Is it possible to construct an efficient frontier without the mean?

Question

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?

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 .