When / how do I vol-scale portfolio weights when optimizing the portfolio?

Question

I have a set of portfolio weights $w$. I'm using cvxpy to optimise the portfolio sharpe, subject to a set of constraints.

$$ \text{maximize} \hspace{10mm} \mu^Tw - w^T\Sigma w $$ $$ \text{subject to:} \hspace{5mm} 1^Tw = 0 \hspace{5mm} \text{(Long/Short cash neutral)} $$ $$ ||w||_1 \leq L_{max} \hspace{5mm} \text{(Leverage limit)} $$ $$ |w_i| \leq c, \forall i $$

where $\mu$ are the expected returns, $w$ are the target weights, $\Sigma$ is the covariance matrix of returns, $1$ is a vector of ones, $c$ is a maximimum concentration limit per asset (e.g. 1%), and $L_{max}$ is the maximum allowable leverage.

The issue now is, I want to scale the position sizes by a portfolio volatility target in this process, and I am not sure how. In other words, I would like to target, say, an 10% annualised volatility.

I could scale $w$ using a standard method:

$$ w_{\text{vol scaled}} = w \times \frac{\text{vol target}}{std(r)} $$

where $std(r)$ is the standard deviation of the returns over the last $n$ periods (e.g. 12 months). The problem with this method is that some weights will break the constraints above after having optimised the portfolio.

How can I meet a volatility target, either before, as part of or after portfolio optimisation?

Answer 1

You can try

$$ \min \frac{1}{2}w^T\Sigma w \quad \mathrm{s.t.} \quad w^T\mu=\mu_c $$

and subject to your other constraints. Then, trace out $\mu_c$ until the optimal solution reaches your target risk level.

You cannot add a constraint on risk to a standard quadratic program as these commonly work only under linear constraints.