Advertisement

Portfolio optimization under Solvency II

  • Marcos Escobar
  • Paul Kriebel
  • Markus Wahl
  • Rudi Zagst
S.I.: Risk in Financial Economics
  • 82 Downloads

Abstract

In the current low interest-rate and highly-regulated environment investing capital efficiently is one of the most important challenges insurance companies face. Certain quantitative parts of regulatory requirements (e.g. Solvency II capital requirements) result in constraints on the investment strategies. This paper mathematically describes the implications of Solvency II constraints on the investment strategies of insurance companies in an expected utility framework with a focus on the market risk module. For this constrained expected utility problem, we define a two-step approach that leads to closed-form approximations for the optimal investment strategies. This proposal circumvents the technical difficulties encountered when applying the convex duality approach or the theory of viscosity solutions. The investment strategies found using the two-step approach can be understood as the optimal investment strategies for constraint problems according to Solvency II. The impact of such constraints on the asset allocation and the performance of these strategies is assessed in a numerical case study.

Keywords

Portfolio optimization Investment strategies Regulatory constraints Market risk Solvency II 

Notes

Acknowledgements

Markus Wahl and Rudi Zagst greatfully acknowledge the support of Allianz Global Investors for this research.

References

  1. Bauer, D., Bergmann, D., & Reuss, A. (2010). Solvency II and nested simulations—A least-squares Monte Carlo approach. In Proceedings of the ICA congress.Google Scholar
  2. Bellman, R. (1957). Dynamic programming. Princeton: Princeton University Press.Google Scholar
  3. Bertsekas, D. P. (1995). Dynamic programming and optimal control (Vol. 1). Belmont: Athena Scientific. No. 2, Chapter 1.Google Scholar
  4. Björk, T. (2009). Arbitrage theory in continuous time (p. 255). Oxford: Oxford University Press.Google Scholar
  5. Braun, A., Schmeiser, H., & Schreiber, F. (2015). Portfolio optimization under solvency II: Implicit constraints imposed by the market risk standard formula. Journal of Risk and Insurance, 82, 1–31.CrossRefGoogle Scholar
  6. Christiansen, M., & Niemeyer, A. (2012). The fundamental definition of the solvency capital requirement in solvency II. Working Paper, Ulm University.Google Scholar
  7. Cox, J. C., & Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory, 49, 33–83.CrossRefGoogle Scholar
  8. Cvitanić, J., & Karatzas, I. (1992). Convex duality in constrained portfolio optimization. The Annals of Applied Probability, 2, 767–818.CrossRefGoogle Scholar
  9. de Soto, J. H. (2009). The fatal error of Solvency II. Economic Affairs, 29, 74–77.CrossRefGoogle Scholar
  10. Devolder, P., & Lebègue, A. (2016). Risk measures versus ruin theory for the calculation of solvency capital for long-term life insurances. Dependence Modeling, 4(1), 306–327.CrossRefGoogle Scholar
  11. Doff, R. (2008). A critical analysis of the Solvency II proposals. Risk Management and Insurance Review, 10, 69–85.Google Scholar
  12. EIOPA (2013). Technical findings on the long-term guarantees assessment appendix 1: Methodology for the calibration of the adaptation (CCP) (as tested in the assessment). EIOPA/13/297.Google Scholar
  13. Eling, M., Schmeiser, H., & Schmit, J. T. (2007). The Solvency II process: Overview and critical analysis. The Geneva Papers on Risk and Insurance Issues and Practice, 33, 193–206.Google Scholar
  14. European Union (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). Official Journal of the European Union, L 335.Google Scholar
  15. European Union (2015). 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) (1). Official Journal of the European Union, L12, Volume 58.Google Scholar
  16. Fischer, K., & Schlütter, S. (2015). Optimal investment strategies for insurance companies when capital requirements are imposed by a standard formula. The Geneva Risk and Insurance Review, 40, 15–40.CrossRefGoogle Scholar
  17. Fitch Ratings (2011). Solvency II set to reshape asset allocation and capital markets. Insurance Rating Group Special Report.Google Scholar
  18. Gatzert, N., & Wesker, H. (2012). A comparative assessment of Basel II/III and Solvency II. Geneva Papers on Risk and Insurance, 37, 539–570.CrossRefGoogle Scholar
  19. He, H. (1990). Convergence from discrete-to continuous-time contingent claims prices. Review of Financial Studies, 3, 523–546.CrossRefGoogle Scholar
  20. Höring, D. (2012). Will Solvency II market risk requirements bite? The impact of Solvency II on insurers’ asset allocation. The Geneva Papers on Risk and Insurance Issues and Practice, 38, 250–273.CrossRefGoogle Scholar
  21. Hürlimann, W. (2009). On the non-life Solvency II model. In M. Cruz (Ed.), The Solvency II handbook, Chapter 13, risk books. London: Incisive Media.Google Scholar
  22. Karatzas, I., Lehoczky, J. P., & Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM Journal on Control and Optimization, 25, 557–1586.CrossRefGoogle Scholar
  23. Kouwenberg, R. (2017). strategic asset allocation and risk budgeting for insurers under Solvency II. Working Paper, Mahidol University and Erasmus University Rotterdam.Google Scholar
  24. Merton, R. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics, 51, 247–257.CrossRefGoogle Scholar
  25. Mittnik, S. (2011). Solvency II calibrations: Where curiosity meets spuriosity, Working Paper.Google Scholar
  26. Pfeifer, D., & Strassburger, D. (2008). Solvency II: Stability problems with the SCR aggregation formula. Scandinavian Actuarial Journal, 2008, 61–77.CrossRefGoogle Scholar
  27. Pliska, S. R. (1986). A stochastic calculus model of continuous trading: Optimal portfolios. Mathematics of Operations Research, 11, 371–382.CrossRefGoogle Scholar
  28. Rieder, U., & Zagst, R. (1994). Monotonicity and bounds for convex stochastic control models. ZOR Methods and Models for Operation Research, 39, 187–207.CrossRefGoogle Scholar
  29. Sandström, A. (2007). Solvency II: Calibration for skewness. Scandinavian Actuarial Journal, 2007, 126–134.CrossRefGoogle Scholar
  30. Schubert, T. (2004). Solvency II = Basel II + X. Program on Regulation, Supervision and Legal Issues, 40 .Google Scholar
  31. Steffen, T. (2008). Solvency II and the work of CEIOPS. Geneva Papers on Risk and Insurance, 33, 60–65.CrossRefGoogle Scholar
  32. Zagst, R. (2002). Interest-rate management. Berlin: Springer.CrossRefGoogle Scholar
  33. Zariphopoulou, T. (1994). Consumption-investment models with constraints. SIAM Journal on Control and Optimization, 32, 59–85.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Statistical and Actuarial SciencesWestern UniversityLondonCanada
  2. 2.Chair of Mathematical FinanceTechnical University of MunichGarching-HochbrückGermany

Personalised recommendations