## Step-by-Step Solution ### **Step 1: Find the Yield to Maturity (YTM) of the Treasury Bond (Risk-Free Rate)** A 10-year **zero-coupon** Treasury bond sells for **$657.22**. Zero-coupon means it pays no interest, just face value at maturity. Let the face value \( F \) be **$1,000** (standard assumption unless otherwise stated). The formula for zero-coupon bond price: \[ P = \frac{F}{(1 + r)^n} \] Where: - \( P = 657.22 \) - \( F = 1000 \) - \( n = 10 \) - \( r = \) risk-free rate (YTM) \[ 657.22 = \frac{1000}{(1 + r)^{10}} \] Rearrange and solve for \( r \): \[ (1 + r)^{10} = \frac{1000}{657.22} \] \[ (1 + r)^{10} = 1.521 \] \[ 1 + r = (1.521)^{1/10} \] \[ 1 + r = 1.0432 \] \[ r = 0.0432 \] So, the **risk-free rate** is **0.0432** (or 4.32%). --- ### **Step 2: Add the Credit Spread for BBB Bonds** From the table: - **BBB spread = 1.17% = 0.0117** The YTM for a 10-year BBB corporate bond: \[ \text{YTM}_{\text{BBB}} = \text{risk-free rate} + \text{credit spread} \] \[ \text{YTM}_{\text{BBB}} = 0.0432 + 0.0117 = 0.0549 \] --- ### **Step 3: Final Answer** \[ \boxed{0.0549} \] --- #### **Summary** - **Risk-free YTM (Treasury):** 0.0432 - **BBB spread:** 0.0117 - **BBB Bond YTM:** **0.0549** (as a decimal, rounded to four places)

# Calculation of Yield to Maturity (YTM) for a 10-Year BBB Corporate Bond ## Objective To determine the yield to maturity (YTM) of a 10-year corporate bond rated BBB, using the price of a 10-year U.S. Treasury zero-coupon bond and its associated credit spread. ## Given Data - Price of 10-year Treasury zero-coupon bond, \( P = 657.22 \) - Face value of Treasury bond, \( F = 1000 \) - Maturity, \( n = 10 \) years - Credit spread for BBB bonds, \( \text{spread} = 1.17\% = 0.0117 \) ## Step 1: Calculate the Risk-Free Rate (YTM of Treasury Bond) Using the formula for a zero-coupon bond: \[ P = \frac{F}{(1 + r)^n} \] Substituting the known values: \[ 657.22 = \frac{1000}{(1 + r)^{10}} \] Rearranging the equation: \[ (1 + r)^{10} = \frac{1000}{657.22} \] Calculating the right side: \[ (1 + r)^{10} = 1.521 \] Taking the tenth root: \[ 1 + r = (1.521)^{1/10} \] Calculating the root: \[ 1 + r \approx 1.0432 \] Solving for \( r \): \[ r \approx 1.0432 - 1 = 0.0432 \] Thus, the risk-free rate is: \[ \text{risk-free rate} = 0.0432 \quad \text{(or 4.32\%)} \] ## Step 2: Calculate the YTM for BBB Corporate Bond The YTM for a corporate bond can be calculated as: \[ \text{YTM}_{\text{BBB}} = \text{risk-free rate} + \text{credit spread} \] Substituting the values: \[ \text{YTM}_{\text{BBB}} = 0.0432 + 0.0117 \] Calculating the YTM: \[ \text{YTM}_{\text{BBB}} = 0.0549 \] ## Final Answer The yield to maturity (YTM) for the 10-year corporate bond with a BBB rating is: \[ \boxed{0.0549} \]

# Calculation of Yield to Maturity (YTM) for a 10-Year BBB Corporate Bond ## Introduction The calculation of yield to maturity (YTM) for a corporate bond involves understanding the relationship between bond prices, interest rates, and credit risk. In this scenario, a 10-year U.S. Treasury zero-coupon bond price is given, from which the risk-free rate can be derived. This rate serves as the basis for calculating the YTM of a corporate bond with a specific credit rating (BBB). The YTM incorporates the additional risk premium associated with the bond's credit spread compared to risk-free securities. **Explanation:** This introduction sets the stage for understanding the significance of YTM in evaluating bond investments. The risk-free rate derived from Treasury bonds reflects the return investors expect from a secure investment, while the credit spread reflects the additional yield required for taking on additional risk associated with corporate bonds. This foundational knowledge is crucial for accurately determining the YTM of a BBB-rated bond. ## Relevant Formulas 1. **Price of a Zero-Coupon Bond:** \[ P = \frac{F}{(1 + r)^n} \] - Where: - \( P \) = Current price of the bond - \( F \) = Face value of the bond - \( r \) = Yield to maturity (risk-free rate) - \( n \) = Number of years to maturity 2. **Yield to Maturity Calculation for Corporate Bond:** \[ \text{YTM}_{\text{Corporate}} = \text{risk-free rate} + \text{credit spread} \] - Where: - \(\text{YTM}_{\text{Corporate}}\) = Yield to maturity of the corporate bond - \(\text{credit spread}\) = Additional yield demanded by investors for taking on credit risk **Explanation:** The first formula is critical for determining the risk-free rate from the Treasury bond price, which serves as a benchmark for all other yields. The second formula allows for the incorporation of credit risk into the YTM calculation, thus reflecting the true return expected from the corporate bond. ## Detailed Step-by-Step Solution ### Step 1: Calculate the Risk-Free Rate (YTM of the Treasury Bond) 1. Given: - Price of Treasury bond, \( P = 657.22 \) - Face value, \( F = 1000 \) - Maturity, \( n = 10 \) 2. Using the zero-coupon bond formula: \[ 657.22 = \frac{1000}{(1 + r)^{10}} \] 3. Rearranging the formula: \[ (1 + r)^{10} = \frac{1000}{657.22} \] 4. Calculating the right side: \[ (1 + r)^{10} \approx 1.521 \] 5. Taking the tenth root: \[ 1 + r = (1.521)^{0.1} \] 6. Calculating the root: \[ 1 + r \approx 1.0432 \] 7. Solving for \( r \): \[ r \approx 1.0432 - 1 = 0.0432 \] **Explanation:** This step calculates the risk-free rate by rearranging the bond pricing formula. The derived risk-free rate of 0.0432 (or 4.32%) signifies the yield an investor would expect from a risk-free investment over 10 years. ### Step 2: Calculate the YTM for the BBB Corporate Bond 1. Given: - Credit spread for BBB bonds = 1.17% = 0.0117 2. Using the YTM formula for corporate bonds: \[ \text{YTM}_{\text{BBB}} = \text{risk-free rate} + \text{credit spread} \] 3. Substituting the values: \[ \text{YTM}_{\text{BBB}} = 0.0432 + 0.0117 \] 4. Calculating the YTM: \[ \text{YTM}_{\text{BBB}} = 0.0549 \] **Explanation:** In this step, the YTM for the BBB corporate bond is calculated by adding the credit spread to the risk-free rate. This reflects the total expected return, accounting for the additional risk associated with the corporate bond. ## Conclusion The calculated yield to maturity (YTM) for a 10-year corporate bond with a BBB rating is: \[ \boxed{0.0549} \] This value indicates the expected return on the corporate bond, incorporating both the risk-free rate and the additional yield demanded by investors for the credit risk associated with the bond's rating.