Arithmetic Mean vs. Geometric Mean: Why Are They Different
If someone says, "My stock was +50% the first year and -50% the next, so the average is 0%," are they really back to break-even? Run the numbers and the money has actually shrunk. The identity of this illusion is exactly the difference between the arithmetic mean and the geometric mean.
Starting with the Definitions of the Two Means
The arithmetic mean is the "plain average" we all know. You add up each year's returns and divide by the number of years. For example, if three years of returns are +10%, +20%, and -10%, the arithmetic mean is (10 + 20 − 10) ÷ 3 = 6.7%.
The geometric mean is calculated by "multiplying." You add 1 to each year's return (1 + return), multiply them all together, then take the nth root (for n years) and subtract 1 again. For the example above, it is the cube root of (1.10 × 1.20 × 0.90) minus 1.
What matters is that the geometric mean is the same concept as CAGR (compound annual growth rate), which we hear about often. It is the true report card that tells you "if it grew at the same rate each year, what percent would it be."
Remember just one key rule. The geometric mean can never be greater than the arithmetic mean. The only time the two are exactly equal is when the return is identical every year, meaning volatility is 0.
+50% Then -50%: Really Break-Even?
Let's go to the most famous trap example. Start with about $740, and a +50% first year makes it about $1,110. A -50% the next year makes it half of $1,110, that is, about $555.
By the arithmetic mean, (+50% − 50%) ÷ 2 = 0%. It looks like break-even. But your wallet holds not $740 but $555—25% has vanished.
When you calculate the geometric mean here, you get about -13.4%. (Taking the square root of 1.5 × 0.5 and subtracting 1 gives -0.134.) If you consider it as compounding down at -13.4% per year for both years, you land exactly at $555. In other words, the arithmetic mean is the "average in feeling," while the geometric mean is the "true result that shows up in your account."
Why does this happen? When a loss occurs, the remaining principal itself shrinks, so the next return applies only to a smaller amount. That's why recovering a -50% requires +100%, not +50%. This asymmetry is why drawdown (MDD) is so frightening.
Volatility Drag: Why More Swinging Means More Loss
The gap between the arithmetic mean and the geometric mean is called "volatility drag (variance drain)." The more returns swing, the wider this gap grows. Conversely, if returns are calm and similar each year, the two means are nearly the same.
There is also a rough approximation. Geometric mean ≈ arithmetic mean − (half the variance). Here, variance is volatility (standard deviation) squared. That is, the greater the volatility, the larger the value subtracted, so the true compound return is shaved down by that much.
So if there's an asset with a "20% average annual return but a yearly roller coaster" and one with a "12% average but calm," the actual compound result left in your account may be larger for the latter. This is why you must be wary of ads that only slap up a large arithmetic mean.
As an empirical example, the return of Australia's benchmark index (ASX/S&P 200) since 1980 had an arithmetic mean of about 13.9% per year but a geometric mean of about 11.6% per year, a gap of about 2.3 percentage points. This is only the value for a specific past period and does not guarantee the future.
So Which Average Should We Look At
When judging long-term investment performance, you should look at the geometric mean (CAGR). It honestly shows "how many times my money actually became." The arithmetic mean is used for specific purposes, such as statistically estimating future expected returns, but when looking at "how much my money became," it causes overestimation.
《Returns of Almost Everything》's calculators also follow this principle. The "average annual return" that comes out of the lump-sum or recurring-investment simulators is not a simple average but a value reflecting the actual compound result, and it shows the maximum drawdown and drawdown period alongside.
The habit of not being fooled by a single number matters. If someone says "an average of X% per year" and you can ask back, "is that the arithmetic mean or the geometric mean?", you're already halfway to being an expert.
Frequently Asked Questions
Q. So is the arithmetic mean just a wrong calculation?
It's not wrong. The arithmetic mean has its use in dealing with future expected values, such as "how much can I expect on average over the coming year." However, when expressing the "actual cumulative performance" of several past years, the geometric mean (CAGR) is closer to the right answer. Think of them as tools for different purposes.
Q. Is high volatility always a loss?
Volatility itself is not bad. It means that, for the same arithmetic mean, the greater the volatility, the more the actual compound result (geometric mean) is shaved down. So when holding an asset that swings widely in pursuit of high returns, you must also weigh how much that swinging eats into the final result.
Q. Why does recovering a -50% loss require +100%?
Starting from about $740, a -50% makes it about $370. To turn this $370 back into $740, you need to earn another $370, and that is +100% based on $370. When a loss occurs, the principal itself shrinks, so the same percent can't bring you back to the original spot. Because of this asymmetry, avoiding large drawdowns is so important.
📋 Results are based on historical data; past returns do not guarantee future returns.
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