All of the examples thus far have assumed that returns were received once a year, or annually. Suppose, however, that you put your $1,000 in a bank that offers to pay 6% interest compounded semiannually. How much will you have at the end of 1 year? Semiannual compounding means that interest is actually paid every 6 months, a fact taken into account in the tabular calculations in Table A.7. Here the annual interest rate is divided by two, but twice as many compounding periods are used because interest is paid twice a year. Comparing the amount on hand at the end of the second 6-month period, $1,060.90, with what would have been on hand under annual compounding, $1,060, shows that semiannual compounding is better from the standpoint of the saver. This result occurs because you earn interest on interest more frequently.

Throughout the economy, different types of investments use different compounding periods. For example, bank and savings and loan accounts generally pay interest quarterly, some bonds pay interest semiannually, and other bonds pay interest annually. Thus, if we are to compare securities with different compounding periods, we need to put them on a common basis. This need has led to the development of the terms nominal, or stated, interest rate and effective annual, or annual percentage rate (APR). The stated, or nominal, rate is the quoted rate; thus, in our example the nominal rate is 6%. The annual percentage rate is the rate that would have produced the final compound value, $1,060.90, under annual rather than semiannual compounding. In this case, the effective annual rate is 6.09%:

Thus, if one bank offered 6% with semiannual compounding, whereas another offered 6.09% with annual compounding, they would both be paying the same effective rate of interest. In general, we can determine the effective annual rate of interest, given the nominal rate, as follows:

• Step 1: Find the FV of $1 at the end of 1 year, using the equation

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