When Mean-Variance Portfolio Weights Are Random Variables
Summary
In the classical mean-variance setup, expected returns and covariances are treated as known inputs, so optimization produces deterministic portfolio weights. The weights become random when the inputs are estimated from return data: changing samples can change the estimated means and covariances, and therefore the resulting weights. With repeated estimation, weights are generally continuous random variables, and future weights are uncertain from today’s perspective.
The document also describes two ways to assess this uncertainty. One can use the inverse Hessian of an optimizer to estimate standard errors, though binding constraints may make those estimates unreliable or near zero. Alternatively, Michaud-style resampled efficiency repeatedly recalculates weights from perturbed covariance matrices. These approaches account for estimation uncertainty; they do not imply that a single optimization with fixed inputs is stochastic. The discussion is conceptual and gives no empirical comparison of the methods or guidance on which is preferable in a particular portfolio setting.
Key ideas
- With fixed, known expected returns and covariances, mean-variance optimization yields deterministic weights.
- Estimated inputs make portfolio weights uncertain across samples and can be modeled as continuous random variables.
- An optimizer’s inverse Hessian may help estimate weight standard errors.
- Binding constraints can undermine standard-error estimates for affected weights.
- Resampled efficiency explores weight variation by repeatedly perturbing the covariance matrix.
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# Are mean-variance efficient portfolio weights random variables with probability distributions?
# Are mean-variance efficient portfolio weights random variables with probability distributions?
The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the individual weights within the portfolio weight vector can take on values that belong to the real number set, but are they random variables? If so, are they discrete or continuous random variables?
If portfolio weights are random variables, is that because portfolio weights have a probability distribution? How can this be if the mean-variance model only provides a static answer upon optimization? A one-off answer (the portfolio weight vector) does not seem stochastic/random whatsoever
## Answer by fes (score 2)
https://quant.stackexchange.com/a/57024
The original mean-variance model was static and assumed that the mean vector $\mu$ and covariance matrix $\Sigma $ are known. These determine the optimal portfolio weights that in this case are deterministic as well.
However, in practice people do two types of modifications. First because these means and covariances are generally time-varying, we instead use the conditional mean $\mu_{t,t+1}$ and covariance matrix $\Sigma_{t,t+1}$ and try to find the optimal portfolio period by period.
Second, and most importantly, these conditional means and covariances need to be estimated so we actually use estimators for the conditional mean and covariance $\hat{\mu}_{t,t+1}(R_{0,t})$ and $\hat{\Sigma}_{t,t+1}(R_{0,t})$, using a sample of return data $R_{0,t}$. Because this return sample is a random variable, these estimators will be random variables as well. Finally, this implies that the weights are generally continuous random variables. E.g. today we don't generally know what the weights will be in the future.
## Answer by kurtosis (score 1)
https://quant.stackexchange.com/a/57025
There are ways that you might think of portfolio weights as estimates and thus random variables. If you are working with the optimizer, you may be able to get the inverse Hessian out of it. If so, that can be used to get you estimates of the standard errors of your portfolio weights.
One big caveat though: any weight with a binding constraint will likely have a tiny or 0 standard error -- because the partial derivative is not defined at the constraint and that may greatly mess up the estimate of standard error. (Note that this can depend on your optimizer, however.)
Do I know of many people who are conversant enough with optimizers to do this much less people who thought to do it? No. That said, it is worth investigating to see if it can help you add value to portfolio construction.
## Answer by nbbo2 (score 1)
https://quant.stackexchange.com/a/57036
In the original Markowitz papers, no.
In the so called 'resampled efficiency' or 'resampling frontier' method by Michaud, the weights are recalculated over and over from perturbed versions of the covariance matrix, to account for the fact that the covariance matrix is not known exactly (estimation error). In this case yes, the weights are random variables.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.