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 portfolios.

Efficient Frontier With Risk Free Assets

We now introduce a risk free asset into the framework. Let \( r_{f}\) and \(w_{f} \) represent the return and weight of the risk free asset, respectively. Let Z represent the vector of excess return over the risk free rate.

$$ r_{p}= R^{'}W + (1-\mathbf{1}^{'}W)r_{f} =r_{f} + \left( R^{'}-r_{f} \mathbf{1}^{'}\right)W = r_{f} + Z^{'}W$$ $$ {\sigma_{p}}^2=W^{'}\Sigma W $$

The portfolio optimisation problem is then again to minimise the variance of the portfolio for a given level of return but there is no budget constraints as the presence of risk free asset allows unlimited borrowing and lending:

$$ L \left( W, \lambda \right)=\frac{1}{2} W^{'}\Sigma W + \lambda \left( r_{p} - r_{f} - Z^{'}W \right)$$

Differentiating and setting the derivatives equal to zero, we get

$$ \frac{\partial L \left( W, \lambda \right)}{\partial W}=\Sigma W - \lambda Z=0 $$ $$ \frac{\partial L \left( W, \lambda \right)}{\partial \lambda}=r_{p} - r_{f} - Z^{'}W=0 $$

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

$$ \Sigma W - \lambda Z=0 $$ $$ W=\lambda \Sigma^{-1}Z $$

And substituting into the second equation,

$$ r_{p} - r_{f} - Z^{'}W=0 $$ $$ r_{p} - r_{f}- \lambda Z^{'} \Sigma^{-1}Z =0 $$ $$ \lambda = \frac{r_{p} - r_{f}}{Z^{'} \Sigma^{-1}Z} $$

Thus:

$$ W=\lambda \Sigma^{-1}Z $$ $$ W=\frac{r_{p} - r_{f}}{Z^{'} \Sigma^{-1}Z}\Sigma^{-1}Z $$