Markowitz portfolio with factor/position constraints

Question

General Markowitz-style optimization (problem objective of $w^T \mu - \lambda w^T \Sigma w$) yields simple optimal weights policy $w \propto \Sigma^{-1} \mu$.

However, I would like to add a series of factor exposure constraints (i.e. $| w^T \beta_i | < k_i$ for factors indexed by $i$, instrument exposure vector to $i^{th}$ factor by $\beta_i$). I may wish to extend to maximum position values as well, which is basically just another constraint with a different beta vector.

Is there a standard way do this quickly, closed form or similar, with the addition of these types of constraints? Or some common, clever transformations/relaxations that would help facilitate practical solutions under time constraints?

I've got to imagine this is a pretty solved problem, since many portfolios have these types of constraints. I'm just not sure if folks throw into an optimizer and don't care about very fast solutions/closed form solution explicitly. Or do something altogether different. Is there some industry standard or set of resources available to follow? Thanks!

Answer 1

Generally speaking, as long as we can accommodate the optimization using quadratic programming, we are still within the realm of Markowitz optimization:

$$ \begin{align} \min&\quad w^Tq+\frac{1}{2}w^TQw\\ \mathrm{s.t.}&\quad Aw=a\\ \mathrm{s.t.}&\quad Bw\leq b\\ \end{align} $$

In your case, $|w^T\beta_i|\leq k_i$ translates to two additional entries in $B$, i.e. $w^T\beta_i\leq k_i$ and $-w^T\beta_i\leq k_i$:

$$ \begin{pmatrix} \ldots & \ldots & \ldots & \ldots \\ \beta_{1i}&\beta_{2i}&\ldots&\beta_{ni}\\ -\beta_{1i}&-\beta_{2i}&\ldots&-\beta_{ni}\\ \ldots & \ldots & \ldots & \ldots \end{pmatrix}w \leq \begin{pmatrix}\ldots\\k_i\\k_i\\\ldots \end{pmatrix} $$