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The numeraire property and long-term growth optimality for drawdown-constrained investments

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  • Constantinos Kardaras
  • Jan Obloj
  • Eckhard Platen

Abstract

We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the numeraire property exist and are unique, but may depend on the financial planning horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained numeraire portfolio is given explicitly through a model-independent transformation of the unconstrained numeraire portfolio. Further, it is established that the asymptotically growth-optimal strategy is obtained as limit of numeraire strategies on finite horizons.

Suggested Citation

  • Constantinos Kardaras & Jan Obloj & Eckhard Platen, 2012. "The numeraire property and long-term growth optimality for drawdown-constrained investments," Papers 1206.2305, arXiv.org, revised Nov 2012.
  • Handle: RePEc:arx:papers:1206.2305
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    References listed on IDEAS

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    1. Dirk Becherer, 2001. "The numeraire portfolio for unbounded semimartingales," Finance and Stochastics, Springer, vol. 5(3), pages 327-341.
    2. John Burr Williams, 1936. "Speculation and the Carryover," The Quarterly Journal of Economics, Oxford University Press, vol. 50(3), pages 436-455.
    3. Harry M. Markowitz, "undated". "Investment for the Long Run," Rodney L. White Center for Financial Research Working Papers 20-72, Wharton School Rodney L. White Center for Financial Research.
    4. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    5. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151.
    6. Henry Allen Latane, 1959. "Criteria for Choice Among Risky Ventures," Journal of Political Economy, University of Chicago Press, vol. 67, pages 144-144.
    7. Paul A. Samuelson, 2011. "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long," World Scientific Book Chapters,in: THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 34, pages 491-493 World Scientific Publishing Co. Pte. Ltd..
    8. Blanchet-Scalliet, Christophette & El Karoui, Nicole & Jeanblanc, Monique & Martellini, Lionel, 2008. "Optimal investment decisions when time-horizon is uncertain," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1100-1113, December.
    9. Stefan Ankirchner & Steffen Dereich & Peter Imkeller, 2005. "The Shannon information of filtrations and the additional logarithmic utility of insiders," Papers math/0503013, arXiv.org, revised May 2006.
    10. Constantinos Kardaras, 2011. "On the closure in the Emery topology of semimartingale wealth-process sets," Papers 1108.0945, arXiv.org, revised Jul 2013.
    11. Bruno Bouchard & Huyên Pham, 2004. "Wealth-path dependent utility maximization in incomplete markets," Finance and Stochastics, Springer, vol. 8(4), pages 579-603, November.
    12. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    13. repec:dau:papers:123456789/1803 is not listed on IDEAS
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    1. repec:eee:spapps:v:127:y:2017:i:8:p:2679-2698 is not listed on IDEAS
    2. Angoshtari, Bahman & Bayraktar, Erhan & Young, Virginia R., 2016. "Minimizing the probability of lifetime drawdown under constant consumption," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 210-223.
    3. Angoshtari, Bahman & Bayraktar, Erhan & Young, Virginia R., 2015. "Minimizing the expected lifetime spent in drawdown under proportional consumption," Finance Research Letters, Elsevier, vol. 15(C), pages 106-114.

    More about this item

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • F3 - International Economics - - International Finance
    • G3 - Financial Economics - - Corporate Finance and Governance

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