Question

You and your partner are saving for your son Sam's (who today turned 10) college expenses. Sam will enter college eight years from today. Annual tuition and other expenses at UCLA are currently $14,500, but they increase at a rate of 3.5 percent every year. Sam, you hope, will graduate in four years. Naturally, you need to pay tuition (and other fees) at the beginning of each year (at t = 8, 9, 10, and 11). So far, you and your partner have been able to save $15,000 (at t = 0). You plan to add an additional $5,000 at the end of each of the next 4 years (at t = 1, 2, 3, and 4). Then your plan is to make three equal (annual) contributions in each of the following years, t = 5, 6, and 7. You are expecting your account to earn 9 percent per year. How large must the annual payments at t = 5, 6, and 7 be to cover Sam's anticipated college tuition and other expenses?

Answer 1

Let's first work out the kitty size required to fund the college expenses in the year 8, 9, 10 & 11 at the end of year 7. Since the college expenses are to be paid at the beginning of the year, we will first calculate the present value of college expenses at the end of year 7.

Current college expenses, at t = 0, E₀ = $ 14,500. Annual escalation, e = 3.5%

Hence, college expenses in:

• the year 8, E₈ = E₀ x (1 + e)⁸ = 14,500 x (1 + 3.5%)⁸ =  19,093.73
• the year 9, E₉ = E₀ x (1 + e)⁹ = 14,500 x (1 + 3.5%)⁹ =   19,762.01
• the year 10, E₁₀ = E₀ x (1 + e)¹⁰ = 14,500 x (1 + 3.5%)¹⁰ =    20,453.68
• the year 11, E₁₁ = E₀ x (1 + e)¹¹ = 14,500 x (1 + 3.5%)¹¹ =  21,169.56

Kitty size required at the end of year 7 = PV of all the above four expenses at t = 7

Discount factor, R = 9%

Kitty size required at the end of year 7 = PV of all the above four expenses at t = 7

= 64,941.52

Let's come back to t = 0,

So far, you and your partner have been able to save PV = $15,000 (at t = 0). You plan to add an additional PMT = $5,000 at the end of each of the next 4 years (at t = 1, 2, 3, and 4)

We need to calculate the FV of these at t = 4 and then at t = 7.

FV at t = 4 will be calculated using FV formula in excel.

FV₄ = FV(Rate, Period, PMT, PV) = FV(9%, 4, -5000, -15000) = $44,039.37

FV at t= 7 will be = FV₄ x (1 + R)^(7-4) = $44,039.37 x (1 + 9%)³ =  57,032.26

We have already calculated the required kitty size to be $ 64,941.52 of which $ 57,032.26 will be received through the existing saving and period savings from t = 1 through t = 4.

Hence, shortfall in kitty = $ 64,941.52 - $ 57,032.26 =  7,909.26
This shortfall has to be met through the annual savings through year t = 5 to t = 7. Let this annual saving amount be A.

Hence, FV of these annuities through t= 4 to t = 7, at the end of t = 7 will be = A / R x [(1+ R)^N - 1]

= A / 0.09 x [(1+ 0.09)^{(7 - 4)} - 1] = 3.2781A

This should be exactly equal to the kitty shortfall amount of 7,909.26 .

Hence, 3.2781A = 7,909.26

Hence, A =  $ 2,412.7574

Hence, the annual payments at t = 5, 6, and 7 must be $ 2,412.7574 be to cover Sam's anticipated college tuition and other expenses. (Please round off your answer as per your requirement).