4.14 Climate & ESG Risk

Basic Prerequisites: Probability & Measure Theory (expectations, quantiles; the VaR/ES machinery is reused verbatim) and 01 · From Zero (this folder). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Climate and ESG risk is not a new kind of risk factor - it is the ordinary risk problem with three properties that break the ordinary machinery: the exposure is a cash-flow externality over decades, the data are non-stationary, sparse and contested, and the loss distribution cannot be estimated from history because the future policy path is a choice, not a sampled outcome. Everything quantitative here is therefore either (a) an exposure metric (how many tonnes, at what price), or (b) a scenario (if policy goes this way, the portfolio loses this much).

This folder separates the two families that the practitioner literature routinely conflates:

Climate risk - physical risk (acute: flood, storm, wildfire; chronic: heat, sea-level, drought) and transition risk (policy/carbon price, technology substitution, preference shifts), which are financial exposures.
ESG risk / double materiality - ratings and metrics that mix financial materiality (how ESG factors move the firm’s cash flows) with impact materiality (how the firm moves the world). The first is a risk model input; the second is a mandate constraint.

This folder is a hub: it (a) gives the fast metric lookup and the carbon-exposure engine below (job #1 of this pillar), and (b) routes to six sub-pages that walk from raw intuition through physical/transition risk and carbon pricing, the scenario stress-testing frameworks (NGFS, CBES, PACTA), ESG scores / temperature alignment / the carbon premium, the failure modes, and the advanced allocation extensions.

The one-sentence essence. “Climate risk is a valuation problem measured in tonnes times price over decades, not a return problem measurable from a price history - so its portfolio number is produced by scenarios, and its dominant error is the data behind the emissions, not the mathematics.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Notation: $w_{i}$ portfolio weight, $R_{i}$ firm revenue, $E_{i}$ firm emissions (tCO2e), $M_{i}$ a valuation multiple (firm value / profit), $Δ p$ a carbon-price step ($/tCO2e), λ the carbon-cost pass-through rate, $EF$ the emission factor per unit of fuel, $p^{*}$ a break-even carbon price.

Quantity	Formula	Verified check (§3)
WACI (TCFD weighted average carbon intensity)	$WACI = i \sum w_{i} \frac{E _{i}}{R _{i}}$	$815.0$ tCO2e per $1m revenue (§3)
Carbon-price P&L (first-order flow-through)	$\frac{Δ V}{V} = - i \sum w_{i} (1 - λ_{i}) M_{i} \frac{E _{i}}{R _{i}} \frac{Δ p}{1 0 ^{6}}$	$+$ $30/t, $λ = 0.30 \Rightarrow - 10.73%$ (§3)
Pass-through sensitivity	multiplier $(1 - λ)$	$λ = 0 \Rightarrow - 15.33%$ , $λ = 0.8 \Rightarrow - 3.07%$ (§3)
Weighted-average multiple	$\overset{ˉ}{M} = \sum_{i} w_{i} M_{i}$	$10.85 \times$ (§3)
Implied temperature rise (benchmark-pathway interpolation)	$ITR = T_{lo} + (T_{hi} - T_{lo}) \frac{C - C _{lo}}{C _{hi} - C _{lo}}$	$C = 19, 020 \Rightarrow 1.9 8^{\circ}$ C (04 §3)
Stranded fraction	$i : m_{i} < EF p \sum v_{i} / i \sum v_{i}$	$p = 100$ USD/t $\Rightarrow 64.4%$ (02 §3)
Scenario ES	$ES_{α} (L) = \frac{1}{1 - α} \int_{α}^{1} VaR_{u} (L) d u$ over the scenario measure	6-scenario $ES_{90%} = 28.02%$ (03 §3)
Transition-jump overlay	Mixture of $N (0, σ^{2})$ and a jump law	$ES_{99%} : 3.066% \to 4.607%$ (03 §3)
Climate beta / carbon premium	Slope $b$ in $r_{i} = a + b ln E_{i} + \dots$	planted $+ 0.018$ recovered as $+ 0.0180$ (04 §3)
Carbon-adjusted PD (Merton)	$PD = Φ (- \frac{ln ( A / D ) + ( μ - σ ^{2} /2 ) T}{σ T})$ , $D \mapsto D + C$	$C = 0 \to 12.19%$ ; $C = 15 \to 37.96%$ (06 §3)

Two axioms of the whole subject. First, carbon is a cost that scales with output, not a return that scales with price - so its portfolio signature is a level shift in margins (the boxed P&L identity above), which is why a carbon price is a slow, compounding hit that no daily VaR sees. Second, the loss distribution is chosen, not measured: NGFS, the Bank of England’s CBES and PACTA all convert narratives into numbers, and every downstream risk number inherits that modelling choice (03 · Climate Scenarios & Stress Testing).

Scope accounting defines the number. The Greenhouse Gas Protocol splits emissions into Scope 1 (direct combustion), Scope 2 (purchased energy) and Scope 3 (value-chain, upstream and downstream). The TCFD recommends Scope 1+2 and (separately) Scope 3; the Partnership for Carbon Accounting Financials (PCAF) standardises how to attribute them to a portfolio. The scope boundary is not a detail - for the same firm it changes the headline intensity by more than an order of magnitude (01 · §3: $16.0 \to 216.0$ tCO2e per $1m).

Carbon pricing, two prices with one purpose. An explicit price is charged (emissions trading systems, carbon taxes); a shadow (internal) carbon price is charged only inside project appraisal, to make long-lived investment decisions robust to a future explicit price. The canonical external corridor is the High-Level Commission on Carbon Prices (Stern & Stiglitz, 2017): $40–$80/tCO2e by 2020 rising to $50–$100/tCO2e by 2030, consistent with the Paris temperature objective. A carbon price becomes stranded-asset risk when the cost per unit of output exceeds the asset’s break-even margin, $p^{*} = m / EF$ .

3. Computational Implementation - the carbon-exposure engine

Standard library only. One portfolio, three outputs: the TCFD intensity metric, the first-order carbon-price P&L, and its pass-through sensitivity. This is the whole “climate risk in the portfolio” calculation in miniature.

Read the last line twice: the same portfolio, the same carbon price, and the answer moves by a factor of five purely on the pass-through assumption. The emissions are a measurement; $λ$ is an economic model. That gap is where climate-risk model risk lives, and it is why the failure modes below are about data and assumptions rather than about formulae.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s full failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Horizon mismatch (Carney’s “tragedy of the horizon”). A $30$ -year $30%$ equity loss is an equivalent daily drift of $- 0.00472%$ - about $986 \times$ smaller than a $1$ -day $99%$ VaR of a $2%$ -vol book. Climate risk is structurally invisible to a return-based VaR, and must be injected as a scenario or a jump (§3 of 03 · Climate Scenarios).
Data-quality failure, not estimation error. The scope boundary, the rating agency and the emissions data vintage move the reported climate number more than any statistical refinement: Scope 1+2 vs 1+2+3 is a $13.5 \times$ swing, and ESG ratings correlate at only $\approx 0.54$ on average (Berg et al. 2022). Greenwashing is the name for this, not a separate phenomenon (05 · §2).
No stationary distribution to estimate. The carbon price path is a policy choice; there is no long, stationary sample of “transition shocks”. Scenario-based measures are therefore the honest instrument - and they cap the resolvable confidence level at the number of scenarios (03 · §4).

5. References

TCFD - Recommendations of the Task Force on Climate-related Financial Disclosures (FSB, June 2017). The physical/transition taxonomy and the recommended metrics (weighted average carbon intensity, absolute emissions, carbon footprint). The primary framework source for this folder.
Greenhouse Gas Protocol - Corporate Accounting and Reporting Standard (2004, rev. 2015) and Corporate Value Chain (Scope 3) Accounting and Reporting Standard (2011)
PCAF - The Global GHG Accounting and Reporting Standard for the Financial Industry
NGFS - Climate Scenarios for Central Banks and Supervisors (Network for Greening the Financial System, 2019 onwards; phases I–V). Orderly, disorderly and hot-house-world scenario families. [REG]
Bank of England - Key Elements of the 2021 Biennial Exploratory Scenario: Financial Risks from Climate Change (2021) and Results of the 2021 Climate Biennial Exploratory Scenario (May 2022). Three scenarios: early action, late action, no additional action. [REG]
BCBS - *Climate-related Financial Risks
PACTA - Paris Agreement Capital Transition Assessment methodology (2° Investing Initiative; stewardship transferred to RMI, 2022). Forward-looking alignment of portfolios with climate scenarios using company production plans.
High-Level Commission on Carbon Prices (Stern & Stiglitz) - Report of the High-Level Commission on Carbon Prices (World Bank, 2017). The $40–$80 (2020) / $50–$100 (2030) corridor.
IPCC - Climate Change 2013: The Physical Science Basis (AR5, WG1
Bolton, P. & Kacperczyk, M. - Do investors care about carbon risk?, Journal of Financial Economics 142(2):517–549 (2021)
Pástor, Ľ., Stambaugh, R.F. & Taylor, L.A. - Sustainable investing in equilibrium, Journal of Financial Economics 142(2):550–571 (2021)
Berg, F., Kölbel, J.F. & Rigobon, R. - Aggregate Confusion: The Divergence of ESG Ratings, Review of Finance 26(6):1315–1344 (2022)
Andersson, M., Bolton, P. & Samama, F. - Hedging Climate Risk, Financial Analysts Journal 72(3):13–32 (2016)

6. Connected Graph Bridges

Foundational base: Probability & Measure Theory · Statistics & Inference · 01 · From Zero
Sub-pages (in-folder): 01 From Zero · 02 Physical & Transition Risk · 03 Climate Scenarios & Stress Testing · 04 ESG Scores & the Carbon Premium · 05 Failure Modes · 06 Advanced Extensions
Sibling topics: Stress Testing & Scenario Analysis (the scenario mathematics this folder specialises) · Value at Risk & Expected Shortfall (the tail machinery reused in 03 and 06) · Model Risk & Validation (scenario and data model risk) · Basel & Regulation (the supervisory layer) · Credit Risk & the Merton Model (the carbon-adjusted PD of 06)
Forward: Robust Optimization (climate is the canonical deep-uncertainty constraint) · Multi-Asset & Factor Allocation · Quantitative Fundamental Investing (ESG as a cross-sectional signal)

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

4.14 Climate & ESG Risk

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the carbon-exposure engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

4.14.1 Climate & ESG Risk from Zero

4.14.2 Physical vs Transition Risk

4.14.3 Climate Scenarios & Stress Testing

4.14.4 ESG Scores, Temperature Alignment & the Carbon Premium

4.14.5 Failure Modes & Real-World Practice

4.14.6 Advanced Extensions