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RISK MARGIN QUANTILE FUNCTION VIA PARAMETRIC AND NON-PARAMETRIC BAYESIAN APPROACHES

Published online by Cambridge University Press:  09 July 2015

Alice X.D. Dong ,
Jennifer S.K. Chan  and
Gareth W. Peters
Alice X.D. Dong*
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia
Jennifer S.K. Chan
Affiliation:
School of Mathematics and Statistics, The University of Sydney, NSW 2006, Australia E-Mail: [email protected]
Gareth W. Peters
Affiliation:
Department of Statistical Science, University College London UCL, London, UK E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

We develop quantile functions from regression models in order to derive risk margin and to evaluate capital in non-life insurance applications. By utilizing the entire range of conditional quantile functions, especially higher quantile levels, we detail how quantile regression is capable of providing an accurate estimation of risk margin and an overview of implied capital based on the historical volatility of a general insurers loss portfolio. Two modeling frameworks are considered based around parametric and non-parametric regression models which we develop specifically in this insurance setting. In the parametric framework, quantile functions are derived using several distributions including the flexible generalized beta (GB2) distribution family, asymmetric Laplace (AL) distribution and power-Pareto (PP) distribution. In these parametric model based quantile regressions, we detail two basic formulations. The first involves embedding the quantile regression loss function from the nonparameteric setting into the argument of the kernel of a parametric data likelihood model, this is well known to naturally lead to the AL parametric model case. The second formulation we utilize in the parametric setting adopts an alternative quantile regression formulation in which we assume a structural expression for the regression trend and volatility functions which act to modify a base quantile function in order to produce the conditional data quantile function. This second approach allows a range of flexible parametric models to be considered with different tail behaviors. We demonstrate how to perform estimation of the resulting parametric models under a Bayesian regression framework. To achieve this, we design Markov chain Monte Carlo (MCMC) sampling strategies for the resulting Bayesian posterior quantile regression models. In the non-parametric framework, we construct quantile functions by minimizing an asymmetrically weighted loss function and estimate the parameters under the AL proxy distribution to resemble the minimization process. This quantile regression model is contrasted to the parametric AL mean regression model and both are expressed as a scale mixture of uniform distributions to facilitate efficient implementation. The models are extended to adopt dynamic mean, variance and skewness and applied to analyze two real loss reserve data sets to perform inference and discuss interesting features of quantile regression for risk margin calculations.

Keywords

Asymmetric Laplace distributionBayesian inferenceMarkov chain Monte Carlo methodsQuantile regressionloss reserverisk margincentral estimate
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 45 , Issue 3 , September 2015 , pp. 503 - 550
Copyright
Copyright © Astin Bulletin 2015 

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References

Aiuppa, A. (1988) Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss. Journal of Risk and Insurance, 55 (3), 425–441.CrossRefGoogle Scholar
Artzner, P. (1999) Application of coherent risk measures to capital requirements in insurance. North American Actuarial Journal, 3 (2), 11–25.CrossRefGoogle Scholar
Australian Prudential Regulatory Authority, Prudential Standard GPS 320, Actuarial and Related Matters. (May 2012) http://www.apra.gov.au/CrossIndustry/Consultations/Documents/Draft-GPS-320-Actuarial-and-Related-Matters-May-2012.pdf.Google Scholar
Bank of England Prodential Regulation Authority, Solvency II: An update on implementation. (August 2014) http://www.bankofengland.co.uk/pra/Documents/solvency2/solvency2updateaugust2014.pdf.Google Scholar
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2006) Statistics of Extremes: Theory and Applications. John Wiley & Sons.Google Scholar
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989) Regular Variation. Cambridge University Press.Google Scholar
Borovkov, A.A. and Borovkov, K.A. (2008) Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions. Cambridge University Press.CrossRefGoogle Scholar
Cai, Y. (2010) Polynomial power-Pareto quantile function models. Extremes, 13 (3), 291–314.CrossRefGoogle Scholar
Carlin, B.P. and Thomas, A.L. (2000) Bayes and Empirical Bayes Methods for Data Analysis. CRC Press.CrossRefGoogle Scholar
Chan, J.S.K., Choy, S.T.B. and Makov, U.E. (2008) Robust Bayesian analysis of loss reserves data using the generalized-t distribution. Astin Bulletin, 38 (1), 207–230.CrossRefGoogle Scholar
Chen, Q., Gerlach, R. and Lu, Z. (2012) Bayesian value-at-risk and expected shortfall forecasting via the asymmetric Laplace distribution. Computational Statistics and Data Analysis, 56 (11), 3498–3516.CrossRefGoogle Scholar
Claeskens, G. and Hjort, N.L. (2008) Model Selection and Model Averaging, vol. 330. Cambridge: Cambridge University Press.Google Scholar
Cruz, M.G., Peters, G.W. and Shevchenko, P.V. (2014) Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk. John Wiley and Sons.Google Scholar
Cummins, J.D., McDonald, J.B. and Craig, M. (2007) Risk loss distributions and modelling the loss reserve pay-out tail. Review of Applied Economics, 3 (1–2), 1–23.Google Scholar
Daouia, A., Gardes, L. and Girard, S. (2012) On kernel smoothing for extremal quantile regression. Bernoulli, 19 (5B), 2557–2589.Google Scholar
Denison, D.G.T., Holmes, C.C., Mallick, B.K. and Smith, A.F.M. (2002) Bayesian Methods for Nonlinear Classification and Regression. John Wiley and Sons.Google Scholar
de Alba, E. (2002) Bayesian estimation of outstanding Claim Reserves. North American Actuarial Journal, 6 (4), 1–20.CrossRefGoogle Scholar
Delbaen, F. (2002) Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics, pp. 1–37. Berlin Heidelberg: Springer.Google Scholar
Dong, X. and Chan, J. (2013) Bayesian analysis of loss reserving using dynamic models with generalized beta distribution. Insurance: Mathematics and Economics, 53 (2), 355–365.Google Scholar
Dowd, K. and Blake, D. (2006) After VaR: The theory, estimation, and insurance applications of quantile-based risk measures. Journal of Risk and Insurance, 73 (2), 193–229.CrossRefGoogle Scholar
Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997) Modeling Extreme Events for Insurance and Finance. Berlin: Springer.Google Scholar
Engle, R. and Manganelli, S. (2004) CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business and Economic Statistics, 22 (4), 367–381.CrossRefGoogle Scholar
Gilchrist, W. (2002) Statistical Modelling with Quantile Functions. CRC Press.Google Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996) Markov Chain Monte Carlo in Practice. London: Chapman and Hall.Google Scholar
Goegebeur, Y., Guillou, A. and Schorgen, A. (2014) Nonparametric regression estimation of conditional tails: The random covariate case. Statistics, 48 (4), 732–755.CrossRefGoogle Scholar
Goovaerts, M.J., Dhaene, J. and De Schepper, A. (2000) Stochastic upper bounds for present value functions. Journal of Risk and Insurance Theory, 67 (1), 1–14.CrossRefGoogle Scholar
Guermat, C. and Harris, R.D.F. (2002) Forecasting value at risk allowing for time variation in the variance and kurtosis of portfolio returns. International Journal of Forecasting, 18 (3), 409–419.CrossRefGoogle Scholar
Gyorgy, S. and Shaw, W.T. (2008) Quantile mechanics. European Journal of Applied Mathematics, 19 (2), 87–112.Google Scholar
Hastings, W.K. (1970) Monte Carlo sampling methods using Markov Chains and their applications. Biometrika, 57 (1), 97–109.CrossRefGoogle Scholar
Hu, Y., Grimacy, R.B. and Lian, H. (2012) Bayesian quantile regression for single-index models. Statistics and Computing, 23 (4), 437–454.CrossRefGoogle Scholar
Kaas, R., Dhaene, J. and Goovaerts, M. (2000) Upper and lower bounds for sums of random variables. Insurance: Mathematics and Economics, 27 (2), 151–168.Google Scholar
Kluppelberg, C. and Mikosch, T. (1998) Large deviations of heavy-tailed random sums with applications in insurance and finance. Extremes, 1 (1), 81–110.Google Scholar
Koenker, R. and Hallock, K. (2001) Quantile regression: An introduction. Journal of Economic Perspectives, 15 (1), 143–156.CrossRefGoogle Scholar
Koenker, R. and Machado, A. F. (1999) Goodness of fit and related inference processes for quantile regression. Journal of the American Statistical Association, 94 (448), 1296–310.CrossRefGoogle Scholar
Marshall, K., Collings, S., Hodson, M. and O'Dowd, C. (2008) A framework for assessing risk margins. Prepared by the Risk Margins Task Force for Institute of Actuaries of Australia, 16th General Insurance Seminar, 9–12*** November 2008, Coolum, Australia.Google Scholar
McDonald, J.B. (1984) Some generalized functions for the size distribution of income. Econometrica, 52 (3), 647–663.CrossRefGoogle Scholar
McDonald, J.B. and Newey, W.K. (1988) Partially adaptive estimation of regression models via the Generalized t distribution. Econometric Theory, 4 (3), 428–457.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N. and Teller, A.H. (1953) Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21 (6), 1087–1091.CrossRefGoogle Scholar
Nelson, C.R. and Siegel, A.F. (1987) Parsimonious modeling of yield curves. Journal of Business, 60 (4), 473–489.CrossRefGoogle Scholar
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian modeling of outstanding claim reservesliabilites incorporating claim count uncertainty. North American Actuarial Journal, 6 (1), 113–128.CrossRefGoogle Scholar
Ori, R. and Cohen, A. (1996) Extreme percentile regression. In Statistical Theory and Computational Aspects of Smoothing, pp. 200–2014. Physica-Verlag HD.CrossRefGoogle Scholar
Paulson, A.S. and Faris, N.J. (1985) A practical approach to measuring the distribution of total annual claims. In Strategic Planning and Modeling in Property-Liability Insurance (ed. Cumins, J.D.), Norwell, MA: Kluwer Academic Publishers.Google Scholar
Peters, G.W., Byrnes, A.D. and Shevchenko, P.V. (2011a) Impact of insurance for operational risk: Is it worthwhile to insure or be insured for severe losses? Insurance: Mathematics and Economics, 48 (2), 287–303.Google Scholar
Peters, G.W., Shevchenko, P.V. and Wuthrich, M.V. (2009) Model uncertainty in claims reserving within Tweedie compound Poisson models. ASTIN Bulletin, 39 (1), 1–33.CrossRefGoogle Scholar
Peters, G.W., Shevchenko, P.V., Young, M. and Yip, W. (2011b) Analytic loss distributional approach models for operational risk from the α-stable doubly stochastic compound processes and implications for capital allocation. Insurance: Mathematics and Economics, 49 (3), 565–579.Google Scholar
Peters, G.W., Targino, R.S. and Shevchenko, P.V. (2013) Understanding operational risk capital approximations: First and second orders. Governance and Regulation (Invited Special Issue 8th International Conference “International Competition in Banking: Theory and Practice'', Sumy, Ukraine), 2 (3), 58–79.Google Scholar
Ramlau-Hansen, H. (1988) A solvency study in non-life insurance. Part 1. analysis of fire, Windstorm, and glass claims. Scandinavian Actuarial Journal, 3–34.Google Scholar
Smith, A.F.M. and Roberts, G.O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B, 55, 3–23.Google Scholar
Spiegelhalter, D., Best, N.G., Carlin, B.P. and Van der Linde, A. (2002) Bayesian measures of model complexity and fit. (with Discussion). Journal of the Royal Statistical Society, Series B, 64 (4), 583–616.CrossRefGoogle Scholar
Stacy, E.W. (1962) A generalization of the gamma distribution. The Annals of Mathematical Statistics, 33 (3), 1187–92.CrossRefGoogle Scholar
Taylor, G.APRA (2006) General insurance risk margins. (February 2006). http://fbe.unimelb.edu.au/__data/assets/pdf_file/0011/806267/136.pdf.Google Scholar
Verrall, R.J. and Wuthrich, M. (2013) Reversible jump Markov chain Monte Carlo method for parameter reduction in claims reserving. To appear in North American Actuarial Journal.CrossRefGoogle Scholar
Yu, K. and Moyeed, R.A. (2001) Bayesian quantile regression. Statists and Probability Letters, 54 (4), 437–447.CrossRefGoogle Scholar
Yu, K. and Zhang, J. (2005) A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics Theory and Methods, 34 (9), 1867–1879.CrossRefGoogle Scholar
Zhang, Y., Dukic, V. and Guszcza, J. (2012) A Bayesian nonlinear model for forecasting insurance loss payments. Journal of the Royal Statistical Society, Series A, 175 (2), 1–20.CrossRefGoogle Scholar