Question

You anticipate having to pay $30,000 per year for your child’s college education starting 10 years from now. You plan to finance four years of college by making quarterly deposits in a savings account starting now. The final deposit is made three months prior to the first college payment, for a total of 40 deposits. Each annual college payment is made in full at the beginning of the school year.

If the savings account earns 8% per annum convertible quarterly, what should your quarterly deposit be?

1. Less than $1,700

2. At least $1,700, but less than $1,710

3. At least $1,710, but less than $1,720

4. At least $1,720, but less than $1,730

5. $1,730 or more

Answer 1

The correct answer is last option showing: $1,730 or more

Let's shift ourselves to the end of t = 10 years.

R = Effective annual discount rate = (1 + 8% / 2)⁴ - 1 = 1.02⁴ - 1 = 8.24%

At this time present value of all the future payments to be done towards college fees = 30,000 + 30,000 / (1 + R) + 30,000 / (1 + R)² + 30,000 / (1 + R)³ = 30,000 + 30,000 / (1 + 8.24%) + 30,000 / (1 + 8.24%)² + 30,000 / (1 + 8.24%)³ = $ 106,974.87

So, the kitty size required at the end of 10 years = $ 106,974.87

Let's now come back to t = 0.

We are making 40 equal quarterly payments starting now. Let the quarterly payment be A.

Discount rate, r = 4 = 8% / 4 = 2%

n = number of payments = 40

FV of these immediate annuity payments = A / R x (1 + r) x [(1 + r)ⁿ - 1] = = A / 2% x (1 + 2%) x [(1 + 2%)⁴⁰ - 1] =  61.61A

This amount should be equal to the desired kitty size of $ 106,974.87

Hence, 61.16A = 106,974.87

Hence, A = 106,974.87 / 61.61 = $  1,736

Hence, the correct answer is last option showing: $1,730 or more