Probability of constraint satisfaction and Robust Optimization

This post explores some of the properties of robust mean-variance optimization.

Consider the classic mean-variance optimization model to select a portfolio $\mathbf{x} \in \mathbb{R}^N$

where $r_{\text{min}}$ is the target return, $\mathbf{Q}$ is the covariance matrix of asset returns, and $\boldsymbol{\mu}$ is the expected return.

We do not know the true values of $\boldsymbol{\mu}$ and $\mathbf{Q}$. Instead, we only have data to estimate these quantities. We form the corresponding estimates $\hat{\boldsymbol{\mu}}$ and $\hat{\mathbf{Q}}$ from a dataset of returns $S = \{\mathbf{r}^{(i)}\}_{i=1}^T$ and solve the MVO model with $\boldsymbol{\mu}$ and $\mathbf{Q}$ replaced with $\hat{\boldsymbol{\mu}}$ and $\hat{\mathbf{Q}}$. Let $\mathbf{x}_{\text{MVO}} \left(\hat{\boldsymbol{\mu}}, \hat{\mathbf{Q}} \right)$ denote an MVO solution obtained using estimates $\hat{\boldsymbol{\mu}}$ and $\hat{\mathbf{Q}}$.

What is random here? The estimates $\hat{\boldsymbol{\mu}}$ and $\hat{\mathbf{Q}}$ are random, since they depend on a random sample $S$. We can then ask, "What is the probability of meeting the constraints?"

$\mathbb{P}_S[\boldsymbol{\mu}^{\intercal} \mathbf{x}_{\text{MVO}} \left(\boldsymbol{\mu}, \mathbf{Q} \right) \geq r_{\text{min}}] = 1$ since $\mathbf{x}_{\text{MVO}}$ is a deterministic quantity and satisfies the return constraint by definition.

Similarly, one can consider the probability of meeting the return constraint when using the estimates: $\mathbb{P}[\boldsymbol{\mu}^{\intercal} \mathbf{x}_{\text{MVO}} \left(\hat{\boldsymbol{\mu}}, \hat{\mathbf{Q}} \right) \geq r_{\text{min}}] = ?$. Unfortunately, this probability does not equal 1. This drawback motivates the use of robust optimization, where we replace the constraint

with

where $\mathcal{U}$ is an uncertainty set for the mean. A popular uncertainty set is the ellipsoid centered at the estimate $\hat{\boldsymbol{\mu}}$ with shape parameter

and radius $\delta$. In this setting, the robust optimization problem can be written as:

Now let $\mathbf{x}_{\text{ROB}} \left(\hat{\boldsymbol{\mu}}, \hat{\mathbf{Q}} \right)$ denote a robust MVO solution obtained using estimates $\hat{\boldsymbol{\mu}}$ and $\hat{\mathbf{Q}}$ and consider the probability of constraint satisfaction $\mathbb{P}[\boldsymbol{\mu}^{\intercal} \mathbf{x}_{\text{ROB}} \left(\hat{\boldsymbol{\mu}}, \hat{\mathbf{Q}} \right) \geq r_{\text{min}}] = ?$. This probability should be higher than the estimated MVO case if the ellipsoid covers a large portion of the support of the distribution of $\hat{\boldsymbol{\mu}}$. However, in practice, we cannot easily evaluate these probabilities.

Google Colab Notebook

The notebook located here explores these facts in a contrived setting where we have sampling access to the return distribution. Indeed, the robust model increases the probability of satisfying the return constraint.