There are numerous ways to measure portfolio performance and determine if the investment strategy is successful. Among all of these metrics, investment professionals most often use means to estimate growth rates and returns on their portfolios. The average growth rate can vary depending on which method is used to calculate it. One of the most common averages used, especially in finance, is the geometric mean which differs from the arithmetic average.

## Arithmetic Average

**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: (60% + 70% + 80% + 90% + 100%) ÷ 5 = 80%.

**The reason we 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 other students' scores. However, there are some instances, particularly in the world of finance, where the 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 financial markets for five years. If your portfolio returns each year were 90%, 10%, 20%, 30% and -90%, what would your average return be during this period? With the arithmetic average, the average return would be 12% which appears at first glance to be impressive.

**However, when it comes to annual investment returns, the numbers are not independent of each other.** If you lose a substantial amount of money in a particular year, you have that much less capital to invest and generate returns in the following years. Because of this reality, we need to calculate the geometric average of your investment returns to arrive at an accurate measurement of what your actual average annual return over the five-year period is.

## Geometric Average

**Geometric average,** more commonly called the geometric mean, takes into account the compounding that occurs from period to period. The geometric mean for a series of numbers is calculated by taking the product of these numbers and raising it to the inverse of the length of the series.

To do this, we 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. Please don't forget to subtract one from the result.

The formula, written in decimals, looks like this: {[(1+Return1) x (1+Return2) x (1+Return3)...)]^(1/n)]} - 1

The formula appears to be quite intense, but on paper, it's 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:

The result gives a geometric average annual return of -20.08%. The result using the geometric average is a lot worse than the 12% arithmetic average we calculated earlier, and unfortunately, it's also the number that represents reality in this case.

## Bottom Line

The geometric mean works best when used with percentage changes. Also, for volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding.

The geometric mean is most appropriate for series that exhibit serial correlation. This is especially true for investment portfolios. Since an investor lost 10% of his portfolio value in year one, he has much less capital to start with in year two and has to earn more than 10% to get back to the original value of his portfolio. The return numbers from year two to year five are not independent events and depend on the amount of capital invested at the beginning of each year.

In fact, most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums. The longer the time horizon, the more critical compounding becomes and the more appropriate the use of geometric mean.

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