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in BUS 4028F - Financial Economics by (240 points)

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.

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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 \(\sigma_A\):

= \(X^{2}_A *\sigma^{2}_A + (1-X^{2}_A) * 2* \sigma^{2}_A + 2*\sqrt{2}*X_A * (1-X_A)*\sigma^{2}_A*p\)