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# 数学代写|金融数学代写Financial Mathematics代考|MATH605 Implications for Investing

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## 数学代写|金融数学代写Financial Mathematics代考|Implications for Investing

As the focus of this book is on trading and investing we want to summarize some main takeaways from the widely used theoretical models. Some implications of these models are listed below:

One-Fund Theorem: There is a single investment of risky assets such that any efficient portfolio can be constructed as a combination of this fund and the risk-free asset. The optimal one-fund is the market fund which implies that the investor should purchase every stock which is not practical; this is a central foundation behind the rise of passive index funds.

Two-Fund Theorem: Two efficient portfolios (funds) can be established so that any other efficient portfolios can be duplicated through these funds; all investors seeking efficient portfolios need only to invest in combinations of these two funds.

No Short-Selling: When the weights, $w_i$, are restricted to be positive, typically many weights tend to be zeros, by contrast with short-selling, most of the weights tend to be non-zeros. The resulting concentrated portfolios are more likely to have higher turnover rates. This has implications for the increase in trading costs.

CAPM: Recall the model is $r_i-r_f=\beta_i\left(r_m-r_f\right)+\epsilon_i$ with $\sigma_i^2=\beta_i^2 \sigma_m^2+\sigma_\epsilon^2 ; \beta_i=\frac{\sigma_{i m}}{\sigma_m^2}$ is the normalized version of covariance. If $\beta_i=0, r_i \sim r_f$ even if $\sigma_i^2$ is large. There is no risk premium; the risk that is uncorrelated with the market can be diversified away. If $\beta_i<0, r_i<r_f$ even if $\sigma_i^2$ is large; such an asset reduces the overall portfolio risk when it is combined with the market portfolio; they provide a form of insurance. Observe that the portfolio ‘Beta’ is $\beta_p=\sum_{i=1}^m w_i \beta_i$. The CAPM can be used also as a pricing formula. Note if the purchase price of an asset is ‘ $P$ ‘ (known) which may be sold at ‘ $Q$ ‘ (unknown), from CAPM, $r=\frac{Q-P}{P}=r_f+\beta\left(r_m-r_f\right)$ which results in the pricing formula
$$P=\frac{Q}{1+r_f+\beta\left(r_m-r_f\right)} .$$
This procedure for evaluating a single asset can be extended to jointly evaluating multiple assets as well.

## 数学代写|金融数学代写Financial Mathematics代考|Investment Pattern

Investment Pattern: The minimum-variance portfolio method is likely to invest into low residual risk and low beta stocks. The CAPM model, (6.11) written in the vector form leads to
$$r_t-r_f 1=\alpha+\beta\left(r_{m t}-r_f\right)+\epsilon_t$$

and with the assumption that the error-covariance matrix is diagonal, the covariance matrix of the excess returns is
$$\Sigma_{r r}=\beta \beta^{\prime} \sigma_m^2+D,$$
where $D$ is diagonal with the element, $\sigma_{\epsilon i}^2$. The inverse of the covariance matrix has a simpler structure:
$$\Sigma_{r r}^{-1}=D^{-1}-\frac{\sigma_m^2}{1+a} \cdot b b^{\prime},$$
where $a=\sigma_m^2 \cdot \sum_{i=1}^m b_i \beta_i$ and $b_i=\beta_i / \sigma_i^2$. We have shown earlier that the portfolio weights under minimum-variance is
$$w=\frac{\Sigma_{r r}^{-1} 1}{1^{\prime} \Sigma_{r r}^{-1} 1}, \quad \sigma_{\mathrm{MV}}^2=\frac{1}{1^{\prime} \Sigma_{r r}^{-1} 1} .$$
Substituting (6.30) in (6.31), notice that the vector weights,
$$w=\sigma_{\mathrm{MV}}^2\left(D^{-1} 1-\frac{\sigma_m^2}{1+a} \cdot b b^{\prime} 1\right),$$
with a typical element, $w_j=\frac{\sigma_{\mathrm{MV}}^2}{\sigma_j^2}\left(1-\beta_j\left(\frac{\sigma_m^2}{1+a} \cdot \sum_{i=1}^m b_i\right)\right)$. It is easy to show that the second term in parenthesis above, $\frac{\sigma_m^2}{1+a} \cdot \sum_{i=1}^m b_i \sim 1$ and so
$$w_j=\frac{\sigma_{\mathrm{MV}}^2}{\sigma_j^2}\left(1-\beta_j\right) .$$
Thus when $\sigma_j^2$ is small and low $\beta_j, w_j$ will be large.

## 数学代写|金融数学代写金融数学代考|投资的启示

CAPM:回想一下，模型是$r_i-r_f=\beta_i\left(r_m-r_f\right)+\epsilon_i$, $\sigma_i^2=\beta_i^2 \sigma_m^2+\sigma_\epsilon^2 ; \beta_i=\frac{\sigma_{i m}}{\sigma_m^2}$是协方差的归一化版本。如果$\beta_i=0, r_i \sim r_f$即使$\sigma_i^2$很大。没有风险溢价;与市场无关的风险可以分散。如果$\beta_i<0, r_i<r_f$即使$\sigma_i^2$很大;当这种资产与市场投资组合结合时，降低了整体投资组合风险;它们提供了一种保险形式。注意到投资组合“Beta”是$\beta_p=\sum_{i=1}^m w_i \beta_i$。CAPM也可以用作定价公式。请注意，如果资产的购买价格是’ $P$ ‘(已知)，可能以’ $Q$ ‘(未知)出售，从CAPM, $r=\frac{Q-P}{P}=r_f+\beta\left(r_m-r_f\right)$，结果是定价公式
$$P=\frac{Q}{1+r_f+\beta\left(r_m-r_f\right)} .$$

## 数学代写|金融数学代写金融数学代考|投资模式

$$r_t-r_f 1=\alpha+\beta\left(r_{m t}-r_f\right)+\epsilon_t$$

$$\Sigma_{r r}=\beta \beta^{\prime} \sigma_m^2+D,$$
，其中$D$与元素$\sigma_{\epsilon i}^2$对角线。协方差矩阵的逆有一个更简单的结构:
$$\Sigma_{r r}^{-1}=D^{-1}-\frac{\sigma_m^2}{1+a} \cdot b b^{\prime},$$
，其中$a=\sigma_m^2 \cdot \sum_{i=1}^m b_i \beta_i$和$b_i=\beta_i / \sigma_i^2$。我们先前已经证明，最小方差下的投资组合权重
$$w=\frac{\Sigma_{r r}^{-1} 1}{1^{\prime} \Sigma_{r r}^{-1} 1}, \quad \sigma_{\mathrm{MV}}^2=\frac{1}{1^{\prime} \Sigma_{r r}^{-1} 1} .$$

$$w=\sigma_{\mathrm{MV}}^2\left(D^{-1} 1-\frac{\sigma_m^2}{1+a} \cdot b b^{\prime} 1\right),$$

$$w_j=\frac{\sigma_{\mathrm{MV}}^2}{\sigma_j^2}\left(1-\beta_j\right) .$$

## 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。