The classical Markowitz objective is: $$ f(w) = w^T \mu - \frac{\lambda}{2} w ^T \Sigma w = \mathbb{E}[r^Tw] - \frac{\lambda}{2} \sigma^2(r^Tw) $$ where $\mu$ is the vector of mean returns. This is a quadratic function, and we can directly/analytically solve for the solution in the unconstrained case, as well as in cases with various reasonable constraints (e.g. the full investment constraint $\sum_{i} w_i = 1)$. It rewards increased expected returns, and at the same rate, depending on $\lambda$, penalizes variance of returns.
I am interested in another objective, the same as above, but instead of penalizing the variance penalizes the standard-deviation of returns. I am interested in this objective mostly because the standard-deviation of returns is much easier to interpret than the variance of returns - particularly because it is in the same units. As such, we would consider: $$ g(w) = w^T \mu - \lambda \sqrt{w ^T \Sigma w} = \mathbb{E}[r^Tw] - \lambda \sigma(r^Tw) $$ Like the Markowitz objective, this function is convex, although not quadratic, so all of the theoretical goodness that we get with convex problems is retained. However, we don't get analytical solutions, but if say a numerical solver converges to a solution we are guaranteed to have a global optimum.
Is there a reason this objective isn't used more frequently? Does it have poor numerical properties?
One reason that I initially thought of is that the problem above with $f$ can be re-formulated as:
- Maximize the mean return
- Given an equality constraint on the variance, which is in turn an equality constraint on volatility (take square-roots). This constrained problem is equivalent to the unconstrained problem with a correctly chosen value of $\lambda$. The constraint can be set in terms of volatility rather than variance, which is intuitive as it is in terms of returns. Thus, if we wish to set a volatility constraint, rather than use a mysterious risk aversion, we could get explicit and interpretable solutions.
