Question

Suppose one of your clients is four years away from retirement and has only \(2,500 in pretax income to devote to either a Roth or traditional IRA. The traditional IRA permits investors to contribute the full \)2,500 since contributions to these accounts are tax deductible , but they must pay taxes on all future distributions. In contrast, contributions to a Roth IRA are not tax-deductible. For example, if a person’s tax rate is 25 percent, an investor is able to contribute only \(1,875 after taxes; however, the earnings of a Roth IRA grow tax-free. Your company has decided to waives the one-time set-up fee of \)50 setup fee. Assuming that your clients anticipates that her tax rate will remain at 19 percent in retirement and will earn a stable 7 percent return on her investments, will she prefer a traditional or Roth IRA?

Short answer

The client should choose Roth IRA

Step-by-step solution

Determining formula.

Use the formula for the future value to calculate the returns of traditional IRA:

$FV=C.{(1+r)}^2$

Where c is the cash contribution, r is interest rate.

Determining which IRA is more profitable for the client.

Since the client has $2500 to devote to the IRA, substitute C=$2500 into the formula:

$FV=2500\cdot {(1+r)}^4$

We have i=7%=0.07,

role="math" localid="1657717439769" $\begin{array}{l}FV=2500{\left(1+0.07\right)}^4\cdot \\ FV=2500\cdot {1.07}^4\\ FV\approx 3277\end{array}$

Therefore, the total amount accumulated at the end of the first year is $3277.

To find the total amount accumulated at the end of the second year, substitute n=3,

$\begin{array}{l}FV=2500\cdot {1.07}^3\\ FV\approx 3063\end{array}$

Therefore the amount accumulated at the end of second year is $3063.

At third year, n=2;

$\begin{array}{l}FV=2500\cdot {1.07}^2\\ FV\approx 2862\end{array}$

Therefore the amount accumulated at the end of third year is $2862.

At fourth year, n=1;

$\begin{array}{l}FV=2500\cdot {1.07}^1\\ FV=2675\end{array}$

Therefore, the amount accumulated t the end of fourth year is $2675.

Returns after four years re given by the sum of returns for each year:

$3277+3063+2862+2675=1187.$

Traditional IRA requires that a client pays tax rate is 19%, hence the amount of money the client will pay on taxes is:

19%.1187=0.19.1187=2257

The client will then have 1187-2257=9620 left after paying the taxes.

Therefore, the total return from the traditional IRA is $9620.

Using the above formula to calculate the Roth IRA:

The contribution to a Roth IRA is non-deductable, therefore subtract 19% from the initial contribution of $2500:

$\begin{array}{l}\$2500-19\%\\ =2500-475\\ =\$2025.\end{array}$

Substitute C=2025 into the formula,

$FV=2025\cdot {(1+r)}^n$

We have i=7%=0.07,

role="math" localid="1657717864713" $\begin{array}{l}FV=2025\cdot {\left(1+0.07\right)}^4\\ FV=2025\cdot {1.07}^4\\ FV\approx 2654\end{array}$

Therefore, the total amount accumulated at the end of first year is $2654.

At the second year, n=3:

$FV=2025\cdot {1.07}^3\approx 2481$

Therefore the total amount accumulated at the end of the second year is $2481.

At the third year, n=2:

$FV=2025\cdot {1.07}^2\approx 2318$

Therefore, the total amount accumulated at the end of third year is $2318.

At the fourth year, =1:

$FV=2025\cdot {1.07}^1\approx 2167$

Therefore the total amount accumulated at the end of fourth year is $2167.

Returns after four years are given by the sum of returns for each year:

$2654+2481+2318+2167=9620$

Therefore, the total returns from Roth IRA are $9620.

Observe that the returns from traditional and Roth IRA are the same; therefore, no conclusion can yet be made. But note that the company has waived the set-up fee for the Roth IRA, whereas the client must pay $50 setup fee for traditional IRA.

Hence, the client should choose Roth IRA.