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JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact email@example.com. 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.