Set out Grinold's fundamental law of active management with its assumptions and limits, and use it to work out the highest information ratio that can be attained.
The idea
Under the ideal assumptions of independent bets and full implementation, Grinold's fundamental law of active management says the highest attainable information ratio, written IR, equals the information coefficient, IC, times the square root of breadth, BR: IR = IC × √BR. The information coefficient is the correlation between forecasts and realised returns, a measure of skill. Breadth is the number of independent bets per year. Skill and breadth trade off, but with diminishing returns: doubling the information ratio at the same skill needs four times as many independent bets. Expected alpha is then the information ratio times the tracking error, written TE, which is the active risk taken: Alpha = IR × TE. The law assumes bets are independent, skill is the same on every bet, forecasts are used optimally and there are no constraints. A transfer coefficient, TC, below one, as with a long-only limit, scales the result down to roughly TC times IC times the square root of BR. Limits include correlated signals, which shrink effective breadth, a skill that is hard to estimate, and costs. The usual error is counting correlated positions as independent bets.
Formulas
Fundamental law of active management
IR = IC × √BR
- IC
- the information coefficient, the correlation between forecasts and realised returns
- BR
- breadth, the number of independent bets a year
Holds for independent bets, equal skill and no constraints. Correlated forecasts shrink effective breadth, so counting every position overstates the ratio.
Expected alpha from the information ratio
α = IR × TE
- IR
- the information ratio
- TE
- the tracking error, the active risk taken
The information ratio is set by skill and breadth; tracking error sets the scale of the alpha.
Fundamental law with a transfer coefficient
IR = TC × IC × √BR
- TC
- the transfer coefficient, below one when constraints such as long-only bind
- IC
- the information coefficient
- BR
- breadth, the number of independent bets a year
Breadth counts each new independent forecast, so stocks followed times revisions a year when information is independent.
A practice question
Grinold's fundamental law of active management states that the highest attainable information ratio is:
- the information coefficient multiplied by breadth.Why not: Drops the square root; breadth has diminishing, not linear, returns.
- the information coefficient divided by the square root of breadth.Why not: Divides instead of multiplies; more independent bets raise the information ratio.
- the square of the information coefficient multiplied by the square root of breadth.Why not: Squares the information coefficient, a skill measure that enters the law linearly.
- the information coefficient multiplied by the square root of breadth. - correct
IR = IC x sqrt(BR). Skill (IC) enters linearly and the number of independent bets (breadth) enters through its square root.
A harder one
A manager has an information coefficient of 0.04 and makes 64 independent bets a year. The portfolio is run with a tracking error of 6%. Using the fundamental law, which is closest to the maximum expected alpha?
- 15.36%Why not: Uses breadth without the square root (IC x BR x TE).
- 1.92% - correct
- 0.32%Why not: Reports the information ratio (0.32) as if it were alpha, forgetting to multiply by tracking error.
- 0.24%Why not: Ignores breadth entirely and multiplies skill by tracking error.
IR = 0.04 x sqrt(64) = 0.32. Alpha = IR x tracking error = 0.32 x 6% = 1.92%.
The hardest tier
A manager's information coefficient is 0.08. She forecasts returns for 100 stocks every year, but the forecasts all come from a few common signals, so the effective number of independent bets is only 25. What is the highest information ratio the fundamental law allows?
- 0.40 - correct
- 0.80Why not: Counts the 100 stocks as 100 independent bets, ignoring the correlation between the forecasts.
- 2.00Why not: Uses breadth of 25 without the square root.
- 0.016Why not: Divides the information coefficient by the square root of breadth instead of multiplying.
Only independent bets count: BR = 25, so IR = 0.08 x sqrt(25) = 0.40. Using 100 would give 0.80, which overstates the information ratio because correlated forecasts add little new information.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Alpha (and the Low-Risk Anomaly)
- Characterise the low-risk anomaly in asset returns and weigh up the case for it.
- Work out alpha, tracking error, the Sharpe ratio and the information ratio from data, after defining each of these four measures.
- Account for how the choice of benchmark shifts measured alpha and set out what makes a benchmark good for measuring it.
- Use a factor regression to build a multifactor benchmark, gauge how sensitive a portfolio is to those factors, and measure alpha relative to the benchmark.
- Show how style analysis can be used to cope with factor exposures that change over time.
- Outline the problems that come up when alphas are measured for strategies with nonlinear payoffs.
- Distinguish the beta anomaly from the volatility anomaly, and examine what the data show for both.
- Lay out the possible explanations offered for the risk anomaly.