Posted on Categories:Financial Mathematics, 数学代写, 金融数学

# 数学代写|金融数学代写Financial Mathematics代考|MAT280 Introduction to Modern Portfolio Theory

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|金融数学代写Financial Mathematics代考|Introduction to Modern Portfolio Theory

Of fundamental interest to financial economists is to examine the relationship between the risk of a financial security and its return. While it is obvious that risky assets can generally yield higher returns than risk-free assets, a quantification of the trade-off between risk and expected return was made through the development of the Capital Asset Pricing Model (CAPM) for which the groundwork was laid by Markowitz (1959) [260]. A central feature of the CAPM is that the expected return is a linear function of the risk. The risk of an asset typically is measured by the covariability between its return and that of an appropriately defined ‘market’ portfolio. Models of expected return-risk relationships include the Sharpe (1964) [301] and Lintner (1965) [245] CAPM, the zero-beta CAPM of Black (1972) [43], the arbitrage pricing theory (APT) due to Ross (1976) [293] and the intertemporal asset pricing model by Merton (1973) [268]. The economy-wide models developed by Sharpe, Lintner and Black are based on the work of Markowitz which assumes that investors would hold a mean-variance efficient portfolio. The main difference between the work of Sharpe and Lintner and the work of Black is that the former assumes the existence of a riskfree lending and borrowing rate whereas the latter derived a more general version of the CAPM in the absence of a risk-free rate.

In this section, we briefly review the CAPM model and the implications for empirical research in the area of portfolio construction and testing.

## 数学代写|金融数学代写Financial Mathematics代考|Mean-Variance Portfolio Theory

A portfolio which is composed of individual assets has its risk and return characteristics based on its composition and how the individual asset characteristics correlate with each other. The optimal combination is designed to produce the best balance between risk and return. For a given level of return, it will provide the lowest risk and for an acceptable level of risk, it will provide the maximum return. The locus of the combination of risk and reward that characterizes the optimal portfolios is called the “Efficient Frontier.”

We follow the same conventions as before to denote the return as $r_t=\ln \left(P_t\right)-$ $\ln \left(P_{t-1}\right)$; assume we have ‘ $m$ ‘ assets in the portfolio with ‘ $w_i$ ‘ denoting the share value Invested in asset, $i$, if $R_t=\left(P_t-P_{t-1}\right) / P_{t-1}$, the portfolio return is $R_{p t}=\sum_{i=1} w_i R_{i t}$ and thus $r_t=\ln \left(1+\sum_{i=1}^m w_i R_{i t}\right) \simeq \sum_{i=1}^m w_i r_{i t}$. If $r_t$ denotes the vector of ‘ $m$ ‘ asset returns and with weights stacked up as a vector, $w$, then the portfolio return, $r_{p t}=w^{\prime} r_t$ resulting in $\mu_p=E\left(r_{p t}\right)=w^{\prime} \mu$ and $\sigma_p^2=w^{\prime} \Sigma w$, where $\Sigma$ is the $m \times m$ variancecovariance matrix of $r_t$. If the returns are uncorrelated or negatively correlated, then observe that
$$\sigma_p^2=w^{\prime} \Sigma w \leq \sum_{i=1}^m w_i \operatorname{Var}\left(r_{i t}\right) \leq \frac{v}{m},$$
where ‘ $v$ ‘ is the maximum of $\operatorname{Var}\left(r_{i t}\right)$, clearly indicating that diversification tends to reduce risk. The power of the diversification can be seen clearly if we assume the covariance matrix, $\Sigma$, has all variances equal and if all off-diagonal covariance elements are the same. Then, $\sigma_p^2=\frac{1}{n} \cdot \sigma^2+\frac{n-1}{n} \cdot \rho \cdot \sigma^2$. Observe that if $\rho=0, \sigma_p^2 \rightarrow 0$ as $n \rightarrow \infty$ and if $\rho=1, \sigma_p^2=\sigma^2$ that results in no benefit. When $\rho<0$ as shown in (6.1), the portfolio variance is less due to diversification. But it should be noted that we cannot completely eliminate portfolio risk when the correlations among the assets are positive. Observe that $\sigma_p^2=\frac{1}{m^2} \sum_{i=1}^m \sigma_i^2+\frac{1}{m^2} \sum_{i \neq j} \sigma_{i j} \leq \frac{\sigma_{\max }^2}{m}+\frac{m-1}{m} \cdot A \rightarrow A$ as $m \rightarrow \infty$. Some amount of risk will remain if the portfolio consists of assets that move with the market.

## 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。