Question 1201043
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<font color=red>Answers</font>
(a) <font color=red>$30,858.38</font>
(b) <font color=red>$2,802,539.87</font> 
(c) interest payment = <font color=red>$28,025.40</font>; principal payment = <font color=red>$2,832.98</font>



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Work Shown for part (a)


Price = $5 million
Down-payment = $2 million
Loan amount = 5-2 = $3 million


We will use this 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 = 3,000,000
i = 0.12/12 = 0.01
n = 30*12 = 360 months


P = (L*i)/( 1-(1+i)^(-n) )
P = (3,000,000*0.01)/( 1-(1+0.01)^(-360) )
P = 30,858.3779077651
P = <font color=red>30,858.38</font>


The answer can be confirmed through the use of a loan calculator such as this one
<a href = "https://www.calculator.net/loan-calculator.html">https://www.calculator.net/loan-calculator.html</a>


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Work Shown for part (b)


Refer to the 2nd formula mentioned on this page
<a href="https://www.mtgprofessor.com/formulas.htm">https://www.mtgprofessor.com/formulas.htm</a>
It calculates the remaining balance at any given month.


That formula looks rather messy. 
The numerator is L*( (1+c)^n - (1+c)^p )
The denominator is (1+c)^n - 1


L = loan amount
c = monthly interest rate in decimal form 
n = number of months of the entire loan
p = current month number


In this case we have
L = 3,000,000
c = 0.01 calculated earlier
n = 360 months total
p = 120


numerator = L*( (1+c)^n - (1+c)^p )
numerator = 3,000,000*( (1+0.01)^360 - (1+0.01)^120 )
numerator = 97,947,763.299334


denominator = (1+c)^n - 1
denominator = (1+0.01)^360 - 1
denominator = 34.949641327685


Divide the results:
97,947,763.299334/34.949641327685 = 2,802,539.87103856


John still owes <font color=red>$2,802,539.87</font> after the 120th payment was made. This is approximately 2.8 million dollars.


The first link I mentioned earlier (the loan calculator) provides a handy amortization table to show the balance for any given month. 
Scroll down to month 120 to confirm that $2,802,539.87 is the ending balance. 
120 months = 120/12 = 10 years


At first a big chunk of the monthly payment is composed of interest. 
As time goes on, more of the monthly payment is devoted to the principal.


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Work Shown for part (c)


The balance at the end of month 120 is $2,802,539.87 as mentioned in part (b)


Multiply this with c = 0.01 mentioned in the previous part. 


0.01*(2,802,539.87) = 28,025.3987 =  <font color=red>$28,025.40</font> is the interest part of the payment


Subtract the interest from the monthly payment to determine the principal.
30,858.38 - 28,025.40 = <font color=red>$2,832.98</font>


For month 121 we have
interest = <font color=red>$28,025.40</font>
principal = <font color=red>$2,832.98</font>


The amortization table can be used to confirm these results. 
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