Today is your child\'s first birthday, and you are planning for your child\'s co

Question

Today is your child's first birthday, and you are planning for your child's college education. When your child was born, your rich aunt opened a college savings account for your child, and deposited $1,400 in it. Now, you would like to make deposits every 26 weeks (half year) in Years 1 through 21, with your first deposit to be made today (a total of 41 deposits), so that your child may make withdrawals in each of the Years 18 through 21 for tuition. Tuition was currently $2,500/year, and is expected to grow at 4%/year through year 10, and then at 5%/year for each of years 11 through 21. The college savings account earns a nominal annual rate of 8.632%, with interest compounded weekly (52-week year). How much must you deposit every 26 weeks? Answer is whole dollars, rounded up to next whole dollar, with no punctuation. For example, if your answer is $824.45, enter "825".

Explanation / Answer

Let us calculate the tuition cost:

2500 * (1+4%)10 = 3700.61 ; after Year 10 the tuition cost will increase at 5% per annum.

Tuition cost in Year 18 = 3700.61*(1+5%)8 = 5467.49 (assuming all cash flows are end of year)

Tuition in Year 19 = 5467.49 * (1+5%) = 5740.86

Tution in Year 20 = 5740.86 * (1+5%) = 6027.90

Tuition in Year 21 = 6027.90 * (1+5%) = 6329.30

Now we will calculate the present value of total tuition cost at Year 18, using the college savings account rate as discount rate. The effective rate for the savings account = (1+8.632%/52)52 = 9.01%

PV of total tuition = 5467.49 + 5740.86/(1+9.01%) + 6027.90/(1+9.01%)2 + 6329.30/(1+9.01%)3 = 20693.15

Hence the future value of savings account should be 20693.15. This is like a semi annual annuity and the FV formula for annuity is = Periodic Cash Flow * [(1+r)t - 1 / r] ; where r is the applicable rate (9.01%/2 in this case) and t is the time period or number of payment (41 in this case - given). Let the periodic cash flow be X.

then we have : 20693.15 = X * [((1+9.01%/2)41 - 1) / (9.01%/2)] - note that we have divided the rate by 2 since the payments are being made semi annually .

solving for X, we get X = 183.20 which is the amount they should invest every 26 weeks. Rounded the answer is 183

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