SIMPLE AND COMPOUND INTEREST

BY

CAROLYN JOHNSON

UNIVERSITY OF GEORGIA

Have you ever had money that you didn't know what to do with it? That doesn't occur very often in real life. Let's pretend that you did have extra money, and you wanted to make it "work" for you. One of the simplest ways to invest your money would be to place it in a savings account at a bank. The bank pays "interest" back to you for the money you keep in the savings account. On the business side of the investment, the bank uses the money you keep in the savings account, then pays you "interest" for the usage of it. "interest" is money paid by a bank, individual, or other sources for using someone else's money. The amount of money used by the bank, etc, is called the "principal." Interest is usually paid at specified intervals of time, usually annually, semiannually, or quarterly. This is the good side of interest. Unfortunately, what most people know of interest is the interest they have to pay to a bank, person, or other source for the money thay have borrowed. When people lack enough money to pay bills or whatever, they may use a bank's money (versus the bank using their money). In this situation the person pays the bani for the usage of the bank's mney. The first scenario is far better than the second one.

We've already talked about what "interest" is. Now let's see what is meant by the "rate of interest." The rate of interest is the ratio of the interest charged in one unit of time to the principal. The unit of time is usually a year or less. The rate is generally expressed as a percentage.

Simple interest is interest computed on the principal for the time it is used. Simple interest "I" on the principal "P" for "t" years at a rate of "r" per year is as follows:

The amount A is the principal plus the interest and is given as

For example, if a person borrows \$800 at a rate of 4% to be paid back in two and a half years, the interest is

So the amount A due at the end of the two and a half years is

Let's look at another example. What is the rate if interest that will yield \$1000 on a principal of \$800 in 5 years? Here we are to find the "rate" instead of the total amount to be paid back. Remember that the amount is the sum of the principal and the interest. So we start out with the formula

and set it up to find "r." So we now have

.

With this we plug in our knowns in order to find our unknown "r."

Our 0.05 could be written as 5%. Therefore the rate would be 5% if we borrowed \$800 and had to pay back a total of \$1000. Now let's use the EXCEL program to compare the interest accrued with various time lengths, keeping the principal and the rate constant. Notice the differences of the interest and the total amount that has to be paid back.
 interest rate time principal amount 60 0.06 1 1000 1060 150 0.06 2.5 1000 1150 240 0.06 4 1000 1240 300 0.06 5 1000 1300 600 0.06 10 1000 1600 510 0.06 8.5 1000 1510 1200 0.06 20 1000 2200

So we see that simple interest is relatively easy to calculate. Compound interest, though not as easy to compute, is much better to receive, but not as nice to have to pay out. How does compound interest work? Suppose the interest due at the end of the first specified number of equal intervals of time is added to the original principal, and then this amount acts as the second principal for the second interval of time. This process continues for the specified length of time that the money is borrowed. In this case the interest has been "compounded" or converted into the principal. That is, the interest has been added to the principal before computing the next unit of time's interest. So the principal used to compute the interest is actually the sum of the previous unit of time's interest plus the principal. This process continues throughout the time the money is borrowed. The sum of the compounded interest and original principal is called the "compound amount." The successive equal intervals of time during which interest is compounded is called the conversion period or interest period. This is usually three or six months or one year. Interest rates are usually based on a yearly basis even though the compound interest is computed for each conversion period. The rate of interest quoted as a yearly rate is called the "nominal rate." So if a nominal rate is 4% compounded quarterly and the conversion period is three months, then the interest rate is 1/4(4%) = 1% for each conversion period.

If "P" is the original principal, "i" the rate of interest per conversion period, and "n" the number of conversion periods, then the compound amount A at the end of these "n" conversion periods is given by

The compound interest is I = A - P.

EXAMPLE: A woman invests \$1000 at 6% compounded semiannually. Find the compound amount A and the compound interest I after 2 years. P = \$1000, the rate = 1/2(6%) = 3%, and n = 4 (because each conversion period is 1/2 year and there are 4 such periods in the 2 years). Then

and

Let's compare the previous table of simple interest with a table of compound interest using the same basic information.

 interest rate time principal amount 60 0.06 1 1000 1060 150 0.06 2.5 1000 1150 240 0.06 4 1000 1240 300 0.06 5 1000 1300 600 0.06 10 1000 1600 510 0.06 8.5 1000 1510 1200 0.06 20 1000 2200

 interest rate time principal amount 60.8999999999999 0.06 2 1000 1060.9 318.27 0.06 5 1060.9 1379.17 662.0016 0.06 8 1379.17 2041.1716 1224.702 0.06 10 2041.17 3265.872 3919.044 0.06 20 3265.87 7184.914 7328.6082 0.06 17 7184.91 14513.5182 34832.4 0.06 40 14513.5 49345.9

Wow! Look at the difference between the amount of interest computed with simple interest versus compound interest. We started with the same principal of \$1000, the same rate of 6%, and the same time-frame. In the compound table the time is listed by the number of conversion periods rather than by the number of years. So twice a year the interest was compounding. I know that I would want to receive an interest amount of \$34,832.40 instead of just \$1200.00. Wouldn't you?

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