| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | 要确定Lore Ltd.股票的当前公允价值，我们将使用股利折现模型（Dividend Discount Model，简称DDM）中的两阶段增长模型。第一阶段涵盖未来五年的高增长期（13%），第二阶段涵盖永续增长期（5%）。我们将计算未来五年的股利，确定这些股利的现值，然后计算高增长期结束时的终值。最后，我们将这些股利和终值的现值相加，得到股票的当前公允价值。 ### 步骤1：计算未来五年的股利 假设当前年度的股利为$14，未来五年股利的增长率为13%。我们可以计算未来五年的股利： - \( D_1 = D_0 \times (1 + g_1) = 14 \times (1 + 0.13) = 14 \times 1.13 = 15.82 \) - \( D_2 = D_1 \times (1 + g_1) = 15.82 \times 1.13 = 17.88 \) - \( D_3 = D_2 \times (1 + g_1) = 17.88 \times 1.13 = 20.21 \) - \( D_4 = D_3 \times (1 + g_1) = 20.21 \times 1.13 = 22.84 \) - \( D_5 = D_4 \times (1 + g_1) = 22.84 \times 1.13 = 25.80 \) ### 步骤2：计算第一阶段股利的现值 使用折现率12%计算未来五年股利的现值： - \( PV(D_1) = \frac{15.82}{(1 + 0.12)^1} = \frac{15.82}{1.12} = 14.13 \) - \( PV(D_2) = \frac{17.88}{(1 + 0.12)^2} = \frac{17.88}{1.2544} = 14.26 \) - \( PV(D_3) = \frac{20.21}{(1 + 0.12)^3} = \frac{20.21}{1.4049} = 14.38 \) - \( PV(D_4) = \frac{22.84}{(1 + 0.12)^4} = \frac{22.84}{1.5735} = 14.52 \) - \( PV(D_5) = \frac{25.80}{(1 + 0.12)^5} = \frac{25.80}{1.7623} = 14.64 \) ### 步骤3：计算终值（Terminal Value） 在第五年结束时，股利将以5%的增长率永续增长。终值计算公式为： \[ TV_5 = \frac{D_6}{r - g_2} = \frac{D_5 \times (1 + g_2)}{r - g_2} = \frac{25.80 \times 1.05}{0.12 - 0.05} = \frac{27.09}{0.07} = 387.00 \] ### 步骤4：计算终值的现值 将终值折现回当前时点： \[ PV(TV_5) = \frac{387.00}{(1 + 0.12)^5} = \frac{387.00}{1.7623} = 219.67 \] ### 步骤5：计算股票的当前公允价值 将所有现值相加： \[ V_0 = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(TV_5) \] \[ V_0 = 14.13 + 14.26 + 14.38 + 14.52 + 14.64 + 219.67 = 291.60 \] 因此，Lore Ltd.股票的当前公允价值为 **$291.60**。 |
291.45 |
| deepseekr1 | The fair value of Lore Ltd. stock is calculated using a two-stage dividend discount model. Here's the step-by-step breakdown: 1. **Calculate Dividends for the First 5 Years:** - Current dividend (D₀) = $14 - Growth rate for the first 5 years (g₁) = 13% - Dividends: - Year 1: $14 × 1.13 = $15.82 - Year 2: $15.82 × 1.13 = $17.8766 - Year 3: $17.8766 × 1.13 = $20.2006 - Year 4: $20.2006 × 1.13 = $22.8267 - Year 5: $22.8267 × 1.13 = $25.7941 2. **Calculate Terminal Value at Year 5:** - Perpetual growth rate (g₂) = 5% - Dividend in Year 6 (D₆) = $25.7941 × 1.05 = $27.0838 - Terminal Value (TV) = D₆ / (Required Rate - g₂) = $27.0838 / (12% - 5%) = $387.6257 3. **Discount Dividends and Terminal Value to Present:** - Required rate of return (r) = 12% - Present Value (PV) of Dividends: - PV Year 1: $15.82 / 1.12 = $14.125 - PV Year 2: $17.8766 / 1.12² = $14.2511 - PV Year 3: $20.2006 / 1.12³ = $14.3784 - PV Year 4: $22.8267 / 1.12⁴ = $14.507 - PV Year 5: $25.7941 / 1.12⁵ = $14.6364 - Total PV of Dividends = $14.125 + $14.2511 + $14.3784 + $14.507 + $14.6364 = **$71.8979** - PV of Terminal Value = $387.6257 / 1.12⁵ = **$219.95** 4. **Sum Present Values:** - Total Fair Value = PV of Dividends + PV of Terminal Value = $71.8979 + $219.95 = **$291.85** **Answer:** The current fair value of Lore Ltd. stock is **\$291.85**. |
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| gpto1 | To determine the current fair value of Lore Ltd.'s stock using the Dividend Discount Model (DDM), we'll follow a multi-stage approach because the dividend growth rate changes after five years. **Given:** - Current dividend (D₀): \$14 - High-growth rate (g₁): 13% for the next 5 years - Sustainable growth rate (g₂): 5% thereafter - Required rate of return (r): 12% **Steps:** 1. **Calculate dividends for Years 1 to 5 during the high-growth phase.** 2. **Calculate the stock price at the end of Year 5 using the Gordon Growth Model (since dividends grow at a constant rate from Year 6 onwards).** 3. **Discount all expected cash flows (dividends and stock price at Year 5) back to present value terms.** 4. **Sum these present values to get the current fair value of the stock.** --- **1. Calculate Dividends for Years 1 to 5:** Using the formula \( D_t = D_0 \times (1 + g_1)^t \): - **Year 1:** \[ D_1 = \$14 \times (1 + 0.13)^1 = \$14 \times 1.13 = \$15.82 \] - **Year 2:** \[ D_2 = \$14 \times (1.13)^2 \] \[ D_2 = \$14 \times 1.2769 = \$17.8766 \] - **Year 3:** \[ D_3 = \$14 \times (1.13)^3 \] \[ D_3 = \$14 \times 1.442897 = \$20.200558 \] - **Year 4:** \[ D_4 = \$14 \times (1.13)^4 \] \[ D_4 = \$14 \times 1.63047361 = \$22.82663054 \] - **Year 5:** \[ D_5 = \$14 \times (1.13)^5 \] \[ D_5 = \$14 \times 1.84243518 = \$25.7940925 \] --- **2. Calculate the Stock Price at Year 5 (P₅):** First, calculate the dividend for Year 6: \[ D_6 = D_5 \times (1 + g_2) \] \[ D_6 = \$25.7940925 \times 1.05 = \$27.08379713 \] Now, use the Gordon Growth Model for P₅: \[ P_5 = \frac{D_6}{r - g_2} \] \[ P_5 = \frac{\$27.08379713}{0.12 - 0.05} = \frac{\$27.08379713}{0.07} \] \[ P_5 = \$386.9114 \] --- **3. Discount Dividends and P₅ back to Present Value:** The present value (PV) of each cash flow is calculated using: \[ PV = \frac{Cash\ Flow}{(1 + r)^t} \] - **PV of D₁:** \[ PV_1 = \frac{\$15.82}{(1.12)^1} = \frac{\$15.82}{1.12} = \$14.125 \] - **PV of D₂:** \[ PV_2 = \frac{\$17.8766}{(1.12)^2} = \frac{\$17.8766}{1.2544} = \$14.25 \] - **PV of D₃:** \[ PV_3 = \frac{\$20.200558}{(1.12)^3} = \frac{\$20.200558}{1.404928} = \$14.379 \] - **PV of D₄:** \[ PV_4 = \frac{\$22.82663054}{(1.12)^4} = \frac{\$22.82663054}{1.573517} = \$14.50 \] - **PV of D₅:** \[ PV_5 = \frac{\$25.7940925}{(1.12)^5} = \frac{\$25.7940925}{1.762341} = \$14.636 \] - **PV of P₅:** \[ PV_{P_5} = \frac{\$386.9114}{(1.12)^5} = \frac{\$386.9114}{1.762341} = \$219.51 \] --- **4. Sum All Present Values:** \[ \text{Total PV} = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 + PV_{P_5} \] \[ \text{Total PV} = \$14.125 + \$14.25 + \$14.379 + \$14.50 + \$14.636 + \$219.51 \] \[ \text{Total PV} = \$291.40 \] --- **Answer:** **The current fair value of Lore Ltd.'s stock is approximately \$291.40.** |
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| GPT-4o | To determine the current fair value of Lore Ltd. stock, we can use the Dividend Discount Model (DDM). Given the details, we will first calculate the dividends for the next five years with a 13% growth rate, and then determine the stock's value when the growth rate drops to a sustainable 5%. Here are the steps: 1. **Calculate the dividends for the next five years:** - Current dividend (D0): $14 - Growth rate for the first five years (g1): 13% Dividends for the next five years can be calculated as follows: \( D1 = D0 \times (1 + g1) \) \( D2 = D1 \times (1 + g1) \) \( D3 = D2 \times (1 + g1) \) \( D4 = D3 \times (1 + g1) \) \( D5 = D4 \times (1 + g1) \) Let's calculate these: \( D1 = 14 \times (1 + 0.13) = 14 \times 1.13 = 15.82 \) \( D2 = 15.82 \times 1.13 = 17.88 \) \( D3 = 17.88 \times 1.13 = 20.21 \) \( D4 = 20.21 \times 1.13 = 22.84 \) \( D5 = 22.84 \times 1.13 = 25.80 \) 2. **Calculate the terminal value at the end of year 5 (when growth becomes constant):** - Sustainable growth rate (g2): 5% - Required rate of return (discount rate, r): 12% The terminal value (TV) at the end of year 5 can be calculated using the Gordon Growth Model (for a perpetuity starting from D6): \( TV = \frac{D6}{r - g2} \) First, calculate \(D6\): \( D6 = D5 \times (1 + g2) = 25.80 \times 1.05 = 27.09 \) Now, calculate the terminal value: \( TV = \frac{27.09}{0.12 - 0.05} = \frac{27.09}{0.07} = 387 \) 3. **Discount the dividends and terminal value back to present value:** We need to discount each dividend and the terminal value back to the present value using the required rate of return (12%). \( PV = \frac{D1}{(1 + r)^1} + \frac{D2}{(1 + r)^2} + \frac{D3}{(1 + r)^3} + \frac{D4}{(1 + r)^4} + \frac{D5}{(1 + r)^5} + \frac{TV}{(1 + r)^5} \) Let's calculate each term: \( PV(D1) = \frac{15.82}{(1 + 0.12)^1} = \frac{15.82}{1.12} = 14.13 \) \( PV(D2) = \frac{17.88}{(1 + 0.12)^2} = \frac{17.88}{1.2544} = 14.26 \) \( PV(D3) = \frac{20.21}{(1 + 0.12)^3} = \frac{20.21}{1.404928} = 14.38 \) \( PV(D4) = \frac{22.84}{(1 + 0.12)^4} = \frac{22.84}{1.57351936} = 14.51 \) \( PV(D5) = \frac{25.80}{(1 + 0.12)^5} = \frac{25.80}{1.7623416832} = 14.64 \) \( PV(TV) = \frac{387}{(1 + 0.12)^5} = \frac{387}{1.7623416832} = 219.64 \) 4. **Sum the present values to get the current fair value of the stock:** \( PV = PV(D1) + PV(D2) + PV(D3) + PV(D4) + PV(D5) + PV(TV) \) \( PV = 14.13 + 14.26 + 14.38 + 14.51 + 14.64 + 219.64 \) \( |
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| llama3.1-70B-Instruct | To find the current fair value of Lore Ltd. stock, we can use the dividend discount model (DDM), which takes into account the expected dividend growth and the required rate of return. We'll break down the solution into steps. ### Step 1: Calculate the Dividends for the First Five Years Given that the current year's dividend is $14 and the growth rate for the next five years is 13%, we calculate the dividends for each of the next five years. - Year 1: $14 * (1 + 0.13) = $15.82 - Year 2: $15.82 * (1 + 0.13) = $17.89 - Year 3: $17.89 * (1 + 0.13) = $20.23 - Year 4: $20.23 * (1 + 0.13) = $22.86 - Year 5: $22.86 * (1 + 0.13) = $25.83 ### Step 2: Calculate the Present Value of the Dividends for the First Five Years Using the required rate of return (discount rate) of 12%, we calculate the present value (PV) of each of the dividends for the first five years. - PV of Year 1 Dividend: $15.82 / (1 + 0.12)^1 = $14.13 - PV of Year 2 Dividend: $17.89 / (1 + 0.12)^2 = $14.23 - PV of Year 3 Dividend: $20.23 / (1 + 0.12)^3 = $14.39 - PV of Year 4 Dividend: $22.86 / (1 + 0.12)^4 = $14.59 - PV of Year 5 Dividend: $25.83 / (1 + 0.12)^5 = $14.83 ### Step 3: Calculate the Present Value of the Perpetual Dividends Starting from Year 6 From Year 6 onwards, the dividend growth rate settles to 5%. We calculate the dividend for Year 6 and then find the present value of the perpetual dividends starting from Year 6. - Year 6 Dividend: $25.83 * (1 + 0.05) = $27.12 - Present Value of Perpetual Dividends from Year 6: $27.12 / (0.12 - 0.05) = $270.00 - Discounted to Present (Year 0): $270.00 / (1 + 0.12)^5 = $170.41 ### Step 4: Calculate the Total Present Value of All Dividends Adding the present values of the dividends for the first five years and the present value of the perpetual dividends starting from Year 6 gives us the total present value. - Total Present Value = ($14.13 + $14.23 + $14.39 + $14.59 + $14.83) + $170.41 - Total Present Value = $72.17 + $170.41 - Total Present Value = $242.58 The final answer is: $242.58 |
291.45 |
| llama3.1-405B-Instruct | To calculate the current fair value of Lore Ltd. stock, we can use the dividend discount model (DDM). The DDM is a method of estimating the present value of a stock based on the expected future dividend payments. First, we need to calculate the expected dividend payments for the next five years. We can do this by multiplying the current year's dividend by the expected growth rate for each year: Year 1: $14 x 1.13 = $15.82 Year 2: $15.82 x 1.13 = $17.88 Year 3: $17.88 x 1.13 = $20.21 Year 4: $20.21 x 1.13 = $22.84 Year 5: $22.84 x 1.13 = $25.81 Next, we need to calculate the present value of these expected dividend payments using the required rate of return (12%). We can do this by dividing each expected dividend payment by (1 + 0.12)^n, where n is the number of years until the dividend payment is made: Year 1: $15.82 / (1 + 0.12)^1 = $14.11 Year 2: $17.88 / (1 + 0.12)^2 = $15.21 Year 3: $20.21 / (1 + 0.12)^3 = $16.43 Year 4: $22.84 / (1 + 0.12)^4 = $17.75 Year 5: $25.81 / (1 + 0.12)^5 = $19.21 Now, we need to calculate the present value of the expected dividend payments beyond the fifth year. We can do this by using the formula for the present value of a perpetuity: PV = D / (r - g) where PV is the present value, D is the expected dividend payment in the fifth year, r is the required rate of return, and g is the expected growth rate. PV = $25.81 / (0.12 - 0.05) = $51.62 Finally, we can calculate the current fair value of Lore Ltd. stock by adding the present value of the expected dividend payments for the next five years to the present value of the expected dividend payments beyond the fifth year: Current fair value = $14.11 + $15.21 + $16.43 + $17.75 + $19.21 + $51.62 = $134.33 Therefore, the current fair value of Lore Ltd. stock is $134.33. |
291.45 |
| DeepSeek-Coder-V2 | To determine the current fair value of Lore Ltd. stock, we will use the two-stage dividend discount model (DDM). This model accounts for the two distinct growth periods: the initial high-growth period and the subsequent stable growth period. ### Step 1: Calculate the dividends for the high-growth period The dividend growth rate for the next five years is 13%, and the current dividend is $14. We will calculate the dividends for each of the next five years and then discount them back to the present value using the required rate of return of 12%. **Year 1:** \[ D_1 = 14 \times (1 + 0.13) = 14 \times 1.13 = 15.82 \] \[ PV_1 = \frac{15.82}{(1 + 0.12)^1} = \frac{15.82}{1.12} \approx 14.12 \] **Year 2:** \[ D_2 = 15.82 \times (1 + 0.13) = 15.82 \times 1.13 = 17.88 \] \[ PV_2 = \frac{17.88}{(1 + 0.12)^2} = \frac{17.88}{1.2544} \approx 14.48 \] **Year 3:** \[ D_3 = 17.88 \times (1 + 0.13) = 17.88 \times 1.13 = 20.20 \] \[ PV_3 = \frac{20.20}{(1 + 0.12)^3} = \frac{20.20}{1.4049} \approx 14.85 \] **Year 4:** \[ D_4 = 20.20 \times (1 + 0.13) = 20.20 \times 1.13 = 22.83 \] \[ PV_4 = \frac{22.83}{(1 + 0.12)^4} = \frac{22.83}{1.5782} \approx 15.23 \] **Year 5:** \[ D_5 = 22.83 \times (1 + 0.13) = 22.83 \times 1.13 = 25.80 \] \[ PV_5 = \frac{25.80}{(1 + 0.12)^5} = \frac{25.80}{1.7726} \approx 15.62 \] ### Step 2: Calculate the terminal value at the end of the high-growth period After the fifth year, the dividend growth rate will stabilize at 5%. We will use the Gordon Growth Model to calculate the terminal value at the end of the high-growth period. \[ D_6 = D_5 \times (1 + 0.05) = 25.80 \times 1.05 = 27.09 \] \[ Terminal Value = \frac{D_6}{(r - g)} = \frac{27.09}{(0.12 - 0.05)} = \frac{27.09}{0.07} \approx 387.00 \] ### Step 3: Discount the terminal value back to the present \[ PV_{Terminal Value} = \frac{387.00}{(1 + 0.12)^5} = \frac{387.00}{1.7726} \approx 217.00 \] ### Step 4: Sum the present values of all dividends and the terminal value \[ Fair Value = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 + PV_{Terminal Value} \] \[ Fair Value = 14.12 + 14.48 + 14.85 + 15.23 + 15.62 + 217.00 \approx 291.30 \] Therefore, the current fair value of Lore Ltd. stock is approximately $291.30. |
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| Qwen2-72B-Instruct | To find the current fair value of Lore Ltd. stock, we'll use the dividend discount model (DDM). The model comprises of two parts: the first part calculates the present value of dividends during the high growth period (the next five years), and the second part calculates the present value of the perpetuity of dividends after the high growth period. First, let's calculate the present value of dividends for the next five years. The formula for calculating the present value of a future dividend is: Present Value of Dividend = Dividend / (1 + Discount Rate)^n where n is the number of years in the future. For the first five years, the dividend growth rate is 13%. Thus, the dividends for each year are: Year 1: $14 * (1 + 0.13) = $15.82 Year 2: $15.82 * (1 + 0.13) = $17.93 Year 3: $17.93 * (1 + 0.13) = $20.28 Year 4: $20.28 * (1 + 0.13) = $22.92 Year 5: $22.92 * (1 + 0.13) = $25.89 The present value of these dividends at a 12% discount rate is: Year 1: $15.82 / (1 + 0.12)^1 = $14.12 Year 2: $17.93 / (1 + 0.12)^2 = $13.95 Year 3: $20.28 / (1 + 0.12)^3 = $13.28 Year 4: $22.92 / (1 + 0.12)^4 = $12.19 Year 5: $25.89 / (1 + 0.12)^5 = $10.94 Summing these present values gives us $64.48. Next, we'll calculate the present value of the perpetuity of dividends after the high growth period. The formula for calculating the present value of a perpetuity is: Present Value of Perpetuity = Dividend / (Discount Rate - Dividend Growth Rate) The dividend at the end of the high growth period is $25.89, and the dividend growth rate after the high growth period is 5%. Thus, the present value of the perpetuity of dividends is: Present Value of Perpetuity = $25.89 / (0.12 - 0.05) = $369.86 However, we need to discount this perpetuity back to the present. The formula for discounting a future value back to the present is: Present Value = Future Value / (1 + Discount Rate)^n where n is the number of years in the future. The perpetuity starts at the end of the fifth year, so we need to discount it back five years. Thus, the present value of the perpetuity is: Present Value of Perpetuity = $369.86 / (1 + 0.12)^5 = $188.37 Finally, we can calculate the current fair value of Lore Ltd. stock by summing the present values of the dividends during the high growth period and the perpetuity of dividends after the high growth period: Current Fair Value = $64.48 + $188.37 = $252.85 Therefore, the current fair value of Lore Ltd. stock is $252.85. |
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| baichuan4 | To calculate the current fair value of Lore Ltd. stock, we can use the dividend discount model (DDM), which takes into account the present value of all future dividends. In this case, we have two phases of dividend growth: an initial five-year period with a growth rate of 13%, followed by a perpetual period with a growth rate of 5%. First, let's calculate the dividends for the next five years: - Year 1 dividend: $14 * (1 + 13%) = $15.82 - Year 2 dividend: $15.82 * (1 + 13%) ≈ $17.88 - Year 3 dividend: $17.88 * (1 + 13%) ≈ $20.20 - Year 4 dividend: $20.20 * (1 + 13%) ≈ $22.83 - Year 5 dividend: $22.83 * (1 + 13%) ≈ $25.79 Next, we need to find the present value of these dividends, using the required rate of return (discount rate) of 12%. - Present value of Year 1 dividend: $15.82 / (1 + 12%) ≈ $14.13 - Present value of Year 2 dividend: $17.88 / (1 + 12%)^2 ≈ $13.53 - Present value of Year 3 dividend: $20.20 / (1 + 12%)^3 ≈ $12.96 - Present value of Year 4 dividend: $22.83 / (1 + 12%)^4 ≈ $12.41 - Present value of Year 5 dividend: $25.79 / (1 + 12%)^5 ≈ $11.88 Now, let's calculate the present value of the perpetual dividends starting from Year 6. To do this, we need to find the Year 6 dividend, which will grow at a constant rate of 5% forever. - Year 6 dividend: $25.79 * (1 + 5%) ≈ $27.13 The present value of the perpetual dividends can be calculated using the formula: \[ \text{PV of Perpetual Dividends} = \frac{\text{Year 6 Dividend}}{\text{Required Rate of Return} - \text{Constant Growth Rate}} \] \[ \text{PV of Perpetual Dividends} = \frac{$27.13}{12\% - 5\%} = \frac{$27.13}{7\%} ≈ $387.57 \] To find the present value of the perpetual dividends starting from Year 6, we discount this amount back to the present: \[ \text{PV of Perpetual Dividends} = $387.57 / (1 + 12\%)^5 ≈ $204.46 \] Finally, we add up the present values of all the dividends to get the current fair value of Lore Ltd. stock: \[ \text{Current Fair Value} = $14.13 + $13.53 + $12.96 + $12.41 + $11.88 + $204.46 ≈ $269.37 \] Therefore, the current fair value of Lore Ltd. stock is approximately $269.37. |
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| doubao | For the first five years: Dividend in year 1 = $14 * (1 + 0.13) = $15.82 Dividend in year 2 = $15.82 * (1 + 0.13) = $17.8766 Dividend in year 3 = $17.8766 * (1 + 0.13) = $20.190158 Dividend in year 4 = $20.190158 * (1 + 0.13) = $22.81487854 Dividend in year 5 = $22.81487854 * (1 + 0.13) = $25.88081475 Present value of these dividends: PV1 = $15.82 / (1 + 0.12) = $14.125 PV2 = $17.8766 / (1 + 0.12)^2 = $14.125 PV3 = $20.190158 / (1 + 0.12)^3 = $14.125 PV4 = $22.81487854 / (1 + 0.12)^4 = $14.125 PV5 = $25.88081475 / (1 + 0.12)^5 = $14.125 Sum of present values for the first five years = 5 * $14.125 = $70.625 For the constant growth period: Dividend in year 6 = $25.88081475 * (1 + 0.05) = $27.17485549 Using the Gordon growth model, the present value of the perpetual dividends starting from year 6 is: PV6 = ($27.17485549 / (0.12 - 0.05)) / (1 + 0.12)^5 = $241.875 The current fair value of the stock = $70.625 + $241.875 = $312.5. |
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| minimax | To calculate the current fair value of Lore Ltd. stock, we need to use the Dividend Discount Model (DDM), which in this case will be a two-stage model because the growth rate changes after five years. The two-stage DDM is calculated as follows: 1. Calculate the present value of dividends during the first five years when the growth rate is 13% per year. 2. Calculate the present value of the stock price at the end of year five, which is when the growth rate becomes constant at 5%. 3. Discount this future stock price back to the present value. Let's calculate each step: **Step 1: Present value of dividends during the first five years** The dividends for the next five years can be calculated using the formula for the future value of a growing annuity: \[ D_{t} = D_{0} \times (1 + g)^{t} \] Where: - \( D_{t} \) is the dividend in year t - \( D_{0} \) is the current dividend, which is $14 - \( g \) is the growth rate, which is 13% or 0.13 - \( t \) is the year The present value of each of these dividends can be calculated using the formula: \[ PV = \frac{D_{t}}{(1 + r)^{t}} \] Where: - \( PV \) is the present value of the dividend - \( r \) is the required rate of return, which is 12% or 0.12 Let's calculate the present value of each dividend: \[ PV(D_1) = \frac{14 \times (1 + 0.13)^1}{(1 + 0.12)^1} \] \[ PV(D_2) = \frac{14 \times (1 + 0.13)^2}{(1 + 0.12)^2} \] \[ PV(D_3) = \frac{14 \times (1 + 0.13)^3}{(1 + 0.12)^3} \] \[ PV(D_4) = \frac{14 \times (1 + 0.13)^4}{(1 + 0.12)^4} \] \[ PV(D_5) = \frac{14 \times (1 + 0.13)^5}{(1 + 0.12)^5} \] Summing these up gives us the total present value of dividends over the first five years. **Step 2: Present value of the stock price at the end of year five** After five years, the growth rate becomes 5% or 0.05. We need to calculate the stock price at the end of year five using the Gordon Growth Model (a version of the DDM for perpetual growth): \[ P_{5} = \frac{D_{6}}{r - g} \] Where: - \( P_{5} \) is the price of the stock at the end of year five - \( D_{6} \) is the dividend in year six, which can be calculated as \( D_{5} \times (1 + g) \) First, we calculate \( D_{6} \): \[ D_{6} = D_{5} \times (1 + 0.05) \] Then we calculate \( P_{5} \): \[ P_{5} = \frac{D_{6}}{0.12 - 0.05} \] **Step 3: Discount the future stock price back to the present value** The present value of the stock price at the end of year five is: \[ PV(P_{5}) = \frac{P_{5}}{(1 + 0.12)^5} \] Now, we add the present value of the dividends from step 1 and the present value of the stock price from step 3 to get the fair value of the stock today. Let's do the math: \[ PV(D_1) = \frac{14 \times 1.13}{1.12} \] \[ PV(D_2) = \frac{14 \times 1.13^2}{1.12^2} \] \[ PV(D_3) = \frac{14 \times 1.13^3}{1.12^3} \] \[ PV(D_4) = \frac{14 \times 1.13^4}{1.12^4} \] \[ PV(D_5) = \frac{14 \times 1.13^5}{1.12^5} \] \[ D_{6} = 14 \times 1.13^5 \times 1.05 \] \[ P_{5} = \frac{D_{6}}{0.12 - 0.05} \] \[ PV(P_{5}) = \frac{P_{5}}{1.12^5} \] Finally, summing up all the present values: \[ \text{Fair Value} = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(P_{5}) \] You can now calculate these values using a financial calculator or spreadsheet software to get the final fair value of Lore Ltd. stock. |
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| kimi | To calculate the current fair value of Lore Ltd. stock, we can use the Gordon Growth Model (also known as the Dividend Discount Model) for the period of constant growth after the initial phase of high growth. The formula for the fair value of a stock is: \[ P_0 = \frac{D_1}{(r - g)} \] where: - \( P_0 \) is the current stock price - \( D_1 \) is the dividend expected to be received at the end of the first period (after the high growth phase) - \( r \) is the required rate of return - \( g \) is the constant growth rate First, we need to calculate the dividend for the end of the fifth year, which is the last year of the high growth phase. We can use the formula for the dividend growth model: \[ D_5 = D_0 \times (1 + g_1)^5 \] where: - \( D_5 \) is the dividend at the end of the fifth year - \( D_0 \) is the current dividend - \( g_1 \) is the growth rate during the high growth phase (13%) \[ D_5 = 14 \times (1 + 0.13)^5 \] \[ D_5 = 14 \times (1.13)^5 \] \[ D_5 = 14 \times 2.0024 \] \[ D_5 = 28.0336 \] Next, we calculate the dividend for the first year after the high growth phase ends (which is the beginning of the constant growth phase) using the formula: \[ D_6 = D_5 \times (1 + g_2) \] where: - \( D_6 \) is the dividend at the beginning of the constant growth phase - \( g_2 \) is the constant growth rate (5%) \[ D_6 = 28.0336 \times (1 + 0.05) \] \[ D_6 = 28.0336 \times 1.05 \] \[ D_6 = 29.33508 \] Now, we can calculate the present value of the dividends during the high growth phase using the formula: \[ PV_{HG} = \frac{D_1}{(1 + r)} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + \frac{D_4}{(1 + r)^4} + \frac{D_5}{(1 + r)^5} \] where: - \( D_1 \) to \( D_4 \) are the dividends for each of the first four years of the high growth phase - \( D_5 \) is the dividend at the end of the fifth year \[ D_1 = D_0 \times (1 + g_1) \] \[ D_1 = 14 \times 1.13 \] \[ D_1 = 15.82 \] \[ D_2 = D_1 \times (1 + g_1) \] \[ D_2 = 15.82 \times 1.13 \] \[ D_2 = 17.8366 \] \[ D_3 = D_2 \times (1 + g_1) \] \[ D_3 = 17.8366 \times 1.13 \] \[ D_3 = 20.09 \] \[ D_4 = D_3 \times (1 + g_1) \] \[ D_4 = 20.09 \times 1.13 \] \[ D_4 = 22.6017 \] Now, we calculate the present value of these dividends: \[ PV_{HG} = \frac{15.82}{(1 + 0.12)} + \frac{17.8366}{(1 + 0.12)^2} + \frac{20.09}{(1 + 0.12)^3} + \frac{22.6017}{(1 + 0.12)^4} + \frac{28.0336}{(1 + 0.12)^5} \] \[ PV_{HG} = \frac{15.82}{1.12} + \frac{17.8366}{1.2544} + \frac{20.09}{1.4054} + \frac{22.6017}{1.5735} + \frac{28.0336}{1 |
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| wenxinyiyan | To calculate the current fair value of Lore Ltd. stock, we can use the dividend discount model (DDM). This model values a stock based on the present value of its future dividends. Step 1: Calculate the dividends for the next five years based on the estimated 13% growth rate. Year 1 dividend = $14 * (1 + 0.13) = $15.82 Year 2 dividend = $15.82 * (1 + 0.13) = $17.8786 Year 3 dividend = $17.8786 * (1 + 0.13) = $20.200218 Year 4 dividend = $20.200218 * (1 + 0.13) = $22.82624634 Year 5 dividend = $22.82624634 * (1 + 0.13) = $25.79365837 Step 2: Calculate the present value of these dividends using the required rate of return (discount rate) of 12%. PV of Year 1 dividend = $15.82 / (1 + 0.12) = $14.125 PV of Year 2 dividend = $17.8786 / (1 + 0.12)^2 = $13.7605 PV of Year 3 dividend = $20.200218 / (1 + 0.12)^3 = $12.7756 PV of Year 4 dividend = $22.82624634 / (1 + 0.12)^4 = $11.3035 PV of Year 5 dividend = $25.79365837 / (1 + 0.12)^5 = $9.9058 Step 3: Calculate the value of the stock after the fifth year, assuming a constant dividend growth rate of 5% and the same discount rate of 12%. Terminal Value = (Year 5 dividend * (1 + 0.05)) / (0.12 - 0.05) = ($25.79365837 * 1.05) / 0.07 = $386.90487555 PV of Terminal Value = $386.90487555 / (1 + 0.12)^5 = $220.1294 Step 4: Sum up the present values of the dividends and the terminal value to get the current fair value of the stock. Current Fair Value = PV of Year 1 dividend + PV of Year 2 dividend + PV of Year 3 dividend + PV of Year 4 dividend + PV of Year 5 dividend + PV of Terminal Value = $14.125 + $13.7605 + $12.7756 + $11.3035 + $9.9058 + $220.1294 = $282.0008 (rounded to two decimal places) Therefore, the current fair value of Lore Ltd. stock is approximately $282.00. |
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| zhipu | To determine the current fair value of Lore Ltd. stock, we will use the Dividend Discount Model (DDM), specifically a two-stage model since there are two distinct growth phases for the dividends. Here's the step-by-step solution: ### Step 1: Calculate the dividends for the next five years Given: - Current dividend (D0) = $14 - Growth rate for the next five years (g1) = 13% or 0.13 - Growth rate after five years (g2) = 5% or 0.05 The dividends for the next five years (D1, D2, D3, D4, D5) can be calculated as follows: - D1 = D0 * (1 + g1) = $14 * (1 + 0.13) = $14 * 1.13 = $15.82 - D2 = D1 * (1 + g1) = $15.82 * 1.13 = $17.83 - D3 = D2 * (1 + g1) = $17.83 * 1.13 = $20.11 - D4 = D3 * (1 + g1) = $20.11 * 1.13 = $22.67 - D5 = D4 * (1 + g1) = $22.67 * 1.13 = $25.56 ### Step 2: Calculate the terminal value at the end of year 5 The terminal value (P5) is the value of all future dividends beyond year 5, discounted back to the end of year 5. It can be calculated using the Gordon Growth Model (DDM for constant growth): \[ P5 = \frac{D5 \times (1 + g2)}{r - g2} \] Where: - D5 = $25.56 - g2 = 5% or 0.05 - r = required rate of return = 12% or 0.12 \[ P5 = \frac{25.56 \times (1 + 0.05)}{0.12 - 0.05} \] \[ P5 = \frac{25.56 \times 1.05}{0.07} \] \[ P5 = \frac{26.878}{0.07} \] \[ P5 = $384.11 \] ### Step 3: Calculate the present value of the dividends and the terminal value The present value of each dividend and the terminal value needs to be discounted back to the present time (year 0). \[ PV = \frac{D1}{(1 + r)^1} + \frac{D2}{(1 + r)^2} + \frac{D3}{(1 + r)^3} + \frac{D4}{(1 + r)^4} + \frac{D5 + P5}{(1 + r)^5} \] Where: - D1 = $15.82 - D2 = $17.83 - D3 = $20.11 - D4 = $22.67 - D5 = $25.56 - P5 = $384.11 - r = 12% or 0.12 \[ PV = \frac{15.82}{(1 + 0.12)^1} + \frac{17.83}{(1 + 0.12)^2} + \frac{20.11}{(1 + 0.12)^3} + \frac{22.67}{(1 + 0.12)^4} + \frac{25.56 + 384.11}{(1 + 0.12)^5} \] \[ PV = \frac{15.82}{1.12} + \frac{17.83}{1.2544} + \frac{20.11}{1.404928} + \frac{22.67}{1.57351936} + \frac{409.67}{1.76234168} \] \[ PV = 14.12 + 14.24 + 14.34 + 14.45 + 231.85 \] \[ PV = 298.00 \] ### Conclusion The current fair value of Lore Ltd. stock is approximately $298.00. |
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| gpto1 | To determine the current fair value of Lore Ltd.'s stock using the Dividend Discount Model (DDM), we'll follow a multi-stage approach because the dividend growth rate changes after five years. **Given:** - Current dividend (D₀): \$14 - High-growth rate (g₁): 13% for the next 5 years - Sustainable growth rate (g₂): 5% thereafter - Required rate of return (r): 12% **Steps:** 1. **Calculate dividends for Years 1 to 5 during the high-growth phase.** 2. **Calculate the stock price at the end of Year 5 using the Gordon Growth Model (since dividends grow at a constant rate from Year 6 onwards).** 3. **Discount all expected cash flows (dividends and stock price at Year 5) back to present value terms.** 4. **Sum these present values to get the current fair value of the stock.** --- **1. Calculate Dividends for Years 1 to 5:** Using the formula \( D_t = D_0 \times (1 + g_1)^t \): - **Year 1:** \[ D_1 = \$14 \times (1 + 0.13)^1 = \$14 \times 1.13 = \$15.82 \] - **Year 2:** \[ D_2 = \$14 \times (1.13)^2 \] \[ D_2 = \$14 \times 1.2769 = \$17.8766 \] - **Year 3:** \[ D_3 = \$14 \times (1.13)^3 \] \[ D_3 = \$14 \times 1.442897 = \$20.200558 \] - **Year 4:** \[ D_4 = \$14 \times (1.13)^4 \] \[ D_4 = \$14 \times 1.63047361 = \$22.82663054 \] - **Year 5:** \[ D_5 = \$14 \times (1.13)^5 \] \[ D_5 = \$14 \times 1.84243518 = \$25.7940925 \] --- **2. Calculate the Stock Price at Year 5 (P₅):** First, calculate the dividend for Year 6: \[ D_6 = D_5 \times (1 + g_2) \] \[ D_6 = \$25.7940925 \times 1.05 = \$27.08379713 \] Now, use the Gordon Growth Model for P₅: \[ P_5 = \frac{D_6}{r - g_2} \] \[ P_5 = \frac{\$27.08379713}{0.12 - 0.05} = \frac{\$27.08379713}{0.07} \] \[ P_5 = \$386.9114 \] --- **3. Discount Dividends and P₅ back to Present Value:** The present value (PV) of each cash flow is calculated using: \[ PV = \frac{Cash\ Flow}{(1 + r)^t} \] - **PV of D₁:** \[ PV_1 = \frac{\$15.82}{(1.12)^1} = \frac{\$15.82}{1.12} = \$14.125 \] - **PV of D₂:** \[ PV_2 = \frac{\$17.8766}{(1.12)^2} = \frac{\$17.8766}{1.2544} = \$14.25 \] - **PV of D₃:** \[ PV_3 = \frac{\$20.200558}{(1.12)^3} = \frac{\$20.200558}{1.404928} = \$14.379 \] - **PV of D₄:** \[ PV_4 = \frac{\$22.82663054}{(1.12)^4} = \frac{\$22.82663054}{1.573517} = \$14.50 \] - **PV of D₅:** \[ PV_5 = \frac{\$25.7940925}{(1.12)^5} = \frac{\$25.7940925}{1.762341} = \$14.636 \] - **PV of P₅:** \[ PV_{P_5} = \frac{\$386.9114}{(1.12)^5} = \frac{\$386.9114}{1.762341} = \$219.51 \] --- **4. Sum All Present Values:** \[ \text{Total PV} = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 + PV_{P_5} \] \[ \text{Total PV} = \$14.125 + \$14.25 + \$14.379 + \$14.50 + \$14.636 + \$219.51 \] \[ \text{Total PV} = \$291.40 \] --- **Answer:** **The current fair value of Lore Ltd.'s stock is approximately \$291.40.** |
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