What is the difference between arithmetic and geometric averages?

By Chris Gallant AAA
A:

An arithmetic average is the sum of a series of numbers divided by the count of that series of numbers.

If you were asked to find the class (arithmetic) average of test scores, you would simply add up all the test scores of the students, and then divide that sum by the number of students. For example, if five students took an exam and their scores were 60%, 70%, 80%, 90% and 100%, the arithmetic class average would be 80%.

This would be calculated as: (0.6 + 0.7 + 0.8 + 0.9 + 1.0) / 5 = 0.8.

The reason you use an arithmetic average for test scores is that each test score is an independent event. If one student happens to perform poorly on the exam, the next student's chances of doing poor (or well) on the exam isn't affected. In other words, each student's score is independent of the all other students' scores. However, there are some instances, particularly in the world of finance, where an arithmetic mean is not an appropriate method for calculating an average.

Consider your investment returns, for example. Suppose you have invested your savings in the stock market for five years. If your returns each year were 90%, 10%, 20%, 30% and -90%, what would your average return be during this period? Well, taking the simple arithmetic average, you would get an answer of 12%. Not too shabby, you might think.

However, when it comes to annual investment returns, the numbers are not independent of each other. If you lose a ton of money one year, you have that much less capital to generate returns during the following years, and vice versa. Because of this reality, we need to calculate the geometric average of your investment returns in order to get an accurate measurement of what your actual average annual return over the five-year period is.

To do this, we simply add one to each number (to avoid any problems with negative percentages). Then, multiply all the numbers together, and raise their product to the power of one divided by the count of the numbers in the series. And you're finished - just don't forget to subtract one from the result!

That's quite a mouthful, but on paper it's actually not that complex. Returning to our example, let's calculate the geometric average: Our returns were 90%, 10%, 20%, 30% and -90%, so we plug them into the formula as [(1.9 x 1.1 x 1.2 x 1.3 x 0.1) ^ 1/5] - 1. This equals a geometric average annual return of -20.08%. That's a heck of a lot worse than the 12% arithmetic average we calculated earlier, and unfortunately it's also the number that represents reality in this case.

It may seem confusing as to why geometric average returns are more accurate than arithmetic average returns, but look at it this way: if you lose 100% of your capital in one year, you don't have any hope of making a return on it during the next year. In other words, investment returns are not independent of each other, so they require a geometric average to represent their mean.

To learn more about the mathematical nature of investment returns, check out Overcoming Compounding's Dark Side.

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