Portfolio: from QP to LP

Someone asked me how to derive the LP in http://yetanothermathprogrammingconsultant.blogspot.com/2010/04/l1-portfolio-formulation.html from the standard QP formulation:

(see also http://www.or-exchange.com/questions/204/overview-of-portfolio-models).

Of course the first reference would be the original article:

“Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market,” Hiroshi Konno and Hiroaki Yamazaki, Management Science, Vol. 37, No. 5 (May, 1991), pp. 519-531 (http://www.jstor.org/pss/2632458)

However, here is simple, intuitive approach:

1. Form separable QP

First we plug in the definition of the covariances q(i,j), using:

Here the r’s are the returns at each t for each instrument i. The μ's are the means. This results in:

where

The a(i,t)’s are the mean adjusted returns.

2. Approximate by LP

Now it is easy to see how this can be approximated by an LP:

So the resulting L1 norm model looks like:

The appropriate way to model the absolute value in this context was the subject of the discussion in http://yetanothermathprogrammingconsultant.blogspot.com/2010/04/l1-portfolio-formulation.html.