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Question 1202788: A. The Taylors have purchased a $240,000 house. They made an initial down payment of $10,000 and secured a mortgage with interest charged at the rate of 5%/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.)
$
B. 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 $
This question has two parts since part b requires part a and I'm confused on what I am doing wrong cause my part a is wrong to which leads my part b wrong too.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part A
Monthly payment formula
P = (L*i)/( 1-(1+i)^(-n) )
where,
P = monthly payment
L = loan amount
i = monthly interest rate in decimal form
n = number of months
In this case- L = 240,000 - 10,000 = 230,000 (don't forget to subtract off the down payment)
- i = 0.05/12 = 0.004166667 approximately
- n = 30*12 = 360 months (equivalent to 30 years)
Let's plug in those values
P = (L*i)/( 1-(1+i)^(-n) )
P = (230000*0.004166667)/( 1-(1+0.004166667)^(-360) )
P = 1234.6897891544
P = 1234.69
Answer:
Monthly Payment = $1234.69
You can use a calculator like this to confirm the answer.
https://www.calculator.net/loan-calculator.html
The calculator will provide the monthly payment.
Also, it provides the amortization table which will come in handy for part B
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Part B
I'll be using the 2nd formula mentioned at this link
https://www.mtgprofessor.com//formulas.htm
That formula is
B = L * [ (1+c)^n - (1+c)^p ] / [ (1+c)^n - 1 ]
it calculates the remaining balance.
It's the amount the Taylors still owe at any given month p.
We have a giant single fraction
numerator = L*((1+c)^n - (1+c)^p)
denominator = (1+c)^n - 1
L = loan amount
c = monthly interest rate in decimal form
n = number of months of entire mortgage
p = current month number
We have
L = 230,000
c = 0.004166667 approximately
n = 360 months
as calculated earlier in part A.
If 5 years, aka 5*12 = 60 months, elapse then p = 60
B = L * [ (1+c)^n - (1+c)^p ] / [ (1+c)^n - 1 ]
B = 230000 * [ (1+0.004166667)^360 - (1+0.004166667)^60 ] / [ (1+0.004166667)^360 - 1 ]
B = 211,206.084993819
B = 211,206.08
The Taylors still owe $211,206.08 after 5 years (aka 60 months) have elapsed.
This can be confirmed with the amortization table (refer to the 1st link mentioned earlier).
Their equity at this point in time is 240,000 - 211,206.08 = $28,793.92
Formula:
Equity = (home value) - (remaining balance)
Side note: A mortgage is considered "underwater" when the remaining balance exceeds the home value; which leads to negative equity.
Let's see how much the Taylors would still owe at the p = 120 month marker (120 months = 120/12 = 10 years)
B = L * [ (1+c)^n - (1+c)^p ] / [ (1+c)^n - 1 ]
B = 230000 * [ (1+0.004166667)^360 - (1+0.004166667)^120 ] / [ (1+0.004166667)^360 - 1 ]
B = 187,086.750587192
B = 187,086.75
Then subtract that from the home value
240,000 - 187,086.75 = 52,913.25
I'll skip the steps for the "20 years" subsection. But it'll be the same idea, except you'll use p = 240 months.
Keep in mind that each equity calculation uses the stagnant home price of $240,000.
This is what your teacher refers to when mentioning the phrase "disregarding appreciation".
Realistically the home price will fluctuate.
Answers:
Equity at 5 years: $28,793.92
Equity at 10 years: $52,913.25
Equity at 20 years: $123,591.78
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