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|THE GENERAL LOAN SCHEDULE

Suppose that at time $t=0$ an investor lends an amount $L$ in return for a series of $n$ payments, the $r$ th payment, of amount $x_r$ being due at time $r(1 \leq r \leq n)$. Suppose further that the amount lent is calculated on the basis of an effective annual interest rate $i_r$ for the rth year $(1 \leq r \leq n)$. In many situations $i_r$ may not depend on $r$, but at this stage it is convenient to consider the more general case.
The amount lent is simply the present value, on the stated interest basis, of the repayments
\begin{aligned} L=& x_1\left(1+i_1\right)^{-1}+x_2\left(1+i_1\right)^{-1}\left(1+i_2\right)^{-1} \ &+x_3\left(1+i_1\right)^{-1}\left(1+i_2\right)^{-1}\left(1+i_3\right)^{-1}+\cdots \ & \quad \cdots+x_n\left(1+i_1\right)^{-1}\left(1+i_2\right)^{-1} \cdots\left(1+i_n\right)^{-1} \end{aligned}
The investor may consider part of each payment as interest (for the latest period) on the outstanding loan and regard the balance of each payment as a capital repayment, which is used to reduce the amount of the loan outstanding. If any payment is insufficient to cover the interest on the outstanding loan, the shortfall in interest is added to the amount of the outstanding loan. In this situation the investor may draw up a schedule which shows the amount of interest contained in each payment and also the amount of the loan outstanding immediately after each payment has been received. It is desirable to consider this schedule in greater detail. The division of each payment into interest and capital is frequently necessary for taxation purposes. Also, in the event of default by the borrower, it may be necessary for the lender to know the amount of loan outstanding at the time of default. In addition, a change in the terms of the loan would prompt a recalculation of the repayment rates based on the loan outstanding at that time.

Let $F_0=L$ and, for $t=1,2, \ldots, n$, let $F_t$ be the loan outstanding immediately after the payment due at time $t$ has been made. The amount of loan repaid at time $t$ is simply the amount by which the payment then made, $x_t$ exceeds the interest then due, $i_t F_{t-1}$. Also, the loan outstanding immediately after the tth payment equals the loan outstanding immediately after the previous payment minus the amount of loan repaid at time $t$. Hence,
$$F_t=F_{t-1}-\left(x_t-i_t F_{t-1}\right) \quad 1 \leq t \leq n$$
Note that this equation holds for $t=1$, since we have defined $F_0=L$. Therefore,
$$F_t=\left(1+i_t\right) F_{t-1}-x_t \quad t \geq 1$$
Hence,
\begin{aligned} F_1 &=\left(1+i_1\right) F_0-x_1 \ &=\left(1+i_1\right) L-x_1 \end{aligned}
Then
\begin{aligned} F_2 &=\left(1+i_2\right) F_1-x_2 \ &=\left(1+i_2\right)\left[\left(1+i_1\right) L-x_1\right]-x_2 \ &=\left(1+i_1\right)\left(1+i_2\right) L-\left(1+i_2\right) x_1-x_2 \end{aligned}
and so on.

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

Consider the particular case where, on the basis of an interest rate of $i$ per unit time, a loan of amount $a_n$ is made at time $t=0$ in return for $n$ repayments, each of amount 1 , to be made at times $1,2, \ldots, n$. The lender may construct a schedule showing the division of each payment into capital and interest.
Immediately after the $t$ th repayment has been made, there remain $(n-t)$ outstanding payments, and the prospective method (Eq. 5.1.5) shows that the outstanding loan is simply $a_{n-1}$. In the notation of Section 5.1,

$$F_{\mathrm{t}}=a_{n-t}$$
then the amount of loan repaid at time $t$ is
\begin{aligned} f_t=F_{t-1}-F_t &=a_{n-t+1}-a_{\overline{n-t}} \ &=v^{n-t+1} \end{aligned}
The lender’s schedule may be presented in the form of Table 5.2.1. More generally, if an amount $L$ is lent in return for $n$ repayments, each of amount $X=L / a_n$, the monetary amounts in the lender’s schedule are simply those in the schedule of Table 5.2.1 multiplied by the constant factor $X$.

金融数学代考

数学代写|金融数学代写Intro to Mathematics of Finance代考|THE GENERAL LOAN SCHEDULE

$$L=x_1\left(1+i_1\right)^{-1}+x_2\left(1+i_1\right)^{-1}\left(1+i_2\right)^{-1} \quad+x_3\left(1+i_1\right)^{-1}\left(1+i_2\right)^{-1}\left(1+i_3\right)^{-1}+\cdots \quad \cdots+x_n$$

$$F_t=F_{t-1}-\left(x_t-i_t F_{t-1}\right) \quad 1 \leq t \leq n$$

$$F_t=\left(1+i_t\right) F_{t-1}-x_t \quad t \geq 1$$

$$F_1=\left(1+i_1\right) F_0-x_1 \quad=\left(1+i_1\right) L-x_1$$
㹜周
$$F_2=\left(1+i_2\right) F_1-x_2 \quad=\left(1+i_2\right)\left[\left(1+i_1\right) L-x_1\right]-x_2=\left(1+i_1\right)\left(1+i_2\right) L-\left(1+i_2\right) x_1-x_2$$

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

$$F_{\mathrm{t}}=a_{n-t}$$

$$f_t=F_{t-1}-F_t=a_{n-t+1}-a_{\overline{n-t}} \quad=v^{n-t+1}$$

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