Understanding portfolio performance, whether for a self-managed discretionary portfolio or a nondiscretionary portfolio, is vital in determining whether the portfolio strategy is working or needs to be changed. There are numerous ways to measure performance and determine if the strategy is successful. One way is to use the geometric mean.

The geometric mean, sometimes called the compound annual growth rate or time-weighted rate of return, is the average rate of return on a set of values calculated using the products of the terms. What does that mean? The geometric mean takes several values and multiplies them and adjusts them to the 1 / nth power. For example, the calculation of the geometric mean can be easily understood with simple numbers, such as 2 and 8. If you multiply 2 and 8, then take the square root (the ½ power since there are only 2 numbers), the answer is 4. However, when there are many numbers, it is more difficult to calculate unless a calculator or computer program is used.

The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most important is that it takes into account the effects of compounding.

## Geometric vs. arithmetic mean performance

The arithmetic mean is commonly used in many facets of everyday life and is easily understood and calculated. The arithmetic mean is achieved by adding all the values and dividing by the number of values (n). For example, finding the arithmetic mean of the following set of numbers: 3, 5, 8, -1, and 10 is accomplished by adding all the numbers and dividing by the number of numbers.

3 + 5 + 8 + -1 + 10 = 25/5 = 5

This is easily accomplished using simple math, but the average return does not take compounding into account. On the contrary, if the geometric mean is used, the average takes into account the impact of the composition, providing a more accurate result.

*Example 1:*

An investor invests $ 100 and receives the following returns:

Year 1: 3%

Year 2: 5%

Year 3: 8%

Year 4: -1%

Year 5: 10%

The $ 100 grew each year as follows:

Year 1: $ 100 x 1.03 = $ 103.00

Year 2: $ 103 x 1.05 = $ 108.15

Year 3: $ 108.15 x 1.08 = $ 116.80

Year 4: $ 116.80 x 0.99 = $ 115.63

Year 5: $ 115.63 x 1.10 = $ 127.20

The geometric mean is: [(1.03*1.05*1.08*.99*1.10) ^ (1/5 or .2)]-1 = 4.93%.

The average annual profitability is 4.93%, slightly lower than the 5% calculated using the arithmetic mean. Actually, as a mathematical rule, the geometric mean will always be equal to or less than the arithmetic mean.

In the example above, the returns did not show a very high variation from year to year. However, if a portfolio or a stock shows a high degree of variation each year, the difference between the arithmetic mean and the geometric mean is much larger.

*Example 2:*

An investor has a stock that has been volatile with returns that varied significantly from year to year. His initial investment was $ 100 in Stock A, and he returned the following:

Year 1: 10%

Year 2: 150%

Year 3: -30%

Year 4: 10%

In this example, the arithmetic mean would be 35%. [(10+150-30+10)/4].

However, the true return is as follows:

Year 1: $ 100 x 1.10 = $ 110.00

Year 2: $ 110 x 2.5 = $ 275.00

Year 3: $ 275 x 0.7 = $ 192.50

Year 4: $ 192.50 x 1.10 = $ 211.75

The resulting geometric mean, or a compound annual growth rate (CAGR), is 20.6%, much lower than the 35% calculated using the arithmetic mean.

One problem with using the arithmetic mean, even to estimate average performance, is that the arithmetic mean tends to exaggerate the actual average performance by an increasing amount the more the inputs vary. In Example 2 above, returns increased by 150% in year 2 and then decreased by 30% in year 3, a year-over-year difference of 180%, which is a staggeringly large variation. However, if the inputs are close together and do not have a high variance, then the arithmetic mean could be a quick way to estimate returns, especially if the portfolio is relatively new. But the longer the portfolio is held, the greater the likelihood that the arithmetic mean will overstate the actual average return.

## The bottom line

Measuring portfolio returns is the key metric for making buy / sell decisions. Using the right measurement tool is critical to determining the correct portfolio metrics. The arithmetic mean is easy to use, quick to calculate, and can be helpful when trying to find the average of many things in life. However, it is an inappropriate metric for determining the true average return on an investment. The geometric mean is a more difficult metric to use and understand. However, it is a much more useful tool for measuring portfolio performance.

When reviewing the annual performance returns provided by a professionally managed brokerage account or when calculating the performance of a self-managed account, there are several considerations to keep in mind. First, if the change in return is small from year to year, then the arithmetic mean can be used as a quick and dirty estimate of the true average annual return. Second, if there is a large variation each year, then the arithmetic average will greatly exaggerate the true average annual return. Third, when performing the calculations, if there is a negative return, be sure to subtract the rate of return from 1, which will result in a number less than 1. Finally, before accepting any performance data as accurate and true , be critical and verify that the average annual yield data presented is calculated using the geometric mean and not the arithmetic mean, since the arithmetic mean will always be equal to or greater than the geometric mean.