Page:Supplement to the fourth, fifth, and sixth editions of the Encyclopaedia Britannica - with preliminary dissertations on the history of the sciences - illustrated by engravings (IA gri 33125011196181).pdf/648

 91. When the value of an annuity on any proposed life or lives for an assigned term is given, it is evident that the value of an annuity on the same life or lives for one year less may be found, by subtracting from the given value, the present value, of the rent to be received upon the proposed life or lives surviving the term assigned.

92. Proposition. The present value of an assurance on any proposed life or lives for a given term, is equal to the excess of the value of an annuity to be paid at the end of each year, which the life or lives proposed may enter upon in the term, above the value of an annuity on them for the same term, but dependent, as usual, upon their surviving each year.

Demonstration. if an annuity payable at the end of each year, which the proposed life or lives may enter upon during the given term, be granted to P, upon condition that he shall pay over what he receives to Q, at the end of each year which the same life or lives may survive; it is manifest that there will only remain to P, the rent for the year in which the proposed life or lives may fail; that is, the assurance of that sum thereon for the given term (77). Which was to be demonstrated.

93. From the last four numbers (89—92) we derive the following

for determining the present value of an assurance on any life or lives for a given term.

Add unity to the value of an annuity on the proposed life or lives for the given term, and from the sum subtract the present value of one pound, to be received upon the same life or lives surviving the term; multiply the remainder by the present value of L. 1, to be received certainly at the end of a year, and from the product subtract the present value of an annuity on the proposed life or lives for the term.

This last remainder will be the value in present money of the assurance of L. 1 during the same term, on the life or lives proposed.

94. It has been shown above (34—39), how the present value of L. 1, receivable upon any single or joint lives surviving an assigned term, may be found. And all that was demonstrated from No. 48. to 53. inclusive, respecting annuities on the last survivor of two, or of three lives, or on the joint continuance of the two last survivors out of three lives, is equally true of any particular year’s rent of those annuities. Hence it is evident, how the present value of L. 1, to be received upon the last survivor of two, or of three lives, or upon the last two survivors out of three lives, surviving any assigned term, may be found.

95. Example. Required the present value of L. 1, to be received at the end of the year, in which a life, now forty-five years of age, may fail, provided that such failure happens before the expiration of fen years.

Here the present value of L. 1, to be received upon the life surviving the term, will be found to be L. 0·528976, and the value of an annuity on the proposed life for the term, is 7·175 (70.)

value of the assurance; and if the sum assured were L. 3000, the value of the assurance in present money would be L. 320, 15s. 7d.

96. By numbers 89, 91, and 95, it appears, that an annuity, payable at the commencement of each of the next ten years that a lite of 45 enters upon, is worth 7·646 years purchase: therefore, $$\frac{0{\cdot}10693}{7{\cdot}646} = \text{L. } 0{\cdot}013985$$ will be the annual premium for the assurance of L. 1 for ten years on that life. For the assurance of L. 3000, it will therefore be L. 41, 19s. 1d.

97. When the term of the assurance is the while duration of the life or lives assured, L. 1 to be received upon their surviving the term is worth nothing; and an annuity on the lives for the term, is also for their whole duration.

Therefore, from No. 93. we derive the following

for determining the present value of an assurance on any life or lives.

Add unity to the value of an annuity on the proposed life or lives; multiply the sum by the present value of L. 1, to be received certainly at the end of a year; and from the product, subtract the value of an annuity on the same life or lives.

The remainder will be the value of the assurance in present money.

98. Example. Required the present value of L. 1, to be received at the end of the year, in which the survivor of two lives may fail, their ages now being 40 and 50 years respectively.

The value of an annuity on these lives was shown in No. 66. to be 15·066.

Multiply 15·066 by 0·952381, from the product 15·3009, subtract 15·066, the remainder L. 0·2349 is the required value, agreeably to No. 87.

And, in all other cases, the values determined by the rule in the preceding number, will be found to agree with those obtained by the method of No. 81.

99. When an assurance on any life or lives has been effected at a constant annual premium, and kept up for some time, by the regular payment of that premium; the annual premium required for a new assurance of the same sum on the same life or lives, will, on account of the increase of age, be greater than that at which the first assurance was effected: Therefore, the present value of all these greater annual premiums, that is, the total present value of the new assurance, will exceed the present value of all the premiums that may hereafter be received under the existing policy. And the excess Rh