Thursday, April 22, 2010

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:

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(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:

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

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where

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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:

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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 http://yetanothermathprogrammingconsultant.blogspot.com/2010/04/l1-portfolio-formulation.html.