Return Calculation5 分钟阅读

Geometric Mean Return — The True Average of Compounding

+50% one year, -50% the next. Averaged out it's 0%, so is it breakeven? In reality, 25% of your principal vanishes. Removing this illusion is exactly the geometric mean return.

The trap of the arithmetic mean

The 'average' we learned in school is the arithmetic mean. You add up the values and divide by the count. For test scores, this is correct.

But with investment returns, this deceives people. Suppose you start with about $740 and it becomes +50% in the first year (about $1,110) and -50% in the second year (about $555). Taking the arithmetic mean of the returns gives (+50% - 50%) ÷ 2 = 0%. It looks as if you broke even.

But your account actually holds only about $555. 25% of the principal evaporated. The arithmetic mean of 0% is plainly a lie.

The reason is simple. The -50% applies to the already-grown $1,110, while the +50% applies to the initial $740. Because the order and size of what's multiplied differ, you can't simply add and divide.

The geometric mean: the true average of compounding

The geometric mean return is a method of finding the single rate that 'produces the actual result when applied equally each year.' Instead of adding, you multiply and then take a root.

Let's calculate the earlier example with the geometric mean. The final multiple is 555 ÷ 740 = 0.75. Since it's 2 years, taking the square root gives √0.75 ≈ 0.866, that is, -13.4%. Compounding -13.4% twice reduces it exactly from 100 to 75.

Not the arithmetic mean (0%) but the geometric mean (-13.4%) is the report card the investor actually experienced. So when speaking of returns over several years, you must use the geometric mean.

Formula: geometric mean return = (final amount ÷ principal)^(1/period) - 1. Here '^(1/period)' means the root by as many as the number of periods. This value is essentially the same concept as what we commonly call CAGR (compound annual growth rate).

Volatility drag: the more it swings, the more it's shaved

The geometric mean is always less than or equal to the arithmetic mean. The two are equal only when the return is the same every year, and the more the return swings, the wider the gap between the two.

This gap is called 'volatility drag (variance drain).' The rough relationship is this.

Geometric mean ≈ arithmetic mean - (volatility²) ÷ 2

That is, the more it swings, the more the return you actually end up with through compounding is shaved. This is why an asset that swings dazzlingly up and down, which looks good by 'average return' alone, actually falls behind a quietly upward-trending asset in real compounded performance.

Why losses hurt more

At the root of volatility drag lies the 'asymmetry of losses.' If it falls -20%, principal 100 becomes 80. Rising +20% from here gives 80 × 1.2 = 96—not breakeven. To break even, you need +25%.

The larger the decline, the crueler this asymmetry. To recover -50% you need +100%, and for -80% you need +400%. This is why, after a big drawdown, you must not be fooled into thinking 'the average return is positive so it's fine' by looking only at the arithmetic mean.

So in long-term investing, reducing drawdowns is as important as a high return. The geometric mean is an honest metric that shows this truth as-is, without hiding it.

In practice: the two averages of the S&P 500

Let's get a feel with a historical example. Looking at the long-term returns of the U.S. S&P 500 index, the arithmetic mean—simply adding and dividing the annual returns—is known to be roughly 11–12% per year.

Meanwhile, the geometric mean (CAGR), which actually compounded while reinvesting dividends, is roughly around 10% per year. The difference of about 2 percentage points is exactly the volatility drag. (Figures vary slightly by period, dividend reflection, and source.)

Even if 2 percentage points looks small, over decades of compounding it splits the final amount greatly. If you use the arithmetic mean when making long-term plans, you end up inflating your future assets in the calculation. You must view it on a geometric-mean basis to get a realistic picture.

The figures here are approximations referencing a specific period/source (e.g., a CAGR of about 10.2% on a 1928–2024 dividend-reinvestment basis) and are not an investment recommendation or a guarantee of future returns. The key is to understand the concept that a 'difference (volatility drag)' between the arithmetic and geometric means exists.

常见问题

Q. Are the geometric mean and CAGR different?

They're essentially the same. Multiplying the returns of several years, then taking the root by as many as the number of periods to find the 'compound annual growth rate,' is the geometric mean, and CAGR is the name for it in practice. Only the expression differs; the calculation and meaning are identical.

Q. So is the arithmetic mean completely useless?

Not at all. The arithmetic mean is suitable for representing the 'expected return of one year (a single period)' or for looking at the average expected value of one year's return going forward. However, when speaking of 'how much it actually became over several years,' you must use the geometric mean. They're tools with different uses.

Q. Why is the geometric mean always smaller?

If the return swings even a little each year, an asymmetry effect arises where a loss reduces the principal itself and makes recovery harder. Because of this, the actual result accumulated through compounding (geometric mean) ends up lower than the simple average (arithmetic mean). The greater the swing, the greater that difference (volatility drag) too.