## What Is the Geometric Mean?

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically 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.

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

### Key Takeaways

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

## The Formula for Geometric Mean

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

## Understanding the Geometric Mean

The geometric mean, sometimes referred to as 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? Geometric mean takes several values and multiplies them together and sets them to the 1/n^{th} power.

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 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 longer the time horizon, the more critical compounding becomes, and the more appropriate the use of geometric mean.

The main benefit of using the geometric mean is 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 looking at two investment options over more than one time period. Geometric means will always be slightly smaller than the arithmetic mean, which is a simple average.

## How to Calculate the Geometric Mean

To calculate compounding interest using the geometric mean of an investment's return, an investor needs to first calculate the interest in year one, which is $10,000 multiplied by 10%, or $1,000. In year two, the new principal amount is $11,000, and 10% of $11,000 is $1,100. The new principal amount is now $11,000 plus $1,100, or $12,100.

In year three, the new principal amount is $12,100, and 10% of $12,100 is $1,210. At the end of 25 years, the $10,000 turns into $108,347.06, which is $98,347.05 more than the original investment. The shortcut is to multiply the current principal by one plus the interest rate, and then raise the factor to the number of years compounded. The calculation is $10,000 × (1+0.1) 25 = $108,347.06.

#### Geometric Mean

## Example of Geometric Mean

If you have $10,000 and get paid 10% interest on that $10,000 every year for 25 years, the amount of interest is $1,000 every year for 25 years, or $25,000. However, this does not take the interest into consideration. That is, the calculation assumes you only get paid interest on the original $10,000, not the $1,000 added to it every year. If the investor gets paid interest on the interest, it is referred to as compounding interest, which is calculated using the geometric mean.

Using the geometric mean allows analysts to calculate the return on an investment that gets paid interest on interest. This is one reason portfolio managers advise clients to reinvest dividends and earnings.

The geometric mean is also used for present value and future value cash flow formulas. The geometric mean return is specifically used for investments that offer a compounding return. Going back to the example above, instead of only making $25,000 on a simple interest investment, the investor makes $108,347.06 on a compounding interest investment.

Simple interest or return is represented by the arithmetic mean, while compounding interest or return is represented by the geometric mean.