Question

A self-employed individual deposits $1000 into a Roth IRA at the end of year for the next 30 years, at a 10% annual return (compounded annually), how much will individual have in the Roth IRA after years and what is the total earned interest?

Answer 1

To calculate the future value of a series of annual deposits (also known as an annuity), we can use the future value of an annuity formula:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period ($1000)
- \( r \) = annual interest rate (10% or 0.10)
- \( n \) = total number of payments (30 years)

Plugging in the values:

\[
FV = 1000 \times \frac{(1 + 0.10)^{30} - 1}{0.10}
\]

Calculating step-by-step:

1. Calculate \( (1 + r)^n \):
\[
(1 + 0.10)^{30} = 1.10^{30} \approx 17.4494
\]

2. Now substitute back into the formula:
\[
FV = 1000 \times \frac{17.4494 - 1}{0.10}
\]
\[
FV = 1000 \times \frac{16.4494}{0.10}
\]
\[
FV = 1000 \times 164.494 = 164,494
\]

The future value of the Roth IRA after 30 years is approximately **$164,494**.

To find the total earned interest, we need to figure out how much was contributed over the 30 years:

- Total contributions = $1000 per year × 30 years = $30,000

Now subtract the total contributions from the future value to find the earned interest:

\[
\text{Total Earned Interest} = FV - \text{Total Contributions}
\]
\[
\text{Total Earned Interest} = 164,494 - 30,000 = 134,494
\]

Thus, the total earned interest is approximately **$134,494**.

In summary:
- Amount in the Roth IRA after 30 years: **$164,494**
- Total earned interest: **$134,494**