Question 1202788
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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<ul><li>L = 240,000 - 10,000 = 230,000  (don't forget to subtract off the down payment)</li><li>i = 0.05/12 = 0.004166667 approximately</li><li>n = 30*12 = 360 months (equivalent to 30 years)</li></ul>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 = <font color=red>$1234.69</font>


You can use a calculator like this to confirm the answer.
<a href="https://www.calculator.net/loan-calculator.html">https://www.calculator.net/loan-calculator.html</a>
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
<a href="https://www.mtgprofessor.com//formulas.htm">https://www.mtgprofessor.com//formulas.htm</a>


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 = <font color=red>$28,793.92</font>


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 = <font color=red>52,913.25</font>


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: <font color=red>$28,793.92</font>
Equity at 10 years: <font color=red>$52,913.25</font>
Equity at 20 years: <font color=red>$123,591.78</font>
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