How do you measure whether an investment process is any good — and how good can it be, given the skill of its signals? This note builds the measurement theory: the Sharpe and Information Ratios at the strategy level, the Information Coefficient at the signal level, and the chain of identities connecting them — culminating in the Fundamental Law of Active Management, , the single most quoted equation in active management. Two ideas deserve special attention: the reinterpretation of IR as a t-statistic (performance measurement is hypothesis testing), and the exact decomposition of realised alpha into five multiplicative components, which tells you precisely which levers a manager can pull.
Learning objectives. By the end of this note you should be able to:
- define SR and IR, apply the institutional yardsticks, and convert IR to a t-statistic;
- define the single-period cross-sectional IC and test whether a factor's mean IC is positive;
- derive the alpha decomposition and interpret all five components;
- derive and, with the sampling-noise assumption, the Fundamental Law;
- state the Law's caveats and its constrained extension .
In this lesson
- 3.1 Strategy-level ratios
- 3.2 Signal-level skill: the information coefficient
- 3.3 Decomposing portfolio alpha
- 3.4 Ex ante IR and the Fundamental Law
The full lesson, with worked examples and exercise solutions, is for members.
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