## 数学代写|金融数学代写Intro to Mathematics of Finance代考|INTEREST PAYABLE pTHLY

Suppose that, as in the preceding chapter, the force of interest per unit time is constant and equal to $\delta$. Let $i$ and $d$ be the corresponding rates of interest and discount, respectively. In Section $3.1$ we showed that $d$ payable at time 0, $i$ payable at time 1 , and $\delta$ payable continuously at a constant rate over the time interval $[0,1]$ all have the same value (on the basis of the force of interest $\delta$ ). Each of these payments may be regarded as the interest for the period $[0,1]$ payable on a loan of 1 made at time $t=0$.

Suppose, however, that a borrower, who is lent 1 at time $t=0$ for repayment at time $t=1$, wishes to pay the interest on his loan in pequal installments over the interval. How much interest should he pay? This question motivates what follows.

We define $i^{(p)}$ to be that total amount of interest, payable in equal installments at the end of each $p$ th subinterval (i.e., at times $1 / p, 2 / p, 3 / p, \ldots, 1$ ), which has the same value as each of the interest payments just described. Likewise, we define $d^{(p)}$ to be that total amount of interest, payable in equal installments at the start of each $p$ th subinterval (i.e., at times $0,1 / p, 2 / p, \ldots \ldots(p-1) / p)$, which has the same value as each of these other payments.

We may easily express $i^{(p)}$ in terms of $i$. Since $i^{(p)}$ is the total interest paid, each interest payment is of amount $i^{(p)} / p$ and, when we consider the present value of the payments at the end of the interval, our definition implies the following
$$\sum_{t=1}^p \frac{i^{(p)}}{p}(1+i)^{(p-t) / p}=i$$ or, if $i \neq 0$,
$$\frac{i^{(p)}}{p}\left[\frac{(1+i)-1}{(1+i)^{1 / p}-1}\right]=i$$
Hence,
$$i^{(p)}=p\left[(1+i)^{1 / p}-1\right]$$
and
$$\left[1+\frac{i^{(p)}}{p}\right]^p=1+i$$
Note that the last two equations are valid even when $i=0$.

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|ANNUITIES PAYABLE pTHLY: PRESENT VALUES

The nominal rates of interest and discount introduced in the preceding section are of particular importance in relation to annuities which are payable more frequently than once per unit time. We shall refer to an annuity which is payable $p$ times per unit time as payable pthly.

If $p$ and $n$ are positive integers, the notation $a_m^{(\rho)}$ is used to denote the present value at time 0 of a level annuity payable $p$ thly in arrears at the rate of 1 per unit time over the time interval $[0, n]$. For this annuity the payments are made at times $1 / p, 2 / p, 3 / p, \ldots, n$, and the amount of each payment is $1 / p$.
It is a simple matter to derive an expression for $a_n^{(\rho)}$ from first principles. However, the following argument, possibly less immediately obvious, is an important illustration of a kind of reasoning which has widespread application.
By definition, a series of $p$ payments, each of amount $i^{(p)} / p$ in arrears at $p$ thly subintervals over any unit time interval, has the same present value as a single payment of amount $i$ at the end of the interval. By proportion, $p$ payments, each of amount $1 / p$ in arrears at $p$ thly subintervals over any unit time interval, have the same present value as a single payment of amount $i / i^{(p)}$ at the end of the interval. Consider now that annuity for which the present value is $a_\pi^{(p)}$. The $p$ payments after time $r-1$ and not later than time $r$ therefore have the same value as a single payment of amount $i / i^{(p)}$ at time $r$. This is true for $r=1,2, \ldots, n$, so the annuity has the same value as a series of $n$ payments, each of amount $i / i^{(p)}$, at times $1,2, \ldots, n$.

# 金融数学代考

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|INTEREST PAYABLE pTHLY

$$\sum_{t=1}^p \frac{i^{(p)}}{p}(1+i)^{(p-t) / p}=i$$

$$\frac{i^{(p)}}{p}\left[\frac{(1+i)-1}{(1+i)^{1 / p}-1}\right]=i$$

$$i^{(p)}=p\left[(1+i)^{1 / p}-1\right]$$

$$\left[1+\frac{i^{(p)}}{p}\right]^p=1+i$$

## 数学代写|金融数学代写Intro to Mathematics of Finance代考|ANNUITIES PAYABLE pTHLY: PRESENT VALUES

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