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

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 7%** This is a simple compound interest scenario. The formula for compound interest is: \[ A = P \times (1 + r)^n \] Where: - \( A \) is the amount after n years - \( P \) 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: \[ A = 1 \times (1 + 0.07)^3 \] \[ A = 1 \times (1.07)^3 \] \[ A = 1 \times 1.225043 \] \[ A \approx \$1.225 \] **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 specifies a single payment of \$1, which implies it's a lump sum rather than an annuity. If it were an annuity, we'd need to know the number of payments. Since it's a single payment, it's essentially the same as Option 1. Therefore, the present value would still be approximately \$1.225. **Option 3: \$1 compounded annually over 3 years at 5%** Using the compound interest formula: \[ A = 1 \times (1 + 0.05)^3 \] \[ A = 1 \times (1.05)^3 \] \[ A = 1 \times 1.157625 \] \[ A \approx \$1.158 \] **Option 4: \$1 annuity compounded annually over 3 years at 5%** Again, assuming it's a lump sum, the present value would be the same as Option 3, approximately \$1.158. **Comparing the Present Values:** - Option 1: \$1.225 - Option 2: \$1.225 - Option 3: \$1.158 - Option 4: \$1.158 Both Option 1 and Option 2 have the highest present value of approximately \$1.225. Option 3 and Option 4 have a lower present value of approximately \$1.158. However, it's important to note that if Option 2 were indeed an annuity with multiple payments, the present value would be different. But based on the information provided, both Option 1 and Option 2 are identical in terms of present value.