# Arithmetic Average Return

Arithmetic average return is the return on investment calculated by simply adding the returns for sub-periods and then dividing it by total number of periods. It overstates the true return and is only appropriate for shorter time periods.

The arithmetic average return is always higher than the other average return measure called the geometric average return. The arithmetic return ignores the compounding effect and order of returns and it is misleading when the investment returns are volatile.

## Formula

Arithmetic average return can be calculated using the following formula:

$$ Arithmetic\ Average\ Return\\=\frac{Sum\ of\ Individual\ Returns}{Total\ Number\ of\ Returns} $$

It can be calculated using Excel AVERAGE function.

## Example

Your university has created a $100 million endowment to fund financial assistance offered on merit and need-basis. The endowment return for first 5 years was 5%, 8%, -2%, 12% and 9% respectively. Let’s imagine all the return in the form of capital gains. The arithmetic average return will equal 6.4% i.e. (5% + 8% + (-2%) + 12% + 9%)/5.

The investment value after 5 years will be $135.67 million as calculated below:

$$ Endowment\ Value\ after\ 5\ Years \\=$100\ million\times(1+5\%)\times(1+8\%)\times(1-2\%)\times(1+12\%)\times(1+9\%) \\=$135.67 million $$

However, the 6.4% arithmetic average return suggest the investment value will be $145.09 million:

$$ Endowment\ Value\ (based\ on\ Arithmetic\ Average\ Return)\\=$100\ million\times{(1+6.4\%)}^5\\=$145.09\ million $$

Arithmetic average return overstates the return because it ignores the order of return. For example, the decline of 2% occurred in the endowment when it had grown by 5% and 8% in the previous years, but arithmetic average return doesn’t accommodate such compounding effect.

Written by Obaidullah Jan