The stated annual rate is the annual rate expressed in the contract of lending or borrowing. It is the contractual annual rate charged by a lender or offered by a borrower. The effective annual rate (EAR) is the annual rate of interest actually paid or received. It is also called the true annual return. The difference between the stated annual rate and the effective annual rate lies in compounding.

The effective annual rate reflects the impact of compounding frequency, while the stated annual rate is based on simple interest. The relationship between the two rates is expressed by the following formula:

EAR = (1 + r/m)^m– 1

The maximum value of the effective annual rate can attained when the stated annual rate is compounded continuously (c.c). In such a case, the effective annual rate can be calculated by applying the following formula

EAR_c.c = e^r – 1

An example will illustrate the calculation of this rate. Suppose the stated annual rate is 7% and interest is compounded annually, semiannually, quarterly and continuously.

For annual compounding (m= 1):

EAR = (1 + 0.07/1)¹– 1= 0.07= 7%

For semiannual compounding (m= 2):

EAR = (1 + 0.07/2)²– 1= 0.071225 = 7.1225%

For quarterly compounding (m= 4):

EAR = (1 + 0.07/4)⁴– 1= 0.071859= 7.1859%

For continuous compounding (m= ∞):

EAR_c.c = e^0.07 – 1= 0.0725 = 7.25%

In addition to the above two rates, banks and financial institutions are typically required to disclose another type of rate on loans and credit cards. This rate is known as the annual percentage rate (APR) which is defined as the stated annual rate charged by a bank on its loans. Technically, it is the periodic rate (the interest rate per period) multiplied by the number of periods in one year. For instance, if a bank charges on its credit card 1.6% per month, its APR would simply be:

APR = stated rate per period × number of periods

APR = 1.6% × 12 = 19.2%

However, this figure may not tell the actual cost of the loan. In order to find out the actual cost, another measure should be used: it is the annual percentage yield (APY). The yield, of course, will go to the lender of funds. This measure reflects the impact of compounding frequency:

APY = (1+ r)¹² – 1 = (1.016)¹²– 1 = 20.98%

A slight increase in the stated rate per period would drive up the annual percentage yield more than proportionally.