| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | 要估算这家公司的市盈率（PE ratio），我们需要计算前五年的预期股息、在高增长期结束时的终端价值，并将这些值折现到现值。最后，我们将预期股息的现值除以收益来得到市盈率。以下是详细的计算步骤： ### 1. 计算前五年的预期股息 假设公司的初始每股收益（Earnings per Share, EPS）为$E_0$，则在前五年中每年的EPS和股息（Dividends per Share, DPS）可以如下计算： #### EPS和DPS计算： - 第一年： \( EPS_1 = E_0 \times (1 + 0.25) \) \( DPS_1 = EPS_1 \times 0.20 \) - 第二年： \( EPS_2 = EPS_1 \times (1 + 0.25) \) \( DPS_2 = EPS_2 \times 0.20 \) - 第三年： \( EPS_3 = EPS_2 \times (1 + 0.25) \) \( DPS_3 = EPS_3 \times 0.20 \) - 第四年： \( EPS_4 = EPS_3 \times (1 + 0.25) \) \( DPS_4 = EPS_4 \times 0.20 \) - 第五年： \( EPS_5 = EPS_4 \times (1 + 0.25) \) \( DPS_5 = EPS_5 \times 0.20 \) ### 2. 计算终端价值 在高增长期结束时，第六年的EPS和DPS可以如下计算： - 第六年： \( EPS_6 = EPS_5 \times (1 + 0.08) \) \( DPS_6 = EPS_6 \times 0.50 \) 终端价值（Terminal Value, TV）是第六年的DPS除以长期增长率和折现率的差值： \[ TV = \frac{DPS_6}{Cost\ of\ Equity - Long-term\ Growth\ Rate} = \frac{DPS_6}{0.115 - 0.08} \] ### 3. 折现预期股息和终端价值 使用11.5%的股本成本（Cost of Equity）将前五年的DPS和终端价值折现到现值： \[ PV_{DPS_1} = \frac{DPS_1}{(1 + 0.115)^1} \] \[ PV_{DPS_2} = \frac{DPS_2}{(1 + 0.115)^2} \] \[ PV_{DPS_3} = \frac{DPS_3}{(1 + 0.115)^3} \] \[ PV_{DPS_4} = \frac{DPS_4}{(1 + 0.115)^4} \] \[ PV_{DPS_5} = \frac{DPS_5}{(1 + 0.115)^5} \] \[ PV_{TV} = \frac{TV}{(1 + 0.115)^5} \] 总现值（PV）为以上各现值的总和： \[ PV = PV_{DPS_1} + PV_{DPS_2} + PV_{DPS_3} + PV_{DPS_4} + PV_{DPS_5} + PV_{TV} \] ### 4. 计算市盈率（PE Ratio） 市盈率为总现值除以初始每股收益（E_0）： \[ PE = \frac{PV}{E_0} \] ### 具体计算 为了具体计算，我们需要假设一个初始的每股收益（E_0）。假设 \( E_0 = $1 \): #### 计算前五年的DPS： - \( EPS_1 = 1 \times 1.25 = 1.25 \) \( DPS_1 = 1.25 \times 0.20 = 0.25 \) - \( EPS_2 = 1.25 \times 1.25 = 1.5625 \) \( DPS_2 = 1.5625 \times 0.20 = 0.3125 \) - \( EPS_3 = 1.5625 \times 1.25 = 1.953125 \) \( DPS_3 = 1.953125 \times 0.20 = 0.390625 \) - \( EPS_4 = 1.953125 \times 1.25 = 2.44140625 \) \( DPS_4 = 2.44140625 \times 0.20 = 0.48828125 \) - \( EPS_5 = 2.44140625 \times 1.25 = 3.0517578125 \) \( DPS_5 = 3.0517578125 \times 0.20 = 0.6103515625 \) #### 计算第六年的DPS和终端价值： - \( EPS_6 = 3.0517578125 \times 1.08 = 3.2962984375 \) \( DPS_6 = 3.2962984375 \times 0.50 = 1.64814921875 \) - 终端价值： \( TV = \frac{1.64814921875}{0.115 - 0.08} = \frac{1.64814921875}{0.035} = 47.089978125 \) #### 折现预期股息和终端价值： - \( PV_{DPS_1} = \frac{0.25}{(1 + 0.115)^1} = 0.2242 \) - \( PV_{DPS_2} = \frac{0.3125}{(1 + 0.115)^2} = 0.2519 \) - \( PV_{DPS_3} = \frac{0.390625}{(1 + 0.115)^3} = 0.2763 \) - \( PV_{DPS_4} = \frac{0.48828125}{(1 + 0.115)^4} = 0.2981 \) - \( PV_{DPS_5} = \frac{0.6103515625}{(1 + 0.115)^5} = 0.3174 \) - \( PV_{TV} = \frac{47.089978125}{(1 + 0.115)^5} = 27.8106 \) #### 总现值： \[ PV = 0.2242 + 0.2519 + 0.2763 + 0.2981 + 0.3174 + 27.8106 = 29.1785 \] #### 市盈率： \[ PE = \frac{29.1785}{1} = 29.18 \] ### 结论 根据上述计算，这家公司的估算市盈率（PE Ratio）为29.18。 |
28.75 |
| deepseekr1 | To estimate the PE ratio for the firm, we use a two-stage dividend discount model considering the high growth phase for the first five years and a stable growth phase thereafter. Here are the key steps: 1. **High Growth Phase (Years 1-5):** - **Dividends:** The payout ratio is 20%, and earnings grow at 25%. Dividends are calculated as 20% of each year's earnings, growing at 25% annually. - **Present Value of Dividends:** Calculate the present value of these dividends using the cost of equity (11.5%). 2. **Terminal Value (Stable Growth Phase):** - **Dividends from Year 6 Onwards:** The payout ratio increases to 50%, and earnings grow at 8%. The dividend in year 6 is calculated and then discounted back to the present using the Gordon Growth Model. - **Terminal Value at Year 5:** Calculated as the dividend in year 6 divided by the difference between the cost of equity and the stable growth rate (11.5% - 8%). 3. **Present Value of Terminal Value:** Discount the terminal value back to the present value. 4. **PE Ratio Calculation:** Sum the present values of the dividends and the terminal value, then divide by the current earnings (E0) to get the PE ratio. **Calculations:** - **Present Value of Dividends (Years 1-5):** \[ \text{PV of Dividends} = \sum_{t=1}^{5} \frac{0.20 \times (1.25)^t \times E0}{(1.115)^t} = 1.4279 \times E0 \] - **Terminal Value at Year 5:** \[ \text{Terminal Value} = \frac{0.50 \times (1.25)^5 \times 1.08 \times E0}{0.115 - 0.08} = 47.0857 \times E0 \] - **Present Value of Terminal Value:** \[ \text{PV of Terminal Value} = \frac{47.0857 \times E0}{(1.115)^5} = 27.3426 \times E0 \] - **Total PE Ratio:** \[ \text{PE Ratio} = \frac{\text{PV of Dividends} + \text{PV of Terminal Value}}{E0} = 1.4279 + 27.3426 = 28.7705 \] **Final Answer:** \[ \boxed{28.77} \] |
28.75 |
| gpto1 | To estimate the PE ratio for the firm, we'll use a two-stage dividend discount model (DDM), which considers an initial high-growth period followed by a stable growth period. Here's a step-by-step solution: **Given:** - **High Growth Period (Years 1-5):** - Growth rate (\( g_1 \)) = 25% per year - Payout ratio (\( PR_1 \)) = 20% - **Stable Growth Period (Year 6 onward):** - Growth rate (\( g_2 \)) = 8% per year - Payout ratio (\( PR_2 \)) = 50% - **Cost of Equity (\( r \))** = 11.5% (calculated as \( r = R_f + \beta \times \text{Risk Premium} = 6\% + 1 \times 5.5\% \)) - **Initial Earnings per Share (\( E_0 \))** = Assume \$1 for simplicity. **Step 1: Project Earnings and Dividends for Years 1-5** For each year, calculate the earnings and dividends: **Year 1:** - Earnings: \( E_1 = E_0 \times (1 + g_1) = 1 \times 1.25 = \$1.25 \) - Dividends: \( D_1 = E_1 \times PR_1 = 1.25 \times 0.20 = \$0.25 \) **Year 2:** - Earnings: \( E_2 = E_1 \times (1 + g_1) = 1.25 \times 1.25 = \$1.5625 \) - Dividends: \( D_2 = E_2 \times PR_1 = 1.5625 \times 0.20 = \$0.3125 \) **Year 3:** - Earnings: \( E_3 = E_2 \times (1 + g_1) = 1.5625 \times 1.25 = \$1.953125 \) - Dividends: \( D_3 = E_3 \times PR_1 = 1.953125 \times 0.20 = \$0.390625 \) **Year 4:** - Earnings: \( E_4 = E_3 \times (1 + g_1) = 1.953125 \times 1.25 = \$2.44140625 \) - Dividends: \( D_4 = E_4 \times PR_1 = 2.44140625 \times 0.20 = \$0.48828125 \) **Year 5:** - Earnings: \( E_5 = E_4 \times (1 + g_1) = 2.44140625 \times 1.25 = \$3.0517578125 \) - Dividends: \( D_5 = E_5 \times PR_1 = 3.0517578125 \times 0.20 = \$0.6103515625 \) **Step 2: Calculate Terminal Value at Year 5** At the end of Year 5, the company enters the stable growth period. The terminal value (\( P_5 \)) is calculated using the Gordon Growth Model: - **Earnings in Year 6:** \( E_6 = E_5 \times (1 + g_2) = 3.0517578125 \times 1.08 = \$3.2958984375 \) - **Dividends in Year 6:** \( D_6 = E_6 \times PR_2 = 3.2958984375 \times 0.50 = \$1.64794921875 \) - **Terminal Value at Year 5:** \( P_5 = \frac{D_6}{r - g_2} = \frac{1.64794921875}{0.115 - 0.08} = \frac{1.64794921875}{0.035} = \$47.0843 \) **Step 3: Calculate Present Value of Dividends and Terminal Value** Discount each dividend and the terminal value back to present value using the cost of equity (\( r = 11.5\% \)): **Discount Factors:** \( DF_t = \frac{1}{(1 + r)^t} \) Compute \( (1 + r)^t \) for \( t = 1 \) to 5: - \( (1 + r)^1 = 1.115 \) - \( (1 + r)^2 = (1.115)^2 = 1.243225 \) - \( (1 + r)^3 = (1.115)^3 = 1.38639375 \) - \( (1 + r)^4 = (1.115)^4 = 1.54565281 \) - \( (1 + r)^5 = (1.115)^5 = 1.72319548 \) **Present Value Calculations:** - \( PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{0.25}{1.115} = \$0.22472 \) - \( PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{0.3125}{1.243225} = \$0.25130 \) - \( PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{0.390625}{1.38639375} = \$0.28172 \) - \( PV(D_4) = \frac{D_4}{(1 + r)^4} = \frac{0.48828125}{1.54565281} = \$0.31602 \) - \( PV(D_5) = \frac{D_5}{(1 + r)^5} = \frac{0.6103515625}{1.72319548} = \$0.35428 \) - \( PV(P_5) = \frac{P_5}{(1 + r)^5} = \frac{47.0843}{1.72319548} = \$27.326 \) **Total Present Value (Price):** \( P_0 = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(P_5) \) \( P_0 = 0.22472 + 0.25130 + 0.28172 + 0.31602 + 0.35428 + 27.326 = \$28.754 \) **Step 4: Calculate the PE Ratio** Since \( E_0 = \$1 \): \( \text{PE Ratio} = \frac{P_0}{E_0} = \frac{28.754}{1} = 28.754 \) **Answer:** The estimated PE ratio for the firm is **approximately 28.75**. |
28.75 |
| GPT-4o | To estimate the Price-to-Earnings (PE) ratio for the firm, we can follow a multi-step approach that involves calculating the present value of future dividends and then using the Gordon Growth Model for the stable growth period. Here are the detailed steps: ### Step 1: Calculate the Dividends for the High Growth Period (First 5 Years) - **Growth Rate (g1):** 25% - **Payout Ratio (PR1):** 20% - **Cost of Equity (r):** 11.5% Assume the firm has an initial earnings per share (EPS) of \( E_0 \). 1. Year 1 Dividend: \[ E_1 = E_0 \times (1 + g1) \] \[ D_1 = E_1 \times PR1 = E_0 \times (1 + 0.25) \times 0.20 \] 2. Year 2 Dividend: \[ E_2 = E_1 \times (1 + g1) = E_0 \times (1 + 0.25)^2 \] \[ D_2 = E_2 \times PR1 = E_0 \times (1 + 0.25)^2 \times 0.20 \] 3. Year 3 Dividend: \[ E_3 = E_2 \times (1 + g1) = E_0 \times (1 + 0.25)^3 \] \[ D_3 = E_3 \times PR1 = E_0 \times (1 + 0.25)^3 \times 0.20 \] 4. Year 4 Dividend: \[ E_4 = E_3 \times (1 + g1) = E_0 \times (1 + 0.25)^4 \] \[ D_4 = E_4 \times PR1 = E_0 \times (1 + 0.25)^4 \times 0.20 \] 5. Year 5 Dividend: \[ E_5 = E_4 \times (1 + g1) = E_0 \times (1 + 0.25)^5 \] \[ D_5 = E_5 \times PR1 = E_0 \times (1 + 0.25)^5 \times 0.20 \] ### Step 2: Calculate the Year 6 Dividend and Beyond (Stable Growth Period) - **Growth Rate (g2):** 8% - **Payout Ratio (PR2):** 50% Dividends will grow at a stable rate after year 5: \[ E_6 = E_5 \times (1 + g2) = E_0 \times (1 + 0.25)^5 \times (1 + 0.08) \] \[ D_6 = E_6 \times PR2 = E_0 \times (1 + 0.25)^5 \times (1 + 0.08) \times 0.50 \] ### Step 3: Calculate the Present Value of Dividends for the High Growth Period \[ PV(D_1) = \frac{D_1}{(1 + r)^1} \] \[ PV(D_2) = \frac{D_2}{(1 + r)^2} \] \[ PV(D_3) = \frac{D_3}{(1 + r)^3} \] \[ PV(D_4) = \frac{D_4}{(1 + r)^4} \] \[ PV(D_5) = \frac{D_5}{(1 + r)^5} \] ### Step 4: Calculate the Terminal Value at the End of Year 5 The terminal value (TV) represents the present value of all dividends from year 6 onwards: \[ TV_5 = \frac{D_6}{r - g2} = \frac{E_0 \times (1 + 0.25)^5 \times (1 + 0.08) \times 0.50}{0.115 - 0.08} \] ### Step 5: Calculate the Present Value of the Terminal Value \[ PV(TV_5) = \frac{TV_5}{(1 + r)^5} \] ### Step 6: Calculate the Total Present Value of Dividends \[ PV(Dividends) = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(TV_5) \] ### Step 7: Estimate the PE Ratio The PE ratio is the present value of the dividends divided by the initial earnings: \[ PE = \frac{PV(Dividends)}{E_0} \] ### Putting It All Together Let's assume \( E_0 = 1 |
28.75 |
| llama3.1-70B-Instruct | To estimate the PE ratio for the firm, we can use the two-stage dividend discount model. Here's the step-by-step solution: **Step 1: Calculate the dividend per share in the first five years** Growth rate in the first five years = 25% Payout ratio in the first five years = 20% Assuming the initial dividend per share is D0, the dividend per share in the first five years will grow at a rate of 25%. The dividend per share in each year can be calculated as: D1 = D0 x (1 + 0.25) = 1.25D0 D2 = D1 x (1 + 0.25) = 1.25^2D0 = 1.5625D0 D3 = D2 x (1 + 0.25) = 1.25^3D0 = 1.953125D0 D4 = D3 x (1 + 0.25) = 1.25^4D0 = 2.44140625D0 D5 = D4 x (1 + 0.25) = 1.25^5D0 = 3.0517578125D0 **Step 2: Calculate the present value of the dividends in the first five years** Cost of equity = 11.5% The present value of the dividends in the first five years can be calculated as: PV(D1) = D1 / (1 + 0.115)^1 = 1.25D0 / 1.115 = 1.1216D0 PV(D2) = D2 / (1 + 0.115)^2 = 1.5625D0 / 1.243 = 1.2569D0 PV(D3) = D3 / (1 + 0.115)^3 = 1.953125D0 / 1.376 = 1.4203D0 PV(D4) = D4 / (1 + 0.115)^4 = 2.44140625D0 / 1.515 = 1.6125D0 PV(D5) = D5 / (1 + 0.115)^5 = 3.0517578125D0 / 1.661 = 1.8385D0 **Step 3: Calculate the present value of the dividends after five years** Growth rate after five years = 8% Payout ratio after five years = 50% Assuming the dividend per share in year 5 is D5, the dividend per share after five years will grow at a rate of 8%. The present value of the dividends after five years can be calculated as: PV(D6) = D5 x (1 + 0.08) / (1 + 0.115)^6 = 3.0517578125D0 x 1.08 / 1.815 = 1.8639D0 PV(D7) = PV(D6) / (1 + 0.115)^7 = 1.8639D0 / 1.973 = 0.9439D0 PV(D8) = PV(D7) / (1 + 0.115)^8 = 0.9439D0 / 2.136 = 0.4421D0 ... The present value of the dividends after five years can be calculated using the formula for the present value of a growing perpetuity: PV(D6) = D5 x (1 + 0.08) / (0.115 - 0.08) = 3.0517578125D0 x 1.08 / 0.035 = 95.5714D0 **Step 4: Calculate the estimated PE ratio** The estimated PE ratio can be calculated as the sum of the present values of the dividends in the first five years and the present value of the dividends after five years, divided by the initial dividend per share: PE ratio = (PV(D1) + PV(D2) + PV(D3) + PV(D6) / D0 = (1.1216 + 1.2569 + 1.4203 + 1.6125 + 1.8385) + 95.5714 / D0 = 7.2508 + 95.5714 / D0 Assuming the initial dividend per share is D0 = 1, the estimated PE ratio is: PE ratio = 7.2508 + 95.5714 = 102.82 Therefore, the estimated PE ratio for the firm is approximately 102.82. |
28.75 |
| llama3.1-405B-Instruct | To estimate the PE ratio for this firm, we can use the following steps: 1. Estimate the present value of the dividends during the high growth period: The firm will pay dividends for the next 5 years, with a growth rate of 25% per year. We can estimate the present value of these dividends using the formula for the present value of a growing annuity: PV = D0 x (1 - (1 + g)^(-n)) / (r - g) where: PV = present value D0 = initial dividend g = growth rate n = number of years r = cost of equity The initial dividend (D0) can be estimated as follows: D0 = Earnings x Payout ratio = Earnings x 0.20 Since we don't know the current earnings, we can assume that the current earnings are equal to the current price (P0). Therefore: D0 = P0 x 0.20 Now we can estimate the present value of the dividends during the high growth period: PV = (P0 x 0.20) x (1 - (1 + 0.25)^(-5)) / (0.115 - 0.25) = 0.20P0 x 3.491 / 0.135 = 5.17P0 2. Estimate the present value of the dividends after the high growth period: After the high growth period, the firm will grow at a rate of 8% per year, and the payout ratio will be 50%. We can estimate the present value of these dividends using the formula for the present value of a perpetuity: PV = D0 / (r - g) The initial dividend (D0) after the high growth period can be estimated as follows: D0 = Earnings x Payout ratio = Earnings x 0.50 Since we don't know the earnings after the high growth period, we can assume that the earnings will grow at a rate of 8% per year. Therefore: D0 = P0 x (1 + 0.25)^5 x 0.50 Now we can estimate the present value of the dividends after the high growth period: PV = (P0 x (1 + 0.25)^5 x 0.50) / (0.115 - 0.08) = 3.38P0 3. Estimate the total present value of the dividends: The total present value of the dividends is the sum of the present value of the dividends during the high growth period and the present value of the dividends after the high growth period: Total PV = 5.17P0 + 3.38P0 = 8.55P0 4. Estimate the PE ratio: The PE ratio is the ratio of the current price (P0) to the total present value of the dividends: PE ratio = P0 / Total PV = P0 / 8.55P0 = 0.117 Therefore, the estimated PE ratio for this firm is 0.117 or 11.7. |
28.75 |
| DeepSeek-Coder-V2 | To estimate the PE ratio for the given firm, we need to follow these steps: 1. **Calculate the terminal value of the firm at the end of the high-growth period (year 5):** - Growth rate in first five years = 25% - Payout ratio in first five years = 20% - Retention ratio = 1 - Payout ratio = 1 - 0.20 = 0.80 - Return on Equity (ROE) = Growth rate / Retention ratio = 0.25 / 0.80 = 0.3125 or 31.25% 2. **Calculate the earnings per share (EPS) at the end of year 5:** - Assume initial EPS = $1 (for simplicity) - EPS at the end of year 5 = $1 * (1 + 0.25)^5 = $1 * 3.0518 = $3.0518 3. **Calculate the terminal value of the firm at the end of year 5:** - Growth rate after five years = 8% - Payout ratio after five years = 50% - Retention ratio = 1 - Payout ratio = 1 - 0.50 = 0.50 - ROE = Growth rate / Retention ratio = 0.08 / 0.50 = 0.16 or 16% - Cost of equity = 11.5% - Terminal value = EPS at the end of year 5 * (1 + Growth rate after five years) / (Cost of equity - Growth rate after five years) - Terminal value = $3.0518 * (1 + 0.08) / (0.115 - 0.08) = $3.0518 * 1.08 / 0.035 = $94.67 4. **Calculate the present value of the terminal value:** - Discount rate = Cost of equity = 11.5% - Present value of terminal value = Terminal value / (1 + Discount rate)^5 - Present value of terminal value = $94.67 / (1 + 0.115)^5 = $94.67 / 1.716 = $55.17 5. **Calculate the present value of the dividends during the high-growth period:** - Dividend per share = EPS * Payout ratio - Dividend per share in year 1 = $1 * 0.20 = $0.20 - Dividend per share in year 2 = $1 * (1 + 0.25)^1 * 0.20 = $0.25 - Dividend per share in year 3 = $1 * (1 + 0.25)^2 * 0.20 = $0.3125 - Dividend per share in year 4 = $1 * (1 + 0.25)^3 * 0.20 = $0.3906 - Dividend per share in year 5 = $1 * (1 + 0.25)^4 * 0.20 = $0.4883 - Present value of dividends = Sum of (Dividend per share / (1 + Discount rate)^year) - Present value of dividends = $0.20 / (1 + 0.115)^1 + $0.25 / (1 + 0.115)^2 + $0.3125 / (1 + 0.115)^3 + $0.3906 / (1 + 0.115)^4 + $0.4883 / (1 + 0.115)^5 - Present value of dividends = $0.179 + $0.217 + $0.263 + $0.318 + $0.384 = $1.361 6. **Calculate the total present value of the firm:** - Total present value = Present value of terminal value + Present value of dividends - Total present value = $55.17 + $1.361 = $56.53 7. **Calculate the PE ratio:** - PE ratio = Total present value / EPS at the end of year 5 - PE ratio = $56.53 / $3.0518 = 18.52 Therefore, the estimated PE ratio for this firm is approximately 18.52. |
28.75 |
| Qwen2-72B-Instruct | To estimate the PE (Price-to-Earnings) ratio for the firm, we need to use the multi-stage dividend discount model. The PE ratio can be derived from the model by dividing the stock price by the earnings per share (EPS). The stock price can be estimated using the present value of future dividends. First, let's calculate the expected dividends for the first five years and the terminal value. 1. **Dividends during the high growth phase (first five years):** - The growth rate is 25%. - The payout ratio is 20%. Let's assume the earnings per share (EPS) in year 0 is $1 (for simplicity). The dividends in each year can then be calculated as follows: - Dividend in Year 1 = $1 * 1.25 * 0.20 = $0.25 - Dividend in Year 2 = $1 * 1.25^2 * 0.20 = $0.3125 - Dividend in Year 3 = $1 * 1.25^3 * 0.20 = $0.390625 - Dividend in Year 4 = $1 * 1.25^4 * 0.20 = $0.488281 - Dividend in Year 5 = $1 * 1.25^5 * 0.20 = $0.610352 2. **Dividends after the high growth phase:** - The growth rate after five years is 8%. - The payout ratio after five years is 50%. The EPS in year 5, before dividends, can be calculated as: - EPS in Year 5 = $1 * 1.25^5 = $3.0517578125 The dividend in year 6 can then be calculated as: - Dividend in Year 6 = $3.0517578125 * 0.50 * 1.08 = $1.6671045455 3. **Terminal value:** - The terminal value is the present value of all future dividends after year 5. - We can use the Gordon Growth model to calculate this, which is P = D / (r - g), where D is the dividend in the first period of stable growth, r is the cost of equity, and g is the stable growth rate. - Terminal value = $1.6671045455 / (0.115 - 0.08) = $59.1968 4. **Present value of future dividends:** - Now, we need to calculate the present value of all the dividends and the terminal value. - The present value of the dividends for the first five years can be calculated using the formula PV = D / (1 + r)^t, where D is the dividend, r is the cost of equity, and t is the time. - The present value of the terminal value can be calculated as PV = Terminal value / (1 + r)^5. 5. **Calculate the stock price:** - The stock price is the sum of the present values of all dividends and the terminal value. - Stock price = PV(D1) + PV(D2) + PV(D3) + PV(D4) + PV(D5) + PV(Terminal value) 6. **Calculate the PE ratio:** - The PE ratio is the stock price divided by the earnings per share in year 0. - PE ratio = Stock price / EPS in Year 0 Now, let's calculate all these values: 1. **Dividends during the high growth phase:** - PV(D1) = $0.25 / (1 + 0.115)^1 = $0.224299 - PV(D2) = $0.3125 / (1 + 0.115)^2 = $0.253736 - PV(D3) = $0.390625 / (1 + 0.115)^3 = $0.278950 - PV(D4) = $0.488281 / (1 + 0.115)^4 = $0.300727 - PV(D5) = $0.610352 / (1 + 0.115)^5 = $0.319553 2. **Terminal value:** - PV(Terminal value) = $59.1968 / (1 + 0.115)^5 = $32.5795 3. **Stock price:** - Stock price = $0.224299 + $0.253736 + $0.278950 + $0.300727 + $0.319553 + $32.5795 = $33.9563 4. **PE ratio:** - PE ratio = $33.9563 / $1 = 33.96 Thus, the estimated PE ratio for this firm is approximately 33.96. |
28.75 |
| baichuan4 | To estimate the PE (Price-to-Earnings) ratio for the firm, we can use the Gordon Growth Model (GGM) which is a formula used to determine the intrinsic value of a stock based on future dividends. The GGM assumes a constant growth rate for dividends. However, since this firm has different growth and payout rates for the first five years and after five years, we need to calculate the present value of the future cash flows in two stages and then sum them up. First, let's calculate the present value of the dividends during the high growth period (years 1-5). We will use the formula for the present value of a growing perpetuity since the growth rate is constant during this period. Present Value of Dividends in High Growth Period = DPS1 / (r - g) Where: DPS1 = Dividend per share in the next period (start of year 1) r = Cost of equity (discount rate) g = Growth rate of dividends We need to calculate DPS1 based on the payout ratio and a notional earnings per share (EPS) value, which we will assume to be $1 for simplicity (since we are calculating a ratio, the actual value doesn't matter). DPS1 = EPS * Payout Ratio = $1 * 20% = $0.20 Now we can calculate the present value of dividends for the high growth period: Present Value of Dividends in High Growth Period = $0.20 / (11.5% - 25%) = $0.20 / (-13.5%) = -$1.48 The negative sign here implies that the present value is actually an amount that would need to be paid to get $1.48 in the future, which is not intuitive. This is because the growth rate is higher than the discount rate, which is unusual. In practical terms, this means the stock price would be expected to rise rapidly. Next, let's calculate the present value of the dividends after the high growth period (year 6 onwards). After year 5, the dividend growth rate becomes 8%, and the payout ratio is 50%. Present Value of Dividends in Stable Growth Period = DPS6 / (r - g_stable) We need to calculate DPS6, which is the dividend per share at the start of year 6: DPS6 = DPS1 * (1 + g_high) ^ 5 * (Payout Ratio_stable / Payout Ratio_high) Assuming an initial EPS of $1: DPS6 = $0.20 * (1 + 25%)^5 * (50% / 20%) = $0.20 * 2.9769 * 2.5 = $1.4885 Now we can calculate the present value of dividends for the stable growth period: Present Value of Dividends in Stable Growth Period = $1.4885 / (11.5% - 8%) = $1.4885 / 3.5% = $42.53 Now, to find the total present value of the dividends, we need to discount the present value of the stable growth period back to year 5 and add it to the present value of the high growth period: Total Present Value of Dividends = -$1.48 + $42.53 / (1 + 11.5%)^5 = -$1.48 + $42.53 / 1.7183 = -$1.48 + $24.75 = $23.27 The estimated PE ratio can be found by dividing the total present value of dividends by the initial EPS (which we assumed to be $1): Estimated PE Ratio = Total Present Value of Dividends / EPS = $23.27 / $1 = 23.27 Therefore, the estimated PE ratio for this firm is approximately 23.27. |
28.75 |
| doubao | First, we need to calculate the dividends for the high growth period. Let D0 be the initial dividend. The dividend in year 1 is D1 = D0 * (1 + 25%) = 1.25D0. The dividend in year 2 is D2 = D1 * (1 + 25%) = 1.25 * 1.25D0 = 1.5625D0. And so on for the next three years. Next, we calculate the terminal value at the end of the high growth period. The dividend in year 6 is D6 = D5 * (1 + 8%) = D5 * 1.08. The terminal value is calculated as D6 / (Cost of equity - Growth rate after five years) = D5 * 1.08 / (11.5% - 8%) = D5 * 1.08 / 3.5%. We then discount the dividends and the terminal value back to the present using the cost of equity. The present value of the dividends for the high growth period is the sum of the discounted dividends. The present value of the terminal value is also discounted back to the present. Finally, we divide the current stock price (which is the sum of the present values of the dividends and the terminal value) by the earnings per share to get the PE ratio. The earnings per share grow at the same rate as the dividends during the high growth period. After five years, the earnings per share grow at 8%. The payout ratio is used to calculate the dividends from the earnings. This is a complex calculation that requires detailed numerical computations to arrive at the final PE ratio. |
28.75 |
| minimax | To estimate the PE ratio for the firm, we need to calculate the present value of the future cash flows and then divide the current price by the earnings. However, we are not given the current price or the earnings directly, so we will need to make some assumptions or use a model that relates the PE ratio to the growth rates, payout ratios, and cost of equity. One common method to estimate the PE ratio is to use the Gordon Growth Model (also known as the Dividend Discount Model for a perpetuity), which is based on the assumption that dividends grow at a constant rate forever. However, this model does not directly apply to our case because we have two different growth phases. Instead, we can use a two-stage dividend discount model, which accounts for the high-growth phase and the subsequent stable growth phase. We will calculate the present value of dividends during the high-growth phase and the terminal value at the end of the high-growth phase, which represents the value of all future dividends beyond that point. First, let's calculate the dividends during the high-growth phase: For the first five years, the dividend growth rate is 25%, and the payout ratio is 20%. The dividend in the first year (D1) can be calculated as follows: D1 = EPS1 * Payout Ratio Since we don't have the EPS1, we'll need to estimate it using the growth rate and the cost of equity. The earnings growth model is: EPS1 = EPS0 * (1 + g) Where: - EPS0 is the current earnings per share (which we don't have) - g is the growth rate in the first five years (25% or 0.25) We can't calculate EPS1 without EPS0, so we'll need to make an assumption or use another approach to estimate the PE ratio. One way to estimate EPS0 is to assume that the current price is equal to the present value of future cash flows, which would imply that the PE ratio at time 0 is the same as the PE ratio we are trying to estimate. This is a simplification and may not hold true in all cases, but without additional information, it's the best we can do. Let's assume that the PE ratio at time 0 is the same as the PE ratio we are trying to estimate. We can then use the Gordon Growth Model to estimate the terminal value at the end of year 5, which is when the stable growth phase begins: Terminal value (TV5) = D6 / (Cost of Equity - Growth Rate after five years) D6 = D5 * (1 + Growth Rate after five years) / Payout Ratio after five years Where: - D5 is the dividend in year 5 - Growth Rate after five years is 8% or 0.08 - Payout Ratio after five years is 50% or 0.5 We can't calculate D5 without knowing D1, and we can't calculate D1 without knowing EPS0. Since we are stuck in a loop without the initial EPS or price, we need to make an assumption or use a different approach. One alternative approach is to use the average PE ratio for similar companies in the industry or the market as a whole, adjusted for the firm's growth prospects and risk profile. However, this requires external data that is not provided in the problem statement. Given the constraints of the problem, we cannot accurately calculate the PE ratio without additional information, such as the current earnings per share (EPS0) or the current stock price. If you can provide either of these, we can proceed with the calculation. Otherwise, we would need to make assumptions or use industry benchmarks to estimate the PE ratio. |
28.75 |
| kimi | To estimate the PE (Price to Earnings) ratio for a firm, we can use the Gordon Growth Model (also known as the Dividend Discount Model) for the period of high growth and then switch to a more stable growth model after the high-growth period ends. The Gordon Growth Model is given by: \[ P_0 = \frac{D_1}{r - g} \] Where: - \( P_0 \) is the current price of the stock. - \( D_1 \) is the dividend expected to be received at the end of the first year. - \( r \) is the required rate of return (cost of equity). - \( g \) is the constant growth rate in dividends. Since we don't have the current dividend amount, we can express it in terms of the earnings per share (EPS) and the payout ratio. Let's assume the earnings per share at the beginning of the high-growth period is \( E_0 \). Then the dividend per share at the end of the first year \( D_1 \) would be \( E_1 \) times the payout ratio in the first five years. The earnings per share at the end of the first year \( E_1 \) can be calculated as \( E_0 \) multiplied by \( 1 + g \) (the growth rate in the first five years). So, \[ D_1 = E_1 \times \text{Payout ratio in first five years} \] \[ D_1 = E_0 \times (1 + 0.25) \times 0.20 \] Now, we can calculate the PE ratio at the end of the high-growth period as: \[ PE_1 = \frac{P_0}{E_0} \] Substituting \( P_0 \) from the Gordon Growth Model: \[ PE_1 = \frac{E_0 \times (1 + 0.25) \times 0.20}{0.115 - 0.25} \] After the high-growth period, the growth rate drops to 8% and the payout ratio increases to 50%. We can use a two-stage dividend discount model to estimate the PE ratio after the high-growth period. For the second stage, the formula becomes: \[ P_5 = \frac{D_6}{r - g_2} \] Where: - \( P_5 \) is the price at the end of the high-growth period. - \( D_6 \) is the dividend expected to be received at the end of the first year after the high-growth period. - \( g_2 \) is the growth rate after the high-growth period. \[ D_6 = E_6 \times \text{Payout ratio after five years} \] \[ D_6 = E_5 \times (1 + 0.08) \times 0.50 \] \[ E_5 = E_0 \times (1 + 0.25)^5 \] (since the earnings grow at 25% for 5 years) \[ P_5 = \frac{E_0 \times (1 + 0.25)^5 \times (1 + 0.08) \times 0.50}{0.115 - 0.08} \] \[ PE_5 = \frac{P_5}{E_5} \] \[ PE_5 = \frac{E_0 \times (1 + 0.25)^5 \times (1 + 0.08) \times 0.50}{E_0 \times (1 + 0.25)^5 \times (0.115 - 0.08)} \] \[ PE_5 = \frac{(1 + 0.25)^5 \times (1 + 0.08) \times 0.50}{(1 + 0.25)^5 \times (0.115 - 0.08)} \] \[ PE_5 = \frac{(1.25)^5 \times 1.08 \times 0.50}{(1.25)^5 \times 0.035} \] \[ PE_5 = \frac{1.08 \times 0.50}{0.035} \] \[ PE_5 = \frac{0.54}{0.035} \] \[ PE_5 = 15.43 \] So, the estimated PE ratio for the firm after the high-growth period is approximately 15.43. This is the PE ratio we would expect to see after five years when the firm transitions to a more stable growth rate |
28.75 |
| wenxinyiyan | To estimate the PE ratio for this firm, we will use the two-stage dividend discount model. This model considers two stages of growth: an initial high-growth stage and a subsequent lower growth stage. First, let's calculate the dividends per share (DPS) during the high-growth stage (first five years) and the lower growth stage (after five years). High-growth stage: Growth rate = 25% Payout ratio = 20% Assuming an initial earnings per share (EPS) of $1, the DPS during the high-growth stage will be: DPS_high_growth = EPS * payout ratio = $1 * 20% = $0.20 Lower growth stage: Growth rate = 8% Payout ratio = 50% The EPS after the high-growth stage will be: EPS_after_high_growth = EPS * (1 + growth rate)^5 = $1 * (1 + 25%)^5 Then, the DPS during the lower growth stage will be: DPS_lower_growth = EPS_after_high_growth * payout ratio = EPS_after_high_growth * 50% Now, let's calculate the present value of the dividends during the high-growth stage and the lower growth stage. Present value of dividends during high-growth stage: PV_high_growth = DPS_high_growth * [(1 - (1 + growth rate)^(-5)) / growth rate] Present value of dividends during lower growth stage: PV_lower_growth = DPS_lower_growth / (cost of equity - growth rate_lower) The total present value of dividends will be the sum of PV_high_growth and PV_lower_growth discounted back to the present value. Total PV of dividends = PV_high_growth + PV_lower_growth / (1 + cost of equity)^5 Finally, the PE ratio is calculated as the total present value of dividends divided by the EPS. PE ratio = Total PV of dividends / EPS Note: The calculations above require numerical values for EPS, growth rates, payout ratios, and the cost of equity. However, since the initial EPS was assumed to be $1 and we have all the other necessary information, we can proceed with the calculations using the provided formulas. Please note that the actual calculation involves more complex mathematics and might require the use of a financial calculator or software. The explanation provided gives you the framework to understand how the PE ratio is estimated using the two-stage dividend discount model. For an accurate PE ratio estimation, you would plug in the specific numbers provided into the formulas and calculate the precise values. Given the complexity of the calculations and the need for precise numerical inputs, the final PE ratio cannot be directly computed without performing the detailed arithmetic operations. |
28.75 |
| zhipu | To estimate the PE ratio for the firm, we need to calculate the present value of its future earnings and divide that by the current earnings. This involves two stages: the high-growth period and the stable-growth period. ### Step 1: Calculate the earnings in the high-growth period Let's assume the current earnings per share (EPS) is \( E_0 \). **Year 1:** \[ E_1 = E_0 \times (1 + g_1) = E_0 \times 1.25 \] **Year 2:** \[ E_2 = E_1 \times (1 + g_1) = E_0 \times 1.25^2 \] **Year 3:** \[ E_3 = E_2 \times (1 + g_1) = E_0 \times 1.25^3 \] **Year 4:** \[ E_4 = E_3 \times (1 + g_1) = E_0 \times 1.25^4 \] **Year 5:** \[ E_5 = E_4 \times (1 + g_1) = E_0 \times 1.25^5 \] ### Step 2: Calculate the dividends in the high-growth period Using the payout ratio (b): **Year 1:** \[ D_1 = E_1 \times b_1 = E_0 \times 1.25 \times 0.20 \] **Year 2:** \[ D_2 = E_2 \times b_1 = E_0 \times 1.25^2 \times 0.20 \] **Year 3:** \[ D_3 = E_3 \times b_1 = E_0 \times 1.25^3 \times 0.20 \] **Year 4:** \[ D_4 = E_4 \times b_1 = E_0 \times 1.25^4 \times 0.20 \] **Year 5:** \[ D_5 = E_5 \times b_1 = E_0 \times 1.25^5 \times 0.20 \] ### Step 3: Calculate the terminal value at the end of Year 5 The terminal value (TV) at the end of Year 5 can be calculated using the Gordon Growth Model (GGM): \[ TV_5 = \frac{E_5 \times (1 + g_2) \times b_2}{r - g_2} \] Where: - \( g_2 \) is the growth rate after Year 5 (8%) - \( b_2 \) is the payout ratio after Year 5 (50%) - \( r \) is the cost of equity (11.5%) \[ TV_5 = \frac{E_0 \times 1.25^5 \times 1.08 \times 0.50}{0.115 - 0.08} \] \[ TV_5 = \frac{E_0 \times 1.25^5 \times 1.08 \times 0.50}{0.035} \] ### Step 4: Calculate the present value of dividends and terminal value The present value (PV) of each dividend and the terminal value is calculated as follows: \[ PV(D_1) = \frac{D_1}{(1 + r)^1} \] \[ PV(D_2) = \frac{D_2}{(1 + r)^2} \] \[ PV(D_3) = \frac{D_3}{(1 + r)^3} \] \[ PV(D_4) = \frac{D_4}{(1 + r)^4} \] \[ PV(D_5) = \frac{D_5}{(1 + r)^5} \] \[ PV(TV_5) = \frac{TV_5}{(1 + r)^5} \] ### Step 5: Sum the present values \[ PV(Earnings) = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(TV_5) \] ### Step 6: Calculate the PE ratio \[ PE = \frac{PV(Earnings)}{E_0} \] ### Numerical Calculation Let's assume \( E_0 = 1 \) for simplicity. **Dividends:** \[ D_1 = 1 \times 1.25 \times 0.20 = 0.25 \] \[ D_2 = 1 \times 1.25^2 \times 0.20 = 0.3125 \] \[ D_3 = 1 \times 1.25^3 \times 0.20 = 0.390625 \] \[ D_4 = 1 \times 1.25^4 \times 0.20 = 0.48828125 \] \[ D_5 = 1 \times 1.25^5 \times 0.20 = 0.6103515625 \] **Terminal Value:** \[ TV_5 = \frac{1 \times 1.25^5 \times 1.08 \times 0.50}{0.035} \] \[ TV_5 = \frac{1 \times 3.0517578125 \times 1.08 \times 0.50}{0.035} \] \[ TV_5 = \frac{1.6728515625}{0.035} \] \[ TV_5 \approx 47.79 \] **Present Values:** \[ PV(D_1) = \frac{0.25}{1.115} \approx 0.224 \] \[ PV(D_2) = \frac{0.3125}{1.115^2} \approx 0.257 \] \[ PV(D_3) = \frac{0.390625}{1.115^3} \approx 0.288 \] \[ PV(D_4) = \frac{0.48828125}{1.115^4} \approx 0.318 \] \[ PV(D_5) = \frac{0.6103515625}{1.115^5} \approx 0.347 \] \[ PV(TV_5) = \frac{47.79}{1.115^5} \approx 31.15 \] **Sum of Present Values:** \[ PV(Earnings) = 0.224 + 0.257 + 0.288 + 0.318 + 0.347 + 31.15 \approx 32.786 \] **PE Ratio:** \[ PE = \frac{32.786}{1} = 32.786 \] Therefore, the estimated PE ratio for the firm is approximately **32.79**. |
28.75 |
| gpto1 | To estimate the PE ratio for the firm, we'll use a two-stage dividend discount model (DDM), which considers an initial high-growth period followed by a stable growth period. Here's a step-by-step solution: **Given:** - **High Growth Period (Years 1-5):** - Growth rate (\( g_1 \)) = 25% per year - Payout ratio (\( PR_1 \)) = 20% - **Stable Growth Period (Year 6 onward):** - Growth rate (\( g_2 \)) = 8% per year - Payout ratio (\( PR_2 \)) = 50% - **Cost of Equity (\( r \))** = 11.5% (calculated as \( r = R_f + \beta \times \text{Risk Premium} = 6\% + 1 \times 5.5\% \)) - **Initial Earnings per Share (\( E_0 \))** = Assume \$1 for simplicity. **Step 1: Project Earnings and Dividends for Years 1-5** For each year, calculate the earnings and dividends: **Year 1:** - Earnings: \( E_1 = E_0 \times (1 + g_1) = 1 \times 1.25 = \$1.25 \) - Dividends: \( D_1 = E_1 \times PR_1 = 1.25 \times 0.20 = \$0.25 \) **Year 2:** - Earnings: \( E_2 = E_1 \times (1 + g_1) = 1.25 \times 1.25 = \$1.5625 \) - Dividends: \( D_2 = E_2 \times PR_1 = 1.5625 \times 0.20 = \$0.3125 \) **Year 3:** - Earnings: \( E_3 = E_2 \times (1 + g_1) = 1.5625 \times 1.25 = \$1.953125 \) - Dividends: \( D_3 = E_3 \times PR_1 = 1.953125 \times 0.20 = \$0.390625 \) **Year 4:** - Earnings: \( E_4 = E_3 \times (1 + g_1) = 1.953125 \times 1.25 = \$2.44140625 \) - Dividends: \( D_4 = E_4 \times PR_1 = 2.44140625 \times 0.20 = \$0.48828125 \) **Year 5:** - Earnings: \( E_5 = E_4 \times (1 + g_1) = 2.44140625 \times 1.25 = \$3.0517578125 \) - Dividends: \( D_5 = E_5 \times PR_1 = 3.0517578125 \times 0.20 = \$0.6103515625 \) **Step 2: Calculate Terminal Value at Year 5** At the end of Year 5, the company enters the stable growth period. The terminal value (\( P_5 \)) is calculated using the Gordon Growth Model: - **Earnings in Year 6:** \( E_6 = E_5 \times (1 + g_2) = 3.0517578125 \times 1.08 = \$3.2958984375 \) - **Dividends in Year 6:** \( D_6 = E_6 \times PR_2 = 3.2958984375 \times 0.50 = \$1.64794921875 \) - **Terminal Value at Year 5:** \( P_5 = \frac{D_6}{r - g_2} = \frac{1.64794921875}{0.115 - 0.08} = \frac{1.64794921875}{0.035} = \$47.0843 \) **Step 3: Calculate Present Value of Dividends and Terminal Value** Discount each dividend and the terminal value back to present value using the cost of equity (\( r = 11.5\% \)): **Discount Factors:** \( DF_t = \frac{1}{(1 + r)^t} \) Compute \( (1 + r)^t \) for \( t = 1 \) to 5: - \( (1 + r)^1 = 1.115 \) - \( (1 + r)^2 = (1.115)^2 = 1.243225 \) - \( (1 + r)^3 = (1.115)^3 = 1.38639375 \) - \( (1 + r)^4 = (1.115)^4 = 1.54565281 \) - \( (1 + r)^5 = (1.115)^5 = 1.72319548 \) **Present Value Calculations:** - \( PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{0.25}{1.115} = \$0.22472 \) - \( PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{0.3125}{1.243225} = \$0.25130 \) - \( PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{0.390625}{1.38639375} = \$0.28172 \) - \( PV(D_4) = \frac{D_4}{(1 + r)^4} = \frac{0.48828125}{1.54565281} = \$0.31602 \) - \( PV(D_5) = \frac{D_5}{(1 + r)^5} = \frac{0.6103515625}{1.72319548} = \$0.35428 \) - \( PV(P_5) = \frac{P_5}{(1 + r)^5} = \frac{47.0843}{1.72319548} = \$27.326 \) **Total Present Value (Price):** \( P_0 = PV(D_1) + PV(D_2) + PV(D_3) + PV(D_4) + PV(D_5) + PV(P_5) \) \( P_0 = 0.22472 + 0.25130 + 0.28172 + 0.31602 + 0.35428 + 27.326 = \$28.754 \) **Step 4: Calculate the PE Ratio** Since \( E_0 = \$1 \): \( \text{PE Ratio} = \frac{P_0}{E_0} = \frac{28.754}{1} = 28.754 \) **Answer:** The estimated PE ratio for the firm is **approximately 28.75**. |
28.75 |
| deepseekv3 | To estimate the PE ratio for the firm, we can use the **Two-Stage Dividend Discount Model (DDM)**, which is well-suited for firms with high growth in the initial period and stable growth thereafter. Here’s the step-by-step solution: --- ### Step 1: Key Inputs - **High growth phase**: - Length: 5 years - Growth rate (\(g_1\)): 25% - Payout ratio (\(PR_1\)): 20% - **Stable growth phase**: - Growth rate (\(g_2\)): 8% - Payout ratio (\(PR_2\)): 50% - **Cost of equity (\(r\))**: 11.5% - **Risk-free rate**: 6% - **Risk premium**: 5.5% --- ### Step 2: Calculate Earnings in Year 0 (\(E_0\)) Assume \(E_0 = 1\) (we can scale the final PE ratio later if needed). --- ### Step 3: Earnings Growth in High Growth Phase Earnings grow at 25% for 5 years. Earnings in Year 5 (\(E_5\)) are: \[ E_5 = E_0 \times (1 + g_1)^5 = 1 \times (1.25)^5 = 3.052 \] --- ### Step 4: Dividends in High Growth Phase Dividends in Year 5 (\(D_5\)) are: \[ D_5 = E_5 \times PR_1 = 3.052 \times 0.20 = 0.610 \] --- ### Step 5: Earnings and Dividends in Stable Growth Phase Earnings in Year 6 (\(E_6\)) grow at 8%: \[ E_6 = E_5 \times (1 + g_2) = 3.052 \times 1.08 = 3.296 \] Dividends in Year 6 (\(D_6\)) are: \[ D_6 = E_6 \times PR_2 = 3.296 \times 0.50 = 1.648 \] --- ### Step 6: Terminal Value at the End of Year 5 The terminal value (\(TV_5\)) is calculated using the Gordon Growth Model: \[ TV_5 = \frac{D_6}{r - g_2} = \frac{1.648}{0.115 - 0.08} = \frac{1.648}{0.035} = 47.086 \] --- ### Step 7: Present Value of Dividends and Terminal Value Discount the dividends and terminal value back to Year 0 using the cost of equity (\(r = 11.5\%\)): \[ PV = \sum_{t=1}^{5} \frac{D_t}{(1 + r)^t} + \frac{TV_5}{(1 + r)^5} \] Since dividends grow at 25% in the high-growth phase, the present value of dividends is: \[ PV_{dividends} = \frac{0.20 \times 1.25^1}{1.115^1} + \frac{0.20 \times 1.25^2}{1.115^2} + \frac{0.20 \times 1.25^3}{1.115^3} + \frac{0.20 \times 1.25^4}{1.115^4} + \frac{0.20 \times 1.25^5}{1.115^5} \] \[ PV_{dividends} = 0.224 + 0.251 + 0.281 + 0.315 + 0.353 = 1.424 \] The present value of the terminal value is: \[ PV_{TV} = \frac{47.086}{1.115^5} = \frac{47.086}{1.744} = 26.99 \] --- ### Step 8: Total Present Value and PE Ratio The total present value of the firm’s cash flows is: \[ PV_{total} = PV_{dividends} + PV_{TV} = 1.424 + 26.99 = 28.414 \] Since \(E_0 = 1\), the PE ratio is: \[ PE = \frac{PV_{total}}{E_0} = 28.414 \] --- ### Final Answer: The estimated PE ratio for the firm is **28.41**. |
28.75 |