What diversification actually removes, and what it cannot

Diversification is often described as reducing risk, which is true in the same imprecise way that a thermometer measures weather. It reduces one of the two components from lesson 2 and does nothing whatever to the other.

The mechanism is averaging. Residuals are specific to companies, so across many holdings they arrive with different signs at different times and partly cancel. The cancellation follows the square root of the count rather than the count: with equal weights and residual volatility of 30 percent per name, four names give 30 ÷ 2 = 15 percent, nine names give 30 ÷ 3 = 10 percent, and forty names give 30 ÷ 6.3246 = 4.74 percent.

The market component does not average away, because it is the same component in every holding. If each name has a beta of 1.0 against a market with 16 percent volatility, the market-attributed volatility is 16 percent for one name and 16 percent for forty. That figure is the floor, and it is reached quickly: most of the available reduction is present by twenty or so holdings.

Combining the two components uses the sum of squares, since they are unrelated by construction. One name: the square root of 16 squared plus 30 squared, which is the square root of 1,156, or 34.0 percent. Forty names: the square root of 256 plus 22.5, which is 16.69 percent. So the whole benefit available was 34.0 − 16.69 = 17.31 points, and the last 16.69 was never available at all.

Diversification is not a general-purpose risk reducer. It is a specific machine that deletes residual variance and leaves market variance untouched, which is why the second half of this track exists.