document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190652>Question 1190652</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the present value of an ordinary annuity having semi-annual payments of 8, 000 pesos for 12 years with an interest rate of 12% compounded annually? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849198 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the present value of the ordinary annuity:\r<BR>\n" );
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document.write( "**1. Determine the effective interest rate per payment period:**\r<BR>\n" );
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document.write( "Since payments are semi-annual, but the interest is compounded annually we need to find the effective semi-annual rate.  We can't simply divide the annual rate by two, instead we find the equivalent rate such that if we compound the semi-annual rate twice we get the annual rate. The formula to convert is:\r<BR>\n" );
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document.write( "(1 + annual rate) = (1 + semi-annual rate)^2\r<BR>\n" );
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document.write( "(1 + 0.12) = (1 + semi-annual rate)^2\r<BR>\n" );
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document.write( "1.12 = (1 + semi-annual rate)^2\r<BR>\n" );
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document.write( "√1.12 = 1 + semi-annual rate\r<BR>\n" );
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document.write( "1.0583 = 1 + semi-annual rate\r<BR>\n" );
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document.write( "semi-annual rate = 1.0583 - 1\r<BR>\n" );
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document.write( "semi-annual rate = 0.0583 or 5.83%\r<BR>\n" );
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document.write( "**2. Determine the number of payment periods:**\r<BR>\n" );
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document.write( "* The annuity lasts for 12 years.<BR>\n" );
document.write( "* Payments are made semi-annually, meaning twice a year.<BR>\n" );
document.write( "* Number of payment periods (n) = 12 years * 2 payments/year = 24 periods\r<BR>\n" );
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document.write( "**3. Use the present value of an ordinary annuity formula:**\r<BR>\n" );
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document.write( "PV = PMT * [1 - (1 + r)^-n] / r\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* PV = Present Value (what we want to find)<BR>\n" );
document.write( "* PMT = Payment amount per period (8,000 pesos)<BR>\n" );
document.write( "* r = Interest rate per period (0.0583 or 5.83% as a decimal)<BR>\n" );
document.write( "* n = Number of periods (24)\r<BR>\n" );
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document.write( "**4. Calculate:**\r<BR>\n" );
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document.write( "PV = 8000 * [1 - (1 + 0.0583)^-24] / 0.0583\r<BR>\n" );
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document.write( "PV = 8000 * [1 - (1.0583)^-24] / 0.0583\r<BR>\n" );
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document.write( "PV = 8000 * [1 - 0.2452] / 0.0583\r<BR>\n" );
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document.write( "PV = 8000 * 0.7548 / 0.0583\r<BR>\n" );
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document.write( "PV = 8000 * 12.9468\r<BR>\n" );
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document.write( "PV ≈ 103,574.27 pesos\r<BR>\n" );
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document.write( "Therefore, the present value of the ordinary annuity is approximately **103,574.27 pesos**.<BR>\n" );
document.write( " </SPAN>\n" );
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