The article explains the difference between arithmetic average return and geometric average return as methods for summarizing investment performance. It highlights how arithmetic average ignores compounding, while geometric average incorporates compounding effects, providing a more accurate reflection of investment growth over time.
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.
Arithmetic & Geometric Average Return Key Takeaways
Understanding the difference between arithmetic and geometric average returns is essential for accurately analyzing and comparing investment performance over time. While the arithmetic average offers a straightforward calculation by simply averaging individual returns, it can be misleading because it does not account for the compounding effect, which is a critical factor in real-world investing. In contrast, the geometric average considers the compounding of returns, making it a more realistic measure of an investment’s actual growth. This distinction is especially important when evaluating long-term investments, where fluctuations in annual returns can significantly impact the final outcome. Because of its accuracy and practical relevance, financial professionals and mutual fund companies rely on the geometric average to report performance, ensuring that investors have a clearer understanding of their potential gains or losses. Therefore, both methods have their place, but the geometric average is generally more reliable for assessing the true performance of an investment.