### 金融代写|量化风险管理代写Quantitative Risk Management代考|MINE7034

statistics-lab™ 为您的留学生涯保驾护航 在代写量化风险管理Quantitative Risk Management方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写量化风险管理Quantitative Risk Management代写方面经验极为丰富，各种代写量化风险管理Quantitative Risk Management相关的作业也就用不着说。

• 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

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Semi-Interquartile Deviation

The semi-interquartile deviation or range ${ }^{2}$ corresponds to one-half of the interquartile range, i.e., the difference between the third quartile $(\mathrm{Q} 3)$ and the first $(\mathrm{Q} 1)$ and the coefficient of quartile variation is the interquartile range divided by the second quartile. Formally, the semi-interquartile range, measuring the dispersion, is expressed as follows:
$$S I=\frac{(Q 3-Q 1)}{2}$$
while the coefficient of quartile variation is expressed as follows:
$$S I=\frac{(Q 3-Q 1)}{Q 2}$$
In a symmetric distribution, contrary to a skewed distribution, an interval stretching from one semi-interquartile range below the median to one semi-interquartile above the median will contain half of the values.

It is interesting to mention that semi-interquartile range is barely affected by extreme values, as a consequence it is a good dispersion measure for skewed distributions. However, it is more subject to sampling fluctuation in the Gaussian case than is the standard deviation and therefore not often used for data that are approximately normally distributed.

However, this class of risk measure exhibits the major drawbacks of assuming distribution with specific characteristics such as symmetry and of not take into account losses occurring with small probabilities.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Mean Absolute Difference

Assuming the probability space defined in introduction, let $X$ and $Y$ be two iid random variables following the same distribution. The mean absolute difference (MAD) is given by the average of the differences of all possible pairs of variatevalues, taken regardless of their sign. It is formally defined as follows:
$$\mathrm{MAD}:=E[|X-Y|] .$$
Let $x_{1}, \ldots, x_{n}$ and $y_{1}, \ldots, y_{n}$ be two sets of respective realisations of random variables $X$ and $Y$. For a random sample of size $n$ of a population uniformly distributed, by the law of total expectation ${ }^{3}$ the mean absolute difference of the sample $y_{i}, i=1$ to $n$ corresponds to the arithmetic mean of the absolute value of all possible differences,
$$\mathrm{MAD}=E[|X-Y|]=E_{X}\left[E_{X|Y|}[|X-Y|]\right]=\frac{1}{n^{2}} \sum_{i=1}^{n} \sum_{j=1}^{n}\left|y_{i}-y_{j}\right| .$$
If $Y$ follows a discrete probability function $f(y)$, where $y_{i}, i=1$ to $n$ are the values with non-zero probabilities:
$$\mathrm{MAD}=\sum_{i=1}^{n} \sum_{j=1}^{n} f\left(y_{i}\right) f\left(y_{j}\right)\left|y_{i}-y_{j}\right|$$
In the continuous case, let $f(x)$ be the probability density function, then,
$$\mathrm{MAD}=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x) f(y)|x-y| d x d y$$
Let $F(x)$, absolutely continuous, be the cumulative distribution function associated with $f(x)$ with quantile function $F^{-1}(x)$, then, since $f(x)=d F(x) / d x$ and $F^{-1}(x)=x$, it follows that:
$$\mathrm{MAD}=\int_{0}^{1} \int_{0}^{1}\left|F_{1}^{-1}-F_{2}^{-1}\right| d F_{1} d F_{2} .$$

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Modern Portfolio Theory

Modern portfolio theory (Markowitz 1952) is a mathematical framework for constituting a portfolio of assets such that the expected return is maximised for a given level of risk, here the variance. The difference with what we discussed before is that both risk and return of an asset should not be assessed on their own, but by how it contributes to a portfolio’s overall risk and return.

Modern portfolio theory assumes that investors are risk averse that for the same expected return given by two portfolios, investors will prefer the less risky one. As a consequence, an investor may only accept on increased exposure if this one is compensated by higher expected returns. Conversely, an investor who is willing higher expected returns must face larger exposures. The exact trade-off will be the same for all investors, but different investors will evaluate the trade-off differently based on individual risk aversion characteristics. This implies that a rational investor would not invest in a portfolio if it exists a second portfolio with a more favourable risk-expected return profile-i.e., if for that level of risk an alternative portfolio exists that has better expected returns.

Under the model:

• Portfolio return is the proportion-weighted combination of the constituent assets’ returns.
• Portfolio volatility is a function of the correlations $\rho_{i j}$ of the component assets, for all asset pairs $(i, j)$.
The expected return is given by the following equation:
$$\mathrm{E}\left(R_{p}\right)=\sum_{i} w_{i} \mathrm{E}\left(R_{i}\right)$$
where $R_{p}$ is the return on the portfolio, $R_{i}$ is the return on asset $i$, and $w_{i}$ is the weighting of component asset $i$ (that is, the proportion of asset ” $i$ ” in the portfolio). The portfolio return variance is provided by the following equation:
$$\sigma_{p}^{2}=\sum_{i} w_{i}^{2} \sigma_{i}^{2}+\sum_{i} \sum_{j \neq i} w_{i} w_{j} \sigma_{i} \sigma_{j} \rho_{i j}$$
where $\sigma_{i}$ is the standard deviation of the returns on asset $i$, and $\rho_{i j}$ is the correlation coefficient between the returns on assets $i$ and $j$. It is also possible to rewrite the expression as:
$$\sigma_{p}^{2}=\sum_{i} \sum_{j} w_{i} w_{j} \sigma_{i} \sigma_{j} \rho_{i j}$$
where $\rho_{i j}=1$ for $i=j$, or
$$\sigma_{p}^{2}=\sum_{i} \sum_{j} w_{i} w_{j} \sigma_{i j}$$
where $\sigma_{i j}=\frac{\sigma_{i} \sigma_{i}}{\rho_{i}}$ is the covariance of the returns of the two assets.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Modern Portfolio Theory

• 投资组合回报是成分资产回报的比例加权组合。
• 投资组合波动率是相关性的函数ρ一世j组件资产的数量，适用于所有资产对(一世,j).
预期回报由以下等式给出：
和(Rp)=∑一世在一世和(R一世)
在哪里Rp是投资组合的回报，R一世是资产回报率一世， 和在一世是组成资产的权重一世（即资产比例”一世”在投资组合中）。投资组合收益方差由以下等式提供：
σp2=∑一世在一世2σ一世2+∑一世∑j≠一世在一世在jσ一世σjρ一世j
在哪里σ一世是资产回报率的标准差一世， 和ρ一世j是资产收益率之间的相关系数一世和j. 也可以将表达式重写为：
σp2=∑一世∑j在一世在jσ一世σjρ一世j
在哪里ρ一世j=1为了一世=j， 或者
σp2=∑一世∑j在一世在jσ一世j
在哪里σ一世j=σ一世σ一世ρ一世是两种资产收益的协方差。

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## MATLAB代写

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