4.14.1 Climate & ESG Risk from Zero

Basic Prerequisites: Probability & Measure Theory (an expectation and a weighted average are all you need).

1. Intuition & Practical Objective

This page builds the why of climate and ESG risk with no prior climate-finance knowledge needed. The objective is one distinction that everything else hangs on: climate risk is a cash-flow exposure you can price; ESG is a preference plus a data product you must audit. A quant needs the first to size the position and the second only to know how reliable its inputs are.

Start with the sentence every analyst has heard: “Our portfolio is net-zero aligned.” Unpack it literally. Aligned to what pathway, measured with which emissions boundary, and priced at what carbon cost? Change any one of the three and the sentence can flip from true to false without a single holding changing. That is the whole subject in one line: climate risk is an input problem before it is a mathematics problem.

Four steps, four “aha”s:

Two risks, one cash flow. Physical risk is damage to assets and supply chains (acute: storm, flood, wildfire; chronic: heat, drought, sea level). Transition risk is the cost of changing the economy (carbon prices, technology substitution, consumer and investor preferences). Both land in the same place - earnings - which is why a single valuation channel can hold both.
Carbon is a cost per unit of output, not a return per unit of price. Raise a carbon price by $50/t and a firm emitting $4$ kg of CO2 per $1 of revenue takes a $0.20 hit per $1 of revenue. That is a margin event, and margins are what shareholders own. It is why the canonical exposure metric - weighted average carbon intensity - is emissions divided by revenue.
The horizon kills the standard machinery. A one-day $99%$ VaR of an equity book asks about tomorrow; climate damage and policy arrive over decades. The annualised equivalent of a $30$ -year $30%$ loss is about $- 0.0047%$ per day - roughly $1000 \times$ smaller than the daily VaR it must compete with (§3, panel A). No historical-simulation engine will ever flag it. The fix is not a better estimator; it is a different instrument: scenarios.
The data boundary defines the answer. “Scope 1+2” (direct + purchased energy) and “Scope 1+2+3” (adding the value chain) are both legitimate, both standards-compliant, and differ by $13.5 \times$ for the same firm and the same year (§3, panel B). ESG ratings are constructed from the same kind of contested inputs, and disagree accordingly. So the first question about any climate number is not “is the model right?” but “what was measured, over what boundary, in what vintage?“

2. Mathematical Ground Truth & Derivations

The horizon-mismatch identity. Let a climate-driven loss of fraction $L$ of value accrue over $T$ years of $252$ trading days. Its equivalent constant daily drift is

d = (1 - L)^{1/ (252 T)} - 1

which is the number a daily risk report would have to show if climate risk were a diffusion. Because $d$ scales like $- L / (252 T)$ for small $L$ , it decays as $1/ T$ : the longer the true horizon, the smaller the daily signature - the opposite of comfort. Comparing it to a normal $99%$ VaR, $VaR_{α} = σ z_{α}$ with $z_{0.99} = 2.326348$ , gives the “invisibility ratio” of §3.

Carbon cost as a margin identity. For a firm with emissions intensity $I_{i} = E_{i} / R_{i}$ (tCO2e per $1 of revenue), a carbon price step $Δ p$ ($/tCO2e) and pass-through $λ \in [0, 1]$ :

\frac{Δ ( cost )}{R _{i}} = (1 - λ) I_{i} Δ p (a fraction of revenue), \frac{Δ V _{i}}{V _{i}} \approx - (1 - λ) M_{i} I_{i} \frac{Δ p}{1 0 ^{6}}

with $M_{i}$ a valuation multiple (firm value / profit). Summing over holdings gives the portfolio P&L identity in the hub: $Δ V / V = - \sum_{i} w_{i} (1 - λ_{i}) M_{i} (E_{i} / R_{i}) Δ p /1 0^{6}$ , where the $1 0^{6}$ converts “per $1m of revenue” into a fraction.

The three scopes (GHG Protocol). For a firm, total emissions are conventionally partitioned as

E^{total} = E_{S1} + E_{S2} + E_{S3},

where Scope 1 is direct combustion, Scope 2 is purchased electricity/heat/steam, and Scope 3 is everything upstream and downstream in the value chain. The TCFD recommends reporting Scope 1+2 always and Scope 3 when material; PCAF standardises attributing financed emissions to a portfolio share. The scope choice is a definition, and definitions do not have error bars - which is precisely why the number moves so much.

Double materiality. Two evaluations of the same firm, both legitimate and useful:

financial materiality: \frac{\partial ( firm cash flows )}{\partial ( climate variable )}, impact materiality: \frac{\partial ( climate variable )}{\partial ( firm activity )} .

A quant uses the first as a risk input (it is a derivative of cash flows). The second is a mandate/constraint (“align the portfolio with 1.5 °C”), implemented as an optimisation constraint - never as a risk number. Conflating the two produces the familiar absurdity of a “high-ESG” portfolio that is also high-carbon: the score measured disclosure quality and controversy management, not emissions.

3. Computational Implementation - the horizon mismatch and the scope boundary

Two panels. (A) shows why a daily risk system cannot see climate risk; (B) shows how far the definition of the exposure moves the reported number; (C) translates a carbon price into a margin event. Standard library only.

Panel (A): the daily VaR is $986 \times$ larger than the climate drift, so the climate signal is below the noise floor by three orders of magnitude. Panel (B): one definitional choice moves the intensity $13.5 \times$ . Panel (C): the same carbon price is either a $62.5%$ EBITDA event or an $18.8%$ one, depending only on pass-through. Together they explain why the rest of this folder is about scenarios (03) and data quality (05) rather than about cleverer estimators.

4. Failure Modes & First-Principles Breakdowns

“Just add it as a risk factor.” A carbon-price factor estimated from a short, non-stationary price history has no tail information; it will understate exactly the events that matter (a disorderly repricing). Climate must enter as a scenario or a jump, not a fitted beta (03 · §3).
Reading a rating as a measurement. An ESG score is a weighted aggregation of contested indicators; the same firm legitimately ranks $2$ nd and $8$ th out of eight across three raters (04 · §3). Never treat it as an observed variable with small noise.
Confusing financial and impact materiality. A portfolio optimised on impact (an emissions-reduction constraint) is not thereby lower-risk; a portfolio optimised on financial materiality is not thereby greener. State which one the mandate means.
Scope shopping. Moving the boundary between Scope 1+2 and 1+2+3 changes the headline by an order of magnitude; a provider that reports only the favourable boundary is not lying, it is choosing. Pin the boundary in the reporting standard, not in the marketing.
Ignoring pass-through. $λ$ is the single most influential and least scrutinised assumption in a carbon-price stress test - a factor of five in §3. It is an economic estimate and should be challenged like one.

5. References

TCFD, Recommendations of the Task Force on Climate-related Financial Disclosures (FSB, 2017)
Carney, M., *Breaking the Tragedy of the Horizon
Greenhouse Gas Protocol, Corporate Standard (2004) and Scope 3 Standard (2011)
PCAF, The Global GHG Accounting and Reporting Standard for the Financial Industry
High-Level Commission on Carbon Prices (Stern & Stiglitz), Report (World Bank, 2017)
Bolton, P. & Kacperczyk, M., Do investors care about carbon risk?, Journal of Financial Economics 142(2) (2021)

6. Connected Graph Bridges

Base: Probability & Measure Theory · Statistics & Inference
Continue: 02 · Physical & Transition Risk · Index Hub
Sibling: Stress Testing & Scenario Analysis (the instrument climate risk actually needs) · Value at Risk & Expected Shortfall (the measure that will not see it)

Kwant Atlas

Explorer