• 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

Most interest payments, from common savings accounts in banks to unpaid balances on credit cards, involve compound interest calculations. Additionally, compound interest forms the primary foundation of finance, investment analyses, and modern portfolio theory. As such, compound interest is an essential topic required for delving into these more advanced financial topics.

The defining property of compound interest is that once interest is paid on the initial principal amount, the interest is immediately added to the principal. This new principal amount, which now consists of the original principal amount plus the interest, earns interest during the next time period. Thus, the interest earned in one time period (referred to as a compounding period) earns interest in succeeding periods; this is known as interest being paid on interest and is the defining characteristic of compound interest calculations.
As an example, consider the deposit of $\$ 2,000$in an account paying an annual interest rate of$3 \%$, with compound interest computed and paid once a year. In the first year the principal earns$3 \%$of$\$2,000$ or $(0.03)(\$ 1,000)=\$60$. The new principal is now $\$ 2,060$(which is the original investment of$\$2,000$ plus the $\$ 60$interest payment). Thus, the second year’s interest payment is now based on this new amount, which becomes$3 \%$of$\$2,060$ or $\$ 61.80$. This makes the balance at the end of the second year$\$2,121,80$. Interest payments for the third year are now computed based on this new balance. The results of all interest computations through the fifth year have been collected in Table 4.1.

The yearly interest payments listed in Table $4.1$ are illustrated in Figure 4.2. Notice that the interest payments for each year is greater than that of the previous year. The reason for this is that each year’s interest is calculated on the sum of the initial principal and all prior interest payments (not just on the initial principal, as in simple interest calculations). Compare Figure $4.2$ with the analogous simple interest payments shown in Figure 4.1.

Interest rates are generally quoted on an annual basis but are typically compounded over shorter intervals of time. The annual rate, referred to as either the nominal interest rate or the stated interest rate is denoted by the symbol $\boldsymbol{r}$. The time between successive interval payments is called the compounding period, or the period, for short. The interest rate per period is denoted by the symbol $\boldsymbol{i}$; it is calculated by dividing the stated annual rate, $\boldsymbol{r}$ by the number of compounding periods in a year, which is denoted as $N$
$$i=r / N$$
If the interest is compounded quarterly, then $N$ is 4 (there are fourquarters in a year, and $i=r / 4$. For interest compounded semiannually, $N=2$ and $i=r / 2$; for interest compounded monthly $N=12$ and $i=r / 2$; and for interest compounded. Weekly $N=52$ and $i=r / 52$. If no compounding period is stated, compounding periods are assumed to be annual and $i=r$. This information is summarized in Table $4.2$, which lists the most commonly used compounding periods and the interest rates that apply to them, where $i$ is the stated annual interest rate. ${ }^{2}$

As seen in the last column of Table $4.2$, the interest rate per compounding period is the annual rate divided by the number of compounding periods in a year.

Equation $4.7$ remains valid for all the compound periods listed in Table $4.2$, as long as we realize that $i$ signifies the interest rate per compound period, and $P_{n}$ is the balance after $n$ compound periods. For example, if the interest is $2 \%$ compounded quarterly, $i=0.02 / 4=0.005$, which is the interest rate per quarter. Also, $P_{10}$, for example, denotes the principal after 10 compounding periods which, in this case, is 10 quarters and corresponds to $2^{1 / 2}$ years.

## 商科代写|商业数学代写business mathematics代考|LUMP-SUM FUTURE AND PRESENT VALUES

A lump-sum is a dollar amount made as a one-time single payment. Examples of lump-sum payments are an initial deposit, a single one-time dollar investment, or a final, single loan repayment. Equation $4.6$ relates a lump-sum principal amount at two points in time-the present, when the principal is first deposited, and its value in the future. The reason these values differ is due to the interest that is earned.

In this section, we rewrite and use Equation $4.6$ in two different ways to emphasize this time relationship. To do this, we will use standard financial notation that emphasizes the two unique usages. The first usage emphasizes determining $P_{n}$, the future value of an initial principal amount, given that we know $P_{0^{*}}$ The second usage emphasizes the equation’s use in determining the initial amount deposited, that is $P_{0}$, given that we know $P_{n}$, its future value. In financial applications, this second usage is typically much more important and the key to comparing investment alternatives.

For convenience, we first reproduce Equation $4.6$ as Equation $4.8$, so that we can rewrite it using standard financial notation. The advantage of this new notation is that it clearly relates the values of the principal amounts at two differing points in time, the present and the future.
$$P_{n}=P_{0}(1+i)^{n}$$
Financially, $P_{0}$, the initial principal, is referred to as the present value of the principal, or present value, for short. The notation used for this quantity is $\boldsymbol{P V}$. Similarly, $P_{n}$, which denotes the value of this principal sometime in the future, is referred to as the future value of the principal, or future value, for short. The notation used for this quantity is $\boldsymbol{F V}$. Note that this notation emphasizes what these quantities actually represent in time (now and in the future), as opposed to their strictly mathematical relationship.

## 有限元方法代写

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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。