document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201043>Question 1201043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question 3 (15 marks) John has just purchased an apartment at a price of $5,000,000. He made a down-payment of $2,000,000 and financed the remaining with a 30-year mortgage at APR 12%, compounded monthly. (a) Determine the size of the fixed month-end payments. (5 marks)<BR>\n" );
document.write( "(b) Calculate the amount John still owes the bank right after the 120th payment was made. (5 marks)<BR>\n" );
document.write( "(c) Calculate the interest payment and the amount of principal paid in the 121st loan repayment. (5 marks) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837746 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=837746\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<font color=red>Answers</font><BR>\n" );
document.write( "(a) <font color=red>$30,858.38</font><BR>\n" );
document.write( "(b) <font color=red>$2,802,539.87</font> <BR>\n" );
document.write( "(c) interest payment = <font color=red>$28,025.40</font>; principal payment = <font color=red>$2,832.98</font>\r<BR>\n" );
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document.write( "Work Shown for part (a)\r<BR>\n" );
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document.write( "Price = $5 million<BR>\n" );
document.write( "Down-payment = $2 million<BR>\n" );
document.write( "Loan amount = 5-2 = $3 million\r<BR>\n" );
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document.write( "We will use this monthly payment formula<BR>\n" );
document.write( "P = (L*i)/( 1-(1+i)^(-n) )<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "P = monthly payment<BR>\n" );
document.write( "L = loan amount<BR>\n" );
document.write( "i = monthly interest rate in decimal form<BR>\n" );
document.write( "n = number of months\r<BR>\n" );
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document.write( "In this case<BR>\n" );
document.write( "L = 3,000,000<BR>\n" );
document.write( "i = 0.12/12 = 0.01<BR>\n" );
document.write( "n = 30*12 = 360 months\r<BR>\n" );
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document.write( "P = (L*i)/( 1-(1+i)^(-n) )<BR>\n" );
document.write( "P = (3,000,000*0.01)/( 1-(1+0.01)^(-360) )<BR>\n" );
document.write( "P = 30,858.3779077651<BR>\n" );
document.write( "P = <font color=red>30,858.38</font>\r<BR>\n" );
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document.write( "The answer can be confirmed through the use of a loan calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.calculator.net/loan-calculator.html\">https://www.calculator.net/loan-calculator.html</a>\r<BR>\n" );
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document.write( "Work Shown for part (b)\r<BR>\n" );
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document.write( "Refer to the 2nd formula mentioned on this page<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.mtgprofessor.com/formulas.htm\">https://www.mtgprofessor.com/formulas.htm</a><BR>\n" );
document.write( "It calculates the remaining balance at any given month.\r<BR>\n" );
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document.write( "That formula looks rather messy. <BR>\n" );
document.write( "The numerator is L*( (1+c)^n - (1+c)^p )<BR>\n" );
document.write( "The denominator is (1+c)^n - 1\r<BR>\n" );
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document.write( "L = loan amount<BR>\n" );
document.write( "c = monthly interest rate in decimal form <BR>\n" );
document.write( "n = number of months of the entire loan<BR>\n" );
document.write( "p = current month number\r<BR>\n" );
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document.write( "In this case we have<BR>\n" );
document.write( "L = 3,000,000<BR>\n" );
document.write( "c = 0.01 calculated earlier<BR>\n" );
document.write( "n = 360 months total<BR>\n" );
document.write( "p = 120\r<BR>\n" );
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document.write( "numerator = L*( (1+c)^n - (1+c)^p )<BR>\n" );
document.write( "numerator = 3,000,000*( (1+0.01)^360 - (1+0.01)^120 )<BR>\n" );
document.write( "numerator = 97,947,763.299334\r<BR>\n" );
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document.write( "denominator = (1+c)^n - 1<BR>\n" );
document.write( "denominator = (1+0.01)^360 - 1<BR>\n" );
document.write( "denominator = 34.949641327685\r<BR>\n" );
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document.write( "Divide the results:<BR>\n" );
document.write( "97,947,763.299334/34.949641327685 = 2,802,539.87103856\r<BR>\n" );
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document.write( "John still owes <font color=red>$2,802,539.87</font> after the 120th payment was made. This is approximately 2.8 million dollars.\r<BR>\n" );
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document.write( "The first link I mentioned earlier (the loan calculator) provides a handy amortization table to show the balance for any given month. <BR>\n" );
document.write( "Scroll down to month 120 to confirm that $2,802,539.87 is the ending balance. <BR>\n" );
document.write( "120 months = 120/12 = 10 years\r<BR>\n" );
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document.write( "At first a big chunk of the monthly payment is composed of interest. <BR>\n" );
document.write( "As time goes on, more of the monthly payment is devoted to the principal.\r<BR>\n" );
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document.write( "Work Shown for part (c)\r<BR>\n" );
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document.write( "The balance at the end of month 120 is $2,802,539.87 as mentioned in part (b)\r<BR>\n" );
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document.write( "Multiply this with c = 0.01 mentioned in the previous part. \r<BR>\n" );
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document.write( "0.01*(2,802,539.87) = 28,025.3987 =  <font color=red>$28,025.40</font> is the interest part of the payment\r<BR>\n" );
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document.write( "Subtract the interest from the monthly payment to determine the principal.<BR>\n" );
document.write( "30,858.38 - 28,025.40 = <font color=red>$2,832.98</font>\r<BR>\n" );
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document.write( "For month 121 we have<BR>\n" );
document.write( "interest = <font color=red>$28,025.40</font><BR>\n" );
document.write( "principal = <font color=red>$2,832.98</font>\r<BR>\n" );
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document.write( "The amortization table can be used to confirm these results. <BR>\n" );
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