Modeling Inequality Views in Black-Litterman
Summary
The discussion considers how to express a relative return belief such as one asset outperforming another without specifying a fixed return spread. A standard Black-Litterman view can encode a specified difference between expected returns, but the proposed alternative is to condition the posterior distribution of expected returns on an inequality constraint.
This approach changes the posterior from the usual normal form and raises practical questions. Computing the conditional expectation under the constraint requires suitable integration or an algorithm for constrained Bayesian inference. The discussion also questions whether conventional mean-variance optimization remains appropriate with that non-normal posterior, while suggesting it may still be usable under further assumptions. The answer is tentative: its author had not implemented the method, and points to an external paper as a possible treatment of computation and uncertainty in the inequality parameters, without presenting results or resolving the optimization issue.
Key ideas
- An inequality view can express that one asset is expected to outperform another without fixing a return spread.
- The proposed method conditions the posterior expected returns on the inequality constraint.
- Computing the constrained conditional expectation is a central implementation challenge.
- The resulting posterior is not generally normal, which complicates the use of standard mean-variance optimization.
- The discussion is exploratory and provides no implementation results.
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Full text
# Can I formulate a general relative view in Black-Litterman?
# Can I formulate a general relative view in Black-Litterman?
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 by vanguard2k (score 1, accepted)
https://quant.stackexchange.com/a/28022
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?)
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.