Black-Litterman Model
Goldman Sachs says that "Fischer Black and Robert Litterman revolutionize portfolio management in 1990 with the creation of the Black-Litterman Global Asset Allocation Model – quickly adopted for optimal portfolio allocation across international equity, fixed income and currency markets".
Black-Litterman (BL) model uses the Theil mixed estimator which is a kind of GLS estimator. You can find more information regarding these estimators in the following previous posts.
Prior Expected Returns
Prior expected returns relies on market equilibrium returns or market clearing returns. To find these returns, a reverse optimization is used. \[\begin{align} \Pi = \lambda\Sigma w_{m} \end{align}\] where
- \(\Pi\) : the implied excess equilibrium return vector (\(N \times 1\))
- \(\lambda\) : the risk aversion coefficient (scalar)
- \(\Sigma\) : the covariance matrix of excess returns (\(N \times N\))
- \(w_{m}\) : the market capitalization weight of assets (\(N \times 1\))
| The above equation can be derived from the efficient market assumption. Market maximizes the following utility function. \[\begin{align} U(w) = w \Pi - \frac{1}{2} \lambda w^{\prime} \Sigma w \end{align}\] The FOC is \(U^{\prime}(w)=0\) so that \(\Pi - \lambda \Sigma w = 0\). This leads to \(\Pi = \lambda \Sigma w \). |
Uncertainty of Prior Expected Returns
"Expected" is always followed by "uncertainty" or error. While the market clearing returns (\(\Pi\)) are "estimates" and are known, the true expected returns (\(\mu\)) are unknown. This leads to the following equation.
\[\begin{align} \Pi = \mu + \epsilon_{\Pi}, \quad \epsilon_{\Pi} \sim N(0,\tau \Sigma) \end{align}\] for some small scale factor parameter \(\tau \ll 1\). A small \(\tau\) means a high confidence in equilibrium estimates. This reflects that the variance in the expected returns is smaller than the variance of the actual returns.
Investor's Views
Views of BL model can be relative or absolute. K views are expressed as K linear combinations of the true excess return vector (\(\mu\)). It is a common practice to set the off diagonal elements of \(\Omega\) to zero due to the independence assumption for each views. \[\begin{align} q = P\mu + \epsilon_q, \quad \epsilon_q \sim N(0,\Omega) \end{align}\] where \(P\) is a \(K \times K\) pick matrix and \(\Omega\) is a \(K \times K\) matrix of confidence in the views.
Matrix Representation
Let's collect two equations and represent it as a matrix form.
\[\begin{align} \Pi &= \color{white}{P}\mu + \epsilon_{\Pi}, &\epsilon_{\Pi} \sim N(0,\tau \Sigma) \\ q &= P\mu + \epsilon_q, &\epsilon_q \sim N(0,\color{white}{\tau}\Omega) \\ \\ \begin{bmatrix} \Pi \\ q \end{bmatrix} &= \begin{bmatrix} I \\ P \end{bmatrix} \mu + \epsilon, \quad &\epsilon_{\Pi} \sim N(0,V) \end{align}\] \[\begin{align} V = \begin{bmatrix} \tau \Sigma & \\ & \Omega \end{bmatrix} \end{align}\] where \(I\) is the \(N \times N\) identity matrix. We are familiar with this matrix representation.
The Posterior Expected Returns
Using Theil mixed estimator (GLS estimor) which is explained in the previous posts, we can get the following posterior returns (\(\mu_{BL}\)).
\[\begin{align} \mu_{BL} &= [(\tau \Sigma)^{-1}+P^{'} \Omega^{-1} P]^{-1} [(\tau \Sigma)^{-1}\Pi +P^{'} \Omega^{-1} Q] \\ Var(\mu_{BL}) &= [(\tau \Sigma)^{-1}+P^{'} \Omega^{-1} P]^{-1} \end{align}\] This is a combined expected returns of the market equilibrium and investor's views.
Concluding Remarks
This post delivers the derivation of the famous the Black-Litterman model using the Theil mixed estimator. For a clear understanding of BL model, I also explained the GLS estimator and Theil mixed estimator in the earlier posts. Real applications will be treated in the next post. \(\blacksquare\)
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