Black-Litterman model ($m=\pi +C{P}^T(PC{P}^T+\Omega {)}^{-1}(q-P\pi )$)

The Black-Litterman model combines equilibrium-implied expected returns with additional views whose uncertainty is stated explicitly. Fischer Black and Robert Litterman introduced it to make portfolio optimization less dependent on unstable unconstrained return estimates.^[ref]

In a standard Gaussian formulation, let the unknown return-mean vector have prior $\mu \sim N(\pi ,C)$, often $C=\tau \Sigma$ for asset-return covariance matrix $\Sigma$. Views take the form $q=P\mu +\epsilon$, with independent $\epsilon \sim N(0,\Omega )$. The Gaussian conditional expectation formula gives posterior meanThe matrix $P$ describes absolute or relative views, while $\Omega$ describes their uncertainty.

For one mean with prior $4\%$, prior variance $0.0004$, and a view of $8\%$ with equal variance, the posterior mean is $6\%$. Increasing the view variance pulls the result toward $4\%$; making it very small pulls toward $8\%$. The uncertainty $C$ about a mean is distinct from the covariance matrix $\Sigma$ of fluctuating returns, and must not be substituted for it without specifying the portfolio model.