Effective Interest Rate Formulation
Understanding effective interest rates requires a definition of a nominal interest rate r as the interest rate per period times the number of periods. In equation form,
A nominal interest rate can be found for any time period that is longer than the compounding period. For example, an interest rate of 1.5% per month can be expressed as a nominal 4.5% per quarter (1.5% per period ×3 periods), 9% per semiannual period, 18% per year, or 36% per 2 years. Nominal interest rates obviously neglect compounding.
The equation for converting a nominal interest rate into an effective interest rate is
where i is the effective interest rate for a certain period, say six months, r is the nominal interest rate for that period (six months here), and m is the number of times interest is compounded in that same period (six months in this case). The term m is often called the compounding frequency. As was true for nominal interest rates, effective interest rates can be calculated for any time period longer than the compounding period of a given interest rate. The next example illustrates the use of Equations (3-1) and (3-2).
1) A Visa credit card issued through Frost Bank carries an interest rate of 1% per month on the unpaid balance. Calculate the effective rate per semiannual and annual periods.
2) If the card’s interest rate is stated as 3.5% per quarter, find the effective semiannual and annual rates.
1) The compounding period is monthly. For the effective interest rate per semiannual period, the r in Equation (3-1) must be the nominal rate per 6 months.
The m in Equation (3-2) is equal to 6, since the frequency with which interest is compounded is 6 times in 6 months. The effective semiannual rate is
For the effective annual rate, r = 12% per year and m = 12. By Equation (3-2),
2) For an interest rate of 3.5% per quarter, the compounding period is a quarter. In a semiannual period, m = 2 and r = 7%.
The effective interest rate per year is determined using r = 14% and m = 4.
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 (3-2) reduces to
Equation (3-3) 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
For national and international chains—retailers, banks, etc.—and corporations that move thousands of items in and out of inventory each day, the flow of cash is essentially continuous. Continuous cash flow is a realistic model for the analyses performed by engineers and others in these organizations. The equivalence computations reduce to the use of integrals rather than summations. The topics of financial engineering analysis and continuous cash flows, coupled with continuous interest rates, are beyond the scope of this text; consult more advanced texts for formulas and procedures.
1) For an interest rate of 18% per year compounded continuously, calculate the effective monthly and annual interest rates.
2) An investor requires an effective return of at least 15%. What is the minimum annual nominal rate that is acceptable for continuous compounding?
Table (3-3): Effective Annual Interest Rates for Selected Nominal Rates
1) The nominal monthly rate is r = 18%/12 = 1.5%,or 0.015 per month. By Equation (3-3), the effective monthly rate is
Similarly, the effective annual rate using r = 0.18 per year is
2) Solve Equation (3-3) for r by taking the natural logarithm.
Therefore, a nominal rate of 13.976% per year compounded continuously will generate an effective 15% per year return.
Comment: The general formula to find the nominal rate, given the effective continuous rate i, is r = ln(1 + i)