Fundamental Law of Active Management

Why do some skilled managers add little value while others with modest skill add a lot? Grinold and Kahn's answer: value added depends on how good each decision is and how many independent decisions are made.

• Skill is the Information Coefficient (IC) — the correlation between forecasts and outcomes.
• Breadth is the number of independent bets — how many times the skill is put to work.

Two equally skilled managers will not be equally productive if one makes 50 independent forecasts and the other makes 5. The first has more breadth and, by the law, a higher Information Ratio.

Breadth = forecasts per period × number of periods. A manager making quarterly forecasts on 11 industry segments has a breadth of 4 × 11 = 44 in a year.

Two conditions from the workbook:

1. Each forecast must be independent.
2. Forecasts must not be correlated. If a manager says two technology stocks will outperform because the technology sector will outperform, that is one bet, not two. Duplications must be removed.

Because breadth enters under a square root, IR responds to it non-linearly. The workbook's illustration: increasing bets from 50 to 100 (2×) raises IR by √2 = 1.414×. IR responds to IC linearly — double the skill, double the IR.

The law also leaves something out, as the workbook points out: more bets mean more transaction costs, which the simple equation ignores and which can materially reduce net return.

IR = IC × √Breadth

Breadth = number of independent forecasts per period × number of periods

Value added (for a given active risk) = IR × active risk

Illustrative figures. Two PMS managers, each with an IC of 0.08, run ₹200 crore books.

On paper, B is more than three times as productive with exactly the same skill.

Now the two corrections the workbook insists on:

1. Independence. If 30 of B's 50 stocks are really bets on the same banking-sector view, B's genuine breadth is nearer 21 × 12 = 252, and IR falls to 0.08 × 15.87 = 1.27.
2. Costs. If B's 600 decisions each mean a trade of about ₹20 lakh at an all-in cost of 0.4%, that is ₹8,000 per trade and ₹48 lakh a year — 0.24% of the book — before a rupee of alpha is counted.

The law explains why breadth matters; it does not say that trading more is free.

Chapter 18 (Equity Portfolio Management Strategies), section 18.3, presents the law, its two inputs (IC in 18.3.1, breadth in 18.3.2) and the 50-to-100 bets illustration. The Chapter 18 caselets ask for breadth and IR from a hit-rate table (IR = 0.575 × √80 = 5.14). Chapter 20 then measures the same Information Ratio from realised returns.

• Square root of breadth, not breadth. Doubling bets raises IR 1.414×, not 2×.
• Correlated bets count once. Two tech stocks picked on one sector view are one bet.
• The workbook's caselet labels 8.94 as "breadth". By the chapter's own definition breadth is the count of forecasts (80); 8.94 is √80. The IR answer (5.14) is unaffected.
• The workbook's illustration uses IC = 4 (IR 28.28 → 40). The multiplication is right; the IC value is outside the −1 to +1 range the chapter defines.
• Transaction costs are not in the equation — the workbook flags this as an over-simplification.
• The law is ex ante (a prediction of productivity). The Chapter 20 IR formula is ex post (measured from results).