On the Use of Conditional Distributions in Portfolio Theory

0. Abstract

It is well known that the result of mean-variance optimization
is extremely sensitive to its inputs. This sensitivity problem 
often causes the resulting optimal portfolio to be overwhelmingly 
concentrated in a few assets thus hampers its application to asset 
allocation problems. The Black-Litterman method alleviates this 
problem by using conditional distribution to adjust the entire 
expected return vector to reflect an investor's view on just a 
few assets. However, the Black-Litterman method is restricted 
to the conditional adjustment of the mean vector only. 
Increasingly, there is need to adjust the covariance matrix 
to reflect an investor's view on the volatilities and correlations 
of selected assets as well because it also has an impact on the 
optimal portfolio.

We present a new unified method to obtain conditional mean 
vector and conditional covariance matrix given an investor's 
view. Both adjustments occur simultaneously through a multivariate 
linear regression implied by conditional distribution of the market 
equilibrium. The new method is simple since we apply the conditional 
distribution directly to the distribution of the asset return rather 
than to an unknown distribution of the mean vector, as in the 
Black-Litterman approach. Yet the method is powerful to provide us 
both conditional adjustments analytically. Especially, it enables 
us to gain insights about how a single volatility shock spreads to 
the rest of the assets and changes the correlation coefficients 
among assets. We provide a detailed two-asset example to illustrate 
these effects. Next, we derive an analytic solution of the 
unconstrained mean-variance optimization for the adjusted mean 
vector and covariance matrix. This solution reveals a special 
condition under which the vector of optimal weights stays 
unchanged despite the difference between the investor's view and 
the market equilibrium. We discuss its implications to asset 
allocation decisions. Finally, we identify other applications of 
our method and areas for further research.