## 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:

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