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Value at Ruin and Tail Value at Ruin of the Compound Poisson Process with Diffusion and Efficient Computational Methods

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  • Riccardo Gatto

    (University of Bern)

  • Benjamin Baumgartner

    (University of Bern)

Abstract

We analyze the insurer risk under the compound Poisson risk process perturbed by a Wiener process with infinite time horizon. In the first part of this article, we consider the capital required to have fixed probability of ruin as a measure of risk and then a coherent extension of it, analogous to the tail value at risk. We show how both measures of risk can be efficiently computed by the saddlepoint approximation. We also show how to compute the stabilities of these measures of risk with respect to variations of probability of ruin. In the second part of this article, we are interested in the computation of the probability of ruin due to claim and the probability of ruin due to oscillation. We suggest a computational method based on upper and lower bounds of the probability of ruin and we compare it to the saddlepoint and to the Fast Fourier transform methods. This alternative method can be used to evaluate the proposed measures of risk and their stabilities with heavy-tailed individual losses, where the saddlepoint approximation cannot be used. The numerical accuracy of all proposed methods is very high and therefore these measures of risk can be reliably used in actuarial risk analysis.

Suggested Citation

  • Riccardo Gatto & Benjamin Baumgartner, 2014. "Value at Ruin and Tail Value at Ruin of the Compound Poisson Process with Diffusion and Efficient Computational Methods," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 561-582, September.
    • Handle: RePEc:spr:metcap:v:16:y:2014:i:3:d:10.1007_s11009-012-9316-5
    DOI: 10.1007/s11009-012-9316-5
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    References listed on IDEAS

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    1. Riccardo Gatto, 2012. "Saddlepoint Approximations to Tail Probabilities and Quantiles of Inhomogeneous Discounted Compound Poisson Processes with Periodic Intensity Functions," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 1053-1074, December.
    2. Patrick Cheridito & Freddy Delbaen & Michael Kupper, 2006. "Coherent and convex monetary risk measures for unbounded càdlàg processes," Finance and Stochastics, Springer, vol. 10(3), pages 427-448, September.
    3. Dufresne, François & Gerber, Hans U., 1989. "Three Methods to Calculate the Probability of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 71-90, April.
    4. Riccardo Gatto, 2010. "A Saddlepoint Approximation to the Distribution of Inhomogeneous Discounted Compound Poisson Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 533-551, September.
    5. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    6. Wang, Suojin, 1995. "One-step saddlepoint approximations for quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 20(1), pages 65-74, July.
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    Cited by:

    1. Riccardo Gatto & Benjamin Baumgartner, 2016. "Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 217-235, March.
    2. Cossette, Hélène & Marceau, Etienne & Trufin, Julien & Zuyderhoff, Pierre, 2020. "Ruin-based risk measures in discrete-time risk models," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 246-261.

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