Respuesta :
Answer:
Time period required to pay off the mortgage = 18 years
Explanation:
Note: This question is incomplete and lacks necessary data to solve. But I have found that necessary data on the internet, which I have written down and solved the question accordingly.
Data Missing:
Buying Cost of House = $320000
Interest rate = 7%
Annual Mortgage Payment = $25525.8
Now, we are required to calculate the time period required to pay off the mortgage.
Solution:
Data Given:
Increase in annual payment percentage = 5%
So,
Formula:
P = C[tex]e^{A-i}[/tex] + C[tex]e^{2(A-i)}[/tex] + C[tex]e^{3(A-i)}[/tex] + ........ + C[tex]e^{n(A-i)}[/tex]
Where,
P = Buying Cost of House = $320000
i = interest rate = 7% = 0.07
A = Increase in annual payment percentage = 5% = 0.05
C = Annual Mortgage Payment = $25525.8
P = C[tex]e^{A-i}[/tex] + C[tex]e^{2(A-i)}[/tex] + C[tex]e^{3(A-i)}[/tex] + ........ + C[tex]e^{n(A-i)}[/tex]
In this formula, we have all the required things expect the value of n, which we have to calculate.
n = Time period required to pay the mortgage.
So,
$320000 = 25525.8 [tex]e^{0.05 - 0.07}[/tex] + 25525.8 [tex]e^{2(0.05 - 0.07)}[/tex] + 25525.8 [tex]e^{3(0.05 - 0.07)}[/tex] + ..... + 25525.8 [tex]e^{n(0.05 - 0.07)}[/tex]
Taking 25525.8 common,
320000 = 25525.8 ( [tex]e^{-0.02}[/tex] + [tex]e^{-0.04}[/tex] + [tex]e^{-0.06}[/tex] + .... + [tex]e^{-0.02n}[/tex] )
320000/25525.8 = ( [tex]e^{-0.02}[/tex] + [tex]e^{-0.04}[/tex] + [tex]e^{-0.06}[/tex] + .... + [tex]e^{-0.02n}[/tex] )
12.536 = ( [tex]e^{-0.02}[/tex] + [tex]e^{-0.04}[/tex] + [tex]e^{-0.06}[/tex] + .... + [tex]e^{-0.02n}[/tex] )
Taking e common:
12.536 = [tex]e^{-0.02 -0.04 - 0.06 + .... -0.02n}[/tex]
Taking Ln to solve for n, we get:
n = 17.89
n ≈ 18
n = 18 years
Hence, Time period required to pay off the mortgage = 18 years
