Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions from prior to posterior

Question

In Black-Litterman we get a new vector of expected returns of the form: \begin{align} \Pi_{BL} = \Pi + \underbrace{\tau \Sigma P^T[P\tau\Sigma P^T+\Omega]^{-1}}_{\text{correction}}[Q-P\Pi] \end{align} where $P$ is the pick matrix and we mix the prior $\Pi$ with the expected value of the views $Q$. $\Sigma$ is the historical covariance matrix and $\Omega$ is the covariance matrix of the views.

Let us assume that $P$ is just the identity matrix and look at the choice $\Omega = \tau\Sigma$, then we see that $$ \Pi_{BL} = \frac12 \Pi + \frac12 Q, $$ thus we have a 50:50 mix and the covariance of the matrix does not affect the posterior at all - it is just a trivial mixture. This is against my intuition. Furthermore optimal weights using this $\Pi_{BL}$ will differ relatively much from optimal weights of the prior (of course depending on $Q$).

If we assume $\Omega = \text{diag}(\tau \Sigma)$ then I can not find a closed form for $\Pi_{BL}$ but appearantly the posterior is more compatible with the prior and the optimal weights are more similar than in the other setting.

My question: how can I choose $\Omega$ best in order to get results that do not deviate too much from my prior? I know that in the literature there are theories (e.g. here The Black-Litterman Model In Detail) but I can't see through. What is used in practice?

Answer 1

In practice, $\Omega$ (the covariance of the investor views) often 'inherits' the market covariance $\Sigma$. A convenient choice is

$ \Omega = \left( 1/c -1 \right) P \Sigma P^T$

where $c$ is a confidence parameter: the case $c \rightarrow 1$ corresponds to a strongly peaked distribution of views (the investor views dominate the market), while $c \rightarrow 0$ gives an infinitely disperse distribution where investor views have no influence. Tuning $c$ allows you to deviate smoothly from the prior $\Pi$.

This choice for $\Omega$ is proposed in Attilio Meucci's Risk and Asset Allocation, chapter 9.2.

Edit: In the example you give ($P$ is the identity matrix and $\Omega = \tau \Sigma$), the investor provides views on each asset with the same uncertainty as the market. In that case, the posterior return $\Pi_{BL}$ is just the average of market prior $\Pi$ and investor expectation $Q$. This seems plausible by symmetry: if you switch market and investor, $\Pi_{BL}$ stays the same.

Answer 2

When I implemented a BL model, I chose to do the omega optimization using the technique Idzorek proposed here:

https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/BlackLitterman.pdf

It's a numerical procedure though.