### 统计代写|金融统计代写Mathematics with Statistics for Finance代考|LIMITED LIABILITY

statistics-lab™ 为您的留学生涯保驾护航 在代写金融统计Mathematics with Statistics for Finance G1GH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。我们的专家在代写金融统计Mathematics with Statistics for Finance G1GH方面经验极为丰富，各种代写金融统计Mathematics with Statistics for Finance G1GH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

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|LIMITED LIABILITY

Another useful feature of log returns relates to limited liability. For many financial assets, including equities and bonds, the most that you can lose is the amount that you’ve put into them. For example, if you purchase a share of $X Y Z$ Corporation for $\$ 100$, the most you can lose is that$\$100$. This is known as limited liability. Today, limited liability is such a common feature of financial instruments that it is easy to take it for granted, but this was not always the case. Indeed, the widespread adoption of limited liability in the nineteenth century made possible the large publicly traded companies that are so important to our modern economy, and the vast financial markets that accompany them.

That you can lose only your initial investment is equivalent to saying that the minimum possible return on your investment is $-100 \%$. At the other end of the spectrum, there is no upper limit to the amount you can make in an investment. The maximum possible return is, in theory, infinite. This range for simple returns, $-100 \%$ to infinity, translates to a range of negative infinity to positive infinity for log returns.
\begin{aligned} &R_{\min }=-100 \% \Rightarrow r_{\min }=-\infty \ &R_{\max }=+\infty \Rightarrow r_{\max }=+\infty \end{aligned}
As we will see in the following chapters, when it comes to mathematical and computer models in finance, it is often much easier to work with variables that are unbounded, that is variables that can range from negative infinity to positive infinity.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|GRAPHING LOG RETURNS

Another useful feature of log returns is how they relate to log prices. By rearranging Equation $1.10$ and taking logs, it is easy to see that:
$$r_{t}=p_{t}-p_{t-1}$$

where $p_{t}$ is the log of $P_{t}$, the price at time $t$. To calculate log returns, rather than taking the log of one plus the simple return, we can simply calculate the logs of the prices and subtract.

Logarithms are also useful for charting time series that grow exponentially. Many computer applications allow you to chart data on a logarithmic scale. For an asset whose price grows exponentially, a logarithmic scale prevents the compression of data at low levels. Also, by rearranging Equation $1.13$, we can easily see that the change in the log price over time is equal to the log return:
$$\Delta p_{t}=p_{t}-p_{t-1}=r_{t}$$
It follows that, for an asset whose return is constant, the change in the log price will also be constant over time. On a chart, this constant rate of change over time will translate into a constant slope. Figures $1.2$ and $1.3$ both show an asset whose price is increasing by $20 \%$ each year. The y-axis for the first chart shows the price; the $y$-axis for the second chart displays the log price.

For the chart in Figure $1.2$, it is hard to tell if the rate of return is increasing or decreasing over time. For the chart in Figure 1.3, the fact that

the line is straight is equivalent to saying that the line has a constant slope. From Equation $1.14$ we know that this constant slope is equivalent to a constant rate of return.

In the first chart, the $y$-axis could just have easily been the actual price (on a log scale), but having the log prices allows us to do something else. Using Equation 1.13, we can easily estimate the log return. Over 10 periods, the log price increases from approximately $4.6$ to $6.4$. Subtracting and dividing gives us $(6.4-4.6) / 10=18 \%$. So the log return is $18 \%$ per period, which-because log returns and simple returns are very close for small values-is very close to the actual simple return of $20 \%$.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|CONTINUOUSLY COMPOUNDED RETURNS

Another topic related to the idea of log returns is continuously compounded returns. For many financial products, including bonds, mortgages, and credit cards, interest rates are often quoted on an annualized periodic or nominal basis. At each payment date, the amount to be paid is equal to this nominal rate, divided by the number of periods, multiplied by some notional amount.

For example, a bond with monthly coupon payments, a nominal rate of $6 \%$, and a notional value of $\$ 1,000$, would pay a coupon of$\$5$ each month: $(6 \% \times \$ 1,000) / 12=\$5$.

How do we compare two instruments with different payment frequencies? Are you better off paying $5 \%$ on an annual basis or $4.5 \%$ on a monthly basis? One solution is to turn the nominal rate into an annualized rate:
$$R_{\text {Arnual }}=\left(1+\frac{R_{\text {Nominal }}}{n}\right)^{n}-1$$
where $n$ is the number of periods per year for the instrument.
If we hold $R_{\text {Annual }}$ constant as $n$ increases, $R_{\text {Nominal gets smaller, but at }}$ a decreasing rate. Though the proof is omitted here, using L’Hôpital’s rule, we can prove that, at the limit, as $n$ approaches infinity, $R_{\text {Nominal converges }}$ to the log rate. As $n$ approaches infinity, it is as if the instrument is making infinitesimal payments on a continuous basis. Because of this, when used to define interest rates the log rate is often referred to as the continuously compounded rate, or simply the continuous rate. We can also compare two financial products with different payment periods by comparing their continuous rates.

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|LIMITED LIABILITY

R分钟=−100%⇒r分钟=−∞ R最大限度=+∞⇒r最大限度=+∞

r吨=p吨−p吨−1

Δp吨=p吨−p吨−1=r吨

## 统计代写|金融统计代写Mathematics with Statistics for Finance代考|CONTINUOUSLY COMPOUNDED RETURNS

R年报 =(1+R标称 n)n−1

## 广义线性模型代考

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## 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。