Explanation

To calculate the dirty and clean prices of the bond:

Given:

• Par value = $1,000
• Coupon rate = 6% (semiannual = 3% per period)
• Coupon payment = $1,000 × 3% = $30
• Market rate = 5% (semiannual = 2.5% per period)
• Remaining payments = 10
• Days to next coupon = 90
• Total days in coupon period = 180 (assuming 30/360 convention)

Step 1: Calculate present value of remaining cash flows

Using the bond pricing formula: $\text{PV} = \sum_{t=1}^{10} \frac{30}{(1.025)^t} + \frac{1000}{(1.025)^{10}}$

This can be calculated as:

• PV of coupons = 30 × [1 - (1.025)^{-10}] / 0.025 = 30 × 8.7521 = 262.56
• PV of principal = 1000 / (1.025)^{10} = 1000 / 1.2801 = 781.20

Total PV = 262.56 + 781.20 = $1,043.76

Step 2: Calculate accrued interest

Accrued interest = Coupon payment × (Days elapsed / Total days) = $30 × (90 / 180) = $30 × 0.5 = $15

Step 3: Calculate clean and dirty prices

• Dirty price = Present value = $1,043.76
• Clean price = Dirty price - Accrued interest = $1,043.76 - $15 = $1,028.76

Therefore, the dirty price is $1,043.76 and the clean price is $1,028.76.