Chapter 11.2, Problem 28E

(a)

To determine

To find:

Themonthly payment pay on a home mortgage and payments made for $30$ years.

Answer to Problem 28E

  $P_0\approx 846.05\text{dollars}$

Explanation of Solution

Given:

The present value of an annuity is $=P_n$

Concept used:

Formula

  $P_0=P\left[\frac{1-{\left(1+i\right)}^{-n}}{i}\right]$

Where $i$ is the interest rate for the period?

Calculation:

The interest rate for the period

  $\begin{array}{l}i=\frac{\text{annual}\text{rate}}{12}\\ i=\frac{0.075}{12}\\ i=0.00625\end{array}$

The period for n

  $\begin{array}{l}n=30\times 12\\ n=360\end{array}$

The monthly payment

  $\begin{array}{l}{P}_n=P_0\left[\frac{1-{\left(1+i\right)}^{-n}}{i}\right]\\ 121,000=P_0\left[\frac{1-{\left(1+0.00625\right)}^{-360}}{0.00625}\right]\\ P_0=\frac{121000}{\frac{1-{\left(1+0.00625\right)}^{-360}}{0.00625}}\\ P_0\approx 846.05\text{dollars}\end{array}$

(b)

To determine

To find:

The monthly payment pay on a home mortgage and payments made for $20$ years.

Answer to Problem 28E

  $P_0\approx 956.21\text{dollars}$

Explanation of Solution

Given:

The present value of an annuity is $=P_n$

Concept used:

Formula

  $P_0=P\left[\frac{1-{\left(1+i\right)}^{-n}}{i}\right]$

Where $i$ is the interest rate for the period?

Calculation:

The interest rate for the period

  $\begin{array}{l}i=\frac{\text{annual}\text{rate}}{12}\\ i=\frac{0.0725}{12}\\ i=0.00604\end{array}$

The period for n

  $\begin{array}{l}n=20\times 12\\ n=240\end{array}$

The monthly payment

  $\begin{array}{l}{P}_n=P_0\left[\frac{1-{\left(1+i\right)}^{-n}}{i}\right]\\ 121,000=P_0\left[\frac{1-{\left(1+0.00604\right)}^{-360}}{0.00604}\right]\\ P_0=\frac{121000}{\frac{1-{\left(1+0.00604\right)}^{-240}}{0.00604}}\\ P_0\approx 956.21\text{dollars}\end{array}$

(c)

To determine

To find:

The interest over the life of each mortgage.

Answer to Problem 28E

$1853578$ , $108490.4$

Explanation of Solution

Given:

The present value of an annuity is $=P_n$

Concept used:

Formula

The interest amount $=$ the monthly payment $\times 12\times$ number of years $121000$

Calculation:

For $30\text{years}$

The interest amount $=$ the monthly payment $\times 12\times$ number of years $121000$

  $\begin{array}{l}=846.05\times 12\times 30-121000\\ =1853578\end{array}$

For $20\text{years}$

The interest amount $=$ the monthly payment $\times 12\times$ number of years $121000$

  $\begin{array}{l}=846.05\times 12\times 20-121000\\ =108490.4\end{array}$

(d)

To determine

To explain:

The borrower might choose each mortgage.

Answer to Problem 28E

$1853578$ , $108490.4$

Explanation of Solution

Given:

The present value of an annuity is $=P_n$

Concept used:

Formula

The interest amount $=$ the monthly payment $\times 12\times$ number of years $121000$

Calculation:

For $30\text{years}$

The interest amount $=$ the monthly payment $\times 12\times$ number of years $121000$

  $\begin{array}{l}=846.05\times 12\times 30-121000\\ =1853578\end{array}$

For $20\text{years}$

The interest amount $=$ the monthly payment $\times 12\times$ number of years $121000$

  $\begin{array}{l}=846.05\times 12\times 20-121000\\ =108490.4\end{array}$

Since monthly payment for $20\text{years}$

$>$ monthly payment for $30\text{years}$

The amount of interest for $20\text{years}$

$<$ the amount of interest for $30\text{years}$

A borrower might choose the $30\text{years}$ mortgage in order to have a lower monthly payment

A borrower might choose the $20\text{years}$ mortgage in order to have a lower interest expense.