This post explores some of the properties of robust mean-variance optimization.
Consider the classic mean-variance optimization model to select a portfolio \(\bf{x} \in \mathbb{R}^N\)
\[ \min_{\bf{x}} \bf{x}^{\intercal} Q \bf{x} \quad \textit{s.t.}\quad \bf{1}^{\intercal}\bf{x} = 1, \ \bf{x} \geq \bf{0},\ \boldsymbol{\mu}^{\intercal}\bf{x} \geq r_{\text{min}}, \]where \(r_{\text{min}}\) is the target return, \(\bf{Q}\) is the covariance matrix of asset returns, and \(\mathbf{\mu}\) is the expected return.
We do not know the true values of \(\boldsymbol{\mu}\) and \(\bf{Q}\). Instead, we only have data to estimate these quantities. We form the corresponding estimates \(\hat{\boldsymbol{\mu}}\) and \(\hat{\bf{Q}}\) from a dataset of returns \(S = \{\mathbf{r}^{(i)}\}_{i=1}^T\) and solve the MVO model with \(\boldsymbol{\mu}\) and \(\bf{Q}\) replaced with \(\hat{\boldsymbol{\mu}}\) and \(\hat{\bf{Q}}\). Let \({\bf x}_{\text{MVO}} \left(\boldsymbol{\hat{\mu}}, \hat{\bf{Q}} \right)\) denote an MVO solution obtained using estimates \(\boldsymbol{\hat{\mu}}\) and \(\hat{\bf{Q}}\).
What is random here? The estimates \(\boldsymbol{\hat{\mu}}\) and \(\hat{\bf{Q}}\) are random, since they depend on a random sample \(S\). We can then ask, "What is the probability of meeting the constraints?"
\(\mathbb{P}_S[\mu^{\intercal} {\bf x}_{\text{MVO}} \left(\boldsymbol{\mu}, \bf{Q} \right) \geq r_{\text{min}}] = 1\) since \({\bf x}_{\text{MVO}}\) 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: \(\mathbb{P}[\mu^{\intercal} {\bf x}_{\text{MVO}} \left(\boldsymbol{\hat{\mu}}, \hat{\bf{Q}} \right) \geq r_{\text{min}}] = ? \). Unfortunately, this probability does not equal 1. This drawback motivates the use of robust optimization, where we replace the constraint
\[ \boldsymbol{\mu}^{\intercal}\bf{x} \geq r_{\text{min}} \]with
\[ \boldsymbol{\mu}^{\intercal}\bf{x} \geq r_{\text{min}} \ \forall \boldsymbol{\mu} \in \mathcal{U}, \]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
\[\Theta = \frac{1}{T}\begin{pmatrix} \hat{Q}_{11} & 0 & \cdots & 0 \\ 0 & \hat{Q}_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \hat{Q}_{nn} \end{pmatrix} \]and radius \(\delta\). In this setting, the robust optimization problem can be written as:
\[ \begin{align*} &\begin{align*} & \min_{\bf{x}} & \bf{x}^{\intercal} \hat{\bf{Q}}\ \bf{x} \end{align*}\\ &\begin{align*} &\ \mathrm{s.t.} & \hat{\boldsymbol{\mu}}^T \bf{x} - \delta \|\bf{\Theta}^{1/2} \bf{x}\|_2 &\geq r_{\text{min}} \\ & & \bf{1}^T \bf{x} &= 1 \end{align*} \end{align*} \]Now let \({\bf x}_{\text{ROB}} \left(\boldsymbol{\hat{\mu}}, \hat{\bf{Q}} \right)\) denote a robust MVO solution obtained using estimates \(\boldsymbol{\hat{\mu}}\) and \(\hat{\bf{Q}}\) and consider the probability of constraint satisfaction \(\mathbb{P}[\mu^{\intercal} {\bf x}_{\text{ROB}} \left(\boldsymbol{\hat{\mu}}, \hat{\bf{Q}} \right) \geq r_{\text{min}}] = ? \). 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.
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.