Let's break down the problem step by step to find the final value of the account in both scenarios.

Given Information:

• Marginal Tax Rate: 35%
• Annual Contribution: $2,400
• Annual Rate of Return: 6%
• Number of Years: 10

We need to calculate the final value of the account in two scenarios:

1. Taxes Deferred (i.e., no taxes are paid during the accumulation period)
2. Taxes Not Deferred (i.e., taxes are paid on the contributions each year before investing)

Part (a): Taxes Deferred

When taxes are deferred, the entire contribution grows at the annual rate of return without any tax being subtracted during the accumulation period.

We can use the future value of an ordinary annuity formula for this situation, since deposits are made at the end of each year:

$FV = P \times \frac{(1 + r)^n - 1}{r}$ Where:

• $FV$ is the future value.
• $P = 2400$ is the annual contribution.
• $r = 0.06$ is the annual rate of return.
• $n = 10$ is the number of years.

Let's calculate:

$FV = 2400 \times \frac{(1 + 0.06)^{10} - 1}{0.06}$

Now calculate:

$FV = 2400 \times \frac{(1.06)^{10} - 1}{0.06}$

$(1.06)^{10} \approx 1.790847$

$FV = 2400 \times \frac{1.790847 - 1}{0.06}$

$FV = 2400 \times \frac{0.790847}{0.06}$

$FV = 2400 \times 13.180783$

$FV \approx 31,563.88$

So, the final value of the account after 10 years, with taxes deferred, is approximately $31,563.88.

Part (b): Taxes Not Deferred

When taxes are not deferred, the contributions are taxed each year before they are invested. Therefore, the contribution after tax is:

$\text{After-tax contribution} = P \times (1 - \text{Tax rate})$

The after-tax contribution is:

$\text{After-tax contribution} = 2400 \times (1 - 0.35) = 2400 \times 0.65 = 1560$

Now, we use the same future value of an ordinary annuity formula, but with the after-tax contribution:

$FV = 1560 \times \frac{(1 + 0.06)^{10} - 1}{0.06}$

We already know that $(1.06)^{10} \approx 1.790847$, so:

$FV = 1560 \times \frac{1.790847 - 1}{0.06}$

$FV = 1560 \times 13.180783$

$FV \approx 20,566.83$

So, the final value of the account after 10 years, with taxes not deferred, is approximately $20,566.83.

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Summary:

• (a) If taxes are deferred, the final value is approximately $31,563.88.
• (b) If taxes are not deferred, the final value is approximately $20,566.83.

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Would you like more details or have any questions on this? Here are some related questions for further exploration:

1. How would the final value change if the tax rate were 25% instead of 35%?
2. How does the number of years affect the final value in both scenarios?
3. What would happen if the rate of return were higher, say 8%?
4. How do different contribution frequencies (monthly, quarterly, etc.) impact the final value?
5. What is the impact of inflation on the future value of the investment?

Tip: The longer the period and the higher the rate of return, the more powerful the effect of compounding becomes, especially in tax-deferred accounts.