Problem with deriving weights of a market portfolio in a mean variance framework

Question

I paste a part of a paper that formulate the MV investment problem. I don't understand how equation 5 has been derived and, in particular, how the expected return on the zero beta portfolio comes up. Many thanks in advance for a reply.

Answer 1

From (3) $$ \pmb \Sigma\pmb x+\lambda\pmb \iota=s\pmb \mu\quad\Longrightarrow\quad \pmb x=s\pmb \Sigma^{-1}\pmb \mu-\lambda\pmb \Sigma^{-1}\pmb \iota\tag a $$ Substituting in (4) $$ \pmb \iota'\pmb x=1 \quad\Longrightarrow\quad\pmb \iota' \pmb x=s\underbrace{\pmb \iota'\pmb \Sigma^{-1}\pmb \mu}_a-\lambda\underbrace{\pmb \iota'\pmb \Sigma^{-1}\pmb \iota}_c=sa-\lambda c=1\tag b $$ From (b) we have $$ sa-\lambda c=s\left(a-\frac{\lambda}{s}c\right)=s\left(a-\bar r_zc\right)=1\quad\Longrightarrow\quad s=\frac{1}{a-\bar r_zc}\tag c $$ where $\frac{\lambda}{s}=\bar r_z$. And then substituting (c) in (a) we have $$ \pmb x_m=s\left(\pmb \Sigma^{-1}\pmb \mu-\frac{\lambda}{s}\pmb \Sigma^{-1}\pmb \iota\right)=\frac{1}{a-\bar r_zc}\left(\pmb \Sigma^{-1}\pmb \mu-\bar r_z\pmb \Sigma^{-1}\pmb \iota\right)\tag 5 $$