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ANALYZING AND PREDICTING CAT BOND PREMIUMS: A FINANCIAL LOSS PREMIUM PRINCIPLE AND EXTREME VALUE MODELING

Published online by Cambridge University Press:  02 November 2017

Gilles Stupfler  and
Fan Yang
Gilles Stupfler
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK, E-Mail: gilles.stupfler@nottingham.ac.uk
Fan Yang*
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada
*
E-Mail: fan.yang@uwaterloo.ca
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Abstract

CAT bonds play an important role in transferring insurance risks to the capital market. It has been observed that typical CAT bond premiums have changed since the recent financial crisis, which has been attributed to market participants being increasingly risk averse. In this work, we first propose a new premium principle, the financial loss premium principle, which includes a term measuring losses in the financial market that we represent here by the Conditional Tail Expectation (CTE) of the negative daily log-return of the S&P 500 index. Our analysis of empirical evidence suggests indeed that in the post-crisis market, instead of simply increasing the fixed level of risk load universally, the increased risk aversion should be modeled jointly by a fixed level of risk load and a financial loss factor to reflect trends in the financial market. This new premium principle is shown to be flexible with respect to the confidence/exceedance level of CTE. In the second part, we focus on the particular example of extreme wildfire risk. The distribution of the amount of precipitation in Fort McMurray, Canada, which is a very important factor in the occurrence of wildfires, is analyzed using extreme value modeling techniques. A wildfire bond with parametric trigger of precipitation is then designed to mitigate extreme wildfire risk, and its premium is predicted using an extreme value analysis of its expected loss. With an application to the 2016 Fort McMurray wildfire, we demonstrate that the extreme value model is sensible, and we further analyze how our results and construction can be used to provide a design framework for CAT bonds which may appeal to (re)insurers and investors alike.

Keywords

CAT bondsConditional Tail Expectationextreme value theoryfinancial crisisfinancial loss principle
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 375 - 411
Copyright
Copyright © Astin Bulletin 2017 

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References

Alvarado, E., Sandberg, D.V. and Pickford, S.G. (1998) Modeling large forest fires as extreme events. Northwest Science, 72, 66–75.Google Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59, 817–858.CrossRefGoogle Scholar
Basel Committee on Banking Supervision (2013) Fundamental Review of the Trading Book: A Revised Market Risk Framework. Basel: Bank for International Settlements.Google Scholar
Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004) Statistics of Extremes: Theory and Applications. Chichester: John Wiley & Sons.Google Scholar
Beirlant, J., Vynckier, P. and Teugels, J. L. (1996) Excess functions and estimation of the extreme value index. Bernoulli, 2 (4), 293–318.Google Scholar
Bradley, B.O. and Taqqu, M.S. (2003) Financial risk and heavy tails. In Handbook of Heavy-Tailed Distributions in Finance (ed. Rachev, S.T.), pp. 35–103. Amsterdam: Elsevier.CrossRefGoogle Scholar
Braun, A. (2016) Pricing in the primary market for CAT bonds: New empirical evidence. Journal of Risk and Insurance, 83 (4), 811–847.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Models in Risk Theory. Berlin, Heidelberg: Springer-Verlag.Google Scholar
Carayannopoulos, P. and Perez, M.F. (2015) Diversification through catastrophe bonds: Lessons from the subprime financial crisis. The Geneva Papers on Risk and Insurance-Issues and Practice, 40 (1), 1–28.Google Scholar
Chavez-Demoulin, V., Embrechts, P. and Sardy, S. (2014) Extreme-quantile tracking for financial time series. Journal of Econometrics, 181, 44–52.CrossRefGoogle Scholar
Cummins, J.D. and Weiss, M. A. (2009) Convergence of insurance and financial markets: Hybrid and securitized risk-transfer solutions. Journal of Risk and Insurance, 76 (3), 493–545.CrossRefGoogle Scholar
Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989) A moment estimator for the index of an extreme-value distribution. Annals of Statistics, 17, 1833–1855.Google Scholar
de Haan, L. and Ferreira, A. (2006) Extreme Value Theory: An Introduction. New York: Springer.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
Drees, H. (1998) On smooth statistical tail functionals. Scandinavian Journal of Statistics, 25, 187–210.Google Scholar
Drees, H. (2003) Extreme quantile estimation for dependent data, with applications to finance. Bernoulli, 9, 617–657.CrossRefGoogle Scholar
El Methni, J. and Stupfler, G. (2017a) Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions. Statistica Sinica, 27 (2), 907–930.Google Scholar
El Methni, J. and Stupfler, G. (2017b) Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions. Available at https://hal.archives-ouvertes.fr/hal-01406342/document.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Ferreira, A. and de Haan, L. (2015) On the block maxima method in extreme value theory: PWM estimators. Annals of Statistics, 43 (1), 276–298.Google Scholar
Fisher, R.A. and Tippett, L.H.C. (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 24 (2), 180–190.CrossRefGoogle Scholar
Froot, K.A. (2001) The market for catastrophe risk: A clinical examination. Journal of Financial Economics, 60 (2), 529–571.Google Scholar
Gabaix, X., Gopikrishnan, P., Plerou, V. and Stanley, H.E. (2003) A theory of power-law distributions in financial market fluctuations. Nature, 423 (6937), 267–270.Google Scholar
Galeotti, M., Gürtler, M. and Winkelvos, C. (2013) Accuracy of premium calculation models for CAT bonds–-an empirical analysis. Journal of Risk and Insurance, 80 (2), 401–421.Google Scholar
Ghosh, S. and Resnick, S. (2010) A discussion on mean excess plots. Stochastic Processes and their Applications, 120 (8), 1492–1517.CrossRefGoogle Scholar
Gnedenko, B.V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Annals of Mathematics, 44 (3), 423–453.Google Scholar
Gürtler, M., Hibbeln, M. and Winkelvos, C. (2016) The impact of the financial crisis and natural catastrophes on CAT bonds. Journal of Risk and Insurance, 83, 579–612.Google Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Statistics, 3, 1163–1174.CrossRefGoogle Scholar
Hoeppe, P. (2016) Trends in weather related disasters – Consequences for insurers and society. Weather and Climate Extremes, 11, 70–79.Google Scholar
Ibragimov, R. and Walden, J. (2007) The limits of diversification when losses may be large. Journal of Banking & Finance, 31 (8), 2551–2569.Google Scholar
Koutsoyiannis, D. (2004) Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records. Hydrological Sciences Journal, 49 (4), 591–610.Google Scholar
Kuan, C.-M., Yeh, J.-H. and Hsu, Y.-C. (2009) Assessing value at risk with CARE, the conditional autoregressive expectile models. Journal of Econometrics, 2, 261–270.CrossRefGoogle Scholar
Lane, M.N. (2000) Pricing risk transfer transactions. ASTIN Bulletin, 30 (2), 259–293.Google Scholar
Lane, M. and Mahul, O. (2008) Catastrophe risk pricing: An empirical analysis. World Bank Policy Research Working Paper Series.CrossRefGoogle Scholar
Lumley, T. and Heagerty, P. (1999) Weighted empirical adaptive variance estimators for correlated data regression. Journal of the Royal Statistical Society: Series B, 61, 459–477.Google Scholar
Major, J.A. and Kreps, R.E. (2002) Catastrophe risk pricing in the traditional market. In Alternative Risk Strategies (ed. Lane, M.), pp. 201–222. London: Risk Books.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton, NJ: Princeton University Press.Google Scholar
McNeil, A.J. and Saladin, T. (1997) The peaks over thresholds method for estimating high quantiles of loss distributions. In Proceedings of 28th International ASTIN Colloquium, pp. 23–43.Google Scholar
Newey, W.K. and Powell, J.L. (1987) Asymmetric least squares estimation and testing. Econometrica, 55, 819–847.Google Scholar
Newey, W.K. and West, K.D. (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55 (3), 703–708.Google Scholar
Resnick, S. (2007) Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. New York: Springer.Google Scholar
Rootzén, H. and Tajvidi, N. (1997) Extreme value statistics and wind storm losses: A case study. Scandinavian Actuarial Journal, 1, 70–94.Google Scholar
Scarrott, C. and MacDonald, A. (2012) A review of extreme value threshold estimation and uncertainty quantification. REVSTAT – Statistical Journal, 10 (1), 33–60.Google Scholar
Simiu, E. and Heckert, N.A. (1996) Extreme wind distribution tails: A “peaks over threshold” approach. Journal of Structural Engineering, 122 (5), 539–547.Google Scholar
Wang, S.S. (2004) Cat bond pricing using probability transforms. Geneva Papers: Etudes et Dossiers, 278, 19–29.Google Scholar
Zimbidis, A.A., Frangos, N.E. and Pantelous, A.A. (2007) Modelling earthquake risk via extreme value theory and pricing the respective catastrophe bonds. ASTIN Bulletin, 37 (1), 163–183.Google Scholar