Question

B) Suppose that regular raises at your job allow you to increase your annual payment by 5% each year. For simplicity, assume this is a nominal rate, and your payment amount increases continuously. How long will it take to pay off the mortgage

Answer 1

Time period required to pay off the mortgage = 18 years

Step-by-step 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
`e^(A-i)` + C
`e^(2(A-i))` + C
`e^(3(A-i))` + ........ + C
`e^(n(A-i))`

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
`e^(A-i)` + C
`e^(2(A-i))` + C
`e^(3(A-i))` + ........ + C
`e^(n(A-i))`

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
`e^(0.05 - 0.07)` + 25525.8
`e^(2(0.05 - 0.07))` + 25525.8
`e^(3(0.05 - 0.07))` + ..... + 25525.8
`e^(n(0.05 - 0.07))`

Taking 25525.8 common,

320000 = 25525.8 (
`e^(-0.02)` +
`e^(-0.04)` +
`e^(-0.06)` + .... +
`e^(-0.02n)` )

320000/25525.8 = (
`e^(-0.02)` +
`e^(-0.04)` +
`e^(-0.06)` + .... +
`e^(-0.02n)` )

12.536 = (
`e^(-0.02)` +
`e^(-0.04)` +
`e^(-0.06)` + .... +
`e^(-0.02n)` )

Taking e common:

12.536 =
`e^(-0.02 -0.04 - 0.06 + .... -0.02n)`

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