By now, you have a clear understanding of simple and compound interest. However, when interest is compounded, for more than one year, the actual interest rate per annum is lesser than the effective rate of interest. In this article, we will look at the definition, formula, and some examples of calculating the effective rate of interest.

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## Effective Rate of Interest

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### Definition

The effective rate of interest is the equivalent annual rate of interest which is compounded annually. Further, the compounding must happen more than once every year. Let’s look at an example for better clarity:

**Example 1: **Peter invests Rs. 10,000 for one year at the rate of 6% per annum. The interest is compounded semi-annually. Let’s calculate the interest earned in the first six months (I_{1}).

Solution: I_{1} = 10,000 x \( \frac {6}{100} \) x \( \frac {6}{12} \) = Rs. 300. Since the interest is compounded, the principal for the next 6 months = 10,000 + 300 = Rs. 10,300. Therefore, the interest earned in the next six months (I_{2}) is,

I_{2} = 10,300 x \( \frac {6}{100} \) x \( \frac {6}{12} \) $$ = Rs. 309.

Hence, the total interest earned during the year I = I_{1} + I_{2} = 300 + 309 = Rs. 609. We know the formula for interest is I = PNR … where ‘I’ is the interest, ‘P’ is the principal amount, ‘N’ is the time period, and ‘R’ is the rate of interest. In the case of this example, R = E or the effective rate of interest. Therefore, we have,

E = \( \frac {I}{PN} \) = \( \frac {609}{10, 000 × 1} \) = 0.0609 or 6.09%.

**Browse more Topics under Time Value Of Money**

- Simple and Compound Interest
- Depreciation
- Present and Net Present Value
- Future Value and Perpetuity
- Annuities and Sinking Funds
- Valuation of Bonds and Calculating EMI
- Calculations of Returns

### Formula for Calculation of Effective Rate of Interest

You can use the following formula to calculate the effective rate of interest:

E = (1 + i)^{n} – 1 … (1)

Where ‘E’ is the effective rate of interest, ‘i’ is the actual rate of interest in decimal, and ‘n’ is the number of conversion periods.

**Example 2: **John invests Rs. 5,000 in a term deposit scheme. The scheme offers an interest rate of 6% per annum, compounded quarterly. How much interest will John earn after one year? Also, what is the effective rate of interest?

Solution: We know that,

- Principal amount = P = Rs. 5,000
- Actual rate of interest = i = 6% p.a. = 0.06 p.a. = 0.015 per quarter
- Number of conversion periods = n = 4 (since we are calculating for one year and compounding happens every quarter)

Therefore, the compound interest (I) is,

I = P x [(1 + i)^{n} – 1] = 5000 x [(1 + 0.015)^{4} – 1] = 5000 x 0.06136355 = 306.82

Hence, after one year, John earns a total interest (I) of Rs. 306.82. Further, the effective rate of interest (E) is,

E = (1 + i)^{n} – 1 = (1 + 0.015)^{4} – 1 = 0.0613 or 6.13%.

## Solved Examples on Effective Rate of Interest

**Example 3:** In a bank, an amount of Rs. 20,000 is deposited for one year. The rate of interest is 8% per annum and is compounded semi-annually. What is the effective rate of interest?

- 8 percent
- 8.08 percent
- 8.16 percent
- 8.22 percent

Solution: To calculate the effective rate of interest, we do not need the amount. As per equation (1) above,

E = (1 + i)^{n} – 1 … where ‘E’ is the effective rate of interest, ‘i’ is the actual rate of interest in decimal, and ‘n’ is the number of conversion periods.

In this problem, we know that,

- The actual rate of interest = i = 8% p.a. = 0.08 p.a. = 0.04 per semi-year (6 months).
- Number of conversion periods = n = 2 (since we are calculating for one year and compounding happens once every six months)

Therefore, the effective rate of interest is,

E = (1 + i)^{n} – 1 = (1 + 0.04)^{2} – 1 = 1.0816 – 1 = 0.0816 or 8.16%.

Hence, the correct answer is option c – 8.16 percent.

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