Reviewing the Portfolio Construction

Portfolio Context

Background

The AILA methodology typically generate allocation opportunities for a given asset, e.g. the WTI Dec25 futures.
These individual asset return streams can be combined into a portfolio based on their expected Sharpe ratio (SR) and correlation.
The allocation weights resulting in the maximum portfolio SR can be determined by a Markowitz like approach.
However, the optimal solution can be strongly influenced by statistical noise when estimating the inputs.
In this short exercise we review three aspects related to constructing an optimal portfolio within the context of the AILA methodology and the associated asset return streams.

Review Aspects

Higher moments of the return dist.
Correlation structure.
Sharpe ratio uncertainty.

References

[Caj] D. Cajas, Semidefinite Relaxation of Higher Portfolio Moments (Oct 2025). Available at SSRN: https://ssrn.com/abstract=5284483
[Lop] M. Lopez de Prado et al., How to use the Sharpe ratio (Sep 2025). Available at SSRN: https://ssrn.com/abstract=5520741
[Ant] A. Antonov et al., Overcoming Markowitz's instability with the help of the HRP (Dec 2023). Available at SSRN: https://ssrn.com/abstract=4748151

Higher Moments

Portfolio Optimization

Optimization of the portfolio Sharpe ratio can be extended to also consider higher moments of the asset returns.
Studies have shown that such methods can impact on the portfolio properties as intended, however, often at the expense of additional approximations and constraints, e.g. see [Caj].
Beside aspects related to increased computational complexity, the indicated potential improvements has to be balanced w.r.t. the implications from realistically estimating the asset moments themselves.
In the AILA strategy context considered here the indicated improvements from including higher moments in the optimization appears small compared to the issues associated with realistically estimate the moments.

➔ We therefore focus on an optimal SR portfolio in the simpler case of only considering the mean and variance, i.e. two first moments.

SR distribution

As pointed out in various studies, e.g. see [Lop], the estimated SR tends, under relatively general assumptions, to be normally distributed, even if the underlying return distribution in not normal.
The variance of the estimated SR is influenced by the higher moments of the returns, however, in the context addressed here the dominant aspect for the variance typically is the sample size (T).
Further we can express the optimal weights to maximize the portfolio SR in terms of asset correlation as well as asset SRs.

➔ We will use this framework to review the impact on the optimal weights associated with the estimation uncertainty of the asset correlation as well as asset SRs.

Max SR portfolio

w = c_R · ρ^-1 E

c_R: risk factor, c_R → Σ|w| = 1
ρ: asset correlation.
E: asset SR (exp. risk adj. return).
Note: all SR below are annual.

Correlation

Spurious Correlations

A common approach to reduce the impact from noise when estimating the asset correlation, is to reduce the rank of the correlation matrix, for example, by clustering assets, e.g. see [Ant].
Here we consider an AILA portfolio of 77 commodity futures, across Ags, Metals and Energy. Including the case of several month contracts for the same commodity.
We cluster the assets w.r.t. the estimated correlation (last 252 days) and find, using a silhouette analysis like in [Ant], that the resulting clusters coincide closely with the traditional commodity sectors.
For commodities, such a cluster structure with weak inter (strong intra) cluster correlation seems to naturally fit intuitive expectations.
The optimal weights obtained when asset SR are kept identical, however, show that even slight differences of the high intra cluster correlations can result in non-trivial spread configurations.

➔ Small estimation noise w.r.t correlation can result in very different ("random") weights, unless theoretical formula is constrained.

Sharpe ratio

SR Uncertainty

The uncertainty w.r.t. asset SRs can cause the same kind of issue regarding the optimal weights as shown for correlation.
To study the effect, we make a MC analysis under assumption that the AILA commodity asset models all have true long-term SRs, consistent with the long-term back test results.
Given a period of time (T) the realized portfolio SR vary because:
- 1) Weight not optimal, i.e. est. asset SRs not the true SRs.
- 2) Weights optimal, but asset SRs vary between periods.
For the study we use two sets of asset SR distributions:
- a) SRs from back test period 2017 to 2021.
- b) SRs from back test period 2022 to 2025.
For a given asset, the dot (error bar) represent the estimated SR (STD variation) for that given period.
We then sample random values from a normal distribution with the corresponding mean and STD, to generate new asset SR values corresponding to a new period of the same length assuming the estimated (mean) SR is true.

Textbook Approach

Formula Weights

To illustrate the issue caused by the SR uncertainty, the optimal weights are calculated directly from the formulas using the estimated SRs from period (a).
Portfolio SRs are then calculated by sampling asset SRs from either the distribution (a) or (b), i.e. mimicking new periods of the same length assuming either (a) or (b) is true.
The portfolio SR distribution (blue) from using period (a) SRs both to calculate the weights and for sampling new periods, correspond to the case of optimal weights, but with random variation for each new time period generated.
In contrast (orange), using period (a) to calculate the weights and sampling new time periods from the (b) SRs, correspond to having different weights (a) than those that are optimal for the true SRs (b), together with the random variation for each time period.
The result demonstrates how the optimal weights from directly using the formula are tailored closely to the relative details of the SR set (a) and how these optimal weights are practically useless when applied to the true SR set (b), generally positive but different to (a).

Practical Approach

Numerical Optimization with Constraints

The issue is expected to be largest in situations where two (or groups of) assets have a strong SR imbalance and high correlation, e.g. cases of SR₁/SR₂ < ρ₁₂ resulting in spread configurations.
Since the AILA allocations are associated with an expected SR>0, it is reasonable to constrain the expected SR to some realistic range, e.g. 0.5 < SR <1.0. Note SR~0 conceptually corresponds to no allocation.
A second constraint could be to prevent weights from flipping sign in the optimization, i.e. retain the original long and short signals.
The optimal weights (SR set (a) and corr) are compared to those with all SR identical (corr only) and equal weights.
Repeating the same MC exercise as before show that with the constraints, knowing the correlation (corr only) improve the portfolio SR when sampling from both (a) and (b) compared to equal weights.
However, when including the estimated asset SRs, period (b) results improve marginally compared to the (overfitted) period (a) results.

➔ Estimated SR often result in optimal weights with large portfolio concentration to few "past winners", not repeating out-of-sample.

Conclusions

The optimal portfolio weights can be calculated theoretically, however, the impact from the uncertainty of the asset level inputs often cause severe distortion to what would be optimal out-of-sample.
In this short study we review three aspects related to determining the optimal portfolio weights in the context of the AILA strategy methodology, i.e. with typical AILA asset portfolios and corresponding allocation opportunity models.
The results shown are in some respects exaggerated, however, demonstrate the effects supporting the following conclusions.
In the AILA context, the potential benefit from including moments higher than the first two (mean and variance) in the weight optimization method appears small compared to overcoming the problems associated with realistically estimating those moments, therefore we give priority to understanding the impact from estimation uncertainty.
For commodities the traditional sector structure naturally coincide with the results and the idea behind clustering of asset correlations, i.e. reduce to a weakly inter (strongly intra) cluster correlation structure.
The tendency of optimal weights corresponding to highly overfitted spread configurations can be addressed with optimization constraints, however, often still result in high portfolio concentration to a "past winners", possibly with minimal out-of-sample benefit and potentially in conflict with risk/allocation limits.