Question

A father is now planning a savings program to put his daughter through college. She just celebrated her 13th birthday, she plans to enroll at the university in 5 years when she turns 18 years old, and she should graduate in 4 years. Currently, the annual cost (for everything – food, clothing, tuition, books, transportation, and so forth) is $15,000, but these costs are expected to increase by 5% annually. The college requires that this amount be paid at the start of the school year. She now has $7,500 in a college savings account that pays 6% annually.

a) How large must each payment be if the father makes five equal annual deposits into her account; the first deposit today and the fifth deposit on the daughter’s 17th birthday? [Hint: Calculate the cost (inflated at 5%) for each year of college and find the PV of these costs, discounted at 6%, as of the day she enters college. Then find the compounded value of her initial $7,500 on that same day. The difference between the PV costs and the amount that would be in the savings account must be made up by the father’s deposits].

b) If instead, her father makes six equal annual deposits into her account; the first deposit today and the sixth on the day she starts college. How large must each of the six payments be?

Answer 1

future value = present value(1+r)^n

where , r = rate of interest

n = number of years

first lets find out cost of college for all 4 years using inflation rate 5%

1st year = 15,000(1.05)^5 = 19,144.22

2nd year = 15,000(1.05)^6 = 20,101.43

3rd year = 15.000(1.05)^7 = 21,106.51

4th year = 15,000(1.05)^8 = 22,161.83

present value of all above payments using discount rate 6%

Year 1 2 3 cash flow PV factor @6% present value 19,144.22 0.94340 18060.58491 20,101.43 0.89000 17890.20114 21,106.51 0.83962 17721.43279 22,161.83 0.79209 17554.24511 total 71226.46 4

7500 of savings account will become after 4 years = 7500(1.06)^4

=9468.58

amount to be required on her 17th birth day = 71,226.46 - 9468.58

= 61,757.89

So future value should be 61,757.89

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

where P = annual deposit

n = number of periods

r = Rate of interest

61,757.89 = (1.06) * P((1.06)^5 - 1 / 0.06]

so P = 61,757.89 / 5.975319

P = 10,335.5

b)

in this case he makes 6 deposits instead of 5 and we have to find present value as on the date of her 18th birthday

so present value of college expenses on her 18th birthday as follows

Year 1 2 3 cash flow PV factor @6% present value 19,144.22 1.00000 1 9144.22 20,101.43 0.94340 18963.61321 21,106.51 0.89000 18784.71876 22,161.83 0.83962 18607.49982 total 75500.05

7500 will become after 5 years = 7500(1.06)^5

= 10,036.69

amount to be required on her 18th birth day = 75,500.05 - 10,036.69

= 65,463.36

So future value should be 65,463.36

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

where P = annual deposit

n = number of periods

r = Rate of interest

65,463.36 = (1.06) * P((1.06)^6 - 1 / 0.06]

so P = 65,463.36 / 7.393838

= $8853.77