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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

## 数学代写|金融数学代写Intro to Mathematics of Finance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
$$a(t)=(1+i)^t$$
This is an exponential function whose graph is very different from the linear function we obtained for simple interest. See Figure 2.3.

## 数学代写|金融数学代写Intro to Mathematics of Finance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.

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Compound Interest

$$a(n)=(1+i)^n$$

$$a(n)=\left{\begin{array}{lll} 1 & 0 \leq n<1 & 1+i \quad 1 \leq n<2(1+i)^2 \quad 2 \leq n<3(1+i)^3 \quad 3 \leq n<4 \text { etc. } \end{array}\right.$$

$t$ 这是方程的连续、可微扩展 $2.3$

$$a(t)=(1+i)^t$$

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Other Accumulation Functions

$$300=A(1)=P_0 \cdot a(1) P_0=\frac{300}{a(1)}=\frac{300}{1.1}$$

$$A(10)=P_0 \cdot a(10)=\frac{300}{a(1)} \cdot a(10)=\frac{300}{1.1} \cdot 11=\ 3,000$$

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## MATLAB代写

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