# 数学代写|金融数学代写Intro to Mathematics of Finance代考|MAT265

## 数学代写|金融数学代写Intro to Mathematics of Finance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 $a_{\text {mp, } i}$. 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
$$v^m a_{\bar{m}, i}=a_{n+m}, i-a_{m, i}$$
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. ## 数学代写|金融数学代写Intro to Mathematics of Finance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$\left(R \cdot a_{\bar{n}, i}\right)$for$m+n$periods $$F V=R a_{\text {万๊, } i}(1+i)^{n+m}$$ b) Accumulating the accumulated value of the annuity just after the last payment$\left(R \cdot s_{\text {пn, }, i}\right)$for$m$periods $$F V=R s_{\eta \eta, i}(1+i)^m$$ 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 $$F V=R\left(s_{\bar{\Pi} \mid n+m, i}-s_{\bar{n}, i}\right)$$ 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$5000 a_{20}, .08299(1.0829995)^{30}$. # 金融数学代考 ## 数学代写|金融数学代写Intro to Mathematics of Finance代考|The Value of an Annuity prior to its Inception 在许多情况下，人们购买年金作为退休收入的来源。在离婚的情况下，通常有必要在年金开始之前确定 其价值，因为年金可能是和解的一部分。正如我们前面所讨论的，人们有时会将剩余的年金款项捐赠给 非营利组织，并在捐赠者去世后开始为非营利组织支付款项。为了确定此类捐赠的税收影响，需要在捐 皆时确定此类递延年金的价值。 为了计算年金在其开始之前的价值，我们只是将递延年金在实际开始付款时的价值按付款开始前的剩余 年数折现。开始时的价值 (付款开始的年份) 是$a_{\mathrm{mp}, i}$. 因此价值$m$成立前的年数由$v^m a_{\eta, i}$. 我们也可以 将其计算为年金$m+n$付款减去第一笔$m$付款给我们 $$v^m a_{\bar{m}, i}=a_{n+m}, i-a_{m, i}$$ 示例$4.14$立即年金将包括十年的每月支付$\$500$ 每个。如果名义年利率为 $6 \%$ 每季度复利?

$$(1+i)^3=1.015$$

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

a) 在开始时㽧积年金的价值 $\left(R \cdot a_{\bar{n}, i}\right)$ 为了 $m+n$ 时期
$$F V=R a_{\text {万 }, i}(1+i)^{n+m}$$
b) 在最后一次付款后㽧积年金的㽧积值 $\left(R \cdot s_{\pi \mathrm{n}, i}\right)$ 为了 $m$ 时期
$$F V=R s_{\eta \eta, i}(1+i)^m$$
c) 将年金视为在整个期间持续支付 $(m+n)$ 然后减去缺失付款的价值
$$F V=R\left(s_{\bar{\Pi} \mid n+m, i}-s_{\bar{n}, i}\right)$$

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