Question

John just turned 12 (at t = 0), and he will be entering college 6 years from now (at t = 6). College tuition and expenses at State U. are currently $16,500 a year, but they are expected to increase at a rate of 3.5% a year. John is expected to graduate in 4 years. Tuition and other costs will be due at the beginning of each school year (at t = 6, 7, 8, and 9). So far, John’s college savings account contains $10,000 (at t = 0). John’s parents plan to add an additional $12,000 in each of the next 4 years (at t = 1, 2, 3, and 4). Then they plan to make 2 equal annual contributions in each of the following two years, t = 5,, and 6. They expect their investment account to earn 5.5%. How large must the annual payments at t = 5, and 6 be to cover John’s anticipated college costs?

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Answer 1

Future value = present value*(1+r)^n

r = increasing rate of tution fee and expenses

n = number of years

Year 1 expenses = 16,500*(1+3.5%)^6 = 20,282.71

Year 2 expenses = 16,500*(1+3.5%)^7 = 20,992.61

Year 3 expenses = 16,500*(1+3.5%)^8 = 21,727.35

Year 4 expenses = 16,500*(1+3.5%)^9 = 22,487.81

inevestment account earns 5.5%, so using this rate we calculate total value of expenses for 4 years at beginning of college (i.e., present value of college expenses (at t = 6))

Present value = 20,282.71 + ( 20,992.61 / (1+5.5%)) + (21,727.35 / (1+5.5%)^2) +  (22,487.81 / (1+5.5%)^3)

= 78,852.83

so john need $78,852.83 at the beginning of his college to cover all his 4 year expenses

John need 78,852.83 at t = 6

Current balance = $10,000

Value of 10,000 in four years (t = 4) = 10,000*(1+5.5%)^4 = 12,388.25.............(i)

future value of annuity = P*[(1+r)^n - 1 / r ]

future value of 12,000 payments = 12,000 * [(1+5.5%)^4 - 1 / 5.5% ] = 52,107.20............(ii)

(i) + (ii) = 12,388.25 + 52,107.20 = 64,495.44

Above value after 2 years = 64,495.44 * (1+5.5%)^2 = 71,785.04

remaining balance required = 78,852.83 - 71,785.04 = 7067.79

let 'x' be the equal amount of deposits at t = 5 and t = 6

so x*(1+5.5%) + x = 7067.79

2.055x = 7067.79

x = 7067.79 / 2.055

x = $3439.32

Annual payments at t = 5 and t = 6 = $3439.32

(in case of any further explanation please comment)