86 CHAPTER 2 Time Value of MoneyYou could also use the interest conversion feature of a financial calculator.^12
In the EAR equation, iNom/m is the periodic rate, and m is the number of
periods per year. For example, suppose you could borrow using either a credit
card that charges 1 percent per month or a bank loan with a 12 percent quoted
nominal interest rate that is compounded quarterly. Which should you choose?
To answer this question, the cost rate of each alternative must be expressed as
an EAR:Credit card loan: EAR (1 0.01)^12 1.0 (1.01)^12 1.0
1.126825 1.0 0.126825 12.6825%.
Bank loan: EAR (1 0.03)^4 1.0 (1.03)^4 1.0
1.125509 1.0 0.125509 12.5509%.Thus, the credit card loan is slightly more costly than the bank loan. This result
should have been intuitive to you—both loans have the same 12 percent nominal
rate, yet you would have to make monthly payments on the credit card versus quar-
terly payments under the bank loan.
The EAR rate is not used in calculations. However, it should be used to com-
pare the effective cost or rate of return on loans or investments when payment pe-
riods differ, as in the credit card versus bank loan example.The Result of Frequent Compounding
Suppose you plan to invest $100 for five years at a nominal annual rate of 10 percent.
What will happen to the future value of your investment if interest is compounded
more frequently than once a year? Because interest will be earned on interest more of-
ten, you might expect the future value to increase as the frequency of compounding
increases. Similarly, you might also expect the effective annual rate to increase with
more frequent compounding. As Table 2-1 shows, you would be correct—the future
value and EAR do in fact increase as the frequency of compounding increases. Notice(^12) Most financial calculators are programmed to find the EAR or, given the EAR, to find the nominal rate.
This is called “interest rate conversion,” and you simply enter the nominal rate and the number of com-
pounding periods per year and then press the EFF% key to find the effective annual rate.
TABLE 2-1 The Inpact of Frequent Compounding
Nominal Effective Future Value of
Frequency of Annual Annual Rate $100 Invested
Compounding Rate (EAR)a for 5 Yearsb
Annual 10% 10.000% $161.05
Semiannual 10 10.250 162.89
Quarterly 10 10.381 163.86
Monthly 10 10.471 164.53
Dailyc 10 10.516 164.86
aThe EAR is calculated using Equation 2-13.
bThe future value is calculated using Equation 2-12.
cThe daily calculations assume 365 days per year.