Thursday, April 22, 2010

Portfolio: from QP to LP

Someone asked me how to derive the LP in from the standard QP formulation:


(see also

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 (

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:




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:

imageThe appropriate way to model the absolute value in this context was the subject of the discussion in