### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|A Simulated Example – Equity Versus Cash

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|A Simulated Example – Equity Versus Cash

In our experiment based on a similar example in Bicksler and Thorp (1973), there are two assets: US equities and US T-bills. 1 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 $\lambda$ of wealth invested in equities ${ }^{2}$ and the corresponding Kelly fraction $f$ are

Bicksler and Thorp used 10 and 20 yearly decision periods, and 50 simulated scenarios. MacLean et al. used 40 yearly decision periods, with 3000 scenarios.
The results from the simulations appear in Table $1.3$ and Figs. 1.7, 1.8 and 1.9. The striking aspects of the statistics in Table $1.3$ are the sizable gains and losses. In his lectures, Ziemba always says when in doubt bet less – that is certainly borne out in these simulations. For the most aggressive strategy ( $1.57 \mathrm{k}$ ), it is possible to lose 10,000 times the initial wealth. This assumes that the shortselling is permissible through the decision period at the horizon $T=40$.

The highest and lowest final wealth trajectories are presented in Fig. 1.7. In the worst case, the trajectory is terminated to indicate the timing of vanishing wealth. There is quick bankruptcy for the aggressive overbet strategies.

The substantial downside is further illustrated in the distribution of final wealth plot in Fig. 1.8. The normal probability plots are almost linear on the upside.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Final Comments

The Kelly optimal capital growth investment strategy is an attractive approach to wealth creation. In addition to maximizing the asymptotic rate of long-term growth of capital, it avoids bankruptcy and overwhelms any essentially different investment strategy in the long run. See MacLean, Thorp, and Ziemba (2010a) for a discussion of the good and bad properties of these strategies. However, automatic use of the Kelly strategy in any investment situation is risky and can be very dangerous. It requires some adaptation to the investment environment: rates of return, volatilities, correlation of alternative assets, estimation error, risk aversion preferences, and planning horizon are all important aspects of the investment process. Chopra and Ziemba (1993) show that in typical investment modeling, errors in the means average about 20 times in importance in objective value than errors in co-variances with errors in variances about double the co-variance errors. This is dangerous enough but they also show that the relative importance of the errors is risk aversion dependent with the errors compounding more and more for lower risk aversion investors and for the extreme log investors with essentially zero risk aversion the errors are worth about $100: 3: 1$. So log investors must estimate means well if they are to survive. This is compounded even more by the observation that when times move suddenly from normal to bad the correlations/co-variances approach 1 and it is hard to predict the transition from good times to bad. Poundstone’s (2005) book, while a very good read with lots of useful discussions, does not explain these important investment aspects and the use of Kelly strategies by advisory firms such as Morningstar and Motley Fools is flawed; see, for example, Fuller $(2006)$ and Lee $(2006)$. The experiments in Bicksler and Thorp (1973). Ziemba and Hausch (1986). and MacLean. Thorp. Zhao, and Ziemba $(2011)$ and that described here represent some of the diversity in the investment environment. By considering the Kelly and its variants we get

a concrete look at the plusses and minuses of the capital growth model. We can conclude that

• 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 $1 .$
• There is a trade-off of risk and return with the fraction invested in the Kelly portfolio. In cases of large uncertainty, from either 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 retums 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 and to be penalized for being below the path.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|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 irvestment. Math of Operations Reseanch 5, 161-166.
Bertocchi, M., S. L. Schwartz, and W. T. Zicmba (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. Sournal of Financial and Quantirative Analysis \& (2), 273-287.
Breiman, L. (1960). Investment policies for expanding businesses optimal in a long run sense. Naval Research Logistics Ouarterly 4 (4), 647-651.
Breiman, L. (1961). Optimal gambling system for favorable games. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability l, 63-68.
Chopra, V. K. and W. T. Ziemba (1993). The effect of crrors in mean, variance and co-variance estimates on optimal portfolio choice. Joumal of Portfolio Management $19,6-11$.
Cover, T. M. and J. Thomas (2006). Elements of Information Theory (2nd cd.). Wiley, New York, NY.
Fuller, J. (2006). Optimize your portfolio with the Kelly formula. morningstar.com, October $6 .$ Hakansson, N. H. and W. T. Zicmba (1995). Capital growth theory. In R. A. Jarrow, V. Maksimovic, and W. T. Zicmba (Eds.), Funance, Handbooks in $O R$ \& $M S$, Pp. 65-86. North Holland, Amstcrdam.
Harville, D. A. (1973). Assigning probabilitics to the outcome of multi-cntry competitions. /ournal of the American Statistical Association 68,312-316.
Hausch, D. B., V. Lo, and W. T. Ziemba (Eds.) (1994). Efficiency of Racetrack Berting Markets. Academic, San Diego.
Hausch, D. B., V. Lo, and W. T. Ziemba (Eds.) (2008). Efficiency of Racetrack Benting Markets ( 2 cd.). World Scientific, Singapore.
Hausch, D. B. and W. T. Ziemba (1985). Transactions costs, extent of inefficiencies, entries and multiple wagcrs in a racetrack betting model. Management Sclence 31, 381-394.
Hausch, D. B., W. T. Zicmba, and M. E. Rubinstein (1981). Efficiency of the market for racetrack betting. Management Science XXYII, 1435-1452.
Kelly, Jr., J. R. (1956). A new interpretation of the information rate. Bell System Techuical .lournal 35, 917-926.
Latané, H. (1978). The geometric-mean principle revisited – a reply. Journal of Banking and Finance 2 (4), 395-398.
Lec, 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 Economuc Dynamics and Control 17, 887-906.
MacLean, L. C., R. Sancgre, Y. Zhao, and W. T. Zicmba (2004). Capital growth with security. Sournal of Economic Dynamics and Control 28 (4), 937-954.
MacLean, L. C., E. O. Thorp, Y. Zhao, and W. T. Zicmba (2011). How does the Fortunes FormulaKelly capital growth model perform? Joumal of Portfolio Management 37 (4).

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|A Simulated Example – Equity Versus Cash

Bicksler 和 Thorp 使用了 10 和 20 年的决策周期，以及 50 个模拟场景。麦克莱恩等人。使用 40 个年度决策周期，包含 3000 个场景。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Final Comments

• 完全凯利策略积累的财富不会随机支配部分凯利财富。不利的一面往往是更有利的一小部分1.
• 投资于凯利投资组合的部分需要权衡风险和回报。在存在较大不确定性的情况下，无论是内在波动性还是估计误差，都可以通过减少凯利投资分数来获得安全性。
• 完整的凯利策略可以高度利用。虽然使用借贷可以有效地产生大量投资回报，但没有理由增加杠杆率超过全部凯利，因为它是增长型证券主导的。过度杠杆投资的回报被越来越大的破产可能性所抵消。
• 凯利策略不仅仅是一种长期的方法。在短期和中期适当使用可以实现财富目标，同时防止回撤。MacLean、Sanegre、Zhao 和 Ziemba（2004 年）以及 MacLean、Zhao 和 Ziemba（2009 年）讨论了一种降低凯利分数以高概率保持在预定财富路径之上并因低于路径而受到惩罚的策略。

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|References

Aase, KK (2001)。关于圣彼得堡悖论。斯堪的纳维亚精算杂志3(1), 69-78。贝尔、RM 和 TM 封面（1980 年）。对数投资的竞争最优性。运维数学研究 5, 161-166。
Bertocchi, M., SL Schwartz 和 WT Zicmba (2010)。优化老龄化、退休、养老金困境。威利，霍博肯，新泽西州。

Breiman, L. (1960)。从长远来看，扩大业务的投资政策是最优的。海军研究后勤 Oarterly 4 (4), 647-651。
Breiman, L. (1961)。有利游戏的最佳赌博系统。第四届伯克利数理统计和概率研讨会论文集，63-68。
Chopra、VK 和 WT Ziemba (1993)。均值、方差和协方差估计中的误差对最优投资组合选择的影响。投资组合管理杂志19,6−11.

DA 哈维尔 (1973)。为多国竞争的结果分配概率。/美国统计协会杂志 68,312-316。
Hausch、DB、V. Lo 和 WT Ziemba (Eds.) (1994)。Racetrack Berting 市场的效率。学术，圣地亚哥。
Hausch、DB、V. Lo 和 WT Ziemba (Eds.) (2008)。赛马场弯曲市场的效率 (2 cd.)。世界科学，新加坡。
Hausch、DB 和 WT Ziemba (1985)。赛道投注模型中的交易成本、低效率程度、条目和多个 wagcr。管理学 31, 381-394。
Hausch、DB、WT Zicmba 和 ME 鲁宾斯坦 (1981)。赛道博彩市场的效率。管理科学 XXYII，1435-1452。

Lec, E. (2006)。如何计算凯利公式。傻瓜网，10 月 31 日。
Luenberger, DG (1993)。投资组合选择问题中对数均值方差标准的偏好基础。经济动力学与控制杂志 17, 887-906。
MacLean, LC, R. Sancgre, Y. Zhao 和 WT Zicmba (2004)。安全的资本增长。经济动态与控制杂志 28 (4), 937-954。
MacLean、LC、EO Thorp、Y. Zhao 和 WT Zicmba (2011)。Fortunes FormulaKelly 资本增长模型的表现如何？投资组合管理杂志 37 (4)。

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