Grinold's fundamental law of active management, developed with Kahn in Active Portfolio Management, states that the information ratio a manager can expect is approximately her skill multiplied by the square root of how often she uses it:
IR ~= IC x sqrt(BR)
IC is the information coefficient, the correlation between a forecast and the outcome it forecasts, measured across the cross-section of things being forecast. BR is the breadth, the number of independent forecasts made per year.
The derivation is worth seeing once even in sketch, because the assumptions are where all the arguments live. Suppose the manager's forecast for security i is a standardised score z_i with unit variance, and her skill converts it into an expected residual return alpha_i = IC x sigma_i x z_i, where sigma_i is the security's residual volatility. That scaling rule is not a convention; it is what an information coefficient of IC means. Optimal active weights are then proportional to alpha_i / sigma_i^2, which is proportional to IC x z_i / sigma_i. Substitute those weights back into the expected active return and the active risk, take the ratio, and the security-specific terms cancel, leaving IC multiplied by the square root of the number of independent bets.
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What does the fundamental law imply?
Three consequences follow, and each is a fact a committee will resist.
Skill and breadth are substitutes at a fixed exchange rate. An information coefficient of 0.05 with 200 independent decisions produces an information ratio of 0.7071. So does 0.50 with 2, and 0.10 with 50. All three managers are equally good in this arithmetic even though the first would be described as almost unskilled and the second as a genius.
The exchange rate is a square root, so breadth is expensive. Doubling an information ratio requires quadrupling breadth: at an information coefficient of 0.05, an information ratio of 0.30 needs 36 independent decisions a year and 0.60 needs 144. A manager who wants to be twice as good by working harder must work four times as hard.
And the required skill falls fast with breadth, which is why systematic strategies exist at all. To reach an information ratio of 0.50 you need an information coefficient of 0.25 with four decisions a year, 0.1118 with twenty, 0.05 with a hundred, 0.02236 with five hundred and 0.01118 with two thousand. The first of those numbers is implausible. Measured information coefficients on real equity signals sit around 0.02 to 0.12, so a manager making four calls a year and claiming an information ratio of 0.50 is claiming a level of per-decision skill that nobody has ever demonstrated.
What are the assumptions of the fundamental law?
The law is an approximation and its assumptions matter. It assumes the forecasts are independent, that each carries the same information coefficient, that the manager implements the optimal portfolio without constraints, and that the alphas are correctly scaled to the skill level. The transfer coefficient and the breadth correction remove the third and the first assumption respectively, and the two removals together typically cut a claimed information ratio by more than half.