### 金融代写|金融数学代写Financial Mathematics代考|Financial Mathematics for Actuarial Science

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• 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

## 金融代写|金融数学代写Financial Mathematics代考|Other Accumulation Functions

In theory any function $a(t)$ which is continuous, increasing, and satisfies $a(0)=1$ can serve as an accumulation function. We now consider a few problems involving non-standard accumulation functions. While these don’t show up in “real life,” they do appear on the actuarial exams.

Example 2.15 Suppose that $a(t)=.1 t^{2}+b$. The only investment is $\$ 300$made at time$t=1$. What is the accumulated value of the investment at time$t=10 ?$Solution: Since it is always required that$a(0)=1$we must have$b=1$, so$a(t)=.1 t^{2}+1$. We are interested in computing$A(t)$at$t=10$. We know that$A(t)=P_{0} a(t)$and so need to find a value for$P_{0}$. The trick is to pretend that the$\$300$ is not the only investment but rather the value of the investment at time $t=1$. We then have
$$\begin{gathered} 300=A(1)=P_{0} \cdot a(1) \ P_{0}=\frac{300}{a(1)}=\frac{300}{1.1} \end{gathered}$$
We then have:
$$A(10)=P_{0} \cdot a(10)=\frac{300}{a(1)} \cdot a(10)=\frac{300}{1.1} \cdot 11=\ 3,000$$
The technique used in this example can be generalized to any problem involving an accumulation function. Suppose we know $A\left(t_{1}\right)$ and want to find $A\left(t_{2}\right)$. Since we do not know the value of $P_{0}$, we can’t compute $A\left(t_{2}\right)$ directly. However, we do know that $A\left(t_{1}\right)=P_{0} \cdot a\left(t_{1}\right)$ and $A\left(t_{2}\right)=P_{0} \cdot a\left(t_{1}\right)$ hence $P_{0}=\frac{A\left(t_{1}\right)}{a\left(t_{1}\right)}=\frac{A\left(t_{2}\right)}{a\left(t_{2}\right)}$. We solve this equation for $A\left(t_{2}\right)$ to obtain a very useful formula which computes the amount function at any time based on the amount at any other time.

## 金融代写|金融数学代写Financial Mathematics代考|Present and Future Value: Equations of Value

We now have formulas which compute the accumulated amount (also called the future value or $F V$ ) of $P_{0}$ at an interest rate of $i$ over $t$ periods. The amount $P_{0}$ is usually called the present value (PV). In many cases, we must answer the converse question: how much must be deposited at an interest rate of $i$ so that it will accumulate to a given future amount $F V$ ? As an example, we might be putting money away toward the purchase of a car or a down payment on a house.

In some cases, money is being saved toward the payment of an annual tax or other obligation. In Chapter 9 we will discuss ways in which investors seek to ensure that the funds saved will suffice to cover a set of future obligations. Banks often require mortgage holders to contribute to an account to provide funds for the payment of property taxes. These sorts of accounts are known as escrow accounts.

We begin with the calculation of the relation between the present and future value of a single deposit for one year. We want to find the amount $P V$ which is sufficient to accumulate to an amount $F V$ at the end of one year. In one period a deposit of $P V$ will accumulate to $P V(1+i)$. We want $P V(1+i)=F V$ and so $P V=F V \cdot \frac{1}{1+i}$. The term $\frac{1}{1+i}$ is called the discount factor and is referenced by the symbol $v$. Thus:
\begin{aligned} &v=\frac{1}{1+i} \ &i=\frac{1-v}{v}=\frac{1}{v}-1 \end{aligned}
Over $t$ periods we have one set of formulas for simple interest and a second for compound interest. In each case, we merely solved the first equation (relating FV to PV) for the present value in order to get an equation with computes $\mathrm{PV}$ in terms of FV.
Present and Future Value: Simple Interest
\begin{aligned} &F V=P V(1+i t) \ &P V=F V \frac{1}{1+i t} \end{aligned} Present and Future Value: Compound Interest
\begin{aligned} &F V=P V(1+i)^{t} \ &P V=F V(1+i)^{-t}=F V v^{t} \end{aligned}
Note that in each case there are four variables (PV, FV, $i$ and $t$ ). Knowing any three of the four enables us to solve for the fourth variable. As we have seen the TI BA II Plus will perform these calculations. The examples below show how this can be done.

## 金融代写|金融数学代写Financial Mathematics代考|Nominal and Effective Rates of Interest

We now consider cases where interest is paid either more or less often than the period used for measuring time. In many cases an annual interest rate is quoted even though interest may be compounded (or converted) more (or less) often than once per year. In order to compare such rates we must convert to a common time unit. This is typically a year, but in some problems another unit will be most useful. If the unit of measurement is one year we are computing what is called the effective annual rate of interest.

Example 2.26 Mammoth Credit offers a credit card with a monthly interest rate of $1.9 \%$. Cards $\mathrm{R}$ Us offers a card with an annual rate of $23 \%$. Which card is the better buy?

Solution: This may seem simple-multiply $1.9$ times 12 to obtain $22.8$. Hence the Mammoth card is the better deal. This is exactly what Mammoth wants us to do. In fact, Mammoth is allowed to describe their card as carrying an “APR” (Annual Percentage Rate) of $22.8 \%$. However, this simple calculation only works if interest is computed using simple interest! Since all credit card interest is computed as compound interest, we need to convert the Mammoth rate to its effective annual interest rate. To do that we need an annual rate of interest, $i$, which is equivalent to our monthly rate of $1.9 \%$. If we invest $\$ 1$at an annual rate of interest of$i$we will have$1+i$after one year. On the other hand,$\$1$ invested at $1.9 \%$ per month for twelve months will accumulate to $1 \cdot(1.019)^{12}=1.25340$. We thus we must have $1+i=1.25340$ so that $i=.25340=25.34 \%$. The Mammoth Card has an effective annual interest rate of $25.34 \%$ making the Cards $\mathrm{R}$ Us card at $23 \%$ the better buy.

## 金融代写|金融数学代写Financial Mathematics代考|Other Accumulation Functions

300=一个(1)=磷0⋅一个(1) 磷0=300一个(1)=3001.1

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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。