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III. Life Assurance.

15. A contract to secure a sum of money payable on the death of a person is called a life assurance.

16. To find the value of an endowment assurance of £1, payable at the end of n years if x shall die in the nth year.

Let d_x=l_x¹/_x denote the number of persons dying between the ages x and x+1, or in the 1st year,... ⁿ⁻¹d_x=ⁿ⁻¹l_x−ⁿl_x, the number dying between the ages x+n−1 and x+n, or in the nth year; then, by Art. 1, the probability of a person whose age is x dying in the nth year is , and, by Art. 10, the present value of £1 payable at the end of n years if x shall die in the nth year is

Example 1. Value of an endowment assurance of £10, payable at the end of 7 years if a person now aged 30 shall die on his 37th year, at 3 per cent., according to the Manchester Unity Table of Mortality for 1866-70.

In formula (1) the numerator of the fraction is the present value of the number of pounds falling due at the end of 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 page 15 which is a more convenient formula for constructing tables of the values of assurances (presently to be explained) than formula (1).

The number in column C of Table IV., opposite each age, is the product of the number dying in the year following that age into the present value of £1 due as many years hence as are equal to that age increased by unity, so that in (2) the numerator of the fraction is equal to C_x+n−1 and the denominator to D_x. Hence the formula becomes finally

By means of columns C and D we are now enabled to compute the values of endowment assurances by simple division.

For example, as above: Value of an endowment assurance of £10, payable at the end of seven years if a person now aged 30 shall die in his 87th year, at 3 per cent.

17. To find, the value of an assurance of £1, payable at the end of the year in which x shall die.

The value of the assurance is the sum of the present values of £1 payable at the end of 1, 2, 3,... years, to the last tabular age, if x shall die in the 1st, 2nd, 3rd,... years.

Let A_x denote the present value of an assurance of £1, payable at the end of the year in which x shall die; then, by Art. 16,

In formula (4) the numerator of the fraction is the sum of the present values of the number of pounds falling 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.

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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

The number in column M of Table IV., opposite each age, is the sum of the numbers in column C opposite that and all the older ages;^* thus

M_x=C_x+ C_x+1+C_x+2+...to the last tabular age;

M_x=C_x+M_x+1;

Example 2. Value of an assurance of £1 on the life aged 30.

18. To find the annual premium for an assurance of £1, payable at the end of the year in which x shall die.

If the annual premium completed at the end of the year be £1, the present value of it, at yearly interest, will be a_x, the value of an annuity of £1 on the life x.

Let P_xA_x. denote the annual premium for an assurance of £1 on the life x; then the value of an annuity of £1 on the life x is to £1 as the value of an assurance of £1 on the life x is to its annual premium, or

Example 3. Annual premium for an assurance of £1 on the life aged 30.

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Table IV. Preparatory Table for finding the Values of Assurance, according to the Manchester Unity Table of Mortality for 1866-70. Interest 3 per Cent.

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Table V. Value of an Assurance of £1, according to the Manchester Unity Table of Mortality for 1866-70. Interest 3 per Cent.