### Nominal and Effective Interest Rate Statements

To understand and correctly handle effective interest rates is important in engineering practice as well as for individual finances. The interest amounts for loans, mortgages, bonds, and stocks are commonly based upon interest rates compounded more frequently than annually. The engineering economy study must account for these effects. In our own personal ﬁ nances, we manage most cash disbursements and receipts on a nonannual time basis. Again, the effect of compounding more frequently than once per year is present. First, consider a nominal interest rate.

A nominal interest rate r is an interest rate that does not account for compounding. By definition,

A nominal rate may be calculated for any time period longer than the time period stated by using Equation [4.1]. For example, the interest rate of 1.5% per month is the same as each of the following nominal rates.

Note that none of these rates mention anything about compounding of interest; they are all of the form “ r % per time period.” These nominal rates are calculated in the same way that simple rates are calculated using Equation [1.7], that is, interest rate times number of periods.

After the nominal rate has been calculated, the compounding period (CP) must be in- cluded in the interest rate statement. As an illustration, again consider the nominal rate of 1.5% per month. If we deﬁ ne the CP as 1 month, the nominal rate statement is 18% per year, compounded monthly, or 4.5% per quarter, compounded monthly . Now we can consider an effective interest rate.

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.

The most common form of interest rate statement when compounding occurs over time periods shorter than 1 year is “% per time period, compounded CP-ly,” for example, 10% per year, compounded monthly, or 12% per year, compounded weekly. An effective rate may not always include the compounding period in the statement. If the CP is not mentioned, it is understood to be the same as the time period mentioned with the interest rate. For example, an interest rate of “1.5% per month” means that interest is compounded each month; that is, CP is 1 month. An equivalent effective rate statement, therefore, is 1.5% per month, compounded monthly.

All of the following are effective interest rate statements because either they state they are effective or the compounding period is not mentioned. In the latter case, the CP is the same as the time period of the interest rate.

All nominal interest rates can be converted to effective rates. The formula to do this is discussed in the next section.

All interest formulas, factors, tabulated values, and spreadsheet functions must use an effective interest rate to properly account for the time value of money.

The term APR (Annual Percentage Rate) is often stated as the annual interest rate for credit cards, loans, and house mortgages. This is the same as the nominal rate . An APR of 15% is the same as a nominal 15% per year or a nominal 1.25% on a monthly basis. Also the term APY (Annual Percentage Yield) is a commonly stated annual rate of return for investments, certiﬁcates of deposit, and saving accounts. This is the same as an effective rate . The names are different, but the interpretations are identical. As we will learn in the following sections, the effective rate is always greater than or equal to the nominal rate, and similarly APY APR.

Based on these descriptions, there are always three time-based units associated with an interest rate statement.

**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 deﬁ ned 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 com-pounding frequency is 1, for example, 1% per month, compounded monthly.

Consider the (nominal) rate of 8% per year, compounded monthly. It has an interest period t of 1 year, a compounding period CP of 1 month, and a compounding frequency m of 12 times per year. A rate of 6% per year, compounded weekly, has t = 1 year, CP = 1 week, and m = 52, based on the standard of 52 weeks per year.

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

As an illustration, assume r = 9% per year, compounded monthly; then m = 12. Equation [4.2] is used to obtain the effective rate of 9% / 12 = 0.75% per month, compounded monthly.

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.

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