### 金融代写|金融数学代写Financial Mathematics代考|MATH3090

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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|Deposits and Withdrawals: Cash Flow Problems

We now look at a class of problems where the CF Worksheet is virtually required. Cash flow problems involve situations where money is deposited and/or withdrawn at various points during the time period of interest. Many of the problems we have just finished looking at involve a sequence of deposits and withdrawals over a specified period of time. Such a sequence of deposits/withdrawals is called an investment scheme. We make no assumptions as to the interest earned by these deposits and withdrawals. They may or may not earn interest and the interest rate earned may or may not be constant.

We are often interested in a number called the internal rate of return (IRR). As you may recall from our earlier work:

The internal rate of return (IR.R) is a single interest rate which represents the effective or average rate of interest earned by the investment scheme. If the IRR is used as the (constant) rate of interest for a given investment scheme the final balance will match the actual final balance.

If we assume that the account is closed by withdrawing all funds available on the date of closure the sequence would have a net present value at inception of $\$ 0$. The IRR is the (constant) interest rate which would also result in a NPV of$\$0 .$ The IRR is used by managers to compare the returns on various schemes. Schemes which yield higher IRR values are deemed to be more profitable than those with smaller IRR values.

We begin with situations where we know the interest rate and then proceed to problems which involve calculating the IRR. Our first examples are just slightly more complicated versions of the problems we have been solving.

## 金融代写|金融数学代写Financial Mathematics代考|Fixed Term Annuities-Immediate with Constant Payments

We start our analysis with an annuity-immediate which provides $n$ payments of $\$ 1$at the end of each period. We assume further that the effective interest rate$^{2}$is$i$per period. For example, if the payments are made every month, we will assume that interest is compounded monthly as well. Later we will deal with cases where the payment period and interest conversion periods are not the same. For example, an annuity with monthly payments is valued using an interest rate convertible quarterly. We compute the value of a fixed term annuity-immediate which pays$\$1$ each period at the time of its inception using techniques developed in Chapter 1:
Since the first payment occurs one period after inception it has a value at inception $(t=0)$ of : $1 \cdot \frac{1}{1+i}=\frac{1}{1+i}=v$. The second payment will be delayed two periods and so has a value of $v^{2}$, and so forth. The total value at inception of this annuity is given by the sum $v+v^{2}+\cdots+v^{n}$. This expression occurs so often that it has a special symbol: $a_{\text {пn, } i}$ If we are clear about the interest rate we will write it as just $a_{\text {기. }}$. Using this notation we have:
$$a_{\text {同, } i}=v+v^{2}+v^{3}+\cdots v^{n}=\sum_{i=1}^{n} v^{i}$$
We can simplify this expression by using the fact that it is a geometric series. We can then use Equation $1.1$ to compute its sum. Here is Equation $1.1$
$$a+a r+a r^{2}+a r^{3}+\cdots+a r^{n-1}=a \frac{1-r^{n}}{1-r}$$
Comparing Equation $4.1$ with Equation $1.1$ we see that $a=v, r=v$ and have:
\begin{aligned} a_{\text {䒣 }, i} &=v+v^{2}+v^{3}+\cdots v^{n} \ &=v\left(1+v+v^{2}+\cdots+v^{n-1}\right) \ &=v \frac{1-v^{n}}{1-v} \ &=v \frac{1-v^{n}}{i v} \ &=\frac{1-v^{n}}{i} \end{aligned}
Present Value $(\mathrm{PV})$ of an Annuity Immediate of $\$ 1$per period for$\mathrm{n}$periods $$a_{\bar{n}, i}=\frac{1-v^{n}}{i}$$ To compute the value of$a_{\text {卫 , } i}$using the TVM keys we use the$P M T$key. Since payments cease at the end of the term of the annuity the future value is$\$0$. If we enter PMT $=-1$ the PV will be reported as positive (Table 4.1).

## 金融代写|金融数学代写Financial Mathematics代考|Fixed Term Annuities-Due

An annuity-due consists of a sequence of payments which commence immediately and are then made at the start of each subsequent period. Recall that annuities-immediate provide payments at the end of the each period. We will use $\ddot{a}{\text {ᄁ }, i}$ for the present value of an annuity-due and $\ddot{s}{\text {万n, } i}$ for the accumulated value of an annuity-due. Since an annuity-due starts one term earlier than an annuity-immediate there is a simple relationship between the present values of these two types of annuities:
Present Value of an Annuity-Due
$$\ddot{a}{\text {出 } i}=a{\text {氞 }, i}(1+i)$$
Accumulated Value of an Annunity-Due
$$\vec{s}{\text {ๆ }, i}=s{\text {ๆ, }, i}(1+i)$$
We can also compute these values directly from their definition. For the present value (PV) of an annuity-due we have:
\begin{aligned} \ddot{a}{\text {ฑ , }, i} &=1+v+v^{2}+v^{3}+\cdots v^{n-1} \ &=\frac{1-v^{n}}{1-v} \ &=\frac{1-v^{n}}{i v} \ &=\frac{1-v^{n}}{d} \end{aligned} Likewise, for the accumulated value we have: \begin{aligned} s{\text {畄 }, i} &=(1+i)+(1+i)^{2}+\cdots+(1+i)^{n} \ &=(1+i) \frac{(1+i)^{n}-1}{(1+i)-1} \ &=\frac{(1+i)^{n}-1}{i v} \ &=\frac{(1+i)^{n}-1}{d} \end{aligned}
Recall that $d$ is the discount rate and that $d=\frac{i}{1+i}$.

## 金融代写|金融数学代写Financial Mathematics代考|Fixed Term Annuities-Immediate with Constant Payments

䒣一个䒣 ,一世=在+在2+在3+⋯在n =在(1+在+在2+⋯+在n−1) =在1−在n1−在 =在1−在n一世在 =1−在n一世

## 金融代写|金融数学代写Financial Mathematics代考|Fixed Term Annuities-Due

ๆๆs→其他 ,一世=s其他， ,一世(1+一世)

ฑ一个¨n , ,一世=1+在+在2+在3+⋯在n−1 =1−在n1−在 =1−在n一世在 =1−在nd同样，对于累积值，我们有：

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