Averaging Scenario-Based Optimal Portfolios Under Constraints

Summary

The document describes a proposed portfolio procedure: draw samples from a random vector of asset returns, optimize weights for each draw subject to linear constraints, and average the resulting allocations. Because the feasible set defined by the constraints is convex, the averaged weights remain feasible. The question is whether this average has a sound statistical interpretation or a practical advantage over optimizing using the mean return vector.

The author suggests that averaging may provide a compromise across extreme return scenarios, but does not establish that it improves expected performance or constitutes a Bayesian method. An update points to related discussion and a paper using a different objective that includes a quadratic risk term. No derivation, empirical comparison, or performance evidence is provided here, so the proposed procedure should be treated as a question about optimization and estimation rather than a validated strategy. Its value would depend on the sampling model, objective, and treatment of risk.

Key ideas

• Optimizing separately across sampled return scenarios produces a set of candidate portfolios.
• Averaging feasible allocations preserves feasibility when the constraint set is convex.
• The document raises, but does not resolve, whether allocation averaging is statistically preferable to using mean returns.
• A referenced related method adds a quadratic risk term, but its details and evidence are not included.

Tags

# Sampling in Portfolio Optimization


# 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)

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