×

Valuation of hybrid financial and actuarial products in life insurance by a novel three-step method. (English) Zbl 1454.91177

Summary: Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

MSC:

91G05 Actuarial mathematics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ballotta, L., Deelstra, G. and Rayée, G. (2017) Multivariate FX models with jumps: Triangles, quantos and implied correlation. European Journal of Operational Research, 260 (3), 1181-1199. · Zbl 1403.91329
[2] Barigou, K. and Dhaene, J. (2019) Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting. Scandinavian Actuarial Journal, 2, 163-187. · Zbl 1411.91264
[3] Bielecki, T.R. and Rutkowski, M. (2004) Credit Risk: Modeling, Valuation and Hedging. Berlin, Heidelberg, Springer. · Zbl 1134.91023
[4] Boyle, P.P. and Schwartz, E.S. (1977) Equilibrium prices of guarantees under equity-linked contracts. Journal of Risk and Insurance, 44(4), 639-660.
[5] Brennan, M.J. and Schwartz, E.S. (1979) Alternative investment strategies for the issuers of equity-linked life insurance with an asset value guarantee. Journal of Business, 52(1), 63-93.
[6] Chen, A., Hieber, P. and Klein, J.K. (2019) Tonuity: A novel individual-oriented retirement plan. ASTIN Bulletin, 49(1), 5-30. · Zbl 1419.91352
[7] Chen, A. and Vigna, E. (2017) A unisex stochastic mortality model to comply with EU gender directive. Insurance: Mathematics & Economics, 73, 124-136. · Zbl 1416.91162
[8] Cont, R. and Tankov, P. (2003) Financial Modelling with Jump Processes. Boca Raton, London, New York, Chapman & Hall/CRC Financial Mathematics Series. · Zbl 1052.91043
[9] Dahl, M. and Møller, T. (2006) Valuation and hedging of life insurance liabilities with systematic mortality risk. Insurance: Mathematics and Economics, 39 (2), 193-217. · Zbl 1201.91089
[10] Delong, Ł., Dhaene, J. and Barigou, K. (2019a) Fair valuation of insurance liability cash-flow streams in continuous time: Applications. ASTIN Bulletin, 49(2), 299-333. · Zbl 1410.91262
[11] Delong, Ł., Dhaene, J. and Barigou, K. (2019b) Fair valuation of insurance liability cash-flow streams in continuous time: Theory. Insurance: Mathematics & Economics, 88, 196-208. · Zbl 1425.91219
[12] Dhaene, J., Stassen, B., Barigou, K., Linders, D. and Chen, Z. (2017) Fair valuation of insurance liabilities: Merging actuarial judgement and market-consistency. Insurance: Mathematics and Economics, 76, 14-27. · Zbl 1395.91249
[13] Eberlein, E., Papapantoleon, A. and Shiryaev, A. (2009) Esscher transform and the duality principle for multidimensional semimartingales. The Annals of Applied Probability, 19, 1944-1971. · Zbl 1233.91268
[14] Engsner, H., Lindensjö, K. and Lindskog, F. (2020) The value of a liability cash flow in discrete time subject to capital requirements. Finance and Stochastics, 24(1), 125-167. · Zbl 1429.91277
[15] Gerber, H.U. (1997) Life Insurance Mathematics, 3rd edition. Springer. · Zbl 0869.62072
[16] Gerber, H.U. and Shiu, E.S.W. (1994) Option pricing by Esscher transform. Transactions of Society of Actuaries, 46, 99-191.
[17] Gerber, H.U. and Shiu, E.S.W. (1996) Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics and Economics, 18(3), 183-218. · Zbl 0896.62112
[18] Hirbod, A. and Gospodinov, N. (2018) Market consistent valuation with financial imperfection. Decision in Economics and Finance, 41, 65-90. · Zbl 1391.91167
[19] Ikeda, N. and Watanabe, S. (2014) Stochastic Differential Equations and Diffusion Processes. Elsevier. · Zbl 0495.60005
[20] Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2008) Modern Actuarial Risk Theory: Using R, Vol. 128. Springer Science & Business Media. · Zbl 1148.91027
[21] Keller, P. and Luder, T. (2004) White paper of the Swiss Solvency Test. Swiss Federal Office of Private Insurance, Switzerland.
[22] Kijima, M. (2006) A multivariate extension of equilibrium pricing transform: The multivariate Esscher and Wang transform for pricing financial and insurance risks. ASTIN Bulletin, 36 (1), 269-283. · Zbl 1162.91418
[23] Laeven, R.J. and Goovaerts, M.J. (2008) Premium calculation and insurance pricing. In Encyclopedia of Quantitative Risk Analysis and Assessment, vol. 3, pp. 1302-1314.
[24] Lin, Y. and Cox, S. (2008) Securitization of catastrophe mortality risks. Insurance: Mathematics & Economics, 42(2), 628-637. · Zbl 1152.91593
[25] Luciano, E., Regis, L. and Vigna, E. (2012) Delta-gamma hedging of mortality and interest rate risk. Insurance: Mathematics and Economics, 50 (3), 402-412. · Zbl 1237.91134
[26] Luciano, E. and Vigna, E. (2008) Mortality risk via affine stochastic intensities: Calibration and empirical relevance. Belgian Actuarial Journal, 8(1), 5-16. · Zbl 1398.91345
[27] Malamud, S., Trubowitz, E. and Wüthrich, M.V. (2008) Market consistent pricing of insurance products. Astin Bulletin, 38(2), 483-526. · Zbl 1256.91018
[28] Möhr, C. (2011) Market-consistent valuation of insurance liabilities by cost of capital. ASTIN Bulletin, 41(2), 315-341. · Zbl 1239.91082
[29] Møller, T. (2002) On valuation and risk management at the interface of insurance and finance. British Actuarial Journal, 8 (4), 787-827.
[30] Pelsser, A. and Ghalehjooghi, A.S. (2016) Time-consistent actuarial valuations. Insurance: Mathematics & Economics, 66, 97-112. · Zbl 1348.91178
[31] Pelsser, A. and Stadje, M. (2014) Time-consistent and market-consistent evaluations. Mathematical Finance, 24 (1), 25-65. · Zbl 1303.91095
[32] Perlman, M.D. (1974) Jensen’s inequality for a convex vector-valued function on an infinite-dimensional space. Journal of Multivariate Analysis, 4, 52-65. · Zbl 0274.28012
[33] Rotar, V.I. (2014) Actuarial Models: the Mathematics of Insurance. Boca Raton, London, New York, CRC Press. · Zbl 1117.62118
[34] Schoutens, W. (2003) Lévy Processes in Finance: Pricing Financial Derivatives. Berlin, Heidelberg, John Wiley & Sons, Ltd.
[35] Tsai, J. and Tzeng, L. (2013) Securitization of catastrophe mortality risks. Astin Bulletin, 43(2), 97-121. · Zbl 1309.91143
[36] Zeddouk, F. and Devolder, P. (2019) Pricing of longevity derivatives and cost of capital. Risks, 7 (2), 1-29.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.