## What is the relationship between effective interest rate and compound interest?

Learn about the relationship between effective annual interest rate and compound interest.

## 1. Definition of effective interest rate and compound interest

Compounding is a powerful application of interest calculation. When compounding is used, nominal (stated) interest rate will result in an effective interest rate that is not the same as the nominal rate. Note that when we talk about a nominal (stated) interest rate we mean the annual rate (e.g., 10% annual rate of return on an investment). When we talk about the effective annual interest rate, we mean the actual rate resulting from interest compounding (e.g., 10.25% annual rate of return on the same investment).

In the context of compound interest, effective annual interest rate (EAR) is an annual interest rate when compounding period differs from one year. In other words, effective interest rate is the actual interest when interest is compounded more than once a year. In this case, interest is compounded on both the principal (initial investment) and the interest that has already accrued. As the result, effective interest rate differs from the nominal (stated) interest rate when compounding occurs more than once a year, and it depends on the frequency of compounding.

## 2. Effective interest rate formulas

In the context of compound interest, the effective annual rate of interest can be determined using the formula below:

 EAR = (1 + i � m)m � 1

where i is the nominal (stated) interest rate; and m is the number of times the interest is compounded per year.

The effective interest rate increases as the frequency of compounding increases. Nevertheless, the effective interest rate almost does not change when it is compounded a large number of times. To see an example, refer to the table below:

 Compounding Periods per Year Effective Interest Rate 1 10.00% 2 10.25% 5 10.41% 10 10.46% 100 10.51% 500 10.52% 100,000 10.52%

When interest is compounded an infinite number of times per year, it is considered to be continuously compounded. For example, many derivative pricing models require continuous compounding. In this case, the effective interest rate is determined as follows:

 EAR = ei x n � 1

where e is the natural base of logarithms (2.7182818); i is the nominal interest rate; and n is the number of years.

When interest is compounded continuously, the following formulas for the present and future values of an investment can be used:

 PV = FV ei x n
 FV = P x ei x n

where PV is the present value; FV is the future value; P is the amount invested; e is the natural base of logarithms; i is the nominal interest rate; and n is the number of years interest is compounded.

Let us look at a few examples to see how effective interest rate is calculated.

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