The Fundamental Law of Active Management (and why breadth is hard to buy)
How does forecast skill translate into active return when bets are correlated?
Estimate forecast accuracy and the number of genuinely independent bets, then account for portfolio constraints through the transfer coefficient. Repeating correlated signals does not increase effective breadth by the raw count, and estimated information coefficients need uncertainty bounds.
Transfer coefficient, empirical information coefficient and independent breadth are assumptions behind the adapted expression.
Evidence map
| Aspect | Finding |
|---|---|
| Idea | Performance decomposes into skill and diversification. |
| Key result / formula | Grinold's law: IR ≈ IC · √BR, where IC is the information coefficient (the correlation between forecast and realised return) and BR the number of independent bets per year. |
| Practical rule | Before building an ensemble, ask which factor is missing. |
Idea
The information ratio a manager can expect grows with the accuracy of each forecast and with the square root of the number of independent forecasts made — which is why adding genuinely new bets helps, and why adding more variants of the same bet does not.
Key result / formula
Clarke, de Silva & Thorley (2002) add the transfer coefficient TC — the correlation between the ideal and the implemented portfolio — giving IR ≈ TC · IC · √BR; constraints such as long-only or turnover caps cut TC and can dominate everything else. The critical term is independent. Breadth counts independent bets, not positions and not signals: k formulas computed from one underlying source share their errors, so their combination has a breadth close to one however many there are. Averaging many correlated weak signals therefore does not deliver the √k improvement the formula seems to promise — and when the individual ICs are indistinguishable from zero, combining them cannot manufacture one, since the product IC · √BR is zero for any BR.
Practical rule
If IC is near zero, more breadth is worthless and the work belongs on the forecast. If IC is real but small, breadth is the lever, and it must come from new sources of information or genuinely different markets; more transformations of the same series add none. Measure the realised correlation between signals, since a count of them overstates breadth: correlated bets are not independent bets, so breadth falls as correlation rises. Under the strong simplification of equicorrelated bets there are closed forms for how far it falls, but they are approximations of a special case — do not carry a single formula across to a real book. Measure TC explicitly whenever constraints bind.
Source
Grinold, "The Fundamental Law of Active Management", Journal of Portfolio Management 15(3), 1989; Grinold & Kahn, Active Portfolio Management, 2nd ed., McGraw-Hill 2000; Clarke, de Silva & Thorley, "Portfolio Constraints and the Fundamental Law of Active Management", Financial Analysts Journal 58(5), 2002. Primary source