4.14.4 ESG Scores, Temperature Alignment & the Carbon Premium

Basic Prerequisites: 02 · Physical & Transition Risk and Econometrics & Time Series (cross-sectional regression, standard errors).

---

1. Intuition & Practical Objective

Two distinct products are routinely confused under the ESG label, and a quant must separate them: a rating (an ordinal summary of a firm’s ESG performance, produced by a private methodology) and a price signal (a return premium or discount in the cross-section, produced by the market). The first is an input you must audit; the second is an equilibrium fact you can measure. This page covers both, plus the third object in the family: temperature alignment, which converts an emissions trajectory into a single number in °C.

Three things a practitioner must know before using any of them:

1. A rating is an aggregation of contested indicators, and the aggregation itself is proprietary. Berg, Kölbel & Rigobon (2022) decompose the divergence between six major raters into scope (which attributes are assessed), measurement (which indicator measures a given attribute) and weight (how indicators are combined), and find the average pairwise rating correlation is only 0.54, ranging $0.38$–$0.71$ - against $\approx 0.99$ for credit ratings. The decomposition attributes 56% of the divergence to measurement, 38% to scope and only 6% to weights. That ordering is the punchline: aligning weighting schemes - the obvious fix - would remove almost nothing. The problem is how the data are generated.
2. Temperature alignment is an interpolation between benchmark pathways, not a measurement. Given a portfolio’s cumulative emissions over a horizon, one finds the two IPCC/NGFS-consistent pathways it sits between and interpolates a temperature. It is a benchmarking device: transparent, monotone, and entirely dependent on the benchmark table and the emissions boundary feeding it.
3. The cross-sectional price of carbon emissions has been documented with both signs. Bolton & Kacperczyk (2021) find that US stocks of firms with higher total CO2 emissions earn higher returns, controlling for size, book-to-market and other predictors - a carbon (brown) premium interpreted as compensation for transition risk. Pástor, Stambaugh & Taylor (2021) show theoretically that green assets command lower expected returns (investors accept a lower return for green holdings - a “greenium”) but can outperform after climate-concern shocks, because they hedge transition risk. These are not contradictory: a level effect (a greenium in expected returns) and a news effect (green outperformance when climate concerns rise) coexist. In the green-bond market the level effect was measured by Zerbib (2019) as a yield premium of roughly $-2$ basis points for green bonds.

The practical objective: build a score, measure its instability, map a portfolio to a temperature, and estimate the carbon premium - with standard errors attached, because a premium without an error bar is a story.

---

2. Mathematical Ground Truth & Derivations

2.1 How an ESG score is built (and why raters disagree)

Following Berg, Kölbel & Rigobon’s notation, rater $k$‘s rating of firm $f$ is a linear aggregation of category scores $C_{fkj}$ with rater-specific weights $w_{kj}$:

$$\boxed{R_{fk}=\sum _{j\in {\mathcal{C}}_k}{C}_{fkj}{w}_{kj},w_{kj}\ge 0}$$

Three divergence channels follow immediately:

$$\underset{\text{scope}}{\underbrace{{\mathcal{C}}_k\ne {\mathcal{C}}_{k^{\prime }}}},\underset{\text{measurement}}{\underbrace{C_{fkj}\ne C_{f{k}^{\prime }j}\text{on the same attribute}}},\underset{\text{weight}}{\underbrace{w_{kj}\ne w_{k^{\prime }j}}}.$$

Taking variances of $D_f^{k,k^{\prime }}=R_{fk}-R_{f{k}^{\prime }}$ and splitting by an arithmetic decomposition gives the reported shares $(38\%,56\%,6\%)$. Consequence for a quant: the rating is a noisy proxy whose noise is not independent across the object being measured - hence a rater fixed effect (a “halo”) beyond idiosyncratic noise.

Rank agreement. Because ratings are ordinal in use, the appropriate agreement statistic is the Spearman rank correlation: replace each $R_{fk}$ by its average rank ${\bar{r}}_{fk}$ (ties share the mean rank) and take the Pearson correlation of the rank vectors,

$${\rho }_s=\frac{{\sum }_f({\bar{r}}_f-\overline{\stackrel{ˉ}{r}})({\bar{r}}_f^{\prime }-\overline{{\stackrel{ˉ}{r}}^{\prime }})}{\sqrt{{\sum }_f({\bar{r}}_f-\overline{\stackrel{ˉ}{r}}{)}^2}\sqrt{{\sum }_f({\bar{r}}_f^{\prime }-\overline{{\stackrel{ˉ}{r}}^{\prime }}{)}^2}}.$$

2.2 Implied temperature rise by benchmark-pathway interpolation

Let a portfolio have cumulative emissions intensity $C$ (tCO2e per $1m of revenue over the horizon), and let {(T_j,C_j)}_{j=1}^m$ be a table of benchmark pathways with increasing temperature outcomes and increasing cumulative intensities. The ITR is the piecewise-linear interpolation

$$\boxed{\mathrm{ITR}(C)=T_j+(T_{j+1}-T_j)\frac{C-C_j}{C_{j+1}-C_j}\text{for }C\in [C_j,C_{j+1}],\mathrm{ITR}=T_1\text{for }C\le C_1}$$

with no extrapolation beyond the last benchmark: a portfolio worse than the worst pathway is reported as “worse than $T_m$”, not as a fabricated $4.1^{\circ }$C. The physical justification for a monotone cumulative-emissions-to-warming map is the TCRE relation (IPCC AR5: $0.8$–$2.5^{\circ }$C per 1000 PgC, i.e. $\approx 0.2$–$0.7^{\circ }$C per 1000 GtCO2); the benchmark table is a modelling choice, and belongs in the disclosure.

2.3 The carbon premium as a cross-sectional regression

The canonical specification regresses realised returns on a measure of emissions exposure with controls:

$$r_f=a+b\ln E_f+{\mathbf{c}}^{\top }{\mathbf{X}}_f+{\epsilon }_f,\hat{b}=({\mathbf{X}}^{\top }\mathbf{X}{)}^{-1}{\mathbf{X}}^{\top }\mathbf{r},\widehat{\mathrm{se}}({\hat{b}}_j)=\sqrt{{\hat{\sigma }}^2[({\mathbf{X}}^{\top }\mathbf{X}{)}^{-1}{]}_{jj}}$$

with ${\hat{\sigma }}^2=\mathrm{RSS}/(n-k)$. Interpretation:

• $\hat{b}>0$ (brown premium): high-emitting firms earned higher realised returns, the Bolton–Kacperczyk finding - compensation for transition risk.
• $\hat{b}<0$ (greenium in realised returns): the low-emission firms outperformed - the Pástor–Stambaugh–Taylor regime after a climate-concern shock. Both regimes are the same estimator on different samples, which is exactly the discipline the carbon-premium literature requires: report the sample, the shock, and the standard error. A premium is a conditional moment, not a constant.

---

3. Computational Implementation - score divergence, temperature alignment and the premium

Three panels: (A) three raters scoring eight firms, with rank correlations; (B) implied temperature rise for a portfolio under two emissions boundaries; (C) an OLS that recovers the carbon premium and then shows its sampling dispersion. Standard library only.

Panel (A). Firm A is ranked $2.5$ by R1 and $8.0$ (last) by R3 - the identical firm, the identical pillars, only different weights and one alternative environmental indicator. Four of eight firms move three or more places. The mean rank correlation is $+0.439$, in the same regime as Berg et al.’s $0.54$ empirical average. A portfolio “constructed on ESG scores” is constructed on a rater, not on a fact.

Panel (B). The ITR interpolation gives $1.98^{\circ }$C for this book - but $82.8\%$ of the cumulative emissions come from the coal and oil holdings. Widen the boundary to Scope 1+2+3 and the same portfolio is beyond the $3.2^{\circ }$C benchmark: the interpolation refuses to extrapolate, which is the correct behaviour (§2.2). The alignment label is a boundary choice, not a property of the holdings.

Panel (C). The OLS recovers the planted premium exactly ($+0.0180$ vs $+0.018$; $-0.0200$ vs $-0.020$) on identical firms under two pricing regimes - the brown-premium and greenium cases of §2.3, in one estimator. But four disjoint subsamples of the same regime give $\hat{b}\in [+0.0114,+0.0217]$: the sign is stable, the magnitude is not. That is the honest report of a carbon premium.

---

4. Failure Modes & First-Principles Breakdowns

1. Treating a rating as a variable with small measurement error. Rating divergence is structural, not noise: measurement contributes $56\%$ and it is correlated across categories within a rater (the halo effect), so averaging raters does not cleanly cancel it (Berg et al. 2022).
2. Rating restatement (the “rewriting history” problem). ESG data providers have retroactively revised historical scores, so a backtest built on a vendor’s as-of-today history is using information that was not available in real time - a look-ahead bias generated by the data vendor rather than by the researcher (Berg, Fabisik & Sautner). Tests of ESG signals must use point-in-time vintages.
3. Weight-shifting as a “fix”. Because weights explain only $6\%$ of divergence, re-weighting the pillars changes the label without changing the disagreement; the binding constraint is measurement and scope.
4. ITR as a measured quantity. An ITR is an interpolation against a chosen benchmark table, computed from a chosen scope boundary, over a chosen horizon. Reporting “$1.98^{\circ }$C” without the table, the boundary and the horizon is reporting a spreadsheet cell as a physical fact.
5. Reading a carbon premium without a sample. $\hat{b}$ flips sign with the regime (panel C) and its magnitude varies by a factor of two across subsamples. A premium is a conditional moment: quote the period, the controls and the standard error.
6. Ignoring the demand-side explanation. A “greenium” in expected returns (Pástor–Stambaugh–Taylor) is an equilibrium consequence of investor preferences, not evidence that green assets are safer; it coexists with a brown risk premium. Confusing the preference-driven and risk-driven components mis-specifies the hedge.
7. Scope/aggregation laundering in alignment metrics. Portfolio-level intensity can be reduced by reweighting rather than by real-world change - the mechanical decomposition between “portfolio mix” and “financed emissions” must be disclosed, or a portfolio’s ITR becomes a portfolio-construction artefact.

---

5. References

• Berg, F., Kölbel, J.F. & Rigobon, R. - Aggregate Confusion: The Divergence of ESG Ratings, Review of Finance 26(6):1315–1344 (2022)
• Berg, F., Fabisik, K. & Sautner, Z. - Rewriting History II: The (Un)Predictable Past of ESG Ratings (ECGI Finance Working Paper 708/2020; circulated as Is History Repeating Itself?)
• 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)
• Zerbib, O.D. - The effect of pro-environmental preferences on bond prices: Evidence from green bonds, Journal of Banking & Finance 98:39–60 (2019)
• Andersson, M., Bolton, P. & Samama, F. - Hedging Climate Risk, Financial Analysts Journal 72(3):13–32 (2016)
• TCFD, Recommendations (2017) and SBTi, Foundations for Science-Based Net-Zero Target Setting
• IPCC, Climate Change 2013: The Physical Science Basis (AR5, WG1

---

6. Connected Graph Bridges

• Back: 02 · Physical & Transition Risk · 03 · Climate Scenarios & Stress Testing · Index Hub
• Forward: 05 · Failure Modes & Practice · 06 · Advanced Extensions
• Sibling: Fundamental Multi-Factor Models (where a carbon factor would be estimated) · Factor Investing & Timing · Quantitative Fundamental Investing · Model Risk & Validation