Ridge Factor Risk Decomposition

The Risk Decomposition sub-pill answers: "how much of my portfolio's return variance is explained by factor exposures, and how much remains as idiosyncratic risk?" The engine fits a ridge regression of portfolio returns onto factor returns and reports an R² plus per-factor variance contributions. This article explains the method, why ridge over OLS, and how to interpret the output.

The regression

For T rebalance periods, with portfolio return r_p,t and factor returns r_i,t for i in {value, quality, momentum, growth}:

The model is estimated via ridge regression with a small penalty (default α_ridge = 0.1) on the β coefficients:

Then R² is the standard variance-explained ratio:

Why ridge instead of OLS

OLS (ordinary least squares) is fine when factor returns are orthogonal and the sample is large. In factor backtests neither is true:

• Factors are correlated. Value and quality have correlation ~0.3; momentum and growth even higher in some periods. OLS with correlated regressors produces unstable, hard-to-interpret coefficients.
• Small T. Most backtests have 20–60 rebalance periods. With 4 regressors plus intercept, degrees of freedom are tight; OLS overfits.

Ridge (Hoerl & Kennard 1970) shrinks the β coefficients toward zero, exchanging a small amount of bias for substantially lower variance. The result is a more stable decomposition. The small α_ridge = 0.1 default means the shrinkage is gentle — only the noisiest coefficients are pulled materially toward zero.

R² interpretation

Per-factor variance contribution

The sub-pill decomposes the explained variance per factor. With factor returns r_i and coefficients β_i, the contribution to variance is:

The sub-pill reports a normalised per-factor share so all contributions sum to the R². The largest-contribution factor is the dominant risk driver — the factor whose performance most determines the strategy's.

Cross-checking with attribution

Risk Decomposition and Factor Attribution use the same inputs but answer different questions:

• Attribution: "Which factor contributed the most to my realised return?"
• Risk Decomposition: "Which factor contributed the most to my realised return variance?"

They usually agree, but not always. A factor that delivered a small steady return over many periods can be the top attribution contributor without being the top variance contributor. Conversely, a factor that delivered one large swing dominates variance without dominating cumulative return.

Caveats

• Small T inflates R². With T = 20 and 4 regressors, R² can hit 0.4 by chance. Always compare against an adjusted R² if making strong claims.
• The factor list is closed. Variance not explained by value/quality/momentum/growth becomes "specific" — even if it's really a sector tilt or country tilt. Add candidate factors to the model and re-run if specific is unexpectedly high.
• Linear-only. Ridge regression assumes the portfolio return is a linear function of factor returns. If the strategy has option-like or non-linear exposure (e.g., stop-losses), the linear fit will misattribute.

When R² is the wrong number to chase

High R² sounds good but means the strategy has surrendered its alpha to factor exposure. Most quant strategies want moderate R² (0.30–0.50): the factor exposures explain the strategy's baseline behaviour, while specific return provides the differentiator vs. just buying a factor ETF.