I need to derive an expression for the share of wealth allocated to Asset

**1**in a portfolio that has the minimum possible variance. The following is known:

Two risky assets:

**1**and

**2**

Returns:

**and**

*r*_{1}

*r*_{2}Variance:

**σ****and**

_{1}^{2}

**σ**

_{2}

^{2}Covariance:

**σ**

_{12}--------

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}+ 2

*w*_{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}+ 2

*w*_{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}+ 2

*w*_{1}

*w*_{2}

**σ****So 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}+ 2

*w*_{1}

*w*_{2}

**σ**

_{12}+*λ*(*w*_{1}+*w*_{2}- 1)such that

**∂**

*y*/**∂**

*w*_{1}= 0**∂**

*y*/**∂**

*w*_{2}= 0**∂**

*y*/∂*λ***= 0**

But, 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 }