Chicago_Kelly - The Kelly criterion and its variants theory...

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The Kelly criterion and its variants: theory and practice in sports, lottery, futures & options trading The symmetric downside Sharpe ratio and the evaluation of great investors & speculators and their use of the Kelly criterion William T Ziemba Alumni Professor at Financial Modeling and Stochastic Optimization, Emeritus, Sauder School of Business, UBC, Vancouver, BC, Canada V6T 1Z2 email: [email protected] Mathematical Finance Seminar University of Chicago April 6, 2007
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2 MacLean, L C and W T Ziemba (2006) The Kelly criterion: theory and practice Thorp, E. O. (2006). The Kelly criterion in blackjack, sports betting and the stock market. in S A Zenios and W T Ziemba, eds., Handbook of Asset and Liability Management, Volume A: Theory and Methodology , North Holland. MacLean, Sanegre, Zhao, Ziemba (2004) Capital growth with security, J.Economic Dynamics and Control, How to calculate the optimal Kelly fraction subject to being above a wealth path with high probability Ziemba, W T (2005) The Symmetric downside Sharpe ratio and the evaluation of great investors and speculators, Journal of Portfolio Management (Fall). Chapter 6 of W T Ziemba (2003) The Stochastic Programming Approach to Asset Liability and Wealth Management, AIMR updated into various chapters in Ziemba and Ziemba (2007), Scenarios for Risk Management and Global Investment Strategies, Wiley, July which is Wilmott columns merged into a book Samuelson, P A and W T Ziemba (2007) Understanding the finite properties of Kelly log betting: a tale of five investors, Tech Report UBC References
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3 The Kelly or capital growth criteria maximizes the expected logarithm as its utility function period by period. It has many desirable properties such as being myopic in that today’s optimal decision does not depend upon yesterday’s or tomorrow’s data, it asymptotically maximizes long run wealth almost surely and it attains arbitrarily large wealth goals faster than any other strategy. Also in an economy with one log bettor and all other essentially different strategy wagers, 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. 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 facing very favorable investment choices. One way to modify the growth-security profile is to use either ad hoc or scientifically computed fractional Kelly strategies that blend the log optimal portfolio with cash. to keep one above the highest possible wealth path with high probability or to risk adjust the wealth with convex penalties for being below the path Abstract
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4 For log normally distributed assets this simply means using a negative power utility function whose risk aversion coefficient is 1:1 determined by the fraction and vice versa.
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