Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 70.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 70.0 when he fully retires, he will wants to have $3,480,239.00 in his retirement account. He he will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 9.00% interest rate.


Answer format: Currency: Round to: 2 decimal places.

To solve this, we can break the problem down into multiple steps involving compound interest, and regular contributions to a retirement account. Here are the steps and the formulas involved:

### Step 1: Understand the final target amount and interest rate
Derek wants to have exactly $3,480,239.00 one year after he turns 70 years old, when he is fully retired. The account will accumulate interest during his part-time working years (ages 65 to 70) without any withdrawals or deposits.

The interest rate is 9% annually.

### Step 2: Calculate the value of the retirement account at age 65
The amount at age 70 is known to be $3,480,239. To find the value at age 65, we need to "discount" this value back to his 65th birthday, using compound interest. The compound interest formula for the future value $FV$ is:

\[
FV = PV \times (1 + r)^n
\]

Where:
- $FV = 3,480,239$ (amount at age 70)
- $PV$ is the amount at age 65 (what we want to find)
- $r = 9\% = 0.09$ (annual interest rate)
- $n = 5$ (number of years from age 65 to 70)

We solve for $PV$:

\[
PV = \frac{FV}{(1 + r)^n}
\]

### Step 3: Calculate the required annual contributions
Derek will make contributions from age 26 to 65, a total of 40 years. We now need to calculate the necessary contributions using the future value of an annuity formula:

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

Where:
- $FV$ is the value of the retirement account at age 65 (from Step 2)
- $P$ is the annual contribution (what we want to find)
- $r = 9\% = 0.09$ (annual interest rate)
- $n = 40$ (number of years of contributions)

Let's calculate this step-by-step. I'll start with Step 2.The value of Derek's retirement account at age 65 must be approximately $2,261,916.56 to grow to $3,480,239.00 by age 70, given a 9% annual interest rate.

Now, let's proceed to Step 3 and calculate the required annual contributions from ages 26 to 65 using the future value of an annuity formula.

\[
FV_{65} = P \times \frac{(1 + r)^{40} - 1}{r}
\]

We will solve for $P$, the annual contribution.Derek needs to make annual contributions of approximately **$6,694.39** to his retirement account from age 26 to age 65 in order to have $3,480,239.00 by the time he fully retires at age 70.

Would you like more details on any of the steps? Here are 5 related questions to consider:

1. How would changes in the interest rate affect the required contributions?
2. What if Derek wanted to contribute monthly instead of annually?
3. How would delaying the contributions by a few years affect his plan?
4. What happens if the target retirement amount changes to a different value?
5. Can this model handle variations in annual contributions (e.g., increasing them over time)?

**Tip:** The earlier you start saving and investing, the more compound interest can work in your favor, lowering the required contributions.

Derek plans to retire on his 65th birthday. However, he plans to work part-time until he turns 70.00. During these years of part-time work, he will neither make deposits to nor take withdrawals from his retirement account. Exactly one year after the day he turns 70.0 when he fully retires, he will want to have $3,480,239.00 in his retirement account. He will make contributions to his retirement account from his 26th birthday to his 65th birthday. To reach his goal, what must the contributions be? Assume a 9.00% interest rate.

Discover how to calculate retirement contributions using compound interest and annuity formulas. Derek aims to save $3,480,239.00 by age 70 with a 9% interest rate. Find out how much he must contribute annually from age 26 to 65 to meet his goal.

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