a. Suppose that between the ages of 22 and 26 , you contribute \( \$ 5000 \) per year to a \( 401(\mathrm{k}) \) and your employer contributes \( \$ 2500 \) per year on your behalf. The interest rate is \( 7.6 \% \) compounded annually. What is the value of the \( 401(\mathrm{k}) \) after 4 years? b. Suppose that after 4 years of working for this firm, you move on to a new job. However, you keep your accumulated retirement funds in the \( 401(\mathrm{k}) \). How much money will you have in the plan when you reach age 65 ? \( \mathbf{c} \). What is the difference between the amount of money you will have accumulated in the \( 401(\mathrm{k}) \) and the amount you contributed to the plan? (i) Click the icon to view some finance formulas. a. The value of the \( 401(\mathrm{k}) \) after 4 years is \( \$ \square \).

To solve part a, we first calculate the total contributions to the 401(k) over 4 years. You contribute \( \$5000 \) annually and your employer contributes \( \$2500 \), making a total of \( \$7500 \) per year. After 4 years, that equals \( 4 \times 7500 = \$30,000 \). Now, to find the future value of this investment, we use the formula for compound interest: \[ FV = P \times \left(1 + r\right)^n \] where \( P = initial \, investment, r = annual \, interest \, rate, n = number \, of \, years \). For the yearly contributions, we have to calculate the future value of each annual contribution separately as they will compound for different durations. The future value of each contribution is: - Year 1: \( 5000 \times (1 + 0.076)^4 \) - Year 2: \( 5000 \times (1 + 0.076)^3 \) - Year 3: \( 5000 \times (1 + 0.076)^2 \) - Year 4: \( 5000 \times (1 + 0.076)^1 \) Calculating these gives: - Year 1: approx. \( 5000 \times 1.3605 \approx 6802.50 \) - Year 2: approx. \( 5000 \times 1.2625 \approx 6312.50 \) - Year 3: approx. \( 5000 \times 1.176 \approx 5880.00 \) - Year 4: approx. \( 5000 \times 1.076 \approx 5380.00 \) Adding these values together: - Total FV = \( 6802.50 + 6312.50 + 5880.00 + 5380.00 = 26375.00 \) Don’t forget to account for the employer's contribution with the same compounding method: - Year 1: approx. \( 2500 \times 1.3605 \approx 3401.25 \) - Year 2: approx. \( 2500 \times 1.2625 \approx 3156.25 \) - Year 3: approx. \( 2500 \times 1.176 \approx 2940.00 \) - Year 4: approx. \( 2500 \times 1.076 \approx 2690.00 \) Adding these values together: - Total FV = \( 3401.25 + 3156.25 + 2940.00 + 2690.00 = 12187.50 \) By combining both contributions after 4 years: \[ Total FV = 26375.00 + 12187.50 = 38562.50 \] Thus, the value of the 401(k) after 4 years is \( \$38,562.50 \). --- For part b, if you keep your money invested until age 65, you have a period from age 26 to age 65, which is 39 years. We will take the total from part a, \( \$38,562.50 \) and apply the compound interest formula again: \[ FV = P \times (1 + r)^n \] Using \( P = 38562.50 \), \( r = 0.076 \), and \( n = 39 \): \[ FV = 38562.50 \times (1 + 0.076)^{39} \] Calculating the maturity: - \( (1.076)^{39} \approx 11.682 \) Thus, \[ FV = 38562.50 \times 11.682 \approx 450,000.00 \] So, the amount of money in the plan at age 65 will be approximately \( \$450,000.00 \). --- For part c, your total contribution over the 4 years is: \[ \$5000 \times 4 + \$2500 \times 4 = 30000 \] The difference between the amount accumulated and your contributions would be: \[ 450,000 - 30,000 = 420,000 \] So, the difference is approximately \( \$420,000.00 \).