document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172988>Question 1172988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the present value of a 2-year deferred annuity at 4% interest compounded quarterly with payments of $1000 made every quarter for 3 years? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850791 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=850791\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this calculation step-by-step:\r<BR>\n" );
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document.write( "**1. Understand Deferred Annuity**\r<BR>\n" );
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document.write( "* A deferred annuity is an annuity whose payments begin at some future date.<BR>\n" );
document.write( "* In this case, the payments begin 2 years from now (deferred period).<BR>\n" );
document.write( "* Payments are $1,000 made quarterly for 3 years.<BR>\n" );
document.write( "* Interest rate is 4% per year, compounded quarterly.\r<BR>\n" );
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document.write( "**2. Calculate the Interest Rate per Period (i)**\r<BR>\n" );
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document.write( "* Annual interest rate: 4% or 0.04<BR>\n" );
document.write( "* Compounding frequency: Quarterly (4 times a year)<BR>\n" );
document.write( "* Interest rate per quarter: 0.04 / 4 = 0.01\r<BR>\n" );
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document.write( "**3. Calculate the Number of Payment Periods (n)**\r<BR>\n" );
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document.write( "* Payment duration: 3 years<BR>\n" );
document.write( "* Payments per year: 4 (quarterly)<BR>\n" );
document.write( "* Total payment periods: 3 * 4 = 12\r<BR>\n" );
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document.write( "**4. Calculate the Present Value of the Ordinary Annuity**\r<BR>\n" );
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document.write( "* We'll use the present value of an ordinary annuity formula:<BR>\n" );
document.write( "    * PV = PMT * [(1 - (1 + i)^-n) / i]<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * PV = Present value of the annuity<BR>\n" );
document.write( "        * PMT = Payment amount ($1,000)<BR>\n" );
document.write( "        * i = Interest rate per period (0.01)<BR>\n" );
document.write( "        * n = Number of periods (12)\r<BR>\n" );
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document.write( "* PV = 1000 * [(1 - (1 + 0.01)^-12) / 0.01]<BR>\n" );
document.write( "* PV = 1000 * [(1 - (1.01)^-12) / 0.01]<BR>\n" );
document.write( "* PV = 1000 * [(1 - 0.887449) / 0.01]<BR>\n" );
document.write( "* PV = 1000 * [0.112551 / 0.01]<BR>\n" );
document.write( "* PV = 1000 * 11.2551<BR>\n" );
document.write( "* PV = $11,255.10\r<BR>\n" );
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document.write( "This $11,255.10 is the present value of the annuity at the *start* of the payment period, which is 2 years from now.\r<BR>\n" );
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document.write( "**5. Calculate the Present Value of the Deferred Annuity**\r<BR>\n" );
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document.write( "* Now, we need to discount this present value back to today (2 years earlier).<BR>\n" );
document.write( "* Deferred period: 2 years<BR>\n" );
document.write( "* Number of deferred periods: 2 * 4 = 8 quarters\r<BR>\n" );
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document.write( "* We'll use the present value formula for a lump sum:<BR>\n" );
document.write( "    * PV_deferred = PV / (1 + i)^deferred_periods<BR>\n" );
document.write( "    * Where:<BR>\n" );
document.write( "        * PV_deferred = Present value of the deferred annuity<BR>\n" );
document.write( "        * PV = Present value of the annuity at the start of payments ($11,255.10)<BR>\n" );
document.write( "        * i = Interest rate per period (0.01)<BR>\n" );
document.write( "        * deferred_periods = Number of deferred periods (8)\r<BR>\n" );
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document.write( "* PV_deferred = 11255.10 / (1 + 0.01)^8<BR>\n" );
document.write( "* PV_deferred = 11255.10 / (1.01)^8<BR>\n" );
document.write( "* PV_deferred = 11255.10 / 1.082857<BR>\n" );
document.write( "* PV_deferred = $10,393.85 (approximately)\r<BR>\n" );
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document.write( "**Therefore, the present value of the 2-year deferred annuity is approximately $10,393.85.**<BR>\n" );
document.write( " </SPAN>\n" );
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