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# IV. Health Assurance

page 19 ### 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 sx 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 lx 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 lx′, lX′, lX′, 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 lx′ 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

page 20 as only lx′ 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 lx′ 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 lx′ 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

page 21 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 sx denote the value, at age x, of an assurance of £1 per week in sickness for the following year, sx+n the value, at age x+n, of the same payment for the following year; then

the value, at age x, of sx being sx

page 22 Let Sx 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 vx, 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.

page 23 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 |nSX denote the sum of the first n terms of the series Sx, and n|Sx 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 ax, the value of an annuity of £1 on the life x.

Let PxSx 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 |nax, the value of a temporary annuity of £1 for the next n years if x shall live so long.

page 24 Let P|nSx 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.

page 25  Table VI. Average Amount of Sickness Yearly in the first Twelve Months' Sickness, and after Twelve Months' continued Sickness, to each Member of the Independent Order of Odd Fellows, Manchester Unity Friendly Society for 1866-70, in the Rural, Town, and City Districts combined.

page 26  Table VII. Preparatory Table for finding the values of Assurances in the first Twelve Months' Sickness, according to the Manchester Unity Tables of Sickness and Mortality for 1866-70. Interest 3 per Cent.

page 27  Table VIII. Value of an Assurance of £1 per Week in the first Twelve Months' Sickness, according to the Manchester Unity Tables of Sickness and Mortality for 1866-70. Interest 3 per Cent.

page 28  Table IX. Preparatory Table for finding the values of Assurances in Sickness after Twelve Months' continued Sickness, according to the Manchester Unity Tables of Sickness and Mortality for 1866-70. Interest 3 per Cent.

page 29  Table X. Value of an Assurance of £1 per Week in Sickness after Twelve Months' continued Sickness, according to the Manchester Unity Tables of Sickness and Mortality for 1866-70. Interest 3 per Cent.

* The method employed in the construction of columns C′ and M′ of Table VII. is described in the Appendix.