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.