document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185513>Question 1185513</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The present value of an annuity of R pesos payable annually for 8 years, with the first payment at the end of 10 years, is 187, 481.25. Draw the cash flow diagram and find the value of R if money is worth 5%. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849689 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849689\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "**1. Cash Flow Diagram:**\r<BR>\n" );
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document.write( "The cash flow diagram will show the payments (R) occurring at the end of each year, but *shifted* by 9 years.  Here's a representation:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Time (Years):  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17<BR>\n" );
document.write( "Cash Flow:    0   0   0   0   0   0   0   0   0   0   R   R   R   R   R   R   R   R<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**2. Present Value Calculation:**\r<BR>\n" );
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document.write( "The present value of the annuity is given as 187,481.25.  Since the first payment occurs at the end of year 10, we first calculate the present value of the annuity *as if it started at the end of year 9*.  Then, we discount that value back to the present (time 0).\r<BR>\n" );
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document.write( "* **Present Value of Annuity (at end of year 9):** Let's call this PV₉.  The formula for the present value of an ordinary annuity is:\r<BR>\n" );
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document.write( "   PV = R * [1 - (1 + i)^-n] / i\r<BR>\n" );
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document.write( "   where:<BR>\n" );
document.write( "   * PV is the present value<BR>\n" );
document.write( "   * R is the payment amount (what we want to find)<BR>\n" );
document.write( "   * i is the interest rate (5% or 0.05)<BR>\n" );
document.write( "   * n is the number of periods (8 years)\r<BR>\n" );
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document.write( "   So, PV₉ = R * [1 - (1.05)^-8] / 0.05\r<BR>\n" );
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document.write( "* **Discounting back to the present (time 0):**  We now treat PV₉ as a single future value and discount it back 9 years to time 0:\r<BR>\n" );
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document.write( "   PV₀ = PV₉ / (1 + i)^9\r<BR>\n" );
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document.write( "   where PV₀ is the present value at time 0, which is given as 187,481.25.\r<BR>\n" );
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document.write( "**3. Solving for R:**\r<BR>\n" );
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document.write( "Now we have the equation:\r<BR>\n" );
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document.write( "187,481.25 = {R * [1 - (1.05)^-8] / 0.05} / (1.05)^9\r<BR>\n" );
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document.write( "We need to solve for R.  Here's the step-by-step process:\r<BR>\n" );
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document.write( "1. Simplify the annuity part: [1 - (1.05)^-8] / 0.05 ≈ 6.7665<BR>\n" );
document.write( "2. Simplify the discounting part: (1.05)^9 ≈ 1.5513<BR>\n" );
document.write( "3. Rewrite the equation: 187,481.25 = (R * 6.7665) / 1.5513<BR>\n" );
document.write( "4. Multiply both sides by 1.5513: 187,481.25 * 1.5513 = R * 6.7665<BR>\n" );
document.write( "5. Divide both sides by 6.7665: R = (187,481.25 * 1.5513) / 6.7665<BR>\n" );
document.write( "6. Calculate R: R ≈ 42,999.99 or approximately 43,000\r<BR>\n" );
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document.write( "**Therefore, the value of R is approximately 43,000 pesos.**<BR>\n" );
document.write( " </SPAN>\n" );
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