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Bounds for Quantile-Based Risk Measures of Functions of Dependent Random Variables

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  • Goncalves Marcelo

    (Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil)

  • Kolev Nikolai

    (Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil)

  • Fabris Antonio Elias

    (Department of Applied Mathematics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil)

Abstract

This paper introduces two techniques for computing bounds for several quantile-based risk measures based on distortion functions. Knowledge about the marginal distribution of the involved random variables is assumed with the optional assumption of some partial information about the structure of dependence. The aim is to derive bounds for risk measures of functions of dependent random variables. Several examples taken from an insurance context are given. We use Embrechts et al. (2003) methodology and the stochastic ordering approach to derive bounds for various risk measures in the bi-dimensional and the multidimensional cases.

Suggested Citation

  • Goncalves Marcelo & Kolev Nikolai & Fabris Antonio Elias, 2008. "Bounds for Quantile-Based Risk Measures of Functions of Dependent Random Variables," Stochastics and Quality Control, De Gruyter, vol. 23(1), pages 55-70, January.
    • Handle: RePEc:bpj:ecqcon:v:23:y:2008:i:1:p:55-70:n:7
    DOI: 10.1515/EQC.2008.55
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    References listed on IDEAS

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    1. Embrechts, Paul & Puccetti, Giovanni, 2006. "Bounds for functions of multivariate risks," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 526-547, February.
    2. Denneberg, Dieter, 1990. "Premium Calculation: Why Standard Deviation Should be Replaced by Absolute Deviation1," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 181-190, November.
    3. Wang, Shaun, 1996. "Premium Calculation by Transforming the Layer Premium Density," ASTIN Bulletin, Cambridge University Press, vol. 26(1), pages 71-92, May.
    4. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    5. Paul Embrechts & Giovanni Puccetti, 2006. "Bounds for Functions of Dependent Risks," Finance and Stochastics, Springer, vol. 10(3), pages 341-352, September.
    6. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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