document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=175411>Question 175411</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How much you should invest now to get $8,000 annually for ever, given an interest rate of 8% for the first 3 years & 10% for the remaining years </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #130576 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=130576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> in order for you to get the money forever, you have to invest enough so that the principal never gets below a minimum figure.<BR>\n" );
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document.write( "what you will be drawing out each year is 8000.<BR>\n" );
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document.write( "assume that you are starting in the 4th year.<BR>\n" );
document.write( "your interest rate is 10% forever.<BR>\n" );
document.write( "your money that you want to take out each year is $8,000<BR>\n" );
document.write( "your equation would be:<BR>\n" );
document.write( "p = principal<BR>\n" );
document.write( "r = interest rate<BR>\n" );
document.write( "i = interest<BR>\n" );
document.write( "---<BR>\n" );
document.write( "p*r = i<BR>\n" );
document.write( "solving for p, we get:<BR>\n" );
document.write( "p = i/r<BR>\n" );
document.write( "if i = 8000 and r = .1, then p = 8000/.1 = 80000<BR>\n" );
document.write( "---<BR>\n" );
document.write( "to see if this is true, then check to see if this holds.<BR>\n" );
document.write( "80000 * .1 = 8000 + 80000 = 88000 at the end of the 4th year.<BR>\n" );
document.write( "withdraw 8000 and you are left with 80000.<BR>\n" );
document.write( "80000 * .1 = 8000 + 80000 = 88000 at the end of the 5th year.<BR>\n" );
document.write( "withdraw 8000 and you are left with 80000.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "it's plain to see that if you have 80000 at the beginning of the 4th year you can withdraw 8000 forever assuming the interest rate remains at 10%.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "now to the first 3 years.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "you need to have 80,000 left in the account at the beginning of the 4th year.<BR>\n" );
document.write( "your future value is therefore 80,000<BR>\n" );
document.write( "you need to withdraw 8000 each year.<BR>\n" );
document.write( "your interest rate is .08 per year.<BR>\n" );
document.write( "your initial investment is x (unknown)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "this winds up being the future value of an annuity formula.<BR>\n" );
document.write( "that formula is so complicated i won't even show it here.<BR>\n" );
document.write( "i used the PV formula from excel.<BR>\n" );
document.write( "that function takes in arguments as follows:<BR>\n" );
document.write( "PV(rate,nper,pmt,fv,type)<BR>\n" );
document.write( "where:<BR>\n" );
document.write( "rate is the interest rate per payment period = .08 per year.<BR>\n" );
document.write( "nper is number of payment periods = 3 years<BR>\n" );
document.write( "pmt is 8000 which is the amount you withdraw each period.<BR>\n" );
document.write( "fv is future value of the investment = 80,000 cause that's what you need to have at the end of the 3 years.<BR>\n" );
document.write( "type is 0 which means the payments are made at the end of each payment period.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "that function gave me the answer.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the answer is:<BR>\n" );
document.write( "initial investment is 84,123.36<BR>\n" );
document.write( "---<BR>\n" );
document.write( "here's how it works:<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the problem:<BR>\n" );
document.write( "How much you should invest now to get $8,000 annually for ever, given an interest rate of 8% for the first 3 years & 10% for the remaining years<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the answer:<BR>\n" );
document.write( "you invest 84,123.36 now.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "interest rate for the first 3 years is 8% annually.<BR>\n" );
document.write( "amount to withdraw each year is 8,000<BR>\n" );
document.write( "---<BR>\n" );
document.write( "end of year 1:<BR>\n" );
document.write( "1.08 * 84123.36 - 8000 = 82853.2288<BR>\n" );
document.write( "end of year 2:<BR>\n" );
document.write( "1.08 * 82853.2288 - 8000 = 81481.4871<BR>\n" );
document.write( "end of year 3:<BR>\n" );
document.write( "1.08 * 81481.4871 - 8000 = 80000.00607<BR>\n" );
document.write( "---<BR>\n" );
document.write( "now you start at 10% a year interest:<BR>\n" );
document.write( "---<BR>\n" );
document.write( "end of year 4:<BR>\n" );
document.write( "1.10 * 80000.00607 - 8000 = 80,000.00668<BR>\n" );
document.write( "end of year 5:<BR>\n" );
document.write( "1.10 * 80000.00668 - 8000 = 80,000.00735<BR>\n" );
document.write( "etc.........<BR>\n" );
document.write( "---<BR>\n" );
document.write( "you will be able to withdraw 8000 forever and the principal will remain the same.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "there is a very slight increase of principal each year due to rounding errors.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "hope this helps.<BR>\n" );
document.write( "i tried deriving the formula for the first 3 years but gave up because it was too complicated and was taking too much time.<BR>\n" );
document.write( "for problems like this, using a financial calculator, or microsoft excel financial functions, is the best way to go.<BR>\n" );
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document.write( " </SPAN>\n" );
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