Question

Advanced Time Value of Money Problems

(Try to work this question WITHOUT using Excel, get calculation in detail)

Question (College planning)

Your child was just born and you are planning for his/her college education. Based on your wonderful experience in Financial Economics you decide to send your child to Hofstra University as well. You anticipate the annual tuition to be $60,000 per year for the four years of college. You plan on making equal deposits on your child’s birthday every year starting today, the day of your child’s birth. No deposits will be made after starting college. The first tuition payment is due in exactly 18 years from today (the day your child turns 18 – no deposit required, i.e. last deposit is on 17th birthday). Assume the annual expected return on your investments is 10% over this period.

1. (i) Calculate the annual deposit.

2. (ii) Calculate the amount needed if only equal annual deposits are made on birthday’s 5-10 inclusive.

3. (iii) Calculate the amount needed if two equal annual deposits are made on birthday’s 5 and 13.

4. (iv) Answer part (i), now assume tuition rises 10% per year.

5. (v) Answer part (i) assuming first deposit will be made on your child’s 1st birthday. All other information is the same. What is the annual tuition payment? How does it compare to part (i)? Is your answer surprising?

Answer 1

(i) Annual Tuition Fees = $ 60000, Tuition Fees is Required on the child's 18th,19th,20th and 21st birthday, Annual Interest Rate = 10%,Annual Deposits start at t=0 (current time) and continue upto t = 17 for a total of 17 deposits

Let the required annual deposits be $ p

Therefore, Future Value of 17 Deposits at the end of year 18 = PV of Annual Tuition Withdrawals at the end of Year 18

p x (1.1)^(18) + p x (1.1)^(17) +.........+ p x (1.1) = 60000 x (1/0.1) x [1-{1/(1.1)^(4)}] x (1.1)

p x [{(1.1)^(18)-1} / {1.1-1}] x (1.1) = 209211.12

p x 50.1591 = 209211.12

p = 209211.12 / 50.1591 = $ 4170.95

(ii) let the required annual deposits be y

If annual deposits are made on birthdays between 5-10 years, then deposits come in at the end of year 5,year 6,year 7,year 8,year 9 and year 10.

Therefore, y x (1.1)^(18-5) + y x (1.1)^(18-6) +.............+ y x (1.1)^(18-10) = 209211.12

y x 16.5391 = 209211.12

y = 209211.12 / 16.5391 = $ 12649.49

(iii) Let the required deposits be $ m

Two Deposits are made, one each at the end of Year 5 and Year 13

Therefore, m x (1.1)^(18-5) + m x (1.1)^(18-13) = 209211.12

m x 5.06278 = 209211.12

m = 41323.36

(iv) If tuition fees rise at a rate of 10 % per annum, the required tuition withdrawals constitute a growing annuity starting immediately.

Therefore, PV of Growing Annuity at the end of Year 18 = 60000 + 60000 x [(1.1)/(1.1)] + 60000 x [(1.1)^(2)/(1.1)^(2)] + 60000 x [(1.1)^(3)/(1.1)^(3)] = $ 240000

Let the required annual deposits be $ n

n x 50.1591 = 240000

n = 240000 / 50.1591 = $ 4784.77

(v) If first deposit is made on the child's 1st birthday, then the deposit stream resembles an ordinary annuity.

Let the required annual deposits be $ k

Therefore, k x (1.1)^(17) + k x (1.1)^(16) +..............+ k = 209211.12

k x [{(1.1)^(18)-1} / {1.1-1}] = 209211.12

k x 45.5992 = 209211.12

k = 209211.12 / 45.5992 = $ 4588.05