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. This incompatibility goes away by noticing that the model used in both approaches , geometric Brownian motion, is a non-ergodic process, in the sense that ensemble-average returns differ from time-average returns in a single realization . The classic papers on portfolio theoryuse ensemble-average returns. The Kelly-result is obtained by considering  time-average returns. The averages differ by a logarithm. In portfolio theory this logarithm can be implemented as a logarithmic utility function. It is important to distinguish between effects of non-ergodicity and genuine utility constraints. For instance, ensemble-average returns depend linearly on leverage. This measure can thus incentivize investors to maximize leverage, which is detrimental to time-average returns and overall market stability  . 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.
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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