To calculate the total amount in the 401(k) after 11 years, we need to consider both the employee's contribution and the employer's matching contribution, as well as the growth of the investment due to compounding interest.
Given:
- Employee's monthly contribution: \( C = 123.25 \)
- Employer's match: \( C_{match} = C = 123.25 \)
- Total monthly contribution: \( C_{total} = C + C_{match} = 123.25 + 123.25 = 246.50 \)
- Annual interest rate (APR): \( r = 6.3% = 0.063 \)
- Monthly interest rate \( r_{monthly} = \frac{0.063}{12} = 0.00525 \)
- Total number of contributions (months): \( n = 11 \times 12 = 132 \)
Future Value of a Series Formula:
The future value of a series (like monthly contributions), compounded monthly, is calculated using the formula:
\[ FV = C \times \frac{(1 + r_{monthly})^n - 1}{r_{monthly}} \]
Where:
- \(FV\) is the future value of the investment.
- \(C\) is the total contribution for each period (in this case, \(C_{total}\)).
- \(r_{monthly}\) is the monthly interest rate.
- \(n\) is the total number of contributions.
Solution Steps:
- Calculate the future value from the total monthly contributions:
\[ FV = 246.50 \times \frac{(1 + 0.00525)^{132} - 1}{0.00525} \]
- First, calculate \( (1 + 0.00525)^{132} \):
\[ (1 + 0.00525)^{132} \approx 1.898977 \]
- Now, substitute this value back into the future value formula:
\[ FV = 246.50 \times \frac{1.898977 - 1}{0.00525} \]
- Calculate \( \frac{1.898977 - 1}{0.00525} \):
\[ \frac{1.898977 - 1}{0.00525} \approx 171.068 \]
- Finally, multiply by the total monthly contribution:
\[ FV \approx 246.50 \times 171.068 \approx 42,157.15 \]
Final Result:
The total amount in the 401(k) when the employee retires after 11 years, with both the employee's and employer's contributions considered, is approximately $42,157.15.