Consider three securities with the following expected returns, standard deviations of returns, and correlations between returns:
$$
\begin{matrix}
\mu_1=0.20, & \sigma_1=0.31, & \rho_{12}=\rho_{21}=-0.2, \\
\mu_2=0.25, & \sigma_2=0.17, & \rho_{23}=\rho_{32}=0.6, \\
\mu_3=0.17, & \sigma_3=0.27, & \rho_{31}=\rho_{31}=0.15, \\
\end{matrix}
$$
Determine the weights in the minimum variance portfolio.
My question is what is a minimum variance portfolio?
For finding the weights I know $0.20\omega_1+0.25\omega_2+0.17\omega_3=$ expected return of portfolio and $\omega_1+\omega_2+\omega_3=1$ but I don't know what the expected return of the portfolio is and there are three variables but only 2 constraints?
Best Answer
first form the covariance matrix, $C,$ its entries are found from the standard deviations and correlations.
Let $e = (1,1,\dots,1).$ In this case, $(1,1,1).$
Solve $Cy = e.$
Normalize the elements of $y$ so they add to $1.$
Done.
(for discussion of why this is the minimal variance portfolio, see Joshi--Paterson, Introduction to Mathematical Portfolio Theory.)