# What Is a Geometric Mean? How to Calculate and Example

![Geometric Mean](https://www.investopedia.com/thmb/leyY8MttX6hkL-ssQVgpd1nEKYM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Geometricmean_rev-0d799eca4eea4d168884a39d617cbcaf.jpg)

*— Investopedia / Michela Buttignol*

## What Is the Geometric Mean?

The geometric mean is the average of a set of products. Analysts, portfolio 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 [compounded annual growth rate](https://www.investopedia.com/terms/c/cagr.asp) or [time-weighted rate of return](https://www.investopedia.com/terms/t/time-weightedror.asp), 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 multiplies several values and sets them to the 1/nth power.

For various reasons, the geometric mean is an important tool for calculating [portfolio performance](https://www.investopedia.com/articles/08/performance-measure.asp). One of the most significant of those reasons is that it takes into account the [effects of compounding](https://www.investopedia.com/terms/c/compounding.asp).

For example, 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, when 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 [arithmetic mean](https://www.investopedia.com/terms/a/arithmeticmean.asp), 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 portfolio 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
*   \[1.05 x 1.03 x 1.06 x 1.02 x 1.04\] - 1
*   \[1.2161\] - 1
*   \[1.2161\] -1 = .0399

Multiply the 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

You can also use the _Geomean_ function of a Google Sheet to calculate the geometric mean of the previous returns.

In an empty cell, enter the following (make sure you click Format> Number> Plain Text):

> \=GEOMEAN(B2:B6)

> **Important**
>
> The longer the time horizon, the more critical compounding becomes, and the more appropriate 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 _n_th 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 multiply 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 whether any adjustments need to be made.

> **Article Sources**
>
> 1.  Envestnet PMC. "[Arithmetic, Geometric and Other Types of Averages](https://www.investpmc.com/sites/default/files/documents/PMC_Types_Of_Averages.pdf)."