Question

James would like to invest some money every year-end for 17 years in a money market account earning 5% per year, and then begin to withdraw money from the account to finance his son’s education for 4 years. James thinks that his son should have $35,000 for the first year of college at the start of the 19th year, and would like to increase that amount by 4% every year to compensate for inflation. How much will James’ first deposit be if he plans to increase the annual deposit by $450?

First Deposit:

Answer 1

ANSWER:

First we will find out the present value of james son fees at the end of the 17th year because james needs $35,000 at the start of the 19th year which means the end of the 18th year.

so the amount needed by james at the end of the 18th , 19th , 20th and 21st year are.

amount at the end of the 18th year = $35,000

amount at the end of the 19th year = $35,000 +$35,000 * 4% = $35,000 + $1,400 = $36,400

amount at the end of the 20th year = $36,400 + $36,400 * 4% = $36,400 + $1,456 = $37,856

amount at the end of the 21st year = $37,856 + $37,856 * 4% = $37,856 + $1,514.24 =$39,370.24

at 5% per year interest we will find out the present value of the amount needed at the end of the 17th year.

npv needed at the end of the 17th year = $35,000(p/f,i,n) + $36,400(p/f,i,n) + $37,856(p/f,i,n) + $39,370.24(p/f,i,n)

npv needed at the end of the 17th year = $35,000(p/f,5%,1) + $36,400(p/f,5%,2) + $37,856(p/f,5%,3) + $39,370.24(p/f,5%,4)

npv needed at the end of the 17th year = $35,000 * 0.9524 + $36,400 * .9070 + $37,856 * 0.8638 + $39,370.24 * .8227

npv needed at the end of the 17th year = $33,334 + $33,014.8 + $32,700.01 + $32,389.9

npv needed at the end of the 17th year = $131,440.64

this amount of the $131,440.64 is the amount that james requires at the end of the 17th year.

so now we will the find the present worth of this amount $131,440.64 at year 0.

pv = fv / ( 1+r) ^ n

n = 17 , i = 5% , fv = $131,440.64

pv = 131,440.64 / ( 1+5%) ^ 17

pv = 131,440.64 / (1.05) ^17

pv =131,440.64 / 2.292

pv = $57,347.12

the present value is $57,347.12 and this will be equal to the present value of the deposits that james will start depositing.

pv = first deposit(p/a,i,n) + gradient(p/g,i,n)

where gradient = $450 , i = 5% , n =17

57,347.12 =  first deposit(p/a,5%,17) + 450(p/g,5%,17)

57,347.12 = first deposit * 11.274 + 450 * 77.14

57,347.12 = 11.274 *first deposit + $34,713

57,347.12 - $34,713 = 11.274 *first deposit

22634.12 = 11.274 *first deposit

first deposit = 22634.12 / 11.274

first deposit = $2,007.638