SOLUTION: Your friend Bill has a daughter who just turned 8 years old. She is planning on starting college on her 18th birthday. Bill has determined he will need $25,000, $26,000, $27,000

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Question 945355: Your friend Bill has a daughter who just turned 8 years old. She is planning on starting college on her 18th birthday. Bill has determined he will need $25,000, $26,000, $27,000 and $30,000 for her freshman, sophomore, junior and senior years of college. He plans on making these amounts available to his daughter at the beginning of each of these years. Bill currently has $7,000 saved and would like to make monthly deposits for ten years at the end of each month until she turns 18. He wants the account to be worth enough just to pay for her college expenses listed above. If he can earn 9% APR on his deposits, and assuming that any balance remaining in the account will only earn 3% APR after she turns 18, how much will Bill have to deposit each month to provide for his daughter�s education?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
you have 4 years of college at the following required amounts.
25000
26000
27000
30000

9% apr divided by 12 = .75% compounded monthly.

3% apr divided by 12 = .25% compound monthly.

the first thing you need to do is find the present value of the yearly college costs brought back to the point where she turns 18.

this present value will be compounded monthly at the monthly interest rate of .25%.

the present value of the first year's tuition = 25000 since that is already at when she turns 18.

the present value of the second year's tuition = 26000 / 1.0025^12 = 25233.

the present value of the third year's tuition = 27000 / 1.0025^24 = 25430.

the present value of the fourth year's tuition = 30000 / 1.0025^36 = 27421.

add them up and the present value of the college costs required at the beginning of each school year totals up to be 103084.

he needs to invest enough for 10 years to have 103184 in her account on her 18th birthday.

the 7000 he already has in the bank will be equal to 7000 * 1.0075^120 = 17159 at the end of 10 years.

subtract that from the 103084 and you get 103084 - 17159 = 85925.

he needs to deposit an amount at the end of each month for 10 years so that the future value of that amount will be equal to 85925.

the payment required each month will be 444.02.

this payment will be made at the end of each month for 120 months because 10 years * 12 months per year is equal to 120 months.

this should be your solution.

i'll do the month by month calculations for you to show you that this is a correct analysis.

i'll also use the unrounded numbers so that the end result will be as close as possible to what is desired.

what is desired is that he has no money left in her account at the point where the final year's tuition is due.

this means that he had invested just enough to cover her tuition costs for the 4 years.

the actual details of the analysis are shown below:

it should be self explanatory.

if you need further details about how the calculations were done, let me know and i'll explain as best i can.

note that the middle time periods from time period 17 to 108 have been omitted because nothing was happening there except the continuation of what went before and after.