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How Does the Fortune's Formula-Kelly Capital Growth Model Perform?

Profile image of Yonggan ZhaoYonggan Zhao

2016, WORLD SCIENTIFIC eBooks

https://doi.org/10.1142/9789813144385_0008
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28 pages

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Abstract

William Poundstone's (2005) book, Fortune's Formula, brought the Kelly capital growth criterion to the attention of investors. But how do full Kelly and fractional Kelly strategies that blend with cash actually preform in practice? To investigate this we revisit three simple investment situations and simulate the behavior of these strategies over medium term horizons using a large number of scenarios. These examples are from Bicksler and Thorp (1973) and Ziemba and Hausch (1986) and we consider many more scenarios and strategies. 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 scenarios can lead to very poor final wealth outcomes, with a loss of most of the investor's initial capital. Hence, in practice, financial engineering is important to deal with the short term volatility and long run situations with a sequence of bad scenarios. But properly used, the strategy has much to commend it, especially in trading with many repeated investments.

FAQs

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AI

What is the primary advantage of the Kelly capital growth model?add

The Kelly capital growth model maximizes the asymptotic long run growth rate while minimizing the time to achieve large investment goals.

How do Kelly strategies differ in terms of risk and return?add

Higher Kelly fractions increase wealth potential but also significantly elevate the risk of substantial losses, with minimum wealth decreasing as fractions increase.

What role does estimation error play in Kelly strategy outcomes?add

Estimation errors in mean, variance, and covariance result in skewed risk assessments, emphasizing the criticality of accurate mean estimations for portfolio success.

How do empirical performances of fractional Kelly strategies compare?add

Fractional Kelly strategies avoid bankruptcy and yield lower volatility, however, they also result in lower growth rates compared to full Kelly strategies.

What criticisms have been posed against the Kelly strategy?add

Critics argue that the Kelly strategy may not account for varied investor utility functions and can lead to substantial losses despite high growth expectations.

Figures (17)
Table 2: Final Wealth Statistics by Kelly Fraction for the Ziemba and Hausch Model skewness, excess kurtosis and the number out of the 2000 scenarios for which the final wealth starting from an initial wealth of $1000 is more than $50, $100, $500 (lose less than half), $1000 (breakeven), $10,000 (more than 10-fold), $100,000 (more than 100-fold), and $1 million (more than a thousand-fold). Table 2 shows these results and illustrates the conclusions listed in the abstract. The final wealth levels are much higher on average, the higher the Kelly fraction. With 1/8 Kelly, the average final wealth is $2072, starting with $1000. It is $4339 with 1/4 Kelly, $19,005 with half Kelly, $70,991 with 3/4 Kelly and $524,195 with full Kelly. So as you approach full Kelly, the typical final wealth escalates dramatically. This is shown also in the maximum wealth levels which for full Kelly is $318,854,673 versus $6330 for 1/8 Kelly. Figure 1 shows the wealth paths of these maximum final wealth levels. Most of the gain is in the final 100 of the 700 decision points. Even with these maximum graphs, there
Table 2: Final Wealth Statistics by Kelly Fraction for the Ziemba and Hausch Model skewness, excess kurtosis and the number out of the 2000 scenarios for which the final wealth starting from an initial wealth of $1000 is more than $50, $100, $500 (lose less than half), $1000 (breakeven), $10,000 (more than 10-fold), $100,000 (more than 100-fold), and $1 million (more than a thousand-fold). Table 2 shows these results and illustrates the conclusions listed in the abstract. The final wealth levels are much higher on average, the higher the Kelly fraction. With 1/8 Kelly, the average final wealth is $2072, starting with $1000. It is $4339 with 1/4 Kelly, $19,005 with half Kelly, $70,991 with 3/4 Kelly and $524,195 with full Kelly. So as you approach full Kelly, the typical final wealth escalates dramatically. This is shown also in the maximum wealth levels which for full Kelly is $318,854,673 versus $6330 for 1/8 Kelly. Figure 1 shows the wealth paths of these maximum final wealth levels. Most of the gain is in the final 100 of the 700 decision points. Even with these maximum graphs, there
Figure 1: Highest Final Wealth Trajectory for the Ziemba and Hausch Model is much volatility in the final wealth with the amount of volatility generally higher with higher Kelly fractions. Indeed with 3/4 Kelly, there were losses from about decision points 610 to 670.
Figure 1: Highest Final Wealth Trajectory for the Ziemba and Hausch Model is much volatility in the final wealth with the amount of volatility generally higher with higher Kelly fractions. Indeed with 3/4 Kelly, there were losses from about decision points 610 to 670.
follows. The trouble is that the penalty for losses, especially with large portfolios in the billions is way too low. Fees are kept and fired traders either retire or get rehired by other funds. Examples include John Merriwether, Victor Niederhoffer and Brian Hunter. Table 3: The Investment Proportions (A) and Kelly Fractions. The investment proportions X’s are full Kelly as in the 5’ column; see Table 1. The other proportions are scaled from the Kelly fractions f in this column
follows. The trouble is that the penalty for losses, especially with large portfolios in the billions is way too low. Fees are kept and fired traders either retire or get rehired by other funds. Examples include John Merriwether, Victor Niederhoffer and Brian Hunter. Table 3: The Investment Proportions (A) and Kelly Fractions. The investment proportions X’s are full Kelly as in the 5’ column; see Table 1. The other proportions are scaled from the Kelly fractions f in this column
Table 4: Wealth Statistics by Kelly Fraction for the Ziemba and Hausch Model The initial wealth for investment was 1000. Table 4 reports final wealth statistics on the for T = 40 periods with the various strategies.
Table 4: Wealth Statistics by Kelly Fraction for the Ziemba and Hausch Model The initial wealth for investment was 1000. Table 4 reports final wealth statistics on the for T = 40 periods with the various strategies.
Figure 4: Final Ln(Wealth) Distributions by Fraction for the Ziemba and Hausch Model
Figure 4: Final Ln(Wealth) Distributions by Fraction for the Ziemba and Hausch Model
Figure 5: Mean-Std Tradeoff for the Ziemba and Hausch Model
Figure 5: Mean-Std Tradeoff for the Ziemba and Hausch Model
Figure 6: Trajectories with Final Wealth ] Extremes for the Bicksler and Thorp | Example I
Figure 6: Trajectories with Final Wealth ] Extremes for the Bicksler and Thorp | Example I
Figure 7: Final Ln(Wealth) Distributions by Fraction: Bicksler-Thorp Example I
Figure 7: Final Ln(Wealth) Distributions by Fraction: Bicksler-Thorp Example I
Figure 8: Mean-Std Trade-off for the Bicksler and Thorp Example I Table 6: Final Wealth Statistics by Kelly Fraction for the Bicksler and Thorp Example I
Figure 8: Mean-Std Trade-off for the Bicksler and Thorp Example I Table 6: Final Wealth Statistics by Kelly Fraction for the Bicksler and Thorp Example I
The highest and lowest final wealth trajectories are presented in Figure 9. In the worst case, the trajectory is terminated to indicate the timing of vanishing wealth. There is quick bankruptcy for the aggressive strategies.
The highest and lowest final wealth trajectories are presented in Figure 9. In the worst case, the trajectory is terminated to indicate the timing of vanishing wealth. There is quick bankruptcy for the aggressive strategies.
Figure 9: Trajectories with Final Wealth ] Extremes for the Bicksler and Thorp |] Example II
Figure 9: Trajectories with Final Wealth ] Extremes for the Bicksler and Thorp |] Example II
Figure 10: Final Ln(Wealth) Distributions by Fraction for the Bicksler and Thorp Example II
Figure 10: Final Ln(Wealth) Distributions by Fraction for the Bicksler and Thorp Example II
Figure 11: Mean-Std Tradeoff: Bicksler and Thorp Example IT
Figure 11: Mean-Std Tradeoff: Bicksler and Thorp Example IT

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