document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170650>Question 1170650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Calculate the present value of the following annuity streams: \r<BR>\n" );
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document.write( "a. $5,000 received each year for 5 years on the last day of each year if your investments pay 6 percent compounded annually.\r<BR>\n" );
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document.write( "b. $5,000 received each quarter for 5 years on the last day of each quarter if your investments pay 6 percent compounded quarterly.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851122 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=851122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the present value of these annuity streams.\r<BR>\n" );
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document.write( "**a) $5,000 received each year for 5 years at 6% compounded annually.**\r<BR>\n" );
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document.write( "This is an ordinary annuity, where payments are made at the end of each period.\r<BR>\n" );
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document.write( "* Payment (PMT) = $5,000<BR>\n" );
document.write( "* Number of periods (n) = 5 years<BR>\n" );
document.write( "* Interest rate per period (r) = 6% or 0.06\r<BR>\n" );
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document.write( "The formula for the present value of an ordinary annuity is:\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( "Plugging in the values:\r<BR>\n" );
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document.write( "PV = 5000 * [1 - (1 + 0.06)^-5] / 0.06<BR>\n" );
document.write( "PV = 5000 * [1 - (1.06)^-5] / 0.06<BR>\n" );
document.write( "PV = 5000 * [1 - 0.747258] / 0.06<BR>\n" );
document.write( "PV = 5000 * [0.252742] / 0.06<BR>\n" );
document.write( "PV = 5000 * 4.21236666667<BR>\n" );
document.write( "PV ≈ $21,061.83\r<BR>\n" );
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document.write( "**b) $5,000 received each quarter for 5 years at 6% compounded quarterly.**\r<BR>\n" );
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document.write( "This is also an ordinary annuity, but with quarterly payments.\r<BR>\n" );
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document.write( "* Payment (PMT) = $5,000<BR>\n" );
document.write( "* Number of periods (n) = 5 years * 4 quarters/year = 20 quarters<BR>\n" );
document.write( "* Interest rate per period (r) = 6% / 4 = 1.5% or 0.015\r<BR>\n" );
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document.write( "The formula for the present value of an ordinary annuity is the same:\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( "Plugging in the values:\r<BR>\n" );
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document.write( "PV = 5000 * [1 - (1 + 0.015)^-20] / 0.015<BR>\n" );
document.write( "PV = 5000 * [1 - (1.015)^-20] / 0.015<BR>\n" );
document.write( "PV = 5000 * [1 - 0.742470] / 0.015<BR>\n" );
document.write( "PV = 5000 * [0.25753] / 0.015<BR>\n" );
document.write( "PV = 5000 * 17.1686666667<BR>\n" );
document.write( "PV ≈ $85,843.33\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "a. The present value of the annual annuity is approximately $21,061.83.\r<BR>\n" );
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document.write( "b. The present value of the quarterly annuity is approximately $85,843.33.<BR>\n" );
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