John and Daphne are saving for their daughter Ellen's college education. Ellen just turned 10 (at t = 0)So far, John and Daphne have accumulated $15,000 in their college savings account (at t = 0). Their long-run financial plan is to add an additional $5,000 in each of the next 4 years (at t = 1, 2, 3, and 4). Then they plan to make 3 equal annual contributions in each of the following years, t = 5, 6, and 7. They expect their investment account to earn 9%. How large must the annual payments at t = 5, 6, and 7 be to cover Ellen's anticipated college costs?
A) $1,965.21
B) $2,068.64
C) $2,177.51
D) $2,292.12
E) $2,412.76

Question

Quizplus · Accepted Answer

First, let's calculate the future value (FV) of the savings account at t=4 when they make additional contributions of $5,000 per year with an interest rate of 9%.

FV = $15,000 x (1 + 0.09)^4 + $5,000 x [(1 + 0.09)^3 + (1 + 0.09)^2 + (1 + 0.09)^1]
FV = $15,000 x 1.4116 + $5,000 x (1.29503 + 1.1881 + 1.0809)
FV = $60,799.46

This means that they will need to have $60,799.46 in their savings account at t=4 to cover their daughter's college education.

Next, we can use the formula for the present value of an annuity to find out how much they need to contribute annually at t=5, 6, and 7.

PV = C x [1 - (1 + r)^-n]/r

Where PV is the present value of the annuity, C is the annual contribution, r is the interest rate, and n is the number of payments.

Since they plan to make 3 equal annual contributions, n = 3, and we know that PV = $60,799.46, r = 9%, and the first payment will be made at t=5. Substituting these values into the formula gives us:

$60,799.46 = C x [1 - (1 + 0.09)^-3]/0.09
C = $2,412.76

Therefore, the annual payments at t=5, 6, and 7 must be $2,412.76 to cover Ellen's anticipated college costs. The answer is E.

John and Daphne Are Saving for Their Daughter Ellen's College