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Optimal dividends under a drawdown constraint and a curious square-root rule

Author

Listed:
  • Hansjörg Albrecher

    (University of Lausanne
    Swiss Finance Institute)

  • Pablo Azcue

    (Universidad Torcuato Di Tella)

  • Nora Muler

    (Universidad Torcuato Di Tella)

Abstract

In this paper, we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e., the dividend rate can never decrease below a given fraction a $a$ of its historical maximum. We solve the resulting two-dimensional optimal control problem and identify the value function as the unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We then derive sufficient conditions under which a two-curve strategy is optimal, and we show how to determine its concrete form using calculus of variations. We establish a smooth-pasting principle and show how it can be used to prove the optimality of two-curve strategies for sufficiently large initial and maximum dividend rates. We also give a number of numerical illustrations in which the optimality of the two-curve strategy can be established for instances with smaller values of the maximum dividend rate and the concrete form of the curves can be determined. One observes that the resulting drawdown strategies nicely interpolate between the solution for the classical unconstrained dividend problem and that for a ratcheting constraint as recently studied in Albrecher et al. (SIAM J. Financial Math. 13:657–701, 2022). When the maximum allowed dividend rate tends to infinity, we show a surprisingly simple and somewhat intriguing limit result in terms of the parameter a $a$ for the surplus level above which, for a sufficiently large current dividend rate, a take-the-money-and-run strategy is optimal in the presence of the drawdown constraint.

Suggested Citation

  • Hansjörg Albrecher & Pablo Azcue & Nora Muler, 2023. "Optimal dividends under a drawdown constraint and a curious square-root rule," Finance and Stochastics, Springer, vol. 27(2), pages 341-400, April.
  • Handle: RePEc:spr:finsto:v:27:y:2023:i:2:d:10.1007_s00780-023-00500-6
    DOI: 10.1007/s00780-023-00500-6
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    References listed on IDEAS

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    1. Philip H. Dybvig, 1995. "Dusenberry's Ratcheting of Consumption: Optimal Dynamic Consumption and Investment Given Intolerance for any Decline in Standard of Living," Review of Economic Studies, Oxford University Press, vol. 62(2), pages 287-313.
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    3. Hansjoerg Albrecher & Nicole Bäuerle & Martin Bladt, 2018. "Dividends: From Refracting to Ratcheting," Swiss Finance Institute Research Paper Series 18-32, Swiss Finance Institute.
    4. Radner, Roy & Shepp, Larry, 1996. "Risk vs. profit potential: A model for corporate strategy," Journal of Economic Dynamics and Control, Elsevier, vol. 20(8), pages 1373-1393, August.
    5. Asmussen, Soren & Taksar, Michael, 1997. "Controlled diffusion models for optimal dividend pay-out," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 1-15, June.
    6. Romuald Elie & Nizar Touzi, 2008. "Optimal lifetime consumption and investment under a drawdown constraint," Finance and Stochastics, Springer, vol. 12(3), pages 299-330, July.
    7. Hans U. Gerber, 1972. "Games of Economic Survival with Discrete- and Continuous-Income Processes," Operations Research, INFORMS, vol. 20(1), pages 37-45, February.
    8. Benjamin Avanzi, 2009. "Strategies for Dividend Distribution: A Review," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(2), pages 217-251.
    9. Julia Eisenberg & Peter Grandits & Stefan Thonhauser, 2014. "Optimal Consumption Under Deterministic Income," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 255-279, January.
    10. Bahman Angoshtari & Erhan Bayraktar & Virginia R. Young, 2018. "Optimal Dividend Distribution Under Drawdown and Ratcheting Constraints on Dividend Rates," Papers 1806.07499, arXiv.org, revised Mar 2019.
    11. Loeffen, Ronnie L. & Renaud, Jean-François, 2010. "De Finetti's optimal dividends problem with an affine penalty function at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 98-108, February.
    12. Albrecher, Hansjörg & Bäuerle, Nicole & Bladt, Martin, 2018. "Dividends: From refracting to ratcheting," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 47-58.
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    Cited by:

    1. Chonghu Guan & Jiacheng Fan & Zuo Quan Xu, 2023. "Optimal dividend payout with path-dependent drawdown constraint," Papers 2312.01668, arXiv.org.
    2. Hansjoerg Albrecher & Pablo Azcue & Nora Muler, 2023. "Optimal dividend strategies for a catastrophe insurer," Papers 2311.05781, arXiv.org.

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    More about this item

    Keywords

    Optimal dividends; Viscosity solution; HJB equation; Optimal control;
    All these keywords.

    JEL classification:

    • G35 - Financial Economics - - Corporate Finance and Governance - - - Payout Policy
    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • G33 - Financial Economics - - Corporate Finance and Governance - - - Bankruptcy; Liquidation

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