Convex bound approximations for sums of random variables under multivariate log-generalized hyperbolic distribution and asymptotic equivalences

We propose convex bound approximations for the sum of log-multivariate generalized hyperbolic random variables. We derive explicit formulas for the distributions of convex bounds and for the frequently-used risk measures such as Value-at-Risk, Conditional Tail Expectation and stop-loss premium. We present numerical results showing that such approximations are not only accurate but also robust. Moreover, we further prove that there exist asymptotic equivalences between the sum and its convex bounds. To further illustrate the potentials of the convex bound approximations, we provide an application to capital allocation. We show that our formulas can be easily applied to precisely approximate capital allocation rule based on the conditional tail expectation.