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 So that the value of an annuity on any life or on the joint continuance of any number of lives for an assigned term, is equal to the excess of the value of an annuity on their whole duration, with immediate possession, above the value of an annuity on them deferred for the term.

59. Whatever has been advanced from No. 48. to 53, inclusive, respecting the values of annuities for the whole duration of the lives whereon they depend, will apply equally to those which are either deferred or temporary; and, therefore, to determine the value of any deferred or temporary annuity, dependent upon the last survivor of two or of three lives; or, upon the joint continuance of the last two survivors out of three lives; we have only to substitute temporary or deferred annuities, as the case may require, for annuities on the whole duration of the lives; and the result will, accordingly, be the value of a temporary or deferred annuity on the life of the last survivor, or on the joint lives of the two last survivors.

60. ''Prop. 5. A and B'' being any two proposed lives now in existence, the present value of an annuity to be payable only while A survives B, is equal to the excess of the value of an annuity on the life of A, above that of an annuity on the joint existence of both the lives.

61. Demons. If an annuity on the life of A, and to be entered upon immediately, were now granted to P, upon condition that he should pay it to B during the joint lives of A and B; it is evident that there would only remain to P, an annuity on the life of A after the decease of B: whence the truth of the proposition is manifest.

62. When any thing is affirmed or demonstrated of any life or lives, it is to be understood as applying equally to any proposed single life, or to the joint continuance of the whole of any number of lives that may be proposed together, or to that of any assigned number of the last survivors of them, or to the last surviving life of the whole.

63. ''Prop. 6''. The present value of the reversion of a perpetual annuity after the failure of any life or lives, is equal to the excess of the present value of the perpetuity with immediate possession, above the present value of an annuity on the same life or lives.

64. Demons. If a perpetual annuity with immediate possession were granted to P, upon condition that he should pay the annual produce to another individual, during the existence of the life or lives proposed; it is evident that there would only remain to P, the reversion after the failure of such life or lives; and the present value of that reversion would manifestly be as stated above.

65. The 6th, 7th, and 8th tables at the end of this article, which have been extracted from the 19th, 21st, and 22d, respectively, in Mr Milne’s Treatise on Annuities, will serve to illustrate the application of these propositions to the solution of questions in numbers.

In all the following examples, we suppose the lives to be such, as the general average of those the Carlisle table of mortality was constructed from, and the rate of interest to be 5 per cent.

66. Ex. 1. What is the present value of an annuity on the joint lives, and the life of the survivor of two persons now aged 40 and 50 years respectively?

According to No. 48, the process is as follows:

And if the annuity be L. 200, its present value will be L. 3013·2, or L. 3013, 4s.

67. Ex. 2. The lives A, B, and C, being now aged 50, 55, and 60 years respectively, an annuity on the joint continuance of all the three, is worth 6·289 years purchase; What is the value of an annuity on the joint existence of the last two survivors of them?

According to No. 50, the process is thus:

Therefore, if the annuity were L. 300, it would be worth L. 3197, 2s. in present money.

68. Ex. 3. Required the value of an annuity on the last survivor of the three lives in the last example.

Proceeding according to No. 52, we have

purchase, the required value. And if the annuity were L. 300, it would now be worth L. 4200, 6s.

69. Ex. 4. What is the present value of an annuity on a life now 45 years of age, which is not to be entered upon until the expiration of ten years; the first payment thereof being to be made at the expiration of eleven years from this tine, if the life survive till then?

Solution.

The present value of an annuity on a life of 55 is 10·347 (Table VI.), and the present value of L. 1 to be received upon the proposed life attaining to the