Geometric Mean vs Arithmetic Mean

Answer 1

We all want to be above average, don’t we? Definitely when discussing average investment results we want higher averages, right? But how do you compute your average rates of return? One might think they’d do the calculation like any other average they would use day to day; start by adding a series of values together and then divide the sum by the number of values in the series. Such a calculation is known as the arithmetic mean and while handy to use for averaging things like gas prices at different filling stations or students’ test results, it isn’t adequate for finding the average rate of return on your investment portfolio.

The reason is that each value in the series that represents your rate of return on your portfolio is itself a product of other numbers. If your investment results over a three-year period were 10%, 30%, and 50%, what this means is that your portfolio value at the beginning of each year was multiplied by 1.10, 1.30, and 1.50 respectively. It wouldn’t be appropriate to use the arithmetic mean here because each years’ result is the product of your portfolio’s value and some rate of return. You would instead use something called the geometric mean.

The way to get to the geometric mean is by first finding the product of the series of data. In this example we would solve for the product of the series, 1.1, 1.3, and 1.5.

1.1 * 1.3 * 1.5 = 2.145

The next step would be to take the nth root of the resulting product, where n = the number of values in the series. In this case we have 3 values in the series, so n = 3. That means we’d take the cube root of 2.145. Another way to do this would be to raise the value 2.145 by the power of 1/n. (1/3 = .333333333), so 2.145.333333 = 1.289662. So the geometric mean would equal .289662 or 28.97% return.

It should be noted that while it is not appropriate to use the arithmetic mean to show investment results, the number will always skew higher than the geometric mean would. (In the above example the arithmetic mean would have shown average rates of return of 30%.)

So when an investment advisor shows you any sort of average rates of return over a number of periods ask them if they’re showing you the arithmetic or geometric mean. They should be using the geometric mean to gauge average rates of return so as to not overstate your returns.