Using copulae to bound the value-at-risk for functions of dependent risks.

*(English)*Zbl 1039.91023The aim of this paper is to give more insight into the problem of managing the Value-at-Risk (VaR) of a joint position resulting from the combination of different dependent risks. The authors review and extend some of the recent results for finding distributional bounds for functions of dependent risks. They describe the dependent structure and copulae, comonotonicity and orthant dependence. The results concerning distributional bounds for VaR are formulated in terms of a function \(\Psi(X_1,\dots,X_n)\), where \(X_1,\dots,X_n\) are the risks, and they are generalized by relaxing the continuity assumptions on \(\Psi\). In contrast to the existing literature it is only required that \(\Psi\) is increasing in each argument and left continuous in the last one. This allows a more flexible application of the results in practice since, especially in the financial derivative world, discontinuous functions abound. Comonotonicity and independence are discussed as a particularly interesting case. Practical computational methods are proposed and an example for a two-dimensional portfolio is considered.

Reviewer: Yuliya Mishura (Kyïv)

##### MSC:

91G10 | Portfolio theory |

60E05 | Probability distributions: general theory |

60E15 | Inequalities; stochastic orderings |