| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | 要确定这张15年期的债券在当前市场利率为4%的情况下的公平市场价值，我们需要计算债券未来现金流的现值。这些现金流包括每六个月支付的票息和到期时的面值。具体步骤如下： 1. **计算总期数（n）**： - 由于债券期限为15年，每年支付两次利息，所以总期数为： \[ n = 15 \text{年} \times 2 = 30 \text{期} \] 2. **确定半年的市场利率（r）**： - 市场年利率为4%，因此半年的市场利率为： \[ r = \frac{4\%}{2} = 2\% = 0.02 \] 3. **计算半年度票息支付的现值**： - 每期票息支付为$25，现值公式为： \[ PV_{\text{coupons}} = \sum_{t=1}^{30} \frac{25}{(1+0.02)^t} \] 4. **计算面值的现值**： - 债券到期时支付的面值为$1000，现值公式为： \[ PV_{\text{face value}} = \frac{1000}{(1+0.02)^{30}} \] 5. **总现值**： - 将票息支付的现值和面值的现值相加，即为债券的公平市场价值： \[ PV_{\text{total}} = PV_{\text{coupons}} + PV_{\text{face value}} \] ### 具体计算 #### 票息支付的现值 我们使用现值公式计算每个票息支付的现值并求和： \[ PV_{\text{coupons}} = \sum_{t=1}^{30} \frac{25}{(1+0.02)^t} \] 这可以通过公式计算为： \[ PV_{\text{coupons}} = 25 \times \left(\frac{1 - (1+0.02)^{-30}}{0.02}\right) \] 计算结果为： \[ PV_{\text{coupons}} = 25 \times 19.292 = 482.30 \text{美元} \] #### 面值的现值 计算面值现值： \[ PV_{\text{face value}} = \frac{1000}{(1+0.02)^{30}} \] 计算结果为： \[ PV_{\text{face value}} = \frac{1000}{(1.02)^{30}} = \frac{1000}{1.81136} = 552.07 \text{美元} \] #### 总现值 将两部分现值相加： \[ PV_{\text{total}} = 482.30 + 552.07 = 1034.37 \text{美元} \] ### 结论 在当前市场利率为4%的情况下，这张15年期、每六个月支付$25票息的$1000债券的公平市场价值为 **1034.37美元**。 |
1111.97 |
| deepseekr1 | The fair market value of the bond is calculated by finding the present value of its future cash flows, which include the semi-annual coupon payments and the face value repaid at maturity. Here's the step-by-step solution: 1. **Determine Variables:** - Face Value (F) = $1,000 - Semi-annual Coupon Payment (C) = $25 - Number of Periods (n) = 15 years × 2 = 30 periods - Semi-annual Market Rate (r) = 4% ÷ 2 = 2% or 0.02 2. **Present Value of Coupons:** \[ PV_{\text{coupons}} = C \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] = 25 \times \left[\frac{1 - (1.02)^{-30}}{0.02}\right] \] - Calculate \((1.02)^{-30} \approx 0.55207\) - \(PV_{\text{coupons}} = 25 \times \frac{1 - 0.55207}{0.02} = 25 \times 22.3965 \approx \$559.91\) 3. **Present Value of Face Value:** \[ PV_{\text{face value}} = \frac{F}{(1 + r)^n} = \frac{1000}{(1.02)^{30}} \approx \frac{1000}{1.81136} \approx \$552.07 \] 4. **Total Bond Value:** \[ \text{Bond Price} = PV_{\text{coupons}} + PV_{\text{face value}} = 559.91 + 552.07 = \$1,111.98 \] **Final Answer:** The fair market value of the bond is approximately **$\boxed{1112}$**. |
1111.97 |
| gpto1 | **Solution:** To find the fair market value of the bond, we need to calculate the present value (PV) of all future cash flows from the bond, discounted at the current market interest rate. The bond provides semiannual coupon payments and returns the face value at maturity. **Given:** - **Face Value (Par Value) of the Bond (F)**: $1,000 - **Annual Coupon Payment**: Since the bond pays $25 every six months, the **semiannual coupon payment (C)** is $25. - **Number of Years to Maturity**: 15 years - **Market Interest Rate (Annual)**: 4% **Calculations:** 1. **Determine the Number of Periods and Periodic Interest Rate:** - **Number of Periods (n)**: Since payments are semiannual over 15 years: \[ n = 15 \text{ years} \times 2 \text{ periods per year} = 30 \text{ periods} \] - **Periodic Market Interest Rate (r)**: The annual rate divided by the number of periods per year: \[ r = \frac{4\%}{2} = 2\% = 0.02 \] 2. **Calculate the Present Value of the Annuity (Coupon Payments):** The present value of the coupon payments is calculated using the **Present Value of an Ordinary Annuity formula**: \[ \text{PV}_{\text{coupons}} = C \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \] Plugging in the values: \[ \text{PV}_{\text{coupons}} = \$25 \times \left[ \frac{1 - (1 + 0.02)^{-30}}{0.02} \right] \] **Calculations:** - Compute \((1 + r)^{-n}\): \[ (1 + 0.02)^{-30} = (1.02)^{-30} = \frac{1}{(1.02)^{30}} \] First, calculate \((1.02)^{30}\): \[ (1.02)^{30} \approx e^{30 \times \ln(1.02)} \] - Compute \(\ln(1.02)\): \[ \ln(1.02) \approx 0.019802 \] - Compute the exponent: \[ 30 \times 0.019802 \approx 0.59406 \] - Compute \((1.02)^{30}\): \[ e^{0.59406} \approx 1.8114 \] Therefore: \[ (1.02)^{-30} \approx \frac{1}{1.8114} \approx 0.5521 \] - Compute the numerator: \[ 1 - (1.02)^{-30} = 1 - 0.5521 = 0.4479 \] - Calculate \(\text{PV}_{\text{coupons}}\): \[ \text{PV}_{\text{coupons}} = \$25 \times \left( \frac{0.4479}{0.02} \right) = \$25 \times 22.395 = \$559.88 \] 3. **Calculate the Present Value of the Face Value (Lump Sum at Maturity):** The present value of the face value is calculated using the **Present Value of a Lump Sum formula**: \[ \text{PV}_{\text{face}} = \frac{F}{(1 + r)^n} \] Plugging in the values: \[ \text{PV}_{\text{face}} = \frac{\$1,000}{(1.02)^{30}} \] We've already calculated \((1.02)^{30} \approx 1.8114\), so: \[ \text{PV}_{\text{face}} = \frac{\$1,000}{1.8114} \approx \$552.07 \] 4. **Calculate the Total Present Value (Fair Market Value of the Bond):** \[ \text{PV}_{\text{total}} = \text{PV}_{\text{coupons}} + \text{PV}_{\text{face}} \] \[ \text{PV}_{\text{total}} = \$559.88 + \$552.07 = \$1,111.95 \] **Answer:** The fair market value of the bond is **approximately \$1,111.95**. --- **Explanation:** The bond's fair market value is higher than its face value because the coupon rate (which provides $50 annually) is higher than the current market interest rate of 4%. This means investors are willing to pay more for the bond since it offers better returns compared to the prevailing market rates. |
1111.97 |
| GPT-4o | To determine the fair market value of the bond, we need to calculate the present value of the bond's future cash flows, which include the semi-annual coupon payments and the face value of the bond at maturity. The steps are as follows: 1. **Identify the bond details:** - Face value (\(F\)) = $1000 - Annual coupon payment = $25 every six months, which means the semi-annual coupon payment (\(C\)) = $25 - Number of years (\(N\)) = 15 years - Market interest rate = 4% annually 2. **Convert the annual market interest rate to a semi-annual rate:** - Semi-annual market interest rate (\(r\)) = 4% / 2 = 2% or 0.02 3. **Determine the number of semi-annual periods:** - Total number of semi-annual periods (\(n\)) = 15 years * 2 = 30 periods 4. **Calculate the present value of the semi-annual coupon payments:** The present value of an annuity formula is used here: \[ PV_{\text{coupons}} = C \times \left(1 - (1 + r)^{-n}\right) / r \] Substituting the values: \[ PV_{\text{coupons}} = 25 \times \left(1 - (1 + 0.02)^{-30}\right) / 0.02 \] \[ PV_{\text{coupons}} = 25 \times \left(1 - (1.02)^{-30}\right) / 0.02 \] Using a financial calculator or a spreadsheet: \[ PV_{\text{coupons}} \approx 25 \times 20.588 \] \[ PV_{\text{coupons}} \approx 514.70 \] 5. **Calculate the present value of the face value of the bond:** The present value of a single sum formula is used here: \[ PV_{\text{face value}} = F / (1 + r)^{n} \] Substituting the values: \[ PV_{\text{face value}} = 1000 / (1.02)^{30} \] \[ PV_{\text{face value}} \approx 1000 / 1.81136 \] \[ PV_{\text{face value}} \approx 552.07 \] 6. **Calculate the total present value of the bond:** \[ PV_{\text{bond}} = PV_{\text{coupons}} + PV_{\text{face value}} \] \[ PV_{\text{bond}} = 514.70 + 552.07 \] \[ PV_{\text{bond}} \approx 1066.77 \] Therefore, the fair market value of the bond is approximately **$1066.77**. |
1111.97 |
| llama3.1-70B-Instruct | To find the fair market value of the bond, we need to calculate the present value of the bond's future cash flows, which include the periodic interest payments and the return of the principal at maturity. Here are the given parameters: * Face value of the bond: $1000 * Annual interest rate: 4% (compounded semiannually, so the semiannual rate is 2%) * Number of periods: 30 (15 years \* 2 periods per year) * Periodic interest payment: $25 We can use the formula for the present value of an annuity to calculate the present value of the interest payments: PV = PMT x [(1 - (1 + r)^(-n)) / r] where: PV = present value PMT = periodic interest payment = $25 r = semiannual interest rate = 2% = 0.02 n = number of periods = 30 PV = $25 x [(1 - (1 + 0.02)^(-30)) / 0.02] PV ≈ $431.19 This is the present value of the interest payments. To find the fair market value of the bond, we need to add the present value of the return of the principal at maturity: FV = $1000 / (1 + 0.02)^30 FV ≈ $548.81 The fair market value of the bond is the sum of the present value of the interest payments and the present value of the return of the principal: Fair Market Value = PV + FV = $431.19 + $548.81 ≈ $980.00 Therefore, the fair market value of the bond is approximately $980.00. |
1111.97 |
| llama3.1-405B-Instruct | To calculate the fair market value of the bond, we need to calculate the present value of the bond's cash flows using the current market interest rate. The bond pays $25 every six months for 15 years, which is a total of 30 payments. The present value of these payments can be calculated using the formula for the present value of an annuity: PV = PMT x [(1 - (1 + r)^(-n)) / r] where: PV = present value PMT = periodic payment = $25 r = current market interest rate = 4%/2 = 2% (since the payments are made every six months, we need to use the semi-annual interest rate) n = number of payments = 30 Plugging in the values, we get: PV = $25 x [(1 - (1 + 0.02)^(-30)) / 0.02] PV = $25 x 24.2953 PV = $607.38 This is the present value of the bond's coupon payments. In addition to the coupon payments, the bond also pays the face value of $1000 at maturity. We need to calculate the present value of this payment as well: PV = FV / (1 + r)^n where: PV = present value FV = face value = $1000 r = current market interest rate = 4%/2 = 2% (since the payment is made at the end of 15 years, we need to use the semi-annual interest rate) n = number of periods = 30 Plugging in the values, we get: PV = $1000 / (1 + 0.02)^30 PV = $1000 / 1.8119 PV = $551.61 This is the present value of the bond's face value. The fair market value of the bond is the sum of the present values of the coupon payments and the face value: FMV = PV of coupon payments + PV of face value FMV = $607.38 + $551.61 FMV = $1158.99 Therefore, the fair market value of the bond is $1158.99. |
1111.97 |
| DeepSeek-Coder-V2 | To determine the fair market value of the bond, we need to calculate the present value of all future cash flows, which include the semi-annual coupon payments and the face value of the bond at maturity. The current market interest rate is 4%, which we will use as the discount rate. Here are the steps to solve the problem: 1. **Identify the cash flows:** - The bond pays $25 every six months for 15 years. - The face value of the bond is $1000, which will be paid at the end of 15 years. 2. **Determine the number of periods:** - Since the bond pays every six months, there are 15 years * 2 = 30 periods. 3. **Calculate the semi-annual interest rate:** - The annual market interest rate is 4%. Therefore, the semi-annual interest rate is 4% / 2 = 2%. 4. **Calculate the present value of the semi-annual coupon payments:** - The formula for the present value of an annuity is: \[ PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] where \( C \) is the coupon payment, \( r \) is the semi-annual interest rate, and \( n \) is the number of periods. - Plugging in the values: \[ PV = 25 \times \left( \frac{1 - (1 + 0.02)^{-30}}{0.02} \right) \] - Calculate the denominator first: \[ (1 + 0.02)^{-30} \approx 0.55207 \] \[ 1 - 0.55207 = 0.44793 \] \[ \frac{0.44793}{0.02} = 22.3965 \] - Now calculate the present value of the annuity: \[ PV = 25 \times 22.3965 \approx 559.91 \] 5. **Calculate the present value of the face value:** - The formula for the present value of a single future amount is: \[ PV = FV \times (1 + r)^{-n} \] where \( FV \) is the face value, \( r \) is the semi-annual interest rate, and \( n \) is the number of periods. - Plugging in the values: \[ PV = 1000 \times (1 + 0.02)^{-30} \] - Calculate the present value: \[ PV = 1000 \times 0.55207 \approx 552.07 \] 6. **Add the present values of the coupon payments and the face value:** - The total fair market value of the bond is: \[ \text{Fair Market Value} = 559.91 + 552.07 = 1111.98 \] Therefore, the fair market value of the bond is approximately $1111.98. |
1111.97 |
| Qwen2-72B-Instruct | To find the fair market value of the bond, we need to calculate the present value of the bond's future cash flows. The bond pays $25 every six months for 15 years, or 30 periods, and has a face value of $1000 that will be paid at maturity. The market interest rate is 4% per year, or 2% per six-month period. First, we calculate the present value of the coupon payments. The present value of an annuity formula is: PV = PMT * [1 - (1 + r)^-n] / r where PV is the present value, PMT is the payment amount, r is the interest rate per period, and n is the number of periods. In this case, PMT = $25, r = 0.02, and n = 30. Plugging in these values, we get: PV = 25 * [1 - (1 + 0.02)^-30] / 0.02 PV = 25 * [1 - (1.02)^-30] / 0.02 PV = 25 * [1 - 0.5117] / 0.02 PV = 25 * 0.4883 / 0.02 PV = 25 * 24.415 PV = 610.38 Next, we calculate the present value of the face value of the bond. The present value of a lump sum formula is: PV = FV / (1 + r)^n where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods. In this case, FV = $1000, r = 0.02, and n = 30. Plugging in these values, we get: PV = 1000 / (1 + 0.02)^30 PV = 1000 / 1.02^30 PV = 1000 / 1.9602 PV = 510.11 Finally, we add the present value of the coupon payments to the present value of the face value to get the fair market value of the bond: FMV = PV of coupon payments + PV of face value FMV = 610.38 + 510.11 FMV = 1120.49 Therefore, the fair market value of the bond is $1120.49. |
1111.97 |
| baichuan4 | To find the fair market value of the bond, we need to calculate the present value of all the future payments it will make, including both the semi-annual interest payments and the principal repayment at maturity. First, let's determine the number of payments: - The bond has a term of 15 years, and it pays every six months. So, there will be 15 * 2 = 30 semi-annual payments. Now, let's calculate the present value of each type of payment: 1. Interest payments: - Each interest payment is $25. - The interest rate is 4%, so the semi-annual interest rate is 4% / 2 = 2%. - We'll use the formula for the present value of an annuity to calculate the present value of these payments: PV_annuity = (C / r) * [1 - (1 + r)^-n] where: - C is the payment amount ($25) - r is the interest rate per period (2% expressed as a decimal, i.e., 0.02) - n is the total number of payments (30) 2. Principal repayment: - The principal repayment (face value of the bond) is $1000. - We'll need to discount this back to present value using the same semi-annual interest rate over the full term of the bond (30 periods). PV_principal = FV / (1 + r)^n where: - FV is the future value ($1000) - r and n are the same as above Now, let's calculate these values: PV_annuity = (25 / 0.02) * [1 - (1 + 0.02)^-30] PV_annuity = 1250 * [1 - (1.02)^-30] PV_annuity ≈ 1250 * [1 - 0.412] PV_annuity ≈ 1250 * 0.588 PV_annuity ≈ $735 PV_principal = 1000 / (1 + 0.02)^30 PV_principal ≈ 1000 / (1.02)^30 PV_principal ≈ $412.06 The total present value of the bond is the sum of these two: Total PV = PV_annuity + PV_principal Total PV ≈ $735 + $412.06 Total PV ≈ $1147.06 Therefore, the fair market value of the bond is approximately $1147.06. |
1111.97 |
| doubao | The bond pays $25 every six months, so the annual coupon payment is $25 x 2 = $50. The number of periods is 15 years x 2 = 30 periods. The market interest rate is 4% per year, so the semi-annual interest rate is 4% / 2 = 2%. To find the present value of the bond, we calculate the present value of the coupon payments and the present value of the face value. The present value of the coupon payments can be found using the formula for the present value of an annuity: PV = C * [(1 - (1 + r)^-n) / r], where C is the coupon payment, r is the interest rate, and n is the number of periods. So the present value of the coupon payments is $50 * [(1 - (1 + 0.02)^-30) / 0.02] ≈ $1135.90. The present value of the face value is simply the face value discounted to the present using the interest rate: $1000 / (1 + 0.02)^30 ≈ $552.07. The fair market value of the bond is the sum of the present values of the coupon payments and the face value, so it is approximately $1135.90 + $552.07 = $1687.97. |
1111.97 |
| minimax | To calculate the fair market value of the bond, we need to discount the future cash flows (interest payments and principal repayment) back to the present value using the market interest rate. Since the bond pays $25 every six months for 15 years and the market interest rate is 4% per annum, we need to use the semi-annual interest rate for our calculations. First, let's find the semi-annual interest rate: Market annual interest rate = 4% Semi-annual interest rate = 4% / 2 = 2% Now, we need to calculate the present value of each of the 30 semi-annual interest payments of $25 and the present value of the $1000 principal repayment at the end of the 15 years. The formula for the present value of a single future payment is: PV = FV / (1 + r)^n Where: PV = Present Value FV = Future Value (payment) r = interest rate per period n = number of periods For the interest payments, we will use the above formula for each of the 30 payments: PV(interest) = $25 / (1 + 0.02)^1 + $25 / (1 + 0.02)^2 + ... + $25 / (1 + 0.02)^30 This is a geometric series, and we can use the formula for the present value of an annuity to simplify the calculation: PV(annuity) = C * [1 - (1 + r)^-n] / r Where: C = semi-annual coupon payment ($25) r = semi-annual interest rate (0.02) n = total number of payments (30) PV(annuity) = $25 * [1 - (1 + 0.02)^-30] / 0.02 Now, let's calculate the present value of the principal repayment at the end of the 15 years: PV(principal) = $1000 / (1 + 0.02)^30 Finally, the fair market value of the bond is the sum of the present value of the interest payments and the present value of the principal repayment: Fair Market Value = PV(annuity) + PV(principal) Let's calculate these values: PV(annuity) = $25 * [1 - (1 + 0.02)^-30] / 0.02 PV(annuity) = $25 * [1 - (1.02)^-30] / 0.02 PV(annuity) = $25 * [1 - 1 / (1.02)^30] / 0.02 PV(annuity) = $25 * [1 - 1 / 1.***] / 0.02 PV(annuity) = $25 * [1 - 0.***] / 0.02 PV(annuity) = $25 * 0.*** / 0.02 PV(annuity) = $25 * 22.*** PV(annuity) = $557.90 PV(principal) = $1000 / (1.02)^30 PV(principal) = $1000 / 1.*** PV(principal) = $553.68 Fair Market Value = PV(annuity) + PV(principal) Fair Market Value = $557.90 + $553.68 Fair Market Value = $1111.58 Therefore, the fair market value of the bond is approximately $1111.58. |
1111.97 |
| kimi | To find the fair market value of a bond, you need to calculate the present value of all future cash flows that the bond will generate. A bond's cash flows consist of its coupon payments and its face value at maturity. In this case, the bond pays $25 every six months for 15 years and then pays its face value of $1000 at the end of the 15th year. The formula to calculate the present value of a bond is: \[ PV = \sum \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n} \] Where: - \( PV \) is the present value of the bond. - \( C \) is the coupon payment per period. - \( r \) is the market interest rate per period. - \( t \) is the time period until the next coupon payment. - \( F \) is the face value of the bond. - \( n \) is the total number of periods until maturity. Given: - Coupon payment \( C = $25 \) - Market interest rate \( r = 4\% \) per year, which is \( 0.04 \) per year. Since the bond pays semi-annually, the semi-annual interest rate is \( \frac{0.04}{2} = 0.02 \). - Face value \( F = $1000 \) - Total number of periods \( n = 15 \) years * 2 periods per year = 30 periods Now, let's calculate the present value: \[ PV = \sum_{t=1}^{30} \frac{25}{(1 + 0.02)^t} + \frac{1000}{(1 + 0.02)^{30}} \] We can calculate this using a financial calculator or a spreadsheet. Here's the step-by-step calculation: 1. Calculate the present value of each coupon payment: \[ PV_{coupon} = \frac{25}{(1 + 0.02)^1} + \frac{25}{(1 + 0.02)^2} + \ldots + \frac{25}{(1 + 0.02)^{30}} \] 2. Calculate the present value of the face value of the bond: \[ PV_{face} = \frac{1000}{(1 + 0.02)^{30}} \] 3. Sum the present values of the coupon payments and the face value to get the total present value of the bond. Let's do the calculation: \[ PV_{coupon} = 25 \left( \frac{1}{1.02} + \frac{1}{1.02^2} + \ldots + \frac{1}{1.02^{30}} \right) \] \[ PV_{face} = \frac{1000}{1.02^{30}} \] The sum of the coupon payments is a series of a geometric progression, which can be calculated using the formula for the sum of a geometric series: \[ S_n = a \frac{1 - r^n}{1 - r} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. In this case: \[ a = \frac{25}{1.02} \] \[ r = \frac{1}{1.02} \] \[ n = 30 \] \[ S_{30} = \frac{25}{1.02} \frac{1 - (1/1.02)^{30}}{1 - 1/1.02} \] Now, let's calculate it: \[ S_{30} = \frac{25}{1.02} \frac{1 - (1/1.02)^{30}}{1 - 1/1.02} \] \[ S_{30} = \frac{25}{1.02} \frac{1 - (1/1.02)^{30}}{0.02} \] \[ S_{30} = \frac{25}{0.02} \left(1 - (1/1.02)^{30}\right) \] \[ S_{30} = 1250 \left(1 - (1/1.02)^{30}\right) \] Now calculate \( (1/1.02)^{30} \) and subtract from 1, then multiply by 1250. \[ PV_{face} = \frac{1000}{1.02^{30}} \] Now calculate \( 1.02^{30} |
1111.97 |
| wenxinyiyan | To find the fair market value of the bond, we need to calculate the present value of the bond's future cash flows. The bond pays $25 every six months for 15 years, and then pays back the principal amount of $1000 at maturity. The current market interest rate is 4%, which is an annual rate. However, since the bond pays every six months, we need to use the semi-annual interest rate in our calculations, which is 4% divided by 2, or 2%. The formula for calculating the present value of an annuity (a series of equal payments) is: PV = PMT * [(1 - (1 + r)^(-n)) / r] where: PV = present value PMT = payment per period ($25 in this case) r = interest rate per period (2% or 0.02 in this case) n = total number of payments (15 years * 2 payments per year = 30 payments) And the formula for calculating the present value of the principal amount is: PV_principal = Face_value / ((1 + r)^n) where: PV_principal = present value of the principal Face_value = face value of the bond ($1000 in this case) r = interest rate per period (2% or 0.02 in this case) n = total number of periods (30 in this case) So, we can calculate the fair market value of the bond by adding the present value of the annuity and the present value of the principal amount. Let's calculate: PV_annuity = $25 * [(1 - (1 + 0.02)^(-30)) / 0.02] ≈ $376.89 PV_principal = $1000 / ((1 + 0.02)^30) ≈ $553.68 Fair market value of the bond = PV_annuity + PV_principal ≈ $376.89 + $553.68 = $930.57 Therefore, the fair market value of the bond is approximately $930.57. |
1111.97 |
| zhipu | To determine the fair market value of the bond, we need to calculate the present value of the future cash flows generated by the bond. These cash flows consist of the periodic interest payments (coupons) and the principal repayment at maturity. Given: - Face value (F) of the bond = $1000 - Coupon payment (C) every six months = $25 - Market interest rate per annum = 4% - Bond maturity (n) = 15 years First, we need to adjust the annual market interest rate to a semi-annual rate since the coupons are paid every six months. The semi-annual interest rate is half of the annual rate. Semi-annual interest rate (r) = 4% / 2 = 2% = 0.02 Next, we need to calculate the total number of coupon payments over the life of the bond. Since the bond pays coupons semi-annually for 15 years: Total number of coupon payments (n) = 15 years * 2 = 30 Now, we can calculate the present value of the coupon payments and the present value of the face value repayment at maturity. The formula for the present value of an ordinary annuity (the series of coupon payments) is: \[ PV_{coupons} = C \times \left(1 - (1 + r)^{-n}\right) / r \] The formula for the present value of a lump sum (the face value at maturity) is: \[ PV_{face\ value} = F / (1 + r)^n \] Where: - \( PV_{coupons} \) is the present value of the coupon payments. - \( PV_{face\ value} \) is the present value of the face value. - \( C \) is the coupon payment. - \( r \) is the semi-annual interest rate. - \( n \) is the total number of coupon payments. - \( F \) is the face value of the bond. Let's calculate each part: \[ PV_{coupons} = 25 \times \left(1 - (1 + 0.02)^{-30}\right) / 0.02 \] \[ PV_{face\ value} = 1000 / (1 + 0.02)^{30} \] Now, let's plug in the numbers and calculate: \[ PV_{coupons} = 25 \times \left(1 - (1.02)^{-30}\right) / 0.02 \] \[ PV_{coupons} = 25 \times \left(1 - \frac{1}{(1.02)^{30}}\right) / 0.02 \] \[ PV_{coupons} = 25 \times \left(1 - \frac{1}{1.806111}\right) / 0.02 \] \[ PV_{coupons} = 25 \times (1 - 0.553676) / 0.02 \] \[ PV_{coupons} = 25 \times 0.446324 / 0.02 \] \[ PV_{coupons} = 25 \times 22.3162 \] \[ PV_{coupons} = 557.905 \] \[ PV_{face\ value} = 1000 / (1.02)^{30} \] \[ PV_{face\ value} = 1000 / 1.806111 \] \[ PV_{face\ value} = 553.676 \] Finally, the fair market value of the bond is the sum of the present value of the coupon payments and the present value of the face value: \[ Fair\ Market\ Value = PV_{coupons} + PV_{face\ value} \] \[ Fair\ Market\ Value = 557.905 + 553.676 \] \[ Fair\ Market\ Value = 1111.581 \] Therefore, the fair market value of the bond is approximately $1111.58. |
1111.97 |
| gpto1 | **Solution:** To find the fair market value of the bond, we need to calculate the present value (PV) of all future cash flows from the bond, discounted at the current market interest rate. The bond provides semiannual coupon payments and returns the face value at maturity. **Given:** - **Face Value (Par Value) of the Bond (F)**: $1,000 - **Annual Coupon Payment**: Since the bond pays $25 every six months, the **semiannual coupon payment (C)** is $25. - **Number of Years to Maturity**: 15 years - **Market Interest Rate (Annual)**: 4% **Calculations:** 1. **Determine the Number of Periods and Periodic Interest Rate:** - **Number of Periods (n)**: Since payments are semiannual over 15 years: \[ n = 15 \text{ years} \times 2 \text{ periods per year} = 30 \text{ periods} \] - **Periodic Market Interest Rate (r)**: The annual rate divided by the number of periods per year: \[ r = \frac{4\%}{2} = 2\% = 0.02 \] 2. **Calculate the Present Value of the Annuity (Coupon Payments):** The present value of the coupon payments is calculated using the **Present Value of an Ordinary Annuity formula**: \[ \text{PV}_{\text{coupons}} = C \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \] Plugging in the values: \[ \text{PV}_{\text{coupons}} = \$25 \times \left[ \frac{1 - (1 + 0.02)^{-30}}{0.02} \right] \] **Calculations:** - Compute \((1 + r)^{-n}\): \[ (1 + 0.02)^{-30} = (1.02)^{-30} = \frac{1}{(1.02)^{30}} \] First, calculate \((1.02)^{30}\): \[ (1.02)^{30} \approx e^{30 \times \ln(1.02)} \] - Compute \(\ln(1.02)\): \[ \ln(1.02) \approx 0.019802 \] - Compute the exponent: \[ 30 \times 0.019802 \approx 0.59406 \] - Compute \((1.02)^{30}\): \[ e^{0.59406} \approx 1.8114 \] Therefore: \[ (1.02)^{-30} \approx \frac{1}{1.8114} \approx 0.5521 \] - Compute the numerator: \[ 1 - (1.02)^{-30} = 1 - 0.5521 = 0.4479 \] - Calculate \(\text{PV}_{\text{coupons}}\): \[ \text{PV}_{\text{coupons}} = \$25 \times \left( \frac{0.4479}{0.02} \right) = \$25 \times 22.395 = \$559.88 \] 3. **Calculate the Present Value of the Face Value (Lump Sum at Maturity):** The present value of the face value is calculated using the **Present Value of a Lump Sum formula**: \[ \text{PV}_{\text{face}} = \frac{F}{(1 + r)^n} \] Plugging in the values: \[ \text{PV}_{\text{face}} = \frac{\$1,000}{(1.02)^{30}} \] We've already calculated \((1.02)^{30} \approx 1.8114\), so: \[ \text{PV}_{\text{face}} = \frac{\$1,000}{1.8114} \approx \$552.07 \] 4. **Calculate the Total Present Value (Fair Market Value of the Bond):** \[ \text{PV}_{\text{total}} = \text{PV}_{\text{coupons}} + \text{PV}_{\text{face}} \] \[ \text{PV}_{\text{total}} = \$559.88 + \$552.07 = \$1,111.95 \] **Answer:** The fair market value of the bond is **approximately \$1,111.95**. --- **Explanation:** The bond's fair market value is higher than its face value because the coupon rate (which provides $50 annually) is higher than the current market interest rate of 4%. This means investors are willing to pay more for the bond since it offers better returns compared to the prevailing market rates. |
1111.97 |
| deepseekv3 | To determine the fair market value of the bond, we need to calculate the present value of its future cash flows, which include the semi-annual coupon payments and the face value of the bond at maturity. Here's the step-by-step solution: --- ### **Given:** - **Face value (FV)** = $1000 - **Coupon payment (C)** = $25 every six months - **Market interest rate (YTM)** = 4% per year (or 2% per six-month period) - **Number of periods (N)** = 15 years × 2 = 30 six-month periods --- ### **Step 1: Calculate the present value of the coupon payments** The coupon payments form an annuity. The present value of an annuity is calculated using the formula: \[ PV_{\text{coupons}} = C \times \left[1 - \frac{1}{(1 + r)^N}\right] / r \] Where: - \( C = 25 \) - \( r = 2\% = 0.02 \) - \( N = 30 \) \[ PV_{\text{coupons}} = 25 \times \left[1 - \frac{1}{(1 + 0.02)^{30}}\right] / 0.02 \] \[ PV_{\text{coupons}} = 25 \times \left[1 - \frac{1}{1.81136}\right] / 0.02 \] \[ PV_{\text{coupons}} = 25 \times \left[1 - 0.55207\right] / 0.02 \] \[ PV_{\text{coupons}} = 25 \times 0.44793 / 0.02 \] \[ PV_{\text{coupons}} = 25 \times 22.3965 \] \[ PV_{\text{coupons}} = 559.91 \] --- ### **Step 2: Calculate the present value of the face value** The face value is a lump sum payment at maturity. Its present value is calculated using the formula: \[ PV_{\text{face value}} = FV / (1 + r)^N \] \[ PV_{\text{face value}} = 1000 / (1 + 0.02)^{30} \] \[ PV_{\text{face value}} = 1000 / 1.81136 \] \[ PV_{\text{face value}} = 552.07 \] --- ### **Step 3: Calculate the fair market value of the bond** The fair market value is the sum of the present value of the coupon payments and the present value of the face value: \[ \text{Fair Market Value} = PV_{\text{coupons}} + PV_{\text{face value}} \] \[ \text{Fair Market Value} = 559.91 + 552.07 \] \[ \text{Fair Market Value} = 1111.98 \] --- ### **Final Answer:** The fair market value of the bond is approximately **$1112**. |
1111.97 |