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Optimal leverage from non-ergodicity

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  • Ole Peters

Abstract

In modern portfolio theory, the balancing of expected returns on investments against uncertainties in those returns is aided by the use of utility functions. The Kelly criterion offers another approach, rooted in information theory, that always implies logarithmic utility. The two approaches seem incompatible, too loosely or too tightly constraining investors' risk preferences, from their respective perspectives. The conflict can be understood on the basis that the multiplicative models used in both approaches are non-ergodic which leads to ensemble-average returns differing from time-average returns in single realizations. The classic treatments, from the very beginning of probability theory, use ensemble-averages, whereas the Kelly-result is obtained by considering time-averages. Maximizing the time-average growth rates for an investment defines an optimal leverage, whereas growth rates derived from ensemble-average returns depend linearly on leverage. The latter measure can thus incentivize investors to maximize leverage, which is detrimental to time-average growth and overall market stability. The Sharpe ratio is insensitive to leverage. Its relation to optimal leverage is discussed. A better understanding of the significance of time-irreversibility and non-ergodicity and the resulting bounds on leverage may help policy makers in reshaping financial risk controls.

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  • Ole Peters, 2009. "Optimal leverage from non-ergodicity," Papers 0902.2965, arXiv.org, revised Aug 2010.
  • Handle: RePEc:arx:papers:0902.2965
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    References listed on IDEAS

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    1. Merton, Robert C. & Samuelson, Paul A., 1974. "Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods," Journal of Financial Economics, Elsevier, vol. 1(1), pages 67-94, May.
    2. Allan G. Timmermann, 1993. "How Learning in Financial Markets Generates Excess Volatility and Predictability in Stock Prices," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 108(4), pages 1135-1145.
    3. Harry M. Markowitz, 2011. "Investment for the Long Run: New Evidence for an Old Rule," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 35, pages 495-508, World Scientific Publishing Co. Pte. Ltd..
    4. J. Tobin, 1958. "Liquidity Preference as Behavior Towards Risk," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 25(2), pages 65-86.
    5. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
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    Cited by:

    1. Stefan Thurner & J. Doyne Farmer & John Geanakoplos, 2012. "Leverage causes fat tails and clustered volatility," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 695-707, February.
    2. J. Doyne Farmer & Spyros Skouras, 2013. "An ecological perspective on the future of computer trading," Quantitative Finance, Taylor & Francis Journals, vol. 13(3), pages 325-346, February.
    3. Matej Uhr'in & Gustav v{S}ourek & Ondv{r}ej Hub'av{c}ek & Filip v{Z}elezn'y, 2021. "Optimal sports betting strategies in practice: an experimental review," Papers 2107.08827, arXiv.org.
    4. Simon Gluzman, 2023. "Market Crashes and Time-Translation Invariance," FinTech, MDPI, vol. 2(2), pages 1-27, March.
    5. Mihail Turlakov, 2016. "Leverage and Uncertainty," Papers 1612.07194, arXiv.org.
    6. Ole Peters & Murray Gell-Mann, 2014. "Evaluating gambles using dynamics," Papers 1405.0585, arXiv.org, revised Jun 2015.
    7. El Mouden, Claire, 2013. "The Sciences Of Risk: Implications For Regulation Of The Financial Sector," INET Oxford Working Papers 2013-01, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford.
    8. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    9. Bell, Peter Newton, 2014. "Properties of time averages in a risk management simulation," MPRA Paper 55803, University Library of Munich, Germany.
    10. Smirnov Alexander D., 2018. "Stochastic Logistic Model of the Global Financial Leverage," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 18(1), pages 1-20, January.
    11. Mariam Thalos & Oliver Richardson, 2014. "Capitalization in the St. Petersburg game," Politics, Philosophy & Economics, , vol. 13(3), pages 292-313, August.
    12. Rosella Giacometti & Sergio Ortobelli & Tomáš Tichý, 2015. "Portfolio Selection with Uncertainty Measures Consistent with Additive Shifts," Prague Economic Papers, Prague University of Economics and Business, vol. 2015(1), pages 3-16.

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