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 will evidently be the value of the policy, supposing the life or lives to be still insurable; that being the only advantage that can now be derived from the premiums already paid.

So that, if the present value of all the future annual premiums to be paid under an existing policy for the assurance of a certain sum upon any life or lives, be subtracted from the present value of the assurance of the same sum on the same life or lives; the remainder will be the value of the policy.

100. Example. L. 1000 has been assured some years, on a life now 50 years of age, for its whole duration, at the annual premium of L. 20, one of which has just now been paid: What is the value of the policy?

The present value of the assurance of L. 1000 on that life, has been shown in No. 82. to be L. 397, 2s. 10d.; and an annuity on the life, being worth 11·66 years purchase (Table VI.), the present value of all the premiums to be paid in future under the existing policy, is 11766 × L. 20 = L. 233, 4s. 0d.; the value of the policy, therefore, is L. 163, 18s. 10d.

Immediately before the payment of the premium, the value of the policy would evidently have been less by the premium then due.

101. In our investigations of the values of annuities on lives, we have hitherto assumed, that no part of the rent is to be received for the year in which the life wherewith the annuity may terminate fails.

But if a part of the annuity is to be received at the end of that year, proportional to the part of the year which may hare elapsed at the time of such failure; as, in a great number of such cases, some of the lives wherewith the annuity terminates will fail in every part of the year, and as many, or very nearly so, in any one part of it as in any other: we may assume, that, upon an average, half a year’s rent will be received at the end of the year in which such failure happens; and, therefore, that by the title to what may be received after the failure of the life or lives whereon the annuity depends, the present value of that annuity will be increased by the present value of the assurance of half a year’s rent on the same life or lives.

102. Thus, for example: the present value of the assurance of L. 1 on a life of 50 years of age, having in No. 82. been shown to be L. 0·397143, the value of an annuity of L. 1 on that life, when payable, till the last moment of its existence, will exceed L. 11·66, its value, if only payable, until the expiration of the last year it survives, by $$\left ( \frac{\text{L. } 0{\cdot}397143}{2} = \right )$$ L. 0·199; it will therefore be L. 11·859.

103. If, at the end of the year in which an assigned life A may fail, Q or his heirs are to receive L. 1; and are, at the same time, to enter upon an annuity of L. 1, to be enjoyed during another life P, to be then fixed upon: the present value of Q’s interest will evidently be the same as that of the assurance on the life of A, of a number of pounds, exceeding by unity the number of years purchase in the value of an annuity on the life of P, at the time of nomination.

104. What is the present value of the next presentation to a living of the clear annual value of L. 500; A, the present incumbent, being now 50 years of age; supposing the age of the clerk presented to be 25, at the end of the year in which the present incumbent dies; also, that he takes the whole produce of the living for that year?

By Table VI. it will be found, that the value of an annuity of L. 1 on a life of 25, is L. 15·303; and in No, 82. it has been shown, that the present value of the assurance of L. 1 on a life of 50, is L. 0·397143. Hence, and by the last number, it appears, that if the annual produce of the living were but L. 1, the present value of the next presentation would be L. 16·303 × 0·397143 = L. 6·47467. The required value, therefore, is L. 3237, 6s. 9d.

105. If, to the value of the succeeding life, determined according to No. 103, the value of the present be added, the sum of these will evidently be the present value of both the lives in succession; and, in the case of the preceding number, will be 6·475 + 11·66 = 18·135 years’ purchase.

106. In No. 103, we proceeded upon the supposition that the annuity on the present life is only payable up to the expiration of the last year it survives; and, consequently, that the succeeding life takes the whole rent for the year in which the present fails.

But, if the succeeding life is only to take a part of that rent, in the same proportion to the whole, as the portion of the year which intervenes between the failure of the present life and the end of the year, is to the whole year, then, by reasoning as in No. 101, it will be found, that the portion of that rent which the succeeding life will receive, may be properly assumed to be one half. And, instead of increasing the number of years’ purchase the annuity on the succeeding life will be worth at the end of the year in which the other fails, by unity, we must only add one half to that number, in order that the present value of the assurance of the sum on the existing life, may be the number of years’ purchase, which all that may be received during the succeeding life, is now worth.

107. The value of the succeeding life, in the case of No. 104, will, upon this hypothesis, be 15·803 × 0·397143 = 6·27605 years’ purchase.

And this appears to be the most correct way of calculating the value of an annuity on a succeeding life; although that of No. 103. proceeds upon the principle on which life interests are generally valued.

108. But the value of two lives in succession, will be the same on both hypotheses. The rent for the year in which the first may fail, being, in the one case, given entirely to the successor; and, in the other, divided equally between the two.

This is also true of any greater number of successive lives.