Let's solve each question step-by-step based on the information provided. --- ### **Given Data** - Last dividend paid, \( D_0 = \$1.80 \) - Dividend growth rate for next 4 years, \( g_1 = 9\% \) - Dividend growth rate after year 4, \( g_2 = 4\% \) - Required rate of return, \( r = 13\% \) - Earnings Per Share (EPS) = \$3.50 - Industry average P/E = 18 --- ### **4. Calculate the stock's value using the Price-to-Earnings (P/E) multiple approach.** **Formula:** \[ \text{Stock Value (P/E Multiple)} = \text{EPS} \times \text{Industry P/E} \] **Calculation:** \[ \text{Stock Value} = \$3.50 \times 18 = \$63.00 \] **Final Answer:** \[ \boxed{\$63.00} \] --- ### **5. If investors become more risk-averse and the required rate of return for Synergy Innovations Inc. increases, how would this impact the intrinsic value calculated using the DDM? Explain your reasoning.** **Explanation:** - **In DDM, Value = PV of future dividends discounted at the required rate of return.** - If the required rate of return (\( r \)) increases, the denominator in the DDM formula increases, which **reduces the present value of future dividends**. - **Therefore, the intrinsic value of the stock decreases** as the required return rises. --- ### **6. Why might the stock value calculated using the P/E multiple approach differ from the intrinsic value calculated using the DDM? Discuss at least one potential reason.** **Explanation:** - The **P/E multiple approach** reflects how the market values similar companies' earnings, incorporating market sentiment, industry trends, and non-dividend factors. - The **DDM** is based purely on expected future dividends and the investor’s required rate of return, focusing strictly on dividend cash flows. - **A key reason for the difference:** - The company may **retain a significant portion of earnings** for growth, leading to low dividends but high earnings. Thus, P/E (based on earnings) could value the company higher than DDM (based on dividends). - Additionally, **market conditions or investor optimism/pessimism** may drive the P/E multiple higher or lower than what intrinsic value (DDM) would suggest. --- ### **Summary Table** | Method | Calculated Stock Value | |--------------------|-----------------------| | P/E Multiple | $63.00 | | DDM | (To be calculated if required) | Let me know if you need the full DDM calculation as well!

Let's proceed with the detailed step-by-step calculation of the **Intrinsic Value of the stock using the Dividend Discount Model (DDM)**, considering the data provided: --- ### **Given Data Recap:** - Last dividend paid, \( D_0 = \$1.80 \) - Growth rate for next 4 years, \( g_1 = 9\% \) - Growth rate after year 4, \( g_2 = 4\% \) - Required rate of return, \( r = 13\% \) --- ### **Step 1: Calculate the dividends for the next 4 years** The dividend in year \( t \), \( D_t \), is given by: \[ D_t = D_0 \times (1 + g_1)^t \] - **Year 1:** \[ D_1 = 1.80 \times (1 + 0.09) = 1.80 \times 1.09 = \$1.962 \] - **Year 2:** \[ D_2 = 1.80 \times (1.09)^2 = 1.80 \times 1.1881 = \$2.139 \] - **Year 3:** \[ D_3 = 1.80 \times (1.09)^3 = 1.80 \times 1.295 = \$2.341 \] - **Year 4:** \[ D_4 = 1.80 \times (1.09)^4 = 1.80 \times 1.411 = \$2.567 \] --- ### **Step 2: Calculate the dividend in Year 5, which marks the start of the perpetual growth at \( g_2 = 4\% \)** \[ D_5 = D_4 \times (1 + g_2) = 2.567 \times 1.04 = \$2.669 \] --- ### **Step 3: Calculate the **Terminal Value** at the end of Year 4, which is the present value of all dividends from Year 5 onwards** The terminal (perpetual) value at Year 4: \[ P_4 = \frac{D_5}{r - g_2} = \frac{2.669}{0.13 - 0.04} = \frac{2.669}{0.09} = \$29.655 \] --- ### **Step 4: Discount all future dividends and the terminal value back to the present (Year 0)** Using the discount factor: \[ PV = \frac{\text{Future value}}{(1 + r)^t} \] - **PV of \( D_1 \):** \[ PV_1 = \frac{1.962}{(1.13)^1} = \frac{1.962}{1.13} \approx \$1.737 \] - **PV of \( D_2 \):** \[ PV_2 = \frac{2.139}{(1.13)^2} = \frac{2.139}{1.2769} \approx \$1.674 \] - **PV of \( D_3 \):** \[ PV_3 = \frac{2.341}{(1.13)^3} = \frac{2.341}{1.4429} \approx \$1.622 \] - **PV of \( D_4 \):** \[ PV_4 = \frac{2.567}{(1.13)^4} = \frac{2.567}{1.629} \approx \$1.575 \] - **PV of Terminal Value \( P_4 \):** \[ PV_{P_4} = \frac{29.655}{(1.13)^4} = \frac{29.655}{1.629} \approx \$18.209 \] --- ### **Step 5: Sum all present values to get the intrinsic stock value** \[ \text{Intrinsic Value} = PV_1 + PV_2 + PV_3 + PV_4 + PV_{P_4} \] \[ = 1.737 + 1.674 + 1.622 + 1.575 + 18.209 = \$24.817 \] --- ### **### Final Answer:** \[ \boxed{ \text{Intrinsic value of the stock} \approx \textbf{\$24.82} } \] --- ### **Summary:** - **P/E based valuation:** \$63.00 - **DDM-based valuation:** \$24.82 **Note:** The discrepancy arises because the P/E approach is based on earnings and market sentiment, whereas DDM is based strictly on expected dividend flows discounted at the required rate. Both methods provide valuable insights, but they often differ due to underlying assumptions and market conditions.