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Compute the weights in a portfolio consisting of two kinds of stocks if the expected return on the portfolio is to be $E(K_v)=10\%$, given the following information on the returns on stock 1 and 2: $$ \begin{matrix} Scenerio & probability & return K_1 & return K_2 \\ \omega_1 & 0.1 & -10\% & 10\% \\ \omega_2 & 0.3 & 0\% & -5\% \\ \omega_3 & 0.6 & 15\% & 20\% \\ \end{matrix} $$

I found $E(K_1)=8\%$ and $(E_2)=11.5\%$ so $0.08\omega_1 + 0.115\omega_2 = 0.1$
But I don't know how to find the weights? I think the covariance will help me so I found that to equal $0.0109$ but I am not sure if it is correct and I don't know how to find the weights?

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  • $\begingroup$ your second constraint is $w_1 + w_2=1$. $\endgroup$ – Neeraj Feb 21 '16 at 17:31
  • $\begingroup$ Nothing is said about variance, so choose weights that minimize your portfolio standard deviation for a expected return of 10% $\endgroup$ – Neeraj Feb 21 '16 at 17:32
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    $\begingroup$ There are two variables and two constraints. Solve them simultaneously. What more do you want? $\endgroup$ – Neeraj Feb 21 '16 at 18:01
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    $\begingroup$ @Neeraj How is this basic? it involves linear algebra and calculus right? $\endgroup$ – BCLC Feb 23 '16 at 18:39
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    $\begingroup$ @Neeraj So this is basic? $\endgroup$ – BCLC Feb 23 '16 at 18:39
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In case of 2 securities, each and every combination of portfolio lies on efficient frontier. In your question, you have given to achieve expected return of exactly 10%. So, we have $$E(R_p)=w_1E(K_1) + w_2E(K_2)=0.10 \tag{1}$$ subject to: $$w_1 + w_2=1 \tag{2}$$ Solve your equation 1 and 2 to get $w_1$ and $w_2$. Resulting weights would lead to minimum variance for given expected return. Variance of portfolio: $$var(R_p)= w_1^2 \sigma_{K_1}^2 + w_2^2 \sigma_{K_2}^2 + 2 w_1 w_2\, cov(K_1, K_2) $$ where, $\sigma_{K_1}$ and $\sigma_{K_2}$ are standard deviation of $K_1$ and $K_2$ respectively.

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