Robust-Bayesian optimization in Markowitz framework

Question

Suppose we are in the mean-variance optimization setting with a vector of returns $\alpha$ and a vector of portfolio weights $\omega$.

In a robust setting, the returns are assumed to lie in some uncertainty region. I came accross a paper which lets this region, call it $U$, be given by the sphere centered at $\alpha$ with radius $\chi|\alpha|$ where $\chi$ lies between 0 and 1.

The authors then turn their attention to:

$\min_U r_{p}$

and end up with the following solution:

$\min_U r_{p}=\alpha^\intercal\omega-\chi|\alpha||\omega|$ ... ... ... (1)

They do not provide much detail as to how they arrive at this but mention the following: "this uncertainty region corresponds to a one-sigma neighborhood under a Bayesian prior of an uncertain $\alpha$ distributed normally about the estimated $\alpha$, with $\sigma=\chi|\alpha|$..."

Does anyone know how they might have arrived at equation (1)??? Thanks in advance!

Answer 1

In robust optimization, the true return is not known, we just have a prior $\alpha$ and you have to take into account a possible misestimate which can lower the true return. This is done under the assumption that the posterior return will be within the prior return $\alpha$ plus minus the error being in some $\sigma$-interval.

Now a try for a more formal answer: The posterior return vector is estimated as

$\vec{\alpha} +\vec{\chi}\cdot|\alpha|$ (1)

with $|\vec{\chi}|\leq 1$, or equivalently $\vec{\chi}^{2} \leq 1$. This exactly describes a sphere around $\vec{\alpha}$. Now the return is the product of the return vector $\vec{\alpha}+\vec{\chi}\cdot|\alpha|$ times the weight vector $\vec{\omega}$ :

$r=(\vec{\alpha}+\vec{\chi}\cdot|\alpha|)\cdot\vec{\omega}=\alpha^{T}\omega+\chi^{T}\omega|\alpha|$. (2)

Here, $\vec{\chi}$ can have any orientation. We want the minimum of the second term. $\alpha^{T}\omega$ is minimal if $\vec{\alpha}$ and $\vec{\omega}$ look in opposite direction (property of the dot product), therefore

$\min_U r_P=\alpha^{T}\omega-\chi|\alpha| |\omega|$. (3)

The first term is just the dot product of $\vec{\alpha}$ and $\vec{\omega}$, so it can be written as $|\alpha||\omega|\cos(\phi)$ where $\phi$ is the angle between the two vectors (in n dimensions). This is the next equation in the Golts and Jones working paper:

$\min_U r_P=|\alpha||\omega|(\cos(\phi)-\chi)$. (4)