3) A father is now planning a savings program to put his daughter through colleg

Question

3) 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 4 years later. 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 year. She now has $7,500 in a college savings account that pays 6% 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 5%) for each year of college and find the total present value of those 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 of these costs and the amount that would be in her 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.

Explanation / Answer

Present value = 19144.22 * P.V. factor for first year @ 6% + 20101.43 * P.V. factor for second year @ 6% + 21106.51 * P.V. factor for third year @ 6% + 22161.83 * P.V. factor for fourth year @ 6%

= 19144.22 * 0.943 + 20101.43 * 0.89 + 21106.51 * 0.84 + 22161.83 * 0.792

= $ 71224.90

Cash available = 7500 * (1.06)5 = $ 10036.69 (Compunded value of intial $ 7500)

Cash still needed = 71224.90 - 10036.69 = $ 61188.21

Annuity amount   [Let Z denotes annuity amount]

Z * (1.06)5 + Z * (1 + 0.06)5 - 1 / 0.06 = 61188.21

1.3382 Z + Z * 0.3382 / 0.06 = 61188.21

1.3382 Z + 5.6367 Z = 61188.21

   Z = 61188.21 / 6.9749 = $ 8772.63 (approx)

Conclusion:- Each of the payment made by father towards deposit into her daughter's college savings account will be of $ 8772.63

Year Cost (A) Inflation effect (B) Total amount required = (A) * (B) 1 15000 (1.05)5 19144.22 2 15000 (1.05)6 20101.43 3 15000 (1.05)7 21106.51 4 15000 (1.05)8 22161.83

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