Multiple Choice Question Which of the following has the highest present value? O \( \$ 1 \) compounded annually over 3 years at \( 5 \% \) O \( \$ 1 \) annuity compounded annually over 3 years at \( 7 \% \) O \( \$ 1 \) annuity compounded annually over 3 years at \( 5 \% \) \$1 compounded annually over 3 years at \( 7 \% \)

Alright, I need to determine which of the given options has the highest present value. Let's break down each option step by step to understand their present values. **Option 1: \$1 compounded annually over 3 years at 5%** This is a simple compound interest scenario. The formula for compound interest is: \[ A = P \times (1 + r)^n \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (\$1). - \( r \) is the annual interest rate (5% or 0.05). - \( n \) is the number of years (3). Plugging in the numbers: \[ A = 1 \times (1 + 0.05)^3 = 1 \times (1.05)^3 \] Calculating \( (1.05)^3 \): \[ 1.05 \times 1.05 = 1.1025 \] \[ 1.1025 \times 1.05 = 1.157625 \] So, the amount after 3 years is \$1.157625. **Option 2: \$1 annuity compounded annually over 3 years at 7%** An annuity involves a series of equal payments made at regular intervals. However, the question seems to refer to a lump sum annuity, which is essentially the same as simple compound interest. The formula remains: \[ A = P \times (1 + r)^n \] Where: - \( P \) is the principal amount (\$1). - \( r \) is the annual interest rate (7% or 0.07). - \( n \) is the number of years (3). Plugging in the numbers: \[ A = 1 \times (1 + 0.07)^3 = 1 \times (1.07)^3 \] Calculating \( (1.07)^3 \): \[ 1.07 \times 1.07 = 1.1449 \] \[ 1.1449 \times 1.07 = 1.225043 \] So, the amount after 3 years is \$1.225043. **Option 3: \$1 annuity compounded annually over 3 years at 5%** Again, this is a lump sum annuity. Using the compound interest formula: \[ A = 1 \times (1 + 0.05)^3 = 1 \times (1.05)^3 \] We've already calculated this to be \$1.157625. **Option 4: \$1 compounded annually over 3 years at 7%** This is straightforward compound interest: \[ A = 1 \times (1 + 0.07)^3 = 1 \times (1.07)^3 \] We've calculated this to be \$1.225043. **Comparing the Options:** - **Option 1:** \$1.157625 - **Option 2:** \$1.225043 - **Option 3:** \$1.157625 - **Option 4:** \$1.225043 From the calculations, **Option 4** (\$1 compounded annually over 3 years at 7%) has the highest present value of \$1.225043.