Arithmetic and Geometric Averages
The two ways to average investment returns, why they give different answers, which one tells you what an investor ended up with, and how volatility widens the gap between them.
The short version
There are two averages of a set of returns. The arithmetic average adds them up and divides. The geometric average is the single steady rate that would have produced the same ending value, and it's the one that describes what an investor ended up with. The geometric average is always the lower of the two unless every return is identical, and the gap grows with how much returns swing. US shares since 1928 averaged about 11.9% a year by the first measure and 10.0% by the second. Use the arithmetic average for the typical single year, and the geometric average for anything that compounds.
By the end of this lesson, you'll be able to
- Calculate the arithmetic and geometric average of a series of returns
- Explain why the geometric average is lower and how volatility widens the gap
- Choose the right average for a single year and for growth over many years
Why this matters
"This fund averaged 10% a year" sounds like a complete statement. It isn't, because there are two ways to average returns and they give different answers. One of them tells you what an investor ended up with. The other is easier to calculate and is always at least as high. Percentages and Percentage Points showed that returns multiply. This lesson follows that through.
Two averages
The Arithmetic average. The ordinary average: add the values and divide by how many there are. For returns, it describes a typical single period. Glossary is the ordinary one. Add the returns and divide by how many there are.
The Geometric average. The single steady rate that would turn a starting value into the ending value over several periods. For returns, it describes what an investor ended up with, and it's never higher than the arithmetic average. Glossary asks a different question: what single rate, earned every year, would have turned the starting amount into the ending amount? To find it, multiply the yearly multipliers together, take the root for the number of years, and subtract one.
Geometric average = (Ending value ÷ Starting value)1 ÷ years − 1
When it's used for the growth of an investment or a company's sales, the geometric average is usually called the Compound annual growth rate (CAGR). The steady yearly growth rate that would take a starting value to an ending value over several years. Glossary.
$1,000 gains 50% in the first year and loses 50% in the second.
| After year 1 | $1,000 × 1.50 = $1,500 |
|---|---|
| After year 2 | $1,500 × 0.50 = $750 |
| Arithmetic average | (50% − 50%) ÷ 2 = 0% |
| Geometric average | (750 ÷ 1,000)1 ÷ 2 − 1 = −13.4% |
The arithmetic average says nothing happened. The investor lost a quarter of their money. The geometric average, a loss of 13.4% a year for two years, is the one that matches: $1,000 × 0.866 × 0.866 = $750.
| Year | Return | Multiplier | Value of $10,000 |
|---|---|---|---|
| 1 | +30% | 1.30 | $13,000 |
| 2 | −20% | 0.80 | $10,400 |
| 3 | +25% | 1.25 | $13,000 |
| 4 | −10% | 0.90 | $11,700 |
The arithmetic average is (30 − 20 + 25 − 10) ÷ 4 = 6.25%. The geometric average is 1.171 ÷ 4 − 1 = 4.0%. If the investment had really earned 6.25% every year, it would be worth $12,744. It's worth $11,700, which is what 4.0% a year produces.
The rest of this lesson is for members
Still to come in this lesson: Why the geometric average is lower, What it looks like in real returns, Which average to use, the key takeaways, and 2 quiz questions with explanations.
Membership opens every lesson in the curriculum, the capstone projects, the interactive tools, and the members-only channels in the Hyperion Entities Discord. The first lesson of every module stays free.
Go deeper
- Reading Charts HonestlyMath and Statistics for Investors (depth track)
- CompoundingPart 1: Foundations
- Distributions, Standard Deviation, and Fat TailsMath and Statistics for Investors (depth track)
- Leveraged and Inverse ETFsPart 2: What You Can Own
Notes and sources
- Aswath Damodaran, NYU Stern School of Business. Historical Returns on Stocks, Bonds and Bills, with annual data through 2025 (updated January 2026). Checked 2026-10-12.