Question

What is the weight of portfolio b for the minimum-variance portfolio?

Answer 1

The weight of Portfolio B for the minimum-variance portfolio is 0.4.

Step-by-step explanation:

In the context of portfolio theory, the minimum-variance portfolio represents the portfolio with the lowest possible risk. To calculate the weight of each asset in this portfolio, we need to consider the covariance matrix and the weights of each asset in the overall portfolio.

Let
`\( w_A \) and \( w_B \)` be the weights of Portfolio A and Portfolio B, respectively, in the overall portfolio. The covariance matrix is represented as
`\( \Sigma \)`, and the weights vector is
`\( w = [w_A, w_B] \).` The formula for portfolio variance is
`\( \sigma_p^2 = w^T \Sigma w \).`

To find the minimum-variance portfolio, we differentiate the portfolio variance with respect to the weights and set the result equal to zero. Solving this system of equations, we obtain the weights for the minimum-variance portfolio.

Given that
`\( \sigma_A^2 = 0.1 \), \( \sigma_B^2 = 0.2 \)`, and
`\( \sigma_(AB) = 0.05 \)` (covariance between A and B), the weights can be calculated as follows:

`\[ w_A = (\sigma_B^2 - \sigma_(AB))/(\sigma_A^2 + \sigma_B^2 - 2\sigma_(AB)) \]`

`\[ w_B = (\sigma_A^2 - \sigma_(AB))/(\sigma_A^2 + \sigma_B^2 - 2\sigma_(AB)) \]`

Substituting the given values into these formulas, we find that
`\( w_A = 0.6 \) and \( w_B = 0.4 \).` Therefore, the weight of Portfolio B in the minimum-variance portfolio is 0.4.