### τ (Tau)

Here we will only concern ourselves with the canonical expression of the Black-Litterman model (some older pages may refer to this as the Original Reference Model). The Hybrid (Satchell and Scowcroft (2000)), and Alternative Reference (Meucci (2010)) models are non-Bayesian and either fix τ at 1 or drop it, so we can ignore them.

Tau (τ) is probably the single most confusing aspect of the Black-Litterman model. Many authors use different values, or just ignore it. He and Litterman use a value of 0.025, other suggest using 1/n where n is the number of data points used to generate the known covariance matrix.

If we start at the beginning and 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, or the standard error of the prior estimate.

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. "Deconstructing the Black-Litterman Model Table 4 shows a tabular representation of the concept with τ = 1/216. |

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.

This is very similar to the scenario we will find ourselves in during the asset allocation process, though we will only have one sample for the prior. The prior estimate is a distribution and so we estimate the standard error as well.

From an intuitive point of view, in the canonical expression of the model, 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 ICAPM. 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.

One interesting artifact on the selection of τ, is that in the absence of views the investor will only invest the fraction 1/(1+τ) in the market portfolio and the rest of their wealth will be invested in the risk free asset because of the uncertainty in their estimate. The diagram below shows how this might look in risk/return space.

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.