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

## 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.

# 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）。

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

$$D=L_0 F k$$

$$X=\frac{L_0+D}{n}$$

$$F=\frac{X n-L_0}{L_0 k}$$

$$F=\frac{24 X-L_0}{2 L_0}$$

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