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FAIR VALUATION OF INSURANCE LIABILITY CASH-FLOW STREAMS IN CONTINUOUS TIME: APPLICATIONS

Published online by Cambridge University Press:  10 April 2019

Łukasz Delong*
Affiliation:
Warsaw School of Economics SGH Collegium of Economic Analysis, Institute of Econometrics Niepodległości162, Warsaw 02-554, Poland E-mail: lukasz.delong@sgh.waw.pl
Jan Dhaene
Affiliation:
KU Leuven Actuarial Research Group, AFI, Faculty of Business and Economics Naamsestraat 69, 3000 Leuven, Belgium E-mail: jan.dhaene@kuleuven.be
Karim Barigou
Affiliation:
KU Leuven Actuarial Research Group, AFI, Faculty of Business and Economics Naamsestraat 69, 3000 Leuven, Belgium E-mail: karim.barigou@kuleuven.be

Abstract

Delong et al. (2018) presented a theory of fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. In this paper, we investigate in detail two practical applications of our theory of fair valuation. In the first example, we consider the fair valuation of a terminal benefit which is contingent on correlated tradeable and non-tradeable financial risks. In the second example, we consider a portfolio of unit-linked contracts contingent on a non-tradeable insurance and a tradeable financial risk. We derive partial differential equations (PDEs) which characterize the continuous-time fair valuation operators in these two examples and we find explicit solutions to these PDEs. The fair values of the liabilities are decomposed into the best estimate of the liability and a risk margin. The arbitrage-free representations of the fair values of the liabilities are derived and the dynamic hedging strategies associated with the continuous-time fair valuation operators are also established. Detailed interpretations of the results, which should be useful both for researchers and practitioners, are provided.

Type
Research Article
Copyright
Copyright © Astin Bulletin 2019 

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References

Barigou, K., Chen, Z. and Dhaene, J. (2018) Fair valuation of insurance liabilities: Merging actuarial judgement with market- and time-consistency. Available at SSRN:3293741. Submitted.CrossRefGoogle Scholar
Barigou, K. and Dhaene, J. (2019) Fair valuation of insurance liabilities via mean-variance hedging in a multi-period setting. Scandinavian Actuarial Journal, 2019(2), 163187.CrossRefGoogle Scholar
Delong, Ł. (2013) Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. London: Springer.CrossRefGoogle Scholar
Delong, Ł., Dhaene, J. and Barigou, K. (2018) Fair valuation of insurance liability cash-flow streams in continuous time: Theory. Submitted.CrossRefGoogle 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
Engsner, H., Lindensjö, K. and Lindskog, F. (2018) The value of a liability cash flow in discrete time subject to capital requirements. Available at arXiv:1808.03328v. Submitted.Google Scholar
Engsner, H., Lindholm, M. and Lindskog, F. (2017) Insurance valuation: A computable multiperiod cost-of-capital approach. Insurance: Mathematics and Economics, 72, 250264.Google Scholar
Engsner, H. and Lindskog, F. (2018) Continuous-time limits of multi-period cost-of-capital valuations. Research reports in Mathematical Statistics, Stockholm University, ISSN 1650–0377. Submitted.Google Scholar
Happ, S., Merz, M. and Wüthrich, M. V. (2015) Best-estimate claims reserves in incomplete markets. European Actuarial Journal, 5, 5577.CrossRefGoogle Scholar
Lindset, S. and Persson, S.-A. (2009) Continuous monitoring: Does credit risk vanish? ASTIN Bulletin: The Journal of the IAA, 39, 577589.CrossRefGoogle Scholar
Möhr, C. (2011) Market-consistent valuation of insurance liabilities by cost of capital. ASTIN Bulletin: The Journal of the IAA, 41, 315341.Google Scholar
Natolski, J. and Werner, R. (2018) Mathematical foundation of the replicating portfolio approach. Scandinavian Actuarial Journal, 2018, 481504.CrossRefGoogle Scholar
Pelsser, A. (2010) Time-consistent and Market-consistent Actuarial Valuations. Available at SSRN: https://ssrn.com/abstract=1551323.CrossRefGoogle Scholar
Pelsser, A. and Ghalehjooghi, A.S. (2016) Time-consistent actuarial valuations. Insurance: Mathematics and Economics, 66, 97112.Google Scholar
Pelsser, A. and Stadje, M. (2014) Time-consistent and market-consistent evaluations. Mathematical Finance, 24, 2565.CrossRefGoogle Scholar
Stadje, M. (2010) Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach. Insurance: Mathematics and Economics, 47, 391404.Google Scholar