Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T19:34:47.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Beta transform and discounted aggregate claims under dependency

Published online by Cambridge University Press:  13 July 2018

Zhehao Zhang  and
Shuanming Li
Zhehao Zhang*
Affiliation:
Department of Economics, Centre for Actuarial Studies, The University of Melbourne, Parkville, VIC 3010, Australia
Shuanming Li
Affiliation:
Department of Economics, Centre for Actuarial Studies, The University of Melbourne, Parkville, VIC 3010, Australia
*
*Correspondence to: Zhehao Zhang, Department of Economics, Centre for Actuarial Studies, The University of Melbourne, Parkville, VIC 3010, Australia. Tel: +61 3 83445616. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper starts with the Beta transform and discusses the stochastic ordering properties of this transform under different parameter settings. Later, the distribution of discounted aggregate claims in a compound renewal risk model with dependence between inter-claim times and claim sizes is studied. Recursive formulas for moments and joint moments are expressed in terms of the Beta transform of the inter-claim times and claim severities. Particularly, our moments formula is more explicit and computation-friendly than earlier ones in the references. Lastly, numerical examples are provided to illustrate our results.

Keywords

Discounted aggregate claimsRenewal risk processCopulaBeta distributional transformStochastic ordering
Type
Paper
Information
Annals of Actuarial Science , Volume 13 , Issue 2 , September 2019 , pp. 241 - 267
Copyright
© Institute and Faculty of Actuaries 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albrecher, H. & Boxma, O.J. (2004). A ruin model with dependence between claim sizes and claim intervals. Insurance: Mathematics and Economics, 35(2), 245–254.Google Scholar
Albrecher, H. & Boxma, O.J. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance: Mathematics and Economics, 37(3), 650–672.Google Scholar
Albrecher, H. & Teugels, J.L. (2006). Exponential behavior in the presence of dependence in risk theory. Journal of Applied Probability, 43(1), 257–273.Google Scholar
Badescu, A.L., Breuer, L., Drekic, S., Latouche, G. & Stanford, D.A. (2005). The surplus prior to ruin and the deficit at ruin for a correlated risk process. Scandinavian Actuarial Journal, 2005(6), 433–445.Google Scholar
Barges, M., Cossette, H., Loisel, S. & Marceau, E. (2011). On the moments of aggregate discounted claims with dependence introduced by a FGM copula. ASTIN Bulletin: The Journal of the IAA, 41(1), 215–238.Google Scholar
Boudreault, M., Cossette, H., Landriault, D. & Marceau, E. (2006). On a risk model with dependence between interclaim arrivals and claim sizes. Scandinavian Actuarial Journal, 2006(5), 265–285.Google Scholar
Bromwich, T.J.I. & MacRobert, T.M. (1947). An Introduction to the Theory of Infinite Series. Macmillan, London.Google Scholar
Chadjiconstantinidis, S. & Vrontos, S. (2014). On a renewal risk process with dependence under a Farlie–Gumbel–Morgenstern copula. Scandinavian Actuarial Journal, 2014(2), 125–158.Google Scholar
Cheung, E.C.K. (2012). A unifying approach to the analysis of business with random gains. Scandinavian Actuarial Journal, 2012(3), 153–182.Google Scholar
Cossette, H., Marceau, E. & Marri, F. (2008). On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula. Insurance: Mathematics and Economics, 43(3), 444–455.Google Scholar
Cossette, H., Marceau, E. & Marri, F. (2010). Analysis of ruin measures for the classical compound Poisson risk model with dependence. Scandinavian Actuarial Journal, 2010(3), 221–245.Google Scholar
Delbaen, F. & Haezendonck, J. (1987). Classical risk theory in an economic environment. Insurance: Mathematics and Economics, 6(2), 85–116.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. & Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, Hoboken, NJ and Chichester.Google Scholar
Jang, J.-W. (2004). Martingale approach for moments of discounted aggregate claims. Journal of Risk and Insurance, 71(2), 201–211.Google Scholar
Janssen, J. & Reinhard, J.-M. (1985). Probabilités de ruine pour une classe de modeles de risque semi-Markoviens. ASTIN Bulletin, 15(2), 123–133.Google Scholar
Kim, B. & Kim, H.-S. (2007). Moments of claims in a Markovian environment. Insurance: Mathematics and Economics, 40(3), 485–497.Google Scholar
Lebedev, N.N. (1972). Special Functions and Their Applications . Dover Publications, New York.Google Scholar
Léveillé, G. & Adékambi, F. (2011). Covariance of discounted compound renewal sums with a stochastic interest rate. Scandinavian Actuarial Journal, 2011(2), 138–153.Google Scholar
Léveillé, G. & Adékambi, F. (2012). Joint moments of discounted compound renewal sums. Scandinavian Actuarial Journal, 2012(1), 40–55.Google Scholar
Léveillé, G. & Garrido, J. (2001 a). Moments of compound renewal sums with discounted claims. Insurance: Mathematics and Economics, 28(2), 217–231.Google Scholar
Léveillé, G. & Garrido, J. (2001 b). Recursive moments of compound renewal sums with discounted claims. Scandinavian Actuarial Journal, 2001(2), 98–110.Google Scholar
Léveillé, G., Garrido, J. & Wang, Y.F. (2010). Moment generating functions of compound renewal sums with discounted claims. Scandinavian Actuarial Journal, 2010(3), 165–184.Google Scholar
Ren, J. (2008). On the Laplace transform of the aggregate discounted claims with Markovian arrivals. North American Actuarial Journal, 12(2), 198–206.Google Scholar
Rodriguez-Lallena, J. & Ãbeda-Flores, M. (2004). A new class of bivariate copulas. Statistics and Probability Letters, 66(3), 315–325.Google Scholar
Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. Springer Science & Business Media, New York.Google Scholar
Taylor, G.C. (1979). Probability of ruin under inflationary conditions or under experience rating. ASTIN Bulletin, 10(2), 149–162.Google Scholar
Tse, Y.-K. (2009). Nonlife Actuarial Models: Theory, Methods and Evaluation. Cambridge University Press, Cambridge and New York.Google Scholar
Wang, S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17(1), 43–54.Google Scholar
Wang, Y.F., Garrido, J. & Léveillé, G. (2016). The distribution of discounted compound PH–renewal processes. Methodology and Computing in Applied Probability, 20(1), 1–28.Google Scholar
Willmot, G.E. (1989). The total claims distribution under inflationary conditions. Scandinavian Actuarial Journal, 1989(1), 1–12.Google Scholar
Wirch, J.L. & Hardy, M.R. (1999). A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics, 25(3), 337–347.Google Scholar
Woo, J.-K. & Cheung, E.C. (2013). A note on discounted compound renewal sums under dependency. Insurance: Mathematics and Economics, 52(2), 170–179.Google Scholar