18 — Factor Risk Contribution¶

Free learning pack¶

1. CFA Analysis of Active Portfolio Management
   https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/analysis-active-portfolio-management
2. resources/active_management.md
3. reference/concepts/factor_risk.md, factor_risk_contribution.md

This continues notebook 07's factor model with the same B, F, and specific — now adding portfolio weights to ask how much of the risk is systematic versus specific.

PREDICT¶

A portfolio is concentrated in a single name with high specific risk but low factor exposure. Another is diversified across many names, each with small specific risk but similar aggregate factor exposure. Which one has a bigger specific variance contribution as a share of the total?

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 should sum to the portfolio's total variance under this factor model.

In [ ]:

import numpy as np

B = np.array([
    [1.0, 0.2],
    [0.8,-0.1],
    [0.1, 1.2],
])
F = np.array([[0.04,0.005],[0.005,0.01]])
specific = np.array([0.02,0.025,0.01])
weights = np.array([0.4, 0.35, 0.25])

# MANUAL FIRST:
# compute the portfolio factor exposure e, the factor variance
# contribution, and the specific variance contribution. Then confirm
# they sum to the same total variance you'd get from reconstructing
# Sigma = B F B^T + diag(specific) directly (see notebook 07).
factor_var = None
specific_var = None
print(factor_var, specific_var)

import numpy as np B = np.array([ [1.0, 0.2], [0.8,-0.1], [0.1, 1.2], ]) F = np.array([[0.04,0.005],[0.005,0.01]]) specific = np.array([0.02,0.025,0.01]) weights = np.array([0.4, 0.35, 0.25]) # MANUAL FIRST: # compute the portfolio factor exposure e, the factor variance # contribution, and the specific variance contribution. Then confirm # they sum to the same total variance you'd get from reconstructing # Sigma = B F B^T + diag(specific) directly (see notebook 07). factor_var = None specific_var = None print(factor_var, specific_var)

Experiment¶

Change weights to concentrate entirely in the third asset (which has the largest factor-2 exposure and the smallest specific variance). Watch the factor/specific split shift.

Reference¶

reference/concepts/factor_risk_contribution.md

Promote¶

Use src/pm/factors.py (portfolio_factor_exposure, factor_variance_contribution, specific_variance_contribution) only after your own implementation.

Test¶

pytest tests/test_factors.py

ORAL CHECK¶

Explain to a PM why two portfolios can have identical total volatility but very different factor-vs-specific splits, and why that difference matters for how each should be hedged.

Try /tutor concept factor risk contribution for an adaptive walkthrough.