Why the Markowitz Efficient Frontier Requires Expected Returns
Summary
The response explains why covariance information alone cannot determine the Markowitz efficient frontier, which describes the trade-off between portfolio risk and expected return. Since expected returns affect optimal portfolio weights, removing them leaves the return dimension of the frontier unspecified.
It examines the zero-beta portfolio analogy and shows that its construction still depends on expected returns, either directly or through the market portfolio. The market portfolio weights are expressed using the covariance matrix and expected-return vector; the zero-beta constraint then reduces to a condition involving mean returns. The response also notes that two efficient portfolios can determine the full frontier, with the minimum-variance and zero-beta portfolios given as an example. No estimation procedure or empirical comparison is provided, so the discussion is a theoretical explanation rather than a solution to mean-estimation bias.
Key ideas
- The Markowitz frontier relates portfolio risk to expected return, so it cannot be defined from covariance alone.
- The zero-beta construction depends on expected returns through the market portfolio or an equivalent constraint.
- The market portfolio weights combine the inverse covariance matrix with the expected-return vector.
- Two efficient portfolios are sufficient to delineate the entire frontier.
- The minimum-variance and zero-beta portfolios are offered as one such pair.
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# Is it possible to construct an efficient frontier without the mean?
# Is it possible to construct an efficient frontier without the mean?
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 by Kermittfrog (score 4, accepted)
https://quant.stackexchange.com/a/71002
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 .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.