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IV. Health Assurance.

19. A contract to secure a weekly payment in sickness, incapacitating from ordinary occupation, is called a health assurance.

20. Tables of Sickness, showing from the experience of large numbers the amount of sickness to which the members of Friendly Societies are liable at each age, have been constructed by Mr. Ansell, Mr. Neison, Mr. Finlaison, and Mr. Ratcliffe, and published in their works on Friendly Societies (see list, page 2). Mr. Ratcliffe's Sickness Tables for 1866-70, constructed from the latest and largest experience, include all sickness experienced, and show the amount of sickness in various periods of sickness, as the Manchester Unity and many other Friendly Societies assure full sick benefits in a temporary period of sickness, and then make a reduction in a continuance of the same sickness.

21. To find the value, at age x, of an assurance of£1 per week in the first twelve months' sickness during lift.

Let s_x denote the number of days' sickness experienced in the year following the age x according to any sickness table, and, assuming that the sickness in the year is uniformly distributed through the year, let denote the number of cases of sickness on any day of the year.

Let l_x denote the number of persons whose age is x living at the beginning of the year, and, assuming that the deaths in the year are uniformly distributed through the year, let l_x′, l_X′, l_X′, denote the numbers living to the end of the 1st, 2nd, 3rd,... days of the year.

Then the probability of a person whose age is x being sick on the first day of the year is .

As each of the l_x′ persons living on the second day of the year has the same chance of being one of those who are sick, the probability of x′ being sick on the second day is ; but

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as only l_x′ persons survive the first day, the probability of x living a clay, so as to be liable to sickness on the second day, is . Therefore the probability, at the beginning of the year, of x being alive and sick on the second day of the year is , which is the same probability as x being sick on the first day of the year.

As each of the l_x′ persons living on the third day of the year has the same chance of being one of those who are sick, the probability of x′ being sick on the third day is ; but as only l_x′ persons survive the second day, the probability of x living two days, so as to be liable to sickness on the third day, is . Therefore the probability, at the beginning of the year, of x being alive and sick on the third day of the year is , which is the same probability as x being sick on the first day of the year.

And the probability, at the beginning of the year, of x being alive and sick on any day of that year is always .

And the sum of the probabilities, at the beginning of the year, of x being alive and sick on each day of that year is , that is the average number of days' sickness to each person in the year following the age x.

The present value, at the beginning of the year, of £1 payable

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The sum of these values is equal to the product of into the sum of the series

The sum of the series is very nearly; therefore the value, at the beginning of the year, of £1 payable on each day of that year if x be then alive and sick, is that is the present value of as many pounds as are equal to the average number of days' sickness to each person in the year following the age x due half a year hence.

Let £ 1/7 denote the daily payment in sickness, then is the value, at age x, of an assurance of £ 1/7 per day in sickness for the following year, or making to denote the average number of weeks' sickness yearly to each person, is the value, at age x, of an assurance of £1 per week payable daily in sickness for the following year.

Let s_x denote the value, at age x, of an assurance of £1 per week in sickness for the following year, s_x+n the value, at age x+n, of the same payment for the following year; then

the value, at age x, of s_x being s_x

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Let S_x denote the value, at age x, of an assurance of £1 per week in sickness during life; then

In formula (1) the numerator of the fraction is the sum of the present values of the number of pounds falling due at the end of 0, 1, 2,... years, to the last tabular age, and the denominator the number now living to contribute equally the present fund.

If (1) 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 C′ of Table VII., opposite each age, is the product of as many pounds as are equal to the average number of weeks' sickness to each person in the year following that age due half a year hence into the number opposite that age in column D of Table II.

The number in column M′ of Table VII., 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 1. Value, at age 18, of an assurance of £1 per week in the first twelve months' sickness during life.

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22. To find the value, at aye x, of an assurance of £1 per week in the first twelve months' sickness for n years

Let |_nS_X denote the sum of the first n terms of the series S_x, and _n|S_x the sum of all the terms after the first n terms; then

Example 2. Value, at age 24, of an assurance of £1 per week in the first twelve months' sickness until arriving at age 65.

23. To find the annual premium, at age x, for an assurance of £1 per week in the first twelve months' sickness during life.

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_xS_x denote the annual premium, at age x, for an assurance of £1 per week in the first twelve months' sickness during life; then the value of an annuity of £1 on the life x is to £1 as the value of an assurance of £1 per week in the first twelve months' sickness during life is to its annual premium, or

Example 3. Annual premium, at age 18, for an assurance of £1 per week in the first twelve months' sickness during life.

24. To find the anuual premium, at age x, for an assurance of £1 per week in the first twelve months' sickness for n years.

If the annual premium completed at the end of the year be £1, the present value of it, at yearly interest, will be |_na_x, the value of a temporary annuity of £1 for the next n years if x shall live so long.

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Let P|_nS_x denote the annual premium, at age x, for an assurance of £1 per week in the first twelve months' sickness for n years; then the value of a temporary annuity of £1 for n years if x shall live so long is to £1 as the value of an assurance of £1 per week in the first twelve months' sickness for n years is to its annual premium, or

Example 4. Annual premium, at age 24, for an assurance of £1 per week in the first twelve months' sickness until arriving at age 65.

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