A share of stock with a beta of 0.75 now sells for $50. Investors expect the stock to pay a year-end dividend of $2. The T-bill rate is 4%, and the market risk premium is 7%.

Suppose investors believe the stock will sell for $52 at year-end. Calculate the opportunity cost of capital. Is the stock a good or bad buy? What will investors do?

At what price will the stock reach an “equilibrium” at which it is perceived as fairly priced today?

### 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%.

---

### 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.

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

---

### 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.

---

### 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**.

---

### 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**.

---

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.

A share of stock with a beta of 0.75 now sells for $50. Investors expect the stock to pay a year-end dividend of $2. The T-bill rate is 4%, and the market risk premium is 7%. Suppose investors believe the stock will sell for $52 at year-end. Calculate the opportunity cost of capital. Is the stock a good or bad buy? What will investors do? At what price will the stock reach an “equilibrium” at which it is perceived as fairly priced today?

Calculate the opportunity cost of capital using CAPM for a stock with a beta of 0.75, a T-bill rate of 4%, and a market risk premium of 7%. Determine if the stock is a good or bad buy, and find the equilibrium price at which it will be fairly valued.

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