Chapter 12.4, Problem 24E

(a)

To determine

To find: The monthly payments of 30-year mortgage and the monthly payments of 15-year mortgage and explain which mortgage has the largest monthly payment.

Answer to Problem 24E

The Monthly payments of 30-year mortgage are $1,896.20 and the Monthly payments of 15-year mortgage are $2,491.23.

The larger monthly payment is 15-year mortgage of $2,491.23

Explanation of Solution

Given:

Mortgage Amount = $300,000

Number of Payments = 30 years (or 360 months)

Interest Rate = 6½% or 6.50% per year (or 0.5417% monthly)

Number of Payments = 15 years (or 180 months)

Interest Rate = 5¾% or 5.75% per year (or 0.4792% monthly)

Formula:

$R=\left[\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}\right]$

Here,

A_p = Present Value

R = Periodic Payment

i = Interest Rate

n = Number of Years/Payments

Mortgage: A mortgage is considered as a guarantee or a security for a loan on property or real estate. The mortgagor enters into an agreement with a mortgagee to receive loan amount initially and repay the loan during a specified time period and interest rate.

Calculation:

The 30-year monthly payment is calculated by multiplying the present value with interest rate and the result is divided by 1 minus, 1 plus interest rate to the power minus number of years.

$\begin{array}{c}R=\left[\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}\right]\\ =\left[\frac{\left(\frac{6.50\%}{12}\right)\times \$300,000}{1-{\left(1+\left(\frac{6.50\%}{12}\right)\right)}^{-30\times 12}}\right]\\ =\left[\frac{0.5417\%\times \$300,000}{1-{\left(1+0.5417\%\right)}^{-360}}\right]\\ =\left[\frac{\$1,625}{1-0.1430}\right]\\ =\left[\frac{\$1,625}{0.8570}\right]\\ =\$1,896.2040or\$1,896.20\end{array}$

Therefore the Monthly payments of 30-year mortgage are $1,896.20.

The 15-year monthly payment is calculated by multiplying the present value with interest rate and the result is divided by 1 minus, 1 plus interest rate to the power minus number of years.

$\begin{array}{c}R=\left[\frac{i{A}_p}{1-{\left(1+i\right)}^{-n}}\right]\\ =\left[\frac{\left(\frac{5.75\%}{12}\right)\times \$300,000}{1-{\left(1+\left(\frac{5.75\%}{12}\right)\right)}^{-15\times 12}}\right]\\ =\left[\frac{0.4792\%\times \$300,000}{1-{\left(1+0.4792\%\right)}^{-180}}\right]\\ =\left[\frac{\$1,437.50}{1-0.4230}\right]\\ =\left[\frac{\$1,437.50}{0.5770}\right]\\ =\$2,491.2303or\$2,491.23\end{array}$

Therefore the Monthly payments of 15-year mortgage are $2,491.23.

The monthly payment of 30-year mortgage is $1,896.20 and the monthly payment of 15-year mortgage is $2,491.23. The results suggest that the 15-year mortgage has the larger monthly payment than the 30-year mortgage. The 15-year mortgage has a larger monthly payment because it has a lower interest rate and term.

(b)

To determine

To find: Total amount paid for each loan and lower total payment for each loan.

Answer to Problem 24E

The total amount paid for 30-year mortgage is $682,633.47 and the total amount paid for 15-year mortgage is $448,421.45.

The lowest total payment over lifetime is 15-year mortgage with $448,421.45.

Explanation of Solution

Given:

Number of Payments = 30 years (or 360 months)

Number of Payments = 15 years (or 180 months)

Formula:

$\text{Total}\text{amount}\text{paid}=\left[\text{R}\times \text{n}\right]$

Calculation:

The total amount paid for each loan is calculated by multiplying the monthly payment of each loan with the number of periods of each loan.

$\begin{array}{c}\text{Total}\text{amount}{\text{paid}}_{\text{30-Year}\text{Mortgage}}=\left[R\times n\right]\\ =\left[\$1,896.2040\times \left(30\times 12\right)\right]\\ =\left[\$1,896.2040\times 360\right]\\ =\$682,633.4654or\$682,633.47\end{array}$

$\begin{array}{c}\text{Total}\text{Amount}{\text{Paid}}_{\text{15-Year}\text{Mortgage}}=\left[R\times n\right]\\ =\left[\$2,491.2303\times \left(15\times 12\right)\right]\\ =\left[\$2,491.2303\times 180\right]\\ =\$448,421.447or\$448,421.45\end{array}$

Therefore the total amount paid for 30-year mortgage is $682,633.47 and the total amount paid for 15-year mortgage is $448,421.45.

The total amount paid of 30-year mortgage is $682,633.47 and the total amount paid of 15-year mortgage is $448,421.45. The results suggest that the 15-year mortgage has the lowest total payments than the 30-year mortgage.