How Beta Estimates Relate to Mean-Variance Portfolio Optimization
Summary
The document asks how differences in estimated asset betas affect a mean-variance optimal portfolio. The optimizer described minimizes portfolio variance subject to a target expected return and a fully invested constraint. In the standard formulation, the required inputs are the expected return vector and the return covariance matrix; beta estimates are not generally independent inputs to this optimization.
One response emphasizes that beta is not ordinarily used directly to construct a mean-variance portfolio, though it may relate to covariance risk. Another explains that under a factor model such as CAPM or APT, expected returns are linked to betas, and model assumptions can also connect betas to covariance estimates. The discussion is conceptual and offers no empirical comparison or implementation guide. Its implication is that beta matters through a chosen return-generating model, rather than serving as a universal substitute for estimating portfolio inputs.
Key ideas
- Mean-variance optimization uses expected returns and the covariance matrix as its main inputs.
- Betas are not usually inserted directly into the standard optimization problem.
- Factor models can link expected returns to estimated betas.
- Any role for beta in covariance estimates depends on the assumptions of the factor model.
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# Estimated betas and optimal portfolio
# Estimated betas and optimal portfolio
I ran a regression on 20 assets to estimate their beta with different methods. I would like to see the differences of these estimation differences in terms of mean-variance optimal portfolio. How can I do that?
The problem is that I do not see clearly the role that the estimated beta plays in portfolio optimization empirically. A mean-variance optimization with N assets should be something like
$min_{\textbf{w}}$ $\textbf{w}^{'}\textbf{V}\textbf{w}$
subject to
$\textbf{w}^{'}E(R)=\bar{R}$,$\textbf{w}^{'}\textbf{I}=1$
Where w are the weights of the N assets, our choice variable. $\textbf{V}$ is the covariance matrix of the returns, $\bar{R}$ is the target return, $\textbf{I}$ is just a vector of ones and $E(R)$ is the vector of expected returns of our assets.
The "unknowns" are V and E(R), should I use historical returns to replace them? If yes, then what is the use of the betas in portfolio optimization? none? or maybe should I use the betas to get the values of E(R) and V?
I do not understand, could someone provide a mini-guide to get from betas to the optimizing portfolio?
## Answer by Chris (score 2)
https://quant.stackexchange.com/a/45942
Betas aren't traditionally used in creating MV optimal portfolios. Insofar as beta is a proxy for risk, as is vol, there's probably some relationship between your betas and the covariance matrix that IS used in MV optimization, but there's not really any guide to give you that shows how betas are used, because they aren't.
## Answer by Vitomir (score 0)
https://quant.stackexchange.com/a/45928
In a factorial model, such as CAPM / APT, you have a linear relationship describing the process generating expected returns.
Therefore, expected returns depend on betas. Therefore, since both the Covariance matrix and expected return vector depend on returns, they will also depend on betas under the assumptions of a factorial model.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.