金融代写|投资组合代写Investment Portfolio代考|MBA6510

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金融代写|投资组合代写Investment Portfolio代考|SOME REMARKS ON THE ESTIMATION OF HIGHER MOMENTS

From a practical point of view, when models involve estimated quantities, it is important to understand how accurate these estimates really are. It is well known that the sample mean and variance, computed via averaging, are very sensitive to outliers. The measures of skew and kurtosis of returns,
$$
\begin{aligned}
&\hat{s}^3=\frac{1}{N} \sum_{i=1}^N\left(R_i-\hat{R}\right)^3 \
&\hat{k}^4=\frac{1}{N} \sum_{i=1}^N\left(R_i-\hat{R}\right)^4
\end{aligned}
$$
where
$$
\hat{R}=\frac{1}{N} \sum_{i=1}^N R_i
$$
are also based upon averages. These measures are therefore also very sensitive to outliers. Moreover, it is well known that the standard error of estimated moments of order $n$ is proportional to the square root of the moment of order $2 n^{58}$ Consequently, the accuracy of moments beyond $n=4$ is often too low for practical purposes.

As a matter of fact, the impact of outliers is magnified in the above measures of skew and kurtosis due to the fact that observations are raised to the third and fourth powers. Therefore, we have to use these measures with tremendous caution. For example, in the data set of MSCI

World Index and United States returns from January 1980 through May 2004 , the skews are $-0.37$ and $-1.22$, respectively. Similarly, the kurtosis for the same period is $9.91$ for the MSCI World Index and $27.55$ for the United States. However, recomputing these measures after removing the single observation corresponding to the October 19, 1987 stock market crash, the skews are $-0.09$ and $-0.04$, while the kurtosis are $6.78$ and $5.07$, for the MSCI World Index and the United States indices, respectively. That is a dramatic change, especially in the U.S. market, after removing a single observation. This simple example illustrates how sensitive higher moments are to outliers. The problem of estimating the higher moments (and even the variance) gets worse in the presence of heavy tails, which is not uncommon in financial data. In practice, it is desirable to use more robust measures of these moments.

In the statistics literature, several robust substitutes for mean and variance are available. However, robust counterparts for skew and kurtosis have been given little attention. Many practitioners eliminate or filter out large outliers from the data. The problem with this approach is that it is done on an ad hoc basis, often by hand, without relying upon methods of statistical inference. Several robust measures of skew and kurtosis are surveyed and compared through Monte Carlo simulations in a paper by Kim and White. ${ }^{59}$ Kim and White’s conclusion is that the conventional measures have to be viewed with skepticism. We recommend that in applications involving higher moments, robust measures should at least be computed for comparison along with traditional estimates.

金融代写|投资组合代写Investment Portfolio代考|THE APPROACH OF MALEVERGNE AND SORNETTE

The mean-variance approach and the generalized formulation with higher moments described earlier in this chapter rely upon empirical estimates of expected returns and risk, that is, centered moments or cumulants. In principle, these could all be estimated empirically. However, the estimation errors of higher moments quickly get very large. In particular, the standard error of the estimated moment of order $n$ is proportional to the square root of the moment of order $2 n$, so that for daily historical times series of returns, which with a decent length amount to about a few thousand observations, moments of order greater than six often become unreasonable to empirically estimate. ${ }^{61}$ One way to proceed is to make stronger assumptions on the multivariate distribution of the asset returns. We describe a technique developed by Malevergne and Sornette for this particular problem. ${ }^{62}$

First, we recall from statistical theory that the dependence between random variables is completely described by their joint distribution. Therefore, for a complete description of the returns and risks associated with a portfolio of $N$ assets we would need the knowledge of the multivariate distribution of the returns. For example, assume that the joint distribution of returns is Gaussian, that is,
$$
p(\mathbf{r})=\frac{1}{(2 \pi)^{N / 2} \sqrt{\operatorname{det}(\boldsymbol{\Sigma})}} \exp \left(-\frac{1}{2}(\mathbf{r}-\boldsymbol{\mu})^{\prime} \mathbf{\Sigma}^{-1}(\mathbf{r}-\boldsymbol{\mu})\right)
$$
with $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ being the mean and the covariance of the returns $\mathbf{r}$. Then we would be back in the mean-variance world described in Chapter 2, because in the Gaussian case the joint distribution is completely described by the mean and the covariance matrix of returns.

In general, the joint distribution of asset returns is not normal. We attempt to represent their multivariate distribution by
$$
p(\mathbf{r})=F\left((\mathbf{r}-\boldsymbol{\mu})^{\prime} \boldsymbol{\Sigma}(\mathbf{r}-\boldsymbol{\mu})\right)
$$
where $F$ is an arbitrary function. We see immediately that if we chose $F(x)=\exp (x)$, we would retrieve the Gaussian distribution. Malevergne and Sornette suggest constructing the function $F$ in such a way that each return $r_i$ is transformed into a Gaussian variable $q_i$.

金融代写|投资组合代写Investment Portfolio代考|MBA6510

金融代写|投资组合代写Investment Portfolio代考|SOME REMARKS ON THE ESTIMATION OF HIGHER MOMENTS

从实践的角度来看,当模型涉及估计数量时,了解这些估计的真实准确度非常重要。众所周知,通过平 均计算的样本均值和方差对异常值非常敏感。回报的偏度和峰度的度量,
$$
\hat{s}^3=\frac{1}{N} \sum_{i=1}^N\left(R_i-\hat{R}\right)^3 \quad \hat{k}^4=\frac{1}{N} \sum_{i=1}^N\left(R_i-\hat{R}\right)^4
$$
在哪里
$$
\hat{R}=\frac{1}{N} \sum_{i=1}^N R_i
$$
也是基于平均值。因此,这些措施对异常值也非常敏感。此外,众所周知,估计阶矩的标准误差 $n$ 与阶 矩的平方根成正比 $2 n^{58}$ 因此,超越时刻的准确性 $n=4$ 对于实际目的来说通常太低了。
事实上,异常值的影响在上述偏度和峰态测量中被放大了,因为观测值被提升到三次方和四次方。因 此,我们必须非常谨㥀地使用这些措施。比如在MSCI的数据集中
从 1980 年 1 月到 2004 年 5 月的世界指数和美国回报率,偏度是 $-0.37$ 和 $-1.22$ ,分别。同样,同期 的峰态是 $9.91$ 对于 MSCI 世界指数和 $27.55$ 为美国。然而,在删除对应于 1987 年 10 月 19 日股市崩盘的 单一观察结果后重新计算这些措施,偏斜是 $-0.09$ 和 $-0.04$ ,而峰度是 $6.78$ 和 $5.07$ ,分别代表 $\mathrm{MSCI}$ 世界 指数和美国指数。这是一个巨大的变化,尤其是在美国市场,除去一个单一的观察结果。这个简单的例 子说明了高阶矩对异常值的敏感程度。在存在重尾的情况下,估计更高矩 (甚至方差) 的问题会变得更 糟,这在金融数据中并不少见。在实践中,布望对这些时刻使用更稳健的度量。
在统计文献中,有几个稳健的均值和方差替代项可用。然而,很少有人关注偏叙和峰态的稳健对应物。 许多从业者从数据中消除或过滤掉较大的异常值。这种方法的问题在于它是钿时完成的,通常是手工完 成的,而不依赖于统计推断的方法。Kim 和 White 在一篇论文中通过蒙特卡洛模拟调查和比较了几种可 靠的偏斜和峰态度量。 ${ }^{59} \mathrm{Kim}$ 和 White 的结论是,必须对传统措施持怀疑态度。我们建议在涉及更高矩 的应用程序中,至少应计算稳健度量以便与传统估计进行比较。

金融代写|投资组合代写Investment Portfolio代考|THE APPROACH OF MALEVERGNE AND SORNETTE

本章前面描述的均值-方差方法和具有更高矩的广义公式依赖于预期收益和风险的经验估计,即中心矩或 略积量。原则上,这些都可以根据经验进行估计。然而,更高矩的估计误差很快变得非常大。特别是, 估计阶矩的标准误差 $n$ 与阶矩的平方根成正比 $2 n$ ,因此对于具有相当长度的大约几千个观察值的日历史 回报时间序列,大于 6 的阶矩通常变得无法根据经验进行估计。 ${ }^{61}$ 一种方法是对资产回报的多元分布做 出更强的假设。我们描述了 Malevergne 和 Sornette 为这个特定问题开发的技术。 ${ }^{62}$
首先,我们回顾一下统计理论,随机变量之间的依赖完全由它们的联合分布来描述。因此,要完整菑述 与投资组合相关的收益和风险 $N$ 资产,我们需要了解回报的多元分布。例如,假设收益的联合分布是高 斯分布,即
$$
p(\mathbf{r})=\frac{1}{(2 \pi)^{N / 2} \sqrt{\operatorname{det}(\mathbf{\Sigma})}} \exp \left(-\frac{1}{2}(\mathbf{r}-\boldsymbol{\mu})^{\prime} \mathbf{\Sigma}^{-1}(\mathbf{r}-\boldsymbol{\mu})\right)
$$
和 $\boldsymbol{\mu}$ 和 $\boldsymbol{\Sigma}$ 是回报的均值和协方差r. 然后我们将回到第 2 章中描述的均值-方差世界,因为在高斯情况下, 联合分布完全由收益的均值和协方差矩阵描述。
一般来说,资产收益的联合分布是不正常的。我们试图通过以下方式表示它们的多元分布
$$
p(\mathbf{r})=F\left((\mathbf{r}-\boldsymbol{\mu})^{\prime} \mathbf{\Sigma}(\mathbf{r}-\boldsymbol{\mu})\right)
$$
在哪里 $F$ 是任意函数。我们立即看到,如果我们选择 $F(x)=\exp (x)$ ,我们将检索高斯分布。 Malevergne 和 Sornette 建议构建函数 $F$ 以这样的方式,每个回报 $r_i$ 转化为高斯变量 $q_i$.

金融代写|投资组合代写Investment Portfolio代考

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