| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | 要计算CheckMate股票的公允价值，我们需要按照以下步骤进行： 1. **计算未来四年的股息**，假设每年的增长率为20%。 2. **计算从第五年开始的终端价值**，假设每年的增长率为8%。 3. **将所有未来的股息折现回现值**，使用预期的回报率15%。 4. **将所有未来股息的现值相加**，得到股票的公允价值。 ### 详细步骤 #### 1. 计算未来四年的股息 我们从2016年的股息$12开始，每年的增长率为20%。 - \( D_1 = D_0 \times (1 + g) = 12 \times (1 + 0.20) = 12 \times 1.20 = \$14.40 \) - \( D_2 = D_1 \times (1 + g) = 14.40 \times (1 + 0.20) = 14.40 \times 1.20 = \$17.28 \) - \( D_3 = D_2 \times (1 + g) = 17.28 \times (1 + 0.20) = 17.28 \times 1.20 = \$20.74 \) - \( D_4 = D_3 \times (1 + g) = 20.74 \times (1 + 0.20) = 20.74 \times 1.20 = \$24.89 \) #### 2. 计算从第五年开始的终端价值 从第五年开始，股息增长率将稳定在8%。我们可以使用股息增长模型（Gordon Growth Model）来计算终端价值： - \( D_5 = D_4 \times (1 + g) = 24.89 \times (1 + 0.08) = 24.89 \times 1.08 = \$26.88 \) 终端价值（\( P_4 \)）为第五年及以后的股息的现值： - \( P_4 = \frac{D_5}{r - g} = \frac{26.88}{0.15 - 0.08} = \frac{26.88}{0.07} = \$384.00 \) #### 3. 将所有未来的股息折现回现值 使用预期的回报率15%来折现未来的股息： - \( PV(D_1) = \frac{14.40}{(1 + 0.15)^1} = \frac{14.40}{1.15} = \$12.52 \) - \( PV(D_2) = \frac{17.28}{(1 + 0.15)^2} = \frac{17.28}{1.3225} = \$13.06 \) - \( PV(D_3) = \frac{20.74}{(1 + 0.15)^3} = \frac{20.74}{1.520875} = \$13.64 \) - \( PV(D_4) = \frac{24.89}{(1 + 0.15)^4} = \frac{24.89}{1.74900625} = \$14.23 \) 将终端价值折现回现值： - \( PV(P_4) = \frac{384.00}{(1 + 0.15)^4} = \frac{384.00}{1.74900625} = \$219.57 \) #### 4. 将所有未来股息的现值相加 股票的公允价值为所有未来股息现值的总和： - \( PV_{total} = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(P_4) \) - \( PV_{total} = 12.52 + 13.06 + 13.64 + 14.23 + 219.57 = \$273.02 \) ### 结论 CheckMate股票的公允价值为**\$273.02**。 |
273.0 |
| deepseekr1 | To determine the fair value of CheckMate's stock, we use a two-stage dividend discount model. Here's the step-by-step solution: 1. **Calculate Dividends for the High-Growth Phase (Years 1-4):** - **D₁ (2017):** $12 × 1.20 = $14.40 - **D₂ (2018):** $14.40 × 1.20 = $17.28 - **D₃ (2019):** $17.28 × 1.20 = $20.736 - **D₄ (2020):** $20.736 × 1.20 = $24.8832 2. **Calculate the Terminal Value (TV) at Year 4:** - **D₅ (2021):** $24.8832 × 1.08 = $26.873856 - **Terminal Value (TV₄):** \[ TV_4 = \frac{D_5}{r - g} = \frac{26.873856}{0.15 - 0.08} = \frac{26.873856}{0.07} \approx 383.9122 \] 3. **Discount Dividends and Terminal Value to Present Value (r = 15%):** - **PV of D₁:** \(\frac{14.40}{1.15^1} \approx 12.52\) - **PV of D₂:** \(\frac{17.28}{1.15^2} \approx 13.07\) - **PV of D₃:** \(\frac{20.736}{1.15^3} \approx 13.64\) - **PV of D₄:** \(\frac{24.8832}{1.15^4} \approx 14.23\) - **PV of TV₄:** \(\frac{383.9122}{1.15^4} \approx 219.50\) 4. **Sum All Present Values:** \[ 12.52 + 13.07 + 13.64 + 14.23 + 219.50 = \boxed{272.96} \] **Answer:** The fair value of the stock is **\$272.96**. |
273.0 |
| gpto1 | To determine the fair value of CheckMate's stock today, we'll use the Dividend Discount Model (DDM), which discounts expected future dividends to their present value. Since the dividends are expected to grow at a non-constant rate initially and then at a constant rate thereafter, we'll proceed in two stages: **Stage 1: Calculate Dividends During the Super-Normal Growth Period (Years 1-4)** The dividends grow at 20% annually for the next four years. 1. **Dividend at Year 1 (D₁):** \[ D₁ = D₀ \times (1 + g_1) = \$12 \times 1.20 = \$14.40 \] 2. **Dividend at Year 2 (D₂):** \[ D₂ = D₁ \times (1 + g_1) = \$14.40 \times 1.20 = \$17.28 \] 3. **Dividend at Year 3 (D₃):** \[ D₃ = D₂ \times (1 + g_1) = \$17.28 \times 1.20 = \$20.736 \] 4. **Dividend at Year 4 (D₄):** \[ D₄ = D₃ \times (1 + g_1) = \$20.736 \times 1.20 = \$24.8832 \] **Stage 2: Calculate the Terminal Value at Year 4** After Year 4, dividends grow perpetually at 8%. The terminal value at Year 4 represents the present value of all dividends from Year 5 onward, discounted back to Year 4. 1. **Dividend at Year 5 (D₅):** \[ D₅ = D₄ \times (1 + g_2) = \$24.8832 \times 1.08 = \$26.873856 \] 2. **Terminal Value at Year 4 (TV₄):** \[ TV₄ = \frac{D₅}{k - g_2} = \frac{\$26.873856}{0.15 - 0.08} = \frac{\$26.873856}{0.07} = \$383.91223 \] **Stage 3: Discount All Cash Flows Back to Present Value** We discount each dividend and the terminal value back to present value using the required rate of return (k = 15%). 1. **Present Value Factors:** \[ (1 + k)^t \text{ for } t = 1 \text{ to } 4: \] \[ (1.15)^1 = 1.15, \quad (1.15)^2 = 1.3225, \quad (1.15)^3 = 1.520875, \quad (1.15)^4 = 1.74900625 \] 2. **Present Value of Dividends:** - **PV of D₁:** \[ PV(D₁) = \frac{\$14.40}{1.15} = \$12.5217 \] - **PV of D₂:** \[ PV(D₂) = \frac{\$17.28}{1.3225} = \$13.0649 \] - **PV of D₃:** \[ PV(D₃) = \frac{\$20.736}{1.520875} = \$13.6347 \] - **PV of D₄:** \[ PV(D₄) = \frac{\$24.8832}{1.74900625} = \$14.2300 \] 3. **Present Value of Terminal Value:** \[ PV(TV₄) = \frac{\$383.91223}{1.74900625} = \$219.4675 \] **Stage 4: Sum All Present Values to Find the Fair Value** \[ \text{Fair Value} = PV(D₁) + PV(D₂) + PV(D₃) + PV(D₄) + PV(TV₄) \] \[ \text{Fair Value} = \$12.5217 + \$13.0649 + \$13.6347 + \$14.2300 + \$219.4675 = \$272.9188 \] **Answer:** The fair value of the stock now is approximately **$272.92** per share. **Answer:** $272.92 |
273.0 |
| GPT-4o | To determine the fair value of the stock now, given the growth rates of the dividends and the required rate of return, we need to calculate the present value of the expected dividends during the high growth phase and the present value of the dividends during the constant growth phase (also known as the terminal value). Here's a step-by-step solution: ### Step 1: Calculate the Dividends for the Next Four Years The dividend is expected to grow at 20% per year for the next four years. Starting with the current dividend (\(D_0\)) of $12, we can calculate the dividends for the next four years (\(D_1\), \(D_2\), \(D_3\), and \(D_4\)): \[ D_1 = D_0 \times (1 + g_1) = 12 \times 1.20 = 14.40 \] \[ D_2 = D_1 \times (1 + g_1) = 14.40 \times 1.20 = 17.28 \] \[ D_3 = D_2 \times (1 + g_1) = 17.28 \times 1.20 = 20.74 \] \[ D_4 = D_3 \times (1 + g_1) = 20.74 \times 1.20 = 24.89 \] ### Step 2: Calculate the Terminal Value at the End of Year Four After the high growth phase, the dividend will grow at a constant rate of 8% forever. The terminal value (\(TV\)) at the end of year four can be calculated using the Gordon Growth Model: \[ TV = \frac{D_5}{r - g_2} \] where \(D_5\) is the dividend at the beginning of the constant growth phase: \[ D_5 = D_4 \times (1 + g_2) = 24.89 \times 1.08 = 26.88 \] Now, calculate the terminal value: \[ TV = \frac{26.88}{0.15 - 0.08} = \frac{26.88}{0.07} = 384 \] ### Step 3: Calculate the Present Value of the Expected Dividends and Terminal Value The present value (PV) of the dividends and the terminal value needs to be discounted back to the present value using the required rate of return (15%). \[ PV(D_1) = \frac{14.40}{(1 + 0.15)^1} = \frac{14.40}{1.15} = 12.52 \] \[ PV(D_2) = \frac{17.28}{(1 + 0.15)^2} = \frac{17.28}{1.3225} = 13.06 \] \[ PV(D_3) = \frac{20.74}{(1 + 0.15)^3} = \frac{20.74}{1.5209} = 13.64 \] \[ PV(D_4) = \frac{24.89}{(1 + 0.15)^4} = \frac{24.89}{1.7490} = 14.23 \] \[ PV(TV) = \frac{384}{(1 + 0.15)^4} = \frac{384}{1.7490} = 219.57 \] ### Step 4: Sum the Present Values The fair value of the stock now is the sum of the present values of the expected dividends and the terminal value: \[ \text{Fair Value} = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(TV) \] \[ \text{Fair Value} = 12.52 + 13.06 + 13.64 + 14.23 + 219.57 \] \[ \text{Fair Value} = 273.02 \] Therefore, the fair value of the stock now is approximately $273.02. |
273.0 |
| llama3.1-70B-Instruct | To find the fair value of the stock, we need to calculate the present value of the expected future dividends. We'll break this down into two parts: the present value of the dividends during the high-growth period (next 4 years) and the present value of the dividends during the constant-growth period (after 4 years). **High-Growth Period (2017-2020)** | Year | Dividend | Growth Rate | Present Value | | --- | --- | --- | --- | | 2017 | $12 x 1.20 = $14.40 | 20% | $14.40 / (1 + 0.15)^1 = $12.48 | | 2018 | $14.40 x 1.20 = $17.28 | 20% | $17.28 / (1 + 0.15)^2 = $13.41 | | 2019 | $17.28 x 1.20 = $20.74 | 20% | $20.74 / (1 + 0.15)^3 = $14.51 | | 2020 | $20.74 x 1.20 = $24.89 | 20% | $24.89 / (1 + 0.15)^4 = $15.83 | **Constant-Growth Period (2021 and beyond)** We'll use the Gordon Growth Model to calculate the present value of the dividends during the constant-growth period: PV = D / (r - g) where: PV = present value D = dividend in 2021 = $24.89 x 1.08 = $26.89 r = expected rate of return = 15% g = constant growth rate = 8% PV = $26.89 / (0.15 - 0.08) = $26.89 / 0.07 = $384.14 **Fair Value of the Stock** The fair value of the stock is the sum of the present values of the dividends during the high-growth period and the constant-growth period: Fair Value = $12.48 + $13.41 + $14.51 + $15.83 + $384.14 = $440.37 Therefore, the fair value of the stock is approximately $440.37. |
273.0 |
| llama3.1-405B-Instruct | To calculate the fair value of the stock, we need to calculate the present value of the expected dividends. We can do this using the dividend discount model (DDM). First, let's calculate the expected dividends for the next four years: Year 1 (2017): $12 x 1.20 = $14.40 Year 2 (2018): $14.40 x 1.20 = $17.28 Year 3 (2019): $17.28 x 1.20 = $20.74 Year 4 (2020): $20.74 x 1.20 = $24.89 After the fourth year, the dividend growth rate is expected to settle down at a constant 8% forever. We can calculate the present value of the expected dividends from year 5 onwards using the formula for the present value of a growing perpetuity: PV = D / (r - g) where: PV = present value D = dividend in year 5 r = expected rate of return (15%) g = growth rate (8%) First, we need to calculate the dividend in year 5: Year 5 (2021): $24.89 x 1.08 = $26.89 Now, we can calculate the present value of the expected dividends from year 5 onwards: PV = $26.89 / (0.15 - 0.08) PV = $26.89 / 0.07 PV = $384.14 However, this is the present value of the expected dividends from year 5 onwards, starting from year 5. We need to discount this value back to the present (year 2016) using the expected rate of return: PV (in 2016) = $384.14 / (1 + 0.15)^4 PV (in 2016) = $384.14 / 1.749 PV (in 2016) = $219.71 Now, we can calculate the present value of the expected dividends for the first four years: PV (2017) = $14.40 / (1 + 0.15) PV (2017) = $12.52 PV (2018) = $17.28 / (1 + 0.15)^2 PV (2018) = $13.33 PV (2019) = $20.74 / (1 + 0.15)^3 PV (2019) = $14.23 PV (2020) = $24.89 / (1 + 0.15)^4 PV (2020) = $15.23 The fair value of the stock is the sum of the present values of the expected dividends: Fair value = $12.52 + $13.33 + $14.23 + $15.23 + $219.71 Fair value = $274.02 Therefore, the fair value of the stock is $274.02. |
273.0 |
| DeepSeek-Coder-V2 | To determine the fair value of the stock, we need to calculate the present value of the dividends over two periods: the first four years with a 20% growth rate and the subsequent years with an 8% growth rate. ### Step 1: Calculate the dividends for the first four years The dividend for the current year (2016) is $12. The growth rate for the next four years is 20%. - **Year 1 (2017):** \( 12 \times 1.20 = 14.40 \) - **Year 2 (2018):** \( 14.40 \times 1.20 = 17.28 \) - **Year 3 (2019):** \( 17.28 \times 1.20 = 20.74 \) - **Year 4 (2020):** \( 20.74 \times 1.20 = 24.88 \) ### Step 2: Calculate the present value of these dividends The expected rate of return is 15%. - **Year 1 (2017):** \( \frac{14.40}{(1 + 0.15)^1} = \frac{14.40}{1.15} \approx 12.52 \) - **Year 2 (2018):** \( \frac{17.28}{(1 + 0.15)^2} = \frac{17.28}{1.3225} \approx 13.07 \) - **Year 3 (2019):** \( \frac{20.74}{(1 + 0.15)^3} = \frac{20.74}{1.520875} \approx 13.64 \) - **Year 4 (2020):** \( \frac{24.88}{(1 + 0.15)^4} = \frac{24.88}{1.74900625} \approx 14.23 \) Sum of present values of dividends for the first four years: \[ 12.52 + 13.07 + 13.64 + 14.23 = 53.46 \] ### Step 3: Calculate the present value of the dividends from Year 5 onwards From Year 5 onwards, the dividend grows at a constant 8% forever. We use the Gordon Growth Model to find the present value of this perpetuity. The dividend in Year 5 (2021) will be: \[ 24.88 \times 1.08 = 26.87 \] Using the Gordon Growth Model: \[ \text{Present value of perpetuity} = \frac{D_5}{r - g} = \frac{26.87}{0.15 - 0.08} = \frac{26.87}{0.07} \approx 383.86 \] ### Step 4: Calculate the present value of this perpetuity as of Year 4 \[ \frac{383.86}{(1 + 0.15)^4} = \frac{383.86}{1.74900625} \approx 197.47 \] ### Step 5: Sum the present values The fair value of the stock now is the sum of the present values of the dividends for the first four years and the present value of the perpetuity: \[ 53.46 + 197.47 = 250.93 \] Therefore, the fair value of the stock now is: \[ \boxed{250.93} \] |
273.0 |
| Qwen2-72B-Instruct | To find the fair value of the stock, we will use the two-stage dividend discount model (DDM). This model calculates the present value of future dividends, considering the initial high growth period and the subsequent constant growth period. First, let's calculate the dividends for the next four years, assuming a 20% growth rate: 1. Dividend for 2017 = $12 * (1 + 20%) = $14.40 2. Dividend for 2018 = $14.40 * (1 + 20%) = $17.28 3. Dividend for 2019 = $17.28 * (1 + 20%) = $20.74 4. Dividend for 2020 = $20.74 * (1 + 20%) = $24.89 Next, we will discount these dividends back to the present value using the expected rate of return of 15%: 1. Present value of dividend for 2017 = $14.40 / (1 + 15%)^1 = $12.52 2. Present value of dividend for 2018 = $17.28 / (1 + 15%)^2 = $12.85 3. Present value of dividend for 2019 = $20.74 / (1 + 15%)^3 = $12.88 4. Present value of dividend for 2020 = $24.89 / (1 + 15%)^4 = $13.66 After the four years of high growth, the dividend will grow at a constant 8% rate. To calculate the present value of the perpetuity, we need to find the dividend for 2021 and then discount it back to the present. The dividend for 2021, assuming an 8% growth rate, is: Dividend for 2021 = $24.89 * (1 + 8%) = $26.93 The present value of the perpetuity can be calculated using the formula: PV = D / (r - g) Where D is the dividend for 2021, r is the expected rate of return (15%), and g is the constant growth rate (8%). PV = $26.93 / (15% - 8%) = $384.71 However, this present value is for the end of the high growth period, so we need to discount it back to the present: Present value of perpetuity = $384.71 / (1 + 15%)^4 = $212.39 Finally, we add the present values of the dividends for the high growth period and the present value of the perpetuity to find the fair value of the stock: Fair value of the stock = $12.52 + $12.85 + $12.88 + $13.66 + $212.39 = $264.30 Therefore, the fair value of the stock is $264.30. |
273.0 |
| baichuan4 | To calculate the fair value of the stock now, we can use the dividend discount model (DDM), which takes into account the expected growth rate of dividends and the required rate of return. First, let's calculate the dividend for each of the next four years, assuming a 20% growth rate: - Year 1 (2017): $12 * (1 + 20%) = $14.40 - Year 2 (2018): $14.40 * (1 + 20%) = $17.28 - Year 3 (2019): $17.28 * (1 + 20%) = $20.736 - Year 4 (2020): $20.736 * (1 + 20%) = $24.8832 After the fourth year, the dividend is expected to grow at a constant rate of 8% forever. To find the present value of these future dividends, we need to discount them back to the present using the expected rate of return of 15%. We'll calculate the present value of each dividend and then sum them up: - Year 1: $14.40 / (1 + 15%) = $12.5217 - Year 2: $17.28 / (1 + 15%)^2 = $13.1656 - Year 3: $20.736 / (1 + 15%)^3 = $13.8366 - Year 4: $24.8832 / (1 + 15%)^4 = $14.5356 Now, let's calculate the present value of the perpetual dividend starting from Year 5. To do this, we'll first need to find the Year 5 dividend, which will be the Year 4 dividend growing at 8%: - Year 5 dividend: $24.8832 * (1 + 8%) = $26.8737 The present value of the perpetual dividend can be calculated as follows: PV_perpetual = D5 / (r - g) Where: - D5 is the dividend in Year 5 - r is the required rate of return (15%) - g is the constant growth rate (8%) PV_perpetual = $26.8737 / (0.15 - 0.08) = $26.8737 / 0.07 = $383.91 Now, we need to discount this perpetual dividend back to the present: PV_perpetual_now = PV_perpetual / (1 + 15%)^4 = $383.91 / (1 + 15%)^4 = $238.77 Finally, we sum up the present values of the first four years' dividends and the present value of the perpetual dividend to get the total fair value of the stock now: Fair Value = $12.5217 + $13.1656 + $13.8366 + $14.5356 + $238.77 = $293.83 Therefore, the fair value of the stock now is approximately $293.83. |
273.0 |
| doubao | For the first four years: Year 1 dividend = $12 × (1 + 20%) = $12 × 1.2 = $14.4 Year 2 dividend = $14.4 × (1 + 20%) = $14.4 × 1.2 = $17.28 Year 3 dividend = $17.28 × (1 + 20%) = $17.28 × 1.2 = $20.736 Year 4 dividend = $20.736 × (1 + 20%) = $20.736 × 1.2 = $24.8832 The present value of these dividends is: PV1 = $14.4 / (1 + 15%) = $12.52 PV2 = $17.28 / (1 + 15%)² = $13.33 PV3 = $20.736 / (1 + 15%)³ = $14.29 PV4 = $24.8832 / (1 + 15%)⁴ = $15.37 After the fourth year, it grows at a constant 8%. We can use the Gordon growth model to find the present value of the perpetual dividends starting from year 5. The dividend in year 5 = $24.8832 × (1 + 8%) = $26.873856 The terminal value at year 4 = $26.873856 / (15% - 8%) = $383.9122 The present value of the terminal value = $383.9122 / (1 + 15%)⁴ = $202.54 The fair value of the stock now is the sum of the present values of the dividends and the terminal value: Fair value = $12.52 + $13.33 + $14.29 + $15.37 + $202.54 = $258.05. |
273.0 |
| minimax | To calculate the fair value of the stock, we can use the Dividend Discount Model (DDM), which in this case requires us to consider two stages of growth: a supernormal growth phase and a terminal growth phase. The first step is to calculate the dividends for the first four years when the growth rate is 20%. We'll then calculate the terminal value at the end of year 4, which represents the present value of all future dividends growing at a constant rate of 8% forever. Finally, we'll discount all these future dividends and the terminal value back to the present value using the expected rate of return of 15%. Let's calculate the dividends for the first four years: Year 1: $D_1 = D_0 \times (1 + g_1) = $12 \times (1 + 0.20) = $12 \times 1.20 = $14.40 Year 2: $D_2 = D_1 \times (1 + g_1) = $14.40 \times 1.20 = $17.28 Year 3: $D_3 = D_2 \times (1 + g_1) = $17.28 \times 1.20 = $20.74 Year 4: $D_4 = D_3 \times (1 + g_1) = $20.74 \times 1.20 = $24.88 Next, we calculate the terminal value at the end of year 4, which is the present value of a perpetuity growing at a constant rate of 8% forever, starting from year 5: Terminal value (TV) = $D_5 / (r - g_2) = $D_4 \times (1 + g_2) / (r - g_2) Where: - $D_5 is the dividend in year 5, which is $D_4 \times (1 + g_2)$ - $r$ is the required rate of return (15% or 0.15) - $g_2$ is the constant growth rate after year 4 (8% or 0.08) So, we have: $D_5 = $24.88 \times (1 + 0.08) = $24.88 \times 1.08 = $26.87 Terminal value (TV) = $26.87 / (0.15 - 0.08) = $26.87 / 0.07 = $383.86 Now, we discount all future dividends and the terminal value back to the present value using the required rate of return: PV of Year 1 dividend: $PV_1 = $14.40 / (1 + 0.15)^1 PV of Year 2 dividend: $PV_2 = $17.28 / (1 + 0.15)^2 PV of Year 3 dividend: $PV_3 = $20.74 / (1 + 0.15)^3 PV of Year 4 dividend: $PV_4 = $24.88 / (1 + 0.15)^4 PV of Terminal value: $PV_{TV} = $383.86 / (1 + 0.15)^4 Let's calculate these present values: $PV_1 = $14.40 / 1.15 = $12.52 $PV_2 = $17.28 / (1.15)^2 = $17.28 / 1.3225 = $13.07 $PV_3 = $20.74 / (1.15)^3 = $20.74 / 1.520875 = $13.64 $PV_4 = $24.88 / (1.15)^4 = $24.88 / 1.*** = $14.23 $PV_{TV} = $383.86 / (1.15)^4 = $383.86 / 1.*** = $219.39 Finally, we sum up all the present values to get the fair value of the stock: Fair value = $PV_1 + $PV_2 + $PV_3 + $PV_4 + $PV_{TV} Fair value = $12.52 + $13.07 + $13.64 + $14.23 + $219.39 Fair value = $272.85 Therefore, the fair value of the stock now, given the provided information, is approximately $272.85. |
273.0 |
| kimi | To find the fair value of a stock based on its expected dividends, we can use the Gordon Growth Model for the growing period and then the Gordon Growth Model for the stable period. The formula for the fair value of a stock is: \[ P_0 = \frac{D_1}{r - g} + \frac{D_5}{(r - g) \times (1 + r)^4} \times \frac{1}{(1 + g)} \] Where: - \( P_0 \) is the current stock price (the fair value we want to find). - \( D_1 \) is the dividend in the first year. - \( D_5 \) is the dividend in the fifth year. - \( r \) is the required rate of return (discount rate). - \( g \) is the growth rate. Since the dividend grows at 20% per year for the next four years, we can calculate the dividend in the fifth year (D5) as follows: \[ D_5 = D_0 \times (1 + g_1)^4 \] Where: - \( D_0 \) is the current dividend ($12). - \( g_1 \) is the growth rate for the first four years (20%). \[ D_5 = 12 \times (1 + 0.20)^4 \] \[ D_5 = 12 \times 1.20^4 \] \[ D_5 = 12 \times 2.0736 \] \[ D_5 = 24.8832 \] After the fourth year, the dividend will grow at a constant rate of 8% forever. So, we can use the Gordon Growth Model to find the present value of the dividend stream after the fourth year: \[ P_4 = \frac{D_5 \times (1 + g_2)}{r - g_2} \] Where: - \( g_2 \) is the constant growth rate after the fourth year (8%). \[ P_4 = \frac{24.8832 \times (1 + 0.08)}{0.15 - 0.08} \] \[ P_4 = \frac{26.798816}{0.07} \] \[ P_4 = 382.84 \] Now, we need to discount \( P_4 \) back to the present value to find the value of the stock now: \[ P_0 = \frac{D_1}{r} + \frac{P_4}{(1 + r)^4} \] \[ P_0 = \frac{12}{0.15} + \frac{382.84}{(1 + 0.15)^4} \] \[ P_0 = 80 + \frac{382.84}{2.07893} \] \[ P_0 = 80 + 183.92 \] \[ P_0 = 263.92 \] So, the fair value of the stock now is approximately $263.92. |
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| wenxinyiyan | To find the fair value of the stock now, we will use the dividend discount model. This model calculates the present value of future dividends to determine the fair value of a stock. First, we forecast the dividends for the next four years, during which the dividend is expected to grow at 20% per year. Current dividend (2016) = $12 Year 1 (2017) dividend = $12 * (1 + 20%) = $14.40 Year 2 (2018) dividend = $14.40 * (1 + 20%) = $17.28 Year 3 (2019) dividend = $17.28 * (1 + 20%) = $20.74 Year 4 (2020) dividend = $20.74 * (1 + 20%) = $24.88 After Year 4, the dividend growth rate settles down to a constant 8% per year. To calculate the terminal value of the stock, we use the Gordon Growth Model. Terminal value (TV) = [D4 * (1 + growth rate)] / (required rate of return - growth rate) TV = [$24.88 * (1 + 8%)] / (15% - 8%) TV = $26.87 / 0.07 TV = $383.89 Now, we discount the terminal value back to present value (PV). PV of TV = TV / (1 + required rate of return)^4 PV of TV = $383.89 / (1 + 15%)^4 PV of TV = $383.89 / 1.749 PV of TV = $219.49 Finally, we add up the present values of the dividends for the next four years and the present value of the terminal value to get the fair value of the stock. Fair value of stock = PV of D1 + PV of D2 + PV of D3 + PV of D4 + PV of TV Fair value of stock = $14.40 / 1.15 + $17.28 / (1.15^2) + $20.74 / (1.15^3) + $24.88 / (1.15^4) + $219.49 Fair value of stock = $12.52 + $13.05 + $13.63 + $14.27 + $219.49 Fair value of stock = $272.96 Therefore, the fair value of the stock now is $272.96. |
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| zhipu | To calculate the fair value of the stock, we can use the Dividend Discount Model (DDM), which takes into account the present value of all future dividends. Given that the dividend growth rate changes after a certain period, we will use a two-stage DDM. Here's the step-by-step solution: ### Step 1: Calculate the dividends for the next four years Given: - Current dividend (D0) = $12 - Growth rate for the first four years (g1) = 20% or 0.20 - Growth rate after four years (g2) = 8% or 0.08 - Required rate of return (r) = 15% or 0.15 Dividends for the next four years: - D1 = D0 * (1 + g1) = $12 * (1 + 0.20) = $12 * 1.20 = $14.40 - D2 = D1 * (1 + g1) = $14.40 * (1 + 0.20) = $14.40 * 1.20 = $17.28 - D3 = D2 * (1 + g1) = $17.28 * (1 + 0.20) = $17.28 * 1.20 = $20.74 - D4 = D3 * (1 + g1) = $20.74 * (1 + 0.20) = $20.74 * 1.20 = $24.88 ### Step 2: Calculate the terminal value at the end of the fourth year The terminal value (P4) is the present value of all future dividends beyond the fourth year, assuming a constant growth rate. \[ P4 = \frac{D4 \times (1 + g2)}{r - g2} \] \[ P4 = \frac{24.88 \times (1 + 0.08)}{0.15 - 0.08} \] \[ P4 = \frac{24.88 \times 1.08}{0.07} \] \[ P4 = \frac{26.83}{0.07} \] \[ P4 = 383.29 \] ### Step 3: Calculate the present value of all future dividends and the terminal value The present value of each dividend and the terminal value must be discounted back to the present. \[ PV = \frac{D1}{(1 + r)^1} + \frac{D2}{(1 + r)^2} + \frac{D3}{(1 + r)^3} + \frac{D4 + P4}{(1 + r)^4} \] \[ PV = \frac{14.40}{(1 + 0.15)^1} + \frac{17.28}{(1 + 0.15)^2} + \frac{20.74}{(1 + 0.15)^3} + \frac{24.88 + 383.29}{(1 + 0.15)^4} \] \[ PV = \frac{14.40}{1.15} + \frac{17.28}{1.15^2} + \frac{20.74}{1.15^3} + \frac{408.17}{1.15^4} \] \[ PV = \frac{14.40}{1.15} + \frac{17.28}{1.3225} + \frac{20.74}{1.520875} + \frac{408.17}{1.74900625} \] \[ PV = 12.52 + 13.04 + 13.57 + 233.36 \] \[ PV = 272.49 \] ### Conclusion The fair value of the stock now is approximately $272.49. This calculation assumes that the dividends are paid at the end of each year and that the required rate of return remains constant. Adjustments may be needed for different dividend payment schedules or changing required rates of return. |
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| gpto1 | To determine the fair value of CheckMate's stock today, we'll use the Dividend Discount Model (DDM), which discounts expected future dividends to their present value. Since the dividends are expected to grow at a non-constant rate initially and then at a constant rate thereafter, we'll proceed in two stages: **Stage 1: Calculate Dividends During the Super-Normal Growth Period (Years 1-4)** The dividends grow at 20% annually for the next four years. 1. **Dividend at Year 1 (D₁):** \[ D₁ = D₀ \times (1 + g_1) = \$12 \times 1.20 = \$14.40 \] 2. **Dividend at Year 2 (D₂):** \[ D₂ = D₁ \times (1 + g_1) = \$14.40 \times 1.20 = \$17.28 \] 3. **Dividend at Year 3 (D₃):** \[ D₃ = D₂ \times (1 + g_1) = \$17.28 \times 1.20 = \$20.736 \] 4. **Dividend at Year 4 (D₄):** \[ D₄ = D₃ \times (1 + g_1) = \$20.736 \times 1.20 = \$24.8832 \] **Stage 2: Calculate the Terminal Value at Year 4** After Year 4, dividends grow perpetually at 8%. The terminal value at Year 4 represents the present value of all dividends from Year 5 onward, discounted back to Year 4. 1. **Dividend at Year 5 (D₅):** \[ D₅ = D₄ \times (1 + g_2) = \$24.8832 \times 1.08 = \$26.873856 \] 2. **Terminal Value at Year 4 (TV₄):** \[ TV₄ = \frac{D₅}{k - g_2} = \frac{\$26.873856}{0.15 - 0.08} = \frac{\$26.873856}{0.07} = \$383.91223 \] **Stage 3: Discount All Cash Flows Back to Present Value** We discount each dividend and the terminal value back to present value using the required rate of return (k = 15%). 1. **Present Value Factors:** \[ (1 + k)^t \text{ for } t = 1 \text{ to } 4: \] \[ (1.15)^1 = 1.15, \quad (1.15)^2 = 1.3225, \quad (1.15)^3 = 1.520875, \quad (1.15)^4 = 1.74900625 \] 2. **Present Value of Dividends:** - **PV of D₁:** \[ PV(D₁) = \frac{\$14.40}{1.15} = \$12.5217 \] - **PV of D₂:** \[ PV(D₂) = \frac{\$17.28}{1.3225} = \$13.0649 \] - **PV of D₃:** \[ PV(D₃) = \frac{\$20.736}{1.520875} = \$13.6347 \] - **PV of D₄:** \[ PV(D₄) = \frac{\$24.8832}{1.74900625} = \$14.2300 \] 3. **Present Value of Terminal Value:** \[ PV(TV₄) = \frac{\$383.91223}{1.74900625} = \$219.4675 \] **Stage 4: Sum All Present Values to Find the Fair Value** \[ \text{Fair Value} = PV(D₁) + PV(D₂) + PV(D₃) + PV(D₄) + PV(TV₄) \] \[ \text{Fair Value} = \$12.5217 + \$13.0649 + \$13.6347 + \$14.2300 + \$219.4675 = \$272.9188 \] **Answer:** The fair value of the stock now is approximately **$272.92** per share. **Answer:** $272.92 |
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| deepseekv3 | To determine the fair value of the stock, we need to calculate the present value of all future dividends using the **Dividend Discount Model (DDM)**. The problem involves two phases: 1. **High-growth phase**: Dividend grows at 20% for the next 4 years. 2. **Constant-growth phase**: Dividend grows at a constant rate of 8% forever after year 4. The expected rate of return is 15%. Here’s the step-by-step solution: --- ### Step 1: Calculate dividends for the high-growth phase The dividend in year \( t \) is: \[ D_t = D_0 \times (1 + g)^t \] Where: - \( D_0 = 12 \) (current dividend in 2016) - \( g = 20\% \) (growth rate for the first 4 years) | Year | Dividend Calculation | Dividend Value ($) | |------|---------------------------------|--------------------| | 1 | \( 12 \times (1 + 0.20)^1 \) | 14.40 | | 2 | \( 12 \times (1 + 0.20)^2 \) | 17.28 | | 3 | \( 12 \times (1 + 0.20)^3 \) | 20.74 | | 4 | \( 12 \times (1 + 0.20)^4 \) | 24.88 | --- ### Step 2: Calculate the terminal value at the end of year 4 At the end of year 4, the stock enters the constant-growth phase. The terminal value is calculated using the **Gordon Growth Model**: \[ P_4 = \frac{D_5}{r - g} \] Where: - \( D_5 = D_4 \times (1 + g) = 24.88 \times (1 + 0.08) = 26.87 \) - \( r = 15\% \) (expected rate of return) - \( g = 8\% \) (constant growth rate) \[ P_4 = \frac{26.87}{0.15 - 0.08} = \frac{26.87}{0.07} = 383.86 \] --- ### Step 3: Discount dividends and terminal value to present value The fair value of the stock is the present value of all future cash flows (dividends and terminal value), discounted at the expected rate of return (15%). \[ \text{Fair Value} = \sum_{t=1}^4 \frac{D_t}{(1 + r)^t} + \frac{P_4}{(1 + r)^4} \] | Year | Cash Flow ($) | Discount Factor \( \frac{1}{(1 + 0.15)^t} \) | Present Value ($) | |------|---------------|-----------------------------------------------|--------------------| | 1 | 14.40 | 0.8696 | 12.52 | | 2 | 17.28 | 0.7561 | 13.06 | | 3 | 20.74 | 0.6575 | 13.63 | | 4 | 24.88 | 0.5718 | 14.23 | | 4 | 383.86 | 0.5718 | 219.50 | **Total Fair Value**: \[ 12.52 + 13.06 + 13.63 + 14.23 + 219.50 = 272.94 \] --- ### Final Answer: The fair value of the stock now is **$272.94**. |
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