You are reading a page from An Essay on Probabilities and their Application to Life Contingencies and Insurance Offices, Augustus de Morgan (1838)
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Term Life Insurance
268    ESSAY ON PROBABILITIeS.
is accompanied by this inconvenience, that the measures adopted, whether of precaution or remedy, may be made to press unequally upon the different classes of insurers. If we take, for instance, a fixed rate of interest, sufficiently below that which can really be obtained, we find that many of those insured must pay their premiums at a time when interest is comparatively higher, and vice versa. With regard to the tables of mortality, most probably (it has always so happened) a table which is generally too high will be unequally too high ; so that some classes of insurers will contribute more largely to the safety fund than others. And even in the distribution of the profits, however good the will may be to apportion them duly, there are yet practical difficulties in selecting an equitable method out of those which do not require calculations of insupportable minuteness.
It will only here be necessary to dwell upon two points, the distribution of the premiums, and the method of appropriating the profits.
In the last chapter, in speaking of the use of too high a table of mortality, as a safeguard, I was merely considering the collective security of the office. There are two different ways of answering the same end : either by using a table of mortality confessedly too high, or constructing premiums from a true table of mortality, and increasing these by such a percentage as will pro-duce the same receipts to the office. For general security, these two plans are equally good ; but they may produce very different consequences upon the relative state of the members. For instance, the Northampton table, which is the basis of most of those now in use, is certainly, as already noticed, too favourable to the older lives. Mr. Morgan gives the following table *, exhibiting the numwho did die, and those who should have died, if the Northampton table had been correct, all in the twelve years preceding 1828.
* View of the Rise and Progress of the Equitable Society, London, I£;!9, page 42.
MANAGEMENT OF AN INSURANCE OFFICE. 269
    Age.    Number.    Of whom did die.    Of whom should have died
    .    h
    20 — 30    4,720    29    68
    30—40    15,951    106    243
    40—50    27,072    201    506
    50 — 60    23,307    339    545
    60—70    14,705    426    502
    70—80    5,0.56    289    290
80 — 95    701    99    94
        
From this comparison, Mr. Morgan concluded that the superior vitality of the young and middle ages was the effect of selection, which wore out, so to speak, after the age at which no new members were admitted; thereby proving, in his opinion, at once the effect of selection, and the excellence of the Northampton table. Now, it obviously cannot prove both of these things : granting the latter, it would certainly go a great way to prove the former ; and granting the former, it does not impugn the latter : which is all that can be said. But, if it should happen that the mortality of the Northampton table is near the truth at the older ages, and very much above it at the younger, the sort of result shown in the preceding comparison would follow of course ; and this circumstance, demonstrated as it is by other and independent tables, is, no doubt, the true explanation.
If such be the case, where is the fairness of using a table which demands premiums very much larger than the real risks from the young, while it admits Older lives on more easy terms? Ought the older lives to enjoy any privilege in this respect ? Quite the reverse ; for, (page 253.) belonging to a class which is less known, and entering also in smallcr numbers, with results there-fore more subject to fluctuation, the percentage, added to the premiums deduced from a true table, ought rather to be larger in the case of old lives than in that of young ones. The best customers, both in number and quality, ought not to come worst off.
270    eSSAY ON PROBABILITIES.
The proposed table of Mr. Finlaison (page 259) affords a striking illustration of this point. It is accompanied by a table representing the average premiums of all the offices. At the age of thirty, Mr. Finlaison proposes to demand 17 per cent. less than the average of what is now asked by the offices ; at the age of 60, this same able and strenuous advocate of reduction would only reduce the average premium of the offices by 3 per cent. I now put down the present value of 1001., pay-able at the end of the year in which a life drops, from the Northampton and Carlisle tables, at 3 per cent., and for different ages, together with the percentage which must be taken from the former to reduce it to the latter.
    
Age.    Northampton.    Carlisle.    Percentage of
            difference.
    20    X. 42.8    £ 33.9    20.8
    30    47.8    40.0    16.1
    40    53.8    47.1    12.5
    45    57.2    50.8    11'2
    50    60.9    55.4    9.0
    55    64.6    60.9    5.7
    60    68.6    66.5    3.1
    65    72.9    71.1    2.5
In offices, then, which continue to use the Northamptable throughout, the safety rate is levied upon those who enter at the age of 20, to the amount of 21 per cent. out of the total sum they pay; while on those aged 65 it only amounts to 2,1 per cent. The Carlisle table represents the experience of the Equitable Society very nearly.
Again, the Amicable Society now charges premiums deduced from its own experience, and in which the fundamental inequality of the Northampton table is corrected. It will be worth while to compare the aver-age of all the offices given by Mr. Finlaison, with the
MANAGEMENT OP AN INSURANCe OFFICe. 271
present premiums charged by the Amicable. The sup-position is for 1001. insured.
    Age.    Average.    Amicable.    Mr. F.'s pro
            posed Premium..
    20    42'02    £2.03    a~ 1 r6
    30    2.50    2.53    2'07
    40    3.26    3.25    2.78
    50    4.47    4.83    4.06
    55    5.38    5.90    5.00
    60    6.58    7.33    6.25
From such comparisons as the preceding, I have long been of opinion that, safe as the offices are, each considered as a whole, the proportions of the premiums demanded at different ages are, in the first instance, inequitable. To a certain extent, the young are made to work for the old ; that is to say, the person who insures early in life, the more prudent of the two, is made to pay a part of the premium of the one who does not begin till he is old.
The evil is not so great as it might at first sight appear, for two reasons : firstly, because those who enter at the older ages are few in number compared with those who begin between 30 and 50 years of age ; secondly, because many offices make compensation to the younger members in the division of the profits. Still, however, the inequality is of a sufficient magnitude to demand alteration, which will be brought about in an obvious way ; namely, by the younger insurers giving the preference to those offices in which, premiums and returns considered together, the inequality is the least.
There is another point, though not of so much consequence, in which an inequality falls more heavily upon the young than upon the old ; namely, the method of paying the expenses of management. The yearly contribution of every member to this fund ought to be the same. Suppose, then, that from every premium a given sum is subtracted, to answer this end, the in-
27    ESSAY ON PROBABILITIES.
equality of the remainders is increased ; it being obvious that any disproportion which exists between two numbers is made larger by taking away the same from both.
The way to correct the inequality, without altering the actual receipts of the office, is as follows. The pro-portions in which the different ages exist in the office at any one time can be pretty nearly found. Let the office table of premiums be taken, and from it let an average premium be formed, by taking into account as well the several premiums, as the numbers who pay them, Suppose, for instance, that A persons pay the premium a, B pay b, &c. &c. ; then the average premium is found by dividing the sum of the products of A and a, B and b, &c., by the sum of A, B, &c. Let the actual average premium be called P ; and let the average premium, formed in the same manner from a true table of mortality (in which a, b, &c. are different, but A, B, &c. the same as before), be Q. Let P exceed Q by k per cent. of Q ; then the premiums given by the true table, increased by k per cent., are those which should be substituted for the existing premiums, in order that all inequalities may be corrected, without diminishing the receipts of the office. It matters nothing, in the pre-ceding rule, whether the premiums of what has been called the true table are correct or not, so long as their proportions are correct; and one office might, by this rule, adopt the proportions of another, without altering its own receipts.
If such a process as the preceding were performed, deducting from the receipts required by the office the whole expense of management, and afterwards adding the last-mentioned item in equal shares to all the polithe distribution of the premiums would be theoretically perfect. It remains to consider the more difficult part of the question,—the method of dividing the profits.
Hitherto, I have had no occasion to speak of a most important difference of system which distinguishes one office from another ; the distinction of mutual and proThe former have no capital, except what arises
MANAGEMENT OF AN INSURANCe OFFICE.    273
from their own accumulations, and each member is a guarantee to the rest for the fulfilment of all engagements. If the office possess a charter, this guarantee operates no further than to pledge the premiums already paid by any member for the discharge of all claims which arise be-fore his own, since a corporation is considered in law as an individual. If, on the other hand, there be no charter, the whole fortune of every member is pledged for the discharge of all claims. The risk, however, at the comis not great in character, and small in amount; and the quantity of risk diminishes so much faster than the amount increases, that it may safely be said there is nothing in the commercial world which approaches, even remotely, to the security of a well established and prumanaged insurance office.
A proprietary insurance office has a capital, the proprietors of which may or may not be insured in the office, and for which such a bonus is paid, in adto the market rate of interest, as is mentioned in p.263. It would perhaps be difficult, at the pre-sent time, to establish a new proprietary office with a very large capital. The public now begins to see that much capital is not necessary, and that nearly all the bonus which is paid for its use is so much taken away from the savings of the insured, without any adequate benefit received in return. One by one, the prooffices must (as some have done) admit the insured to a share in the profits: the necessity for which will be taught by the decline of business, if not previously learnt.
The question as to how profits should be divided, is of the same nature in both species of offices; the difference being, that the offices which are partly proprietary have less to distribute among the insured than those which are mutual. The first inquiry must be, What is the profit of an insurance office; and how is the amount to be ascertained? Firstly, as to the profit which an insurance office may be expected to realise, judging by the premiums they receive, and the mortality they have
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274    ESSAY ON PROBABILITIES.
hitherto experienced. Certain limits may be obtained which may sometimes serve as a useful check.
Perhaps the average age of admission to an insurance office is about 40, as many entering younger as older. The average premium charged by the offices at that age is 3.261. per cent. Now the most extreme supposition which can be imagined in favour of the insured is, that the Carlisle table should be taken as the law of mortality, and 4 per cent. as the interest of money. Upon these suppositions, the accumulations of the office would amount, upon a premium of 3.261., at the death of parties aged 40, one with another, to 1371. But this pushes cvery favourable supposition to its extreme, and moreover allows nothing for expenses of management. I am inclined to think, however, that the usual prewill, as long as the rate of mortality continues at its present amount, yield about 1251. for 1001. nominally insured, and perhaps something more.
It must not be left out of sight, that the offices consider every person as having the age which he will attain at his next birth-day. If, for instance, a person who attains 40 years of age on the 12th of March were to insure his life on the 13th, he would be said to be 41 years of age, and would have to pay accordingly. The effect of this very proper * regulation is, that, one party with another, all are half a year younger than their office age. Again, all the tables are computed on the supposition that interest is made yearly, whereas in fact it is made quarterly. Circumstances of this sort, trivial as they appear, do nevertheless produce a sensible effect in a large number of years. To the above we must add the profits arising from the purchase of policies, which is always done by the offices on terms very favourable to themselves ; fines for non-payment of premiums; the profits of lapsed policies; and so on.
Leaving all speculation as to the probable profits, I now proceed to show how to ascertain, from the


* Proper as long as there is no subdivision of a year. I think the offices might very rationally divide the year into quarters.
MANAGEMENT OF AN INSURANCE OFFICe. 275
actual statistics of an office, what its real condition is. And here I must observe, that though in the construction of premiums, a table of more than the real mortality must be used, yet no such thing is absolutely necessary in the valuation of its liabilities and assets. Here truth, and not security, is the object ; and if by any means a true table can be obtained, its results should be calculated ; though I do not say that in the declaration of profit, such results should be admitted to their full extent. The most simple theoretical way of conducting the process, is to ascertain the value of every policy, as in page 218.; that is, to ascertain how much should be given to the holder of each policy to renounce his claim, the office also abandoning the future premiums. 1 hen this is done, it is obvious that the office is not solvent, unless the assets arising from the accumulations of former years be sufficient to pay the values of all the policies, and thus to buy them all up. Supposing the office able to do this, with a capital remaining larger than would be necessary to create a permanent fund for the expenses of management, the surplus of that capital is profit. Otherwise, calculate the present value of all premiums due to the office, and also the present value of all claims to which it is liable. To the former add the sum total of the assets of the office, and to the latter add the present value of a perequal to the expenses of management. Thus, let
P = present value of all premiums.
C = present value of all claims.
A = total assets of the office.
M = present value of all expenses of management.
If then P and A together exceed C and M together, the office is solvent, and the excess is profit.
On each of these items a few remarks may be made.
(P.) All the parties who are of the same office age, may have their several policies considered as one collective policy, in respect of which the sum of the premiums is paid as one premium, and the sum of the
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276    ESSAY ON PROBABILITIES.
possible claims is one claim. But as these premiums are payable at all periods of the year, they may be considered as, one with another, due at six months after the valuation, at which time the present office age of the parties may be considered to be their real age.
(C.) All bonuses which have actually been added to policies (if any) must be included in the claims; and the value of each claim must be carefully found, with reference to the time after death at which it is paid. (See Appendix the Second.)
(A.) The principal of the assets must be deduced entirely by means of the income it yields, and must be ascertained from the income by means of the rate of interest assumed. On this subject, which contains a difficulty of a peculiar character, see the Sixth Appendix.
(M.) Against the expenses of management may be set, as far as they go, the incidental profits, when they can be tolerably well ascertained.
The profit being thus found, and that share of it which belongs to the insured (if the office be not mutual), it remains to inquire, What principle of division should be adopted? And, firstly, it may be doubted whether the whole of the profit is immediately divisible, consistently with prudence. To use an astronomical phrase, the increase of the surplus is partly secular, and partly periodic ; that is to say, instead of a steady and uniform increase, there is a fluctuating rate of augcompounded of that permanent rate which the largeness of the premiums necessarily gives, and the alternate accelerations and retardations occasioned by the departures of the incidents of the several years from the average. The only way of obtaining the permanent part of the surplus is by estimating it on the average of a considerable number of past years, regard being had to the relative, not the absolute, surplus. Let us suppose, for instance, that the present value of all claims is ascertained to be one million, and the present value of all premiums 700,0001., the office possessing besides (clear of charges of management) 500,0001.:
MANAGEMENT OF AN INSURANCE OFFICE. 277
there is then a surplus of 200,0001. ; which having been accumulated out of premiums, and profits having been regularly paid up to the present time, it may be presumed that the premiums themselves are capable of maintaining this rate of surplus. The office must then be presumed able to pay 1201. for every 1001. insured.
But it is important to note, that the present rate of profit must not always be assumed as that which can be permanently maintained. Suppose, for instance, an office which begins for the first time to divide profits : its accumulations are therefore, in part, the reserves of profit which should have been added to former claims, had any division of surplus previously existed. The same remark may be necessary when any change is made in the way of dividing profits, since the surplus existing at the moment of the change is the result of a former state of things. Thus, an office which has proceeded injudiciously, in making too large divisions, may possibly, when it adopts a more prudent system, be justified in forming a system which would require a larger surplus than the one which it actually possesses at the time of discovering the error ; for the then existing surplus has been unduly weakened, and is not to be considered as reprethe permanent effect of the improved mode of proceeding.
I have stated, that the percentage which can be added to each 1001. insured should be determined by the average of a number of years. If this number be too great, the incidental fluctuations of mortality may be compensated ; but at the same time the real and secular changes of mortality may be prevented from producing their proper effect. As long as the value of life is in-creasing, too long an average is a defect on the safe side : but if it were diminishing, it might happen that the mean of a number of preceding years would present a higher result than would be consistent with security. As yet the offices have had nothing to encounter except the diminution of mortality, and its consequences ; but
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2 78    ESSAY ON PROBABILITIES.
in constructing rules for their own guidance, they should be careful not to fall into such errors on the safe side as become errors on the wrong side when circumstances change. I hold an opinion which I think, from his writings, was also that of the late Mr. Morgan; namely, that an insurance office must consider the last half century as having been a period of circumstances singularly favourable for the formation and growth of such institutions, more so than it would be wise to expect for the future. Perhaps from five to ten years is the length of time for which the preceding average should be computed.
The valuations should, if possible, be made yearly. No check which can be devised is so likely to be useful as yearly valuation ; and it is absolutely necessary to any system which gives the real amount of their premiums to the insured. In a mutual insurance office, starting without much capital, it would be madness to rest upon any tables and to neglect valuations ; unless, as before remarked, the returns made to the insured are meant to be very much below their payments. And in conjunction with yearly valuations should come yearly divisions of profits, or something equivalent. There is, I believe, a prejudice against frequent divisions in the minds of many who have derived their ideas on the subject from the former practice of offices. But surely, provided that the proper amount of profit be divided yearly, and no more, it matters nothing whether the apportionment he made seven times in seven years, or once only, as far as security is concerned. For it is to be remembered, that yearly division of profits does not imply an annual expenditure, but only an annual distribution of future expenditure. In septennial divisions, one of two things always takes place : either the profits are made contingent upon a party surviving one or more periods of division, which creates great inequalities between the lot of different persons (the very thing an insurance office was intended to avoid); or it declares beforehand, what the profits shall be during periods of seven years. In the latter
MANAGEMENT OF AS INSURANCE OFFICE. 279
case the annual division is unquestionably the more safe; since it is easier to predict the capabilities of one year than of seven.
In writing upon any point connected with insurance, the practice of the Equitable Society naturally suggests itself. Nevertheless, 1 always consider that society as a distinct and anomalous establishment, existing at this moment under circumstances of an unique character. It is the result of an experiment which it was most important to try ; but which having been tried, need not be repeated. Its history is briefly this :—The AmiSociety, which, in the year 1760, was the only one existing, was originally founded rather on principles of mutual benevolence, than of mutual insurance, as now understood. A certain number of persons (the only restriction being that their ages should be between twelve and forty-five), each paying the same sum yearly, the whole fund of each year (or the greater part) was divided among the representatives of those who died within the year.* The Equitable Society was founded upon the principle of apportioning the payments to the risk of life. The tables were constructed by Dodson, who, as Mr. Morgan remarks, "for greater security assumed the probabilities of life in London, during a period of twenty years ; which, including the year 1740, when the mortality was almost equal to that of a plague, rendered such premiums much higher than they ought to have been, even ac-cording to the ordinary probabilities of life in London itself." The truth of this remark will sufficiently appear, from comparing the average of the present office premiums with the original Equitable premiums, as given in the following table. And even these prewere increased on the most frivolous pretexts. Thus female life and young life were considered as more than usually hazardous, and paid for accordingly.
* The Amicable Society now retains only one of its original characters ; namely, that all members, whatever may be their age at death, or the term of their continuance in the society, participate equally in the profits.
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280    eSSAY ON PROBaBILITIES
Equitable 1 Equitable    Average present Premium, 1771. Premium, 1779. Premium.
    £    s.    d.    £    s.    r1.    .£    s.    d.
    14    2    17    0    2    5    5            
    20    3    9    4    2    12    10    2    0    0
    25    3    14    0    3    0    6    2    4    0
    30    3    18    7    3    8    11    2    10    0
    40    4    17    9    4    7    11    3    5    0
    49    6    2    5    5    10    2    4    6    0
Mr. Morgan says, "that for the first twenty years, the society possessed such an excess of income, that being suffered to accumulate without interruption, it contributed, in a great measure, to form the basis of its future opulence." This circumstance, with the great number of policies which were abandoned* in the early stages of its career, and the increase of interest during the war, are quite sufficient to explain the wealth which the Equitable Society has accumulated : to these must be added the parsimony with which, at first, additions were made to the policies. The whole was an experiment, on a graduated scale of premiums, made with a caution, which, though it turned out to be superfluous, could not be known to be such, except by the result. It was at the same time a venture, and by many considered as a hazardous one ; for instance, the law officers of the Crown refused a charter, on account of the lowness of the premiums. 'f he hazard having been run, and having turned out profitably, the proceeds be-long to those who ran it, and to those who, by their own free consent, became their lineal successors. Nor is it the least remarkable circumstance connected with this society, that the immense funds at its disposal have been always opened, though under restrictions, to the public. Though this has been done in a way which renders the participation of the new insurer in the
Perhaps Mr. Morgan's statement on this point may have led to the statement alluded to in page 266.
Age.
MANAGEMENT OF AN INSURANCE OFFICE. 281
previous accumulations a remote contingency, still it is done, and by a body who might without any bar, legal or moral, immediately close their doors, and divide the whole among themselves.
I have made the preceding remarks, in order that it may be clear how little the history or practice of the Equitable Society should have any direct authoritative bearing on the spirit in which the management of a more modern office should be carried on. The general lesson taught by it is,— be cautious ; but, among other things, be cautious of carrying caution so far as to leave a part of your own property for the benefit of those who are in no way related to you. If there be a Charybdis in an insurance office, there is also a Scylla : the mutual insurer, who is too much afraid of dispensing the profits to those who die before him, will have to leave his own share for those who die after him. Reversing the fable of Spenser, we should write upon the door of every mutual office but one, be wary ; but upon that one should be written, be not too wary, and over it, "EquitSociety."
An insurance office has no existence separate from that of its insurers ; and no public duty to fulfil, exto collect, improve, and equalize their premiums (p. 238.) : therefore, their most important object, next to the fulfilment of their guaranteed engagements, is the distribution of their profits in such manner that every one may obtain his due share. The question now becomes, What is the due share of each party ? This is, in some measure, a question of previous contract, though there are those who consider that there must be a right and a wrong way. For instance, Mr. M°Kean, the compiler of the tables alluded to in page 191., and of a useful work * which accompanies them, says, " Our conclusion, and a most important one, lies conspicuous on the very surface. It is impossible that ALL the
" Exposition of the practical Life Tables, &c. London : Butterworth, Richardson, &c. 1837." This work is, I believe, sold separately.
282    eSSAY ON PROBABILITIES.
offices above mentioned can be correct or just in their aws for dividing the surplus. If the plan of the Equitable is right, then most unquestionably the plan of the Atlas is wrong, and great injustice is done to the younger members, and so vice versa. But, is this a state of things in which so important a system as that of life insurance, based, as that system is, on mathematical science, ought or can continue to exist ? Certainly not."
On this I observe, that though life insurance be an application of (not based upon) mathematical science, yet that the entrance of exact numerical reasoning is subsequent to the admission of certain principles, and the experimental acquisition of certain facts. It is not by mathematics we learn that life is uncertain in individual cases, but nearly certain in the mass — that it is the duty of every one to provide for his family — and that this can be done without contingency, if those who survive the average term agree to surrender a part of their substance to those who do not. Calculation will point out the amount which, upon any given principle of division, belongs to one or another of the insured; but before we can come to this point, it must be settled with what intention the surplus was paid ; which may be different in different offices. The following considerations might be addressed to any person who in-tends to insure his life : — You are aware that the predemanded of you is, avowedly, more than has hitherto been found sufficient for the purpose, the reason being, that it is impossible to settle the exact amount, on account of our not knowing whether the future and the past will coincide in giving the same law of mortality, and the same interest of money. The surplus arising from this overcharge, for the future existence of which it is hundreds to one, is now at your own disposal, and you must choose between one office and another, according to your intentions with regard to its ultimate destination. Firstly, if you doubt the general security of the plan of insurance, and are desirous of an absolute guarantee, independently of accumulations from pre-
MANAGEMENT OF AN INSURANCE OFFICE. 283
miums, there are offices which will, in consideration of the surplus aforesaid, pledge their proprietary capitals for the satisfaction of your ultimate demand upon them. Secondly, if, being of the opinion aforesaid, you think the whole surplus too much to pay for the guarantee, there are proprietary offices which retain a part of the profit in consideration of the risk of their capital, and return the remainder. Thirdly, if you wish the surplus premium, as fast as it is proved to be such, to be applied in obviating the necessity of any further over-charges, there are offices which divide the profits during the life of the insured, by means of a reduction of preFourthly, if you wish the surplus to accumulate, and, feeling confidence in your own life, are willing to risk losing it (the surplus, remember) entirely if you die young, on condition of having it proportionally in-creased if you live to be old, there are offices which divide all or most of the profits among old members. Fifthly, if you would prefer a certainty of profit, die when you may, there are offices which at once admit new members who die early to a full participation in all advantages. The choice between these several modes must be made by yourself, according to your own inclinations, views of fairness, or particular circumstances.
There are three modes of division which deserve parnotice; namely, periodical additions to the policies, periodical diminutions of premium, and addition to the policy at death to an amount depending upon the assets of the office, without reference to the time during which the insured has paid premiums. I may, perhaps, be thought to treat this subject with prolixity; notwithstanding, knowing that tins part of the subject has created more discussion of late years than any other, I think an attempt to compare the principles of different plans not out of place.
The considerations which follow will apply to all offices which divide any profits whatever : the inquiry being, not how much surplus should be divided, but in
284    eSSAY ON PROBABILITIeS.
what proportions a given sum should be divided among the insured.
Let us return to the original constitution of an insurance office (page 238.), derived from the statement of its main object; namely, that it is a savings' bank with a power of equalizing those results in which the different durations of life would cause differences. Sup-pose that such an office sets out with premiums imagined to be no more than sufficient, but which are afterwards found to be more than sufficient, leaving an admitted amount of surplus in hand. The first thought would be of restitution; namely, rendering back to each individual the amount which he had bona fide contributed towards the surplus. To do this properly, it must first be settled whether the insurance office is one or many. Does each age insure itself, or do the separate ages in-sure both themselves and each other ? If the premiums were properly proportioned, there would be no occasion to ask this question : but if the incomers of one age pay unduly as compared with those of another, then it is but fair that they should receive in proportion. In the disof premiums, which I have described in p. 270., it is equitable that a remedy should be provided, by virtue of which those who enter the office young should receive more than the rest. And it is, for this reason, desirable that the proportions of the division should be regulated by a true table of mortality.
Let P be the real premium, and P +p the office pre; and let the death of an individual take place after he has been n years insured, and just before the (n + 1)th premium is paid. If the office had been a compound interest savings' bank, the deceased would, at his death, have been entitled to the following amount.
P + p improved at compound interest for n years
    P +p     n-1
    .    .    .    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .    .    .    .    .
    P + p     year
But under the conditions of insurance, the part P, with
MANaGEMENT OF AN INSURANCE OFFICE. 285
its accumulations, is the consideration for the sum insured ; the remaining part p, with its accumulations, is due under the name of profit or restitution, in a strictly mutual office.
The application of the preceding method would require that a calculation should be made once in every year of the quantity p and its accumulations, for every individual insured. This having been done, and the surplus A+P—C having been calculated from a true table of mortality, it is then known in what proportion any two individuals insured are claimants upon this fund. Sup_. pose that p and its accumulations amount, in the case of the persons X and Y, to 1001. and 1501. Suppose that A+P—C is 100,0001, and that the sum of all the excesses of premium with their accumulations, of which the 1001. and 1501. just mentioned are items, is 120,0001. It matters nothing that the last sum is greater than 100,0001., since we are not speaking of a fund on which there are definite claims, but of one the nature of which it is to be of uncertain amount. The use of the items 1001. and 1501., and of the sum total of 120,0001., is to enable us to divide the real fund of 100,0001. among those who raised it, in the proportions in which they contributed towards it. Thus if X and Y were to die in the year of the valuation, it would be fair that they should receive such proportions of the 100,0001. as 1001. and 1501. are of 120,0001.; that is, five-sixths of 1001. and 1501. This method proceeds upon the principle that all the excess of premium is taken in trust as a guarantee for the main fund, and is to he returned if not wanted, or such proportion of it as is not wanted. It confines the insurance, or provision against the uncertainty of life, entirely to a stipulated sum, and regards all that part of the premium which is not really wanted to provide this sum, for one man with another, as paid into a common savings' bank, in which no equalization is supposed.
The labour of making the calculations would, 1 imaprevent any office from adopting the preceding plan, so as to carry it into execution yearly. With a
286    ESSAY ON PROBABILITIES.
good system, however, the difficulty of managing the details of such a scheme would not be so great as at first sight might be supposed. Upon its principle hang the two first plans of division mentioned ; namely, periodical additions to the policies, and periodical diminutions of premuim. In both of these, the advantage of the insured is increased by the length of his life ; that is to say, the excesses of his premiums are placed to his credit in the first, and considered as having been prospective payments of his future premiums in the second. But nevertheless there runs through the offices which adopt these plans more or less of a practice which prevents the surplus from being divided among the insured in equitable proportions. Suppose that there is a septcnnial bonus, as it is called, which was declared in the year 1830. Immediately after the award, two persons, A and B, aged 30 and 60, enter the office each upon a policy of 1001., and were both alive when the bonus of 1837 was declared. This bonus is generally a percentage, not upon the amount of premiums paid, but upon the sum insured, and both would have the same addition made to the 1001. for which they have insured. But have both contributed to the accumulations of the office in the proportion which would render this mcde of division equitable ? To consider this point, remember that a promise to pay, say 51., at the death of a person aged 67, is of much more value than the same at the death of a person aged 37. The older life therefore receives much more than the younger life. But he has paid much more. That is true ; but at the same time he has occasioned a greater risk to the office, and it is the excess of his premium above the risk (and not the whole premium) which the office acknowledges in declaring the bonus. From page 270. it sufficiently apthat the premiums of the older ages are already too small in comparison with those of the younger : this mode of dividing the surplus, therefore, only tends to increase the existing injustice. The only remedy is, to make use of the process laid down in the preceding
MANAGEMENT OF AN INSURANCE OFFICE. 287
page ; and having ascertained the amount of what each person has paid over and above what was necessary, to consider each person as entitled to the sum which his overplus would purchase at his death, if the bonus be made by addition to his policy ; or to a diminution of premium answering to the annuity on his life, which the overplus would buy, if the bonus be made by diminuof premium.
The knowledge, therefore, of the real premium is necessary for an equitable distribution of the surplus, upon the supposition that the said distribution is made on the principle of dividing the surplus fund among the contributors in proportion to their contributions. Every plan which, ceteris paribus, makes equal additions to the policies of different ages, is inequitable. I repeat again, that in the preceding cases, the principle of division ought to be considered as arising from the combination of an insurance office and a savings' bank ; the porcf premium which covers the risk of life being paid to the former, and the remainder to the latter.
The third method of division supposes the establishto be entirely an insurance office, and not at all a savings' bank. Its object is to make the returns to the different members both equal and equitable. Considering that the real risk of life is not perfectly ascertained, and that if it were it would not be safe to re-duce the premiums to the lowest theoretical safety-point, such an office, instead of demanding a premium avowedly too high for the sum insured, and engaging to return all or part of the surplus, considers the sum insured as indefinite, except only in so far as a minimum is named, below which it is not to fall. Thus such an office, receiving, say 31. of premium, from a person aged 34, for what is called, in compliance with custom, a policy of 1001., does in fact make the following bargain:—The office engages to return, at the death of the party, let that take place when it may, such a sum as will represent the average accumulation of an annuity of 31. continued during the life of a person aged 34, be that sum more
288    ESSAY ON PROBABILITIES.
or less ; with this additional limitation, that the office undertakes that the said accumulation shall not be less than 1001. This last guarantee, though necessary for the satisfaction of the public, is in truth so certain, from the amount of the premium demanded, that a person acquainted with the subject looks upon the possibility of the funds of the society suffering from it as an extremely remote chance.
In order, however, to make the proceedings of such an office equitable, the proportions of the premiums paid by parties of different ages must be fairly regulated. On the supposition that thc inequality pointed out in page 270. is allowed to exist, the preceding methods of division may be (I do not say are) adjusted so that every interest shall be consulted. But in the present plan, it is impracticable to remedy any such defect of proportion, at least without dividing the establishment into as many different offices as there are ages, which would not be easy, and perhaps not very safe. The simple rule for determining the relative premiums is to make them proportional to the real premiums, with the exception of a given addition to each (not premium, but) policy, for expenses of management. In a large office, how-ever, the expenses of management may be made a part of the percentage addition to the premiums.
The method of division in such an office is extremely simple, and has been already described in page 276. Subtracting the present value of all the claims, that is, of all the minimum claims, reckoned as 1001. for each tabular premium paid, from the sum of the present values of all premiums, and of the assets of the office, the proportion which this remainder is of the present value of all the claims expresses the fraction of 1001. which may be added to each 1001. insured.
Let A, the assets * of the office, be 500,0001.; P, the present value of all premiums, 600,0001.; and C, the present value of all claims, 850.0001.: then A + P — C,
Diminished for the expenses of management, as in page 275.
MANAGEMENT OF AN INSURANCE OFFICE. 289
the surplus, is 250,0001., which being 25 parts out of 85 of the whole claims, or 29 it per cent., will afford 129j 71. for every 1001., which is guaranteed. Those who he in the year of this valuation, may therefore receive that sum.
The principle on which the preceding division is made, is, that if the same state of things continue, every one will in turn receive the same dividend. But, can such a prediction be made? Undoubtedly not, for the fluctuations, both of those who come into the office, and those who go out, will tend to produce variations. It is very unlikely that any office should maintain itself for a long series of years nearly in the same position ; and, since the idea of allowing any perdiminution of the surplus must not be admitted, there is no alternative except an arrangement for a gradual increase, which it is the object of this mode of division to make as slow as is consistent with the certainty of having it. But in this case, it may seem as if the old system were revived, and a fund instituted by the present insurers, for the sole benefit of those who come after them. There is, however, an imdifference between never paying more than the guaranteed minimum, so that all the surplus goes to_ wards that fund, and drawing upon the surplus nearly to the full amount which safety would allow, leaving only such a trifle to augment the fund as is requisite to avoid too large an outgoing. The old principle, then, which formerly prevented any bonus whatsoever, is here merely applied to such an extent as to keep the bonus within proper limits.
If the tables of mortality by which the profits are divided, be actual representatives of existing mortality, and if the number of members remain nearly the same, the indications of these tables, implicitly followed, would soon reduce the surplus of the office to that which is barely necessary for the extreme payment which the premiums will admit. To take a case: sup-pose that the premiums will in the long run pay 1251.
'290    ESSAY ON PROBABILITIES.
for every 1001. guaranteed ; the present value of all the claims is 1,000,0001., that of all the premiums 700,0001., and the value of the assets of the office 600,0001. The surplus is therefore 300,0001., and, going upon real tables, the office begins to pay 1301. for every 1001. guaranteed ; and this it would be able to do in favour of all who are insured at the time of the preceding valuation. But part of this dividend does not, and, by hypothesis, cannot, arise from the pre: it is therefore paid entirely out of surplus, and will gradually disappear. The dividend will be reduced to 1251., about which it will fluctuate, being sometimes a little less and sometimes a little more. An increase of business in such an office would make the surplus disappear more rapidly, since each new comer brings in an equivalent to 1251. and those of the new comers who die receive 1301. A diminution of busiwould produce a contrary effect ; and a total cesof new comers would allow the dividend to re-main at 1301. As far as any danger from fluctuations of mortality is concerned, I do not see any objection to such a division as the preceding : but when it is re-membered that the possible diminution of the rate of interest must also be provided for, I think it would be prudent to reserve a small proportion of the surplus for accumulation.
There are two ways in which this reserve may be made ; firstly, by employing a table of less than the real mortality in the valuation of the claims and presecondly, by calculating the surplus from a real table, and dividing as upon the supposition that a given fraction of this surplus, say one eighth or one tenth, should be expunged in the calculation. The latter plan is the best of the two, in every respect but one, as follows. The mutual insurance office must be a republic, and many of its members have very little in-formation upon the questions which are, from time to time, submitted to them. They are easily dazzled by the appearance of surplus, and are quick to believe that
MANAGEMENT OF AN INSURANCE OFFICE. 291
a larger division might be made in their favour. Add to this that the older members carry with them in the discussion of questions, that influence which age naand properly gives in the management of imporaffairs; and as to which the conduct of an inoffice only forms an exception, because questions arise in which the interests of the old and young clash" with each other, which is nowhere else the case. Under such circumstances the disposition to break in upon the surplus is the fault to which the body has a tendency, and it is not a bad thing to place some small difficulties in the way of doing this. Now if a fraction of the surplus be withdrawn from the calculation of the dividend, it is very easy to change one fraction into another. A vote of the general meeting, and a few strokes of the actuary's pen, and the thing is done. But when the requisite fraction of the surplus is de-ducted by the supposition of a lower rate of vitality (or of interest) than actually prevails, no change can be made without the entrance of a large number of important considerations, the discussion of which occusome time, and places a useful check in the way of the restless.
But is it then proposed that every office shall be provided with a fund, which, though slowly, is yet indefinitely, to increase? Not necessarily; for the reserved portion of one year is not put aside, and considered as inalienable, but enters into the surplus of the next year. There may be, then, a limit to the increase of the suras follows. Suppose the office to be in a stationary state, having arrived at the point where the influx of the new members compensates the efflux occasioned by death or surrender. The receipts of the office consist entirely in premiums and produce of capital, the expenditure in management and payment of claims. As long as the surplus increases, the sum of the first pair
The members of a mutual insurance office are not properly reprein their list of directors, unless the individuals composing it are of very different ages.
u 2
'292    eSSAY ON PROBABILITIES.
of items will exceed that of the second; and, whatever may be laid by in each year, it produces a larger surplus, and larger payments on account of claims, in the next year. If, then, the surplus could increase without limit, so would the dividends ; but if the surplus have a limit, the dividends also have a limit: and it is plain that the limit arrives, when the yearly outgoings from claims and management are equal to the receipts from premiums and interest of capital. A mathematical inof the conditions necessary in order that the fund may increase, but not without limit, gives the following result :
Suppose an insurance office, constructed upon the preceding principles, to have arrived at its stationary state, with respect to influx and efflux of members, and make the following suppositions :
A The assets of the office, for precision, say January 1, 1838.
P The real present value of all premiums from members then in existence.
C The real present value of all claims (net including additions) to which the office is then liable.
m The expenses of management till January 1, 1839.
p The amount which will accrue from premiums and interest of premiums by January 1, 1839.
c The amount of claims (not including additions from the surplus fund), which will be paid before January 1, 1839.
r The interest of one pound for one year.
t The fraction which is taken of the tabular surplus fund in the computation of the dividend.
'We suppose (as must be the case in an old office), C greater than P, and (as must be the case in a solvent office) A and P together greater than C.
1. In order that there may be a surplus fund increasing, but not without limit, find the fraction which a year's interest on C is of c. Then t, or the fraction of the surplus fund (or of A+ P — C), which enters into the formation of the dividend, must exceed that
MANAGEMENT OF AN INSURANCE OFFICE 29
fraction which a year's interest on C is of c, otherwise the fund would increase without limit.
  • Neither can there be such a fund unless the sum of m and c should fall short of the sum of p, and of a year's interest on the excess of C over P. But, when this is the case, the limiting surplus capital is found by dividing the excess of the second total just mentioned over the first, by a divisor obtained as follows : — multiply together t and c, divide the product by C, and subtract r from the quotient. To this surplus capital, add the excess of C over P, and the limiting capital is obtained.
  • If it should happen that the limiting surplus capital is less than the actually existing surplus, it is a sign that the action of the preceding plan would diminish the surplus towards that limit instead of increasing it. In such a case, the surplus is already too large for the value of t to increase it ; and if t be not diminished, that is, if less of the tabular surplus be not taken into the computation of the dividend, the fund will diminish.
    • It is not to be supposed that any office will ever reach a stationary state ; but the approach may be near enough to make the preceding process of some use in the determination of the dividends due to the insured. If, following the plan which the preceding problem supposes, we were to inquire what value should be given to the fraction t, the answer to the question must depend on the reduction of interest which is supposed within the bounds of probability. Suppose the present rate of interest to be 31 per cent. and that the extreme limit is supposed to be 2 per cent, in such a case the value of P and C must be calculated at 2', per cent., and such a limiting surplus must be fixed upon as will, at that rate of interest, enable the office to pay at least its guaranteed claims. But it is impossible to lay down an entire system of rules for the regulation of a species of undertaking which depends on the flue, tuations of the state of society. Whatever maximsu 3294eSSAY ON PROBABILITIES.may be     collected, and however sound they may be, skill and judgment will always be requisite to apply them to the cases which arise. In this respect the offices resemble the individual problems which arise in life contingencies. Many as are the cases which have been described in books upon the subject, almost every application of them requires attention to some circumstance peculiar to the instance in question.CHAPTER XIII.MISCELLANEOUS SUBJeCTS CONNECTED WITH IN
    SURANCE, ETC.THE limits of this treatise will only afford a few words on several points of interest, which I will therefore condense into one chapter, taking the subjects as they arise.The management of annuity offices is somewhat more easy than that of insurance establishments ; and the maxims of security in the former are, of course, the direct reverse of those in the latter, so far as any considerations of mortality are concerned. Tables must be assumed of higher than the real vitality, and a rate of interest somewhat below, or at least not above, that which can actually be obtained.Those who wish to buy annuities on the firmest possible basis, may deal with the government. The commissioners for the reduction of the national debt are empowered to grant annuities in lieu of stock, on terms calculated from the government tables (page 168). The rates are high ; and though a private office may really be as solvent as the nation, yet confidence springs