Mean–Variance Portfolios: Minimum Variance and Sharpe Ratio Claims
Summary
The document examines formulas for two portfolios defined using a return vector, a covariance matrix, and a budget vector. One portfolio is described as maximizing the Sharpe ratio, while the other is described as minimizing variance. The author asks how these interpretations follow from the formulas and provides context from an active-portfolio optimization: maximize expected active return subject to a variance target and a zero-sum active-weight constraint. The resulting expression combines inverse-covariance-weighted returns with an adjustment involving inverse-covariance-weighted ones.
The included answer reframes mean–variance optimization as minimizing variance at a target expected return, subject to full-investment and return constraints. It says the minimum-variance portfolio is simpler to establish and points to external derivations for the efficient frontier and maximum-Sharpe result. The document does not supply those proofs, and the active-weight setup differs from the unconstrained fully invested formulation. Its claims should therefore be interpreted in the context of the stated constraints and definitions.
Key ideas
- The document asks how covariance-based portfolio formulas relate to minimum variance and maximum Sharpe ratio.
- Its active portfolio setup maximizes expected active return at a specified variance while requiring active weights to sum to zero.
- The answer presents minimum variance at a target return as an equivalent mean–variance optimization framing.
- The included material does not provide a full proof of the maximum-Sharpe claim or reconcile all constraint differences.
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# Definition of sharpe ratio maximising and variance minimising portfolios
# 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.
## Answer by Sebapi (score 1)
https://quant.stackexchange.com/a/44369
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.pdfShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.