Question

(1) A couple have a 3-year-old child; they decided to make annual deposits into a savings account to fund his 4-year university education. With the first deposit being made on his fourth birthday and the last deposit being made on his 15th birthday. Then, starting on his 18th birthday, 4 withdrawals are required, starting at $4000 and increasing at a rate of 11%. If the effective annual interest rate is 25% during the whole period of time, what are the annual deposits in years 4 through 15?

Answer 1

For this we assume that the annual deposits are equal and denoted by A.

To solve this we jusr equate the present value of all withdrawals with the present value of all deposits. (Let the present day be the fourth birthday of the child)

The present value of withdrawals = PV_W

PV_W = 4000/(1.25)¹⁴ + 4000(1.11)/(1.25)¹⁵ + 4000(1.11)²/(1.25)¹⁶ + 4000(1.11)³/(1.25)¹⁷

PV_W = (4000/(1.25)¹⁴) [ 1 + (1.11/1.25) + (1.11/1.25)² + (1.11/1.25)³]

PV_W = 594.04785

The present value of the deposits = PV_D

PV_D = A + A/1.25 + A/1.25² + ... A/1.25¹¹

PV_D = A * [1 - (1/1.25)¹²] / [1 - (1/1.25)]

PV_D = A *([1.25¹² - 1] / 1.25¹² ) / [0.25/1.25]

PV_D = 5A * [1.25¹² - 1] / 1.25¹²

PV_D = 19.65640A

PV_D = PV_W

19.65640 A = 594.04785

A = 594.04785 / 19.65640 = 30.22156

A = $30.22156 = Annual deposits to be made everly year from 4th birthday to the 15th birthday