### SINGLE PAYMENT COMPOUND INTEREST FORMULAS - TIME VALUE OF MONEY.

Suppose a present sum of money P is invested for one year

**l**at interest rate i. At the end of the year, we should receive back our initial investment P, together with interest equal to iP, or a total amount P + iP. Factoring P, the sum at the end of one year is P(1 + i).

Let us assume that, instead of removing our investment at the end of one year, we agree to let it remain for another year. How much would our investment be worth at the end of the second year? The end-of-first-year sum P(1 + i) will draw interest in the second year of. i P(1 + i). Thismeans that, at the end of the second year, the total investmentwill become

**1**A more general statement is to specify "one interest period" rather than "one year." Since, however,

it is easier to visualize one year, the derivation will assume that one year is the interest period.

This may be rearranged by factoring P(1 + i), which gives us

In other words, a present sum P increases in n periods to P(1 + i)^n. We therefore have a relationship between a present sum P and its equivalentfuture sum, F.

This is the single payment compound amount formula and is written in functional notation as

The notation in parentheses (F/P, i, n) can be read as follows:

To find a future sum F, given a present sum, P, at an interest rate i per interest period,

and n interest periods hence. .

Functional notation is designed so that the compound interest factors may be written in an equation in an algebraically correct form. In Equation 3-4, for example, the functional mnotation is interpreted as

which is dimensionally correct.Without proceeding further, we can see that when we derive. a compound interest factor to find a present sum P, given a future sum F, the factor will

be (P/F, i, n); so, the resulting equation would be

which is dimensionally correct.

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