I thought I understood elliptical distributions, but then I staggered over the following problem:
Let d financial returns be modeled as the components $X_1,...X_d$ of a d-dimensional random vector $X$. Assume that $X$ has an elliptical distribution such that $E[X_i^2]$ finite for all $i=1,,,d$ and $E[X_1]=...=E[X_d]$. We wand to show that the minimum variance portfolio also minimizes Value-at-Risk. More precisely, denote
$Q:=\{w \in R^d : \sum_{i=1}^d w_i=1\}$
and fix a level $\alpha \in (1/2, 1). Then show that the minimization problems
$\min_{w\in Q} Var(\sum_{i=1}^d w_i X_i)$ and $\min_{w\in Q} VaR_{\alpha}(-\sum_{i=1}^d w_i X_i)$
have the same minimizer $w^*\in Q$.
I would be glad if somebody could guide me through the solution. I think, I am missing a piece of the properties of elliptical distributions to solve the problem. I would have guessed that it is the uncorrelatedness of the components, but I am not sure how to implement that into the solution for general elliptical distributions as only the standard multivariate normal has truly independent components.
Many thanks.