What's the disadvantage of using linear programming for portfolio optimization?

Question

I am a MFE student and we have project on the Markowitz portfolio optimization problem.

i am wondering how much impact there will be, if I use a simpler linear optimizater instead of a quadratic one.

Say, i have a target portfolio $x$, my alpha is $a$. I will try to maximize $xa$, and apply a factor exposure limit:

$$l_0 < Ax < l_1$$

while $A$ is my factor exposure

What's the biggest disadvantage of above approach, compared with the classic quadratic approach widely used in Markowitz portfolio optimization.

Can anyone explain to me a bit?

Answer 1

The Markowitz model for portfolio optimization (http://www.princeton.edu/~rvdb/542/lectures/lec17.pdf) is formulated as a quadratic programming (QP) problem, not an LP one.

You cannot use an LP solver to solve a QP problem.

Answer 2

The Markowitz setup assumes agents have mean-variance preferences (CARA utility when returns are normally distributed yields the same). So the standard markowitz optimization maximizes risk-return tradeoff, where risk is measured by variance and return by mean. It penalizes risk depending on the degree of risk aversion.

If instead you use linear programming, that is ok, but you should have a strong theory behind on why those constraints matter. As I said before the markowitz setup is based on a strong theory.

Answer 3

After Markowitz published his famous paper, William F. Sharpe published "A Linear Programming Algorithm for Mutual Fund Portfolio Selection" (1967). I haven't re-read it 20 years, but AFAIR it relies on a special structure for the covariance matrix and some assumptions about the utility. Maybe reading this paper would tell you if you are onto something new or not. http://pubsonline.informs.org/doi/abs/10.1287/mnsc.13.7.499