📈 economics

The choice of probability mapping in ESG questionnaire scoring: a structural risk-metric perspective for venture-capital and private-equity due diligence, with a dynamical-systems foundation

This paper argues that the choice of probability mapping in ESG scoring is a critical structural risk decision rather than a technical afterthought, demonstrating how different mappings can drastically alter rating outcomes and proposing a falsifiable, dynamics-based criterion to ensure defensible and comparable ESG assessments for venture capital and private equity due diligence.

Original authors: Francesco Eusepi

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francesco Eusepi

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

In the world of high-stakes finance, investors often face a difficult puzzle: how to measure the character of a company that has no history. When venture capitalists and private equity firms look to invest in new startups, they cannot rely on past credit scores or decades of financial reports. Instead, they must rely on a structured questionnaire, a series of yes-or-no questions about how the company treats the environment, its workers, and its own governance. The answers are tallied into a single number, a score meant to represent the company's alignment with ethical standards. But a raw number is hard to interpret on its own. To make it useful, analysts must translate that score into a probability—a percentage chance that the company is a good investment or, conversely, a risk of failure. This translation step is the invisible engine behind every rating, yet until now, the rules governing how that translation happens have been treated as a minor technical detail, chosen more for convenience than for deep reasoning.

A new study by researcher Francesco Eusepi challenges this casual approach, arguing that the method used to convert a questionnaire score into a probability is not just a mathematical preference, but a decision that fundamentally alters the outcome of an investment. The paper demonstrates that two different, yet equally valid, mathematical methods can take the exact same set of answers and produce completely different ratings for the same company. In one scenario, a company might be flagged as a high-risk investment requiring immediate fixes; in the other, using a different but defensible method, that same company could be rated as a top-tier, low-risk opportunity. The research suggests that without a strict, agreed-upon rule for this translation, the entire system of ethical investing becomes inconsistent, making it impossible to compare companies fairly or to trust the ratings provided by different firms.

The core of the problem lies in the shape of the curve used to perform the translation. Imagine a line that starts at zero and rises to one, representing the shift from total risk to total safety. The study compares two specific shapes for this line: one that rises steadily and smoothly, and another that shoots up very quickly at the beginning and then flattens out. When the researchers applied these two shapes to the same set of hypothetical companies, the results were startling. Using the smooth, steady curve, a company with a moderate score of fifty out of a possible maximum was rated as a compromised investment, needing significant work before it could be considered safe. However, when the same score was run through the curve that rises sharply at the start, that exact same company was rated as a top-tier, safe investment. The difference was not a small adjustment; it was a shift of seven steps on a ten-step rating scale. This means that two analysts, looking at the same data and using two methods that both pass the basic tests of logic, could reach opposite conclusions about whether to invest.

The paper draws a parallel to how banks assess the risk of a company going bankrupt. In that field, experts use a concept called "distance to default," which measures how far a company is from financial failure. They then use a specific mathematical tool to turn that distance into a probability of failure. The study argues that the ESG questionnaire works in the same way: the score is the distance, and the probability map is the tool. Just as a bank would not arbitrarily switch the tool it uses to measure bankruptcy risk without recalibrating its entire system, ESG raters should not switch their probability maps without understanding the consequences. The research shows that the choice of tool changes where the "danger zones" and "safe zones" fall on the score axis. One method spreads the ratings out evenly across the full range of possible scores, allowing for fine distinctions between companies. The other method, while mathematically valid, compresses almost all the ratings into a tiny sliver of the score range, making it nearly impossible to tell the difference between a good company and a great one until the score is nearly perfect.

Beyond the immediate confusion it causes for investors, the study highlights a deeper issue of trust and governance. If the method used to calculate these ratings is not explicitly stated and justified, the resulting numbers are not truly comparable. A rating from one firm might mean something entirely different than a rating from another, simply because they used different curves to do the math. The paper proposes a set of strict rules, or axioms, that any acceptable method must follow to ensure it is fair and consistent. It suggests that the most reliable method is the one that behaves like a steady, predictable force, where every small improvement in a company's score leads to a proportional improvement in its rating. This approach, known as the logistic sigmoid, ensures that the rating system remains sensitive to changes across the entire spectrum of performance, rather than ignoring most of the data.

The findings serve as a warning to the financial industry that the tools used to measure ethical responsibility must be chosen with the same rigor as the financial models used to measure profit. The study does not claim to have discovered a new law of nature, but rather to have exposed a hidden variable in a system that many assumed was settled. By showing that the choice of probability map can move a company from a failing grade to a passing one, the research forces a re-evaluation of how due diligence is conducted. It argues that for the system to be defensible, auditable, and truly useful, the method of translation must be disclosed and standardized. Without this clarity, the ratings that guide billions of dollars in investment decisions remain vulnerable to the arbitrary choices of the modeler, potentially misdirecting capital away from the companies that truly deserve it.

Technical Summary: The Choice of Probability Mapping in ESG Questionnaire Scoring

Problem Statement
In Venture Capital (VC) and Private Equity (PE) due diligence, Environmental, Social, and Governance (ESG) assessments for data-poor targets (e.g., start-ups) often rely on additive scoring models derived from Yes/No questionnaires. These models generate a raw, unbounded score ( $m$ ) which must be normalized into a dimensionless probability ( $p$ ) of ESG alignment, with the complement ( $\psi = 1-p$ ) representing ESG risk.

The paper identifies a critical, previously under-examined methodological gap: the choice of the probability mapping function (the transformation from score $m$ to probability $p$ ) is often treated as a technical afterthought. However, this choice is not neutral. Different admissible mapping functions, even when calibrated to identical boundary conditions (e.g., the probability at a score of zero and the probability at the maximum possible score), can produce radically different rating boundaries. This leads to a "comparability problem" where two analysts applying different but equally defensible maps to the same questionnaire data can assign targets to rating classes that differ by several notches, fundamentally altering investment decisions, covenant structures, and capital allocation.

Methodology
The paper employs a three-pronged methodological approach:

Axiomatic Formulation: The author defines a set of natural axioms for admissible probability maps:
- Monotonicity: Higher scores must yield higher alignment probabilities.
- Boundedness: The output must lie strictly within $(0, 1)$ .
- Normalizability: The map must be expressible as the integral of a non-negative, integrable density function.
  This leads to Theorem 1, which states that an admissible map $\pi(s)$ must satisfy the autonomous first-order differential equation $\pi' = f(\pi)$ , where $f$ is an integrable function.
Structural Risk-Metric Analogy: The core innovation reframes the normalization problem through the lens of structural credit-risk models (specifically Distance-to-Default models like Merton/KMV).
- The additive ESG score ( $m$ ) is analogized to the "distance to default."
- The probability map ( $\pi$ ) is analogized to the cumulative distribution function (e.g., the normal CDF) used to convert distance into a probability of default.
- This analogy allows for an economic interpretation of calibration parameters: the intercept parameter ( $\beta$ ) represents the base rate (analogous to a firm at the default boundary), and the slope parameter ( $\alpha$ or $a$ ) represents the "volatility" or sensitivity of the rating scale.
Dynamical-Systems Foundation: To provide a rigorous mathematical underpinning for the admissibility criterion, the paper draws an analogy to one-dimensional conservative dynamical systems.
- The score $s$ is treated as "time," the probability $\pi$ as "position," and the density function as "momentum."
- The condition for admissibility ( $\pi' = f(\pi)$ ) is shown to be equivalent to the condition for a conservative force field (where momentum depends only on position, not explicitly on time), ensuring the existence of a conserved energy potential. This derivation is detailed in the appendices to maintain accessibility for a business audience.

Key Contributions

Formalization of the Normalization Problem: The paper moves the choice of probability map from a heuristic decision to an axiomatic choice, providing a general existence criterion (Theorem 1) for admissible maps.
Quantification of Model Risk: Through a calibrated numerical example, the paper demonstrates that two economically reasonable maps—the logistic sigmoid and the arctangent—can assign the same questionnaire outcome to rating classes separated by multiple notches.
- Example: A company with a score of $m=50$ is rated C (requiring remediation) under the sigmoid map but A (no remediation needed) under the arctangent map, despite both maps being calibrated to the same boundary points ( $p(0)=0.5$ and $p(580)=0.998$ ).
Economic Interpretation of Parameters: The paper provides a structural reading of calibration parameters, interpreting the steepness of the map as a measure of "risk aversion" or "volatility" in the rating design. A "nervous" scale (high sensitivity) vs. a "forgiving" scale (low sensitivity) is shown to be a direct consequence of the chosen map's functional form.
Governance Implications: The analysis highlights that without an explicit, disclosed, and economically motivated criterion for selecting the map, ESG ratings lack comparability across different questionnaires and methodologies, undermining their utility for regulatory frameworks (e.g., Basel III/IV) and investment committees.

Results

Comparative Statics: Under identical boundary conditions, the arctangent map compresses the vast majority of the rating scale (from AA down to D) into the bottom 15% of the score axis. In contrast, the sigmoid map distributes ratings more evenly across the full score range.
Marginal Sensitivity: The arctangent map exhibits a sharp sensitivity peak near zero scores (where a few positive answers drastically change the rating) but becomes nearly insensitive to further improvements. The sigmoid map maintains a more gradual, informative marginal sensitivity across a wider range of scores.
Numerical Illustration: The paper provides a reproducible example showing that for low-to-medium scores, the choice of map is the dominant factor in determining the final rating, often overriding the specific questionnaire answers.

Significance and Claims
The paper claims that the choice of probability mapping is not a mere technical detail but a structural design choice with material economic consequences.

Defensibility: A poorly justified map can shift a target across a full rating notch, affecting investment decisions and covenants.
Comparability: Heterogeneous ESG metrics can only be compared if they are expressed on a common probability scale derived from a disclosed, defensible mapping criterion.
Model Risk: The divergence in ratings caused by map selection is a form of model risk that is currently unaddressed in standard ESG due diligence practices.

The author argues that the logistic sigmoid is a robust choice for VC/PE due diligence because it maintains non-negligible marginal sensitivity over a wide range of scores, whereas alternatives like the arctangent may over-reward initial improvements and fail to discriminate effectively across the full spectrum of ESG profiles. The paper concludes by calling for explicit disclosure of the probability map and its calibration parameters to ensure the auditability and comparability of ESG ratings.

Technical Summary: The Choice of Probability Mapping in ESG Questionnaire Scoring

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