Compare nominal and effective interest rates

Nominal annual interest rate is a quoted annual rate applied in equal periodic portions according to a stated compounding frequency, whereas the effective annual rate measures the actual one-year percentage growth after compounding. The relationship $i_{\text{eff}}=(1+j/m)^m-1$ supports comparing rates on a common annual basis and recognizing that, for a positive nominal rate, more frequent compounding produces a larger effective rate; continuous compounding and irregular compounding periods are outside this scope.

Detailed Explanation: Compare nominal and effective interest rates

A nominal annual interest rate is a quoted yearly rate that is divided among the compounding periods. An effective annual rate shows the actual growth over one year after all the compounding is included.

Use

i_{\text{eff}}=\left(1+\frac{j}{m}\right)^m-1

where:

$j$ is the nominal annual rate written as a decimal,
$m$ is the number of compounding periods per year,
$i_{\text{eff}}$ is the effective annual rate.

Example

Which investment has the greater actual one-year return?

Investment A: $12\%$ nominal interest compounded monthly
Investment B: $12\%$ nominal interest compounded annually

Step 1: Find the effective rate for Investment A

For Investment A,

j=0.12 \qquad\text{and}\qquad m=12

because there are $12$ months in a year.

Substitute into the formula:

i_{\text{eff}}=\left(1+\frac{0.12}{12}\right)^{12}-1

The monthly rate is

\frac{0.12}{12}=0.01

i_{\text{eff}}=(1.01)^{12}-1

i_{\text{eff}}\approx 0.1268

Convert to a percentage:

0.1268=12.68\%

Investment A has an effective annual rate of approximately $12.68\%$ .

Step 2: Find the effective rate for Investment B

Investment B compounds annually, so $m=1$ :

i_{\text{eff}}=\left(1+\frac{0.12}{1}\right)^1-1

i_{\text{eff}}=0.12=12\%

Step 3: Compare the effective rates

Although both investments have the same nominal rate of $12\%$ , their effective rates are different:

Investment A: $12.68\%$
Investment B: $12\%$

Therefore, Investment A gives the greater actual one-year return because its interest is compounded more frequently.

Learn by doing: Compare nominal and effective interest rates

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Financial Literacy - Nominal and Effective Interest Rate - Formula and Nominal to Effective

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