To calculate the new price of the stock after the announcement, we can use the Gordon Growth Model (also known as the Dividend Discount Model). The formula for this model is:
\[ P = \frac{D_1}{r - g} \]
Where:
- \( P \) = Price of the stock
- \( D_1 \) = Dividend in the next year
- \( r \) = Required rate of return
- \( g \) = Growth rate of dividends
Step 1: Identify the new dividend and growth rate
From the information provided:
- The new dividend (D₁) is $2.05.
- The new growth rate (g) is 3.53% or 0.0353 in decimal.
Step 2: Estimate the required rate of return (r)
Before the announcement, the stock price was $17.50, and the dividend was initially $3.14. Using the initial growth rate of 2.50% (or 0.025 in decimal), we can calculate the required rate of return \( r \) prior to the announcement:
\[ P = \frac{D_1}{r - g} \]
Using \( D_1 = 3.14 \), \( g = 0.025 \), and \( P = 17.50 \):
\[ 17.50 = \frac{3.14}{r - 0.025} \]
Cross-multiplying gives us:
\[ 17.50(r - 0.025) = 3.14 \]
Expanding and solving for r:
\[ 17.50r - 0.4375 = 3.14 \]
\[ 17.50r = 3.14 + 0.4375 \]
\[ 17.50r = 3.5775 \]
\[ r = \frac{3.5775}{17.50} \approx 0.2045 \text{ or } 20.45% \]
Step 3: Calculate the new price of the stock
Now that we have \( r \approx 0.2045 \) and \( g = 0.0353 \), we can use the Gordon Growth Model again to find the new price \( P \):
\[ P = \frac{D_1}{r - g} = \frac{2.05}{0.2045 - 0.0353} \]
Calculating the denominator:
\[ r - g \approx 0.2045 - 0.0353 \approx 0.1692 \]
Now substituting in the values:
\[ P = \frac{2.05}{0.1692} \approx 12.13 \]
Conclusion
Therefore, the new price of the stock after the announcement is approximately $12.13.
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