Question

$5,000 is deposited into an IRA earning 5% at the beginning of each year. At the end of 30 years, determine the amount in the IRA.

Answer 1

To calculate the final amount in an IRA after 30 years with annual investments and varying net returns due to fees, one would apply the future value of a series formula for an ordinary annuity. Comparing the final amounts after adjusting for a slight difference in annual return rates due to fees allows us to determine the additional amount one investor has over the other.

Step-by-step explanation:

To determine the final amount in an IRA after 30 years with an annual deposit of $5,000, we will use the future value of a series formula for an ordinary annuity. However, since the question also introduces a scenario involving two investors with different net annual returns due to an administrative fee, we need to calculate the final amount for each investor and then find the difference.

Let's start by calculating the future value for each investor using the following formula:

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

Where:

• FV is the future value of the annuity
• P is the annual payment/deposit ($5,000)
• r is the annual interest rate (expressed as a decimal)
• n is the number of periods/deposits (30 years)

Alexx's Investment Calculation:

Using a rate of 5%, Alexx's future value can be calculated as follows:

FV = $5,000 * [((1 + 0.05)^30 - 1) / 0.05]

Calculating the above expression will give us Alexx's total after 30 years.

Spenser's Investment Calculation:

Spenser's effective annual rate after the administrative fee is 4.75% (5% - 0.25%).

FV = $5,000 * [((1 + 0.0475)^30 - 1) / 0.0475]

Calculating this expression will provide us with Spenser's total after 30 years.

Finally, to determine how much more Alexx will have than Spenser, we subtract Spenser's final amount from Alexx's final amount:

Difference = FV

Alexx

- FV

Spenser

Where FVAlexx and FVSpenser are the future values we calculated for each investor. This will give us the amount Alexx has more than Spenser after 30 years of investment.

Answer 2

To determine the amount in the IRA after 30 years, we can use the formula for compound interest. Given the initial deposit of $5,000 and the interest rate of 5%, the amount in the IRA after 30 years is approximately $176,947.17

Step-by-step explanation:

To determine the amount in the IRA after 30 years, we can use the formula for compound interest:

`A = P(1 + r)^n`

Where:

• A is the final amount
• P is the principal amount (initial deposit)
• r is the rate of interest
• n is the number of years

Given that $5,000 is deposited into the IRA at the beginning of each year and the interest rate is 5%, we have:

`A = $5,000(1 + 0.05)^30`

Solving this equation, we find that the amount in the IRA after 30 years is approximately $176,947.17