
12.434 |
11.182 |
J0.133 |
||
35 |
12.010 |
10.838 |
9.S54 |
|
40 |
11.502 |
10.428 |
9-514 |
|
45 |
10.898 |
9-93^ |
9.1 12 |
|
30 |
50 |
10.183 |
9'345 |
8.620 |
55 |
9-33S |
8.634 |
8.018 |
|
60 |
S.33S |
7779 |
7.280 |
|
65 |
y.ibi |
6.748 |
^-373 |
|
70 |
5-777 |
5.505 |
5.254.. |
323
Y 2
324
APPENDIX.
TABLE Vir. Continued.
u |
CO . |
Tj- . |
«^ . |
|
o c |
§0 |
« g |
II |
|
eo 5 |
35 |
« 0 |
{> ex. |
►SS. |
11.632 |
10.530 |
9.600 |
||
40 |
II. 175 |
10.157 |
9.291 |
|
35 |
45 |
10.622 |
9.702 |
8.913 |
50 |
9-955 |
9.149 |
8.450 |
|
55 |
9.156 |
8.476 |
7.879 |
|
60 |
8.202 |
7.658 |
7.172 |
|
65 |
7.066 |
6.662 |
6.294 |
|
70 40 |
5.718 |
5-450 |
5.203 |
|
10.777 |
9.826 |
9.014 |
||
45 |
10.283 |
9.418 |
8.671 |
|
40 |
50 |
9.677 |
8. 911 |
8.244 |
55 |
8.936 |
8.283 |
7.710 |
|
60 |
8.038 |
7.510 |
7.039 |
|
65 |
6.951 |
6.556 |
6.198 |
|
|
70 45 |
5.646 |
5-383 |
5.141 |
9.863 |
9.063 |
8.370 |
||
50 |
9-331 |
8.619 |
7-987 |
|
45 |
55 |
8.662 |
8.044 |
7.500 |
60 |
7-831 |
7-332 |
6.S75 |
|
65 |
6.807 |
6.425 |
6.080 |
|
70 50 |
5-55<^ |
5-300 |
5.063 |
|
8.892 |
8.235 |
7.660 |
||
55 |
8.312 |
7-738 |
7.230 |
|
50 |
60 |
7.56S |
7.091 |
6.664 |
65 |
6.623 |
6.258 |
5.926 |
|
70 55 |
5.442 |
5-'93 |
4.964 |
|
7.8^9 |
7-33^ |
6.873 |
||
55 |
60 |
7.220 |
6.781 |
6.386 |
^•5 |
6.379 |
6.036 |
5.724 |
|
1 70 |
5.291 |
5-053 |
4.833 |
APPENDIX. 32^
TABLE VII. Continued.
•5<ti 'Si H 60 65 70 |
u ^« < 60 65 70 65 70 70 |
n g SO |
so |
Value at 5 per Cent. |
^'737 6.043 5.081 |
6-351 5-73"^ 4.858 |
6.001 4-653 |
||
5-547 4-773 |
5-^77 4-57^ |
5.031 4-385 |
||
4.270 |
4.104 |
3-952 |
T A-
336
APPENDIX.
TABLE VIII.
Shewing the Probability of the Duration of Life in London, deduced by Mr. Simpfo'4 from obfer- vations on the bills of mortality in London for I o years, from 1728 to 1737.
1 |
PerfiMis |
Deer. |
Per Ions |
Deer. |
A 1 |
I'trion- |
Uecr. |
|
Ages 1 |
living. |
of Life. |
Ages. 27 |
living. < |
jf Life. |
Ages.j 54 |
living |
,f Life. |
0 |
1000 |
320 |
321 |
6 |
^35 |
6 |
||
I |
680 |
^33 |
28 |
3^5 |
7 |
55 |
129 |
6 |
2 |
547 |
51 |
29 |
308 |
7 |
56 |
123 |
6 |
0 |
496 |
27 |
30 |
301 |
7 |
57 |
117 |
5 |
4 |
469 |
J7 |
31 |
294 |
7 |
58 |
112 |
5 |
5 |
452 |
12 |
32 |
287 |
7 |
59 |
107 |
5 |
6 |
440 |
10 |
33 |
280 |
7 |
60 |
102 |
5 |
7 |
430 |
s |
34 |
273 |
7 |
61 |
97 |
5 |
8 |
422 |
7 |
35 |
266 |
7 |
62 |
92 |
5 |
9 |
415 |
5 |
36 |
259 |
7 |
63 |
87 |
5 |
10 |
410 |
5 |
37 |
252 |
7 |
64 |
82 |
5 |
' II |
405 |
5 |
38 |
245 |
8 |
65 |
77 |
5 |
12 |
400 |
5 |
39 |
237 |
8 |
66 |
72 |
5 |
13 |
395 |
5 |
40 |
229 |
7 |
67 |
67 |
5 |
. 14 |
390 |
5 |
41 |
222 |
8 |
68 |
62 |
4 |
15 |
385 |
5 |
42 |
214 |
8 |
69 |
58 |
4 |
16 |
380 |
5 |
43 |
206 |
7 |
70 |
54 |
4 |
17 |
375 |
5 |
44 |
199 |
7 |
71 |
50 |
4 |
iS |
370 |
5 |
45 |
192 |
7 |
72 |
46 |
4 |
19 |
3^5 |
5 |
46 |
185 |
7 |
73 |
42 |
3 |
20 |
360 |
5 |
47 |
178 |
7 |
74 |
39 |
3 |
21 |
355 |
^ |
48 |
171 |
6 |
75 |
36 |
3 |
22 |
35^ |
5 |
49 |
165 |
6 |
76 |
33 |
3 |
23 |
345 |
6 |
50 |
^59 |
6 |
77 |
30 |
3 |
24 |
339 |
6 |
51 |
^53 |
6 |
78 |
27 |
2 |
25 |
323 |
6 |
52 |
147 |
6 |
79 |
25 |
|
26 |
3^7 |
6 |
53 |
141 |
6 |
APPENDIX.
327
TABLE IX.
Shewing the ExpeBations of Life in London, ac- cording to the preceding Table. See Mr. Simp- fort's Se(e£i Exercifes, p. 255.
( Age. |
ExpeflatioD. |
Age. 28 |
£xpe£lation. |
\ Age. |
Expectation. |
I |
27.0 |
24.6 |
55 |
14.2 |
|
2 |
32.0 |
29 |
24.1 |
56 |
13-8 , |
3 |
34.0 |
30 |
23.6 |
57 |
13-4 |
4 |
35-^ |
31 |
23.1 |
5« |
13-1 |
5 |
36.0 |
32 |
22.7 |
59 |
12.7 |
6 |
36.0 |
33 |
22.3 |
60 |
12.4 |
7 |
35-8 |
34 |
21.9 |
6r |
12.0 |
8 |
35'^ |
35 |
21.5 |
62 |
II. 6 |
9 |
35-2 |
36 |
21. 1 |
63 |
II. 2 |
10 |
34-8 |
37 |
20.7 |
64 |
10.8 |
II |
34'3 |
3^^^ |
20.3 |
65 |
10.5 |
12 |
33-7 |
39 |
19.9 |
66 |
10. 1 |
13 |
33'"^ |
40 |
19.6 |
67 |
.j.^ |
H |
3^-5 |
41 |
19.2 |
68 |
9,4 |
15 |
31-9 |
42 |
1S.8 |
69 |
q.I |
16 |
31-3 |
43 |
18.5 |
70 |
8.8 |
17 |
307 |
44 |
iS.i |
71 |
8.4 |
18 |
30.1 |
45 |
17.8 |
72 |
8.1 |
^9 |
29.5 |
46 |
17-4 |
73 |
7.8 |
20 |
28.9 |
47 |
17.0 |
74 |
7-5 |
2l |
28.3 |
48 |
16.7 |
75 |
7.2 |
22 |
27.7 |
49 |
16.3 |
76 |
6.^ |
23 |
27.2 |
50 |
16.0. ! |
77 |
6.4 |
24 |
26.6 |
51 |
15.6 |
7« |
6.0 . |
25 |
26.1 |
52 |
15.2 |
79 |
5-5 • |
26 |
25.6 |
53 |
149 |
80 |
5.0 |
27 |
25.1 |
54 |
^^'5 |
i |
Y 4
128
APPENDIX.
TABLE X.
Shewing the Value of an Annuity on One Life, ac- cording to the probabilities of life in London.
See Mr. Simpfon's Sek£l Exercifes, p. 260.
« c
1 ""
ii <-^
>^ 15
18. 8 16.2114.1
18.9116.3114.2
19.0 16.4 14.3'
19. 016.414.3 19.016.414.31
II I9.016.41I4.3'
12 18.9I16.3J14.2
I3!i8.7'i6.2|i4.i
14J18.5J16.014.0
?^ rt
31 14.8
32 14.6
33H-4 3414.2
3514-1
3613-9 13713-7 3^13-5 391I3-3 40113.2
2 ^
12.9 12.7 12.6 12.4 12.3
i6'i8.i 15.6J13.7
17', 1 7-9; 1 5 ■41 1 3- 5 1817.615.2,13.4
I9,i7.4|i5. 013.2
20'l7'.2 14.8 I3.C
II. 4
"•3
II. 2!
II. o
10.9
3 b
K?* «
12. 1 10-8
II. 9 10.6 11.8 10,5
II. 6 10.4
11.5 10.3
56 1 0.1
57 9-9
58! 9.6
59 9-4
60 9-2
9.1
8.9
8.7 8.6
8.4
3 b
^ rt
8.4
8.2
8.1
8.0
7-9
21 17.0 14.7
22 16.8 14.5 2316.5I14.3 241 1 6.3 14 25 16.1 14.0
4ri3.o]ii.4|id.2J
42I12.8
43!i2.6
I4'i2.5
4512.3
12.7 12.6 12.4 12.3
I 1. 2110.1 I I.IIIO.© II. O
10.8
461 1 2.1
47! 1 1.9
48 1 1.8
49 1 1. 6
50 II. 4
10.7 10.5 10.4 10.2
10. I
9.9 9.8
61 62
63 64
65
66
67 68
69
70
8.9
8-7 8.5
8.3
8.0
7-9 7-7 7-5
7-7 7.6
7-4 7-3 7-1
7.8
7.6
7-4|
7-1
6.9
26i5.9'i3.8 27 i5.6[i3.6
28] 1 5-4 13-4 29,15.213.2
30 15-0, '3-^
12.1 12.0 II. 8 11.7 II. 6
II. 2
II. o
10.7 54'io.5|
5510-31
9-9 9.8
9.6
9.4
9-3
9-7 9-5 9-4 93
Q.2
71
7^1 73 74'
75i
67 6.5 6.2
9.0 8.9 8.8 8.6 8.5
7-3 7-1 6.9 6.7 6.5
6-3 6.1
5-9
6.9
6.7 6.6 6.4. 6.2
5-91 5-^ 5.6I 5.4
6.0 5.8 5-6 5-4 5.2
APPENDIX.
329
TABLE XI.
Shewing the Value of an Annuity on the joint con- tinuance of IVo Lives, according to the proba- bilities of life in London. See Mr. Simpfon's Sele£l Exercifes, p. 266.
u . V 1 m 1 |
* \ "^ 1 ' |
u ,0 1 rr 1 |
■^ |
ir, |
||||||
U 3 |
J3 S2 |
So |
til 0 &£J |
eof th Ideft. lueat Cent. |
3 0 |
Jo |
||||
ba 0 |
to V |
(4 li |
fei) 3 |
00 0 ™ r; |
rg >>4 |
|||||
< ^ |
< |
> s. |
> s. |
> s. |
< ^ |
< > a. |
»i-3 |
lO.I |
||
10 |
14.7 |
13.0 II. 6 |
20 12.8 |
|||||||
15 |
14-31 |
12.7J |
II-3 |
25 12.2 |
10.8 |
97 |
||||
20 |
13.8 |
12.2 10.8 |
30 II. 6 |
10.3 |
9.2 |
|||||
25 |
I3-I |
II. 6 10.2 |
35 10.9 |
9.8 |
8.8 |
|||||
30 |
12.3 |
10.9 1 |
9-7 |
40 |
10.2 |
9.2 |
8.4 |
|||
10 |
35 |
II. 5 |
10.2! 1 |
9.1 |
20 |
45 |
9-5 |
8.6 |
7-9 |
|
40 |
10.7 |
9.6 |
8.6 |
50 |
8.8 |
8.0 |
7-4 |
|||
45 |
10.0 |
9.0 |
8.1 |
55 |
8.1 |
7-5 |
6.9 |
|||
50 |
9-3 |
8.4 |
7.6 |
60 |
7-4 |
6.9 |
6.4 |
|||
55 |
8.6 |
7.8 |
7-1 |
65 |
6.7 |
6.3 |
5-9 |
|||
60 |
7.8 |
7.2 |
6.6 |
70 |
6.0 |
57 |
5-4 |
|||
65 70 |
6.9 6.1 |
6.5 58 |
6.1 5-5 |
75 |
5.2 |
5.0 |
4.8 |
|||
i |
||||||||||
75 |
5-3 |
5.1 |
4-9 |
25 30 35 |
IX. 8 II. 3 10.7 |
10.5 10. 1 |
9.4 9.0 8.^ |
|||
15 |
13-9 |
12.3 |
II.O |
g.6 |
||||||
20 |
^3-3 |
II. 8 10.5 |
40 |
10. 0 |
9.1 |
8.2 |
||||
25 |
12.6 |
1 1.2 lO.I |
45 |
9.4 |
8.5 |
7.8 |
||||
30 |
II. 9 |
10.6 |
9-5 |
25 |
50 |
8.7 |
7-9 |
7-3 |
||
35 |
II. 2 |
10,0 |
, 9-0 |
55 |
8.0 |
7-4 |
6.8 |
|||
40 |
10.4 |
9.4 |
8.5 |
60 |
7-3 |
6.8 |
6.3 |
|||
15 |
45 |
96 |
8.8 |
8.0 |
65 |
66 |
6.2 |
5-8 |
||
50 |
8.9 |
8.2 |
7-5 |
70 |
5-9 |
5-6 |
5-3 |
|||
55 |
8.2 |
7.6 |
7.0 |
75 |
5-1 |
4.9 |
47 |
|||
60 |
7-5 |
7.0 |
6.5 |
|||||||
65 |
6.8 |
6.4 |
6.0 |
30 |
10.8 |
9. 61 8.6 |
||||
70 |
6.0 |
5-7 |
5-4 |
30 |
35 10.3 |
9.2 8.3 |
||||
75 |
5.2 |
5.0! 4-8 |
40 9.7 |
1 S.8l 8.o| |
53'^
APPENDIX.
TABLE XL Continued.
V |
l-« |
f^ . |
x^ |
4- A^ |
- c: |
•^ ^ |
n C |
« c |
|
o ta |
i;:i |
%U |
M ^ |
Jo |
<s. |
<:"■ |
> s. |
> ^ |
> £L |
45 |
9.1 |
8.:; |
7.6 |
|
50 |
^•5 |
7.8 |
7.2 |
|
55 |
7-9 |
.7-3 |
6.7 |
|
30 |
60 |
7.2 |
6.7 |
6,2 |
65 |
6.5 |
6.1 |
5-7 |
|
70 |
5.0 |
5-5 |
5.2 |
|
75 |
5-1 |
4.9 |
4-7 |
|
35 |
9-9 |
8.8 |
8.0 |
|
40 |
9.4 |
«-5 |
7-7 |
|
45 |
8.9 |
8.1 |
7-4 |
|
50 |
8.3 |
7.6 |
7.0 |
|
35 |
55 |
7-7 |
7-1 |
6.6 |
60 |
7-1 |
6.5 |
6.1 |
|
65 |
6.4 |
6.0 |
5-6 |
|
70 |
5-7 |
5-4 |
5-1 |
|
75 |
5.0 |
4.8 |
4.6 |
|
40 |
9.1 |
8.1 |
7-3 |
|
45 |
H.y |
7.8 |
7-1 |
|
50 |
8.2 |
7-4 |
6.8 |
|
40 |
55 |
7.6 |
69 |
64 |
60 |
7.0 |
6.4 |
6.0 |
|
•35 |
6.4 |
59 |
5-5 |
|
70 |
5-7 |
5-A- |
5-1 |
|
75 |
5.0 |
4.8 |
4.6 |
|
45 |
8-3 |
7-4 |
6.7 |
|
45 |
50 |
7-9 |
7-1 |
6.5 |
55 |
"7 /I / T |
6.7 |
6.2 |
|
60 |
6.S |
6.3 |
5.8 |
•5 *■ ;■
45
65 70
6.3 5.6
75 |
1 4.9 |
|
50 |
i 7.6 |
|
55 |
7.2 |
|
50 |
60 |
6.7 |
65 |
6.2 |
|
70 |
5-5 |
|
75 |
4.8 |
|
55 |
6.9 |
|
60 |
6.^ |
|
55 |
65 |
6.0 |
70 |
5.4 |
|
75 |
4-7 |
|
60 |
6.1 |
|
60 |
65 |
5-7 |
70 |
5-2 |
|
75 |
4-6 5-4 |
|
65 |
||
65 |
70 |
4.9 |
75 |
4.4 4.6 |
|
70 |
70 |
|
75 |
42 |
|
75 |
75 |
3-8 |
> s.
5.8
5-3
4-7
6.8
6.5 6.1
5-7 5.2
if
6.2
5-9
5.6
5-1
4-:;
5-6
5-3 4.9
4.4
5-0 4.6
4.2
4.4 4.0
3-7
54- 5-0 4.5
6.2 6.0
5-7' 5-3 4-9 4-4' |
5-7 5-5 5.2 4.8 4-3 |
5'2 4-9 4.6 4.2 |
4-7 4.4 4.0 |
4.2 3-9 |
3.6 |
APPENDIX.
33^
TABLE XII.
Shewing the Probabilities of Life in London, on the fuppofition, that all who die in London were born there. Formed from the Bills, for ID years, from 1759 to 1768. Sec page 245.
Perfons |
Deer. |
Perfons |
Deer. |
' Ages 62 |
Perfons |
Deer. ( |
||
Ages 0 |
living. |
of Life. |
Ages 3^ |
tivinj);. .^04 |
of Life. |
living. |
of Life.' |
|
1000 |
240 |
9 |
132 |
7 |
||||
I |
760 |
99 |
32 |
395 |
9 |
;63 |
125 |
7 |
2 |
661 |
42 |
33 |
386 |
9 |
64 |
118 |
7 |
3 |
619 |
29 |
34 |
377 |
9 |
'65 |
III |
7 |
4 |
59^ |
21 |
35 |
36S |
9 |
66 |
104 |
7 |
5 |
569 |
II |
36 |
359 |
9 |
67 |
97 |
7 |
6, |
55^ |
10 |
37 |
350 |
9 |
68 |
90 |
7 |
7* |
548 |
7 |
38 |
341 |
9 |
'6g |
^3 |
7 |
8 |
541 |
6 |
39 |
332 |
10 |
70 |
76 |
6 |
9 |
535 |
5 |
40 |
322 |
10 |
i7i |
70 |
6 |
10 |
53^ |
4 |
41 |
312 |
10 |
172 |
64 |
6 |
II |
526 |
4 |
42 |
302 |
10 |
h^^ |
58 |
5 |
12 |
522 |
4 |
43 |
292 |
10 |
174 |
53 |
5 |
13 |
51S |
3 |
44 |
282 |
10 |
|75 |
48 |
5 |
14. |
51.5 |
3 |
45 |
272 |
10 |
:76 |
43 |
5 |
15 |
512 |
0 0 |
46 |
262 |
10 |
i77 |
3^ |
5 |
16 |
509 |
3 |
47 |
252 |
10 |
j78- |
0 0 0 J |
4 |
17 |
506 |
3 |
48 |
242 |
9 |
'79 |
29 |
4 |
18 |
5^3 |
4 |
49 |
233 |
9 |
1 80 |
25 |
3 |
19 |
499 |
5 |
50 |
224 |
9 |
jSi |
0 .'-> |
3 |
20 |
494 |
7 |
51 |
215 ^ |
9 |
1 Si |
^9 |
3 |
21 |
487 |
8 |
52 |
206 |
8 |
83 |
16 |
3 |
22 |
479 |
8 |
53 |
198 |
8 |
84 |
13 |
2 |
471 |
8 |
54 |
190 |
7 |
85 |
II |
2 |
|
7 1 |
4'^3 |
8 |
55 |
183 |
7 |
86 |
9 |
2 |
25 |
455 |
8 |
56 |
176 |
7 |
87 |
7 |
2 |
26 |
447 |
8 |
57 |
169 |
7 |
8S |
5 |
I |
27 |
43 Q |
8 |
58 |
162 |
7 |
89 |
4 |
r |
28 |
4-3 i |
9 |
59 |
^55 |
8 |
90 |
3 |
I |
29 |
422 |
9 |
60 |
H7 |
8 |
|||
30 |
413 |
9 ' |
61 |
139 |
7 |
^z^
APPENDIX.
TABLE Xlir.
Shewing the true Probabilities of Life in London till the Age of 19. See page 249.
Age. |
Perfons liv- |
Decrements ' |
0 |
ing. |
ofjife. 240 |
75^ |
||
I |
510 |
99 |
2 |
411 |
42 |
3 |
3% |
29 |
4 |
340 |
21 |
5 |
319 |
II |
6 |
308 |
10 |
»7 / |
298 |
7 |
8 |
291 |
6 |
9 |
285 |
5 |
10 |
280 |
4 |
II |
276 |
4 |
12 |
272 |
4 |
13 |
268 |
3 |
14 |
265 |
3 |
15 |
262 |
1 3 |
16 |
259 |
3 |
17 |
256 |
3 |
18 |
'2-53 |
4 . |
19 |
249 |
|
20 |
494 |
|
21 |
487 |
|
&c. |
&c. |
The numbers in the fecond column to be con- tinued as in the lall Table.
333
APPENDIX.
TABLE XIV.
Shewing the true Probabilities of Life in Londow for all Ages. Formed from the bills for lo years, from 1759 to 1768. See page 251.
Perfons 1 |
Deer, |
. Perfons 1 |
Deer. |
; Perfons |
Deer. |
|||
Ages, |
living. |
of Life. |
Ages. 31 |
living. |
of Life. |
Ages. 62 |
' living. |
of Life. |
O |
1518 |
486 |
404 |
9 |
132 |
7 |
||
I |
1032 |
200 |
32 |
395 |
9 |
63 |
125 |
7 |
2 |
832 |
85 |
33 |
386 |
9 |
64 |
118 |
7 |
3 |
747 |
59 |
34 |
377 |
9 |
65 |
III |
? |
4 |
688 |
42 |
35 |
368 |
9 |
66 |
104 |
7 |
5 |
646 |
23 |
36 |
359 |
9 |
67 |
97 |
7 |
6 |
623 |
20 |
37 |
350 |
9 |
68 |
90 |
7 |
7 |
603 |
H |
38 |
341 |
9 |
69 |
83 |
7 |
8 |
589 |
12 |
39 |
332 |
lO |
70 |
76 |
6 |
9 |
577 |
10 |
40 |
322 |
10 |
71 |
70 |
6 |
lO |
5^7 |
9 |
41 |
312 |
10 |
72 |
64 |
6 |
II |
55^ |
9 |
42 |
302 |
10 |
73 |
58 |
5 |
12 |
549 |
8 |
43 |
292 |
lO |
74 |
53 |
5 |
i3 |
541 |
7 |
44 |
282 |
10 |
75 |
48 |
5 |
H |
534 |
6 |
45 |
272 |
lO |
76 |
43 |
5 |
15 |
528 |
6 |
46 |
262 |
jo |
77 |
38 |
5 |
i6 |
522 |
7 |
47 |
252 |
lO |
7S |
33 |
4 |
'7 |
5^5 |
7 |
48 |
242 |
9 |
79 |
29 |
4 |
i8 |
508 |
7 |
49 |
233 |
9 |
80 |
25 |
3 |
'9 |
501 |
7 |
50 |
224 |
9 |
81 |
22 |
3 |
20 |
494 |
7 |
51 |
215 |
9 |
82 |
19 |
3 |
21 |
487 |
8 |
52 |
206 |
8 |
83 |
16 |
3 |
22 |
479 |
8 |
53 |
198 |
8 |
84 |
13 |
2 |
23 |
471 |
8 |
54 |
190 |
7 |
S5 |
1 1 |
2 |
24 |
463 |
8 |
55 |
1^3 |
7 |
'86 |
9 |
2 |
25 |
455 |
8 |
56 |
176 |
7 |
187 |
7 |
2 |
26 |
447 |
8 |
57 |
169 |
7 |
i88 |
5 |
I |
27 |
439 |
8 |
58 |
162 |
7 |
139 |
4 |
I |
28 |
431 |
9 |
59 |
^55 |
8 |
90 |
3 |
1 |
29 |
422 |
9 |
60 |
H7 |
8 |
! 1 |
||
jO |
413 |
9 |
61 |
^9 |
7 |
1 1 |
334
APPENDIX.
OBSERVATIONS
ON TABLES I. and II.
^ I ^'HESE Tables may be met with in J[ moft of the books that treat of com- pound intereft and annuities j but there has been, in this work, fo much occafion for re- ferring to them, that it was necefTary to fave the reader the trouble of turning to other books for them.
The 2d, 3d, 4th, &c. numbers in the Second Table, are only the fums of the firft 2, 3, 4, &c. numbers in the Firjl Table. This Table, therefore, is the foundation of the Second.', and, indeed, of all the common tables of compound interefl j and, with the help of it, almoft all the queflions in com- pound intereft may be eafily anfwered.
The following fpecimen of this may, I think, be of confiderable ufe.
Question I. " To what fum or annuity *' will any given fum or annuity, now to be " laid up for improvement, at a given rate *^ of compound intereft, increafe, in a given " number of years ?"
5 An^
APPENDIX. 335
Answer. Divide the given fum or an- nuity by the value of i/. payable at the end of the given number of years, and the quo- tient will be the anfvver.
Example. Let the given fum be 50 /. and the given time 18 years. The rate of intereft ^ per cent. — The prefent value, at \ per cent. of i/. payable at the end of 18 years is, by Table I. .4936; and 50/. divided by this value, gives/. 101.296, or 101/. ^s. \.\\Qfu?nlo which 50 /. will increafe in 1 8 years. In like manner; 2 I. per anttum, the firfl payment of which is to be made a year hence, will increafe (intereft fuppofed the fame) in 18 years, to an annuity of /.4.05 : for 2/. the given annuity, divided by .4936, gives /. 4.05, or 4/. I J-.
Question II. " To what fiun will a " given annuity amount, in confequence of ** being forborn and improved, at a given ** rate of compound intereft, for a given ** number of years ?"
Answer. From the hicreafed annuity, found by the laft Queftion, fubtradt the givefi annuity ; and multiply the remainder by the PERPETUITY, and the product \\\\\ be the anfwer.
Example. 2/. per ann. Improved at 4 per cent, compound intereft, will, by the laft Qiieftion, increafe, in 18 years; to l.^.o^per ann. 2/. fubtradted from 4.05, leaves 2.05,
which.
336 appendix:*
%vhich, multiplied by 25, th^ perpetuity, gives /. 51.25, or 51/. 5J-, the amount in 18 years. In the lame manner it may be found, that \ol. per ann. (intereft being the fame) will amount, in 41 years, to 998/.
It fhould be remembered, that the per- petuity is 33.33> — 28.57,-25 — 20,— or 16.666, according as intereft is reckoned at 3, — 31,-4, — 5 or 6 per cent : And that the annuity meant in all thefe Queftions is an annuity, the firii: payment of which is to be made at the end of a year.
Question III. *' In what number of ** years will a given fum or annuity increafe " to another givtnfum or annuity, in confe- *' quence of being improved at a given rate *' of intereft r"
Answer. Divide the original fum or ari'^ nuity by the increafed fum or annuity ; and look for the quotient, or the number neareft to it, in Table I ; and the number of years correfponding to it will be the anfwer.
Example. Let ih^fum be 50/. The in- creafed fum /. 101.29. The rate of intereft, /^per cent. The former fum divided by the latter gives .49 '^6, which ftands oppofite in the Table to 1 8 years, or the time in which
50/. will gain the required increafe. In
like manner, it may be found, that 18 years is the time in which 2/. p'er ann. will increafe to /.4.05 per ann.
Question
APPENDIX. -^17
Question IV. " In what time will any ** given f /z 72 z///y amount to a given fum^ in " confequence of being forborn and im- *' proved, at a given rate of compound in- " tereft ?"
Ansv/er. Divide the g\wQ.nfum to which the annuity muft amount by the perpetu- ity. -Add the given annuity to the quo- tient ', and by the quotient fo increafed, di- vide the given annuity; and this fecond quo- tienty found in Table I. will fhevv the an- fwer.
Example. A perfon owes iooo/. and re- folves to appropriate lo/. -per annum of his income towards difcharging it. In what time will fuch an appropriation, in confe- quence of being improved at 4 per ce?2t.
amount to a fum equal to the debt ?
1000/. divided by 25 gives 40/. 10/. added to 40/. makes 50/i and 10/. divided by 50/, gives .20C0, which in the Table flands op- pofite to 41 years, the required time.
In the fame manner it will appear, that the fame annuity, if improved at 5 per cent. will amount to 1000/. in 37 years.
Question V. ^' In what time will a '* given principal be annihilated, by taking ** out of it, at the end of a year, a given fum, *' and after that, the fame fum annually, to- ** gether with its growing interefts ?"
Z An-'
338 APPENDIX.
Answer. In the fame time plainly in which an equal annuity would amount to the given principal.
A perfon, therefore, poflefs'd of 1000/. capital, bearing interefl at 4 per cent, would, by Queftion IV. reduce it to nothing in 41 years, by taking out of it 10/. the firft year, and as much more every following year, as would be ncceffary, together with the intereft of the remaining capital, to make his annual income conftantly 50/.
Remark. The fum to which a given an- nuity \vill amount in a given time, is the fame with the value of an annuity for the given time, equal to the given annuity in- creafed by the yearly intereft of the amount. That is, 1000/. is the value of 50/. per ann. for 41 years, at 4 per cent : And the fame fum is iikewife the value of 60/. per anminti for 37 years, at 5 per cent. The reafon is plain: 1000/. it has appeared, would, in confequence of being put out to thefe dif- ferent rates of intereft, be juft fufficient to pay the annuities.
I have been the more explicit in thefe rules, becaufe they point out a very eafy method of deducing and examining all I have faid, in different parts of this work, and particu- larly in Chap. III. concerning the increafe of
money at intereft. 1 will jud mention one
inltance.
400,000/,
APPENDIX. 339
400,000/. per annum i applied in the man- ner fuppofed in Queflions IV and V. would annihilate 55 millions, bearing intereft at 5 fer cent, in 42 years.
In 1716, when tho. Jinking fund \v2.s cfla- blifhed, the public debts were near this fum, and bore 5 per cent, intereft. This fund then, had but 400,000/. of it been inviolably ap- plied to the annihilation of the public debts, would, in iyfS, have dilcharged all the debts contradled before 1716. — - — And it may be further found very eafily, by the anfvver to Queftion IV. that had it been fuffered to go on in its operation, and been applied, Jince 1758, to the redemption of only 3 per cents at par, it would by this time have difcharged 104 millions; and feven years hence, 140
millions. The aiTertion, therefore, in
page 164, is ftridly true. But the following proof of that aflertion will, perhaps, be more clear and ftrikin?.
Suppofe an annuity of 400^000/, beginning in 1716, to have been applied till 1730, to the annihilation of debts bearing intereft at 5 percent', from 1730 to 1748, to the anni- hilation of debts bearing intereft at ^per cent 6 and from 1748 to 1771, to the annihilation ot debts bearing intereft at 3 per cent. In the firft of thele oeriods the annuity would have incrcafed to 8co,coo/. ; in the /(fr:?;;<3', to
1, 600,000/. ; in the laft, to 3,200,000/. *
In the prefent year, therefore, the nation
/j 7. might
-wLO APPENDIX.
might have been eafed of above t/jree 77nUions per annum in taxes. And, at the fame time, (fuppofmg all the fame meafures taken in other refpeds) it would have enjoyed that very Jinking fund it now has ; and no detri- ment could have arifen to the public, from any applications of it to current expences.
As I am now again on the national debt, I will beg leave to add the following proof of another obfervation on this fubjedl-, in page 140.
The diiburfements, on account of any loans, will be the fame, whatever different interefls they bear, fuppoiing a provifion made for difcharging the principal, by applying to that purpofe furpluiTes bearing to one another the fame proportions with thofe interefts.
For Example. Let a million be borrowed at 3 per cent, and let a fund be charged with it. which brings in 6j". per cent, per annum more than the interefiij or 3-^^,000/. inftead ot 30,000/. per anmifn. This furplu?, ap- plied in the manner I have explained, will annihilate the principal in 81 years, as may be gathered from Quellion V. At the end of this time, the difnurfements on account of the loan will be 33,000/. multiplied by 8i j that is, 2,673,000/.
Let us again fuppofe a million borrowed at 6 per cent, and let a fund be char.o-ed with It producing a furplus of 12 s. per cent, per
annum.
appendix:. 341
anmitn. Such a fund, befides paying the in- tcreft, will difcharge the principal in 41 years ; and the difburfements on account of the k)an, will be 66,000/. multiplied by 41; that is, 2,706,000/. or very nearly the fame with the difburfements on account of an equal loan at 3 per cent.
It appears, therefore, agreeably to the ob- fervation to which I have referred, that were the public, in railing money, to adopt the plan I have propofed, it would be of little confequence what intereft was given" for money. The pradicability of fuch a plan is felf-evident, for it cannot be lefs eafy to apply the intereft of a fum, to the payment of a debt,, than the fum itfelf : and this plan requires no more. — One particular advantage attending it, already hinted, I will beg leave here to repeat. By keeping \\\QftQcks fteadily at or near^^r, that flu^fluation in them would be in a great meafure prevented, which now produces fo many evils; and which, with the aid of annual lotteries, will, I fear, in time, ruin all honefl: induftry, and turn us into a nation of fliarpers and gamblers.
D I-
^42 APPENDIX.
Directions for finding the Va--^ LUES of Two Joint Lives of given Ages, by Table VII.
F both the ages are given in the Table^ the value v/anted Vv'^ill be found imme- diately by infpedion.
If the ages are not given in the; Table, it will be beft to proceed in the following manner. Suppofe the rate of intereft 4 per cent, and
the propofed ages 40 and 66. It will
appear, from infpecfling the Table, that the value fought would be 6.556, were the age of the elder life 65; and 5.383, were it 70. Since, therefore, it is 66, the value muft be the Jirjl of four arithmetical means between
6.556 and 5.383, or 6.322. For the
fame reafon, had the ages of the elder been 68, the value would have been the 3d arith- metical mean between 6.556 and 5.383, or
5.854.- In like manner, were the propofed
ages 43 and 65, the value Vv^ould be the 3d arithmetical mean between 6.556 (the value of two joint lives v/hofe ages are 40 and 65) and 6.425, (the value of two joint lives wiiofe ages are 45 and 65) or 6.478.
Again, let the ages be 43 and 66. That is, let it be fuppofed, that neither of the propofed ages is given in the Table.
The values correfponding to the ages C407 , C667 C6.3227
i^sS 166S' Z6.2GQS
The
APPENDIX. 343
The value, therefore, correfponding to the ages 43 and 66, muft be the 3d mean be- jtwcen 6.322 and 6.200, or 6.250.
A^. B. The I ft, 2d, 3d, and 4th of four arithmetical means between two numbers are found by fubtraCting 4> t> t> ^nd ± of the difference between the two numbers, from the great ejl of them.
Thus. The difference between 6.556, and 5.383, is 1.173. One fifth of this dif- ference is .234; which, fubtraded from 6.556, leaves 6.3223 the firft of 4 means between 6.556 and 5.383.— —In like man- ner i the difference between 6.322 and 6.200 is .122. One fifth of this difference is .024; and, therefore, three-fifths of this difference is .072, which, fubtradied from 6.322, leaves 6.250, the //6/r^/ arithmetical mean between 6.322 and 6.200.
In order to avoid trouble, if the ages are nearly equal, a year or two may be added to the lead, and as much fubtracited from tlie greateft ; and the value taken by infpcvftlon. But if one of them much exceeds the other, it v/ill in general be fufficient to take the peareft number in the Table for the leffer.
The mean between the values at 3 per cent, and 4 fer cent, may be taken for the value at 3I: per cent, without any error of confequence. And the like may be faid of the values at 4I per cent.
The values of the longeji of two lives Is
found by fubtrading the value of the joint
4 lives
544 APPENDIX.
lives from the fiim of the values of the two
fingle lives. -Thus, the values of two
iiDgle lives, v/hofe ages are 25 and 30, are by Table VI. (intereft reckoned at 4 per cent.) 15.31 and 14.68. The fum of thefe two values is 29.99 * ^^ value of the joint lives is (by Table VII) 11.46; and this va- lue, fubtrailed from 29.99, gives 18.53, qr the value of an annuity on the longeft of the two lives.
The value of two joint lives being given, the value of three ]o\nt lives may be found by the following rule, taken from Mr. Simpfons Sele5l Exercijesy page 279.
Let A be the youngeft, and C the olded of the three propofed lives. Take the value of the two joint lives B and C, and find the age of a Jingle life D of the fime value. Then find the value of the joint lives A and D, which Vk'iil be the anfwer.
Example. Let the three given ages be 25, 30, and 40, and let the rate of interell be A^ per cent. Then the value of the two oldeil: joint lives B and C, will (by Tab. VII.) be 10.428, anfwering, in Tab. VI. to a fingle life D of 54 years of age. And the value of the joint lives A and D, which is 8.917 year's purchafe, will be the value fought.
FINIS, .^
\
4
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