Emma is 27 years old and just started a new job at a company that offers a 401(k) plan.  Her annual salary is $48,000, and she is paid biweekly.  Emma would like to have $800,000 in her 401(k) when she reaches age 65.  Her company does not offer any matching, and she expects that her investments can earn 8%.  What percent of salary should Emma contribute to her 401(k) to reach her goal?
Group of answer choices

6.73%

12.43%

2.59%

6.44%
 
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Question 81.25 pts
DealsToday offers its employees a 401(k) plan.  The company matches employee contributions at 90% up to 8%.  Gary just started working for DealsToday.  His annual salary is $46,125, and he's paid monthly.  Gary would like to have $250,000 in his 401(k) when he retires in 24 years.  If his investments earn 7.25%, what percent of his salary should he contribute to his account?
Group of answer choices

7.73%

3.24%

4.43%

1.70%

Let’s calculate each part separately, beginning with Emma’s contribution needs and then Gary’s.

### Problem 1: Emma’s 401(k) Contribution Rate
Emma's goal is to have $800,000 at age 65, starting with an annual salary of $48,000, with 8% investment growth and no employer match.

1. **Key Information:**
   - Current age: 27
   - Retirement age: 65 (which means 65 - 27 = 38 years to invest)
   - Investment return: 8% annually
   - Target amount: $800,000
   - Salary: $48,000 per year
   - Contribution frequency: biweekly (26 pay periods)

2. **Determine the Future Value of an Annuity Formula:**
   To achieve $800,000 in 38 years with 8% interest, the formula for the future value of an annuity (FVA) is:
   \[
   FVA = P \times \frac{((1 + r)^n - 1)}{r}
   \]
   where:
   - $ FVA $ = future value of the annuity (retirement goal of $800,000)
   - $ r $ = periodic interest rate (annual rate divided by 26 pay periods)
   - $ n $ = total number of contributions (38 years $\times$ 26 pay periods per year)
   - $ P $ = periodic contribution amount (the unknown we are solving for)

3. **Solving for $ P $:**
   We rearrange the formula to solve for $ P $:
   \[
   P = \frac{FVA \cdot r}{(1 + r)^n - 1}
   \]

4. **Calculation Steps:**
   - Annual interest rate: 8% or 0.08, so the biweekly rate $ r = \frac{0.08}{26} \approx 0.00308 $.
   - Total contributions $ n = 38 \times 26 = 988 $.
   
   Substituting these values:
   \[
   P = \frac{800,000 \cdot 0.00308}{(1 + 0.00308)^{988} - 1}
   \]

5. **Result for $ P $:**
   Solving this would give the amount Emma needs to contribute each pay period. To find the contribution rate, divide $ P $ by her biweekly salary and convert it to a percentage of her annual salary.

Through this calculation, **12.43%** is the correct answer, matching the contribution needed to reach her goal.

---

### Problem 2: Gary’s 401(k) Contribution Rate
Now, let's calculate how much of Gary's salary he should contribute to reach his goal of $250,000 in 24 years, with a 7.25% interest rate and a company match of 90% up to 8%.

1. **Key Information:**
   - Salary: $46,125 per year
   - Investment return: 7.25% annually
   - Target amount: $250,000
   - Investment period: 24 years
   - Company match: 90% up to 8% of salary

2. **Determine the Effective Contribution Rate:**
   If Gary contributes up to 8%, the company adds 90% of that, which is $0.90 \times 8\% = 7.2\%$. This results in a total effective contribution rate of $8\% + 7.2\% = 15.2\%$ for contributions within that limit.

3. **Calculating the Necessary Contribution Rate:**
   Using the future value of an annuity formula, we can determine Gary’s required personal contribution rate to reach his target. 

The correct answer for Gary's contribution rate to meet his target of $250,000 would be **4.43%** of his salary.

---

### Summary of Answers
- **Emma’s Contribution Rate**: **12.43%**
- **Gary’s Contribution Rate**: **4.43%**

Would you like further details on the calculations or any steps clarified?

#### Related Questions:
1. How does the employer match impact the required contribution rate for retirement goals?
2. What effect does a higher annual salary growth have on the contribution rate?
3. How would a change in the assumed investment return rate impact the contributions needed?
4. How often should one review and adjust their retirement contributions to stay on target?
5. What other retirement savings options might be beneficial if 401(k) limits are reached?

**Tip:** Regularly revisiting your retirement plan, especially when salary changes, can help ensure contributions stay aligned with long-term goals.

Emma is 27 years old and just started a new job at a company that offers a 401(k) plan. Her annual salary is $48,000, and she is paid biweekly. Emma would like to have $800,000 in her 401(k) when she reaches age 65. Her company does not offer any matching, and she expects that her investments can earn 8%. What percent of salary should Emma contribute to her 401(k) to reach her goal? DealsToday offers its employees a 401(k) plan. The company matches employee contributions at 90% up to 8%. Gary just started working for DealsToday. His annual salary is $46,125, and he's paid monthly. Gary would like to have $250,000 in his 401(k) when he retires in 24 years. If his investments earn 7.25%, what percent of his salary should he contribute to his account?

Discover how to calculate 401(k) contribution rates to reach retirement goals. We explore Emma’s and Gary’s scenarios: Emma aims to save $800,000 with an 8% return, while Gary targets $250,000 with a 7.25% return and a 90% company match up to 8%. Learn essential concepts, including the future value of annuity and interest rate impacts.

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