About new approximations of the Value at Risk, Expected Shortfall and the distribution of bivariate aggregate claims

Approximations of the Value at Risk (VaR), the Expected Shortfall (ES) as well as the bivariate aggregate claims amount distribution in the Classical Risk model are discussed. The proposed approximations are based on the values of scaled Laplace transforms of the underlying distributions. The uniform rates of approximations are established and their asymptotic performance is illustrated in a simulation study. Some properties of the smoothed versions of approximations based on Bernstein and Szasz-Mirakyan operators are presented. Applications to the problem of estimating the ruin probability, the VaR and ES from corresponding empirical counterparts are discussed as well.