# Is an arbitrary prior for Black-Litterman valid? Or do we need a market implied one?

I went through The Black-Litterman Approach: Original Model and Extensions - see also. The BL approeach starts with a prior on the expected returns vector derived from the hypothesis that the market is in equilibrium. Then we take a bunch of views and get a posterior model with a new mean vector and covariance matrix. The step that is necessary to do so is mainly straigtht forward Bayes reasoning.

So my question: If my prior is some set of mean vectors (from some statistical methods e.g. based on Sharpe-ratios and historical volatilties) and then mixed with views - can I use just the same formulas? As far as I see it does not really matter where my prior is derived from.

Of course you can choose the prior. As far as I understand the literature, the BL-model is characterized by using the equilibrium implied returns. Otherwise it would just be a Bayesian model.

If you estimate the returns in a different way (not taking implied returns from the market portfolio), you could lose the stabilizing inverse optimization step required in computing the implied returns. (multiplication by $\Sigma$. Optimization - loosely spoken - divides by $\Sigma$) Using a different return estimate as prior, it is my intuition that you will lose this effect and, as a result, some robustness.

One example of a BL-like portfoilio construction technique is found in The Black Litterman Model in Detail - section Active Management and the Black-Litterman Model. In this case of active management one just optimizes the active portfolio, the implied returns (the prior) are assumed to be $0$. Later in the text (Two-Factor Model Black Litterman), a different method for the calculation of implied returns is presented too, so there are definitely different methods to calculate the return prior.

Of course one can also take the risk parity portfolio as a benchmark with this approach. The main reason for this is that the optimization step does not need any return estimates to calculate the ERC portfolio. Implicitly though, this corresponds to the tangency portfolio of an optimization problem: If all assets have equal sharpe ratio and the correlation is constant across all assets, Maillard, Roncalli and Teiletche show, that the ERC portfolio corresponds to the tangency portfolio. Of course, the correlation assumption is strong, but it gives some intuition about the implicit assumptions and it somehow relates to your idea of using return estimates from sharpe ratios...

Apart from the applications there is a small caveat: To arrive at the BL-Formula you calculate the posterior distribution explicitly and extract the parameters. To do this, one assumes a multivariate normal distribution of your assets (the calculation is rather lengthy and I think its in the appendix of your link). Thats where Meucci's entropy pooling idea plays its part - it is just a way to calculate the posterior distribution for the general case but everything comes at a price - it just exists in the machine, not on paper...

• Hi M. ;) Do you have any reference for the riskparty + views part? and/or for the Benchmark + views part? This would be great, thanks – Richard Dec 1 '14 at 15:08
• @Richard Hallo, I have to check the details about the risk parity portfolio agai and will catch up on that tomorrow. – vanguard2k Dec 1 '14 at 15:43
• @Richard I tried to clarify my answer a little and added an additional point. – vanguard2k Dec 2 '14 at 10:26
• Very nice summary and links - thanks a lot! Maybe we meet before Xmas for lunch? – Richard Dec 2 '14 at 10:29
• Hope it helps. Lets try to find a day - check your inbox in a few hours! – vanguard2k Dec 2 '14 at 10:39