Suppose that Ramos contributes $6000/year into a traditional IRA earning interest at the rate of 2%/year compounded annually, every year after 35 until his retirement age at age 65. At the same time, his wife Vanessa deposits $4900/year into a Roth IRA earning interest at the same rate as that of Ramos and also for a period of 30 years. Suppose that the investments of both Ramos and Vanessa are in a marginal tax bracket of 35% at the time of their retirement and they both wish to withdraw all fo the money in their IRAs at that time. (a) After all the taxes are paid, who will have the larger amount? (b) How much larger will that amount be? (need answer rounded to the nearest cent)
a) Vanessa will have the larger amount.
b) The amount will be $3,845.00 larger.
To answer these questions, we'll calculate the final amounts in the IRAs of both Ramos and Vanessa.
For Ramos:
Contribution per year: $6000
Interest rate: 2% compounded annually
Number of years: 65 - 35 = 30
Using the compound interest formula, we can calculate the final amount in Ramos' traditional IRA:
Final Amount in Ramos' Traditional IRA = Contribution per year * ((1 + interest rate)^number of years - 1) / interest rate
Final Amount in Ramos' Traditional IRA = $6000 * ((1 + 0.02)^30 - 1) / 0.02 ≈ $255,173.22
For Vanessa:
Contribution per year: $4900
Interest rate: 2% compounded annually
Number of years: 30
Using the same compound interest formula, we can calculate the final amount in Vanessa's Roth IRA:
Final Amount in Vanessa's Roth IRA = Contribution per year * ((1 + interest rate)^number of years - 1) / interest rate
Final Amount in Vanessa's Roth IRA = $4900 * ((1 + 0.02)^30 - 1) / 0.02 ≈ $128,802.87
Now, we need to calculate the taxes on the traditional IRA for both Ramos and Vanessa. Since they are in a 35% tax bracket, they will have to pay 35% taxes on the amount withdrawn.
Tax on Ramos' Traditional IRA = 35% * $255,173.22 = $89,305.65
Tax on Vanessa's Roth IRA = 0% (Roth IRA withdrawals are tax-free)
Finally, let's calculate the post-tax amount for both Ramos and Vanessa:
Post-Tax Amount for Ramos = Final Amount in Ramos' Traditional IRA - Tax on Ramos' Traditional IRA
Post-Tax Amount for Ramos = $255,173.22 - $89,305.65 = $165,867.57
Post-Tax Amount for Vanessa = Final Amount in Vanessa's Roth IRA
Post-Tax Amount for Vanessa = $128,802.87
(a) After all the taxes are paid, Ramos will have the larger amount.
(b) Ramos' post-tax amount will be $165,867.57 - $128,802.87 = $37,064.70 larger (rounded to the nearest cent).
To calculate the amounts in both Ramos' traditional IRA and Vanessa's Roth IRA after taxes, we need to follow these steps:
(a) Calculate the amount in Ramos' traditional IRA after 30 years:
Ramos contributes $6000/year and earns an interest rate of 2% compounded annually. This is a regular annuity investment.
We can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value
P = Annual contribution
r = Interest rate
n = Number of years
Using the given values:
P = $6000/year
r = 2% = 0.02
n = 30 years
Plugging these values into the formula, we can calculate the future value of Ramos' traditional IRA after 30 years.
FV_Ramos = $6000 * [(1 + 0.02)^30 - 1] / 0.02
Now, let's calculate the amount in Vanessa's Roth IRA after 30 years:
Vanessa contributes $4900/year and earns an interest rate of 2% compounded annually, just like Ramos.
Using the same formula, we can calculate the future value of Vanessa's Roth IRA after 30 years.
FV_Vanessa = $4900 * [(1 + 0.02)^30 - 1] / 0.02
(b) Now, let's calculate the final amounts after taxes are paid.
Both Ramos and Vanessa are in a marginal tax bracket of 35% at retirement.
To calculate the post-tax amount, we need to subtract the taxes paid from the total amount. The tax rate of 35% applies to the entire balance of their IRAs.
Ramos' post-tax amount = FV_Ramos - (35% * FV_Ramos)
Vanessa's post-tax amount = FV_Vanessa
Now we compare the post-tax amounts to determine who has the larger amount:
If Ramos' post-tax amount is greater than Vanessa's post-tax amount, then Ramos will have the larger amount. Otherwise, Vanessa will have the larger amount.
To find out how much larger the amount is, we subtract Vanessa's post-tax amount from Ramos' post-tax amount:
Difference = Ramos' post-tax amount - Vanessa's post-tax amount
Finally, to get the answer rounded to the nearest cent, we round the Difference to the nearest cent.