Multi-factor models and active management

**Fama-French 3-factor**: R – Rf = α + β_MKT(MKT) + β_SMB(SMB) + β_HML(HML) + ε MKT = market risk premium; SMB = small-minus-big; HML = high-book-to-market minus low (value). FF 5-factor adds: RMW (profitability) and CMA (investment). **Fundamental Law of Active Management**: IR = IC × √Breadth Where: - IR = information ratio = α / tracking error. - IC = information coefficient = correlation(forecast, realised return). - Breadth = number of independent bets per year. Implication: small IC × high breadth can match high IC × low breadth. Both pay.

Tracking error (TE) = σ(R_p – R_b). Active risk. Active return / TE = IR. Above 0.5 strong, 1.0 elite. Sources of active return: - **Asset allocation**: deviation from benchmark in asset classes. - **Security selection**: stock picking within asset class. - **Currency**: for international portfolios. - **Timing**: tactical asset allocation. Active risk decomposition: Brinson-Hood-Beebower attribution. Macro factor models: GDP, inflation, credit spread, term spread. BARRA-style: combines fundamental + statistical factors.

L2 trap: assuming bets are independent. In practice, "20 sector views" may be 5 macro factor views in disguise. Effective breadth often << headline count. Transfer coefficient (TC): how well forecasts translate into portfolio positions. Constraints (long-only, position limits) reduce TC. Full formula: IR = TC × IC × √breadth. For true active management: maximise all three. Most managers fail because (a) IC is hard to sustain (b) breadth is overcounted (c) constraints crush TC.

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