Consider an economy with two types of firms, S and I. S firms always move together, but I firms move independently of each other. For both types of firms there is a 40% probability that the firm will have a 20% return and a 60% probability that the firm will have a -30% return. The standard deviation for the return on a portfolio of 20 type I firms is closest to: O A. - 10% OB. 24.49% OC. 5.48% OD. 12.25%
Consider an economy with two types of firms, S and I. S firms always move together, but I firms move independently of each other. For both types of firms there is a 40% probability that the firm will have a 20% return and a 60% probability that the firm will have a -30% return. The standard deviation for the return on a portfolio of 20 type I firms is closest to: O A. - 10% OB. 24.49% OC. 5.48% OD. 12.25%
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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![### Understanding Portfolio Risk in an Economy with Two Types of Firms
In this exercise, we consider an economy with two types of firms: S and I. Here's a brief overview of the problem and the elements involved.
#### Description of Firms:
- **Type S firms:** These firms always move together, indicating that their returns are perfectly correlated.
- **Type I firms:** These firms move independently, meaning their returns are uncorrelated.
#### Probabilities for Firms' Returns:
- **For both types of firms:**
- There is a 40% probability that the firm will have a **+20% return**.
- There is a 60% probability that the firm will have a **–30% return**.
Given this setup, the task is to determine the standard deviation of the return on a portfolio that consists of 20 type I firms.
#### Question:
The standard deviation for the return on a portfolio of 20 type I firms is closest to:
- **A.** –10%
- **B.** 24.49%
- **C.** 5.48%
- **D.** 12.25%
Here are the detailed components and calculations necessary to solve the problem:
1. **Mean Return (μ):**
- The mean return can be calculated using the following formula:
\[
\mu = (0.40 \times 0.20) + (0.60 \times -0.30) = 0.08 - 0.18 = -0.10 = -10\%
\]
2. **Variance (σ²):**
- The variance for each firm is calculated as:
\[
\sigma^2 = (0.40 \times (0.20 - (-0.10))^2) + (0.60 \times (-0.30 - (-0.10))^2)
= (0.40 \times 0.09) + (0.60 \times 0.04) = 0.036 + 0.024 = 0.06
\]
3. **Standard Deviation (σ):**
- The standard deviation for one firm is:
\[
\sigma = \sqrt{0.06} \approx 0.2449 = 24.49\%
\]
However, since we are focusing on a portfolio of 20 independent firms](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c0554f3-a7d8-4c12-8db7-464a17f00206%2F20923aba-1393-481e-97c5-56083955810c%2Fntbag9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Portfolio Risk in an Economy with Two Types of Firms
In this exercise, we consider an economy with two types of firms: S and I. Here's a brief overview of the problem and the elements involved.
#### Description of Firms:
- **Type S firms:** These firms always move together, indicating that their returns are perfectly correlated.
- **Type I firms:** These firms move independently, meaning their returns are uncorrelated.
#### Probabilities for Firms' Returns:
- **For both types of firms:**
- There is a 40% probability that the firm will have a **+20% return**.
- There is a 60% probability that the firm will have a **–30% return**.
Given this setup, the task is to determine the standard deviation of the return on a portfolio that consists of 20 type I firms.
#### Question:
The standard deviation for the return on a portfolio of 20 type I firms is closest to:
- **A.** –10%
- **B.** 24.49%
- **C.** 5.48%
- **D.** 12.25%
Here are the detailed components and calculations necessary to solve the problem:
1. **Mean Return (μ):**
- The mean return can be calculated using the following formula:
\[
\mu = (0.40 \times 0.20) + (0.60 \times -0.30) = 0.08 - 0.18 = -0.10 = -10\%
\]
2. **Variance (σ²):**
- The variance for each firm is calculated as:
\[
\sigma^2 = (0.40 \times (0.20 - (-0.10))^2) + (0.60 \times (-0.30 - (-0.10))^2)
= (0.40 \times 0.09) + (0.60 \times 0.04) = 0.036 + 0.024 = 0.06
\]
3. **Standard Deviation (σ):**
- The standard deviation for one firm is:
\[
\sigma = \sqrt{0.06} \approx 0.2449 = 24.49\%
\]
However, since we are focusing on a portfolio of 20 independent firms
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