Chapter 8.3, Problem 66E

a.

To determine

To solve both of the repayment equations for L.

Answer to Problem 66E

  $L=\frac{M}{1+i}$ , $L=M\left[\frac{1}{1+i}+\frac{1}{{(1+i)}^2}\right]$

Explanation of Solution

Given:

The equation for repayment is $L\left(1+i\right)-M=0$ . For a 2-month loan, $t=2$

  $t=2$, the equation is $[L\left(1+i\right)-M]\left(1+i\right)-M=0$ .

Calculation:

Solving the first equation as follows-

  $\begin{array}{l}L\left(1+i\right)-M=0\\ L=\frac{M}{1+i}\end{array}$

Solving the second equation as follows-

  $\begin{array}{l}[L\left(1+i\right)-M]\left(1+i\right)-M=0\\ [L\left(1+i\right)-M]\left(1+i\right)=M\\ [L\left(1+i\right)-M]=\frac{M}{1+i}\\ L=M\left[\frac{1}{1+i}+\frac{1}{{(1+i)}^2}\right]\end{array}$

b.

To determine

To write a repayment equation for a t month loan.

Answer to Problem 66E

  $L=M\left[\frac{1}{1+i}+\frac{1}{{(1+i)}^2}+\mathrm{...........}+\frac{1}{{(1+i)}^t}\right]$

Explanation of Solution

Given:

The equation for repayment is $L\left(1+i\right)-M=0$ . For a 2-month loan, $t=2$

  $t=2$, the equation is $[L\left(1+i\right)-M]\left(1+i\right)-M=0$ .

Calculation:

After observing the first two equations closely, the equation for a t month loan can be written as follows-

  $L=M\left[\frac{1}{1+i}+\frac{1}{{(1+i)}^2}+\mathrm{...........}+\frac{1}{{(1+i)}^t}\right]$

c.

To determine

To show that the formula in part (b) is equivalent to $M=L\left(\frac{1}{1-{\left(1+i\right)}^{-1}}\right)$

Explanation of Solution

Given:

The equation for repayment is $L\left(1+i\right)-M=0$ . For a 2-month loan, $t=2$

  $t=2$, the equation is $[L\left(1+i\right)-M]\left(1+i\right)-M=0$ .

Calculation:

Solving the equation as follows-

  $\begin{array}{l}L=M\left[\frac{1}{1+i}+\frac{1}{{(1+i)}^2}+\mathrm{...........}+\frac{1}{{(1+i)}^t}\right]\\ L=M\cdot \frac{1}{1+i}\cdot \frac{1-\frac{1}{{(1+i)}^t}}{1-\frac{1}{1+i}}\\ L=M\left[1-\frac{1}{{(1+i)}^t}\right]\cdot \frac{1}{i}\\ M=\frac{Li}{1-\frac{1}{{(1+i)}^t}}\\ M=L\left(\frac{1}{1-{\left(1+i\right)}^{-1}}\right)\end{array}$