Expressions for Marginal Mean Excess and Marginal Expected Shortfall Measures under Bivariate Scale Mixture of Normal Distribution

Here two important risk measures–marginal expected shortfall (MES) and marginal mean excess (MME)–for bivariate risk vectors (Y1,Y2)$$(Y_{1},Y_{2})$$ are studied. Usually, deriving explicitly these measures is challenging and is done through asymptotic methods. In this paper, we derive explicit expressions for these measures when the joint risk factor (Y1,Y2)$$(Y_{1},Y_{2})$$ follows a bivariate normal distribution. As risk factors commonly exhibit heavy-tailed behavior, we extend our findings to attain exact expressions for MES and MME, under scale mixture of normal (SMN) risk factors. This class include important distributions, such as symmetric generalized hyperbolic (SGH) and Student-t$$t$$ distributions, and the established results are extended to include these subclasses.