### 金融代写|金融数学代写Financial Mathematics代考|Measuring Interest

statistics-lab™ 为您的留学生涯保驾护航 在代写金融数学Financial Mathematics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数学Financial Mathematics代写方面经验极为丰富，各种代写金融数学Financial Mathematics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

## 金融代写|金融数学代写Financial Mathematics代考|The Effective Rate of Interest

Definition: The effective rate of interest for a given period is the amount of money that one unit of principal invested at the start of a particular period will earn during that one period. It is assumed that interest is paid at the end of the period in question 2 .

If the period is one year, the effective rate of interest is also referred to as the effective annual rate of interest. This is sometimes called the APR (annual percentage rate) when expressed as a percentage. An annual effective rate of .05 would correspond to an APR of $5 \%$. As we shall see, the APR reported by various consumer credit companies is not always the same as the effective annual rate. As a result, we will use the term effective annual interest rate (or effective monthly interest rate or …) except during our discussion of the Truth in Lending Laws at the end of Chapter $6 .$

We will use the symbol $i$ for the effective rate of interest. Again, unless stated otherwise, the period is assumed to be one year. In terms of $a(t)$ we have $i=a(1)-a(0)=a(1)-1$ so we have:
Effective Rate of Interest, $i_{1}$ in period 1
$$i_{1}=a(1)-1$$
We can express $i$ in terms of $A(t)$ as well. Since $A(t)=P_{0} a(t)$ we have:
\begin{aligned} i_{1} &=\frac{a(1)-a(0)}{a(0)} \ &=\frac{\frac{A(1)}{P_{0}}-\frac{A(0)}{P_{0}}}{\frac{A(0)}{P_{0}}} \ &=\frac{A(1)-A(0)}{A(0)} \end{aligned}
Example 2.4: A deposit of $\$ 550$earns$\$45$ in interest at the end of one year.
What is the effective annual interest rate?
Solution: We use Equation $2.7$
$$i_{1}=\frac{A(1)-A(0)}{A(0)}=\frac{595-550}{550}=.08182$$
This would typically be expressed as a percentage: $8.182 \%$.

## 金融代写|金融数学代写Financial Mathematics代考|Simple Interest

The first method of computing interest we will consider assumes that $a(t)$ is a linear function of $t$. This method is known as simple interest. It was in common use prior to the advent of calculators and computers since it is very easy to compute. It is still used in a few cases, most notably in the case of fractional time periods.

In the case of simple interest we know that $a(t)$ is a linear function and so its graph is a line. We are given two points on this line so it is easy to obtain an expression for the value of $a(t)$ at any time $t$. The slope of our line is given by
$$\frac{(1+i)-1}{1-0}=i$$
Since the line contains the point $(0,1)$ it’s equation is given by:
Accumulation Function for Simple Interest
$$a(t)=1+i t$$
In the case of simple interest, the accumulation function is a linear function with slope $i$. Figure $2.1$ is a plot of the accumulation function for the case that $i=.05$ :The amount function is just as easy to compute:
Amount Function for Simple Interest
$$A(t)=F V=P V \cdot(1+i t)$$
This is a line with y-intercept $=P V$ and slope $=P V_{i}$.

## 金融代写|金融数学代写Financial Mathematics代考|Compound Interest

Simple interest computes the interest in each period based solely on the amount of the initial deposit. If there are no withdrawals during the life of the transaction, the amount on deposit increases over time while the amount of interest paid at the end of each period remains constant. As we just saw this results in a declining effective rate of interest when we use simple interest. If $\$ 400$is deposited at$5 \%$simple interest the amount on deposit after three years is$\$460$. However (using simple interest) the interest paid at the end of year 4 is still only $\$ 20$(5\% of$\$400)$. If interest was paid based on the amount on deposit at the end of year 3 , the interest at the end of year 4 would be $(.05) \cdot \$ 460=\$23$

When the interest paid at the end of a given period is based on the accumulated value of the principal at the start of that period rather than on the amount of the original deposit we obtain what is known as compound interest.

To find the formula for the accumulation function in the case of compound interest, we compute the earned interest at the end of each period, add this to the previous balance, and use that number as the principal in computing the interest for the next period. See Table 2.1.

This leads us to believe that the following formula holds for the accumulation function in the case of compound interest
Accumulation Function: Compound Interest
$$a(n)=(1+i)^{n}$$
You can prove Equation $2.13$ for whole numbers using mathematical induction. We have calculated this result using only integral values for the time on deposit. If interest is only paid at integral multiples of the period, $a(n)$ is a step function:
$$a(n)= \begin{cases}1 & 0 \leq n<1 \ 1+i & 1 \leq n<2 \ (1+i)^{2} & 2 \leq n<3 \ (1+i)^{3} & 3 \leq n<4 \ \text { etc.. } & \end{cases}$$
If we assume that interest can be collected at any time in the life of the investment, it’s natural to extend this step function to include real $t \geq 0$. We thus obtain the function $a(t)=(1+i)^{t}$ defined for all real numbers $t$ which is the continuous, differentiable extension of Equation $2.3$
Accmulation Function for Compound Interest
$$a(t)=(1+i)^{t}$$

## 金融代写|金融数学代写Financial Mathematics代考|Simple Interest

(1+一世)−11−0=一世

Amount Function for Simple Interest

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。