To find the required rate of return for a stock, you can use the Gordon Growth Model (also known as the Dividend Discount Model for constant growth). This model states that the price of a stock is equal to the next expected dividend divided by the difference between the required rate of return and the growth rate of the dividend.
The formula is as follows:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \( P_0 \) = Current stock price
- \( D_1 \) = Dividend expected next year
- \( r \) = Required rate of return
- \( g \) = Growth rate of dividends
First, we need to calculate \( D_1 \), the dividend expected next year. Since the last dividend paid was $4.64 and it grows at a rate of 2.25%, we calculate \( D_1 \) as follows:
\[ D_1 = D_0 \times (1 + g) = 4.64 \times (1 + 0.0225) = 4.64 \times 1.0225 \approx 4.748 \]
Now we have:
- \( D_1 \approx 4.748 \)
- \( P_0 = 41.47 \)
- \( g = 0.0225 \)
We can now plug these values into the Gordon Growth Model formula to solve for \( r \):
\[ 41.47 = \frac{4.748}{r - 0.0225} \]
Rearranging this equation to solve for \( r \):
\[ 41.47(r - 0.0225) = 4.748 \]
\[ 41.47r - 0.933075 = 4.748 \]
\[ 41.47r = 4.748 + 0.933075 \]
\[ 41.47r = 5.681075 \]
\[ r = \frac{5.681075}{41.47} \approx 0.1375 \text{ or } 13.75% \]
Thus, the required rate of return for the stock is approximately 13.75%.
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