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:

R_i = \alpha_i + \sum_{k=1}^{K} \beta_{i,k} f_k + \epsilon_i

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:

\beta_{P,k} = \sum_{i=1}^{N} w_i \cdot \beta_{i,k}

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 $Y$ :

X^{\perp} = X - \frac{\sum_i w_i X_i Y_i}{\sum_i w_i Y_i^2} \cdot 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:

\hat{f} = (X^T W X)^{-1} X^T W R

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:

\Sigma = \lambda \cdot \Sigma_{\text{short}} + (1-\lambda) \cdot \Sigma_{\text{long}}

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:

w_t = \left(\frac{1}{2}\right)^{t / \tau}

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.

Size (SIZE)

Natural log of market capitalization

\text{SIZE}_i = \ln(\text{MarketCap}_i)

Interpretation: Positive exposure means the portfolio tilts toward larger companies.

Size (Non-Linear) (SIZENL)

Captures non-linear size effects, orthogonalized against SIZE

\text{SIZENL}_i = (\ln(\text{MarketCap}_i))^3

Interpretation: Captures mid-cap effects not explained by linear size exposure.

Book-to-Price (BTOP)

Book value divided by market price (value factor)

\text{BTOP}_i = \frac{\text{BookValue}_i}{\text{MarketCap}_i}

Interpretation: High positive exposure indicates a value tilt.

Earnings Yield (EARNYILD)

Trailing 12-month earnings per share divided by price

\text{EARNYILD}_i = \frac{\text{EPS}_{\text{TTM},i}}{\text{Price}_i}

Interpretation: Captures cheapness on an earnings basis. High exposure = cheap stocks.

Dividend Yield (DIVYILD)

Annual dividend per share divided by price

\text{DIVYILD}_i = \frac{\text{DPS}_{\text{annual},i}}{\text{Price}_i}

Interpretation: High exposure indicates income-oriented holdings.

Growth (GROWTH)

5-year earnings growth rate

\text{GROWTH}_i = \left(\frac{\text{EPS}_{t}}{\text{EPS}_{t-5}}\right)^{1/5} - 1

Interpretation: Positive exposure indicates growth stock tilt.

Leverage (LEVERAGE)

Total debt divided by market capitalization

\text{LEVERAGE}_i = \frac{\text{TotalDebt}_i}{\text{MarketCap}_i}

Interpretation: High exposure indicates holdings with higher debt levels.

Beta (BETA)

Sensitivity to market returns over trailing 252 days

\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}

Interpretation: Beta > 1 means more volatile than the market; < 1 means less volatile.

Beta (Non-Linear) (BETANL)

Captures non-linear beta effects, orthogonalized against BETA

\text{BETANL}_i = \beta_i^2

Interpretation: Captures convexity in market sensitivity not explained by linear beta.

Residual Volatility (RESVOL)

Volatility unexplained by market factor

\text{RESVOL}_i = \sqrt{\text{Var}(R_i - \beta_i R_m)}

Interpretation: High exposure indicates holdings with high idiosyncratic risk.

Liquidity (LIQUIDTY)

Average daily dollar volume over trailing 21 days

\text{LIQUIDTY}_i = \frac{1}{21}\sum_{t=1}^{21} \text{Volume}_t \times \text{Price}_t

Interpretation: Higher exposure means more liquid, easily traded positions.

3-Week Momentum (MOM3WKZS)

Z-scored 15-day cumulative return

\text{MOM3WK}_i = \frac{R_{i,[-15,0]} - \mu}{\sigma}

Interpretation: Captures short-term price momentum.

11-Month Momentum (MOM11MNZS)

Z-scored cumulative return from month -12 to month -2

\text{MOM11MN}_i = \frac{R_{i,[-252,-22]} - \mu}{\sigma}

Interpretation: Classic Carhart momentum factor. Positive = recent winners.

5-Day Return (RET5DZS)

Z-scored 5-day return (short-term reversal)

\text{RET5D}_i = \frac{R_{i,[-5,0]} - \mu}{\sigma}

Interpretation: Captures very short-term price movements, often mean-reverting.

Short Interest (SHORT_INTEREST)

Shares sold short as percentage of float

\text{SI}_i = \frac{\text{SharesShort}_i}{\text{FloatShares}_i}

Interpretation: High short interest may indicate bearish sentiment or squeeze potential.

Hedge Fund Ownership (HFOWN)

Percentage of shares held by hedge funds (from 13F)

\text{HFOWN}_i = \frac{\sum_{\text{HF}} \text{Shares}_{\text{HF},i}}{\text{SharesOut}_i}

Interpretation: High exposure indicates positions favored by hedge funds (potential crowding).

Passive Ownership (PASSOWN)

Percentage held by index funds and ETFs

\text{PASSOWN}_i = \frac{\sum_{\text{Passive}} \text{Shares}_{\text{Passive},i}}{\text{SharesOut}_i}

Interpretation: High passive ownership may reduce price discovery efficiency.

Shim Market Cap (SHIMCAPZS)

Market cap z-score for additional size effects

\text{SHIMCAP}_i = \frac{\ln(\text{MarketCap}_i) - \mu}{\sigma}

Interpretation: Captures residual size effects after orthogonalization.

Alpha/Beta Decomposition

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

R_P = \alpha_P + \beta_P \cdot R_M + \epsilon_P

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

Disclaimer

Factor exposures are estimates based on historical data and may not accurately predict future returns. Factor models have known limitations including estimation error, model misspecification, and changing factor dynamics. This documentation is for educational purposes and does not constitute investment advice.