4.14.2 Physical vs Transition Risk

Basic Prerequisites: 01 · From Zero and Probability & Measure Theory.

---

1. Intuition & Practical Objective

The TCFD’s central contribution was taxonomic: it split climate-related financial risk into physical risk and transition risk, and insisted both are financial risks that belong in the ordinary risk process. This page makes that taxonomy operational: how to turn a warming path into a physical drag, how to turn a carbon price into a portfolio P&L, and how to decide when an asset is stranded.

The two families behave very differently, and the difference drives every modelling choice:

• Physical risk is a damage process. Acute events (floods, storms, wildfires) are tail-like, local and insurable-ish; chronic shifts (mean temperature, precipitation, sea level) are slow, cumulative and effectively uninsurable. Physical risk is measured from climate (not financial) data and enters the P&L as a damage function or a scenario haircut.
• Transition risk is a policy and technology process. It is a repricing of an asset’s cash flows when a carbon price, a mandate, or a substitute technology arrives - fast (disorderly) or slow (orderly). Transition risk is measured from emissions and policy data and enters the P&L as a cost per tonne (or an abatement-cost curve).

The bridge between them is the carbon price. An explicit price is charged (emissions trading systems, carbon taxes). A shadow (internal) carbon price is not charged - it is inserted into project appraisal so that long-lived assets are sized for a future price that may become explicit. Corporate and public practice both use shadow prices precisely because the explicit price is a political variable, not a market observable with a long history.

Stranding is the terminal case of transition risk: a capital asset whose operating cost exceeds its revenue at the prevailing (or expected) carbon price, before the end of its useful life. It is not a peak-oil story or a moral judgement - it is the arithmetic $p^{\ast }=m/\mathrm{EF}$ of §2 applied to a whole reserve or plant base.

---

2. Mathematical Ground Truth & Derivations

2.1 The TCFD taxonomy as a decomposition

A firm’s value is exposed to climate through two channels only (TCFD, 2017, Figure 1):

$$\underset{\text{damage}\to \text{assets, costs, revenue}}{\underbrace{\Delta V/V_{\text{physical}}}}+\underset{\text{policy, tech, preference}\to \text{costs, demand, valuation}}{\underbrace{\Delta V/V_{\text{transition}}}}.$$

Both are ultimately cash-flow shocks; both are cumulative and non-stationary, which is why neither can be mimicked by adding a stationary risk factor to a return model (01 · §4).

2.2 Carbon pricing: explicit price, shadow price, pass-through

Let a firm emit $E_i$ tCO2e on revenue $R_i$, so its intensity is $I_i=E_i/R_i$. A carbon price step $\Delta p$ ($/tCO2e) creates a gross cost

$$C_i=\Delta p\cdot E_i=\Delta p{I}_i{R}_i.$$

With pass-through ${\lambda }_i\in [0,1]$ (the fraction the firm can put into its own selling prices), the retained cost is $(1-{\lambda }_i)C_i$, i.e. a margin hit of $(1-{\lambda }_i)I_i\Delta p$ on every unit of revenue. The shadow carbon price is the same formula with $\lambda =0$ and $\Delta p$ set to a policy corridor rather than a market price - the canonical external corridor being the High-Level Commission on Carbon Prices (Stern & Stiglitz, 2017): $40–$80/tCO2e by 2020, $50–$100/tCO2e by 2030, consistent with the Paris Agreement objective.

Pass-through is the crux and it is heterogeneous: a regulated utility with a cost-of-service tariff passes most of a carbon price through; a commodity chemicals producer facing import competition passes almost none. A single portfolio-level $\lambda$ is therefore a strong - and strongly consequential - assumption (hub §3: a $30/t shock is $-15.33\%$ at $\lambda =0$ and $-3.07\%$ at $\lambda =0.8$).

2.3 Stranded assets: the break-even carbon price

For asset $i$ with a break-even net margin $m_i$ ($/unit of fuel) and an emission factor \mathrm{EF} (tCO2e per unit of fuel), the carbon cost per unit is $\mathrm{EF}\cdot p$. The asset is uneconomic when

$$\boxed{p^{\ast }=\frac{m_i}{\mathrm{EF}}\text{equivalently}\mathrm{EF}\cdot p>m_i}$$

so for a reserve base $\{(v_i,m_i)\}$ the stranded fraction at carbon price $p$ is

$$\mathrm{SF}(p)=\frac{{\sum }_{i:m_i<\mathrm{EF}p}{v}_i}{{\sum }_i{v}_i}.$$

$\mathrm{SF}$ is a cumulative distribution function of break-even margins, rescaled - the classic “cost curve” used by PACTA-style alignment tools and by Carbon Tracker’s unburnable-carbon framing. It is monotone, bounded in $[0,1]$, and its derivative is the reserve density at the margin. Crucially it is not a probability: it is a scenario-conditional fraction.

2.4 Physical risk: from cumulative emissions to damage

The physical channel has a defensible three-link chain (IPCC AR5, WG1 Ch. 12):

1. Emissions → warming. The transient climate response to cumulative carbon emissions (TCRE) is approximately linear in cumulative CO2: $\Delta T\approx \mathrm{TCRE}\times ({\text{cumulative GtCO}}_2/1000)$, with the AR5 likely range $0.8$–$2.5^{\circ }$C per 1000 PgC, i.e. $\approx 0.2$–$0.7^{\circ }$C per 1000 GtCO2.
2. Warming → damage. Integrated-assessment damage functions are conventionally convex in $\Delta T$: $D(\Delta T)=\theta \Delta T^2$ (the DICE-family functional form). The convexity is the substantive claim: each additional degree costs more than the last.
3. Damage → value. $V\mapsto V(1-D(\Delta T))$, applied to the exposed asset base rather than the whole portfolio.

Honest labelling. $\mathrm{TCRE}$ is a published, assessed quantity. $\theta$ is not a constant of nature and not a supervised calibration; it is a model parameter, and the code below treats it as an explicitly illustrative value so the arithmetic is verifiable without pretending to a calibration it does not have.

---

3. Computational Implementation - the stranded-asset cost curve and the physical drag

Panel (A) integrates a reserve cost curve into a stranded fraction at four carbon prices and compares the book’s committed emissions with the remaining carbon budget. Panel (B) applies the convex damage function. Standard library only.

Two readings. First, the stranded fraction is violently convex in the carbon price: $0\%\to 14.4\%\to 64.4\%\to 92.2\%$ across $0\to$150/t. A portfolio’s “stranding risk” is therefore not a number but a curve, and reporting a single point on it is a hidden assumption about the policy path. Second, the physical drag at $+3^{\circ }$C ($5.40\%$) is the same order as the transition hit - the two families are comparable in magnitude, and in a hot-house scenario you pay the physical one instead of the transition one, not instead of both.

Cross-check against the earlier identity: at $p=$ $100/t the threshold is \mathrm{EF}\cdot p=0.43\times100= $43/bbl, so $43 separates economic from stranded reserves - which is exactly the $64.4\%$ cut in the table.

---

4. Failure Modes & First-Principles Breakdowns

1. Modelling transition risk as a return factor. A regression of returns on a carbon-price change gives a contemporaneous pass-through coefficient, not a stranding exposure; the stranding loss is a level shift that a short sample cannot contain (01 · §4.1).
2. A single pass-through for the whole portfolio. $\lambda$ is firm-, sector- and jurisdiction-specific and is the most influential assumption in any carbon-price stress test (hub §3: $5\times$). Use a range and report the range.
3. Treating the cost curve as a probability. $\mathrm{SF}(p)$ is the fraction stranded conditional on $p$; it carries no probability attached to $p$. Multiplying it by a subjective scenario probability silently converts a scenario into a forecast.
4. Ignoring the emission factor’s uncertainty. $\mathrm{EF}$ and $E_i$ come from disparate reporting regimes with different boundaries and vintages; a $\pm 20\%$ error in $E_i$ is a $\pm 20\%$ error in the stranding threshold $\mathrm{EF}\cdot p$ and thus moves $p^{\ast }$ materially (05 · §2).
5. Damage-function laundering. Publishing a physical-risk number computed from a quadratic $D(\Delta T)=\theta \Delta T^2$ without disclosing $\theta$ turns a parameter choice into an apparent measurement. Always report $\theta$, its range, and whether damage is applied to the exposed base or the whole book.
6. Double-counting. Transition and physical risk are correlated in the scenario set (hot-house worlds have a low transition price and high physical damage; net-zero worlds the reverse), so summing independently-stressed transition and physical losses overstates the total. They must be drawn from a single coherent scenario (03 · §2).

---

5. References

• TCFD, Recommendations of the Task Force on Climate-related Financial Disclosures (FSB, 2017), Figure 1
• High-Level Commission on Carbon Prices (Stern, N. & Stiglitz, J., chairs), Report of the High-Level Commission on Carbon Prices (World Bank, 2017)
• World Bank, *Shadow Price of Carbon in Economic Analysis
• IPCC, Climate Change 2013: The Physical Science Basis (AR5, WG1
• Carbon Tracker Initiative, Unburnable Carbon: Are the World’s Financial Markets Carrying a Carbon Bubble? (2011)
• PACTA / RMI, Paris Agreement Capital Transition Assessment methodology notes
• NGFS, Climate Scenarios for Central Banks and Supervisors
• Bolton, P. & Kacperczyk, M., Do investors care about carbon risk?, Journal of Financial Economics 142(2):517–549 (2021)

---

6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Climate Scenarios & Stress Testing · 04 · ESG Scores & the Carbon Premium
• Sibling: Stress Testing & Scenario Analysis · Credit Risk & the Merton Model (the Merton route from a carbon liability to a PD, extended in 06)