Answer: USD 103.35

## Explanation To solve this arbitrage pricing problem, we need to find the fair price of the 8% coupon bond using the information from the other two bonds. **Given:** - Zero-coupon bond (0% coupon): USD 96.12 - 10% coupon bond (semi-annual): USD 106.20 - 8% coupon bond (semi-annual): ? **Step 1: Set up cash flow equations** Let's assume a face value of USD 100 for all bonds. For the 10% coupon bond: - Semi-annual coupon = 10%/2 × 100 = USD 5 - Cash flows: USD 5 at 6 months, USD 105 at 12 months - Price = USD 106.20 For the 8% coupon bond: - Semi-annual coupon = 8%/2 × 100 = USD 4 - Cash flows: USD 4 at 6 months, USD 104 at 12 months **Step 2: Find discount factors** Let d₁ = discount factor for 6 months Let d₂ = discount factor for 12 months From zero-coupon bond: 100 × d₂ = 96.12 ⇒ d₂ = 0.9612 From 10% coupon bond: 5 × d₁ + 105 × d₂ = 106.20 5 × d₁ + 105 × 0.9612 = 106.20 5 × d₁ + 100.926 = 106.20 5 × d₁ = 5.274 d₁ = 1.0548 **Step 3: Price the 8% coupon bond** Price = 4 × d₁ + 104 × d₂ Price = 4 × 1.0548 + 104 × 0.9612 Price = 4.2192 + 99.9648 = 104.184 However, this gives USD 104.18, which matches option D, but let me verify the calculation. **Correction:** I made an error in the discount factor calculation. Let me recalculate properly. From zero-coupon bond: 100 × d₂ = 96.12 ⇒ d₂ = 0.9612 ✓ From 10% coupon bond: 5 × d₁ + 105 × 0.9612 = 106.20 5 × d₁ + 100.926 = 106.20 5 × d₁ = 5.274 d₁ = 1.0548 Wait, this can't be right because discount factors should be less than 1. The issue is that I'm treating the 6-month cash flow incorrectly. The zero-coupon bond gives us the 1-year discount factor, but we need to find the 6-month discount factor from the coupon bond. Let me use a different approach: Let P₀ = price of zero-coupon bond = 96.12 Let P₁₀ = price of 10% coupon bond = 106.20 We can create a synthetic 8% coupon bond using the other two bonds: 8% coupon bond = 0.8 × (10% coupon bond) + 0.2 × (zero-coupon bond) Let me verify: - 0.8 × 10% bond: 0.8 × 5 = 4 (6-month coupon), 0.8 × 105 = 84 (principal + final coupon) - 0.2 × zero-coupon bond: 0.2 × 100 = 20 (principal) - Total: 4 + 84 + 20 = 108 (but should be 104) This doesn't work. Let me use the correct approach: We need to find weights w₁ and w₂ such that: w₁ × (cash flows of 10% bond) + w₂ × (cash flows of zero-coupon bond) = cash flows of 8% bond At 6 months: w₁ × 5 + w₂ × 0 = 4 ⇒ w₁ = 0.8 At 12 months: w₁ × 105 + w₂ × 100 = 104 0.8 × 105 + w₂ × 100 = 104 84 + 100w₂ = 104 100w₂ = 20 w₂ = 0.2 So the 8% bond can be replicated by: 0.8 units of 10% bond + 0.2 units of zero-coupon bond Price = 0.8 × 106.20 + 0.2 × 96.12 Price = 84.96 + 19.224 = 104.184 ≈ USD 104.18 Therefore, the correct answer is **D. USD 104.18**.