Harmonic and Geometric Averaging of Ratios

A portfolio holds two stocks, one with a P/E of 10 and one with a P/E of 50, in equal dollar amounts. What's the portfolio's average P/E? The tempting answer is $(10+50)/2 = 30$, but that's wrong — it overstates the true portfolio multiple by a wide margin, because a plain average of ratios ignores that the two stocks contribute unequally to the combined earnings. The correct tool here is the harmonic mean, and a sibling problem — averaging a sequence of annual returns — needs a different tool again, the geometric mean. Both exist because a simple arithmetic average silently assumes something about the underlying quantities that ratios and growth rates don't satisfy.

An analogy: averaging speeds over the same distance

If you drive to a city at 30 mph and drive home over the identical route at 60 mph, your average speed for the round trip is not 45 mph. You spend more time at the slower speed, so the trip is weighted toward 30 mph. The correct average is the harmonic mean, $2/(1/30 + 1/60) = 40$ mph — below the naive average, because it correctly weights by time spent. P/E ratios behave the same way: dollars invested is the fixed "distance," and P/E is the "speed" — the harmonic mean respects the fixed base.

The mechanics, one piece at a time

For ratios like P/E, where the numerator (price, i.e. the dollar amount invested) is what's held equal or aggregated, the harmonic mean of $n$ values $x_1, \dots, x_n$ is

$$H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}.$$

In plain English: take the reciprocal of each ratio, average those reciprocals, then take the reciprocal of that average. This is exactly equivalent to computing the portfolio's aggregate earnings and aggregate price separately and dividing — the harmonic mean of P/Es equals aggregate price divided by aggregate earnings when weights are equal, which is the economically meaningful quantity.

For compounding growth rates like annual returns, where each period's ending value becomes the next period's starting value, the geometric mean of returns $1+r_1, \dots, 1+r_n$ (as growth factors) is

$$G = \left( \prod_{i=1}^{n} (1+r_i) \right)^{1/n} - 1.$$

In plain English: multiply the growth factors together, take the $n$-th root, and subtract 1 to convert back to a rate. This is the constant annual rate that would produce the same total compounded growth as the actual sequence of varying returns — the arithmetic mean of returns is always at least as large, and typically overstates true compounded growth.

Worked example 1: the portfolio P/E

Back to the two stocks, equal dollar-weighted: P/E of 10 and 50. Harmonic mean: $H = 2 / (1/10 + 1/50) = 2 / (0.1 + 0.02) = 2/0.12 \approx 16.7$. Check this directly: put $100 in each stock. The first has earnings of $100/10 = 10$, i.e. $10; the second has earnings of $100/50 = 2$, i.e. $2. Combined: $200 invested, $12 total earnings, giving a true portfolio P/E of $200/12 \approx 16.7$ — matching the harmonic mean exactly, and far below the naive arithmetic average of 30, which would badly overstate how expensive the combined portfolio actually is relative to its earnings.

Worked example 2: compounding three years of returns

A fund returns +50% in year 1, −40% in year 2, and +20% in year 3. The arithmetic mean is $(50 - 40 + 20)/3 = 10\%$, suggesting steady 10% annual growth. But the actual compounded outcome: $1.5 \times 0.6 \times 1.2 = 1.08$, meaning $1 grew to just $1.08 over three years — total growth of only 8%, not the $(1.10)^3 - 1 \approx 33\%$ the arithmetic mean would imply. The geometric mean is $G = (1.08)^{1/3} - 1 \approx 2.6\%$ per year — the number that actually describes the fund's real annual compounding rate, and it's meaningfully lower than the naive 10% arithmetic average because a large loss (−40%) hurts compounded growth more than an equal-sized gain helps it.

The explorer below compares simple versus compound growth over time — watch how the two curves separate as the horizon lengthens, the same gap driving worked example 2.

What this means in practice

Fund fact sheets report geometric ("compound annual growth rate") returns precisely because arithmetic averages mislead whenever returns are volatile. Harmonic means show up wherever a ratio needs aggregating across a fixed base: portfolio-level valuation multiples, fuel-economy-style efficiency ratios, or average cost per share under dollar-cost averaging. Getting the wrong average isn't a rounding error — worked example 1 showed it can overstate a portfolio's valuation multiple by nearly double.