Biometric solvency risk for portfolios of general life contracts. I. the single-life multiple decrement case. (English) Zbl 1219.91069

Summary: Solvency II splits life insurance risk into seven risk classes consisting of three biometric risks (mortality risk, longevity risk, and disability/morbidity risk) and four nonbiometric risks (lapse risk, expense risk, revision risk, and catastrophe risk). The best estimate liabilities for the biometric risks are valued with biometric life tables (mortality and disability tables), while those of the nonbiometric risks require alternative valuation methods. The present study is restricted to biometric risks encountered in traditional single-life insurance contracts with multiple causes of decrement. Based on the results of quantitative impact studies, process risk was deemed to be not significant enough to warrant an explicit calculation. It was therefore assumed to be implicitly included in the systematic/parameter risk, resulting in a less complex standard formula. For the purpose of internal models and improved risk management, it appears important to capture separately or simultaneously all risk components of biometric risks. Besides its being of interest for its own sake, this leads to a better understanding of the standard approach and its application extent. Based on a total balance sheet approach we express the liability risk solvency capital of an insurance portfolio as value-at-risk and conditional value-at-risk of the prospective liability risk understood as random present value of future cash flows at a given time. The proposed approach is then applied to determine the biometric solvency capital for a portfolio of general life contracts. Using the conditional mean and variance of a portfolio’s prospective liability risk and a gamma distribution approximation we obtain simple solvency capital formulas as well as corresponding solvency capital ratios. To account for the possibility of systematic/parameter risk, we propose either to shift the biometric life tables or to apply a stochastic biometric model, which allows for random biometric rates. A numerical illustration for a cohort of immediate life annuities in arrears reveals the importance of process risk in the assessment of longevity risk solvency capital.


91B30 Risk theory, insurance (MSC2010)
91G50 Corporate finance (dividends, real options, etc.)
Full Text: DOI


[1] Ammeter H., Scandinavian Actuarial Journal (1948)
[2] Ammeter H., Bulletin of the Swiss Association of Actuaries pp 35– (1949)
[3] Bowers N. L., Actuarial Mathematics, 2. ed. (2000)
[4] Bühlmann H., Mathematical Methods in Risk Theory (1970) · Zbl 0209.23302
[5] De Pril N. L., Astin Bulletin 19 pp 9– (1989)
[6] Furman E., Insurance: Mathematics and Economics 37 pp 635– (2005) · Zbl 1129.91025
[7] Gerber H. U., Lebensversicherungsmathematik (1986)
[8] Gerber H. U., Life Insurance Mathematics, 3. ed. (1997) · Zbl 0869.62072
[9] Gerber H. U., North American Actuarial Journal 7 (1) pp 38– (2003) · Zbl 1084.62529
[10] Gompertz B., Philosophical Transactions of the Royal Society 36 pp 513– (1825)
[11] Hattendorff K., Maisius’ Rundschau der Versicherungen 18 pp 169– (1868)
[12] Heligman L., Journal of the Institute of Actuaries 107 (1) pp 49– (1980)
[13] Hürlimann W., Astin Bulletin 23 pp 55– (1993)
[14] Hürlimann W., Astin Bulletin 31 pp 107– (2001) · Zbl 1060.91081
[15] Hürlimann W., Astin Bulletin 32 (2) pp 235– (2002) · Zbl 1094.91032
[16] Hürlimann W., Insurance: Mathematics and Economics 30 (1) pp 27– (2002) · Zbl 1055.91026
[17] Hürlimann W., Blatter der Deutschen Gesellschaft für Versicherungsmath-ematik 25 (4) pp 885– (2002)
[18] Hürlimann W., Journal of Applied Mathematics 3 (3) pp 141– (2003) · Zbl 1012.62110
[19] Hürlimann, W. 2009. Actuarial Analysis of the Multiple Life Endowment Insurance Contract. 3rd IAA LIFE Colloquium. September6–92009, Munich.
[20] Hürlimann W., Actuarial Analysis of Dependent Lives in Solvency. II (2010)
[21] Hürlimann W., Astin Bulletin (2010)
[22] Ibrahim R. I., Matematika 24 (1) pp 1– (2008)
[23] MaroCCo, P. and PitaCCo, E. Longevity Risk and Life Annuity Reinsurance. Transactions of the 26th International Congress of Actuaries. Birmingham. vol. 6, pp.453–479.
[24] Olivieri A., Astin Bulletin 39 (2) pp 541– (2009) · Zbl 1179.91108
[25] Panjer H. H., Insurance Risk Models (1992)
[26] PitaCCo E., Giornale dell’Istituto Italiano degli Attuari 67 (1) pp 17– (2004)
[27] Willemse W. J., Scandinavian Actuarial Journal pp 168– (2000) · Zbl 0971.62073
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.