Nominal and Effective Interest Rates

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2 Nominal and Effective Interest Rates 4.1 Introduction In all engineering economy relations developed thus far, the interest rate has been a constant, annual value. For a substantial percentage of the projects evaluated by professional engineers in practice, the interest rate is compounded more frequently than once a year; frequencies such as semiannual, quarterly, and monthly are common. In fact, weekly, daily, and even continuous compounding may be experienced in some project evaluations. Here we discuss nominal and effective interest rates. The difference here is that the concepts of nominal and effective are used when interest is compounded more than once each year. For example, if an interest rate is expressed as 1% per month, the terms nominal and effective interest rates must be considered. 4.2 Nominal and Effective Interest Rate Statements A nominal interest rate( r) is an interest rate that does not account for compounding. By definition, r = interest rate per time period number of periods [4.1] For example, the interest rate of 1.5% per month is the same as each of the following nominal rates. An effective interest rate ( i ) is a rate wherein the compounding of interest is taken into account. Effective rates are commonly expressed on an annual basis as an effective annual rate; however, any time basis may be used.

3 Necessary Definitions: Interest period (t) The period of time over which the interest is expressed. This is the t in the statement of r % per time period t, for example, 1% per month. The time unit of 1 year is by far the most common. It is assumed when not stated otherwise. Compounding period (CP) The shortest time unit over which interest is charged or earned. This is defined by the compounding term in the interest rate statement, for example, 8% per year, compounded monthly. If CP is not stated, it is assumed to be the same as the interest period. Compounding frequency (m) The number of times that compounding occurs within the interest period t. If the compounding period CP and the time period t are the same, the compounding frequency is 1, for example, 1% per month, compounded monthly. Some examples on these definitions In previous chapters, all interest rates had t and CP values of 1 year, so the compounding frequency was always m _ 1. This made them all effective rates, because the interest period and compounding period were the same. Now, it will be necessary to express a nominal rate as an effective rate on the same time base as the compounding period. An effective rate can be determined from a nominal rate by using the relation:

4 Note that changing the interest period t does not alter the compounding period, which is 1 month in this illustration. Therefore, r = 9% per year, compounded monthly, and r = 4.5% per 6 months, compounded monthly, are two expression of the same interest rate. 4.3 Effective Annual Interest Rates In this section, effective annual interest rates are calculated. Therefore, the year is used as the interest period t, and the compounding period CP can be any time unit less than 1 year. For example, we will learn that a nominal 18% per year, compounded quarterly is the same as an effective rate of % per year.

5 The effective annual interest rate formula for i a is: i a = (1 + i ) m - 1 [4.3] where i = r m i a = (1 + r m )m 1 If we allow compounding to occur more and more frequently, the compounding period becomes shorter and shorter. Then m, the number of compounding periods per payment period, increases. This situation occurs in businesses that have a very large number of cash flows every day, so it is correct to consider interest as compounded continuously. As m approaches infinity, the effective interest rate in Equation [4.3] reduces to i a = e r 1 [4.4] Equation [4.4] is used to compute the effective continuous interest rate. The time periods on i and r must be the same. As an illustration, if the nominal annual r = 15% per year, the effective continuous rate per year is i a % = e = % Table 4 3 summarizes the effective annual rate for frequently quoted nominal rates using Equation [4.3] and [4.4].

6 Example 4.2 Sherry expects to deposit $1000 now, $ years from now, and $ years from now and earn at a rate of 12% per year compounded semiannually through a company-sponsored savings plan. What amount can she withdraw 10 years from now? Solution Two ways to solve this problem: 1. An alternative solution strategy is to find the effective annual rate by Equation [4.3] and express n in annual payment periods as stated in the problem. Where r = 12% per year and CP = six months(semiannually) so m = 2 i a = (1 + r m )m 1 i a = ( )2 1 = = 12.36% F = P 0 (F/P, 12.36%, 10) + P 4 (F/P, 12.36%, 6) + P 6 (F/P, 12.36%, 4) = 1000(F/P, 12.36%, 10) (F/P, 12.36%, 6) (F/P, 12.36%, 4) = 1000( ) ( ) ( ) 4 = $11634

7 2. Use an effective rate of 6% per semiannual compounding period and semiannual payment periods. The future worth is calculated as: F = 1000(F/P,6%,20) (F/P,6%,12) (F/P,6%,8) = $ Equivalence Relations: Payment Period and Compounding Period Now that the procedures and formulas for determining effective interest rates with consideration of the compounding period are developed, it is necessary to consider the payment period (PP). The payment period (PP) is the length of time between cash flows (inflows or outflows). It is common that the lengths of the payment period and the compounding period (CP) do not coincide. It is important to determine if PP = CP, PP > CP, or PP < CP. As we learned earlier, to correctly perform equivalence calculations, an effective interest rate is needed in the factors. Therefore, it is essential that the time periods of the interest rate and the payment period be on the same time basis. The next sections describe procedures to determine correct i and n values for engineering economy factors and spreadsheet functions. First, compare the length of PP and CP, then identify the cash flows as only single amounts ( P and F ) or as a series ( A, G, or g ). The section references are the same when PP = CP and PP > CP, because the procedures to determine i and n are the same.

8 4.5 Equivalence Relations: Single Amounts with PP CP With only P and F estimates defined, the payment period is not specifically identified. In virtually all situations, PP will be equal to or greater than CP. The length of the PP is defined by the interest period in the stated interest rate. When PP CP, the procedures to perform equivalence computations are the same for both situations, as explained below: Note that Example 4.2 explains these methods. 4.6 Equivalence Relations: Series with PP CP When uniform or gradient series are included in the cash flow sequence, the procedure is basically the same as method 2 above, except that PP is now defined by the length of time between cash flows. This also establishes the time unit of the effective interest rate. For example, if cash flows occur on a quarterly basis, PP is 1 quarter and the effective quarterly rate is necessary. The n value is the total number of quarters. If PP is a quarter, 5 years translates to an n value of 20 quarters. This is a direct application of the following general guideline: Table 4 6 shows the correct formulation for several cash flow series and interest rates. Note

9 that n is always equal to the total number of payment periods and i is an effective rate expressed over the same time period as n.

10 4.7 Equivalence Relations: Single Amounts and Series with PP < CP If a person deposits money each month into a savings account where interest is compounded quarterly, do the so-called inter-period deposits earn interest? The usual answer is no. The timing of cash fl ow transactions between compounding points introduces the question of how inter-period compounding is handled. Fundamentally, there are two policies: interperiod cash flows earn no interest, or they earn compound interest. If PP < CP and inter-period compounding is earned, then the cash flows are not moved, and the equivalent P, F, or A values are determined using the effective interest rate per payment period. The effective interest rate formula will have an m value less than 1, because there is only a fractional part of the CP within one PP. For example, weekly cash flows and quarterly compounding require that m = 1/13 of a quarter. When the nominal rate is 12% per year, compounded quarterly (the same as 3% per quarter, compounded quarterly), the effective rate per PP is: Effective weekly i % = (1.03 ) 1/13-1 = 0.228% per week

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