MATHEMATICAL MODELING OF ESG RISKS IN INVESTMENT PORTFOLIOS

Introduction

In recent years, environmental, social, and governance factors have become an essential part of investment analysis. While the ESG agenda was initially viewed primarily as a tool for evaluating corporate social responsibility and long-term environmental impact, the focus has increasingly shifted toward assessing the direct influence of ESG factors on the financial characteristics of assets and investment portfolios.

It is fundamental to distinguish between an ESG factor and an ESG risk. An environmental, social, or governance metric in itself does not constitute a financial risk. An ESG risk arises when a given factor can induce changes in future cash flows, cost of capital, issuer creditworthiness, asset market values, liquidity, or other financial parameters. Consequently, a modern approach to ESG integration must be grounded in establishing causal linkages between non-financial factors and financial outcomes.

The principle of ESG integration implies the systematic inclusion of material ESG factors into investment analysis and decision-making. This approach aligns with responsible investment practices, wherein ESG factors are evaluated through their potential impact on risk and return.

Simultaneously, the growth of ESG investing is accompanied by heightened disclosure requirements. In June 2023, the International Sustainability Standards Board (ISSB) published IFRS S1 and IFRS S2, with IFRS S2 focusing specifically on climate-related risks and opportunities disclosures. The standard applies to annual reporting periods beginning on or after January 1, 2024.

However, the abundance of ESG data does not automatically solve the challenge of quantitative risk assessment. In practice, numerous methodological hurdles emerge:

• Different ESG rating providers employ non-uniform methodologies;
• Individual ESG indicators are measured at varying frequencies;
• A significant portion of the data is estimated or qualitative;
• Historical time series for certain ESG indicators remain short;
• Information may be incomplete or non-comparable across sectors and jurisdictions;
• The impact of ESG factors often manifests with a substantial time lag;
• Individual ESG factors can be interdependent;
• Shifts in ESG characteristics can simultaneously affect multiple traditional financial risk categories.

The Basel Committee notes that climate-related financial risks exhibit unique characteristics, requiring granular data and forward-looking evaluation methods. Key obstacles include data deficits, risk classification complexities, and uncertainty surrounding the nature of future climate impacts.

Thus, the problem of mathematically modeling ESG risks extends beyond constructing a formula for a single ESG index. It represents a comprehensive challenge of building a framework that links ESG factors to financial variables and assesses their effect on portfolio risk and return.

The goal of this study is to develop a theoretical and mathematical framework for assessing investment portfolio ESG risks and to formulate a model that integrates ESG factors into classical portfolio management systems.

To achieve this goal, the following objectives are addressed:

1. Define the economic essence of ESG risk;
2. Establish a taxonomy of investor-relevant ESG factors;
3. Formulate a mathematical model for aggregating ESG risks;
4. Establish transmission mechanisms between ESG factors and traditional financial risks;
5. Develop a methodology for incorporating ESG factors into portfolio optimization;
6. Investigate the capabilities of scenario analysis and stress testing;
7. Identify key sources of model risk and uncertainty;
8. Formulate practical implementation guidelines for the proposed model.

Theoretical Foundations of ESG Risk

The acronym ESG encompasses three core categories of factors:

• E — Environmental;
• S — Social;
• G — Governance.

Environmental factors include:

• Greenhouse gas (GHG) emissions;
• Carbon intensity;
• Energy consumption;
• Fossil fuel dependency;
• Water usage;
• Environmental pollution;
• Waste management;
• Impact on biodiversity;
• Climate vulnerability;
• Physical climate risks;
• Low-carbon transition risks.

Social factors include:

• Occupational health and safety;
• Workforce turnover;
• Labor rights compliance;
• Human capital quality;
• Community relations;
• Product safety;
• Personal data protection;
• Accessibility of goods and services;
• Reputational and social conflicts.

Governance factors include:

• Board structure and independence;
• Internal control quality;
• Shareholder rights;
• Transparency and disclosures;
• Anti-corruption mechanisms;
• Executive compensation structures;
• Enterprise risk management;
• Financial reporting quality.

From an investment analysis standpoint, all these factors can be modeled as random or quasi-random variables that influence an issuer's financial parameters.

Let the ESG status of asset i at time t be described by the vector:

X_i,t=(E_i,t,S_i,t,G_i,t)^T [cite: 1]

However, a three-component vector is overly simplified. A more realistic representation of an ESG profile uses a multidimensional vector:

X_i,t=(x_i1,t,x_i2,t,…,x_im,t)^T [cite: 1]

where m is the number of underlying ESG metrics, such as:

(CO_2,i,t,Water_i,t,Waste_i,t,Safety_i,t,Labor_i,t,Board_i,t,Corruption_i,t,Disclosure_i,t)^T [cite: 1]

This relaxes the assumption that a single aggregate ESG score fully captures a firm's sustainability.

ESG Risk as Financial Risk

The central methodological challenge lies in modeling how an ESG factor translates into financial risk.

Let the value of an asset be defined by the function:

P_i=P_i(F_i,X_i) [cite: 1]

where P_i is the market price, F_i represents traditional financial factors, and X_i represents ESG factors.

The change in asset value can be approximated using a Taylor series expansion:

 [cite: 1]

The term  captures the direct impact of ESG factor variations on the asset's price.

In practice, indirect effects also occur. For example, stricter environmental regulations can lead to:

• Increased capital expenditures (CAPEX);
• Higher operating expenses (OPEX);
• Margin compression and reduced profitability;
• Credit rating downgrades;
• Higher borrowing costs;
• Reduced investment appeal;
• Elevated stock price volatility.

A comprehensive model must capture the full transmission chain:

ESG→Financial Variables→Risk→Asset Price [cite: 1]

For instance:

ΔE→ΔCAPEX→ΔFCFF→ΔEV→ΔP [cite: 1]

Here, environmental deterioration causes capital expenditure spikes, altering free cash flows to the firm (FCFF), enterprise value (EV), and ultimately the market price P.

Financial Materiality of ESG Factors

Not all ESG metrics hold equal significance for an investor. Hence, the concept of financial materiality must be integrated.

Let M_j denote the financial materiality weight of ESG factor j . A materiality-adjusted ESG risk metric can be defined as:

 [cite: 1]

If an environmental indicator heavily influences cash flows, M_j will be high. Conversely, if a metric has negligible financial impact on a given issuer, its weight should be reduced.

Furthermore, materiality varies across industries and issuers (M_ij≠M_j). Water stress is a critical factor for agricultural companies, whereas it carries significantly less weight for software enterprises.

The Problem of External ESG Ratings

Relying directly on third-party ESG ratings as risk proxies introduces notable distortion. Rating agencies frequently report divergent scores for the same entity due to:

• Scope and definition variances;
• Differences in underlying indicator selection;
• Divergent weighting schemes;
• Industry normalization methodologies;
• Treatment of controversies and qualitative events;
• Rebalancing frequencies;
• Discrepancies between policy evaluations and operational outputs.

Mathematical models should not operate under the assumption that:

ESG Rating=True ESG Risk [cite: 1]

It is more accurate to define observed ratings as:

Observed ESG Rating=True ESG State+Measurement Error [cite: 1]

Y_i=X_i+ε_i [cite: 1]

where Y_i is the observable rating, X_i is the latent true ESG state, and ε_i is measurement noise. Modeling ESG risk thus transforms into a latent variable estimation problem.

ESG Risk and Discounted Cash Flow Valuation

For equity and corporate fixed income, ESG factors modulate both projected cash flows and discount rates.

Enterprise valuation:

 [cite: 1]

Assuming FCFF_t=FCFF_t(X_t) and r=r(X_t):

 [cite: 1]

ESG factors impact valuation simultaneously across dual channels:

1. ESG→FCFF (cash flow impact);
2. ESG→Cost of Capital (r) (risk premium impact).

For example, governance failures increase investor risk premiums (Δr>0), depressing asset valuations  even when expected cash flows remain unchanged.

Machine Learning Approaches to ESG Risk

Where extensive datasets exist, machine learning (ML) models can be applied.

Let Y_i=1(Loss_i>L*), where Y_i=1 if losses exceed a designated threshold L*. The conditional probability can be estimated as:

P(Y_i=1∣X_i) [cite: 1]

Feature vectors X_i include:

• ESG scores and raw metrics;
• Financial ratios;
• Sectoral and macroeconomic indicators;
• Market pricing and volatility metrics;
• News sentiment, textual disclosures, and climate data.

Applicable ML algorithms include logistic regression, random forests, gradient boosted decision trees, neural networks, survival analysis, clustering, and time-series architectures. However, ML implementations in financial risk management must balance predictive power with explainability: the mapping Y=f(X) must remain economically interpretable.

Comparative Evaluation of ESG Modeling Approaches

Five primary methodologies can be identified:

1. Rating-Based Approach: Relies directly on third-party ESG ratings. Pros: High ease of implementation. Cons: Lack of methodology transparency and rating divergence.
2. Factor-Based Approach: Incorporates ESG factors directly into return-generating models (R_i=f(F,ESG)). Pros: Economic interpretability. Cons: Requires long, consistent time series.
3. Scenario-Based Approach: Evaluates portfolio sensitivity across distinct ESG trajectories. Pros: Captures long-term structural shifts. Cons: High uncertainty in assigning scenario probabilities.
4. Optimization-Based Approach: Integrates ESG metrics into objective utility functions or constraint sets. Pros: Direct link to portfolio execution. Cons: Highly sensitive to parameter selection.
5. Integrated Approach: Combines Ratings, Factor Models, Scenarios, Stress Testing, and Optimization. This unified approach offers the most robust framework for institutional adoption.

Model Risk in ESG Frameworks

Total model risk in ESG systems can be decomposed into four components:

R_Model=R_Data+R_{Specification}+R_Parameter+R_Scenario [cite: 1]

• R_Data: Data quality, gaps, and estimation risk.
• R_{Specification}: Structural misspecification of functional forms.
• R_Parameter: Estimation error in calibrated parameters.
• R_Scenario: Errors in scenario assumptions and probabilities.

Consequently, total portfolio risk comprises both financial ESG risk and system model risk:

Total ESG Risk=Financial ESG Risk+Model Risk [cite: 1]

Practical Architecture of an ESG Risk Model

Based on the analysis, a ten-tier modeling architecture is structured as follows:

• Tier 1. Data Ingestion: Financial filings, ESG reports, climate data, market quotes, macro statistics, and third-party ratings.
• Tier 2. Data Cleansing: Imputation of missing values, outlier detection, normalization, and cross-comparability adjustments.
• Tier 3. Factor Construction: X→(E,S,G) component mapping.
• Tier 4. Materiality Assessment: Computation of issuer-specific materiality coefficients M_ij.
• Tier 5. Portfolio Exposure Assessment: ∑_iw_iExposure_i aggregation.
• Tier 6. Scenario Analysis: Stress testing across climate/ESG trajectories (S₁,…,S_K).
• Tier 7. Financial Integration: Mapping ESG factors onto cash flow, market, credit, and liquidity drivers.
• Tier 8. Quantitative Risk Estimation: Computation of VaR, CVaR, and Stress Loss metrics.
• Tier 9. Portfolio Optimization: Solving w*=argmaxU(w) subject to risk and ESG constraints.
• Tier 10. Dynamic Monitoring: Rebalancing triggered by changes in financial data, ESG scores, regulatory frameworks, or climate projections.

International Regulatory Landscape and Risk Transmission

International standards enhance data availability. IFRS S2 establishes reporting mandates across four pillars: Governance, Strategy, Risk Management, and Metrics & Targets. Nonetheless, regulatory bodies (e.g., the Basel Committee in 2025) continue to emphasize that data consistency and quality constraints remain significant hurdles for quantitative modeling.

Regulators increasingly view climate and ESG risks not as standalone risk types, but as drivers that amplify traditional financial risks—credit, market, operational, and liquidity. The European Central Bank (ECB) highlights physical and transition risks as key drivers of asset prices, financial stability, inflation, and growth. Recent research (e.g., BIS 2026) explores advanced portfolio techniques that treat emissions dynamics as "carbon yield" while explicitly embedding carbon trajectory uncertainty into portfolio optimization algorithms.

Conclusion

Mathematical modeling of ESG risks is a critical component of modern investment management. ESG factors exhibit a multidimensional nature that cannot be captured by a single aggregate score. A robust modeling framework must evaluate:

ESG Risk=f(Exposure,Materiality,Sensitivity,Correlation,Tail Risk,Uncertainty)

The proposed model integrates ESG factors into classical portfolio theory by adjusting the mean-variance utility framework:

Supplementing mean-variance optimization with VaR, CVaR, scenario analysis, and stress testing ensures that both average trends and tail risk events are captured. Given the non-linear trajectories and long time horizons inherent in climate and social risks, forward-looking scenario modeling and robust optimization techniques must replace pure reliance on historical data. Ultimately, treating ESG factors as endogenous drivers of financial distributions allows institutional managers to shift from qualitative compliance toward quantitatively rigorous, risk-adjusted portfolio construction.