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RISK MEASURES DERIVED FROM A REGULATOR’S PERSPECTIVE ON THE REGULATORY CAPITAL REQUIREMENTS FOR INSURERS

Published online by Cambridge University Press:  22 July 2020

Jun Cai
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ONN2L 3G1, Canada, E-Mail: [email protected]
Tiantian Mao*
Affiliation:
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui, China, E-Mail: [email protected]

Abstract

In this study, we propose new risk measures from a regulator’s perspective on the regulatory capital requirements. The proposed risk measures possess many desired properties, including monotonicity, translation-invariance, positive homogeneity, subadditivity, nonnegative loading, and stop-loss order preserving. The new risk measures not only generalize the existing, well-known risk measures in the literature, including the Dutch, tail value-at-risk (TVaR), and expectile measures, but also provide new approaches to generate feasible and practical coherent risk measures. As examples of the new risk measures, TVaR-type generalized expectiles are investigated in detail. In particular, we present the dual and Kusuoka representations of the TVaR-type generalized expectiles and discuss their robustness with respect to the Wasserstein distance.

Type
Research Article
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

Ang, A., Chen, J. and Xing, Y. (2006) Downside risk. The Review of Financial Studies, 19(4), 11911239.CrossRefGoogle Scholar
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9(3), 203228.CrossRefGoogle Scholar
Baroni, P., Pelessoni, R. and Vicig, P. (2009) Generalizing Dutch risk measures through imprecise previsions. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17(2), 153177.CrossRefGoogle Scholar
BCBS (2016) Minimum Capital Requirements for Market Risk. Basel Committee on Banking 6 Supervision. Basel: Bank for International Settlements. https://www.bis.org/bcbs/publ/d352.htm (January 2016).Google Scholar
BCBS (2019) Minimum Capital Requirements for Market Risk. Basel Committee on Banking 8 Supervision. Basel: Bank for International Settlements. https://www.bis.org/bcbs/publ/d457.htm (February 2019).Google Scholar
Belles-Sampera, J., Guillén, M. and Santolino, M. (2014a) GlueVaR risk measures in capital allocation applications. Insurance: Mathematics and Economics, 58, 132137.Google Scholar
Belles-Sampera, J., Guillén, M. and Santolino, M. (2014b) Beyond value-at-risk: GlueVaR distortion risk measures. Risk Analysis, 34(1), 121134.CrossRefGoogle ScholarPubMed
Bellini, F., Klar, B., Müller, A. and Gianin, E.R. (2014) Generalized quantiles as risk measures. Insurance: Mathematics and Economics, 54, 4148.Google Scholar
Bickel, P.J. and Freedman, D.A. (1981) Some asymptotic theory for the bootstrap. Annals of Statistics, 9(6), 11961217.CrossRefGoogle Scholar
Cai, J. and Weng, C.G. (2016) Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 7, 624645.CrossRefGoogle Scholar
Cont, R., Deguest, R. and Scandolo, G. (2010) Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 10, 593606.CrossRefGoogle Scholar
Delbaen, F. (2012) Monetary Utility Functions. Osaka: Osaka University Press.Google Scholar
Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014) An academic response to Basel 3.5. Risks, 2(1), 2548.CrossRefGoogle Scholar
Emmer, S., Kratz, M. and Tasche, D. (2015) What is the best risk measure in practice? A comparison of standard measures. Journal of Risk, 18(2).CrossRefGoogle Scholar
Jansen, D.W., Koedijk, K.G. and De Vries, C.G. (2000) Portfolio selection with limited downside risk. Journal of Empirical Finance, 7(3–4), 247269.CrossRefGoogle Scholar
Glasserman, P. and Xu, X. (2014) Robust risk measurement and model risk. Quantitative Finance, 14(1), 2958.CrossRefGoogle Scholar
Goovaerts, M.J., Kaas, R., Dhaene, J. and Tang, Q.H. (2003) A unified approach to generate risk measures. ASTIN Bulletin, 33(2), 173191.CrossRefGoogle Scholar
Goovaerts, M.J., Kaas, R., Dhaene, J. and Tang, Q.H. (2004) Some new classes of consistent risk measures. Insurance: Mathematics and Economics, 34, 505516.Google Scholar
Heras, A., Balbás, B. and Vilar, J.L. (2012) Conditional tail expectation and premium calculation. ASTIN Bulletin, 42(1), 325342.Google Scholar
Kaluszka, M. and Krzeszowiec, M. (2012a) Mean-value principle under cumulative prospect theory. ASTIN Bulletin, 42(1), 103122.Google Scholar
Kaluszka, M. and Krzeszowiec, M. (2012b) Pricing insurance contracts under cumulative prospect theory. Insurance: Mathematics and Economics, 50(1), 159166.Google Scholar
Kuan, K.-C., Yeh, J.-H. and Hsu, H.-C. (2009) Assessing value at risk with CARE, the conditional autoregressive expectile models. Journal of Econometrics, 150, 261270.CrossRefGoogle Scholar
Kusuoka, S. (2001) On law invariant coherent risk measures. Advances in Mathematical Economics, 8395.CrossRefGoogle Scholar
Krätschmer, V., Schied, A. and Zähle, H. (2012) Qualitative and infinitesimal robustness of tail dependent statistical functionals. Journal of Multivariate Analysis, 103, 3547.CrossRefGoogle Scholar
Krokhmal, P., Palmquist, J. and Uryasev, S. (2002) Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4, 4368.CrossRefGoogle Scholar
Landsman, Z. and Sherris, M. (2001) Risk measures and insurance premium principles. Insurance: Mathematics and Economics, 29, 103115.Google Scholar
Mao, T. and Cai, J. (2018) Risk measures based on the behavioural economics theory. Finance and Stochastics, 22, 367393.CrossRefGoogle Scholar
Newey, W.K. and Powell, J.L. (1987) Asymmetric least squares estimation and testing. Econometrica, 55(4), 819847.CrossRefGoogle Scholar
Pyle, D. and Turnovsky, S. (1970) Safety-first and expected utility maximization in mean-standard deviation portfolio analysis. The Review of Economics and Statistics, 52(1), 7581.CrossRefGoogle Scholar
Quiggin, J. (1982) A theory of anticipated utility. Journal of Economic Behavior and Organization, 3, 323343.CrossRefGoogle Scholar
Roy, A.D. (1952) Safety first and the holding of assets. Econometrica, 431449.CrossRefGoogle Scholar
Taylor, J.W. (2008) Estimating value at risk and expected shortfall using expectiles. Journal of Financial Econometrics, 6, 231252.CrossRefGoogle Scholar
Tversky, A. and Kahneman, D. (1992) Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297323.CrossRefGoogle Scholar
Van Heerwaarden, A.E. and Kaas, R. (1992) The Dutch premium principle. Insurance: Mathematics and Economics, 11(2), 129133.Google Scholar
Vicig, P. (2008) Financial risk measurement with imprecise probabilities. International Journal of Approximate Reasoning, 49(1), 159174.CrossRefGoogle Scholar
Wang, S. (1995) Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17, 4354.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 7192.CrossRefGoogle Scholar
Yaari, M.E. (1987) The dual theory of choice under risk. Econometrica, 55, 95115.CrossRefGoogle Scholar