# Sampling in Portfolio Optimization

I recently came across the following method for portfolio optimization: Let $$Y$$ be a random variable that describes the returns of $$n$$ assets. Fix a constraint matrix $$A \in \mathbb{R}^{m \times n}$$ and $$b \in \mathbb{R}^m$$. Then, we sample from $$Y$$ and calculate the weights $$x$$ that maximize returns under the constraints $$Ax \ge b$$. Finally, we average over all optimal allocations to compute the final weights $$\bar x$$.

Since the set $$\{ Ax \ge b\}$$ is convex, we can be sure that $$\bar x$$ also satisfies the constraints. Thus, I think the practical idea is that $$\bar x$$ will be a good feasible compromise between extreme cases of $$Y$$ while still somewhat maximizing the objective function.

But from a statistical point of view, it is unclear to be why this would be a good procedure. Why don't we use the mean of $$Y$$ directly? Is this some kind of Bayesian approach?

Update

I think my statistical concerns were essentially adressed here and the method was discussed in the paper mentioned there (though with a different objective function including a quadratic term that reflects risk)

• Interesting. Where did you see ths algorithm? Commented Apr 9, 2022 at 15:37
• Such linear constraints will not reflect risk. So this will miss one of the two interesting aspects of portfolio optimization.
– g g
Commented Apr 9, 2022 at 17:51
• Hi Claudio: This is the paper that I was referring to. I don't remember if it's enough for learning how to go about dealing with a specific problem but I remember it being quite helpful for someone without previous exposure. I hope it helps some. www2.stat.duke.edu/~scs/Courses/Stat376/Papers/Basic/… Commented Apr 10, 2022 at 10:55
• The approach is also called Michaud portfolio resampling. en.wikipedia.org/wiki/Resampled_efficient_frontier
– John
Commented Apr 11, 2022 at 15:11
• The argument against resampling is typically 1) it generates less expected utility than a classical mean-variance optimal portfolio, 2) the smooth behavior of the resampled frontier is only produced when inequality constraints are used (most common being the long only)
– John
Commented Apr 11, 2022 at 15:13