Alpha Correlation and Effective Breadth

The fundamental law of active management says expected performance scales with skill times the square root of breadth — the number of independent bets made. It's tempting to count signals and call that breadth: ten signals, ten bets. But if those ten signals are all reading the same underlying economic story with slightly different data, they aren't ten bets, they're closer to one.

What "independent" actually buys

Breadth is valuable because independent errors cancel. If ten signals are perfectly uncorrelated and each has a small edge, averaging them cancels out most of each one's noise while the shared, true signal remains — performance scales roughly with the square root of the count. If the ten signals are perfectly correlated, averaging them changes nothing: the noise doesn't cancel because it's the same noise, ten times over. Real signal libraries sit somewhere in between, and effective breadth is the number that says where.

A workable approximation for $N$ signals with average pairwise correlation $\bar\rho$ is

$N_{\text{eff}} = \frac{N}{1 + (N-1)\bar\rho}$

In words: start with the raw count $N$, and divide by a penalty term that grows with both the number of signals and how correlated they are on average. At $\bar\rho = 0$ this returns $N_{\text{eff}} = N$ — full credit. As $\bar\rho \to 1$, $N_{\text{eff}} \to 1$ — no matter how many signals you have, you effectively have one.

A worked example

A researcher has $N = 10$ signals with an average pairwise correlation of $\bar\rho = 0.3$. Effective breadth is $10 / (1 + 9 \times 0.3) = 10 / 3.7 \approx 2.7$. Ten signals are behaving, for diversification purposes, like fewer than three independent ones — the expected performance benefit from combining them is much closer to what you'd get from 3 good uncorrelated ideas than from 10.

Compare a second library: 4 signals with $\bar\rho = 0.05$. Effective breadth is $4 / (1 + 3 \times 0.05) = 4/1.15 \approx 3.5$ — nearly full credit for all four. The smaller, less-correlated library has more effective breadth than the larger, more-correlated one, even though it has fewer raw signals.