Portfolio optimization under solvency II: a multi-objective approach incorporating market views and real-world constraints


We propose a new approach to handle the problem of portfolio optimization for non-life insurance company incorporating the solvency capital requirement (SCR), market views and their confident levels, several equality and inequality real-world constraints and transaction costs. We analyze two case studies: first, we consider a tri-objective optimization problem in which we minimize the Market SCR, the variance of the so-called basic own funds (BOF) and maximize the return of portfolio; secondly, we consider bi-objective optimization problem in which we minimize the variance of BOF and maximize the return of portfolio while considering the Market SCR as a constraint. We introduce a scenario-based framework in which the reference model is given by an internal model. By entropy pooling approach, we blended market views and their confident levels with the reference model to build the posterior distribution. The latter is used to compute the variance of BOF and the portfolio return. In both case studies, we obtain good results in term of risk-reward tradeoff and diversification.

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    The difference between company’s Assets and Liabilities.


  1. Anagnostopoulos, K.P., Mamanis, G.: The mean-variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms. Expert Syst. Appl. 38(11), 14208–14217 (2011)

    Google Scholar 

  2. Black, F., Litterman R.: Asset allocation: combining investor views with market equilibrium. Goldman Sachs Fixed Income Res. (1990)

  3. Braun, A., Schmeiser, H., Schreiber, F.: Portfolio optimization under solvency II: implicit constraint imposed by the market risk standard formula. J Risk Insur (2015)

  4. Cont, R.: Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223–36 (2001)

    Article  Google Scholar 

  5. Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186(2–4), 311–338 (2000)

    Article  Google Scholar 

  6. Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms. Wiley, New York (2001)

    Google Scholar 

  7. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm. IEEE Trans. Evolut. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  8. Di Tollo, G., Roli, A.: Metaheuristics for the portfolio selection problem. Int. J. Oper. Res. 5(1), 13–35 (2008)

    Google Scholar 

  9. Embrechts, P.: Actuarial versus financial pricing of insurance. J. Risk Finance 1(4), 17–26 (2000)

    Article  Google Scholar 

  10. European Insurance and Occupational Pensions Authority (EIOPA), Technical Specifications for the Preparatory Phase (Part I). Available at: https://eiopa.europa.eu (2014)

  11. Fitch Ratings: Solvency II Set to Reshape Asset Allocation and Capital Markets. Insurance Rating Group Special Report (2011)

  12. Gilli, M., Këllezi, E., Hysi, H.: A data-driven optimization heuristic for downside risk minimization. J. Risk 8(3), 1–16 (2006)

    Article  Google Scholar 

  13. Gilli, M., Maringer, D., Schumann, E.: Numerical Methods and Optimization in Finance. Academic Press, New York (2011)

    Google Scholar 

  14. Kaucic, M., Daris, R.: Multi-objective stochastic optimization programs for a non-life insurance company under solvency constraint. Risks 2015(3), 390–419 (2015). https://doi.org/10.3390/risks3030390

    Article  Google Scholar 

  15. Kaucic, M., Mojtaba, M., Mohmmad, M.: Portfolio optimization by improved NSGA-II and SPEA 2 based on different risk measures. Financ. Innov. 5(1), 5–34 (2019)

    Article  Google Scholar 

  16. Konno, H., Hiroshi, S., Hiroaki, Y.: A mean-absolute deviation-skewness portfolio optimization model. Ann. Oper. Res. 45, 205–220 (1993)

    Article  Google Scholar 

  17. Krink, T., Paterlini, S.: Multiobjective optimization using differential evolution for real-world portfolio optimization. Comput. Manag. Sci. 8, 157–179 (2011)

    Article  Google Scholar 

  18. Markowitz, H.M.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  19. Meghwani, S.S., Thakur, M.: Multi-objective heuristic algorithms for practical portfolio optimization and rebalancing with transaction cost. Appl. Soft. Comput. 67, 865–894 (2018)

    Article  Google Scholar 

  20. Metaxiotis, K., Liagkouras, K.: Multiobjective evolutionary algorithms for portfolio management: a comprehensive literature review. Expert Syst. Appl. 39(14), 11685–11698 (2012)

    Article  Google Scholar 

  21. Meucci, A.: Beyond Black-Litterman in practice: a five-step recipe to input views on non-normal markets. Risk 19, 114–119 (2006)

    Google Scholar 

  22. Meucci, A.: Fully flexible views: theory and practice. Risk 21, 97–102 (2008)

    Google Scholar 

  23. Meucci, A.: The Black–Litterman approach: original model and extensions. In: The Encyclopedia of Quantitative Finance, vol .1, pp. 196–199. Wiley, New York (2010)

  24. Mishra, S.K., Panda, G., Majhi, R.: A comparative performance assessment of a set of multiobjective algorithms for constrained portfolio assets selection. Swarm Evol. Comput. 16, 38–51 (2014)

    Article  Google Scholar 

  25. Pareto, V., Cours, D.: Economie Politique, Vols. I and II. F. Rouge, Lausanne (1986)

  26. Pezier, J.: Global portfolio optimization revisited: A least discrimination alternantive to Black–Litterman. ICMA Centre Discussion Papers in Finance (2007)

  27. Qian, E., Gorman, S.: Conditional distribution in portfolio theory. Financ. Anal. J. 57, 44–51 (2001)

    Article  Google Scholar 

  28. Roy, A.D.: Safety first and the holding of asset. Econometria 20, 431–499 (1952)

    Article  Google Scholar 

  29. Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under condition of risks. J. Finance 19, 425–442 (1964)

    Google Scholar 

  30. Tobin, J.: Liquidity preference as behavior towards risk. Rev. Econ. Stud. 25(1), 65–86 (1958)

    Article  Google Scholar 

  31. Zenios, S.A.: Practical Financial Optimization. Blackwell Publishing Ltd, New York (2007)

    Google Scholar 

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The author acknowledges the editor and two anonymous reviewers for their constructive feedback. It is his belief that the manuscript is substantially improved after making the suggested edits. A special thanks to author’s colleagues who allow him to write this paper. In particular, the author wishes to thank Daniele Della Rossa, for correcting and improving his English language, and Riccardo Casalini, aka Mister Validation, for several helpful conversation on the Solvency II Directive. Finally, the author would like to thank professor Andrea Pascucci, Sergio Polidoro and Paolo Foschi for some useful remarks.

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Correspondence to Marco Di Francesco.

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In this appendix, we report the two correlation matrices for Solvency II Standard Formula as reported in EIOPA (2014). In Table 12, we report the correlation matrix used to compute the SCR as the aggregate capital charge of these modules

Table 12 Solvency II: correlation matrix for solvency II standard formula

In Table 13, we report the correlation matrix used to compute the capital requirement for market risk (\(\text {Market}_{\text {SCR}}\)) as the aggregate capital charge of these sub module where the factor A is equal to 0 in the upward stress scenario and equal to 0.5 in the downward stress scenario.

Table 13 Solvency II: market correlation matrix for solvency II standard formula

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Di Francesco, M. Portfolio optimization under solvency II: a multi-objective approach incorporating market views and real-world constraints. Decisions Econ Finan (2021). https://doi.org/10.1007/s10203-021-00320-3

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  • Portfolio theory
  • Solvency II
  • Multi-objective evolution algorithm
  • Real-world constraints
  • Non-life insurance company