Question

A father is now planning a savings program to put his daughter through college. She is 13, she plans to enroll at the university in 5 years, 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 6% annually. The college requires that this amount be paid at the start of the year. She now has $10,000 in a college savings account that pays 7% annually. Her father will make six equal annual deposits into her account; the first deposit today and sixth on the day she starts college. How large must each of the six payments be? Round your answer to the nearest dollar. [Hint: Calculate the cost (inflated at 6%) for each year of college and find the total present value of those costs, discounted at 7%, as of the day she enters college. Then find the compounded value of her initial $10,000 on that same day. The difference between the PV of costs and the amount that would be in the savings account must be made up by the father's deposits, so find the six equal payments (starting immediately) that will compound to the required amount.]

Answer 1

From this timeline, ordinary annuity formula is used to determine PMT

Annual cost for year of college

CF at the 5^th Year = 15000 (1+0.06)⁵ = 20073.38

CF at the 6^th Year = 15000 (1+0.06)⁶   = 21277.79

CF at the 7^th Year = 15000 (1+0.06)⁷   = 22554.45

CF at the 8^th Year = 15000 (1+0.06)⁸   = 23907.72

Multiplying the Cash Flow with Discount Factor 7%

CF at the 5^th Year = 15000 (1+0.06)⁵ x 1           = 20073.38

CF at the 6^th Year = 15000 (1+0.06)⁶   x 0.9346 = 19886.22

CF at the 7^th Year = 15000 (1+0.06)⁷    x 0.8734 = 19699.06

CF at the 8^th Year = 15000 (1+0.06)⁸   x 0.8163 = 19515.87

Total Cash Flows = 79174.43

PV (at the beginning of 1^st year of college) = Sum of the cash flows = 79174.43

Compounded Value of Initial $10,000 in savings account = Initial amount x (1+r)^n

                                                                                                       = 10000 (1+0.07)⁵

                                                                                                       = 14,025.51

Difference between PV costs and Compounded Value = 79174.43 – 14025.51

                                                                                                 = 65149.02

PMT (Annual Equal Instalments) = PV x ( i / [1-(1+i)^n]

                                                           = 65149.02 x ( 0.07 / -0.5)

                                                           = 65149.02 x (-0.14)

                                                           = 9120.86

Note:

• i= 7%
• n = 6