18-Factor Risk Model Methodology

A Barra USE4-style factor model for decomposing portfolio risk and return into systematic and idiosyncratic components.

Overview

Factor models explain security returns as a linear combination of common factor returns plus an idiosyncratic component. The general form is:

Where:

• $R_i$ is the return of security $i$
• $\alpha_i$ is the security-specific alpha (skill)
• $\beta_{i,k}$ is the exposure of security $i$ to factor $k$
• $f_k$ is the return of factor $k$
• $\epsilon_i$ is the idiosyncratic (stock-specific) return

Portfolio Factor Exposures

Portfolio-level factor exposure is the weighted sum of security-level exposures:

Where $w_i$ is the portfolio weight of security $i$ (position value divided by total portfolio value).

Factor Orthogonalization

To ensure factors capture independent information, we orthogonalize them using the Gram-Schmidt process. Each factor is projected against all preceding factors and the residual becomes the orthogonalized factor.

For a target factor $X$ being orthogonalized against a base factor $Y$:

The weights $w_i$ are typically the square root of market capitalization, following Barra convention. This ensures larger companies have proportionally more influence on the regression.

Orthogonalization Order: The order matters. We follow the Barra USE4 convention: SIZE, SIZENL, BTOP, EARNYILD, DIVYILD, GROWTH, LEVERAGE, BETA, BETANL, RESVOL, LIQUIDTY, MOM3WKZS, MOM11MNZS, RET5DZS, SHORT_INTEREST, HFOWN, PASSOWN, SHIMCAPZS.

Factor Return Computation

Daily factor returns are estimated using weighted cross-sectional regression:

Where $X$ is the matrix of factor exposures, $W$ is a diagonal weight matrix (sqrt market cap), and $R$ is the vector of security returns.

Factor Covariance Matrix

We estimate the factor covariance matrix using an Exponentially Weighted Moving Average (EWMA) with dual half-lives for robustness:

• Short half-life: 32 days (responsive to recent volatility)
• Long half-life: 128 days (stable, long-term structure)
• Blend weight: $\lambda = 0.5$ (equal weight by default)

The EWMA weight for observation $t$ days ago is:

Where $\tau$ is the half-life in days.

The 18 Style Factors

Our model includes the market factor plus 18 style factors. Each factor captures a systematic source of return that has been documented in academic literature and used in practice by institutional investors.

Alpha/Beta Decomposition

Portfolio return can be decomposed into market exposure (beta return) and alpha (skill):

• Beta Return: $\beta_P \cdot R_M$ - Return explained by market exposure
• Alpha: $\alpha_P$ - Return from security selection skill
• Factor Return: Sum of non-market factor contributions
• Specific Return: $\epsilon_P$ - Unexplained (idiosyncratic) return

Data Sources

• Price data: Daily adjusted close prices from market data providers
• Fundamentals: Quarterly financial statements from SEC EDGAR
• Ownership: 13F filings for institutional holdings
• Short interest: FINRA short interest reports

References

• MSCI Barra USE4 Handbook (2011) - Equity Risk Model Methodology
• Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of Financial Economics
• Carhart, M. M. (1997). On persistence in mutual fund performance. Journal of Finance
• Frazzini, A., & Pedersen, L. H. (2014). Betting against beta. Journal of Financial Economics