Formulating a Zero-Beta Minimum-Variance Portfolio

Summary

The document asks how to solve a constrained portfolio optimization problem in R. The proposed objective minimizes portfolio variance while requiring zero covariance with a representative market index, fully invested weights, and an expected return above a threshold. The answers focus on interpreting the covariance constraint and translating it into a form suitable for a quadratic optimizer such as solve.QP.

One response questions the notation and says that with fixed return observations, the covariance expression may reduce to an affine constraint. Another uses covariance’s linearity to express portfolio beta as the weighted sum of individual asset betas, making a zero-beta requirement linear in the portfolio weights. That response also cautions that the target-return constraint may make the problem infeasible if expected returns are all positive, and suggests that long and short positions may be needed. No complete R implementation or numerical solution is supplied. Feasibility depends on the input estimates, asset universe, and any additional weight restrictions, which the question does not specify.

Key ideas

• Portfolio beta can be written as a weighted sum of asset betas, making a zero-beta constraint linear in the weights.
• The objective in the question minimizes portfolio variance subject to covariance, budget, and return requirements.
• The covariance notation needs clarification to determine how the constraint should be represented.
• The return target may be infeasible with positive expected returns unless the available portfolio construction permits suitable long and short positions.
• The discussion gives no complete solver implementation or numerical result.

Tags

# Zero Beta Portfolio in R


# Zero Beta Portfolio in R












I am trying to solve the zero portfolio problem in R. Given n assets, the objective function is to minimize the variance of the portfolio $$Min_x\;\; \frac{1}{2}x^T\Sigma x$$ subject to $$COV\left(x^T R, R^Tm \right) =0 $$ and $$x^T \mathbb{1}=1$$ and $$\mu^Tx \geq \tau$$ Where $x$ are portfolio weights, $\tau$ is a required return and $R_m$ is a representative market index.

I have seen the relevant discussion here and the code provided but it does not fit the above specification. Is it possible to solve that problem either with solve.QP or any other function?

## Answer by Attack68 (score 1)

https://quant.stackexchange.com/a/41068

To be honest I'm a little confused by your notation so this is a preliminary answer which I'll either delete or amend depending on your responses.

If $x^TR$ is a deterministic scalar how are you defining the covariance?

What I think you might be defining is the discrete case of covariance:

$$ Cov(x_iR_{m,i}, R_{m,j}) = \frac{1}{n} \sum_i (x_iR_{m,i} - E[x_iR_{m,i}]) (R_{m,i} - E[R_{m,i}]) = 0$$

In which case I'm assuming that $R_{m,i}$ are fixed, known values, in which case this constraint probably reduces to an affine one.

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/41069

The beta of your portfolio $\beta_P$ is given by $$ \beta_P = \sum_{i=1}^n w_i \beta_i. $$ This can be seen from bilinearity of the covariance: $$ Cov(\sum_{i=1}^n w_i r_i, r_M) = \sum_{i=1}^n w_i Cov(r_i, r_M). $$

Thus the constraint is linear and given above with the $\beta_i$ as constants and the weights are the variables that you want to optimize.

I assume that the solution will only be possible with a long/short portfolio. Thus, if your $\mu$ is a positive vector the condition on expected values that you have will render the problem infeasible.

Shown 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.