Chapter 4, Problem 16P

a.

Summary Introduction

Examine the better choice of loan if the loan is repaid after 20 years.

Explanation of Solution

To calculate the effective interest rate of the loan the balance after 20 years needed to be calculated, the formula to calculate balance is as follows:

$\text{Mortgage balance}=\frac{\text{Principle}\left[{\left(1+\text{r}\right)}^{\text{n}}-{\left(1+\text{r}\right)}^{\text{m}}\right]}{\left[{\left(1+\text{r}\right)}^{\text{n}}-1\right]}$

Substituting equation with $70,500 for principle, 0.005 for r, 240 for n, that is, number of months and 240 for m, that is, number of installments to calculate mortgage balance.

$\begin{array}{c}\text{Mortgage balance}=\frac{\$70,500\left[{\left(1+0.005\right)}^{240}-{\left(1+0.005\right)}^{240}\right]}{\left[{\left(1+0.005\right)}^{240}-1\right]}\\ =\$40,502.43\end{array}$

Hence, the loan balance of loan A at the end of 20 years will be $40,502.43.

$\text{Mortgage balance}=\frac{\text{Principle}\left[{\left(1+\text{r}\right)}^{\text{n}}-{\left(1+\text{r}\right)}^{\text{m}}\right]}{\left[{\left(1+\text{r}\right)}^{\text{n}}-1\right]}$

Substituting equation with $73,500 for principle, 0.0058 for r, 240 for n, that is, number of months and 240 for m, that is, number of installments to calculate mortgage balance.

$\begin{array}{c}\text{Mortgage balance}=\frac{\$73,500\left[{\left(1+0.0058\right)}^{240}-{\left(1+0.0058\right)}^{240}\right]}{\left[{\left(1+0.0058\right)}^{240}-1\right]}\\ =\$42,975.33\end{array}$

Hence, the loan balance of loan A at the end of 20 years will be $42,975.33.

If the loan is repaid after 20 years then the effective interest rate in loan A is 6.63% with the mortgage balance of $40,502.43 and in case of loan B it is 7.21% with mortgage balance of $42,975.33. Hence, as the effective interest rate in case of loan A is lesser than the loan B; therefore, loan A would be a better choice.

b.

Summary Introduction

Determine the better choice of loan if the loan is repaid after 5 years.

Explanation of Solution

To calculate the effective interest rate of the loan the balance after 20 years needed to be calculated, the formula to calculate balance is as follows:

$\text{Mortgage balance}=\frac{\text{Principle}\left[{\left(1+\text{r}\right)}^{\text{n}}-{\left(1+\text{r}\right)}^{\text{m}}\right]}{\left[{\left(1+\text{r}\right)}^{\text{n}}-1\right]}$

Substituting equation with $70,500 for principle, 0.005 for r, 60 for n, that is, number of months and 60 for m, that is, number of installments to calculate mortgage balance.

$\begin{array}{c}\text{Mortgage balance}=\frac{\$70,500\left[{\left(1+0.005\right)}^{60}-{\left(1+0.005\right)}^{60}\right]}{\left[{\left(1+0.005\right)}^{60}-1\right]}\\ =\$69,790.32\end{array}$

Hence, the loan balance of loan A at the end of 5 years will be $69,790.32.

$\text{Mortgage balance}=\frac{\text{Principle}\left[{\left(1+\text{r}\right)}^{\text{n}}-{\left(1+\text{r}\right)}^{\text{m}}\right]}{\left[{\left(1+\text{r}\right)}^{\text{n}}-1\right]}$

Substituting equation with $73,500 for principle, 0.0058 for r, 60 for n, that is, number of months and 60 for m, that is, number of installments to calculate mortgage balance.

$\begin{array}{c}\text{Mortgage balance}=\frac{\$73,500\left[{\left(1+0.0058\right)}^{60}-{\left(1+0.0058\right)}^{60}\right]}{\left[{\left(1+0.0058\right)}^{60}-1\right]}\\ =\$70,599.14\end{array}$

Hence, the loan balance of loan A at the end of 5 years will be $70,599.14.

If the loan is repaid after 5 years then the effective interest rate in loan A is 7.49% with the mortgage balance of $69,790.32 and in case of loan B it is 7.49% with mortgage balance of $70,599.14. Hence, as the effective interest rate in both the case is same, the borrower is at indifference point; therefore, the borrower is ‘indifferent’ if the loan is repaid after 5 years.