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A FORM OF MULTIVARIATE PARETO DISTRIBUTION WITH APPLICATIONS TO FINANCIAL RISK MEASUREMENT

Published online by Cambridge University Press:  31 August 2016

Jianxi Su  and
Edward Furman
Jianxi Su
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
Edward Furman*
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
*
E-Mail: [email protected]
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Abstract

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A new multivariate distribution possessing arbitrarily parametrized and positively dependent univariate Pareto margins is introduced. Unlike the probability law of Asimit et al. (2010), the structure in this paper is absolutely continuous with respect to the corresponding Lebesgue measure. The distribution is of importance to actuaries through its connections to the popular frailty models, as well as because of the capacity to describe dependent heavy-tailed risks. The genesis of the new distribution is linked to a number of existing probability models, and useful characteristic results are proved. Expressions for, e.g., the decumulative distribution and probability density functions, (joint) moments and regressions are developed. The distributions of minima and maxima, as well as, some weighted risk measures are employed to exemplify possible applications of the distribution in insurance.

Keywords

Multivariate Pareto distributionscharacterizationsdependenceweighted risk measuresminimamaxima
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 1 , January 2017 , pp. 331 - 357
Copyright
Copyright © Astin Bulletin 2016 

References

Arnold, B.C. (1983) Pareto Distributions. Fairland: International Cooperative Publishing House.Google Scholar
Asimit, V., Furman, E. and Vernic, R. (2010) On a multivariate Pareto distribution. Insurance: Mathematics and Economics, 46 (2), 308–316.Google Scholar
Asimit, V., Vernic, R. and Zitikis, R. (2013) Evaluating risk measures and capital allocations based on multi-losses driven by a heavy-tailed background risk: The multivariate Pareto-II model. Risk, 1 (1), 14–33.Google Scholar
Asimit, V., Vernic, R. and Zitikis, R. (2016) Background risk models and stepwise portfolio construction. Methodology and Computing in Applied Probability, in press.Google Scholar
Balkema, A. and de Haan, L. (1974) Residual life time at great age. Annals of Probability, 2 (5), 792–804.Google Scholar
Benson, D.A., Schumer, R. and Meerschaert, M.M. (2007) Recurrence of extreme events with power-law interarrival times. Geophysical Research Letters, 34 (16), 1–5.Google Scholar
Boucher, J.P., Denuit, M. and Guillén, M. (2008) Models of insurance claim counts with time dependence based on generalization of Poisson and negative binomial distributions. Variance, 2 (1), 135–162.Google Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997) Actuarial Mathematics, 2nd ed. Schaumburg: Society of Actuaries.Google Scholar
Cebrián, A.C., Denuit, M. and Lambert, P. (2003) Generalized Pareto fit to the society of actuaries large claims database. North American Actuarial Journal, 7 (3), 18–36.Google Scholar
Chavez-Demoulin, V., Embrechts, P. and Hofert, M. (2015) An extreme value approach for modeling operational risk losses depending on covariates. Journal of Risk and Insurance, in press.Google Scholar
Cherian, K.C. (1941) A bivariate correlated gamma-type distribution function. Journal of the Indian Mathematical Society 5, 133–144.Google Scholar
Chiragiev, A. and Landsman, Z. (2009) Multivariate flexible Pareto model: Dependency structure, properties and characterizations. Statistics and Probability Letters, 79 (16), 1733–1743.Google Scholar
Choo, W. and de Jong, P. (2009) Loss reserving using loss aversion functions. Insurance: Mathematics and Economics, 45 (2), 271–277.Google Scholar
Choo, W. and de Jong, P. (2010) Determining and allocating diversification benefits for a portfolio of risks. ASTIN Bulletin, 40 (1), 257–269.Google Scholar
Embrechts, P., McNeil, A. and Straumann, D. (2002) Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond (ed. Dempster, M.), Cambridge: Cambridge University Press.Google Scholar
Engelmann, B. and Rauhmeier, R. (2011) The Basel II Risk Parameters: Estimation, Validation, Stress Testing - with Applications to Loan Risk Management. Berlin: Springer.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. New York: John Wiley and Sons.Google Scholar
Franke, G., Schlesinger, H. and Stapleton, R.C. (2006) Multiplicative background risk. Management Science, 52 (1), 146–153.Google Scholar
Furman, E. (2008) On a multivariate gamma distribution. Statistics and Probability Letters, 78 (15), 2353–2360.Google Scholar
Furman, E. and Landsman, Z. (2005) Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics and Economics, 37 (3), 635–649.Google Scholar
Furman, E. and Landsman, Z. (2010) Multivariate Tweedie distributions and some related capital-at-risk analysis. Insurance: Mathematics and Economics, 46 (2), 351–361.Google Scholar
Furman, E. and Zitikis, R. (2008a) Weighted premium calculation principles. Insurance: Mathematics and Economics, 42 (1), 459–465.Google Scholar
Furman, E. and Zitikis, R. (2008b) Weighted risk capital allocations. Insurance: Mathematics and Economics 43 (2), 263–269.Google Scholar
Furman, E. and Zitikis, R. (2010) General Stein-type covariance decompositions with applications to insurance and finance. ASTIN Bulletin 40 (1), 369–375.Google Scholar
Gabaix, X., Gopikrishnan, P., Plerou, V. and Stanley, H.E. (2003) A theory of power-law distributions in financial market fluctuations. Nature, 423, 267–270.Google Scholar
Gollier, C. and Pratt, J.W. (1996) Weak proper risk aversion and the tempering effect of background risk. Econometrica 64 (5), 1109–1123.Google Scholar
Gradshteyn, I.S. and Ryzhik, I.M. (2007) Table of Integrals, Series and Products. 7th ed. New York: Academic Press.Google Scholar
Koedijk, K.G., Schafgans, M.M.A. and de Vries, C.G. (1990) The tail index of exchange rate returns. Journal of International Economics, 29 (1–2), 93–108.Google Scholar
Kotz, S., Balakrishnan, N. and Johnson, N.L. (2000) Continuous Multivariate Distributions. 2nd ed. New York: Wiley.Google Scholar
Longin, F.M. (1996) The asymptotic distribution of extreme stock market returns. Journal of Business, 69 (3), 383–408.Google Scholar
Mathai, A.M. and Moschopoulos, P.G. (1991) On a multivariate gamma. Journal of Multivariate Analysis 39 (1), 135–153.Google Scholar
Mathai, A.M. and Moschopoulos, P.G. (1992) A form of multivariate gamma distribution. Annals of the Institute of Statistical Mathematics, 44 (1), 97–106.Google Scholar
Meyers, G.G. (2007) The common-shock model for correlated insurance losses. Variance, 1 (1), 40–52.Google Scholar
Moschopoulos, P.G. (1985) The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics, 37, 541–544.Google Scholar
Pareto, V. (1897) The new theories of economics. Journal of Political Economy, 5 (4), 485–502.Google Scholar
Pfeifer, D. and Nešlehová, J. (2004) Modeling and generating dependent risk processes for IRM and DFA. ASTIN Bulletin, 34 (2), 333–360.Google Scholar
Pickands, J. (1975) Statistical inference using extreme order statistics. The Annals of Statistics, 3 (1), 119–131.Google Scholar
Ramabhadran, V. (1951) A multivariate gamma-type distribution. Journal of Multivariate Analysis 38, 213–232.Google Scholar
Soprano, A., Crielaard, B., Piacenza, F. and Ruspantini, D. (2009) Measuring Operational and Reputational Risk: A Practitioner's Approach. Chichester: Wiley.Google Scholar
Standard & Poor's (2015) Default, transition and recovery: 2014 annual global corporate default study and rating transitions. Technical report, Standard and Poor's, New York.Google Scholar
Tsanakas, A. (2008) Risk measurement in the presence of background risk. Insurance: Mathematics and Economics, 42 (2), 520–528.Google Scholar
Vernic, R. (1997) On the bivariate generalized Poisson distribution. ASTIN Bulletin, 27 (1), 23–32.Google Scholar
Vernic, R. (2000) A multivariate generalization of the generalized Poisson distribution. ASTIN Bulletin, 30 (1), 57–67.Google Scholar
Vernic, R. (2011) Tail conditional expectation for the multivariate Pareto distribution of the second kind: Another approach. Methodology and Computing in Applied Probability, 13 (1), 121–137.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26 (1), 71–92.Google Scholar