The geometric mean—sometimes called compounded annual growth rate or time-weighted rate of return—is the average rate of return of a set of values, calculated using the products of the terms.

What does that mean? The geometric mean takes several values, multiplies them, and sets them to the 1/nth power, with *n* representing the number of values. The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most significant is that it considers the effects of compounding.

### Key Takeaways

- A geometric mean is the root of the product of any number of values and considers compounding.
- The arithmetic mean can be used to evaluate data, but it doesn’t consider compounding.
- A geometric mean helps you evaluate investment returns based on the number of periods you’ve held it and how much it has returned over time.
- The geometric and arithmetic mean may be similar if investment returns do not fluctuate greatly.
- If investment returns vary highly, the arithmetic and geometric means will be very different.

## Geometric vs. Arithmetic Mean Return

The arithmetic mean is commonly used in many facets of everyday life, and it is easily understood and calculated. The arithmetic mean is achieved by adding all 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 achieved by adding all the numbers and dividing by the quantity of numbers:

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

This is easily accomplished using simple math, but the average return fails to consider compounding. Conversely, if the geometric mean is used, the average considers the impact of compounding, providing a more accurate result.

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The geometric mean may be easier to understand as the root of a number, such as a square root, third root, or fourth root, and so on. The n^{th} root you use depends on the number of periods you’re evaluating.

The geometric mean may be easier to understand as the root of a number, such as a square root, third root, or fourth root, and so on. The n^{th} root you use depends on the number of periods you’re evaluating.

### 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 return is 4.93%, slightly less than the 5% computed using the arithmetic mean.

### Example 2

An investor holds a stock that has been volatile, with returns that varied significantly from year to year. His initial investment was $100 in stock A, and it 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].

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As a mathematical rule, the geometric mean will always be equal to or less than the arithmetic mean.

As a mathematical rule, the geometric mean will always be equal to or less than the arithmetic mean.

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 compounded annual growth rate (CAGR), is 20.6%, much lower than the 35% calculated using the arithmetic mean.

## Limitations of Using the Geometric Mean in Analysis

One problem with using the arithmetic mean, even to estimate the average return, is that the arithmetic mean tends to overstate the actual average return by a greater and greater amount the more the inputs vary. In Example 2, the returns increased by 10% in year one, 150% in year two and then decreased by 30% in year 3. This is an increase of 130% from the original investment.

However, if the returns are close together, then the arithmetic mean could be a quick way to estimate the returns, especially if the portfolio is relatively new. But the longer the portfolio is held, the higher the chance the arithmetic mean will overstate the actual average return.

## Special Considerations

When reviewing the annual performance returns provided by a professionally managed brokerage account or calculating the performance to a self-managed account, you should be aware of several considerations.

First, if the returns do not vary much from year to year, then the arithmetic mean can be used as a quick and dirty estimate of the actual average annual return. Second, if the returns vary greatly each year, then the arithmetic average will overstate the actual average annual return by a large amount.

Third, when performing the calculations, if there is a negative return make sure to subtract the return rate from 1, which will result in a number less than 1. Last, before accepting any performance data as accurate and true, be critical and check that the average annual return data presented is calculated using the geometric average and not the arithmetic average, since the arithmetic average will always be equal to or higher than the geometric average.

## What Is the Geometric Mean of 2 and 4 and 8?

To get the answer, multiply the numbers—2 x 4 x 8 = 64. You have to use the third root because you have three values—which looks like ^{3}√64. You invert the three, resulting in a fraction of 1/3, and raise 64 to its power. So, 64 raised to the power of the inverse of three, or 64^{.333} = 3.99.

## How Do You Find the Geometric Mean of Two Numbers?

To find the geometric mean of two numbers, you multiply them and determine their square root. You can also raise the result to the power of the inverse of the number of values—for example, if there are four values, the inverse of the number of values is 1/4 or 0.25.

## How Is Geometric Mean Calculated for Data?

The geometric mean is the root of the product of all values. The root must be the n^{th} root—so if there are 100 values, they need to be multiplied together and the one-hundredth root found. In this instance, it would look like ^{100}√a * b* c…

You can also count the number of values, invert it, and raise ( a* b * c…) to the power of that number in decimal form. The decimal form of 1/100 is .01, so the calculation would look like ( a * b * c )^{.01}.

## The Bottom Line

Measuring portfolio returns is the key metric in making buy/sell decisions. Using the appropriate measurement tool is critical to ascertaining the correct portfolio metrics. The arithmetic mean is easy to use, quick to calculate, and useful when finding the average for many things.

However, it is an inappropriate metric to determine an investment’s actual average return because compounding is not considered. The geometric mean is a more difficult metric to determine, but it is much more useful for measuring portfolio performance.