# 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.

commented Apr 16, 2016 by (4,220 points)

What are your thoughts so far?

answered May 30, 2017 by (3,180 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 $$\sigma_A$$:
= $$X^{2}_A *\sigma^{2}_A + (1-X^{2}_A) * 2* \sigma^{2}_A + 2*\sqrt{2}*X_A * (1-X_A)*\sigma^{2}_A*p$$