VaR as the CVaR sensitivity: Applications in risk optimization

VaR minimization is a complex problem playing a critical role in many actuarial and financial applications of mathematical programming. The usual methods of convex programming do not apply due to the lack of sub-additivity. The usual methods of differentiable programming do not apply either, due to the lack of continuity. Taking into account that the CVaR may be given as an integral of VaR, one has that VaR becomes a first order mathematical derivative of CVaR. This property will enable us to give accurate approximations in VaR optimization, since the optimization VaR and CVaR will become quite closely related topics. Applications in both finance and insurance will be given.