To find the new price of the stock after the announcement, we can use the Gordon Growth Model (also known as the Dividend Discount Model) which states that the price of a stock is equal to the expected dividend next year divided by the difference between the required rate of return and the growth rate.
The formula is given by:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \( P_0 \) = price of the stock today
- \( D_1 \) = dividend expected next year
- \( r \) = required rate of return
- \( g \) = growth rate of dividends
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Determine the expected dividend next year, \( D_1 \): The company announced that the dividend will be \( D_1 = 2.02 \).
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Determine the growth rate, \( g \): The new growth rate of dividends is \( g = 4.05% = 0.0405 \).
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Determine the required rate of return, \( r \): We need to estimate the required rate of return. However, we can derive it using the initial stock price and the old growth rate.
The initial stock price was $18.05 with a dividend of $3.12 and growth rate of 2.50% or \( 0.025 \). We can rearrange the formula to find \( r \):
\[ 18.05 = \frac{3.12}{r - 0.025} \]
Solving for \( r \):
\[ 18.05(r - 0.025) = 3.12 \] \[ 18.05r - 0.45125 = 3.12 \] \[ 18.05r = 3.12 + 0.45125 \] \[ 18.05r = 3.57125 \] \[ r = \frac{3.57125}{18.05} \approx 0.1971 \text{ or } 19.71% \]
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Now apply the new dividend and growth rate: Using \( D_1 = 2.02 \), \( g = 0.0405 \), and \( r = 0.1971 \):
\[ P_0 = \frac{2.02}{0.1971 - 0.0405} \] \[ P_0 = \frac{2.02}{0.1566} \approx 12.88 \]
The new price of the stock after the announcement is approximately $12.88.
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