Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/93

Rh a single age, when we do not require a complete table of annuities. The following method was demonstrated by Mr Lubbock (afterwards Sir J. W. Lubbock) in a paper " On the Comparison of Various Tables of Annuities " in the Cambridge Philosophical Transactions for the year 1829. Instead of calculating the value of each payment of the annuity to be received at the ages x+1, x + 2, to the extremity of life, it will be sufficient to calculate the values of the payments to be received at a series of equidistant ages, say at the ages x + n, x + 2n, x + 3n, Then, if V w denote the payment to be received at the age x + m, and A 1? A 2, A,, denote the lead ing differences of V, V,, V,,, Vy, the value of the annuity is approximately A, + _ 24u 2 720n 3

Here V &#61; 1, V, &#61; -j-v*, V 2)l &#61; -y v- n, &c. 480?i 3. As an example, we will apply this formula to calculate the value of an annuity on a nominee of 40, according to the English Table, No. 3, Males, at 3 per cent, interest.

First, taking n &#61; 7, we find

V &#61;1-0000 - 2654 V 7 &#61; 7346 + -0521 2133 - -0115 V 14 &#61; -5213 -0406 +-0029 1727 -0086 V.,&#61; -3486 -0320 1407 V 88 &#61; -2079 V 35 &#61; -0990 V 42 &#61; -0318 V 49 &#61; -0055 V&#61; 0004 Sum&#61;2-9491

Hence A x &#61; -2654, A 2 &#61; 0521, A 3 &#61; - 0115, A 4 - 0029 ; and the value of the annuity is approximately

4 2 &#61; 7 + 2-9491- 4-yx-2654-yX 0521- -1808 x -0115 - 1283 x -0029. &#61;20-6437-4-0000 - -1517 - -0149 - -0021 - -0004 &#61; 20-6437-4-1691 &#61; 16-4746.

Next, taking &#61;11, we have

V &#61;1-0000 - -3924 V u &#61; -6076 +-1115 2809 - -0310 V 2a &#61; -3267 -0805 + 0453 2004 + -0143 V 33 &#61; -1263 -0948 1056 V 44 &#61; -0207 V 56 &#61; -0006 2-0819

Hence, the value of the annuity is approximately

10 5 &#61;11 x 2 -0819 - 6 - x -3924 - x -1115 - 2878 x -0310 &#61; 22-9009-6-0000 - -3567 - -0507 - Od89 - -0093 &#61; 22-9009-6-4256 &#61; 16-4753. -2044x -0453

The value of the annuity calculated ia the ordinary way is, as we have seen (page 80), 16-4744. An improved form of this method was given by Mr W. S. B. Woolhouse in the Ass. Nag., xi. 321. In order to explain this, we must introduce the reader to a term which is of recent origin, but which the application of improved mathematical methods to the science of life con tingencies has rendered of great importance tJie force of mortality at a given age. This may be defined as the pro portion of the persons of that age who would die in the course of a year, if the intensity of the mortality remained constant for a year, and the number of persons under obser vation also remained constant, the places of those who die being constantly replaced by fresh lives. More briefly, it is the instantaneous rate of mortality. A very full explanation of this term is given by Mr W. M. Makeharn, in his paper "On the Law of Mortality, "Ass. Mag., xiii. 325. The value of the function can be approximately found by dividing the number of persons who die in a year by the number alive in the middle of the year. Thus, if l x denote the number of persons living at the age x, d x the number dying between the ages x and x+1, and d x _ x the number dying between the ages x 1 and x, then the number dying between the ages x-- and x + - will be approximately x i + d,, and the force of mortality is approximately a -^ -. Thus, in the English Table, No. 3, Males, the * 3465 + 3529 value of the force of mortality at age 40 is - &#61; 012853. This quantity is usually denoted by the Greek letter p., while 8 is used to denote the quantity log (l+i), which Woolhouse has called the force of discount. This being premised, Woolhouse s formula for the approximate value of an annuity is [ math ] where it will be noticed that, since V &#61; 1, the two first terms are exactly equal in value to those in Lubbock s formula. Taking the same example as above, we have seen that

^o&#61; -01 285 3 also 8 &#61;-029558 ^40 + 8 &#61; 042411

Making n &#61; 7, we have the value of the annuity

&#61; 16-6437 -4 x -042411 &#61; 16-6437- -169644 &#61; 16-4741.

Making n &#61; 11, we have the value

&#61; 16-9009- 10 x -042411 &#61; 16-9009- -4241 &#61; 16-4768. Comparing the two processes, we see that when we have the values of p. and 8 already computed, Woolhouse s L&gt; decidedly the shorter. On the other hand, it is easy to see that Lubbock s formula applies, not only to annuities, but to other benefits; and that it will be applicable to find the values of such quantities as contingent annuities, the values of which cannot be found exactly except by a very long series of calculations. (See Davies, p. 354.) The reader who refers to Lubbock s paper (which is reprinted in the Ass. Mag., v. 277), or to the short account of it given in the Treatise on Probability, issued by the Useful Know ledge Society, and often bound up with D. Jones s work on annuities, will see that the terms involving A 2, A 3 , A 4 are not given there ; and it may assist the student who is desirous of working out the formula fully, to be referred to De Morgan s expansion of r- r; -, Diff. Calc., p. 314, 184. Lubbock not only considered it unnecessaiy to calculate the terms involving A 2, A 3 , &c., but thought that