# Portfolio optimization - maximize variance with exposure to risk factors equal to zero

Optimize a portfolio such that the exposure to risk factors is zero and the variance is maximized (instead of traditional minimization problem).

so the optimization problem look like:

$$maximize\;w^T\,\Sigma\,w$$

With following constraints:

$$\beta_0\,w=0$$ $$\beta_1\,w=0$$ $$\beta_2\,w=0$$ $$\beta_3\,w=0$$

Where

$$\Sigma - covariance\,matrix$$ $$\beta_0..._3 - factor\,exposure$$

I am told that this is a non-convex problem. Can I convert this into an SDP with some relaxations? I could convert the objective function into a quadratic constraint such as below:

$$w^T\,\Sigma\,w > Min$$ Please advise.

There is no solution. If $w$ is a solution to the original problem, then consider $aw$ with $a>1$
$$\beta_i(aw) = a(\beta_i w) = 0$$
$$(aw)^T\Sigma(aw) = a^2 (w^T\Sigma w) > w^T\Sigma w$$
so the original solution $w$ was not a maximum.
• You can set up upper bound on variance, but still any portfolio with non-zero variance can be arbitrarily scaled so that it hits the upper bound. Your question is like asking "what values of $x$ and $y$ maximize the function $f(x,y)=x^2+y^2$". Without an upper bound on $f$ there is no solution at all. With an upper bound on $f$ there is no unique solution. – Chris Taylor Apr 14 '16 at 9:27