Multi-Index and Mixed Models

Multi-index models extend the single-index framework to $L > 1$ common factors — market, industry groups, macro variables, fundamental characteristics — preserving the tractable rank-$(L+\text{diagonal})$ covariance structure while capturing patterns of co-movement the market alone cannot explain. This note covers the three main families:

1. Industry index models — adding sector indexes alongside the market.
2. Average-correlation (constant-correlation) models — replacing the structural approach with a smoothed historical estimate.
3. Fundamental multi-index models — using firm characteristics (size, book-to-market) or macro factors as the indexes.

• write down the general multi-index model and state its key assumption;
• derive the formulas for expected return, variance, and covariance under a multi-index structure, and count the inputs required;
• describe the industry index model and state the covariance formulas for same-industry vs cross-industry pairs;
• explain the average-correlation model and interpret its empirical superiority over single-index models as a forecasting tool;
• describe the mixed-model approach and the Rosenberg (BARRA) methodology;
• state the Fama-French three-factor model structure and identify what SMB and HML capture.

1. The general multi-index model
2. Industry index models
3. Average-correlation models
4. Mixed models (Rosenberg/BARRA)
5. Fundamental multi-index models