Let's solve the yield to maturity (YTM) for the PBJ Corporation bond step by step. ## **Given Data** - **Face Value (\(FV\))**: \$1,000 - **Coupon Rate**: 8.5% (paid semiannually) - **Coupon Payment (\(PMT\))**: \( \$1,000 \times \frac{8.5\%}{2} = \$42.50 \) per period - **Issue Date**: Jan 1, 2005 - **Maturity Date**: Jan 1, 2035 (30 years) - **Market Price on Jan 1, 2016 (\(PV\))**: \$930 - **Number of years from 2016 to 2035**: \(2035 - 2016 = 19\) years - **Number of periods (\(N\))**: \(19 \text{ years} \times 2 = 38\) semiannual periods We are to solve for the yield to maturity (YTM), which is the semiannual rate, and then annualize it. --- ## **Step 1: Set Up the YTM Equation** The price of a bond is the present value (PV) of all future cash flows (coupon payments + face value): \[ PV = \sum_{t=1}^{N} \frac{PMT}{(1 + r)^t} + \frac{FV}{(1 + r)^N} \] Where: - \(PV\) = Current bond price (\$930) - \(PMT\) = Coupon payment (\$42.50) - \(N\) = Number of periods (38) - \(FV\) = Face value (\$1,000) - \(r\) = semiannual yield (YTM/2), to solve for --- ## **Step 2: Plug in the Values** \[ 930 = \sum_{t=1}^{38} \frac{42.50}{(1 + r)^t} + \frac{1,000}{(1 + r)^{38}} \] This is a bond price equation. We can use the formula for the present value of an ordinary annuity for the coupon payments: \[ PV_{\text{coupons}} = PMT \times \left[ \frac{1 - (1 + r)^{-N}}{r} \right] \] And the present value of the face value: \[ PV_{\text{face}} = \frac{FV}{(1 + r)^N} \] So, \[ 930 = 42.50 \times \left[ \frac{1 - (1 + r)^{-38}}{r} \right] + \frac{1,000}{(1 + r)^{38}} \] --- ## **Step 3: Solve for \(r\) (semiannual YTM) Numerically** This equation cannot be solved algebraically, so let's use **trial and error** (or a financial calculator/excel). We'll try several values for \(r\) to zero in on the correct rate. ### Let's try r = 0.05 (5% semiannual, 10% annual) - \((1 + r)^{38} = (1.05)^{38} \approx 7.04\) - \(1/(1+r)^{38} \approx 1/7.04 \approx 0.142\) - So, - \(PV_{\text{face}} = 1,000 \times 0.142 = 142\) - \(PV_{\text{coupons}} = 42.5 \times \left[ \frac{1 - 0.142}{0.05} \right] = 42.5 \times \left[\frac{0.858}{0.05}\right] = 42.5 \times 17.16 = 729.3\) - Total PV = 729.3 + 142 = 871.3 (Too low, need a lower rate) ### Try r = 0.04 (4% semiannual, 8% annual) - \((1.04)^{38} \approx 5.26\) - \(1/5.26 \approx 0.190\) - \(PV_{\text{face}} = 1,000 \times 0.190 = 190\) - \(PV_{\text{coupons}} = 42.5 \times \left[ \frac{1 - 0.190}{0.04} \right] = 42.5 \times 20.25 = 861\) - Total PV = 861 + 190 = 1,051 (Too high, need a higher rate) ### Try r = 0.045 (4.5% semiannual, 9% annual) - \((1.045)^{38} \approx 6.07\) - \(1/6.07 \approx 0.165\) - \(PV_{\text{face}} = 1,000 \times 0.165 = 165\) - \(PV_{\text{coupons}} = 42.5 \times \left[ \frac{1 - 0.165}{0.045} \right] = 42.5 \times 18.56 = 789.8\) - Total PV = 789.8 + 165 = 954.8 (Still a bit high) ### Try r = 0.0475 (4.75% semiannual, 9.5% annual) - \((1.0475)^{38} \approx 6.52\) - \(1/6.52 \approx 0.153\) - \(PV_{\text{face}} = 1,000 \times 0.153 = 153\) - \(PV_{\text{coupons}} = 42.5 \times \left[ \frac{1 - 0.153}{0.0475} \right] = 42.5 \times 17.86 = 759.1\) - Total PV = 759.1 + 153 = 912.1 (Slightly low) ### Try r = 0.046 (4.6% semiannual, 9.2% annual) - \((1.046)^{38} \approx 6.23\) - \(1/6.23 \approx 0.160\) - \(PV_{\text{face}} = 1,000 \times 0.160 = 160\) - \(PV_{\text{coupons}} = 42.5 \times \left[ \frac{1 - 0.160}{0.046} \right] = 42.5 \times 18.26 = 776.1\) - Total PV = 776.1 + 160 = 936.1 (Close to \$930) ### Try r = 0.0465 (4.65% semiannual, 9.3% annual) - \((1.0465)^{38} \approx 6.32\) - \(1/6.32 \approx 0.158\) - \(PV_{\text{face}} = 1,000 \times 0.158 = 158\) - \(PV_{\text{coupons}} = 42.5 \times \left[ \frac{1 - 0.158}{0.0465} \right] = 42.5 \times 18.13 = 770.5\) - Total PV = 770.5 + 158 = 928.5 (Very close to \$930) ### Interpolation \[ \text{At } r_1 = 0.046, PV_1 = 936.1 \\ \text{At } r_2 = 0.0465, PV_2 = 928.5 \] \[ r = r_1 + \frac{PV_1 - PV}{PV_1 - PV_2} \times (r_2 - r_1) \] \[ r = 0.046 + \frac{936.1 - 930}{936.1 - 928.5} \times (0.0465 - 0.046) \] \[ r = 0.046 + \frac{6.1}{7.6} \times 0.0005 = 0.046 + 0.401 \times 0.0005 \approx 0.046 + 0.0002 = 0.0462 \] So, the **semiannual yield** is **0.0462** (4.62%). **Annual yield (YTM) = 2 × semiannual yield = 9.24%** --- ## **Step 4: Final Answer Formatting** The question asks: > If your answer is (2.258%), write it as 2.26. Drop the % sign. So: - Our answer is **9.24** (rounded to two decimals), **drop the percent sign.** --- ## **Final Answer** \[ \boxed{9.24} \] --- ### **Summary Table** | Step | Value | |----------------------|-------------------------| | Face Value (\$FV) | \$1,000 | | Coupon Payment (\$) | \$42.50 (semiannual) | | Periods (N) | 38 | | Price (\$PV) | \$930 | | YTM (annual, %) | **9.24** | --- **The yield to maturity for the PBJ Corporation bond on Jan 1, 2016, if the market price is \$930, is:** **9.24** (rounded to two decimals, percent sign dropped)

Let's clarify the calculation and ensure accuracy. **Given Data Recap:** - Face Value (FV): \$1,000 - Coupon Rate: 8.5% annually, paid semiannually - Coupon Payment per period: \$1,000 * 8.5% / 2 = \$42.50 - Maturity date: Jan 1, 2035 - Issue date: Jan 1, 2005 - Time from Jan 1, 2016 to Jan 1, 2035: 19 years - Number of periods (N): 19 years * 2 = 38 - Market Price on Jan 1, 2016: \$930 --- ### Step-by-step solution: **1. Set up the bond valuation formula:** \[ PV = \left( \sum_{t=1}^{N} \frac{PMT}{(1 + r)^t} \right) + \frac{FV}{(1 + r)^N} \] Where: - \(PV = 930\) - \(PMT = 42.50\) - \(FV = 1000\) - \(N = 38\) - \(r\) = semiannual yield (unknown) --- **2. Approximate \(r\) using trial and error:** We've tried various \(r\) values and found that around **4.6% semiannual** gives a PV close to \$930. - At \(r = 4.6\%\) (0.046): \[ (1 + r)^{38} \approx 6.23 \] - Present value of face: \[ PV_{face} = \frac{1000}{6.23} \approx 160.4 \] - Present value of coupons: \[ PV_{coupons} = 42.50 \times \left[\frac{1 - (1 + r)^{-N}}{r}\right] \] \[ = 42.50 \times \left[\frac{1 - 1/6.23}{0.046}\right] = 42.50 \times \left[\frac{1 - 0.160}{0.046}\right] = 42.50 \times 18.13 \approx 771.0 \] - Total PV: \[ 771.0 + 160.4 \approx 931.4 \] which is very close to \$930. --- ### 3. Find the semiannual yield: Since at 4.6%, PV ≈ \$931, very close to market price \$930, the semiannual yield is approximately **4.6%**. ### 4. Convert semiannual yield to annual yield: \[ \text{YTM} = 2 \times 4.6\% = 9.2\% \] ### **Final answer:** **YTM = 9.2%** --- ### **Answer formatting per instructions:** - Drop the % sign and round to two decimal places: \[ \boxed{9.20} \] --- ### **Conclusion:** The **correct final answer** for the bond's yield to maturity on January 1, 2016, is: **9.20** --- ### **Note about the given answer (2.258):** The value **2.258** (or 2.26) appears to be a **semiannual yield in percent**, or possibly a misunderstanding. But based on calculations, the **annualized YTM** is approximately **9.20**. **Therefore, the answer is:** **9.20**

Let's carefully re-express the problem and verify the calculations step by step. --- ### **Given Data:** - **Face Value (\(FV\))**: \$1,000 - **Coupon Rate**: 8.5% annually - **Coupon Payment per period (\(PMT\))**: \[ \$1,000 \times \frac{8.5\%}{2} = \$42.50 \] - **Issue Date**: January 1, 2005 - **Maturity Date**: January 1, 2035 - **Remaining Periods as of January 1, 2016**: From Jan 1, 2016 to Jan 1, 2035: 19 years Since coupons are semiannual, total periods remaining: \[ 19 \text{ years} \times 2 = 38 \text{ periods} \] - **Market Price on Jan 1, 2016 (\(PV\))**: \$930 --- ### **Step 1: Set Up the Bond Price Equation** The bond's present value: \[ PV = \text{PV of coupons} + \text{PV of face value} \] \[ PV = PMT \times \left[\frac{1 - (1 + r)^{-N}}{r}\right] + \frac{FV}{(1+r)^N} \] Where: - \(r\) = semiannual YTM (unknown) - \(N=38\) --- ### **Step 2: Approximate \(r\) via iterative trial** Trying key guesses based on earlier rough calculations: **At \(r = 4.6\%\) (semiannual):** - Compute \((1 + r)^{38}\): \[ (1.046)^{38} \approx e^{38 \times \ln(1.046)} \approx e^{38 \times 0.045} \approx e^{1.71} \approx 5.53 \] - Present value of face: \[ PV_{face} = \frac{1000}{5.53} \approx \$180.88 \] - Present value of coupons: \[ PV_{coupons} = 42.50 \times \left[\frac{1 - (1.046)^{-38}}{0.046}\right] \] Calculate \((1.046)^{-38}\): \[ (1.046)^{-38} = \frac{1}{(1.046)^{38}} \approx \frac{1}{5.53} \approx 0.181 \] So, \[ PV_{coupons} = 42.50 \times \frac{1 - 0.181}{0.046} = 42.50 \times \frac{0.819}{0.046} \approx 42.50 \times 17.80 \approx \$756.50 \] Total PV: \[ 756.50 + 180.88 = \$937.38 \] This is very close to the market price (\$930), slightly above. Adjust \(r\) slightly upward. --- **Try \(r = 4.65\%\):** - Compute \((1.0465)^{38}\): \[ (1.0465)^{38} \approx e^{38 \times 0.0458} \approx e^{1.74} \approx 5.70 \] - PV of face: \[ \frac{1000}{5.70} \approx \$175.44 \] - \((1.0465)^{-38} = 1/5.70 \approx 0.175\) - PV of coupons: \[ 42.50 \times \frac{1 - 0.175}{0.0465} = 42.50 \times \frac{0.825}{0.0465} \approx 42.50 \times 17.73 \approx \$754.0 \] Total PV: \[ 754.0 + 175.44 = \$929.44 \] Very close to \$930. The PV slightly less, so the actual rate is just slightly below 4.65%. **At 4.6% the PV was ~\$937.38, at 4.65% it's ~\$929.44**, so the YTM is approximately **4.62% semiannually**. --- ### **Step 3: Convert semiannual YTM to annual YTM** \[ \text{Annual YTM} = 2 \times 4.62\% = 9.24\% \] --- ### **Final Step: Formatting the answer** The problem asks to **write the answer as 2.258 if it were a percentage**, but in our case, the **YTM is approximately 9.24%**. **Drop the percent sign and round to two decimals:** \[ \boxed{9.24} \] --- ### **Summary:** | Description | Value | |--------------|---------| | Semiannual YTM | approximately **4.62%** | | Annual YTM | **9.24%** | **Thus, the yield to maturity on January 1, 2016, is approximately:** **9.24** --- ### **Regarding the 2.258 figure:** - The number **2.258** likely represents the semiannual yield **in percent** (i.e., 2.258%), which corresponds to a **4.516%** semiannual rate, or **9.03%** annualized, close to our estimate. - But based on our calculations, **the precise YTM is about 9.24%**. --- ### **Final answer:** **9.24** --- **Note:** The small differences are due to rounding during the iterative process, but 9.24 is a reliable estimate based on the calculations.