Compounding Intervals

<BODY LANG="EN"> <HR><center> <table border=1> <td><A NAME="tex2html54" HREF="node3.html"><img height=36 width=36 alt="PREVIOUS" src="http://www.cs.utah.edu/icons/latex2html/previous.gif"></A></td> <td><A NAME="tex2html60" HREF="uces-4.html"><img height=36 width=36 alt="UP" src="http://www.cs.utah.edu/icons/latex2html/up.gif"></A></td> <td><A NAME="tex2html62" HREF="node5.html"><img height=36 width=36 alt="NEXT" src="http://www.cs.utah.edu/icons/latex2html/next.gif"></A></td> <td><a href="contents-node4.html"><img height=36 width=36 alt="CONTENTS" src="http://www.cs.utah.edu/icons/latex2html/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="http://www.cs.utah.edu/icons/latex2html/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="http://www.cs.utah.edu/icons/latex2html/copyright.gif"></a></td> <td><a href="mailto:hamlet@cs.utah.edu"><img height=36 width=36 alt="FEEDBACK" src="http://www.cs.utah.edu/icons/latex2html/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="http://www.cs.utah.edu/icons/latex2html/about.gif"></a></td> </table> </center> <HR><H1><A NAME="SECTION00020000000000000000">Compounding Intervals</A></H1> <P> In the real world, interest is credited to your account more often than once a year. A checking account paying 5% annual interest, for example, may earn interest every day. This means that every day, interest is paid into your account at the rate of 1/365 of 5%. In other words, the effect of the interest is spread across the entire year. <P> Let's see what would happen if annual interest were compounded bi-annually. For example, if instead of getting 6% interest at the end of the year, what would happen if you got 3% interest in the middle of the year and then another 3% at the end of the year? <P> Let's repeat the analysis that we did when we first considered compound interest. If we invest an amount <I>P</I> at an annual rate of <I>R</I>, then after six months we would have <I>P</I>(1+<I>R</I>/2). At the end of a year, then, we would have <I>P</I>(1+<I>R</I>/2)(1+<I>R</I>/2). Try to come up with a formula for the amount of money in the account after <I>n</I> full years. <P> <P><A NAME="tex2html6" HREF="answer60.html">Click here for the answer</A><P> <P> Now suppose that compounding is quarterly: four times per year. Can you come up with a formula for the amount of money in the account after <I>n</I> full years? <P> <P><A NAME="tex2html7" HREF="answer64.html">Click here for the answer</A><P> <P> You should notice a pattern building up. Using Maple, define a function called <TT>intervals</TT> that takes four parameters: the initial balance, the interest rate, the number of years for which the balance will be invested, and the number of times per year that interest is compounded. The function should calculate the amount of money that will be in the account at the end of the specified interval. <P> <P><A NAME="tex2html8" HREF="answer69.html">Click here for the answer</A><P> <P> Now we can calculate the effects of more frequent compounding. Let's compare receiving 6% interest paid once per year vs. 6% interest compounded daily: <P> <table><td valign=top><a href="send12.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>compound(1000, .06, 1);</code></td></table> <P> <table><td valign=top><a href="send13.hamlet"><img height=24 width=24 alt="EXECUTE" src="http://www.cs.utah.edu/icons/latex2html/execute.gif"></a></td><td valign=top><code>intervals(1000, .06, 1, 365);</code></td></table> <P> Can you explain why the second result is larger than the first? <P> <HR><center> <table border=1> <td><A NAME="tex2html54" HREF="node3.html"><img height=36 width=36 alt="PREVIOUS" src="http://www.cs.utah.edu/icons/latex2html/previous.gif"></A></td> <td><A NAME="tex2html60" HREF="uces-4.html"><img height=36 width=36 alt="UP" src="http://www.cs.utah.edu/icons/latex2html/up.gif"></A></td> <td><A NAME="tex2html62" HREF="node5.html"><img height=36 width=36 alt="NEXT" src="http://www.cs.utah.edu/icons/latex2html/next.gif"></A></td> <td><a href="contents-node4.html"><img height=36 width=36 alt="CONTENTS" src="http://www.cs.utah.edu/icons/latex2html/toc.gif"></a></td> <td><a href="reset.hamlet"><img height=36 width=36 alt="RESET" src="http://www.cs.utah.edu/icons/latex2html/stop.gif"></a></td> <td><a href="copyright.html"><img height=36 width=36 alt="COPYRIGHT" src="http://www.cs.utah.edu/icons/latex2html/copyright.gif"></a></td> <td><a href="mailto:hamlet@cs.utah.edu"><img height=36 width=36 alt="FEEDBACK" src="http://www.cs.utah.edu/icons/latex2html/feedback.gif"></a></td> <td><a href="http://www.cs.utah.edu/%7Ehamlet/icons.html"><img height=36 width=36 alt="ABOUT" src="http://www.cs.utah.edu/icons/latex2html/about.gif"></a></td> </table> </center> <HR><P><ADDRESS> <I><A NAME="tex2html1" HREF="http://www.cs.utah.edu/~zachary">Joseph L. Zachary</A> <BR> <A NAME="tex2html2" HREF="http://www.cs.utah.edu/~hamlet">Hamlet Project</A> <BR> <A NAME="tex2html3" HREF="http://www.cs.utah.edu/">Department of Computer Science</A> <BR> <A NAME="tex2html4" HREF="http://www.utah.edu/ HTML_Docs/UofUHome.html">University of Utah</A></I> </ADDRESS> </BODY>