# 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|FNCE463

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes-lab™ 为您的留学生涯保驾护航 在代写风险和利率理论Market Risk, Measures and Portfolio 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和利率理论Market Risk, Measures and Portfolio Theory代写方面经验极为丰富，各种代写风险和利率理论Market Risk, Measures and Portfolio 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
couryes™为您提供可以保分的包课服务

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Risk and return

A portfolio constructed from $n$ different securities can be described by IIleans of the vector of weights
$$\mathbf{w}=\left(w_1, \ldots, w_n\right),$$
with the constraint $\sum_{j=1}^n w_j=1$. Denoting by $\mathbf{1}$ the $n$-dimensional vector
$$\mathbf{1}=(1, \ldots, 1),$$
the constraint can conveniently be written as
$$\mathbf{w}^{\mathrm{T}} \mathbf{1}=1 .$$

The attainable set is the set of all weight vectors $w$ that satisfy this constraint.

If short-selling is not possible, the condition $w_j \geq 0$ is added to the constraint, so in that case the attainable set becomes
$$\left{\mathbf{w}: \mathbf{w}^{\mathrm{T}} \mathbf{1}=1, w_j \geq 0 \text { for all } j \leq n\right} .$$
Unless stated otherwise, we shall assume availability of short sales.
Alternatively a portfolio is described by the vector of positions taken in particular components (numbers of units of assets)
$$\mathbf{x}=\left(x_1, \ldots, x_n\right) \text {. }$$
We have the following relations between the weights, prices and the numbers of shares:
$$w_j=\frac{x_j S_j(0)}{V(0)}, \quad j=1, \ldots, n$$
where $x_j$ is the number of shares of security $j$ in the portfolio, $S_j(0)$ is the initial price of security $j$, and $V(0)$ is the total money invested.

Denote the random returns on the securities by $K_1, \ldots, K_n$, and the vector of expected returns by
$$\boldsymbol{\mu}=\left(\mu_1, \ldots, \mu_n\right)$$
with
$$\mu_j=\mathbb{E}\left(K_j\right), \quad \text { for } j=1, \ldots, n$$

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Three risky securities

The purpose of this section is to provide geometric intuition as to the shape of the attainable set.

In the case when we have three risky assets, the third weight of a portfolio can be computed from the first two weights
$$w_3=1-w_2-w_1,$$
meaning that the attainable set is parameterised by $w_1$ and $w_2$. We can write the formulae for $\mu_{\mathrm{w}}$ and $\sigma_{\mathrm{w}}$ with respect to these two parameters as
\begin{aligned} \mu_{\mathrm{w}} &=w_1 \mu_1+w_2 \mu_2+w_3 \mu_3 \ &=w_1 \mu_1+w_2 \mu_2+\left(1-w_1-w_2\right) \mu_3, \end{aligned}
and
\begin{aligned} \sigma_w^2=& w_1^2 \sigma_1^2+w_2^2 \sigma_2^2+w_3^2 \sigma_3^2+2 w_1 w_2 \sigma_{12}+2 w_1 w_3 \sigma_{13}+2 w_2 w_3 \sigma_{23} \ =& w_1^2 \sigma_1^2+w_2^2 \sigma_2^2+\left(1-w_2-w_1\right)^2 \sigma_3^2+2 w_1 w_2 \sigma_{12} \ &+2 w_1\left(1-w_2-w_1\right) \sigma_{13}+2 w_2\left(1-w_2-w_1\right) \sigma_{23} . \end{aligned}
The plots of $\mu_{\mathbf{w}}$ and $\sigma_{\mathbf{w}}$ are given in Figure 4.1. The lines on the graphs represent the level sets $\left{\mu_{\mathrm{w}}=m\right}$ and $\left{\sigma_{\mathrm{w}}=c\right}$ for several values of $m$ and $c$

Since the third weight can be computed from the first two, the attainable set is represented as the $\left(w_1, w_2\right)$-plane in Figure 4.2. The vertices of the grey triangle represent investments in single assets. The point $(1,0)$ represents the first asset, $(0,1)$ the second asset, and since $w_3=1-w_1-w_2$, the point $(0,0)$ represents the third asset. The grey triangle consists of the points
$$\left{\left(w_1, w_2\right) \mid w_1, w_2 \geq 0, w_1+w_2 \leq 1\right},$$ and contains portfolios attainable without short-selling.
The level sets $\left{\mu_w=m\right}$ and $\left{\sigma_w=c\right}$ from Figure $4.1$ can be projected onto the $\left(w_1, w_2\right)$-plane in Figure 4.2. These are the straight lines and ellipses in Figure 4.2, respectively. The middle point of the ellipses is the minimum variance portfolio. In this particular figure, since the point lies outside of the triangle, we see that the minimum variance portfolio requires short selling. In Figure $4.2$ we also see that if short-selling is not allowed, then the smallest attainable $\sigma_w$ lies on the ellipse which is tangent to the grey triangle. The minimum variance portfolio without short-selling is the tangency point.

We now discuss the shape that the set of attainable portfolios takes in the $(\sigma, \mu)$-plane. We start with Figure $4.3$, where we see the plane corresponding to portfolios with $\mu_{\mathrm{w}}=m$, together with the plot of $\sigma_{\mathrm{w}}$. We see that there is a single point that has smallest attainable variance under the constraint $\mu_{\mathrm{w}}=m$. This is the point at the bottom of the intersection of the plane with the hyperbola. From the plot we also see that for $\mu_{\mathrm{w}}=m$ we can have portfolios with arbitrarily large $\sigma$. This leads to the conclusion that in the $(\sigma, \mu)$-plane, the set of portfolios with $\mu_{\mathrm{w}}=m$ is a horizontal half line, which is depicted in Figure 4.4. Intuitively one can think of Figure $4.4$ as the leftmost graph from Figure 4.3, rotated clockwise by ninety degrees, and projected onto the plane. Since the plot of $\sigma_w$ is a hyperbola, one is led to believe that the boundary of the attainable set on the $(\sigma, \mu)$-plane should also be a hyperbola. This is just a geometric intuition, and is by no means meant as a proof. We shall prove this fact later on.

# 风险和利率理论代写

## 金融代写|风险和利率理论代写市场风险、措施和投资组合理论代考|风险和回报

$$\mathbf{w}=\left(w_1, \ldots, w_n\right),$$

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)