07 — Factor Risk Model¶

Learning objectives¶

Reconstruct a covariance matrix from a factor model by hand; split portfolio variance into factor (systematic) and specific (idiosyncratic) pieces and confirm they sum to the total; understand why factor models scale to large universes where a full covariance matrix doesn't.

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

Do not search for more material until these are insufficient.

PREDICT¶

A full covariance matrix for n assets needs n*(n+1)/2 numbers. A factor model with k factors needs n*k exposures, k*(k+1)/2 factor covariances, and n specific variances. For n=3, k=2 (the example below), which needs more numbers? What about at n=500, k=10 — a realistic large-universe risk model?

Model¶

r = Bf + epsilon

Sigma = B F B^T + D

where B is exposures (assets x factors), F is the factor covariance matrix, and D = diag(specific variances).

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])

# MANUAL FIRST: reconstruct Sigma.
# Step 1: B @ F  (a 3x2 intermediate)
BF = None
# Step 2: (B @ F) @ B.T, then add diag(specific)
Sigma = None
print(Sigma)

# CHECK (uncomment after your attempt):
# assert np.allclose(Sigma, Sigma.T), "a covariance matrix must be symmetric"
# assert np.allclose(np.diag(Sigma), [0.0624, 0.0499, 0.026], atol=1e-3)

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]) # MANUAL FIRST: reconstruct Sigma. # Step 1: B @ F (a 3x2 intermediate) BF = None # Step 2: (B @ F) @ B.T, then add diag(specific) Sigma = None print(Sigma) # CHECK (uncomment after your attempt): # assert np.allclose(Sigma, Sigma.T), "a covariance matrix must be symmetric" # assert np.allclose(np.diag(Sigma), [0.0624, 0.0499, 0.026], atol=1e-3)

Parameter count, for real¶

Confirm your PREDICT answer with the actual formulas: n*(n+1)//2 for the full covariance vs. n*k + k*(k+1)//2 + n for the factor model, at both the toy size above and a realistic institutional universe.

In [ ]:

# MANUAL FIRST:
def full_covariance_params(n):
    return None

def factor_model_params(n, k):
    return None

print("n=3, k=2  -> full:", full_covariance_params(3), " factor:", factor_model_params(3, 2))
print("n=500, k=10 -> full:", full_covariance_params(500), " factor:", factor_model_params(500, 10))

# CHECK (uncomment after your attempt):
# assert full_covariance_params(3) == 6 and factor_model_params(3, 2) == 12
# assert full_covariance_params(500) == 125250 and factor_model_params(500, 10) == 5555

# MANUAL FIRST: def full_covariance_params(n): return None def factor_model_params(n, k): return None print("n=3, k=2 -> full:", full_covariance_params(3), " factor:", factor_model_params(3, 2)) print("n=500, k=10 -> full:", full_covariance_params(500), " factor:", factor_model_params(500, 10)) # CHECK (uncomment after your attempt): # assert full_covariance_params(3) == 6 and factor_model_params(3, 2) == 12 # assert full_covariance_params(500) == 125250 and factor_model_params(500, 10) == 5555

Interpretation¶

At n=3, the factor model actually needs more numbers than the full covariance — factor models don't pay off at toy scale. At n=500, the factor model needs about 4% of the full covariance's parameter count. That gap is the entire reason factor models exist: not because they're mathematically prettier, but because estimating 125,250 independent covariance entries from noisy historical data is far less reliable than estimating 5,555.

Split variance into factor vs. specific¶

MANUAL FIRST: for a portfolio w = [0.5, 0.3, 0.2], compute the factor variance contribution (exposure' F exposure, where exposure = w' B) and the specific variance contribution (sum(w_i^2 * specific_i)). Confirm they sum to total portfolio variance.

In [ ]:

from pm.risk import portfolio_variance

w = np.array([0.5, 0.3, 0.2])

# MANUAL FIRST:
exposure = None            # w @ B
factor_variance = None     # exposure @ F @ exposure
specific_variance = None   # sum(w**2 * specific)
print("exposure:", exposure)
print("factor var:", factor_variance, " specific var:", specific_variance)

total_variance = portfolio_variance(w, Sigma)
print("sum:", factor_variance + specific_variance, " total:", total_variance)

# CHECK (uncomment after your attempt):
# assert np.allclose(exposure, [0.76, 0.31], atol=1e-3)
# assert np.isclose(factor_variance + specific_variance, total_variance)
# assert factor_variance / total_variance > 0.7, "most of this portfolio's risk should be factor-driven, not stock-specific"

from pm.risk import portfolio_variance w = np.array([0.5, 0.3, 0.2]) # MANUAL FIRST: exposure = None # w @ B factor_variance = None # exposure @ F @ exposure specific_variance = None # sum(w**2 * specific) print("exposure:", exposure) print("factor var:", factor_variance, " specific var:", specific_variance) total_variance = portfolio_variance(w, Sigma) print("sum:", factor_variance + specific_variance, " total:", total_variance) # CHECK (uncomment after your attempt): # assert np.allclose(exposure, [0.76, 0.31], atol=1e-3) # assert np.isclose(factor_variance + specific_variance, total_variance) # assert factor_variance / total_variance > 0.7, "most of this portfolio's risk should be factor-driven, not stock-specific"

Reference¶

reference/concepts/factor_risk.md reference/concepts/factor_risk_contribution.md

Promote¶

Compare with src/pm/factors.py (factor_model_covariance, portfolio_factor_exposure, factor_variance_contribution, specific_variance_contribution).

Test¶

pytest tests/test_factors.py

ORAL CHECK¶

Explain to a PM what "common" and "specific" risk mean using this portfolio's actual 70/30-ish split, and why a factor model is the only practical way to build a risk model for a 500-name universe.

Try /tutor factor risk for an adaptive walkthrough. This notebook uses the same B, F, and specific as notebook 18 (factor risk contribution) — that notebook picks up where this one leaves off.