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A variational inequality approach to financial valuation of retirement benefits based on salary

Author

Listed:
  • Avner Friedman

    (University of Minnesota, Department of Mathematics, Minneapolis, MN 55455, USA)

  • Weixi Shen

    (Department of Mathematics, Fudan University, Shanghai 200433, China Manuscript)

Abstract

We consider a pension plan with the option of early retirement, and paid benefits $\Psi (S,t)$ based on salary S at the time of retirement, but with guaranteed minimum; $S=S(t)$ is a Markov process. Denote by V(S,t) the financial value of the retirement benefits; its formal definition is given in (1.16). Then $\Psi (S,t) = V(S,t)$ at the end period T, while $\Psi (S,t)\leq V(S,t)$ if early retirement is exercised. We prove that V is the unique solution of a variational inequality, and that the set $\{\Psi = V\}$, which corresponds to the optimal time to retire, consists of either one or two continuous curves $S = S_i(t)$, depending on the parameters of the model.

Suggested Citation

  • Avner Friedman & Weixi Shen, 2002. "A variational inequality approach to financial valuation of retirement benefits based on salary," Finance and Stochastics, Springer, vol. 6(3), pages 273-302.
    • Handle: RePEc:spr:finsto:v:6:y:2002:i:3:p:273-302
    Note: received: January 2001; final version received: August 2001
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    Citations

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    Cited by:

    1. Moshe A. Milevsky & Kristen S. Moore & Virginia R. Young, 2006. "Asset Allocation And Annuity‐Purchase Strategies To Minimize The Probability Of Financial Ruin," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 647-671, October.
    2. Shen, Weixi & Xu, Huiping, 2005. "The valuation of unit-linked policies with or without surrender options," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 79-92, February.
    3. Moshe Milevsky, 2004. "A diffusive wander through human life," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 21-23.
    4. Calvo-Garrido, Maria del Carmen & Pascucci, Andrea & Vázquez Cendón, Carlos, 2012. "Mathematical analysis and numerical methods for pricing pension plans allowing early retirement," MPRA Paper 36494, University Library of Munich, Germany.
    5. Tiziano Angelis & Gabriele Stabile, 2019. "On the free boundary of an annuity purchase," Finance and Stochastics, Springer, vol. 23(1), pages 97-137, January.
    6. Moore, Kristen S., 2009. "Optimal surrender strategies for equity-indexed annuity investors," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 1-18, February.
    7. E. Chevalier, 2006. "Optimal Early Retirement Near the Expiration of a Pension Plan," Finance and Stochastics, Springer, vol. 10(2), pages 204-221, April.

    More about this item

    Keywords

    Retirement benefits; variational inequality; free boundary; stochastic differential equations; optimal time;
    All these keywords.

    JEL classification:

    • G23 - Financial Economics - - Financial Institutions and Services - - - Non-bank Financial Institutions; Financial Instruments; Institutional Investors

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