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

statistics-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

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

Consider the three assets described in Example 1.4. Although $\sigma_1=\sigma_3$, the third asset carries no ‘downside risk’, since neither outcome for $S_3(1)$ involves a loss for the investor. Similarly, although $\sigma_2>\sigma_1$, the downside risk for the second asset is the same as that for the first (a 50\% chance of incurring a loss of 10), but the expected return for the second asset is $15 \%$, making it the more attractive investment even though, as measured by variance, it is more risky. Since investors regard risk as concerned with failure (i.e. downside risk), the following modification of variance is sometimes used. It is called semi-variance and is computed by a formula that takes into account only the unfavourable outcomes, where the return is below the expected value
$$\mathbb{E}(\min {0, K-\mu})^2 .$$
The square root of semi-variance is denoted by semi- $\sigma$. However, this notion still does not agree fully with the intuition.
Example $1.6$
Assume that $\Omega=\left{\omega_1, \omega_2\right}, P\left(\left{\omega_1\right}\right)=P\left(\left{\omega_2\right}\right)=\frac{1}{2}$ and
\begin{aligned} &K\left(\omega_1\right)=10 \%, \ &K\left(\omega_2\right)=20 \% . \end{aligned}

Consider a modification $K^{\prime}$ with
\begin{aligned} &K^{\prime}\left(\omega_1\right)=10 \%, \ &K^{\prime}\left(\omega_2\right)=30 \% . \end{aligned}
Then $K^{\prime}$ is definitely better than $K$ but the semi-variance and the variance for $K^{\prime}$ are both higher than for $K$.

If variance or semi-variance are to represent risk, it is illogical that a better version should be regarded as more risky. This defect can be rectified by replacing the expectation by some other reference point, for instance the risk-free return with the following modification of (1.1),
$$\mathbb{E}(\min {0, K-R})^2,$$
which eliminates the above unwanted feature. Instead of the risk-free rate, one can also consider the return required by the investor.

These versions are not very popular in the financial world, the variance being the basic measure of risk. In our presentation of portfolio theory we follow the historical tradition and take variance as the measure of risk. It is possible to develop a version of the theory for alternative ways of measuring risk. In most cases, however, such theories do not produce neat analytic formulae as is the case for the mean and variance.

We will return to a more general discussion of risk measures in the final chapters of this volume. An analysis of the popular concept of Value at Risk (VaR), which has been used extensively in the banking and investment sectors since the 1990s, will lead us to conclude that, despite its ubiquity, this risk measure has serious shortcomings, especially when dealing with mixed distributions. We will then examine an alternative which remedies these defects but still remains mathematically tractable.

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Portfolios consisting of two assets

We begin our discussion of portfolio risk and expected return with portfolios consisting of just two securities. This has the advantage that the key concepts of mean-variance portfolio theory can be expressed in simple geometric terms.

For a given allocation of resources between the two assets comprising the portfolio, the mean and variance of the return on the entire portfolio are expressed in terms of the means and variances of, and (crucially) the covariance between, the returns on the individual assets. This enables us to examine the set of all feasible weightings of (in other words, allocations of funds to) the different assets in the portfolio, and to find the unique weighting with minimum variance. We also find the collection of efficient portfolios – ones that are not dominated by any other. Finally, adding a risk-free asset, we find the so-called market portfolio, which is the unique portfolio providing an optimal combination with the risk-free asset.

We denote the prices of the securities as $S_1(t)$ and $S_2(t)$ for $t=0,1$. We start with a motivating example.

Example $2.1$
Let $\Omega=\left{\omega_1, \omega_2\right}, S_1(0)=200, S_2(0)=300$. Assume that
$$P\left(\left{\omega_1\right}\right)=P\left(\left{\omega_1\right}\right)=\frac{1}{2},$$
and that
$$\begin{array}{ll} S_1\left(1, \omega_1\right)=260, & S_2\left(1, \omega_1\right)=270 \ S_1\left(1, \omega_2\right)=180, & S_2\left(1, \omega_2\right)=360 \end{array}$$
The expected returns and standard deviations for the two assets are
$$\begin{array}{ll} \mu_1=10 \%, & \mu_2=5 \%, \ \sigma_1=20 \%, & \sigma_2=15 \% . \end{array}$$
Assume that we spend $V(0)=500$, buying a single share of stock $S_1$ and a single share of stock $S_2$. At time 1 we will have
\begin{aligned} &V\left(1, \omega_1\right)=260+270=530 \ &V\left(1, \omega_2\right)=180+360=540 \end{aligned}
The expected return on the investment is $7 \%$ and the standard deviation is just $1 \%$. We can see that by diversifying the investment into two stocks we have considerably reduced the risk.

# 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、度量和投资组合理论代考|半方差

$$\mathbb{E}(\min {0, K-\mu})^2 .$$

\begin{aligned} &K\left(\omega_1\right)=10 \%, \ &K\left(\omega_2\right)=20 \% . \end{aligned}

\begin{aligned} &K^{\prime}\left(\omega_1\right)=10 \%, \ &K^{\prime}\left(\omega_2\right)=30 \% . \end{aligned}

$$\mathbb{E}(\min {0, K-R})^2,$$

## 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|由两种资产组成的投资组合

$$P\left(\left{\omega_1\right}\right)=P\left(\left{\omega_1\right}\right)=\frac{1}{2},$$$$\begin{array}{ll} S_1\left(1, \omega_1\right)=260, & S_2\left(1, \omega_1\right)=270 \ S_1\left(1, \omega_2\right)=180, & S_2\left(1, \omega_2\right)=360 \end{array}$$

$$\begin{array}{ll} \mu_1=10 \%, & \mu_2=5 \%, \ \sigma_1=20 \%, & \sigma_2=15 \% . \end{array}$$

\begin{aligned} &V\left(1, \omega_1\right)=260+270=530 \ &V\left(1, \omega_2\right)=180+360=540 \end{aligned}

## 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|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|FE630

statistics-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

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

While we may allow any probability space, we must make sure that negative values of the random variable $S(1)$ are excluded since negative prices make no sense from the point of view of economics. This means that the distribution of $S(1)$ has to be supported on $[0,+\infty)$ (meaning that $P(S(1) \geq 0)=1)$.

The return (also called the rate of return) on the investment $S$ is a random variable $K: \Omega \rightarrow \mathbb{R}$, defined as
$$K=\frac{S(1)-S(0)}{S(0)} .$$
By the linearity of mathematical expectation, the expected (or mean) return is given by
$$\mathbb{E}(K)=\frac{\mathbb{E}(S(1))-S(0)}{S(0)} .$$
We introduce the convention of using the Greek letter $\mu$ for expectations of various random returns
$$\mu=\mathbb{E}(K),$$
with various subscripts indicating the context, if necessary.
The relationships between the prices and returns can be written as
\begin{aligned} S(1) &=S(0)(1+K), \ \mathbb{E}(S(1)) &=S(0)(1+\mu), \end{aligned}
which illustrates the possibility of reversing the approach: given the returns we can find the prices.

The requirement that $S(1)$ is nonnegative implies that we must have $K \geq-1$. This in particular excludes the possibility of considering $K$ with Gaussian (normal) distribution.

At time 1 a dividend may be paid. In practice, after the dividend is paid, the stock price drops by this amount, which is logical. Thus we have to determine the price that includes the dividend; more precisely, we must distinguish between the right to receive that price (the cum dividend price) and the price after the dividend is paid (the ex dividend price).

## 金融代写|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|Variance as a risk measure

The concept of risk in finance is captured in many ways. The basic and most widely used one is concerned with risk as uncertainty of the unknown future value of some quantity in question (here we are concerned with return). This uncertainty is understood as the scatter around some reference point. A natural candidate for the reference value is the mathematical expectation (though other benchmarks are sometimes considered). The extent of scatter is conveniently measured by the variance. This notion takes care of two aspects of risk:
(i) The distances between possible values and the expectation.
(ii) The probabilities of attaining the various possible values.
Definition $1.3$
By (the measure of) risk we mean the variance of the return
$$\operatorname{Var}(K)=\mathbb{E}(K-\mu)^2=\mathbb{E}\left(K^2\right)-\mu^2,$$
or the standard deviation $\sqrt{\operatorname{Var}(K)}$.
The variance of the return can be computed from the variance of $S(1)$,
\begin{aligned} \operatorname{Var}(K) &=\operatorname{Var}\left(\frac{S(1)-S(0)}{S(0)}\right) \ &=\frac{1}{S(0)^2} \operatorname{Var}(S(1)-S(0)) \ &=\frac{1}{S(0)^2} \operatorname{Var}(S(1)) \end{aligned}
We use the Greek letter $\sigma$ for standard deviations of various random returns
$$\sigma=\sqrt{\operatorname{Var}(K)}$$
qualified by subscripts, as required.

# 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|预期收益

$$K=\frac{S(1)-S(0)}{S(0)} .$$

$$\mathbb{E}(K)=\frac{\mathbb{E}(S(1))-S(0)}{S(0)} .$$

$$\mu=\mathbb{E}(K),$$
，如果必要的话，各种下标表示上下文。

\begin{aligned} S(1) &=S(0)(1+K), \ \mathbb{E}(S(1)) &=S(0)(1+\mu), \end{aligned}
，这说明了反转方法的可能性:给定收益，我们可以找到价格

$S(1)$非负的要求意味着我们必须有$K \geq-1$。这尤其排除了将$K$考虑为高斯(正态)分布的可能性

## 金融代写|风险和利率理论代写市场风险、度量和投资组合理论代考|方差作为风险度量

(i)可能值与期望之间的距离
(ii)达到各种可能值的概率。

$$\operatorname{Var}(K)=\mathbb{E}(K-\mu)^2=\mathbb{E}\left(K^2\right)-\mu^2,$$

\begin{aligned} \operatorname{Var}(K) &=\operatorname{Var}\left(\frac{S(1)-S(0)}{S(0)}\right) \ &=\frac{1}{S(0)^2} \operatorname{Var}(S(1)-S(0)) \ &=\frac{1}{S(0)^2} \operatorname{Var}(S(1)) \end{aligned}

$$\sigma=\sqrt{\operatorname{Var}(K)}$$
，按要求由下标限定

## 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|风险和利率理论代写Market Risk, Measures and Portfolio Theory代考|MATH0094

statistics-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

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

Financial investors base their activity on the expectation that their investment will increase over time, leading to an increase in wealth. Over a fixed time period, the investor seeks to maximise the return on the investment, that is, the increase in asset value as a proportion of the initial investment. The final values of most assets (other than loans at a fixed rate of interest) are uncertain, so that the returns on these investments need to be expressed in terms of random variables. To estimate the return on such an asset by a single number it is natural to use the expected value of the return, which averages the returns over all possible outcomes.

Our uncertainty about future market behaviour finds expression in the second key concept in finance: risk. Assets such as stocks, forward contracts and options are risky because we cannot predict their future values with certainty. Assets whose possible final values are more ‘widely spread’ are naturally seen as entailing greater risk. Thus our initial attempt to measure the riskiness of a random variable will measure the spread of the return, which rational investors will seek to minimise while maximising their return.

In brief, return reflects the efficiency of an investment, risk is concerned with uncertainty. The balance between these two is at the heart of portfolio theory, which seeks to find optimal allocations of the investor’s initial wealth among the available assets: maximising return at a given level of risk and minimising risk at a given level of expected return.

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

We are concerned with just two time instants: the present time, denoted by 0 , and the future time 1 , where 1 may stand for any unit of time. Suppose we make a single-period investment in some stock with the current price $S(0)$ known, and the future price $S(1)$ unknown, hence assumed to be represented by a random variable
$$S(1): \Omega \rightarrow[0,+\infty),$$
where $\Omega$ is the sample space of some probability space $(\Omega, \mathcal{F}, P)$. The members of $\Omega$ are often called states or scenarios. (See [PF] for basic definitions.)
When $\Omega$ is finite, $\Omega=\left{\omega_1, \ldots, \omega_N\right}$, we shall adopt the notation
$$S\left(1, \omega_i\right)=S(1)\left(\omega_i\right) \text { for } i=1, \ldots, N,$$
for the possible values of $S(1)$. In this setting it is natural to equip $\Omega$ with the $\sigma$-field $\mathcal{F}=2^{\Omega}$ of all its subsets. To define a probability measure $P$ : $\mathcal{F} \rightarrow[0,1]$ it is sufficient to give its values on single element sets, $P\left(\left{\omega_i\right}\right)=$ $p_i$, by choosing $p_i \in(0,1]$ such that $\sum_{i=1}^N p_i=1$. We can then compute the expected price at the end of the period
$$\mathbb{E}(S(1))=\sum_{i=1}^N S\left(1, \omega_i\right) p_i,$$
and the variance of the price
$$\operatorname{Var}(S(1))=\sum_{i=1}^N\left(S\left(1, \omega_i\right)-\mathbb{E}(S(1))\right)^2 p_i .$$

# 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|预期收益

$$S(1): \Omega \rightarrow[0,+\infty),$$

$$S\left(1, \omega_i\right)=S(1)\left(\omega_i\right) \text { for } i=1, \ldots, N,$$

## 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。