Reconstructing the Black-Litterman Model

Here are the high points of Reconstructing the Black-Litterman Model, a response to Michaud et al's recent paper, "Deconstructing the Black-Litterman Model", in the Journal of Investment Management.

We will begin with a quote. When asked about the proper usage of the parameter tau, Dr Michaud responded, "... the paper presents two formulas with tau and includes a fully worked out example...".

We have three main critiques of "Deconstructing the Black-Litterman Model". First, the Black-Litterman model is constructed around uncertain estimates (essentially a Bayesian model). Their paper uses a non-Bayesian approach to the model and this yields many inconsistencies in their arguments. Second, they argue that reverse optimization to generate the prior estimates of returns is an invalid statistical technique, but this argument is easily shown to be without merit. Third, they incorrectly assert that the Black-Litterman model requires the use of unconstrained mean variance optimization for the portfolio choice step.First, they apply frequentist arguments to show the Black-Litterman model is incorrect because it does not incorporate uncertainty. This is quite odd, since the Black-Litterman model is built around uncertain estimates. It appears likely that they did not consult the canonical Black-Litterman literature, but instead chose to use other references which deal with frequentist derivatives of the Black-Litterman model. This manifests itself in table 4 of "Deconstructing the Black-Litterman Model" where they show the impact of a tau of 1/216 on the uncertainty of the prior mean estimates, asserting how much uncertainty the model should pick up. We note that tau is usually on the order of 1/20 to 1/60 and thus the uncertainty in the implemented Black-Litterman model is typically sqrt(3) higher than shown in their table 4.

Second, as for reverse optimization and the prior estimate. Their endnote 5 covers the most interesting set of arguments in this regard. Here they throw up many different arguments:

- Implied estimates have embedded errors because of errors in the covariance matrix.
- Reverse optimization of an optimal portfolio reduces the statistical significance of the optimization.
- Neutral portfolio must be maximal Sharpe-ratio portfolio, but cannot be.

Essentially all of these arguments miss the point that the model is Bayesian and so τΣ represents the uncertainty in the estimate (or the error term that Michaud et al think is missing). While the neutral portfolio is mean variance efficient and the MSR portfolio on the efficient frontier that corresponds to the point estimates of the prior, this is not the true parameters (because of uncertainty) so there is no requirement that this portfolio is MSR on the true efficient frontier.

As for the assertion that somehow the Black-Litterman model requires the use of unconstrained mean variance optimization I only point to Black and Litterman (1991) where they clearly state that they allow for constraints and for the user to specify an arbitrary utility function. They further state "that the only requirement for portfolio choice is that the tradeoff between risk and return be reasonably smooth." Many portfolio optimization techniques meet this definition.

MATLAB code to reproduce the Black-Litterman model exhibits in the paper (tables 1, 2 and 4) is available here. The file michaud.m and meuccibl.m are needed to reproduce the results. The fidelity is not 100% because data in the paper is shown with only 2 digits of precision. There is also a mistake in the paper in table 2 and 3(return and risk of BL* Optimla Portfolio). With the correction the Sharpe Ratio of the BL* Optimal Portfolio exceeds that of the Michaud Optimized Portfolio, but it is unclear if this is statistically significant.