ON LIFE ANNUITIES. The value of an annuity for life depends not only on the interest of money, but also on the probability of the continuance of life, it may therefore be considered such a sum as will be sufficient to enable the person who grants the annuity to pay it without loss, allowing for the chances of mortality. If money is supposed to bear no interest, the value of an annuity is always equal to the expectation of life; but as money can be improved by putting it out to interest, the value of an annuity will not be worth so many years purchase as are equal to the expectation of life, and the higher the rate of interest is, the fewer years purchase the annuity will be worth. PROBLEM I. To find the expectation of any single life. RULE. Make the number of persons living opposite the given age (in Table 1.) a divisor, and the sum of the number of persons living from that age to 96 inclusive, a dividend; the quotient diminished by 5 will be the answer. Or, the expectation of life may be found in Table II. opposite to the given age. Examples. (1.) How many years may a person of 60 expect to live? Against 60 is 2038, the sum of this number, and those above it to 1 inclusive is 27953: then (279532038)-5=13·21 years. Answer, see Table II. (2.) Required the expectation of a life of 45, of 75, and of 80. PROBLEM II. To find the probability that a person of a given age shall live a certain number of years. RULE. Make the number opposite to the given age (in Table L.) the denominator of a fraction, and the number opposite to the proposed age the numerator. In the case of joint lives, the product of the fractions found as above will shew the probabilities. Examples. (1.) What is the probability that a person of the age of 60 shall live 10 years? Against 60 in Table I. stands 2038, and against 70 you will find 1232; so the probability is 1232 2038 The probability of a person aged 60 being dead in 10 years, is 1 1232 806 2038 2038 (2.) What is the probability that each of three sons, separately, whose ages are 20, 30, and 40, shall live 15 years; and what is the probability that they shall -all live 15 years? PROBLEM III. To find the probability that either the one or the other of two persons of different ages shall live a certain number of years. RULE. Find the probability that each of the persons shall live the proposed number of years, and the probability that they shall jointly live the said number of years: the latter result subtracted from the sum of the former will give the answer. Examples. (1.) Suppose there are two persons, the one aged 20, and the other 40 years, what is the probability that one of them will be alive after 30 years have elapsed? 2837 5132' and that they Hence The probability that a man of 20 shall attain the age of 50, is 1232 that a person aged 40 shall live to 70, is 3635 2837 1232 3495184 5132 8635 18654820 3495184 13139935 Answer. 18654820 18654820 shall jointly live 30 years, is X 2837 1232 (2.) What is the probability that of two persons, the one aged 50, the other 65, one of them shall be living at the expiration of 12 years? PROBLEM IV. To find the value of an annuity on any single life. RULE. Multiply the number in Table III. against the given age, by the proposed annuity, and the product will be the answer. Examples. (1.) What must be given for an annuity of £60 during the life of a person aged 46, reckoning interest at 4 per cent.? Against 46, and under 4 per cent. yon will find 12.089, that is, the annuity is worth 12 years purchase, hence 12.089 × 60=£725·34. (2.) What is the value of an annuity of £200 payable during the life of a person aged 25 years, reckoning interest at 5 per cent.? (3.) What is the value of the life interest of a person aged 56 in £3000 stock in the 3 per cent. consolidated annuities. Interest at 5 per cent.? (4.) What is the difference in value between an annuity of £80 during the life of a person aged 36, and an annuity of the same amount, certain for 20 years. Interest at 5 per cent.? PROBLEM V. To find what annuity any given sum will purchase during the life of a person of a given age. RULE. Divide the given sum by the number opposite to the given age, and under the given rate per cent. in Table III. and the quotient will shew the annuity. Examples. (1.) A person of 50 years of age wishes to lay out £1500 in an annuity for his life. Interest at 5 per cent. What annuity will it purchase? Against 50 years, and under 5 per cent. you will find 10.269 hence 1500-10.269 £146.07, the annuity required. (2.) When the 3 per cent, consols sell for 773 per cent, what annuity for life should be granted to a person aged 58 for £6000 stock? (3.) A gentleman aged 60, who receives an annuity of £200 for life, wishes to exchange it for an annuity of the same sum to continue during the life of his wife, whose age is 34, what sum ought he to give for the exchange, calculating at 4 per cent.? (4.) A person has an annuity of £150 during the life of a gentleman aged 30, but being advanced in age, and wanting money, he is willing to exchange it for an equivalent annuity to continue during the life of a person aged 50; what annuity should be granted him? Interest at 5 per cent. (5.) A person aged 30 is possessed of £80 a year in the government long annuities, which will terminate in January 1860; this he is willing to relinquish for an annuity during his life, to commence in January 1820; what annuity ought he to receive, reckoning interest at 5 per cent.? PROBLEM VI. To find the present value of a given sum to be received at the death of a person of any age: or to find what sum must be paid annually by a person of any age, that his heirs muy receive a given sum of money at his death. RULE. Multiply the number in Table III. against the given age, by the interest of £1 for a year, and subtract the product from an unit; divide the remainder by the amount of £1 for 1 year, the quotient multiplied by the given sum will give the value required. To find the value in annual payments to the number in Table III. opposite to the given age add an unit, and divide the value found above by this result, the quotient will be the answer. See TABLE IV. Examples. (1.) What ought a person, aged 45, to pay down, that his children may receive £1000 at his death, or what sum ought he to pay annually for the same advantage, reckoning interest at 4 per cent.? In Table III against 45, and under 4 per cent, stands 12-283; (12 283 × 04 X1000 £489.115 £489 2 3 the value in 1.04 then 1 a single payment; and 489.115 1+12.283 =£36·82263=£36 16 5 the annual payment; the first being paid immediately, and the remaining ones at the beginning of every subsequent year. (2.) What is the present value of £1000 to be re 串。 ceived on the death of a person aged 60, interest being reckoned at 3 per cent. *; and what ought to be paid annually to insure the same sum. (3.) What sum must be paid annually that the heirs of a person aged 30, may receive £1000 at his decease, reckoning interest at 5 per cent. ? PROBLEM VII. To find what sum a person ought to receive, who has insured his life to a given amount, in order that he may relinquish his claim. RULE. Multiply the annual payment which has been made since the insurance commenced by the value of an annuity on the life at its present age from Table III.; subtract the product from the value of the insurance of the given sum on the life at its present age, (Prob. VI.) the remainder will be the answer. * Examples. (1.) A person whose present age is 50 has been pay The rates of insurances for lives, at all the different offices established in London, are calculated from the Northampton Tables, at 3 per cent. interest; viz. at the lowest rate of interest, and the lowest probabilities of living. See Table IV. |