τ (Tau)
Tau is probably the single most confusing aspect of the Black-Litterman model. All the central papers of Black-Litterman research use different values, or just ignore it. He and Litterman use a value of 0.025 in their paper whereas Satchell and Scowcroft remark that many people use a value of τ close to 1. Several other authors, Meucci, and Krishnan and Mains, completely eliminate τ.
 | If we look at Theil's paper on the Mixed Estimation model, the source of the Black-Litterman mixing model, we see that τΣ is the error in the estimated mean, the standard error. We can empirically demonstrate this concept as shown in the histogram to the left. The histogram shows the distribution of the mean estimated from 1000 simulations drawing 60 samples from a normal distribution with mean 1 and standard deviation 1. As you can see, the sample means form a normal distribution about the population mean. This is very similar to the scenario we will find ourselves in during the asset allocation process, though we will only have one sample. Note how dispersed the estimated means are around the actual mean of 1. The blue band across the histogram shows the sample error of the estimated mean. It is computed as τΣ where τ = 1/n. |
He and Litterman propose a version of the model which generates both an estimate of the return, and an estimate of the variance of the posterior distribution. The variance of the posterior distribution is strongly impacted by the value of τ, and is the reason for having multiple parameters for the mixing, τ and Ω. If we are not interested in the variance of the posterior distribution, then we can eliminate τ and vary Ω as required to facilitate the mixing of the prior and the conditional distributions. Generally we will find that authors who use values of τ close to 1 are ignoring the variance of the posterior distribution and could just ignore τ all together.
From an intuitive point of view, and as described by most authors, τ is usually much less than 1 as it reflects the fact that the uncertainty in the mean of the distribution is much smaller than the variance in the returns. This indicates that the portfolio manager is reasonably confident in the estimates from CAPM. If we multiply the variance of the views by τ as well, that indicates the confidence level is similar (assuming the view confidences are of the same order as the return variance) and the blending should be fairly even. If the view variance term does not explicitly include τ, then changes in τ will cause the blending proportions to change.
For a more thorough discussion of just tau see my paper on the factor tau in the Black-Litterman model or my paper on the Black-Litterman model.