1. Calculating the Opportunity Cost of Capital

The opportunity cost of capital can be calculated using the Capital Asset Pricing Model (CAPM):

$r = r_f + \beta (r_m - r_f)$

Where:

• $r_f = 4\%$ (T-bill rate, the risk-free rate),
• $r_m - r_f = 7\%$ (market risk premium),
• $\beta = 0.75$.

Substituting the values into the formula:

$r = 4\% + 0.75 \times 7\%$ $r = 4\% + 5.25\%$ $r = 9.25\%$

Thus, the opportunity cost of capital is 9.25%.

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2. Stock Value Assessment

Now, let's determine if the stock is a good or bad buy by comparing the expected rate of return with the opportunity cost of capital.

Expected Rate of Return (E[R]):

The expected rate of return can be calculated using the dividend and the expected price increase:

$\text{Expected rate of return} = \frac{\text{Dividend} + (\text{Expected price} - \text{Current price})}{\text{Current price}}$

Where:

• Dividend = $2,
• Expected year-end price = $52,
• Current price = $50.

$\text{Expected rate of return} = \frac{2 + (52 - 50)}{50} = \frac{2 + 2}{50} = \frac{4}{50} = 8\%$

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3. Good or Bad Buy?

• The opportunity cost of capital is 9.25%.
• The expected return on the stock is 8%.

Since the expected return (8%) is lower than the opportunity cost of capital (9.25%), the stock is overvalued and is not a good buy. Investors will likely sell the stock, as they can find better opportunities elsewhere.

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4. Equilibrium Price Calculation

At equilibrium, the expected rate of return should equal the opportunity cost of capital (9.25%). We can use the same formula for the expected rate of return and solve for the price that will make the expected return match the opportunity cost of capital.

$9.25\% = \frac{2 + (P - 50)}{50}$

First, convert 9.25% to a decimal:

$0.0925 = \frac{2 + (P - 50)}{50}$

Now solve for $P$:

$0.0925 \times 50 = 2 + (P - 50)$ $4.625 = 2 + (P - 50)$ $4.625 - 2 = P - 50$ $2.625 = P - 50$ $P = 52.625$

Thus, the stock will reach equilibrium at a price of $52.625.

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Summary:

• The opportunity cost of capital is 9.25%.
• The stock’s expected return is 8%, which is less than the opportunity cost of capital. Therefore, the stock is not a good buy, and investors are likely to sell.
• The equilibrium price at which the stock will be fairly priced today is $52.625.

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Would you like more details on any part of the solution?

Here are 5 related questions you might explore:

1. How does beta impact the opportunity cost of capital in CAPM?
2. What would happen if the market risk premium increased?
3. How do changes in the T-bill rate affect stock valuation?
4. What does equilibrium pricing in the stock market mean?
5. Can stocks be undervalued based on CAPM alone?

Tip: When assessing stock value, always compare the expected return to the opportunity cost of capital to make informed decisions.