Black Litterman and Bayesian portfolio views

Definition The Black-Litterman model is a Bayesian model for the unknown vector $\mu \in \mathbb{R}^d$ of one-period expected excess returns, the expected returns above the risk-free rate. All distributions on this page belong to this model of the physical measure $\mathbb{P}$, in which $\mu$ is itself random, and none of them is a pricing measure. The model assumes

with $\eta$ independent of $\mu$. Here $\pi \in \mathbb{R}^d$ is the prior mean, $C$ the $d \times d$ prior covariance of $\mu$, often $C = \tau \Sigma$ with a scalar $\tau > 0$, $P$ a $k \times d$ view matrix, $q \in \mathbb{R}^k$ the stated views and $\Omega$ the $k \times k$ covariance of the view errors. Future excess returns are $R = \mu + \varepsilon$ with $\varepsilon \sim N(0, \Sigma)$ independent of $(\mu, \eta)$, so $\Sigma$ is the covariance of returns given their mean. Then $(\mu, q)$ is jointly Gaussian, and when $PCP^{\top} + \Omega$ is invertible the posterior of $\mu$ given $q$ is the Gaussian $N(m, C_{\text{post}})$ of the posterior mean card. Three covariance matrices play three roles, $\Sigma$ for return noise, $C$ for prior uncertainty about $\mu$ and $\Omega$ for view error, and only $\Sigma$ describes how realized returns fluctuate around their mean. The prior, $\tau$ and $\Omega$ are modeling inputs chosen by the modeler, not facts verified from data.

Definition Suppose a risk-free asset is available, and an investor with risk aversion $\delta > 0$ chooses risky weights $w$, with the rest of the wealth in the risk-free asset, to maximize $w^{\top} \mu - \tfrac{\delta}{2} w^{\top} \Sigma w$ without other constraints. With $\Sigma$ positive definite this objective is strictly concave, and its unique maximizer solves $\mu = \delta \Sigma w$. Reverse optimization runs this condition backward. Given reference weights $w_{\text{mkt}}$, usually market capitalization weights, the equilibrium prior mean is

the expected excess returns for which $w_{\text{mkt}}$ is optimal in this problem, and the prior is $\mu \sim N(\pi, C)$. Multiplying by $w_{\text{mkt}}^{\top}$ gives $w_{\text{mkt}}^{\top} \pi = \delta\, w_{\text{mkt}}^{\top} \Sigma w_{\text{mkt}}$, so $\delta$ is the ratio of the reference portfolio’s prior expected excess return to its variance. This is a modeling construction. It assumes that the reference portfolio is mean-variance optimal for the chosen $\delta$ and $\Sigma$, and it does not show that market weights reveal true expected returns.

Definition A view matrix is a $k \times d$ matrix $P$ whose row $i$ picks a linear combination $(P\mu)_i$ of the expected excess returns, about which the investor states the view $q_i$. The view model is

with $\eta$ independent of $\mu$. An absolute view on asset $j$ has the row $e_j^{\top}$, and a relative view that asset $i$ outperforms asset $j$ by $q_i$ has the row $e_i^{\top} - e_j^{\top}$. Before $q$ is seen, the prior predicts

The surprise is $q - P\pi$, the difference between what the views say and what the prior predicts for them. The views are statements about expected returns, not about the returns that will be realized.

Definition View confidence is expressed by the view-error covariance $\Omega$, where a smaller variance means a more confident view, measured against the prior uncertainty $C$, which with $C = \tau\Sigma$ is set by $\tau$. Write $\Psi = PCP^{\top}$ for the prior covariance of the viewed combinations. When $\Psi + \Omega$ is invertible, the posterior mean $m$ of the next card satisfies

so for a single view with $\Psi > 0$ and $\Omega > 0$ the viewed quantity moves the fraction $\kappa = \Psi/(\Psi + \Omega)$ of the way from $P\pi$ to $q$, with $0 < \kappa < 1$. As $\Omega$ grows without bound, the gain tends to zero and the posterior returns to the prior. As $\Omega$ tends to zero with $\Psi$ invertible, the posterior satisfies the views exactly. An exact view is this limit, or equivalently conditioning on a degenerate Gaussian, and the precision formula of the next card, which inverts $\Omega$, cannot be used for it. Several exact views need more care. With $\Psi$ invertible the limit formula applies, while views that repeat one another need a rank-aware treatment, and views that contradict one another cannot all hold. If $\Omega = \tau\Omega_0$ scales with $\tau$, then $\tau$ cancels from the posterior mean, though not from the posterior covariance. Common conventions, such as $\Omega = \operatorname{diag}(PCP^{\top})$, are modeling choices.

Definition In the Black-Litterman model with $PCP^{\top} + \Omega$ invertible, the posterior distribution of $\mu$ given $q$ is Gaussian with mean $m$ and covariance $C_{\text{post}}$, where

This is Gaussian conditioning. For the jointly Gaussian pair $(\mu, q)$ with the moments of the view matrix card, $E[\mu \mid q] = E[\mu] + \operatorname{Cov}(\mu, q) \operatorname{Cov}(q)^{-1} (q - E[q])$, and $\operatorname{Cov}(\mu \mid q)$ is $\operatorname{Cov}(\mu)$ minus $\operatorname{Cov}(\mu, q) \operatorname{Cov}(q)^{-1} \operatorname{Cov}(q, \mu)$. When $C$ and $\Omega$ are both positive definite, the densities give the same answer. With $a = \mu - \pi$ and $b = q - P\mu$, the negative log posterior is, up to a constant,

and collecting the quadratic and linear terms in $\mu$ gives the posterior precision and mean

with $C_{\text{post}} = J^{-1}$. The two forms agree by the Woodbury identity. Since $J$ is positive definite, the quadratic is strictly convex, and $m$ is both the posterior mode and the posterior mean. The gain $K$ multiplies the surprise, and its columns $CP^{\top}$ record how each expected return co-varies with the viewed combinations under the prior. The reduction $C - C_{\text{post}} = CP^{\top}(PCP^{\top} + \Omega)^{-1}PC$ is positive semidefinite, so in this model the views never increase the uncertainty about the mean in any direction, and $C_{\text{post}}$ does not depend on the value of $q$.

Definition In the Black-Litterman model with future excess returns $R = \mu + \varepsilon$ and $\varepsilon \sim N(0, \Sigma)$ independent of $(\mu, \eta)$, and hence of $(\mu, q)$, the predictive distribution of $R$ given the views is $N(m, \Sigma + C_{\text{post}})$. Its covariance, the predictive covariance, follows from the law of total covariance,

because given $(\mu, q)$ the return has mean $\mu$ and covariance $\Sigma$, and the conditional covariance of $\mu$ given $q$ is $C_{\text{post}}$. Using $\Sigma$ alone treats $m$ as the known mean, and using $C_{\text{post}}$ alone leaves out the return noise. With the risk-free asset and the mean-variance objective of the equilibrium prior card applied to the predictive distribution, the optimal risky weights given the views are

with the remainder $1 - \mathbf{1}^{\top} w^{*}$ in the risk-free asset. A different model for future returns, for example with noise that depends on $\mu$, needs its own predictive covariance computed from that model.