What Is a Geometric Mean? How to Calculate and Example

Updated October 22, 2024

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What Is the Geometric Mean?

The geometric mean is the average of a set of products. Analysts, portfo🌳lio managers, and others commonly use the calculation of the geometric mean to determine the performance results of an 📖investment or portfolio.

Technically, a geometric mean is defined as "the nth root product of n numbers." The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

Key Takeaways

The geometric mean is the average of a set of values calculated using the products of the terms.
The geometric mean is most appropriate for series that exhibit serial correlation—especially investment portfolios.
Most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums.
For volatile numbers, the geometric mean provides a far more accurate measurement of the true return by considering year-over-year compounding, which smooths the mean.

Understanding the Geometric Mean

The geometric mean, sometimes referred to as 澳洲幸运5开奖号码历史查询:compounded annual growth rate or 澳洲幸运5开奖号码历史查询: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 multꦏiplies several values and sets them to the 1/nth power.

For various reasons, the geometric mean is an important tool for calculating 澳洲幸运5开奖号码历史查询:portfolio performance. One of the most significant of those reasons is that it takes into account the 澳洲幸运5开奖号码历史查询:effects of compounding.

For exam✅ple, the geometric mean calculation 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 two numbers), the answer is 4. However, wh🍸en there are many numbers, it is more difficult to calculate unless a calculator or computer program is used.

The main benefit of using the geometric mean is that the actual amounts invested do not need to be known. The calculation focuses entirely on the return figures themselves and presents an "apples-to-apples" comparison when comparing two investment options over more than one time period.

The geometric mean will always be slightly smaller than the 澳洲幸运5开奖号码历史查询:arithmetic mean, which is a simple average.

Formula and Calculation With Example

Formula for the Geometric Mean

$\begin{aligned} &\mu _{\text{geometric}} = [(1+R _1)(1+R _2)\ldots(1+R _n)]^{1/n} - 1\\ &\textbf{where:}\\ &\bullet R_1\ldots R_n \text{ are the returns of an asset (or other}\\ &\text{observations for averaging)}. \end{aligned}$

Calculating Geometric Mean

Imagine that your portfoꦡlio returned the following amounts each 🐭year for five years:

Year one: 5%
Year two: 3%
Year three: 6%
Year four: 2%
Year five: 4%

You would use the formula with those values:

[ ( 1 + .05)(1 + .03)(1 + .06)(1 + .02)(1 + .04) ] ^1/5 - 1
[1.05 x 1.03 x 1.06 x 1.02 x 1.04]^1/5 - 1
[1.2161]^1/5 - 1
[1.2161]^.2 -1 = .0399

Multiply t🌟he result by 100%, and your portfolio returned ♒a geometric mean of 3.99% over five years, slightly less than the arithmetic mean of (5+3+6+2+4) ÷ 5 = 4.

Calculate the Geometric Mean in a Spreadsheet

Using a spreadsheet, you'll get a slightly different result. Use the Geomean function to calculate the geometric mean of the previous returns.

	A	B
1	Period	Return
2	Year one	5%
3	Year two	3%
4	Year three	6%
5	Year four	2%
6	Year five	4%

In an empty cell, enter:

=GEOMEAN(B2:B6)

The result s💛hould be close to the one you got using a calculator. Google Sheets gave us 0.0373, or 3.73% (make sure you click Format> Number> Plain Text).

Important

The 𓃲longer the time horizon, the more critical compounding becomes, and the more appro𓂃priate the use of geometric mean.

What Is the Geometric Mean of N Terms?

The geometric mean of n terms is the product of the terms to the nth root where n represents the number of terms.

Can You Calculate the Geometric Mean With Negative Values?

You cannot—it is impossible to calculate a geometric mean that includes negative numbers. To use negative numbers in a geometric mean calculation, you have to convert them to a proportion. For example, if you had an investment that returned -3%, you would use 1 - 0.03 = 0.97 as your value.

How Do You Find the Geometric Mean Between Two Numbers?

To calculate the geometric mean of two numbers, you would mul꧅tiply the numbers together and take the square root of the result.

The Bottom Line

In mathematics, the geometric mean is an average or mean that demonstrates the central tendency of a group of numbers. In investing, it is a statistical metric that can determine an investment portfolio's returns by considering the effects of compounding.

The geometric mean can help investors understand how their portfolio is performing and ꦕwhether any adjustments need to be made.🎶

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