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Asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting

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  • Delong, Łukasz
  • Chen, An

Abstract

The present paper studies an optimal withdrawal and investment problem for a retiree who is interested in sustaining her retirement consumption above a pre-specified minimum consumption level. Apparently, the withdrawal and investment policy depends substantially on the retiree’s health condition and her time preferences (subjective discount factor). We assume that the health of the retiree can worsen or improve in an unpredictable way over her lifetime and model the retiree’s mortality intensity by a stochastic process. In order to make the decision about the consumption and investment policy more realistic, we assume that the retiree applies a non-exponential discount factor (an exponential discount factor with a small amount of hyperbolic discounting) to value her future income. In other words, we consider an optimization problem by combining four important aspects: asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting. Due to the non-exponential discount factor, we have to solve a time-inconsistent optimization problem. We derive a non-local HJB equation which characterizes the equilibrium optimal investment and consumption strategy. We establish the first-order expansions of the equilibrium value function and the equilibrium strategies by applying expansion techniques. The expansion is performed on the parameter controlling the degree of discounting in the hyperbolic discounting that is added to the exponential discount factors. The first-order equilibrium investment and consumption strategies can be calculated in a feasible way by solving PDEs.

Suggested Citation

  • Delong, Łukasz & Chen, An, 2016. "Asset allocation, sustainable withdrawal, longevity risk and non-exponential discounting," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 342-352.
    • Handle: RePEc:eee:insuma:v:71:y:2016:i:c:p:342-352
    DOI: 10.1016/j.insmatheco.2016.10.002
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Bäuerle Nicole & Chen An, 2019. "Optimal retirement planning under partial information," Statistics & Risk Modeling, De Gruyter, vol. 36(1-4), pages 37-55, December.
    2. Huang, H. & Milevsky, M.A. & Salisbury, T.S., 2017. "Retirement spending and biological age," Journal of Economic Dynamics and Control, Elsevier, vol. 84(C), pages 58-76.
    3. Łukasz Delong, 2018. "Time-inconsistent stochastic optimal control problems in insurance and finance," Collegium of Economic Analysis Annals, Warsaw School of Economics, Collegium of Economic Analysis, issue 51, pages 229-254.
    4. Milevsky, Moshe A., 2020. "Calibrating Gompertz in reverse: What is your longevity-risk-adjusted global age?," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 147-161.
    5. Maria Alexandrova & Nadine Gatzert, 2019. "What Do We Know About Annuitization Decisions?," Risk Management and Insurance Review, American Risk and Insurance Association, vol. 22(1), pages 57-100, March.
    6. Chen Shou & Xiang Shengpeng & He Hongbo, 2019. "Do Time Preferences Matter in Intertemporal Consumption and Portfolio Decisions?," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 19(2), pages 1-13, June.

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    More about this item

    Keywords

    Hyperbolic discounting; Time-inconsistent optimization problem; Non-local HJB equation; Equilibrium strategies; PDE;
    All these keywords.

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • G1 - Financial Economics - - General Financial Markets
    • D9 - Microeconomics - - Micro-Based Behavioral Economics

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