An investment company pays \( 8 \% \) compounded semiannually You want to have \( \$ 15,000 \) in the future (A) How much should you deposit now to have that amount 5 years from now? \( \$ 10,133.46 \) (Round to the nearest cent) B) How much should you deposit now to have that amount 10 years from now? (Round to the nearest cent)

To determine how much you should deposit now to have \( \$ 15,000 \) in the future with an interest rate of \( 8\% \) compounded semiannually, we can use the formula for the present value of a future sum: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( PV \) = present value (the amount to deposit now) - \( FV \) = future value (the amount you want in the future, which is \( \$ 15,000 \)) - \( r \) = annual interest rate (as a decimal, so \( 0.08 \)) - \( n \) = number of times interest is compounded per year (for semiannual compounding, \( n = 2 \)) - \( t \) = number of years until the future value is reached ### Part A: 5 Years from Now 1. **Identify the variables:** - \( FV = 15000 \) - \( r = 0.08 \) - \( n = 2 \) - \( t = 5 \) 2. **Plug the values into the formula:** \[ PV = \frac{15000}{(1 + 0.08/2)^{2 \times 5}} \] \[ PV = \frac{15000}{(1 + 0.04)^{10}} \] \[ PV = \frac{15000}{(1.04)^{10}} \] 3. **Calculate \( (1.04)^{10} \):** \[ (1.04)^{10} \approx 1.48024 \] 4. **Calculate \( PV \):** \[ PV = \frac{15000}{1.48024} \approx 10133.46 \] So, the amount to deposit now for 5 years is approximately \( \$ 10,133.46 \). ### Part B: 10 Years from Now 1. **Identify the variables:** - \( FV = 15000 \) - \( r = 0.08 \) - \( n = 2 \) - \( t = 10 \) 2. **Plug the values into the formula:** \[ PV = \frac{15000}{(1 + 0.08/2)^{2 \times 10}} \] \[ PV = \frac{15000}{(1 + 0.04)^{20}} \] \[ PV = \frac{15000}{(1.04)^{20}} \] 3. **Calculate \( (1.04)^{20} \):** \[ (1.04)^{20} \approx 2.20840 \] 4. **Calculate \( PV \):** \[ PV = \frac{15000}{2.20840} \approx 6781.46 \] So, the amount to deposit now for 10 years is approximately \( \$ 6,781.46 \). ### Summary: - **A)** \( \$ 10,133.46 \) - **B)** \( \$ 6,781.46 \)