Derek can deposit $11,951.00 on each birthday beginning with his 27.00th and ending with his 69.00th. What will the rate on the retirement account need to be for him to have $3,830,722.00 in it when he retires?

To determine the interest rate needed for Derek to accumulate $3,830,722.00 by his retirement, we need to model the problem as the future value of a series of annuity payments. Derek makes 43 annual deposits starting on his 27th birthday and ending on his 69th birthday.

### Step-by-step solution:

1. **Identify the known values**:
   - Derek will deposit $\$11,951$ annually.
   - There will be 43 deposits (from age 27 to 69).
   - He wants to accumulate $\$3,830,722.00$ by his 69th birthday (retirement).
   - We are solving for the unknown interest rate $r$.

2. **Use the future value of an annuity formula**:

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

Where:
- $FV$ is the future value (Derek's goal: \$3,830,722).
- $P$ is the annual deposit (\$11,951).
- $r$ is the annual interest rate (the unknown we are solving for).
- $n$ is the number of deposits (43).

3. **Substitute the known values into the formula**:

\[
3,830,722 = 11,951 \times \frac{(1 + r)^{43} - 1}{r}
\]

4. **Solve for $r$**:
This equation requires iterative methods or a financial calculator to solve for $r$. I'll calculate the interest rate now.The required annual interest rate for Derek to accumulate \$3,830,722 by the time he retires is approximately **7.90%**.

Would you like more details or have any questions about this calculation?

Here are some related questions for deeper exploration:
1. How does the interest rate affect the future value of an annuity?
2. What happens if Derek starts saving earlier or later in life?
3. What is the impact of making larger deposits each year?
4. How would compound interest affect a monthly deposit scenario?
5. Can you calculate the total amount deposited by Derek over the 43 years?

**Tip**: Compounding frequency (e.g., monthly vs. annually) can have a significant impact on the total interest earned in a retirement account.

Explore how to calculate the interest rate required for Derek to accumulate $3,830,722.00 in his retirement account by making 43 annual deposits of $11,951.00, beginning at age 27 and ending at age 69. This solution uses the future value of an annuity formula and applies key concepts in compound interest.

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