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Optimal leverage from non-ergodicity


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  • Ole Peters


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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Paper provided by in its series Papers with number 0902.2965.

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Date of creation: Feb 2009
Date of revision: Aug 2010
Publication status: Published in Quant. Fin., Vol. 11, Issue 11, 1593--1602, 2011 (open access)
Handle: RePEc:arx:papers:0902.2965

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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. Markowitz, Harry M, 1976. "Investment for the Long Run: New Evidence for an Old Rule," Journal of Finance, American Finance Association, vol. 31(5), pages 1273-86, December.
  3. Timmermann, Allan G, 1993. "How Learning in Financial Markets Generates Excess Volatility and Predictability in Stock Prices," The Quarterly Journal of Economics, MIT Press, vol. 108(4), pages 1135-45, November.
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Cited by:
  1. Stefan Thurner & J. Doyne Farmer & John Geanakoplos, 2009. "Leverage Causes Fat Tails and Clustered Volatility," Papers 0908.1555,, revised Jan 2010.
  2. Bell, Peter Newton, 2014. "Properties of time averages in a risk management simulation," MPRA Paper 55803, University Library of Munich, Germany.
  3. Ole Peters & Murray Gell-Mann, 2014. "Evaluating gambles using dynamics," Papers 1405.0585,


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