Or 'Solving materiality's Three Body Problem'
Materiality assessments - both single and double, ISSB and ESRS - have a 3 body problem. When preparing an assessment, the guidance is clear: compare the entity in question to a peer group and an index to give context. This is correct and sensible. But it's remarkably hard to do objectively, in a data-driven methodology, with an auditable process.
Bringing order to chaos
The reason is well-known in classical mechanics or astrophysics. It is called "The three body problem". It is the mathematical study of how three celestial objects (like stars or planets) interact purely through their gravitational pull. While a "two body" system (like the Earth and Sun) has predictable orbits, adding a third body (like Earth's moon) creates a chaotic, unpredictable system with no general mathematical formula to solve it.
In our terms, the reporting company is the Earth, the peer group is Moon and the index is the Sun. We have already determined that we can take closing day share price as a reasonable dependent variable to help us isolate which sustainability drivers are having a real world impact on investors' capital allocation decisions. And we have already solved the one-body problem with standard machine learning processes. So the final step is to scale up the dataset, run the algorithm across multiple entities, and we should have an answer...
The problem with Machine Learning
When we tested the ML solution at scale with real data, we found several fatal problems:
we couldn't remove the sector and index impacts to isolate only the entity's ESG drivers
the high multicollinearity between ESG drivers made the system unstable and rankings volatile
we still had arbitrary thresholds for materiality
as materiality is dynamic, we would need to constantly cross-validate and adjust the hyperparameters
machine learning is black-box by its nature so struggles at audit time.
Time for a Rethink
So we needed a different approach. The one we settled on involves convex optimization and linear algebra instead. Without wanting to pre-empt the academic research paper we are now preparing, there are three key problems which this approach effectively overcomes:
Multicollinearity. The Ledoit–Wolf solution is an elegant way to deal with the overfitting that comes when combining large numbers of sustainability driver data points with limited historical time periods. This solution computes an analytically optimal shrinkage intensity automatically without needing cross-validation, so preventing portfolio optimization algorithms from blowing up on extreme error.
Arbitrary Thresholds. We need to zero out non-material drivers without assuming any arbitrary threshold. There is a concept in transformers (LLMs) called Sparsemax which projects the weight into a probability simplex and gets a threshold, below which all values will be zeroed out.
Removing sector and index effects. Here we use the Frisch–Waugh–Lovell (FWL) theorem which states that multiple regression coefficients can be calculated by first removing ("partialling out") the effect of control variables from both the dependent and explanatory variables. This gives us the correlation coefficients for each of the three bodies.
BP plc example
Using our revised approach, we ran a 52-week window to 28 March 2025 using the industry peer group (per SASB taxonomy) as the peer group and the FTSE 100 index as the benchmark. We found:
Firm-specific sentiment (Earth) carries the single largest share of variance (47.09%)
Materiality is concentrated, not diffuse: only 9 of 26 SASB dimensions are active under Sparsemax regularization, and just 3 of those 9 clear statistical
significance — Access & Affordability (p=0.0177), GHG Emissions (p=0.0211) and Water & Hazardous Materials (p=0.0305). Employee Health & Safety sits just
outside the threshold (p=0.0592).
Both macro factors clear significance — unusually. The sector peer-group index (Moon) is the dominant and most significant driver (p<0.0001, 39.99% share);
the FTSE 100 benchmark (Sun) is also significant, though far weaker (p=0.0159, 12.92% share). Seeing both clear the 5% threshold at once is uncommon.
Context: a separate 2-body baseline for the sector itself, estimated over the same 52-week window (5 Apr 2024 – 28 Mar 2025), finds the sector's own returns are 74.55% explained by its own ESG drivers, led by Competitive Behavior (19.60% share, the sector's only significant driver, p=0.0242). This suggests the sector's pull on BP is itself partly a sustainability-sentiment signal, not a purely macro one.
Source: Maxwell Data
To give more context to the summary slide above, we also include the supporting diagnostics.
Signal diagnostics support genuine, if modest, information content: Rank IC (+0.467) and Pearson IC (+0.487) both comfortably exceed common institutional
thresholds, and the 65.38% directional hit rate is well above a 50% coin-flip — but this remains an in-sample attribution framework, not a forecasting tool.
Source: Maxwell Data
An Industry Standard?
Maxwell Data and the Department of Mathematics at Brunel University of London will present these findings and those from a number of proof-of-concepts at an industry workshop at UCL Centre for Sustainable Business on 18th September. If there is consensus, we will work with stakeholders to roll out a data-driven approach to financial materiality for sustainability matters as an industry standard ahead of reporting season in Q1 2027.