Detailed Explanation

We need to calculate the present value (PV) of each investment using a 10% discount rate.

Investment 1:

• 20 annual payments of $50,000
• Starting one year from today (ordinary annuity)
• PV = PMT × [1 - (1 + r)^(-n)] / r
• PV₁ = $50,000 × [1 - (1.10)^(-20)] / 0.10
• PV₁ = $50,000 × [1 - 0.148644] / 0.10
• PV₁ = $50,000 × [0.851356] / 0.10
• PV₁ = $50,000 × 8.51356
• PV₁ = $425,678

Investment 2:

• 25 annual payments of $45,000
• Starting today (annuity due)
• PV = PMT × [1 - (1 + r)^(-n)] / r × (1 + r)
• PV₂ = $45,000 × [1 - (1.10)^(-25)] / 0.10 × (1.10)
• PV₂ = $45,000 × [1 - 0.092296] / 0.10 × 1.10
• PV₂ = $45,000 × [0.907704] / 0.10 × 1.10
• PV₂ = $45,000 × 9.07704 × 1.10
• PV₂ = $45,000 × 9.984744
• PV₂ = $449,313

Investment 3:

• Infinite annual payments of $40,000
• Starting one year from today (perpetuity)
• PV = PMT / r
• PV₃ = $40,000 / 0.10
• PV₃ = $400,000

Comparison:

• PV₁ = $425,678
• PV₂ = $449,313 (highest)
• PV₃ = $400,000

Therefore, Investment 2 has the highest present value.

Key Formulas Used:

1. Ordinary Annuity: PV = PMT × [1 - (1 + r)^(-n)] / r
2. Annuity Due: PV = PMT × [1 - (1 + r)^(-n)] / r × (1 + r)
3. Perpetuity: PV = PMT / r