What is the Geometric Mean?
The geometric mean is the average of a set of products, a crucial metric often utilized for evaluating the performance results of an investment or portfolio.
Technically, the geometric mean is defined as ’the nth root product of n numbers.’ This type of mean becomes particularly useful when working with percentages derived from values, while the standard arithmetic mean directly calculates with the values themselves.
Key Highlights
- The geometric mean calculates the average value using the products of the data points.
- It is highly appropriate for series demonstrating serial correlation, such as those in investment portfolios.
- Many returns in finance, like bond yields, stock returns, and market risk premiums, exhibit correlation.
- For volatile figures, the geometric mean provides a more precise measure of true return by considering year-over-year compounding, which smooths out the average.
Formula for the Geometric Mean
μ_{geometric} = [(1+R_1)(1+R_2)...(1+R_n)]^{1/n} - 1
Where:
- $R_1 … R_n$ are the returns of an asset (or other observations for averaging).
Calculating a Geometric Mean
Suppose your portfolio had the following annual returns over five years:
- Year one: 5%
- Year two: 3%
- Year three: 6%
- Year four: 2%
- Year five: 4%
You would use the formula as follows:
1[(1 + 0.05)(1 + 0.03)(1 + 0.06)(1 + 0.02)(1 + 0.04)]^{1/5} - 1
2= [1.05 * 1.03 * 1.06 * 1.02 * 1.04]^{1/5} - 1
3= [1.2161]^{1/5} - 1
4= 1.0399 - 1 = 0.0399
Multiply the result by 100%, and the geometric mean return over five years is approximately 3.99%, slightly less than the arithmetic mean of 4%.
Calculate the Geometric Mean in a Spreadsheet
Below is an example spreadsheet setup:
|—|————|——–| | | Period | Return | | 1 | Year one | 5% | | 2 | Year two | 3% | | 3 | Year three | 6% | | 4 | Year four | 2% | | 5 | Year five | 4% |
In an empty cell, enter: =GEOMEAN(B2:B6)
Google Sheets gives a result of 0.0373, or 3.73%.
Understanding the Geometric Mean
The geometric mean, also known as the compounded annual growth rate (CAGR) or time-weighted rate of return, averages a set of values by multiplying them and setting them to the 1/n power. This calculation is crucial for evaluating portfolio performance because it accounts for the effects of compounding.
For example, to familiarize yourself with the concept, consider simple numbers like 2 and 8. Multiplying 2 and 8, then taking the square root (the ½ power since there are two numbers), yields 4. For many numbers, the calculation can be cumbersome without a calculator or a computer program.
The geometric mean is exceptionally beneficial for investment analysis because it presents a return-focused approach, unaffected by the actual amounts invested. This method retains a fair comparison basis (apples-to-apples) for different investment options across multiple periods. Generally, geometric means are minimally smaller than arithmetic means, providing a more conservative measure of longer-term returns.
Frequently Asked Questions
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?
No, geometric mean calculations do not support negative values. To include negative numbers, convert them to a proportion. For example, convert a return of -3% to value 0.97 (1 - 0.03).
How Do You Find the Geometric Mean Between Two Numbers?
You calculate the geometric mean of two numbers by multiplying them and taking the square root of the result.
The Bottom Line
The geometric mean is a powerful tool for averaging a group of numbers and plays a pivotal role in the sphere of investment evaluation. By accounting for compounding effects, it helps investors monitor portfolio performance accurately, guiding necessary adjustments for continuous improvement.
Related Terms: Arithmetic Mean, Compound Interest, CAGR, Time-Weighted Rate of Return.
References
- Envestnet PMC. “Arithmetic, Geometric and Other Types of Averages”.
