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Methods for generating coherent distortion risk measures

Published online by Cambridge University Press:  29 August 2018

Ranadeera G.M. Samanthi  and
Jungsywan Sepanski
Ranadeera G.M. Samanthi*
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA
Jungsywan Sepanski
Affiliation:
Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA
*
*Correspondence to: Ranadeera G.M. Samanthi, Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA. E-mail: saman1rg@cmich.edu
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Abstract

This paper presents methods for generating new distortion functions utilising distribution functions and composite distribution functions. To ensure the coherency of the corresponding distortion risk measures, the concavity of the proposed distortion functions is established by restricting the parameter space of the generating distribution. Closed-form expressions for risk measures are derived for some cases. Numerical and graphical results are presented to demonstrate the effects of parameter values on the risk measures for exponential, Pareto and log-normal losses. In addition, we apply the proposed distortion functions to derive risk measures for a segregated fund guarantee.

Keywords

Risk measureDistortion functionKumaraswamy distortionTruncated normal distortionExponential-exponential distortion

Information

Type
Review
Information
Annals of Actuarial Science , Volume 13 , Issue 2 , September 2019 , pp. 400 - 416
Copyright
© Institute and Faculty of Actuaries 2018 

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References

Alzaatreh, A., Lee, C. & Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71, 63–79.Google Scholar
Alzaatreh, A., Lee, C. & Famoye, F. (2014). T-normal family of distributions: a new approach to generalize the normal distribution. Journal of Statistical Distributions and Applications, 2014, 1–16.Google Scholar
Alzaatreh, A., Lee, C. & Famoye, F. (2015). Family of generalized gamma distributions: properties and applications. Journal of Mathematical Statistics, 45, 869–886.Google Scholar
Artzner, P., Delbaen, F., Eber, J.M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.Google Scholar
Belles-Sampera, J., Guillen, M. & Santolino, M. (2016). What attitudes to risk underlie distortion risk measure choices? Insurance, Mathematics and Economics, 68, 101–109.Google Scholar
Castellares, F., Montenegro, L.C. & Cordeiro, G.M. (2013). The beta log-normal distribution. Journal of Statistical Computation and Simulation, 83, 203–228.Google Scholar
Cordeiro, G.M. & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81, 883–898.Google Scholar
Cotter, J. & Dowd, K. (2006). Extreme spectral risk measures: an application to futures clearinghouse margin requirements. Journal of Banking & Finance, 30, 3469–3485.Google Scholar
Eugene, N., Lee, C. & Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics – Theory and Methods, 31, 497–512.Google Scholar
Goovaerts, M., Kaas, R., Dhaene, J. & Tang, Q. (2004). Some new classes of consistent risk measures. Insurance: Mathematics and Economics, 34, 505–516.Google Scholar
Johnson, N.L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd edition. Wiley, New York.Google Scholar
Kaiser, T. & Brazauskas, V. (2006). Interval estimation of actuarial risk measures. North American Actuarial Journal, 10, 249–268.Google Scholar
Kotz, S. & Van Dorp, J.R. (2004). Beyond Beta: Other Continuous Families of Distributions With Bounded Support and Applications. World Scientific, London.Google Scholar
Nadarajah, S. & Rocha, R. (2016). Newdistns: an R package for new families of distributions. Journal of Statistical Software, 69(10), 1–32.Google Scholar
Pflug, G. (2009). On distortion functionals. Statistics and Decisions, 27, 201–209.Google Scholar
Sereda, E.N., Bronshtein, E.M., Rachev, S.T., Fabozzi, F.J., Sun, W. & Stoyanov, S.V. (2010). Distortion Risk measures in portfolio optimization in handbook of portfolio construction. Business and Economics, 3, 649–673.Google Scholar
Tsukahara, H. (2014). Estimation of distortion risk measures. Journal of Financial Econometrics, 12, 213–235.Google Scholar
Wang, S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17, 43–54.Google Scholar
Wang, S. (2000). A class of distortion operations for pricing financial and insurance risks. Journal of Risk and Insurance, 67, 15–36.Google Scholar
Wirch, J. & Hardy, M. (1999). A synthesis of risk measures for capital adequacy. Insurance: Mathematics and Economics, 25, 337–347.Google Scholar