Struggling with tau in Black-Litterman

Question

According to the omega formula in B-L tau is used in the Omega estimation to determine the degree of uncertainty given to views of the investor:

So, if tau is given a low value then the inverse of omega will be large and therefore suppose a lot of uncertainty in the investors view and so giving more importance to implicit returns in contrast to investors. In short, and based on this assumption (don't hesitate in correct it) tau can be used to calibrate the importance I give to investors views in opposition to implicit returns and vice versa.

Supposing the above statement is correct, according to Thomas M. Idzorek paper on B-L model regarding page 15 he comments that

"When the covariance matrix of the error term ( Ω ) is calculated using this method, the actual value of the scalar ( τ ) becomes irrelevant because only the ratio τ ω / enters the model. For example, changing the assumed value of the scalar ( τ ) from 0.025 to 15 dramatically changes the value of the diagonal elements of Ω , but the new Combined Return Vector ( ] [ R E ) is unaffected. "

I have made some calculations assigning tau a 0.025 and a 1 and the new Combined Return Vector is unaffected

Then I understand that there is no way I can assign investors views a degree of uncertainty because it does not matter the value I assign to tau, that the return vector will be the same.

Therefore, my questions are: Are my statements above correct? If so , whats the point of the existence of tau? Are there alternative methods to determine investors uncertainty / weights given to implicit or investor returns ?

Thank you in advance for taking your time reading my post.

Answer 1

Your statement about the properties of $\tau$ is correct. $\tau$ is a measure of uncertainty. I think the problem you are having is because in most practical situations nobody really knows what values should be used for $\tau$ and/or $\Omega$.

There is plenty of practical advice out there, and some of it is very confusing! For example, Jay Walters has written an entire paper on $\tau$ (see https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1701467). That paper discusses the origin of $\tau$ as a measure of an investors uncertainty in their prior returns. Unfortunately, this does not translate into practical advice and it refers to different papers giving different values for $\tau$ (0.05 in one paper and 1.0 in another).

Nobody really knows what $\Omega$ should be either because that would require them to have an estimate of the uncertainty of the views. Most human-generated forecasts don’t come with an associated covariance matrix. Hence, most people use an $\Omega$ that is derived from their estimate of the variance matrix. So, for example, Idzorek’s formula for $\Omega$ assumes that the forecasts are independent when they are not likely to be in practice.

Idzorek’s formula has rolled $\tau$ in to the estimate of $\Omega$. That inclusion means that the effect of $\tau$ is cancelled out in the Combined Return Vector. Hence, as you have noticed, you cannot use $\tau$ as a single scalar to alter the weight of the prior in the final Combined Return Vector if you use Idzorek’s formula. Including $\tau$ has the advantage of giving one single result, but that is not necessarily what you need.

A simple modification to Idzorek’s formula would be to set $\Omega = diag(P\Sigma P’)$. The Combined Return Vector would then change as $\tau$ changes.

Another popular choice for $\Omega$ is given in chapter 9 of Meucci’s Risk And Asset Allocation (and also discussed at Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions form prior to posterior). Meucci suggests that people use: $\Omega = (\frac{1}{c} -1) P\Sigma P’$