# Effective Interest Rate

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The Effective Interest Rate (EIR) is the rate you actually earn on your investment after taking into account the compounding frequency.

Explanation

If you deposit $100 into an account that pays 12% compounded once per year, at the end of the year you will have 6. This 0.36, or 6% of$6.

Your total earnings would then be $6 each half year plus$0.36 or 1 after 1 year and then subtract the initial dollar. What’s left is the interest for the year, which is called effective interest rate (EIR).

The effective interest rate is computed by subtracting 1. Equation 1 computes the effective interest rate:

$\begin{matrix}\text{Effective interest rate=}{{\left( 1+\frac{i}{m} \right)}^{m}}-1 & {} & \left( 1 \right) \\\end{matrix}$

Where

i=the quoted annual rate

m= the number of compounding intervals

Example 1 Effective Interest Rate Calculation

What is the effective interest rate for a nominal rate of 12%, which is compounded monthly?

Solution

In this example, we know that

\begin{align}& i=0.12 \\& m=12 \\\end{align}

Substitute these numbers into EIR Formula:

$\text{EIR}={{\left( 1+\frac{0.12}{12} \right)}^{12}}-1=12.68%$

An annually compounded interest rate of 12.68% is equivalent to earning a 12% annual rate compounded 12 times per year.

Table 1 shows the effective rate at different compounding intervals, when the annual rate is 12%.

Table 1 Effective Interest Rates with 12% Annual Rate

 Compounding Interval Equation 1 Effective Rate Annual FV=(1.12)1 − 1 12.00% Semiannual FV=(1.06)2 − 1 12.36 Quarterly FV=(1.03)4 − 1 12.55 Monthly FV=(1.01)12 − 1 12.68 Weekly FV=(1.0023)52 − 1 12.73 Daily FV=(1.0003288)365 − 1 12.7475

Suppose you are attempting to choose between two bank savings accounts. The first pays 5% compounded annually, and the second pays 4.9% compounded monthly. Which would you prefer?

In order to answer, you must first calculate the effective rate of each alternative and choose the largest one. The effective rate of the first is unaffected, 5%. The effective rate of the second is 5.01%. So, you would pick the bank proposing 4.9% compounded monthly.

$EIR={{\left( 1+\frac{0.049}{12} \right)}^{12}}-1=5.01%$

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