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Fundamental Law of Active Management

One-line definition

Grinold's equation linking expected skill (information ratio) to raw forecasting ability (IC), how many independent bets that ability gets applied to (breadth), and how well real-world constraints let those bets reach the portfolio (transfer coefficient).

Formula

IR ~= IC * sqrt(BR) * TC

  • IC: information coefficient — skill
  • BR: breadth — number of independent bets per year (use effective_breadth(n_bets, average_correlation) when bets aren't actually independent — correlated bets are worth less than their raw count)
  • TC: transfer coefficient — how much of the unconstrained-optimal active weights actually get implemented, 1.0 = no constraint cost (the original Grinold form of the law omits TC entirely, i.e. assumes TC=1)

Why PMs care

It explains why a modestly-skilled manager (IC around 0.05, not obviously impressive on its own) can still produce a strong track record by applying that skill across many independent opportunities — and, symmetrically, why a manager with genuinely excellent IC can still post a mediocre IR if constraints (long-only, position limits, turnover caps) prevent the signal from actually reaching the portfolio. It's the standard framework for decomposing "why did this fund's IR change" into skill vs. opportunity-set vs. implementation.

Worked example

IC = 0.05, BR = 100 independent bets/year, TC = 1.0 (unconstrained): IR ~= 0.05 * sqrt(100) * 1.0 = 0.5. Halve the transfer coefficient to 0.5 (heavier constraints) and expected IR halves to 0.25 — the law is linear in TC, so constraint cost shows up directly and proportionally.

Effective breadth: correlated bets are worth less

effective_breadth(n_bets, average_correlation) implements BR_eff = n / (1 + (n-1)*rho): at rho=0 (truly independent bets) it equals n unchanged; at rho=1 (every "independent" bet is actually the same bet) it collapses to 1, no matter how large n is. This is why adding more names to a book doesn't help if they're all really the same macro or factor bet in disguise.

Common mistakes

  • treating IC and breadth as independent levers — in practice, relaxing a signal's threshold to catch more "bets" (raising breadth) usually dilutes IC at the same time, since the added bets are lower-conviction
  • forgetting the transfer coefficient term entirely and comparing a constrained real-world portfolio's IR against the unconstrained IC * sqrt(BR) prediction — the gap between them is the cost of the constraints, not a modeling error
  • treating the transfer coefficient itself as scale-sensitive: because transfer_coefficient is a correlation, a portfolio that captures every bet's direction correctly but only half its magnitude (e.g. a uniform leverage cap) still scores TC = 1.0 — correlation doesn't see the shrinkage. TC measures whether constraints changed which bets you took, not how big they ended up.

Limitations

The law is an approximation (~=, not =) that assumes bets are made independently over time and across assets, forecasts are unbiased, and risk is proportional across the portfolio. Real portfolios violate all three to some degree, which is exactly why realized IR (see information ratio) and the law's predicted IR often diverge.

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