26 — Covariance Shrinkage and Black-Litterman¶

Free learning pack¶

1. MIT OCW Portfolio Theory I–III
2. reference/concepts/covariance_shrinkage.md, black_litterman.md

Do not search for more material until these are insufficient.

PREDICT¶

Full shrinkage (shrinkage=1.0) toward a diagonal target removes all off-diagonal covariance. Would that make minimum_variance treat all assets as if uncorrelated? Is that obviously good or obviously bad?

Formula¶

Sigma_shrunk = shrinkage * Target + (1 - shrinkage) * Sigma_sample

Target = diag(diag(Sigma_sample))

In [ ]:

import numpy as np

sample_cov = np.array([
    [0.040, 0.018, 0.006],
    [0.018, 0.030, 0.010],
    [0.006, 0.010, 0.020],
])

# MANUAL FIRST:
# shrink sample_cov toward its diagonal at shrinkage=0.3 and shrinkage=1.0.
# Confirm variances (the diagonal) never change, only off-diagonals shrink.
shrunk_30pct = None
shrunk_full = None
print(shrunk_30pct)
print(shrunk_full)

import numpy as np sample_cov = np.array([ [0.040, 0.018, 0.006], [0.018, 0.030, 0.010], [0.006, 0.010, 0.020], ]) # MANUAL FIRST: # shrink sample_cov toward its diagonal at shrinkage=0.3 and shrinkage=1.0. # Confirm variances (the diagonal) never change, only off-diagonals shrink. shrunk_30pct = None shrunk_full = None print(shrunk_30pct) print(shrunk_full)

PREDICT (Black-Litterman)¶

The market-implied prior return for Asset A is 5%. You have a highly confident view that Asset A will return 10%. Should the Black-Litterman posterior for Asset A land exactly at 10%, or somewhere between 5% and 10%? What would make it land closer to one end?

Formula¶

pi = risk_aversion * Sigma @ w_market (the prior)

posterior_cov = [(tau*Sigma)^-1 + P'*Omega^-1*P]^-1

posterior_mean = posterior_cov @ [(tau*Sigma)^-1 @ pi + P'*Omega^-1*Q]

In [ ]:

Sigma = np.array([[0.04, 0.01], [0.01, 0.09]])
w_market = np.array([0.6, 0.4])
risk_aversion = 3.0

# MANUAL FIRST (step 1):
# compute the market-implied prior pi = risk_aversion * Sigma @ w_market.
pi = None
print(pi)

Sigma = np.array([[0.04, 0.01], [0.01, 0.09]]) w_market = np.array([0.6, 0.4]) risk_aversion = 3.0 # MANUAL FIRST (step 1): # compute the market-implied prior pi = risk_aversion * Sigma @ w_market. pi = None print(pi)

In [ ]:

P = np.array([[1.0, 0.0]])   # a view purely on asset 1
Q = np.array([0.10])          # asset 1 will return 10%
Omega = np.array([[0.0001]])  # small = high confidence
tau = 0.05

# MANUAL FIRST (step 2):
# using `pi` from above, compute posterior_cov and posterior_mean per the
# formula. Confirm posterior_mean[0] moved toward 0.10 from pi[0], and
# notice whether asset 2's posterior moved at all despite having no
# direct view - and why.
posterior_mean = None
print(posterior_mean)

P = np.array([[1.0, 0.0]]) # a view purely on asset 1 Q = np.array([0.10]) # asset 1 will return 10% Omega = np.array([[0.0001]]) # small = high confidence tau = 0.05 # MANUAL FIRST (step 2): # using `pi` from above, compute posterior_cov and posterior_mean per the # formula. Confirm posterior_mean[0] moved toward 0.10 from pi[0], and # notice whether asset 2's posterior moved at all despite having no # direct view - and why. posterior_mean = None print(posterior_mean)

Experiment¶

Rerun the Black-Litterman step with Omega = [[10.0]] (very low confidence) instead. Confirm the posterior stays close to pi, not Q.

Reference¶

reference/concepts/covariance_shrinkage.md
reference/concepts/black_litterman.md

Promote¶

Use src/pm/robust.py (shrink_covariance, market_implied_returns, black_litterman_posterior) only after your own implementation.

Test¶

pytest tests/test_robust.py

ORAL CHECK¶

Explain to a PM why Black-Litterman is generally preferred over feeding raw historical average returns straight into mean_variance.

Try /tutor portfolio-construction black litterman for an adaptive walkthrough.