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The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. I: Foundations

Author

Listed:
  • Martin Herdegen

    (University of Warwick)

  • David Hobson

    (University of Warwick)

  • Joseph Jerome

    (University of Liverpool)

Abstract

The goal of this article is to provide a detailed introduction to infinite-horizon investment–consumption problems for agents with preferences described by Epstein–Zin (EZ) stochastic differential utility (SDU). In the setting of a Black–Scholes–Merton market, we seek to describe all parameter combinations that lead to a well-founded problem in the sense that the problem is not just mathematically well posed, but the solution is also economically meaningful. The key idea is to consider a novel and slightly different description of EZ SDU under which the aggregator has only one sign. This new formulation clearly highlights the necessity for the coefficients of relative risk aversion and of elasticity of intertemporal complementarity (the reciprocal of the coefficient of intertemporal substitution) to lie on the same side of unity.

Suggested Citation

  • Martin Herdegen & David Hobson & Joseph Jerome, 2023. "The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. I: Foundations," Finance and Stochastics, Springer, vol. 27(1), pages 127-158, January.
  • Handle: RePEc:spr:finsto:v:27:y:2023:i:1:d:10.1007_s00780-022-00495-6
    DOI: 10.1007/s00780-022-00495-6
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    1. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
    2. Jose A. Scheinkman & Wei Xiong, 2003. "Overconfidence and Speculative Bubbles," Journal of Political Economy, University of Chicago Press, vol. 111(6), pages 1183-1219, December.
    3. Weil, Philippe, 1989. "The equity premium puzzle and the risk-free rate puzzle," Journal of Monetary Economics, Elsevier, vol. 24(3), pages 401-421, November.
    4. Robert J. Shiller, 2014. "Speculative Asset Prices (Nobel Prize Lecture)," Cowles Foundation Discussion Papers 1936, Cowles Foundation for Research in Economics, Yale University.
    5. George Chacko & Luis M. Viceira, 2005. "Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets," The Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1369-1402.
    6. Anna Scherbina & Bernd Schlusche, 2014. "Asset price bubbles: a survey," Quantitative Finance, Taylor & Francis Journals, vol. 14(4), pages 589-604, April.
    7. Martin Herdegen & David Hobson & Joseph Jerome, 2021. "An elementary approach to the Merton problem," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1218-1239, October.
    8. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    9. Dumas, Bernard & Uppal, Raman & Wang, Tan, 2000. "Efficient Intertemporal Allocations with Recursive Utility," Journal of Economic Theory, Elsevier, vol. 93(2), pages 240-259, August.
    10. repec:bla:jfinan:v:59:y:2004:i:4:p:1481-1509 is not listed on IDEAS
    11. Martin Herdegen & David Hobson & Joseph Jerome, 2020. "An elementary approach to the Merton problem," Papers 2006.05260, arXiv.org, revised Mar 2021.
    12. Schroder, Mark & Skiadas, Costis, 1999. "Optimal Consumption and Portfolio Selection with Stochastic Differential Utility," Journal of Economic Theory, Elsevier, vol. 89(1), pages 68-126, November.
    13. Robert J. Shiller, 2014. "Speculative Asset Prices," American Economic Review, American Economic Association, vol. 104(6), pages 1486-1517, June.
    14. Herdegen, Martin & Muhle-Karbe, Johannes, 2019. "Sensitivity of optimal consumption streams," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1964-1992.
    15. Matoussi, Anis & Xing, Hao, 2018. "Convex duality for Epstein-Zin stochastic differential utility," LSE Research Online Documents on Economics 82519, London School of Economics and Political Science, LSE Library.
    16. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    17. Martin Herdegen & David Hobson & Joseph Jerome, 2021. "Proper solutions for Epstein-Zin Stochastic Differential Utility," Papers 2112.06708, arXiv.org.
    18. Diba, Behzad T & Grossman, Herschel I, 1988. "The Theory of Rational Bubbles in Stock Prices," Economic Journal, Royal Economic Society, vol. 98(392), pages 746-754, September.
    19. Loewenstein, Mark & Willard, Gregory A., 2000. "Rational Equilibrium Asset-Pricing Bubbles in Continuous Trading Models," Journal of Economic Theory, Elsevier, vol. 91(1), pages 17-58, March.
    20. Anis Matoussi & Hao Xing, 2018. "Convex duality for Epstein–Zin stochastic differential utility," Mathematical Finance, Wiley Blackwell, vol. 28(4), pages 991-1019, October.
    21. Kraft, Holger & Seifried, Frank Thomas, 2014. "Stochastic differential utility as the continuous-time limit of recursive utility," Journal of Economic Theory, Elsevier, vol. 151(C), pages 528-550.
    22. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    23. Hao Xing, 2017. "Consumption–investment optimization with Epstein–Zin utility in incomplete markets," Finance and Stochastics, Springer, vol. 21(1), pages 227-262, January.
    24. repec:hal:spmain:info:hdl:2441/8686 is not listed on IDEAS
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    Cited by:

    1. Kexin Chen & Kyunghyun Park & Hoi Ying Wong, 2024. "Robust dividend policy: Equivalence of Epstein-Zin and Maenhout preferences," Papers 2406.12305, arXiv.org.
    2. Martin Herdegen & David Hobson & Joseph Jerome, 2023. "The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. II: Existence, uniqueness and verification for ϑ ∈ ( 0 , 1 ) $\vartheta \in (0,1)$," Finance and Stochastics, Springer, vol. 27(1), pages 159-188, January.
    3. Zixin Feng & Dejian Tian & Harry Zheng, 2024. "Consumption-investment optimization with Epstein-Zin utility in unbounded non-Markovian markets," Papers 2407.19995, arXiv.org.

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    More about this item

    Keywords

    Epstein–Zin stochastic differential utility; Lifetime investment and consumption; Backward stochastic differential equations; Discounted aggregator;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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