# 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

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|SOME REMARKS ON THE ESTIMATION OF HIGHER MOMENTS

$$\hat{s}^3=\frac{1}{N} \sum_{i=1}^N\left(R_i-\hat{R}\right)^3 \quad \hat{k}^4=\frac{1}{N} \sum_{i=1}^N\left(R_i-\hat{R}\right)^4$$

$$\hat{R}=\frac{1}{N} \sum_{i=1}^N R_i$$

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

$$p(\mathbf{r})=\frac{1}{(2 \pi)^{N / 2} \sqrt{\operatorname{det}(\mathbf{\Sigma})}} \exp \left(-\frac{1}{2}(\mathbf{r}-\boldsymbol{\mu})^{\prime} \mathbf{\Sigma}^{-1}(\mathbf{r}-\boldsymbol{\mu})\right)$$

$$p(\mathbf{r})=F\left((\mathbf{r}-\boldsymbol{\mu})^{\prime} \mathbf{\Sigma}(\mathbf{r}-\boldsymbol{\mu})\right)$$

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