Question

Suppose you wish to plan for your newborn’s college tuition payment. You intend to make equal quarterly deposits into an account offering annual rate of 6% compounded quarterly on the child’s 1st through 12th birthdays. You expect that tuition payments will be $40,000 every six months by the time the child is ready to enter college. Therefore, your goal is to make eight semi-annual withdrawals of $40,000 each starting on the child’s 18th birthday to be used for his tuition payments. How much must each of the quarterly deposits be such that there will be enough money accumulated in the account to exactly meet the goal? Assume the account offers 6% per year compounded quarterly until the child’s 18th birthday and 6% per year compounded semi- annually after that. [Suggestion: Draw the cash flow diagram to aid you in solving this problem.

Answer 1

Step 1: Fund required for the Tuition fee at the end of 18 years,

Number of withdrawal ( nper) = 8

amount per withdrawal (PMT) = 40,000

Interest rate (rate) =6% (annual) = 3% (semi annual)

value of the fund = PV (3%,8,40000,0,1)  = $ 289,211.32.

Step 2: Value of fund at end of 12 years of child

Value = FV / (1+r/n)^ n*t

=$ 289,211.32 / ( 1+ 1.5%) ^24 = $ 202,316.02

Step 3": For the calculation of Quarterly deposit ,

Future value (FV) =$ 202,316.02.

Number of withdrawal ( nper) = 11*4= 44 ( from 1 year to 12 year = 11 year

Interest rate (rate) =6% (annual) = 1.5% (quaterly)

amount per withdrawal (PMT) = ?

value of the fund = PMT(1.5%,48,0,-202316.02,1) = $ 28,65.31.

Cash flow at time : as

Year	Cash flow	Year	Cash flow	Year	Cash flow
0	0	5.75	2865.31	11.25	2865.31
0.25	0	6	2865.31	11.5	2865.31
0.5	0	6.25	2865.31	11.75	2865.31
1	2865.31	6.5	2865.31	12	2865.31
1.25	2865.31	6.75	2865.31	12.25	0
1.5	2865.31	7	2865.31	13	0
1.75	2865.31	7.25	2865.31	14	0
2	2865.31	7.5	2865.31	15	0
2.25	2865.31	7.75	2865.31	16	0
2.5	2865.31	8	2865.31	17	0
2.75	2865.31	8.25	2865.31	18	-40,000
3	2865.31	8.5	2865.31	18.5	-40,000
3.25	2865.31	8.75	2865.31	19	-40,000
3.5	2865.31	9	2865.31	19.5	-40,000
3.75	2865.31	9.25	2865.31	20	-40,000
4	2865.31	9.5	2865.31	20.5	-40,000
4.25	2865.31	9.75	2865.31	21	-40,000
4.5	2865.31	10	2865.31	21.5	-40,000
4.75	2865.31	10.25	2865.31
5	2865.31	10.5	2865.31
5.25	2865.31	10.75	2865.31
5.5	2865.31	11	2865.31