Karen has been depositing $150 at the end of each month in a tax-free retirement account since she was 30. Matt, who is the same age as Karen, started depositing $300 at the end of each month in a tax-free retirement account when he was 40. Assuming that both accounts have been and will be earning interest at the rate of 7%/year compounded monthly, who will end up with the larger retirement account at the age of 65, Karen or Matt?
A) Matt will end up with the larger retirement account.
B) Karen will end up with the larger retirement account.
C) Karen and Matt will end up with the equal retirement accounts.

Question

Karen has been depositing $150 at the end of each month in a tax-free retirement account since she was 30. Matt, who is the same age as Karen, started depositing $300 at the end of each month in a tax-free retirement account when he was 40. Assuming that both accounts have been and will be earning interest at the rate of 7%/year compounded monthly, who will end up with the larger retirement account at the age of 65, Karen or Matt? ​
A) Matt will end up with the larger retirement account.
B) Karen will end up with the larger retirement account.
C) Karen and Matt will end up with the equal retirement accounts.

Quizplus · Accepted Answer

We can approach this problem by using the formula for the future value of an annuity (regular series of deposits) with monthly compounding:

FV = PMT * [(1 + r/12)^(n*12) - 1] / (r/12)

where FV is the future value of the annuity, PMT is the monthly deposit, r is the monthly interest rate (in decimal form), and n is the number of years.

For Karen:
PMT = 150
r = 0.07/12
n = 35 (since she started at age 30 and deposits for 35 years until age 65)

Using the formula, we get:

FVKaren = 150 * [(1 + 0.07/12)^(35*12) - 1] / (0.07/12)
FVKaren = $413,499.46

For Matt:
PMT = 300
r = 0.07/12
n = 25 (since he started at age 40 and deposits for 25 years until age 65)

Using the formula, we get:

FVMatt = 300 * [(1 + 0.07/12)^(25*12) - 1] / (0.07/12)
FVMatt = $386,234.95

Therefore, Karen will end up with the larger retirement account.

Karen Has Been Depositing $150 at the End of Each