Two managers run the same benchmark. One is right about individual stocks 52% of the time across 400 names; the other is right about the market direction 60% of the time, four times a year. The second sounds far more skilled. The fundamental law says the first will almost certainly add more value.
Active management is judged against a passive alternative, so the raw number that matters is active return, the managed portfolio return minus the benchmark return: . A useful benchmark is representative of the investable opportunity set, replicable at low cost, with weights verifiable in advance and returns available promptly. Float-adjusted capitalization weighting dominates because such indexes self-rebalance and can be held by everyone at once.
That last property has a consequence. If the benchmark spans the whole market, active management is a zero-sum game before costs and negative-sum after fees. For a narrower benchmark it is not zero-sum, because managers can buy assets outside the index.
Common mistakes
- Adding breadth linearly. Breadth enters as . Going from 100 to 400 independent decisions doubles the information ratio, it does not quadruple it.
- Counting correlated bets as independent. One valuation model applied to 400 names is not BR = 400. With pairwise correlation of 0.10 across decisions, effective breadth is closer to 40, and expected value added falls by roughly two-thirds.
- Believing more aggressiveness means more skill. Multiplying every active weight by 2 doubles the 1.20% expected active return to 2.40% and doubles active risk from 4.0% to 8.0%. The information ratio stays at 0.30.
Bottom line
- Value added: active return = portfolio return − benchmark return = sum of active weights times active security returns; active weights sum to zero
- Decomposition: allocation term = active class weights × benchmark class returns; selection term = portfolio class weights × within-class value added
- Sharpe ratio measures total risk-adjusted return; information ratio measures benchmark-relative return per unit of active risk; both come in ex ante and ex post forms
- Invariance: Sharpe ratio is unaffected by cash or leverage; information ratio is unaffected by scaling active weights or by mixing with the benchmark, but is affected by cash
Exam shortcut
Write on the scratch sheet before reading the vignette; three of the four inputs are usually given and the fourth is the answer. If a stem mentions long-only, position caps, sector bounds, or turnover limits, it is signaling the transfer coefficient. If it counts securities, rebalancing frequency, or timing calls per year, it is signaling breadth, and remember to multiply securities by rebalances per year.
The full lesson (about 2,917 words, 19 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- active portfolio management
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