Statistical measures of standalone risk Remember, the expected value of a probability distribution is a statistical measure of the average (mean) value expected to occur during all possible circumstan

Question

Statistical measures of standalone risk Remember, the expected value of a probability distribution is a statistical measure of the average (mean) value expected to occur during all possible circumstances. To compute an asset's expected return under a range of possible circumstances (or states of nature), multiply the anticipated return expected to result during each state of nature by its probability of occurrence. Consider the following case: Ethan owns a two-stock portfolio that invests in Celestial Crane Cosmetics Company (CCC) and Lumbering Ox Truckmakers (LOT). Three-quarters of Ethan's portfolio value consists of CCC's shares, and the balance consists of LOT's shares. Each stock's expected return for the next year will depend on forecasted market conditions. The expected returns from the stocks in different market conditions are detailed in the following table: Calculate expected returns for the individual stocks in Ethan's portfolio as well as the expected rate of return of the entire portfolio over the three possible market conditions next year. - The expected rate of return on Celestial Crane Cosmetics's stock over the next year is - The expected rate of return on Lumberin

VIPSolutions · Accepted Answer

Let's walk through the calculations step-by-step.

**Given Data:**

- Ethan's portfolio:
  - 75% in Celestial Crane Cosmetics (CCC)
  - 25% in Lumbering Ox Truckmakers (LOT)

- Expected returns under 3 market conditions:
  
| Market Condition | Probability | CCC Return (%) | LOT Return (%) |
|---------------------|--------------|----------------|----------------|
| Boom                | $ p_1 $    | $ r_{CCC1} $ | $ r_{LOT1} $ |
| Normal              | $ p_2 $    | $ r_{CCC2} $ | $ r_{LOT2} $ |
| Recession           | $ p_3 $    | $ r_{CCC3} $ | $ r_{LOT3} $ |

*(Note: You need to provide the specific probabilities and returns under each condition for accurate calculations; if these are missing, I will use hypothetical values for demonstration.)*

---

### Step 1: Expected Return for Each Stock

The **expected return** for each stock is calculated as:

$$
E(r) = \sum_{i=1}^{n} p_i \times r_i
$$

Where:
- $ p_i $ = probability of market condition $ i $
- $ r_i $ = return of the stock in condition $ i $

---

### Step 2: Expected Return of the Portfolio

The **expected return of the portfolio** is:

$$
E(r_{portfolio}) = w_{CCC} \times E(r_{CCC}) + w_{LOT} \times E(r_{LOT})
$$

Where:
- $ w_{CCC} = 0.75 $
- $ w_{LOT} = 0.25 $

---

### Example Calculation (using hypothetical data):

Suppose the following data:

| Market Condition | Probability $ p_i $ | CCC Return $ r_{CCC,i} $ | LOT Return $ r_{LOT,i} $ |
|---------------------|------------------------|---------------------------|---------------------------|
| Boom                | 0.25                   | 20%                       | 10%                       |
| Normal              | 0.50                   | 10%                       | 5%                        |
| Recession           | 0.25                   | -10%                      | -5%                       |

---

### Step 3: Calculate Expected Returns for Each Stock

**CCC:**

$$
E(r_{CCC}) = (0.25 \times 20\%) + (0.50 \times 10\%) + (0.25 \times -10\%) \
= 0.25 \times 0.20 + 0.50 \times 0.10 + 0.25 \times (-0.10) \
= 0.05 + 0.05 - 0.025 = 0.075 \text{ or } 7.5\%
$$

**LOT:**

$$
E(r_{LOT}) = (0.25 \times 10\%) + (0.50 \times 5\%) + (0.25 \times -5\%) \
= 0.025 + 0.025 - 0.0125 = 0.0375 \text{ or } 3.75\%
$$

---

### Step 4: Calculate Portfolio's Expected Return

$$
E(r_{portfolio}) = 0.75 \times 7.5\% + 0.25 \times 3.75\% \
= 0.75 \times 0.075 + 0.25 \times 0.0375 \
= 0.05625 + 0.009375 = 0.065625 \text{ or } 6.56\%
$$

---

### **Final Results:**

- **Expected return on CCC stock:** **7.5%** (based on hypothetical data)
- **Expected return on LOT stock:** **3.75%**
- **Expected return on Ethan's portfolio:** **6.56%**

---

### **Note:**
Please replace the hypothetical probabilities and returns with the actual data you have for precise calculations.

---

**Summary:**

| Measure | Value |
|---------|--------|
| Expected return on CCC | **[Your calculated value]%** |
| Expected return on LOT | **[Your calculated value]%** |
| Expected return of portfolio | **[Your calculated value]%** |

Feel free to provide the actual probabilities and returns for a more accurate calculation!

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