📈 economics

Designing Stable ESG Grade Bands: Cut-Point Optimisation and a Quantisation Perspective

This paper demonstrates through computational optimization and stability analysis that the incumbent equal-width ESG grade scale offers superior stability against score perturbations compared to non-uniform alternatives, while establishing a framework showing that optimal boundary design depends jointly on score distribution and the magnitude of expected revisions.

Original authors: Hira Mahmood, Minh N. Dao, Claudio Conversano

Published 2026-10-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hira Mahmood, Minh N. Dao, Claudio Conversano

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Investors increasingly rely on environmental, social, and governance scores to decide where to put their money. These scores attempt to measure how responsibly a company operates, turning complex data about carbon emissions, worker treatment, and board oversight into a single number. To make these numbers easier to read, rating agencies often convert them into letter grades, similar to school report cards. A company might receive an "A" for excellent performance or a "C" for average. However, the line between an "A" and a "B" is not a natural law; it is a human-made boundary drawn on a continuous scale. If a company's score improves by just a tiny fraction, it might cross that line and jump a whole letter grade, or it might stay put. This raises a critical question for the people who manage billions of dollars: where should these lines be drawn to ensure that small, noisy changes in a score do not cause wild swings in a company's rating?

A team of researchers set out to answer this by examining the historical scoring system used by a major provider, Refinitiv/LSEG. In this system, companies received a score from zero to one hundred, which was then sliced into twelve letter grades, ranging from D-minus to A-plus. The researchers focused on the four main categories—D, C, B, and A—which were already fixed at specific points on the scale. Their goal was to test whether the smaller slices within those categories, which separated the pluses and minuses, were drawn in the best possible way. They asked if moving these internal lines could make the ratings more stable, meaning less likely to change due to minor fluctuations in the data.

The study began with a surprising discovery about the data itself. The researchers looked at over a thousand records of large companies from 2018 to 2023 and found that the published scores were heavily rounded. Almost all the scores landed exactly in the middle of the existing grade bands. Because of this rounding, the researchers found that for five out of six alternative ways they tried to draw the lines, the result was identical to the current system. If they shifted the boundaries slightly, the companies did not move to a new grade because the scores were already sitting safely in the center of the old ones. This meant that simply looking at the published numbers could not tell them which design was better; the data was too coarse to distinguish between the options.

To get around this, the researchers used a method called simulation. They took the existing scores and added random, small amounts of noise to them, mimicking the kind of tiny errors or updates that happen in real life. They then watched to see how often a company's grade would change under different boundary designs. They tested a standard design where every grade band was the same width, and they compared it against six other designs where the middle bands were wider and the edge bands were narrower. The logic was that wider middle bands might act as a buffer, keeping companies in their current grade even if their score wobbled.

The results showed that the standard, equal-width design was the most reliable for the kinds of small changes that typically occur. When the researchers introduced small to moderate disturbances, the equal-width system produced the fewest grade changes. It also kept the observed scores furthest away from the boundaries, giving the ratings the most room to breathe before a reclassification was triggered. The researchers tested this using four different mathematical optimization tools, and all of them pointed to the same conclusion: the current equal-width approach was the best choice for stability.

However, the study also found that this preference is not absolute. When the researchers simulated very large, chaotic changes to the scores, the situation flipped. Under these extreme conditions, a design with a very wide central band actually performed better, keeping more companies in their original grades. This suggests that the best way to draw the lines depends entirely on what kind of changes the system is expected to face. If the goal is to protect against small, everyday noise, equal bands are superior. If the goal is to withstand massive shocks, a different shape might be needed.

Ultimately, the paper concludes that the current equal-width system is a robust choice for the data it examined, but it is not a universal rule for all time. The researchers also highlighted a broader lesson for the industry: because published scores are often rounded, it can be impossible to tell if a new rating system is truly better just by looking at the final numbers. The stability of a rating depends on the hidden mechanics of how the boundaries are set, a detail that is often invisible to the public but crucial for the reliability of the financial markets.

Technical Summary: Designing Stable ESG Grade Bands

Problem Statement
Environmental, Social, and Governance (ESG) ratings are frequently communicated to investors as discrete letter grades derived from continuous numerical scores via cut-points. While extensive literature exists on the divergence of scores between different rating providers, the methodological design of the cut-points themselves—specifically the placement of boundaries within letter grade bands—remains under-explored. The placement of these boundaries is a discrete design problem with direct implications for rating stability; small score revisions that cross a boundary trigger reclassification, while movements within a band leave the reported grade unchanged.

This study focuses on the legacy Refinitiv/LSEG ESG scale (0–100 score, mapped to 12 grades from D− to A+). The primary problem is determining whether alternative internal cut-point structures (specifically non-uniform band widths) offer superior stability compared to the incumbent equal-width structure, given the constraints of published data and the nature of score perturbations.

Methodology
The authors formulate the grade-band design as a constrained scalar-quantisation problem. The analysis retains the main A/B/C/D boundaries fixed at 25, 50, and 75, while varying the widths of the three sub-grades within each quartile.

Candidate Scales: The incumbent scale uses equal widths ( $e = c \approx 8.33$ ). Six alternative scales (S1–S6) were constructed by narrowing the edge sub-grades ( $e$ ) and widening the central sub-grade ( $c = 25 - 2e$ ).
Data: The study utilizes a panel of 177 large-capitalisation firms observed from 2018 to 2023 (1,048 firm-year observations). A critical characteristic of the data is high quantisation: 99.9% of scores correspond exactly to the midpoints of the incumbent grade bands, resulting in only 11 distinct score values.
Static Identification Test: The authors first test if alternative scales can be distinguished from the incumbent using static data. They prove that if no observed score lies between an incumbent boundary and a candidate boundary, the two scales produce identical grade assignments.
Stability Evaluation (Monte Carlo): Since static data cannot distinguish most alternatives, stability is evaluated via Monte Carlo simulations. Observed scores are perturbed with mean-zero Gaussian noise ( $\epsilon \sim N(0, \sigma^2)$ ) for $\sigma$ ranging from 0.5 to 5.0. The "migration rate" (probability of a grade change) is calculated for each scale.
Optimisation: The study employs four numerical optimisation procedures (grid search, golden-section search, simulated annealing, and genetic algorithm) to minimize a composite loss function $L(e)$ . This function weights average migration (at $\sigma = 1, 2, 3$ ) at 60% and midpoint distortion at 40%.
Boundary Safety: The minimum distance between any observed score and the nearest cut-point is calculated to measure "buffer" against reclassification.

Key Contributions

Methodological Distinction: The paper isolates the design of grade boundaries as a distinct methodological choice separate from the construction of the underlying ESG score or the divergence between providers.
Quantisation Invariance: It demonstrates that alternative grading rules may be statistically unidentifiable using published scores if those scores are already rounded to the incumbent grid. The authors provide a formal proposition (Proposition 1) and a test for when alternative scales cannot be distinguished on static data.
Quantisation Framework: The study bridges ESG grading with scalar quantisation theory (Lloyd–Max conditions), suggesting that for percentile-rank scores (which are approximately uniform within peer groups), equal-width bands are theoretically optimal for minimizing distortion.

Results

Static Indistinguishability: Five of the six alternative scales (S1–S5) produced identical grade assignments to the incumbent scale for all 1,048 observations. Only the most extreme alternative (S6, with a very wide center band) resulted in reclassifications. This confirms that static data alone cannot validate alternative designs due to the quantisation of published scores.
Migration Stability: Under small to moderate perturbations ( $\sigma \leq 4$ ), the incumbent equal-width scale exhibited the lowest or near-lowest grade migration rates. For instance, at $\sigma = 2$ , the incumbent migration rate was 3.90%, compared to 4.14% for S3 and significantly higher rates for scales with narrower edge bands (e.g., S6 at 32.96%).
Boundary Safety: The incumbent scale provided the largest minimum buffer (4.14 points) between observed scores and boundaries. Non-uniform scales with wider center bands moved cut-points closer to the observed mass points, reducing safety.
Optimisation Convergence: All four optimisation algorithms converged on an edge width ( $e$ ) very close to the incumbent value of $25/3$ (approx. 8.33).
Large Disturbance Exception: At the largest disturbance considered ( $\sigma = 5$ ), the most extreme wide-center scale (S6) exhibited lower migration than the incumbent. This indicates that the "optimal" design depends on the magnitude of expected score revisions.

Significance and Claims
The paper claims to provide a computational framework for designing discrete ESG grade boundaries. Its primary conclusion is that the preferred partition depends jointly on the score distribution and the magnitude of expected score revisions.

Local Robustness: For the observed quantised distribution and for small to moderate score revisions (up to ~4 points), the uniform (equal-width) scale is the most stable design. It maximizes the distance of scores from boundaries and minimizes migration.
Conditional Preference: The preference for uniform bands is conditional. If the design objective shifts to protecting against very large shocks (e.g., $\sigma = 5$ ), a non-uniform design with a wider central band may become more stable.
Limitations: The authors explicitly state that these findings are based on a historical sample of large-cap firms and quantised published scores. The study does not claim that the existing score is a complete measure of sustainability, nor does it examine the financial impact of grade changes. The results are presented as methodological evidence for the legacy system, noting that similar design questions could be applied to newer 0–5 scales if unrounded data were available.

In summary, the study argues that while the placement of cut-points is a critical design choice, the incumbent equal-width structure is robust for the specific data characteristics and disturbance magnitudes analyzed, and alternative designs cannot be validated using the currently available rounded ESG scores.

Technical Summary: Designing Stable ESG Grade Bands

More like this