A Response to Professor Paul A. Samuelson’s Objections to Kelly Capital Growth Investing

The Kelly capital growth investment strategy maximizes the expected utility of final wealth with a Bernoulli logarithmic utility function. In 1956, Kelly showed that static expected log maximization yields the maximum asymptotic long-run growth. Good properties include minimizing the time to large asymptotic goals, maximizing the median, and being ahead (on average) after the first period. Bad properties include extremely large bets for short-term favorable investment situations because the Arrow-Pratt risk aversion index is essentially zero. Paul Samuelson was a critic of this approach, and his criticism is partially responsible for the fact that most finance academics and investment professionals do not recommend Kelly strategies. Samuelson’s points are theoretically correct and sharpen the theory, cautioning Kelly investors to understand the strategy’s true characteristics, including ways to lower investment exposure. His objections help us better understand the theory without detracting from its numerous valuable applications.