
  Investment
Forecasted Returns for Each Economy
Boom
Stable 
Growth
Stagnant
Recession
  Stock
22​%
10​%
4​%
negative 12​%
  Corporate bond
9​%
7​%
5​%
4​%
  Government bond
8​%
6​%
4​%
3​%
 

Bacon and​ Associates, a famous Northwest think​ tank, 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 21 %​, the probability of a stable growth economy is 43​%, the probability of a stagnant economy is 20 %​, and the probability of a recession is 16​%.  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 five decimal​ places.)

To calculate the variance and standard deviation of each investment (stock, corporate bond, and government bond), we will follow these steps:

1. **Calculate the Expected Return (Mean)** for each investment by multiplying the return in each economic state by the probability of that state and then summing up the results.

\[
\text{Expected Return} = \sum (\text{Return in state} \times \text{Probability of state})
\]

2. **Calculate the Variance** for each investment by using the following formula:

\[
\text{Variance} = \sum \left( \text{Probability of state} \times \left( \text{Return in state} - \text{Expected Return} \right)^2 \right)
\]

3. **Standard Deviation** is the square root of the variance.

### Step 1: Calculate the Expected Return (Mean) for Stock
Using the given probabilities and returns for stock:

- Boom: \( 22\% \) with a probability of \( 0.21 \)
- Stable Growth: \( 10\% \) with a probability of \( 0.43 \)
- Stagnant: \( 4\% \) with a probability of \( 0.20 \)
- Recession: \( -12\% \) with a probability of \( 0.16 \)

Expected Return for stock:
\[
E(\text{Stock}) = (0.21 \times 22) + (0.43 \times 10) + (0.20 \times 4) + (0.16 \times -12)
\]

### Step 2: Calculate the Variance for Stock
The variance formula involves subtracting the expected return from each individual return, squaring the difference, and multiplying by the respective probabilities:

\[
\text{Variance}(\text{Stock}) = 0.21 \times (22 - E(\text{Stock}))^2 + 0.43 \times (10 - E(\text{Stock}))^2 + 0.20 \times (4 - E(\text{Stock}))^2 + 0.16 \times (-12 - E(\text{Stock}))^2
\]

Now let's compute these steps.The calculations for the stock investment are as follows:

- **Expected Return (Mean):** 7.80000%
- **Variance:** 110.04000% (rounded to five decimal places)
- **Standard Deviation:** 10.49000% (rounded to five decimal places)

Would you like me to proceed with the calculations for the corporate bond and government bond investments? Let me know if you have any questions!

Here are 5 follow-up questions you might want to consider:
1. How do the variances of the corporate and government bonds compare to stocks?
2. How does risk, represented by standard deviation, affect investment choices?
3. Should an investor prioritize expected return or risk when choosing an investment?
4. How would changing the probabilities of the economic states affect these calculations?
5. How can diversification impact risk in a portfolio containing these investments?

**Tip:** Variance measures how far the returns spread from the average, while standard deviation provides insight into investment volatility.

Bacon and Associates, a famous Northwest think tank, has provided probability estimates for the four potential economic states for the coming year. 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 stock, corporate bond, and government bond investments using probability estimates for different economic states. Explore key financial concepts like expected return, variance, and standard deviation to assess risk and return. Suitable for finance and statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation