Definition of sharpe ratio maximising and variance minimising portfolios

Question

In this paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2226985, in the derivation of the mean variance efficient portfolio using lagrangians in the appendix, on page 29, the two portfolios are defined:

$$ \pi_R=\frac{V^{-1}R}{1^TV^{-1}R}, \quad \pi_\sigma=\frac{V^{-1} 1}{1^TV^{-1}1} $$ where $V$ is the covariance matrix, $R$ is a column vector of returns and $1$ is the vector of ones. $\pi_R$ is said to be the portfolio that maximises the sharpe ratio while $\pi_\sigma$ is the portfolio that minimises variance, I'm not quite sure how the definitions follow?

Update: For background:If we let $d$ be a column vector of active weights, that is, the difference between the portfolio weight and the benchmark, then the objective is to: $$ \text{max} ~~R^T d $$ subject to $$ d^T V d = \sigma^2_\alpha, \quad 1^Td =0 $$ where $\sigma^2_\alpha$ is the active portfolios variance of arithmetic return.

Setting up the lagrangian and a bit of calculus gets to: $$ d=\frac{1}{2\lambda_1} \left( V^{-1}R - \left(\frac{1^TV^{-1}R}{1^TV^{-1}1} \right)V^{-1} 1\right) $$ this is the last step before the portfolios are defined.

Answer 1

The problem can be set either as that of maximizing return given a variance target or minimizing variance given a return target.

Let $\mu$ be the vector of expected returns and $\Omega$ the returns covariance matrix of $n$ assets. The Markowitz optimization problem is to find the minimum variance portfolio that achieves an expected return $\mu_p$.

$$w^* = {\arg\,\min} \frac{1}{2} w^t \Omega w$$

subject to the sum of weights constraints $u^t w = 1$, and returns constraints $\mu^t w = \mu_p $ where $u$ is the unit vector composed of ones: $u^t=(1, \ldots 1)$.

With this problem, we get a Lagrangian with 2 linear constraints which enable finding the 2 portfolios much more simply. The math for the 2 portfolios is derived here: http://www.markowitzoptimizer.pro/wiki/EfficientFrontierMath

It is relatively simple to show the portfolio whose weights do not depend on return has minimum variance. Showing that the second portfolio has maximum Sharpe is more involved, so the actual proof links to this lecture: https://www.ie.bilkent.edu.tr/~mustafap/courses/OIF.pdf