The Black–Litterman model explained

Abstract

Active portfolio management is about leveraging information. The Black and Litterman Global Portfolio Optimisation Model (BL) sets information processing in a Bayesian analytic framework. In this framework, the portfolio manager needs only produce views and the model translates the views into security return forecasts. As a portfolio construction tool, the BL model is appealing both in theory and in practice. Although there has been no shortage of literature exploring it, the model still appears somehow mysterious, and suffers from practical issues. This article is dedicated to enabling a better understanding of this model, and features:

• an economic interpretation;

• a clarification of the model's assumptions and formulation;

• implementation guidance;

• a dimension-reduction technique to enable large portfolio applications; and

• a full proof of the main result in the Appendix.

We also provide a checklist of other practical issues that we aim to address in our forthcoming articles.

Appendix

Proof

The full proof of Theorem 1 is given below.

Theorem 1 (Posterior Return Estimates)

• The posterior return vector is normally distributed, that is, where the updated mean vector is:

  

  and the updated variance-covariance matrix is

  

Proof

• We first deal with a general case and then apply it into the BL context. In general, assume the following linear relationship:

  

  where Q is a known/observable matrix; is an observable vector; is unobservable and needs to be estimated; and is the error vector.

  Also, assume the following Gaussian distributions for regression errors and the prior:

  
  

  We therefore have conditional on a realisation of

  

  In terms of probability density function (pdf), distributions (A3) and (A4) can be equivalently written as:

  

  where ∣·∣ gives the determinant of the matrix it applies to; and

  

  respectively.

  We try to imply the probability distribution of from the joint pdf:

  

  where

  
  

  We hope, based on the Bayes’ Rule, that it is possible to express (A7) in terms of as we are interested in the posterior estimation of given . This translates into a need to replace the dependence of the mean estimates of on (As in (A4)) with a dependence of the mean estimates of on . In other words, through some transformation, we need to get rid of from the lower part of the error vector in (A8), but allow to enter the upper part. To this end, we construct the following matrix (Hamilton, 1994, Ch.12):

  

  where note ∣A∣=1.

  Using A as the transform matrix, we define:

  

  where and

  

  where V̂(Q, Σ _x , Λ)=[(Σ _x )⁻¹+Q^TΛ⁻¹Q]⁻¹.

  Therefore, (A7) can be rearranged, noting ∣A⁻¹∣=∣A∣=1, as follows:

  
  

  where

  From (A1), (A2) and (A3), we have the unconditional distribution:

  

  With (A15), it is easy to imply from (A14) the following:

  

  To assess in (A16), we use our best knowledge regarding and .

  Recall that in the model setting our best knowledge based on the public information G leads to the following prior belief:

  

  After examining the private information H, we form the following updated views:

  

  where P is the view structure; is the view forecast vector; and we assume .

  Therefore, conditional on a realisation of

  

  Substituting for and for into (A16), we reach:

  

  Finally, using our eventual conviction about the mean of we reach the following posterior belief:

  

  This completes the proof. □