Answer:
a. Tuition and housing costs today = $65,000 per year
Inflation rate = 4%
Tuition and housing costs in 13 years = 65,000 * (1 + 0.04)^13
Tuition and housing costs in 13 years = $108,229.78
b. Amount to be in the savings account can be calculated using the present value of a growing annuity due formula
After tax rate of return = 10 * (1 - 0.3) = 7%, Growth rate = 4%, Number of year = 4
PV = P x (1 + r) * [1 - (1 + g)^n * (1 + r)^-n] / (r - g)
PV = 108,229.78 * (1 + 0.07) * [1 - (1 + 0.04)^4 * (1 + 0.07)^-4] / (0.07 - 0.04)
PV = $415,050.16
c. Amount of the first payment can be calculated using FV of a growing annuity
FV = $415,050.16, Number of years = 13, Growth rate = 2%, Rate of return = 10%
FV = P * [(1 + r)^n - (1 + g)^n] / (r - g)
415,050.16 = P * [(1 + 0.07)^13 - (1 + 0.02)^13] / (0.07 - 0.02)
P = $18,591.47
d. If the investments are tax free, the rate of return = 10%
Amount to be in the savings account = PV = P * (1 + r) * [1 - (1 + g)^n * (1 + r)^-n] / (r - g)
= 108,229.78 * (1 + 0.1) * [1 - (1 + 0.04)^4 * (1 + 0.1)^-4] / (0.1 - 0.04)
= $398,768.92
FV = P * [(1 + r)^n - (1 + g)^n] / (r - g)
398,768.92 = P * [(1 + 0.1)^13 - (1 + 0.02)^13] / (0.1 - 0.02)
P = $14,778.36
