Challenge question. In the chapter text, we dealt exclusively with a single lump sum, but often we may be looking at several lump-sum values simultaneously. Let's consider the retirement plan of a couple. Currently, the couple has four different investments: a \( 401(\mathrm{k}) \) plan, two pension plans, and a personal portfolio. The couple is 6 years away from retirement. They believe they have sufficient money in their plans today so that they do not have to contribute to the plans over the next 6 years and will still meet their \( \$ 1.5 \) million retirement goal. Here are the current values and growth rates of their plans: \( 401(\mathrm{k}): \$ 93,000 \) growing at \( 6 \% \). Pension Plan One: \( \$ 287,000 \) growing at \( 6.75 \% \). Pension Plan Two: \( \$ 250,000 \) growing at \( 7.5 \% \). Personal Portfolio: \( \$ 141,000 \) growing at \( 10 \% \). What is the value of the Pension Plan Two 6 vears from now?

To find the value of the Pension Plan Two 6 years from now, we can use the formula for compound interest: \[ A = P(1 + r)^n \] where: - \( A \) is the future value of the investment - \( P \) is the present value of the investment - \( r \) is the annual growth rate - \( n \) is the number of years Given: - Present value of Pension Plan Two: \( P = \$250,000 \) - Annual growth rate: \( r = 7.5\% = 0.075 \) - Number of years: \( n = 6 \) Substitute the values into the formula to find the future value of Pension Plan Two 6 years from now. Calculate the value by following steps: - step0: Calculate: \(250000\left(1+0.075\right)^{6}\) - step1: Add the numbers: \(250000\times 1.075^{6}\) - step2: Convert the expressions: \(250000\left(\frac{43}{40}\right)^{6}\) - step3: Simplify: \(250000\times \frac{43^{6}}{40^{6}}\) - step4: Rewrite the expression: \(40\times 6250\times \frac{43^{6}}{40^{6}}\) - step5: Reduce the numbers: \(6250\times \frac{43^{6}}{40^{5}}\) - step6: Rewrite the expression: \(10\times 625\times \frac{43^{6}}{40^{5}}\) - step7: Rewrite the expression: \(10\times 625\times \frac{43^{6}}{10^{5}\times 4^{5}}\) - step8: Reduce the numbers: \(625\times \frac{43^{6}}{10^{4}\times 4^{5}}\) - step9: Rewrite the expression: \(625\times \frac{43^{6}}{5^{4}\times 2^{4}\times 4^{5}}\) - step10: Rewrite the expression: \(5^{4}\times \frac{43^{6}}{5^{4}\times 2^{4}\times 4^{5}}\) - step11: Reduce the numbers: \(1\times \frac{43^{6}}{2^{4}\times 4^{5}}\) - step12: Multiply: \(\frac{43^{6}}{2^{14}}\) - step13: Simplify: \(\frac{43^{6}}{16384}\) The value of the Pension Plan Two 6 years from now is approximately $385,825.38.