Two Averages, Two Different Answers
Given a series of yearly returns, there are two ways to average them, and they disagree:
- Arithmetic mean — add the returns, divide by the count. It answers "what was a typical year?"
- Geometric mean — compound the returns and take the nth root. It answers "what did I actually earn?"
The geometric mean is the honest one for measuring past performance, because it is the only rate that turns the starting balance into the ending balance.
The Classic Demonstration
An investment gains 50% and then loses 50%. The arithmetic average is 0%. The reality:
| Year | Return | $10,000 becomes |
|---|---|---|
| 1 | +50% | $15,000 |
| 2 | −50% | $7,500 |
| Average | Arithmetic 0% / Geometric −13.4% | $7,500 |
A 25% loss described as a 0% average return. The geometric mean of −13.4% is the figure that reproduces the actual outcome, and it is why fund performance must legally be reported on a compound basis.
Volatility Drag
The gap between the two averages grows with volatility, approximately by half the variance:
Geometric ≈ Arithmetic − σ²/2
| Arithmetic average | Volatility | Approximate geometric | Drag |
|---|---|---|---|
| 8% | 5% | 7.9% | 0.1% |
| 8% | 15% | 6.9% | 1.1% |
| 8% | 25% | 4.9% | 3.1% |
| 8% | 40% | 0.0% | 8.0% |
Two portfolios with identical average returns produce very different wealth if one is far more volatile. This is the mathematical case for diversification: reducing volatility raises compound return even when it does not raise the average return.
Which One to Use
Use the geometric mean to report or compare historical performance — it is what happened. Use the arithmetic mean when estimating a single future year from historical data, because it is the unbiased expected value for one period. Reporting arithmetic averages as past performance is the most common way investment returns are overstated.
Frequently Asked Questions
Why is the geometric mean always lower?
Because losses need larger gains to recover: a 20% loss requires a 25% gain to break even. Any variation in returns therefore drags the compound result below the simple average. The two are equal only when every return is identical.
Is CAGR the same as the geometric mean?
Yes, when computed over the same series. CAGR is usually derived from just the starting and ending values, which gives the same answer as compounding all the yearly returns.
Can the geometric mean be calculated with a −100% year?
No. A total loss makes the product zero and the account is gone; no rate reproduces the outcome. The calculator rejects returns of −100% or worse.