= Black-Litterman model
{c}
{title2=$m=\pi+CP^T(PCP^T+\Omega)^{-1}(q-P\pi)$}
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.<Global Portfolio Optimization by Fischer Black and Robert Litterman>[1992, equilibrium reference returns and confidence-weighted views]{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 $q=P\mu+\varepsilon$, with independent $\varepsilon\sim N(0,\Omega)$. The <Gaussian conditional expectation> formula gives posterior mean
$$
m=\pi+CP^T(PCP^T+\Omega)^{-1}(q-P\pi).
$$
The 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.
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