7.d Effective Rate¶

Introduction¶

The effective interest rate, also known as the annual equivalent rate (AER) or effective annual rate (EAR), represents the actual interest earned or paid on a loan or investment over a year, taking into account the effect of compounding. Unlike the nominal rate, which does not account for the frequency of compounding, the effective rate provides a true reflection of the financial impact by incorporating the compounding periods.

The formula for calculating the effective rate when the nominal annual interest rate is compounded \( n \) times per year is:

\[ \text{Effective Rate} (ER) = \left(1 + \frac{R}{n}\right)^n - 1 \]

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

The result is usually expressed as a percentage.

Example 1: Calculating the Effective Rate for Quarterly Compounding¶

Problem¶

A bank offers a loan with a nominal annual interest rate of 10% compounded quarterly. Calculate the effective annual rate.

Solution¶

Given: - \( R = 10\% = 0.10 \) (Nominal annual interest rate) - \( n = 4 \) (Compounded quarterly)

Using the effective rate formula:

\[ ER = \left(1 + \frac{0.10}{4}\right)^4 - 1 \]

\[ ER = \left(1 + 0.025\right)^4 - 1 \]

Calculating \( 1.025^4 \):

\[ ER \approx 1.10381289 - 1 = 0.10381289 \]

Converting to a percentage:

\[ ER \approx 10.38\% \]

Explanation¶

In this example, a nominal rate of 10% compounded quarterly translates to an effective annual rate of approximately 10.38%. The increase from the nominal rate reflects the effect of quarterly compounding over the entire year.

Example 2: Calculating the Effective Rate for Monthly Compounding¶

Problem¶

An investment offers a nominal annual interest rate of 12% compounded monthly. Calculate the effective annual rate.

Solution¶

Given: - \( R = 12\% = 0.12 \) (Nominal annual interest rate) - \( n = 12 \) (Compounded monthly)

Using the effective rate formula:

\[ ER = \left(1 + \frac{0.12}{12}\right)^{12} - 1 \]

\[ ER = \left(1 + 0.01\right)^{12} - 1 \]

Calculating \( 1.01^{12} \):

\[ ER \approx 1.12682503 - 1 = 0.12682503 \]

Converting to a percentage:

\[ ER \approx 12.68\% \]

Explanation¶

For this example, a nominal interest rate of 12% compounded monthly results in an effective annual rate of approximately 12.68%. The effective rate is higher than the nominal rate due to the impact of monthly compounding throughout the year.

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