27 — Risk Parity and Scenario-Robust Optimization¶

Free learning pack¶

1. MIT OCW Portfolio Theory I–III
2. reference/concepts/risk_parity.md, scenario_robust_optimization.md

Do not search for more material until these are insufficient.

PREDICT¶

Asset A has volatility 20%, Asset B has volatility 10%, uncorrelated. A risk-parity allocation gives each equal risk contribution. Will Asset A or Asset B get the larger capital weight?

Formula¶

Solved via: minimize 0.5*w'*Sigma*w - sum(log(w_i)), then renormalize w to sum to 1.

In [ ]:

import numpy as np
import cvxpy as cp
from pm.risk import component_risk_contribution

Sigma = np.array([[0.04, 0.0], [0.0, 0.01]])  # 20% and 10% vol, uncorrelated

# MANUAL FIRST:
# set up and solve the convex risk-parity problem with cvxpy, then
# renormalize the raw weights to sum to 1. Confirm with
# component_risk_contribution that both assets end up with equal risk
# contribution despite unequal weights.
w_rp = None
print(w_rp)

import numpy as np import cvxpy as cp from pm.risk import component_risk_contribution Sigma = np.array([[0.04, 0.0], [0.0, 0.01]]) # 20% and 10% vol, uncorrelated # MANUAL FIRST: # set up and solve the convex risk-parity problem with cvxpy, then # renormalize the raw weights to sum to 1. Confirm with # component_risk_contribution that both assets end up with equal risk # contribution despite unequal weights. w_rp = None print(w_rp)

PREDICT (scenario-robust)¶

Two assets. Scenario A: Asset 1 +10%, Asset 2 -5%. Scenario B: Asset 1 -5%, Asset 2 +10%. By symmetry, what weights maximize the worst-case return across both scenarios?

Formula¶

maximize t subject to R @ w >= t, sum(w) = 1, w >= 0

In [ ]:

R = np.array([
    [0.10, -0.05],
    [-0.05, 0.10],
])

# MANUAL FIRST:
# set up and solve the minimax scenario-robust problem with cvxpy.
# Confirm both scenarios bind at the same worst-case return t at the
# optimal weights (R @ w should give two equal values).
w_robust = None
print(w_robust)

R = np.array([ [0.10, -0.05], [-0.05, 0.10], ]) # MANUAL FIRST: # set up and solve the minimax scenario-robust problem with cvxpy. # Confirm both scenarios bind at the same worst-case return t at the # optimal weights (R @ w should give two equal values). w_robust = None print(w_robust)

Not implemented here: hierarchical risk parity, regime-aware allocation¶

Both need machinery beyond this repo's scope (clustering + recursive bisection; regime detection). Read reference/concepts/hierarchical_risk_parity.md and reference/concepts/regime_aware_allocation.md for what they are and why.

Experiment¶

Break the symmetry in the scenario-robust example (make Scenario A more extreme: Asset 1 +20%, Asset 2 -5%). Does the optimal weight shift toward or away from Asset 1? Why might that feel counterintuitive at first?

Reference¶

reference/concepts/risk_parity.md
reference/concepts/scenario_robust_optimization.md

Promote¶

Use src/pm/robust.py (risk_parity_weights, scenario_robust_weights) only after your own implementation.

Test¶

pytest tests/test_robust.py

ORAL CHECK¶

Explain to a PM the difference between risk parity (equalizing risk contribution under one covariance estimate) and scenario-robust optimization (maximizing the worst case across several discrete scenarios) - when would you reach for each?

Try /tutor portfolio-construction risk parity for an adaptive walkthrough.