# Definition of sharpe ratio maximising and variance minimising portfolios

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.

• You may need to provide more details such as the objective function and the constraints. For people to go through the whole paper is a lot of work. – Gordon Sep 1 '15 at 13:05
• @Gordon sorry about that, I've updated now to hopefully give a better picture – dimebucker91 Sep 1 '15 at 13:11
• @dimebucker91, does your question only refers to why the minimum variance portfolio takes the form $\pi_\sigma$? If yes, the derivation of the portfolio weights has already been given couple of times in this forum. – muffin1974 Sep 2 '15 at 16:35

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: $$u^t=(1, \ldots 1)$$.