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.