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How does the Fortune ’ s Formula- Kelly Capital Growth Model perform?.

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How does the Fortune’s Formula-Kelly Capital Growth Model perform?

Professor William T ZiembaAlumni Professor of Financial Modeling and Stochastic Optimization (Emeritus), UBCICMA Financial Markets Centre, University of ReadingVisiting Professor Mathematical Institute, Oxford University andPresident, William T Ziemba Investment Management Inc

Kaist Lecture Program

August 2011


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Summary of Key Points

  • 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 as shown in this paper 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 a long sequence of very favorable investment choices.


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Key points (cont’d)

  • 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

  • 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.

  • f=1/(1- where u(w)=w, <0.

  • For other asset returns this is an approximate solution, but it  can be  quite inaccurate.

  • Thus one moves the risk aversion away from zero to a higher level.

  • This results in a smoother wealth path but usually has less growth as the example below shows.


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Key points (cont’d)

  • This talk reviews the good and bad properties of the Kelly and fractional Kelly strategies and discusses their use in practice by great investors and speculators most of whom have become centi-millionaires or billionaires by isolating profitable anomalies and betting on them well with these strategies.

  • The latter include Bill Bentor the Hong Kong racing guru, Ed Thorp , the inventor of blackjack card counting who compiled one of the finest hedge fund records and James Simon of the Renaissance Hedge Fund, arguably the best hedge fund in the world.

  • They all had very smooth, low variance wealth paths.

  • Additionally legendary investors such as John Maynard Keynes (0.8 Kelly) running the King’s College Cambridge endowment, George Soros (? Kelly) running the Quantum funds and Warren Buffett running Berkshire Hathaway also had very good results but had much more variable wealth paths.

  • The difference seems to be in the choice of Kelly fraction and other risk control measures that relate to true diversification and position size relative to liquid assets under management.


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  • Success in investments has two key pillars:

  • devising a strategy with positive expectation and

  • betting the right amount to balance growth of one’s fortune against the risk of losses.

  • This talk discusses the Kelly or capital growth log utility criteria for investing.

  • A strategy which has wonderful asymptotic long run properties

  • the log bettor will dominate other strategies with probability one and

  • accumulate unbounded amount more wealth.

    But in the short run the strategy can be very risky since it has very low Arrow-Pratt risk aversion.


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Fractional Kelly strategies provide more security but with less growth.

  • Examples from blackjack, horseracing, lotteries and futures, options and high frequency trading illustrate the theory and its use in practice.

  • I have been fortunate to work/consult with seven individuals who turned a humble beginning with essentially zero wealth into hundreds of millions (at least five are billionaires) using security market imperfections and anomalies in racing, futures trading and options mispricings.

  • All of them used Kelly or fractional Kelly betting strategies.


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Some points to learn from this research

  • Means are by far the most important aspect of any portfolio problem.

  • You must have the mean right to have good performance.

  • If you have the mean right and do not overbet you should do well.

  • In levered bets, it’s the left tail that can lead to trouble so you must not overbet or you can have a large disaster occurring without warning.

  • Behavioral and other anomalies can yield strategies that have positive means.

  • These biases yield ideas that yield profitable positive mean strategies in racing, sports betting and options markets.

  • The capital growth or Kelly criterion strategy yields the most wealth in the long run and dominates all other essentially different strategies.

  • But in the short run, the expected log criterion with its essentially zero Arrow-Pratt risk aversion index is very risky and can have substantial losses.

  • The most you should ever bet is the log optimal amount; betting more is suboptimal and betting double yields a zero growth rate.

  • Negative power utility, which blends cash with the expected log maximizing portfolio provides more security but has less long run growth.

  • These fractional Kelly strategies are attractive for many investment situations; determination of what fraction to use depends on constrained optimization models.


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Growth versus Security: Tradeoffs in Dynamic Investment Analysis

  • One is faced with a sequence of investments in periods 1, …, nsome favorable, some unfavorable

  • Given an initial fortune, how should one invest over time to have long-run growth of their fortune while at the same time maintaining its security?

  • Develop computational schemes so that the investor can have a desired growth and security tradeoff.

  • Find simple operational policies that achieve these tradeoffs

  • Transactions and market impact costs are crucial in practice so stochastic programming optimization is needed

  • Use results to analyze favorable investment situations


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POLAR APPROACHES Analysis

, Ziemba (2003)

Markowitz (1976), Hausch, Ziemba and Rubinstein (1981), Hausch and Ziemba (1985), Luenberger (1993) and othersSee MacLean, Thorp and Ziemba (2010) for more references


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Laplace (17xy) and others including Bhulmann Analysis

Our goal is to marry growth and security and achieve tradeoffs.


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If you bet on a horse, that Analysis’s gambling. If you bet you can make three spades, that’s entertainment. If you bet cotton will go up three points, that’s business. See the difference? Blackie Sherrod


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Games: favorable or unfavorable Analysis

  • Blend growth versus security to your risk tolerance and the situation at hand


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Medium Term Simulations of The Full Kelly and Fractional Kelly Investment Strategies

  • Using three simple investment situations, we simulate the behavior of the Kelly and fractional Kelly proportional betting strategies over medium term horizons using a large number of scenarios. We extend the work of Bicksler and Thorp (1973) and Ziemba and Hausch (1986) to more scenarios and decision periods.

  • The results show:

    • (1) the great superiority of full Kelly and close to full Kelly strategies over longer horizons with very large gains a large fraction of the time;

    • (2) that the short term performance of Kelly and high fractional Kelly strategies is very risky;

    • (3) that there is a consistent tradeoff of growth versus security as a function of the bet size determined by the various strategies; and

    • (4) that no matter how favorable the investment opportunities are or how long the finite horizon is, a sequence of bad results can lead to poor final wealth outcomes, with a loss of most of the investor's initial capital.


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Kelly and half Kelly medium time simulations: Kelly Investment StrategiesZiemba-Hausch (1986)

These were independent


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Final Wealth Statistics by Kelly Fraction: Kelly Investment StrategiesZiemba-Hausch (1986) Model


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Highest Final Wealth Trajectory: Kelly Investment StrategiesZiemba-Hausch (1986) Model


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Lowest Final Wealth Trajectory: Kelly Investment StrategiesZiemba-Hausch (1986) Model




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Mean- Model (1986)Std Tradeoff: Ziemba-Hausch Model (1986)


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Case I - Uniform Returns ( Model (1986)Bicksler-Thorp, 1973)

  • There is one risky asset R with mean return of +12.5%, that is uniformly distributed between 0.75 and 1.50 for each dollar invested.

    • Assume we can lend or borrow capital at a risk free rate r=0.0.

    • Let be the proportion of capital invested in the risky asset, where  ranges from 0.4 to 2.4.

  • So =2.4$ means $1.4 is borrowed for each $1 of current wealth and $2.40 is invested in the risky asset.

    • The Kelly optimal growth investment in the risky asset is x=2.8655.

    • The Kelly fractions for the different values of  are in the next slide.

    • The formula relating and f for this experiment appears in a later slide.

  • Bicksler and Thorp used 10 and 20 yearly decision periods, and 50 simulated

  • scenarios.

  • We use 40 yearly decision periods, with 3000 scenarios.





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Mean- Model (1986)Std Trade-off


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Case II - Equity Market Returns ( Model (1986)Bicksler-Thorp, 1973)

  • In the third experiment there are two assets: US equities and US T-bills.

  • According to Siegel (2002), during 1926-2001 US equities returned 10.2% with a yearly standard deviation of 20.3%, and the mean return was 3.9% for short term government T-bills with zero standard deviation.

  • We assume the choice is between these two assets in each period.

    • The Kelly strategy is to invest a proportion of wealth x=1.5288 in equities and sell short the T-bill at 1-x=-0.5228 of current wealth.

    • With the short selling and levered strategies, there is a chance of substantial losses.

  • For the simulations, the proportion:  of wealth invested in equities and the corresponding Kelly fraction f and the formula relating  and f are shown below.

  • Bicksler and Thorp used 10 and 20 yearly decision periods, and 50 simulated scenarios.

  • We use 40 yearly decision periods, with 3000 scenarios.





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Mean-Std Tradeoff Model (1986)


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Conclusions from the results in experiment 3 Model (1986)

  • The statistics describing the end of the horizon (T=40)$wealth are again monotone in the fraction of wealth invested in the Kelly portfolio. Specifically

    (i) the maximum terminal wealth and the mean terminal wealth increase in the Kelly fraction; and

    (ii) the minimum wealth decreases as the fraction increases and the standard deviation grows as the fraction increases.

    The growth and decay are pronounced and it is possible to have extremely large losses. The fraction of the Kelly optimal growth strategy exceeds $1$ in the most levered strategies and this is very risky. There is a trade-off between return and risk, but the mean for the levered strategies is growing far less than the standard deviation. The disadvantage of leveraged investment is illustrated with the cumulative distributions. The log-normality of final wealth does not hold for the levered strategies.

    2. The maximum and minimum final wealth trajectories show the return - risk of levered strategies. The worst and best scenarios are the not same for all Kelly fractions. The worst scenario for the most levered strategy shows the rapid decline in wealth. The mean-standard deviation trade-off confirms the extreme riskyness of the aggressive strategies.


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The main points from the Model (1986)Bicksler and Thorp (1973) and Ziemba and Hausch (1986) studies are confirmed:

  • The wealth accumulated from the full Kelly strategy does not stochastically dominate fractional Kelly wealth. The downside is often much more favorable with a fraction less than one.

  • There is a tradeoff of risk and return with the fraction invested in the Kelly portfolio. In cases of large uncertainty, either from intrinsic volatility or estimation error, security is gained by reducing the Kelly investment fraction.

  • The full Kelly strategy can be highly levered. While the use of borrowing can be effective in generating large returns on investment, increased leveraging beyond the full Kelly is not warranted as it is growth-security dominated. The returns from over-levered investment are offset by a growing probability of bankruptcy.

  • The Kelly strategy is not merely a long term approach. Proper use in the short and medium run can achieve wealth goals while protecting against drawdowns.

  • MacLean, Sanegre, Zhao and Ziemba (2004) and MacLean, Zhao and Ziemba (2009) discuss a strategy to reduce the Kelly fraction to stay above a prespecified wealth path with high probability.




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