document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=854689>Question 854689</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If $39,500 is invested at 6.3% for 30 years, find the future value if the interest is compounded the following ways. (Round your answers to the nearest cent.)<BR>\n" );
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document.write( "(b) semiannually<BR>\n" );
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document.write( "(c) quarterly<BR>\n" );
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document.write( "(d) monthly<BR>\n" );
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document.write( "(e) daily<BR>\n" );
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document.write( "(f) every minute (N = 525,600)<BR>\n" );
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document.write( "(g) continuously<BR>\n" );
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document.write( "(h) simple (not compounded)<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 #514818 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=514818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If $39,500 is invested at 6.3% for 30 years, find the future value if the interest is compounded the following ways. (Round your answers to the nearest cent.)\r<BR>\n" );
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document.write( "the formula for future value is:\r<BR>\n" );
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document.write( "f = p * (1+i)^n where:\r<BR>\n" );
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document.write( "f = future value<BR>\n" );
document.write( "p = present value<BR>\n" );
document.write( "i = interest rate per time period<BR>\n" );
document.write( "n = number of time periods.\r<BR>\n" );
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document.write( "time period can be yearly, semi-annually, quarterly, monthly, daily, every minute, or continuous.\r<BR>\n" );
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document.write( "if you are given the annual interest rate and the number of years, then you adjust your figures based on the time period used.\r<BR>\n" );
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document.write( "if the time period is annual, then the interest rate given per year and the number of years are used as is with no adjustment.\r<BR>\n" );
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document.write( "if the time period is quarterly, then the annual interest rate is divided by 4 and the number of years is multiplied by 4.\r<BR>\n" );
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document.write( "if the time period is monthly, then the annual interest rate is divided by 12 and the number of years is multiplied by 12.\r<BR>\n" );
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document.write( "essentially, you divide the annual interest rate by the number of compounding periods per year and you multiply the number of years by the number of compounding periods per year.\r<BR>\n" );
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document.write( "you will see this as we go through the individual cases.\r<BR>\n" );
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document.write( "(a) annually<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500.\r<BR>\n" );
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document.write( "no adjustment is necessary.\r<BR>\n" );
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document.write( "formula of f = p * (1+i)^n becomes f = 39500 * (1.063)^30 = 246942.04\r<BR>\n" );
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document.write( "(b) semiannually<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500.\r<BR>\n" );
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document.write( "annual interest rate is divided by 2 to get .0315.<BR>\n" );
document.write( "number of time periods is equal to 2 * 30 = 60.\r<BR>\n" );
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document.write( "formula of  f = p * (1+i)^n becomes f = 39500 * (1.0325)^60 = 253951.65<BR>\n" );
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document.write( "(c) quarterly<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500.  \r<BR>\n" );
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document.write( "annual interest rate is divided by 4 to get .01575.<BR>\n" );
document.write( "number of years is multiplied by 4 to get 120.\r<BR>\n" );
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document.write( "formula of f = p * (1+i)^n becomes f = 39500 * (1.01575)^120 = 257642.09<BR>\n" );
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document.write( "(d) monthly<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500. \r<BR>\n" );
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document.write( "annual interest rate is divided by 12 to get .00525.<BR>\n" );
document.write( "number of years is multiplies by 12 to get 360.\r<BR>\n" );
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document.write( "formula of f = p * (1+i)^n becomes f = 39500 * (1.00525)^360 = 260175.58.\r<BR>\n" );
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document.write( "(e) daily<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500. \r<BR>\n" );
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document.write( "annual interest rate is divided by 365 to get .0001726027397.<BR>\n" );
document.write( "number of years is multiplied by 365 to get 10950.\r<BR>\n" );
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document.write( "formula of f = p * (1+i)^n becomes f = 39500 * (1.0001726027397)^10950 = 261422.4238\r<BR>\n" );
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document.write( "(f) every minute (N = 525,600)<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500.  \r<BR>\n" );
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document.write( "annual interest rate is divided by 525600 to get .0000001198630137.<BR>\n" );
document.write( "number of years is multiplied by 525600 to get 15768000.\r<BR>\n" );
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document.write( "formula of f = p * (1+i)^n becomes f = 39500 * (1.0000001198630137)^15768000 = 261464.98\r<BR>\n" );
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document.write( "(g) continuously<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500.\r<BR>\n" );
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document.write( "a different formula is used here.<BR>\n" );
document.write( "the formula is f = p * e^(in) where:\r<BR>\n" );
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document.write( "i = annual interest rate<BR>\n" );
document.write( "n = number of years.\r<BR>\n" );
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document.write( "the formula of f = p * e^(in) becomes f = 39500 * e^(.063*30) = 261465.06\r<BR>\n" );
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document.write( "(h) simple (not compounded)<BR>\n" );
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document.write( "annual interest rate is .063.<BR>\n" );
document.write( "number of years is 30.<BR>\n" );
document.write( "p = 39500.  \r<BR>\n" );
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document.write( "with simple interest there is no compounding.<BR>\n" );
document.write( "the formula is f = p * i * n + p.\r<BR>\n" );
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document.write( "i is the annual interest rate<BR>\n" );
document.write( "n is the number of years\r<BR>\n" );
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document.write( "the formula of f = p * i * n + p becomes f = 39500 * .063 * 30 + 39500 = 114155\r<BR>\n" );
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document.write( "the continuous compounding rate is the best that you can do.<BR>\n" );
document.write( "as you increase the number of compounding periods per year, you approach the continuous compounding rate.\r<BR>\n" );
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document.write( "this is why you see progressively higher values when the compounding periods per year are increased.\r<BR>\n" );
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document.write( "the simple interest rate gives you the lowest value because there is no compounding.\r<BR>\n" );
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