In 1738, Daniel Bernoulli postulated that marginal utility should be proportional to the reciprocal of wealth and hence log utility was born. The Kelly or capital growth criterion maximizes the expected logarithm as its utility function period by period. That max E log maximizes long run asymptotic growth dates to Kelly (1956). Breiman (1960, 1961), Algoet and Cover (1988) and Thorp (1969, 2007) cleaned up the math. Thorp (1960) used the strategy in his blackjack card counting and named it Fortune¿s Formula. Hausch, Ziemba and Rubinstein (1981) used it in a betting system for the inefficient place & show horse race betting markets.
Max E log has many desirable properties such as being myopic in that today¿s optimal decision does not depend upon yesterday¿s or tomorrow¿s data assuming no frictions. It asymptotically maximizes long run wealth almost surely and it attains arbitrarily large wealth goals faster than any other strategy. In an economy with one log bettor and all others essentially different strategy wagerers, the log bettor will eventually get all the economy¿s wealth.
The drawback of log with its essentially zero Arrow-Pratt absolute risk aversion is that in the short run it is the most risky utility function one would ever consider. Since there is essentially no risk aversion, the wagers it suggests are very large and typically undiversified. Examples include the stock portfolios of noted investors George Soros and Warren Buffett who are interested in long term growth and do not care about monthly losses and highly variable wealth paths.
Simulations show that log bettors have much more final wealth most of the time than those using other strategies but can essentially go bankrupt a small percentage of the time, even when facing a long sequence of very favorable investment choices.
One can modify the growth-security profile with ad hoc or scientifically computed fractional Kelly strategies that blend the log optimal portfolio with cash to keep one above an exogenous prespecified wealth path with high probability and to risk adjust the wealth with convex penalties for being below the path. For log normally distributed assets this means using a negative power utility function whose risk aversion coefficient is determined by the fraction and vice versa. For other asset returns this is an approximate solution that can be inaccurate. Thus moving the risk aversion away from zero to a higher level results in a smoother wealth path with less growth.
This lecture reviews the good and bad properties of the Kelly and fractional Kelly strategies and a discussion of their use in practice by great investors some of whom have become billionaires by isolating profitable anomalies and betting on them with these strategies. The latter include several I have worked with such as Bill Bentor the Hong Kong racing guru, Ed Thorp, who compiled one of the finest hedge fund records and James Simon of the Renaissance Hedge Fund, who runs arguably the best hedge fund in the world. They had smooth, low variance wealth paths.
A critic of the approach was Professor Paul Samuelson. Largely because of that, the academic finance, and some of the business, world has remained mean-variance oriented. The response by Thorp and Ziemba is discussed and all the Samuelson points are in agreement with our thinking.