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Sagot :
To find the expected rate of return and standard deviation for Maria's portfolio, let's go through the problem step by step.
### 1. Determine the total investment:
First, calculate the total amount Maria has invested in each stock.
- Investment in stock A:
[tex]\[ \text{Investment in stock A} = 7 \, \text{shares} \times \$70 \, \text{per share} = \$490 \][/tex]
- Investment in stock B:
[tex]\[ \text{Investment in stock B} = 4 \, \text{shares} \times \$100 \, \text{per share} = \$400 \][/tex]
- Total investment:
[tex]\[ \text{Total investment} = \$490 + \$400 = \$890 \][/tex]
### 2. Calculate the expected rate of return for the portfolio:
Using the weights of the investments in stocks A and B, calculate the expected return of the portfolio.
- Weight of stock A:
[tex]\[ w_A = \frac{\$490}{\$890} = 0.5506 \][/tex]
- Weight of stock B:
[tex]\[ w_B = \frac{\$400}{\$890} = 0.4494 \][/tex]
- Expected rate of return of portfolio ([tex]\(E(R_p)\)[/tex]):
[tex]\[ E(R_p) = (w_A \times \text{return}_A) + (w_B \times \text{return}_B) \][/tex]
[tex]\[ E(R_p) = (0.5506 \times 0.02) + (0.4494 \times 0.15) \][/tex]
[tex]\[ E(R_p) \approx 0.0784 \][/tex]
So, the expected rate of return after 1 year for Maria's portfolio is approximately 0.0784.
### 3. Calculate the standard deviation for the portfolio for different correlation coefficients:
#### Variance and standard deviation of each stock:
- Variance of stock A ([tex]\(\sigma_A^2\)[/tex]): 0.04
- Variance of stock B ([tex]\(\sigma_B^2\)[/tex]): 0.18
- Standard deviation of stock A ([tex]\(\sigma_A\)[/tex]):
[tex]\[ \sigma_A = \sqrt{0.04} = 0.2 \][/tex]
- Standard deviation of stock B ([tex]\(\sigma_B\)[/tex]):
[tex]\[ \sigma_B = \sqrt{0.18} \approx 0.4243 \][/tex]
#### Portfolio variance and standard deviation for different correlations ([tex]\(\rho\)[/tex]):
Using the formula:
[tex]\[ \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho \][/tex]
1. For [tex]\(\rho = -0.4\)[/tex]:
[tex]\[ \sigma_p^2 = (0.5506^2 \times 0.04) + (0.4494^2 \times 0.18) + (2 \times 0.5506 \times 0.4494 \times 0.2 \times 0.4243 \times (-0.4)) \][/tex]
[tex]\[ \sigma_p \approx 0.1780 \][/tex]
2. For [tex]\(\rho = 0.0\)[/tex]:
[tex]\[ \sigma_p^2 = (0.5506^2 \times 0.04) + (0.4494^2 \times 0.18) + (2 \times 0.5506 \times 0.4494 \times 0.2 \times 0.4243 \times 0) \][/tex]
[tex]\[ \sigma_p \approx 0.2202 \][/tex]
3. For [tex]\(\rho = 1.0\)[/tex]:
[tex]\[ \sigma_p^2 = (0.5506^2 \times 0.04) + (0.4494^2 \times 0.18) + (2 \times 0.5506 \times 0.4494 \times 0.2 \times 0.4243 \times 1.0) \][/tex]
[tex]\[ \sigma_p \approx 0.3008 \][/tex]
### Completing the table:
| Coefficient of Correlation ([tex]\(\rho\)[/tex]) | Standard Deviation ([tex]\(\sigma_p\)[/tex]) |
|---------------------------------------|-----------------------------------|
| [tex]\(\rho = -0.4\)[/tex] | 0.1780 |
| [tex]\(\rho = 0.0\)[/tex] | 0.2202 |
| [tex]\(\rho = 1.0\)[/tex] | 0.3008 |
Thus, we have successfully calculated the expected rate of return for Maria's portfolio and the standard deviations for different coefficients of correlation between the returns of stocks A and B.
### 1. Determine the total investment:
First, calculate the total amount Maria has invested in each stock.
- Investment in stock A:
[tex]\[ \text{Investment in stock A} = 7 \, \text{shares} \times \$70 \, \text{per share} = \$490 \][/tex]
- Investment in stock B:
[tex]\[ \text{Investment in stock B} = 4 \, \text{shares} \times \$100 \, \text{per share} = \$400 \][/tex]
- Total investment:
[tex]\[ \text{Total investment} = \$490 + \$400 = \$890 \][/tex]
### 2. Calculate the expected rate of return for the portfolio:
Using the weights of the investments in stocks A and B, calculate the expected return of the portfolio.
- Weight of stock A:
[tex]\[ w_A = \frac{\$490}{\$890} = 0.5506 \][/tex]
- Weight of stock B:
[tex]\[ w_B = \frac{\$400}{\$890} = 0.4494 \][/tex]
- Expected rate of return of portfolio ([tex]\(E(R_p)\)[/tex]):
[tex]\[ E(R_p) = (w_A \times \text{return}_A) + (w_B \times \text{return}_B) \][/tex]
[tex]\[ E(R_p) = (0.5506 \times 0.02) + (0.4494 \times 0.15) \][/tex]
[tex]\[ E(R_p) \approx 0.0784 \][/tex]
So, the expected rate of return after 1 year for Maria's portfolio is approximately 0.0784.
### 3. Calculate the standard deviation for the portfolio for different correlation coefficients:
#### Variance and standard deviation of each stock:
- Variance of stock A ([tex]\(\sigma_A^2\)[/tex]): 0.04
- Variance of stock B ([tex]\(\sigma_B^2\)[/tex]): 0.18
- Standard deviation of stock A ([tex]\(\sigma_A\)[/tex]):
[tex]\[ \sigma_A = \sqrt{0.04} = 0.2 \][/tex]
- Standard deviation of stock B ([tex]\(\sigma_B\)[/tex]):
[tex]\[ \sigma_B = \sqrt{0.18} \approx 0.4243 \][/tex]
#### Portfolio variance and standard deviation for different correlations ([tex]\(\rho\)[/tex]):
Using the formula:
[tex]\[ \sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \sigma_A \sigma_B \rho \][/tex]
1. For [tex]\(\rho = -0.4\)[/tex]:
[tex]\[ \sigma_p^2 = (0.5506^2 \times 0.04) + (0.4494^2 \times 0.18) + (2 \times 0.5506 \times 0.4494 \times 0.2 \times 0.4243 \times (-0.4)) \][/tex]
[tex]\[ \sigma_p \approx 0.1780 \][/tex]
2. For [tex]\(\rho = 0.0\)[/tex]:
[tex]\[ \sigma_p^2 = (0.5506^2 \times 0.04) + (0.4494^2 \times 0.18) + (2 \times 0.5506 \times 0.4494 \times 0.2 \times 0.4243 \times 0) \][/tex]
[tex]\[ \sigma_p \approx 0.2202 \][/tex]
3. For [tex]\(\rho = 1.0\)[/tex]:
[tex]\[ \sigma_p^2 = (0.5506^2 \times 0.04) + (0.4494^2 \times 0.18) + (2 \times 0.5506 \times 0.4494 \times 0.2 \times 0.4243 \times 1.0) \][/tex]
[tex]\[ \sigma_p \approx 0.3008 \][/tex]
### Completing the table:
| Coefficient of Correlation ([tex]\(\rho\)[/tex]) | Standard Deviation ([tex]\(\sigma_p\)[/tex]) |
|---------------------------------------|-----------------------------------|
| [tex]\(\rho = -0.4\)[/tex] | 0.1780 |
| [tex]\(\rho = 0.0\)[/tex] | 0.2202 |
| [tex]\(\rho = 1.0\)[/tex] | 0.3008 |
Thus, we have successfully calculated the expected rate of return for Maria's portfolio and the standard deviations for different coefficients of correlation between the returns of stocks A and B.
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