## Calculating and Comparing Effective Annual Rates

To see why it is important to work only with effective rates, suppose you've shopped around and come up with the following three rates:

Bank A: 15 percent compounded daily Bank B: 15.5 percent compounded quarterly Bank C: 16 percent compounded annually

Which of these is the best if you are thinking of opening a savings account? Which of these is best if they represent loan rates?

To begin, Bank C is offering 16 percent per year. Because there is no compounding during the year, this is the effective rate. Bank B is actually paying .155/4 = .03875 or 3.875 percent per quarter. At this rate, an investment of $1 for four quarters would grow to:

The EAR, therefore, is 16.42 percent. For a saver, this is much better than the 16 percent rate Bank C is offering; for a borrower, it's worse.

Bank A is compounding every day. This may seem a little extreme, but it is very common to calculate interest daily. In this case, the daily interest rate is actually:

This is .0411 percent per day. At this rate, an investment of $1 for 365 periods would grow to:

The EAR is 16.18 percent. This is not as good as Bank B's 16.42 percent for a saver, and not as good as Bank C's 16 percent for a borrower.

This example illustrates two things. First, the highest quoted rate is not necessarily the best. Second, compounding during the year can lead to a significant difference between the quoted rate and the effective rate. Remember that the effective rate is what you get or what you pay.

If you look at our examples, you see that we computed the EARs in three steps. We first divided the quoted rate by the number of times that the interest is compounded. We then added 1 to the result and raised it to the power of the number of times the interest is compounded. Finally, we subtracted the 1. If we let m be the number of times the interest is compounded during the year, these steps can be summarized simply as:

stated interest rate

The interest rate expressed in terms of the interest payment made each period. Also, quoted interest rate.

effective annual rate (EAR)

The interest rate expressed as if it were compounded once per year.

For example, suppose you are offered 12 percent compounded monthly. In this case, the interest is compounded 12 times a year; so m is 12. You can calculate the effective rate as:

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

PART THREE Valuation of Future Cash Flows

EAR = [1 + (Quoted rate/m)]" = [1 + (.12/12)]12 - 1 = 1.0112 - 1 = 1.126825 - 1 = 12.6825%

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