Optimal leverage from non-ergodicity

TL;DR

This paper explores how non-ergodicity in multiplicative models affects optimal leverage, contrasting ensemble and time-average returns, and discusses implications for investment strategies and financial regulation.

Contribution

It clarifies the impact of non-ergodicity on leverage optimization and connects theoretical insights to practical risk management and policy implications.

Findings

Optimal leverage is derived from time-average growth rates.

Ensemble-average returns can mislead leverage incentives.

Non-ergodicity explains the divergence between ensemble and time averages.

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.