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Modern Tontine with Transaction Costs

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  • Lin He
  • Zongxia Liang
  • Sheng Wang

Abstract

In this paper, we propose a new type of reversible modern tontine with transaction costs. The wealth of the retiree is divided into a bequest account and a tontine account. And consumption can only be withdrawn from the bequest account. Each transaction between the two accounts incurs fixed and proportional transaction costs depending on the transaction volume. The retiree dynamically controls the allocation policy between the two accounts and the consumption policy to maximize the consumption and bequest utilities. We formulate the optimization problem as a combined stochastic and impulse control problem with infinite time horizon, and characterize the value function as the unique viscosity solution of a HJBQVI (Hamilton-Jacobi-Bellmen quasi-variational inequality). The numerical results exhibit the V-shaped transaction region which consists of two stages. In the former stage, since longevity credits increase gradually, the retiree decreases the proportion of wealth in the tontine account to smooth the wealth and reduce the volatility. However, in the latter stage, the retiree increases the proportion of wealth in the tontine account to gamble for the considerable longevity credits.

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  • Lin He & Zongxia Liang & Sheng Wang, 2022. "Modern Tontine with Transaction Costs," Papers 2209.09709, arXiv.org, revised Jun 2023.
    • Handle: RePEc:arx:papers:2209.09709
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    References listed on IDEAS

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    1. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    2. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    3. Bernhardt, Thomas & Donnelly, Catherine, 2019. "Modern tontine with bequest: Innovation in pooled annuity products," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 168-188.
    4. Yuri Kabanov & Claudia Klüppelberg, 2004. "A geometric approach to portfolio optimization in models with transaction costs," Finance and Stochastics, Springer, vol. 8(2), pages 207-227, May.
    5. Seydel, Roland C., 2009. "Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3719-3748, October.
    6. Marianne Akian & Agnès Sulem & Michael I. Taksar, 2001. "Dynamic Optimization of Long‐Term Growth Rate for a Portfolio with Transaction Costs and Logarithmic Utility," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 153-188, April.
    7. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    8. Dagpunar, John, 2021. "Closed-form solutions for an explicit modern ideal tontine with bequest motive," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 261-273.
    9. Christoph Belak & Sören Christensen, 2019. "Utility maximisation in a factor model with constant and proportional transaction costs," Finance and Stochastics, Springer, vol. 23(1), pages 29-96, January.
    10. Horneff, Wolfram J. & Maurer, Raimond H. & Mitchell, Olivia S. & Stamos, Michael Z., 2010. "Variable payout annuities and dynamic portfolio choice in retirement," Journal of Pension Economics and Finance, Cambridge University Press, vol. 9(2), pages 163-183, April.
    11. Stamos, Michael Z., 2008. "Optimal consumption and portfolio choice for pooled annuity funds," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 56-68, August.
    12. Framstad, Nils Chr. & Oksendal, Bernt & Sulem, Agnes, 2001. "Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 233-257, April.
    13. Ransom, Roger L. & Sutch, Richard, 1987. "Tontine Insurance and the Armstrong Investigation: A Case of Stifled Innovation, 1868–1905," The Journal of Economic History, Cambridge University Press, vol. 47(2), pages 379-390, June.
    14. Wolfram J. Horneff & Raimond H. Maurer & Michael Z. Stamos, 2008. "Optimal Gradual Annuitization: Quantifying the Costs of Switching to Annuities," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 75(4), pages 1019-1038, December.
    15. Hainaut, Donatien & Devolder, Pierre, 2006. "Life Annuitization: Why and how Much?," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 629-654, November.
    16. John Dagpunar, 2020. "Closed-form Solutions for an Explicit Modern Ideal Tontine with Bequest Motive," Papers 2005.00715, arXiv.org, revised Jun 2021.
    17. Thomas Bernhardt & Catherine Donnelly, 2019. "Modern tontine with bequest: innovation in pooled annuity products," Papers 1903.05990, arXiv.org.
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