Table of Links Introduction
Mean-field Libor market model
Life insurance with profit participation
Numerical ALM modeling
Phenomenological assumptions and numerical evidence
Estimation of future discretionary benefits
Application to publicly available reporting data
Conclusions
Declarations
References Introduction Introduction Mean-field Libor market model Mean-field Libor market model Life insurance with profit participation Life insurance with profit participation Numerical ALM modeling Numerical ALM modeling Phenomenological assumptions and numerical evidence Phenomenological assumptions and numerical evidence Estimation of future discretionary benefits Estimation of future discretionary benefits Application to publicly available reporting data Application to publicly available reporting data Conclusions Conclusions Declarations Declarations References References Conclusions In the present paper we have concerned ourselves with three strands of thought all connected to the market consistent valuation of realistic life insurance portfolios: (1) interest rate scenario generation suitable for long term modelling; (2) numerical ALM modelling; (3) algebraic formulae to estimate numerical results. From a practical point of view we may draw the following conclusions concerning the calculation of best estimates or future discretionary benefits: (1) Theorem 1.4 provides sufficient conditions for the existence and uniqueness of solutions to the mean-field Libor market model. If the MF-LMM is specified by (1.3) these assumptions are straightforward to check. This allows to extend the Libor market model, which is very popular in the insurance industry, to the mean-field setting, and thus reduce the probability of explosion (cf. [7]). (2) The representation formula (2.32) allows to represent the F DB without any policyholder cash-flows. Since F DB0 is known, one expects that this formula should lead to a reduced Monte Carlo error in any numerical simulation. To apply this formula it should be checked that the evolution equation (2.27) and the no-leakage principle are complete, i.e. contain all relevant flows, else these should be amended to arrive at an augmented representation. Numerical evidence for the reduced Monte Carlo error is provided in Table 2 where it can be seen that F DBmce rep , corresponding to representation (2.32), is significantly smaller than F DBmce CF . (3) The representation formula also gives conditions on the suitable market consistency of an interest rate scenario generator: the expressions E[B −1 t cogt] should be priced as accurately as possible. To connect this to market data we can use the simplified model (4.47), acting as an over-estimate (cf. (4.49)), to arrive at the caplet prices O − t defined in (5.54). The market implied value, COG \, can now be compared to the model implied value, COG \MC . These values are contained in Table 2, and hence it can be observed that the MF-LMM provides an acceptable level of market consistency (compared to MV0) but certainly not an exact fit. (4) Section 3 contains a detailed description of management rules. These rules are designed to maximize shareholder value while meeting certain constraints (most prominently the surplus fund constraint (3.42)) to realistically remain a competitive insurance provider. However, this maximization is only speculative and not proven in any formal sense. In this regard it would be interesting to analyze the connection to the optimal control problem stated in Section 2.4, this is left as an open problem for future investigation. (5) Section 4 contains the assumptions and numerical study providing evidence for their validity so that we may derive lower and upper bound estimates in Section 5. The relevant formulae are (5.58), (5.59) and (5.60) for LBd, UBd and F DB \, respectively. The crucial point of these formulae is that their calculation is very easy, it is purely algebraic and depends only on a few numbers. Table 2 shows that the estimates hold over a relatively wide range of parameters. We have considered 9 scenarios corresponding to different levels of initial unrealized gains and scalings of premium payments. The latter is equivalent to varying the relationship between prevailing yield curve and level of minimum guarantee rate. (6) Section 6.2 contains an application of the estimation formulae to public data of a real life insurance company for 6 different accounting years. These results are summarized in Table 5, and it can be observed that the estimation is successful to a remarkable degree of accuracy – given that the 6 accounting years represent quite different economic conditions and that all the relevant information is taken from publicly available records. These sources are provided en d´etail in Tables 3 and 4, and have to be complimented by the prevailing yield curve ([35]). Declarations Funding. This work is not supported by external funding. Funding. Conflicts of interest/Competing interests. The authors declare no conflicts of interest or competing interests. Conflicts of interest/Competing interests. Availability of data and material. All data used in this work is either publicly available (sources are given in the References) or aggregated and anonymized. Data can be provided upon request to the corresponding author. Availability of data and material. Authors’ contributions. Florian Gach analyzed most of the public data. Simon Hochgerner designed the project and wrote most of the manuscript. Eva Kienbacher and Gabriel Schachinger carried out most of the programming. All authors reviewed the manuscript. Authors’ contributions. References [1] Albrecher, H., Bauer, D., Embrechts, P. et al. Asset-liability management for long-term insurance business. Eur. Actuar. J. 8, 9–25 (2018). https://doi.org/10.1007/s13385-018-0167-5 [2] Anna Rita Bacinello, Thorsten Sehner and Pietro Millossovich, On the Market-Consistent Valuation of Participating Life Insurance Heterogeneous Contracts under Longevity Risk, Risks 2021, 9(1), 20. https://doi.org/10.3390/risks9010020 [3] Bao, J, Ren, P & Wang, F-Y, Bismut formula for Lions derivative of distribution-path dependent SDEs, Journal of Differential Equations, vol. 282, pp. 285-329. https://doi.org/10.1016/j.jde.2021.02.019 [4] F. Black, The pricing of commodity contracts, J. Financial Economics 3 (1976), pp. 167-179. https://doi.org/10.1016/0304- 405X(76)90024-6 [5] D. Brigo, F. Mercurio, Interest rate models – theory and practice, Springer 2006. [6] P. Cardaliaguet, Notes on mean-field games, P.-L. Lions’ lecture at Coll`ege de France (2012). [7] S. Desmettre, S. Hochgerner, S. Omerovic, S.Thonhauser, A mean-field extension of the Libor market model International J. 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(2017), mean-field Limit for Stochastic Particle Systems, In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 1 . Modeling and Simulation in Science, Engineering and Technology. Birkh¨auser, Cham. https://doi.org/10.1007/978-3-319-49996-3 10 [21] B. Jourdain, S. M´el´eard, W.A. Woyczynski, Nonlinear SDEs driven by L´evy processes and related PDEs, Alea 4:1-29, 2008. [22] Laurent, J.-P., Norberg, R., and Planchet, F., Modelling in Life Insurance – A Management Perspective, EAA Series (2016), Springer International Publishing. [23] H.P. McKean, A class of Markov processes associated with nonlinear parabolic equations, PNAS 56 (1966) [24] C. O’Brien, Valuation of Life Insurance Liabilities on a Market-Consistent Basis: Experience from the United Kingdom, Actuarial Practice Forum (2009). [25] R. Rebonato, K. McKay & R. White (2009), The SABR/LIBOR Market Model, Wiley, 1. Edition (2009) [26] T. Sheldon, A. Smith, Market Consistent Valuation of Life Assurance Business, British Actuarial Journal 10(3) (2004), 543-605. [27] A.-S. Sznitman, Topics in propagation of chaos, Springer Lecture Notes 1991 [28] J. Vedani, N. El Karoui, S. Loisel, J.-L. Prigent, Market inconsistencies of market-consistent European life insurance economic valuations: pitfalls and practical solutions, Eur. Actuar. J. 7 (2017). https://doi.org/10.1007/s13385-016-0141-z [29] Mario V. W¨uthrich, Market-Consistent Actuarial Valuation, Springer EAA Series 3rd Ed. 2016. [30] Bundesministerium der Finanzen (BMF), Verordnung ¨uber die Mindestbeitragsr¨uckerstattung in der Lebensversicherung. [31] Verordnung der Finanzmarktaufsichtsbeh¨orde (FMA) ¨uber die Gewinnbeteiligung in der Lebensversicherung. [32] 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). [33] Commission Delegated Regulation (EU) 2015/35 of 10 October 2014 supplementing Directive 2009/138/EC of the European Parliament and of the Council on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II). [34] Commission Implementing Regulation (EU) 2015/2450 of 2 December 2015 laying down implementing technical standards with regard to the templates for the submission of information to the supervisory authorities according to Directive 2009/138/EC of the European Parliament and of the Council [35] Risk-free interest rate term structure: https://eiopa.europa.eu/regulation-supervision/insurance/solvency-ii-technicalinformation/risk-free-interest-rate-term-structures. Accessed 14 Sep 2023. [36] Allianz Lebensversicherungs-AG Bericht ¨uber Solvabilit¨at und Finanzlage. https://www.allianzdeutschland.de/berichte-uebersolvabilitaet-und-finanzlage. Accessed 14 Sep 2023 [37] Allianz Lebensversicherungs-AG Gesch¨aftsbericht. https://www.allianzdeutschland.de/geschaeftsberichte-der-allianzdeutschland-ag. Accessed 14 Sep 2023 Authors:
FLORIAN GACH
SIMON HOCHGERNER
EVA KIENBACHER
GABRIEL SCHACHINGER Authors: FLORIAN GACH SIMON HOCHGERNER EVA KIENBACHER GABRIEL SCHACHINGER This paper is available on arviv under CC by 4.0 Deed (Attribution 4.0 International) license This paper is available on arviv under CC by 4.0 Deed (Attribution 4.0 International) license available on arviv

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Mean-Field LIBOR Models Offer Reliable Bounds for Life Insurance Valuation, Study Finds

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