## The Parameter Tau in Idzorek’s Version of the Black-Litterman Model

While on my sabbatical I spent some time working with Idzorek’s version [1] of the Black-Litterman asset allocation model.  I like Idzorek’s version because it gives an intuitive method for specifying one of the most complicated parts of the Black-Litterman model, the parameter $\Omega$.

Another interesting aspect of Idzorek’s version is that the parameter $\tau$ disappears from the combined expected return vector, the model’s primary output.  This is nice, as specifying $\tau$ is another of the model’s more complicated aspects.  Because of this, at least one author I know of concludes that $\tau$ is irrelevant in Idzorek’s version of Black-Litterman.

However, it turns out that $\tau$ is not irrelevant to Idzorek’s version of Black-Litterman.  I discovered this while trying to understand how the posterior covariance matrix works in Idzorek’s version.  While $\tau$ does disappear from the combined expected return vector, it remains in the formula for the posterior covariance matrix.

(As an aside, Idzorek never explicitly discusses the posterior covariance matrix, other than to cite its formula derived in Satchell and Scowcroft [3].  Instead, his focus is on the combined expected return vector.)

Here’s the derivation.

If R denotes the excess return on the market, the Black-Litterman model assumes

$\displaystyle R \sim N(\mu, \Sigma).$

The original version of Black-Litterman also takes $\mu$ to be a random variable having the following distribution:

$\displaystyle \mu \sim N(\pi, \tau \Sigma).$

The parameter $\pi$ is the implied excess return vector from the capital asset pricing model.  The distribution of $\mu$ given here is taken to be a Bayesian prior.

As shown in Meucci [2], the views, together with the assumed distribution of $\pi$, produce the following posterior distribution for R:

$\displaystyle R|v; \Omega \sim N(\mu_{BL}, \Sigma_{BL}),$

where

$\displaystyle \mu_{BL} = \pi + (\tau \Sigma) P^T (P (\tau \Sigma) P^T + \Omega)^{-1}(v - P \pi),$

$\displaystyle \Sigma_{BL} = \Sigma + \left(I - (\tau \Sigma) P^T (P (\tau \Sigma) P^T + \Omega)^{-1} P \right) (\tau \Sigma),$

v is the vector of view returns, and P is the “pick” matrix specifying which views are applied to which assets.

Following Idzorek’s approach of estimating $\Omega' = \Omega/\tau$ rather than $\Omega$ and $\tau$ separately, the equation for $\mu_{BL}$ becomes

$\displaystyle \mu_{BL} = \pi + (\tau \Sigma) P^T \left(P (\tau \Sigma) P^T + \tau \frac{\Omega}{\tau} \right)^{-1} (v - P \pi) \\ = \pi + (\tau \Sigma) P^T \tau^{-1} \left(P \Sigma P^T + \frac{\Omega}{\tau} \right)^{-1} (v - P \pi) \\ = \pi + \Sigma P^T (P \Sigma P^T + \Omega')^{-1} (v - P \pi).$

As we can see, estimating $\Omega'$ rather than $\Omega$ means that $\tau$ is eliminated from the formula for the combined expected return vector.

However, the same calibration does not result in the removal of $\tau$ from the formula for the posterior covariance matrix $\Sigma_{BL}$.  Instead, the equation for $\Sigma_{BL}$ becomes

$\displaystyle \Sigma_{BL} = \Sigma + \left(I - (\tau \Sigma) P^T \left( P (\tau \Sigma) P^T + \tau \frac{\Omega}{\tau} \right)^{-1} P \right) (\tau \Sigma) \\ = \Sigma + \left(I - (\tau \Sigma) P^T \tau^{-1} \left(P \Sigma P^T + \frac{\Omega}{\tau} \right)^{-1} P \right) (\tau \Sigma) \\ = \Sigma + \left(I - \Sigma P^T (P \Sigma P^T + \Omega')^{-1} P \right) (\tau \Sigma).$

This shows that only some of the factors of $\tau$ in the expression for $\Sigma_{BL}$ are eliminated by choosing to estimate $\Omega'$ rather than $\Omega$.  The parameter $\tau$ remains in the formula for the posterior covariance matrix $\Sigma_{BL}$.

References

1. Thomas Idzorek, “A Step-by-Step Guide to the Black-Litterman Model: Incorporating User-Specified Confidence Levels,” Working paper, Zephyr Associates, 2004.
2. Attilio Meucci, “The Black-Litterman Approach: Original Model and Extensions,” in The Encyclopedia of Quantitative Finance, Wiley, 2010.
3. Stephen Satchell and Alan Scowcroft, “A Demystification of the Black-Litterman Model: Managing Quantitative and Traditional Portfolio Construction,” Journal of Asset Management, September 2000, pp. 138–150.