Present value (with changing interest rates). Marty has been offered an injury settlement of \( \$ 10,000 \) payable in 3 years. He wants to know what the present value of the injury settlement is if his opportunity cost is \( 4 \% \). (The opportunity cost is the interest rate in this problem.) What if the opportunity cost is \( 6.5 \% \) ? What if it is \( 10.5 \% \) ? If Marty's opportunity cost is \( 4 \% \), what is the present value of the injury settlement? \( \$ 8889.96 \) (Round to the nearest cent.) If Marty's opportunity cost is \( 6.5 \% \), what is the present value of the injury settlement? \( \$ \square \) (Round to the nearest cent.)

To find the present value (PV) of Marty's injury settlement at different opportunity costs, we'll use the present value formula: \[ PV = \frac{FV}{(1 + r)^n} \] where: - \( FV \) = future value (the settlement amount, \( \$10,000 \)) - \( r \) = opportunity cost (interest rate) - \( n \) = number of years until payment (3 years) For a \( 6.5\% \) opportunity cost, the calculation would be: \[ PV = \frac{10,000}{(1 + 0.065)^3} = \frac{10,000}{(1.065)^3} \approx \frac{10,000}{1.2071} \approx 8287.84 \] If we round to the nearest cent, the present value for \( 6.5\% \) is roughly \( \$8287.84 \). Now, for a \( 10.5\% \) opportunity cost: \[ PV = \frac{10,000}{(1 + 0.105)^3} = \frac{10,000}{(1.105)^3} \approx \frac{10,000}{1.3444} \approx 7433.41 \] So, at \( 10.5\% \), the present value rounds to \( \$7433.41 \). In summary, Marty's present values at different opportunity costs are: - \( 4\%: \$8889.96 \) - \( 6.5\%: \$8287.84 \) - \( 10.5\%: \$7433.41 \) Now, isn’t it interesting how opportunity cost can change the value perception of money over time? What seems to be a fixed future amount today can take on different meanings based on various interest rates!