## Investing for More Than One Period

Going back to our \$100 investment, what will you have after two years, assuming the interest rate doesn't change? If you leave the entire \$110 in the bank, you will earn \$110 X .10 = \$11 in interest during the second year, so you will have a total of \$110 + 11 = \$121. This \$121 is the future value of \$100 in two years at 10 percent. Another way of looking at it is that one year from now you are effectively investing \$110 at 10 percent for a year. This is a single-period problem, so you'll end up with \$1.10 for every dollar invested, or \$110 X 1.1 = \$121 total.

This \$121 has four parts. The first part is the \$100 original principal. The second part is the \$10 in interest you earned in the first year, and the third part is another \$10 you earn in the second year, for a total of \$120. The last \$1 you end up with (the fourth part) is interest you earn in the second year on the interest paid in the first year: \$10 X .10 = \$1.

This process of leaving your money and any accumulated interest in an investment for more than one period, thereby reinvesting the interest, is called compounding. Compounding the interest means earning interest on interest, so we call the result compound interest. With simple interest, the interest is not reinvested, so interest is earned each period only on the original principal.

### Interest on Interest

Suppose you locate a two-year investment that pays 14 percent per year. If you invest \$325, how much will you have at the end of the two years? How much of this is simple interest? How much is compound interest?

At the end of the first year, you will have \$325 X (1 + .14) = \$370.50. If you reinvest this entire amount, and thereby compound the interest, you will have \$370.50 X 1.14 = \$422.37 at the end of the second year. The total interest you earn is thus \$422.37 - 325 = \$97.37.

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