Let's break down and solve each part step by step: --- ### **Part 1a: Premium or Discount?** #### **Given:** - **Coupon Rate:** 13% (annual, paid semiannually) - **Market Yield (YTM):** 12% (annual, paid semiannually) - **Par Value:** $1,000 - **Years to Maturity:** 16 years **Question:** Are these premium or discount bonds? #### **Solution:** - **If the coupon rate is greater than the yield to maturity, the bond sells at a premium.** - Here, **Coupon Rate (13%) > YTM (12%)**. **Best answer:** **A. The bonds should be selling at a premium because the bond's coupon rate is greater than the yield to maturity of similar bonds.** --- ### **Part 1b: Price of the Bonds** #### **Step 1: Convert to Semiannual Periods and Rates** - **Semiannual coupon payment = (13% of $1,000) / 2 = $130 / 2 = $65** - **Number of semiannual periods = 16 years × 2 = 32** - **Semiannual market yield = 12% / 2 = 6%** #### **Step 2: Plug Into the Bond Price Formula** The price of the bond is the present value of all future cash flows: \[ \text{Bond Price} = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^n} \] Where: - \( C \) = semiannual coupon = $65 - \( r \) = semiannual yield = 0.06 - \( n \) = total periods = 32 - \( F \) = face value = $1,000 #### **Step 3: Calculate Present Value of Coupons** \[ PV_{\text{coupons}} = C \times \left[1 - (1+r)^{-n}\right] / r \] \[ PV_{\text{coupons}} = 65 \times \left[1 - (1+0.06)^{-32}\right] / 0.06 \] First, calculate \((1+0.06)^{-32}\): \[ (1+0.06)^{-32} = (1.06)^{-32} \approx 0.1581 \] So, \[ PV_{\text{coupons}} = 65 \times \frac{1 - 0.1581}{0.06} \] \[ = 65 \times \frac{0.8419}{0.06} \] \[ = 65 \times 14.0317 \] \[ = 912.06 \] #### **Step 4: Calculate Present Value of Par Value** \[ PV_{\text{par}} = \frac{1000}{(1.06)^{32}} \] \[ = \frac{1000}{6.3266} \] \[ = 158.08 \] #### **Step 5: Add Both Parts** \[ \text{Bond Price} = PV_{\text{coupons}} + PV_{\text{par}} \] \[ = 912.06 + 158.08 \] \[ = \$1,070.14 \] --- ## **Final Answers:** ### **Part 1a:** **A. The bonds should be selling at a premium because the bond's coupon rate is greater than the yield to maturity of similar bonds.** ### **Part 1b:** **The price of the bond is: \(\boxed{\$1,070.14}\)** Let me know if you need further explanation!