This paper studies the approximation of extreme quantiles of random sums of heavy-tailed random variables, or, more specifically, subexponential random variables. A key application of this approximation is the calculation of operational value-at-risk (VaR) for financial institutions in order to determine operational risk capital requirements. This paper follows work by Böcker, Klüppelberg and Sprittulla and makes several advances. These include two new approximations of VaR and an extension to multiple loss types where the VaR relates to a sum of random sums, each of which is defined by different distributions. The proposed approximations are assessed via a simulation study.