To find the new price of the stock after the announcement using the Gordon Growth Model (also known as the Dividend Discount Model), we can use the formula:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \( P_0 \) = Price of the stock today
- \( D_1 \) = Dividend next year
- \( r \) = Required rate of return
- \( g \) = Growth rate of dividends
From the information given:
- The new dividend \( D_1 \) is $2.09.
- The new growth rate \( g \) is 3.74%.
Step 1: We need to find the required rate of return \( r \).
We can estimate \( r \) based on the old dividend, assuming the old stock price was justified at the old growth rate (2.50%):
Using the old dividend \( D_0 = 3.04 \) and price \( P_0 = 16.75 \): \[ P_0 = \frac{D_0 (1 + g)}{r - g} \]
Substituting the known values before the announcement:
- \( D_0 \) = 3.04
- \( g \) = 0.025 (2.50%)
Calculating \( D_1 \) for the old dividend: \[ D_1 = D_0 (1 + g) = 3.04 \times (1 + 0.025) = 3.04 \times 1.025 = 3.11 \]
Now, substituting into the price formula to find \( r \): \[ 16.75 = \frac{3.11}{r - 0.025} \] \[ 16.75(r - 0.025) = 3.11 \] \[ 16.75r - 0.41875 = 3.11 \] \[ 16.75r = 3.11 + 0.41875 = 3.52875 \] \[ r = \frac{3.52875}{16.75} \approx 0.2104 \text{ or } 21.04% \]
Step 2: Now we can use this \( r \) to find the new stock price after the announcement.
Now substituting the new \( D_1 = 2.09 \) and new growth \( g = 0.0374 \):
\[ P_0 = \frac{2.09}{0.2104 - 0.0374} \] \[ P_0 = \frac{2.09}{0.173} \] \[ P_0 \approx 12.05 \]
Therefore, the new price of the stock after the announcement is approximately $12.05.
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