log (1 +a) when that of log a is known. In other words, the argument of the table is log a, and the tabular result is log(l+a). When ordinary logarithmic tables are used,
the formulas beinglog a, &eq log vp x + log (1 + a -+1 ),
log a_, &eq log />_! + log (1 + a.) ;
we have to find a t by means of an inverse entry into the table before log (1 +a,) can be found; but when Gauss s table is used (as recomputed and extended by Gray), "All the entries of the same kind direct and inverse are brought together, the whole of the logarithms being found before a single natural number is taken out. We con sequently proceed right through the table ; and as we proceed, we find two, three, four, and even as many as six and eight entries on the same opening. At the close, moreover, the taking out of the numbers may, if necessary, be turned over to an assistant. On the other hand, when the common tables are used, direct and inverse entries alternate with each other, and involve likewise a continual turning of the leaves backwards and forwards, by which the process is rendered exceedingly irksome." Page 1G5, second issue, 1870.
When the only object is to form a complete table of immediate annuities, the above is the simplest and most expeditious mode of procedure ; but when it is desired to have the means of obtaining readily the values of deferred and temporary annuities, it is better to employ a wholly different method.
The value of a deferred annuity may be found as follows:— If it were certain that the nominee, whose age is sup posed to be now x, would survive n years, so as to attain the age of x + n, the value of the annuity on his life being then a *+> its present value would be v*a f+n . But as the nominee may die before attaining the age of x + ?i, the above value must be multiplied by the probability of his living to that age, which is - , and we thus get the present value of the x deferred annuity, t>" . -y 15 . *+ We may arrive at this result otherwise. Thus, as we have seen above, the pre sent value of the first payment of the annuity, that is, of 1 to be received if the nominee shall be alive at the end of *+H n+ I years, is -y- i> n+1 . The present value of the next payment is similarly seen to be -j-v n+2 , and so on. The value It of the deferred annuity is therefore
B+37 -_V+3 + ........ n? / 7 &eq 7 v" . "a, (or ^ . v n . v \ *
(We may here mention that this formula holds good, not only for ordinary annuities, but also for annuities payable half-yearly or quarterly, and for continuous annuities; also for complete annuities.)
A temporary annuity is, as explained above, an annuity which is to continue for a term of years provided the- nominee shall so long live. Hence it is clear that if the value of a temporary annuity for n years is added to that of an annuity on the same life deferred n years, this sum must be equal to an annuity for the whole continuance of the same life ; the value of a temporary annuity for n years will therefore be equal to the difference between the value of a whole term annuity and that of an annuity deferred n years, or to
o- j-tr.. a, (ora a -^ v n a* ijC
We are now in a position to explain the method of calculating the value of annuities above referred to. We have seen that the value of an annuity for the life of a nominee whose age is x, is
[ math ]
which, multiplying both numerator and denominator by the same quantity v*, becomes
l m v
In the same way, the value of an annuity on the same life, deferred n years, is
[ math ]
If, then, we calculate in the first instance the values of the product l x v x for all values of x, and then form their sums, beginning at the highest age, we shall have the means of obtaining by a single division the value of any immediate or deferred annuity we wish.
It is convenient to arrange these results in a tabular form, as shown in the appended tables (3) and (4). The quantity l x v* is placed in the column headed D, oppo site the age x, and is denoted by D,,,; while the sum ^+i v * +l + e+i y * +2 + . . . . + l x +,v* + is placed in the column headed N, opposite the same age x, and is denoted by N a ; so that the value of an immediate annuity on a life x is equal to ; the letters N and D being chosen as the first letters of the words Numerator and Denominator. Then it is easy to see that the value of an annuity on x deferred n years is equal to ; whence by subtraction the value of a temporary annuity for n years on the same life is * * +n
If, for example, we wish to find the value of an annuity on a male life of 40 according to the English Table No. 3, with interest at 3 per cent., we find from table (3) appended to this article, N 40 = 1374058, D 40 = 83406, and by division we get the value of = 1 6 4744, which agrees L) 40 with the value contained in the table (5), also appended to this article.
Next, suppose we wish to find the value of a deferred annuity on a life of 30 to commence at the end of 10 years. From what precedes, we see that the value of this n -u i.- * N 40 1374058 annuity will be equal to the quotient or which will be found to be equal to 10 9518.
If we wish to find the value of this deferred annuity without using the D and N table, the formula for it will be r^ v 10 a 40 , v being equal to . But we have seen above that the value of ^-(1*03)- 10 = 664779, and that a w = 16*4744; and multiplying these together, we get the value of the deferred annuity, 10 9518, as before.
auxiliary table a " D and N table." It is also called a
" commutation table," a name proposed by De Morgan