84 MATHEMATICAL THEORY OF INVESTMENT If the annual rent of an annuity is S, then K will be a factor of every term in the progression whose sum is given by formula (1) or formula (3). Hence, K will be a factor of the sum, and we have as formulas expressing the present value A of an annuity whose annual rent is S, A = Ita^ = R ~ v , (Q\ i ' ' and A=^=R———=J__. ^ p (1+i^-l The formulas are similar when expressed in terms of the nom- inal rate j, payable m times a year. EXAMPLES 1. A man buys a house, agreeing to pay $1000 cash and $1000 at the end of each year for five years. What would be the cash price of the house (a) if money is worth 6% effective? (6) if money is worth 6% nominal, payable semiannually ? Solution. The first $1000 constitutes a cash payment whose present value is (1000, and the other five payments constitute an annuity of $1000 per annum. For the first case the present value of the annuity, as given by formula (1), is $1000 ^^"^ N212.36. .08 The cash price would then be $5212.36. For the second case the present value of the annuity, as given by for- mula (2), is 1— <1 03^-10 $loool(i^)^^^=$420207' and the cash price would be $5202.07. 2. Find the present value of an annuity of $395.47 per annum, payable twice a year, for 10 years, with interest at 6% nominal, convertible twice a year. 33. Annuities due and deferred annuities. An annuity due has been denned in § 30 as one whose first payment is made at the beginning instead of the end of the first interval. The present value of an annuity due may be expressed easily in terms of the present value of an ordinary annuity, for the ANNUITIES 85 first payment of an annuity due is cash, and the remainder of the payments constitute an ordinary annuity with one inter- val less. We have therefore a,n=l+a,.-TT (r> Similarly, for an annuity due, payable p times a year,
The amount of an annuity due is used by continental writers in place of the amount of an ordinary annuity. Denoting the amount of an annuity due by s^, and remembering that the first payment is at interest for n years, the second for n — 1 years, and so on, we have ^^^,..^+,y.-i+ ...+(!+,) ^l+^l^l+zy-'+Cl+O-2 +...+1]; so that s^ = (1 + i) s^ = s,^ — 1. (3) Similarly, for an annuity payable p times a year we have ^=(i+iys^ (4) A deferred annuity has been defined as one whose term does not begin until the expiration of a fixed number of years. If an annuity payable for n years be deferred for m years, m + n years will elapse before the last payment is made. The annuity may be looked upon as an annuity for m + n years, for which the pay- ments for the first m years are withheld. The present value of the annuity for n years deferred m years is, then the present value of the annuity for m + n years diminished by the present value of the annuity for m years under the same conditions. In symbols, n^^a^-a^ . (5) I n\ m+ nj m) - y Formula (5) holds for all values of p, whatever the conversion interval for the interest may be. The amount of a deferred * Ordinarily the last payment of an annuity is looked upon as closing the trans- action. On the date of the last payment of an annuity due the amount would be an or s^, according as the payments are made annually or p times a year.