## 金融代写|风险理论代写Risk theory代考|MATH4128

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory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。我们的专家在代写风险理论Risk theory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|风险理论代写Risk theory代考|Bayes and Empirical Bayes

Let $\boldsymbol{X}=\left(X_1, \ldots, X_n\right)$ be a vector of r.v.s describing the outcome of a statistical experiment. For example, in the insurance context, $n$ can be the number of persons insured for losses due to accidents in the previous year, and $X_i$ the payment made to the $i$ th.

A traditional (frequentists’) model is to assume the $X_i$ to be i.i.d. with a common distribution $F_\theta$ where $\theta$ is an unknown parameter (possibly multidimensional). F.g. in the accident insurance example, one could let $b$ denote the probability that a person has an accident within one year, $b=\mathbb{P}\left(X_i>0\right)$, and one could assume that the cost of the accident has a $\operatorname{gamma}(\alpha, \lambda)$ distribution. Thus the density of $X_i$ is
$$f_{b, \alpha, \lambda}(x)=b \mathbb{1}{x=0}+(1-b) \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)} \mathrm{e}^{-\lambda x_1} \mathbb{1}{x>0}$$
w.r.t. the measure defined as Lebesgue measure $\mathrm{d} x$ on $(0, \infty)$ with an added atom of unit size at $x=0$. Then $\theta=(b, \alpha, \lambda)$, and the conventional statistical procedure would be to compute estimates $\widehat{b}, \widehat{\alpha}, \widehat{\lambda}$ of $b, \alpha, \lambda$. These estimates could then be used as basis for computing first the expectation
$$\mathbb{E}{\widehat{\theta}} X=\mathbb{E}{\widehat{b}, \widehat{\alpha}, \widehat{\lambda}} X=(1-\widehat{b}) \widehat{\alpha} / \widehat{\lambda}$$
of $X$ under the estimated parameters, and next one could use $\mathbb{E}_{\widehat{\theta}} X$ as the net premium and add a loading corresponding to one of the premium rules discussed in Sect. I.3. For example, the expected value principle would lead to the premium
$$p=(1+\eta)(1-\widehat{b}) \widehat{\alpha} / \widehat{\lambda}$$

We now turn to the general implementation of Bayesian ideas in insurance. Here one considers an insured with risk parameter $Z^1$ and an r.v. with distribution $\pi^{(0)}(\cdot)$, with observable past claims $X_1, \ldots, X_n$ and an unobservable claim amount $X_{n+1}$ for year $n+1$. The aim is to assert which (net) premium the insured is to pay in year $n+1$

For a fixed $\zeta$, let $\mu(\zeta)=\mathbb{E}\zeta X{n+1}$, where $\mathbb{E}\zeta[\cdot]=\mathbb{E}[\cdot \mid Z=\zeta]$. The (net) collective premium $H{\mathrm{Coll}}$ is $\mathbb{E} \mu(\boldsymbol{Z})=\mathbb{E} X_{n+1}$. This is the premium we would charge without prior statistics $X_1, \ldots, X_n$ on the insured. The individual premium is $H_{\text {Ind }}=\mathbb{E}\left[X_{n+1} \mid \boldsymbol{Z}\right]=\mu(\boldsymbol{Z})$. This is the ideal net premium in the sense of supplying the complete relevant prior information on the customer. The Bayes premium $H_{\text {Bayes }}$ is defined as $\mathbb{E}\left[\mu(\boldsymbol{Z}) \mid X_1, \ldots, X_n\right]$. That is, $H_{\text {Bayes }}$ is the expected value of $X_{n+1}$ in the posterior distribution.

Note that the individual premium is unobservable because $\boldsymbol{Z}$ is so; the Bayes premium is ‘our best guess of $H_{\text {Ind }}$ based upon the observations’. To make this precise, let $H^$ be another premium rule, that is, a function of $X_1, \ldots, X_n$ and the prior parameters. We then define its loss as $$\ell_{H^}=\mathbb{E}\left[\mu(\boldsymbol{Z})-H^\right]^2=\left|\mu(\boldsymbol{Z})-H^\right|^2$$
where $|X|=\left(\mathbb{E} X^2\right)^{1 / 2}$ is the $L_2$-norm (in obvious notation, we write $\ell_{\text {Coll }}=\ell_{H_{\text {Coll }}}$ etc). In mathematical terms, the optimality property of the Bayes premium is then that it minimizes the quadratic loss:
Theorem 1.3 For any $H^, \ell_{\text {Bayes }} \leq \ell_{H^}$. That is,
$$\mathbb{E}\left(H_{\text {Bayes }}-H_{\text {Ind }}\right)^2 \leq \mathbb{E}\left(H^*-H_{\text {Ind }}\right)^2$$

## 金融代写|风险理论代写Risk theory代考|Bayes and Empirical Bayes

$$f_{b, \alpha, \lambda}(x)=b 1 x=0+(1-b) \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)} \mathrm{e}^{-\lambda x_1} 1 x>0$$
wrt 定义为 Lebesgue 度量的度量 $\mathrm{d} x$ 上 $(0, \infty)$ 添加一个单位大小的原子 $x=0$. 然后 $\theta=(b, \alpha, \lambda)$ ，而传统的统 计程序是计算估计值 $\hat{b}, \widehat{\alpha}, \widehat{\lambda}$ 的 $b, \alpha, \lambda$. 然后可以将这些估计值用作首先计算期望值的基础
$$\mathbb{E} \hat{\theta} X=\mathbb{E} \hat{b}, \widehat{\alpha}, \hat{\lambda} X=(1-\hat{b}) \widehat{\alpha} / \widehat{\lambda}$$

$$p=(1+\eta)(1-\hat{b}) \widehat{\alpha} / \widehat{\lambda}$$

$$\mathbb{E}\left(H_{\text {Bayes }}-H_{\text {Ind }}\right)^2 \leq \mathbb{E}\left(H^*-H_{\text {Ind }}\right)^2$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|风险理论代写Risk theory代考|STAT4901

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory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。我们的专家在代写风险理论Risk theory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|风险理论代写Risk theory代考|Reinsurance

Reinsurance means that the company (the cedent or first insurer) insures a part of the risk at another insurance company (the reinsurer). The purposes of reinsurance are to reduce risk and/or to reduce the risk volume of the company.

We start by formulating the basic concepts within the framework of a single risk $X \geq 0$. A reinsurance arrangement is then defined in terms of a function $r(x)$ with the property $0 \leq r(x) \leq x$. Here $r(x)$ is the amount of the claim $x$ to be paid by the reinsurer and $s(x)=x-r(x)$ the amount to be paid by the cedent. The function $s(x)$ is referred to as the retention function. The most common examples are the following two:

• Proportional reinsurance $r(x)=(1-\theta) x, s(x)=\theta x$ for some $\theta \in(0,1)$. Also called quota share reinsurance.
• Stop-loss reinsurance $r(x)=(x-b)^{+}$for some $b \in(0, \infty)$, referred to as the retention limit. The retention function is $x \wedge b$.

Concerning terminology, note that in the actuarial literature the stop-loss transform of $F(x)=\mathbb{P}(X \leq x)$ (or, equivalently, of $X)$ is defined as the function
$$b \mapsto \mathbb{E}(X-b)^{+}=\int_b^{\infty}(x-b) F(\mathrm{~d} x)=\int_b^{\infty} \bar{F}(x) \mathrm{d} x$$
(the last equality follows by integration by parts, see formula (A.1.1) in the Appendix). It shows up in a number of different contexts, see e.g. Sect. VIII.2.1, where some of its main properties are listed.

The risk $X$ is often the aggregate claims amount $A=\sum_1^N V_i$ in a certain line of business during one year; one then talks of global reinsurance. However, reinsurance may also be done locally, i.e. at the level of individual claims. Then, if $N$ is the number of claims during the period and $V_1, V_2, \ldots$ their sizes, then the amounts paid by reinsurer, resp. the cedent, are
$$\sum_{i=1}^N r\left(V_i\right), \text { resp. } \sum_{i=1}^N s\left(V_i\right)$$

## 金融代写|风险理论代写Risk theory代考|The Poisson Process

By a (simple) point process $\mathscr{N}$ on a set $\Omega \subseteq \mathbb{R}^d$ we understand a random collection of points in $\Omega$ [simple means that there are no multiple points]. We are almost exclusively concerned with the case $\Omega=[0, \infty)$. The point process can then be specified by the sequence $T_1, T_2, \ldots$ of interarrival times such that the points are $T_1, T_1+T_2, \ldots$ The associated counting process ${N(t)}_{t \geq 0}$ is defined by letting $N(t)$ be the number of points in $[0, t]$. Write
$$\mathscr{N}(s, t]=N(t)-N(s)=#\left{n: s<T_1+\cdots+T_n \leq t\right}$$
for the increment of ${N(t)}$ over $(s, t]$ or equivalently the number of points in $(s, t]$.
Definition 5.2 $\mathscr{N}$ is a Poisson process on $[0, \infty)$ with rate $\lambda$ if ${N(t)}$ has independent increments and $N(t)-N(s)$ has a Poisson $(\lambda(t-s))$ distribution for $s<t$.

Here independence of increments means independence of increments over disjoint intervals.

It is not difficult to extend the reasoning hehind example 1) ahnve to conclude. that for a large insurance portfolio, the number of claims in disjoint time intervals are independent Poisson r.v.s, and so the times of occurrences of claims form a Poisson process. There are, however, different ways to approach the Poisson process. In particular, the infinitesimal view in part (iii) of the following result will prove useful for many of our purposes.

## 金融代写|风险理论代写Risk theory代考|Reinsurance

• 比例再保险 $r(x)=(1-\theta) x, s(x)=\theta x$ 对于一些 $\theta \in(0,1)$. 也称为配额份额再保险。
• 止损再保险 $r(x)=(x-b)^{+}$对于一些 $b \in(0, \infty)$ ，称为保留限制。保留函数为 $x \wedge b$.
关于术语，请注意在精算文献中，止损变换 $F(x)=\mathbb{P}(X \leq x)$ (或者，等效地， $X$ )被定义为函数
$$b \mapsto \mathbb{E}(X-b)^{+}=\int_b^{\infty}(x-b) F(\mathrm{~d} x)=\int_b^{\infty} \bar{F}(x) \mathrm{d} x$$
（最后一个等式后面是分部积分，见附录中的公式 (A.1.1) )。它出现在许多不同的上下文中，例如参见 Sect。 VIII.2.1，其中列出了它的一些主要属性。
风险 $X$ 通常是总索赔额 $A=\sum_1^N V_i$ 一年内从事某项业务；然后有人谈到全球再保险。然而，再保险也可以在当 地进行，即在个人索赔层面。那么，如果 $N$ 是该期间的索赔数量，并且 $V_1, V_2, \ldots$ 他们的规模，然后是再保险公 司支付的金额，resp。分出商是
$$\sum_{i=1}^N r\left(V_i\right), \text { resp. } \sum_{i=1}^N s\left(V_i\right)$$

## 金融代写|风险理论代写Risk theory代考|The Poisson Process

Poisson rvs，因此理赔发生的次数构成一个 Poisson 过程。然而，有不同的方法来处理泊松过程。特别是，以下 结果的 (iii) 部分中的无穷小视图将证明对我们的许多目的有用。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|风险理论代写Risk theory代考|STAT553

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory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。我们的专家在代写风险理论Risk theory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|风险理论代写Risk theory代考|Actuarial Versus Financial Pricing

The last decades have seen the areas of insurance mathematics and mathematical finance coming closer together. One reason is the growing linking of pay-outs of life insurances and pension plans to the current value of financial products, another that certain financial products have been designed especially to be of interest for the insurance industry (see below). Nevertheless, some fundamental differences remain, and the present section aims at explaining some of these, with particular emphasis on the principles for pricing insurance products, resp. financial products.

In insurance, expected values play a major role. For example, let a claim $X \geq 0$ be the amount of money the insurance company has to pay out for a fire insurance on a given house next year (of course, $\mathbb{P}(X=0)$ is close to 1 !). The insurance company then ideally charges $H(X)=\mathbb{E} X$ in premium plus some loading, that is, an extra amount to cover administration costs, profit, risk etc. (different rules for the form of this loading are discussed in Sect. 3). The philosophy behind this is that charging premiums smaller than expected values in the long run results in an overall loss. This is a consequence of the law of large numbers (LLN). In its simplest form it says that if the company faces $n$ i.i.d. claims $X_1, \ldots, X_n$ all distributed as $X$, then the aggregate claim amount $A=X_1+\cdots+X_n$ is approximately $n \mathbb{E} X$ for $n$ large. Therefore, if the premium $H$ is smaller than $\mathbb{E} X$, then with high probability the total premiums $n H$ are not sufficient to cover the total aggregate claims $A$.

This argument carries far beyond this setting of i.i.d. claims, which is of course oversimplified: even in fire insurance, individual houses are different (the area varies, a house may have different types of heating, thatched roof or tiles, etc), and the company typically has many other lines of business such as car insurance, accident insurance, life insurance, etc. Let the claims be $X_1, X_2, \ldots$ Then the asymptotics
$$\frac{X_1+\cdots+X_n}{\mathbb{E} X_1+\cdots+\mathbb{E} X_n} \rightarrow 1$$
holds under weak conditions. For example, the following elementary result is sufficiently general to cover a large number of insurance settings

The standard setting for discussing premium calculation in the actuarial literature is in terms of a single risk $X \geq 0$ and does not involve portfolios, stochastic processes, etc. Here $X$ is an r.v. representing the random payment (possibly 0 ) to be made from the insurance company to the insured. A premium rule is then a $\lfloor 0, \infty)$-valued function $H$ of the distribution of $X$, often written $H(X)$, such that $H(X)$ is the premium to be paid, i.e. the amount for which the company is willing to insure the given risk. From an axiomatic point of view, the concept of premium rules is closely related to that of risk measures, to which we return in Sect. X.1.

The standard premium rules discussed in the literature (not necessarily the same as those used in practice!) are the following:

• The net premium principle $H(X)=\mathbb{E} X$ (also called the equivalence principle). As follows from a suitable version of the CLT that this principle will lead to a substantial loss if many independent risks are insured. This motivates that a loading should be added, as in the next principles:
• The expected value principle $H(X)=(1+\eta) \mathbb{E} X$, where $\eta$ is a specified safety loading. For $\eta=0$, we are back to the net premium principle. A criticism of the expected value principle is that it does not take into account the variability of $X$. This leads to:
• The variance principle $H(X)=\mathbb{E} X+\eta \operatorname{Var}(X)$. A modification (motivated by $\mathbb{E} X$ and $\operatorname{Var}(X)$ not having the same dimension) is
• The standard deviation principle $H(X)=\mathbb{E} X+\eta \sqrt{\operatorname{Var}(X)}$.

## 金融代写|风险理论代写Risk theory代考|Actuarial Versus Financial Pricing

$$\frac{X_1+\cdots+X_n}{\mathbb{E} X_1+\cdots+\mathbb{E} X_n} \rightarrow 1$$

• 净保费原则 $H(X)=\mathbb{E} X$ (也称为等价原则) 。从一个合适的 $\mathrm{CLT}$ 版本可以看出，如果许多独立风险被投 保，这一原则将导致重大损失。这促使应该添加负载，如下面的原则:
• 期望值原则 $H(X)=(1+\eta) \mathbb{E} X$ ，在哪里 $\eta$ 是指定的安全载荷。为了 $\eta=0$ ，我们又回到了净溢价原则。 对期望值原则的一个批评是它没有考虑到 $X$. 这将导致:
• 方差原理 $H(X)=\mathbb{E} X+\eta \operatorname{Var}(X)$. 修改 (动机是 $\mathbb{E} X$ 和 $\operatorname{Var}(X)$ 尺寸不同) 是
• 标准差原则 $H(X)=\mathbb{E} X+\eta \sqrt{\operatorname{Var}(X)}$.

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|CONSTRAINED OPTIMIZATION

In constructing optimization problems solving practical issues, it is very often the case that certain constraints need to be imposed in order for the optimal solution to make practical sense. For example, long-only portfolio optimization problems require that the portfolio weights, which represent the variables in optimization, should be nonnegative and should sum up to one. According to the notation in this chapter, this corresponds to a problem of the type,
$$\begin{array}{rl} \min {x} & f(x) \ \text { subject to } & x^{\prime} e=1 \ & x \geq 0, \end{array}$$ where: $f(x)$ is the objective function. $e \in \mathbb{R}^{n}$ is a vector of ones, $e=(1, \ldots, 1)$. $x^{\prime} e$ equals the sum of all components of $x, x^{\prime} e=\sum{i}^{n} x_{i}$.
$x \geq 0$ means that all components of the vector $x \in \mathbb{R}^{n}$ are nonnegative.
In problem (2.10), we are searching for the minimum of the objective function by varying $x$ only in the set
$$\mathbf{X}=\left{x \in \mathbb{R}^{n}: \begin{array}{l} x^{\prime} e=1 \ x \geq 0 \end{array}\right}$$
which is also called the set of feasible points or the constraint set. A more compact notation, similar to the notation in the unconstrained problems, is sometimes used,
$$\min _{x \in \mathrm{X}} f(x)$$
where $\mathbf{X}$ is defined in equation (2.11).
We distinguish between different types of optimization problems depending on the assumed properties for the objective function and the constraint set. If the constraint set contains only equalities, the problem is easier to handle analytically. In this case, the method of Lagrange multipliers is applied. For more general constraint sets, when they are formed

by both equalities and inequalities, the method of Lagrange multipliers is generalized by the Karush-Kuhn-Tucker conditions (KKT conditions). Like the first-order conditions we considered in unconstrained optimization problems, none of the two approaches leads to necessary and sufficient conditions for constrained optimization problems without further assumptions. One of the most general frameworks in which the KKT conditions are necessary and sufficient is that of convex programming. We have a convex programing problem if the objective function is a convex function and the set of feasible points is a convex set. As important subcases of convex optimization, linear programming and convex quadratic programming problems are considered.

In this section, we describe first the method of Lagrange multipliers, which is often applied to special types of mean-variance optimization problems in order to obtain closed-form solutions. Then we proceed with convex programming that is the framework for reward-risk analysis. The mentioned applications of constrained optimization problems is covered in Chapters 8,9 , and 10 .

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

Consider the following optimization problem in which the set of feasible points is defined by a number of equality constraints,
$$\begin{array}{rl} \min {x} & f(x) \ \text { subject to } & b{1}(x)=0 \ & b_{2}(x)=0 \ \cdots \ & b_{k}(x)=0 \end{array}$$
The functions $h_{i}(x), i=1, \ldots, k$ build up the constraint set. Note that even though the right-hand side of the equality constraints is zero in the classical formulation of the problem given in equation $(2.12)$, this is not restrictive. If in a practical problem the right-hand side happens to be different than zero, it can be equivalently transformed, for example,
$$\left{x \in \mathbb{R}^{n}: v(x)=c\right} \quad \Longleftrightarrow \quad\left{x \in \mathbb{R}^{n}: h_{1}(x)=v(x)-c=0\right} .$$
In order to illustrate the necessary condition for optimality valid for (2.12), let us consider the following two-dimensional example:
\begin{aligned} \min _{x \in \mathbb{R}^{2}} & \frac{1}{2} x^{\prime} C x \ \text { subject to } & x^{\prime} e=1, \end{aligned}

where the matrix is
$$C=\left(\begin{array}{cc} 1 & 0.4 \ 0.4 & 1 \end{array}\right) .$$
The objective function is a quadratic function and the constraint set contains one linear equality. In Chapter 8, we see that the mean-variance optimization problem in which short positions are allowed is very similar to (2.13). The surface of the objective function and the constraint are shown on the top plot in Figure 2.7. The black line on the surface shows the function values of the feasible points. Geometrically, solving problem (2.13) reduces to finding the lowest point of the black curve on the surface. The contour lines shown on the bottom plot in Figure $2.7$ imply that the feasible point yielding the minimum of the objective function is where a contour line is tangential to the line defined by the equality constraint. On the plot, the tangential contour line and the feasible points are in bold. The black dot indicates the position of the point in which the objective function attains its minimum subject to the constraints.

Even though the example is not general in the sense that the constraint set contains one linear rather than a nonlinear equality, the same geometric intuition applies in the nonlinear case. The fact that the minimum is attained where a contour line is tangential to the curve defined by the nonlinear equality constraints in mathematical language is expressed in the following way: The gradient of the objective function at the point yielding the minimum is proportional to a linear combination of the gradients of the functions defining the constraint set. Formally, this is stated as
$$\nabla f\left(x^{0}\right)-\mu_{1} \nabla h_{1}\left(x^{0}\right)-\cdots-\mu_{k} \nabla h_{k}\left(x^{0}\right)=0 .$$
where $\mu_{i}, i=1, \ldots, k$ are some real numbers called Lagrange multipliers and the point $x^{0}$ is such that $f\left(x^{0}\right) \leq f(x)$ for all $x$ that are feasible. Note that if there are no constraints in the problem, then (2.14) reduces to the first-order condition we considered in unconstrained optimization. Therefore, the system of equations behind (2.14) can be viewed as a generalization of the first-order condition in the unconstrained case.

The method of Lagrange multipliers basically associates a function to the problem in $(2.12)$ such that the first-order condition for unconstrained optimization for that function coincides with (2.14). The method of Lagrange multiplier consists of the following steps.

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

The general form of convex programming problems is the following:
$$\begin{array}{rl} \min {x} & f(x) \ \text { subject to } & g{i}(x) \leq 0, \quad i=1, \ldots, m \ & h_{j}(x)=0, \quad j=1, \ldots, k, \end{array}$$

where:
$f(x)$ is a convex objective function.
$g_{1}(x), \ldots, g_{m}(x)$ are convex functions defining the inequality constraints. $h_{1}(x), \ldots, h_{k}(x)$ are affine functions defining the equality constraints.
Generally, without the assumptions of convexity, problem $(2.17)$ is more involved than $(2.12)$ because besides the equality constraints, there are inequality constraints. The KKT condition, generalizing the method of Lagrange multipliers, is only a necessary condition for optimality in this case. However, adding the assumption of convexity makes the KKT condition necessary and sufficient.

Note that, similar to problem (2.12), the fact that the right-hand side of all constraints is zero is nonrestrictive. The limits can be arbitrary real numbers.
Consider the following two-dimensional optimization problem;
\begin{aligned} \min {\substack{x \in \mathbb{R}^{2}}} & \frac{1}{2} x^{\prime} \mathrm{C} x \ \text { subject to } &\left(x{1}+2\right)^{2}+\left(x_{2}+2\right)^{2} \leq 3 \end{aligned}
in which
$$C=\left(\begin{array}{cc} 1 & 0.4 \ 0.4 & 1 \end{array}\right) \text {. }$$
The objective function is a two-dimensional convex quadratic function and the function in the constraint set is also a convex quadratic function. In fact, the boundary of the feasible set is a circle with a radius of $\sqrt{3}$ centered at the point with coordinates $(-2,-2)$. The top plot in Figure $2.8$ shows the surface of the objective function and the set of feasible points. The shaded part on the surface indicates the function values of all feasible points. In fact, solving problem (2.18) reduces to finding the lowest point on the shaded part of the surface. The bottom plot shows the contour lines of the objective function together with the feasible set that is in gray. Geometrically, the point in the feasible set yielding the minimum of the objective function is positioned where a contour line only touches the constraint set. The position of this point is marked with a black dot and the tangential contour line is given in bold.

Note that the solution points of problems of the type $(2.18)$ can happen to be not on the boundary of the feasible set but in the interior. For example, suppose that the radius of the circle defining the boundary of the feasible set in $(2.18)$ is a larger number such that the point $(0,0)$ is inside the feasible

set. Then, the point $(0,0)$ is the solution to problem $(2.18)$ because at this point the objective function attains its global minimum.

In the two-dimensional case, when we can visualize the optimization problem, geometric reasoning guides us to finding the optimal solution point. In a higher dimensional space, plots cannot be produced and we rely on the analytic method behind the KKT conditions.

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

X≥0表示向量的所有分量X∈Rn是非负的。

\mathbf{X}=\left{x \in \mathbb{R}^{n}: \begin{array}{l} x^{\prime} e=1 \ x \geq 0 \end{array}\对}\mathbf{X}=\left{x \in \mathbb{R}^{n}: \begin{array}{l} x^{\prime} e=1 \ x \geq 0 \end{array}\对}

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

\left{x \in \mathbb{R}^{n}: v(x)=c\right} \quad \Longleftrightarrow \quad\left{x \in \mathbb{R}^{n}: h_{1 }(x)=v(x)-c=0\right} 。\left{x \in \mathbb{R}^{n}: v(x)=c\right} \quad \Longleftrightarrow \quad\left{x \in \mathbb{R}^{n}: h_{1 }(x)=v(x)-c=0\right} 。

C=(10.4 0.41).

∇F(X0)−μ1∇H1(X0)−⋯−μķ∇Hķ(X0)=0.

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

F(X)是一个凸目标函数。
G1(X),…,G米(X)是定义不等式约束的凸函数。H1(X),…,Hķ(X)是定义等式约束的仿射函数。

C=(10.4 0.41).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|UNCONSTRAINED OPTIMIZATION

When there are no constraints imposed on the set of feasible solutions, we have an unconstrained optimization problem. Thus, the goal is to maximize or to minimize the objective function with respect to the function arguments without any limits on their values. We consider directly the $n$-dimensional case; that is, the domain of the objective function $f$ is the $n$-dimensional space and the function values are real numbers, $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. Maximization is denoted by
$$\max f\left(x_{1}, \ldots, x_{n}\right)$$
and minimization by
$$\min f\left(x_{1}, \ldots, x_{n}\right)$$
A more compact form is commonly used, for example
$$\min {x \in \mathbb{R}^{n}} f(x)$$ denotes that we are searching for the minimal value of the function $f(x)$ by varying $x$ in the entire $n$-dimensional space $\mathbb{R}^{n}$. A solution to equation (2.1) is a value of $x=x^{0}$ for which the minimum of $f$ is attained, $$f{0}=f\left(x^{0}\right)=\min {x \in \mathbb{R}^{\pi}} f(x) .$$ Thus, the vector $x{0}$ is such that the function takes a larger value than $f_{0}$ for any other vector $x$,
$$f\left(x^{0}\right) \leq f(x), x \in \mathbb{R}^{n}$$
Note that there may be more than one vector $x^{0}$ satisfying the inequality in equation (2.2) and, therefore, the argument for which $f_{0}$ is achieved may not be unique. If (2.2) holds, then the function is said to attain its global minimum at $x^{0}$. If the inequality in $(2.2)$ holds for $x$ belonging only to a small neighborhood of $x^{0}$ and not to the entire space $\mathbb{R}^{n}$, then the objective function is said to have a local minimum at $x^{0}$. This is usually denoted by
$$f\left(x^{0}\right) \leq f(x)$$

for all $x$ such that $\left|x-x^{0}\right|_{2}<\epsilon$ where $\left|x-x^{0}\right|_{2}$ stands for the Euclidean distance between the vectors $x$ and $x^{0}$,
$$\left|x-x^{0}\right|_{2}=\sqrt{\sum_{i=1}^{n}\left(x_{i}-x_{i}^{0}\right)^{2}}$$
and $\epsilon$ is some positive number. A local minimum may not be global as there may be vectors outside the small neighborhood of $x_{0}$ for which the objective function attains a smaller value than $f\left(x_{0}\right)$. Figure $2.2$ shows the graph of a function with two local maxima, one of which is the global maximum.

There is a connection between minimization and maximization. Maximizing the objective function is the same as minimizing the negative of the objective function and then changing the sign of the minimal value,
$$\max {x \in \mathbb{R}^{n}} f(x)=-\min {x \in \mathbb{R}^{n}}[-f(x)] .$$
This relationship is illustrated in Figure 2.1. As a consequence, problems for maximization can be stated in terms of function minimization and vice versa.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Minima and Maxima of a Differentiable Function

If the second derivatives of the objective function exist, then its local maxima and minima, often called generically local extrema, can be characterized.

Denote by $\nabla f(x)$ the vector of the first partial derivatives of the objective function evaluated at $x$,
$$\nabla f(x)=\left(\frac{\partial f(x)}{\partial x_{1}}, \ldots, \frac{\partial f(x)}{\partial x_{n}}\right) .$$
This vector is called the function gradient. At each point $x$ of the domain of the function, it shows the direction of greatest rate of increase of the function in a small neighborhood of $x$. If for a given $x$, the gradient equals a vector of zeros,
$$\nabla f(x)=(0, \ldots, 0)$$
then the function does not change in a small neighborhood of $x \in \mathbb{R}^{n}$. It turns out that all points of local extrema of the objective function are characterized by a zero gradient. As a result, the points yielding the local extrema of the objective function are among the solutions of the system of equations,
\mid \begin{aligned} &\frac{\partial f(x)}{\partial x_{1}}=0 \ &\cdots \ &\frac{\partial f(x)}{\partial x_{n}}=0 \end{aligned}
The system of equation $(2.3)$ is often referred to as representing the first-order condition for the objective function extrema. However, it is only a necessary condition; that is, if the gradient is zero at a given point in the $n$-dimensional space, then this point may or may not be a point of a local extremum for the function. An illustration is given in Figure 2.2. The top plot shows the graph of a two-dimensional function and the bottom plot contains the contour lines of the function with the gradient calculated at a grid of points. There are three points marked with a black dot that have a zero gradient. The middle point is not a point of a local maximum even though it has a zero gradient. This point is called a saddle point since the graph resembles the shape of a saddle in a neighborhood of it. The left and the right points are where the function has two local maxima corresponding to the two peaks visible on the top plot. The right peak is a local maximum that is not the global one and the left peak represents the global maximum.

This example demonstrates that the first-order conditions are generally insufficient to characterize the points of local extrema. The additional condition that identifies which of the zero-gradient points are points

of local minimum or maximum is given through the matrix of second derivatives,
$$H=\left(\begin{array}{cccc} \frac{\partial^{2} f(x)}{\partial x_{1}^{2}} & \frac{\partial^{2} f(x)}{\partial x_{1} \partial x_{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{1} \partial x_{e}} \ \frac{\partial^{2} f(x)}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f(x)}{\partial x_{2}^{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{2} \partial x_{n}} \ \vdots & \vdots & \ddots & \vdots \ \frac{\partial^{2} f(x)}{\partial x_{n} \partial x_{1}} & \frac{\partial^{2} f(x)}{\partial x_{n} \partial x_{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{n}^{2}} \end{array}\right)$$

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

In section 2.2.1, we demonstrated that the first-order conditions are insufficient in the general case to describe the local extrema. However, when certain assumptions are made for the objective function, the first-order conditions can become sufficient. Furthermore, for certain classes of functions, the local extrema are necessarily global. Therefore, solving the first-order conditions, we obtain the global extremum.

A general class of functions with nice optimal properties is the class of convex functions. Not only are the convex functions easy to optimize but they have also important application in risk management. In Chapter 6, we discuss general measures of risk. It turns out that the property which guarantees that diversification is possible appears to be exactly the convexity

property. As a consequence, a measure of risk is necessarily a convex functional. ${ }^{1}$

Precisely, a function $f(x)$ is called a convex function if it satisfies the property: For a given $\alpha \in[0,1]$ and all $x^{1} \in \mathbb{R}^{n}$ and $x^{2} \in \mathbb{R}^{n}$ in the function domain,
$$f\left(\alpha x^{1}+(1-\alpha) x^{2}\right) \leq \alpha f\left(x^{1}\right)+(1-\alpha) f\left(x^{2}\right)$$
The definition is illustrated in Figure 2.3. Basically, if a function is convex, then a straight line connecting any two points on the graph lies “above” the graph of the function.

There is a related term to convex functions. A function $f$ is called concave if the negative of $f$ is convex. In effect, a function is concave if it

satisfies the property: For a given $\alpha \in[0,1]$ and all $x^{1} \in \mathbb{R}^{n}$ and $x^{2} \in \mathbb{R}^{n}$ in the function domain,
$$f\left(\alpha x^{1}+(1-\alpha) x^{2}\right) \geq \alpha f\left(x^{1}\right)+(1-\alpha) f\left(x^{2}\right) .$$
We use convex and concave functions in the discussion of the efficient frontier in Chapter 8 .

If the domain $D$ of a convex function is not the entire space $\mathbb{R}^{n}$, then the set D satisfies the property,
$$\alpha x^{1}+(1-\alpha) x^{2} \in D$$
where $x^{1} \in D, x^{2} \in D$, and $0 \leq \alpha \leq 1$. The sets that satisfy equation (2.6) are called convex sets. Thus, the domains of convex (and concave) functions should be convex sets. Geometrically, a set is convex if it contains the straight line connecting any two points belonging to the set. Rockafellar (1997) provides detailed information on the implications of convexity in optimization theory.
We summarize several important properties of convex functions:

• Not all convex functions are differentiable. If a convex function is two times continuously differentiable, then the corresponding Hessian defined in equation $(2.4)$ is a positive semidefinite matrix. ${ }^{2}$
• All convex functions are continuous if considered in an open set.
• The sublevel sets
$$L_{c}={x: f(x) \leq c}$$
where $c$ is a constant, are convex sets if $f$ is a convex function. The converse is not true in general. Section $2.2 .3$ provides more information about non-convex functions with convex sublevel sets.
• The local minima of a convex function are global. If a convex function $f$ is twice continuously differentiable, then the global minimum is obtained in the points solving the first-order condition,
$$\nabla f(x)=0 .$$
• A sum of convex functions is a convex function:
$$f(x)=f_{1}(x)+f_{2}(x)+\ldots+f_{k}(x)$$
is a convex function if $f_{i}, i=1, \ldots, k$ are convex functions.

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

F(X0)≤F(X),X∈Rn

F(X0)≤F(X)

|X−X0|2=∑一世=1n(X一世−X一世0)2

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Minima and Maxima of a Differentiable Function

∇F(X)=(∂F(X)∂X1,…,∂F(X)∂Xn).

∇F(X)=(0,…,0)

∣∂F(X)∂X1=0 ⋯ ∂F(X)∂Xn=0

H=(∂2F(X)∂X12∂2F(X)∂X1∂X2⋯∂2F(X)∂X1∂X和 ∂2F(X)∂X2∂X1∂2F(X)∂X22⋯∂2F(X)∂X2∂Xn ⋮⋮⋱⋮ ∂2F(X)∂Xn∂X1∂2F(X)∂Xn∂X2⋯∂2F(X)∂Xn2)

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

F(一种X1+(1−一种)X2)≤一种F(X1)+(1−一种)F(X2)

F(一种X1+(1−一种)X2)≥一种F(X1)+(1−一种)F(X2).

• 并非所有凸函数都是可微的。如果一个凸函数是两次连续可微的，则等式中定义的相应 Hessian(2.4)是一个半正定矩阵。2
• 如果在开集中考虑，所有凸函数都是连续的。
• 子级集
大号C=X:F(X)≤C
在哪里C是一个常数，如果是凸集F是一个凸函数。反之亦然。部分2.2.3提供有关具有凸子水平集的非凸函数的更多信息。
• 凸函数的局部最小值是全局的。如果一个凸函数F是两次连续可微的，则在求解一阶条件的点中获得全局最小值，
∇F(X)=0.
• 凸函数之和是一个凸函数：
F(X)=F1(X)+F2(X)+…+Fķ(X)
是一个凸函数，如果F一世,一世=1,…,ķ是凸函数。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Chebyshev’s Inequality

Some of the topics discussed in the book concern a setting in which we are not aware of the particular distribution of a random variable or the particular joint probability distribution of a pair of random variables. In such cases, the analysis may require us to resort to general arguments based on certain general inequalities from the theory of probability. In this section, we give an account of such inequalities and provide illustration where possible.

Chebyshev’s inequality provides a way to estimate the approximate probability of deviation of a random variable from its mean. Its most simple form concerns positive random variables.

Suppose that $X$ is a positive random variable, $X>0$. The following inequality is known as Chebyshev’s inequality,
$$P(X \geq \epsilon) \leq \frac{E X}{\epsilon},$$
where $\epsilon>0$. In this form, equation $(1.5)$ can be used to estimate the probability of observing a large observation by means of the mathematical expectation and the level $\epsilon$. Chebyshev’s inequality is rough as demonstrated geometrically in the following way. The mathematical expectation of a positive continuous random variable admits the representation,
$$E X=\int_{0}^{\infty} P(X \geq x) d x,$$

which means that it equals the area closed between the distribution function and the upper limit of the distribution function. This area is illustrated in Figure $1.9$ as the shaded area above the distribution function. On the other hand, the quantity $\epsilon P(X \geq \epsilon)=\epsilon\left(1-F_{X}(x)\right)$ is equal to the area of the rectangle in the upper-left corner of Figure $1.9$. In effect, the inequality
$$\epsilon P(X \geq \epsilon) \leq E X$$
admits the following geometric interpretation-the area of the rectangle is smaller than the shaded area in Figure 1.9.

For an arbitrary random variable, Chebychev’s inequality takes the form
$$P\left(|X-E X| \geq \epsilon \sigma_{X}\right) \leq \frac{1}{\epsilon^{2}}$$
where $\sigma_{X}$ is the standard deviation of $X$ and $\epsilon>0$. We use Chebyshev’s inequality in Chapter 6 in the discussion of dispersion measures.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Frechet-Hoeffding Inequality

Consider an $n$-dimensional random vector $Y$ with a distribution function $F_{Y}\left(y_{1}, \ldots, y_{n}\right)$. Denote by
$$W\left(y_{1}, \ldots, y_{n}\right)=\max \left(F_{Y_{1}}\left(y_{1}\right)+\cdots+F_{Y_{n}}\left(y_{n}\right)+1-n, 0\right)$$

and by
$$M\left(y_{1}, \ldots, y_{n}\right)=\min \left(F_{Y_{1}}\left(y_{1}\right), \ldots, F_{Y_{n}}\left(y_{n}\right)\right)$$
in which $F_{Y_{i}}\left(y_{i}\right)$ stands for the distribution function of the $i$-th marginal. The following inequality is known as Fréchet-Hoeffding inequality,
$$W\left(y_{1}, \ldots, y_{n}\right) \leq F_{Y}\left(y_{1}, \ldots, y_{n}\right) \leq M\left(y_{1}, \ldots, y_{n}\right) .$$
The quantities $W\left(y_{1}, \ldots, y_{n}\right)$ and $M\left(y_{1}, \ldots, y_{n}\right)$ are also called the Fréchet lower bound and the Fréchet upper bound. We apply FréchetHoeffding inequality in the two-dimensional case in Chapter 3 when discussing minimal probability metrics.

Since copulas are essentially probability distributions defined on the unit hypercube, Fréchet-Hoeffding inequality holds for them as well. In this case, it has a simpler form because the marginal distributions are uniform. The lower and the upper Fréchet bounds equal
and
$$\begin{gathered} W\left(u_{1}, \ldots, u_{n}\right)=\max \left(u_{1}+\cdots+u_{n}+1-n, 0\right) \ M\left(u_{1}, \ldots, u_{n}\right)=\min \left(u_{1}, \ldots, u_{n}\right) \end{gathered}$$
respectively. Fréchet-Hoeffding inequality is given by
$$W\left(u_{1}, \ldots, u_{n}\right) \leq C\left(u_{1}, \ldots, u_{n}\right) \leq M\left(u_{1}, \ldots, u_{n}\right) .$$
In the two-dimensional case, the inequality reduces to
$$\max \left(u_{1}+u_{2}-1,0\right) \leq C\left(u_{1}, u_{2}\right) \leq \min \left(u_{1}, u_{2}\right) .$$
In the two-dimensional case only, the lower Fréchet bound, sometimes referred to as the minimal copula, represents perfect negative dependence between the two random variables. In a similar way, the upper Fréchet bound, sometimes referred to as the maximal copula, represents perfect positive dependence between the two random variables.

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

We considered a number of concepts from probability theory that will be used in later chapters in this book. We discussed the notions of a random variable and a random vector. We considered one-dimensional and multidimensional probability density and distributions functions, which completely characterize a given random variable or random vector. We discussed statistical moments and quantiles, which represent certain characteristics of a random variable, and the sample moments which provide a way of estimating the corresponding characteristics from historical data. In the multidimensional case, we considered the notion of dependence between the components of a random vector. We discussed the covariance matrix versus the more general concept of a copula function. Finally, we described two probabilistic inequalities, Chebychev’s inequality and Fréchet-Hoeffding inequality.

ε磷(X≥ε)≤和X

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|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))

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Conditional Probability

A useful concept in understanding the relationship between multiple random variables is that of conditional probability. Consider the returns on the stocks of two companies in one and the same industry. The future return $X$ on the stocks of company 1 is not unrelated to the future return $Y$ on the stocks of company 2 because the future development of the two companies is driven to some extent by common factors since they are in one and the same industry. It is a reasonable question to ask, what is the probability that the future return $X$ is smaller than a given percentage, e.g. $X \leq-2 \%$, on condition that $Y$ realizes a huge loss, e.g. $Y \leq-10 \%$ ? Essentially, the conditional probability is calculating the probability of an event provided that another event happens. If we denote the first event by $A$ and the second event by $B$, then the conditional probability of $A$ provided that $B$ happens, denoted by $P(A \mid B)$, is given by the formula,
$$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$$
which is also known as the Bayes formula. According to the formula, we divide the probability that both events $A$ and $B$ occur simultaneously, denoted by $A \cap B$, by the probability of the event $B$. In the two-stock example, the formula is applied in the following way,
$$P(X \leq-2 \% \mid Y \leq-10 \%)=\frac{P(X \leq-2 \%, Y \leq-10 \%)}{P(Y \leq-10 \%)}$$
Thus, in order to compute the conditional probability, we have to be able to calculate the quantity
$$P(X \leq-2 \%, Y \leq-10 \%)$$
which represents the joint probability of the two events.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Definition of Joint Probability Distributions

A portfolio or a trading position consists of a collection of financial assets. Thus, portfolio managers and traders are interested in the return on a portfolio or a trading position. Consequently, in real-world applications, the interest is in the joint probability distribution or joint distribution of more than one random variable. For example, suppose that a portfolio consists of a position in two assets, asset 1 and asset 2 . Then there will be a probability distribution for (1) asset 1 , (2) asset 2, and (3) asset 1 and asset 2. The first two distributions are referred to as the marginal probability distributions or marginal distributions. The distribution for asset 1 and asset 2 is called the joint probability distribution.

Like in the univariate case, there is a mathematical connection between the probability distribution $P$, the cumulative distribution function $F$, and the density function $f$ of a multivariate random variable (also called a random vector) $X=\left(X_{1}, \ldots, X_{n}\right)$. The formula looks similar to the equation we presented in the previous chapter showing the mathematical connection between a probability density function, a probability distribution, and a cumulative distribution function of some random variable $X$ :
\begin{aligned} P\left(X_{1} \leq t_{1}, \ldots, X_{n} \leq t_{n}\right) &=F_{X}\left(t_{1}, \ldots, t_{n}\right) \ &=\int_{-\infty}^{t_{1}} \ldots \int_{-\infty}^{t_{n}} f_{X}\left(x_{1}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n} \end{aligned}
The formula can be interpreted as follows. The joint probability that the first random variable realizes a value less than or equal to $t_{1}$ and the second less than or equal to $t_{2}$ and so on is given by the cumulative distribution function $F$. The value can be obtained by calculating the volume under the density function $f$. Because there are $n$ random variables, we have now $n$ arguments for both functions: the density function and the cumulative distribution function.

It is also possible to express the density function in terms of the distribution function by computing sequentially the first-order partial derivatives of the distribution function with respect to all variables,
$$f_{X}\left(x_{1}, \ldots, x_{n}\right)=\frac{\partial^{n} F_{X}\left(x_{1}, \ldots, x_{n}\right)}{\partial x_{1} \ldots \partial x_{n}}$$

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Dependence of Random Variables

Typically, when considering multivariate distributions, we are faced with inference between the distributions; that is, large values of one random variable imply large values of another random variable or small values of a third random variable. If we are considering, for example, $X_{1}$, the height of a randomly chosen U.S. citizen, and $X_{2}$, the weight of this citizen, then large values of $X_{1}$ tend to result in large values of $X_{2}$. This property is denoted as the dependence of random variables and a powerful concept to measure dependence will be introduced in a later section on copulas.

The inverse case of no dependence is denoted as stochastic independence. More precisely, two random variables are independently distributed if and only if their joint distribution given in terms of the joint cumulative distribution function $F$ or the joint density function $f$ equals the product of their marginal distributions:
\begin{aligned} &F_{X}\left(x_{1}, \ldots, x_{n}\right)=F_{X_{1}}\left(x_{1}\right) \ldots F_{X_{n}}\left(x_{n}\right) \ &f_{X}\left(x_{1}, \ldots, x_{n}\right)=f_{X_{1}}\left(x_{1}\right) \ldots f_{X_{n}}\left(x_{n}\right) \end{aligned}
In the special case of $n=2$, we can say that two random variables are said to be independently distributed, if knowing the value of one random variable does not provide any information about the other random variable. For instance, if we assume in the example developed in section 1.6.1 that the two events $X \leq-2 \%$ and $Y \leq-10 \%$ are independent, then the conditional probability in equation (1.1) equals
\begin{aligned} P(X \leq-2 \% \mid Y \leq-10 \%) &=\frac{P(X \leq-2 \%) P(Y \leq-10 \%)}{P(Y \leq-10 \%)} \ &=P(X \leq-2 \%) \end{aligned}
Indeed, under the assumption of independence, the event $Y \leq-10 \%$ has no influence on the probability of the other event.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Definition of Joint Probability Distributions

FX(X1,…,Xn)=∂nFX(X1,…,Xn)∂X1…∂Xn

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Dependence of Random Variables

FX(X1,…,Xn)=FX1(X1)…FXn(Xn) FX(X1,…,Xn)=FX1(X1)…FXn(Xn)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Probability Distribution Function, Probability

If the random variable can take on any possible value within the range of outcomes, then the probability distribution is said to be a contimuous random variable. ${ }^{7}$ When a random variable is either the price of or the return on a financial asset or an interest rate, the random variable is assumed to be continuous. This means that it is possible to obtain, for example, a price of $95.43231$ or $109.34872$ and any value in between. In practice, we know that financial assets are not quoted in such a way. Nevertheless, there is no loss in describing the random variable as continuous and in many times treating the return as a continuous random variable means substantial gain in mathematical tractability and convenience. For a continuous random variable, the calculation of probabilities is substantially different from the discrete case. The reason is that if we want to derive the probability that the realization of the random variable lays within some range (i.e., over a subset or subinterval of the sample space), then we cannot proceed in a similar way as in the discrete case: The number of values in an interval is so large, that we cannot just add the probabilities of the single outcomes. The new concept needed is explained in the next section.

A probability distribution function $P$ assigns a probability $P(A)$ for every event $A$, that is, of realizing a value for the random value in any specified subset $A$ of the sample space. For example, a probability distribution function can assign a probability of realizing a monthly return that is negative or the probability of realizing a monthly return that is greater than $0.5 \%$ or the probability of realizing a monthly return that is between $0.4 \%$ and $1.0 \%$

To compute the probability, a mathematical function is needed to represent the probability distribution function. There are several possibilities of representing a probability distribution by means of a mathematical function. In the case of a continuous probability distribution, the most popular way is to provide the so-called probability density function or simply density function.

In general, we denote the density function for the random variable $X$ as $f_{X}(x)$. Note that the letter $x$ is used for the function argument and the index denotes that the density function corresponds to the random variable $X$. The letter $x$ is the convention adopted to denote a particular value for the random variable. The density function of a probability distribution is always nonnegative and as its name indicates: Large values for $f_{X}(x)$ of the density function at some point $x$ imply a relatively high probability of realizing a value in the neighborhood of $x$, whereas $f_{X}(x)=0$ for all $x$ in some interval $(a, b)$ implies that the probability for observing a realization in $(a, b)$ is zero.

Figure $1.1$ aids in understanding a continuous probability distribution. The shaded area is the probability of realizing a return less than $b$ and greater than $a$. As probabilities are represented by areas under the density function, it follows that the probability for every single outcome of a continuous random variable always equals zero. While the shaded area

in Figure $1.1$ represents the probability associated with realizing a return within the specified range, how does one compute the probability? This is where the tools of calculus are applied. Calculus involves differentiation and integration of a mathematical function. The latter tool is called integral calculus and involves computing the area under a curve. Thus the probability that a realization from a random variable is between two real numbers $a$ and $b$ is calculated according to the formula,
$$P(a \leq X \leq b)=\int_{a}^{b} f_{X}(x) d x$$
The mathematical function that provides the cumulative probability of a probability distribution, that is, the function that assigns to every real value $x$ the probability of getting an outcome less than or equal to $x$, is called the cumulative distribution function or cumulative probability function or simply distribution function and is denoted mathematically by $F_{X}(x)$. A cumulative distribution function is always nonnegative, nondecreasing, and as it represents probabilities it takes only values between zero and one. ${ }^{8} \mathrm{An}$ example of a distribution function is given in Figure 1.2.

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

The class of normal distributions, or Gaussian distributions, is certainly one of the most important probability distributions in statistics and due to some of its appealing properties also the class which is used in most applications in finance. Here we introduce some of its basic properties.

The random variable $X$ is said to be normally distributed with parameters $\mu$ and $\sigma$, abbreviated by $X \in N\left(\mu, \sigma^{2}\right)$, if the density of the random

$$f_{X}(x)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}}, x \in \mathbb{R} \text {. }$$
The parameter $\mu$ is called a location parameter because the middle of the distribution equals $\mu$ and $\sigma$ is called a shape parameter or a scale parameter. If $\mu=0$ and $\sigma=1$, then $X$ is said to have a standard normal distribution.

An important property of the normal distribution is the location-scale invariance of the normal distribution. What does this mean? Imagine you have random variable $X$, which is normally distributed with the parameters $\mu$ and $\sigma$. Now we consider the random variable $Y$, which is obtained as $Y=$ $a X+b .$ In general, the distribution of $Y$ might substantially differ from the distribution of $X$ but in the case where $X$ is normally distributed, the random variable $Y$ is again normally distributed with parameters and $\bar{\mu}=a \mu+b$ and $\bar{\sigma}=a \sigma$. Thus we do not leave the class of normal distributions if we multiply the random variable by a factor or shift the random variable. This fact can be used if we change the scale where a random variable is measured: Imagine that $X$ measures the temperature at the top of the Empire State Building on January 1, 2008, at 6 A.M. in degrees Celsius. Then $Y=\frac{9}{5} X+32$ will give the temperature in degrees Fahrenheit, and if $X$ is normally distributed, then $Y$ will be too.

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

The exponential distribution is popular, for example, in queuing theory when we want to model the time we have to wait until a certain event takes place. Examples include the time until the next client enters the store, the time until a certain company defaults or the time until some machine has a defect.

As it is used to model waiting times, the exponential distribution is concentrated on the positive real numbers and the density function $f$ and the cumulative distribution function $F$ of an exponentially distributed random variable $\tau$ possess the following form:
$$f_{\mathrm{r}}(x)=\frac{1}{\beta} e^{-\frac{x}{\beta}}, x>0$$
and
$$F_{\mathrm{r}}(x)=1-e^{-\frac{x}{\beta}}, x>0 .$$

In credit risk modeling, the parameter $\lambda=1 / \beta$ has a natural interpretation as hazard rate or default intensity. Let $\tau$ denote an exponential distributed random variable, for example, the random time (counted in days and started on January 1, 2008) we have to wait until Ford Motor Company defaults. Now, consider the following expression:
$$\lambda(\Delta t)=\frac{P(\tau \in(t, t+\Delta t] \mid \tau>t)}{\Delta t}=\frac{P(\tau \in(t, t+\Delta t])}{\Delta t P(\tau>t)} .$$
where $\Delta t$ denotes a small period of time.
What is the interpretation of this expression? $\lambda(\Delta t)$ represents a ratio of a probability and the quantity $\Delta t$. The probability in the numerator represents the probability that default occurs in the time interval $(t, t+\Delta t]$ conditional upon the fact that Ford Motor Company survives until time $t$. The notion of conditional probability is explained in section 1.6.1.

Now the ratio of this probability and the length of the considered time interval can be denoted as a default rate or default intensity. In applications different from credit risk we also use the expressions hazard or failure rate.
Now, letting $\Delta t$ tend to zero we finally obtain after some calculus the desired relation $\lambda=1 / \beta$. What we can see is that in the case of an exponentially distributed time of default, we are faced with a constant rate of default that is independent of the current point in time $t$.

Another interesting fact linked to the exponential distribution is the following connection with the Poisson distribution described earlier. Consider a sequence of independent and identical exponentially distributed random variables $\tau_{1}, \tau_{2}, \ldots$ We can think of $\tau_{1}$, for example, as the time we have to wait until a firm in a high-yield bond portfolio defaults. $\tau_{2}$ will then represent the time between the first and the second default and so on. These waiting times are sometimes called interarrival times. Now, let $N_{t}$ denote the number of defaults which have occurred until time $t \geq 0$. One important probabilistic result states that the random variable $N_{t}$ is Poisson distributed with parameter $\lambda=t / \beta$.

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

Fr(X)=1b和−Xb,X>0

Fr(X)=1−和−Xb,X>0.

λ(Δ吨)=磷(τ∈(吨,吨+Δ吨]∣τ>吨)Δ吨=磷(τ∈(吨,吨+Δ吨])Δ吨磷(τ>吨).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合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。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|BASIC CONCEPTS

An outcome for a random variable is the mutually exclusive potential result that can occur. The accepted notation for an outcome is the Greek letter $\omega$. A sample space is a set of all possible outcomes. The sample space is denoted by $\Omega$. The fact that a given outcome $\omega_{i}$ belongs to the sample space is expressed by $\omega_{i} \in \Omega$. An event is a subset of the sample space and can be represented as a collection of some of the outcomes. ${ }^{3}$ For example, consider Microsoft’s stock return over the next year. The sample space contains outcomes ranging from $100 \%$ (all the funds invested in Microsoft’s stock will be lost) to an extremely high positive return. The sample space can be partitioned into two subsets: outcomes where the return is less than or equal to $10 \%$ and a subset where the return exceeds $10 \%$. Consequently, a return greater than $10 \%$ is an event since it is a subset of the sample space. Similarly, a one-month LIBOR three months from now that exceeds $4 \%$ is an event. The collection of all events is usually denoted by $\mathfrak{A}$. In the theory of probability, we consider the sample space $\Omega$ together with the set of events $\mathfrak{A}$, usually written as ( $\Omega, \mathfrak{A})$, because the notion of probability is associated with an event. ${ }^{4}$

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

$$X=\left{1 如果 C 默认值 一世 0 别的。 \对。$$

(n ķ)=n!(n−ķ)!ķ!

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。