7.c Equivalent Rate¶

Introduction¶

The equivalent rate is a concept in commercial arithmetic used to compare different interest rates that have different compounding periods. It allows for the conversion of an interest rate compounded over one period to an equivalent interest rate for a different period. This is useful for comparing financial products with different compounding frequencies.

The formula to convert an annual interest rate compounded \( n \) times a year to an equivalent annual rate compounded once is:

\[ \text{Equivalent Annual Rate} (EAR) = \left(1 + \frac{R}{n}\right)^n - 1 \]

Where: - \( R \) = Nominal annual interest rate (as a decimal) - \( n \) = Number of compounding periods per year

The result is expressed as a percentage.

Example 1: Converting a Quarterly Compounded Rate to an Equivalent Annual Rate¶

Problem¶

A bank offers a savings account with a nominal interest rate of 8% compounded quarterly. Calculate the equivalent annual rate (EAR).

Solution¶

Given: - \( R = 8\% = 0.08 \) (Nominal annual interest rate) - \( n = 4 \) (Compounded quarterly)

Using the equivalent annual rate formula:

\[ EAR = \left(1 + \frac{0.08}{4}\right)^4 - 1 \]

\[ EAR = \left(1 + 0.02\right)^4 - 1 \]

Calculating \( 1.02^4 \):

\[ EAR \approx 1.08243216 - 1 = 0.08243216 \]

Converting to a percentage:

\[ EAR \approx 8.24\% \]

Explanation¶

In this example, a nominal rate of 8% compounded quarterly is equivalent to an 8.24% interest rate when compounded annually. The slightly higher equivalent rate reflects the impact of quarterly compounding over the year.

Example 2: Converting a Monthly Compounded Rate to an Equivalent Annual Rate¶

Problem¶

An investment offers a nominal interest rate of 6% compounded monthly. Calculate the equivalent annual rate (EAR).

Solution¶

Given: - \( R = 6\% = 0.06 \) (Nominal annual interest rate) - \( n = 12 \) (Compounded monthly)

Using the equivalent annual rate formula:

\[ EAR = \left(1 + \frac{0.06}{12}\right)^{12} - 1 \]

\[ EAR = \left(1 + 0.005\right)^{12} - 1 \]

Calculating \( 1.005^{12} \):

\[ EAR \approx 1.0616778 - 1 = 0.0616778 \]

Converting to a percentage:

\[ EAR \approx 6.17\% \]

Explanation¶

For this example, a nominal interest rate of 6% compounded monthly results in an equivalent annual rate of approximately 6.17%. The higher equivalent rate illustrates the effect of monthly compounding over a year.

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