### 金融代写|金融数学代写Financial Mathematics代考|ACTL20001

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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
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• 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|Annuities Payable at Different Frequencies than Interest Is Convertible

Method One: Convert the conversion period of the interest to the period of the payments.

The simplest way to deal with the case in which the annuity payments are made at a different frequency than the interest conversion frequency is to convert the interest to the equivalent rate convertible at the period of the annuity payments. After that is done, we do all of our calculations using the period of the annuity payments.

We begin with an annuity-immediate paying $\$ R$at the end of each five periods for a total of$n$interest conversion periods. To compute the present value, we sum the present value of each payment. Since it is delayed by$k$interest conversion periods, its present value is$v^{k}$. Each subsequent payment is delayed by an additional$kconversion periods. The present value of the annuity at inception is thus: \begin{aligned} P V &=R\left(v^{k}+v^{2 k}+\cdots v^{\frac{n}{k} k}\right) \ &=R v^{k}\left(1+v^{k}+\cdots+v^{n-k}\right) \ &=R v^{k} \frac{1-v^{n}}{1-v^{k}} \ &=R \frac{a_{\text {꾜 }, i}}{s_{\text {ㅈ, }, i}} \end{aligned} The accumulated value immediately after the last payment is made is computed by multiplying the expression in4.36$by$(1+i)^{n}$We can use this same technique to find the present and future (accumulated) values of annuities for which payments are made at the beginning of each$k$interest conversion periods. ## 金融代写|金融数学代写Financial Mathematics代考|Annuities Paid More Frequently than Interest Is Converted We already know how to do these problems by converting the interest rate to the period of the annuity payments. In this section we develop an alternative method which avoids the conversion problem. We assume that$m$payments are made during each interest conversion period and that there are a total of$n$interest conversion periods. There are thus$m n$payments in all. We assume that the amount of each payment is$\frac{R}{m}$(for a total of$\$R$ each period!).

The symbol $a_{\bar{n}}^{(m)}$ we will used to denote the present value of an annuityimmediate in this situation. Note the similarity to the symbol $i^{(m)}$ which represents the nominal annual rate of interest compounded $m$ times per year.

Remember: $n$ represents the period of the annuity in interest conversion periods and $m$ is the number of payments per interest conversion period.

We compute the present value by summing the appropriate geometric series:
\begin{aligned} a_{\bar{n}}^{(m)} &=\frac{1}{m}\left(v^{\frac{1}{m}}+v^{\frac{2}{m}}+\cdots+v^{n}\right) \ &=\frac{1}{m} \frac{v^{\frac{1}{m}}-v^{n+\frac{1}{m}}}{1-v^{\frac{1}{m}}} \ &=\frac{1}{m} \frac{v^{\frac{1}{m}}}{v^{\frac{1}{m}}}\left(\frac{1-v^{n}}{\left(\frac{1}{v^{\frac{1}{m}}}\right)-1}\right) \ &=\frac{1-v^{n}}{m\left((1+i)^{\frac{1}{m}}-1\right)} \ &=\frac{1-v^{n}}{i^{(m)}}=i \cdot\left(\frac{a_{\text {m, }, i}}{i^{(m)}}\right) \end{aligned}
Here we have used the symbol $i^{(m)}=m\left((1+i)^{\frac{1}{m}}-1\right)$ from an earlier chapter. The accumulated value of this annuity immediately after the last payment is made is denoted by $s_{\bar{n}}^{(m)}$ and is simply the value of $a_{\bar{n}}^{(m)}$ accumulated over the $n$ interest conversion periods.
\begin{aligned} s_{\bar{n}}^{(m)} &=a_{\bar{n}}^{(m)}(1+i)^{n} \ &=\frac{(1+i)^{n}-1}{i(m)} \ &=i \cdot\left(\frac{s_{\text {司, } i}}{i^{(m)}}\right) \end{aligned}

## 金融代写|金融数学代写Financial Mathematics代考|Annuities Payable at Different Frequencies than Interest Is Convertible

TI BA II Plus 解决方案：（表 4.44）我们使用 TVM 键计算利率和未来价值。

## 金融代写|金融数学代写Financial Mathematics代考|Alternative Method: Annuities Payable Less Frequently than Interest Is Convertible

꾜ㅈ磷在=R(在ķ+在2ķ+⋯在nķķ) =R在ķ(1+在ķ+⋯+在n−ķ) =R在ķ1−在n1−在ķ =R一个꾜 ,一世s兆， ,一世

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## MATLAB代写

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