Question 1197529: The Taylors have purchased a 290000 house. they made an initial payment of 10000 and secured a mortgage with interest charged at the rate of 6%/year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized of 30 years, what monthly payment will the Taylor's be required to make? What is their equity after 5 years, 10 years and 20 years?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Home value = $290,000
Down payment = $10,000
Amount loaned = 290,000 - 10,000 = $280,000
The monthly payment formula is
M = (L*i)/( 1-(1+i)^(-n) )
where,
L = loan amount
i = interest rate per month, in decimal form
n = number of months
In this case:
L = 280,000
i = 0.06/12 = 0.005
n = 360 months (equivalent to 30 years)
Let's compute the monthly payment
M = (L*i)/( 1-(1+i)^(-n) )
M = (280000*0.005)/( 1-(1+0.005)^(-360) )
M = 1678.74147042771
M = 1678.74
You can use a specialized online calculator such as these
https://www.calculator.net/loan-calculator.html
https://www.bankrate.com/loans/loan-calculator/
to confirm the correct monthly payment.
Alternatively, you can type this command into a spreadsheet
=PMT(0.005,360,280000,0,0)
or
=PMT(0.005,360,-280000,0,0)
to compute the monthly payment.
Yet another alternative is to use a TI83 or TI84 calculator.
Press the button labeled "APPS". Select "Finance" then select "TVM Solver".
TVM = time value of money
This will be the set of inputs
N = 360
I% = 0.5
PV = 280000
After those inputs are entered, move the cursor down to PMT. Then press the key labeled "Alpha" and then "Enter" and -1678.74147 should show up
Using PV = -280000 is equally valid. All we care about really is the absolute value of the result.
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Next, I'll use the remaining balance formula mentioned on this page
https://www.mtgprofessor.com//formulas.htm
That formula is
B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )
where the L, i, and n are the same as before.
I'm using the letter 'i' to replace the letter 'c' mentioned in the link.
The p refers to the number of months you are into the loan.
Let's find the remaining balance after 5 years, aka 5*12 = 60 months
B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )
B = 280000*( (1+0.005)^360 - (1+0.005)^60 )/( (1+0.005)^360 - 1 )
B = 260552.199103894
B = 260552.20
The remaining balance after 5 years is $260,552.20
Assuming the home value stays at $290,000, then the home equity value is the difference of these items.
equity = homeValue - remainingBalance
equity = 290000 - 260552.20
equity = 29447.80
This is the amount of money that the home owner can get out of the home after the debts are paid off.
Now find the remaining balance after 10 years, aka 10*12 = 120 months
B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )
B = 280000*( (1+0.005)^360 - (1+0.005)^120 )/( (1+0.005)^360 - 1 )
B = 234320.029898434
B = 234320.03
then,
equity = homeValue - remainingBalance
equity = 290000 - 234320.03
equity = 55679.97
The steps will be the same for 20 years, but this time we use p = 20*12 = 240
B = remaining balance after p months
B = L*( (1+i)^n - (1+i)^p )/( (1+i)^n - 1 )
B = 280000*( (1+0.005)^360 - (1+0.005)^240 )/( (1+0.005)^360 - 1 )
B = 151210.041484951
B = 151210.04
So,
equity = homeValue - remainingBalance
equity = 290000 - 151210.04
equity = 138789.96
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Answers:
Monthly payment: $1,678.74
Equity after 5 years: $29,447.80
Equity after 10 years: $55,679.97
Equity after 20 years: $138,789.96
Note: this assumes the home's value stays at $290,000. Realistically of course this home value will fluctuate over time.
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