Environmental Modeling & Assessment

, Volume 19, Issue 3, pp 207–220 | Cite as

Frequency and Severity Modelling Using Multifractal Processes: An Application to Tornado Occurrence in the USA and CAT Bonds

Article

Abstract

This paper proposes a statistical model for insurance claims arising from climatic events, such as tornadoes in the USA, that exhibit a large variability both in frequency and intensity. To represent this variability and seasonality, the claims process modelled by a Poisson process of intensity equal to the product of a periodic function, and a multifractal process is proposed. The size of claims is modelled in a similar way, with gamma random variables. This method is shown to enable simulation of the peak times of damage. A two-dimensional multifractal model is also investigated. The work concludes with an analysis of the impact of the model on the yield of weather bonds linked to damage caused by tornadoes.

Keywords

Multifractal process Claims process Poisson Gamma Dependence CAT bonds 

References

  1. 1.
    Barrieu, P., & Scaillet, O. (2010). “A primer on weather derivatives.” In: Filar, Jerzy A. and Haurie, Alain, (eds.), Uncertainty and environmental decision making: A handbook of research and best practice. International series in operations research & management science (138) (pp. 155-176). Springer US, doi: 10.1007/978-1-4419-1129-2_5.
  2. 2.
    Schmock, U. (1999). “Estimating the value of the Wincat coupons of the Winthertur insurance convertible bond.” A study of the model risk. ASTIN Bulletin, 29(1), 101–163.CrossRefGoogle Scholar
  3. 3.
    Barrieu, P., & Dischel, R. (2002).“Financial weather contracts and their application in risk management.” Chapter in Climatic Risk and the Weather Market, Risk Book, Risk Waters Group.Google Scholar
  4. 4.
    Lopez Cabrera, B. (2010). “Weather Risk Management: CAT bonds and weather derivatives.” PhD Dissertation, Humboldt Universitaet, Berlin. http://edoc.hu-berlin.de/dissertationen/lopez-cabrera-brenda-2010-04-27/PDF/lopez-cabrera.pdf.
  5. 5.
    Vaugirard, V. (2003). Valuing catastrophe bonds by Monte Carlo simulations. Applied Mathematical Finance, 10, 75–90.CrossRefGoogle Scholar
  6. 6.
    Lee, J. P., & Yub, M. T. (2007). Valuation of catastrophe reinsurance with catastrophe bonds. Insurance: Mathematics and Economics, 41(2), 264–278.Google Scholar
  7. 7.
    Alaton, P., Djehiche, B., & Stillberger, D. (2002). On modelling and pricing weather derivatives. Applied Mathematical Finance, 9, 1–20.CrossRefGoogle Scholar
  8. 8.
    Campbell, S., & Diebold, F. (2005). Weather forecasting for weather derivatives. Journal of American Statistical Association, 100(469), 6–16.CrossRefGoogle Scholar
  9. 9.
    Dornier, F., & Queruel, M. (2000). "Caution to the wind." Risk Magazine, EPRM, July, pp. 30-32.Google Scholar
  10. 10.
    Benth, F. E., & Benth, J. S. (2007). The volatility of temperature and pricing of weather derivatives. Quantitative Finance, 7(5), 553–561.CrossRefGoogle Scholar
  11. 11.
    Benth, F. E., & Benth, J. S. (2009). The volatility of temperature and pricing of weather derivatives. Quantitative Finance, 7(5), 553–561.CrossRefGoogle Scholar
  12. 12.
    Hainaut, D. (2012). “Pricing of an insurance bond, with stochastic seasonal effects.” Bulletin Français d'actuariat. Juillet 2012.Google Scholar
  13. 13.
    Chiera, B. A., Filar, J. A., Zachary, D. S., & Gordon, A. H. (2010). “Comparative Forecasting and a test of Persistence in the El Nino Southern Oscillation” In: Filar, Jerzy A. and Haurie, Alain, (eds.), Uncertainty and environmental decision making: A handbook of research and best practice. International series in operations research & management science (138) (pp. 255-273) Springer.Google Scholar
  14. 14.
    Lovejoy, S., & Schertzer, D. (2007). Scale, scaling and multifractals in geophysics: Twenty years on. Nonlinear dynamics. In: A. A. Tsonis, J. Elsner (Ed.), Geosciences (pp. 311-337). Springer.Google Scholar
  15. 15.
    Sachs, D., Lovejoy, S., & Schertzer, D. (2002). The multifractal scaling of cloud radiances from 1m to 1km. Fractals, 10, 253–265.CrossRefGoogle Scholar
  16. 16.
    Tchiguirinskaia, I., Schertzer, D., Lovejoy, S., & Veysseire, J. M. (2006). Wind extremes and scales: multifractal insights and empirical evidence, In: EUROMECH Colloquium on Wind Energy, edited by P. S. Eds. J. Peinke, Springer-Verlag.Google Scholar
  17. 17.
    Calvet, L., & Fisher, A. J. (2008). "Multifractal volatility. Theory, forecasting and pricing". Academic Press. Elsevier.Google Scholar
  18. 18.
    Resnick, S., & Samorodnitsky, G. (2003). Limits of on/off hierarchical product models for data transmission. The Annals of Applied Probability, 13(4), 1355–1398.CrossRefGoogle Scholar
  19. 19.
    Mu, M., & Zheng, Q. (2005). Zigzag oscillation in variational data assimilation with physical "on-off" processes. Monthly Weather Review, 133.Google Scholar
  20. 20.
    Brody, D. C., Syroka, J., & Zervos, M. (2002). “Dynamical pricing of weather derivatives”. Quantitative Finance 2, p. 189-198.Google Scholar
  21. 21.
    Mandlebrot, B. (1982). "The fractal geometry of nature". Freeman.Google Scholar
  22. 22.
    Mandelbrot, B. (1997). Fractals and scaling in finance: Discontinuity, concentration, risk. Berlin: Springer Verlag.CrossRefGoogle Scholar
  23. 23.
    Mandlebrot, B. (2001). Scaling in financial prices: Multifractals and the star equation. Quantitative Finance, 1, p124–p130.CrossRefGoogle Scholar
  24. 24.
    Major, J. A., & Lantsman, Y. (2001). "Actuarial applications of multifractal modeling part 1: Introduction and spatial applications." Non-refereed Paper of the Casualty Actuarial Society Forum.Google Scholar
  25. 25.
    Jung, R. C., & Tremayne, A. R. (2011). Useful models for time series of counts or simplywrong ones? AStA Advances in Statistical Analysis, 95(1), 59–91.CrossRefGoogle Scholar
  26. 26.
    Bremaud, P. (1981). Point processes and queues martingales dynamics. New York: Springer Verlag.CrossRefGoogle Scholar
  27. 27.
    Lecki, T. R., & Rutkowski, M. (2004). "Credit risk: Modeling, valuation and hedging". Springer Finance.Google Scholar
  28. 28.
    Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica, 57(2), 357–384.CrossRefGoogle Scholar
  29. 29.
    Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82(1), 35–45.CrossRefGoogle Scholar
  30. 30.
    Calvet, L., & Fisher, A. J. (2001). Forecasting multifractal volatility. Journal of Econometrics, 105, 27–58.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Rennes Business School-CRESTRennesFrance
  2. 2.Département de mathématiquesUQAMQuébecCanada

Personalised recommendations