ANNUITIES. AN annuity, strictly speaking and practically, is a certain sum of money by the year; payable, usually, either in a single payment yearly, or in half, half-yearly, quarter, quarter-yearly, &c., and for a succession of years, greater or less, or forever. Pensions, awards, bequests, and the like, that are made payable in fixed sums for a succession of payments, are commonly rated by the year, and denominated annuities. A current annuity that has already commenced, or that is to commence after an interval of time not greater than that between the stipulated payments, is said to be in possession. One that is to commence or cease on the occurrence of an indeterminate event, as upon the death of an individual, is a reversionary, contingent, or life annuity. One that is to commence at a given period, and to continue for a given number of years or payments, is a certain annuity. One that is to continue from a given time, forever, is a perpetual annuity, or a perpetuity. Annuity payments do not exist fractionally: they mature, and exist only in that state, and are then due. A current annuity commences with a payment, and terminates with a payment. One current in the past is measured from a present included payment, closes with an included payment, and is said to be in arrears or forborne, from a supposed cancelled payment one regular interval or time beyond. One current in the future is measured from the present to the first included payment of the series, and from thence is said to continue to the close; but if the interval from the present to the first included payment is equal to that between the successive payments, it is supposed to continue from the present. Annuities in negotiation are adjusted, with regard to time, by interest, or discount, or both. The TABLES applicable to compound interest and compound discount are applicable in adjusting annuities at compound rates. To find the Amount of a Current Annuity in Arrears. LEMMA. The amount of an annuity that has been forborne for a given time is equal to the sum of the several payments that have become due in that time, plus the interest on each, from the time it became due, until the close of the time. Then the amount of an annuity of $100, payable in a single payment annually, but delayed of payment 4 years, allowing simple interest at per cent. on the payments, is And at 6 per cent. compound interest on the payments, it is And at 6 per cent. compound interest PER ANNUM, when payable in half-yearly instalments, it is From the foregoing, we derive the following general RULES: Let Pannuity or yearly sum, rrate of interest per annum, n or "nominal time of the annuity in full years, D=present worth for the full years. When the annuity is payable in a single payment yearly, r A=Pn(1+"(1)), Simple Interest. A= P(1+r)-1, Compound Interest. When payable in equal half-yearly instalments, A=Pn (1++), Simple Interest. 2 A=Px1+x(1+), Compound Interest. When payable in equal third-yearly instalments, A=p (1+r)*-1 2 r 1+), Compound Interest. When payable in quarter-yearly instalments, r 3r A=Pn (1+(-1)+3), Simple Interest. 2 3r A=P(1+r)^-1(1+ 3), Compound Interest. 8 When there are odd payments, to find the amount, S. When 1 half-yearly, For 1 third-yearly, quarter-yearly, S=A(1 + 66 2. P. S=A(1+r) + P(8+r)÷16. any number of equal and regular payments at compound interest per interval between the payments, S = P' (1+)-1), and for any number of equal and regular payments at simple interest per interval between the payments, SP'n' (1+1); P' being a payment, n' or " the number of payments, and r' the rate of interest per interval between the payments. But this must not be confounded with compound interest annually, on payments occurring semi-annually, quarterly, &c. EXAMPLE. What is the amount of an annuity of $150, payable in half, half-yearly, but delayed of payment 2 years and 72 days, allowing compound interest per annum at 7 per cent. ? = $310.50, the amount for 2 years, if payable in yearly payments, and 310.50× (1.-)=$315.93, the amount for 2 years, if payable in half-yearly payments, and 315.93 × (1. .07 x 72 $320.29, the amount for 2 years and 72 days, if payable in half-yearly payments. Ans. EXAMPLE. What is the amount of an allowance, pension, or award, of $100 a year, payable quarterly, but forborne 3 years, interest compound per annum at 6 per cent.? x(1+06x3)=$325.52, the amount for 3 years, 325.52 (1 +.03)+100×8.0616=$385.66. Ans. EXAMPLE. - What is the amount of $100 a year, payable in quarterly payments, and in arrears 4 years, interest being compound per quarter-year, at 6 per cent. a year? .06\16 4 25 [(1+0)-1]x. By tabular powers of (1+r), page 125, $448.30. Ans. To find the Present Worth of an Annuity Current. LEMMA. The present worth of an annuity that is to continue for a given time is equal to that sum of money, which, if put at interest from the present time to the close of the payments, will amount to the amount of the payments at that time; and therefore, the times being full, is equal to the sum of the several payments, discounted, respectively, at the rate of interest for their respective times. NOTE.-If the foregoing proposition is tenable, it follows, since simple interest is due and payable annually, that the true present worth of an annuity having more than one year to run cannot be found by simple interest and discount. By simple interest and discount, at 6 per cent., predicating the rule upon the foregoing lemma, the amount of $100, payable annually, and in arrears for 4 years, is $436; and the present worth, at 6 per cent., is 100 100 100 100 But $349 at 6 per cent. interest for 4 years, with the payments of interest annually, will amount to $440.60; and at interest simply for 4 years it will amount to only $432.76. Then the present worth of an annuity of $100, payable in a single payment yearly, and to continue 4 years, or to become due 1, 2, 3, and 4 years hence, interest and discount being compound per annum, and each at 6 per cent. — 100 × (1.06)3 =119.10 100.00 437.46 (1.06)*= $346.51. And interest at 6 per cent. and discount at 10, both compound, it is Therefore, when the annuity is payable in a single payment yearly from the present time, D=P {+r)2 = 1, = Α r (1+a)" (1+r)" when r and a are equal. When payable in half-yearly payments, D=P×(1+r)2 = 1 × (1+{r). r(1+a)" When there are odd payments, to find the present worth, S. For any number of equal payments, at equal intervals between the payments, S=P' x ; P' being a payment, n' the |