# Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica/Annuities, Addendum to

ADDENDUM.
ANNUITIES. As an addition to the article Annuities, we beg to insert here an expeditious method of calculating the values of annuities on single or joint lives, from any tables or bills of mortality, with sufficient accuracy for all practical purposes.
We must begin by determining the mean complement of life, according to the average number of deaths during a certain period, which must vary according to the nature of the proposed calculation; being shorter as the rate of interest is higher; and as the number of lives concerned is greater; but not requiring to be very accurately defined. If the rate of interest be $r$, we must find the time in which the number of deaths is expressed by the fraction ${\frac {3}{r+3}}$ of the whole number of survivors at the given age, for a single life: for two lives, the fraction must be ${\frac {3}{r+5}}$, and for three, ${\frac {3}{r+7}}$; and, in each of these cases, the time determined from the age of the oldest life must be employed for finding the complements of both the others.
Having thus calculated the complements for each of the ages, we may, in most instances, save ourselves the trouble of further computation, by employing tables of the value of annuities on one and two lives, according to Demoivre’s hypothesis. For this purpose, we have only to subtract the complement from 86, and we obtain an equivalent life on this hypothesis. If we take, for example, the age of 20, the number of survivors in the Northampton tables ts 5132; and, for a single life, at 3 and at 6 per cent. we must find the time at which they are reduced ${\frac {3}{6}}$ and ${\frac {3}{9}}$ respectively; that is, to about 2566 and 3421: now at 54 and 43, the numbers are 2530 and 3404; and ${\frac {5132\times 34}{5132-2530}}=67{\cdot }07$, and ${\frac {5132\times 23}{5132-3404}}=68{\cdot }3$; whence the equivalent ages in Demoivre’s tables are 18·93 and 17·7, giving 18·62 and 12·43, for the value of the annuity; while Dr Price’s table, deduced from the actual decrements at all ages, gives 18·64 and 12·40.
The utility of this mode of calculation will be still further illustrated by a comparison of the very different values of lives, as indicated by different tables. Taking, for example, the age of 30, and the interest at 5 per cent. we may find the value of the annuity, by this approximation, in different situations, for which correct tables have been published by Dr Price, and may thence infer how much nearer it approaches to the truth than the generality of the results approach to each other:
| London, 1730 Compl. | 41·52 | 11·22 | 11·6 |
| Northampton | 57·05 | 13·09 | 13·07 |
| Sweden, males | 67·93 | 14·04 | 13·89 |
| Deparcieux | 71·11 | 14·28 | 14·72 |
| Sweden, females | 75·60 | 14·58 | 14·27 |
According to the bills of mortality of London for 1815, out of 9472 survivors at 30, 5573 lived to 50, and this is near enough to ${\frac {3}{8}}$ for our purpose: hence the complement is 48·58, and the value of an annuity at 5 per cent. 12·16 years’ purchase. Where the age is much greater, the approximation is somewhat less accurate, though not often materially erroneous; thus, at 70, the values, according to the Northampton tables, at 3 and 6 per cent, are 6·23 and 5·35, instead of 6.73 and 5.72 respectively.
In the values of joint lives, there is more difference, according to the different tables employed, than in those of single lives: thus, at 30, the value of an annuity, at 4 per cent. on a single life, differs at Northampton, and in Sweden, in the proportion of 14·78 to 16·00, or of 12 to 13; but, for two joint lives at 30, in that of 11·31 to 12·96, or of 7 to 8; and for three lives, the disproportion would be still greater.
In the absence of Demoivre’s tables, or for cases to which they do not extend, it becomes necessary to calculate the value of the annuity for each particular instance. Calling then the complement, as already determined, $a$, the number of survivors after $x$ years will be represented by $a-x$, and the present value of any sum to be paid to each of them by $av^{x}-xv^{x}$, $v$ being the present value of a unit payable at the end of a year: and if we suppose such payments to be made continually, their whole present value may be found by multiplying this expression by the fluxion of $x$, and finding the fluent, which will be $-pv^{x}(a-x-p)$, $p$ being $=-{\frac {1}{\operatorname {HL} v}}$, or the reciprocal of the hyperbolical logarithm of the amount of a unit after a year. When $x$ vanishes, this fluent becomes $-p(a-p)$, and when $x=a$, $p^{2}v^{a}$; the difference, divided by $a$, gives the present value of the annuity, $p-{\frac {p^{2}}{a}}+{\frac {p^{2}v^{a}}{a}}$; from which, when the annuity is supposed to become due and to be paid periodically, we must subtract in all cases half a payment; that is, ½ for yearly payments, and ¼ for quarterly; and if, at the same time, we choose to assume that money is capable of being improved by laying out the interest more frequently than once ​a year at the given rate, we must alter the value of $v$ accordingly.
For two joint lives, the complement of the elder, determined from the fraction ${\frac {3}{r+5}}$, being $a$, and that of the younger, deduced from the deaths in an equal number of years, $b$, we have for the binary combinations of the survivors, after x years, $(a-x)(b-x)$, and the fluent will be $-pv^{x}(ab-(a+b)(x+p)+x^{2}+2px+2p^{2})$, which, corrected and divided by $ab$, gives the value of the annuity $p-{\frac {p^{2}}{ab}}(a+b-2p)-{\frac {p^{2}v^{a}}{ab}}(a-b+2p)$; and this, with the deduction of half a payment, agrees with the tables calculated on Demoivre’s hypothesis, taking the same complements of life.
But for three lives we have no such tables, and this method of calculation becomes therefore of still greater importance. Employing here the fraction ${\frac {3}{r+7}}$ for the oldest life, we must determine the complement $a$ for this life, and those of the two younger, $b>$ and $c$, from an equal period. The combinations will then be $(a-x)(b-x)(c-x)=abc-(ab+ac+bc)x+(a+b+c)x^{2}-x^{3}$, which we may call $d-ex+fx^{2}-x^{3}$; hence the fluent is found $-pv^{x}\left(d-e(x+p)+f(x^{2}+2px+2p^{2})-(x^{3}+3px^{2}+6p^{2}x+6p^{3})\right)$ this, when $x$ vanishes, becomes $-p(d-ep+fp^{2}-6p^{3})$, and calling this $-pg$, the corrected fluent will give the value of the annuity ${\frac {pg}{d}}-{\frac {pv^{a}}{d}}(g-en+fa^{a}+2fpa-a^{3}-3pa^{2}-6p^{2}a)$. Thus, if the ages are 10, 20, and 30, and the rate of interest 4 per cent. we find, in the Northampton tables, the survivors at 30 4385, $^8$⁄$_{11}$ of which are 3199; and at 46, the survivors are 3170; whence $a=57{\cdot }7$, and $b$ and $c$ found also from periods of 16 years after the respective ages, are 68·5 and 91·7. Calculating with these numbers, we find the value of the annuity $10{\cdot }954-{\cdot }5=10{\cdot }454$. Dr Price’s short table gives it 10·438; and Simpson’s approximation from the tables of two joint lives 10·563, which is less accurate in this instance, even supposing such tables to have been previously calculated.
It would, indeed, be easy to form, by this mode of computation, a table of the corrections required at different ages for Simpson’s approximation, since these corrections must be very nearly the same, whether Demoivre’s hypothesis, or the actual decrements of lives be employed, both for the two joint lives, and for the correct determination of the three. But the value thus found would still be less accurate, with respect to any other place, or perhaps even any other time, than the immediate result of the mode of calculation here explained.
It may, perhaps, save some trouble to subjoin a table of the values of $p$ and their logarithms.
| 3 per cent. | $p$ = 33·831 | log. $p$ = 1·5293132 |
| 4 | 25·497 | 1·4064846 |
| 5 | 20·497 | 1·3116680 |
| 6 | 17·162 | 1·2345630 |
(RT.)
End of Volume First.