To find the new price of the stock after the dividend cut, we can use the Gordon Growth Model (also known as the Dividend Discount Model). The formula is:
\[ P = \frac{D_1}{r - g} \]
Where:
- \(P\) = price of the stock
- \(D_1\) = dividend for the next year
- \(r\) = required rate of return
- \(g\) = growth rate of the dividend
Step 1: Identify the new dividend \(D_1\)
After the announcement, the new dividend (year 1) is given as \(D_1 = 2.19\).
Step 2: Identify the new growth rate \(g\)
The expected growth rate after the announcement is given as \(g = 3.90% = 0.039\).
Step 3: Determine the required rate of return \(r\)
To find \(r\), we can use the old dividend and stock price before the announcement. The old dividend was \(D_0 = 3.09\), and it was expected to grow at a rate of \(g = 2.50% = 0.025\). The stock price before the announcement was \(P_0 = 15.62\).
Using the Gordon Growth Model for the original situation:
\[ P_0 = \frac{D_1}{r - g} \]
We need to find \(D_1\) (the dividend after one year based on the old growth rate):
\[ D_1 = D_0 \times (1 + g) = 3.09 \times (1 + 0.025) = 3.09 \times 1.025 \approx 3.16 \]
Now we plug \(P_0\) into the Gordon Growth Model:
\[ 15.62 = \frac{3.16}{r - 0.025} \]
Now we can solve for \(r\):
\[ 15.62(r - 0.025) = 3.16 \]
\[ r - 0.025 = \frac{3.16}{15.62} \]
\[ r - 0.025 \approx 0.2025 \]
\[ r \approx 0.2025 + 0.025 = 0.2275 \text{ or } 22.75% \]
Step 4: Calculate the new stock price \(P\) using the new growth rate
Now, we have \(r = 0.2275\) and the new growth rate \(g = 0.039\). Now we can find the new price \(P\):
\[ P = \frac{2.19}{0.2275 - 0.039} \]
Calculating the denominator:
\[ 0.2275 - 0.039 = 0.1885 \]
Now substituting back into the formula:
\[ P = \frac{2.19}{0.1885} \approx 11.63 \]
Thus, the new price of the stock after the announcement is approximately $11.63.
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