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Permutation Monotone Functions of Random Vectors with Applications in Financial and Actuarial Risk Management

Part of: Distribution theory - Probability Mathematical economics

Published online by Cambridge University Press:  04 January 2016

Xiaohu Li  and
Yinping You
Xiaohu Li*
Affiliation:
Stevens Institute of Technology
Yinping You*
Affiliation:
Huaqiao University
*
∗ Postal address: Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA. Email address: mathxhli@hotmail.com, xiaohu.li@stevens.edu
∗∗ Postal address: School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, China.
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Abstract

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In this paper we develop two permutation theorems on argument increasing functions of a multivariate random vector and a real parameter vector. We use the unified approach of our two theorems to provide some important theoretical results on the capital allocation in actuarial science, the deductible and upper limit allocations in insurance policy, and portfolio allocation in financial engineering. Our results successfully improve or extend the corresponding works in the literature.

Keywords

Archimedean copulaarrangement increasingdeductibledistortion measurestochastic orderupper limitutility function

MSC classification

Primary: 91B16: Utility theory 91B30: Risk theory, insurance
Secondary: 60E15: Inequalities; stochastic orderings
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 270 - 291
Copyright
© Applied Probability Trust 

References

Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Boland, P. J. and Proschan, F. (1988). Multivariate arrangement increasing functions with applications in probability and statistics. J. Multivariate Anal. 25, 286–298.Google Scholar
Boland, P. J., Proschan, F. and Tong, Y. L. (1988). Moment and geometric probability inqualities arising from arrangement increasing functions. Ann. Prob. 16, 407–413.Google Scholar
Chen, Z. and Hu, T. (2008). Asset proportions in optimal portfolios with dependent default risks. Insurance Math. Econom. 43, 223–226.Google Scholar
Cheung, K. C. (2006). Optimal portfolio problem with unknown dependency structure. Insurance Math. Econom. 38, 167–175.Google Scholar
Cheung, K. C. (2007). Optimal allocation of policy limits and deductibles. Insurance Math. Econom. 41, 382–391.CrossRefGoogle Scholar
Cheung, K. C. and Yang, H. (2004). Ordering optimal proportions in the asset allocation problem with dependent default risks. Insurance Math. Econom. 35, 595–609.CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005). Actuarial Theory for Dependent Risks. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Dhaene, J., Tsanakas, A., Valdez, E. A. and Vanduffel, S. (2012). Optimal capital allocation principles. J. Risk Insurance 79, 1–28.Google Scholar
Dhaene, J. et al. (2006). Risk measures and comonotonicity: a review. Stoch. Models 22, 573–606.CrossRefGoogle Scholar
Esary, J. D. and Proschan, F. (1972). Relationships among some concepts of bivariate dependence. Ann. Math. Statist. 43, 651–655.Google Scholar
Frostig, E., Zaks, Y. and Levikson, B. (2007). Optimal pricing for a heterogeneous portfolio for a given risk factor and convex distance measure. Insurance Math. Econom. 40, 459–467.Google Scholar
Furman, E. and Zitikis, R. (2008). Weighted risk capital allocations. Insurance Math. Econom. 43, 263–269.Google Scholar
Genest, C. and MacKay, J. (1986a). Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14, 145–159.Google Scholar
Genest, C. and MacKay, J. (1986b). The Joy of copulas: bivariate distributions with uniform marginals. Amer. Statistician 40, 280–283.Google Scholar
Genest, C. and Rivest, L.-P. (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88, 1034–1043.Google Scholar
Hennessy, D. A. and Lapan, H. E. (2002). The use of Archimedean copulas to model portfolio allocations. Math. Finance 12, 143–154.Google Scholar
Hollander, M., Proschan, F. and Sethuraman, J. (1977). Functions decreasing in transportation and their applications in ranking problems. Ann. Statist. 5, 722–733.Google Scholar
Hua, L. and Cheung, K. C. (2008a). Stochastic orders of scalar products with applications. Insurance Math. Econom. 42, 865–872.CrossRefGoogle Scholar
Hua, L. and Cheung, K. C. (2008b). Worst allocations of policy limits and deductibles. Insurance Math. Econom. 43, 93–98.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Google Scholar
Kijima, M. and Ohnishi, M. (1996). Portfolio selection problems via the bivariate characterization of stochastic dominance relations. Math. Finance 6, 237–277.Google Scholar
Kimberling, C. H. (1974). A probabilistic interpretation of complete monotonicity. Aequationes Math. 10, 152–164.CrossRefGoogle Scholar
Laeven, R. J. A. and Goovaerts, M. J. (2004). An optimization approach to the dynamic allocation of economic capital. Insurance Math. Econom. 35, 299–319.Google Scholar
Landsberger, M. and Meilijson, I. (1990). Demand for risky financial assets: a portfolio analysis. J. Econom. Theory 50, 204–213.Google Scholar
Landsman, Z. M. and Valdez, E. A. (2003). Tail conditional expectations for elliptical distributions. N. Amer. Actuarial J.. 7, 55–71.Google Scholar
Li, H. and Li, X. (eds) (2013). Stochastic Orders in Reliability and Risk. Springer, New York.Google Scholar
Li, X. and You, Y. (2014). A note on allocation of portfolio shares of random assets with Archimedean copula. Ann. Operat. Res. 212, 155–167.Google Scholar
Lu, Z. Y. and Meng, L. L. (2011). Stochastic comparisons for allocations of policy limits and deductibles with applications. Insurance Math. Econom. 48, 338–343.Google Scholar
Marshall, A. W. and Olkin, I. (1988). Families of multivariate distributions. J. Amer. Statist. Assoc. 83, 834–841.Google Scholar
Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York.Google Scholar
McNeil, A. J. and Neslehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l 1-norm symmetric distributions. Ann. Statist. 37, 3059–3097.Google Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Myers, S. C. and Read, J. A. Jr. (2001). Capital allocation for insurance companies. J. Risk Insurance 68, 545–580.CrossRefGoogle Scholar
Nelsen, R. B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.Google Scholar
Panjer, H. H. (2001). Measurement of risk, solvency requirements, and allocation of capital within financial conglomerates. Res. Rep., Department of Res. Rep..Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Tsanakas, A. (2004). Dynamic capital allocation with distortion risk measures. Insurance Math. Econom. 35, 223–243.Google Scholar
Tsanakas, A. (2009). To split or not to split: capital allocation with convex risk measures. Insurance Math. Econom. 44, 268–277.Google Scholar
Xu, M. and Hu, T. (2012). Stochastic comparisons of capital allocations with applications. Insurance Math. Econom. 50, 293–298.Google Scholar
Zhuang, W., Chen, Z. and Hu, T. (2009). Optimal allocation of policy limits and deductibles under distortion risk measures. Insurance Math. Econom. 44, 409–414.CrossRefGoogle Scholar