### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Using the Kelly Criterion for Investing

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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|William T. Ziemba and Leonard C. MacLean

The Kelly capital growth strategy is defined as allocate your current wealth to risky assets so that the expected logarithm of wealth is maximized period by period. So it is a one-period static calculation that can have transaction costs and other market imperfections considered. Log utility dates to Daniel Bernoulli in 1738 who postulated that marginal utility was monotone increasing but declined with wealth and, specifically, is equal to the reciprocal of wealth, $w$, which yields the utility of wealth $u(w)=\log w$. Prior to this it was assumed that decisions were made on an expected value or linear utility basis. This idea ushered in declining marginal utility or risk aversion or concavity which is crucial in investment decision making. In his chapter, in Latin, he also discussed the St. Petersburg paradox and how it might be analyzed using $\log w$.

The St. Petersburg paradox actually originates from Daniel’s cousin, Nicolas Bernoulli, a professor at the University of Basel where Daniel was also a professor of mathematics. In 1708 , Nicolas submitted five important problems to Professor Pierre Montmort. This problem was how much to pay for the following gamble:
A fair coin with $\frac{1}{2}$ probability of heads is repeatedly tossed until heads oecurs, ending the game. The investor pays $c$ dollars and receives in return $2^{k-1}$ with probability $2^{-k}$ for $k=1,2, \ldots$ should a head occur. Thus, after each succeeding loss, assuming a head does not appear, the bet is doubled to $2,4,8, \ldots$ ete. Clearly the expected value is $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\ldots$. or infinity with linear utility.
Bell and Cover (1980) argue that the St. Petersburg gamble is attractive at any price $c$, but the investor wants less of it as $c \rightarrow \infty$. The proportion of the investor’s wealth invested in the St. Petersburg gamble is always positive but decreases with the $\cos t c$ as $c$ increases. The rest of the wealth is in cash.

Bernoulli offers two solutions since he felt that this gamble is worth a lot less than infinity. In the first solution, he arbitrarily sets a limit to the utility of very large payoffs. Specifically, any amount over 10 million is assumed to be equal to $2^{24}$. Under that bounded utility assumption, the expected value is
$$\frac{1}{2}(1)+\frac{1}{4}(2)+\frac{1}{8}(4)+\cdots+\left(\frac{1}{2}\right)^{24}\left(2^{24}\right)+\left(\frac{1}{2}\right)^{25}\left(2^{24}\right)+\left(\frac{1}{2}\right)^{2 h}\left(2^{24}\right)+\ldots=12+\text { the original } 1=13 .$$

When utility is log the expected value is
$$\frac{1}{2} \log 1+\frac{1}{4} \log 2+\cdots+\frac{1}{8} \log 4+\cdots=\log 2=0.69315$$
Use of a concave utility function does not eliminate the paradox.
For example, the utility function $U(x)=x / \log (x+A)$, where $A>2$ is a constant, is strictly concave, strictly increasing, and infinitely differentiable yet the
As Menger (1967) pointed out in 1934 , the log, the square root, and many other, but not all, concave utility functions eliminate the original St. Petersburg paradox but it does not solve one where the payoffs grow faster than $2^{n}$. So if log is the utility function, one creates a new paradox by having the payoffs increase at least as fast as $\log$ reduces them so one still has an infinite sum for the expected utility. With exponentially growing payoffs one has
$$\frac{1}{2} \log \left(e^{1}\right)+\frac{1}{4} \log \left(e^{2}\right)+\cdots=\infty$$

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

We can break risk aversion, both absolute and relative, into categories of investors as Ziemba (2010) has done in his response to letters he received from Professor Paul A Samuelson $(2006,2007,2008$ ) (Table 1.1).

Ziemba named Ida after Ida May Fuller who paid $\$ 24.75$into US social security and received her first social security check numbered 00-000-001 on January 31 , 1940 , the actual first such check. She lived in Ludlow, Vermont, to the age of 100 and collected$\$22,889$. Such are the benefits and risks of this system; see Bertoccchi, Schwartz, and Ziemba (2010) for more on this. Victor is named for the hedge fund trader Victor Niederhoffer who seems to alternate between very high returns and blowing up; see Ziemba and Ziemba (2007) for some but not all of his episodes. The other three investors are the overbetting Tom who is growth-security dominated in the sense of MacLean, Ziemba, and Blazenko (1992), our $E$ log investor Dick and Harriet, approximately half Kelly, who Samuelson says fits the data well. We agree that in practice, half Kelly is a toned down version of full Kelly that provides a lot more security to compensate for its loss in long-term growth. Figure 1.l shows this behavior in the context of Blackjack where Thorp first used Kelly strategies.
The edge for a successful card counter varies from about $-5$ to $+10 \%$ depending upon the favorability of the deck. By wagering more in favorable situations and less or nothing when the deck is unfavorable, an average weighted edge is about $2 \%$. An approximation to provide insight into the long-run behavior of a player’s fortune is to assume that the game is a Bernoulli trial with a probability of success $=0.51$ and probability of loss $1=0.49$.

Figure $1.1$ shows the relative growth rate $f \ln (1+p)+(1-f) \ln (1-p)$ versus the fraction of the investor’s wealth wagered, $f$. This is maximized by the Kelly $\log$ bet $f^{}-p-q-0.02$. The growth rate is lower for smaller and for larger bets than the Kelly bet. Superimposed on this graph is also the probability that the investor doubles or quadruples the initial wealth before losing half of this initial wealth. Since the growth rate and the security are both decreasing for $f>f^{}$, it follows that it is never advisable to wager more than $f^{*}$.

Observe that the $E$ log investor maximizes long-run growth and that the investor who wagers exactly twice this amount has a growth rate of zero plus the risk-free rate of interest. The fractional Kelly strategies are on the left and correspond to.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Understanding the Behavior of E log Strategies

There are many possible investment situations and $E$ log Kelly wagering is useful for some of them. Good uses of the strategy are in situations with many repeated bets that approximate an infinite sequence as in the Breiman, etc., theory. See the papers in MacLean, Thorp, and Ziemba $(2010 b)$ for such extensions; MacLean, Thorp, and Ziemba (2010a) for good and bad Kelly and fractional Kelly properties; and MacLean, Thorp, Zhao, and Ziemba $(2011)$ for simulations of typical behavior. Luenberger (1993) looks at long-run asymptotic behavior. Futures and options trading. sports hetting, including horseracing, are gond examples. The policies tend to non-diversify, plunge on a small number of the best assets, have a lot of volatility,

and produce more total wealth in the end than other strategies. Notable investors who use such strategies are Warren Buffett of Berkshire Hathaway, George Soros of the Quantum funds, and John Maynard Keynes who ran the King’s College Cambridge endowment from 1927 to 1945 . Figure $1.2 \mathrm{a}$, b shows the best and worst months for the Buffett and Soros funds. Observe that Buffett and Soros are asymptotically equivalent in both the left and right tails. Figure $1.3$ shows their wealth graphs. These correspond to typical Kelly behavior. Some Kelly advocates with a gambling background have produced nice smooth graphs such as those of Hong Kong racing guru Bill Benter, famed hedge fund traders Ed Thorp and Jim Simons; seẽ Fig. 1.4ã-d for thẽ vărioùs wealth grạhhs.

According to Ziemba (2005), Keynes was approximately an $80 \%$ Kelly bettor with a utility function of $-w^{-0.25}$. In Ziemba (2005) it is argued that Buffett and Soros are full Kelly bettors. They focus on long run wealth gains, not worrying about short term monthly losses. They tend to have few positions and try not to lose on any of them and not focusing on diversification. Table $1.2$ supports this showing their top 10 equity holdings on September 30,2008 . Soros is even more of a plunger with more than half his equity portfolio in just one position.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|William T. Ziemba and Leonard C. MacLean

Bell 和 Cover (1980) 认为圣彼得堡的赌局不惜任何代价都具有吸引力C，但投资者想要的更少，因为C→∞. 投资于圣彼得堡赌博的投资者财富比例始终为正，但随着某物⁡吨C作为C增加。其余的财富是现金。

12(1)+14(2)+18(4)+⋯+(12)24(224)+(12)25(224)+(12)2H(224)+…=12+ 原本的 1=13.

12日志⁡1+14日志⁡2+⋯+18日志⁡4+⋯=日志⁡2=0.69315

12日志⁡(和1)+14日志⁡(和2)+⋯=∞

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

Ziemba 以 Ida May Fuller 的名字命名 Ida$24.75进入美国社会保障并于 1940 年 1 月 31 日收到了她的第一张社会保障支票，编号为 00-000-001，实际上是第一张这样的支票。她住在佛蒙特州的勒德洛，直到 100 岁$22,889. 这就是这个系统的好处和风险；有关这方面的更多信息，请参见 Bertoccchi、Schwartz 和 Ziemba (2010)。Victor 以对冲基金交易员 Victor Niederhoffer 的名字命名，他似乎在高回报和暴涨之间交替出现；请参阅 Ziemba 和 Ziemba (2007) 了解他的部分但不是全部剧集。其他三个投资者是过度投注的汤姆，他在 MacLean、Ziemba 和 Blazenko (1992) 的意义上是增长安全主导的，我们的和日志投资者迪克和哈里特，大约一半凯利，萨缪尔森说他很适合数据。我们同意，在实践中，半凯利是全凯利的低调版本，它提供了更多的安全性来弥补其在长期增长中的损失。图 1.l 显示了索普首次使用凯利策略的二十一点背景下的这种行为。

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## MATLAB代写

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