# Compounding Periods and Effective Interest Rates - Financial Management

The frequency with which interest rates are compounded (for example, annually, semiannually, quarterly, and so on) affects both the present and future values of cash flows as well as the effective interest rates being earned or charged.

Effect of Compounding Periods on Present and Future Values

Thus far, it has been assumed that compounding (and discounting) occurs annually. Recall the general compound interest equation

FVn = PV0(1 + i)n

where PV0 is the initial deposit, i is the annual interest rate, n is the number of years, and FVn is the future value that will accumulate from the annual compounding of PV0. An interest rate of i percent per year for n years is assumed. In the remainder of this section, this annual nominal interest rate will be designated inom to differentiate it from the annual effective interest rate, ieff.

In some circumstances, interest on an account is compounded semiannually instead of annually; that is, half of the nominal annual interest rate, inom/2, is earned at the end of six months. The investor earns additional interest on the interest earned before the end of the year, or (inom/2)PV0. In calculating interest compounded semiannually, Equation is rewritten as follows: The same logic applies to interest compounded quarterly: In general, the compound interest for any number of periods during a year may be computed by means of the following equation: where mis the number of times during the year the interest is compounded and n is the number of years. (The limiting case of continuous compounding and discounting is discussed.)

Table contains the future value, FV1, of $1,000 earning a nominal interest of 10 percent for several different compounding frequencies. For example, the future value (FV1) of$1,000 compounded semiannually (m = 2) at a nominal interest rate (inom) of 10 percent per year by Equation

As Table shows, the more frequent the compounding, the greater the future value of the deposit and the greater the effective interest rate. Effective interest, in contrast to nominal interest, is the actual rate of interest earned by the lender and is generally the most economically relevant definition of interest rates.

The relationship between present values and compound values suggests that present values will also be affected by the frequency of compounding. In general, the present value of a sum to be received at the end of year n, discounted at the rate of inom percent and compounded m times per year, is as follows:

Table contains a number of present values, PV0, for $1,000 received one year in the future discounted at a nominal interest rate of 10 percent with several different compounding frequencies. For example, the present value (PV0) of$1,000 compounded quarterly (m = 4) at a nominal interest rate (inom) of 10 percent per year by Equation is $1,000 As shown in Table, the more frequent the compounding, the smaller the present value of a future amount. Throughout the text, much of the analysis assumes annual compounding instead of compounding for more frequent periods because it simplifies matters and because the differences between the two are small. Similarly, unless otherwise stated, cash flows from a security or investment project are assumed to be received in a lump sum at the beginning or end of each period. More frequent compounding periods require more extensive tables or the use of a financial calculator. Regardless of the frequency of compounding, it is important to recognize that effective rates of interest are the relevant rates to use for financial and economic analysis. The next section considers the calculation of effective interest rates in more detail for those cases where compounding is done more than one time a year. Effects of Different Compounding Frequencies on Future Values of$1,000 at a 10 Percent Interest Rate Effects of Different Compounding Frequencies on Present Values of Effective Rate Calculations

The previous section illustrated the fact that the more frequently an annual nominal rate of interest is compounded, the greater is the effective rate of interest being earned or charged. Thus, if you were given the choice of receiving (1) interest on an investment, where the interest is compounded annually at a 10 percent rate, or (2) interest on an investment,where the interest is compounded semiannually at a 5 percent rate every 6 months, you would choose the second alternative, because it would yield a higher effective rate of interest. Given the annual nominal rate of interest (inom), the effective annual rate of interest (ieff) can be claculated as follows: where m is the number of compounding intervals per year.
For example, suppose a bank offers you a loan at an annual nominal interest rate of 12 percent compounded quarterly. What effective annual interest rate is the bank charging you? Substituting inom = 0.12 and m = 4 into Equation yields 0.124 or 12.55 percent. There are also situations in finance where one is interested in determining the interest rate during each compounding period that will provide a given annual effective rate of interest. For example, if the annual effective rate is 20 percent and compounding is done quarterly, you may wish to know what quarterly rate of interest will result in an effective annual rate of interest of 20 percent.

In general, the rate of interest per period (where there is more than one compounding period per year), im, which will result in an effective annual rate of interest, ieff, if compounding occurs m times per year, can be computed as follows:

im = (1 + ieff)1/m – 1

In this example, the quarterly rate of interest that will yield an annual effective rate of interest of 20 percent is11

im = (1 + 0.20)0.25 – 1 = (1.04664) – 1 = 0.04664 or 4.664%

Thus, if you earn 4.664 percent per period and compounding occurs four times per year, the effective annual rate earned will be 20 percent. This concept is encountered in Chapter in the discussion of the valuation of bonds that pay interest semiannually.

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