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Annuity

value, cent, sum, annuities, time, table and annual

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ANNUITY is the term applied, in its most general sense, to a fixed sum of money paid yearly or in certain portions at fixed periods of a year. There are two classes of annuities : certain, and life annuities ; the latter of which are those most popularly known as annuities. We will deal first with annuities certain. Such an annuity is a fixed sum payable yearly, or at certain periods in a year, during and for a specific time only. Thus the lessor of a house let at .£50 per annum, and having a residue of ten years of the term to run, would bo in receipt and the owner of an annuity certain of £50 for ten years. Se, too, may a salary be an annuity certain. Again, where a capitalist lends a sum of money to be repaid with interest at the expiration of a certain period, by equal animal payments, he is in effect purchasing by that loan an annuity certain. Intimately bound up with a consideration of annuities is the subject of interest. If capital could not earn interest, the value of an annuity certain would be simply the sum of the annual payments. Thus, the value of the above-mentioned lease would be £500—ten times its annual value ; and the value of a life annuity would be found by simply multiplying the annual payment by the average number of years to which individuals of the age of the annuitant live. But in the case of the loan we have instanced there is a question of interest. The sum paid annually is always the same, but the two elements which compose that sum vary necessarily at each time of paym6nt. The interest constantly diminishes ; and the amount of capital repaid as constantly increases. In the calculation of annuities certain there are therefore four things to be considered : the capital amount of the annuity ; the rate of interest ; the amount of the periodical payments ; and lastly, the duration of the time for payment. Any one of these four quantities can be calculated, if the other three are known. The method of calculation is, however, except in simple cases, too complicated and tedious for the general reader unversed in mathematical processes. Accordingly tables have been at various times prepared, in order to afford a facile method of solution of all prob lems connected with this class of annuities. Such tables are based on this principle : the duration of the annuity and the rate of interest not varying, the amounts of the annual payments must be proportionate to the capital sum they are designed to extinguish. Of these tables, we will refer to and set out selec

tions from two. Table I. shows the present value or purchase price of an annuity of £1 during one, two, three, &c., years, the interest being compound, at the rate of 2, 2i, 3, 4, or 5 per cent. Table II. gives the amount of an annuity of L1 at the end of one, two, three, &c., years, the interest being compound, at the rate of 27i, 3, 4, 41, or 5 per cent.

To illustrate the use of Table I. we will refer to the above example of a lease. In order to receive its value, interest being at 5 per cent., the lessor should look in the column headed 5 per cent., and there opposite to 10 in the first column will be found the value of £1 —£7.7217, wherefore the value of £386, Is. 8id. =the value of the lease. To illustrate the use of Table II., we will take the case of an investor or other person who wishes to create a sinking fend [see AMORTISATION] to provide a sum of, say, £1087, 5s. 7d. in fifteen years at 5 per cent. compound interest. Let him look opposite to fifteen years in Table II., and under 5 per cent. is 21.578, the amount of £1 for the given time and rate. Dividing £1087, 5s. 7d. (reduced into decimals, 1087.279) by this sum, the quotient 50.387 = £50, 7s. 9d.= the annual provision required. Or again, we will assume A. owes £1000, and desires to set aside £10 per annum for its repayment ; in what time will the debt be repaid, reckoning compound interest at 4 per cent.? 1000 divided by 10 gives 100. The number in Table II. under 4 per cent. nearest to 100 is 99.826, opposite forty-one years, the required time. It should bo noticed that the above tables and calculations are based upon the assumption that the payments are annual. If half-yearly or quarterly, their present value would be somewhat greater than in the former case. This is so, because certain portions of the annuity are received sooner, affording thereby a gain in interest. Tables such as the above are sufficient to meet these latter cases. Thus £100 per annum, payable quarterly for ten years, interest being 4 per cent., is equivalent to an annuity of .-S25, payable yearly for forty years, interest being at 1 per cent. Other equivalents can be worked out on the same principle.

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