document.write( "Question 1207430: A property worth $35 OOO is purchased for 10% down and semi-annual payments of $2100 for 12 years. What is the nominal annual rate of interest if interest is compounded quarterly?
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Algebra.Com's Answer #852186 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "A property worth $35,000 is purchased for 10% down and semi-annual payments of $2100 for 12 years.
\n" ); document.write( "What is the nominal annual rate of interest if interest is compounded quarterly?
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\n" ); document.write( "\n" ); document.write( "        The solution by tutor @Theo has an error (or a typo). \r
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\n" ); document.write( "\n" ); document.write( "        His answer 8.035524952% is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "        I came to bring a correct solution, right numbers and proper answer.\r
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document.write( "Down payment is 10% of $35,000, i.e. 0.1*35000 = 3500 dollars.\r\n" );
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document.write( "Hence, the loan is the rest amount of  $35,000 - $3,500 = $31,500.\r\n" );
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document.write( "    +------------------------------------------------------------+\r\n" );
document.write( "    |   Notice that, as it is given in the problem, semi-annual  |\r\n" );
document.write( "    |   payments are desynchronized with quarterly compounding.  |\r\n" );
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document.write( "Nevertheless, we can synchronize payments and compounding by considering an EQUIVALENT scheme \r\n" );
document.write( "with semi-annual compounding with the effective growth coefficient  'r'  semi-annually.\r\n" );
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document.write( "This coefficient 'r' is not known now, and we should find it from the problem.\r\n" );
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document.write( "We then have a loan of $31,500 with semi-annual payments of $2100 and semi-annual compounding\r\n" );
document.write( "with the effective semi-annual rate of  r.\r\n" );
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document.write( "Write the standard loan equation for such a loan\r\n" );
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document.write( "     = \r\n" );
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document.write( "     = \r\n" );
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document.write( "     15 = \r\n" );
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document.write( "Solve this equation numerically to find 'r'.\r\n" );
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document.write( "I used online calculator https://www.wolframalpha.com/calculators/equation-solver-calculator/\r\n" );
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document.write( "It found the approximate real solution  r = 0.0416015.\r\n" );
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document.write( "Thus, in the equivalent scheme, the semi-annual effective rate is 0.0416015.\r\n" );
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document.write( "    //  Notice that till point my solution coincides with that by @Theo.\r\n" );
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document.write( "Hence, the effective semi-annual growth factor is  1+r = 1.0416015.\r\n" );
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document.write( "It implies that in the basic scheme, the effective quarterly growth factor is the square root of that\r\n" );
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document.write( "     = 1.020589.\r\n" );
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document.write( "Hence, the effective quarterly rate is  0.020589.\r\n" );
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document.write( "Then the annual effective rate is four times this, or  4*0.020589 = 0.082388.\r\n" );
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document.write( "Thus the nominal annual compounding interest  is 8.2388%.    <<<---===  ANSWER\r\n" );
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