Effective Annual Rates and Compounding

If a rate is quoted as 10 percent compounded semiannually, then what this means is that the investment actually pays 5 percent every six months. A natural question then arises: Is 5 percent every six months the same thing as 10 percent per year? It's easy to see that it is not. If you invest $1 at 10 percent per year, you will have $1.10 at the end of the year. If you invest at 5 percent every six months, then you'll have the future value of $1 at 5 percent for two periods, or:

This is $.0025 more. The reason is very simple. What has occurred is that your account was credited with $1 X .05 = 5 cents in interest after six months. In the following six months, you earned 5 percent on that nickel, for an extra 5 X .05 = .25 cents.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

III. Valuation of Future Cash Flows

6. Discounted Cash Flow Valuation

© The McGraw-Hill Companies, 2002

CHAPTER 6 Discounted Cash Flow Valuation

As our example illustrates, 10 percent compounded semiannually is actually equivalent to 10.25 percent per year. Put another way, we would be indifferent between 10 percent compounded semiannually and 10.25 percent compounded annually. Anytime we have compounding during the year, we need to be concerned about what the rate really is.

In our example, the 10 percent is called a stated, or quoted, interest rate. Other names are used as well. The 10.25 percent, which is actually the rate that you will earn, is called the effective annual rate (EAR). To compare different investments or interest rates, we will always need to convert to effective rates. Some general procedures for doing this are discussed next.

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