Example 69 I Quoting a Rate

-.—' Now that you know how to convert a quoted rate to an EAR, consider going the other way. As a lender, you know you want to actually earn 18 percent on a particular loan. You want to quote a rate that features monthly compounding. What rate do you quote?

In this case, we know the EAR is 18 percent and we know this is the result of monthly compounding. Let q stand for the quoted rate. We thus have:

EAR = [1 + (Quoted rate/m)]m - 1 .18 = [1 + (q/12)]12 - 1 1.18 = [1 + (q/12)]12

We need to solve this equation for the quoted rate. This calculation is the same as the ones we did to find an unknown interest rate in Chapter 5:

1.18(1/12) = 1 + (q/12) 1.1808333 = 1 + (q/12) 1.0139 = 1 + (q/12)

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 q = .0139 X 12 = 16.68%

Therefore, the rate you would quote is 16.68 percent, compounded monthly.

EARs and APRs

Sometimes it's not altogether clear whether or not a rate is an effective annual rate. A case in point concerns what is called the annual percentage rate (APR) on a loan. Truth-in-lending laws in the United States require that lenders disclose an APR on virtually all consumer loans. This rate must be displayed on a loan document in a prominent and unambiguous way.

Given that an APR must be calculated and displayed, an obvious question arises: Is an APR an effective annual rate? Put another way, if a bank quotes a car loan at 12 percent APR, is the consumer actually paying 12 percent interest? Surprisingly, the answer is no. There is some confusion over this point, which we discuss next.

The confusion over APRs arises because lenders are required by law to compute the APR in a particular way. By law, the APR is simply equal to the interest rate per period multiplied by the number of periods in a year. For example, if a bank is charging 1.2 percent per month on car loans, then the APR that must be reported is 1.2% X 12 = 14.4%. So, an APR is in fact a quoted, or stated, rate in the sense we've been discussing. For example, an APR of 12 percent on a loan calling for monthly payments is really 1 percent per month. The EAR on such a loan is thus:

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