Can I formulate a general relative view in Black-Litterman?

Question

We have often discussed the Black-Litterman approach/model in the forum. What I was wondering is: is it possible to formulate relative views in the model. Relative in the sense $$ \mu_1 > \mu_2 $$ where $\mu_i$ is the expected return (as a view) of asset $i$.

It is clear how to formulate relative views of the form $$ \mu_1 - \mu_2 = v, $$ where the view is that I expect asset 1 to have a return $v$ percent greater as the return of asset 2. However I don't want to fix $v$.

Answer 1

Indeed I think you can, but it comes at a price. To be clear: Neither have I done this myself nor have I attempted it.

The main idea is typical Bayesian: In computing the posterior return parameters for the random variable $\mu$, you can calculate the conditional expectation subject to your inequality constraints, say $A\mu \leq b$).

There are three, fundamentally different, problems I see with this approach:

• Computational Complexity: How do we solve the integral $\mathbb{E}[\mu|A\mu \leq b]$? This should be a standard Bayes problem with plenty of literature (stochastic integration?)

• View Formulation: What does it mean to have uncertainty attached to the views in the view matrix while constraining the return distribution to fulfill the inequalities all the time?

• Following Optimization: What I'm not completely sure about: Considering the fact that the distribution of the posterior is not normal anymore, is it still appropriate to proceed with standard mean-variance optimization? (I would guess the answer is yes - at least if we assume a normal market)

There is a paper that claims to solve the first point by providing an algorithm and also presents a method to add uncertainty to inequality parameters. From what I saw so far I think its definitely worth a look and should fill in all the gaps there are still in the answer.