Why do we use the harmonic mean for averaging P/E ratios instead of the regular mean?

Great question — this is a subtle but important concept that the CFA exam loves to test. The harmonic mean is preferred for averaging ratios like P/E because arithmetic averaging of ratios creates an upward bias.

The Problem with Arithmetic Mean for P/E

When you arithmetically average P/E ratios, you give disproportionate weight to high-P/E stocks. One stock with a P/E of 500 (a nearly-profitless company) can dominate the average and distort the picture.

Harmonic Mean Formula

H = n / (1/x1 + 1/x2 + ... + 1/xn)

For P/E ratios, this is equivalent to computing the weighted average E/P (earnings yield) and then inverting it.

Worked Example — Lakewood Equity Index

Consider a 4-stock equally weighted index:

Arithmetic mean P/E = (20 + 20 + 50 + 100) / 4 = 47.5x

Harmonic mean P/E = 4 / (1/20 + 1/20 + 1/50 + 1/100) = 4 / (0.05 + 0.05 + 0.02 + 0.01) = 4 / 0.13 = 30.8x

The arithmetic mean is inflated by Drumwell's extreme 100x multiple. The harmonic mean effectively computes the portfolio's earnings yield first (0.13/4 = 3.25% each = 13% total / 4 stocks) and inverts it, which is economically what you actually care about.

Inequality of Means

Harmonic <= Geometric <= Arithmetic

This always holds for positive values. The three means converge when all observations are identical.

Exam tip: If a question asks you to calculate the "average P/E" for an equal-weighted portfolio, and the answer choices include both arithmetic and harmonic, the correct answer is the harmonic mean. The CFA curriculum is explicit about this preference for price multiples.

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