You are reading a page from An Essay on Probabilities and their Application to Life Contingencies and Insurance Offices, Augustus de Morgan (1838)
Part of the American Term Life Insurance History Project
Term Life Insurance
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begin after n years, signified by n In -l- t, or by nl t, is de-. termined by the same formula as an annuity on the life of B, to begin after the death of A, signified by AIB.
nln+t=ln+t—l(nn+t)    AIB=IB-lAB
but in the first I write In instead of !(n n-1--t) since the joint continuance of n and n+t years must be n years.
The only difficulty of this notation is the necessity of remembering the distinction by which it appears whether nln+t means that a payment is to be made at the end of the nth year or of the (n+ 1)th. In the case of AIB the first payment is to be made at the end of the year in which A dies, which is always intelligible, since it is considered as an infinitely small probability that A should die at the moment which divides two years. But since the term of n years does expire at such a moment, the analogy which connects the symbols of terms certain and contingencies points out no rule. Let nI stand for a perpetuity of 11., the first payment to be due at the end of n+ 1 years, then n—11 stands for a similar perpetuity due at the end of n years. But since the introduction of a new figure may turn the attention off n, the datum of the problem, and since analogy does not require us to strike off a whole year from the term, let n-I signify a perpetuity deferred for a term (no matter how little) short of n years, that is, payable in n years. And by analogy -IA will signify an annuity due on the life of A. The symbol n-In+t must be treated as if it were
n—lln+torn-Ilt+1
The symbol of a perpetuity of 11. to commence from the present time (that is, payment at the end of a year), is simply I. This is found as in p. 184., and by subtracting IA, the value of an annuity on the life of A, we obtain I — I A, the value of the reversion of a perpetuity, of which payment is made at the end of the year in which A dies. At the end of that year, the holder of the reversion will have in possession and
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211    eSSaY ON PROBABILITIES.
expectation, an equivalent to -I or 1+ I, or the value of a perpetuity due. Consequently, since the present value of 1 + 1 to be received at the death of A is 1 — IA, that of 11. will be found by dividing the latter by the former : or,
The present value of ll. to be received at the end of the year in which A dies, is found by subtracting the present value of an annuity on the life of A from that of a perpetuity, and dividing the remainder by the present value of a perpetuity due, or one year's purmore than the present value of a perpetuity.
EXAMPLE. (Northampton tables, 3 per cent.) What is the value of Il., to be received at the end of the year in which a life of 30 shall fail ?
    
Perpetuity of ,    l at 3 per cent.    X33.3
    Value of annuity on life of 30    116.9
    34'3)    16.4(478
In the preceding rule any status may be substituted for a single life, and the value of the annuity which is to be paid as long as the status lasts is connected with the present value of 11. to be received at the end of she year in which the status fails, by the preceding timple rule.
The premium which should be paid (first down, and afterwards at the end of each year), is an annuity due upon the life or status, and is therefore worth _I A or 1 + IA year's purchase. Consequently the premium which should be paid for the 11. above described is the pre-ceding present value divided by one year's purchase more than the annuity is worth. In the example, divide 478 by 1 +16.9 or 179, which gives 0267, so that 21. 13s. 6d. is the premium for insuring 1001. at the end of the year in which a life of 30 fails.
The following rule is somewhat shorter, in the case in which the premium only is required, and not the present value.
QUESTION. To find the premium which should be paid (first down, &c.), during the continuance of a
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status, to insure 11. at the end of the year in which that status drops.
RULE I. From the quotient of a perpetuity divided by a perpetuity due, subtract that of an annuity on the status divided by an annuity due.
54.3) 33.3 (9708    17.9) 16.9 (9441 9441
0267 Answer, as before.
RULE II. Divide 1 by the value of an annuity due, and by that of a perpetuity due, the difference of the quotients is the premium required.
34.3)1( 0292    17.9)1(0559 0292
0267 Answer, as before.
A perpetuity divided by a perpetuity due, is the present value of ll. to be received a year hence, and may be taken from the following table : —
    
p. C.        p. c.    P. c.    
    2    9804    4    9569    9    9174
    2i    9756    5    9524    10    9091
    3    9709    6    9434        
    3i    9662    7    9436        
    4    9615    8    9259        
From the preceding rule an illustration of the reason of it may be derived, which I give professedly as an exercise of ingenuity to those who may be beginners in the subject. Let there be two persons, one of whom holds a perpetuity and the other a life annuity, each of 11. Both the perpetuitant  and the annuitant desire


If the holder of an annuity be an annuitant, the extension of lanis justifiable, by which the holder of a perpetuity may be called a perpetuitant.
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216    ESSaY ON PROBABILITIES.
to commute their interests for interests due : that is, the perpetuitant, instead of ll. a year hence and so on, desires to receive a fraction of a pound now, and the same fraction at the end of every year ; and the same for the annuitant. Say the value of money is four per cent., then the perpetuitant desires to change an interest which is worth twenty-five years' purchase into an equivalent interest worth twenty-six years' purchase (or income) ; consequently his year's income (now due, &c.) must be only ael. Say that the annuity is worth ten years' purchase ; then by the same reasoning the yearly income of the annuitant (now due, &c.) must be only sil. The second is less than the first; whereas the original incomes were the same, both ll. But there must be some consideration which the commutation gives to the annuitant, and for which this greater diminution of his income is the payment; and it is as follows : —Since the commutation forestalls each successive payment, giving it (or the substitute for it) a year before it becomes due, the annuitant would receive, if his income were made equal to that of the perpetuating, the 11. which, had he lived, would have become due at the end of his last year, but which his death hinders from becoming due. This difference of income (=e -4! )1. is therefore equivalent to preventing his receiving 11. at the end of the year in which he dies, and it is taken from him now and in every succeeding year of his life. Consequently it is the premium which such an annuitant should pay to receive 11. at the end of the year in which he dies ; and it is also the result of the first preceding rule.
The second rule may receive an explanation of a similar kind. I now reverse the problem, and ask the following
QUESTION. If an office charge the premium p for insuring 11. at the end of the year in which a life (or other terminable status) drops, what should we infer that they suppose to be the greatest possible value of an annuity to continue during the remainder of that life
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or status ; that is, what is the value of an annuity on that status, which is such that the office must be ruined if the truth falls below it?
RULE. Take the rate of interest which money really makes ", and subtract the premium for 11. from the present value of 11. to be received at the end of a year (see last table) : divide the remainder by the excess of unity over the remainder, and the quotient is the number of years' purchase in the present value of the annuity.
EXaMPLE. A society professes to insure lives of 35 at a premium of 3 per cent. on the nominal sum insured ; what is the lowest value of the annuity on such lives at which this can be done ?


At 4 per cent. I1    ='9615    1'0000
(page 200.), A 11 1    = 0300    9315
'0685)9315(13.6    0685
ANSWER. Such an office cannot permanently stand (as far as this one species of bargain is concerned), unless the value of an annuity on lives of 35 (at 4 per cent.) be more than 13.6 years' purchase.
Generally speaking, contracts of insurance are not made for the end of the year in which the party dies, but for payment at a given number of months after the parties' death is proved and the claim made. If this agreement were always made for six months after the real death, the office would, one party with another, neither gain nor lose, while for every month less than six, the office gives that month's interest to the parties' executor, while for every month more than six by which payment is deferred, the office takes a month's interest. I believe no office defers its payments more than six months after the claim is made ; and the difference is rendered imby the probable errors of the tables, which require too large a covering profit to make it worth while to take such a circumstance into account.
This is an essential element, but cannot be very accurately determined : something above the truth should be assumed.
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The premium demanded by an office is that charged by their tables at the age which the party will attain at his next birthday ; thus if a person desire to insure his life the day after he attains 31 years complete, he will be required to pay the same as if he had deferred completing the insurance till the day before his thirty_ second birthday. This is, one party with another, a gain of half a year to the office. Thus, the Northampton table at 3 per cent. giving 16.7 and 16.5 as the values of annuities at the above-mentioned ages, all parties who have passed 31 years at their last birth day are considered as having lives worth 16.5, whereas they are worth, one with another, 16.6. The tables are not sufficiently accurate to make the effect worth caring for.
A party having made an insurance, and paid one or more premiums, the instrument by which the right to receive the stipulated sum at death on payment of a stipulated premium is conveyed, is called a policy of insurance. The value of this policy is then easily determined ; at least what we may call its office value, supposing the tables of the office to be perfectly correct. A person aged thirty insures for 1001., for which he pays, say Si. ; he continues to pay this premium until the age of fifty, at which time, if he had began to insure, the annual premium would have been, say 51. Suppose that the holder of the policy wishes to sell his interest just before he would otherwise have had to pay another premium, it is plain that he then offers for an insurance on the life of 50, a better bargain than the office would offer, since the buyer of the policy (who pays all future premiums) will acquire, in consideration of an annuity due of 31. upon the life of A, that which the office would not sell for less than an annuity due of 51. upon the same life. The difference, or an annuity due of 21. upon the same life, is the value of the policy.
RULE. To find the present value of a policy of insur ante, at the moment before a premium becomes due, sub-tract the premium which is to be paid from the premium
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which would be paid if the same party made the same insurance at the present time. Find the present value of an annuity on the life of the party insured, of the same yearly amount as the preceding difference, and this value, increased by one year's purchase, is the present value of the policy.
To find the value of the policy immediately after a premium is paid, add the premium just paid to the result of the preceding rule. It would not be worth while, in the present work, to give a rule for any intermediate value. (See Milne, p. 283.)
But in finding the real value of a policy, there are one or two circumstances to be considered, of which no mention is made in the preceding rule. The buyer of the policy, being uncommitted by any previous act of his own, is not bound to consider the premium of any one office as a standard. Suppose that in the preceding example, another office of equal solvency can be found, which will insure a life of 50 at 4 per cent. instead of 5: the buyer, therefore, may consider that the seller offers him for 31. a year during his life a benefit which he might buy elsewhere for 41., and that he should therefore pay only the value of an annuity due of 11. instead of 21. But since the two offices cannot be together parties to any transfer of policy, the pre-ceding case will only serve to show that it may be more prudent for a person who has money to invest, to lay it out at once in insuring lives in a cheap office, than in buying existing policies iii a dear one. It is to be remembered that the lower premium in the preceding rule is to be paid, by bargain already existing, while the higher one is hypothetical, depending on the buyer's opinion of tables of mortality. That the office which demanded and obtained the Si. would demand the 51. for an insurance now to commence, must be no consideration for a person who is merely thinking how to lay out his money to the best advantage ; it may be by buying the policy which is offered to him, or by insuring his own life, or that of some one else, in the
220    ESSAY ON PROBABILITIES.
same or another office. It is his business to consider what he is likely to have to pay, in the shape of future premiums, and not what an office, which must be on the safe side, has thought fit to suppose it will have to receive. Putting out of view the state of health r of the party insured, I should think it most advisable to calculate the value of policies by finding the present value of the sum insured, and also that of the premiums to be paid, from the tables which best represent healthy life, and using the rate of interest which money will really obtain, rather above than below ; that is, I should use the Carlisle tables at 4 per cent. The profits guaranteed by the office, if any, should be duly considered. Thus suppose a person at the age of 30 had insured for 10001. in an office which demands 251. premium for that insurance, and returns no profits, and suppose that twenty years have elapsed, so that the life insured is now at the age of 50, what is the real value of his policy? The value of 11. to be received at the death of a person aged fifty, by the Carlisle tables at 4 per cent., is (p. 214.), 25 — 12'9 divided by 25 + 1 or 465: that of 10001. is therefore 4651. If a premium be just becoming due, the present value of all the premiums is therefore 1 -1 12.9 or 13.9 years' purchase; and 13.9 x 251. is 347.51. Consequently 465— 3471-, or 117y., is what I should consider to be the value of that policy. But if I took the tables of the same office, which require a premium of 471. at the age of fifty, and which, with some variation, are derived from the Northampton tables at 3 per cent., I should find by the rule in p.218., (1+12.4)x(47—25), or 2951. nearly. So great is the difference between policies valued by the nearest approximation which exists to the actual truth, and then valued by the tables which offices adopt for their own security.
The office itself, which takes an advantage of the buyer when the policy is first created, may reasonably
* Of course the policy of a person whose health has very much declined since he effected the insurance, is of higher value on that account: but this cannot be made the subject of calculation.
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allow that advantage to the insured, if he afterwards desire to sell his policy to the office itself. I am not aware of the exact rule which is followed by the offices in this respect, except in one or two cases, in which the plan is, or was, to follow their own tables, with a certain deduction from the result, and to give the difto an insured party who desired to sell his policy. This is well enough in the case of offices which return profits ; but if such a rule be followed by those which do not, it may amount to a contradiction of their profession in the case of the sale of policies; and may become in effect an allowance of that share in the profits to those who desire to leave the office, which they re-fuse to grant to those who continue. To prevent such a result, I believe the offices who would be liable to it, make a large deduction from the value of policies, as indicated by their tables.
All the preceding rules apply to any given status as well as to a given life. Thus, to effect an insurance on the survivor of two lives, the present value and the premium (payable as long as either is alive) are to be found by using IA +1B — I AB for the value of the aninstead of IA. I now proceed to some simple cases of insurance, where the payment on one party's death is made conditional upon another party being alive to receive it.
The symbol Al (1B) denotes the value of an annuity upon the joint continuance of one year and the life of B, payment being made at the end of the year in which A dies. It is therefore necessary that B should be alive at the end of the year in which A dies. But in the usual conditions of contingent insurances, it is sufficient that B should be alive at the moment in which A dies.
Let this be expressed by AIiB; it is then evident that
A'I1B is greater than AI1B. The following preliminary considerations will be necessary.
PROBLEO.—Required the value of an annuity on the joint lives of A and B, to be paid at every end of a year at which B shall be alive, provided A were alive
222    ESSAY ON PROBABILITIES.
at the beginning of the year. This may be denoted by I (A...B).
The condition that a life shall be alive at the beginof the year must be, in tables of averages, the same as that of a life a year younger being alive at the end of the year. For example, suppose that of 500 persons of the age of n—1, 493 attain to that of n, then 500 annuities granted on the lives of A,t_t will be equivalent to 493 granted to An, if the latter be payable at the end of any year in which A shall have been alive at the beginning ; and the same for any joint lives combined with both. Thus 500 annuities granted on the joint lives of An_ i and B, payable as usual, are equivalent to 493 on the joint lives of A
and B, payable as long as B is alive at the end, and A was alive at the beginning, of a year. Hence 1 (A...B) is ;!1 of IA,B, where A, stands for a life a year younger than A. Hence the following RULE. To solve the preceding question, multiply the value of a joint anon B and one year younger than A by the number alive at that younger age in the table, and divide by the number alive at A's age: the result is the present value required. Or more concisely, divide 1 (A,B) by the chance which A, has of living a year.
Now let us ask, by how much does I(A...B) as above described exceed CAB. The only possible case in which a payment will ever be made upon the first annuity, and not upon the second, is when A dies before B, for both are determined by the death of B. When A dies be-fore B, the annuity will be paid at the end of the year upon :(A....B), if B be alive, but not upon CAB. Conthe excess of the first over the second is the value of £1 to be received at the end of the year in which A dies, provided B be then alive.
Or,
AI1B=i(A...B)—IAB,    BI1A=I(B...A)-I AB
Also the present value of an annuity to be paid at the end of any year in which both A and B were alive at the
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beginning, or lAB..., is evidently IAB increased by ABI 1, the present value of an insurance of one pound on the joint lives.
Problem. Required the present value of XI to be paid at the end of any year, provided that both A and B die in that year ; which has been signified by A : B5,
Grant the following annuities ; —
IA B, on the joint lives of A and B.
I(AB...) the same as IAB, but to be also paid at the end of the ycar in which the joint existence fails.
Take ip exchange the following annuities : —
I(A...B) and I(B....A) two annuities to be paid during the joint lives, and also at the end of the year in which the joint existence fails, provided B in the first, and A in the second, be alive at the end of the year.
The balance of this transaction will be £1 to be paid at the end of any year, provided B and A both die in the year. For as long as both are alive, two annuities are payable each way ; if A die and B remain alive till the end of the year, CAB has ceased, !(AB...) is payable, but I(B...A)hasceased, and (A...B) is payable; similarly in the case of B dying and A remaining alive. If both die in one year IAB has ceased, but I(AB...) is payable, while IB...A and IA...B have both ceased. Consequently, the only possible payment which the grantor has to make, over and above those which he receives, is the L1 in the question proposed; or
A:—CAB+CAB...—~A...B-I B...A
We are now in a condition to solve the final PROBLEM. Required the value of to be paid at the end of the year in which A dies, if B should have been alive at the
moment of A's death. This is denoted by A1B. When A dies before B, either B survives till the end of the year, or dies in the intermediate time. The
insurance on the first risk is worth All B determined in p. 222.; on the second it is worth half the result of the last problem, if it be considered that the chances of
224    ESSAY ON PROBABILITIES.
A dying before and after B, in any one given year, are equal. We have therefore
IA...B—IAB+{IAB+IAB...—IA...B—IB...A}
Or,
2{IAB...—IAB+IA...B-IB...A}
But, (p.223.)        _ IAB...—IAB isABII
Whence the final result is,
1{ABII +IA...B—IB...A}
RULE. For determining the value of 11. payable at the end of the year of the death of A, provided B be alive at the moment of A's death.
IA...B=8115
All B=522, which in work cited
I AB=7 593    [is52556. 0.522
The present value of a pound, to be received at the end of the year in which A dies, provided he survives B, or B:AI1, is readily found from the preceding:
i    2
for since A must either die before or after B, the sum of the two must be the present value of 11. to be received at the death of A, independently of B. That is,
B:AI l —Ail — A I I B = Ali — All B very nearly.
The present value of an insurance is also that of the reversion of a fixed sum ; since it is the same thing whell. is to be received from an office, or conditionally under a will, or in any other way. The reversion of a
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226    ESSAY ON PROBABILITIES.
perpetuity should be treated as that of the value of a perpetuity due at the end of the year in which the life drops.
PROBLEM. Required the present value of 11. to be received at the end of the year in which A dies, provided that event take place before the expiration of t years
from the present time : signified by A 1 1 t.
Suppose a person to have the certain reversion of a perpetuity due at the end of t years, or sooner, if A die before t years are expired : the reversion of the perpetuity after the failure of the joint existence of A and t years, is 1 -' A t, which can be found from page 206. But this is more than that fraction of a perpetuity due at the end of the year in which A dies, which will pay for the chance of entering on it before t years are expired: for part of it expresses the value of a perpetuity due, which, though A should be alive, is to be entered on by the failure of t years. If t t a be the chance that A is alive at the end of t years, then t t a x t ! must be deducted, as being expressed twice in the preceding : consequently
1—~At — tta x tl
is the present value of a perpetuity due, or -I, to he entered upon at the end of the year in which A dies, if before t years. The preceding then divided by -I gives the present value of It. to be received under the conditions of the question. But a perpetuity created at the end of t years, or tI, divided by a perpetuity now due, gives the present value of I1. to be received at the end of t + I years ; which gives the following
RULE. From the value of a perpetuity subtract that of an annuity on the given life for t years, and divide by the value of a perpetuity due. From the quotient sub-tract the present value of one pound to be received at the end of t 4- 1 years, if the life be in being at the end of t years: the difference is the present value of It. to be received at the end of the year in which the life drops, if before t years have expired. The present value just
ON THE VALUE OF REVERSIONS.    227
found, divided by that of an annuity due on the given life for t years, gives the requisite premium.
And since the present value of such an insurance as the preceding, together with the present value of 11. to be received at the end of the year in which A dies, if after t years, make up the present value of 11. to be received in any case at the death of A, the third diminished by the first will give the second. But it will be prefer-able to make an independent investigation of this case.
PROBLEM. Required t : All the present value of 11.

to be received at the end of the year in which A dies, provided t years shall have previously expired.
If from the present value of a perpetuity deferred for t years, we deduct that of an annuity on the life of A deferred for t years, we have the value of a deferred perpefurther suspended during the term by which A out-lasts t years, and to commence at the end of the year in which A dies : or not to be suspended at all if A should die in less than t years. Take away the value of a perbeginning from the end of t years, if A should have died in the interval, and we have remaining the present value of a perpetuity due at the end of the year in which A dies, if that be deferred beyond t years. This last is therefore
11— tIA —(1—t t a) x
where t t a is the chance of A living t years. This can be reduced to
ttaxtl —tIA
Divide this by the value of a perpetuity due, and we have the present value of 11. receivable on the same con
ditions.    But tl divided by -1 gives t 1, as before; whence the following
RULE. Multiply the present value of 11. receivable at the end of t+ 1 years by the chance which A has of living t years ; and from the product subtract the quotient
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228    ESSaY ON PROBABILITIeS.
of a deferred annuity on A's life, divided by a perpetuity due : the remainder is the present value of 11. to be received, if A outlive t years, at the end of the year in which he dies.
The preceding pages contain all those cases which most usually occur in practice; but it is to be noticed that various modifying circumstances will present themselves in different problems, for which no general rule can be given. I now proceed to problems in which successions occur ; that is, in which a benefit depends upon the continuance of a status which does not begin until another status is finished.
In the case of an annuity for a certain term of years t, to begin payment at the end of the year in which A dies, we have obviously to consider a benefit which, at the end of the said year, will amount to an annuity due, of t payments ; or 11. augmented by the present (as it will be then) value of t — 1 future payments. This then value can be found as in page 1 85., and, being multiplied by the present value of 11. receivable at the end of the year in which A dies (found in page 214.), gives the present value of this contingently deferred annuity.
In the case of AFB, an annuity on the life of B after the death of A, we certainly have a succession, but it is one which may never exist. To make a problem which may come under the present division of our subject, we must imagine that, at the end of the year in which A dies, a new life may be nominated at pleasure, which is then to be of a given age. If P be the value of an annuity upon such a life, then, according as the benefit is an annuity, or an annuity due at the end of A's year of death, we find the present value of P or 1 + P to be received at the end of that year. The result is the pre-sent value of the succession. This problem includes that of finding the value of the next presentation to a living. The patron of a living of 5001. a year may consider that he gives the clergyman whom he presents 1001. a year (or whatever may be called liberal remunerfor a curate) for work and labour, and the remain-
ON THe VALUE OF REVERSIONS.    229
ing 4001. as a free gift. If he sell the next presentation lie must therefore consider that he sells 4001. a year (not 5001., since that would be to allow the clergyman no salary* for his labour), to be paid yearly during the conof a life to be named by the buyer, at the de-cease of the present incumbent. And, since the right to name new incumbents of 24 years of age is part of the bargain, the patron will require a sum corresponding to the value of an annuity upon a life of that age. De-ducting a sum for first fruits, probability of expenses from dilapidations, &c., which must be determined by the circumstances of each case, the remainder is the net present value of the living. It would probably be most fair to value the interest of the purchaser as if°the new incumbent would come into half a year's revenue at the end of the year in which the present incumbent dies.
QUESTION. What is the present value of the next presentation to a living of which the average annual income is £s, the salary of a curate'' being iv, and f the estimated expenses at entry. Let A be the value of the incumbent's life, and P that of a life of 24 years of age. Find the present value of P + i, to be received at the end of the year in which A dies (p. 214.), and multiply the result by the excess of s over v ; from this deduct the present value of f, to be received at the end of the year in which A dies, and the remainder is the net present value of the next presentation.
The perpetual value of an advowson (that is, of the right to nominate the incumbent in all time to come, after the decease of the present one) is generally valued as the reversion of the net income after the death of the present incumbent. But the expenses of entry, first fruits, &c., should he considered as a fine levied on the property at the death of every tenant, in diminution of
* The right of selling livings is therefore a bond fide right to alienate all the church property which is in private hands, with the exception only of that minimum which will obtain a curate.
t If the living be one on which a curate must he kept by the incumbent, the salary of two curates should be deducted from the yearly revenue in the valuation,
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230    eSSAY ON PROBABILITIES.
the total value. To the problems connected with this subject I now pass.
A great many interests are held in this country on the consideration of rents, fines, or whatever they may be called, which are not paid at any fixed time, but at the deaths of successive lives which are named, each life being nominated, and the rent or fine paid, at the death of the preceding nominee. Leases held under ecclesiand other corporations, copyholds, &c., are in-stances. By a statute of Henry VIII., corporations are permitted to lease lands for three lives, or twenty-one years; so that it may be suspected the legislature imagined the average term of the duration of three lives to he 21 years ; or that, any three mature lives being named in one set, and a large number of such sets being taken, and each set being considered as a status to last as long as any one of its lives was in being, the average duration of such a status was 21 years. If this were the opinion, and grounded upon any thing like experithe value of life in that day must have been incredibly below what it is at present: but it must be remembered that in that day of insecurity few people would venture on the life of a child or a woman ; and that in all probability the lives contemplated were those of men of middle age. However this may be, since that time the tenure of lease upon lives has become excommon, it being understood that the lives which drop are renewable upon the payment of a fine, either fixed or at the discretion of the lessor.
It is, of course, the interest of the lessor that the lives should be as bad, and of the lessee that they should be as good, as it is possible : but the lessee, having the noof the lives, will choose the best the tables afford. The rate of interest being settled, the highest life annuity in the tables gives the age which the life nominated ought to have. The best age in the Northtables is 8 years, and in the Carlisle 7 years: for which ages I subjoin the values of annuities at various rates of interest, adding also the age of 24, which
ON THE VALUE OF REVERSIONS.    231
will be useful in the calculation of the values of advowwith the correction above proposed.
    Northampton.    Carlisle.
        i
    Ages.    3p.c    4p. c.    5 p. c    6 p. c.'3 p.c.    4 p.c.    5p. c.    6 p. c.    Ages)
    8    09    17.7    15'2    13 3    23'9    19'8    16.8    14'5    7
    24    18'0    15.6    13'7    12'1    20.9    17.8    15'4    13'5    24
QUESTION. At the end of the year in which A dies, a fine of 11. is to be paid, and a new life nominated, of which the value will then be P : at the end of the year in which P dies, another fine of 11. is to be paid, and a new life P nominated, and so on for ever: what is the present value of all the fines, or what present money must a person be considered as paying who receives an estate charged with the preceding liabilities?
This problem, as will be more fully explained in the second appendix, was incorrectly solved by every writer on the subject, down to the time of Mr. Milne, whose solution, though perfectly correct, is in a difficult form. The coincidence of the rule I now give with that of Mr. Milne will be shown in the appendix cited.
Let us suppose a fine of 11. per annum, first payable at the end of the year in which A dies. If, then, a receiver P were appointed for his life, his interest in the fines, at the end of the year in which A dies, would be 1 +1P; and if at his death a second receiver were appointed, of the same age at which the first was when his term began, the interest of this second receiver at his entrance would also be 1 +1P, and so on. But if the tenant compounded with each receiver on his entrance, for the rents payable during the life of that receiver,. it would evidently be equivalent to paying a fine of 1+ IP at the end of the year in which each dies, and also at the end of the year in which A dies. But the present value of all the rents is a perpetuity diminished by the value of an annuity on A's life, or 1— IA. And if this be the value of a fine of I +1P, then I— IA, divided by
Q 4
232    eSSAY ON PROBABILITIES
1+gives the value of a fine of 11. in the same cirHence the following
RULe. From the value of a perpetuity subtract that of an annuity on A's life, and divide the remainder by the value of an annuity due on the renewal life at the time of renewal.
If there be several lives in the lease, apply the pre-ceding rule to each life, and add the results : for the several contingencies do not interfere with or depend upon each other, nor will the case of more lives than one in one lease differ from that of several leases each on one life. The most convenient method is as follows:
RULE (for several lives). Multiply the value of a perpetuity by the number of the lives, and subtract the sum of the values of the annuities on the different lives: divide the result by the value of an annuity due on the renewal life at the time of renewal.
QUESTION. An estate of the clear annual value of £a per annum is to be leased on n lives, A, B, C, &c., with liberty to renew at the end of each year in which a life drops, the best life in the tables being P : what fine should be paid, on the supposition that the puris to have a given rate of interest for his money?
RULe. Find the value of the perpetuity of in per annum; multiply it by the value of an annuity due on the renewal life at the time of renewal, and divide by the excess of n times the value of a perpetuity of 11. over the sum of the values of annuities on the lives of A, B, C, &c. : the quotient is the value of each fine required. But if a sum s be paid down, and the rest of the value of the estate is to be paid in fines, then subtract s from the perpetuity of La per annum, before using it in the preceding rule.
EXAMPLE 1. The lives in possession, A, B, and C, are 35, 48, and 60 years of age, and the fine paid on renewal is 3001. 'What is the present value of all the fines, using the Carlisle tables, and interest at 4 per cent.? *
+ I have taken Mr. Milne's example, in order to show the accordance of
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1=25    IA=16.041    IP=19.792
3    IB=13.419    1
IC= 9'663
75    20.792)35.877(1'7255
39.123    39'123    300
35.877    517.65
In Milne    517.6296
The present value of every single pound of the fine is P72551., which, multiplied by 300, gives 517.651.
EXAMPLE 2. All things remaining as above, the pre-ceding lease, worth 1201. per annum, is purchased for 25001. and a contract for fines on renewal. What should the fine be?
    
I= 25    1+IP=20.792    
    120    500    
    3000    35.877)10396(289.77    
    2500    289.782    in Milne.
    500        
The answer is 289.771.
QUESTION. What yearly rental should the fines be considered as amounting to ; and what should be paid by the lessee annually to an insurance office which would undertake to pay all the fines as they become due ?
These two questions are the same, and the answer to both is, —the yearly interest upon the present value of the fines. Thus, in the first preceding example, the lessor's interest, at 4 per cent., is worth 20.71. per annum for ever; which the lessee might either pay to his landlord, as a commutation of fines, or to an insuroffice, which should take them upon itself.
QUESTION. What is the present value of the next fine upon the renewal of the first life which drops of the three; A, B, C ?
the rules. The slight difference arises from Mr. Milne's rule requiring an interpolation, which he has very properly thought it not worth while to make. I have taken more decimal places than those previously given, in order to show the accordance more clearly. (Milne, page 365.)
234    ESSAY ON PROBABILITIES.
This is evidently the present value of 11. to be received upon the failure of the joint existence of A, B, and C, and is to be found (page 214.) by subtracting ABC, the value of an annuity on the joint lives, from that of a perpetuity, and dividing by the present value of a perpetuity due.
QUESTION. If the tenant wish to exchange one life for another and a better, how much should he pay to be allowed to do so ?
RULE. If the fine be 11., subtract the value of the inferior life from that of the better one, and divide the difference by the value of a perpetuity due on the relife at the time of renewal. To exchange two lives, or three lives, use the sums of the values of the better lives, diminished by that of the inferior ones.
QUESTION.* If the lessor have only a life interest in the estate on which he grants leases for lives, what is the value of his interest ?
In strict justice to future holders, it ought not to be worth more than the rental calculated in the last quesbut three, continued for his life. But it is the nature of this species of property, that the life interest of the holder is subject to considerable fluctuations of value, the preceding annuity being at one time less and at another greater. A lessor, for instance, who enters when the lives in possession are very old, himself being very young, has nearly a certainty of one fine on account of each, and not much less than an even chance of a second, while his prospect of a third may be worth calculating. But a lessor who enters at an advanced age, against lives which are very young, has a present interest in each coming fine, which may be determined by finding the present value of 11., to be paid when the life drops,on condition the lessor survives(page 223). Indeed, whatever may be the lives in the lease, provided the lessor enters at an advanced age, his interest is deter
This problem, properly treated, would be of extreme complication, and I do not remember having seen it proposed. The method in the text is an approximation.
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mined with sufficient accuracy by finding the values of insurances on the lives in possession, on condition of the lessor surviving. But when the lessor is young, I am not aware of any rule to which I would trust, as making as good an approximation to the value of his life interest as can be made in other cases. Each case must be deter-mined by its own details ; and it will always be safe to begin by calculating what we may call the mean value, namely, the annuity first mentioned ; which may, for any thing I see to the contrary, be a perfectly correct mode of proceeding in all cases.
QUESTION. If the lessor should refuse to renew, and if it be pretty certain that his successor will adopt the same course, what is the present value of the tenant's interest ?
Evidently the clear annual value of the estate, considered as an annuity upon the longest of the three lives, the value of which is determined in page 208.
This awkward contrivance for limiting the rights of corporators over property is prejudicial in its effects, both upon the tenants and the lessors. The former, holding an interest of a comparatively precarious character, have not the same inducement to improve their property which is felt by leaseholders for fixed terms of years. On the other hand, the lessor, in all cases in which he has a personal interest in the proceeds of the estate, has two distinct periods of temptation to an act of equivocal morality. If he be young, he may, as it is called, run his life against those in possession ; that is, refuse all renewals, upon the prospect of a large ultimate gain from the falling in of the old leases : if he be old, he may induce the tenants, by offering easy terms, to change their old lives for young ones, thus impoverishing the successor, by leaving him nothing but long leases, or leases on young lives.
It must very often be a question for the lessee, whether it would not be his wisest plan to refuse all renewal, and to insure a certain sum upon the last surof the three lives by which he now holds. The
236    eSSAY ON PROBABILITIES.
prudence of such a step must depend upon the fine demanded for a renewal. In the case of church leases, I believe it could not often be desirable : and certainly not if they are let so much below their value as has been asserted. But if the fine demanded should be exorbitant, it would then become cheaper to insure the longest of the lives in possession than to pay the demand, The premium for such insurance would be found as in page 214, the value of an annuity on the longest of the lives having been previously found in page 208.
QUESTION. The average value* of a living is £s per annum, and the proper allowance to the incumbent for the performance of the duties is £v; the unavoidable expenses at entrance are if for each new incumbent; and the value of the life of the present incumbent is IA, while that of the new incumbent will be 'P. What is the value'of the perpetual advowson of such living?
RULE. The value of a reversion of l 1. per annum after the death of A being found (by subtracting !A from the value of a perpetuity), and multiplied by the excess of s over v, will give the value of the perpetual advowson, as it would be but for the expenses at entry. For these, deduct the present value of a fine f, payable at the end of the year in which each incumbent dies, the value of each pound of which determined by dividing the reversion aforesaid by one year's purchase more than P. The difference is the net present value of the advowson required. According to a frequent practice of valuing advowsons, in which the expenses at entrance are neglected, the buyer pays them twice over.
* This should include all real profits : for instance, the value of the parhouse as a residence, considered as taken on a strict repairing lease.