An Application of Exponential Functions: Compound Interest

Simple interest, a topic you probably studied in grade 7 or 8, involves the computation of interest once during the completion of a financial transaction. The interest, I, is found by multiplying the amount of money borrowed, or the principal, P, the rate of interest, r, and the amount of time, t. The terms of the interest rate and the amount of time must agree in units. If the rate of interest is given in years, then the time must be given in years. For example, if $1000 is borrowed for 18 months at a rate of 6% per year, the amount of interest due is given by the equation I = (1000)(0.06)(1.5) = $90. The time of 18 months is expressed as 1.5 years, and the interest rate is converted to a decimal for computational purposes.

Most people have an understanding of borrowing money from a financial institution. Savings accounts and other investments are also loans in that the consumer is lending money to the institution. The big difference in these cases is the rate of interest given.

Compound interest is the mechanism by which most of the world's financial transactions take place. With compound interest, the interest is computed periodically throughout the term of the loan. In the example just used, suppose the $1,000 is borrowed for 18 months at 6% annual interest compounded quarterly. That is to say, the interest on the loan will be computed each quarter. The rate of interest is an annual 6%. Dividing this into four equal amounts (quarterly), the quarterly rate of interest (or periodic rate of interest) is 1.5%. To determine the amount of interest due at the end of the term of the loan, you will need to work through the process at each quarter. It is important to realize that the time unit in each of these calculations will be 1. The interest being computed is the interest for 1 quarter.

   The amount of interest due after 18 months (or 6 quarters of a year) is $168.12, much more than the $90 collected with simple interest. If you were doing the calculations, you may have noticed that the amount of interest due was not rounded by applying the usual rules for rounding but, rather, that the numbers were always rounded up to the next penny. The bank also keeps the fractional parts of a penny.

Look back on these calculations. The $1,827 due at the end of the first quarter was computed by 1800 + 1800(0.015). Factor the 1800 to get 1800(1.015). At the end of the second quarter, the 1800(1.015), or 1827, becomes 1800(1.015) + 1800(1.015)(0.015). Factoring the 1800(1.015), you get 1800(1.015)(1.015), or 1800(1.015)2. If you continue in this manner, the amount due at the end of the sixth quarter would be 1800(1.015)6 = 1868.197, or 1868.20. The difference of 8 cents are all those fractions of a penny that earned interest!

The language of time divisions: annually—once per year; semiannually—twice per year; quarterly—4 times per year; monthly—12 times per year.

When P dollars are invested for t years at 100r % per year compounded n times per year, the value of the investment at the end of the term is given by

.

$2000 is invested for 25 years at 3.4% compounded monthly. How much will the investment be worth at the end of 25 years?

P = 2000, r = 0.034, n = 12, and t = 25. The value of the investment is

(Note: Type the information into the calculator as it is written. Do not try to compute and enter the decimal for 0.034/12, and be sure that there are parentheses about 12*25.)

Which investment will be worth more at the end of 10 years, $1000 invested at 3.8% compounded semiannually or $1000 invested at 3.6% compounded monthly?

P = 1000 and t = 10 for both investments. The first investment has r = 0.038 and n = 2. It will be worth

.

The second investment has r = 0.036 and n = 12. It will be worth:

. The first investment will yield the larger amount.

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