Chapter 12.4, Problem 23E

a.

To determine

The amount paid monthly.

Answer to Problem 23E

The monthly payment is $9,020.60.

Explanation of Solution

Monthly payment is the amount a borrower needs to pay every month until a debt is paid off. Usually monthly payments are specified in the loan document regarding how they are calculated, their due date and what happens if the situation is made as pre-agreed.

Formula of loan amount:

$\text{Monthly}\text{payment}\text{=}\frac{i\times \left(\text{Loan amount}\right)}{1-{\left(1+i\right)}^{-n}}$

Where,

i= interest amount

$A_p$ = loan amount

$n$= period

Given:

Loan amount: $100,000

Period: 30 years

Total number of payment: $30\times 12=360\text{times}$

Interest rate: $9\frac{3}{4}\%$

Calculation:

Calculate of the monthly payment for car:

$\begin{array}{c}\text{Monthly}\text{payment}\text{=}\frac{i\times \left(A_p\right)}{1-{\left(1+i\right)}^{-n}}\\ =\frac{\frac{9\frac{3}{4}\%}{12}\times 100,000}{1-{\left(1+\frac{9\frac{3}{4}\%}{12}\right)}^{-30\times 12}}\\ =\frac{\frac{9.75\%}{12}\times 100,000}{1-{\left(1+\frac{9.75\%}{12}\right)}^{-30\times 12}}\\ =\frac{0.008125\times 100,000}{1-{\left(1+0.008125\right)}^{-30\times 12}}\end{array}$

On further simplification as follows,

$\begin{array}{c}=\frac{0.008125\times 100,000}{1-{\left(1+0.008125\right)}^{-30\times 12}}\\ =\frac{\$812.5}{0.945697}\\ =\$859.15\end{array}$

Therefore, monthly payment is $859.15.

b.

To determine

The total amount payable over 30 year’s period.

Answer to Problem 23E

The total amount pays over 30 years of period $309,295.59.

Explanation of Solution

Calculation of total amount pay over 30 year’s period:

$\begin{array}{c}\text{Total}\text{amount}\text{=}\text{Monthly}\text{payment }\times \text{ payment period }\\ \text{=\$859}\text{.15}\times \left(30\times 12\right)\\ =\$309,295.59\end{array}$

Therefore, the total amount pay over 30 year’s period is $309,295.59.

c.

To determine

The amount at the end of 30 year periods.

Answer to Problem 23E

The amount at the end of 30-year period is $1,841.528.75.

Explanation of Solution

Amount of annuity is a sum of all individual payment which is paid for given period with interest. Annuity is sum of money which paid in regular equal payment.

Formula:

Amount of annuity formula is

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

Where,

A = amount of annuity

i = interest rate

n = period

 R = regular equal payment

Calculation:

Calculate the amount at the end of 30-year period:

$\begin{array}{c}\text{Amount of annuity =}R\frac{{\left(\text{1+i}\right)}^{\text{n}}-1}{i}\\ =859.19\times \frac{{\left(1+\frac{9.75\%}{12}\right)}^{30\times 12}-1}{\frac{9.75\%}{12}}\\ =859.19\times \frac{{\left(1+0.008125\right)}^{360}-1}{0.008125}\\ =859.19\times \frac{18.41528752-1}{0.008125}\end{array}$

On further simplification as follows,

$\begin{array}{c}=\$859.19\times \frac{17.41528752}{0.008125}\\ =\$859.19\times 2,143.42\\ =\$1,841,528.75\end{array}$

Therefore, the amount at the end of 30 year periods is $1,841.528.75.