Fundamental Law of Active Management Formula
The fundamental law of active management states that a manager's information ratio equals their skill per bet (the information coefficient) times the square root of the number of independent bets (breadth). The refined version multiplies by a transfer coefficient capturing how much of the theoretical skill survives real-world constraints. It explains why breadth, not just skill, drives risk-adjusted performance.
Formula
Copy the exact expression or work through it step by step below.
IR = IC x sqrt(breadth)
Refined: IR = TC x IC x sqrt(breadth) Variables
IR
Information ratio
Annualized active return over tracking error, the quantity the law predicts. A higher IR means more consistent benchmark-relative outperformance.
IC
Information coefficient
Correlation between forecasts and realized returns, the manager's skill on a single bet. Realistic ICs are small (0.02 to 0.10); the law shows even a small IC compounds through breadth.
breadth
Breadth
Number of independent bets per year. Doubling breadth raises IR by only the square root of 2, so independence matters as much as count: correlated bets inflate apparent breadth without raising true IR.
TC
Transfer coefficient
Correlation between the ideal active positions the forecasts imply and the actual positions taken after constraints (long-only limits, turnover caps, risk budgets). It is the fraction of paper skill that reaches the live portfolio, typically well below 1.
Step By Step
- 1
Estimate the information coefficient, the correlation between forecasts and outcomes.
Backtested IC of 0.06.
- 2
Count the number of genuinely independent bets per year.
Rebalancing 100 names monthly, with low cross-sectional correlation, gives breadth near 200 per year.
- 3
Multiply IC by the square root of breadth for the unconstrained information ratio.
0.06 x sqrt(200) = 0.06 x 14.142 = 0.849.
- 4
Multiply by the transfer coefficient to reflect implementation losses.
With TC = 0.6 (long-only and turnover limits), IR = 0.6 x 0.849 = 0.509.
Worked Example
Quant equity strategy, constrained long-only
Information coefficient (IC)
0.06
Breadth
200 independent bets/year
Transfer coefficient (TC)
0.6
Unconstrained IR = IC x sqrt(breadth) = 0.06 x sqrt(200) = 0.06 x 14.142 = 0.8485. Refined IR = TC x IC x sqrt(breadth) = 0.6 x 0.8485 = 0.5091.
A theoretical information ratio of 0.85 falls to 0.51 once the 0.6 transfer coefficient prices in long-only and turnover constraints. This is why high-IC strategies can disappoint live: roughly 40% of the paper skill never reaches the portfolio. The law's practical message is to raise breadth and transfer coefficient, not just chase a higher IC.
Common Variations
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Sources & References
- The Fundamental Law of Active Management — Richard C. Grinold, Journal of Portfolio Management (1989)
- Breadth, Skill, and Time — Roger Clarke, Harindra de Silva, Steven Thorley, Journal of Portfolio Management (2005)
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Information Coefficient Formula
The information coefficient formula: correlation between forecast and realized returns. How to measure quant forecasting skill, with a worked example.
Information Ratio Formula
The information ratio formula: active return over a benchmark divided by tracking error. Measures the consistency of excess return, with an example.
Tracking Error Formula
The tracking error formula: the standard deviation of a portfolio's active return versus its benchmark. How tightly a fund tracks its index.
Alpha
Alpha as risk-adjusted excess return: definition, the beta-adjustment math, and why most claimed alpha disappears once you adjust for the right factors.