### 金融代写|金融数学代写Financial Mathematics代考|STAT2032

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• 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 Value of an Annuity prior to its Inception

In many cases, persons purchase an annuity as a source of retirement income. In the case of a divorce it is often necessary to determine a value of an annuity prior to its inception since the annuity will likely be part of the settlement. As we discussed earlier, persons sometimes donate the remaining payments of an annuity to nonprofit organizations with payments to begin for the nonprofit upon the death of the person making the donation. In order to determine the tax implications of such a donation, it is required to determine the value of such a deferred annuity at the time of its donation.

To compute the value of an annuity prior to its inception, we merely discount the value of the deferred annuity at the time payments actually begin by the number of years remaining until payments begin. The value at inception (the year payments begin) is $a_{\text {匀,i. }}$. Hence the value $m$ years prior to inception is given by $v^{m} a_{\eta, i}$. We can also compute this as an annuity of $m+n$ payments minus the first $m$ payments giving us
$$v^{m} a_{\text {司,i }}=a_{n+m, i}-a_{m, i}$$
Example 4.14 An annuity immediate will consist of ten years of monthly payments of $\$ 500$each. What is the value of this annuity seven months prior to its first payment if the nominal annual interest rate is$6 \%$compounded quarterly? Solution: We first observe that the period for compounding interest does not match the period for payments. The rate of interest per quarter is$\frac{.06}{4}=.015$. We need to convert this to a monthly rate so that the compounding period is equal to the payment period. Let$i$be the monthly interest rate. We then have: $$(1+i)^{3}=1.015$$ Solving for$i$, we obtain$i=.004975206$. We can now keep track of time in months. Since we are interested in the value seven months prior to inception$m=7$. The annuity pays monthly for ten years so$n=120$. The required value is then $$500 \cdot v^{7} \cdot a \prod_{120},-00497=\ 43,557.61$$ ## 金融代写|金融数学代写Financial Mathematics代考|The Value of an Annuity after the Final Payment Is Made Suppose that an annuity consists of$n$payments with an interest rate per payment period of$i$and we want the value of the annuity$m$periods after the final payment is made. We can do this in at least three different ways. a) Accumulating the value of the annuity at inception$\left(R \cdot a_{\rightarrow, i}\right)$for$m+n$periods $$F V=R a_{\text {п, } i}(1+i)^{n+m}$$ b) Accumulating the accumulated value of the annuity just after the last payment$\left(R \cdot s_{\text {त⿴囗二 }, i}\right)$for$m$periods $$F V=R s_{\text {円 }, i}(1+i)^{m}$$ c) Treating the annuity as if the payments had continued for the entire period$(m+n)$and then subtracting the value of the missing payments $$F V=R\left(s_{\text {囯 } n+m, i}-s_{\text {ri, }, i}\right)$$ Example 4.15 An annuity consists of level payments of$\$5,000$ at the end of each year for twenty years. If the prevailing interest rate is a nominal rate of annual interest of $8 \%$ per year compounded monthly, how much must be deposited in five years as a single payment in order that the accumulated value of the annuity and that of the single deposit are equal at the end of thirty years? That is: accumulated value of annuity $=$ accumulated value of the single payment when measured at thirty years.

Solution. The annual interest rate is found on the TI BA II Plus (Table 4.27) using
The effective annual interest rate is $8.2999507=8.3 \%$.
The accumulated value of a single deposit of $\$ x$made at the end of year 5 after thirty years is$x(1+i)^{25}$. The accumulated value of the annuity (using Equation 4.15) is$5000 a$20,.08299$(1.0829995)^{30}$. ## 金融代写|金融数学代写Financial Mathematics代考|The Value of an Annuity at any Time between the First and Last Payments We suppose that the annuity consists of$n$payments with an interest rate per payment period of$i$and want to compute the value of the annuity$m$periods after the inception of the annuity. In this case, we assume further that$m<n$. We also assume that we are computing the value of all payments, not just those remaining at time of our calculation. We have several methods: a) Accumulate the value of the annuity at inception for$m$periods to obtain $$R a_{\text {ๆ }, i}(1+i)^{m}$$ b) Add the accumulated value of the payments already made ($m$of them) to the present value of the payments yet to be made$(n-m$of these). $$R\left(s_{\text {mm, }}, i+a \overline{n-m, i}\right)$$ c) Deflate the accumulated value of the annuity just after the final payment by$n-m$years $$R s_{\text {无 }, i} v^{n-m}$$ We can also compute the value of the remaining payments (as opposed to the value of all the payments). If we want the value of the remaining$n-m$payments just after the$m^{\text {th }}$payment is made, we use the formula for the value at inception of an annuity of$m-n$payments:$a \overline{n-m, i}$– Example 4.16 Bob and Carol are divorcing. Among other assets which must be split up is a forty-year annuity-immediate which they purchased ten years ago with level payments of$\$1,000$ per month. Bob would like a lump sum payment while Carol would prefer to continue to receive monthly payments for the thirty years remaining on the annuity. If the prevailing interest rate is a nominal $11 \%$ annually compounded monthly, what lump sum payment to Bob would be fair? What will Carol’s new payments be?

Solution: The present value of the remaining payments is $1000 a \frac{360, \frac{11}{12}}{=}=$ $\$ 105,006.35$Since they are to split this as a lump sum to Bob plus an annuity for Carol, the present value for each of them must be one-half of the total or$\$52,503.17$. Since Bob has elected a lump sum, this is the amount he receives.

Carol has elected to continue to receive monthly payments – how much should she get each month? That’s easy! Had they elected to share the annuity payments equally, they would each receive $\$ 500$per month. Since the scheme proposed also shares the value equally, Carol’s annuity should be the same in either case and hence her share must be$\$500$ per month – no calculations required.

## 金融代写|金融数学代写Financial Mathematics代考|The Value of an Annuity after the Final Payment Is Made

a) 在开始时累积年金的价值(R⋅一个→,一世)为了米+n时期

пF在=R一个n, 一世(1+一世)n+米
b) 在最后一次付款后累积年金的累积值त⿴囗二(R⋅sत⿴囗Ⅱ ,一世)为了米时期

c) 将年金视为在整个期间持续支付(米+n)然后减去缺失付款的价值

## 金融代写|金融数学代写Financial Mathematics代考|The Value of an Annuity at any Time between the First and Last Payments

a) 在开始时累积年金的价值米获得时间

ๆR一个其他 ,一世(1+一世)米
b) 加上已付款的累计值 (米其中）到尚未支付的款项的现值(n−米这些）。

R(s毫米， ,一世+一个n−米,一世¯)
c) 在最后一次付款后将年金的累积值缩减为n−米年

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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。