# Effective Annual Return

Effective annual return (EAR) is the annual rate that captures the magnifying effect of multiple compounding periods per year of an investment. When return on investment is compounded more than once in a year, return earned in each compounding period in added to the investment value which results in a return higher than nominal rate of return in subsequent compounding periods. Due to this phenomenon, the future value of investment is higher than the future value arrived at by simply applying the nominal rate of return to the initial investment value. Effective annual return is the rate that when applied to the initial investment will give a future value equal to the value arrived at after the compounding process.

Let’s say you put $10,000 in a fixed-income investment that pays 8% per annum compounded semiannually. The value of your investment after the first six month will be$10,400 [=$10,000 × (1 + 4%)]. After the second compounding period, the investment value will rise to$10,816 [=$10,400 × (1 + 4%)]. You can see that if we apply 8% to$10,000, we get $10,800 after the first year. The 8% nominal annual return doesn’t capture the effect of compounding. This is where the concept of effective annual return becomes relevant. We need a single annual rate that when applied to the initial value of the investment will give$10,816. This rate is 8.16% [=($10,816 −$10,000) ÷ $10,000] which is exactly the effective annual return.  Nominal Rate of Return % Compounding Periods Effective Annual Return % ## Formula $$\text{Effective Annual Return}\ (\text{r}) \\= \left[\text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n} - \text{1}$$ Where n is the number of compounding periods per year. Let’s derive the formula above. FV of$1 investment after n compounding periods

$$= \text{\1} \times \left[ \frac{\text{1} + \text{nominal rate of return}}{\text{n}} \right] ^\text{n}$$

Where r is the effective annual return that captures the effect of compounding, another formula to arrive at the future value is $1 × (1 + r). This can be written in equation form as: $$\text{\1} \times (\text{1} + \text{r}) = \text{\1} \times \left[ \text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n}$$ Removing$1 from both sides, we get:

$$\text{1} + \text{r} = \left[\text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n}$$

Subtracting 1 from both sides:

$$\text{r} = \left[\text{1} + \frac{\text{Nominal Rate of Return}}{\text{n}} \right] ^ \text{n} - \text{1}$$

## Example

Mark is a university student whose parents set up a trust to finance his university education. 5% of the money which equals \$15,000 is put invested in a bank time deposit paying 6% compounded monthly. Calculate the effective annual return.

$$\text{Effective Annual Return}\ (\text{monthly compounding}) \\= \left[\text{1} + \frac{\text{6%}}{\text{12}}\right]^{\text{12}} - \text{1} \\= \text{6.168%}$$

EAR can also be calculated using Microsoft Excel EFFECT function. The formula requires two inputs: (a) nominal_rate which is nominal annual rate on the investment and (b) npery which is the number of compounding periods per year.

The formula you need to enter to work out effective annual return = EFFECT(6%, 12).