Quantitative Finance
Fundamental Law of Active Management Formula (Grinold IR, IC, Breadth & TC)
Calculate ex-ante Information Ratio IR = IC * SQRT(Breadth) * TC and expected active alpha E[R_A] = IR * Tracking_Error under the Clarke-de Silva-Thorley extended Fundamental Law.
Link signal skill (IC), independent bets (Breadth), and portfolio constraints (TC)!
Grinold’s (1989) Fundamental Law of Active Management states that a portfolio’s attainable Information Ratio grows linearly with forecasting skill (Information Coefficient IC = rank correlation between forecasts and realized returns) and with the square root of independent annual bets (Breadth BR). Clarke, de Silva & Thorley (2002) added the Transfer Coefficient TC in [0, 1] to capture real-world long-only, turnover, and sector-cap constraints: IR = IC * SQRT(BR) * TC!
Multiplying ex-ante IR by target active risk (Tracking Error TE) yields Expected Annual Active Return E[R_A] = IR * TE—letting quant PMs compare a high-IC concentrated strategy against a modest-IC high-breadth systematic strategy on equal footing.
| Strategy Profile | Skill IC | Breadth BR / TC | Ex-Ante IR & Alpha (TE = 4.5%) |
|---|---|---|---|
| Constrained Quant Equity | 0.06 | 225 bets / TC = 0.75 | IR = 0.6750 | Alpha = +3.0375% |
| Unconstrained Long-Short | 0.06 | 225 bets / TC = 1.00 | IR = 0.9000 | Alpha = +4.0500% |
| Concentrated Fundamental | 0.12 | 25 bets / TC = 0.85 | IR = 0.5100 | Alpha = +2.2950% |
Ex-Ante Information Ratio (IR = IC * SQRT(BR) * TC)
Evaluates constrained ex-ante IR from IC (B2), annual independent bets B3, and Transfer Coefficient B4 in [0, 1].
=IF(OR(ABS(B2) > 1, B3 <= 0, B4 < 0, B4 > 1, B5 <= 0), "Invalid Law Inputs", ROUND(B2 * SQRT(B3) * B4, 4))Expected Annual Active Alpha (% E[R_A] = IR * TE)
Scales ex-ante Information Ratio by annualized percentage Tracking Error B5.
=IF(OR(ABS(B2) > 1, B3 <= 0, B4 < 0, B4 > 1, B5 <= 0), "Invalid Law Inputs", ROUND((B2 * SQRT(B3) * B4) * B5, 4))Try the example
Use the original values or change them to test your understanding.