Factor Risk Contribution¶

One-line definition¶

How much of total portfolio variance comes from factor (systematic) exposure versus specific (idiosyncratic) risk.

Formula¶

Portfolio factor exposure: e = w^T B

Factor variance contribution: e^T F e

Specific variance contribution: sum(w_i^2 * d_i)

These two sum to total portfolio variance under the factor model.

Why PMs care¶

Two portfolios with identical total volatility can have very different splits — one dominated by factor bets, the other by idiosyncratic, single-name risk. That split changes how you'd hedge or diversify it.

Code recipe¶

def portfolio_factor_exposure(weights, exposures):
    return np.asarray(weights, float) @ np.asarray(exposures, float)

def factor_variance_contribution(weights, exposures, factor_covariance):
    e = portfolio_factor_exposure(weights, exposures)
    F = np.asarray(factor_covariance, float)
    return float(e @ F @ e)

def specific_variance_contribution(weights, specific_variance):
    w = np.asarray(weights, float)
    return float(np.sum(w**2 * np.asarray(specific_variance, float)))

Common mistakes¶

forgetting the two contributions only sum to total variance under the same exposures/factor_covariance/specific_variance used to build the portfolio's covariance matrix in the first place
comparing factor exposure across portfolios without checking they share the same factor definitions