Extending Black–Litterman to Non-Normal Return Distributions
Summary
The document considers whether Black–Litterman portfolio analysis can use return distributions that are not normal. It separates two roles often combined in implementations: deriving implied-return priors from a market portfolio and covariance structure, and updating views through Bayesian inference. The answer says a Bayesian treatment can be extended to non-normal distributions, but generally loses the closed-form posterior available in the familiar setup. Reverse-engineering implied returns may also be difficult, with elliptical distributions mentioned as a possible case to explore.
For portfolio optimization after the update, the chosen investor utility matters: mean–variance methods have their usual theoretical interpretation under normality, while quadratic utility makes higher moments irrelevant. A second answer mentions copula-based opinion pooling as a way to model non-normal returns, but flags an independence assumption about priors. The discussion is conceptual and supplies no implementation details, empirical results, or resolution of that limitation.
Key ideas
- Bayesian updating in Black–Litterman can be formulated for non-normal returns, though a closed-form posterior may be lost.
- Deriving market-implied return priors is less straightforward without normal-market assumptions.
- The utility function used in the later optimization affects how non-normality matters.
- Copula-based opinion pooling is offered as an approach, with prior independence identified as a limitation.
Tags
Full text
# Is it possible to deal with non-normal distribution in Black-Litterman model?
# Is it possible to deal with non-normal distribution in Black-Litterman model?
Suppose that I know that the normality assumption about my data is unrealistic (as it is very frequently): is it possible to apply any distribution that I judge the right one to the Black-Litterman model? Does it make sense, or does it lose some of its usefulness?
## Answer by vanguard2k (score 7, accepted)
https://quant.stackexchange.com/a/23008
Well there are two main things to consider here.
Many implementation of Black-Litterman use the market portfolio and the ex post volatility and correlation structure to back out implied returns to use as prior. As far as I know, there is no standard way to reverse-engineer the optimization problem in the presence of nonnormal markets. (the first guess is that it could be possible for elliptical markets) So in that respect its probably at least hard (maybe not possible).
If you think of Black-Litterman as a Bayesian method, there is no reason why you shouldn't do it for nonnormal markets. You will lose the closed form posterior though. Check out Meucci's work on this. ("The Black Litterman approach: Original Model and Extensions" on SSRN or maybe you find something in his book., Also check out "Bayesian Portfolio Analysis" by Avramov and Zhou)
So the final question is: "Does it make sense?" Well, it depends how you proceed from there. If you want to do an optimization there is the topic: either the market is normal (then, in theory, you have an optimal solution with mean-variance optimization) or you have a quadratic utility function (which says that higher moments dont matter for you as an investor). You have to think about the utility function you use.
Just to sum this up the things to think about are:
- What are "implied returns" in a nonnormal market? (or do you want to use a different prior)
- How to calculate the posterior distrubution (Bayes' formula)
- If optimizing afterwards: What about the utility function? (Full-scale optimization)
## Answer by Alejandro Andrade (score 0)
https://quant.stackexchange.com/a/23078
Yes there is a method to deal with non-normal market distributions in Black Litterman optimization. It is call Black Litterman Copula Opinion Pooling which uses copulas to model the market returns and therefore solve the non-normality problem. It was propose by Attilio Meucci and it can be implemented in R or Matlab. Never the less there is an other problem with the model and it is that it assumes that the priors are independent between them which in some cases it is not realistic.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.