- William T. Ziemba
(University of British Columbia
Mathematical Institute, Oxford University
ICMA Centre, University of Reading
University of Bergamo)
- Leonard C. MacLean
(Herbert Lamb Chair School of Business, Dalhousie University)
This chapter describes the use of the Kelly capital growth model. This model, dubbed Fortune’s Formula by Thorp and used in the title by Poundstone (Fortune’s Formula: The Untold Story of the Scientific System That Beat the Casinos and Wall Street, 2005), has many attractive features such as the maximization of asymptotic long-run wealth; see Kelly (Bell System Technical Journal 35:917–926, 1956), Breiman (Proceedings of the 4th Berkely Symposium on Mathematical Statistics and Probability 1:63–68, 1961), Algoet and Cover (Annals of Probability 16(2):876–898, 1988) and Thorp (Handbook of Asset and Liability Management, 2006). Moreover, it minimizes the expected time to reach asymptotically large goals (Breiman, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability 1:63–68, 1961) and the strategy is myopic (Hakansson, Journal of Business 44:324–334, 1971). While the strategy to maximize the expected logarithm of expected final wealth computed via a nonlinear program has a number of good short- and medium-term qualities (see MacLean, Thorp, and Ziemba, The Kelly Capital Growth Investment Critria, 2010b), it is actually very risky short term since its Arrow–Pratt risk aversion index is the reciprocal of wealth and that is essentially zero for non-bankrupt investors. The chapter traces the development and use of this strategy from the log utility formulation in 1738 by Bernoulli (Econometrica 22:23–36, 1954) to current use in financial markets, sports betting, and other applications. Fractional Kelly wagers that blend the E log maximizing strategy with cash tempers the risk and yield smoother wealth paths but with generally less final wealth. Great sensitivity to parameter estimates, especially the means, makes the strategy dangerous to those whose estimates are in error and leads them to poor betting and possible bankruptcy. Still, many investors with repeated investment periods and considerable wealth, such as Warren Buffett and George Soros, use strategies that approximate full Kelly which tends to place most of one’s wealth in a few assets and lead to many monthly losses but large final wealth most of the time. A simulation study is presented that shows the possibility of huge gains most of the time, possible losses no matter how good the investments appear to be, and possible extreme losses from overbetting when bad scenarios occur. The study and discussion shows that Samuelson’s objections to E log strategies are well understood. In practice, careful risk control or financial engineering is important to deal with short-term volatility and the design of good wealth paths with limited drawdowns. Properly implemented, the strategy used by many billionaires has much to commend it, especially with many repeated investments.
William T. Ziemba & Leonard C. MacLean, 2011.
"Using the Kelly Criterion for Investing,"
International Series in Operations Research & Management Science, in: Marida Bertocchi & Giorgio Consigli & Michael A. H. Dempster (ed.), Stochastic Optimization Methods in Finance and Energy, edition 1, chapter 0, pages 3-20,
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