Estimating Breadth With Correlated Bets

The fundamental law of active management says your information ratio grows with the square root of "breadth" — the number of independent bets you place per year. A stock-picker running 500 names looks, on paper, like they have far more breadth than a macro trader with six positions. That comparison is only fair if the 500 stock bets are actually independent of one another, and in practice they rarely are: two banks move together, two miners move together, a whole sector moves on one interest-rate surprise. Counting correlated names as separate bets is the single most common way a paper information ratio ends up flattering the strategy.

Why correlation eats bets

Picture betting on ten coin flips. If the coins are truly independent, ten flips genuinely give you ten chances for luck to average out. Now imagine the ten coins are wired together so that when one lands heads, the rest lean heads too. You still see ten outcomes, but you no longer have ten independent pieces of evidence — you have something closer to two or three. The averaging-out that breadth is supposed to buy you never fully happens, because the "bets" keep failing together.

Stocks in a portfolio are wired together the same way, through shared factor exposures: sector, country, size, momentum. The raw count of positions, $N$, overstates the number of genuinely separate bets whenever the average pairwise correlation between them, $\bar\rho$, is above zero.

The formula

The effective, or diversified, number of bets is

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

In words: start with the raw count of positions, then shrink it by a factor that grows with both the number of positions and how correlated they are. When $\bar\rho = 0$, $N_{\text{eff}} = N$ — genuinely independent bets cost you nothing. As $\bar\rho$ rises even slightly, $N_{\text{eff}}$ collapses toward $1/\bar\rho$ and stays there no matter how large $N$ gets, because adding another correlated name barely adds new information.

Worked example

A long-short equity book holds 500 stocks with an average pairwise return correlation of $\bar\rho = 0.02$ — a genuinely diversified, low-correlation book.

$$N_{\text{eff}} = \frac{500}{1 + (499)(0.02)} = \frac{500}{10.98} = 45.5.$$

Roughly 46 effective bets from 500 positions. Plug that into the fundamental law with an IC of 0.04: $IR = IC\sqrt{N_{\text{eff}}} = 0.04\sqrt{45.5} = 0.27$, annualised. Using the raw count instead, $IR = 0.04\sqrt{500} = 0.89$ — more than three times too optimistic.

Now compare a sector-concentrated version of the same book: still 500 names, but clustered so the average pairwise correlation is $\bar\rho = 0.10$. $N_{\text{eff}} = 500/(1+49.9) = 9.8$. The IC-implied information ratio falls to $0.04\sqrt{9.8} = 0.125$ — worse than half the low-correlation book's figure, from the same signal and the same headcount, purely because the bets stopped being separate.

A subtler point: $N_{\text{eff}}$ is bounded even as $N$ grows without limit. Take the limit of the effective-bets formula as $N \to \infty$ with $\bar\rho$ held fixed: the $N$ in the numerator and the $(N-1)\bar\rho$ term in the denominator both grow together, and the ratio settles near $1/\bar\rho$. At $\bar\rho = 0.02$ that ceiling is 50 effective bets; at $\bar\rho = 0.10$ it is 10. A book could hold five thousand correlated names instead of five hundred and its effective breadth would barely move, because past a certain point every new name is mostly buying correlated exposure it already had, not new independent information.

In practice

• A quick diagnostic for any "we hold hundreds of names" pitch: ask for the average pairwise correlation, or the number of statistically independent factors the book actually loads on, before trusting a breadth-based information ratio.
• Diversify across signals and horizons, not only names. Ten uncorrelated signals on the same 50 stocks often buy more effective breadth than 500 stocks on one signal.
• Check the ceiling, not just the current count. Since $N_{\text{eff}}$ tops out near $1/\bar\rho$, a book already close to that ceiling gets little diversification benefit from adding more correlated names, and the research effort is better spent lowering $\bar\rho$ than growing $N$.
• Sector- and factor-neutralising a book is, among other things, a direct way to push $\bar\rho$ down and $N_{\text{eff}}$ up.
• The same shrinkage shows up under multiple names. It is the reasoning behind Factor Crowding concerns (crowded trades correlate exactly when you need them not to) and behind why Portfolio-Level Versus Signal-Level Significance treats a bundle of correlated signals as far less than the sum of its parts.