document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170648>Question 1170648</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Calculate the present value of $5,000 received five years from today if your investments pay <BR>\n" );
document.write( "a. 6 percent compounded annually<BR>\n" );
document.write( "b. 8 percent compounded annually<BR>\n" );
document.write( "c. 10 percent compounded annually<BR>\n" );
document.write( "d. 10 percent compounded semiannually<BR>\n" );
document.write( "e. 10 percent compounded quarterly<BR>\n" );
document.write( "What do your answers to these questions tell you about the relation between present values and interest rates and between present values and the number of compounding periods per year?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851123 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=851123\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's calculate the present value of $5,000 received five years from today under different compounding scenarios.\r<BR>\n" );
document.write( "\n" );
document.write( "**Formula for Present Value (PV):**\r<BR>\n" );
document.write( "\n" );
document.write( "PV = FV / (1 + r)^n\r<BR>\n" );
document.write( "\n" );
document.write( "Where:\r<BR>\n" );
document.write( "\n" );
document.write( "* PV = Present Value<BR>\n" );
document.write( "* FV = Future Value ($5,000)<BR>\n" );
document.write( "* r = Interest rate per period<BR>\n" );
document.write( "* n = Number of periods\r<BR>\n" );
document.write( "\n" );
document.write( "**a) 6% Compounded Annually**\r<BR>\n" );
document.write( "\n" );
document.write( "* r = 0.06<BR>\n" );
document.write( "* n = 5 years\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 5000 / (1 + 0.06)^5<BR>\n" );
document.write( "PV = 5000 / (1.06)^5<BR>\n" );
document.write( "PV = 5000 / 1.3382255776<BR>\n" );
document.write( "PV ≈ $3,736.30\r<BR>\n" );
document.write( "\n" );
document.write( "**b) 8% Compounded Annually**\r<BR>\n" );
document.write( "\n" );
document.write( "* r = 0.08<BR>\n" );
document.write( "* n = 5 years\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 5000 / (1 + 0.08)^5<BR>\n" );
document.write( "PV = 5000 / (1.08)^5<BR>\n" );
document.write( "PV = 5000 / 1.4693280768<BR>\n" );
document.write( "PV ≈ $3,402.92\r<BR>\n" );
document.write( "\n" );
document.write( "**c) 10% Compounded Annually**\r<BR>\n" );
document.write( "\n" );
document.write( "* r = 0.10<BR>\n" );
document.write( "* n = 5 years\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 5000 / (1 + 0.10)^5<BR>\n" );
document.write( "PV = 5000 / (1.10)^5<BR>\n" );
document.write( "PV = 5000 / 1.61051<BR>\n" );
document.write( "PV ≈ $3,104.61\r<BR>\n" );
document.write( "\n" );
document.write( "**d) 10% Compounded Semiannually**\r<BR>\n" );
document.write( "\n" );
document.write( "* r = 0.10 / 2 = 0.05 (semiannual interest rate)<BR>\n" );
document.write( "* n = 5 years * 2 = 10 periods\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 5000 / (1 + 0.05)^10<BR>\n" );
document.write( "PV = 5000 / (1.05)^10<BR>\n" );
document.write( "PV = 5000 / 1.6288946268<BR>\n" );
document.write( "PV ≈ $3,069.50\r<BR>\n" );
document.write( "\n" );
document.write( "**e) 10% Compounded Quarterly**\r<BR>\n" );
document.write( "\n" );
document.write( "* r = 0.10 / 4 = 0.025 (quarterly interest rate)<BR>\n" );
document.write( "* n = 5 years * 4 = 20 periods\r<BR>\n" );
document.write( "\n" );
document.write( "PV = 5000 / (1 + 0.025)^20<BR>\n" );
document.write( "PV = 5000 / (1.025)^20<BR>\n" );
document.write( "PV = 5000 / 1.6386164403<BR>\n" );
document.write( "PV ≈ $3,051.35\r<BR>\n" );
document.write( "\n" );
document.write( "**Observations**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Relation between Present Values and Interest Rates:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * As the interest rate increases (from 6% to 10%), the present value decreases. This is because a higher interest rate means that money today can grow to a larger amount in the future, so the present value of a future sum is lower.\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Relation between Present Values and Compounding Periods:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * For a given annual interest rate (10% in this case), as the number of compounding periods per year increases (annually, semiannually, quarterly), the present value decreases. This is because more frequent compounding means that the money earns interest on interest more often, making the future value grow faster, and thus requiring a lower present value.\r<BR>\n" );
document.write( "\n" );
document.write( "**Summary**\r<BR>\n" );
document.write( "\n" );
document.write( "* Higher interest rates lead to lower present values.<BR>\n" );
document.write( "* More frequent compounding leads to lower present values.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );