70 MATHEMATICAL THEORY OF INVESTMENT ILLUSTRATIVE EXAMPLE. Find the present value of $379.25 with in- terest payable annually at 4% and due 3 years hence when money is worth 5% nominal, payable semiannually. Solution. By formula (4), ., -.„ V, = $379.25 ( ) , " " /I nrtC\<
(1.025)< where the nominal rate has been introduced by means of the relation . (l+f)2^^). The result is found by logarithms to be ^367.86. The expression ,——. occurs so often and is of such importance 1 +z ' in what follows that it is customary to denote it by a single letter, v. We have, then, as an equation denning v, v=-^=d+ir1. (5) Replacing (1 + i)-1 by v, the formulas (2) and (4) become F,.=JV (6) and F; = P (1 + ^)"V. (7) The expression ®''=(l+z')-" is sometimes called the discounting factor, just as (1 + iy has been called an accumulation factor (§ 22). A given sum is discounted by multiplying it by v". Formulas (3) and (4), or the equivalent formulas (6) and (7), are sufficient for the solution of all ordinary problems requir- ing the determination of present values or discounts. In prac- tice the work is usually done by means of a table of values of t>"=(l+i)-». Table IV gives the present values of 1 at rates 1^ 1^., 2, 2^, 3, 31-, 4, 4^ 5, and 6% for times up to one hundred years. 27. The rate of discount. The measure of the discount on unit principal for unit time is called the rate of discount. Since present value has been denned as the principal diminished by the discount, we have, as a denning equation, v=l-d, (1) where d denotes the rate of discount. INTEREST 71 From equation (1) we may easily find the relation between the rate of discount and the rate of interest, for, replacing v by its value as given by (1) (§ 26), 1 . ,
1+z Solving this equation for d,
=l-d.
d=w=iv<2)
Solving the same equation for i, ^~~\~~d'
(3)
Formulas (2) and (3) furnish the means for the direct com- putation of i when d is known, and vice versa. For example, if i = .06, then, by simple division, d=-06 =.05660 +.... 1.06 In many cases, however, it is even easier to express the one quantity in terms of the other by means of a power series, and to use the power series as a means of computation. By division we obtain from (2) the formula d=i-is+is-ii+.... (4) Similarly, from (3), i=d+d2+ds+.... (5) ILLUSTRATIVE EXAMPLE. What is the rate of discount when the rate of interest is .06 ? Solution. By (4), d = .06 - (.06)2 + (.06)3 - (.06)4 + ... = .05660 + .. Additional terms would not change this result unless more decimal places were used. The computation of i by means of (6) is even simpler, since all the terms of the series are positive. ^F NOTE. The so-called true discount is denned as the difference between the principal and the sum which, put at simple interest, would amount to the princi- pal when the latter becomes due. 72 MATHEMATICAL THEOEY OF INVESTMENT Denoting the present value of 1 discounted by true discount by »„, the definition gives .;. (1 + m) = 1; so that ^TTr.i' (6)
B" '"'- " 1 + ni Similarly, if V^ denote the present value of P, ^_ p
K=
1+m
(6)
(7)
True discount is rarely used in practice. The fact that it cannot be used in problems where it is assumed that interest is paid promptly is brought out by means of Example 8, below. In the second part of this example the payments must be discounted by compound and not by true discount, in order to make the purchase price of the farm $5000, as it should be.
EXAMPLES 1. Find the present value of $1239 due in 2 years without interest when money is worth 5%. 2. Find the present value- of a debt of $1239 with interest payable annually at 6% for 2 years, when money is worth 5%. 3. A father wishes to set aside, at the birth of his son, a sum that will accumulate to $5000 by the time the son reaches his majority. If money is worth 5%, what is the sum required ? 4. Find the present value of $1225 due 3 years and 6 months hence without interest when money is worth o%. 5. What is the rate of discount when the rate of interest is .05 ? 6. What is the rate of interest corresponding to the rate of discount .05 ? 7. A wholesale merchant sells his goods on 90 days' time or 3% off for cash. What rate of interest is equivalent to this discount ? In other words, what would be the highest rate the buyer could afford to pay to borrow money to pay cash ? 8. One man buys a farm for $5000, agreeing to pay $1000 cash and $1000 with interest at 6% on all sums remaining due at the end of each year, until the whole amount is paid. How much does he pay each year ? Another man buys a farm, agreeing to pay $1000 cash, $1240 at the end of the first year, $1180 at the end of the second, $1120 at the end of ' the third, and $1060 at the end of the fourth. If money is worth 6%, what is the cash price of the second man's farm? Explain the significance of these two problems in the light of the note at the end of § 27. 9. Find the rate of discount -when the nominal rate of interest is .06, convertible semiannually.