Risk-Free Investments and Their Comparison with Simple Risky Strategies in Pension Insurance Model: Solving Singular Problems for Integro-Differential Equations

Abstract

A collective pension insurance (life annuity) model is investigated in the case of risk-free investments, i.e., when the whole surplus of an insurance company at each time is invested in risk-free asset (bank account). This strategy is compared with previously studied simple risky investment strategies, according to which, irrespective of the surplus of an insurance company, a constant positive fraction of this surplus at each time consists of risky assets (stocks), while the remaining fraction is invested in a bank account. The strategies are compared in terms of a traditional solvency criterion, namely, the survival probability. The original insurance model is dual to the classical Cramér–Lundberg model: the variation in the surplus over the portfolio of same-type contracts is described by the sum of a decreasing deterministic linear function corresponding to total pension payments and a compound Poisson process with positive jumps corresponding to the income gained by the company at the moments of transferring policyholders' property. In the case of an exponential jump size distribution and risk-free investments, it is shown that the survival probability regarded as a function of the initial surplus defined on the nonnegative real half-line is a solution of a singular problem for an integro-differential equation with a non-Volterra integral operator. The solution of the stated problem is obtained, its properties are analytically examined, and numerical examples are given. Examples are used to compare the influence exerted by risky and risk-free investments on the survival probability in the given model.

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REFERENCES

  1. 1

    T. A. Belkina, N. B. Konyukhova, and B. V. Slavko, “Solvency of an insurance company in a dual risk model with investment: Analysis and numerical study of singular boundary value problems,” Comput. Math. Math. Phys. 59 (11), 1904–1927 (2019).

    MathSciNet  Article  Google Scholar 

  2. 2

    J. Grandell, Aspects of Risk Theory (Springer-Verlag, New York, 1991).

    Google Scholar 

  3. 3

    S. Asmussen and H. Albrecher, Ruin Probabilities, Advanced Series on Statistical Science and Applied Probability, Vol. 14 (World Scientific, Singapore, 2010).

  4. 4

    H. Albrecher, A. Badescuc, and D. Landriaultd, “On the dual risk model with tax payments,” Insurance Math. Econ. 42 (3), 1086–1094 (2008).

    MathSciNet  Article  Google Scholar 

  5. 5

    B. Avanzia, H. U. Gerber, and E. S. W. Shiub, “Optimal dividends in the dual model,” Insurance Math. Econ. 41 (1), 111–123 (2007).

    MathSciNet  Article  Google Scholar 

  6. 6

    Yu. Kabanov and S. Pergamenshchikov, “In the insurance business risky investments are dangerous: The case of negative risk sums,” Finance Stoch. 20 (2), 355–379 (2016).

    MathSciNet  Article  Google Scholar 

  7. 7

    T. A. Belkina and Yu. M. Kabanov, “Viscosity solutions of integro-differential equations for nonruin probabilities,” Theory Probab. Appl. 60 (4), 671–679 (2016).

    MathSciNet  Article  Google Scholar 

  8. 8

    T. A. Belkina and N. B. Konyukhova, “Survival probability in the life annuity insurance model as a viscosity solution to an integro-differential equation,” Vestn. Tsentr. Ekon.-Mat. Inst. Ross. Akad. Nauk, No. 1, 1–9 (2018). https://doi.org/10.33276/S0000097-9-1

    Article  Google Scholar 

  9. 9

    T. A. Belkina, N. B. Konyukhova, and B. V. Slavko, “Analytic-numerical investigations of singular problems for survival probability in the dual risk model with simple investment strategies,” in Analytical and Computational Methods in Probability Theory and Its Applications, Ed. by V. V. Rykov et al., Lect. Notes Comput. Sci. 10684, 236–250 (2017). https://doi.org/10.1007/978-3-319-71504-9

  10. 10

    T. A. Belkina and N. B. Konyukhova, “On sufficient conditions for survival probability in the life annuity insurance model with risk-free investment income,” in Proceedings of the 9th Moscow International Conference on Operations Research (ORM-2018), Moscow, October 22–27,2018, Ed. by F. Ereshko (MAKS, Moscow, 2018), Vol. 1, pp. 213–218.

  11. 11

    T. Belkina, “Risky investment for insurers and sufficiency theorems for the survival probability,” Markov Processes Relat. Fields 20, 505–525 (2014).

    MathSciNet  MATH  Google Scholar 

  12. 12a

    N. B. Konyukhova, “Singular Cauchy problems for singularly perturbed systems of nonlinear ordinary differential equations,” I: Differ. Equations 32 (1), 54–63 (1996);

    MathSciNet  MATH  Google Scholar 

  13. 12b

    II: Differ. Equations 32 (4), 491–500 (1996).

    MathSciNet  Google Scholar 

  14. 13

    T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Dynamical insurance models with investment: Constrained singular problems for integrodifferential equations,” Comput. Math. Math. Phys. 56 (1), 43–92 (2016).

    MathSciNet  Article  Google Scholar 

  15. 14

    C. Hipp and M. Plum, “Optimal investment for insurers,” Insurance Math. Econ. 27 (2), 215–228 (2000).

    MathSciNet  Article  Google Scholar 

  16. 15

    C. Hipp and M. Plum, “Optimal investment for investors with state dependent income, and for insurers,” Finance Stoch. 7 (3), 299–321 (2003).

    MathSciNet  Article  Google Scholar 

  17. 16

    P. Azcue and M. Muler, “Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints,” Insur. Math. Econ. 44 (1), 26–34 (2009).

    MathSciNet  Article  Google Scholar 

  18. 17

    T. Belkina, C. Hipp, S. Luo, and M. Taksar, “Optimal constrained investment in the Cramer–Lundberg model,” Scand. Actuarial J., No. 5, 383–404 (2014).

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Correspondence to T. A. Belkina or N. B. Konyukhova or B. V. Slavko.

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Translated by I. Ruzanova

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Belkina, T.A., Konyukhova, N.B. & Slavko, B.V. Risk-Free Investments and Their Comparison with Simple Risky Strategies in Pension Insurance Model: Solving Singular Problems for Integro-Differential Equations. Comput. Math. and Math. Phys. 60, 1621–1641 (2020). https://doi.org/10.1134/S096554252010005X

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Keywords:

  • pension insurance
  • dual risk model
  • survival probability
  • investments
  • risk-free assets
  • exponential premium size distribution
  • integro-differential equation
  • singular problem