EXPECTATION OF LIFE. From the Bills of Mortality in different places, tables have been constructed which shew how many persons, upon an average, out of a certain number born, are left at the end of each year to the extremity of life. From such tables, which, as we have seen, are founded on the doctrine of Chances, the probability of the continuance of a life, of any proposed age is known. TABLE I. Shewing the Probabilities of the Duration of Human Life, deduced from the Register of Mortality at Northampton. Persons Decrem. Persons Decrem. Persons Decrem. Age. living. of Life. Age. living. of Life. Age. living. of Life. Note. 1. Here it must be observed that, of 11650 infants born, 3000 will die in the first year. Of the 8650 who live to be one year old, CASE I. To find, by this Table, the expectation of any single life. RULE. Divide the sum of all the living in the table, at the age whose expectation is required, and at all greater ages, by the sum of all that die annually at that age, and above it, or, which is the same thing, by the number in the table of the living at that age, and half unity, or .5 subtracted from the quotient will be the expectation required. Ex. 1. What is the expectation of a life at 60? The sum of the living at the age of 60 and upwards, by the table, is 27947, which divided by 2038, the number of living at that age, gives 13.71, from which subtract`.5, and the expectation of a life at 60 is equal to 13.21, or 13 years, 11 weeks nearly.† Ex. 2. What is the expectation of a life 70 years of age, one of 80, and one of 90 ? CASE II. To find the probability that a given life shall continue any number of years, or attain a given age. RULE. Make the number in the table, opposite to the proposed age, the numerator of the fraction, and for the denominator take the number opposite the present age. Ex. What is the probability that I, who am 45, shall live to 60 ? The number against 60 or as - 5: 3. 2038 2038 Therefore the chances in my favour are 20: 12 nearly, 3248, the chance of 3248 1367 will die in the course of the second year. Therefore, of the 11650 new born infants, the chance of living to the end of the year, is to that of dying within that period, as 8650: 300, or almost 3 to 1. Again, the chance which an infant, just born, has of living two years, is as the number of living at the end of two yea s, is to the number that have died in that time, or as 7283 to (3000+ 1367) 4367, or nearly 2 to 1. * This number is found by adding all the numbers up from 2038 to 1 inclusive. + Lest the youth should mistake the meaning of the phrase "Expectation of Life," let him be warned that no more is meant by it than that a set of lives, as 100, aged 60, will, one with another, enjoy 13 years 11 weeks each of existence, some of them enjoying a duration as much longer as others fall short of it. denominators being the same, the chance of life is to the probability of dying as 2038 to 1210, or as 20 to 12, or as 5 to 3 nearly. Ex. 2. What is the probability that a person aged 21, shall attain to 54? Ex. till 70 ? 3. What is the probability that a person aged 15 should live Ex. 4. What chance has a person aged 70 of living 10 years longer? From the foregoing table is formed TABLE II. Shewing the Expectation of Human Life at every Age, according to the Probabilities found by Table I. To find the expectation of any given life. RULE. Seek in the table the given age, and opposite to it is the expectation. Thus, the chance of life to an infant just bòrm is 25.18, or rather more than 25 years; to a person of 45 years of age 20.52, as we have found before, see p. 192; and to a person of 69, just 9 years. Upon these tables is founded the doctrine of LIFE ANNUITIES. LIFE ANNUITIES are annual payments to continue during any life or lives. These are generally purchased or sold for a present sum of money. "The present value of a life annuity" is the sum that would be sufficient (allowing for the chance of life failing, which has been considered in the preceding pages) to pay the annuity without loss. If money bore no interest, the value of an annuity of 17. would be equal to the expectation of life. Thus, Table II. P. 202, the value of an annuity for a life of 20 years of age, if money bore no interest, would be equal to nearly 33 years and a half purchase; that is, 337. 10s. in hand for each life, would be sufficient to pay to any number of such lives 17. per annum. If money is capable of being improved by being put out to interest, the sum just mentioned would be more than the value, because it would be more than sufficient to pay the annuity; and it will be as much more than sufficient as the interest is greater. As an example, If money can be improved at 5 per cent. compound interest, the half of 331. 10s., or 16. 15s., will, as we have seen, p. 192, in little more than 14 years, produce the 337. 10s. required. It must not however be supposed, that 167. 15s. is the true value of an annuity of 17. during a life of 20. The value of an annuity certain for a term equal to the expectation, always exceeds the true value, because, in a number of life annuities, many of the payments would not be to be made till a much more remote period than the term equal to the expectation. Upon this principle the following table. is computed, from which it appears that the present value of an annuity of 17. on a life of 20 years of age, is equal to 147. and a small fraction only; that is, 147. in hand for each life, improved at compound interest, will be sufficient to pay to any number of such lives 17. per annum. TABLE I. Shewing the Value of an Annuity of 14. on a Single Life, at every Age, according to the probabilities of the Duration of Human Life at Northampton, reckoning interest at 5 per cent. To find the value of an annuity for a person of age. any given RULE. Multiply the number in the table against the given age, by the sum, and the product is the answer. Ex. 1. What should a person, aged 45, give to purchase an annuity of 607. per annum during life, interest being reckoned 5 per cent ?. The value in the table against 45 years is 11.105, and this multiplied by 60 gives the answer, 6661. 6s. Ex. 2. A person aged 69 years would purchase an annuity of 2001. for life, what must he pay for it in ready money at the same rate of interest ? |