# Full text of "Letter on life insurance to Fred S. Winston [microform] .."

## See other formats

COLUMBIA UNIVERSITY LIBRARIES PRESERVATION DIVISION BIBLIOGRAPHIC MICROFORM TARGET ORIGINAL MATERIAL AS FILMED - EXISTING BIBUOGRAPHIC RECORD qn-e>4oo"7>z MASTER NEGATIVE # 308 Box 3 Bartlett, Wcllliamj Hcolmes, Cchambers, 1804-1893. Letter on life insurance to Fred S. Winston... New York, 1871. 29 p. 17i om. RESTRICTIONS ON USE: Reproductions may not be mad, wtt^pmrnMon Oom Columbia Unim^ubrarios. TECHNICAL MICBOFOHM DATA FILM SIZE: REDUCTION RATIO an IMAGE PLACEMENT : lA IB IIB DATE FILMED: INITIALS: TRACKING # : FILMED BY PRESERVATION RESOURCES, BETHLEHEM, PA. BIBLIOGRAPHIC IRREGULARITIES MAIN ENTRY: Bart l ett . W illia m Ho l mes Chambers Letter on life insurance to Fred S, Winston Bibliographic Irregularities in the Original Document: List all volumes and pages affected; include name of institution if filming borrowed text Page(s) missing/not available: ^Volume(s) missing/not available: ^Illegible and/or damaged page(s): Page(s) or volume(s) misnumbered: Bound out of sequence: Page(s) or volume(s) filmed from copy borrowed from: X pagination begins with page [3] Other: Inserted material: TRACKING*: MSH20022 LETTER soi" -7 Life AssuRAi^CE TO Feed. S. WmsTOif, Esq., PBEUDBNT MUTUAIi lilFE INSUBANCE COMFAKT OF NEW XORK. ; Pkof. WM. Urc. BAETLETT. POWEEb & MACGOWAN, PRINTERS, VUTSC JOB PRINTING HOUSE. 1871. t LETTER ^ ON - LIFE ASSURANCE. } — ^ — West Point, January 2, 1871. To Fbed. S. Winston, President Mutual Life Lis. Co. of New York. Bear Sir: The recent failures of some well-known Insurance Companies, the variety of methods practiced by those still in existence, and other circumstances not important to designate, have caused a good deal of discussion in and out of the public prints ; and ttie character of the discussion has seemed to me better suited to engender and stimnlate to rapid growth a dis- trust of the principles of Life Assurance, than to vindicate their just claims to general confidence. PubUc opinion is not always founded in sufficient knowledge to be just, but is too often the result of honest ignorance misled by misrepresentation. Nothing seems too valuable or sacred, nowadays, for the schemes of daring empirics or the whims of thoughtless mediocrity. It would be a crime to deceive the large class of persons for whose benefit the institutiou of lofe Assurance was specially devised, or to do anything which might have the effect to beckon them away from its ftdvantages. It is the duty of all to do what they may to de- fend this noble charity, for such it is ; and I have, therefore, thought it proper to address to you, os the head of the largest and most prosperous company in the country, perhaps in the world, this letter, in which I hope to explain the few and simple princi- ples of life Assurance in a way to bring them within the easy comprehension of all who need or desire this sort of information, and to demonstrate by implication the utter unsoundness of the assertions, I will not say arguments, too frequently arrayed against them. The object of Life Assurance, as you well know, is to protect 4 LETTER ON LIFE ASSURANCE from ivant those who lose their means of sapport by death. Vast numbers of widows and orphans are now drawing their onlj mefU3is of subsistence from its provident care ; and still greater numbers are reposing in confidenoe upon its promises of future aid in time of need. It is one of the most beneficent gifts of civilization to the working classes, especially to salaried men, and is emphatically the poor man's friend. It shields his family from the chances of want in case of his sudden death, and only requires, as an antecedent condition, that he give it the temporary custody of a very small shwe of his current yearly earnings. It is to him a savings bank, and to them much more — an open-handed friend in time of bereavement. PLAN. The leading idea of Life Assurance may be thus briefly stated, viz. : Each of a number of persons pays into a common fund, either at once or by instalments, a certain sum called o, jivemium^ determined in amount by the condition that at death, or before, according to the agreement, his or her heira shall receive from this fund, and its interest earnings, another specified sum, called a reversion. In case of early death, the sum paid to the heirs greatly exceeds that paid by the deceased, but this disparity con- tracts more and more as life continues, till, finally, the common fund becomes exempt from the chances of loss by accretions to the individual payments. It must not be supposed, howevw, that the heirs of ever}/ member will receive more than he pays ; on the contrary, the members agree in effect, among themselves, that those whose good fortune it may be to have more than an average longevity, shall give of the excess of their payment and its gains to the support of the heirs of those who have less. Though based upon self-interest, the scheme is, as Professor de Morgan justly remarks, the most enlightened and beneficent form which the projects of self-interest ever took. To show how this idea may be put into practical execution with entire certainty, I shall be obhged to employ some elementary mathematics, but my letter shall be none the less easy of comprehension to ibe general reader on that aoooont, since aU its lesults will be translate into x)lain English. LETTER ON LIFE ASSURANCE. 5 SJI\IGLE NET PREMIUM. Take the following notation, viz. : r= the number of persons, of age ^, that unite to form a Mu* tnal Life Company. d^=^ the number of those that die within any given portion of time, say a year. i =r rate of interest, or what one dollar unll earn by being loaned out during the same unit of time. : = the amount which each survivor, at the end of this unit of time, has to his <mdit at the b^[inning of the second unit of time. 4^.1 3= the number living at the beginning of the second unit of time, or seoond year. 11= the present value of a single and only payment of each member to secure a reversion of one dollar at death, called a single nel premium. Then the amount with which the company begins its business will be times n, or Z..n; and this put out at interest will, at the end of the year, amount to /,.n. (1+0; firom this is to be snbtmetod doUan, for each death requires the payment of one dollar, and the sum with which the company b^ins its^cond year wiU be and this must be equal to the amount of each survivor's credit at the beginniug of the second year, multiphed by the number of survivors ; whence tiie equation : The amount l,^\.R^:^x, or its equal /, .n, (l-f-i) — (/^ being put out at interert, mil amount, at the end of the second year, to and this, diminished by the payment of the death claims, amount- 6 LETTJER 02i LIFE ASSUliA^CK iug to f^^i, this latter denoting the number that die during tlie second year, will give, as before, the amoiiut of funds witii which the company begins the third year. But this is equal to the amount standing to the credit of each survivor at the be- ginning of the third year, denoted by multiplied by th« number of survivors, denoted by 4+s> and we ifaaU have [/, . II . (1+0— ^4] E^*, 4 . 11 . (1+/)-'— <4 (1+0— <4-n=4+8 • E,^^ • (2-) Again, putting out at interest the sum or its equal, in the first member, and subtractii^ from its amount at the end of the third year the death claims of that year, denoted by d,+» we have, by suitably changing the subscript notation, 4.n.(l+i)^— rf. (1+i)^— <4+i (i+0-<+*=/.+«-^.+» ; (3-) in which the htw is developed and manifest ; so that if x-^ de- note the oldest age to which any member of the ccmipany may reach, we may write, generally, 4.n.(l+»)»+'— <4(l+i)"— <4+.i(l+i)— * . . . ~<U«=^.t*+i.-BH^». But being the greatest age, there can be none of a;+w+l, and hence 4+»f i=o ; which will reduce the preceding equation to 4.n.(i+0" --^4{i+0''-^«+i(i+0''"'- .-<4+,»-i{i+0-<4,«-o. (4.) Now, just as certainly as the mortality conforms to the rates d„ dSn-ij ^4+2, &c., and the interest to the rate i, will the last dollar be paid to the heirs of the last survivor, aged se-^, and the fund and the company become extinct together — after paying the claims of the heirs of the other members as they mature in tiie lapse of time ; the only condition on the port of the as- sured being the payment by each, at the outset, of the single premium II. Of the certainty of these rates, more presently. To find the value of n, scdve equation (4), and w« have 1 ^4 1 <4-.i 1 <4fj 1 f4+, n.= — .—-I 1 — I- • • • — - ; l+i 4 (l+^r 4 (H-»r 4 4 LETTER ON LIFE ASSURANCE 7 or making 1 <4 d,+i d,-f <4fi. n^r. — — f- • • • ^ ■ ' (5-) 4 4 4 4 Or, multiplying both numerator and denominator of all the terms in the second member by v', which will not alter the value, if^i . <I,-\-v'^- . r4+i+t^+" -d,^, +f^+*^» . d^ (6.) which is the formula employed in the construction of what tan called commxfaiiou levies, for the easy and expeditious determina- tion of singte net pieminms for assnzanoes of <me dollar at dei^ For any other reversion this premium has only to be multiplied by the number of dollars in the reversion. Alii /il UAL NET PREMIUM. So much for a single premium. But far the more common mode of making payments, by the assured, is by what are called net amraal praniums daring life, or for ft shorter period — ^ihe X)rinciple being the same in both. Take the case of annual pay- ments during life, and denote the annual joremium by tt. Then, from what has ahready been expkdned in detail, ii will be easily seen that the mathematics will stand thus, viz. : For the first year, as before, * 4 . ^ . (1+0— ^4==^*f 1 • There being /^^i, living at the banning of the second year, and as each pays another x^remium tt, the company will begin the second year with the amount, or with its equal, and this at interest during the second year will amount, at its close, to 8 LETTER g?^ UFE A^SUBA^fCE. [4.^.(l+i)+U.T-.4].(l+0 ; and ibis, diminkhed by the death dainis, amonntiiig to d^i dol- lars, during the year, will give ^ -[Ul-^iY+U (l+t)]-^4 (l+0-<?.+i=4+2.i2.+.2. In like maimer, the amount of 4+2--Ka:^o-f4+2.7r, or its equal, ob- tained by replacing by its value in the first member of the laafc equation, diminished by the death daisM of ^le thiid year, amounting to dollars, will give (1+0^+44-2 (i+^->-^4+«= and liere, the law being manifest, we may write, generally, *P.(l+0"^'+4+i(]+0"-K+2(l4-0"-^ . .+4+»-i .(l+01-[c?.(l+ but, as before, 4h»+i~o, and staving with respect to n-. ■n dividing the ternis in both numerator and d^imaaaaibox by 4-(l and wzitixig 1 9 for we find '4 <4+-S ^x-l-n-l ^4+11 4 4 4 t t 4 1 4+2 4-f« hi.* (7.) 1+0. — 4-«'"- — I I I. I which gives the ordinary rule for finding the net annual pre- mium. And here, it may.be remarked, as in the oaae of a single ptreminm, it only requires the anticipated rates of mortality and of interest to be realized, to secure to the heirs of the assured Mie prompt payment ol the company's obligations as they matmre. LETTER ON LIU! ASSUBANCEL 9 RATES OF MORTALIU AHD OF mEREST. Now, an examination of equations (5) and (7) shows that the premiums increase in valae as the rate of mortality increases and that of interest decreases. Safety to the company is, therefore, secured by assuming, in the computation of premiums, a rate of mortality hi^^ier, and of interest lower, than tfiose men likely to bereali2»d. Careful records of births and deaths, extended through a long series of years, in dijOferent countries, have revealed the laws of mortality among people oomposed of all classes and possessed of the ordinary means and comforts of life. These laws are defined in what are called mortuary tables, and the ratios of <l« to and l^^x to for all ages, tabulated for easy reference and use. These laws being employed iu the computations for cissured lives, which are always selectedy and of which the rates of mortaliiy are always Jess tiian those of the communis at large, will satisfy one of the conditions of safety. The rate of interest is also suggested by experience; and being taken below that received on the actual loans and otiier businem operations of the community where the funds of the companjaxe to be employed, will secure the other condition of safety. Thus, assuming as a basis of computation a higher rate of mortality than tiie company will realize among its members, and a lower rate of interest than it will get upon its loans, the premiums will be higher than necessary, and the objects sought by the f <»mula secured beyond all reasonable doubt ■ PRESENT AND FUTURE OBLIGATIONS. Let me apply formula (6) to an hypothetical case for the pur- pose of illustration. Suppose a company of 1000 persons aged 20; and take the rates of your own office, in whidi t=:0.04 ; . 20+w=95; or w=76, and we find II«F^$0.24776 ; and, therefore, 4d.I1i»=^1000x0.24776 =$247.70 ; so that, with a capital of two hundred and forty-seven d^i^^^^^ and 10 LETTER 02i LIFE ASSURANCE. seventy-six cents, at the outset, the company engages to pay one thousand dollars in the comae oi «e¥wty-fiye years, which it does with its original capital and its interest earnings. Again, if the membei's of the same company engage to pay by instahueuts a net annual premium during life, which is the more common case, we shall find. and 4o . ^a)=1000xO. 01267=$12. 67. That is, one thousand persons beginning business at the age of 20, with a capital of twelve doUais and sizly'seyen crats, and each paying into the common fund but a trifle over a cent and a quarter annually, will give to the heirs of each member at death, one dollar. The last payment, which will be made in the seveniy- fifth year of the company's existence, will just exhaust the funds, and at the very moment the company becomes extinct from the limitation of human existence. The actual value of the company's obligations, at any time, is measured by the amount of death claims then matured, and can never be as great as the company's assets, except tiie death of the last surviving member, when they become enctly equal. You often hear persons, who talk without thought, employ language which has the effect to condense, as it were, into a single day the successive obligations of a company that can, by the very terms which bring them into existence, only mature through a long series of years, and no more rapidly than the assets to meet them, and thus produce the most enroneons and damaging impressions. Such persons will, without any regard to the ob- jects and ofi^ces of verbal tense, say, for instance, that my hypothetical company has outstanding obligatimis amounting to a thousand dollars, while its assets are but a little over twelve; and assert, with equal emphasis, as a consequence, that it is help- lessly bankrupt — a eonduaion, to use no harsher terms, utteily illogical, and true in no sense whatever. The fact is, no one but those having access to a company's books can know anything of its aciual obligations. Your own company has now assured to the amount of two hundred and forty-two milHons of dollars, and its present assets are under forty-five millions, and to infer that LETTER OX LIFE ASSURANCE 11 you are, therefore, bankrupt, would be about as wise as to con- clude that the Crotou river is unreliable because the amount of water now running between its bimks is insuflScient for the /tdure as well as present supply of your city. Again, and I write from experience, these i^ersons will argue that local epidemic diseases suggest the advent of others so wide-spread and devastating as to render all computations founded upon the observed laws of human vitaUty unreliable and worthless. And so do rain showers suggest another Noachiaa deluge to drown all the living, and earthquakes another general OTuption of the internal fires to burn them up! The one sugges- tion is about as significant as the other. Such people have no belief that a hiw of nature may be detected by observation and experience, however well directed and long continued. ANNUITIES. It is unnecessary to my purpose to give the demonstrations for the various rules by which are computed the premiums apfoo- priate to the great variety of assnrance policies. They are equally dependent upon the same elementary formula with which I began. But as many companies take annuity risks, I will add the demonstration of the rule for finding the -pteuetA value or price of an annuUy, An annuity is a specific sum of money paid to an individual at stated periods of time, say at the end of every year, in con- sideration of another sum paid down by the recipient, called an annuitant, or other person in the annuitant's behalf. Suppose a company of persons, all of the same age and con- sisting of /, m^BUien, to pay into a eommon fund a sum suflSoient to give each an annuity of one dollar, to be paid annually at tihe end of the year during their natural life. What price each pay, the rate of interest being t, and the mortality rates those adopted by any oranpany ? Denote this price by the usual nota- tion a„ the number living at the age x by x+l by by 4-r2, etc. The funds of the company, at the end of the first year, will amount, according to what has already been explained, to 12 LETTER OX L.1FE ASSUJiANCK and at the end of the second year to or 4 . a^^l+Z)'— 44.1(1+/)— 4-i-j=4H-« . -R.+ii, at the end of the third year, 4.0,. — 4+i(i-l-i) — 4+s(l-f i)— 4^^==4+* . ; ci" generally, since the law of the series is manifest, 4.r/..(l+/r-4^i(l+0"-'— ^.-aCl+O"-" — 4+n-i(l+0 But US none will be alive after the (.c-^-ny^ year of age, the paj- meixt to those who reach that age must be the last, aud will ex- haust the fands, so that i2,4.«=so ; and we find and solving with respect to a,» and making 1 we get h^t 4-1-3 4-H» ^-t^.—f^. ^fF^ ; (a) /, h h I or multiplying both nrunerator and denominator of each term in the second member by tf^ which will not alter its valne, we have a,= ; (9.) which is the price sought, and under the form emx>loyed in com- puting the commutation tables for annuities on single lives. THE TEMPORARy COMPANY MADE PERPETUAL Now analyze formulas (5) and (8) : taking the iirst of tiiese, the first term of the second member is <4 v.— ; 4 LETTER ON LIFE ASSUliANCE. 13 in which v denotes the present vidue of one dollar at the end of a year ; that is, it is the sum which, put out at interest at the rate i, will grow to one dollar in one year. But the payment of this dollar is contingent upon the death of the assured, and the dollar itself is called the conliugeat gain to the heirs of the as- sured, or contingent loss to the comx>any. The numerator of the fraction into which it is multiplied, to wit, d^, is the number out of 4 that die at age x, and when divided by I,, gives the ratio of the number that favor death to the whole number that favor and oppose it, and is, therefore, the probability that a par- ticular person named in advance will die at the age x. The pro- duct of the contingent gain by the probability that the event will liappen to cause the loss or gain, is proeeiit value of what is called the expectation. In the same way may the second term of the second member be shown to be the present value of the expectation of loss w gain at the end of the second year, and so on ; so that we have this rule for finding tlie present value of a single net premium of assunmce, viz. : Find the present value of the expectations of loss to the company for all the years from date of policy to end of human life, as given by the mortuary tables, and take their sum. Again, — is the m^wure of the probability that any desig- l nated individual will live through his year to receive his an- nuity. And a similar analysis of formula (8) tells us that the rule by which to find the present value of an annuity for life, is to find the present values of the expectations of the annuitant from date to the end of human life, and take the sum. And a reference to formula (7) will show this rule for finding the an- nual net premium, viz. : Divide the single net premium by the value of an annuity of one dollar increased by unity. These rules are perfectly independent of the actual number of m^bers in a company, and are rtrietfy applicable to single cases ; so that it is not necessary, as I assumed at the outset, that the members shall be of the same age, but they may be of all ages and admitted at all times, a sufficient numbw only fure required to make sure of the laws of human mortality when the 14 LETTER ON LIFE ASSURANCE company begins its business. My hypotlietical and temporary company becomes, therefore, converted into a perpetual one. The rates of mortality and of interest are prepared in advance, and have ouly to be ai:)plied in the manner just explained, to as- certain the sum or sums to be paid on tiie adnuanon of any member. In this -way ideas of probability and of chance came to be associated with life assurance, and gave it, to those unacquainted with its real character, an aspect of nncertednty wUch it does not, in reality, possess. The uncertainty attaches to individuals, not to large collections of individuals. It would be very unwise, for instance, to assure one individual, and no more, upon the conditions of formulas (5) and (7), and yet perfectly safe to as- sure a hundred thousand, each on those conditions. RESERVES. The reserve of any membw is his wcnrking insoranoe capital At the date of joining the company, it is his net premium. It augments as time laj^ses by the accumulation of interest, and diminishes by the loss of its share of the death daims. It continually varying in value, but, on the whole, expands more than it contracts, till finally, at the close of the longest life of the table, should it pertain to such life, it amounts to the sum assured. The sum total of all the reserves is a ^e measure of what the company's assets ought to be. If the actual assets fall below this sum, the company is insolvent and should be wound up by re-assuring in some responnble company to tiie extent of the reserves available for this purpose. If the assets be in excess of the sum total of the reserves, it is a sign of healthy condition, and that the members have overpaid, and the excess should be distributed in the manner to be described presently. By a study of the mathematical processes I have given, it Avill be apparent that a mutual company performs both the f an<^ons of a savings bank and an insurance institution. It receives, cares for, and adds interest to the premiums ; rnd converts, from time to time, this interest, in part, into capital ; while it also takes the vital risks of the assured, and, in case of death, j^ays to his heirs his individual reserve, and, in addition, the excess of the sum assured LETTER ON LIFE ASSURANCE 15 above this reserve, from the accumulating funds of the company, of which the bidance goes to make np the reserves of the rarviv- ing members. An individual reserve, which is the trae money value of the policy, and in which the holder and the company have a joint interest, has been very properly called the amount of a member's a^-idsnrance. It is, so to speak, the tangible substance of the policy, and should be at all times within certain reach of the company. All persons intoMted wi& the inteirert of others shonld be held to a stringent accountability, and be made to render, at short and stated intervals, an accoui^t of their stewardship ; and this is emphatically true of the officers of an insurance company. The Legislature of this State, acting upon this principle, has very wisely required, as one of the conditions of your own charter, for instance, that yon strike a balance on a given day of every year, which shall exhibit the actual state of every member's account at that time, and show the precise amount of your re- serves, actual assets and future liabilities, that <be members of the company, and those who may desire to become members, may form a just estimate of your actual condition. Then, every member's reserve should be accurately computed up to this day, the watm of the vHboki taken, and this som compared with the actual and well-ascertained assets. To compute these reserves, resume the elementary formula (1), viz. : . (1+0— ^4=«?.+i.i2,+i ; in which R^^x represents the reserve of each member that begins his (.^:+l)*'year ; and had all the members entered the company at the same age, and on the legal day of settlement, it woold only be necMsary to find the value of R^^ i from this equation in terms of the known values which compose it. But we have just shown that members join at idmost all ages, uid at all times, and as- suming that the rate of mortality is continuous, and denoting by /the fractional part of the year from date of policy to the day of reckoning, we may write the above equation thus, viz. : whence 16 LETTER ON LIFE ASSURANCE. or E,^r=- — -\n ./.-[. (10.) l—t.- or l-H-i ( 1 d, ,=:- . II .t. B^t=^- -11 .t.— \. [XL) d, { I,) 1— /.— h But for the use we sliaU. make of the reserve formula for the fractional part of the year, the form (10) will perhaps be the more convenient. MakiTig i==^i we have B,^r^ .3 11— -4; (12.J d. { 1+/: lA 1 aod generally for a single net premium, i2.,,= -. \ \ ; ■ (13.) —■ A J^x^n-l • r 5 ( 1+'" ) and for an annual premium, 1 For any fractional part of the year at any distance, as » — 1, from that of entrance, eqii^tion (10) may be written i2,^+^ — . J — [ (15.) LETTER ON LIFE ASSURANCE. 11 for a single net premium ; and for an tomual premium, (1 + 0' j 1 ^^-r^l d^^^, { (1+/)' «-i From the equations (13) or (14), according as the case pertains to a single or annm^] premium, compute the reserves from anni- yeraary to anniyersaiy of the polioj, and onter them in a table arranged in the form of double entry, the ailments being the ages of the members at time of entrance, in a horizontal series of Gaptkm -Spaces, separated by vertical lines when the sheet is held upright, and the i^ies of the policies arranged in tiae left hand vertical column, each pair of consecutive whole years being sep- arated by nine horizontal lines for the reception of interpolated ▼alnes for tidnths of years, determined by the use of equations (15) or (IG), as the case may be. On the right of every vertical oohmm of reserve values there should be a column for difier- enees, which should abo be entered. With the aid of this table, the reserve on any policy may be as- certained for any -day of the policy's existence, with the same ease and aoenracy thai t^ logaritiim of a number may be taken from a talile of logarithms. The sum total of all the reserves may 'oe determined long before the close of the fiscal year, ready for comparison with the assets as sooa as known, and the question of solvency answered. If, i)erohanee, some of the included re- serves may have disappeared by post-mortem settlement, these may easily be subtracted from the result. Such a table will last as long as the company adheres to any given or assumed set of rates, both of interest and mortality, and should be re-computed as soon as these are changed. EXPENSES AND LOADING. • The only payments the company has thus far been required to make have been for death claims, and the explanations have been 0(mfined to that branch of a company's operations which pertains strictly to assurance. But thOT© are many incidental expenses, such as t)fiico rent, salaries of officers, lawyers' and doctors* fees, 18 LETTER ON LIFE ASSURANCE clerk hire, fuel, stationery, and the like, to be oaied far. Tliese are decayed from a fund created by adding to each net premium a certain percentage of its own value, and requiring the assured to pay it at the time of paying his premium. This addition, called ioading, amounts to from 20 to 40 per cent., according to circumstances. The funds of the company, above the amount required for immediate death claims and current expenses, are loaned out at the market rate of interest, upon undoubted secu- rities, of which the intrinsic vidue is generally double the sum loaned. I do not here speak of the expenses of agencies, because they have nothing to do with the zeal business of assurance. They will be referred to further on. SURPLUS AMD ITS mTRIBUTION. If the expenses of a company be just equal to the sum total of the loadings ; if the rate of interest on its loans and losses from death be just equal to those assumed in the computations of net- {orauums, the assets, at the <dose of the fiscal year, will be just equal to the sum total of the reserves. But this, in well-managed companies, rarely happens, and almost always the gross loadmgs exceed the expenses, the rate of interest is hif^ier than that an- ticipated, and the losses from mortality less ; and hence the assets are generally greater than the reserves. The excess of the former over the latter is what is called surplus, and is the amount by which the members, collectively, have been overcharged ; and is therefore returned, in equitable proportions, not in money, but in the form of credito for paid up policies, of which the re^ yeraonary amounts are determined hy the principles already explained for single net premiums and loadings. l%us, make X, ,=the share of a member, aged x-^i, ; m»4he percentage of which goes to the loading. n,4 «,=net single premium to assure one dollar at a^e x-y., ; ^.^».=reveraionary value of paid up policy ; Then, „ — mX^^,—{\ — m) . Jr,^.,,=net sum to pay for policy ; LfiTTiai LIFE AaSUSANCE, 19 and as there must be as many dollars in the are net dollar premiums in the net purchase money, we hi^ve S,,,=(L—m).X,^^. . (17.) This new policy, the property of an old member, will, at the close of eveiy fiscal year of its existence, have its reserves, and, also, its credits or debits, according as the company is prospering or the reverse. In some years the expenses may be small, the interest on its loans high, and the mortality below that of tiie tables, in wiuch case the snrplns would be lai^ and the reversion- ary credits correspondingly large. In other years just the reverse may happen, and, indeed, there might be a deficiency instead of a sacplus, and this should be met by a reversionary debit to tiiis fluctuating account, always leaving the original policy intact. MODE OF DISTRIBUTION. The only equitable method of distribution is that known as the contribidio7i plan. As its name indicates, it aims to return to each member the amount of his actual contribution to the sor- plua It has been a good deal discussed, much praised, and, in my opinion, not too much ; and yet it is susceptible of very easy mkapplication, and of being made the meaiia of wrong as well as of right. The f<^owing demonstration will explain it : BerasM Hie elem^tary equation t . IT . (14*t) — ttssB^^i . -B,^,i=(4— <4) . . This embraces <me unit of time, as a year. And for any feao- tional portion of a year, denoted by or which is the uame thing, ir.(l-f/.i)— (1— i?,^,=o; (18.) that is, the amount of net premium at end of time ^ diminished by its share of death daims and the reserve at same epochs nlttst leave nothing. LETTER ON lAFE AaHUHAHCE. Kowmake A=4oadiiig; «s=individiial yearly expense ; / =rate of interest realized ; flT .^earij number of deaihs realized at age a; ; then substitute n-j-A— for n ; *' for » ; rf . for d„ and ve get (TT+X-te) (19.) in "whicli X,^ t is the contribution at the end of t. Subtracting equation (18) from this, and diminishing the difference by the reserve loading (1— we find t\ </--i)-f./_^-f(A_./)/.f(i )._(i_i2,^,)Ux,,„ (20.) and using tite sign 2 to denote the operation of summing, 2A:2e::X:e, or e=A. — ; and denoting the cKhial loss from deaths by a, and that of the tables or impiUed by we may write d,^e* and substituting in equation (20), we get ( ( 2e ) a d^ ) t |<i'-i)^-A |l+,'__.(i4.i i)^ -j-(i__)._..(i_i2,^,) L (21.) or, perhaps, sufficiently near t, -J ^(i_i)4.x.(l_^)(l^-^•')^-(l_!).^.(l_i^,^,)l=x^^ 2X el, ) (21.)' LETTER ON LIFE ASSURANCE. 31 The sum of the actual expenses and that of the loadings will be known ttom tbe books of tiie ocnnpuiy. The ratio 2X le will therefore be known, and the factor — becomes constant for n d, a every polu^ for same settlement. The ratio — for which - is d, e snbstitated, is tiie ratio of the actual to tbe expected deaths at age x, or the ratio of the actual to the expected loss from deatii, sopposing each assured for one dollar. In strictness, the ratio — mi^t be employed, but it would work great hard- d. skip, if not downright injustice, in many cases. Supposing, for ixistimoe, an epidanic disease should ptevoil partienlarly fa^ to persons of a certain age, and not to those who are younger and older, the loss from death would fall heavily upon the survivors of that age. Indeed, in their case the ackud loss from death might be greater than ^ expected, and make tbe factor 1 . whidi will present^ i^pear, nc^iative ; while if ^ actual nnm- ber of deaths in the whole company shonld be less than the ex- pected, this factor, for other members, would be positive, thus deranging the g^ieral rates of mortality assumed at the outsek On the principle of protection implied in the mutual system, the actual losses and benefits arising from temporaiy departures from general laws should be shared alike by the whole company, and it would be much more in acc(»daiice with the principles of a equity to make the ratio — a general term, applicable to ail pol- icies. To do this, a should denote the actual loss from death of the wJude ixmixmy during the year, and e ^ eoqiected loss from same cause. The actual loss of the whole company during the year is the 22 LETTEK ON LIFE ASSURANCE Mnonnt of death claims diminished by that of the reserves of the deceased members, and is therefore known from the books. Thus, denoting a single death claim, at age by D„ and the re- serve of same at end of year, on one dollar, by E^^i, and making similar notation for other ages, we have ««2LD.(1— (22.) The expected loss is also easily found thus : denote by L, the sum assured on any life of age .r, and by the corresponding reserve, and make similar notation for other ages, ^ea will or more generally and concisely, f=2-.2i.,(l— (28.) . That is, take the sum of the amounts assured for each age, dimimah this sum by that of the reserves lor that age, and mul- tiply the remainder by the probability of death at that age, then take the sum of the products — the result will be the expected loss. Equation (21) mQ give to a member his appropriate share of surplus at the close of the fiscal year of entrance, t being the fractional portion of the yeax from date of poli<7 to that epoch, lifoldng that equation becomes 2^ a ^{i' . (1 ) . a+t )-Kl— -) — . (l-^*^-x)=X,+i, (24) IX e 4 which is the appropriate formula for individual simxe of surplns at the end of the first policy year, provided no settlement be made before. But this formula should never be used unless the company is at liberty to have just as many days of g^ieral settlement as there are poHcy anniversaries. Now the account which was settled at thts end of the fiscal year, or at the &xd of begins the next entire fiscal year with the re- LETTEB ON UEE A88URANGE. a«ve JB.^ and with the loading (1— and has added to its credit at the end of the policy year, and at the distance t in time, from the close of the next fiscal year, the Bom tr^ in case of an annual pramimn, or only A, if a single premium. Supposing the former, for the present— the latter will be considered presently— the formula becomes but employing the net premium and rates of interest and of mortality of the table, subtracting this, and diminishing the difference by the reserve loading we find 2e ad, ^].(i'-»>Hl.(l ). (lH.i>Kl ) .^l-R el, which gives the snrplns due to the assumed policy at the close of the first entire fiscal year of membership ; tliat is, if the day of settlement be the 31st of December, it will give the snrplns at tho end of the first calendar year after that of entrance. And generally, for the surplus at the the close of the w*" calendar year after entrance, we may write [H^^i+t-H-^] .(» — i)-H^.(l— — ).(l-f-i>H3— -)• ^^.(1- (25.) in which tiiefaotoTB (t"— /), (1 ) and (1 ) are constant ; that is, the same for all policies for the same day of reckoning. When a member dies, and sufficient time has been allowed to cdUect the evidence of the cLrenmstances of death, the amount LETTER ON LIFE ASSURANCE. standing to his credit at the date of last settlement is usuaiijp paid to his hein. But this does not exhaust his interest in the company. At the next day of general adjustment of accounts, his special account is closed, and the balance which may stand to his credit on that day is paid« Eormnlas for these post*mortem eredite and payments may be easily deduced from equation (15). ANNUAL LQAOm FOR SINGLE PREMIUM. It has been the practice in some quarters to load a single pre- mium by the addition of a percentage, and to make this loading contribute to expenses for one year and no more. To do this according to the method required by equation (25) would be very unjust, and a gross violation of the prinmples of equity. Ob- Tiously, the proper way would be, in all cases of single i^remiums, to ascertain the value of the corresponding annual premium to assure the same sum, take the percraitage on this, uid then load tiie single premiums by a sum which would give an annuity for life equal to this percentage. This percentage would be the value of Ay to be used annually, in equation (25)» during the pol* ley's existence If the assurance be purchased by a limited number of annual payments, the case would be somewhat different, though the principle would be the same. The following process will lead to the rule to suit all cases of whole life policies, whether the pay- ment be by single, several, or mmual premiums. Make A=the loading on each of the limited number of payments ; Assthe equivalent loading on an annmd immimn for life * 2/=greate8t age of the tables ; x=B^e of the assured at entrance ; y — ^QB^Bsn; msmumber of annual x^ayments. Then (A — A) (l-f/)=A, (!-}-/) — ^A(l-f-0 = accumulation at end of first year; - A(l4-i)+A. 'K{Xj^i)—\ — \ .(l+i), or LETTER ON LIFE ASSURANCE. 25 A I (l4-i)24-ll\ (14.,) I I (14.^/4^' (14.,) I « aoOTmuhUdon at end of second year. and generally, stopping the accumulation at the end of the (»— 1)^ year, and omitting the terms involving A, after the payment, and because the last annual loading, exhausts the loadings A, and their accumulations, we have A j(lH.i)-^4.(i4-£)-*,_4. . . , _x |(l+/r-4^ I. I, s or A. . 1+ .—+... .— t-- A. . -1 + 1 4+1 1 omitting tlie common factor, transposing and multiplying the niuaenilor aikl deaiomiiuto of oMdi term, ensoept tiie fizst in eaeh 1 bracket, by , we find A. \ \->tif^\ — -^t-v'-^-. — \ \ — h (4 4 4 i ' 4 4+* '«+"-! i 4 4 ^ But 4fi 4+« 4+— 1) * — + — . . .+ v'^"-* \ = a^— i=ithe price of a 4 4 4 ) temporary annuity of one dollar for m — 1 years ; and 26 LETTfiB on ASSURANCE, tr'-H 4.„--s 1_. . . 1 =tlie price of a tern 4 4 4 porary anunity of one dolkr for «— 1 years. Whence or A=:a— . (26,) If ?«=1, wliicli is the case of a single pi:muum» then 'i*^i='^c^=o, and If fn=n, which gives the case of au auuual loading on an annual premium for life, then l+",.;:=I=l+'v,.^, and As before remarbwl , iSm annoal premium should first be found, and its appropriate loading determined. This will give 'a, which, substituted in equation (26), will give the value of A to be charged npon a angle or limited number of pr^ninms, as the case may be. The factor , which measures the probability that any one designated member, whose age is .r-f-w, will die in the current year, is only true in proportion as the mortality tables are true. But these, at best, are but approximations to the truth — dose, to be sure, but still, only ai^proximatione. And again, there is no absolute certainty that in all cases the assured knows his own age. Here, then, is a range of unoertainiy quite broad enough for the asBumpticm that any slight change in this factor, less than half a year, would as probably prove an approach to as a departure from the real truth. And as it is desirable to free the equation for sufplns from all ambiguity in its application, and to diminish LETTEE ON I.IFE ASSUKANCE. 27 the number of its terms to save labor, it is suggested that in all cases the anniversary of the date of natural birth, as designated by the assured, be made coincident with that of the anniversary of settlement nearest thereto. This would have the effect to increase the affix numbers to the subscript x by unity, at the close of every calendar year. DISTRmnON INTEREST FACTOR. One of the most important op^ntions an actuary has to per- form is, as we have just seen, to divide a given surplus among the pohcy holders of a company upon the principles of equity ; that is, upon the contribution plan. The formulas for this have just been explained. But there is a very important feature about these formulas yet to be considered. An actuary may attempt the division, and carry his work to a conclusion, and yet find that he has not attained his objeefc. He may not have divi- ded all his surplus, or he may have divided more than his sur- plus. The (q»e«ition has, from its very nature, been tedious to the computers and vyeiisive to the company, and failure be- comes a serious matter ; because, in addition, it produces uneasi- ness and dissatisfaction among the policy holdan. Thequesticm now is, how shall &dlure be prevented with certainty? answer is, by finding a suitable value for the interest factor, i' , before the process of division is begun. To this end it is re- marked, that if the surplus arose only from diminidied mortal- ity and expenses, and increased and uniform interest on loans, all the quantities iu the first member of equation (25) would be known ; and, therefore, the second member or individual share <rf Burplua would be known. But there are other sources that pour their proceeds into the general surplus, such as forfeit- ures, rents, sales of investments, etc., etc., and as these addi- tional contributions may be regaided as inteiesfe gained upon the company's working capital, a rate of interest should be found which will embrace them as well as the earnings on money actu- ally loaned. This rate once found, and placed in equation (25) for i" , will secure the exact division of the surplus. The equa- tion (25) considers the sum assured to be one dollar, and when multiplied by the sum assured, called A and performing the op- 28 LETTER ON LIFE ASSUKAJiCE. erations indicated, as regards the factor / , it is fonnd, after transposing aU the terms which axe independent of i to the sec- ond member, that -(1^ ).X.x_(i__). ^ L.il-S^-); and conceiving aU the equations for aU the poUcies to be imtfe-n <rat in this way and added together, we get, by employing the sign S to indicate, as before, the process of summation. But vZ X, ^ ,=X=sum to be divided. Substitating this, and solving with respect to »', there will lesnlt t 2^ ^ / , le ) SZ. j i2.+,_,+e-f-^ f -f Ml )[ ' 2a ) (27.) whence this role to find the interest factor that will divide a given surplus, viz : Increase the sum to be divided by the tabu- lar interest on t^^^eser^s at the beginning of the year and the net premiums ^ m S^T Siwriiig the year ; diminish the sum by the surplus loadings and charges for mortality losses ; divide the difference by the sum of the reserves at the beginning of the year, increased by the surplus loadings and the sum of the frac- tional premiums that carry the policies from their anniversary dates to end of current fiscal year. The value of X is found from tiie equation LETTER ON LIFE ASSUBANGE 29 jr=il-2i?.— 2.(1-/) (W-O-^A (23.) in which A denotes the gross assets ; ^R, the sum of the reserves to end oLfiscal year ; 2 (1 — t) (ks^) the sum of the unexpended Cpmnn l^mBSu3m ^' ^^^o ta mma that carry the policies to the ends of their anniversary years ; and 2Z) the sum of unijaid death chums, or other claims that may not have been adjusted. I have dwdt the longer on this mode of dividing sur^^lus — the oontribntioii plan — because of its intrinsic merits, and of the ease with which it may, without great caation^ be ap* plied to BggTAyBte the evils it was intended, and is so admirably suited, to remedy. I have now finished the task I had proposed to myself, to wit : that of running two of the more common and popular policies of assurance through a mutual company for the purposes of expla- nation and illustration. It was not in my intention to write a treatise on life assuiancey but merely to demonstni^ the relia- bility of its simple principles by actual examples ; and I think I am justified in the conclusion, that no one can study and under- stand what I have written, and avoid the conohision, that of all kinds of business, tiiat of life assurance is, when properly man- aged, the safest and surest ; and that the wisest and most pru- dent disposition a person of moderate income, and having a fam- ily to care for, can make of his little savings, is to invest them in a i^olicy of assurance with some company known to conduct its affairs upon true assurance principles. If a life assurance oompany fail, you may with certainty look for the cause in some gross actuarial mistakes, wasteful and un- necessary expenditures, or gross mismanagement of its funds — pOTliax>s all of these combined. A oompany suffering under all of these ills has but one alternative, and that is, to wind up by reassuring, as far as its assets will permit, in some other respon- sible company, and pass to the category of things that were. But this remark is just as applicable to all corporations as to insurance companies.