Skill: 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

$$\frac{0.12}{12}=0.01$$

so

$$i_{\text{eff}}=(1.01)^{12}-1$$ $$i_{\text{eff}}\approx 0.1268$$

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.