If we assume the estimator for a sample mean is biased and if the optimal portfolio weights vary with the estimated mean, is there a way (similar to the zero beta portfolio approach wrt the risk free rate) to construct the Markowitz efficient frontier only from the covariance matrix?

  • $\begingroup$ Maybe it would make more sense to use a more appropriate estimator for the mean? $\endgroup$
    – g g
    May 23, 2022 at 20:19

1 Answer 1


The Markowitz efficient frontier maps the trade-off between risk (volatility or variance) and (expected) return. As such, there exists no way to construct the frontier without resorting to expected returns in one way or another.

Let's consider your idea of using something along the line of the zero beta approach. The ZB portfolio solves

$$ \min_{w} \frac{1}{2}w^T\Sigma w \quad \mathrm{s.t.}\quad w^T\mathbf{1}=1,w^T\Sigma m=0 $$

where $m$ is the vector of market portfolio weights. It is exactly this vector $m$, which entails the market's tradeoff between risk and return, as $m$ is calculated as (without proof)

$$ m^*=\frac{\Sigma^{-1}\mu}{\mathbf{1}^T\Sigma^{-1}\mu} $$

Inserting the optimal portfolion in the ZB ansatz yields the condition

$$ w^T\Sigma m=0\rightarrow w^T\Sigma\frac{\Sigma^{-1}\mu}{\mathbf{1}^T\Sigma^{-1}\mu}=\frac{w^T\mu}{\mathbf{1}^T\Sigma^{-1}\mu}\rightarrow w^T\mu=0 $$

I.e. the zero beta weights must be orthogonal to the mean returns, again requiring mean returns, at least implicitly thru the market consensus portfolio.

Do note, however, that knowledge of any two efficient portfolios suffice to delineate the efficient frontier in its entirety. As such, knowledge of the minimum variance portfolio and the Zb portfolio suffice .

  • $\begingroup$ Thanks for answering my silly question. Of course it doesn't make sense but I quickly played with the idea to change to a risk neutral framework as in Vilkov El. AL. Where the risk free return and the implied vola and correlations would suffice (thus the idea not to use the mean) But that's under Q, not under P, which I need to allocate my assets.. $\endgroup$
    – T123
    May 25, 2022 at 14:56
  • $\begingroup$ I was refering to this paper here: onlinelibrary.wiley.com/doi/full/10.1111/jofi.12778 $\endgroup$
    – T123
    Feb 17, 2023 at 13:48

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