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# 金融代写|金融工程代写FINANCIAL ENGINEERING代写|IE2097 Time Value of Money

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## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Simple Interest

Suppose that an amount is paid into a bank account, where it is to earn interest. The future value of this investment consists of the initial deposit, called the principal and denoted by $P$, plus all the interest earned since the money was deposited in the account.

To begin with, we shall consider the case when interest is attracted only by the principal, which remains unchanged during the period of investment. For example, the interest earned may be paid out in cash, credited to another account attracting no interest, or credited to the original account after some longer period.

After one year the interest earned will be $r P$, where $r>0$ is the interest rate. The value of the investment will thus become $V(1)=P+r P=(1+r) P$. After two years the investment will grow to $V(2)=(1+2 r) P$. Consider a fraction of a year. Interest is typically calculated on a daily basis: the interest earned in one day will be $\frac{1}{365} r P$. After $n$ days the interest will be $\frac{n}{365} r P$ and the total value of the investment will become $V\left(\frac{n}{365}\right)=\left(1+\frac{n}{365} r\right) P$. This motivates the following rule of simple interest: The value of the investment at time $t$, denoted by $V(t)$, is given by
$$V(t)=(1+t r) P,$$
where time $t$, expressed in years, can be an arbitrary non-negative real number; see Figure 2.1. In particular, we have the obvious equality $V(0)=P$. The number $1+r t$ is called the growth factor. Here we assume that the interest rate $r$ is constant. If the principal $P$ is invested at time $s$, rather than at time 0 , then the value at time $t \geq s$ will be
$$V(t)=(1+(t-s) r) P .$$

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Periodic Compounding

Once again, suppose that an amount $P$ is deposited in a bank account, attracting interest at a constant rate $r>0$. However, in contrast to the case of simple interest, we assume that the interest earned will now be added to the principal periodically, for example, annually, semi-annually, quarterly, monthly, or perhaps even on a daily basis. Subsequently, interest will be attracted not just by the original deposit, but also by all the interest earned so far. In these circumstances we shall talk of discrete or periodic compounding.

## 金融代写|金融工程代写FINANCIAL ENGINEERING代写|Simple Interest

$$V(t)=(1+t r) P$$

$$V(t)=(1+(t-s) r) P \text {. }$$

## 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。