document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179819>Question 1179819</A>: <I><SPAN STYLE=\"word-break: break-all;\"> f.	Find the present values of the following annuities\r<BR>\n" );
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document.write( "i.	RM6,000 every year for 8 years at 12% compounded annually<BR>\n" );
document.write( "ii.	RM800 every month for 2 years 5 months at 5% compounded monthly<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 #850168 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=850168\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the present values of these annuities:\r<BR>\n" );
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document.write( "**i. RM6,000 every year for 8 years at 12% compounded annually**\r<BR>\n" );
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document.write( "* **Payment (PMT):** RM6,000<BR>\n" );
document.write( "* **Number of periods (n):** 8 years<BR>\n" );
document.write( "* **Interest rate per period (r):** 12% or 0.12\r<BR>\n" );
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document.write( "We will 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( "PV = 6000 * [1 - (1 + 0.12)^(-8)] / 0.12<BR>\n" );
document.write( "PV = 6000 * [1 - (1.12)^(-8)] / 0.12<BR>\n" );
document.write( "PV = 6000 * [1 - 0.403883] / 0.12<BR>\n" );
document.write( "PV = 6000 * [0.596117] / 0.12<BR>\n" );
document.write( "PV = 6000 * 4.96764<BR>\n" );
document.write( "PV ≈ RM29,805.84\r<BR>\n" );
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document.write( "**ii. RM800 every month for 2 years 5 months at 5% compounded monthly**\r<BR>\n" );
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document.write( "* **Payment (PMT):** RM800<BR>\n" );
document.write( "* **Number of periods (n):** (2 years * 12 months) + 5 months = 24 + 5 = 29 months<BR>\n" );
document.write( "* **Annual interest rate:** 5% or 0.05<BR>\n" );
document.write( "* **Monthly interest rate (r):** 0.05 / 12 ≈ 0.00416667\r<BR>\n" );
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document.write( "We will 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( "PV = 800 * [1 - (1 + 0.00416667)^(-29)] / 0.00416667<BR>\n" );
document.write( "PV = 800 * [1 - (1.00416667)^(-29)] / 0.00416667<BR>\n" );
document.write( "PV = 800 * [1 - 0.887649] / 0.00416667<BR>\n" );
document.write( "PV = 800 * [0.112351] / 0.00416667<BR>\n" );
document.write( "PV = 800 * 26.96424<BR>\n" );
document.write( "PV ≈ RM21,571.39\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "i. The present value of RM6,000 every year for 8 years at 12% compounded annually is approximately **RM29,805.84**.\r<BR>\n" );
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document.write( "ii. The present value of RM800 every month for 2 years 5 months at 5% compounded monthly is approximately **RM21,571.39**.<BR>\n" );
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