Question

In 4 years, your son will be entering college and you would like to help financially. You decide to create a college fund and make four annual deposits, starting now. In four years, you would like your son to be able to make four annual withdrawals of $7,500 from the fund (at the beginning of each year) that will cover his annual tuition. If the college fund earns 3.6% compounded annually, how much must you deposit at the beginning of each year? Assume tuition remains the same for the four years your son is attending college.

Answer 1

The formula you can use for the withdrawals is that of an annuity. You have interest adding to the balance at the same time withdrawals are reducing the balance.

The formula I remember for annuities is
.. A = Pi/(1 -(1 +i)^-n) . . . . . i is the interest for each of the n intervals; A is the withdrawal, P is the initial balance.
This formula works when the withdrawal is at the end of the interval. To find the principal amount required at the time of the first withdrawal, we will compute for 3 withdrawals and then add the 7500 amount of the first withdrawal.
.. 7500 = P*.036/(1 -1.036^-3)
.. 7500 = P*0.357616
.. 7500/0.0347616 = P = 20,972.20
so the college fund balance in 4 years needs to be
.. 20,972.20 +7,500 = 28,472.20

Since the last payment P into the college fund earns interest, its value at the time of the first withdrawal is P*1.036. Each deposit before that earns a year's interest, so the balance in the fund after 4 deposits is
.. B = P*1.036*(1.036^4 -1)/(1.036 -1)
We want this balance to be the above amount, so the deposit (P) is
.. 28,472.20*0.036/(1.036*(1.036^4 -1)) = 6510.62

You must make 4 annual deposits of $6,510.62 starting now.