The average value-at-risk of a quadratic form , where is given by a particularly complex closed-loop formula which I’ll describe below.
Even the computation of the value-at-risk is given by a formula which is not very simple to work with either.
We know that if , then . Then, the CDF of the quadratic form is given by
where is the regularised Gamma function. For convenience, let us define the function . The value-at-risk is then given by
For ease of notation, let us denote the value-at-risk by . Now the average value-at-risk is given by
Let us call the integral which appears in the last equation . For even values of this integral is given by the product of a polynomial by an exponential. For odd values, it involved the complementary error function.
For instance, for
For we have
In case , then follows a univariate Wishart distribution with scale matrix and degrees of freedom and similar manipulations can be done to derive the average value-at-risk formula.
When is not centered at zero but its components are independently distributed then we need to use a non-central distribution or a non-central version of the Wishart distribution. In that case derivations become tough and even more so when is not centered at zero and we do not have the assumption of component-wise independence.