To find the required rate of return for the stock, you can use the Gordon Growth Model (also known as the Dividend Discount Model). The formula for the price of a stock in this model is:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \( P_0 \) is the current stock price ($49.85)
- \( D_1 \) is the expected dividend next year
- \( r \) is the required rate of return
- \( g \) is the growth rate of the dividend (2.68%, or 0.0268 as a decimal)
First, we need to calculate \( D_1 \), the dividend expected next year. Since the dividend just paid (\( D_0 \)) is $5.15, we can find the next year's dividend using the growth rate:
\[ D_1 = D_0 \times (1 + g) = 5.15 \times (1 + 0.0268) = 5.15 \times 1.0268 \approx 5.28382 \]
Now, we can substitute \( P_0 \), \( D_1 \), and \( g \) into the Gordon Growth Model formula and solve for \( r \):
\[ 49.85 = \frac{5.28382}{r - 0.0268} \]
Rearranging this equation gives:
\[ 49.85 \times (r - 0.0268) = 5.28382 \]
\[ 49.85r - 1.33538 = 5.28382 \]
\[ 49.85r = 5.28382 + 1.33538 \]
\[ 49.85r = 6.6192 \]
Now solving for \( r \):
\[ r = \frac{6.6192}{49.85} \approx 0.1324 \text{ or } 13.24% \]
Therefore, the required rate of return for the stock is approximately 13.24%.
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