How to Gamble or Invest If You Must: A Large Deviations Approach听
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The Kelly Criterion (a.k.a. expected log utility maximization) is a well-known criterion function used to select optimal repeated gambles or long-term portfolio investments.听 It is often rationalized by its asymptotic properties as the investment horizon grows to infinity.听听 One criticism is that use of this criterion leads to excessively volatile wealth paths that can lead to uncomfortably high, finite-time probabilities of underperformance.听
The Statistical Theory of Large Deviations and its key object 鈥� the Rate Function -- provide a tractable framework for introducing considerations of risk-control into the asymptotic rationale.听 Instead of maximizing the expected log utility of wealth, I maximize the asymptotic decay rate of the probability that wealth will fall short of user-selected targets, contrast this criterion with the Kelly Criterion, and empirically implement the idea to select long-term optimal portfolios of stocks and bonds.听
Doing so in a realistic manner requires calculation of the large deviations rate function for time averages generated by Markov Switching (a.k.a. Hidden Markov) processes, which may also be of interest to those in other fields.听