Summary and Valuation of the Policies of the Series of 1846, as at 31st December, 1870.

Valuation Balance Sheet, as at 31st December, 1870.

The average rate of Interest on the Investments of the three Series, at the close of each year since the last investigation was made, is as follows:

Prior to 1870 there were no separate investments for the "Old Series," the Fund was charged against the general assets of the Corporation, interest at the rate of 4 per cent. being credited annually

thereon.

The minimum value allowed for the surrender of assurances for the whole Term of Life, and for Endowments and Endowment Assurances, on which three or more annual Premiums have been paid, is 25 per cent. of the amount of the Premiums received thereon exclusive of extra Premiums. The actual value allowed is the result of a calculation made for each particular case.

Policies issued at other than European rates, are subject to the condition, that on the return home of the persons whose lives are assured, the Premiums are to be reduced to the European rates. They, therefore, form part of the ordinary risks, and are valued on the same principles, except that the amount of the extra Premium does not enter into the calculation.

In the case of Policies on unhealthy lives, the additional Premiums charged are regarded as a source of income, out of which the extra claims arising annually from the increased mortality of this class are defrayed. So that the "net liability" is the same, but the difference between the Office and net Premiums is greater, in these Policies than in similar Policies on healthy lives.

CORRESPONDENCE.

ON THE INTEGRAL OF GOMPERTZ'S FUNCTION.

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

SIR,-On reperusing my paper in the last Number of the Journal, I observe a slight misdescription of the capabilities of the table appended thereto, which I shall be glad to be allowed the opportunity of correcting.

The characteristic property of the gamma-function, expressed by the equation femdv=—e ̄vvm+mse ̄vm-dv, attaches also to the transformed expression 10-10-nzdz; but in the latter case, the equation, instead of holding between successive integer values of n, or between successive values differing by less than unity (which latter supposition I had, by some means, erroneously entertained), applies to values of n proceeding by differences of loge 10(=2·302 ....). Consequently, in order to obtain the power of deducing the integral for all values of n, the tabulated matter must cover an interval equal to 2.302....—that is to say, to be theoretically complete, the table must be extended in the ratio of 2:302 . . . . to 1.

The property in question may be readily demonstrated as follows:

dv

In Jdzu =uv - Sdzv

dz

du

dz'

the general formula for integration by

ERRATUM. In the paper "On the Integral of Gompertz's Function," above referred to, there is a misprint. For the second formula on p. 309,

ON THE RELATION BETWEEN THE NET PREMIUM AND THE RATE OF INTEREST.

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

SIR,-In the current volume of the Journal, p. 227, I gave a demonstration in reference to the relation between the value of a policy and the rate of interest according to which it is calculated, and in the present letter I propose to examine in a similar way the relation between the net premium and the rate of interest.

Thus, since P increases or decreases, when v increases, according as

x

is positive or negative, we have only to examine whether

Now, (1+a)2=(1+ax)+pxv(1+αx)+2Pxv2(1+ax) + ....

dP

dv

Hence, under the same condition, we shall certainly have

that is, if the value of a perpetuity of 1 is greater than the value of a life annuity of 1, the rate of interest being the same in both cases. In other words, since the value of the perpetuity is necessarily the dP

greater, is positive; therefore Pa, the net premium, increases as dv the rate of interest decreases, provided that ar is not less than ɑx+1,

ON THE FORMULA FOR THE MARKET VALUE OF A
COMPLETE ANNUITY.

To the Editor of the Journal of the Institute of Actuaries. SIR, The usefulness of the expression for the value of a life annuity in terms of d and p, first proposed by the late Griffith Davies, is obvious, whether from a theoretical or practical point of view. From the theoretical, in that it shows the elements of which the value consists; and from the practical, in that it is of universal application, equally valid whether p and d be based on the same rate of interest or not, or when p is a purely arbitrary quantity.