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document.write( "You want to be able to withdraw $45,000 from your account each year for 15 years after you retire.
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document.write( "You expect to retire in 25 years.
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document.write( "If your account earns 4% interest, how much will you need to deposit each year until retirement to achieve your retirement goals?
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document.write( " I will solve this problem in two steps.\r
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document.write( " Step 1.\r
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document.write( "First, let's determine how much money X should be accumulated on the account during 25 years when you make your annual deposits,\r\n" );
document.write( "in order for to have enough to withdraw $45,000 each year for 15 years of your retirement.\r\n" );
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document.write( "Withdrawing $45,000 each year, your account (the remaining money) still earns 4% per annum compounded at the end of each year. \r\n" );
document.write( "So, during this period of 15 years, the annuity works as a sinking fund.\r\n" );
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document.write( "If you withdraw at the beginning of each year for living, the formula for the starting value X of sinking fund is\r\n" );
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document.write( " X = 
, (1)\r\n" );
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document.write( "where W is the annual withdraw amount; r is the annual rate of compounding as the decimal,\r\n" );
document.write( "and n is the number of withdrawals.\r\n" );
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document.write( "In this problem, W = $45,000, r = 0.04; n = 15. Substitute these values into the formula (1) and calculate X\r\n" );
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document.write( " X = 
= $520340.53. (2)\r\n" );
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document.write( "So, after 25 years of accumulating money, the fund should have $520340.53.\r\n" );
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document.write( "Thus, the first step of calculations is complete.\r\n" );
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document.write( " Step 2. \r
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document.write( "Now we are in position to determine how much should be deposited each year during 25 years when you make annual deposits.\r\n" );
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document.write( "I will assume that the annual deposits are made at the end of each of 25 years.\r\n" );
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document.write( "Then the fund works as a standard Ordinary Annuity saving plan.\r\n" );
document.write( "So, we use this formula for the future value\r\n" );
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document.write( " FV = 
, (3)\r\n" );
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document.write( "where D is the annual deposit amount; r is the annual rate of compounding as the decimal,\r\n" );
document.write( "and m is the number of deposits.\r\n" );
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document.write( "In this problem, FV = $500327.43 as we determined above; D is the unknown value of the deposit to find from the equation; \r\n" );
document.write( "r = 0.04; m = 25. Substitute these values into the formula (3) \r\n" );
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document.write( " 520340.53 = 
. (4)\r\n" );
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document.write( "The multiplier 
is 41.64591, \r\n" );
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document.write( "It implies from equation (4) that D = 
= $12494.40.\r\n" );
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document.write( "It is your ANSWER: During 25 years of the accumulating period you should deposit $12494.40 at the end of each year to your account.\r\n" );
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document.write( " Then you will be able to withdraw $45000 at the beginning of every year during 15 years.\r\n" );
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document.write( "For clarity, notice that in 25 years, you deposit in the bank 25*12494.40 = 312360 dollars.\r\n" );
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document.write( "During following 15 years, you withdraw from the bank 15*45000 = 675000 dollars.\r\n" );
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document.write( "The difference 675000 - 312360 = 362640 dollars is the interest, which your account earned \r\n" );
document.write( "during 25 + 15 = 40 years.\r\n" );
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document.write( "Solved.\r
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