## Effective and Nominal Interest Rates

Interest rates may be stated for some period, like a year, while the computation of interest is based on shorter compounding subperiods such as months. In this section we consider the relation between the nominal interest rate that is stated for the full period and the effective interest rate that results from the compounding based on the subperi-ods. This relation between nominal and effective interest rates must be understood to answer questions such as: How" would you choose between two investments, one bearing 12% per year interest compounded yearly and another bearing 1% per month interest compounded monthly? Are they the same?

Nominal interest rate is the conventional method of stating the annual interest rate. It is calculated by multiplying the interest rate per compounding period by the number of compounding periods per year. Suppose that a time period is divided into vi equal subperiods. Let there be stated a nominal interest rate, r, for the full period. By convention, for nominal interest, the interest rate for each subperiod is calculated as = rim. For example, a nominal interest rate of 18% per year, compounded monthly, is the same as

Effective interest rate is the actual but not usually stated interest rate, found by converting a given interest rate with an arbitrary compounding period (normally less than a year) to an equivalent interest rate with a one-year compounding period. What is the effective interest rate, ic, for the fall period that will yield the same amount as compounding at the end of each subperiod, if: If we compound interest even7 subpe-riod, we have

We want to find the effective interest rate, i,, that yields the same future amount F at the end of the full period from the present amount P. Set

Note that Equation (2.3) allows the conversion between the interest rate over a compounding subperiod, /„ and the effective interest rate over a longer period, i. by using the number of subperiods, m, in the longer period.

EXAMPLE 2.4

"What interest rate per year, compounded yearly, is equivalent to 1% interest per month, compounded monthly?

Since the month is the shorter compounding period, we let /, = 0.01 and m = 12. Then ie refers to die effective interest rate per year. Substitution into Equation (2.3) then gives ie = (1 + if - 1

An interest rate of 1% per month, compounded monthly, is equivalent to an effective rate of approximately 12.7% per year, compounded yearly. The answer to our previously posed question is that an investment bearing 12% per year interest, compounded yearly, pays less than an investment bearing 1 % per month interest, compounded monthly.B

Interest rates are normally given as nominal rates. We may get the effective (yearly) rate by substituting i, - rim into Equation (2.3). We then obtain a direct means of computing an effective interest rate, given a nominal rate and the number of compounding periods per year: 