Question

Jack’s friend plans to buy a boat 45 years from now, when he retires. Today’s price for the boat is $300,000. The price is expected to rise 3% per year. The friend also wants to send his child to BC in 12 years. College is expected to cost $117,000 in the child’s first year, growing at 4% per year while in school. College lasts for 4 years and the first payment is due the first day of school. The friend has $25000 saved now and expects to put a certain fraction of his salary away each year, starting in 1 year with the final payment on the retirement date. His salary will grow by 2% per year. He currently makes $120,000. If Jack can earn 6% per year on the friend’s investments, what fraction of the friend’s salary must be saved?

Answer 1

All financials below are in $.

Today’s price for the boat is $300,000. The price is expected to rise 3% per year.

Price of the boat at the end of 45 years = 300,000 x (1 + 3%)⁴⁵ =  1,134,479

i = interest rate = 6%

PV of the price of boat today (t = 0) = 1,134,479 / (1 + i)⁴⁵ = = 1,134,479 / (1 + 6%)⁴⁵ =  82,419.97

The friend also wants to send his child to BC in 12 years. College is expected to cost $117,000 in the child’s first year, growing at 4% per year while in school. College lasts for 4 years and the first payment is due the first day of school.

PV of growing annuity at t = 11 years will be = 117,000 / ( i - g) x [1 - (1+g)ⁿ / (1 + i)ⁿ] = 117,000 / (6% - 4%) x [1 - (1+4%)⁴ / (1 + 6%)⁴] =  429,170.32

Hence, PV of the education cost today (i. e. at t = 0) = 429,170.32 / (1 + i)¹¹ = 429,170.32 / (1 + 6%)¹¹ = 226,081.57

Hence, total PV required today = PV of boat price + PV of college expenses = 82,419.97 + 226,081.57 =  308,501.54

Savings available now = 25,000

Shortfall =  308,501.54 - 25,000 =  283,501.54

This shortfall has to be met by putting a certain fraction of his salary away each year, starting in 1 year with the final payment on the retirement date. His salary will grow by 2% per year. He currently makes $120,000.

Let's assume he puts aside a fraction F of his salary.

So, first amount put aside = F x 120,000 x (1 + 2%) =  122,400F

Hence, PV of this growing annuity over 45 years = 122,400F / (i - g) x / ( i - g) x [1 - (1+g)^N / (1 + i)^N] = 122,400F / (6% - 4%) x [1 - (1+4%)⁴⁵ / (1 + 6%)⁴⁵​​​​​​​] =  3,522,905.49F

Hence, Shortfall = 283,501.54 =  3,522,905.49F

Hence, F = 283501.54 /  3,522,905.49 = 8.05% = 0.0805