数学代写|金融数学代写Intro to Mathematics of Finance代考|ACFl1003

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数学代写|金融数学代写Intro to Mathematics of Finance代考|THE LOAN SCHEDULE FOR A pTHLY ANNUITY

No new principles are involved in the loan schedule for a pthly annuity, since this is simply a particular example of the general schedule discussed in the preceding sections. For a loan repayable by a level annuity payable $p$ thly in arrears over $n$ time units and based on an interest rate $i$ per unit time, the schedule is best derived by working with an interest rate of $i^{(p)} / p$ per time interval of length $1 / p$. Therefore, the interest due at time $r / p(r=1,2, \ldots, n p)$ is $i^{(p)} / p$ times the loan outstanding at time $(r-1) / p$ (immediately after the repayment then due has been received).

For example, in relation to a loan of $a_n^{(p)}$ (at rate $i$ ), it is simple to show that the capital repaid in the $r$ th annuity payment $(r=1,2, \ldots, n p)$ is $(1 / p) v^{n-(r-1) / p}$ and that the loan outstanding immediately after the $r$ th payment has been received is $a \frac{(p)}{n-r / \rho}$ (at rate $i$ ). This is simply the value of the outstanding payments from the prospective method.

Various countries around the world have enacted laws aimed at making people who borrow money or buy goods or services on credit more aware of the true cost of credit; in particular, the laws enable people to compare the true interest rates implicit in various lending schemes. Examples of laws of this type are the Consumer Credit Act 1974 (revised in 2006) in the UK and the Consumer Credit Protection Act 1968 (which contains the “truth in lending” provisions) in the USA.

Regulations made under powers introduced in the UK Consumer Credit Act 1974 lay down what items should be treated as entering into the total charge for credit and how the rate of charge for credit should be calculated. The rate is known as the Annual Percentage Rate of Charge (APR) and is defined in such a way as to be the effective annual rate of interest on the transaction, obtained by solving the appropriate equation of value for $i$, taking into account all the items entering into the total charge for credit. The $\mathrm{APR}$ is therefore closely associated with the internal rate of return for the loan that we will cover in Chapter 6 . In all cases the APR is to be quoted to the lower one-tenth of $1 \%$. For example, if the rate $i$ is such that $0.155 \leq i<0.156$, the quoted APR is $15.5 \%$. The total charge for credit and the APR have to be agreements.

Regulation $Z$ of the US Consumer Credit Protection Act 1968 requires the disclosure of the “finance charge” (defined as the excess of the total repayments over the amount lent) and the “annual percentage rate”, which is the nominal rate of interest per annum convertible as often as the repayments are made (e.g., monthly or weekly). The value quoted must be accurate to onequarter of $1 \%$.

数学代写|金融数学代写Intro to Mathematics of Finance代考|Flat Rates of Interest

In many situations in which a loan is to be repaid by level installments at regular intervals, it is occasionally common commercial practice to calculate the amount of each repayment installment by specifying a flat rate of interest for the transaction. The operation of flat rates of interest is as follows.

Consider a loan of $L_0$. Which is to be repayable over a certain period by $n$ level installments. Suppose that the flat rate of interest for the transaction is $F$ per specified time unit. (Note that the time unit used to specify $F$ need not be the time interval between repayments; in practice the time unit used to specify $F$ is generally a year.) The total charge for credit for the loan is defined to be
$$
D=L_0 F k
$$
where $k$ is the repayment period of the loan, measured in units of time used in the definition of $F$.

The total amount repaid is defined to be the amount of the loan plus the charge for credit, i.e., $\left(L_0+D\right)$. Each installment is therefore of amount
$$
X=\frac{L_0+D}{n}
$$
This together with Eq. 5.4.1 leads to the definition of the flat rate
$$
F=\frac{X n-L_0}{L_0 k}
$$
which defines the flat rate as the total interest paid per unit time, per unit borrowed.

One can calculate the flat rate from the repayment installments and the amount of the loan using Eq. 5.4.3. For example, if the loan $L_0$ is repaid over 2 years by level monthly installments of $X$, the total paid is $24 X$. This amount includes the total capital and interest paid, so the interest is $24 X-L_0$. The flat rate is therefore
$$
F=\frac{24 X-L_0}{2 L_0}
$$
The flat rate is a simple calculation that ignores the details of gradual repayment of capital over the loan. For this reason it is only useful for comparing loans of equal term. Since the flat rates ignore the repayment of capital over the term of the loan, it will be considerably lower than the true effective rate of interest charged on the loan.

数学代写|金融数学代写Intro to Mathematics of Finance代考|ACFl1003

金融数学代考

数学代写|金融数学代写Intro to Mathematics of Finance代考|THE LOAN SCHEDULE FOR A pTHLY ANNUITY

pthly 年金的贷款时间表没有涉及新的原则,因为这只是前面几节中讨论的一般时间表的一个特定示例。以应付水平年金偿还的贷款p欠款超过n时间单位和基于利率一世每单位时间,时间表最好通过使用利率一世(p)/p每个长度的时间间隔1/p. 因此,按时到期的利息r/p(r=1,2,…,np)是一世(p)/p乘以当时未偿还的贷款(r−1)/p(在收到到期还款后立即)。

例如,关于贷款一个n(p)(按速率一世),很容易证明,在r年金支付(r=1,2,…,np)是(1/p)在n−(r−1)/p并且在贷款之后立即未偿还的贷款r已收到的付款是一个(p)n−r/r(按速率一世)。这只是预期方法中未偿付款的价值。

世界各国已颁布法律,旨在让以信贷方式借钱或购买商品或服务的人更加了解信贷的真实成本;特别是,法律使人们能够比较各种贷款计划中隐含的真实利率。这类法律的例子是英国的 1974 年消费者信用法(2006 年修订)和美国的 1968 年消费者信用保护法(其中包含“贷款真相”条款)。

根据 1974 年《英国消费者信贷法》中引入的权力制定的法规规定了哪些项目应被视为计入信贷总收费以及应如何计算信贷收费率。该利率被称为年费率(APR),其定义方式为交易的有效年利率,通过求解适当的价值方程获得一世, 考虑到所有进入信贷总额的项目。这一个磷R因此,它与我们将在第 6 章中介绍的贷款的内部收益率密切相关。在所有情况下,APR 的报价都应为较低的十分之一1%. 例如,如果利率一世是这样的0.155≤一世<0.156, 引用的 APR 是15.5%. 信贷的总费用和 APR 必须是协议。

规定从1968 年美国消费者信用保护法案要求披露“财务费用”(定义为总还款额超过借出金额的部分)和“年利率”,即每年可兑换的名义利率通常在还款时(例如,每月或每周)。报价必须精确到四分之一1%.

数学代写|金融数学代写Intro to Mathematics of Finance代考|Flat Rates of Interest

在许多需要定期分期偿还佮款的情况下,通常的商业惯例是通过指定交易的固定利率来计算每期还款的金 额。固定利率的操作如下。
考虑贷款 $L_0$. 应在一定期限内偿还 $n$ 水平分期付款。假设交易的固定利率为 $F$ 每个指定的时间单位。(请注 意,用于指定的时间单位 $F$ 不必是还款之间的时间间隔;在实践中用于指定的时间单位 $F$ 通常是一年。) 侙款的信贷总额定义为
$$
D=L_0 F k
$$
在哪里 $k$ 是代款的还款期,以以定义中使用的时间单位计量 $F$.
偿还的总金额定义为贷款金额加上信㑔费用,即 $\left(L_0+D\right)$. 因此,每一期的金额
$$
X=\frac{L_0+D}{n}
$$
这与方程式一起。5.4.1引出统一岪率的定义
$$
F=\frac{X n-L_0}{L_0 k}
$$
它将统一利率定义为每单位时间支付的总利息,每单位借入。
可以使用方程式从分期付款和侣款金额计算统一利率。5.4.3. 例如,如果贷款 $L_0$ 在 2 年内按月分期偿还 $X$ ,支付的总金额为 $24 X$. 该金额包括支付的总资本和利息,因此利息为 $24 X-L_0$. 因此,统一费率是
$$
F=\frac{24 X-L_0}{2 L_0}
$$
统一利率是一个简单的计算,忽略了逐步偿还侙款资本的细节。出于这个原因,它仅用于比较等额偘款。 由于统一利率忽略了喪款期限内的资本偿还,因此它将大大低于贷款的直实有效利率。

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