**Investment**

**Forecasted Returns for Each Economy**

**Boom**

**Stable**

**Growth**

**Stagnant**

**Recession**

Stock

2323​%

1111​%

33​%

negative 13−13​%

Corporate bond

1010​%

88​%

55​%

33​%

Government bond

99​%

77​%

44​%

22​%

Hull​ Consultants, a famous think tank in the​ Midwest, has provided probability estimates for the four potential economic states for the coming year in the following​ table:

LOADING...

.

The probability of a boom economy is

12 %12%​,

the probability of a stable growth economy is

1717​%,

the probability of a stagnant economy is

49 %49%​,

and the probability of a recession is

2222​%.

Calculate the variance and the standard deviation of the three​ investments: stock, corporate​ bond, and government bond.  If the estimates for both the probabilities of the economy and the returns in each state of the economy are​ correct, which investment would you​ choose, considering both risk and​ return?

​Hint: Make sure to round all intermediate calculations to at least seven​ (7) decimal places. The input​ instructions, phrases in parenthesis after each answer​ box, only apply for the answers you will type.

Question content area bottom

Part 1

What is the variance of the stock​ investment?

enter your response here​%

​(Round to six decimal​ places.)

To calculate the variance and standard deviation of the three investments (stock, corporate bond, and government bond), we need to follow these steps:

### 1. **Expected Return for Each Investment**

The expected return \( E(R) \) is calculated using the formula:

\[
E(R) = \sum \text{(Probability of economic state)} \times \text{(Return in that state)}
\]

Let's calculate the expected return for each investment.

### **Stock**
\[
E(R_{stock}) = (0.12 \times 23) + (0.17 \times 11) + (0.49 \times 3) + (0.22 \times -13)
\]

\[
E(R_{stock}) = 2.76 + 1.87 + 1.47 - 2.86 = 3.24\%
\]

### **Corporate Bond**
\[
E(R_{corp}) = (0.12 \times 10) + (0.17 \times 8) + (0.49 \times 5) + (0.22 \times 3)
\]

\[
E(R_{corp}) = 1.2 + 1.36 + 2.45 + 0.66 = 5.67\%
\]

### **Government Bond**
\[
E(R_{gov}) = (0.12 \times 9) + (0.17 \times 7) + (0.49 \times 4) + (0.22 \times 2)
\]

\[
E(R_{gov}) = 1.08 + 1.19 + 1.96 + 0.44 = 4.67\%
\]

### 2. **Variance of Each Investment**

Variance \( \sigma^2 \) is calculated using the formula:

\[
\sigma^2 = \sum \text{(Probability of economic state)} \times (\text{Return in that state} - E(R))^2
\]

Now, let's calculate the variance for each investment.

### **Variance of Stock**
\[
\sigma^2_{stock} = (0.12 \times (23 - 3.24)^2) + (0.17 \times (11 - 3.24)^2) + (0.49 \times (3 - 3.24)^2) + (0.22 \times (-13 - 3.24)^2)
\]

\[
\sigma^2_{stock} = (0.12 \times 390.2976) + (0.17 \times 60.0576) + (0.49 \times 0.0576) + (0.22 \times 261.1776)
\]

\[
\sigma^2_{stock} = 46.8357 + 10.2098 + 0.0282 + 57.4582 = 114.5319\%
\]

### **Variance of Corporate Bond**
\[
\sigma^2_{corp} = (0.12 \times (10 - 5.67)^2) + (0.17 \times (8 - 5.67)^2) + (0.49 \times (5 - 5.67)^2) + (0.22 \times (3 - 5.67)^2)
\]

\[
\sigma^2_{corp} = (0.12 \times 18.7209) + (0.17 \times 5.4329) + (0.49 \times 0.4489) + (0.22 \times 7.1289)
\]

\[
\sigma^2_{corp} = 2.2465 + 0.9236 + 0.2190 + 1.5684 = 4.9575\%
\]

### **Variance of Government Bond**
\[
\sigma^2_{gov} = (0.12 \times (9 - 4.67)^2) + (0.17 \times (7 - 4.67)^2) + (0.49 \times (4 - 4.67)^2) + (0.22 \times (2 - 4.67)^2)
\]

\[
\sigma^2_{gov} = (0.12 \times 18.5856) + (0.17 \times 5.3969) + (0.49 \times 0.4489) + (0.22 \times 7.1289)
\]

\[
\sigma^2_{gov} = 2.2303 + 0.9175 + 0.2190 + 1.5684 = 4.9352\%
\]

### 3. **Standard Deviation**

Standard deviation \( \sigma \) is the square root of variance:

\[
\sigma = \sqrt{\sigma^2}
\]

For **Stock**:
\[
\sigma_{stock} = \sqrt{114.5319} = 10.7024\%
\]

For **Corporate Bond**:
\[
\sigma_{corp} = \sqrt{4.9575} = 2.2260\%
\]

For **Government Bond**:
\[
\sigma_{gov} = \sqrt{4.9352} = 2.2215\%
\]

### 4. **Conclusion**

Given the results, the **Stock** has the highest expected return but also the highest risk (variance and standard deviation). The **Corporate Bond** and **Government Bond** have lower returns but also significantly lower risk.

**Which Investment to Choose:**
- If you prioritize higher returns and are willing to accept higher risk, **Stock** would be the choice.
- If you prefer lower risk, **Corporate Bond** or **Government Bond** are safer choices.

Would you like a deeper explanation of any part, or do you have any questions?

---

### Related Questions:
1. How is the expected return of an investment calculated?
2. What are the advantages and disadvantages of investing in stocks versus bonds?
3. How can investors use variance and standard deviation to assess risk?
4. What factors might influence an investor's decision to choose one investment over another?
5. How do economic forecasts impact investment strategies?

### Tip:
Diversifying your portfolio can help balance risk and return, minimizing the impact of any single investment's volatility.

Calculate the variance and the standard deviation of the three investments: stock, corporate bond, and government bond. If the estimates for both the probabilities of the economy and the returns in each state of the economy are correct, which investment would you choose, considering both risk and return?

Learn how to calculate the variance and standard deviation of three investment options (stock, corporate bond, government bond) based on economic state probabilities. This problem involves probability, expected value, and risk analysis. Suitable for undergraduate students.

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