Abstract
In this paper we consider the problem of studying the gap between bounds of risk measures of sums of non-independentrandom variables. Owing to the choice of the context of where to set the problem, namely that of distortion risk measures, we first deduce an explicit formula for the risk measure of a discrete risk by referring to its writing as sum of layers. Then, we examine the case of sums of discrete risks with identical distribution. Upper and lower bounds for risk measures of sums of risks are presented in the case of concave distortion functions. Finally, the attention is devoted to the analysis of the gap between risk measures of upper and lower bounds, with the aim of optimizing it.
This research was partially supported by MIUR.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Campana, A., Ferretti, P.: On distortion risk measures for sums of dis-crete and identically distributed risks. Giornale dell’Istituto Italiano degli At-tuari, LXVIII: 89–104 (2005)
Dhaene, J., Denuit, M.: The safest dependence structure among risks. Insur-ance: Mathematics and Economics, 25: 11–21 (1999)
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vynckce, D.: The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathe-matics and Economics, 31: 3–33 (2002)
Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R., Vynckce, D.: The concept of comonotonicity in actuarial science and finance: applications. Insurance: Mathematics and Economics, 31: 133–161 (2002)
Dhaene, J., Vanduffel, S., Tang, Q. H., Goovaerts, M. J., Kaas, R., Vyncke, D.: Solvency capital, risk measures and comonotonicity: a review. Research Re-port OR 0416, Department of Applied Economics, K.U. Leuven (2004)
Kaas, R., Dhaene, J., Goovaerts, M. J.: Upper and lower bounds for sums of random variables. Insurance: Mathematics and Economics, 27: 151–168 (2000)
Wang, S.: Insurance pricing and increased limits ratemaking by proportional hazard transforms. Insurance: Mathematics and Economics, 17: 43–54 (1995)
Wang, S.: Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26: 71–92 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer, Milan
About this paper
Cite this paper
Campana, A., Ferretti, P. (2008). Bounds for Concave Distortion Risk Measures for Sums of Risks. In: Perna, C., Sibillo, M. (eds) Mathematical and Statistical Methods in Insurance and Finance. Springer, Milano. https://doi.org/10.1007/978-88-470-0704-8_6
Download citation
DOI: https://doi.org/10.1007/978-88-470-0704-8_6
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-0703-1
Online ISBN: 978-88-470-0704-8
eBook Packages: Business and EconomicsEconomics and Finance (R0)