Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Shot-Noise Cox Process

Affiliation(s)

Department of Applied Finance & Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, Australia.

Benelec Pty Ltd., Sydney, Australia.

Department of Applied Finance & Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney, Australia.

Benelec Pty Ltd., Sydney, Australia.

ABSTRACT

In this paper, we introduce tail dependene measures for collateral losses from catastrophic events. To calculate these measures, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses. A homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Joint Laplace transform of the distribution of the aggregate collateral losses is derived and joint Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses. For numerical illustrations, a member of Farlie-Gumbel-Morgenstern copula with exponential margins is used. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are also provided.

In this paper, we introduce tail dependene measures for collateral losses from catastrophic events. To calculate these measures, we use bivariate compound process where a Cox process with shot noise intensity is used to count collateral losses. A homogeneous Poisson process is also examined as its counterpart for the case where the catastrophic loss frequency rate is deterministic. Joint Laplace transform of the distribution of the aggregate collateral losses is derived and joint Fast Fourier transform is used to obtain the joint distributions of aggregate collateral losses. For numerical illustrations, a member of Farlie-Gumbel-Morgenstern copula with exponential margins is used. The figures of the joint distributions of collateral losses, their contours and numerical calculations of risk measures are also provided.

Cite this paper

J. Jang and G. Fu, "Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Shot-Noise Cox Process,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2191-2204. doi: 10.4236/am.2012.312A300.

J. Jang and G. Fu, "Measuring Tail Dependence for Aggregate Collateral Losses Using Bivariate Compound Shot-Noise Cox Process,"

References

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[2] H. Cossette, P. Gaillardetz, E. Marceau and J. Rioux, “On Two Dependent Individual Risk Models,” Insurance: Mathematics and Economics, Vol. 30, No. 2, 2002, pp. 153-166. doi:10.1016/S0167-6687(02)00094-X

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[5] M. L. Centeno, “Dependent Risks and Excess of Loss Reinsurance,” Insurance: Mathematics and Economics, Vol. 37, No. 2, 2005, pp. 229-238. doi:10.1016/j.insmatheco.2004.12.001

[6] F. Lindskog and A. J. McNeil, “Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling,” ASTIN Bulletin, Vol. 33, No. 2, 2003, pp. 209-238. doi:10.2143/AST.33.2.503691

[7] D. Pfeifer and J. Neslehová, “Modeling and Generating Dependent Risk Processes for IRM and DFA,” ASTIN Bulletin, Vol. 34, No. 2, 2004, pp. 333-360. doi:10.2143/AST.34.2.505147

[8] V. Chavez-Demoulin, P. Embrechts and J. Neslehová, “Quantitative Models for Operational Risk: Extremes, Dependence and Aggregation,” Journal of Banking and Finance, Vol. 30, No. 10, 2006, pp. 2635-2658. doi:10.1016/j.jbankfin.2005.11.008

[9] R. Cont and P. Tankov, “Financial Modelling with Jump Processes,” Champan & Hall, Boca Raton, 2004.

[10] Victorian Bushfires Royal Commission, “Final Report-Summary, Parliament of Victoria,” Parliament of Victoria, Melbourne, 2010.

[11] M. L. Burton and M. J. Hicks, “Hurricane Katrina: Preliminary Estimates of Commercial and Public Sector Damages,” Marshall University, Huntington, 2005.

[12] G. Makinen, “The Economic Effects of 9/11: A Retrospective Assessment,” Congressional Research Service, Washington DC, 2002.

[13] R. B. Nelsen, “An Introduction to Copulas,” Springer-Verlag, New York, 1999.

[14] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” Elsevier, New York, 1983.

[15] C. Klüppelberg and T. Mikosch, “Explosive Poisson Shot Noise Processes with Applications to Risk Reserves,” Bernoulli, Vol. 1, No. 1-2, 1995, pp. 125-147. doi:10.2307/3318683

[16] A. Dassios and J. Jang, “Pricing of Catastrophe Reinsurance & Derivatives Using the Cox Process with Shot Noise Intensity,” Finance & Stochastics, Vol. 7, No. 1, 2003, pp. 73-95. doi:10.1007/s007800200079

[17] A. Dassios and J. Jang, “Kalman-Bucy Filtering for Linear System Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Contracts,” Journal of Applied Probability, Vol. 42, No. 1, 2005, pp. 93-107. doi:10.1239/jap/1110381373

[18] A. Dassios and J. Jang, “The Distribution of the Interval between Events of a Cox Process with Shot Noise Intensity,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 2008, 2008, Article ID: 367170. doi:10.1155/2008/367170

[19] J. Jang and Y. Krvavych, “Arbitrage-Free Premium Calculation for Extreme Losses Using the Shot Noise Process and the Esscher Transform,” Insurance: Mathematics and Economics, Vol. 35, No. 1, 2004, pp. 97-111. doi:10.1016/j.insmatheco.2004.05.002

[20] J. Jang and G. Fu, “Transform Approach for Operational Risk Management: VaR and TCE,” Journal of Operational Risk, Vol. 3, No. 2, 2008, pp. 45-61.

[21] D. R. Cox and V. Isham, “Point Processes,” Chapman & Hall, London, 1980.

[22] A. McNeil, R. Frey and P. Embrechts, “Quantitative Risk Management: Concepts, Techniques and Tools,” Princeton University Press, Princeton, 2005.

[23] M. H. A. Davis, “Piecewise Deterministic Markov Processes: A General Class of Non Diffusion Stochastic Models,” Journal of the Royal Statistical Society Series B, Vol. 46, No. 3, 1984, pp. 353-388.

[24] A. Dassios and P. Embrechts, “Martingales and Insurance Risk,” Communications in Statistics. Stochastic Models, Vol. 5, No. 2, 1989, pp. 181-217. doi:10.1080/15326348908807105

[25] T. Rolski H. Schmidli, V. Schmidt and J. L. Teugels, “Stochastic Processes for Insurance and Finance,” John Wiley & Sons, Hoboken, 1998.

[26] K. R. Castleman, “Digital Image Processing,” Prentice Hall, Englewood Cliffs, 1996.

[27] R. C. Gonzalez and R. E. Woods, “Digital Image Processing,” 2nd Edition, Prentice Hall, Upper Saddle River, 2002.

[28] R. C. Gonzalez, R. E. Woods and S. L. Eddins, “Digital Image Processing Using MATLAB,” Prentice Hall, Upper Saddle River, 2004.

[1] N. Bauerle and A. Müller, “Modeling and Comparing Dependencies in Multivariate Risk Portfolios,” ASTIN Bulletin, Vol. 28, No. 1, 1998, pp. 59-76. doi:10.2143/AST.28.1.519079

[2] H. Cossette, P. Gaillardetz, E. Marceau and J. Rioux, “On Two Dependent Individual Risk Models,” Insurance: Mathematics and Economics, Vol. 30, No. 2, 2002, pp. 153-166. doi:10.1016/S0167-6687(02)00094-X

[3] C. Genest, E. Marceau and M. Mesfioui, “Compound Poisson Approximations for Individual Models with Dependent Risks,” Insurance: Mathematics and Economics, Vol. 32, No. 1, 2003, pp. 73-91. doi:10.1016/S0167-6687(02)00205-6

[4] N. Bauerle and R. Grübel, “Multivariate Counting Processes: Copulas and Beyond,” ASTIN Bulletin, Vol. 35, No. 2, 2005, pp. 379-408. doi:10.2143/AST.35.2.2003459

[5] M. L. Centeno, “Dependent Risks and Excess of Loss Reinsurance,” Insurance: Mathematics and Economics, Vol. 37, No. 2, 2005, pp. 229-238. doi:10.1016/j.insmatheco.2004.12.001

[6] F. Lindskog and A. J. McNeil, “Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling,” ASTIN Bulletin, Vol. 33, No. 2, 2003, pp. 209-238. doi:10.2143/AST.33.2.503691

[7] D. Pfeifer and J. Neslehová, “Modeling and Generating Dependent Risk Processes for IRM and DFA,” ASTIN Bulletin, Vol. 34, No. 2, 2004, pp. 333-360. doi:10.2143/AST.34.2.505147

[8] V. Chavez-Demoulin, P. Embrechts and J. Neslehová, “Quantitative Models for Operational Risk: Extremes, Dependence and Aggregation,” Journal of Banking and Finance, Vol. 30, No. 10, 2006, pp. 2635-2658. doi:10.1016/j.jbankfin.2005.11.008

[9] R. Cont and P. Tankov, “Financial Modelling with Jump Processes,” Champan & Hall, Boca Raton, 2004.

[10] Victorian Bushfires Royal Commission, “Final Report-Summary, Parliament of Victoria,” Parliament of Victoria, Melbourne, 2010.

[11] M. L. Burton and M. J. Hicks, “Hurricane Katrina: Preliminary Estimates of Commercial and Public Sector Damages,” Marshall University, Huntington, 2005.

[12] G. Makinen, “The Economic Effects of 9/11: A Retrospective Assessment,” Congressional Research Service, Washington DC, 2002.

[13] R. B. Nelsen, “An Introduction to Copulas,” Springer-Verlag, New York, 1999.

[14] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” Elsevier, New York, 1983.

[15] C. Klüppelberg and T. Mikosch, “Explosive Poisson Shot Noise Processes with Applications to Risk Reserves,” Bernoulli, Vol. 1, No. 1-2, 1995, pp. 125-147. doi:10.2307/3318683

[16] A. Dassios and J. Jang, “Pricing of Catastrophe Reinsurance & Derivatives Using the Cox Process with Shot Noise Intensity,” Finance & Stochastics, Vol. 7, No. 1, 2003, pp. 73-95. doi:10.1007/s007800200079

[17] A. Dassios and J. Jang, “Kalman-Bucy Filtering for Linear System Driven by the Cox Process with Shot Noise Intensity and Its Application to the Pricing of Reinsurance Contracts,” Journal of Applied Probability, Vol. 42, No. 1, 2005, pp. 93-107. doi:10.1239/jap/1110381373

[18] A. Dassios and J. Jang, “The Distribution of the Interval between Events of a Cox Process with Shot Noise Intensity,” Journal of Applied Mathematics and Stochastic Analysis, Vol. 2008, 2008, Article ID: 367170. doi:10.1155/2008/367170

[19] J. Jang and Y. Krvavych, “Arbitrage-Free Premium Calculation for Extreme Losses Using the Shot Noise Process and the Esscher Transform,” Insurance: Mathematics and Economics, Vol. 35, No. 1, 2004, pp. 97-111. doi:10.1016/j.insmatheco.2004.05.002

[20] J. Jang and G. Fu, “Transform Approach for Operational Risk Management: VaR and TCE,” Journal of Operational Risk, Vol. 3, No. 2, 2008, pp. 45-61.

[21] D. R. Cox and V. Isham, “Point Processes,” Chapman & Hall, London, 1980.

[22] A. McNeil, R. Frey and P. Embrechts, “Quantitative Risk Management: Concepts, Techniques and Tools,” Princeton University Press, Princeton, 2005.

[23] M. H. A. Davis, “Piecewise Deterministic Markov Processes: A General Class of Non Diffusion Stochastic Models,” Journal of the Royal Statistical Society Series B, Vol. 46, No. 3, 1984, pp. 353-388.

[24] A. Dassios and P. Embrechts, “Martingales and Insurance Risk,” Communications in Statistics. Stochastic Models, Vol. 5, No. 2, 1989, pp. 181-217. doi:10.1080/15326348908807105

[25] T. Rolski H. Schmidli, V. Schmidt and J. L. Teugels, “Stochastic Processes for Insurance and Finance,” John Wiley & Sons, Hoboken, 1998.

[26] K. R. Castleman, “Digital Image Processing,” Prentice Hall, Englewood Cliffs, 1996.

[27] R. C. Gonzalez and R. E. Woods, “Digital Image Processing,” 2nd Edition, Prentice Hall, Upper Saddle River, 2002.

[28] R. C. Gonzalez, R. E. Woods and S. L. Eddins, “Digital Image Processing Using MATLAB,” Prentice Hall, Upper Saddle River, 2004.