document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167429>Question 1167429</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Kenneth ran into some money and decides to invest it for retirement. He has $75,000 to invest over<BR>\n" );
document.write( "40 years. Find the effective rates given:<BR>\n" );
document.write( "(a) 4.5% growth compounded monthly.<BR>\n" );
document.write( "(b) 4.45% growth compounded continuously.<BR>\n" );
document.write( "(c) Should Kenneth invest in option (a) or option (b)? Why? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792090 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=792090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the present value of the investment is 75,000.<BR>\n" );
document.write( "the investment period is 40 years.<BR>\n" );
document.write( "the discrete compounding growth rate is 4.5% compounded monthly.<BR>\n" );
document.write( "the continuous compounding growth rate is 4.45% compounded continuously.\r<BR>\n" );
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document.write( "the formula for discrete compounding is f = p * (1 + r) ^ n\r<BR>\n" );
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document.write( "f is the future value<BR>\n" );
document.write( "p is the present value<BR>\n" );
document.write( "r is the interest rate per time period<BR>\n" );
document.write( "n is the number of time periods.\r<BR>\n" );
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document.write( "the formula for continuous compounding is f = p * e ^ (r * t)\r<BR>\n" );
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document.write( "f is the future value<BR>\n" );
document.write( "p is the present value<BR>\n" );
document.write( "r is the interest rate per time period<BR>\n" );
document.write( "t is the number of time periods.\r<BR>\n" );
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document.write( "when your initial investment is 75,000 and your time period is 40 years, your future value will be calculated as follows:\r<BR>\n" );
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document.write( "with discrete monthly compounding, you get:\r<BR>\n" );
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document.write( "f = 75000 * (1 + .045/12) ^ (40 * 12) = 452198.6287.\r<BR>\n" );
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document.write( "with continuous compounding, you get:\r<BR>\n" );
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document.write( "f = 75000 * e ^ (.0445 * 40) = 444739.2314\r<BR>\n" );
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document.write( "4.5% compounded monthly gives you more future value than 4.45% compounded continuously.\r<BR>\n" );
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document.write( "if you look at the effective annual rates, you will see why this occurs.\r<BR>\n" );
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document.write( "with discrete compounding, the effective annual growth factor becomes:\r<BR>\n" );
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document.write( "f = (1 + .045/12) ^ 12 = 1.045939825.\r<BR>\n" );
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document.write( "with continuous compounding, the effective annual growth factor becomes:<BR>\n" );
document.write( "f = e ^ (.0445) = 1.045504977.\r<BR>\n" );
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document.write( "with discrete compounding, 75,000 * 1.045939825 ^ 40 = 452198.6287\r<BR>\n" );
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document.write( "with continuous compounding, 75,000 * 1.045504977 = 444739.2314\r<BR>\n" );
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document.write( "the effective annual growth rate tells you which will give you a greater future value.\r<BR>\n" );
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document.write( "the answers to your questions are:\r<BR>\n" );
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document.write( "(a) 4.5% growth compounded monthly.\r<BR>\n" );
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document.write( "effective annual growth rate is .045939825 or 4.5939825%\r<BR>\n" );
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document.write( "(b) 4.45% growth compounded continuously.\r<BR>\n" );
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document.write( "effective annual growth rate is .045504977 or 4.5504977%\r<BR>\n" );
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document.write( "(c) Should Kenneth invest in option (a) or option (b)? Why?\r<BR>\n" );
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document.write( "invest in option a because the effective annual interest rate is higher.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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