Black-Litterman¶

One-line definition¶

A way to build expected returns by blending a market-implied equilibrium (the "prior") with the PM's own views, weighted by how confident each view is — rather than plugging raw historical-mean returns into an optimizer.

Formula¶

Prior (market-implied, via reverse optimization):

pi = risk_aversion * Sigma @ w_market

Posterior (combining prior with views P, Q, Omega):

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

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

posterior_cov here is the covariance of the estimated mean itself, not a posterior return covariance — don't pass it straight into an optimizer in place of Sigma (it's smaller by roughly a factor of tau, which would make the optimizer far too confident). Use Sigma itself, or Sigma + posterior_cov, as the optimizer's covariance input.

Why PMs care¶

Feeding raw historical average returns straight into a mean-variance optimizer is exactly what produces the extreme, unstable weights mean_variance is known for — small differences in estimated returns swing weights wildly. Black-Litterman anchors to a defensible market prior and only tilts away from it where the PM has an actual view, sized by how confident they are.

Worked example¶

Verified in tests/test_robust.py: a view with near-infinite uncertainty (Omega huge) leaves the posterior equal to the prior; a highly confident view pulls the posterior toward the view (and spills over to correlated assets through Sigma).

Common mistakes¶

setting Omega too small (overconfidence) and effectively overriding the market prior with an unvalidated view
forgetting views on one asset spill over to correlated assets via Sigma — a single-asset view can move the whole posterior
passing black_litterman_posterior's second return value straight into an optimizer as the covariance — see the note above
reverse-optimizing a prior with one risk_aversion and then re-optimizing with a different one — the round trip back to market weights only holds at the same risk_aversion on both sides, see mean-variance optimization