Let's break down the information and help you answer the questions. --- ### **Given Data** - **Ragan, Inc.** - EPS (last year): $4.54 - Dividend paid per sibling: $60,000 each - Shares per sibling: 50,000 each (total 100,000 shares) - ROE (last year): 16% - Required return (siblings’ assumption): 14% - Industry average required return: 11.67% - **Competitors**: Data is in the table for context. #### **Dividends per Share (D₀):** Total dividend paid = $60,000 × 2 = $120,000 Total shares = 100,000 Dividends per share (D₀) = $120,000 / 100,000 = **$1.20** --- ## **1. Assuming the company continues its current growth rate, what is the value per share of the company’s stock?** ### **Step 1: Calculate Growth Rate (g)** Using the ROE and dividend payout ratio: - **Dividend payout ratio** = Dividends / Earnings = $1.20 / $4.54 ≈ **0.2645 (26.45%)** - **Retention ratio (b)** = 1 - payout ratio = 0.7355 (73.55%) - **Growth rate (g)** = ROE × Retention ratio = 0.16 × 0.7355 = **0.1177 (11.77%)** ### **Step 2: Use the Gordon Growth Model (Dividend Discount Model)** \[ P_0 = \frac{D_1}{r - g} \] Where: - \( D_1 \) = Next year’s dividend = \( D_0 \times (1+g) = 1.20 \times 1.1177 = \$1.341 \) - \( r \) = Required return = 14% (0.14) - \( g \) = Growth rate = 11.77% (0.1177) \[ P_0 = \frac{1.341}{0.14 - 0.1177} = \frac{1.341}{0.0223} = \$60.17 \] **Stock value per share: \$60.17** --- ## **2. Josh’s Scenario:** - **Technological advantage lasts only 5 years, then growth slows to industry average.** - **Industry required return: 11.67%** - **Industry average growth rate:** Not directly given, but can approximate using the table’s industry averages: - Industry ROE = 12.33% - Industry payout ratio = $0.40 / $0.60 = 66.67% - Retention = 33.33% - Industry growth rate = 12.33% × 33.33% ≈ 4.11% ### **Step 1: Calculate Dividends for the Next 5 Years** - D₀ = $1.20, growth for 5 years at 11.77% - D₁ = $1.20 × 1.1177 = $1.341 - D₂ = $1.341 × 1.1177 = $1.500 - D₃ = $1.500 × 1.1177 = $1.677 - D₄ = $1.677 × 1.1177 = $1.875 - D₅ = $1.875 × 1.1177 = $2.099 ### **Step 2: Calculate Terminal Value at Year 5** - After year 5, growth slows to **4.11%** - Terminal value at year 5: \[ TV_5 = \frac{D_6}{r - g_{industry}} = \frac{2.099 \times 1.0411}{0.1167 - 0.0411} \] - \( D_6 = 2.099 \times 1.0411 = 2.186 \) - \( r = 0.1167 \), \( g = 0.0411 \) - \( TV_5 = \frac{2.186}{0.0756} = \$28.92 \) ### **Step 3: Discount Dividends and Terminal Value to Present** \[ P_0 = \frac{D_1}{(1+r)^1} + \frac{D_2}{(1+r)^2} + \frac{D_3}{(1+r)^3} + \frac{D_4}{(1+r)^4} + \frac{D_5}{(1+r)^5} + \frac{TV_5}{(1+r)^5} \] Where \( r = 0.1167 \): - \( (1.1167)^1 = 1.1167 \) - \( (1.1167)^2 = 1.247 \) - \( (1.1167)^3 = 1.393 \) - \( (1.1167)^4 = 1.555 \) - \( (1.1167)^5 = 1.736 \) Now, discount each: 1. \( \frac{1.341}{1.1167} = 1.201 \) 2. \( \frac{1.500}{1.247} = 1.203 \) 3. \( \frac{1.677}{1.393} = 1.204 \) 4. \( \frac{1.875}{1.555} = 1.206 \) 5. \( \frac{2.099}{1.736} = 1.209 \) 6. \( \frac{28.92}{1.736} = 16.67 \) Sum: \( 1.201 + 1.203 + 1.204 + 1.206 + 1.209 + 16.67 = \$22.693 \) **Stock value per share under Josh's assumptions: \$22.69** --- ## **Summary Table** | Scenario | Stock Value per Share | |----------------------------------------------|----------------------| | Continue current growth & required return | $60.17 | | Josh's scenario (slower growth, lower r) | $22.69 | --- If you need the calculations shown step by step or a spreadsheet format, let me know!

Introduction: To determine the intrinsic value of a company's stock, financial analysts employ valuation models that incorporate key financial metrics, growth assumptions, and investor-required returns. The primary concept underpinning these models is the present value of expected future dividends or earnings, discounted at an appropriate rate that reflects the risk profile of the investment. The Gordon Growth Model (also known as the Dividend Discount Model for a perpetually growing dividend stream) is a widely used approach for valuing stocks that are expected to grow at a stable rate indefinitely. This model assumes that dividends grow at a constant rate and that the required rate of return exceeds this growth rate. In the context of Ragan, Inc., understanding its current earnings, dividends, growth prospects, and the required rate of return allows for an estimation of its stock value. The company's recent earnings, dividend payout, and return on equity inform the calculation of its sustainable growth rate, which is essential for applying the Gordon Growth Model. Furthermore, variations in assumptions about growth duration and required return impact valuation significantly, highlighting the importance of scenario analysis in stock valuation. **Explanation:** This introduction emphasizes the fundamental principles of stock valuation, particularly the relevance of dividends, growth rates, and discount rates. It clarifies that the Gordon Growth Model relies on stable dividend growth assumptions, which are central to the problem. Recognizing how earnings, dividends, and return on equity relate to growth provides the foundation for subsequent calculations. This conceptual background ensures a comprehensive understanding of how the valuation process operates and why specific financial metrics are used. --- ### **Presentation of Relevant Formulas Required To Solve The Question:** 1. **Dividend Growth Rate (g):** \[ g = ROE \times Retention\,Ratio \] *This formula calculates the sustainable growth rate of dividends based on the company's profitability and dividend payout policy.* 2. **Dividend Next Year (D₁):** \[ D_1 = D_0 \times (1 + g) \] *This computes the expected dividend in the upcoming year, which serves as the numerator in the valuation formula.* 3. **Stock Valuation via Gordon Growth Model:** \[ P_0 = \frac{D_1}{r - g} \] *This fundamental formula estimates the present value of all future dividends assuming constant growth, where \( r \) is the required rate of return.* 4. **Terminal Value Calculation (for scenarios with finite growth periods):** \[ TV_{n} = \frac{D_{n+1}}{r - g_{terminal}} \] *This determines the value at the end of a high-growth period, discounted back to the present.* 5. **Present Value of Future Dividends and Terminal Value:** \[ P_0 = \sum_{t=1}^{n} \frac{D_t}{(1 + r)^t} + \frac{TV}{(1 + r)^n} \] *This aggregates the discounted expected dividends and the discounted terminal value to arrive at the current stock value.* **Explanation:** Each formula plays a critical role in valuation. The growth rate formula links profitability and dividend policy to future earnings and dividends, providing a basis for projecting dividends. The Gordon Growth Model directly relates dividends, growth, and required return to estimate intrinsic value. Terminal value calculations extend valuation when growth is expected to change after a certain period, accommodating more realistic scenarios. Discounting future cash flows aligns with the time value of money principle, ensuring that future expectations are appropriately weighted to their present worth. --- ### **A Detailed Step-by-Step Solution:** **Step 1: Calculate the Dividend Payout and Retention Ratios** - **Total dividends paid:** \[ \text{Dividends per sibling} = \$60,000 \] \[ \text{Total dividends} = 2 \times \$60,000 = \$120,000 \] - **Total shares outstanding:** \[ 50,000 \text{ shares per sibling} \times 2 = 100,000 \text{ shares} \] - **Dividends per share (D₀):** \[ D_0 = \frac{\$120,000}{100,000} = \$1.20 \] - **Earnings per share (EPS):** \[ EPS = \$4.54 \] - **Dividend payout ratio:** \[ \text{Payout ratio} = \frac{D_0}{EPS} = \frac{\$1.20}{\$4.54} \approx 0.264 \] (26.4%) - **Retention ratio:** \[ b = 1 - 0.264 = 0.736 \] (73.6%) **Explanation:** This step quantifies how much of the earnings are retained for growth versus paid out as dividends. The retention ratio directly influences the company's sustainable growth rate, vital for valuation. --- **Step 2: Calculate the Sustainable Growth Rate (g)** \[ g = ROE \times b = 0.16 \times 0.736 = 0.1178 \text{ or } 11.78\% \] **Explanation:** The sustainable growth rate reflects how rapidly earnings and dividends can grow indefinitely, assuming the current return on equity and payout policy remain constant. --- **Step 3: Calculate the Expected Dividends Next Year (D₁)** \[ D_1 = D_0 \times (1 + g) = \$1.20 \times (1 + 0.1178) = \$1.20 \times 1.1178 = \$1.341 \] **Explanation:** Estimating next year's dividend is necessary for applying the Gordon Growth Model, which uses the expected dividend in the valuation formula. --- **Step 4: Apply the Gordon Growth Model to Find Stock Price (P₀)** \[ P_0 = \frac{D_1}{r - g} = \frac{\$1.341}{0.14 - 0.1178} = \frac{\$1.341}{0.0222} \approx \$60.27 \] **Explanation:** This formula calculates the intrinsic value of the stock based on the perpetuity of dividends growing at rate \( g \), discounted at the required rate \( r \). The relatively high valuation reflects the company's growth prospects and payout policy. --- **Step 5: Summary of Scenario 1 Result** **The estimated intrinsic value per share, assuming constant growth and a 14% required return, is approximately \$60.27.** --- **Step 6: Scenario 2—Adjustments Based on New Assumptions** - Growth duration limited to 5 years, after which growth slows to industry average (~4.11%). - Required return for the industry is 11.67%. **Step 6.1: Calculate Dividends for the Next 5 Years at 11.78% growth** - Year 1: \$1.341 - Year 2: \$1.341 × 1.1178 = \$1.500 - Year 3: \$1.500 × 1.1178 = \$1.677 - Year 4: \$1.677 × 1.1178 = \$1.876 - Year 5: \$1.876 × 1.1178 = \$2.099 **Explanation:** Forecasting dividends over the high-growth period provides the basis for calculating the terminal value after this period, capturing the value of the stock when growth stabilizes. --- **Step 6.2: Calculate Terminal Value at Year 5** - Terminal growth rate post-year 5: g_terminal = 4.11% - Next year's dividend after year 5: \[ D_6 = D_5 \times (1 + g_{terminal}) = 2.099 \times 1.0411 = 2.186 \] - Terminal value at year 5: \[ TV_5 = \frac{D_6}{r_{industry} - g_{terminal}} = \frac{2.186}{0.1167 - 0.0411} = \frac{2.186}{0.0756} \approx \$28.89 \] **Explanation:** The terminal value estimates the ongoing value of the stock after the high-growth period, discounted back to the valuation date. --- **Step 6.3: Discount All Cash Flows to Present** - Discount factors for 5 years at 11.67%: \[ (1 + r)^t \] Year 1: 1.1167 Year 2: 1.247 Year 3: 1.393 Year 4: 1.555 Year 5: 1.736 - Present value of dividends: \[ PV_{dividends} = \frac{1.341}{1.1167} + \frac{1.500}{1.247} + \frac{1.677}{1.393} + \frac{1.876}{1.555} + \frac{2.099}{1.736} \approx \$1.201 + \$1.203 + \$1.204 + \$1.206 + \$1.209 = \$6.023 \] - Present value of terminal value: \[ PV_{TV} = \frac{28.89}{1.736} \approx \$16.65 \] - **Total stock value:** \[ P_0 = PV_{dividends} + PV_{TV} = \$6.023 + \$16.65 = \$22.67 \] **Explanation:** Discounting both the projected dividends during high growth and the terminal value back to the present provides an estimate of the stock's fair value under the revised scenario. --- ### **Conclusion:** - **Under the assumption of perpetual growth at 11.78% and a required return of 14%, the stock value is approximately \$60.27.** - **Adjusting for a finite high-growth period of five years, after which growth slows to industry average (~4.11%), and using a lower required return of 11.67%, the estimated stock value decreases to approximately \$22.67.** These valuations demonstrate how different growth assumptions and discount rates significantly influence stock valuation, emphasizing the importance of scenario analysis for investment decision-making.