Let's break down the problem step by step: ### Given Data: - **Par value (Face Value), \( F \) = $1,000** - **Coupon rate = 10% per year, paid quarterly** - **Life of bond = 6 years** - **Purchaser's desired yield = 8% per year, compounded quarterly** - **Payments per year = 4 (quarterly)** - **Total number of periods, \( n \) = 6 years × 4 = 24 quarters** ### 1. **Find the periodic coupon payment (\( PMT \)):** Annual coupon payment = 10% × $1000 = **$100 per year** Quarterly coupon payment = $100 / 4 = **$25 per quarter** ### 2. **Find the periodic (quarterly) market rate (\( i \)):** Annual desired yield = 8% Quarterly desired yield = 8% / 4 = **2% per quarter or 0.02** ### 3. **Calculate Present Value of Coupons (Annuity):** The present value of the coupons is: \[ PV_{\text{coupons}} = PMT \times \frac{1 - (1 + i)^{-n}}{i} \] Plug in the values: - \( PMT = 25 \) - \( i = 0.02 \) - \( n = 24 \) Calculate \( (1 + i)^{-n} \): \[ (1 + 0.02)^{-24} = (1.02)^{-24} \] Calculate \( 1.02^{24} \): \[ 1.02^{24} = e^{24 \cdot \ln(1.02)} = e^{24 \cdot 0.01980} = e^{0.4752} \approx 1.60703 \] Thus, \[ (1.02)^{-24} = \frac{1}{1.60703} \approx 0.62228 \] Now substitute: \[ PV_{\text{coupons}} = 25 \times \frac{1 - 0.62228}{0.02} \] \[ = 25 \times \frac{0.37772}{0.02} \] \[ = 25 \times 18.886 \] \[ = 472.15 \] ### 4. **Present Value of Par Value (Face Value):** \[ PV_{F} = F \times (1 + i)^{-n} \] \[ PV_{F} = 1000 \times 0.62228 = 622.28 \] ### 5. **Total Present Value (Price of the bond):** \[ PV_{\text{total}} = PV_{\text{coupons}} + PV_{F} \] \[ = 472.15 + 622.28 \] \[ = 1094.43 \] ### 6. **Final Answer (rounded to nearest whole dollar):** \[ \boxed{1094} \] --- ## **Final Answer** **$1,094** --- **Explanation:** - We discounted each coupon and the par value at the market's required rate (8% compounded quarterly). - The sum is the maximum a purchaser would be willing to pay to achieve an 8% (quarterly) return.

Let's carefully analyze and solve the problem step-by-step. --- ### **Problem Restatement:** - **Par value of bond, \(F\) = \$1,000** - **Coupon rate = 10% per year, paid quarterly** - **Bond term = 6 years** - **Desired yield by purchaser = 8% per quarter** - **Payments are quarterly** --- ### **Step 1: Determine the periodic coupon payment** - **Annual coupon rate = 10%** - **Annual coupon payment = 10% of \$1,000 = \$100** Since payments are quarterly: \[ \text{Quarterly coupon payment} = \frac{\$100}{4} = \$25 \] --- ### **Step 2: Determine the total number of periods** \[ \text{Number of quarters} = 6 \text{ years} \times 4 \text{ quarters/year} = 24 \text{ quarters} \] --- ### **Step 3: Determine the quarterly discount rate** - **Desired yield per quarter = 8%** Expressed as decimal: \[ i = 0.08 \] --- ### **Step 4: Calculate the present value of the coupons (annuity)** Use the Present Value of an Ordinary Annuity formula: \[ PV_{\text{coupons}} = PMT \times \frac{1 - (1 + i)^{-n}}{i} \] Where: - \(PMT = \$25\) - \(i = 0.08\) - \(n = 24\) --- ### **Step 5: Calculate \((1 + i)^{-n}\)** First, find \((1 + 0.08)^{24}\): \[ (1.08)^{24} \] Using a calculator or TVM table: \[ (1.08)^{24} \approx 6.102 \] Then: \[ (1.08)^{-24} = \frac{1}{6.102} \approx 0.164 \] --- ### **Step 6: Calculate the PV of coupons** \[ PV_{\text{coupons}} = 25 \times \frac{1 - 0.164}{0.08} \] \[ = 25 \times \frac{0.836}{0.08} \] \[ = 25 \times 10.45 \] \[ = 261.25 \] --- ### **Step 7: Calculate the present value of the face value** \[ PV_{F} = F \times (1 + i)^{-n} \] \[ PV_{F} = 1000 \times 0.164 = 164.00 \] --- ### **Step 8: Calculate the total price the purchaser is willing to pay** \[ PV_{\text{total}} = PV_{\text{coupons}} + PV_{F} = 261.25 + 164.00 = 425.25 \] --- ### **Final step: Round to the nearest whole dollar** \[ \boxed{\$425} \] --- ## **Final Answer:** **$425** --- ### **Summary:** The purchaser would be willing to pay approximately **\$425** for the bond to achieve an 8% quarterly return over the 6-year life of the bond.

Let's carefully analyze and solve the problem step-by-step. --- ### **Problem Restatement:** - **Par value of bond, \(F\) = \$1,000** - **Coupon rate = 10% per year, paid quarterly** - **Bond term = 6 years** - **Desired yield by purchaser = 8% per quarter** - **Payments are quarterly** --- ### **Step 1: Determine the quarterly coupon payment** Annual coupon payment: \[ \$1,000 \times 10\% = \$100 \] Since payments are quarterly: \[ \text{Quarterly coupon payment} = \frac{\$100}{4} = \$25 \] --- ### **Step 2: Determine the total number of payments (periods)** \[ n = 6 \text{ years} \times 4 \text{ quarters/year} = 24 \text{ quarters} \] --- ### **Step 3: Determine the quarterly discount rate** Desired yield per quarter: \[ i = 8\% = 0.08 \] --- ### **Step 4: Calculate the present value of the coupons (annuity)** Use the Present Value of an Ordinary Annuity formula: \[ PV_{\text{coupons}} = PMT \times \frac{1 - (1 + i)^{-n}}{i} \] Where: - \(PMT = \$25\) - \(i = 0.08\) - \(n = 24\) --- ### **Step 5: Calculate \((1 + i)^{-n}\)** Compute \((1 + 0.08)^{24}\): \[ (1.08)^{24} \] Using a calculator or TVM table: \[ (1.08)^{24} \approx 6.10203 \] Therefore: \[ (1.08)^{-24} = \frac{1}{6.10203} \approx 0.16407 \] --- ### **Step 6: Calculate the PV of coupons** \[ PV_{\text{coupons}} = 25 \times \frac{1 - 0.16407}{0.08} \] \[ = 25 \times \frac{0.83593}{0.08} \] \[ = 25 \times 10.44913 \approx 261.228 \] --- ### **Step 7: Calculate the present value of the face value (par value)** \[ PV_{F} = F \times (1 + i)^{-n} \] \[ PV_{F} = 1000 \times 0.16407 \approx 164.07 \] --- ### **Step 8: Sum the present values to find the maximum price** \[ PV_{\text{total}} = PV_{\text{coupons}} + PV_{F} \approx 261.228 + 164.07 = 425.298 \] ### **Step 9: Round to the nearest whole dollar** \[ \boxed{\$425} \] --- ## **Final Answer:** **$425** --- ### **Summary:** The purchaser would be willing to pay approximately **\$425** for the bond to earn an 8% quarterly return over 6 years.