document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090542>Question 1090542</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A principal of &#8203;$7000 is invested in an account paying an annual rate of 4&#8203;%. Find the amount in the account after 3 years if the account is compounded&#8203; semiannually, quarterly, and monthly.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705032 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=705032\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula to use is:\r<BR>\n" );
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document.write( "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 interest rate per time period is determined as follows.\r<BR>\n" );
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document.write( "the interest rate per semi-annual time period is equal to the interest rate per year divided by 2.\r<BR>\n" );
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document.write( "the interest rate per quarterly time period is equal to the interest rate per year divided by 4.\r<BR>\n" );
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document.write( "the interest rate per monthly time period is equal to the interest rate per year divided by 12.\r<BR>\n" );
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document.write( "the number of time periods is determined as follows:\r<BR>\n" );
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document.write( "the number of semi-annual time periods is equal to the number of years times 2.\r<BR>\n" );
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document.write( "the number of quarterly time periods is equal to the number of years times 4.\r<BR>\n" );
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document.write( "the number of monthly time periods is equal to the number of years times 12.\r<BR>\n" );
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document.write( "interest rate is equal to interest rate percent divided by 100.\r<BR>\n" );
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document.write( "in your problem, the interest rate percent, otherwise known as the apr, is equal to 4%.\r<BR>\n" );
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document.write( "divide that by 100 to get an annual interest rate of .04.\r<BR>\n" );
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document.write( "the interest rate per year is equal to .04.<BR>\n" );
document.write( "the interest rate per semi-annual time periods is equal to .04/2<BR>\n" );
document.write( "the interest rate per quarterly time periods is equal to .04/4<BR>\n" );
document.write( "the interest rate per monthly time periods is equal to .04/12\r<BR>\n" );
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document.write( "the number of yearly time periods is equal to 3.<BR>\n" );
document.write( "the number of semi-annual time periods is equal to 2*3<BR>\n" );
document.write( "the number of quarterly time periods is equal to 4*3<BR>\n" );
document.write( "the number of monthly time periods is equal to 12*3\r<BR>\n" );
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document.write( "your equation of f = p * (1 + r) ^ n is set up as follows:\r<BR>\n" );
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document.write( "for annual time periods:\r<BR>\n" );
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document.write( "p = 7000<BR>\n" );
document.write( "r = .04<BR>\n" );
document.write( "n = 3<BR>\n" );
document.write( "f = 7000 * (1 + .04) ^ 3 = 7874.048\r<BR>\n" );
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document.write( "for semi-annual time periods:\r<BR>\n" );
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document.write( "p = 7000<BR>\n" );
document.write( "r = .04/2<BR>\n" );
document.write( "n = 3*2<BR>\n" );
document.write( "f = 7000 * (1 + .04/2) ^ (3*2) = 7883.136935\r<BR>\n" );
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document.write( "for quarterly time periods:\r<BR>\n" );
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document.write( "p = 7000<BR>\n" );
document.write( "r = .04/4<BR>\n" );
document.write( "n = 3*4<BR>\n" );
document.write( "f = 7000 * (1 + .04/4) ^ (3*4) = 7887.775211\r<BR>\n" );
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document.write( "for monthly time periods:L\r<BR>\n" );
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document.write( "p = 7000<BR>\n" );
document.write( "r = .04/12<BR>\n" );
document.write( "n = 3*12<BR>\n" );
document.write( "f = 7000 * (1 + .04/12) ^ (3*12) = 7890.903122\r<BR>\n" );
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document.write( "the more time periods per year, the greater the future value will be, up to the point where you have continuous compounding.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "with daily compounding, the number of time periods per year is 365.\r<BR>\n" );
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document.write( "p = 7000<BR>\n" );
document.write( "r = .04 / 365<BR>\n" );
document.write( "n = 3 * 365<BR>\n" );
document.write( "f = 7000 * (1 + .04/365) ^ (3*365) = 7892.426069\r<BR>\n" );
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document.write( "the continuous compounding formula is f = p * e^(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( "when p = 7000 and r = .04 and n = 3, the future value will be equal to 7000 * e^(.04 * 3) = 7892.477961\r<BR>\n" );
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document.write( "with continuous compounding, the number of time periods per year becomes irrelevant, since all time periods per year will give you the same answer.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "with 365 time periods a year, the continuous compounding formula becomes f = 7000 * e^(.04 / 365 * 3 * 365) which is equal to 7892.477962.\r<BR>\n" );
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document.write( "this is because .04 / 365 * 3 * 365 becomes .04 * 3 * 365 / 365 which becomes .04 * 3.\r<BR>\n" );
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document.write( "with discrete compounding, which is the f = p * (1 + r) ^ n formula, 365 time periods a year gets you pretty close to what the continuous compounding formula gives you.\r<BR>\n" );
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document.write( "the future value curve for discrete compounding flattens as you approach continuous compounding and can go no further than that.\r<BR>\n" );
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document.write( "this means that the incremental change gets less and less, the more time periods you use.\r<BR>\n" );
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document.write( "1000 time periods a year gets you closer to continuous compounding.\r<BR>\n" );
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document.write( "10000 time periods a year gets you closer still.\r<BR>\n" );
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document.write( "the limit is continuous compounding.\r<BR>\n" );
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document.write( "your future value can't get any higher than that.\r<BR>\n" );
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