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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.

Portfolio optimization chooses asset allocations to optimize an objective under specified constraints. In a one-period mean-variance optimization model, the return vector $R$ has mean $\mu$ and covariance matrix $\Sigma$. Portfolio weights $w$ give expected return ${\mu }^{T}w$ and return variance $w^T\Sigma w$. A standard problem isThe constraints describe a fully invested portfolio, a target expected return and no short selling. Different assumptions produce different optimization problems. Harry Markowitz made the dependence on covariance and the tradeoff between mean and variance central to portfolio selection.^[ref]

For two uncorrelated assets with expected returns $4\%$ and $8\%$, each with standard deviation $20\%$, a weight $w$ in the first gives variance $0.04(w^2+(1-w{)}^2)$. Its unconstrained minimum is at $w=1/2$, yielding mean $6\%$ and standard deviation $\sqrt{0.02}\simeq 14.14\%$. A target of at least $7\%$ instead requires $w\le 1/4$, making the constrained minimum $w=1/4$. The same covariance matrix therefore gives a different solution when the objective's feasible set changes.

Estimated means and covariances are model inputs, rather than known future outcomes. Sampling uncertainty, transaction costs and additional allocation restrictions can materially change the resulting weights.