High Dimensional Modelling and Simulation with Asymmetric Normal Mixtures

A family of multivariate distributions, based on asymmetric normal mixtures, is introduced in order to model the dependence among insurance and financial risks. The model allows for straight-forward parameterisation via a correlation matrix and enables the modelling of radially asymmetric dependence structures, which are often of interest in risk management applications. Dependence is characterised by showing that increases in correlation values produce models which are ordered in the supermodular order sense. Explicit expressions for the Spearman and Kendall rank correlation coefficients are derived to enable calibration in a copula framework. The model is adapted to simulation in very high dimensions by using Kronecker products, enabling specification of a correlation matrix and an increase in computational speed.

[1]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[2]  Michel Denuit,et al.  Actuarial Theory for Dependent Risks: Measures, Orders and Models , 2005 .

[3]  Laura Ballotta,et al.  Pricing and capital requirements for with profit contracts: modelling considerations , 2007 .

[4]  Steven Haberman,et al.  On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling , 2008 .

[5]  Mark A. McComb.  Comparison Methods for Stochastic Models and Risks , 2003, Technometrics.

[6]  Paola Sebastiani,et al.  The Use of Exogenous Knowledge to Learn Bayesian Networks from Incomplete Databases , 1997, IDA.

[7]  Emiliano A. Valdez,et al.  Understanding Relationships Using Copulas , 1998 .

[8]  A. McNeil,et al.  The t Copula and Related Copulas , 2005 .

[9]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[10]  O. Barndorff-Nielsen,et al.  Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes , 2005 .

[11]  Martin Karlsson,et al.  In Sickness and in Health? Dynamics of Health and Cohabitation in the United Kingdom , 2009 .

[12]  S. Gupta.  Probability Integrals of Multivariate Normal and Multivariate $t^1$ , 1963 .

[13]  Andrew D Smith.  Dependent Tails , 2002 .

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

[15]  Steven Haberman,et al.  The income drawdown option: quadratic loss , 2004 .

[16]  C. Loan.  The ubiquitous Kronecker product , 2000 .

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

[18]  F. Lindskog,et al.  Multivariate extremes, aggregation and dependence in elliptical distributions , 2002, Advances in Applied Probability.

[19]  Christine M. Anderson-Cook,et al.  Book review: quantitative risk management: concepts, techniques and tools, revised edition, by A.F. McNeil, R. Frey and P. Embrechts. Princeton University Press, 2015, ISBN 978-0-691-16627-8, xix + 700 pp. , 2017, Extremes.

[20]  Collin Carbno,et al.  Actuarial Theory for Dependent Risks: Measures, Orders, and Models , 2007, Technometrics.