Answer :
### Step-by-Step Solution:
To analyze the stock's returns, we need to calculate several statistical measures: the expected return, the standard deviation, and the coefficient of variation. Below is the detailed step-by-step process for each calculation.
#### 1. Expected Return
The expected return is calculated as the weighted average of all possible returns, where each return is weighted by the probability of its occurrence. The formula for expected return [tex]\( E(R) \)[/tex] is:
[tex]\[ E(R) = \sum_{i=1}^{n} P_i \times R_i \][/tex]
where [tex]\( P_i \)[/tex] is the probability of occurrence of return [tex]\( R_i \)[/tex].
Given:
- Probabilities: [tex]\( [0.1, 0.2, 0.4, 0.2, 0.1] \)[/tex]
- Returns: [tex]\( [-0.50, -0.05, 0.16, 0.25, 0.60] \)[/tex]
Let's calculate the expected return:
[tex]\[ E(R) = (0.1 \times -0.50) + (0.2 \times -0.05) + (0.4 \times 0.16) + (0.2 \times 0.25) + (0.1 \times 0.60) \][/tex]
[tex]\[ E(R) = -0.05 + -0.01 + 0.064 + 0.05 + 0.06 \][/tex]
[tex]\[ E(R) = 0.114 \][/tex]
So, the expected return is 0.114 or 11.4%.
#### 2. Variance and Standard Deviation
Variance measures the dispersion of returns around the expected return. The formula for variance [tex]\( \sigma^2 \)[/tex] is:
[tex]\[ \sigma^2 = \sum_{i=1}^{n} P_i \times (R_i - E(R))^2 \][/tex]
To find the standard deviation, we take the square root of the variance:
[tex]\[ \sigma = \sqrt{\sigma^2} \][/tex]
Let's first calculate the individual squared deviations:
[tex]\[ \begin{aligned} &\text{For } -0.50: (P = 0.1) \, (R - E(R))^2 = (-0.50 - 0.114)^2 = (-0.614)^2 = 0.376 \\ &\text{For } -0.05: (P = 0.2) \, (R - E(R))^2 = (-0.05 - 0.114)^2 = (-0.164)^2 = 0.0269 \\ &\text{For } 0.16: (P = 0.4) \, (R - E(R))^2 = (0.16 - 0.114)^2 = (0.046)^2 = 0.002116 \\ &\text{For } 0.25: (P = 0.2) \, (R - E(R))^2 = (0.25 - 0.114)^2 = (0.136)^2 = 0.018496 \\ &\text{For } 0.60: (P = 0.1) \, (R - E(R))^2 = (0.60 - 0.114)^2 = (0.486)^2 = 0.236196 \\ \end{aligned} \][/tex]
[tex]\[ \sigma^2 = (0.1 \times 0.376) + (0.2 \times 0.0269) + (0.4 \times 0.002116) + (0.2 \times 0.018496) + (0.1 \times 0.236196) \][/tex]
[tex]\[ \sigma^2 = 0.0376 + 0.00538 + 0.0008464 + 0.0036992 + 0.0236196 \][/tex]
[tex]\[ \sigma^2 \approx 0.0711452 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex]:
[tex]\[ \sigma = \sqrt{0.0711452} \approx 0.2669 \][/tex]
So, the standard deviation is approximately 0.267 or 26.7%.
#### 3. Coefficient of Variation
The coefficient of variation (CV) is a measure of relative risk per unit of return. It is calculated as:
[tex]\[ CV = \frac{\sigma}{E(R)} \][/tex]
Given:
- Standard deviation [tex]\( \sigma = 0.267 \)[/tex]
- Expected return [tex]\( E(R) = 0.114 \)[/tex]
[tex]\[ CV \approx \frac{0.267}{0.114} \approx 2.341 \][/tex]
So, the coefficient of variation is approximately 2.341.
### Summary:
- Expected Return: 11.4%
- Standard Deviation: 26.7%
- Coefficient of Variation: 2.341
These calculations give us a detailed understanding of the stock's expected return, its risk in terms of standard deviation, and how much risk is present per unit of return as measured by the coefficient of variation.
To analyze the stock's returns, we need to calculate several statistical measures: the expected return, the standard deviation, and the coefficient of variation. Below is the detailed step-by-step process for each calculation.
#### 1. Expected Return
The expected return is calculated as the weighted average of all possible returns, where each return is weighted by the probability of its occurrence. The formula for expected return [tex]\( E(R) \)[/tex] is:
[tex]\[ E(R) = \sum_{i=1}^{n} P_i \times R_i \][/tex]
where [tex]\( P_i \)[/tex] is the probability of occurrence of return [tex]\( R_i \)[/tex].
Given:
- Probabilities: [tex]\( [0.1, 0.2, 0.4, 0.2, 0.1] \)[/tex]
- Returns: [tex]\( [-0.50, -0.05, 0.16, 0.25, 0.60] \)[/tex]
Let's calculate the expected return:
[tex]\[ E(R) = (0.1 \times -0.50) + (0.2 \times -0.05) + (0.4 \times 0.16) + (0.2 \times 0.25) + (0.1 \times 0.60) \][/tex]
[tex]\[ E(R) = -0.05 + -0.01 + 0.064 + 0.05 + 0.06 \][/tex]
[tex]\[ E(R) = 0.114 \][/tex]
So, the expected return is 0.114 or 11.4%.
#### 2. Variance and Standard Deviation
Variance measures the dispersion of returns around the expected return. The formula for variance [tex]\( \sigma^2 \)[/tex] is:
[tex]\[ \sigma^2 = \sum_{i=1}^{n} P_i \times (R_i - E(R))^2 \][/tex]
To find the standard deviation, we take the square root of the variance:
[tex]\[ \sigma = \sqrt{\sigma^2} \][/tex]
Let's first calculate the individual squared deviations:
[tex]\[ \begin{aligned} &\text{For } -0.50: (P = 0.1) \, (R - E(R))^2 = (-0.50 - 0.114)^2 = (-0.614)^2 = 0.376 \\ &\text{For } -0.05: (P = 0.2) \, (R - E(R))^2 = (-0.05 - 0.114)^2 = (-0.164)^2 = 0.0269 \\ &\text{For } 0.16: (P = 0.4) \, (R - E(R))^2 = (0.16 - 0.114)^2 = (0.046)^2 = 0.002116 \\ &\text{For } 0.25: (P = 0.2) \, (R - E(R))^2 = (0.25 - 0.114)^2 = (0.136)^2 = 0.018496 \\ &\text{For } 0.60: (P = 0.1) \, (R - E(R))^2 = (0.60 - 0.114)^2 = (0.486)^2 = 0.236196 \\ \end{aligned} \][/tex]
[tex]\[ \sigma^2 = (0.1 \times 0.376) + (0.2 \times 0.0269) + (0.4 \times 0.002116) + (0.2 \times 0.018496) + (0.1 \times 0.236196) \][/tex]
[tex]\[ \sigma^2 = 0.0376 + 0.00538 + 0.0008464 + 0.0036992 + 0.0236196 \][/tex]
[tex]\[ \sigma^2 \approx 0.0711452 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex]:
[tex]\[ \sigma = \sqrt{0.0711452} \approx 0.2669 \][/tex]
So, the standard deviation is approximately 0.267 or 26.7%.
#### 3. Coefficient of Variation
The coefficient of variation (CV) is a measure of relative risk per unit of return. It is calculated as:
[tex]\[ CV = \frac{\sigma}{E(R)} \][/tex]
Given:
- Standard deviation [tex]\( \sigma = 0.267 \)[/tex]
- Expected return [tex]\( E(R) = 0.114 \)[/tex]
[tex]\[ CV \approx \frac{0.267}{0.114} \approx 2.341 \][/tex]
So, the coefficient of variation is approximately 2.341.
### Summary:
- Expected Return: 11.4%
- Standard Deviation: 26.7%
- Coefficient of Variation: 2.341
These calculations give us a detailed understanding of the stock's expected return, its risk in terms of standard deviation, and how much risk is present per unit of return as measured by the coefficient of variation.
