Question

Starting at age 35, you deposit $2000 a year into an IRA account for retirement. Treat the yearly deposits into the

account as a continuous income stream. If money in the account earns 7%, compounded continuously, how much

will be in the account 30 years later, when you retire at age 65? How much of the final amount is interest?

What is the value of the IRA when you turn 65?

Answer 1

To calculate the future value of your IRA account when you retire at age 65, you can use the formula for continuous compounding. The formula is:

\[A = P e^{rt}\]

Where:
- \(A\) is the future value of the investment.
- \(P\) is the initial deposit or the annual deposit.
- \(r\) is the annual interest rate (decimal).
- \(t\) is the number of years.

In this case:
- \(P = $2000\) (annual deposit)
- \(r = 7%\) or \(0.07\) (annual interest rate in decimal)
- \(t = 65 - 35 = 30\) years (the number of years until retirement)

Now, plug these values into the formula:

\[A = 2000 e^{0.07 \cdot 30}\]

Let's calculate that:

\[A = 2000 e^{2.1}\]

Using the value of \(e^{2.1}\) (approximately 8.166):

\[A ≈ 2000 \cdot 8.166 ≈ $16,332\]

So, when you retire at age 65, your IRA account will have approximately $16,332.

To find out how much of this final amount is interest, you can subtract the total amount deposited over the years (30 years x $2000 per year) from the final amount:

Total deposits = $2000 x 30 = $60,000

Interest = $16,332 (final amount) - $60,000 (total deposits) = $16,332 - $60,000 = -$43,668

The negative value indicates that you've paid more into the account than the interest you've earned. In other words, you've contributed $60,000 in deposits, and the interest earned is $43,668 less than that.