Answer :
Let's break down the solution step-by-step for each part of the question:
### Part (a): Calculate the expected return and standard deviation of Stock A and Stock B.
#### Expected Return:
First, let's calculate the expected return for each stock. The expected return [tex]\( E(R) \)[/tex] is found using the weighted average of returns under different states of the economy.
For Stock A:
[tex]\[ E(R_A) = (0.3 \times 10) + (0.4 \times 15) + (0.3 \times 20) = 3 + 6 + 6 = 15\% \][/tex]
For Stock B:
[tex]\[ E(R_B) = (0.3 \times 30) + (0.4 \times 20) + (0.3 \times 10) = 9 + 8 + 3 = 20\% \][/tex]
#### Standard Deviation:
Next, we calculate the standard deviation, which measures the dispersion of returns from the expected return. The formula for variance [tex]\( \sigma^2 \)[/tex] is:
[tex]\[ \sigma^2 = \sum p_i (R_i - E(R))^2 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance.
For Stock A:
[tex]\[ \sigma_A^2 = 0.3 \times (10 - 15)^2 + 0.4 \times (15 - 15)^2 + 0.3 \times (20 - 15)^2 = 0.3 \times 25 + 0.4 \times 0 + 0.3 \times 25 = 15 \][/tex]
[tex]\[ \sigma_A = \sqrt{15} \approx 3.87\% \][/tex]
For Stock B:
[tex]\[ \sigma_B^2 = 0.3 \times (30 - 20)^2 + 0.4 \times (20 - 20)^2 + 0.3 \times (10 - 20)^2 = 0.3 \times 100 + 0.4 \times 0 + 0.3 \times 100 = 30 \][/tex]
[tex]\[ \sigma_B = \sqrt{30} \approx 7.75\% \][/tex]
### Part (b): Calculate the covariance and correlation coefficient between Stock A and Stock B.
#### Covariance:
Covariance measures how two stocks move together. The formula is:
[tex]\[ \text{Cov}_{AB} = \sum p_i (R_{A,i} - E(R_A))(R_{B,i} - E(R_B)) \][/tex]
[tex]\[ \text{Cov}_{AB} = 0.3 \times (10 - 15)(30 - 20) + 0.4 \times (15 - 15)(20 - 20) + 0.3 \times (20 - 15)(10 - 20) \][/tex]
[tex]\[ = 0.3 \times (-5) \times 10 + 0.4 \times 0 + 0.3 \times 5 \times (-10) = -15 - 15 = -30 \][/tex]
#### Correlation Coefficient:
The correlation coefficient [tex]\( \rho \)[/tex] is found using:
[tex]\[ \rho_{AB} = \frac{\text{Cov}_{AB}}{\sigma_A \sigma_B} \][/tex]
[tex]\[ \rho_{AB} = \frac{-30}{3.87 \times 7.75} \approx -1 \][/tex]
### Part (c): Calculate the risk and return of a portfolio with 40% in Stock A and 60% in Stock B.
#### Expected Return:
The expected return of the portfolio [tex]\( E(R_P) \)[/tex] is:
[tex]\[ E(R_P) = w_A \times E(R_A) + w_B \times E(R_B) \][/tex]
[tex]\[ E(R_P) = 0.4 \times 15 + 0.6 \times 20 = 6 + 12 = 18\% \][/tex]
#### Standard Deviation:
The portfolio variance [tex]\( \sigma_P^2 \)[/tex] is:
[tex]\[ \sigma_P^2 = (w_A \times \sigma_A)^2 + (w_B \times \sigma_B)^2 + 2 \times w_A \times w_B \times \text{Cov}_{AB} \][/tex]
[tex]\[ \sigma_P^2 = (0.4 \times 3.87)^2 + (0.6 \times 7.75)^2 + 2 \times 0.4 \times 0.6 \times (-30) \][/tex]
[tex]\[ = 2.4 + 21.6 - 14.4 = 9.6 \][/tex]
[tex]\[ \sigma_P = \sqrt{9.6} \approx 3.1\% \][/tex]
### Part (d): Coefficient of Variation (CV):
The coefficient of variation [tex]\( CV \)[/tex] for each stock and the portfolio is calculated as:
[tex]\[ CV = \frac{\sigma}{E(R)} \][/tex]
For Stock A:
[tex]\[ CV_A = \frac{3.87}{15} \approx 0.258 \][/tex]
For Stock B:
[tex]\[ CV_B = \frac{7.75}{20} \approx 0.387 \][/tex]
For the Portfolio:
[tex]\[ CV_P = \frac{3.1}{18} \approx 0.172 \][/tex]
### Conclusion:
Given that the portfolio has the lowest coefficient of variation (0.1722), it indicates that the portfolio has the best return per unit of risk compared to the individual stocks. Therefore, investing in the portfolio is the best option due to its lower risk relative to the expected return.
The detailed solution with numerical results confirms that the portfolio, with a combination of Stocks A and B, provides a favorable balance of return and risk.
### Part (a): Calculate the expected return and standard deviation of Stock A and Stock B.
#### Expected Return:
First, let's calculate the expected return for each stock. The expected return [tex]\( E(R) \)[/tex] is found using the weighted average of returns under different states of the economy.
For Stock A:
[tex]\[ E(R_A) = (0.3 \times 10) + (0.4 \times 15) + (0.3 \times 20) = 3 + 6 + 6 = 15\% \][/tex]
For Stock B:
[tex]\[ E(R_B) = (0.3 \times 30) + (0.4 \times 20) + (0.3 \times 10) = 9 + 8 + 3 = 20\% \][/tex]
#### Standard Deviation:
Next, we calculate the standard deviation, which measures the dispersion of returns from the expected return. The formula for variance [tex]\( \sigma^2 \)[/tex] is:
[tex]\[ \sigma^2 = \sum p_i (R_i - E(R))^2 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance.
For Stock A:
[tex]\[ \sigma_A^2 = 0.3 \times (10 - 15)^2 + 0.4 \times (15 - 15)^2 + 0.3 \times (20 - 15)^2 = 0.3 \times 25 + 0.4 \times 0 + 0.3 \times 25 = 15 \][/tex]
[tex]\[ \sigma_A = \sqrt{15} \approx 3.87\% \][/tex]
For Stock B:
[tex]\[ \sigma_B^2 = 0.3 \times (30 - 20)^2 + 0.4 \times (20 - 20)^2 + 0.3 \times (10 - 20)^2 = 0.3 \times 100 + 0.4 \times 0 + 0.3 \times 100 = 30 \][/tex]
[tex]\[ \sigma_B = \sqrt{30} \approx 7.75\% \][/tex]
### Part (b): Calculate the covariance and correlation coefficient between Stock A and Stock B.
#### Covariance:
Covariance measures how two stocks move together. The formula is:
[tex]\[ \text{Cov}_{AB} = \sum p_i (R_{A,i} - E(R_A))(R_{B,i} - E(R_B)) \][/tex]
[tex]\[ \text{Cov}_{AB} = 0.3 \times (10 - 15)(30 - 20) + 0.4 \times (15 - 15)(20 - 20) + 0.3 \times (20 - 15)(10 - 20) \][/tex]
[tex]\[ = 0.3 \times (-5) \times 10 + 0.4 \times 0 + 0.3 \times 5 \times (-10) = -15 - 15 = -30 \][/tex]
#### Correlation Coefficient:
The correlation coefficient [tex]\( \rho \)[/tex] is found using:
[tex]\[ \rho_{AB} = \frac{\text{Cov}_{AB}}{\sigma_A \sigma_B} \][/tex]
[tex]\[ \rho_{AB} = \frac{-30}{3.87 \times 7.75} \approx -1 \][/tex]
### Part (c): Calculate the risk and return of a portfolio with 40% in Stock A and 60% in Stock B.
#### Expected Return:
The expected return of the portfolio [tex]\( E(R_P) \)[/tex] is:
[tex]\[ E(R_P) = w_A \times E(R_A) + w_B \times E(R_B) \][/tex]
[tex]\[ E(R_P) = 0.4 \times 15 + 0.6 \times 20 = 6 + 12 = 18\% \][/tex]
#### Standard Deviation:
The portfolio variance [tex]\( \sigma_P^2 \)[/tex] is:
[tex]\[ \sigma_P^2 = (w_A \times \sigma_A)^2 + (w_B \times \sigma_B)^2 + 2 \times w_A \times w_B \times \text{Cov}_{AB} \][/tex]
[tex]\[ \sigma_P^2 = (0.4 \times 3.87)^2 + (0.6 \times 7.75)^2 + 2 \times 0.4 \times 0.6 \times (-30) \][/tex]
[tex]\[ = 2.4 + 21.6 - 14.4 = 9.6 \][/tex]
[tex]\[ \sigma_P = \sqrt{9.6} \approx 3.1\% \][/tex]
### Part (d): Coefficient of Variation (CV):
The coefficient of variation [tex]\( CV \)[/tex] for each stock and the portfolio is calculated as:
[tex]\[ CV = \frac{\sigma}{E(R)} \][/tex]
For Stock A:
[tex]\[ CV_A = \frac{3.87}{15} \approx 0.258 \][/tex]
For Stock B:
[tex]\[ CV_B = \frac{7.75}{20} \approx 0.387 \][/tex]
For the Portfolio:
[tex]\[ CV_P = \frac{3.1}{18} \approx 0.172 \][/tex]
### Conclusion:
Given that the portfolio has the lowest coefficient of variation (0.1722), it indicates that the portfolio has the best return per unit of risk compared to the individual stocks. Therefore, investing in the portfolio is the best option due to its lower risk relative to the expected return.
The detailed solution with numerical results confirms that the portfolio, with a combination of Stocks A and B, provides a favorable balance of return and risk.
