So I have a question:

I need to derive an expression for the share of wealth allocated to Asset1in a portfolio that has the minimum possible variance. The following is known:

Two risky assets:1and2

Returns:andr_{1}r_{2}

Variance:σand_{1}^{2}σ_{2}^{2}

Covariance:σ_{12}

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So thus far, I have portfolio return as:

E[r] =_{p}w_{1}E[r_{1}] +w_{2}E[r_{2}]

=w_{1}E[r_{1}] + (1 -w_{1})

And portfolio variance :

σ_{p}^{2}=w_{1}^{2}σ_{1}^{2}+w_{2}^{2}σ_{2}^{2}+ 2w_{1}w_{2}σ_{1}σ_{2}ρ_{1}_{2}

Substituting a formula for Corr_{12}:ρ_{1}_{2}=σ_{12}/σ_{1}σ:_{2}

σ_{p}^{2}=w_{1}^{2}σ_{1}^{2}+w_{2}^{2}σ_{2}^{2}+ 2w_{1}w_{2}σ_{1}σ_{2}*(σ_{12}/σ_{1}σ_{2})

=w_{1}^{2}σ_{1}^{2}+w_{2}^{2}σ_{2}^{2}+ 2w_{1}w_{2}σSo to minimise said portfolio, I would set up a Lagrangian Constrained Optimization as follows:_{12}

y=w_{1}^{2}σ_{1}^{2}+w_{2}^{2}σ_{2}^{2}+ 2w_{1}w_{2}σ_{12}+λ(w_{1}+w_{2}- 1)

such that∂y/∂w_{1}= 0

∂y/∂w_{2}= 0

∂y/∂λ= 0But, from here, I am a tad lost as to how I can reconstruct this to derive a formula for

with minimum variance. Any help appreciated.w_{1 }