To find the present value of an an nuity of IL per annum continued for 10 years, interest being at 5 per cent., look in the column headed 5 p. c., and there, opposite to 10 in the first column, will be found the value 71221., or 71. 14s. 64d. This would be commonly said to be 7.722 years' purchase of the annuity. For a convenient rule for reducing decimals of a pound to shillings and pence, and the converse, see the Penny Magazine,' No. 52. It may also be done by the following table :- These conversions are not made with perfect exactness, as only three decimal places are taken. The error will never be more than one farthing.
To use Table I. where the number of years is not in the table, but is interme diate between two of those in the table, such a mean must be taken between the annuities belonging to the nearest years above and below the given year, as the given year is between those two years. This will give the result with sufficient nearness. We must observe, that no tables which we have room to give are sufficient for more than a first guess, so to speak, at the value required, such as may enable any one who is mas ter of common arithmetic, not to form a decisive opinion on the case before him, but to judge whether it is worth his while to make a more exact inquiry, either by taking professional advice or consulting larger tables. As an example of the case mentioned, suppose we ask for the value of an annuity of 11., continued for 12 years, interest being at 4 per cent. We find in Table I., column 4 per cent.
For 10 years 8.111 ,, 15 „ 11•118 Difference 3'007 Since 5 years adds 3.007 to the value of the annuity, every year will add about one-fifth part of this, or •601, and 2 years will add about 1.202. This, added to 8•111, gives 9'313. The real value is more near to 9'385, and the error of our table is '07 out of 9'313, or about the 133rd part of the whole. The higher we go in the table, the less proportion of the whole will this error be.
The last line in Table I. gives the va lue of the annuity of II. continued for ever : for example, at 5 per cent., the value of 11. for ever, or, as it is called, a perpetuity of 11., is 201. This is the sum which at 5 per cent. yields 11. a-year in interest only, without diminution of the principal. We see that an annuity for a long term of years differs very little in present value from the same continued for ever: for example, 11. continued for 70 years at 4 per cent. is worth 23•3951., while the per petuity at the same rate is worth only 251. Hence the present value of an annuity which is not to begin to be paid till 70 years have elapsed, but is afterwards to be continued for ever, is 1.605 at 4 per cent. : which sum improved during the 70 years, would yield the 251. necessary to pay the annuity for all years succeeding.
In this Table we see what would be possessed by the receiver of an annuity at the end of his term, if he put each years annuity out at interest so soon as he received it. For example, an annuity of 11., in 40 years, at 5 per cent., amounts to 120.81., which includes 401. received altogether at the end of the different years, and 80.81. the compound interest arising from the first year's annuity, which has been 39 years at interest, the second year's annuity, which has been 38 years at interest, and so on, down to the last year's annuity, which has only just been received. When the annuity is
payable half-yearly or quarterly, its pre sent value is somewhat greater than that given in the preceding Table. For the annuitant, receiving certain portions of his annuity sooner than in the case of yearly payments, gains an additional por tion of interest. Since 4 per cent. is 2 per cent. half-yearly and 1 per cent. quarterly, and since every term contains twice as many half-years as years, and four times as many quarters, it is evident that an annuity of 100/. a-year, payable half-yearly, at 4 per cent., for 10 years, is the same in present value as one of 501. Ter annum, payable yearly, at 2 per cent., for 20 years. Again, 100/. a-year, pay tale quarterly for 10 years, money being at 4 per cent., is equivalent to an annuity of 251., payable yearly for 40 years, money being at 1 per cent.
The principles on which the calculation of life annuities depends will be ex plained in the articles Lira INSURANCE and BILLS Or MORTALITY. Let us sup pose 100 persons, of the same age, buy a life annuity at the same office. Let us also suppose it has been found out, that of 100 persons at that age, 10 die in the first year, on the average, 10 more in the se cond year, and so on. If then it can be relied upon that 100 persons will die nearly in the same manner as the average of mankind, or at least that in such a number the longevity of' some will be compensated by the unexpected death of others, the fair estimation of the value of a life annuity to be granted to each may be made as follows:—To make the ques tion more distinct, let us suppose the bar gain to be made on the 1st of January, 1844, so that payment of the annuities is due to the survivors on new-year's day of each year. Moreover let each year's an nuity be made the subject of a separate contract. The first question is, what ought each individual to pay in order that he may receive the annuity of IL, if he survives in 1845. By the general law of mortality, we suppose that only 90 will remain to claim, who will, therefore, re ceive 901. among them, the remaining 10 having died in the interval. It is suffi cient, therefore, to meet the claims of 1845, that the whole 100 pay among them, January 1, 1844, such a sum as will, when put out at interest 4 per cent) amount to 90/. on anuary 1, 1845. This sum is 86.654/., and its hun dredth part is •866541., which is, there fore, what each should pay to entitle him self to receive the annuity in 1845. There will be only 80 to claim in 1846, and, therefore, the whole 100 must among them pay as much as will, put out at 4 per cent. for 2 years, amount to 80/. This sum is 73.968L, and its hundreth part is '739681., which is, therefore, what each must pay, in order to receive the annuity, if he lives, in 1846. The remaining years are treated in the same way, and the sum of the shares of each individual for the different years, is the present value of an annuity for his life. We must observe, that in the term value ofan annuity it is always implied that the first annuity becomes payable at the expira ton of a year after the payment of the purchase-money.