SURVIVORSHIP

SURVIVORSHIP, is that branch of mathematics which treats of reversion payments provided one or more particular persons survive certain others. By reversion are meant payments not to take place till some future period. Survivorship forms one of the most difficult and complicated parts of the doctrine of reversion and life-annuities. It has been very fully treated of by Mr Thomas Simpson in his Select Exercises, and considerably improved by Dr Price and Mr Morgan, who have bestowed a great deal of attention on this subject; though some parts of their principles are erroneous.

The calculations are founded on the expectation of lives at different ages, deduced from tables formed from bills of mortality, of which see several examples under the article Bills of MORTALITY. By the expectation of life is meant the mean time that any single or joint lives

at a given age is found to continue; that is, the number of years which, taking one with another, they actually enjoy, and may be considered as sure of enjoying; those who survive that period enjoying as much more time in proportion to their number as those who fall short of it enjoy less. Thus, supposing 46 persons alive all 40 years of age, and that one will die every year till they are all dead in 46 years, half 46 or 23 will be the expectation of each of them. If M. de Moivre's hypothesis were true, that men always decrease in an arithmetical progression, the expectation of a single life is always half its complement ($\backslash lambda$), and the expectation of two joint lives one-third of their common complement. Thus, supposing a man 40, his expectation would be 23, the half of 46, his complement; the expectation of two joint lives, each 40, would be 15 years 4 months, or the third part of 46.

The number expressing the expectation, multiplied by the number of single or joint lives (of which it is the expectation), added annually to a society, gives the whole number living together, to which such an annual addition would in time grow. Thus, since 19, or the third of 57, is the expectation of two joint lives, whose common age is 29, twenty marriages every year between persons of this age would in 57 years grow to 20 times

(A) By the complement of a life is meant what it wants of 86, which M. de Moivre makes the boundary of human life. Thus if a man be 30, the complement of his life is 56.

Survivorship. 19. or 380 marriages, always existing together. And since the expectation of a single life is always half its complement, in 57 years 20 single persons added annually to a town will increase to 20 times 28.5, or 570; and when arrived at this number, the deaths every year will just equal the accessions, and no further increase be possible. It appears from hence, that the particular proportion that becomes extinct every year, out of the whole number constantly existing together of single or joint lives, must, wherever this number undergoes no variation, be exactly the same with the expectation of those lives, at the time when their existence commenced. Thus, was it found that a 19th part of all the marriages among any bodies of men, whose numbers do not vary, are dissolved every year by the deaths of either the husband or wife, it would appear that 19 was, at the time they were contracted, the expectation of these marriages. In like manner, was it found in a society, limited to a fixed number of members, that a 28th part dies annually out of the whole number of members, it would appear that 28 was their common expectation of life at the time they entered. So likewise, were it found in any town or district, where the number of births and burials are equal, that a 20th or 30th part of the inhabitants die annually, it would appear that 20 or 30 was the expectation of a child just born in that town or district. These expectations, therefore, for all single lives, are easily found by a table of observations, showing the number that die annually at all ages out of a given number alive at those ages; and the general rule for this purpose is, to divide the sum of all the living in the table, at the age whose expectation is required, and at all greater ages, by the sum of all that die annually at that age and above it; or, which is the same, by the number (in the Table) of the living at that age; and half unity subtracted from the quotient will be the required expectation. Thus, in Dr Halley's table, given in the article ANNUITY, the sum of all the living at 20 and upwards is 20,724, which, divided by 598, the number living at the age of 20, and half unity subtracted from the quotient, gives 34.15 for the expectation of 20.

In calculating the value or expectation of joint lives, M. de Moivre had recourse to the hypothesis, that the probabilities of life decrease in a geometrical progression; believing that the values of joint lives, obtained by rules derived from it, would not deviate much from the truth. But in this he was greatly mistaken; they generally give results which are near a quarter of the true value too great in finding the present value of one life after it has survived another in a single payment, and about two-fifths too great when the value is sought in annual payments during the joint lives. They ought therefore to be calculated on the hypothesis (if they are calculated on hypothesis at all), that the probabilities of life decrease in arithmetical progression, which is not very far from the truth. Even this hypothesis never corresponds with the fact in the first and last periods of life, and in some situations not in any period of life. Dr Price and Mr Morgan therefore have given tables of the value of lives, not founded on any hypothesis, but deduced from bills of mortality themselves. Some of these we shall give at the end of this article. Mr Morgan has likewise given rules for calculating values of lives in this manner.

M. de Moivre has also fallen into mistakes in his rules

for calculating the value of reversions depending on survivorship: these have been pointed out by Dr Price in the third essay in the first volume of his Treatise on Reversionary Payments; who has also given proper rules for calculating these values, the most important of which are comprehended in the following paragraphs.

Survivorship. Suppose a set of married men to enter into a society in Method of, order to provide annuities for their widows, and that it is finding the number of members, and constantly annuitants kept up to that number by the admission of new members that will be as the old ones are lost; it is of importance, in the case on a first place, to know the number of annuitants that after society. some time will come upon the establishment. Now since every marriage produces either a widow or widower; and since all marriages taken together would produce as many widows as widowers, were every man and his wife of the same age, and the chance equal which shall die first; it is evident, that the number of widows that have ever existed in the world, would in this case be equal to half the number of marriages. And what would take place in the world must also, on the same suppositions, take place in this society. In other words, every other person in such a society leaving a widow, there must arise from it a number of widows equal to half its own number. But this does not determine what number, all living at one and the same time, the society may expect will come to be constantly on it. It is, therefore, necessary to determine how long the duration of survivorship between persons of equal ages will be, compared with the duration of marriage. And the truth is, that, supposing the probabilities of life to decrease uniformly, the former is equal to the latter; and consequently that the number of survivors, or (which is the same, supposing no second marriages) of widows and widowers alive together, which will arise from any given set of such marriages constantly kept up, will be equal to the whole number of marriages; or half of them (the number of widows in particular) equal to half the number of marriages. Now it appears that in most towns the decrease in the probabilities of life is in fact nearly uniform. According to the Breslaw Table of Observations (see ANNUITY), almost the same numbers die every year from 20 years of age to 77. After this, indeed, fewer die, and the rate of decrease in the probabilities of life is retarded. But this deviation from the hypothesis is inconsiderable; and its effect, in the present case, is to render the duration of survivorship longer than it would otherwise be. According to the London Table of Observations, the numbers dying every year begin to grow less at 50 years of age; and from hence to extreme old age there is a constant retardation in the decrease of the probabilities of life. Upon the whole, therefore, it appears that, according to the Breslaw Table, and supposing no widows to marry, the number inquired after is somewhat greater than half the number of the society; but, according to the London Table, a good deal greater. This, however, has been determined on the supposition that the husbands and wives are of equal ages, and that then there is an equal chance who shall die first. But in reality husbands are generally older than wives, and males have been found to die sooner than females, as appears incontestably from several of the tables in Dr Price's Treatise on Reversions. It is therefore more than an equal chance that the husband will die before his wife. This will increase considerably the duration of

of survivorship on the part of the women, and consequently the number which we have been inquiring after. The marriage of widows will diminish this number, but not so much as the other causes will increase it.

If the society comprehends in it from the first all the married people of all ages in any town, or among any class of people where the numbers always continue the same, the whole collective body of members will be at their greatest age at the time of the establishment of the society; and the number of widows left every year will at a medium be always the same. The number of widows will increase continually on the society, till as many die every year as are added. This will not be till the whole collective body of widows are at their greatest age, or till there are among them the greatest possible number of the oldest widows; and therefore not till there has been time for an accession to the oldest widows from the youngest part.

Let us, for the sake of greater precision, divide the whole medium of widows that come on every year into different classes according to their different ages, and suppose some to be left at 56 years of age, some at 46, some at 36, and some at 26. The widows, constantly in life together, derived from the first class, will come to their greatest age, and to a maximum, in 30 years, supposing, with M. de Moivre, 86 to be the utmost extent of life. The same will happen to the second class in 40 years, and to the third in 50 years. But the whole body composed of these classes will not come to a maximum till the same happens to the fourth or youngest class; that is, not till the end of 60 years. After this the affairs of the society will become stationary, and the number of annuitants on it of all ages will keep always nearly the same.

If a society begins with its complete number of members, but at the same time admits none above a particular age: If, for instance, it begins with 200 members all under 50, and afterwards limits itself to this number, and keeps it up by admitting every year, at all ages between 26 and 50, new members as old ones drop off; in this case, the period necessary for bringing on the maximum of annuitants will be just doubled.

To determine the sum that every individual ought to pay in a single present payment, in order to intitle his widow to a certain annuity for her life, let us suppose the annuity 31. per annum, and the rate of interest four per cent. It is evident, that the value of such an expectation is different, according to the different ages of the purchasers, and the proportion of the age of the wife to that of the husband. Let us then suppose that every person in such a society is of the same age with his wife, and that one with another all the members when they enter may be reckoned 40 years of age, as many entering above this age as below it. It has been demonstrated by M. de Moivre and Mr Simpson, that the value of an annuity on the joint continuance of any two lives, subtracted from the value of an annuity on the life in expectation, gives the true present value of annuity on what may happen to remain of the latter of the two lives after the other.

In the present case, the value of an annuity to be enjoyed during the joint continuance of two lives, each 40, is, by Table II. 9.826, according to the probabilities of life in the Table of Observations formed by Dr Halley from the bills of mortality of Breslaw in Silesia.

The value of a single life 40 years of age, as given by M. de Moivre, agreeably to the same table, is 13.20; and the former subtracted from the latter, leaves 3.37, or the true number of years purchase, which ought to be paid for any given annuity, to be enjoyed by a person 40 years of age, provided he survives another person of the same age, interest being reckoned at four per cent. per annum. The annuity, therefore, being 301, the present value of it is 30 multiplied by 3.37, or 101.11.

If, instead of a single present payment, it is thought preferable to make annual payments during the marriage; what these annual payments ought to be is easily determined by finding what annual payments during two joint lives of given ages are equivalent to the value of the reversionary annuity in present money. Suppose, as before, that the joint lives are each 40, and the reversionary annuity 301. per annum. An annual payment during the continuance of two such lives is worth (according to Table II.) 9.82 years purchase. The annual payment ought to be such as, being multiplied by 9.82, will produce 101.11, the present value of the annuity in one payment. Divide then 101.11 by 9.82, and 10.3 the quotient will be the annual payment. This method of calculation supposes that the first annual payment is not to be made till the end of a year. If it is to be made immediately, the value of the joint lives will be increased one year's purchase; and therefore, in order to find the annual payments required, the value of a present single payment must be divided by the value of the joint lives increased by unity. If the society prefer paying part of the value in a present single payment on admission, and the rest in annual payments; and if they fix these annual payments at a particular sum, the present single payment paid on admission is found by subtracting the value of the annual payment during the joint lives from the whole present value of the annuity in one payment. Suppose, for instance, the annual payments to be fixed at five guineas, the annuity to be 301, the rate of interest four per cent. and the joint lives each 40; the value of the annuity in one present single payment is 101.11. The value of five guineas or 5.25 per annum, is (5.25 multiplied by 9.82 the value of the joint lives) 51.55; which, subtracted from 101.11, gives 49.56, the answer.

If a society takes in all the marriages among persons of a particular profession within a given district, and subjects them for perpetuity to a certain equal and common tax or annual payment, in order to provide life annuities for all the widows that shall result from these marriages; since, at the commencement of such an establishment, all the oldest, as well as the youngest, marriages are to be intitled equally to the proposed benefit, a much greater number of annuitants will come immediately on it than would come on any similar establishment which limited itself in the admission of members to persons not exceeding a given age. This will check that accumulation of money which should take place at first, in order to produce an income equal to the disbursements at the time when the number of annuitants comes to a maximum; and therefore will be a particular burden upon the establishment in its infancy. For this some compensation must be provided; and the equitable method of providing it is, by levying fines at the beginning of the establishment on every member exceeding

Survivorship. ceeding a given age, proportioned to the number of years which he has lived beyond that age. But if such fines cannot be levied, and if every payment must be in equal and common, whatever disparity there may be in the value of the expectations of different members, the fines must be reduced to one common one, answering as nearly as possible to the disadvantage, and payable by every member at the time when the establishment begins. After this, the establishment will be the same with one that takes upon it all at the time they marry; and the tax or annual payment of every member adequate to its support will be the annual payment during marriage due from persons who marry at the mean age at which, upon an average, all marriages may be considered as commencing. The fines to be paid at first are, for every particular member, the same with the difference between the value of the expectation to him at his present age, and what would have been its value to him had the scheme begun at the time he married. Or, they are, for the whole body of members, the difference between the value of the common expectation, to persons at the mean age of all married persons taken together as they exist in the world, and to persons at that age which is to be deemed their mean age when they marry.

Method of finding the present value of an annuity to be enjoyed by one life after the expiration of another. Suppose we wish to know the present value of an annuity to be enjoyed by one life, for what may happen to remain of it beyond another life, after a given term; that is, provided both lives continue from the present time to the end of a given term of years; the method of calculating is this: Find the value of the annuity for two lives, greater by the given term of years than the given lives; discount this value for the given term; and then multiply by the probability, that the two given lives shall both continue the given term; and the product will be the answer. Thus, let the two lives be each 30, the term seven years, the annuity 10l. interest four per cent. The given lives, increased by seven years, become each 37. The value of two joint lives, each 37, is (by Table II.) 10.25. The value of a single life at 37 is (by the table under the article ANNUITY) 13.67. The former subtracted from the latter is 3.42, or the value of an annuity for the life of a person 37 years of age, after another of the same age, as has been shown above. 3.42 discounted for seven years (that is, multiplied by 0.76 the value of 1l. due at the end of seven years) is 2.6. The probability that a single life at 30 shall continue seven years is $\backslash frac\{2\}\{3\}$ (B). The probability, therefore, that two such lives shall continue seven years, is $\backslash frac\{4\}\{9\}$, or in decimals 0.765; and 2.6 multiplied by 0.765 is 1.989, the number of years purchase which ought to be given for an annuity

to be enjoyed by a life now 30 years of age, after a life of the same age, provided both continue seven years. The annuity then being 10l. its present value is 19.89l.

Method of finding the value of an annuity for what may happen to remain of one life after another, provided the life in expectation continues a given time. 1. Find the present value of the annuity for what may remain of the life in expectation after the given time, which is done in this manner: Multiply the present value of the life at the given time by the present value of 1l. to be received at that time, and multiply thereto, the product again by the probability that the life in expectation will continue so long. Let the given time be 15 years, and let the person then be arrived at 50 years of age. A life at fifty, according to M. de Moivre's valuation term. of lives, and reckoning interest at four per cent, is worth 11.34 years purchase. The present value of 1l. to be received at the end of 15 years, is 0.5553, and the probability that a life at 35 will continue 15 years is $\backslash frac\{1\}\{2\}$. These three values multiplied into one another give 4.44l. for the present value of the life in expectation. 2. Find the value of the reversion, provided both lives continue the given time, by the rule given in parag. 5th. 3. Add these values together, and the sum will be the answer in a single present payment. We shall now illustrate this rule by an example.

An annuity of 10l. for the life of a person now 30, is to commence at the end of 11 years, if another person now 40 should be then dead; or, if this should not happen at the end of any year beyond 11 years in which the former shall happen to survive the latter: What is the present value of such an annuity, reckoning interest at four per cent, and taking the probabilities of life as they are in Dr. Halley's table, given in the article MORTALITY?

The value of 10l. per annum, for the remainder of the life of a person now 30, after 11 years is 69.43l. The probability that a person 40 years of age shall live 11 years, is, by Dr. Halley's table $\backslash frac\{1\}\{2\}$. The probability, therefore, that he will die in 11 years, is $\backslash frac\{1\}\{2\}$ subtracted from unity (C), or $\backslash frac\{1\}\{2\}$; which multiplied by 69.43l. gives 17.16l.—The value of the reversion, provided both live 11 years, is 17l. and this value added to the former, makes 34.16l. the value required in a single present payment; which payment divided by 11.43l. the value of two joint lives, aged 30 and 40, with unity added, gives 3l.; or the value required in annual payments during the joint lives, the first payment to be made immediately.

(B) The probability that a given life shall continue any number of years, or reach a given age, is (as is well known) the fraction, whose numerator is the number of the living in any table of observations opposite to the given age, and denominator, the number opposite to the present age of the given life.

(C) For the difference between unity and the fraction expressing the probability that an event will happen, gives the probability that it will not happen.

The values of joint lives in these tables have been computed for only one rate of interest; and of single lives in Table III. for only two rates of interest. The following rules will show, that it would be a needless labour to compute these values (in strict conformity to the observations) for any other rates of interest.

ACCOUNT of a method of deducing, from the correct values (according to any observations) of any single or joint Lives at one rate of Interest, the same values at other rates of Interest.

PROB. I. The expectation given of a single life by any table of observations, to find its value, supposing the decrements of life equal, at any given rate of interest.

Solution. Find the value of an annuity certain for a number of years equal to twice the expectation. Multiply this value by the perpetuity increased by unity, and divide the product by twice the expectation: The quotient subtracted from the perpetuity will be the value required.

Example. The expectation of a male life aged 10, by the Sweden observations, is 43.94. Twice this expectation is 87.88. The value of an annuity certain for 87.88 years is (reckoning interest at 4 per cent.) 24.200. The product of 24.200 into 26 (the perpetuity increased by unity) is 629.2, which, divided by 87.88, gives 7.159. And this quotient subtracted from 25 (the perpetuity) gives 17.84 years purchase, the value of a life aged ten, deduced from the expectation of life at that age, according to the Sweden observations. (See the Tables in Dr Price on Reversions, vol. ii.)

PROB. II. Having the expectations given of any two lives by any table of observations, to deduce from thence the value of the joint lives at any rate of interest, supposing an equal decrement of life.

Solution. Find the difference between twice the expectation of the youngest life and twice the expectation of the oldest life increased by unity and twice the perpetuity. Multiply this difference by the value of an annuity certain for a time equal to twice the expectation of the oldest life; and by twice the same expectation divide the product, reserving the quotient.

From twice the perpetuity subtract the reserved quotient, and multiply the remainder by the perpetuity increased by unity. This last product divided by twice the expectation of the youngest life, and then subtracted from the perpetuity, will be the required value.

When twice the expectation of the youngest life is greater than twice the expectation of the oldest life increased by unity and twice the perpetuity, the reserved quotient, instead of being subtracted from twice the perpetuity, must be added to it, and the sum, not the difference, multiplied by the perpetuity increased by unity.

Example. Let the joint lives proposed be a female life aged 10, and a male life aged 15; and let the table of observations be the Sweden table for lives in general, and the rate of interest 4 per cent. Twice the expectations of the two lives are 90.14 and 83.28.

Twice the expectation of the oldest life, increased by unity, and twice the perpetuity, is 134.28, which lessened by 90.14 (twice the expectation of the youngest life),

leaves 44.14 for the reserved remainder. This remainder multiplied by 24.045 (the value of an annuity certain for 83.28 years), and the product divided by 83.28 (twice the expectation of the oldest life), gives 12.744, the quotient to be reserved; which subtracted from double the perpetuity, and the remainder (or 37.255) multiplied by the perpetuity increased by unity (or by 26) gives 968.630, which divided by 90.14 (twice the expectation of the youngest life) and the quotient subtracted from the perpetuity, we have 14.254 for the required value.

The value of an annuity certain, when the number of years is a whole number with a fraction added (as will be commonly the case) may be best computed in the following manner. In this example the number of years is 83.28. The value of an annuity certain for 83 years is 24.035. The same value for 84 years is 24.072. The difference between these two values is 0.037; which difference multiplied by .28 (the fractional part of the number of years), and the product (.0103) added to the least of the two values, will give 24.045 the value for 83.28 years.

General Rule. Call the correct value (supposed to be computed for any rate of interest) the first value. Call the value deduced (by the preceding problems) from the expectations at the same rate of interest, the second value. Call the value deduced from the expectations for any other rate of interest the third value.

Then the difference between the first and second values added to or subtracted from the third value, just as the first is greater or less than the second, will be the value at the rate of interest for which the third value has been deduced from the expectations.

The following examples will make this perfectly plain.

Example I. In the two last tables the correct values are given of two joint lives among mankind at large, without distinguishing between males and females, according to the Sweden observations, reckoning interest at 4 per cent. Let it be required to find from these values the values at 3 per cent. and let the ages of the joint lives be supposed 10 and 10.

The correct value by Table IV. (reckoning interest at 4 per cent.) is 16.141. The expectation of a life aged 10 is 43.07. The value deduced from this expectation at 4 per cent. by Prob. II. is 14.539. The value deduced by the same problem from the same expectation at 3 per cent. is 16.808. The difference between the first and second values is 1.602, which, added to the third value (the first being greater than the second), makes 18.410, the value required.

Example II. Let the value be required of a single male life aged 10, at 3 per cent. interest, from the correct value at 4 per cent. according to the Sweden observations.

First, or correct value at 4 per cent. (by Table III.) is 18.674. The expectation of a male life aged 10 is 43.94.

The second value (or the value deduced from this expectation by Prob. I.) is 17.838.

The third value (or the value deduced from the same expectation at 3 per cent.) is 21.277.

The difference between the first and second is .836; which (since the first is greater than the second) must be added to the third; and the sum (that is, 22.113) will be the value required.

The third value at 5 per cent. is 15.286; and the difference added to 15.286 makes 16.122 the value of a male life aged 10 at 5 per cent. according to the Sweden observations. The exact value at 5 per cent. is (by Table III.) 16.014.

Again: The difference between 16.014 (the correct value at 5 per cent.), and 15.286 (the value at the same interest deduced from the expectation), is .728; which, added (because the first value is greater than the second) to 13.335 (the value deduced at 6 per cent. from the expectation) gives 14.063, the value of the same life, reckoning interest at 6 per cent.

These deductions, in the case of single lives particularly, are so easy, and give the true values so nearly, that it will be scarcely ever necessary to calculate the exact values (according to any given observations) for more than one rate of interest.

If, for instance, the correct values are computed at 4 per cent. according to any observations, the values at 3, 3½, 4½, 5, 6, 7, or 8 per cent. may be deduced from them by the preceding rules as occasion may require, without much labour or any danger of considerable errors. The values thus deduced will seldom differ from the true values so much as a tenth of a year's purchase. They will not generally differ more than a 20th or 30th of a year's purchase. In joint lives they will differ less than in single lives, and they will come equally near to one another whatever the rates of interest are.

The preceding tables furnish the means of determining the exact differences between the values of annuities, as they are made to depend on the survivorship of any male or female lives; which hitherto has been a consideratum of considerable consequence in the doctrine of life annuities. What has made this of consequence is chiefly the multitude of societies lately established in this and foreign countries for providing annuities for widows. The general rule for calculating from these tables the value of such annuities is the following.

Rule. "Find in Table III. the value of a female life at the age of the wife. From this value subtract the value in Table IV. of the joint continuance of two lives at the ages of the husband and wife. The remainder will be the value in a single present payment of an annuity for the life of the wife, should she be left a widow. And this last value divided by the value of the joint lives increased by unity, will be the value of the same annuity in annual payments during the joint lives, and to commence immediately."

Example. Let the age of the wife be 24, and of the husband 30. The value in Table III. (reckoning interest at 4 per cent.) of a female life aged 24, is 17.252. The value in Table IV. of two joint lives aged 24 and 30, is 13.455, which subtracted from 17.252 leaves 3.797, the value in a single present payment of an annuity of 11. for the life of the wife after the husband; that is, for the life of the widow. The annuity, therefore, being supposed 201. its value in a single payment is 20 multiplied by 3.797, that is, 75.941. And this last value divided by 14.455 (that is, by the value of the joint lives increased by unity), gives 5.25, the value in annual payments beginning immediately, and to be continued during the joint lives of an annuity of 201. to a wife aged 24 for her life, after her husband aged 30.