Deriving a Worst-Case Return in Robust Mean-Variance Optimization
Summary
The document derives the worst-case portfolio return when estimated asset returns are uncertain within a spherical region around their estimated vector. The uncertainty radius is proportional to the magnitude of the estimate, with a parameter controlling its size. A candidate return vector is represented as the estimate plus a perturbation whose length is bounded by the sphere's radius. Portfolio return is the dot product of this perturbed vector and the portfolio weights.
To minimize that return, the perturbation points opposite the weight vector, producing the estimated return minus the uncertainty radius times the magnitudes of the return and weight vectors. Equivalently, the worst-case result depends on the angle between those vectors and the uncertainty parameter. The explanation connects the sphere to a one-standard-deviation neighborhood under a normal Bayesian prior, as described in the question. It clarifies the geometric minimization, but does not assess whether this uncertainty model is well calibrated or how it performs in portfolio applications.
Key ideas
- Represent return uncertainty as a sphere centered on the estimated return vector.
- Portfolio return under a perturbed estimate is the dot product of the perturbed vector and portfolio weights.
- The worst-case perturbation points opposite the portfolio weight vector.
- The resulting return penalty scales with uncertainty, return-vector magnitude, and weight-vector magnitude.
- The derivation explains the geometry but does not establish that the uncertainty assumptions fit a given application.
Tags
Full text
# Robust-Bayesian optimization in Markowitz framework
# Robust-Bayesian optimization in Markowitz framework
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 by philippe (score 3, accepted)
https://quant.stackexchange.com/a/7087
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)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.