Institute and Faculty of Actuaries

ON THE VALUE OF A POLICY ON THE LONGEST OF TWO LIVESAuthor(s): Thomas CarrSource: Journal of the Institute of Actuaries and Assurance Magazine, Vol. 14, No. 5(OCTOBER, 1868), pp. 415-416Published by: Cambridge University Press on behalf of the Institute and Faculty of ActuariesStable URL: http://www.jstor.org/stable/41135187 .

Accessed: 13/05/2014 19:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Cambridge University Press and Institute and Faculty of Actuaries are collaborating with JSTOR to digitize,preserve and extend access to Journal of the Institute of Actuaries and Assurance Magazine.

http://www.jstor.org

This content downloaded from 195.78.108.168 on Tue, 13 May 2014 19:30:18 PMAll use subject to JSTOR Terms and Conditions

1868.] Correspondence. 415

obscure, the substantial merits of the book - its comprehensiveness, com- pleteness, fairness and soundness. The Author shows himself ever jealous for the honour of Assurance Institutions, and always, in intention, loyal to their best interests. He is earnest alike in his denunciation of irregulari- ties within and in his exposure of delusions without, as well as honest in his eager preference for that sober moderation which in all things puts aside a present, if a doubtful, good, for a permanent, though deferred, success. The cause of Assurance cannot be the worse, it ought to be greatly the better for this work. How much it has been bettered in fact, it would be exceedingly difficult to say. After all the labour of so many able and honest men, and all the accumulated experience of both the true and the false in the practice of Assurance, it is at once a matter for sur- prise and mortification that there should have been brought about so small an abatement of public credulity ; but, though matter for mortification, it is no matter for surprise that, the credulity remaining, official malpractice, designed to impose on it, should remain also.

COKRESPONDENCE.

ON THE VALUE OF A POLICY ON THE LONGEST OF TWO LIVES.

To the Editor of the Journal of the Institute of Actuaries.

Sir, - It has been demonstrated in the Journal, vol. xi., page 104, that the Value of the Policies existing in an Office is equal to the accu- mulated net Premiums less the claims. The application of this result to an ordinary Policy on a single life is shown in the formula - -±=r ± = =r - -±- , vol. xi., page 107, where- - =<**.

Mr+w ^ar-l Mr+n -«ar-1 This formula will also apply, by simply changing the values of M, N, D, and **, to Policies of some other descriptions, viz., those where no alteration takes place in their status till their expiry or till claims arise on them, as for instance, Policies on Joint Lives, Endowment Assurances and Term Policies. But the formula will not be applicable, at least by a like simple, modification, to the case of Policies, the status of which may be changed during their continuance, as that of Policies on the Longest of two lives, which may be changed by the dropping of one life.

I have thought a similar demonstration in the case of a Policy on the Longest of two lives to the one above referred to might not be without interest.

Let there be lx lives of the age x and ly of the age y, then the number of pairs which can be formed out of them, each containing a life aged x and a life aged y, is lxxly. Let this be expressed by lXmyi so that in the following demonstration lx.y signifies lx x ly. Let each of these lxy pairs be assured for £l payable on the last death, at an annual net Premium m. Then the number of claims (that is, of pairs in which both lives become extinct) in the first year, will be lx.y - lx.y+i - lx+'.y+h+'.y+' ; and the sum reserved at the end of the first year for existing Policies will be

¿r.yn<l +i) - (k,- k.y+i - ¿r+l.y + WžM-l) (1) But of the latter Policies there will be ¿r+i.y- ¿r+i.y+i? by which only sur-

This content downloaded from 195.78.108.168 on Tue, 13 May 2014 19:30:18 PMAll use subject to JSTOR Terms and Conditions

416 Correspondence. [Oct. 1868.

viving single lives aged #+1 are assured, and lx.y+' - ¿r+i.?/+i> by which only single lives aged y+1 are assured. The value of these Policies will be

(Wir-t+i.if+i){1-(rf+wX1 + «*+i)} + fe+i~Wi,+i){l-(^^)(l + «,+i)} (2)

which deducted from (1) will give the value of the Policies on the lx+'.y+' surviving pairs of lives.

The investigation of this for the first year is as follows:

In(ír+l.y-í«r+l.If+l){l--(^+*X1+^+l)}

and -fc.y+1{ 1 -(<*+ w)(l + 0^+1)}= -fe.H-1 + (1 + i)h.y(d+<n)ay The sum of these is

- 4+l.y- ¿r.H-1 + 0 + *%/«* + wXa* + ö|f) and the other terms of (2), with their signs changed, are equal to

ir+l.y+lC^p+l + AžM-l) - lx+'.y+'(2 + «#+1 + ay+l)>ar. Again, Aa.+1.If+1(P+1.H.,=Aar.yir.lf(l + i )- (Z^- 4+1 .y+1) (a) and (%i.y+i + l)Wi.y+i=^W1+O (¿)

If all the terms of (1), and (2) as above modified, are collected and Az+i.y+ilx+uj+i and («x+i.y+i + l)(r+i.y+i are added and subtracted according to equations (a) and (b), the result is

- lx.y + h+l.y + h.y+' - 4+1 .y+l "~ h+l.y - h.y+l 4- ( 1 + O4.y <«Ä + a,y) + (1 + i)lx.yAx.y- lx.y + 4+1.y+1 + (1 + O^-sr + (1 + i) lxy{ax + ay)<m- (1 + 1 ) lx.yax.y.*j*

+ lx+].y+l(Ax+l + Ay+1 - A#+1 y+1) - 4+Ly+1(l +a#+1 +ťfy+1 - «ar+i.y+ijw. Now lx.y=(l+i)vlx.y and A^rsv- (1- 1?)^=^- ^

.-. -2/^ + (l + t)(ryi(air + flr) + (l +t)/^A^ = -(1 +0W2t;-<ai+ßy)-A^} = -(1 + OWA** Ay- A,,)

Also ^7= A^+Ay-A£iL

(1 + i)lx.y(} + «*+ fly- fl^)«r=(l + 1%.^* + A,- A*.y) Therefore the sum of the terms in the above result before the asterisk (*) vanishes, and the remaining terms divided by 4+i.y+i are equal to

A#+i + Ay+l - Ax+Ly+l - (1 + ax+l + ay+l - fl^+i.y+i}»1, the formula for the value of a Policy on the longest of two lives.

I am, Sir, Your obedient servant,

7, Royal Exchange, June, 1868.

THOS. CARR.

This content downloaded from 195.78.108.168 on Tue, 13 May 2014 19:30:18 PMAll use subject to JSTOR Terms and Conditions