Tuesday, 27 May 2014

Understanding Black/Litterman Posterior Returns

In our experience, many people still struggle to understand the Black/Litterman model. The available literature does not really help: it consists of either highly mathematical papers requiring strong priors (pun intended) in Bayesian statistics and matrix algebra or then "intuitive" papers mainly working with examples. We think that the basic idea can be expressed with rather simple algebra. The trick is to assume that the investment universe consists of one asset only.

The core of the Black/Litterman model is the equation for posterior returns below...

\$ r_{posterior} = [(\tau\cdot\Sigma)^{-1}+P'\cdot\Omega\cdot P]^{-1}\cdot[(\tau\cdot\Sigma)^{-1}\cdotr_{prior}+P'\cdot\Omega \cdot r_{forecast}] \$

...which calculates posterior returns \$ r_{posterior} \$ from numerous input variables and parameters: \$ r_{prior} \$ as prior returns (e.g. implied returns derived from a reverse optimization), \$ \tau \$ as the famous and rather dubious "confidence" parameter, \$ \Sigma \$ as asset covariances, \$ \Omega \$ as estimation risk (expressed as a covariance matrix, P as a view matrix defining the view portfolios, \$ r_{forecast \$ as the forecasted returns of the views.

Quite a bit of insight is already gained by reshuffling terms...

 \$ r_{posterior} = \frac{(\tau\cdot\Sigma)^{-1}+P'\cdot\Omega\cdot P}{\tau\cdot\Sigma}\cdot r_{prior}+\frac{P'\cdot\Omega}{(\tau\cdot\Sigma)^{-1}+P'\cdot\Omega\cdot P}\cdot r_{forecast} \$

It becomes immediately clear that posterior returns are a weighted sum of prior and forecast returns. If we now assume that there exists one asset only, the matrix expressions simplify rather dramatically to...

\$ r_{posterior} = (1-w) \cdot r_{prior}+ w \cdot r_{forecast} \$

\$ w = \frac{\sigma}{(1/\tau) \cdot \sigma_{forecast}+ \sigma} \$

Therefore, Black/Litterman posterior returns can be understood as a diversified portfolio consisting of prior and forecast returns, with the relative weight between prior and forecast returns is determined by the percentage of confidence-adjusted forecast risk relative to asset risk. This is the shortest and most precise explanation of Black/Litterman posterior returns possible. It creates the space for the relevant discussion, which should centre around how to specify the inputs (asset risk, asset return priors, asset return forecasts, forecast errors & confidence).