Let's break down the problem step-by-step: ### **Given Data** Zero-coupon yields for years 1-5: - \( r_1 = 2.57\% \) - \( r_2 = 2.82\% \) - \( r_3 = 2.94\% \) - \( r_4 = 3.01\% \) - \( r_5 = 3.05\% \) We are to find the **yield to maturity (YTM)** for 5-year bonds with annual coupon rates of **0%, 3%, 6%, and 9%**. --- ### **Step 1: Find Present Value Factors (Discount Factors)** Using the zero rates, the present value (PV) discount factors for each year are: \[ DF_t = \frac{1}{(1 + r_t)^t} \] Where \( r_t \) is the zero rate for year \( t \) (expressed as a decimal). Let's compute the discount factors: #### **Year 1** \[ DF_1 = \frac{1}{(1 + 0.0257)^1} = \frac{1}{1.0257} = 0.97493 \] #### **Year 2** \[ DF_2 = \frac{1}{(1 + 0.0282)^2} = \frac{1}{(1.0282)^2} = \frac{1}{1.0571} = 0.94600 \] #### **Year 3** \[ DF_3 = \frac{1}{(1 + 0.0294)^3} = \frac{1}{(1.0294)^3} = \frac{1}{1.0907} = 0.91683 \] #### **Year 4** \[ DF_4 = \frac{1}{(1 + 0.0301)^4} = \frac{1}{(1.0301)^4} = \frac{1}{1.12657} = 0.88752 \] #### **Year 5** \[ DF_5 = \frac{1}{(1 + 0.0305)^5} = \frac{1}{(1.0305)^5} = \frac{1}{1.16294} = 0.85944 \] --- ### **Step 2: Calculate Price of Bond for Each Coupon Rate** Let’s assume all bonds have a face value of $100. #### **Bond Price Formula** \[ P = \sum_{t=1}^{5} \left( \frac{C}{(1 + r_t)^t} \right) + \frac{F}{(1 + r_5)^5} \] Where: - \( C \) = Annual coupon payment - \( F \) = Face value ($100) - \( r_t \) = Zero rate for year \( t \) #### **Coupon Rate 0% (Zero Coupon Bond)** \[ P = \frac{100}{(1.0305)^5} = 100 \times 0.85944 = \$85.94 \] #### **Coupon Rate 3%** \( C = 3 \) \[ P = 3 \times (DF_1 + DF_2 + DF_3 + DF_4 + DF_5) + 100 \times DF_5 \] \[ P = 3 \times (0.97493 + 0.94600 + 0.91683 + 0.88752 + 0.85944) + 100 \times 0.85944 \] \[ = 3 \times 4.58472 + 85.944 \] \[ = 13.75416 + 85.944 = \$99.70 \] #### **Coupon Rate 6%** \( C = 6 \) \[ P = 6 \times 4.58472 + 85.944 = 27.50832 + 85.944 = \$113.45 \] #### **Coupon Rate 9%** \( C = 9 \) \[ P = 9 \times 4.58472 + 85.944 = 41.26248 + 85.944 = \$127.21 \] --- ### **Step 3: Find the Yield to Maturity (YTM)** The YTM is the rate \( y \) that solves: \[ P = \sum_{t=1}^{5} \frac{C}{(1 + y)^t} + \frac{100}{(1 + y)^5} \] This is, in general, solved numerically. We'll estimate the YTM for each scenario: --- #### **1. 0% Coupon (Zero Coupon Bond):** \[ 85.944 = \frac{100}{(1 + y)^5} \] \[ (1 + y)^5 = \frac{100}{85.944} = 1.16294 \] \[ 1 + y = (1.16294)^{1/5} = 1.0305 \] \[ y = 3.05\% \] **This matches the 5-year zero rate as expected.** --- #### **2. 3% Coupon** Using the bond price $99.70: \[ 99.70 = \sum_{t=1}^{5} \frac{3}{(1 + y)^t} + \frac{100}{(1 + y)^5} \] Guess and check (since the price is almost $100 and coupons are close to the market rates, YTM should be about 3.05%). Let's estimate at \( y = 3.05\% \): \[ \text{PV of coupons:} \quad 3 \times 4.58472 = 13.75416 \] \[ \text{PV of principal:} \quad 100 \times 0.85944 = 85.944 \] \[ \text{Total: } 99.70 \] So, **YTM ≈ 3.05%**. --- #### **3. 6% Coupon** Price = $113.45 Let’s estimate YTM. Since price is above par, YTM will be below the coupon rate and below the 5-year zero rate. Try \( y = 3.0\% \): \[ DF = \frac{1}{(1.03)^t} \text{ for t = 1 to 5} \] Calculate: - \( DF_1 = 0.97087 \) - \( DF_2 = 0.94260 \) - \( DF_3 = 0.91514 \) - \( DF_4 = 0.88849 \) - \( DF_5 = 0.86261 \) Total coupons: \[ 6 \times (0.97087 + 0.94260 + 0.91514 + 0.88849 + 0.86261) = 6 \times 4.57971 = 27.4783 \] PV of principal: \[ 100 \times 0.86261 = 86.261 \] Total: \[ 27.4783 + 86.261 = 113.74 \] This is very close to the calculated price ($113.45). So, **YTM ≈ 3.0\%** --- #### **4. 9% Coupon** Price = $127.21 Try \( y = 3.0\% \): PV of coupons: \[ 9 \times 4.57971 = 41.2174 \] PV of principal: \[ 86.261 \] Total: \[ 41.2174 + 86.261 = 127.48 \] Again, this is very close to the calculated price ($127.21). So, **YTM ≈ 3.0\%** --- ## **Summary Table** | Coupon Rate | Price | YTM | |-------------|----------|---------| | 0% | $85.94 | 3.05% | | 3% | $99.70 | 3.05% | | 6% | $113.45 | 3.00% | | 9% | $127.21 | 3.00% | --- ### **Final Answers** 1. **YTM for 0% coupon:** 3.05% 2. **YTM for 3% coupon:** 3.05% 3. **YTM for 6% coupon:** 3.00% 4. **YTM for 9% coupon:** 3.00% --- If you need a more precise value for the 6% and 9% bonds, you could interpolate between 3.00% and 3.05%, but the difference will be very small (since the bond prices are very close at these yields). Let me know if you'd like to see that calculation!

Certainly! Let's go through each part of the problem step-by-step with detailed calculations, explanations, and the final answers. --- ## **Given Data** - Zero-coupon (discount) yields for years 1-5: - \( r_1 = 2.57\% = 0.0257 \) - \( r_2 = 2.82\% = 0.0282 \) - \( r_3 = 2.94\% = 0.0294 \) - \( r_4 = 3.01\% = 0.0301 \) - \( r_5 = 3.05\% = 0.0305 \) - **Objective:** Find the **Yields to Maturity (YTM)** for 5-year bonds with annual coupons of: - 0% - 3% - 6% - 9% --- ## **Part 1: Calculate Discount Factors for Each Year** The discount factor for year \( t \): \[ DF_t = \frac{1}{(1 + r_t)^t} \] ### Year 1: \[ DF_1 = \frac{1}{(1 + 0.0257)^1} = \frac{1}{1.0257} \approx 0.97493 \] ### Year 2: \[ DF_2 = \frac{1}{(1 + 0.0282)^2} = \frac{1}{(1.0282)^2} \approx \frac{1}{1.0571} \approx 0.94600 \] ### Year 3: \[ DF_3 = \frac{1}{(1 + 0.0294)^3} = \frac{1}{(1.0294)^3} \] Calculate denominator: \[ (1.0294)^3 \approx 1.0907 \] So: \[ DF_3 \approx \frac{1}{1.0907} \approx 0.91683 \] ### Year 4: \[ DF_4 = \frac{1}{(1 + 0.0301)^4} = \frac{1}{(1.0301)^4} \] Calculate denominator: \[ (1.0301)^4 \approx 1.12657 \] Thus: \[ DF_4 \approx \frac{1}{1.12657} \approx 0.88752 \] ### Year 5: \[ DF_5 = \frac{1}{(1 + 0.0305)^5} = \frac{1}{(1.0305)^5} \] Calculate denominator: \[ (1.0305)^5 \approx 1.16294 \] Thus: \[ DF_5 \approx \frac{1}{1.16294} \approx 0.85944 \] --- ## **Part 2: Calculate Present Value (Price) of Bonds** Assuming **face value = \$100**. ### **a) Zero Coupon Bond (0% coupon):** Price equals the discounted face value: \[ P = \frac{F}{(1 + r_5)^5} = 100 \times DF_5 = 100 \times 0.85944 = \boxed{\$85.94} \] **YTM:** Equal to the 5-year zero rate, \( 3.05\% \). --- ### **b) 3% Coupon Bond:** - Coupon payment \( C = 3 \) - Price \( P \) (from earlier calculations): \[ P = \sum_{t=1}^{5} \frac{C}{(1 + y)^t} + \frac{F}{(1 + y)^5} \] But since we have the market price (from the calculation), we can estimate \( y \): First, calculate the present value of coupons using the discount factors: \[ PV_{\text{coupons}} = C \times (DF_1 + DF_2 + DF_3 + DF_4 + DF_5) \] \[ PV_{\text{coupons}} = 3 \times (0.97493 + 0.94600 + 0.91683 + 0.88752 + 0.85944) \] Sum of discount factors: \[ 0.97493 + 0.94600 = 1.92093 \] \[ 1.92093 + 0.91683 = 2.83776 \] \[ 2.83776 + 0.88752 = 3.72528 \] \[ 3.72528 + 0.85944 = 4.58472 \] Thus: \[ PV_{\text{coupons}} = 3 \times 4.58472 = \boxed{\$13.75} \] PV of face value: \[ PV_{\text{principal}} = 100 \times DF_5 = 100 \times 0.85944 = \boxed{\$85.94} \] Total present value (price at current market YTM): \[ P \approx 13.75 + 85.94 = \$99.69 \] which is very close to \$100 (par). Therefore, the **YTM \( y \)** is approximately **3.05%**, matching the zero rate. **Final answer for 3% coupon bond: \(\boxed{YTM \approx 3.05\%}\).** --- ### **c) 6% Coupon Bond:** - Coupon \( C = 6 \) Calculate the present value of coupons: \[ PV_{\text{coupons}} = 6 \times 4.58472 = \$27.51 \] PV of face value: \[ PV_{\text{principal}} = 100 \times 0.85944 = \$85.94 \] Total: \[ PV \approx 27.51 + 85.94 = \$113.45 \] The bond's price exceeds \$100, indicating YTM is **less than** the coupon rate (since bond is trading above par). Estimate YTM: Try \( y = 3.00\% \): Calculate PV of coupons at \( y=3.00\% \): \[ DF_t = \frac{1}{(1.03)^t} \] - \( DF_1 = 0.97087 \) - \( DF_2 = 0.94260 \) - \( DF_3 = 0.91514 \) - \( DF_4 = 0.88849 \) - \( DF_5 = 0.86261 \) Sum: \[ 0.97087 + 0.94260 + 0.91514 + 0.88849 + 0.86261 = 4.57971 \] PV of coupons: \[ 6 \times 4.57971 = \$27.48 \] PV of face: \[ 100 \times 0.86261 = \$86.26 \] Total: \[ 27.48 + 86.26 = \$113.74 \] Close to \$113.45, so **YTM ≈ 3.00%**. --- ### **d) 9% Coupon Bond:** - Coupon \( C=9 \) Calculate PV: \[ PV_{\text{coupons}} = 9 \times 4.57971 = \$41.22 \] PV of face: \[ 100 \times 0.86261 = \$86.26 \] Total: \[ 41.22 + 86.26 = \$127.48 \] which is very close to the market price \$127.21, indicating **YTM ≈ 3.00%**. --- ## **Final Summary of Yields to Maturity:** | Coupon Rate | Approximate Price | Estimated YTM | |--------------|---------------------|---------------| | 0% | \$85.94 | **3.05%** | | 3% | \$99.70 | **3.05%** | | 6% | \$113.45 | **3.00%** | | 9% | \$127.21 | **3.00%** | --- ## **Key Takeaways:** - The YTM for the 0% coupon bond matches the 5-year zero rate. - Slight differences in prices and YTM estimates for coupons above 3% are due to approximation, but they are very close to the zero rate. - For bonds with coupons close to the zero rates, the YTM converges near the zero-coupon yield. --- If you'd like, I can provide interpolated values or more precise calculations, but this detailed approach covers all key steps thoroughly.