Turning Information Coefficient Into PnL

Information coefficient (IC) — the correlation between a signal's forecasts and the returns that actually followed — is a natural way to score how skillful a signal is. But two signals with the same IC can produce wildly different amounts of money: an IC of 0.05 applied once a year to a handful of stocks is nearly worthless, while the same 0.05 IC applied daily across a thousand largely independent stocks compounds into a strong track record. IC measures skill per bet; it says nothing about how many independent bets you get to make.

The Fundamental Law of Active Management connects the two: information ratio (return per unit of risk, roughly analogous to Sharpe ratio) scales with IC multiplied by the square root of breadth, where breadth is the number of independent bets made per year.

$$IR \approx IC \times \sqrt{Breadth}$$

In words: a modest edge repeated across many independent opportunities compounds into a much stronger overall result than the same edge applied rarely — because the noise in each individual bet tends to cancel out across many uncorrelated bets, while the skill component adds up.

Visualizing the tradeoff

Worked example

A signal has an IC of 0.04, tested daily across 200 stocks whose forecasts are roughly independent of each other. Over a year (250 trading days), breadth is approximately $250 \times 200 = 50{,}000$ independent bets. Applying the Fundamental Law:

$$IR \approx 0.04 \times \sqrt{50{,}000} \approx 0.04 \times 223.6 \approx 8.9$$

That IR looks implausibly high, which is the point of the example: real signals rarely have 50,000 genuinely independent bets, because stocks within a sector or a common risk factor move together, so the effective breadth is far lower than the raw count of stocks times days. If those 200 stocks are grouped into just 10 effectively independent clusters (correlated returns within each cluster), true breadth is closer to $250 \times 10 = 2{,}500$, giving $IR \approx 0.04 \times 50 = 2.0$ — still strong, but a far more believable number, and the gap between the two calculations is exactly why overstating breadth is one of the most common ways a paper backtest overstates real expected performance.