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  will be the value, in present money, of the assurance of one pound on the same life or lives. (63)

82. Ex. 1. What is the present value of L. 1, to be received at the end of the year, in which a life now 50 years of age may fail?

The rate of interest being 5 per cent. the value of the perpetuity is 20 years purchase, and that of the life 11·66; the answer therefore is $$\frac{20-11{\cdot}66}{20+1} = \frac{8{\cdot}34}{21} = \text{L. } 0{\cdot}397143$$. And if the sum assured were L. 1000, the present value of the assurance would be L. 397, 2s. 10d.

83. When the term of a life assurance exceeds one year, its whole value is hardly ever paid down at the time that the contract is entered into, but, in the instrument (called a Policy) whereby the assurance is effected, an equivalent annual premium is stipulated for, payable at the commencement of each year during the term, but subject to failure with the life or lives assured.

84. But by reasoning as in No. 74, it will be found, that an annual premium payable at the commencement of each year in the whole duration of the life or lives assured, will be worth one year’s purchase more, than an annuity on them payable at the end of each year; and, consequently, that if the value in present money of an assurance on any life or lives, be divided by the number of years purchase an annuity on the same life or lives is worth, increased by unity, the quotient will be the equivalent annual premium for the same assurance.

85. Ex. 2. Required the annual premium for the assurance of L. 1, on a life of 50 years of age.

In No. 82, the single premium for that assurance was shown to be 0·397143, and the value of an annuity on the life is 11·66, therefore, by the preceding number, the required annual premium will be $$\frac{0{\cdot}397143}{12{\cdot}66} = {\cdot}0313699$$ for the assurance of L. 1; and for the assurance of L. 1000, it will be L. 31, 7s. 5d.

86. Ex. 3. Let both the single payment in present money, and the equivalent annual premium be required for the assurance of L. 1, on the joint continuance of two lives of the respective ages of 45 and 50 years.

The value of an annuity of L. 1 on the joint continuance of these two lives, appears by Table VII. to be L. 9·737, therefore $$\frac{20-9{\cdot}737}{20+1} = \frac{10{\cdot}263}{21} = \text{L. } 0{\cdot}488714$$ is the single premium, and $$\frac{0{\cdot}488714}{10{\cdot}737} = \text{L. } 0{\cdot}0455168$$, the equivalent annual one for the assurance of L. 1 to be paid at the end of the year, in which that life becomes extinct which may happen to fail the first of the two.

Therefore, if the sum assured were L. 500, the total present value of the assurance would be L. 244, 7s. 2d. and the equivalent annual premium L. 22, 15s. 2d.

87. Ex. 4. Let both the single and the equivalent annual premium be required for the assurance of L. 1, on the life of the survivor of two persons now aged 40 and 50 years respectively.

The value of an annuity on the survivor of these two lives was shown in No. 66, to be 15·066, therefore, by No. 81, the single premium will be $$\frac{20-15{\cdot}066}{20+1} = \frac{4{\cdot}934}{21} = \text{L. } 0{\cdot}234952$$; and by No. 84, the annual one will be $$\frac{\text{L. } 0{\cdot}234952}{16{\cdot}066} = \text{L. } 0{\cdot}0146242$$.

That is, for the assurance of L. 1 to be received at the end of the year, in which the last surviving life of the two becomes extinct.

Therefore, for the assurance of L. 5000, the single premium will be L. 1174, 15s. 2d. the equivalent annual one L. 73, 2s. 5d.

88. Ex. 5. What should the single and equivalent annual premiums be for an assurance on the last survivor of three lives, of the respective ages of 50, 55, and 60 years.

The value of an annuity on the last survivor of them, was shown in No. 68, to be 14·001, the single premium should therefore be $$\frac{20-14{\cdot}001}{20+1} = \frac{5{\cdot}999}{21} = \text{L. } 0{\cdot}285666$$, and the annual $$\frac{\text{L. } 0{\cdot}285666}{15{\cdot}001} = \text{L. } 0{\cdot}0190431$$, for the assurance of L. 1, to be received at the end of the year, in which the last surviving life of the three may fail.

For the assurance of L. 2000, the single premium would therefore be L. 571, 6s. 8d. the annual one L. 38, 1s. 9d.

89. Lemma. If an annuity be payable at the commencement of each year, which some assigned life or lives may enter upon in a given term; the number of years purchase in its present value, will exceed by unity the number of years purchase, in the present value of an annuity on the same life or lives for one year less than the given term, but payable as annuities generally are, at the end of each year.

Demonstration. Since the proposed life or lives can only enter upon any year after the first, by surviving the year that precedes it; the receipt of each of the payments after the first, that are to be made at the commencement of the year, will take place at the same time, and upon the same conditions as that of the rent for the year then expired of the life-annuity, for a term one year less than the term proposed: this last mentioned annuity, will therefore, be worth in present money, just the same number of years’ purchase as all the payments subsequent to the first, which may be made at the commencements of the several years.

And, since the first of these is to be made immediately, the present value of the whole of them, will evidently exceed the number of years purchase last mentioned, by unity, which was to be demonstrated.

90. If, while the rest remains the same, the payment of the annuity which depends upon the assigned life or lives entering upon any year, is not to be made until the end of that year; as the condition upon which every payment is to be made, will remain the same, but each of them will be one year later; the only alteration in the value of the whole, will arise from this increase in the remoteness of payment, by which it will be reduced in the ratio of L. 1, to the present value of L. 1, receivable at the end of a year (2).