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 the model is:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \(P_0\) is the price of the stock today.
- \(D_1\) is the expected dividend in the next year.
- \(r\) is the required rate of return (discount rate).
- \(g\) is the growth rate of dividends.
Step 1: Determine \(D_1\)
The new dividend \(D_1\) is given as $2.14.
Step 2: Establish the growth rate \(g\)
The new growth rate \(g\) after the announcement is 3.88%, or 0.0388 in decimal form.
Step 3: Find the required rate of return \(r\)
To find the required rate of return, we can estimate it using the previous stock price and the expected future dividends before the announcement. Prior to the announcement, the dividends grew at 2.50%, which gives us a model:
The old dividend is \(D_0 = 3.01\). The expected dividend for next year based on the old growth rate (2.50%) is:
\[ D_1 = D_0 \times (1 + g) = 3.01 \times (1 + 0.025) = 3.01 \times 1.025 = 3.07925 \text{ (old D1)} \]
Using the old price before the announcement:
\[ 15.34 = \frac{D_1}{r - 0.025} \]
Substituting the old dividend for \(D_1\):
\[ 15.34 = \frac{3.07925}{r - 0.025} \]
Now solve for \(r\):
\[ 15.34 \times (r - 0.025) = 3.07925 \]
\[ 15.34r - 0.3835 = 3.07925 \]
\[ 15.34r = 3.07925 + 0.3835 \]
\[ 15.34r = 3.46275 \]
\[ r = \frac{3.46275}{15.34} \approx 0.2255 \text{ or } 22.55% \]
Step 4: Calculate the new price \(P_0\)
Now we can plug in the new dividend \(D_1 = 2.14\), the calculated \(r \approx 0.2255\), and the new growth rate \(g = 0.0388\) into the Gordon Growth Model formula:
\[ P_0 = \frac{2.14}{0.2255 - 0.0388} \]
Calculating the denominator:
\[ 0.2255 - 0.0388 = 0.1867 \]
Calculating \(P_0\):
\[ P_0 = \frac{2.14}{0.1867} \approx 11.396 \]
Conclusion
The new price of the stock after the announcement is approximately $11.40.
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