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THE
ASSUEANCE MAGAZINE,
AND
JOURNAL
OF THE
INSTITUTE OF ACTUARIES.
On the Improvement of Life Contingency Calculation. By Edwin
James Farren, Esq., one of the Vice Presidents of the Institute
of Actuaries.
[Read before the Institute 8th January, 1855, and ordered by the Council to
be printed.]
J.HE prevailing system of life contingency calculation is one not
of variable but of invariable quantities. At the very threshold the
admission of two such important assumptions is asked for, as that
•the rate of mortality is always invariable at the same age, whether
old or young, and that the rate of interest is equally invariable for
all periods, whether long or short. Upon these admissions of in-
variability a system is formed for assessing the relative values of
different cases, thereby necessarily in every instance indicating an
invariable answer; and with such indications the system rests
content. Whether such assessments, however logically fair in
connection with agreed postulates of invariability, are themselves
eventually justified by the same invariability of actual result as was
a priori assumed, has not hitherto been commonly brought within
the general scope of the actuary's studies. Directly, however, he
is called on to take upon himself the practical responsibility of
upholding this theory of invariability, he is somewhat surprised to
find that, good as the mere logic of his studies may have been, it
is by no means an easy task to connect such logic with the nature
of the events he may see passing around him. In place of a
definite rate of mortality at each age, he may find a perfect series
vol. v. o
186 On the Improvement of [April
of such rates — in place' of one uniform rate of interest, he may
find one portion of the funds yielding no interest whatever, while
other portions may he lent upon classes of security, fruitful indeed
in interest, but not as equally fruitful in obvious expedients for
realization of the principal, should such be desirable. The con-
viction is thus forced upon him at a very early period of his career,
that, as a practical man, he must either henceforth shut his eyes
altogether to the prevailing theory of the subject, or else open his
eyes much wider than as a student he was taught to expect would
be necessary in the application of rules professedly based upon
invariable elements.
I am aware it may be said, that although the rates both of
mortality and interest, as used by actuaries under the prevailing
system, are certainly specific rates, yet that they are to be con-
sidered as averages, and therefore typical, as well in theory as in
practice, of a diversity of rates. But I would ask, has the subject
ever been so treated? Can we find, for instance, in the treatises
of those who are termed by common consent, and therefore rightly
termed, our standard writers, not merely a single chapter, but even
a single page, on the proper calculation and employment of average
results, as applicable to insurance transactions? So long then as
this omission exists, so long I think we may without presumption
assert that the prevailing system of life contingency calculation is
susceptible of considerable improvement. But in alluding to this
omission, I particularly wish to guard myself from the inference
that I thereby desire to stigmatize the labours of such men as
Price, Morgan, Baily, and Milne, or of our late excellent cotem-
porary David Jones, as not worthy of the favourable reception they
have received. On the contrary, I believe (and it is one of the
objects of the present paper to illustrate such belief) that the style
of treatment adopted by these pioneers of our literature constituted
the only style that was really applicable to the nature of insurance,
as then publicly understood. Indeed, in all branches of natural
philosophy we find a similar assumption of invariability at an early
period of their history, as preparatory to the more comprehensive
study of deviations required by modern science. One of these
branches presents us with so close a parallelism to the line of pro-
gress I wish to illustrate as advisable in insurance calculations,
that I cannot consider it wholly a digression to allude to it even in
some detail.
In the early history of navigation, we find it taken almost as
the basis of the science that the compass needle pointed in a fixed
1855.] Life Contingency Calculation. 187
direction, and that such direction was due north. The utility of so
simple an assumption in early days can scarcely be overrated; the
more especially as it was by this simplicity then rendering itself
acceptable that the real properties of the magnet have been reserved
for more elaborate study in our own time. Indeed, had variability
in place of invariability been originally taken for its prevailing
attribute, the important uses of the mariner's compass would
probably have been lost to modern science, inasmuch as our sim-
plicity-loving ancestors might have considered such indications too
uncertain and complex either to be useful to themselves or to their
posterity. In further illustration of this, we find that for a long
time the simplicity of such a theory of invariability completely
overrode the nature of the actual facts. Gradually, however, the
proverbial stubbornness of facts developed itself in this as in other
sciences, and brought about the admission that the compass needle
did not really and in fact point due north. There was as yet,
notwithstanding, no absolute surrender of the principle of fixity of
direction itself; such would have been too sudden a change either
to have been expected or even desired. A variety of adjustments
were accordingly advocated, each claiming some favourite point as
that of the real fixity. The correct theory, nevertheless (started, I
believe, by Gunter, and now admitted by all conversant with the
subject, because the only theory that can interpret all the facts), is,
that the direction of the compass needle is variable even in the
same localities, and must be so apprehended by mariners, if safety
based upon truth, and not -merely upon simplicity of theory, be
their aim. It is, then, by the known variability of the compass
needle, and not by a pseudo invariability, that the triumphs of
modern over ancient navigation have been achieved.
The use I seek to make of this as a parallelism in illustration
of the subject before us is, I presume, sufficiently obvious. I con-
sider the formerly universal adoption of the Northampton Table
and 3 per cent, as typified by the assumption that the compass
needle pointed fixedly due north, and the various petty controversies
for other fixed points as equally typified by the various pros and
cons for the Carlisle and other tables. Further, that the true
theory, in this as in the former subject, is one strictly of variation,
both as to mortality and interest, and that it must be so accepted
as the only guide to safe practice, if we would avoid those rocks
and shoals which a purblind adherence to a fixed in place of a
variable course might unpreparedly develope. Indeed, the distinc-
tion between the proper treatment of variable and invariable ele-
o 2
188 On the Improvement of [April
ments is precisely the distinction that characterizes the vocation of
an actuary as compared with that of an accountant. Thus the
actuary who should take probabilities, because fairly assessed now,
as necessary certainties hereafter, would be virtually an accountant,
because he allows no range for the possibly conflicting evidences of
the future. The accountant, on the other hand, who endeavours
to put estimates upon fluctuating things to come, is virtually striv-
ing to be an actuary ; for he cannot but allow that no estimate of
to-day, unless professedly subject to variation, can pretend to also
fulfil the condition of being an equally good estimate for a change-
able to-morrow. In some Offices, I believe, this distinction between
fixed and variable estimates is already sufficiently carried out — in
the first case, by the actual amount of assets at one period, as
compared with the actual amount at another, being illustrated by
Dr. and Cr. after the manner of accountants ; and in the second,
by the difference between the amount of the life valuation at one
period and another being substantiated, actuary-wise, by taking
into consideration the accrued contingencies of the past as compared
with the range of contingencies to be provided against for the
future. The actuaries, then, of what I shall venture to call the
old school, were essentially accountants in the modern sense, for it
was only with fixed quantities they professed to deal, as is suffi-
ciently proved by their assuming a fixity when they found it not.
The results of such a system have been exactly those to be
expected. Where exorbitancy secretly existed, as in the rates
required by the Insurance Societies, there the errors of a fixed and
affected precision eventually came to light, in the shape of bonuses
added to the sums assured, in varying amounts from time to time,
in strange contrast with the declared formality of the original
fixedness of calculation. Where no such exorbitancy of charge
was allowable or even possible, and the fixed calculations had to
stand or fall by their own merits, there the dangers of professing
to deal with variable quantities as if they were absolutely invariable
were unfortunately not so easily neutralized. To the numerous
Friendly and Annuity Societies of the last century throughout the
country, a reliance on fixed tables and on such tables alone has at
once proved a delusion and a snare ; for it tempted them to appro-
priate the temporary surplus of a day, in the vain expectation that
the fixed nature of the tabular values assigned to the future would
necessarily be sufficient guarantee for the fulfilment of impending
engagements.
The nature, then, of the improvement I seek to introduce into
1855.] Life Contingency Calculation. 18&
life contingency calculation is to openly take as our guide not
merely a calculus of averages, but of their fluctuations; and to
thereby declaredly characterize our methods, not as composing a
system of specific and precise results, whatever it may be of prices,
but of results expected to vary between limits of assigned ranges
of probability. By such a declaration it would at once become
manifest that our expected gain by computation would not be to
find even averages themselves invariable, but that their fluctuations,
being considerably less, would therefore be more readily dealt with
than the fluctuations of the elements of which they may be com-
posed. The phrases therefore of the prevailing system implying
" a true table," or " a true rate of interest," would under such a
calculus have to give way to average tables, with their probable
limits and the per centages of their expected deviations.
What experienced actuary, for instance, can read without feel-
ing the truth of the following reflection, extracted from the article
on " Probability" in the Encyclopedia Metropolitana : —
" Not being well able to decide upon the relative importance of small
details, calculators on this subject (life contingencies) have hitherto judici-
ously presented their results such as they ought to be if the tables were
mathematically exact, and to the nearest farthing. But more extended
views on the subject of probabilities, and on the nature of observations in
general, would have caused the time which has been wasted in carrying out
annuities to many more decimals than the data are good for, to be employed
in apportioning the risks of fluctuation by estimation of the mean risks of
the tables."
Or the following, from the equally excellent article on "Proba-
bility" in the Encyclopedia Britannica : —
" We may remark that, although English writers have 'almost without
exception confined themselves to the explanation of the methods of com-
puting annuity tables and of determining from them the values of sums
depending on life contingencies, the aid which this branch of economy
derives from the general theory of probabilities is by no means confined to
the consideration of such elementary questions. The number of observa-
tions necessary to inspire confidence in the tables, the extent to which
risks may be safely undertaken, the comparative weights of different sets of
observations, and the probable limits of departure from the average results
of previous observations in a given number of future instances, are all
questions of the utmost importance, which come within the scope of the
calculus, and cannot in fact be justly appreciated by any other means."
In the concluding part of the latter extract we have, indeed,
the real explanation of the formal and what I may venture to
call the " wooden" cast that has been given to the subject by our
standard writers already referred to : for we are to remember that,
190 On the Improvement of [April
though complete masters of their art as then understood, yet that
they were all teachers or disciples of a school and of a day when
the differential and integral calculus was but little employed by
English writers on any branch of science. The omission of such
processes has now, however, become the exception and not the rule.
Thus, for instance, if we look into any of our modern treatises on
mechanics, engineering, or navigation, all of them essentially prac-
tical subjects, we find every aid that the calculus can give or has a
chance of giving sedulously pressed into the service. By these
means the great discovery of Newton and Leibnitz is brought
home — vicariously indeed, but still effectually — to the uses of the
humblest mechanic, engineer, or mariner, whenever he has to avail
himself of what can be done for him, by way of previous calcula-
tion, in guiding him to the simplest and most trustworthy results.
Indeed, the modern improvements of the Nautical Almanack alone
form at once a sufficient and striking illustration of what benefit
can be achieved by the calculus in devising the best forms for
practical computation. Whatever therefore may have been the
opinion of our elder school of writers, I think the time has now
come for our students when, as in other subjects, the more search-
ing investigations of the calculus should also be brought to openly
and commonly bear upon that of life contingencies.
Indeed, without this or some other extra aid, bow is it possible
for us to intelligibly explain to a modern public those differences
of results in various Offices, which, when judged by a hypothetical
standard of invariability, appear rather to proclaim the failure of
all methods whatever, than to justify the indications of any par-
ticular one? So long as this diversity remains unexplained by
having no proper limits assigned to it, so long assuredly may
any amount of diversity appear justifiable to boards of manage-
ment, and actuaries continue to be exposed to the risk of having
their opinions only treated with respect when not obstructive of
other money arrangements. That the calculus, especially con-
sidered as a calculus of averages, contains within itself the means
of dealing with and explaining these diversities, has been too often
asserted both by continental and native writers on probability, to
be strengthened by mere reassertion on my part. But as I am not
aware that any very ready example, in a professional sense, has
been given of the sort of assistance to be derived by actuaries from
this calculus, when treated as a calculus of averages, I shall beg
leave to conclude this paper by offering at least one such illustra-
tion, hoping it may prove an incentive to other actuaries to look
1855.] Life Contingency Calculation. 191
farther into the subject than perhaps they have hitherto done.
Before, however, giving such an example, I should wish to state
that I have purposely selected such an one as will show that I by
no means pretend, as a practical man, that a more general study of
the differential and integral calculus by actuaries would materially
alter the external appearance of insurance results and rates, as
at present accepted by the public. On the contrary, I believe that
no actuarial theorizing would or ought to induce the public to be
otherwise than mainly led by their own experience of the past,
already somewhat extensive, and every day becoming more and
more patent to themselves. But there is considerable difference,
in a professional point of view, between venturing on general
assumptions, however plausible, and the cautious adoption of
approximations based upon elaborate investigations. Were then
the calculus capable of no more than pointing out to us convenient
approximations, and referring us to its own processes for its justi-
fication of them, it would still, I think, be" an ally obviously well
worthy of the actuary's seeking. It is to illustrate the calculus
in this character that has decided the kind of example I have
selected.
Example. — A hundred pounds has to be put out at compound
interest for twenty years, at rates indefinitely fluctuating between
3 and 4 per cent, per annum. What is the general average of all
the possible sums, even to infinity, to which the hundred pounds
may be thus made to amount ?
Putting this into the form of a definite integral, we have
which, when »i=100, a=3, 5=100, c=l, and w=20, as in the
case before us, becomes innl9 ( ^ J = 199'273l, which
is the general average amount required.
Having thus determined what would appear as the more recon-
dite question of the average amount of a sum at fluctuating rates
of interest, it may be well, in order to show the ductility of the
calculus when studied as an extensive system of averages, to also
determine by its means the more simple question of what is the
average rate of interest between 3 and 4 per cent., so obviously
determinable by other means as 3£ ? This extra illustration, how-
ever simple, is considered advisable; because there may be many
minds, even in our own profession, so framed, that it is only by
192 On the Improvement of [April
treating well known examples, having obvious solutions by the
current methods, that the reliability of any new method of solution
is considered admissible by them in more difficult cases. To deter-
mine, then, the average rate of interest between 3 and 4 per cent,
by means of the calculus, we have to consider the definite integral of
xdx = ;
42 — 32
which, when a =3, becomes — s — =3|; in exact equality with
the obvious result of sheer mental arithmetic. I have already
hinted that the first example is purposely chosen as one suscep-
tible of an easy approximation, and such has just been portrayed :
for if £100 be invested at 3J per cent, per annum for twenty years
throughout, it will amount to £198-9789, a close approximation
to the general average, or £199-2731, as determined by the cal-
culus. But should we therefore be justified in saying that the
£100 must necessarily be considered as having to be invested at
3£ per cent, per annum throughout ? Decidedly not ; and the less
so, because all the supposed advantages of such a misstatement
are more readily obtained by adhering to the scientific truth, and
saying that the proposed calculation, being really one of indefinite
fluctuation, has been accordingly so dealt with, and the general
average ascertained to be £199-2731; without, however, guarantee-
ing either that or any other as the precise result, that experience
alone can determine. It is moreover manifest, that 3J as a rate of
interest could not be connected as such with the average amount
at the end of the term, for it is as obvious by common arithmetic
as it is by the calculus that the accumulations between 3J and 4,
considered as fixed quantities, would more than counterbalance
those between 3 and 3^ : and hence, whether 3£ would have to
be considered as affording a good or bad approximation is not
matter for assumption, but for demonstration ; and it is precisely
these demonstrations that are beyond the reach of the common
methods.
The instances I have given will, I think, sufficiently portray,
so far as isolated instances can do, the nature of the improvement
I am advocating in life contingency calculations. It will be seen
that, though I seek to deprive the prevailing system of its pre-
tensions to an invariability that does not really belong to it, yet
that at the same time I propose a similar equivalent, by the
adoption of the calculus and its limits, to that which has already
1855.] Life Contingency Calculation. 193
been accepted; in place of a similarly false invariability, in other
branches of natural philosophy. It is true, indeed, that before
being exactly adapted to our wants as actuaries, the calculus must
be moulded into one of averages ; but this is a transformation so
legitimate, that I consider no better method of studying the cal-
culus exists even for the more general mathematical student.
In concluding this paper, I am perfectly well aware it is con-
sidered by many as dangerous, in an official and commercial sense,
for any actuary td show he has been studying other books and
productions than directories, prospectuses, and advertisements ;
but I trust that a better spirit, is beginning to prevail, and that,
within the walls of this Institute at least, any advocacy for the
improvement of the theory of our subject will be immediately seen
as also implying a desire to improve its practical aspects. Speak-
ing for myself, I have long considered that the wants of the public
are daily forcing upon actuaries the investigation of subjects which
the incompleteness of the prevailing theory renders it too powerless
to sufficiently grapple with ; and it is the hope of exciting atten-
tion to this view of the matter that has induced me to offer the
present paper. Considerable difficulty, as may be imagined from
its tenor, has been felt in keeping it within due bounds ; for, had
examples of general limits been chosen, the subject in this form
appears to be at present so little understood in its practical bear-
ings, that at least the range of a lengthened essay, if not of a
volume, might have been required to treat the matter with that
fulness of illustration which the importance of it demands. It
may therefore be allowable for me to attempt to reinforce the
object of so circumscribed a paper by a general declaration on my
part that, after having devoted considerable attention, and indeed
some years, to the subject, I feel confident the proposed change
from an invariable to a variable calculus as the basis of our
calculations will be beneficial in every respect. We shall thereby
be able to wholly dismiss the ancient doctrine of chances, with its
fixed equalities of paper cards, wooden dice, and similar mechanical
illustrations, and rely upon the more modern doctrine of pro-
bability, as the science of observation based upon experience. The
actuarial adaptations of this doctrine, aided by the calculus, will
assuredly ultimately bring a class of problems involving averages
and their fluctuations within reach of our solutions which at pre-
sent are merely statistically guessed at, even by the most experi-
enced actuary, the most cautious finance minister, or the most
learned political economist. To improve our own science, more-
194 On the Improvement of [April
over, is virtually a step towards the improvement of others, and
thereby the better helps us to substantiate the claims of our stu-
dies to those designations of learned and liberal so duly prized by
other professions.
Postscript. — The writer is glad to avail himself of the interval
between the reading and printing of his paper to state, that he does
not by such paper claim for his views — as might perhaps hastily,
without this disclaimer, be inferred — the merit of perfect originality
as regards the proposed improvement of life contingency calculation.
Lacroix has long since glanced at the differential and integral cal-
culus as essentially a calculus of averages, and the calculus itself
has already been often employed in connection with life contin-
gencies in England, as by De Moivre, Waring, Young, Gompertz,
Lubbock, De Morgan, Galloway, Edmonds, and indeed by the
present writer himself, in his last publication on Life Contingency
Tables. It is necessary further to remark, in the same spirit, that
even the terms " true table " and " true rate of interest/' though
commonly used in the prevailing system, have also been frequently
associated with the notion of a margin for fluctuations, or accom-
panied with the qualification that it is only by neglecting variations
that the epithet of" true " becomes allowable, and that, if it be pro-
posed to include such variations, that modification of the ordinary
language should ensue. Reference may be made to Mr. Jellicoe's
paper, in Vol I. p. 172 of the present Journal, for instances of this.
It is, then, Tather to excite renewed attention to the subject of
variability, than to propose it as wholly new, that has been the
writer's real aim ; and he has accordingly treated the matter in the
preceding paper in that mixed style of pleading and demonstration,
as appearing to him the most suitable for such a purpose in its
more general form. As however it may aid the illustrations already
given in the paper itself, if a tabular form be presented, and may
also tend to better satisfy many minds to whom tabular forms are
more acceptable than even the most earnest disquisitions upon
principles, such a table is now appended. It may be taken as a
temporary specimen of the proposed improved manner of dealing
with such subjects — without, however, the writer's wishing such
table to be understood as having the exact form, even in his own
opinion, that may ultimately be best adapted for the purpose
under consideration.
1855.]
IAfe Contingency Calculation.
195
Table of the Average Amounts of £100 at Fluctuating
Bates of Interest.
Average Amounts of £100 at Compound Interest from 1 to 100
years, at rates fluctuating between the limits ofO and 6 per cent.
per annum.
N.B. The maximum rate of interest is taken at 6, rather than at 5, per cent.; because
5 per cent., if payable by half yearly, quarterly, monthly, or smaller instalments, can be
made to exceed 5 per cent, per annum.
Term
Term
Term
Term
of
Average
of
Average.
of
Average
of
Average
Years.
Amount.
Years.
Amount
Years.
Amount.
Years.
Amount.
1
1030000
26
235-9473
51
631-3104
76
1901-0467
2
106-1200
27
244-7433
52
658-4496
77
1990-5568
3
1093654
28
2539303
53
686-8833
78
2084-5473
4
112-7419
29
263-5273
54
716-6764
79
2183-2499
5
116-2553
30
273-5538
55
747-8971
80
2286-9085
6
119-9120
31
284-0306
56
780-6170
81
2395-7800
7
123-7183
32
294-9793
57
8149117
82
25101350
8
127-6813
33
306-4428
58
850-8605
83
2630-2580
9
131-8080
34
318-3851
59
888-5470
84
2756-4491
10
136-1058
35
330-8913
60
9280588
85
2889-0240
11
140-5829
36
343-9679
61
969-4884
86
3028-3153
12
145-2472
37
357-6426
62
1012-9330
87
3174-6731
13
150-1076
38
371-9448
63
1058-4948
88
3328-4664
14
155-1731
39
386-9049
64
1106-2813
89
34900835
15
160-4533
40
402-5553
65
1156-4058
90
3659-9337
16
165-9581
41
4189299
66
1208-9873
91
3838-4479
17
171-6981
42
436-0641
67
1264-1512
92
40260800
18
177-6842
43
453-9955
68
13220292
93
4223-3082
19
183-9280
44
472-7634
69
1382-7602
94
4430-6361
20
190-4416
45
492-4090
70
1446-4903
95
4648-5943
21
197-2377
46
512-9758
71
1513-3730
96
4877-7418
22
204-3297
47
534-5094
72
1583-5703
97
5118-6675
23
211-7316
48
557-0580
73
1657-2523
98
5371-9917
24
219-4580
49
580-6718
74
1734-5982
99
5638-3681
25
227-5245
50
605-4040
75
1815-7968
100
5918-4853
Example. — The average amount of all the amounts possible,
even to infinity, to which £100 can be made to accumulate in
twenty years, at rates of interest fluctuating between and 6 per
cent, per annum, is £190*4416 ; as may be seen set forth in the
table opposite to the term of 20 years.
When it is remembered that money is more likely to remain
unproductive, or at per cent., for short than for long periods, it is
obvious that the relative effect of unproductiveness must be more
operative when considering brief than enlarged cycles of finance.
The preceding table, by its averages, properly represents this effect
among others ; and shows that while on the one hand the average
amount at the end of the first period or year from the original
times of the deposits may be taken as sufficiently defined by the
common mean rate of interest between the limits, or in the present
196 On the Improvement of Life Contingency Calculation. [April
case by 3 per cent., yet that,- on the other hand, the period of a
century is allied by its average amount, in connection with the
same limits, to the amount productive by a uniform rate through-
out of about 4 per cent. (4'165). To deal with such wide limits
as from to 6, and with such durations as a century, is obviously
to strain the calculus to its utmost; but even in this extreme
state it will be seen to keep closely attendant upon the incidents
of practical insurance, for it certainly appears consistent with even
popular justice that, as a matter of calculation, those who may
remain longest insured should be also rated as those to be re-
latively assigned the higher ratios in the general appropriation
of accumulations of interest.
The table has been virtually calculated by the aid of the same
definite integral as that given in the paper, viz. —
in which n varies by units from 1 to 100 ; a varies indefinitely to
and from a+g, or from to 6; J = 100; m=100; c=l; and
g = 6. Or, putting such expressions into the form of a rule, we
have the following extremely simple one whenever is the lower
limit : —
Subtract £100 from its amount, when improved for the whole
term and one year beyond, at the maximum rate of interest con-
sidered as uniform as by the common tables. Divide the remainder
by the product of such maximum rate and the number of years
including the year beyond, and the quotient will be the average
amount required.
Example for Twenty Years. — £100, put out at 6 per cent, per
annum uniform interest for twenty years and one year beyond, will
amount by the ordinary tables (Smart's) to £339 - 95636; from
which if the £100 be subtracted, the remainder is £239*95636 •
which, divided by 6 times 20 and the year beyond, or 126, gives a
quotient of £190'4416, which is. the average amount required
between the assigned limits in conformity with the result as given
by the table in question.
The reasons for the trustworthiness of the rule can of course
only be explained by aid of the calculus itself, or by some allied
process of reasoning which it would be here out of place to
dilate upon.