Question

The difference between a Roth IRA and a traditional IRA is that in a Roth IRA taxes are paid on the income that is contributed but the withdrawals at retirement are tax-free. In a traditional IRA, however, the contributions reduce your taxable income, but the withdrawals at retirement are taxable. Assume you plan to devote $5,000 to retirement savings in each year. You will retire in 30 years and expect to live for an additional 20 years after retirement.

a. Assume the before-tax interest rate is 5%. What will be your after-tax 20-year retirement consumption stream if you choose to save in a traditional IRA? Assume your tax rate is fixed at 30%.

b. What will be your 20-year retirement consumption stream if you choose to save in a Roth IRA?

c. Which provides better expected results if you expect your tax rate to decrease from 30% today to 25% at retirement?

Answer 1

a]

Amount in account after 30 years

Future value of annuity = P * [(1 + r)n - 1] / r,

where P = periodic payment. This is $5,000

r = periodic rate of interest. This is 5%

n = number of periods. This 30

Amount in account after 30 years   = $5,000 * [(1 + 5%)30 - 1] / 5%

Amount in account after 30 years   = $332,194.24

Pretax consumption stream

Present of annuity = P * [1 - (1 + r)-n] / r,

where P = periodic payment. We need to calculate this.

r = interest rate per period. This is 5%

n = number of periods. This is 20.

$332,194.24 = Pretax consumption stream * [1 - (1 + 5%)-20] / 5%

Pretax consumption stream = $26,656.13

After-tax consumption stream

After-tax consumption stream = Pretax consumption stream * (1 - tax rate)

After-tax consumption stream = $26,656.13 * (1 - 30%)

After-tax consumption stream = $18,659.29

b]

Post-tax contribution = pretax contribution * (1 - tax rate)

Post-tax contribution = $5,000 * (1 - 30%) = $3,500

Amount in account after 30 years

Future value of annuity = P * [(1 + r)n - 1] / r,

where P = periodic payment. This is $3,500

r = periodic rate of interest. This is 5%

n = number of periods. This 30

Amount in account after 30 years   = $3,500 * [(1 + 5%)30 - 1] / 5%

Amount in account after 30 years   = $232,535.97

After-tax consumption stream

Present of annuity = P * [1 - (1 + r)-n] / r,

where P = periodic payment. We need to calculate this.

r = interest rate per period. This is 5%

n = number of periods. This is 20.

$232,535.97 = After-tax consumption stream * [1 - (1 + 5%)-20] / 5%

After-tax consumption stream = $18,659.29

c]

Roth IRA

With the Roth IRA, there is no change in after-tax consumption stream because taxes are paid at the time of contribution and withdrawals are tax-free. As the tax rate is changed during retirement, there is no effect on the after-tax consumption stream

Traditional IRA

After-tax consumption stream

After-tax consumption stream = Pretax consumption stream * (1 - tax rate)

After-tax consumption stream = $26,656.13 * (1 - 25%)

After-tax consumption stream = $19,992.09

A traditional IRA offers better results as the after-tax consumption stream is higher