# 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

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.

• If portfolio weights are essentially random variables due to the temporal nature of the mean-variance problem, can a static in-sample estimate of optimal portfolio weights also be described as probabilities such that weight $n$ for asset $n$ is a probability? If so, what is a better interpretation of portfolio weights being probabilities? – develarist Aug 5 '20 at 7:54
• @develarist If you do the optimization one time today, at time $t$, your weights $w(t)$ are known and not random. But in a dynamic setting you likely want to change your weights at some point in the future to $w(t+T)$. From the perspective of time $t$ this $w(t+T)$ is a random variable because it depends on information known only in the future at time $t+T$. – fesman Aug 5 '20 at 8:57
• so portfolio weights can be thought of us probabilities? after all, their sum is also 1 like probabilities – develarist Aug 5 '20 at 9:26
• If you assume the weights are positive they can satisfy the technical definition of probabilities. But I don't know why you'd want to make that interpretation, what would be the economic interpretation? I can assign probabilities to different values of the future weights $w(t+T)$. If you think the values of $w(t+T)$ itself are probabilities I would be assigning probabilities to probabilities. – fesman Aug 5 '20 at 9:56
• @develarist I think they are random variables but still have hard time seeing why you'd want to view them as probabilities. This also doesn't work if you allow for short sales / negative weights because probabilities must be positive. – fesman Aug 5 '20 at 14:28

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.

• Given that there are closed-form analytical solutions for frontier portfolios, does this rule out portfolio weights from being random variables since random variables cannot possibly be derived as analytical solutions? Or am i overthinking, and that random variables can in fact be derived as analytical solutions? – develarist Sep 29 '20 at 20:54
• Absolutely not. That just means you could (possibly) find the distributions in closed-form. A non-constant function of a random variable is still a random variable. – kurtosis Sep 30 '20 at 3:08

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.

• so I am referring to the classical mean-variance model (and not resamples of it), but your answer (the first line of it, which refers to the classical case that I am also talking about) is the first here to say that portfolio weight estimates are not random variables. Is this directly said by Markowitz in his papers, that they are not random variables? – develarist Aug 1 '20 at 2:51