document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=286378>Question 286378</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You enter into a contract that requires you to pay $100 000 at the end of every four months(three times per year) for four years.At the end of the fourth year you will also be required to pay a lump sum of $500 000.The interest rate is 24% per annum compounded monthly.How much have you borrowed?How much do you still owe after two years </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #207786 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=207786\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You pay $100,000 at the end of every 4 months.<BR>\n" );
document.write( "You do that for 4 years.<BR>\n" );
document.write( "At the end of the 4th year you are paying a lump sum of $500,000.<BR>\n" );
document.write( "Interest rate is 24% per year compounded monthly.<BR>\n" );
document.write( "How much have you borrowed?<BR>\n" );
document.write( "How much do you still owe after 2 years.\r<BR>\n" );
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document.write( "The easiest way to solve part of this problem is to simulate what happens in a spreadsheet or some such other mechanized program.\r<BR>\n" );
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document.write( "That's the part for the remaining balance after 2 years.\r<BR>\n" );
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document.write( "I used Microsoft Excel Spreadsheet.\r<BR>\n" );
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document.write( "The answer I got is as follows:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Time Period	\r\n" );
document.write( "    Present Value of Loan	\r\n" );
document.write( "                  Payment	\r\n" );
document.write( "                              Interest Rate	\r\n" );
document.write( "                                     Borrower Balance\r\n" );
document.write( "                                                        Lender Balance\r\n" );
document.write( "0   $937,471.542              0.02   $937,471.542	$0.000\r\n" );
document.write( "1				     $956,220.973	$0.000\r\n" );
document.write( "2				     $975,345.392	$0.000\r\n" );
document.write( "3				     $994,852.300	$0.000\r\n" );
document.write( "4		  $100,000.000       $914,749.346	$100,000.000\r\n" );
document.write( "5				     $933,044.333	$102,000.000\r\n" );
document.write( "6				     $951,705.220	$104,040.000\r\n" );
document.write( "7				     $970,739.324	$106,120.800\r\n" );
document.write( "8		  $100,000.000       $890,154.110	$208,243.216\r\n" );
document.write( "9				     $907,957.193	$212,408.080\r\n" );
document.write( "10				     $926,116.337	$216,656.242\r\n" );
document.write( "11				     $944,638.663	$220,989.367\r\n" );
document.write( "12		  $100,000.000       $863,531.437	$325,409.154\r\n" );
document.write( "13				     $880,802.065	$331,917.337\r\n" );
document.write( "14				     $898,418.107	$338,555.684\r\n" );
document.write( "15				     $916,386.469	$345,326.798\r\n" );
document.write( "16		  $100,000.000       $834,714.198	$452,233.334\r\n" );
document.write( "17				     $851,408.482	$461,278.000\r\n" );
document.write( "18				     $868,436.652	$470,503.560\r\n" );
document.write( "19				     $885,805.385	$479,913.631\r\n" );
document.write( "20		  $100,000.000       $803,521.492	$589,511.904\r\n" );
document.write( "21				     $819,591.922	$601,302.142\r\n" );
document.write( "22				     $835,983.761	$613,328.185\r\n" );
document.write( "23				     $852,703.436	$625,594.749\r\n" );
document.write( "24		  $100,000.000	     $769,757.505	$738,106.644\r\n" );
document.write( "25				     $785,152.655	$752,868.777\r\n" );
document.write( "26				     $800,855.708	$767,926.152\r\n" );
document.write( "27				     $816,872.822	$783,284.675\r\n" );
document.write( "28		  $100,000.000	     $733,210.278	$898,950.369\r\n" );
document.write( "29				     $747,874.484	$916,929.376\r\n" );
document.write( "30				     $762,831.974	$935,267.964\r\n" );
document.write( "31				     $778,088.613	$953,973.323\r\n" );
document.write( "32		  $100,000.000	     $693,650.385	$1,073,052.789\r\n" );
document.write( "33				     $707,523.393	$1,094,513.845\r\n" );
document.write( "34				     $721,673.861	$1,116,404.122\r\n" );
document.write( "35				     $736,107.338	$1,138,732.204\r\n" );
document.write( "36		  $100,000.000	     $650,829.485	$1,261,506.848\r\n" );
document.write( "37				     $663,846.075	$1,286,736.985\r\n" );
document.write( "38				     $677,122.996	$1,312,471.725\r\n" );
document.write( "39				     $690,665.456	$1,338,721.160\r\n" );
document.write( "40		  $100,000.000	     $604,478.765	$1,465,495.583\r\n" );
document.write( "41				     $616,568.341	$1,494,805.494\r\n" );
document.write( "42				     $628,899.707	$1,524,701.604\r\n" );
document.write( "43				     $641,477.701	$1,555,195.636\r\n" );
document.write( "44		  $100,000.000	     $554,307.256	$1,686,299.549\r\n" );
document.write( "45				     $565,393.401	$1,720,025.540\r\n" );
document.write( "46				     $576,701.269	$1,754,426.051\r\n" );
document.write( "47				     $588,235.294	$1,789,514.572\r\n" );
document.write( "48		  $600,000.000	     $0.000     	$2,425,304.863\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "I calculated the Present Value of the loan as follows:\r<BR>\n" );
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document.write( "Annual Interest Rate of 24% compounded monthly is equivalent to a monthly interest rate of 2%.\r<BR>\n" );
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document.write( "The Effective Annual Interest Rate of a monthly 2% interest rate is equal to 1.02^12 = a factor of 1.268241795 which is equivalent to an Effective Annual Interest Rate of 26.8241795%.\r<BR>\n" );
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document.write( "To get an equivalent interest rate every 4 months, you need to solve the equation of:\r<BR>\n" );
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document.write( "(1+x)^3 = 1.268241795.\r<BR>\n" );
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document.write( "Take the cube root of both sides of this equation to get:\r<BR>\n" );
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document.write( "1+x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C1.268241795%29 ALIGN=MIDDLE  ALT=\"root%283%2C1.268241795%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals 1.08243216.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Subtract 1 from both sides of this equation to get x = .08243216 which is equal to an interest rate of 8.243216% every 4 months.\r<BR>\n" );
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document.write( "What you have so far is payments every 4 months at an interest rate of 8.243216% every 4 months.\r<BR>\n" );
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document.write( "Now that the payment schedule and the compound interest rate schedule is synchronized, you can use the financial calculators to determine the Present Value of the $100,000 payments every 4 months for the number of time periods required by the loan.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The number of time periods required by the loan is 4 years multiplied by 3 time periods per year to get a total of 12 time periods, each of which is 4 months long.\r<BR>\n" );
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document.write( "The inputs to the financial calculator are:\r<BR>\n" );
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document.write( "Present Value = 0<BR>\n" );
document.write( "Future Value = 0<BR>\n" );
document.write( "Payment = $100,000<BR>\n" );
document.write( "Interest Rate = 8.243216%<BR>\n" );
document.write( "Number of Time Periods is equal to 12.\r<BR>\n" );
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document.write( "The calculator tells you that:\r<BR>\n" );
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document.write( "Present Value of the loan payments = $744,202.7377<BR>\n" );
document.write( "Future Value of the loan payments = $1,925,304.863.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "$744,202.7377 is what you owe today.<BR>\n" );
document.write( "$1,925,304.863 is what the lender receives by the end of the loan period.\r<BR>\n" );
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document.write( "That includes principal of the loan, interest you paid on the loan, additional interest that the lender received by re-investing the interest received from the loan.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "On top of that you have to pay the lender $500,000 at the end of the loan period.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The total the lender receives is $1,925,304.863 + $500,000 = $2,425,304.863.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The present value of $2,425,304.863 for 48 time periods at 2% per month is equal to $937,471.5419.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The present value of $2,425,304.863 for 12 time periods at 8.243216% every 4 months is also equal to $937,471.5419.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The interest rates are equivalent and yield the same Present Value as they should.\r<BR>\n" );
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document.write( "The Present Value of the loan, taking into account the payments you make plus the $500,000 you give at the end of the loan period is equal to $937,471.5419.\r<BR>\n" );
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document.write( "That's the amount of the loan.\r<BR>\n" );
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document.write( "The remaining balance of the loan after 2 years was more difficult to determine using just formulas, so I resorted to the Excel Spreadsheet which provides you with the remaining balance on the loan after 2 years.\r<BR>\n" );
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document.write( "That remaining balance is equal to $769,757.505\r<BR>\n" );
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document.write( "That is the amount that you still owe after 2 years.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );