document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177706>Question 1177706</A>: <I><SPAN STYLE=\"word-break: break-all;\"> three-year treasury securities currently yield 6%, while 4-year treasury securities currently yield 6.5%. assume that the expectations theory holds. what does the market believe the rate will be on 1-year treasury securities three years from now? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850404 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=850404\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using the expectations theory.\r<BR>\n" );
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document.write( "**Understanding the Expectations Theory**\r<BR>\n" );
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document.write( "The expectations theory states that long-term interest rates are determined by the market's expectations of future short-term interest rates. In other words, the yield on a long-term bond is the average of the expected yields on short-term bonds over the same period.\r<BR>\n" );
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document.write( "**Given Information**\r<BR>\n" );
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document.write( "* 3-year Treasury yield: 6%<BR>\n" );
document.write( "* 4-year Treasury yield: 6.5%\r<BR>\n" );
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document.write( "**Applying the Expectations Theory**\r<BR>\n" );
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document.write( "Let's denote:\r<BR>\n" );
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document.write( "* r_3 = the yield on the 3-year Treasury security (6% or 0.06)<BR>\n" );
document.write( "* r_4 = the yield on the 4-year Treasury security (6.5% or 0.065)<BR>\n" );
document.write( "* f_4 = the expected 1-year Treasury yield three years from now (what we want to find)\r<BR>\n" );
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document.write( "According to the expectations theory:\r<BR>\n" );
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document.write( "* (1 + r_4)^4 = (1 + r_3)^3 * (1 + f_4)\r<BR>\n" );
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document.write( "**Solving for f_4**\r<BR>\n" );
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document.write( "1.  Plug in the given values:<BR>\n" );
document.write( "    * (1 + 0.065)^4 = (1 + 0.06)^3 * (1 + f_4)\r<BR>\n" );
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document.write( "2.  Calculate the terms:<BR>\n" );
document.write( "    * (1.065)^4 = (1.06)^3 * (1 + f_4)<BR>\n" );
document.write( "    * 1.2864388126 = 1.191016 * (1 + f_4)\r<BR>\n" );
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document.write( "3.  Isolate (1 + f_4):<BR>\n" );
document.write( "    * 1 + f_4 = 1.2864388126 / 1.191016<BR>\n" );
document.write( "    * 1 + f_4 ≈ 1.07928\r<BR>\n" );
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document.write( "4.  Solve for f_4:<BR>\n" );
document.write( "    * f_4 ≈ 1.07928 - 1<BR>\n" );
document.write( "    * f_4 ≈ 0.07928\r<BR>\n" );
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document.write( "5.  Convert to percentage:<BR>\n" );
document.write( "    * f_4 ≈ 7.928%\r<BR>\n" );
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document.write( "**Therefore, the market believes the rate will be approximately 7.93% on 1-year Treasury securities three years from now.**<BR>\n" );
document.write( " </SPAN>\n" );
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