We can use the bootstrap method to derive the discount factors.

1. **For $0.5$years (Tranche 1):** Price = `$99-00 = \$99`$. Semi-annual coupon = $3.5% / 2 = 1.75%$. Cash flow at 0.5 years =$`100` + \`1.75 = \$101.75`$. $99 = 101.75 \times d(0.5) \Rightarrow d(0.5) = \frac{99}{101.75} \approx 0.97297 \approx 0.9730$2. **For $1$year (Tranche 2):** Price = `$100-16 = 100 + \frac{16}{32} = \$100.50`$. Semi-annual coupon = $4% / 2 = 2%$. `$100.50= 2 \times d(0.5) + 102 \times d(1)$$100.50` = 2(0.97297) + 102 \times d(1)$ $100.50 = 1.9459 + 102 \times d(1) \Rightarrow 102 \times d(1) = 98.5541 \Rightarrow d(1) \approx 0.9662$3. **For $1.5$years (Tranche 3):** Price = `$101-04 = 101 + \frac{4}{32} = \$101.12`5$. Semi-annual coupon = $5% / 2 = 2.5%$. `$101.125 = 2.5 \times d(0.5) + 2.5 \times d(1) + 102.5 \times d(1.5)$ $101.12`5 = 2.5(0.9730) + 2.5(0.9662) + 102.5 \times d(1.5)$ $101.125 = 2.4325 + 2.4155 + 102.5 \times d(1.5) \Rightarrow 102.5 \times d(1.5) = 96.277$$d(1.5) = \frac{96.277}{102.5} \approx 0.9393$