Skip to main content
SpringerLink
Log in

Pension funds with a minimum guarantee: a stochastic control approach

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

In this paper we propose and study a continuous-time stochastic model of optimal allocation for a defined contribution pension fund with a minimum guarantee. We adopt the point of view of a fund manager maximizing the expected utility from the fund wealth over an infinite horizon. In our model the dynamics of wealth takes directly into account the flows of contributions and benefits, and the level of wealth is constrained to stay above a “solvency level.” The fund manager can invest in a riskless asset and in a risky asset, but borrowing and short selling are prohibited. We concentrate the analysis on the effect of the solvency constraint, analyzing in particular what happens when the fund wealth reaches the allowed minimum value represented by the solvency level.

The model is naturally formulated as an optimal stochastic control problem with state constraints and is treated by the dynamic programming approach. We show that the value function of the problem is a regular solution of the associated Hamilton–Jacobi–Bellman equation. Then we apply verification techniques to get the optimal allocation strategy in feedback form and to study its properties. We finally give a special example with explicit solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Battocchio, P., Menoncin, F.: Optimal pension management in a stochastic framework. Insur. Math. Econ. 34, 79–95 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Battocchio, P., Menoncin, F., Scaillet, O.: Optimal asset allocation for pension funds under mortality risk during the accumulation and decumulation phases. Ann. Oper. Res. 152, 141–165 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M., Martellini, L.: Optimal investment decisions when time-horizon is random. J. Math. Econ. 44, 1100–1113 (2008)

    Article  MATH  Google Scholar 

  4. Bouchard, B., Pham, H.: Wealth-path dependent utility maximization in incomplete markets. Finance Stoch. 8, 579–603 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Boulier, J.F., Huang, S.J., Taillard, G.: Optimal management under stochastic interest rates: the case of a protected pension fund. Insur. Math. Econ. 28, 173–189 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cadenillas, A., Sethi, S.: Consumption-investment problem with subsistence consumption, bankruptcy, and random market coefficients. J. Optim. Theory Appl. 93, 243–272 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cairns, A.J.G., Blake, D., Dowd, K.: Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans. J. Econ. Dyn. Control 30, 843–877 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choulli, T., Taksar, M., Zhou, X.Y.: A diffusion model for optimal dividend distribution for a company with constraints on risk control. SIAM J. Control Optim. 41, 1946–1979 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crandall, M., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (New Ser.) 27, 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deelstra, G., Grasselli, M., Koehl, P.F.: Optimal investment strategies in the presence of a minimum guarantee. Insur. Math. Econ. 33, 189–207 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deelstra, G., Grasselli, M., Koehl, P.F.: Optimal design of a guarantee for defined contribution funds. J. Econ. Dyn. Control 28, 2239–2260 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Duffie, D., Fleming, W., Soner, H.M., Zariphopoulou, T.: Hedging in incomplete markets with HARA utility. J. Econ. Dyn. Control 21, 753–782 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. El Karoui, N., Meziou, A.: Max-plus decomposition of supermartingales and convex order. Applications to American options and portfolio insurance. Ann. Probab. 36, 647–697 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. El Karoui, N., Jeanblanc, M., Lacoste, V.: Optimal portfolio management with American capital guarantee. J. Econ. Dyn. Control 29, 449–468 (2005)

    Article  MATH  Google Scholar 

  15. Federico, S.: A pension fund in the accumulation phase: a stochastic control approach. Banach Cent. Publ. Adv. Math. Finance 83, 61–83 (2008)

    Article  MathSciNet  Google Scholar 

  16. Federico, S.: Stochastic optimal control problems for pension funds management. Ph.D. Thesis, Scuola Normale Superiore (2009)

  17. Federico, S.: A stochastic control problem with delay arising in a pension fund model. Finance Stoch. (2010, forthcoming)

  18. Feller, W.: The parabolic differential equations and the associated semi-group of transformations. Ann. Math. 55, 468–519 (1952)

    Article  MathSciNet  Google Scholar 

  19. Feller, W.: Diffusion processes in one dimension. Trans. Am. Math. Soc. 77, 1–31 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fuhrman, M., Tessitore, G.: Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Ann. Probab. 32, 607–660 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Goetzmann, W.N., Ingersoll, J.E., Ross, S.A.: High-water marks and hedge fund management contracts. J. Finance 58, 1685–1718 (2003)

    Article  Google Scholar 

  22. Goldys, B., Gozzi, F.: Second order Hamilton–Jacobi equations in Hilbert spaces and stochastic optimal control: L 2 approach. Stoch. Process. Appl. 116, 1932–1963 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gozzi, F., Marinelli, C.: Stochastic optimal control of delay equations arising in advertising models. In: Da Prato, G., et al. (eds.) Stochastic Partial Differential Equations and Applications VII. Papers of the 7th Meeting, Levico Terme, Italy, 5–10 January 2004. Lecture Notes in Pure and Applied Mathematics, vol.  245, pp. 133–148. Chapman & Hall/CRC Press, London/Boca Raton (2006)

    Google Scholar 

  24. Haberman, S., Vigna, E.: Optimal investment strategies and risk measures in defined contribution pension schemes. Insur. Math. Econ. 31, 35–69 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ishii, H., Loreti, P.: A class of stochastic optimal control problems with state constraint. Indiana Univ. Math. J. 51, 1167–1196 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  27. Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)

    MATH  Google Scholar 

  28. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

    MATH  Google Scholar 

  29. Karatzas, I., Lehoczky, J.P., Sethi, S., Shreve, S.E.: Explicit solution of a general consumption/investment problem. Math. Oper. Res. 11, 261–294 (1986)

    Article  MathSciNet  Google Scholar 

  30. Katsoulakis, M.: Viscosity solutions of second order fully nonlinear elliptic equations with state constraints. Indiana Univ. Math. J. 43, 493–519 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Merton, R.C.: Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econ. Stat. 51, 247–257 (1969)

    Article  Google Scholar 

  32. Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413 (1971)

    Article  MathSciNet  Google Scholar 

  33. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999)

    MATH  Google Scholar 

  34. Sbaraglia, S., Papi, M., Briani, M., Bernaschi, M., Gozzi, F.: A model for the optimal asset-liability management for insurance companies. Int. J. Theor. Appl. Finance 6, 277–299 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Schwartz, E.S., Tebaldi, C.: Illiquid assets and optimal portfolio choice. NBER Working Paper No. W12633 (2006). http://www.nber.org/papers/w12633

  36. Sethi, S., Taksar, M.: Infinite-horizon investment consumption model with a nonterminal bankruptcy. J. Optim. Theory Appl. 74, 333–346 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sethi, S., Taksar, M., Presman, E.: Explicit solution of a general consumption/portfolio problem with subsistence consumption and bankruptcy. J. Econ. Dyn. Control 16, 747–768 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. Soner, H.M.: Optimal control with state-space constraint I. SIAM J. Control Optim. 24, 552–561 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  39. Soner, H.M.: Stochastic Optimal Control in Finance. Cattedra Galileiana, Scuola Normale Superiore, Pisa (2004)

    MATH  Google Scholar 

  40. Starks, L.T.: Performance incentive fees: an agency theoretic approach. J. Financ. Quant. Anal. 22, 17–32 (1987)

    Article  Google Scholar 

  41. Vigna, E., Haberman, S.: Optimal investment strategy for defined contribution pension schemes. Insur. Math. Econ. 28, 233–262 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    MATH  Google Scholar 

  43. Zariphopoulou, T.: Consumption-investment models with constraints. SIAM J. Control Optim. 32, 59–85 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento Istituzioni, Metodi Quantitativi e Territorio, Facoltà di Economia, Università degli studi di Cassino, via Sant’Angelo s.n.c., 03043, Cassino, FR, Italy

    Marina Di Giacinto

  2. Dipartimento di Scienze Economiche ed Aziendali, Facoltà di Economia, Libera Università Internazionale degli Studi Sociali “Guido Carli”, viale Romania 32, 00197, Roma, Italy

    Salvatore Federico & Fausto Gozzi

Authors
  1. Marina Di Giacinto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salvatore Federico
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fausto Gozzi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fausto Gozzi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di Giacinto, M., Federico, S. & Gozzi, F. Pension funds with a minimum guarantee: a stochastic control approach. Finance Stoch 15, 297–342 (2011). https://doi.org/10.1007/s00780-010-0127-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-010-0127-7

Keywords

Mathematics Subject Classification (2000)

JEL Classification

Search

Navigation