If you have a sample of returns (say a mutual fund’s past 10 years of returns), then there are two alternatives for summarizing the fund’s past performance: arithmetic average return and geometric (compound) average return.
Arithmetic Average Return
In this method, the sum of the returns in the sample is divided by the size of sample (n). The arithmetic average return does not take into account the compounding effect.
Geometric Average Return
The return computed that recognizes that interest or earnings are paid on accumulated interest or earnings. It is also called a compound return because it incorporates the compounding effect.
Formula
We can use the following formula to compute the arithmetic average return
\[\begin{matrix}\text{Arithmetic Average Return=}\frac{\text{1}}{\text{n}}\sum\limits_{\text{i=1}}^{\text{n}}{{{\text{k}}_{\text{i}}}} & {} & \left( 1 \right) \\\end{matrix}\]
Where
n= number of samples
ki= series of samples
Whereas the formula for geometric average return is:
\[\begin{matrix} \text{Geometric Average Return=}{{\left( \frac{\text{ending value}}{\text{beginning value}} \right)}^{\frac{\text{1}}{\text{ }\text{no of years}}}}\text{-1} & {} & \left( \text{2} \right) \\\end{matrix}\]
Example
For a simple example, consider the data below for the Explosive Mutual Fund, Inc. The first return is 100% and the second is −50%.
Table 1 Explosive Mutual Fund’s Annual Returns
| Year | Price | Return |
|---|---|---|
| 0 | $10 | |
| 1 | $20 | 100% |
| 2 | $10 | −50% |
The table titled “Explosive Mutual Fund’s Annual Returns” has 3 columns namely “Year,” “Price,” and “Return.” The data depicted in the table are as follows: For the Year 0, Price is 10 dollars; for the Year 1, Price is 20 dollars, Return is 100 percent; and for the Year 2, Price is 10 dollars, Return is minus 50 percent. “
For the above example, the arithmetic average return is 25%.
\[{\rm{Arithmetic Average Return = }}\frac{{\rm{1}}}{{\rm{n}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{{\rm{k}}_{\rm{i}}}} = \frac{{\left( {100\% + \left( { – 50\% } \right)} \right)}}{2} = 25\% \]
And the geometric average return is 0%.
\[\text{Geometric Average Return=}{{\left( \frac{\$10}{\$10}\right)}^{\frac{\text{1}}{2}}}\text{-1}=0\]
The example of Explosive shows clearly why the geometric average is better than the arithmetic average for summarizing investment returns.
If you look at the price of the fund, you see that it started at $10 and ended at $10. An investor who bought at Year 0 and sold at Year 2 would have earned nothing! Yet, the average return is 25%!
The geometric average is the annual return that would have made $10 grow into $10 over 2 years with compounding. The only return that can do that is 0%, so 0% is the geometric average return for Explosive.
Mutual fund companies are required by law to report the compound average, not the arithmetic average.