Question

The Taylors have purchased a $310,000 house. They made an initial down payment of $30,000 and secured a mortgage with interest charged at the rate of 7%/year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over 30 years, what monthly payment will the Taylors be required to make? (Round your answer to the nearest cent.)
$

What is their equity (disregarding appreciation) after 5 years? After 10 years? After 20 years? (Round your answers to the nearest cent.)

5 years	$
10 years	$
20 years	$

Answer 1

House price = $ 310,000

Down payment = $ 30,000

Loan amount (present value) = House price – down payment

= $ 310,000 - $ 30,000 = $280,000

Term in year = 30 years

Number of month in 30 years = 30*12 = 360

Annual interest rate = 7%

Monthly interest rate = 7/12 = 0.005833

Present value of Annuity = Annuity * [ 1 - 1/(1+Periodic interest rate)^{number of periods} / Periodic interest rate]

$ 280000 = Annuity * [ 1 - 1/(1+0.005833)³⁶⁰ / 0.005833 ]

$ 280000 = Annuity * 150.3059

Annuity = $280000 / 150.3059 = $ 1862.87

Therefore, the monthly payment on the loan is $1862.87

a)

Equity value after 5 years:

Monthly payment (Annuity) =$1862.87

Time left in years = 25 years (30 –5)

Number of period in 25 years = 25 * 12 = 300

Present value of Annuity = Annuity * [ 1 - 1/(1+Periodic interest rate)^{number of periods} / Periodic interest rate]

= 1862.87 * [ 1 - 1/(1+0.005833)³⁰⁰ / 0.005833 ]

= 1862.87 * 141.4847

= 263581.22

So, the equity after 5 years is $310000 - 263581.22 = 46418.78

b)

Equity value after 10 years:

Monthly payment (Annuity) =$1862.87

Time left in years = 20 years (30 –10)

Number of period in 20 years = 20 * 12 = 240

Present value of Annuity = Annuity * [ 1 - 1/(1+Periodic interest rate)^{number of periods} / Periodic interest rate]

= 1862.87 * [ 1 - 1/(1+0.005833)²⁴⁰ / 0.005833 ]

= 1862.87 * 128.9865

= 240285.08

So, the equity after 10 years is $310000 - 240285.08 = 69714.92

c)

Equity value after 20 years:

Monthly payment (Annuity) =$1862.87

Time left in years = 10 years (30 –20)

Number of period in 10 years = 10 * 12 = 120

Present value of Annuity = Annuity * [ 1 - 1/(1+Periodic interest rate)^{number of periods} / Periodic interest rate]

= 1862.87 * [ 1 - 1/(1+0.005833)¹²⁰ / 0.005833 ]

= 1862.87 * 86.12788

= 160445.05

So, the equity after 20 years is $310000 - 160445.05 = 149554.95

5 years	$ 46,418.78
10 years	$ 69,714.92
20 years	$149,554.95