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II. Life Annuities.

9. When an annuity (or annual payment) is payable during life, it is called a life annuity.

10. To find the value of an endowment of £1 payable in n years if x be then alive.

By Art. 5, the probability of a person whose age is x living n years is

Let denote the present value of £1 due n years hence; then the present value of £1 payable in n years if x be then alive is

Example 1. Value of an endowment of £10 payable if a person aged 18 attain the age of 25, at 3 per cent., according to the Manchester Unity Table of Mortality for 1860-70.

In formula (1) the numerator of the fraction is the present value of the number of pounds payable in n years, and the denominator the number now living to contribute equally the present fund.

If (1) be transformed by multiplying both numerator and denominator by t^x, by which the value of the fraction is not altered, it becomes which is a more convenient formula for constructing tables of the values of annuities (presently to be explained) than formula (1).

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The number in column D of Table II., opposite each age, is the product of the number living at that age into the present value of £1 due as many years hence as are equal to that age, so that in (2) the numerator of the fraction is equal to D_x+n, and the denominator to D_x. Hence the formula becomes finally

By means of colum D we are now enabled to compute the values of endowments by simple division.

For example, as above: Value of an endowment of £10 payable if a person aged 18 attain the age of 25, at 3 per cent.

11. To find the value of an annuity of £1 due at the end of every year through which x shall live.

The value of the annuity is the sum of the present values of £1 due at the end of 1, 2, 3, ... years, to the last tabular age, if x shall live so long.

Let a_x denote the present value of an annuity of £1 due at the end of every year through which x shall live; then, by Art, 10,

In formula 4 the numerator of the fraction is the sum of the present values of the number of pounds due at the end of 1,2,3,... years, to the last tabular age, and the denominator the number now living to contribute equally the present fund.

If (4) be transformed by multiplying both numerator and denominator by v^x, by which the value of the fraction is not altered, the equation becomes

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The number in column N of Table II., opposite each age, is the sum of the numbers in column D opposite all the older ages;^* thus

N_x=D_x+1+D_x+2+D_x+3+... to the last tabular age;

N_x=D_x+1+N_x+1;

therefore

Example 2.—Value of an annuity of £1 or the life aged 18.

By this method may be constructed columns 2 and 7 of Table 3, showing the value of an immediate annuity of £1 during life, according to the Manchester Unity Table of Mortality for 1866-70.

12. To find the value of an annuity of £1 deferred n years and then to continue so long as x shall live.

The value of the annuity is the sum of the present values of £1 due at the end of n+1, n+2, n+3,... years, to the last tabular age, if x shall live so long.

Let _n|^a_x denote the present value of an annuity of £1 deferred n years and then to continue so long as x shall live; then

The number in column N of Table II., opposite the age x+n, is, by Art. 11, the numerator of the fraction, and the number in column D, opposite the age x, is, by Art. 10, the denominator;

therefore

Example 3. Value of an annuity of £1 to be entered upon at the expiration of 47 years on the life aged 18.

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By this method may be constructed column 4 of Table III., showing the value of a deferred annuity of £1, to commence after age 65 and then to continue during life, according to the Manchester Unity Table of Mortality for 1866-70.

13. To find the value of a temporary annnity of £1 for the next n years if x shall live so long.

The value of the annuity is the sum of the present values of £1 due at the end of 1, 2, 3,... n years if x shall live so long.

Let |_na_x denote the present value of a temporary annuity of £1 for the next n years if x shall live so long; then since the value of a temporary annuity for the next n years, together with that of an annuity deferred n years, is equal to the value of an annuity during life,

Example 4. Value of an annuity of £1 for the next 47 years on the life aged 18.

This added to 1.004, the value of an annuity of £1 to be entered upon at the expiration of 47 years on the life aged 18 (Example 3), gives 22.235, the value of an annuity of £1 on the life aged 18 (Example 2).

By this method may be constructed column 3 of Table III., showing the value of a temporary annuity of £1 for a term of years if the life shall be in existence so long, according to the Manchester Unity Table of Mortality for 1866-70.

14. To find the annual premium for an annuity of £1 deferred n years and then to continue so long as x shall live.

The first annual premium made by members of Friendly Societies is generally completed at the end of the year, and the remaining premiums at the end of 2, 3,... n years: so that if the annual premium be £1, the present value of it, at yearly interest, will be |_nax, the value of a temporary annuity of £1 for the next n years on the life x.

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Let P_n|a_x denote the annual premium for an annuity of £1 deferred n years on the life x; then the value of a temporary annuity of £1 for n years on the life x is to £1 as the value of an annuity of £1 deferred n years on the life x is to its annual premium, or

Example 5. Annual premium for an annuity of £1 to be entered upon at the expiration of 47 years on the life aged 18.

By this method may be constructed column 5 of Table III., showing the annual premium for a deferred annuity of £1, to commence after age 65 and then to continue during life, according to the Manchester Unity Table of Mortality for 1866-70.

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