Markowitz Modern Portfolio Theory / Mean Variance Framework

We derive the efficient frontier, with and without risk free asset, formulae below. We also give characterization of the minimum variance portfolio.

Global Minimum Variance Portfolio

Let W represent the vector of the portfolio weights, R the vector of the returns, and \( \Sigma \) the Variance-Covariance matrix. Then, as before, the mean and variance of the portfolio can be expressed in matrix notation as follows:

$$ r_{p}= W^{'}R = R^{'}W $$ $$ {\sigma_{p}}^2=W^{'}\Sigma W $$

To find the portfolio which has the lowest variance, we minimise the variance of the portfolio subject only to the budget constraint (sum of weights=1):

$$ L \left( W, \gamma \right)=\frac{1}{2} W^{'}\Sigma W +\gamma \left(1- \mathbf{1}^{'} W \right)$$

Differentiating and setting the derivatives equal to zero, we get

$$ \frac{\partial L \left( W, \gamma \right)}{\partial W}=\Sigma W -\gamma \mathbf{1}=0 $$ $$ \frac{\partial L \left( W, \gamma \right)}{\partial \gamma}=1- \mathbf{1} ^{'} W =0 $$

We need to solve the two equations for the two unknowns, with W being the ultimate prize. Solving the first equation for W, we get

$$ \Sigma W -\gamma \mathbf{1}=0 $$ $$ W=\gamma \Sigma^{-1} \mathbf{1} $$

And substituting into the second equation,

$$ 1- \mathbf{1}^{'}W =0 $$ $$ 1- \mathbf{1}^{'} \left( \gamma \Sigma^{-1} \mathbf{1} \right) =0 $$ $$ \gamma=\frac{1}{ \mathbf{1}^{'} \Sigma^{-1} \mathbf{1} } $$

Thus:

$$ W=\gamma \Sigma^{-1} \mathbf{1} $$ $$ W=\frac{1}{ \mathbf{1}^{'} \Sigma^{-1} \mathbf{1} } \Sigma^{-1} \mathbf{1} $$