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Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin

Author

Listed:
  • Stéphane Loisel

    (LSAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Christian Mazza

    (Département de Mathématiques - Albert-Ludwigs-Universität Freiburg = University of Freiburg)

  • Didier Rullière

    (LSAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

We consider the classical risk model and carry out a sensitivity and robustness analysis of finite-time ruin probabilities. We provide algorithms to compute the related influence functions. We also prove the weak convergence of a sequence of empirical finite-time ruin probabilities starting from zero initial reserve toward a Gaussian random variable. We define the concepts of reliable finite-time ruin probability as a Value-at-Risk of the estimator of the finite-time ruin probability. To control this robust risk measure, an additional initial reserve is needed and called Estimation Risk Solvency Margin (ERSM). We apply our results to show how portfolio experience could be rewarded by cut-offs in solvency capital requirements. An application to catastrophe contamination and numerical examples are also developed.

Suggested Citation

  • Stéphane Loisel & Christian Mazza & Didier Rullière, 2008. "Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin," Post-Print hal-00168714, HAL.
    • Handle: RePEc:hal:journl:hal-00168714
    DOI: 10.1016/j.insmatheco.2007.08.007
    Note: View the original document on HAL open archive server: https://hal.science/hal-00168714v1
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    References listed on IDEAS

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    1. repec:cup:astinb:v:23:y:1993:i:02:p:227-258_01 is not listed on IDEAS
    2. Marceau, Etienne & Rioux, Jacques, 2001. "On robustness in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 29(2), pages 167-185, October.
    3. Mazza, Christian & Rulliere, Didier, 2004. "A link between wave governed random motions and ruin processes," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 205-222, October.
    4. repec:cup:astinb:v:16:y:1986:i:s1:p:s81-s90_01 is not listed on IDEAS
    5. Rulliere, Didier & Loisel, Stephane, 2004. "Another look at the Picard-Lefevre formula for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 187-203, October.
    6. Ignatov, Zvetan G. & Kaishev, Vladimir K. & Krachunov, Rossen S., 2001. "An improved finite-time ruin probability formula and its Mathematica implementation," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 375-386, December.
    7. Croux, Kristof & Veraverbeke, Noel, 1990. "Nonparametric estimators for the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 9(2-3), pages 127-130, September.
    8. repec:cup:astinb:v:19:y:1989:i:01:p:57-70_00 is not listed on IDEAS
    9. Picard, Philippe & Lefevre, Claude, 1998. "The moments of ruin time in the classical risk model with discrete claim size distribution," Insurance: Mathematics and Economics, Elsevier, vol. 23(2), pages 157-172, November.
    10. repec:cup:astinb:v:12:y:1981:i:01:p:22-26_00 is not listed on IDEAS
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    Citations

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    Cited by:

    1. Claude Lefèvre & Stéphane Loisel, 2009. "Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities," Methodology and Computing in Applied Probability, Springer, vol. 11(3), pages 425-441, September.
    2. Stéphane Loisel & Nicolas Privault, 2009. "Sensitivity analysis and density estimation for finite-time ruin probabilities," Post-Print hal-00201347, HAL.
    3. Touazi, A. & Benouaret, Z. & Aissani, D. & Adjabi, S., 2017. "Nonparametric estimation of the claim amount in the strong stability analysis of the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 78-83.
    4. Li Qin & Susan M. Pitts, 2012. "Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 919-936, December.
    5. Asimit, Alexandru V. & Badescu, Alexandru M. & Verdonck, Tim, 2013. "Optimal risk transfer under quantile-based risk measurers," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 252-265.
    6. Loisel, Stéphane & Mazza, Christian & Rullière, Didier, 2009. "Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 374-381, December.
    7. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.
    8. Claude Lefèvre & Stéphane Loisel, 2008. "On Finite-Time Ruin Probabilities for Classical Risk Models," Post-Print hal-00168958, HAL.
    9. Roman Adillon & Lambert Jorba & Maite Mármol, 2025. "An Extension of Interval Probabilities using Modal Interval Theory and its Application to Non-life Insurance," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-17, June.

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