Let's break down the problem step by step. **Given Data:** - Both bonds mature in 4 years. - Face value (\(FV\)) = $1,000 - Yield to maturity (\(YTM\)) = 9.7% - **Bond C** (Coupon Bond): Annual coupon rate = 11% - **Bond Z** (Zero Coupon): No coupons - Price needed at times: 0, 1, 2, 3, 4 years from now. - **Assume YTM remains constant at 9.7% throughout.** --- ## Step 1: Calculate the Cash Flows ### 1. **Bond C (Coupon Bond)** - Coupon payment = 11% of $1,000 = **$110** (every year for 4 years, plus $1,000 at maturity) ### 2. **Bond Z (Zero Coupon)** - Pays nothing until maturity, then pays $1,000 in 4 years. --- ## Step 2: Bond Pricing Formula For any bond: \[ \text{Price} = \sum_{t=1}^{N}\frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^N} \] Where: - \(C\) = annual coupon payment - \(N\) = number of years to maturity - \(r\) = yield to maturity (as a decimal) - \(FV\) = face value --- ## Step 3: Calculate Prices for Each Year We'll calculate for each bond at time 0, 1, 2, 3, and 4. ### **A. Bond C: Coupon Bond** #### **At Time 0 (Today):** - 4 years to maturity, so 4 coupon payments left. - \(C = \$110\), \(N=4\), \(r=0.097\) \[ \text{Price}_0 = \frac{110}{(1.097)^1} + \frac{110}{(1.097)^2} + \frac{110}{(1.097)^3} + \frac{110 + 1000}{(1.097)^4} \] Calculate each term: - \((1.097)^1 = 1.097\) - \((1.097)^2 = 1.203409\) - \((1.097)^3 = 1.319883\) - \((1.097)^4 = 1.447367\) Plug in values: - \(110/1.097 = 100.27\) - \(110/1.203409 = 91.41\) - \(110/1.319883 = 83.34\) - \((110+1000)/1.447367 = 1,110/1.447367 = 767.12\) Sum: \[ P_0 = 100.27 + 91.41 + 83.34 + 767.12 = \boxed{1,042.14} \] --- #### **At Time 1:** - 3 years to maturity, 3 coupons left. \[ P_1 = \frac{110}{(1.097)^1} + \frac{110}{(1.097)^2} + \frac{110+1000}{(1.097)^3} \] - \((1.097)^1 = 1.097\) - \((1.097)^2 = 1.203409\) - \((1.097)^3 = 1.319883\) - \(110/1.097 = 100.27\) - \(110/1.203409 = 91.41\) - \(1,110/1.319883 = 841.10\) Sum: \[ P_1 = 100.27 + 91.41 + 841.10 = \boxed{1,032.78} \] --- #### **At Time 2:** - 2 years to maturity, 2 coupons left. \[ P_2 = \frac{110}{(1.097)^1} + \frac{110+1000}{(1.097)^2} \] - \((1.097)^1 = 1.097\) - \((1.097)^2 = 1.203409\) - \(110/1.097 = 100.27\) - \(1,110/1.203409 = 922.37\) Sum: \[ P_2 = 100.27 + 922.37 = \boxed{1,022.64} \] --- #### **At Time 3:** - 1 year to maturity, 1 coupon left. \[ P_3 = \frac{110+1000}{(1.097)^1} = \frac{1,110}{1.097} = \boxed{1,011.39} \] --- #### **At Time 4 (Maturity):** - Only the face value and last coupon: \(1,110\) \[ P_4 = \$1,110 \] --- ### **Summary Table for Bond C** | Time (years) | Price of Bond C | |--------------|-----------------| | 0 | \$1,042.14 | | 1 | \$1,032.78 | | 2 | \$1,022.64 | | 3 | \$1,011.39 | | 4 | \$1,110.00 | --- ### **B. Bond Z: Zero Coupon Bond** #### **Bond Z only pays $1,000 at maturity.** \[ \text{Price at time } t = \frac{1,000}{(1.097)^{N-t}} \] - \(N = 4\) #### **At Time 0:** \[ P_0 = \frac{1,000}{(1.097)^4} = \frac{1,000}{1.447367} = \boxed{690.71} \] #### **At Time 1:** \[ P_1 = \frac{1,000}{(1.097)^3} = \frac{1,000}{1.319883} = \boxed{757.51} \] #### **At Time 2:** \[ P_2 = \frac{1,000}{(1.097)^2} = \frac{1,000}{1.203409} = \boxed{830.97} \] #### **At Time 3:** \[ P_3 = \frac{1,000}{(1.097)^1} = \frac{1,000}{1.097} = \boxed{911.03} \] #### **At Time 4:** \[ P_4 = \$1,000 \] --- ### **Summary Table for Bond Z** | Time (years) | Price of Bond Z | |--------------|-----------------| | 0 | \$690.71 | | 1 | \$757.51 | | 2 | \$830.97 | | 3 | \$911.03 | | 4 | \$1,000.00 | --- ## **Final Answers Table** | Time (years) | Price of Bond C | Price of Bond Z | |--------------|-----------------|-----------------| | 0 | \$1,042.14 | \$690.71 | | 1 | \$1,032.78 | \$757.51 | | 2 | \$1,022.64 | \$830.97 | | 3 | \$1,011.39 | \$911.03 | | 4 | \$1,110.00 | \$1,000.00 | --- ## **Summary** - **Bond C** (coupon bond) starts above par and moves to \$1,110 at maturity. - **Bond Z** (zero coupon) starts below par and moves to \$1,000 at maturity. - As time passes, both bonds' prices move toward their face value plus final coupon (if any). --- **If you need the step-by-step calculations in a table format, just let me know!**