# 数学代写|金融数学代写Intro to Mathematics of Finance代考|MATH424

## 数学代写|金融数学代写Intro to Mathematics of Finance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 Cash flow problems involve situations where money is deposited and/or withdrawn at various points during the time period of interest.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 (IRR) 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.

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|The Price of an Annuity

Many financial institutions sell annuities. These are often purchased in advance as part of a retirement plan. Suppose we wish to purchase an annuity which pays $\$ R$at the end of each period for$n$periods at an effective interest rate of$i$per period. If the annuity is purchased at the start of the first period its purchase price (not including any profit for the financial institution) is the present value of the annuity: $$\text { Price }=P V=R a_{ض, i}$$ We can compute the payment of an annuity in terms of its price (PV) as well: $$R=\frac{P V}{a_{\bar{\eta}, i}}$$ In most cases an additional fee will be charged by the financial institution. The fee may be built into the PV calculation or billed up front. In the case of home loans these up front fees are called points. Each point is valued at$1 \%$of the loan amount. A loan of$\$200,000$ with an up front charge of 1 point would incur a fee of $.01 \cdot 200,000=\$ 2,000$to be paid at closing. Since this fee is considered interest, it is usually a deductible expense for income tax purposes. We will discuss this further when we consider Truth in Lending Laws in Chapter 5. Example 4.5 A person has$\$10,000$ to invest and wishes to purchase an annuity which will provide constant monthly payments for a period of ten years. If the interest rate used to price the annuity is a nominal rate of annual interest of $12 \%$ compounded monthly, what will the monthly payment be?
Solution: We have $i=\frac{12}{12}=.01, n=120$, and $P V=10,000$. Using Equation $4.9$, we obtain $R=\frac{10000}{a 1200.01}=\$ 143.47$. # 金融数学代考 ## 数学代写|金融数学代写Intro to Mathematics of Finance代考|Deposits and Withdrawals: Cash Flow Problems 我们现在来看一类实际上需要 CF 工作表的问题。现金流问题涉及在感兴趣的时间段内的不同时间点存入和/或提取资金的情况。我们刚刚完成的许多问题都涉及到在特定时间段内的一系列存款和取款。这种存款/取款顺序称为投资计划。我们不对这些存款和取款所赚取的利息做任何假设。他们可能会或可能不会赚取利息，所赚取的利率可能会或可能不会是恒定的。 我们经常对一个称为内部收益率 (IRR) 的数字感兴趣。您可能还记得我们早期的工作： 内部收益率 (IRR) 是单一利率，代表投资计划赚取的有效或平均利率。如果 IRR 用作给定投资计划的（恒定）利率，则最终余额将与实际最终余额相匹配。 如果我们假设账户在关闭之日通过提取所有可用资金而关闭，则序列在开始时将具有净现值$0. IRR 是（恒定）利率，它也将导致 NPV$0. 管理人员使用内部收益率来比较各种计划的回报。产生较高 IRR 值的计划被认为比具有较小 IRR 值的计划更有利可图。 我们从知道利率的情况开始，然后处理涉及计算 IRR 的问题。我们的第一个示例只是我们一直在解决的问题的稍微复杂的版本。 ## 数学代写|金融数学代写Intro to Mathematics of Finance代考|The Price of an Annuity 许多金融机构出售年金。这些通常是作为退休计划的一部分提前购买的。假设我们苃望购买支付的年金$\$R$ 在每个时期结束时 $n$ 有效利率为 $i$ 每个时期。如果在第一期开始时购买年金，则其购买价格（不包括金 融机构的任何利润) 是年金的现值:
$$\text { Price }=P V=R a_{\omega, i}$$

$$R=\frac{P V}{a_{\bar{\eta}, i}}$$

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