Effective Annual Rate (Definition, EAR Formula)

What is the Effective Annual Rate (EAR)?

Effective annual rate or the annual equivalent rate is the rate actually earned on investment or paid on the loan after compounding over a given period of time. It is used to compare financial products with different compounding periods i.e. weekly, monthly, annually, etc. As the compounding periods are increased, the effective annual rate increases.

Formula of Effective Annual Rate (EAR)

The EAR is calculated as follows:

Effective Annual Rate Formula = (1 + i/n)ⁿ – 1

Where n = number of compounding periods
i = nominal rate or the given annual rate of interest

The EAR is equal to the nominal rate only if the compounding is done annually. As the number of compounding periods increase, the effective annual rate increases. If it is continuous compounding, the EAR is as follows:

Effective Annual Rate Formula (in case of continuous compounding) = eⁱ – 1

Hence, calculation of Effective annual rate depends on two factors:

The nominal rate of interest
The number of compounding periods

The number of compounding periods is the major factor as the EAR increases with the number of periods.

How to Calculate the Effective Annual Rate?

Let’s see some examples to understand the calculation of the effective annual rate.

Example #1

Let us consider the following example:

Consider a nominal rate of 12%. Let us calculate effective annual rate when the compounding is done annually, semi-annually, quarterly, monthly, weekly, daily and continuously compounded.

Annual Compounding:

EAR = (1 + 12%/1)¹ – 1 = 12%

Semi – Annual Compounding:

EAR = (1 + 12%/2)² – 1 = 12.36%

Quarterly Compounding:

EAR = (1 + 12%/4)⁴ – 1 = 12.55%

Monthly Compounding:

EAR = (1 + 12%/12)¹² – 1 = 12.68%

Weekly Compounding:

EAR = (1 + 12%/52)⁵² – 1 = 12.73%

Daily Compounding:

EAR = (1 + 12%/365)³⁶⁵ – 1 = 12.747%

Continuous Compounding:

EAR = e^12% – 1 = 12.749%

Thus, as can be seen from the above example, the calculation of effective annual rate is highest when it continuously compounded and the lowest when the compounding is done annually.

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Example #2

The calculation of the effective annual rate is important while comparing two different investments. Let us consider the following case.

An investor has $ 10,000 which he can invest in a financial instrument A which has an annual rate of 10% compounded semi-annually or he could invest in a financial instrument B which has an annual rate of 8% compounded monthly. We need to find which financial instrument is better for the investor and why?

To find which instrument is better, we should find the amount he will get after one year from each of the investments:

Amount after one year in Investment A = P * (1 + i/n)ⁿ

Where P is the principal, I is the nominal rate and n is the number of periods of compounding which is 2 in this case

Hence, amount after one year in investment A = 10000 * (1 + 10%/2)²A = $ 11025

Amount after one year in Investment B = P * (1 + i/n)ⁿ

Where P is the principal, I is the nominal rate and n is the number of periods of compounding which is 12 in this case

Hence, amount after one year in investment A = 10000 * (1 + 8%/12)¹²= B = $ 10830

Thus, in this case, investment A is a better option for the investor since the amount earned after one year is more in investment A.

If the interest is compounded, it results in higher interest in the subsequent periods, the highest being in the last period. Till now, we have considered total amounts at the end of the year.

Example #3

Let us see the following example to find interest at the end of each period.

A financial instrument had an initial investment of $ 5000 with an annual rate of 15% compounded quarterly. Let us calculate the quarterly interest received on the investment.

The rate is compounded quarterly, hence the interest rate for each quarter = 15%/4 = 3.75%

Interest earned in first quarter = P(1+i/n)ⁿ – P = 5000*(1+15%/4) – 5000 = $ 187.5

Now, the new principal is 5000 + 187.5 = $ 5187.5

Thus, Interest earned in second quarter = P(1+i/n)ⁿ – P = 5187.5*(1+15%/4) – 5187.5= $ 194.53

Now, the new principal is 5187.5+ 194.53 = $ 5382.03

Thus, Interest earned in third quarter = P(1+i/n)ⁿ – P = 5382.03*(1+15%/4) – 5382.03= $ 201.82

Now, the new principal is 5382.03+ 201.82 = $ 5583.85

Thus, Interest earned in fourth quarter = P(1+i/n)ⁿ – P = 5583.85*(1+15%/4) – 5583.85= $ 209.39

Hence, the final amount after one year will be 5583.85 + 209.39 = $ 5793.25

From the above example, we have seen that the interest earned in the fourth quarter is the highest.

Conclusion

The effective annual rate is the actual rate which the investor earns on his investment or the borrower pays to the lender. It depends on the number of compounding periods and the nominal rate of interest. The Effective annual rate of interest increases if the number of compounding periods increases for the same nominal rate, highest being if the compounding is done continuously.