# Determine the values of ρ which allow the possibility of constructing a zero-risk portfolio

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Consider a mean-variance portfolio model with two securities, S(A) and S(B) , where the expected return and the variance of return for S(B) are twice the corresponding values for S(A) .  Suppose the correlation between the returns on the two securities is ρ

answer the question by calculating the variance of the return on a portfolio with weights x(A) and x(B) invested in the two asset.

by (4.2k points)

What are your thoughts so far?

by (3.2k points)

We are given that 2* $$\sigma^{2}_A = \sigma^{2}_B$$ , p = cor(A, B) , cov(A,B) = $$\sigma_A*\sigma_B*p$$
As per normal, $$Portfolio = X_A *A + X_B *B$$
Thus, $$var(P) = X^{2}_A *var(A) + X^{2}_B *var(B) + 2*X_A*X_B*cov(A,B)$$
= $$X^{2}_A *\sigma^{2}_A + X^{2}_B *\sigma^{2}_B + 2*X_A * X_B*\sigma_A*\sigma_B*p$$
This can be simplified in many ways such as using the fact that $$X_A +X_B =1$$ and 2* $$\sigma^{2}_A = \sigma^{2}_B$$ . For example, in terms of $$X_A$$ and $$\sigma_A$$:
= $$X^{2}_A *\sigma^{2}_A + (1-X^{2}_A) * 2* \sigma^{2}_A + 2*\sqrt{2}*X_A * (1-X_A)*\sigma^{2}_A*p$$