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Part 1

**a.** Suppose that between the ages of 22 and

4040​,

you contribute

​$40004000

per year to a​ 401(k) and your employer contributes

​$20002000

per year on your behalf. The interest rate is

8.28.2​%

compounded annually. What is the value of the​ 401(k) after

1818

​years? **b.** Suppose that after

1818

years of working for this​ firm, you move on to a new job.​ However, you keep your accumulated retirement funds in the​ 401(k). How much money will you have in the plan when you reach age​ 65? **c.** What is the difference between the amount of money you will have accumulated in the​ 401(k) and the amount you contributed to the​ plan?

Let's break this problem into manageable parts to calculate the value of your 401(k) over time. We'll follow these steps for each part.

### **Part 1a: Value of the 401(k) after 18 years of contributions**

Given:
- Annual personal contribution: $ 4000 $ dollars
- Annual employer contribution: $ 2000 $ dollars
- Total annual contribution: $ 4000 + 2000 = 6000 $ dollars
- Interest rate: $ 8.2\% $ (compounded annually)
- Contribution period: $ 18 $ years

The future value of contributions compounded annually is given by the **Future Value of an Annuity** formula:

\[
FV = C \times \left( \frac{(1 + r)^t - 1}{r} \right)
\]

Where:
- $ C $ is the annual contribution
- $ r $ is the interest rate (expressed as a decimal)
- $ t $ is the number of years

For this part:
- $ C = 6000 $
- $ r = \frac{8.2}{100} = 0.082 $
- $ t = 18 $

Substitute these values to calculate the future value after 18 years.

### **Part 1b: Value of the 401(k) at age 65 (no further contributions)**

Now, after 18 years, you stop contributing but leave the accumulated amount in the 401(k) for more growth. The interest rate remains at 8.2%, and the time period is the number of years until you reach age 65.

- Time period: $ 65 - 40 = 25 $ years

We can use the **compound interest formula** to calculate the growth over these 25 years:

\[
FV = P \times (1 + r)^t
\]

Where:
- $ P $ is the amount after 18 years (from Part 1a)
- $ r = 0.082 $ (interest rate)
- $ t = 25 $ years

### **Part 1c: Difference between accumulated amount and contributions**

The total contributions over the 18 years are:

\[
\text{Total Contributions} = 6000 \times 18
\]

Finally, subtract this total contribution from the amount calculated in Part 1b to find the difference between the accumulated amount and what was contributed.

I will now calculate each part.### **Results:**

**Part 1a: Value of the 401(k) after 18 years of contributions**  
The value of your 401(k) after 18 years of contributions, with annual contributions of $6,000 at an 8.2% interest rate, is approximately **$229,122.27**.

**Part 1b: Value of the 401(k) at age 65**  
If you stop contributing after 18 years and let the accumulated amount grow at the same interest rate for 25 more years, the value of your 401(k) at age 65 will be approximately **$1,643,420.96**.

**Part 1c: Difference between accumulated amount and contributions**  
The total contributions over the 18 years are $ 6000 \times 18 = 108,000 $. The difference between the accumulated amount at age 65 and the total contributions is approximately **$1,535,420.96**.

---

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. How would the final amount change if the interest rate was 6% instead of 8.2%?
2. What happens if contributions were made monthly instead of yearly?
3. How does the future value change if the contribution period was 25 years instead of 18?
4. What if the employer stopped contributing after 10 years but the employee continued?
5. How would inflation impact the real value of the retirement fund?

**Tip:** Regularly increasing contributions, even slightly, can significantly boost the long-term value of your retirement fund due to compound interest.

Suppose that between the ages of 22 and 40, you contribute $4000 per year to a 401(k) and your employer contributes $2000 per year on your behalf. The interest rate is 8.2% compounded annually. What is the value of the 401(k) after 18 years? Suppose that after 18 years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the 401(k). How much money will you have in the plan when you reach age 65? What is the difference between the amount of money you will have accumulated in the 401(k) and the amount you contributed to the plan?

Learn how to calculate the future value of a 401(k) after 18 years of contributions, including employer matching, at an 8.2% interest rate compounded annually. Discover the value at age 65 with no further contributions and the difference between the accumulated amount and total contributions. Suitable for Grades 10-12.

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