03 — Marginal and Component Risk¶

Learning objectives¶

Compute marginal risk contribution (MCR) and component risk contribution (CR) by hand on a 2-asset example, then on a realistic 4-asset portfolio; confirm the identity sum(CR) = sigma_p; explain why capital weight and risk weight can differ sharply.

Free learning pack¶

1. MIT OCW Portfolio Theory I–III https://ocw.mit.edu/courses/15-401-finance-theory-i-fall-2008/pages/video-lectures-and-slides/portfolio-theory/
2. CFA Portfolio Risk and Return Part I https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/portfolio-risk-return-part-1
3. resources/portfolio_foundations.md
4. reference/concepts/risk_contribution.md

Do not search for more material until these are insufficient.

PREDICT¶

All capital weights in a 4-asset portfolio are 25%. Must all risk shares also be 25%? If one asset has much higher volatility and is positively correlated with everything else, will its risk share be above or below 25%?

Formula¶

MCR_i = (Sigma w)_i / sigma_p

CR_i = w_i * MCR_i

Invariant: sum(CR_i) = sigma_p

HAND CALCULATION¶

Two uncorrelated assets, equal 50/50 weights, Sigma = [[0.04, 0], [0, 0.01]] (asset 1 variance 0.04, asset 2 variance 0.01, zero covariance).

sigma_p = sqrt(w' Sigma w), MCR = (Sigma @ w) / sigma_p, CR = w * MCR. Work this out by hand before the next cell — with zero correlation, Sigma @ w is just [w1*var1, w2*var2].

In [ ]:

import numpy as np

w2 = np.array([0.5, 0.5])
cov2 = np.array([[0.04, 0.0], [0.0, 0.01]])

# MANUAL FIRST:
sigma_p_2 = None
mcr_2 = None
cr_2 = None
print(sigma_p_2, mcr_2, cr_2)

# CHECK (uncomment after your attempt):
# assert np.isclose(sigma_p_2, 0.1118, atol=1e-3)
# assert np.allclose(mcr_2, [0.1789, 0.0447], atol=1e-3)
# assert np.allclose(cr_2, [0.0894, 0.0224], atol=1e-3)
# assert np.isclose(cr_2.sum(), sigma_p_2)

import numpy as np w2 = np.array([0.5, 0.5]) cov2 = np.array([[0.04, 0.0], [0.0, 0.01]]) # MANUAL FIRST: sigma_p_2 = None mcr_2 = None cr_2 = None print(sigma_p_2, mcr_2, cr_2) # CHECK (uncomment after your attempt): # assert np.isclose(sigma_p_2, 0.1118, atol=1e-3) # assert np.allclose(mcr_2, [0.1789, 0.0447], atol=1e-3) # assert np.allclose(cr_2, [0.0894, 0.0224], atol=1e-3) # assert np.isclose(cr_2.sum(), sigma_p_2)

Now the realistic case¶

Four assets, unequal capital weights all at 25%, unequal volatilities, and real correlation (including one negative pair).

In [ ]:

weights = np.array([0.25, 0.25, 0.25, 0.25])
vols = np.array([0.18, 0.06, 0.10, 0.15])
corr = np.array([
    [1.0, -0.2, 0.6, 0.2],
    [-0.2, 1.0, 0.1, 0.0],
    [0.6, 0.1, 1.0, 0.2],
    [0.2, 0.0, 0.2, 1.0],
])
cov = np.outer(vols, vols) * corr

# MANUAL FIRST:
# calculate sigma_p, MCR, CR for this 4-asset portfolio.
sigma_p = None
mcr = None
cr = None
risk_share = None  # cr / sigma_p, for comparing against the 25% capital weights
print("sigma_p:", sigma_p)
print("risk share:", risk_share)

# CHECK (uncomment after your attempt):
# assert np.isclose(sigma_p, 0.0805, atol=1e-3)
# assert np.isclose(cr.sum(), sigma_p)
# assert risk_share[0] > 0.25, "asset 0 (highest vol, positively correlated with most others) should carry MORE than its 25% capital weight in risk"
# assert risk_share[1] < 0.25, "asset 1 (lowest vol, negatively correlated with asset 0) should carry LESS"

weights = np.array([0.25, 0.25, 0.25, 0.25]) vols = np.array([0.18, 0.06, 0.10, 0.15]) corr = np.array([ [1.0, -0.2, 0.6, 0.2], [-0.2, 1.0, 0.1, 0.0], [0.6, 0.1, 1.0, 0.2], [0.2, 0.0, 0.2, 1.0], ]) cov = np.outer(vols, vols) * corr # MANUAL FIRST: # calculate sigma_p, MCR, CR for this 4-asset portfolio. sigma_p = None mcr = None cr = None risk_share = None # cr / sigma_p, for comparing against the 25% capital weights print("sigma_p:", sigma_p) print("risk share:", risk_share) # CHECK (uncomment after your attempt): # assert np.isclose(sigma_p, 0.0805, atol=1e-3) # assert np.isclose(cr.sum(), sigma_p) # assert risk_share[0] > 0.25, "asset 0 (highest vol, positively correlated with most others) should carry MORE than its 25% capital weight in risk" # assert risk_share[1] < 0.25, "asset 1 (lowest vol, negatively correlated with asset 0) should carry LESS"

Reference¶

reference/concepts/risk_contribution.md

Promote¶

Use src/pm/risk.py (portfolio_volatility, marginal_risk_contribution, component_risk_contribution) only after your own implementation.

Test¶

pytest tests/test_risk.py

ORAL CHECK¶

Explain capital weight vs. risk weight to a PM without using matrix notation — use the actual risk shares you just computed as your example.

Try /tutor risk contribution for an adaptive walkthrough.