Question

You are a financial advisor helping a young family create a college fund to provide for their daughter Mary’s education. Mary just turned 5. Her parents expect that she will go to Harvard University, starting school on her 18th birthday. She will only take four years to complete her degree. Currently, tuition and housing costs total $65,000 per year, which are paid at the beginning of each school year. These expenses are expected to increase at the rate of inflation which will run 4% annually for the next 25 years. In each year until Mary enters Harvard University, Mary’s parents will make a deposit into the college fund to exactly provide for all of the costs of her education when she enters college (the first deposit will be made one year from today). Since her parents expect their income to grow at the rate of 2% annually, they would like the amount that they put each year into the fund to increase in nominal terms at the rate of 2% annually. Mary’s parents can earn a rate of 10% annually on their investments and they face a 30% tax rate.

A. What do you expect tuition and housing to cost during Mary’s first year at Harvard University?
B. How much must be in the savings account on Mary’s 18th birthday after the last deposit has been made but before the first payment to Harvard University?
C. What should be the amount of the first payment?
D. Alternatively, Mary’s parents could open a 529 account that allows college savings to grow tax-free. If they saved in this account rather than in a normal investment fund, and still earned the same pre-tax return, how much should be their first deposit?

Answer 1

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