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 volume 16, pages561–582(2014)Cite this article

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.

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Affiliations

  1. Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012, Bern, Switzerland

    Riccardo Gatto & Benjamin Baumgartner

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  1. Riccardo Gatto
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  2. Benjamin Baumgartner
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Correspondence to Riccardo Gatto.

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Gatto, R., Baumgartner, B. Value at Ruin and Tail Value at Ruin of the Compound Poisson Process with Diffusion and Efficient Computational Methods. Methodol Comput Appl Probab 16, 561–582 (2014). https://doi.org/10.1007/s11009-012-9316-5

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Keywords

  • Coherent measure of risk
  • Daniels’ exponent
  • Fast Fourier transform
  • Lundberg’s exponent
  • Probability of ruin
  • Probabilites of ruin due to claim and to oscillation
  • Richardson’s extrapolation
  • Saddlepoint approximation
  • Stability
  • Upper and lower bounds

Keywords

  • 41A60
  • 65C50
  • 60G51