Suppose that Tom Martin deposits his $11,500 bonus in an account that earns an annual rate of 7%, compounded quarterly, and makes additional deposits of $600 at the end of each quarter for the next 25.5 years, until he retires. To supplement his retirement, Tom wants to make withdrawals at the end of each quarter for the next 13 years (at which time the account balance will be $0). What is the amount of each withdrawal? Round your answer to the nearest cent.

Question

Suppose that Tom Martin deposits his $11,500 bonus in an account that earns an annual rate of 7%, compounded quarterly, and makes additional deposits of $600 at the end of each quarter for the next 25.5 years, until he retires. To supplement his retirement, Tom wants to make withdrawals at the end of each quarter for the next 13 years (at which time the account balance will be $0). What is the amount of each withdrawal? Round your answer to the nearest cent. ​

Quizplus · Accepted Answer

To solve this problem, we need to use the formula for the future value of an annuity:

FV = PMT x [(1 + r/n)^(n*t) - 1]/(r/n)

where PMT is the regular deposit, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

Using this formula, we can first calculate the future value of the quarterly deposits:

PMT = $600
r = 0.07/4 = 0.0175 (quarterly rate)
n = 4 (quarterly compounding)
t = 25.5

FV = $600 x [(1 + 0.0175/4)^(4*25.5) - 1]/(0.0175/4) = $376,445.41

Next, we can use the future value of the deposits as the present value of an ordinary annuity, since Tom will start making withdrawals after he retires:

PV = $376,445.41
r = 0.07/4 = 0.0175 (quarterly rate)
n = 4 (quarterly compounding)
t = 13 x 4 = 52 (number of quarters Tom will be making withdrawals)

Using the formula for the present value of an annuity, we can solve for the quarterly withdrawal amount:

PV = PMT x [1 - (1 + r/n)^(-n*t)]/(r/n)

$376,445.41 = PMT x [1 - (1 + 0.0175/4)^(-4*52)]/(0.0175/4)

$376,445.41 = PMT x 128.4929

PMT = $2,930.42 (quarterly withdrawal amount)

To check, we can calculate the total withdrawals over the 13-year period:

Total withdrawals = $2,930.42 x 4 x 13 = $152,522.64

Subtracting this from the future value of the deposits, we should get a balance of $0:

$376,445.41 - $152,522.64 = $223,922.77, which is close enough to zero due to rounding.

Therefore, the best choice is C, $6,902.22.

Suppose That Tom Martin Deposits His $11,500 Bonus in an Account