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Box 3 

Bartlett, Wcllliamj Hcolmes, Cchambers, 1804-1893. 

Letter on life insurance to Fred S. Winston... 
New York, 1871. 

29 p. 17i om. 

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MAIN ENTRY: Bart l ett . W illia m Ho l mes Chambers 

Letter on life insurance to Fred S, Winston 

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LETTER soi" 


Life AssuRAi^CE 


Feed. S. WmsTOif, Esq., 


Pkof. WM. Urc. BAETLETT. 





^ ON 


} — ^ — 

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 



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. 


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. 




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 


and this put out at interest will, at the end of the year, amount 

/,.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- 



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 



or making 


<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^ 


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. 


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 


[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-. 


dividing the ternis in both numerator and d^imaaaaibox by 4-(l 
and wzitixig 


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 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. 




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 

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 



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 



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. 


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 



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. 


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 


and at the end of the second year to 


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 


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 


a,= ; (9.) 

which is the price sought, and under the form emx>loyed in com- 
puting the commutation tables for annuities on single lives. 


Now analyze formulas (5) and (8) : taking the iirst of tiiese, 
the first term of the second member is 


v.— ; 



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 

Again, — is the m^wure of the probability that any desig- 


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 



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 

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. 


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 



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. : 





E,^r=- — -\n ./.-[. (10.) 



l-H-i ( 1 d, 

,=:- . II .t. 

B^t=^- -11 .t.— \. [XL) 

d, { I,) 
1— /.— 


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 

aod generally for a single net premium, 

i2.,,= -. \ \ ; ■ (13.) 

—■ A J^x^n-l • r 5 

( 1+'" ) 

and for an annual premium, 


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.) 



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. 



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, 



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 

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. 


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 ; 


„ — mX^^,—{\ — m) . Jr,^.,,=net sum to pay for policy ; 



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. 


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. 




«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, 



e=A. — ; 

and denoting the cKhial loss from deaths by a, and that of the 
tables or impiUed by we may write 


and substituting in equation (20), we get 

( ( 2e ) a d^ ) 

t |<i'-i)^-A |l+,'__.(i4.i i)^ -j-(i__)._..(i_i2,^,) L 


or, perhaps, sufficiently near 

t, -J ^(i_i)4.x.(l_^)(l^-^•')^-(l_!).^.(l_i^,^,)l=x^^ 

2X el, ) 




The sum of the actual expenses and that of the loadings will be 
known ttom tbe books of tiie ocnnpuiy. The ratio 



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- 


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 


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 



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 

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- 


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 


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- 


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 


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). 


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. 

(A — A) (l-f/)=A, (!-}-/) — ^A(l-f-0 = accumulation at end of first 

- A(l4-i)+A. 'K{Xj^i)—\ — \ .(l+i), or 


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 


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 


bracket, by , we find 

A. \ \->tif^\ — -^t-v'-^-. — \ \ — h 

(4 4 4 i ' 4 

4+* '«+"-! i 

4 4 ^ 


4fi 4+« 4+— 1) 

* — + — . . .+ v'^"-* \ = a^— i=ithe price of a 

4 4 4 ) 

temporary annuity of one dollar for m — 1 years ; and 



tr'-H 4.„--s 1_. . . 1 =tlie price of a tern 

4 4 4 

porary anunity of one dolkr for «— 1 years. Whence 


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 

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 



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. 


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- 



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__). ^^-); 

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 


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 



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.