# Effective Annual Rate of Interest (EAR)

The effective annual rate of interest (EAR) refers to the rate of return earned by an investor in a year, taking the effects of compounding into account. Remember, compounding is the process by which invested funds grow exponentially as a result of both the principal and the already accumulated interest earning more interest. In other words, interest earned itself earns more interest. Mathematically, we may define EAR as follows:

$$\text{EAR} = \left(1+ \text{periodic rate} \right)^\text{m} – 1$$

Where, $$\text{periodic rate} = \frac{\text{stated annual rate}}{m}$$

And m is the number of compounding periods per year.

## Example 1

Calculate the EAR, given a stated annual rate of 10% compounded semi-annually. You would be expected to directly apply the above formula.

$$\text{EAR} = \left( 1+ \text{periodic rate} \right)^\text{m} – 1$$

Stated annual rate = 0.1;

m = 2

Periodic rate = 0.1/2 = 0.05

Hence, \begin{align*} \text{EAR} & = (1+ 0.05)^2 – 1 \\ & = 10.25\% \end{align*}

## Example 2: A range of Compounding Frequencies

Using a stated annual rate of 12%, compute the effective rates for daily, monthly, quarterly, and semi-annual compounding periods.

\begin{align*} & \text{Semi-annual compounding} = (1+0.06)^2 -1= 0.1236 = 12.36\% \\ & \text{Quarterly compounding} = (1+0.03)^4 -1 = 0.12551 = 12.55\% \\ & \text{Monthly compounding} = (1+ 0.01)^{12} -1 = 0.12683 = 12.68\% \\ & \text{Daily compounding} = (1+0.00032877)^{365} -1 = 12.75\% \\ \end{align*}

First, you should note that as the compounding frequency increases, so does the EAR.

Furthermore, the stated rate is equal to the EAR only when the interest is compounded annually.

## Why is the Effective Annual Rate of Interest So Important?

The EAR is an important concept in financial management as it is used to compare two or more projects that calculate compound interest differently. For example, assume that you have two projects, X and Y. Project X pays 5% interest compounded monthly, while project Y pays 5% interest compounded quarterly.  By calculating the EAR represented by each of these two rates, you would be able to pick the most profitable project of the two. Furthermore, the higher the EAR, the higher the return offered by an investment.

## Question

John Ross, a financial analyst, would like to have $20,000 saved in his bank account at the end of 5 years. The bank offers a return of 10% per annum compounded semi-annually. How much should Ross invest at the beginning so as to attain his goal? A.$0.61

B. $12,279 C.$10,000

Solution

The question asks us to find the present value of an accumulation of \$20,000 that has been earned over a five-year period. As you will recall, the PV formula is:

$$PV = FV (1+r)^{-n}$$

Where FV is the future value,

r is the rate of return and,

n is the term of the investment

First, we have to determine the value of r, which will be the EAR

\begin{align*} EAR & = \left\{ 1+ \left(\frac {0.1}{2} \right) \right\}^2 – 1 \\ & = (1+0.05)^2 -1 = 10.25\% \\ \end{align*}

Lastly, we calculate the PV.

$$PV = 20,000(1.1025)^{-5} = 12,279$$

Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding.

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