Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This example was modified from one in MacLean, Thorp, Zhao, and Ziemba (2011).
- 2.
The formula relating λ and f for this example is as follows. For the problem
$$\mathrm{Max}_{x}\left\{ E(\ln(1+r+x(R-r)\right\} ,$$where R is assumed to be Gaussian with mean μ R and standard deviation σ R , and r = the risk-free rate. The solution is given by Merton (1990) as
$$x=\frac{\mu_{R}-r}{\sigma_{R}}.$$Since \(\mu_{R}=0.102,\sigma_{R}=0.203,r=0.039\), the Kelly strategy is \(x=1.5288\).
References
Aase, K. K. (2001). On the St. Petersburg Paradox. Scandinavian Actuarial Journal 3 (1), 69–78.
Bell, R. M. and T. M. Cover (1980). Competitive optimality of logarithmic investment. Math of Operations Research 5, 161–166.
Bertocchi, M., S. L. Schwartz, and W. T. Ziemba (2010). Optimizing the Aging, Retirement, Pensions Dilemma. Wiley, Hoboken, NJ.
Bicksler, J. L. and E. O. Thorp (1973). The capital growth model: an empirical investigation. Journal of Financial and Quantitative Analysis 8 (2), 273–287.
Breiman, L. (1960). Investment policies for expanding businesses optimal in a long run sense. Naval Research Logistics Quarterly 4 (4), 647–651.
Breiman, L. (1961). Optimal gambling system for favorable games. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability 1, 63–68.
Chopra, V. K. and W. T. Ziemba (1993). The effect of errors in mean, variance and co-variance estimates on optimal portfolio choice. Journal of Portfolio Management 19, 6–11.
Cover, T. M. and J. Thomas (2006). Elements of Information Theory (2nd ed.). Wiley, New York, NY.
Fuller, J. (2006). Optimize your portfolio with the Kelly formula. morningstar.com, October 6.
Hakansson, N. H. and W. T. Ziemba (1995). Capital growth theory. In R. A. Jarrow, V. Maksimovic, and W. T. Ziemba (Eds.), Finance, Handbooks in OR & MS, pp. 65–86. North Holland, Amsterdam.
Harville, D. A. (1973). Assigning probabilities to the outcome of multi-entry competitions. Journal of the American Statistical Association 68, 312–316.
Hausch, D. B., V. Lo, and W. T. Ziemba (Eds.) (1994). Efficiency of Racetrack Betting Markets. Academic, San Diego.
Hausch, D. B., V. Lo, and W. T. Ziemba (Eds.) (2008). Efficiency of Racetrack Betting Markets (2 ed.). World Scientific, Singapore.
Hausch, D. B. and W. T. Ziemba (1985). Transactions costs, extent of inefficiencies, entries and multiple wagers in a racetrack betting model. Management Science 31, 381–394.
Hausch, D. B., W. T. Ziemba, and M. E. Rubinstein (1981). Efficiency of the market for racetrack betting. Management Science XXVII, 1435–1452.
Kelly, Jr., J. R. (1956). A new interpretation of the information rate. Bell System Technical Journal 35, 917–926.
Latané, H. (1978). The geometric-mean principle revisited – a reply. Journal of Banking and Finance 2 (4), 395–398.
Lee, E. (2006). How to calculate the Kelly formula. fool.com, October 31.
Luenberger, D. G. (1993). A preference foundation for log mean-variance criteria in portfolio choice problems. Journal of Economic Dynamics and Control 17, 887–906.
MacLean, L. C., R. Sanegre, Y. Zhao, and W. T. Ziemba (2004). Capital growth with security. Journal of Economic Dynamics and Control 28 (4), 937–954.
MacLean, L. C., E. O. Thorp, Y. Zhao, and W. T. Ziemba (2011). How does the Fortunes Formula-Kelly capital growth model perform? Journal of Portfolio Management 37 (4).
MacLean, L. C., E. O. Thorp, and W. T. Ziemba (2010a). The good and properties of the Kelly and fractional Kelly capital growth criterion. Quantitative Finance (August–September), 681–687.
MacLean, L. C., E. O. Thorp, and W. T. Ziemba (2010b). The Kelly Capital Growth Investment Criteria. World Scientific, Singapore.
MacLean, L. C., Y. Zhao, and W. T. Ziemba (2009). Optimal capital growth with convex loss penalties. Working paper, Oxford University.
MacLean, L. C., W. T. Ziemba, and G. Blazenko (1992). Growth versus security in dynamic investment analysis. Management Science 38, 1562–1585.
MacLean, L. C., W. T. Ziemba, and Li (2005). Time to wealth goals in capital accumulation and the optimal trade-off of growth versus security. Quantitative Finance 5 (4), 343–357.
Menger, K. (1967). The role of uncertainty in economics. In M. Shubik (Ed.), Essays in Mathematical Economics in Honor of Oskar Morgenstern. Princeton University Press, Princeton, NJ.
Merton, R. C. (1990). Continuous Time Finance. Basil Blackwell, Oxford, UK.
Merton, R. C. and P. A. Samuelson (1974). Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods. Journal of Financial Economics 1, 67–94.
Pabrai, M. (2007). The Dhandho Investor. Wiley, Hoboken, NJ.
Poundstone, W. (2005). Fortune’s Formula: The Untold Story of the Scientific System That Beat the Casinos and Wall Street. Hill and Wang, New York, NY.
Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics 51, 239–246.
Samuelson, P. A. (1971). The fallacy of maximizing the geometric mean in long sequences of investing or gambling. Proceedings National Academy of Science 68, 2493–2496.
Samuelson, P. A. (1977). St. Petersburg paradoxes: Defanged, dissected and historically described. Journal of Economic Literature 15 (1), 24–55.
Samuelson, P. A. (1979). Why we should not make mean log of wealth big though years to act are long. Journal of Banking and Finance 3, 305–307.
Samuelson, P. A. (2006–2009). Letters to William T. Ziemba, correspondence December 13, 2006, May 7, 2008, and May 12, 2008.
Siegel, J. (2002). Stocks for the Long Run. Wiley, McGraw-Hill, New York.
Thorp, E. O. (2008). Understanding the Kelly criterion. Wilmott, May and September.
Ziemba, W. T. (2005). The symmetric downside risk Sharpe ratio and the evaluation of great investors and speculators. Journal of Portfolio Management Fall, 32 (1), 108–120.
Ziemba, W. T. (2010). A tale of five investors: response to Paul A. Samuelson letters. Working Paper, University of Oxford.
Ziemba, W. T. and D. B. Hausch (1986). Betting at the Racetrack. Dr Z Investments, San Luis Obispo, CA.
Ziemba, W. T. and D. B. Hausch (1987). Dr Z’s Beat the Racetrack. William Morrow, New York, NY.
Ziemba, R. E. S. and W. T. Ziemba (2007). Scenarios for Risk Management and Global Investment Strategies. Wiley, Chichester.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Ziemba, W.T., MacLean, L.C. (2011). Using the Kelly Criterion for Investing. In: Bertocchi, M., Consigli, G., Dempster, M. (eds) Stochastic Optimization Methods in Finance and Energy. International Series in Operations Research & Management Science, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9586-5_1
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9586-5_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9585-8
Online ISBN: 978-1-4419-9586-5
eBook Packages: Business and EconomicsBusiness and Management (R0)