Effective versus Annualized Rates of Interest Another way of converting stated interest rates to a common basis is the

Effective versus Annualized Rates of Interest Another way of converting stated interest rates to a common basis is the

effective rate of interest. The effective annual rate (EAR) is the true eco- nomic return for a given time period—it takes into account the compound- ing of interest—and is also referred to as the effective rate of interest.

Using our Lucky Break example, we see that we must pay $12,500 interest on the loan of $10,000 for one year. Effectively, we are paying 125% annual interest. Thus, 125% is the effective annual rate of interest. In this example, we can easily work through the calculation of interest and interest on interest. But for situations where interest is compounded more frequently, we need a direct way to calculate the effective annual rate. We can calculate it by resorting once again to our basic valuation equation:

FV = PV (1 + i) n

Next, we consider that a return is the change in the value of an investment over a period and an annual return is the change in value over a year. Using our basic valuation equation, the relative change in value is the difference between the future value and the present value, divided by the present value:

FV PV –

PV 1

EAR = ---------------------- = ---------------------------

PV

PV

Canceling PV from both the numerator and the denominator,

(7-15) Let’s look how the EAR is affected by the compounding. Suppose

EAR = (1 + i) n −1

that the Safe Savings and Loan promises to pay 6% interest on accounts, compounded annually. Since interest is paid once, at the end of the year, the effective annual return, EAR, is 6%. If the 6% interest is paid on a semiannual basis—3% every six months—the effective annual return is larger than 6% since interest is earned on the 3% interest earned at the end of the first six months. In this case, to calculate the EAR, the interest rate per compounding period—six months—is 0.03 (that is, 0.06/2) and the number of compounding periods in an annual period is 2:

EAR = (1 + 0.03) 2 − 1 = 1.0609 − 1 = 0.0609 or 6.09% Extending this example to the case of quarterly compounding with a

nominal interest rate of 6%, we first calculate the interest rate per period,

i, and the number of compounding periods in a year, n:

FOUNDATIONS

i = 0.06/4 = 0.015 per quarter n = 4 quarters in a year

The EAR is: EAR = (1 + 0.015) 4 − 1 = 1.0614 − 1 = 0.0614 or 6.14% As we saw earlier in this chapter, the extreme frequency of com-

pounding is continuous compounding. Continuous compounding is when interest is compounded at the smallest possible increment of time. In con- tinuous compounding, the rate per period becomes extremely small:

i APR = ----------- ∞

And the number of compounding periods in a year, n, is infinite. The EAR is therefore:

(7-16) where e is the natural logarithmic base.

EAR = e APR −1

For the stated 6% annual interest rate compounded continuously, the EAR is:

EAR = e 0.06 − 1 = 1.0618 − 1 = 0.0618 or 6.18% The relation between the frequency of compounding for a given stated

rate and the effective annual rate of interest for this example indicates that the greater the frequency of compounding, the greater the EAR.

Frequency of Compounding

Calculation

Effective Annual Rate

Annual (1 + 0.060) 1 − 1 6.00% Semiannual

(1 + 0.030) 2 − 1 6.09% Quarterly

(1 + 0.015) 4 − 1 6.14% Continuous

e 0.06 − 1 6.18% Figuring out the effective annual rate is useful when comparing

interest rates for different investments. It doesn’t make sense to compare the APRs for different investments having a different frequency of com- pounding within a year. But since many investments have returns stated in terms of APRs, we need to understand how to work with them.

Mathematics of Finance

To illustrate how to calculate effective annual rates, consider the rates offered by two banks, Bank A and Bank B. Bank A offers 9.2% com- pounded semiannually and Bank B other offers 9% compounded daily. We can compare these rates using the EARs. Which bank offers the high-

est interest rate? The effective annual rate for Bank A is (1 + 0.046) 2 −1= 9.4%. The effective annual rate for Bank B is (1 + 0.000247) 365 −1= 9.42%. Therefore, Bank B offers the higher interest rate.