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# 数学代写|金融数学代写Financial Mathematics代考|Math373 Multifactor Models

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## 数学代写|金融数学代写Financial Mathematics代考|Multifactor Models

Ross (1976) [293] introduced the Arbitrage Pricing Theory (APT) which is more general than CAPM; the cross-sectional variation in the asset returns is postulated to be due to multiple risk factors. There is no need to identify a market portfolio nor the risk-free return. The model can be stated as:
$$r_{i t}=\alpha_{i}+\beta_{i}^{\prime} f_{t}+\epsilon_{t}$$
and given $f_{t}$ is a $n \times 1$ vector of factors, $E\left(\epsilon_{i t}\right)=0, E\left(\epsilon_{i t} \epsilon_{j t}\right)=\sigma_{i j}, i, j=1,2, \ldots, m$ and $t=1,2, \ldots, T$. In the absence of arbitrage, Ross (1976) [293] shows that the following relationship should hold:
$$\mu_{i} \sim r_{f}+\beta_{i}^{\prime} \lambda_{k},$$
where $\lambda_{n}$ is a $n \times 1$ vector of factor risk premia.
Generally, there are two approaches to empirically model (6.14) and test if (6.15) holds. In the first approach, it is assumed that the factors $\left(f_{t}\right)$ are unknown and are estimated from the $(m \times m)$ covariance matrix of the returns via principal components. The number of factors, $n$, is generally determined via subjective considerations. In the second approach, it is taken that the factors are determined by macroeconomic variables that reflect systematic risk and by asset characteristics. Chen, Roll and Ross (1986) [76] consider expected inflation, spread between high and low grade bond yields, spread between long and short interest rates for US government bonds, etc., as known factors. The asset characteristics that are noted to be useful are: Ratio of book to market value, price-earnings ratio, momentum, Fama-French factors. These factor models with known or unknown factors tend to fare better than CAPM. As we will demonstrate, they are also useful for the construction of portfolios as these factors are likely to represent different risk dimensions.

## 数学代写|金融数学代写Financial Mathematics代考|Tests Related to CAPM and APT

To test if any particular portfolio is ex ante mean-variance efficient, Gibbons, Ross and Shanken (1989) [166] provide a multivariate test statistic and study its small sample properties under both null and alternative hypotheses. This follows from the multivariate regression model and the associated test procedures discussed in Chapter 3 . Recall the null hypothesis of interest in CAPM is:
$$H_{0}: \alpha_{i}=0 \quad \forall i=1,2, \ldots, m .$$

The test statistic is a multiple of Hotelling’s $T^{2}$ stated as,
$$F=\frac{T-m-1)}{m} \cdot \frac{\hat{\alpha}^{\prime} \hat{\Sigma}{\epsilon \epsilon}^{-1} \hat{\alpha}}{1+\hat{\theta}{p}^{2}} \sim F(m, T-m-1),$$
where $\hat{\theta}{p}=\bar{r}{p} / s_{p}$, the ratio of sample average and standard deviation of $r_{p t}$. The noncentrality parameter depends on $\alpha^{\prime} \Sigma_{\epsilon \epsilon}^{-1} \alpha$ which is zero under $H_{0}$. The statistical power of the $F$-test thus can be studied. The test statistic in (6.17) can be easily derived from the results on multivariate regression in Section 3.1. Thus, we can compute the residual covariance matrices both under the null and alternative hypotheses to arrive at the LR statistic.

## 数学代写|金融数学代写Financial Mathematics代考|Multifactor Models

Ross (1976) [293] 介绍了套利定价理论 (APT)，它比 CAPM 更通用；假设资产回报的横截面变化是由于多种 风险因素造成的。无需确定市场投赕组合或无风险回报。该模型可以表述为:
$$r_{i t}=\alpha_{i}+\beta_{i}^{\prime} f_{t}+\epsilon_{t}$$

$$\mu_{i} \sim r_{f}+\beta_{i}^{\prime} \lambda_{k}$$

## 数学代写|金融数学代写Financial Mathematics代考|Tests Related to CAPM and APT

$$H_{0}: \alpha_{i}=0 \quad \forall i=1,2, \ldots, m .$$

$$F=\frac{T-m-1)}{m} \cdot \frac{\hat{\alpha}^{\prime} \hat{\Sigma} \epsilon \epsilon^{-1} \hat{\alpha}}{1+\hat{\theta} p^{2}} \sim F(m, T-m-1)$$

## 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。