### 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Covariance and Correlation

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## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Covariance and Correlation

There are two strongly related measures among many that are commonly used to measure how two random variables tend to move together, the covariance and the correlation. Letting:

$\sigma_{X}$ denote the standard deviation of $X$.
$\sigma_{Y}$ denote the standard deviation of $Y$.
$\sigma_{X Y}$ denote the covariance between $X$ and $Y$.
$\rho_{X Y}$ denote the correlation between $X$ and $Y$.
The relationship between the correlation, which is also denoted by $\rho_{X Y}$ $=\operatorname{corr}(X, Y)$, and covariance is as follows:
$$\rho_{X Y}=\frac{\sigma_{X Y}}{\sigma_{X} \sigma_{Y}} .$$
Here the covariance, also denoted by $\sigma_{X Y}=\operatorname{cov}(X, Y)$, is defined as
\begin{aligned} \sigma_{X Y} &=E(X-E X)(Y-E Y) \ &=E(X Y)-E X E Y \end{aligned}
It can be shown that the correlation can only have values from $-1$ to $+1$. When the correlation is zero, the two random variables are said to be uncorrelated.

If we add two random variables, $X+Y$, the expected value (first central moment) is simply the sum of the expected value of the two random variables. That is,
$$E(X+Y)=E X+E Y .$$
The variance of the sum of two random variables, denoted by $\sigma_{X+Y}^{2}$, is
$$\sigma_{X+Y}^{2}=\sigma_{X}^{2}+\sigma_{Y}^{2}+2 \sigma_{X Y} .$$
Here the last term accounts for the fact that there might be a dependence between $X$ and $Y$ measured through the covariance. In Chapter 8, we consider the variance of the portfolio return of $n$ assets which is expressed by means of the variances of the assets’ returns and the covariances between them.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Multivariate Normal Distribution

In finance, it is common to assume that the random variables are normally distributed. The joint distribution is then referred to as a multivariate normal

distribution. ${ }^{13}$ We provide an explicit representation of the density function of a general multivariate normal distribution.

Consider first $n$ independent standard normal random variables $X_{1}, \ldots$, $X_{n}$. Their common density function can be written as the product of their individual density functions and so we obtain the following expression as the density function of the random vector $X=X_{1}, \ldots, X_{n}$ :
$$f_{\mathrm{X}}\left(x_{1}, \ldots, x_{n}\right)=\frac{1}{(\sqrt{2 \pi})^{n}} e^{-\frac{x^{\prime} x}{2}},$$
where the vector notation $x^{\prime} x$ denotes the sum of the components of the vector $x$ raised to the second power, $x^{\prime} x=\sum_{i=1}^{n} x_{i}^{2}$.

Now consider $n$ vectors with $n$ real components arranged in a matrix $A$. In this case, it is often said that the matrix $A$ has a $n \times n$ dimension. The random variable
$$Y=A X+\mu,$$
in which $A X$ denotes the $n \times n$ matrix $A$ multiplied by the random vector $X$ and $\mu$ is a vector of $n$ constants, has a general multivariate normal distribution. The density function of $Y$ can now be expressed as ${ }^{14}$
where $|\Sigma|$ denotes the determinant of the matrix $\Sigma$ and $\Sigma^{-1}$ denotes the inverse of $\Sigma$. The matrix $\Sigma$ can be calculated from the matrix $A, \Sigma=A A^{\prime}$. The elements of $\Sigma=\left{\sigma_{i j}\right}_{i, j-1}^{n}$ are the covariances between the components of the vector $Y$,
$$\sigma_{i j}=\operatorname{cov}\left(Y_{i}, Y_{j}\right) .$$
Figure $1.5$ contains a plot of the probability density function of a two-dimensional normal distribution with a covariance matrix,
$$\Sigma=\left(\begin{array}{cc} 1 & 0.8 \ 0.8 & 1 \end{array}\right)$$

and mean $\mu=(0,0)$. The matrix $A$ from the representation given in formula (1.3) equals
$$A=\left(\begin{array}{cc} 1 & 0 \ 0.8 & 0.6 \end{array}\right)$$
The correlation between the two components of the random vector $Y$ is equal to $0.8, \operatorname{corr}\left(Y_{1}, Y_{2}\right)=0.8$ because in this example the variances of the two components are equal to 1 . This is a strong positive correlation, which means that the realizations of the random vector $Y$ clusters along the diagonal splitting the first and the third quadrant. This is illustrated in Figure 1.6, which shows the contour lines of the two-dimensional density function plotted in Figure 1.5. The contour lines are ellipses centered at the mean $\mu=(0,0)$ of the random vector $Y$ with their major axes lying along the diagonal of the first quadrant. The contour lines indicate that realizations of the random vector $Y$ roughly take the form of an elongated ellipse as the ones shown in Figure 1.6, which means that large values of $Y_{1}$ will correspond to large values of $Y_{2}$ in a given pair of observations.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Copula Functions

Correlation is a widespread concept in modern finance and risk management and stands for a measure of dependence between random variables. However, this term is often incorrectly used to mean any notion of dependence. Actually, correlation is one particular measure of dependence among many. In the world of multivariate normal distribution and more generally in the world of spherical and elliptical distributions, it is the accepted measure.
A major drawback of correlation is that it is not invariant under nonlinear strictly increasing transformations. In general,
$$\operatorname{corr}(T(X), T(Y)) \neq \operatorname{corr}(X, Y)$$
where $T(x)$ is such transformation. One example which explains this technical requirement is the following: Assume that $X$ and $Y$ represent the continuous return (log-return) of two assets over the period $[0, t]$, where $t$ denotes some point of time in the future. If you know the correlation of these two random variables, this does not imply that you know the dependence structure between the asset prices itself because the asset prices $\left(P\right.$ and $Q$ for asset $X$ and $Y$, respectively) are obtained by $P_{t}=P_{0} \exp (X)$ and $Q_{t}=Q_{0} \exp (Y)$, where $P_{0}$ and $Q_{0}$ denote the corresponding asset prices at time 0 . The asset prices are strictly increasing functions of the return but the correlation structure is not maintained by this transformation. This observation implies that the return could be uncorrelated whereas the prices are strongly correlated and vice versa.

A more prevalent approach that overcomes this disadvantage is to model dependency using copulas. As noted by Patton (2004, p. 3), “The word copula comes from Latin for a ‘link’ or ‘bond,’ and was coined by Sklar (1959), who first proved the theorem that a collection of marginal distributions can be ‘coupled’ together via a copula to form a multivariate distribution.” The idea is as follows. The description of the joint distribution of a random vector is divided into two parts:

1. The specification of the marginal distributions.
2. the specification of the dependence structure by means of a special function, called copula.
The use of copulas ${ }^{19}$ offers the following advantages:
• The nature of dependency that can be modeled is more general. In comparison, only linear dependence can be explained by the correlation.
• Dependence of extreme events might be modeled.
• Copulas are indifferent to continuously increasing transformations (not only linear as it is true for correlations).

From a mathematical viewpoint, a copula function $C$ is nothing more than a probability distribution function on the $n$-dimensional hypercube $I_{n}=[0,1] \times[0,1] \times \ldots \times[0,1]:$
\begin{aligned} C: I_{n} & \rightarrow[0,1] \ \left(u_{1}, \ldots, u_{n}\right) & \rightarrow C\left(u_{1}, \ldots, u_{n}\right) \end{aligned}
It has been shown ${ }^{20}$ that any multivariate probability distribution function $F_{Y}$ of some random vector $Y=\left(Y_{1}, \ldots, Y_{n}\right)$ can be represented with the help of a copula function $C$ in the following form:
\begin{aligned} F_{Y}\left(y_{1}, \ldots, y_{n}\right) &=P\left(Y_{1} \leq y_{1}, \ldots, Y_{n} \leq y_{n}\right)=C\left(P\left(Y_{1} \leq y_{1}\right), \ldots, P\left(Y_{n} \leq y_{n}\right)\right) \ &=C\left(F_{Y_{1}}\left(y_{1}\right), \ldots, F_{Y_{n}}\left(y_{n}\right)\right) \end{aligned}
where $F_{Y_{i}}\left(y_{i}\right), i=1, \ldots, n$ denote the marginal distribution functions of the random variables $Y_{i}, i=1, \ldots, n$.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Covariance and Correlation

σX表示标准差X.
σ是表示标准差是.
σX是表示之间的协方差X和是.
ρX是表示之间的相关性X和是.

ρX是=σX是σXσ是.

σX是=和(X−和X)(是−和是) =和(X是)−和X和是

σX+是2=σX2+σ是2+2σX是.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Multivariate Normal Distribution

FX(X1,…,Xn)=1(2圆周率)n和−X′X2,

σ一世j=这⁡(是一世,是j).

Σ=(10.8 0.81)

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Copula Functions

1. 边际分布的规范。
2. 通过称为 copula 的特殊函数指定依赖结构。
copula 的使用19提供以下优势：
• 可以建模的依赖性的性质更普遍。相比之下，相关性只能解释线性相关性。
• 极端事件的依赖性可能会被建模。
• Copulas 对不断增加的转换无动于衷（不仅是线性的，因为它对相关性也是如此）。

C:一世n→[0,1] (在1,…,在n)→C(在1,…,在n)

F是(是1,…,是n)=磷(是1≤是1,…,是n≤是n)=C(磷(是1≤是1),…,磷(是n≤是n)) =C(F是1(是1),…,F是n(是n))

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## MATLAB代写

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