An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 18% and a standard deviation of return of 18.0%. Stock B has an expected return of 14% and a standard deviation of return of 5%. The correlation coefficient between the returns of A and B is 0.50. The risk-free rate of return is 9%. The proportion of the optimal risky portfolio that should be invested in stock A is __________blank.

To determine the proportion of the optimal risky portfolio that should be invested in Stock A, we can use the following formula, based on the concept of the optimal weights in a two-asset portfolio:

\[
w_A = \frac{\sigma_B^2 (E(R_A) - R_f) - \sigma_A \sigma_B \rho (E(R_B) - R_f)}{\sigma_A^2 (E(R_B) - R_f) + \sigma_B^2 (E(R_A) - R_f) - \sigma_A \sigma_B \rho (E(R_A) - R_f + E(R_B) - R_f)}
\]

where:
- \( w_A \) = the proportion invested in Stock A
- \( \sigma_A \) = standard deviation of Stock A's return
- \( \sigma_B \) = standard deviation of Stock B's return
- \( \rho \) = correlation coefficient between A and B
- \( E(R_A) \) = expected return of Stock A
- \( E(R_B) \) = expected return of Stock B
- \( R_f \) = risk-free rate of return

### Given Data
1. \( E(R_A) = 0.18 \) (18%)
2. \( E(R_B) = 0.14 \) (14%)
3. \( R_f = 0.09 \) (9%)
4. \( \sigma_A = 0.18 \)
5. \( \sigma_B = 0.05 \)
6. \( \rho = 0.5 \)

### Step-by-Step Solution
1. **Calculate the numerator**:
   \[
   \sigma_B^2 (E(R_A) - R_f) - \sigma_A \sigma_B \rho (E(R_B) - R_f)
   \]
   Substituting the values:
   \[
   (0.05)^2 \times (0.18 - 0.09) - (0.18 \times 0.05 \times 0.5) \times (0.14 - 0.09)
   \]

2. **Calculate the denominator**:
   \[
   \sigma_A^2 (E(R_B) - R_f) + \sigma_B^2 (E(R_A) - R_f) - \sigma_A \sigma_B \rho (E(R_A) - R_f + E(R_B) - R_f)
   \]
   Substituting the values:
   \[
   (0.18)^2 \times (0.14 - 0.09) + (0.05)^2 \times (0.18 - 0.09) - (0.18 \times 0.05 \times 0.5) \times (0.18 - 0.09 + 0.14 - 0.09)
   \]

After computing these expressions, we can determine the exact proportion for \( w_A \). I'll perform these calculations to find the precise value.The calculations indicate that the proportion of the optimal risky portfolio that should be invested in Stock A is 0%. This result suggests that, given the risk-return trade-off between Stock A and Stock B, the optimal portfolio would consist entirely of Stock B.

Would you like further details or have any additional questions?

Here are five related questions to expand on this concept:

1. How does the correlation coefficient affect the portfolio composition?
2. What changes would alter the optimal allocation to include Stock A?
3. How would the Sharpe ratio be calculated for this portfolio?
4. What role does the risk-free rate play in determining optimal weights?
5. How does adding more assets affect portfolio optimization?

**Tip:** Lower-risk stocks often dominate the allocation when they offer a reasonable return, especially if the correlation is moderate to low.

An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 18% and a standard deviation of return of 18.0%. Stock B has an expected return of 14% and a standard deviation of return of 5%. The correlation coefficient between the returns of A and B is 0.50. The risk-free rate of return is 9%. The proportion of the optimal risky portfolio that should be invested in stock A is __________.

Calculate the optimal allocation between two risky assets, Stock A and Stock B, given their expected returns, standard deviations, and correlation coefficient. This finance problem demonstrates portfolio optimization, risk-return trade-offs, and includes step-by-step calculations to determine the weight of Stock A in the portfolio. Suitable for undergraduate finance students.

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