Question

Jim and Elsie are saving for their granddaughter Amy’s college education. Amy just turned 12 (at t = 0), and she will be entering college 6 years from now (at t = 6). College tuition and expenses at Sam Houston State University are currently $15,000 a year, but they are expected to increase at a rate of 2% a year. Amy should graduate in 4 years--if she takes longer or wants to go to graduate school, she will be on her own. Tuition and other costs will be due at the beginning of each school year (at t = 6, 7, 8, and 9).

So far, Jim and Elsie have accumulated $30,000 in their college savings account (at t = 0). Their long-run financial plan is to add an additional $5,000 in each of the next 3 years (at t = 1, 2, and 3). Then they plan to make 3 equal annual contributions in each of the following years, t = 4, 5, and 6. They expect their investment account to earn 6%. How large must the annual payments at t = 4, 5 and 6 be to cover Amy's anticipated college costs? (Note: 1.5x Credit)

		$375.85
		$730.68
		$881.22
		$4,063.96
		$12,191.88

Answer 1

The scenario considers three different cash flows. The first cash flow stream are the deposits at t=1, t=2 and t=3 each worth $ 5000. The second cash flow stream are the planned equal installment deposits at t=4, t=5 and t=6. The third cash flow stream are the withdrawals for college expenses at t=6, t=7, t=8 and t=9

The first two cash flows are inflows whereas the last one is an outflow. The sum of the first two cash flow streams plus the accmulated $ 30000, should equal the third cash flow stream at any point in time discounted/compounded at the given interest rate of 6 %.

Current College Fees = $ 15000, Annual growth rate = 2 % and let current time be t=0

College Fees at t=6 will be : 15000 x (1.02)^(6) = $ 16892.44,

At t=7, will be: 15000 x (1.02)^(7) = $ 17230.29

At t=8, will be: 15000 x (1.02)^(8) = $ 17574.89

At t=9, will be : 15000 x (1.02)^(9) = $ 17926.39

Total Present Value of Expected College Fees at t=0 (current time) = 16892.44 / (1.06)^(6) + 17239.29 / (1.06)^(7) + 17574.89 / (1.06)^(8) + 17926.39 / (1.06)^(9) = $ 45004.93

Total Present Value of Deposits between t=1 and t=3, at t=0 (current time) = 5000 x (1/0.06) x [1-{1/(1.06)^(3)}] = $ 13365.06

Accumulated Amount = $ 30000

Required Present Value of Equal Installments between t=4 and t=6, at t=0 (current time) = 45004.93 - 30000 - 13365.06 = $ 1639.87

Let the required equal installment deposits be $ N

1639.87 = N / (1.06)^(4) + N / (1.06)^(5) + N / (1.06)^(6)

1639.87 = N x [0.79209+0.74726+0.70496]

1639.87 = N x 2.24431

N = 1639.87 / 2.24431 = $ 730.679 ~ $ 730.68

Hence, the correct option is (b)