Question

Required Annuity Payments. A father is now planning a savings program to put his daughter through college. She is 13, plans to enroll at the university in 5 years, and should graduate 4 years later. Currently, the annual cost (for everything—food, clothing, tuition, books, transportation, and so forth) is $12,000, but these costs are expected to increase by 6% annually. The college requires total payment at the start of the year. She now has $10,000 in a college savings account that pays 9% annually. Her father will make 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? (Hint: Calculate the cost (inflated at 6%) for each year of college and find the total present value of those costs, discounted at 9%, 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 that will compound to the required amount.)

Please solve step by step. My book says the answer is $6,147 but I don´t know how it got that number. Thank you!

Answer 1

Current Cost of College = $ 12000, Time to College = 5 years and Growth Rate of Cost = 6 %

Therefore, Cost for Year 1 = C1 = 12000 x (1.06)^(5) = $ 16058.71, C2 = 16058.71 x 1.06 = $ 17022.23, C3 = 17022.23 x 1.06 = $ 18043.56 and C4 = 18043.56 x 1.06 = $ 19126.18

Account Interest Rate = 9 %

Present Value of Total College Cost 5 years from now = PVc = 16058.71 + 17022.23 / 1.09 + 18043.56 / (1.09)^(2) + 19126.18 / (1.09)^(3) = $ 61631.255

Current Amount in Savings Account = $ 10000

Future Value of Current Amount 5 years from now = FV = 10000 x (1.09)^(5) = $ 15386.24

Let the six annual deposits to be made be $ P

Therefore, Total Future Value of Annual Deposits = FVt = P x (1.09)^(5) + P x (1.09)^(4) +..........+ P x (1.09) + P = PVc - FV = 61631.255 - 15386.24 = $ 46245.02

P x [{(1.09)^(6) - 1} / {1.09 -1}] = 46245.02

P x 7.523335 = 46245.02

P = 46245.02 / 7.523335 = $ 6146.878 ~ $ 6147