×

zbMATH — the first resource for mathematics

Stress scenario generation for solvency and risk management. (English) Zbl 1401.91117
Summary: We derive worst-case scenarios in a life insurance model in the case where the interest rate and the various transition intensities are mutually dependent. Examples of this dependence are that (a) surrender intensities and interest rates are high at the same time, (b) mortality intensities of a policyholder as active and disabled, respectively, are low at the same time, and (c) mortality intensities of the policyholders in a portfolio are low at the same time. The set from which the worst-case scenario is taken reflects the dependence structure and allows us to relate the worst-case scenario-based reserve, qualitatively, to a value-at-risk-based calculation of solvency capital requirements. This brings out perspectives for our results in relation to qualifying the standard formula of Solvency II or using a scenario-based approach in internal models. Our results are powerful for various applications and the techniques are non-standard in control theory, exactly because our worst-case scenario is deterministic and not adapted to the stochastic development of the portfolio. The formalistic results are exemplified in a series of numerical studies.

MSC:
91B30 Risk theory, insurance (MSC2010)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bailey, P. B., Shampine, L. F. & Waltman, P. E. (1968). Nonlinear two point boundary value problems. New York: Academic Press. · Zbl 0169.10502
[2] Bertsekas, D. P. (2005). Dynamic programming and optimal control. 3rd ed., Vol. 1. Belmont, MA: Athena Scientific. · Zbl 1125.90056
[3] Börger, M. (2010). Deterministic shock vs. stochastic value-at-risk - an analysis of the Solvency II standard model approach to longevity risk. Blätter der DGVFM31 (2), 225-259. Available online at: http://dx.doi.org/10.1007/s11857-010-0125-z. · Zbl 1232.91341
[4] Christiansen, M. C. (2008). A sensitivity analysis concept for life insurance with respect to a valuation basis of infinite dimension. Insurance: Mathematics and Economics42 (2), 680-690. Available online at: http://ideas.repec.org/a/eee/insuma/v42y2008i2p680-690.html. · Zbl 1152.91573
[5] Christiansen, M. C. & Steffensen, M. (2013). Safe-side scenarios for financial and biometrical risk. ASTIN Bulletin, 1-35. Available online at: http://www.journals.cambridge.org/abstract_S0515036113000160. · Zbl 1290.91083
[6] Devineau, L. & Loisel, S. (2009). Risk aggregation in Solvency II: how to converge the approaches of the internal models and those of the standard formula? Post-Print hal-00403662. HAL. Available online at: http://ideas.repec.org/p/hal/journl/hal-00403662.html.
[7] Doff, R. (2008). A critical analysis of the solvency II proposals. The Geneva Papers on Risk and Insurance Issues and Practice33, 193-206.
[8] EIOPA. (2013). Technical specification on the long term guarantee assessment (part I). Technical report. European Insurance and Occupational Pensions Authority.
[9] Li, J. & Szimayer, A. (2011). The uncertain mortality intensity framework: pricing and hedging unit-linked life insurance contracts. Insurance: Mathematics and Economics49 (3), 471-486. · Zbl 1228.91041
[10] Li, J. & Szimayer, A. (2014). The effect of policyholders’ rationality on unit-linked life insurance contracts with surrender guarantees. Quantitative Finance14 (2), 327-342. Available online at: http://www.tandfonline.com/doi/abs/10.1080/14697688.2013.825922. · Zbl 1294.91079
[11] Norberg, R. (1999). A theory of bonus in life insurance. Finance and Stochastics3 (4), 373-390. Available online at: http://dx.doi.org/10.1007/s007800050067. · Zbl 0939.62108
[12] Olivieri, A. & Pitacco, E. (2008). Assessing the cost of capital for longevity risk. Insurance: Mathematics and Economics42 (3), 1013-1021. Available online at: http://ideas.repec.org/a/eee/insuma/v42y2008i3p1013-1021.html. · Zbl 1141.91540
[13] Orava, P. & Lautala, P. (1976). Back-and-forth shooting method for solving two-point boundary-value problems. Journal of Optimization Theory and Applications18 (4), 485-498. Available online at: http://dx.doi.org/10.1007/BF00932657. · Zbl 0304.34019
[14] Steffen, T. (2008). Solvency II and the work of CEIOPS. The Geneva Papers on Risk and Insurance Issues and Practice33, 60-65.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.