Using copulae to bound the Value-at-Risk for functions of dependent risks

Abstract. The theory of copulae is known to provide a useful tool for modelling dependence in integrated risk management. In the present paper we review and extend some of the more recent results for finding distributional bounds for functions of dependent risks. As an example, the main emphasis is put on Value-at-Risk as a risk measure.

[1]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[2]  B. Schweizer,et al.  Operations on distribution functions not derivable from operations on random variables , 1974 .

[3]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[4]  G. D. Makarov.  Estimates for the Distribution Function of a Sum of Two Random Variables When the Marginal Distributions are Fixed , 1982 .

[5]  L. Rüschendorf.  Random variables with maximum sums , 1982, Advances in Applied Probability.

[6]  M. J. Frank,et al.  Best-possible bounds for the distribution of a sum — a problem of Kolmogorov , 1987 .

[7]  C. Genest,et al.  A characterization of gumbel's family of extreme value distributions , 1989 .

[8]  J. Angus.  The Asymptotic Theory of Extreme Order Statistics , 1990 .

[9]  Robert C. Williamson,et al.  Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds , 1990, Int. J. Approx. Reason..

[10]  Marc Goovaerts,et al.  Dependency of Risks and Stop-Loss Order , 1996, ASTIN Bulletin.

[11]  A. Müller.  Orderings of risks: A comparative study via stop-loss transforms , 1996 .

[12]  A. Müller.  Stop-loss order for portfolios of dependent risks , 1997 .

[13]  A. Müller.  Stochastic Orders Generated by Integrals: a Unified Study , 1997, Advances in Applied Probability.

[14]  Alfred Müller,et al.  MODELING AND COMPARING DEPENDENCIES IN MULTIVARIATE RISK PORTFOLIOS , 1998 .

[15]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[16]  Stephen P. Boyd,et al.  Applications of semidefinite programming , 1999 .

[17]  P. Embrechts,et al.  Correlation: Pitfalls and Alternatives , 1999 .

[18]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[19]  C. Genest,et al.  Stochastic bounds on sums of dependent risks , 1999 .

[20]  A. Müller,et al.  Some Remarks on the Supermodular Order , 2000 .

[21]  Jan Dhaene,et al.  Modern Actuarial Risk Theory , 2001 .

[22]  P. Embrechts,et al.  Risk Management: Correlation and Dependence in Risk Management: Properties and Pitfalls , 2002 .

[23]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[24]  Jan Dhaene,et al.  A Simple Geometric Proof that Comonotonic Risks Have the Convex-Largest Sum , 2002, ASTIN Bulletin.

[25]  A. Juri,et al.  Copula convergence theorems for tail events , 2002 .

[26]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[27]  Michel Denuit,et al.  Distributional bounds for functions of dependent risks , 2002 .

[28]  Thorsten Rheinländer.  Risk Management: Value at Risk and Beyond , 2003 .

[29]  P. Embrechts,et al.  Chapter 8 – Modelling Dependence with Copulas and Applications to Risk Management , 2003 .

[30]  Xiaohong Chen,et al.  Estimation of Copula-Based Semiparametric Time Series Models , 2006 .