• 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

## 商科代写|商业数学代写business mathematics代考|CASH FLOW NET PRESENT VALUES

In Sections $4.1$ through $4.3$, we concerned ourselves with single lump-sum payments. Thus, we either calculated the future value of a lump-sum invested now, or we calculated the present value of a lump sum payment to be made in the future. In this section we consider investments consisting of a set of payments due at different times, a situation known as a cash flow.

As an example of a cash flow, consider an investment that returns $\$ 500$in 1 year, another$\$300$ in 3 years, and a final $\$ 400$in 4 years, with interest rates of$5 \%$compounded annually. What is the present value of such an opportunity? That is, what is the cash equivalent now of the entire transaction? A simple approach is to compute the present value of each of the individual payments using Equation 4.10, repeated below as Equation$4.11$for convenience, and then sum the individual present values to obtain the present value of the entire cash flow. $$P V=F V(1+i)^{-\mathrm{n}}$$ Example 1 Compute the present value of the cash flow that returns$\$500$ in 1 year, another $\$ 300$in 3 years, and a final$\$400$ in 4 years, with interest rates of $5 \%$ compounded annually.

Solution The first payment of $\$ 500$is due in 1 year. The present value of this amount, computed using Equation$4.11$is $$P V_{1}=(\ 500)(1+0.05)^{-1}=\ 476.19$$ The second payment of$\$300$ is due in 3 years. Again using Equation 4.11, we find its present value as:
$$P V_{2}=(\ 300)(1+0.05)^{-3}=\ 259.15 .$$
Similarly, the present value of the last payment is
$$P V_{3}=(\ 400)(1+0.05)=\ 329.08$$
Summing these three present values, we obtain the present value of the entire investment as:
$$P V=P V_{1}+P V_{2}+P V_{3}=\ 476.19+\ 259.15+\ 329.08=\ 1,064.42$$
In most present-value problems, a time diagram illustrating the contributions to the total present value from the individual payments is helpful. The time diagram for the cash flow given in Example 1 is shown as Figure 4.4.

The present and future values of a cash flow can always be determined by calculating the present or future values, respectively, of each individual payment using the appropriate equation – either Equation $4.9$ or 4.10, repeated below as Equations $4.13$ and $4.14$ for convenience – and then summing the results.
$$F V=P V(1+i)^{n}$$
or
$$P V=F V(1+i)^{-n}$$
For a specific type of investment, however, known as an annuity, the final sum can be calculated using a single formula.

Definition 4.1 An annuity is a set of equal payments made at equal intervals of time.

Car loans, mortgages, life insurance premiums, social security payments, and bond coupon payments are all examples of annuities. In each, one party, be it an individual, company, or government, pays to another party a set of equal payments, called periodic installments or payments, denoted as $P M T$, at equal periods of time, called the rent period, payment period, payment interval, or compounding period. Each of these terms can be used interchangeably.
Annuities are classified as either ordinary or due. With an ordinary annuity, payments are made at the end of each payment period, whereas with an annuity due, payments are made at the beginning of each period. Examples of ordinary annuities are car loan payments, mortgages, and bond coupon payments. Examples of annuities due are typically savings plans, pension plans, and lottery winnings that are paid over time.

An annuity is simple if the compounding period at which interest is paid coincides with the payment dates. In this section, we consider simple ordinary annuities; simple annuities due are presented in Section 4.7.

One of the most common types of ordinary annuities is a mortgage on a house or land. The mortgage is a loan used to pay for the property, with the property serving as collateral for the loan. This gives the lender, known as the mortgagor, a claim on the property should the borrower, known as the mortgagee, default on paying the mortgage. Full title to the property is only transferred to the mortgagee when the loan is fully paid.

In a traditional fixed-rate mortgage the monthly payment and interest rate are fixed for the life of the mortgage. Each payment is used to pay both the interest and principal for the loan. First, the monthly interest charge on loan is determined and paid, with the remaining portion of the monthly payment applied to paying off the loan.

Although the monthly payment is fixed, the interest due changes each month, decreasing with every payment. This occurs because the interest is computed each month anew on the unpaid loan balance. As the loan gets paid off, the unpaid balance decreases, which means that the interest due each month also decreases. Thus, each month more and more of the payment gets applied to paying off the loan. This method of payment is commonly referred to as the United States Rule.

The main consideration with mortgages is to determine the amount of the monthly payment, which depends on the original amount of the loan, the interest rate, and the length of the loan. For all mortgages that adhere to the United States Rule, the payment, PMT, is determined as
$$P M T=\frac{P V}{\left[\frac{1-(1+i)^{-n}}{i}\right]}$$
where:
$P M T=$ the monthly payment
$P V=$ the original amount of the loan
$i=$ the monthly interest rate $=$ (the annual interest rate) $/ 12$
$n=$ the length of the loan, in months, $=12$ * (the number of years of the loan)

Notice that Equation $4.18$ is the same as Equation 4.15, except that it is used to solve for the value of $P M T$ given $P V$, rather than solving for $P V$ given the PMT amount.

## 商科代写|商业数学代写business mathematics代考|CASH FLOW NET PRESENT VALUES

F在=磷在(1+一世)n

n=贷款期限，以月为单位，=12*（贷款年数）

## 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。