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Published online by Cambridge University Press:  03 December 2021

Karim Barigou*
Univ Lyon, Université Claude Bernard Lyon 1, Laboratoire de Sciences Actuarielle et Financière, Institut de Science Financière et d’Assurances (50 Avenue Tony Garnier, F-69007 Lyon, France) E-Mail:
Valeria Bignozzi
Department of Statistics and Quantitative Methods University of Milano-Bicocca 20126 Milan, Italy E-Mail:
Andreas Tsanakas
Bayes Business School City, University of LondonLondon EC1V 0HB, UK E-Mail:


Current approaches to fair valuation in insurance often follow a two-step approach, combining quadratic hedging with application of a risk measure on the residual liability, to obtain a cost-of-capital margin. In such approaches, the preferences represented by the regulatory risk measure are not reflected in the hedging process. We address this issue by an alternative two-step hedging procedure, based on generalised regression arguments, which leads to portfolios that are neutral with respect to a risk measure, such as Value-at-Risk or the expectile. First, a portfolio of traded assets aimed at replicating the liability is determined by local quadratic hedging. Second, the residual liability is hedged using an alternative objective function. The risk margin is then defined as the cost of the capital required to hedge the residual liability. In the case quantile regression is used in the second step, yearly solvency constraints are naturally satisfied; furthermore, the portfolio is a risk minimiser among all hedging portfolios that satisfy such constraints. We present a neural network algorithm for the valuation and hedging of insurance liabilities based on a backward iterations scheme. The algorithm is fairly general and easily applicable, as it only requires simulated paths of risk drivers.

Research Article
© The Author(s), 2021. Published by Cambridge University Press on behalf of The International Actuarial Association

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Albrecher, H., Bauer, D., Embrechts, P., Filipović, D., Koch-Medina, P., Korn, R., Loisel, S., Pelsser, A., Schiller, F., Schmeiser, H. and Wagner, J. (2018) Asset-liability management for long-term insurance business. European Actuarial Journal, 8(1), 925.CrossRefGoogle Scholar
Assa, H. and Gospodinov, N. (2018) Market consistent valuations with financial imperfection. Decisions in Economics and Finance, 41(1), 6590.CrossRefGoogle Scholar
Augustyniak, M., Godin, F. and Simard, C. (2017) Assessing the effectiveness of local and global quadratic hedging under garch models. Quantitative Finance, 17(9), 13051318.CrossRefGoogle Scholar
Barigou, K., Chen, Z. and Dhaene, J. (2019) Fair dynamic valuation of insurance liabilities: Merging actuarial judgement with market-and time-consistency. Insurance: Mathematics and Economics, 88, 1929.Google Scholar
Barigou, K. and Delong, L. (2020) Pricing equity-linked life insurance contracts with multiple risk factors by neural networks. Journal of Computational and Applied Mathematics, 404, 113922. Scholar
Bellini, F., Klar, B., Müller, A. and Gianin, E.R. (2014) Generalized quantiles as risk measures. Insurance: Mathematics and Economics, 54, 4148.Google Scholar
Ben-Tal, A. and Teboulle, M. (2007) An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3), 449476.CrossRefGoogle Scholar
Breckling, J. and Chambers, R. (1988) M-quantiles. Biometrika, 75(4), 761771.CrossRefGoogle Scholar
Carbonneau, A. (2021) Deep hedging of long-term financial derivatives. Insurance: Mathematics and Economics, 99, 327340.Google Scholar
Carbonneau, A. and Godin, F. (2021) Equal risk pricing of derivatives with deep hedging. Quantitative Finance, 21(4), 593608.CrossRefGoogle Scholar
Černý, A. (2004) Dynamic programming and mean-variance hedging in discrete time. Applied Mathematical Finance, 11(1), 125.CrossRefGoogle Scholar
Černý, A. and Kallsen, J. (2009) Hedging by sequential regressions revisited. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 19(4), 591617.CrossRefGoogle Scholar
Chen, Z., Chen, B. and Dhaene, J. (2020) Fair dynamic valuation of insurance liabilities: A loss averse convex hedging approach. Scandinavian Actuarial Journal, 2020(9), 792818.CrossRefGoogle Scholar
Coleman, T.F., Li, Y. and Patron, M.-C. (2006) Hedging guarantees in variable annuities under both equity and interest rate risks. Insurance: Mathematics and Economics, 38(2), 215228.Google Scholar
Danelsson, J. (2002) The emperor has no clothes: Limits to risk modelling. Journal of Banking & Finance, 26(7), 12731296.CrossRefGoogle Scholar
Deelstra, G., Devolder, P., Gnameho, K. and Hieber, P. (2020) Valuation of hybrid financial and actuarial products in life insurance by a novel three-step method. ASTIN Bulletin: The Journal of the IAA, 50(3), 709742.CrossRefGoogle Scholar
Delbaen, F., Bellini, F., Bignozzi, V. and Ziegel, J. F. (2016) Risk measures with the cxls property. Finance and Stochastics, 20(2), 433453.CrossRefGoogle Scholar
Delong, Ł., Dhaene, J. and Barigou, K. (2019a) Fair valuation of insurance liability cash-flow streams in continuous time: Applications. ASTIN Bulletin: The Journal of the IAA, 49(2), 299333.CrossRefGoogle Scholar
Delong, Ł., Dhaene, J. and Barigou, K. (2019b) Fair valuation of insurance liability cash-flow streams in continuous time: Theory. Insurance: Mathematics and Economics, 88, 196208.Google Scholar
Deprez, O. and Gerber, H.U. (1985) On convex principles of premium calculation. Insurance: Mathematics and Economics, 4(3), 179189.Google Scholar
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, 1427.Google Scholar
Embrechts, P. (2000) Actuarial versus financial pricing of insurance. The Journal of Risk Finance, 1(4), 1726. Scholar
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), 125167.CrossRefGoogle Scholar
Engsner, H., Lindholm, M. and Lindskog, F. (2017) Insurance valuation: A computable multi-period cost-of-capital approach. Insurance: Mathematics and Economics, 72, 250264.Google Scholar
Engsner, H., Lindskog, F. and Thoegersen, J. (2021) Multiple-prior valuation of cash flows subject to capital requirements. arXiv preprint arXiv:2109.00306.Google Scholar
European Commission (2009) Directive 2009/138/ec of the european parliament and of the council of 25 november 2009 on the taking-up and pursuit of the business of insurance and reinsurance (solvency ii).Google Scholar
Fécamp, S., Mikael, J. and Warin, X. (2019) Risk management with machine-learning-based algorithms. arXiv preprint arXiv:1902.05287.Google Scholar
Föllmer, H. and Leukert, P. (1999) Quantile hedging. Finance and Stochastics, 3(3), 251273.CrossRefGoogle Scholar
Föllmer, H. and Leukert, P. (2000) Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, 4(2), 117146.Google Scholar
Föllmer, H. and Schied, A. (2002) Convex measures of risk and trading constraints. Finance and Stochastics, 6(4), 429447.CrossRefGoogle Scholar
Föllmer, H. and Schweizer, M. (1988) Hedging by sequential regression: An introduction to the mathematics of option trading. ASTIN Bulletin: The Journal of the IAA, 18(2), 147160.CrossRefGoogle Scholar
François, P., Gauthier, G. and Godin, F. (2014) Optimal hedging when the underlying asset follows a regime-switching markov process. European Journal of Operational Research, 237(1), 312322.CrossRefGoogle Scholar
Ghalehjooghi, A.S. and Pelsser, A. (2021) Time-consistent and market-consistent actuarial valuation of the participating pension contract. Scandinavian Actuarial Journal, 2021(4), 266–294.CrossRefGoogle Scholar
Gneiting, T. (2011) Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494), 746762.CrossRefGoogle Scholar
Godin, F. (2016) Minimizing CVaR in global dynamic hedging with transaction costs. Quantitative Finance, 16(3), 461475.CrossRefGoogle Scholar
Goodfellow, I., Bengio, Y. and Courville, A. (2016) Deep Learning. Cambridge, MA: MIT Press.Google Scholar
Happ, S., Merz, M. and Wüthrich, M.V. (2015) Best-estimate claims reserves in incomplete markets. European Actuarial Journal, 5(1), 55–77.CrossRefGoogle Scholar
Henderson, V. and Hobson, D. (2004) Utility Indifference Pricing-an Overview. Princeton, NJ, USA: Princeton University Press.Google Scholar
Hornik, K., Stinchcombe, M. and White, H. (1989) Multilayer feedforward networks are universal approximators. Neural Networks, 2(5), 359366.CrossRefGoogle Scholar
Koenker, R. (2005) Quantile Regression , Econometric Society Monographs. Cambridge: Cambridge University Press.Google Scholar
Koenker, R. and Bassett, G. Jr (1978) Regression quantiles. Econometrica: Journal of the Econometric Society, 46(1), 3350.CrossRefGoogle Scholar
Luciano, E., Regis, L. and Vigna, E. (2017) Single-and cross-generation natural hedging of longevity and financial risk. Journal of Risk and Insurance, 84(3), 961–986.CrossRefGoogle Scholar
Luciano, E. and Vigna, E. (2008) Mortality risk via affine stochastic intensities: Calibration and empirical relevance. Belgian Actuarial Bulletin, 8(1), 516.Google Scholar
Möhr, C. (2011) Market-consistent valuation of insurance liabilities by cost of capital. ASTIN Bulletin: The Journal of the IAA, 41(2), 315–341.Google Scholar
Møller, T. (2001a) Hedging equity-linked life insurance contracts. North American Actuarial Journal, 5(2), 7995.CrossRefGoogle Scholar
Møller, T. (2001b) Risk-minimizing hedging strategies for insurance payment processes. Finance and Stochastics, 5(4), 419–446.CrossRefGoogle Scholar
Møller, T. (2003) Indifference pricing of insurance contracts in a product space model. Finance and Stochastics, 7(2), 197217.CrossRefGoogle Scholar
Newey, W.K. and Powell, J.L. (1987) Asymmetric least squares estimation and testing. Econometrica: Journal of the Econometric Society, 55(4), 819847.CrossRefGoogle Scholar
Pelkiewicz, A., Ahmed, S., Fulcher, P., Johnson, K., Reynolds, S., Schneider, R. and Scott, A. (2020) A review of the risk margin–Solvency II and beyond. British Actuarial Journal, 25, E1. Scholar
Pelsser, A. (2011) Pricing in incomplete markets. Panel paper 25.CrossRefGoogle Scholar
Pelsser, A. and Schweizer, J. (2016) The difference between LSMC and replicating portfolio in insurance liability modeling. European Actuarial Journal, 6(2), 441494.CrossRefGoogle ScholarPubMed
Pelsser, A. and Stadje, M. (2014) Time-consistent and market-consistent evaluations. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 24(1), 2565.CrossRefGoogle Scholar
Rockafellar, R.T. and Uryasev, S. (2013) The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys in Operations Research and Management Science, 18(1–2), 3353.CrossRefGoogle Scholar
Rockafellar, R.T., Uryasev, S. and Zabarankin, M. (2008) Risk tuning with generalized linear regression. Mathematics of Operations Research, 33(3), 712729.CrossRefGoogle Scholar
Schweizer, M. (1995) Variance-optimal hedging in discrete time. Mathematics of Operations Research, 20(1), 1–32.CrossRefGoogle Scholar
Tsanakas, A., Wüthrich, M.V. and Černý, A. (2013) Market value margin via mean–variance hedging. ASTIN Bulletin: The Journal of the IAA, 43(3), 301322.CrossRefGoogle Scholar
Varnell, E. (2011) Economic scenario generators and Solvency II. British Actuarial Journal, 16(1), 121159.CrossRefGoogle Scholar
Wüthrich, M.V. (2016) Market-Consistent Actuarial Valuation , EEA Series. Cham, Switzerland: Springer.Google Scholar
Wüthrich, M.V. and Merz, M. (2013) Financial Modeling, Actuarial Valuation and Solvency in Insurance. Berlin, Heidelberg: Springer.CrossRefGoogle Scholar