# 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?

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)
• Thank you. Ok for the first two points (I'll use a different prior, probably, and plug it in the Bayes's formula, ending up with a very complicated new formula, i guess ;)). It is not clear to me the utility function/full-scale optimization argument you used. – simmy Jan 29 '16 at 15:09
• @vanguard2k I've tried to do the reverse-engineer assuming that prices were multivariate log normal instead of multivariate normal. I wasn't that happy with it. I think it can't hurt to think about what is an appropriate prior, but once you start mixing in all kinds of different assets (equities, fixed income, options), the reverse engineer method of constructing the prior becomes a bit unwieldy I think. – John Jan 29 '16 at 16:26
• @John I suppose so. But I guess this is one of the main points of the BL method. What was your problem with the lognormal version? For multi asset portfolios I dont even have an idea how to get the market cap (e.g. for commodities) even if there are no derivatvies though one could argue there are some sources. – vanguard2k Feb 1 '16 at 8:30
• @SimoneBortolato The output of the BL method is a vector of means. If you want to optimize based on that means, you have to either assume something about the investor or about the market. Usually, its quadratic utility (only first two moments are relevant) or normality of the market. You say that your market is not normal and you care about it so you probably have to change these assumptions. You could incorporate higher moments into your function or do a full-scale optimization for example. – vanguard2k Feb 1 '16 at 8:51
• @vanguard2k Suppose MVN(u, sigma) represents the multivariate normal distribution and MVLN(u*, sigma*) is what you get if you transform it to a multivariate log normal. Both u* and sigma* depend on u. This makes writing the optimization problem more challenging. – John Feb 1 '16 at 20:55

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.