To determine the new price of the stock after the announcement, we can use the Gordon Growth Model (Dividend Discount Model) which states:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \( P_0 \) = price of the stock today
- \( D_1 \) = dividend in year 1
- \( r \) = required rate of return
- \( g \) = growth rate of dividends
From the information provided:
- The new dividend \( D_1 \) is $2.09 (which is the dividend in year 1).
- The new growth rate \( g \) is 4.16% or 0.0416.
To use the formula, we first need the required rate of return \( r \). We can estimate it using the previous dividend and growth information. Initially, dividends were expected to grow at 2.50% (or 0.025) and the dividend was $3.14. The price was $18.88. We can arrange the Gordon Growth Model to find \( r \):
\[ P_0 = \frac{D_0(1+g)}{r - g} \]
Substituting in the known values for \( P_0 = 18.88 \), \( D_0 = 3.14 \), and \( g = 0.025 \):
\[ 18.88 = \frac{3.14(1 + 0.025)}{r - 0.025} \]
Calculating \( D_1 \):
\[ D_1 = 3.14(1 + 0.025) = 3.14 \times 1.025 = 3.2155 \]
Now substitute back into the equation:
\[ 18.88 = \frac{3.2155}{r - 0.025} \]
Cross-multiplying gives:
\[ 18.88(r - 0.025) = 3.2155 \]
Simplifying:
\[ 18.88r - 0.472 = 3.2155 \]
\[ 18.88r = 3.2155 + 0.472 \]
\[ 18.88r = 3.6875 \]
\[ r = \frac{3.6875}{18.88} \approx 0.1954 \text{ or } 19.54% \]
Now with the new growth rate \( g = 4.16% \) or \( 0.0416 \) and the new \( D_1 = 2.09 \):
\[ P_0 = \frac{2.09}{0.1954 - 0.0416} \]
Calculating the denominator:
\[ 0.1954 - 0.0416 = 0.1538 \]
Now substituting into the formula:
\[ P_0 = \frac{2.09}{0.1538} \approx 13.59 \]
Therefore, the new price of the stock after the announcement is approximately $13.59.
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