4.14.6 Advanced Extensions

Basic Prerequisites: 03 · Climate Scenarios & Stress Testing and ES · 06 · Advanced Extensions (Euler’s homogeneous-function theorem, spectral measures).

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1. Intuition & Practical Objective

Once climate risk has become a measurable exposure (02), a scenario distribution (03) and a data-quality problem (05), three advanced questions remain open:

1. How do you allocate it? A firm needs a portfolio-level climate risk budget split across desks, sectors or funds so the parts sum to the whole - without an arbitrary “emissions share” rule. The answer is Euler allocation of a homogeneous risk measure, exactly as for market risk, with the transition-shock factor as one of the factors.
2. How does it reach credit? A carbon liability is a claim that ranks behind nothing - it is a cash outflow, so it behaves like additional debt. Capitalising it into the Merton model turns a climate exposure into a probability of default, which is the currency counterparty and credit risk speak.
3. What if the distribution itself is unknowable? Climate has deep uncertainty: the scenario set is a modelling choice, not a sampling distribution. The correct response is not a better point estimate but an ambiguity-averse (robust) risk measure - and this is where the folder returns to the coherent-risk-measure theory it started from.

The unifying observation: the worst-case loss over a finite scenario set is already a coherent risk measure. Artzner’s scenario representation says $\rho (X)={\sup }_{P\in \mathcal{P}}{\mathbb{E}}_P[-X/r]$ is coherent for any family $\mathcal{P}$ of probability measures (ES · 03 · §2.3). An NGFS scenario set with weights is such a family. So the “worst scenario” and the “probability-weighted scenario ES” are not ad hoc stress numbers - they are points in the theory of coherent risk measures, and the difference between them is exactly the difference between an ambiguity set and a distribution.

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2. Mathematical Ground Truth & Derivations

2.1 Factor Euler allocation of a portfolio ES

Let the loss be a linear function of two factors, $L=b_M{f}_M+b_T{f}_T$, with factor covariance $\Sigma =\left(\begin{array}{cc}{\sigma }_M^2& \rho {\sigma }_M{\sigma }_T\\ \rho {\sigma }_M{\sigma }_T& {\sigma }_T^2\end{array}\right)$, where $f_M$ is a market-loss factor and $f_T$ a transition-shock factor ($\mathrm{Corr}=\rho >0$: a disorderly transition is a market event too). Then ${\sigma }_L=\sqrt{{\mathbf{b}}^{\top }\Sigma \mathbf{b}}$, and for a zero-mean normal loss

$$\boxed{{\mathrm{ES}}_{\alpha }={\sigma }_L{k}_{\alpha },k_{\alpha }=\frac{\phi (z_{\alpha })}{1-\alpha },{\mathrm{EC}}_i=\underset{\text{Euler contribution}}{\underbrace{b_i\frac{(\Sigma \mathbf{b}{)}_i}{{\sigma }_L}}}\cdot k_{\alpha },\sum _i{\mathrm{EC}}_i={\mathrm{ES}}_{\alpha }}$$

The additivity is Euler’s theorem for the positively homogeneous ${\sigma }_L$; the multiply-by-$k_{\alpha }$ step works because $\mathrm{ES}$ is positively homogeneous with the same degree (ES · 04 · §2.1). Explicitly,

$${\mathrm{EC}}_M=\frac{b_M^2{\sigma }_M^2+b_M{b}_T\rho {\sigma }_M{\sigma }_T}{{\sigma }_L}{k}_{\alpha },{\mathrm{EC}}_T=\frac{b_T^2{\sigma }_T^2+b_M{b}_T\rho {\sigma }_M{\sigma }_T}{{\sigma }_L}{k}_{\alpha }.$$

Three facts fall out of the cross term $b_M{b}_T\rho {\sigma }_M{\sigma }_T$:

• A carbon-heavy book ($b_T>0$) carries a strictly positive transition contribution, and total ES rises above the market-only book.
• A green-tilted book ($b_T<0$) has a negative transition contribution: the tilt is a risk reducer whenever $\rho >0$, because it pays off in exactly the states where the market factor is bad. This is the mathematics behind the “hedging climate risk” result of Andersson, Bolton & Samama (2016).
• The contribution is not monotone in the exposure: it is quadratic in $b_T$ with a linear cross term, so its sign depends on the ratio $b_T{\sigma }_T/(b_M\rho {\sigma }_M)$. Sizing a hedge by “emissions saved” is therefore not sizing a risk reduction (05 · §4.8).

2.2 Carbon-adjusted probability of default (Merton)

In the structural model, equity is a call on assets $A$ with strike equal to the debt $D$; default occurs if $A_T<D$. The distance to default and default probability are

$$\boxed{\mathrm{DD}=\frac{\ln (A/D)+(\mu -\frac{1}{2}{\sigma }_A^2)T}{{\sigma }_A\sqrt{T}},\mathrm{PD}=\Phi (-\mathrm{DD})}$$

To make it carbon-aware, capitalise the present value of the firm’s future carbon costs into the liability, $D\mapsto D+C$ - a first-order, transparent and conservative adjustment. Because $\mathrm{DD}$ is concave in $D$ while $\Phi (-\cdot )$ is convex, equal carbon-liability increments produce increasing PD increments: the credit channel amplifies a linearly-growing carbon exposure (verified in §3(B)). This is the natural bridge from climate risk to Credit Risk & the Merton Model, and it is the mechanism by which a transition scenario becomes loan losses in a supervisory stress test.

2.3 Robust and spectral measures: the deep-uncertainty layer

Three nested refinements of “just use ES”:

• Ambiguity aversion (worst-case over a scenario set). If $\mathcal{P}$ is the scenario family, $\rho (X)={\sup }_{P\in \mathcal{P}}{\mathbb{E}}_P[-X]$ is coherent (Artzner Prop. 4.1) and requires no probabilities at all. This is the formal justification for stress-testing disciplines that refuse to weight NGFS scenarios.
• Spectral weighting of the tail. Acerbi’s family $M_{\phi }(L)={\int }_0^1\phi (u){\mathrm{VaR}}_u(L)du$ with $\phi$ non-decreasing lets a firm weight the very worst quantiles more heavily than ES does (${\phi }_{\mathrm{ES}}$ is a step at $\alpha$), which is the defensible way to express climate tail aversion without inventing a distribution (ES · 06 · §2.1). Kusuoka’s representation says every law-invariant coherent measure is a mixture of expected shortfalls, so the choice of climate risk measure is exactly the choice of a mixing measure over ES levels - a policy, not a fact.
• Carbon-efficient portfolios. Constraining the transition factor, $\mathrm{Var}(L)$ minimised subject to $b_T\le {\bar{b}}_T$ (or subject to a carbon-intensity cap), traces a carbon-efficient frontier that is not the mean–variance frontier; Andersson–Bolton–Samama show it can be achieved at negligible tracking error (Constraints & Transaction Costs, Robust Optimization).

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3. Computational Implementation - allocating a climate risk budget and carbon-adjusting a PD

Panel (A) splits a portfolio ES into market and transition contributions for three books, verifying Euler additivity to machine precision. Panel (B) capitalises a carbon liability into the Merton model. Standard library only.

Panel (A). The contributions sum to the portfolio ES to machine precision ($6.9\times 10^{-18}$ for the carbon-heavy book) - the additivity that makes a climate risk budget possible. The carbon-heavy book carries $27.6\%$ of its ES from the transition factor despite being only $20\%$ emissions-adjacent in construction; the green-tilted book shows a negative transition contribution ($-1.5\%$) and a lower total ES ($0.025233$) than the market-only book ($0.025716$) - the hedging result of §2.1, made numeric. Note the green book’s market contribution exceeds total ES ($101.5\%$): the parts are additive, but an individual contribution may exceed the whole when another contribution is negative - a fact worth stating before someone reports a “negative risk budget” as an error.

Panel (B). The carbon liability is concave-in-$\mathrm{DD}$ and convex-in-$\mathrm{PD}$: each $5 increment of capitalised carbon cost raises PD by 7.2$,$8.8$,$9.8$ points - increasing increments from a linear exposure. A transition scenario therefore converts into credit losses faster than proportionally, which is why supervisory exercises report both a market-risk and a credit-risk leg of the same scenario.

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4. Failure Modes & First-Principles Breakdowns

1. Euler needs homogeneity - liquidity and convexity break it. Axiom PH fails for large positions and for non-linear (option-like) climate payoffs, so the additive split understates the marginal risk of scaling up a carbon exposure. Liquidity-adjust before allocating (ES · 06 · §4.2).
2. Reading contribution as exposure. The transition contribution is quadratic in $b_T$ plus a cross term; a larger carbon exposure can lower the contribution, and a hedge can show a positive one for small sizes. Allocate on contributions, report exposures separately.
3. A negative contribution treated as an error. A green tilt legitimately produces a negative risk contribution (§3(A)). Risk-budget systems that clamp contributions at zero destroy the hedging signal.
4. Capitalising carbon costs into debt without a horizon convention. $C$ is a present value of a cost path, so its magnitude depends on the discount rate and the path - publish both; a PD shift of 20 points from unspecified discounting is not a result.
5. Merton’s own limitations imported silently. The distance-to-default model assumes lognormal assets, a single zero-coupon liability and a constant ${\sigma }_A$; none improves by adding a carbon term (Credit Risk & the Merton Model).
6. Ambiguity dressed as probability. If you compute a probability-weighted scenario ES, you have chosen a measure; if you compute a worst-case, you have chosen an ambiguity set. Both are coherent, both are defensible, and they answer different questions - never mix them inside one reported number (05 · §4.5).
7. Spectral choice as a governance decision. Because climate tail aversion has no agreed weight function, the choice of $\phi$ (or the mixture over ES levels in Kusuoka’s representation) is a policy that should be owned by the risk committee, published, and stress-tested for sensitivity - exactly as the regulator’s and the firm’s $\phi$ differ in market risk.
8. Optimising against a climate constraint without a scenario-consistent objective. A carbon cap imposed inside a mean–variance optimiser can trade one factor for another without changing the true scenario exposure; validate the resulting portfolio on the scenario engine of 03, not on the constraint you imposed.

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5. References

• Acerbi, C. & Tasche, D. - On the Coherence of Expected Shortfall, J. Banking & Finance 26(7):1487–1503 (2002)
• Acerbi, C. - Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion, J. Banking & Finance 26(7):1505–1518 (2002)
• Kusuoka, S. - On Law Invariant Coherent Risk Measures, Advances in Mathematical Economics 3:83–95 (2001)
• Artzner, Delbaen, Eber & Heath - Coherent Measures of Risk, Mathematical Finance 9(3):203–228 (1999)
• Andersson, M., Bolton, P. & Samama, F. - Hedging Climate Risk, Financial Analysts Journal 72(3):13–32 (2016)
• Merton, R.C. - On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29(2):449–470 (1974)
• Bolton, P. & Kacperczyk, M. - Do investors care about carbon risk?, JFE 142(2):517–549 (2021)
• BCBS - *Climate-related Financial Risks

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · 03 · Climate Scenarios & Stress Testing · Index Hub
• Base: ES · 06 · Spectral & Euler · Multivariable Calculus (Euler’s theorem)
• Sibling: Credit Risk & the Merton Model · Systemic Risk & Aggregation · Liquidity Risk & Funding
• Forward: Portfolio Risk Constraints & Mean–Variance · Risk Parity & Equal Risk Contribution (the same Euler idea, different budget rule) · Robust Optimization