AdvancedPortfolio ManagementPython
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Black-Litterman Portfolio Optimization¶
The Black-Litterman (1990) model addresses the instability of mean-variance optimization by blending market equilibrium returns with investor views using Bayesian updating.
Functions¶
| Function | Description |
|---|---|
market_implied_returns(cov, weights, lambda) |
Reverse optimize: implied returns from market portfolio |
black_litterman(cov, weights, P, Q, omega, tau, lambda) |
Bayesian blend of equilibrium + views |
bl_optimal_weights(bl_returns, cov, lambda) |
Mean-variance weights from BL posterior |
Key Concepts¶
- Problem with MVO: Small changes in expected return inputs produce wildly different (often extreme) optimal portfolios.
- Equilibrium returns (Pi):
Pi = lambda * Sigma * w_mkt— back out what the market is "pricing in." - Views matrix P: Each row encodes one view.
[1, -1, 0, 0]= "asset 1 outperforms asset 2." - tau: Scales uncertainty of equilibrium priors. Typically 0.01–0.05.
- Omega: Diagonal matrix of view uncertainty. Larger = less confident in that view.
Example¶
import numpy as np
from black_litterman import black_litterman, bl_optimal_weights
# View: US equity outperforms international by 2%
P = np.array([[1, -1, 0, 0]])
Q = np.array([0.02])
result = black_litterman(cov, market_weights, P, Q)
weights = bl_optimal_weights(result["posterior_returns"], result["posterior_covariance"])
Why It Works¶
By starting from market-cap weights (which are efficient by definition if markets are efficient), BL produces sensible portfolios even with few views. Views only tilt allocations where the manager has genuine insight.
Continue in Portfolio Management¶
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