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Effective Annual Rate Formula
The effective annual rate (EAR) is the usage rate that a borrower actually pays on a loan, credit card, or any other debt amount. Also, the effective annual rate is the real interest return rate on a savings account when the effects of compounding over time are taken into account. It may be considered the market rate of interest or the yield to maturity. It is also called the effective rate, or the annual equivalent rate. The effective annual rate is normally higher than the nominal rate because the nominal rate quotes a yearly percentage rate regardless of compounding. If we Increase the number of compounding periods, it makes the effective annual interest rate increase as time goes by.
What is Effective Annual Rate Formula?
Effective interest rate formula can be expressed as,
\(r = (1 + \dfrac{i}{n})^n 1\)
where,
 r = The effective interest rate
 i = The stated interest rate
 n = The number of compounding periods per year
Let's take a quick look at a couple of examples to understand the effective annual rate formula, better.
Examples Using Effective Annual Rate Formula

Example 1
A loan document contains a stated interest rate of 12% and mandates quarterly compounding. Find effective interest rates on a loan document.
Solution:
To find: Effective interest rate on a loan document.
Given:
Interest rate = 12%
Number of compounding periods per year = 4
Using Effective Interest Rate Formula,
\(r = \left(1 + \dfrac{i}{n}\right)^n 1\)
\(r = \left(1 + \dfrac{12\%}{4}\right)^4 1 \)
\(r = \left(1 + \dfrac{0.12}{4}\right)^4 1\)
= 12.55%
Answer: Effective Interest Rate on the loan document is 12.55%.

Example 2:
An American Bank offers a nominal interest rate of 10% on a deposit amount. If the client initially invested $1,200 and agreed to have the interest compounded monthly for one full year. What is an effective annual rate and how much amount will the client receive at the end of the year?
Solution:
To find: Effective annual rate and the amount client will receive at the end of the year.
Given:
Interest rate = 10%
Number of compounding periods per year = 12
Using Effective Interest Rate Formula,
\(r = \left(1 + \dfrac{i}{n}\right)^n 1\)
\(r = \left(1 + \dfrac{10\%}{12}\right)^{12} 1 \)
\(r = \left(1 + \dfrac{0.10}{12}\right)^{12} 1 \)
= \(\left(1 + 0.00833\right)^{12} 1 \)
= 10.47%
The amount will the client receive at the end of the year is \( $1200 \times(1 + 10.47\%)\) = $1325.64
Answer: Effective interest rate 10.47% and the amount will the client receive at the end of the year is $1325.64.
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