Calculating Effective Annual Rate Given Stated Annual Interest Rate and Compounding Frequency

The effective annual rate of interest (EAR) refers to the rate of return an investor earns in a year, taking the effects of compounding into account. Remember, compounding is the process by which invested funds grow exponentially due to 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: Semi-annual Compounding

Imagine that you have been tasked to calculate the EAR, given a stated annual rate of 10% compounded semi-annually. You would be expected to apply the above formula directly.

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

Establishing the components already known,

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 the compounding frequency and the EAR increase concurrently.

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

Why is the Effective Annual Rate of Interest Important?

The EAR is an important concept in financial management because it compares 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 more 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. The annual effective rate of return applicable to Ross’ investment is closest to:

1. 5.00%.
2. 10.25%.
3. 10.47%.

Solution

The correct answer is B.

The question asks us to find the effective rate of return. We, therefore, have to determine the value of \(r\), which will be the EAR:

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

A is incorrect. It assumes the following calculation:

$$\text{EAR}=1+\left[\frac{0.1}{2}\right]-1=5\%$$

C is incorrect. It assumes monthly compounding and not semi-annual compounding in determining the EAR as follows:

$$\text{EAR}=\left[1+\frac{0.1}{12}\right]^{12}-1=10.47\%$$