Risk measures and portfolio analysis in the paradigm of climate finance: a review

Abstract

Climate change brings with it new risks for the finance sector, which in turn provides new opportunities to mitigate this risk, emanating from climate change. To invest sustainably and move away from firms that have disproportionately high carbon footprints, investors need suitable risk measures and appropriate portfolio management approaches. In this paper, we conduct a review of the mathematical models used to measure carbon risk. Subsequently, we review portfolio optimization models based on modern portfolio theory and the incorporation of risk measures into portfolio optimization strategies. We find that there is a lack of consensus about the existence of a carbon premium or an equity greenium in stock prices. We also find that the literature on portfolio optimization techniques is comparatively nascent.

Introduction

The recently published report of the Intergovernmental Panel on Climate Change (IPCC 2022) cautions that societal choices and actions implemented in the next decade will play a crucial rule in determining the pathway that climate change will take in the future. This warning and the facts it is based on have severe implications for investors. Further, the low-carbon transition is inevitable, regardless of whether it comes through voluntary efforts of private corporations or implementation of strict regulations by the policymakers. The implementation of ambitious climate policies can often lead to unintended consequences such as macroeconomic instability (Carattini et al. 2021). In particular, for the stakeholders in the finance sector, this percolates in risks arising from multiple factors like asset stranding (Curtin et al. 2019), cost of equity capital (Kim et al. 2015), the decline of certain sectors (Di Cosmo and Hyland 2013), and induced effects like unemployment and its consequent effects on consumption (Semieniuk et al. 2021). Some researchers suggest that to reduce environmental impacts, consumption patterns will have to move towards cleaner goods (Schandl et al. 2016), while others suggest that environmental damage cannot be decoupled with economic growth and economies must therefore undergo significant change to make “degrowth” feasible (Kallis 2011). Both of these viewpoints mean that financial systems will have to undergo radical transformations to fit the new paradigm, and investors need to find appropriate strategies to invest sustainably, to reduce their own risks while helping the economy move away from unclean production.

The impacts of extending restrictions on fossil fuels has been studied for a long time, with several studies investigating the effect of carbon tax rates (Rutherford 1992; Lin and Li 2011; Callan et al. 2009; Tol 2007), carbon rationing (Fawcett 2004; Howell 2012; Cohen 2011) and other carbon policy related consequences. In the presence of such policies, firms will strongly benefit from restructuring their businesses to move to cleaner production. Accordingly, efforts are already being made in some of the sectors, particularly with the automobile industry and the textile industry exploring electric vehicles and a transition to a circular economy, respectively. However, these kinds of transitions are still in a rather nascent stage, in the case of most firms and industries. It is pertinent to note that the firms engaged in renewable energy projects, in spite of being in need of access to capital markets, are still disproportionately reliant on self-financing and debt financing (Tian 2018). It is imperative for investors and financial institutions to thus become a part of the movement to decarbonize economies around the world. But for investors to appropriately determine which firms to invest in, there need to be suitable indicators and measures that describe not just the qualitative, but also the quantitative impact of firms’ activities on the environment. Given appropriate measures, investors then require strategies that can help optimize their portfolios to hedge against the risk as estimated by the risk measures.

In this article, we conduct a literature review to study the existing literature on mathematical models used to estimate climate-related risks in financial markets, and mathematical models to optimize portfolios to hedge against carbon risk. We review 32 articles on the estimation of carbon risk in financial markets, and 9 articles on hedging strategies that use Markowitz’s mean-variance formulation or Capital Asset Pricing Model (CAPM) approaches. Among risk measures, we classify risk measures into predominantly two broad groups, namely, Fama-French Extended (FFE) models, and Linear Risk Models (LRM). We also study if the type of model plays a role in the relationship between how clean a firm or a portfolio is and stock prices. These are important questions as they give insights not just about whether existing models and techniques are adequate but also about other factors which could play into empirical results. The review also helps us identify gaps in the literature with respect to the development of models and their utility in empirical research. Moreover, models of risk measures are not just important in studying whether markets price risks but they can also be instrumental tools in hedging climate risk in portfolios or guiding portfolio divestments. Therefore, we also study the risk measures used in portfolio optimization strategies to hedge against climate risk to study if there exists a parity between market measures of carbon risk and risk measures used in portfolio optimization.

We find that there is significant scope for research both in the development and estimation of risk measures, particularly in the context of models which establish robust causal relationships and empirical studies in the Global South. We find scope for more studies in mathematical finance for the modeling of long-term risks associated with climate change and low-carbon transitions. All these factors together contribute to the dearth of consensus in the literature on whether the relationship between emissions and returns is positive, negative, or neutral. Our survey shows that different families of models show equal fractions of studies that estimate positive premia. Still, one family estimates negative premia more often than the other, although this may be because of different sample sizes. We also find that existing models of hedging climate risk rarely use risk measures used in empirical studies such as premia, and consider the risk purely in terms of raw measures such as absolute emissions or emissions intensity.

Emissions and stock returns

According to a landmark study (Krueger et al. 2020) of how institutional investors perceive climate risk, the authors found that close to 90% institutional investors who participated in a survey expected carbon transition risk to materialize within 10 years (from 2020). Thus, in a bid to hedge against transition risk, close to a third of the investors claimed to try and reduce the carbon footprints of their portfolios and close to a fourth of the investors attempted to reduce stranded asset risks. However, reducing the carbon footprint portfolio can prove risky for investors, as cleaner assets often give fewer returns and tend to be more volatile, as is evident from the movements of clean energy indices (Andersson et al. 2016). Theoretically, this is unsurprising. In the mean-variance framework, in a sample of N stocks, suppose the portfolio has been optimized for $\mu$ (expected returns) and ${\sigma }^2$ (variance). Then removing polluting stocks from the population of stocks or choosing a small sample of clean energy stocks will only lead to returns that are less than equal to the returns of the original portfolio for a given variance, or variance greater than equal to that of the original portfolio for a given level of returns. This is the fundamental problem of investing in the paradigm of climate change and market transitions, namely that, hedging climate risk comes at a cost.

However, this need not imply that cleaner stocks would underperform dirty stocks, especially in the long run, and it only implies that at present, a smaller set of stocks, be it green or brown, would perform worse than a better-diversified portfolio. There is also evidence that investing in cleaner businesses can reduce risks, particularly risks from stranded assets. This is particularly visible in banks having a lower default rate when they invest in clean energy (Choudhury et al. 2023). Moreover, governments around the world are already introducing policies in the direction of a slow transition. For instance, the European Union Emissions Trading Scheme (EU ETS), for the purpose trading of carbon emissions, was introduced in 2005. The mechanism capped emissions and provided a framework, where firms wanting to go beyond their assigned allowances, could do so by purchasing unused allowances from firms, which had used less than their assigned allowances. A similar policy was also introduced in China in 2017. Policies such as emissions trading could significantly increase operating costs for firms that are overdependent on fossil fuel consumption and could have an impact on their revenue and profits, which could in turn affect their returns and volatility (Bushnell et al. 2013).

Thus, the question of how stock prices interact with the market at present is an extremely important question as it not only describes the state of affairs at the moment but can also provide insights into how risks will be priced in the future. Broadly speaking, there are three ways in which climate risk could show up in the stock prices of polluting or non-polluting firms. It could have no impact on the price of stocks whatsoever, and it would indicate that markets do not perceive climate risk to be a significant factor at the current moment or in the foreseeable future. The second alternative would be for polluting firms to give lesser returns than cleaner firms. One would expect this to be the case if the market is in transition or has already transitioned, such that there are high financial costs or restrictions to running an environmentally unfriendly business. This difference between the value of green and brown stocks due to the overperformance of cleaner assets has been dubbed the “equity greenium” (Pastor et al. 2022), borrowed from the commonly used term in bond valuation. The third case is the most interesting, where firms with higher emissions outperform similar firms which differ only in terms of how polluting they are. This abnormal excess return is called the carbon premium or the carbon transition premium and is thought of as an incentive for investors to invest in risky firms, due to the possibility of a transition in the future. The three situations described above can be thought of as zero, negative and positive carbon premia, respectively. Theoretically speaking,

$$\begin{array}{c}\hfill \text{Carbon Premium}=-\text{Equity Greenium}.\end{array}$$ 1

The existence of one or the other type of premium will have a considerable impact on the way in which investors choose to hedge their portfolios. For instance, if there exists a robust equity greenium, then it is better for investors both in the short and the long term to prefer green stocks in their portfolios. However, if a carbon premium exists, then brown stocks are better for the investor in the short-term but green stocks are better for the investor in the long term, and therefore the investor needs to develop an optimum strategy which does not lead to too modest returns in the short term while simultaneously not running into heavy losses in the long run. Therefore, climate risk itself requires specially catered portfolio optimization techniques to address the above problem. Thus in the following sections of the article we briefly describe our review methodology in “Methodology and sample”, followed by a thorough survey of mathematical models for risk measures in “The risk measures: carbon premium and other measures ” and the portfolio optimization literature in climate finance in “Hedging climate risk: portfolio methods”.

Methodology and sample

Our work is centered around reviewing the methods used in estimating and hedging climate-related risks IN financial markets. Thus we broadly split the review into two sections, one dedicated to reviewing the methods used in estimating risk, and another to review the methods used in managing portfolios. In “The risk measures: carbon premium and other measures ”, we survey the literature for studies which propose/use explicit mathematical or statistical models for estimating carbon-related risks in stock markets. In “Hedging climate risk: portfolio methods”, we review the literature for hedging strategies based on Markowitz’s mean-variance approach and CAPM approaches. The choice for these two strategies, in particular, is because they are fundamental portfolio optimization strategies that are popularly used in financial markets. This allows us to review portfolio optimization strategies in literature and study if there exists a parity between risk measures surveyed in “The risk measures: carbon premium and other measures ” and the models for hedging in “Hedging climate risk: portfolio methods”.

For our review of methods of risk measurement, we begin with a literature search. To avoid using a biased sample of articles chosen arbitrarily, we opt for a systematic review as opposed to a traditional literature review. While traditional literature reviews use subjective methods, which may lead to a bias in the inclusion and exclusion of articles, systematic reviews have an explicit search methodology and inclusion criteria, which reduces bias in the sample of papers surveyed and makes the survey reproducible. We use Google Scholar, and search using keywords that we would expect to find in papers relating carbon emissions to stock prices (Table 1). Therefore, we search the following strings: (“Carbon Risk” OR “Carbon Premium” OR “Carbon Exposure” OR “Greenium”) AND (“Equity” OR “Stocks”). Due to the growing literature on climate finance, we limit our search to the first 30 search results on each combination of strings. Since climate finance is a contemporary growing discipline, we also restrict our survey to papers published in the last 10 years i.e., from 2013 to 2023. After the search, we are left with 132 unique articles. To avoid publication bias, we include published, as well as unpublished articles, and reports. Among these 132 articles, we further filtered the list based on titles and abstracts to include articles relevant to stock returns and climate risk. Several articles were excluded because they investigated risks in other financial instruments such as bonds, or studied the impact of carbon emissions of firms on other financial parameters such as ease of getting loans, and returns on equity, among others. Since this is a survey of methods, we also excluded papers that did not explicitly detail mathematical or statistical methodologies in the estimation of the risk. We are left with a final set of 32 articles that propose or use an environmental or climate risk measure for stock returns.

Table 1.

List of surveyed articles on risk measures showing the authors, year, model and result. 26 articles out of 32 are from 2020 or later

Authors	Year	Model	Premium
Oestreich and Tsiakas (2015)	2015	FFE	Positive
Maddox (2018)	2018	LRM	None
Zhang et al. (2018)	2018	FFE	None
Del Prado (2019)	2019	FFE	Positive
Ravina and Hentati Kaffel (2019)	2019	FFE	Negative
Görgen et al. (2020)	2020	LRM	None
Kuang and Liang (2020)	2020	FFE	Negative
Monasterolo and De Angelis (2020)	2020	FFE	–
Wen et al. (2020)	2020	FFE	Positive
Alessi et al. (2021)	2021	FFE	Positive
Bernardini et al. (2021)	2021	FFE	Negative
Bolton and Kacperczyk (2021a)	2021	LRM	Positive
Bolton and Kacperczyk (2021b)	2021	LRM	Positive
Brunborg and Haldorsen (2021)	2021	FFE	None
Jacob and Nerlinger (2021)	2021	LRM	–
Ji et al. (2021)	2021	FFE	Negative
Mazzarano et al. (2021)	2021	FFE	Mixed
Muñoz (2021)	2021	FFE	Negative
Santi and Moretti (2021)	2021	Other	Mixed
Walkshäusl (2021)	2021	FFE	Positive
Witkowski et al. (2021)	2021	FFE	Mixed
Aswani et al. (2022)	2022	LRM	None
Gurvich and Creamer (2022)	2022	FFE	Positive
Huij et al. (2022)	2022	FFE	–
Lo et al. (2022)	2022	FFE	Mixed
Meyer (2022)	2022	FFE	Negative
Park et al. (2022)	2022	FFE	Positive
Pastor et al. (2022)	2022	FFE	Negative
Tham and Kang (2022)	2022	LRM	–
Vymyatnina and Chernykh (2022)	2022	FFE	None
Zhang (2022)	2022	LRM	None
Nag et al. (2023)	2023	Other	–

There is little consensus in the results of studies that conduct empirical analyses regarding the presence of carbon premia, with nine articles finding a significant positive carbon premium, seven articles finding a negative premium, four articles finding mixed results, and seven articles finding no premium. Weak or statistically insignificant results have been considered as no results. Five articles do not conduct empirical analyses on the existence of the premium or study the premium under special circumstances and events, and their results have not been described in the table. (FFE Fama-French extended models, LRM linear risk models)

For our review of portfolio optimization and hedging strategies, we resort to a traditional literature review. This is because articles often do not explicitly mention the strategy used in their title and abstracts which makes a systematic review inefficient, and reduces the likelihood of including important papers in the field. We survey 9 papers that use mean-variance or CAPM methods to hedge climate risk. The list of articles surveyed and the methods used can be found in Table 2.

Table 2.

List of the surveyed articles on portfolio methods displaying the authors, years, method and risk measures used. 9 articles have been surveyed in this section. 7 articles use mean-variance approaches and only 3 employ CAPM. Most of the risk measures used are basic measures such as emissions or intensity or ESG scores. The Brown-Green-Score and the carbon beta are the only composite measures used. We find that the state of the literature on portfolio methods to hedge climate risk is nascent, but growing

Authors	Year	Method	Measure
Andersson et al. (2016)	2016	Mean-variance	Absolute emissions
Fang et al. (2019)	2019	Mean-variance	Carbon intensity
Pedersen et al. (2021)	2021	Mean-variance	ESG
Roncalli et al. (2021)	2021	Mean-variance	BMG factor
Roston (2021)	2021	Mean-variance	Carbon intensity
Ahmadi and Peivandizadeh (2022)	2022	Mean-variance	Sustainable practices
		Mean-variance	Carbon beta
Acerbi (2022)	2022	and	Absolute emissions
		CAPM	Emissions intensity
Pástor et al. (2021)	2021	CAPM	ESG
Roncalli et al. (2021)	2021	CAPM	BMG factor

The risk measures: carbon premium and other measures

Premia can be measured at the level of the portfolio or the firm. At the level of the portfolio, the carbon premium is sometimes defined as the abnormal excess returns ($\alpha$) of the “dirty-minus-clean” portfolio (Oestreich and Tsiakas 2015). To quantify the carbon premia in the portfolios, Oestreich and Tsiakas perform three regressions to estimate ${\alpha }_j$, where portfolio j is one of either “dirty”, “clean” or “dirty-minus-clean” portfolios. The first regression is based on the CAPM framework,

$$\begin{array}{c}\hfill r_{j,t}-r_{f,t}={\alpha }_j+{\beta }_j(r_{M,t}-r_{f,t})+{ϵ}_{j,t},\end{array}$$ 2

where $r_{j,t}$ is the portfolio log-return, $r_{f,t}$ and $r_{M,t}$ are the monthly riskless return and monthly return to the market portfolio, respectively, and ${ϵ}_{j,t}$ is the idiosyncratic error term, all at time t. The second regression follows the three-factor model (Fama and French 1993)

$$\begin{array}{c}\hfill r_{j,t}-r_{f,t}={\alpha }_j+{\beta }_{j1}(r_{M,t}-r_{f,t})+{\beta }_{j2}SM{B}_t+{\beta }_{j3}HM{L}_t+{ϵ}_{j,t},\end{array}$$ 3

where two additional terms in the regression are included, namely, ${\beta }_{j2}SM{B}_t$ and ${\beta }_{j3}HM{L}_t$, which denote the contributions of the monthly return to the “small-minus-big” size factor, and the monthly return to the “high-minus-low” value factor, respectively. The final regression used is based on the Carhart (1997) four-factor model

$$\begin{array}{c}\hfill r_{j,t}-r_{f,t}={\alpha }_j+{\beta }_{j1}(r_{M,t}-r_{f,t})+{\beta }_{j2}SM{B}_t+{\beta }_{j3}HM{L}_t+{\beta }_{j4}MO{M}_t+{ϵ}_{j,t},\end{array}$$ 4

with a term ${\beta }_{j4}MO{M}_t$ controlling for the monthly return to the momentum factor. Furthermore, some authors use the Fama-French five-factor model.

$$\begin{array}{c}\hfill r_{j,t}-r_{f,t}={\alpha }_j+{\beta }_{j1}(r_{M,t}-r_{f,t})+{\beta }_{j2}SM{B}_t+{\beta }_{j3}HM{L}_t+{\beta }_{j4}RM{W}_t+{\beta }_{j5}CM{A}_t+{ϵ}_{j,t},\end{array}$$ 5

with inclusions to the three-factor model, namely, ${\beta }_{j4}RM{W}_t$ and ${\beta }_{j5}CM{A}_t$, which denote the contributions of the monthly return to the “robust-minus-weak” profitability factor, and the monthly return to the “conservative-minus-aggressive” strategy factor, respectively. Apart from these classical models, the so-called “environmentally-extended Fama French model” (Ravina and Hentati Kaffel 2019) includes a sixth “Green-minus-Carbon” factor ${\beta }_{j6}GM{C}_t$, which makes the final model of the form

$$\begin{array}{cc}\hfill r_{j,t}-r_{f,t}=& {\alpha }_j+{\beta }_{j1}(r_{M,t}-r_{f,t})+{\beta }_{j2}SM{B}_t\hfill \\ \hfill & +{\beta }_{j3}HM{L}_t+{\beta }_{j4}RM{W}_t+{\beta }_{j5}CM{A}_t+{\beta }_{j6}GM{C}_t+{ϵ}_{j,t}.\hfill \end{array}$$ 6

Other factors like a liquidity factor (Wen et al. 2020; Pastor et al. 2022), a transparency factor (Alessi et al. 2021), climate shock factors (Lo et al. 2022), volatility and market timing (Ji et al. 2021), event factors (such as the Paris Agreement) (Monasterolo and De Angelis 2020), and an expected growth factor (Pastor et al. 2022) have also been used. These methods have been the most popular among the articles in our survey, with researchers adopting them for the estimation of carbon premia in various countries like BRICS nations (Ji et al. 2021), China (Zhang et al. 2018; Wen et al. 2020; Lo et al. 2022), the European Union (Oestreich and Tsiakas 2015; Ravina and Hentati Kaffel 2019; Monasterolo and De Angelis 2020; Alessi et al. 2021; Bernardini et al. 2021; Witkowski et al. 2021; Meyer 2022), Korea (Park et al. 2022), Russia (Vymyatnina and Chernykh 2022), the United States of America (Del Prado 2019; Kuang and Liang 2020; Monasterolo and De Angelis 2020; Muñoz 2021; Brunborg and Haldorsen 2021; Huij et al. 2022; Lo et al. 2022; Pastor et al. 2022), and for data across the globe (Monasterolo and De Angelis 2020; Walkshäusl 2021; Gurvich and Creamer 2022). This set of models, which we call the Fama-French extended models, seems to be the most common in literature, in measuring the carbon premium. However, most of the premia (or lack thereof) measured in each study depends on the variables used which quantify the brown-ness or green-ness of firms and the regulatory circumstances in the markets where the data is from.

The next set of models is those where firm-level returns are studied as a function of a carbon factor. We call these linear risk models, where the excess returns at the firm level are linear functions of a risk variable such as absolute carbon emissions or carbon intensity. Unlike the Fama-French extended models which factor in portfolio-level controls such as strategies, these models control for confounding variables at the level of the firm and have the potential to hint at causal relationships between environmental governance and stock returns. Bolton and Kacperczyk, in their seminal work (Bolton and Kacperczyk 2021a, b) with cross-sections of firms, investigate the relationship between (changes in) carbon emissions and stock returns. The premia are defined as “compensation for holding stocks of disproportionately high CO${}_2$ emitters”. The modeling was accomplished through a cross-sectional regression for the return on the i-th stock,

$$\begin{array}{c}\hfill r_{\mathit{it}}={\alpha }_0+{\beta }_{1}log{(em)}_{i,t}+{\beta }_2{c}_{i,t-1}+{\gamma }_t+{ϵ}_{i,t},\end{array}$$

where em is the total emissions, $c_{i,t-1}$ is the control vector of several accounting factors, ${\gamma }_t$ accounts for year/month fixed effects and ${ϵ}_{i,t}$ is the idiosyncratic error. The analysis concluded that the higher returns on investments in polluting firms are suggestive of the existence of a carbon premium, which is consistent with the increasing expectations of investors to be compensated for their carbon emission exposure. Further, to capture the impact of the percentage changes of the firm’s emissions on stock return, the authors used the regression model,

$$\begin{array}{c}\hfill r_{\mathit{it}}={\alpha }_0+{\beta }_1\mathrm{\Delta }{(em)}_{i,t}+{\beta }_2{c}_{i,t-1}+{\gamma }_t+{ϵ}_{i,t}.\end{array}$$

Also considered was a regression analysis for emission intensity as,

$$\begin{array}{c}\hfill r_{\mathit{it}}={\alpha }_0+{\beta }_1{(ei)}_{i,t}+{\beta }_2{c}_{i,t-1}+{\gamma }_t+{ϵ}_{i,t},\end{array}$$

where ei is the emission intensity. The two studies, although significantly different in their empirical questions and research design, brought forward a single unique perspective on how carbon emissions could drive up stock prices as a form of incentive given to investors to invest in stocks that could prove risky in the long term. The control variables in the studies are different as one looked at investors in the United States (Bolton and Kacperczyk 2021a) while the other looked at the premium in various countries which required controlling across different markets along the world apart from financial and sectoral differences. Characterizing firm-level and macroeconomic variables in the model is an important step in robustly demonstrating that the excess return is a function of the emissions alone and not due to correlations arising out of other factors. Recent studies (Aswani et al. 2022; Zhang 2022), however, find contradicting evidence using similar models. These studies find that the carbon premium disappears after accounting for different financial variables and data release lag. Some studies use modified or alternate linear models with introductions of different terms such as explicit and implicit risk factors (Mazzarano et al. 2021), factors capturing the impact of COVID-19 (Jacob and Nerlinger 2021), and news sentiment (Tham and Kang 2022). Based on Koch and Bassen’s work with electric utilities (Koch and Bassen 2013), Maddox studies a linear model with premia each for the market, energy, and carbon emissions (Maddox 2018). Görgen et al. study carbon risk under a different linear model where the risk factor is the compound “Brown Green Score (BGS)” (Görgen et al. 2020). The computation of the BGS, in terms of three indicators, is as defined below,

$$\begin{array}{c}\hfill BG{S}_t:=0.7\times \text{Value}{\text{Chain}}_t+0.15\times \text{Public}{\text{Perception}}_t+0.15\times {\text{Adaptibility}}_t,\end{array}$$

where $0\le BG{S}_t\le 1$, with the lower bound 0 denoting a green firm and the upper bound 1 denoting a brown firm. Based on the BGS, the yearly return of an asset i is given by,

$$\begin{array}{c}\hfill r_{i,t}={\alpha }_i+{\beta }_{i1}BG{S}_t+{\beta }_{i2}(BG{S}_t-BG{S}_{t-1})+{\gamma }_i+v_t,\end{array}$$

where $v_t$ is the error term. The regression controls are for time-specific, firm-specific, industry-specific, and region-specific factors.

While each of these methods estimates the relationship between carbon emissions and excesses and deficits in returns, all of these studies are correlation studies. Despite convincing heuristic arguments on why there should exist a carbon premium or why there should exist an equity greenium, the results do not strongly imply causality. For instance, several results find that higher carbon emissions are correlated with better stock returns. Since several variables have been controlled it is assumed that this surplus return cannot be attributed to any other factor. This, however, does not rule out the possibility of reverse causality, i.e. it is possible that firms that receive higher returns are laxer about pollution, and are less motivated to transition away from fossil fuels.

In theory, the only robust way to estimate whether an increase in a firm’s emissions yields higher returns for investors would be to compare it with the returns of the same firm in an alternate universe where the firm’s emissions have not increased, but such a counterfactual does not exist in the real world and researchers have to use proxies and approximations for counterfactuals. Santi and Moretti make an attempt at comparing firms with similar firms through propensity score matching (Santi and Moretti 2021). The authors use the propensity score which is the probability of getting assigned a particular kind of treatment (here, high emissions), and compare firms. This addresses a major concern of selection bias in the sample, i.e. which are the firms that received the treatment. Once matched, firms are compared with similar firms which did not receive the treatment. However, it would be crucial to note that propensity scores are typically used when the selection bias is attributed to observable characteristics, which in this study are assumed to be market-to-book value, market capitalization, capital expenditure, return on assets, and asset growth, while it is not necessary that these are the factors which govern the likelihood of a firm receiving treatment. Several other variables such as the industry that the firm belongs to or unobservable factors such as management decisions are unaccounted for in this method and may need other econometric methods to analyze (Tucker 2010).

While the empirical studies discussed so far succeed in studying differences in returns at present, a crucial question is to ask what the differences in returns will be in the future, i.e., the problem of quantifying expected returns with ex-post data. One method of calculating the expected returns is by extrapolating the current means, assuming no variation over time. Pastor et al. describe another approach (Pastor et al. 2022), where a tertiary variable that has a relationship with the returns, but has zero mean, is used to find the expected returns using ex-post data. In their study, Pastor et al. consider the variable $x_t$ which is the unanticipated change in climate concerns. Since the change is unanticipated, its expected value is 0. Thus, the returns can be written as,

$$\begin{array}{c}\hfill r_{i,t}={\mu }_i+\beta x_t+{ϵ}_{i,t},\end{array}$$ 7

where $r_{i,t}$ is the sample average return and ${\mu }_i$ is the unconditional expected return. Thus the estimate for ${\mu }_i$ can be found to be,

$$\begin{array}{c}\hfill \widehat{{\mu }_i}=\overline{r_{i,t}}-\widehat{\beta }\overline{x_t}.\end{array}$$ 8

This approach is useful as it tunes the expected returns with changes in $x_t$.

Another approach to quantifying long-term climate risk is through the recently introduced single-event framework (Nag et al. 2023). This approach attempts to integrate measures like the carbon premium and the time-horizon for when climate risk will materialize to develop a proxy measure for long-term carbon exposure. The single-event framework is an approximating technique that assumes that the pricing of carbon risk at some future time will occur over an infinitesimal period, and can be treated like a single shock that will remove the value associated with carbon emissions. Based on the assumption that the carbon premium is fairly priced in real-world markets, and under the no-arbitrage assumption, the authors derive the following relationship between long-term carbon risk and contemporary estimates of carbon premium:

$$\begin{array}{c}\hfill \underset{t_0}{\overset{\infty }{\int }}f(t)\underset{t_0}{\overset{t}{\int }}p(s)dsdt=\underset{t_0}{\overset{\infty }{\int }}\varphi (t)f(t)dt,\end{array}$$ 9

where p(s) is the carbon premium measured at time s, $\varphi (t)$ is the long-term carbon risk or the change in the value of the stock if the transition occurs at time t, and f(t) is the probability density distribution for the time of occurrence of the transition. Here the left-hand side of the equations quantifies the expected cumulative earnings from risk premia earned until the time of transition averaged over all transition scenarios, and the right-hand side is the expected losses incurred at the time of transition averaged over all transition scenarios. Based on this no-arbitrage condition, and based on the mathematical form of the premium as a function of time (constant, exponentially growing, etc.) an estimate of the long-term risk can be arrived at.

Hedging climate risk: portfolio methods

Efforts at mitigating financial risk, emanating from climate change, can be primarily classified into two categories, namely, pure-play portfolios and decarbonized portfolios (Andersson et al. 2016). While the former pertains solely to renewable energy and clean technology, along with an environment-friendly service setup, the latter is driven by an active process of decarbonization. The basic premise of decarbonization has its genesis in the asymmetric concentration of the carbon risk, to a handful of firms (Koch and Bassen 2013), the under-pricing of carbon risk, and the likely better financial returns on a decarbonized index (as compared to a benchmark index), once the market will have a mechanism in place, for pricing carbon risk. There are several approaches to the optimization of a decarbonized index, including elimination or significant reduction of stocks with high carbon footprint firms, either through a dynamic process or at the onset itself. To this end, the authors (Andersson et al. 2016) present two alternative portfolio optimization formulations. For the first approach, a benchmark index with n stocks is considered, with the weight

$$\begin{array}{c}\hfill w_i^{(b)}:=\frac{\text{Market cap of the}i\text{-th asset}}{\text{Total market cap of all}n\text{assets}},\end{array}$$ 10

for $i=1,2,3,\cdots ,n$. Let the carbon intensity of these n firms be arranged in descending order (highest to lowest), with the intensity being $c_i^{(m)}$, where $m=1$ and $m=N$, denote the highest and the lowest carbon intensity respectively. Let the weights of the green decarbonized economy for the i-th stock be denoted by $w_i^{(g)}$. Then the minimization of the Tracking Error (TE) is mathematically represented as,

$$\begin{array}{c}\hfill minTE=\text{Std.Dev}\left(r^{(g)},-,r^{(b)}\right).\end{array}$$ 11

If the worst k carbon intensity firms are first excluded from the green portfolio, and the weights of the remaining firm (to be included in the green index) are re-scaled (to add up to 1), then,

$$\begin{array}{cc}\hfill w_j^{(g)}=& 0,\text{for}j=1,2,\cdots ,k,\hfill \\ \hfill w_j^{(g)}\ge & 0,\text{for}j=k+1,k+2,\cdots ,n.\hfill \end{array}$$

Also “Std.Dev” donates the standard deviation. Another approach would be to (initially) replace the constraint,

$$\begin{array}{c}\hfill w_j^{(g)}=0,\forall j=1,2,\cdots ,k,\end{array}$$ 12

by the condition,

$$\begin{array}{c}\hfill \sum _{m=1}^n{c}_i^{(m)}{w}_i^{(g)}\le C,\end{array}$$ 13

that is, the carbon intensity of the portfolio is not permitted to exceed the threshold of C. As one observes, from the preceding discussion, a key driver in the formulation of both approaches, adopted for the optimization of a green portfolio, is the carbon intensity of each of the firms being considered for inclusion in the green portfolio. This approach has also been used in the context of palm oil companies (Ahmadi and Peivandizadeh 2022) where firms have been ranked according to sustainable production and other practices and the mean-variance portfolio is then constructed for the best k firms.

Another study (Acerbi 2022) uses the mean-variance approach but instead of excluding firms specifically, it sets a constraint on the carbon risk and then considers a three-dimensional optimization strategy with mean, variance, and carbon risk. The constraint for carbon risk is written as

$$\begin{array}{c}\hfill \sum _{i=1}^n{w}_i{C}_i\le C^{+},\end{array}$$ 14

where $w_i$ are the weights, $C_i$ is the firm-level carbon risks and $C^{+}$ is the maximum permissible carbon risk of the portfolio. Acerbi also uses the CAPM approach (Acerbi 2022) and proceeded with an investigation of the efficient frontier. The author used carbon-related constraints - for three risk measures - carbon emissions, carbon intensity, and carbon beta. Based on the mean-variance framework described above, Acerbi found the efficient frontier for an array of $C^{+}$ values, and using the value of $C^{+}$ the weights were found, and the means and variances were calculated.

While the studies reviewed so far use the conventional mean-variance approach, Roston uses a different optimization strategy (Roston 2021) based on Markowitz’s mean-variance portfolio. The idea behind mean-variance optimization is that investors can never eliminate risk, and therefore for a given level of risk, investors should try and maximize the possible returns. This can be stated conversely as, for a given level of desired returns, investors should minimize the risk. This is extended to the problem of hedging climate risk, where instead of a desired level of returns, the authors fix a desired level of carbon reduction from the portfolio, and for the given level of carbon intensity reduction ($\mathrm{\Delta }CI$), the investor tries to minimize the tracking error (TE). Mathematically, the optimization statement can be written as

$$\begin{array}{c}\hfill max\mathrm{\Delta }CI-TE.\end{array}$$ 15

The broader consideration of the problem of sustainable portfolio management is studied in the paradigm of the risk-adjusted performance of three sectors with high carbon intensity, namely, energy, utility, and materials, which is presented in Fang et al. (2019). We begin with the observation that the carbon intensity of a firm, is a key indicator of carbon risk exposure, for investors. It may be noted that the carbon intensity may be revenue-based or profit based, in the context of portfolio management. If Q, R, and P denote the annual emissions (in metric tons), annual revenue, and annual profits, respectively, then the revenue based and the profit-based carbon intensities are respectively defined as,

$$\begin{array}{c}\hfill C_R:=\frac{E(Q)}{E(R)}\text{and}{C}_P:=\left\{\begin{array}{c}\frac{E(Q)}{E(P)},E(P)>0,\hfill \\ \text{Undefined},E(P)\le 0,\hfill \end{array}\right)\end{array}$$ 16

where the expected values (denoted by $E(·)$) are estimated based on the historical data. For a portfolio of n assets, with weights, $w_i,i=1,2,\cdots ,n$, the revenue-based, and profit-based carbon intensities are given by,

$$\begin{array}{c}\hfill C_R^{(p)}:=\sum _{i=1}^n{w}_i{C}_{R,i}\text{and}{C}_P^{(p)}:=\sum _{i=1}^n{w}_i{C}_{P,i},\end{array}$$ 17

where $C_{R,i}$ and $C_{P,i}$ are the revenue and portfolio based carbon intensity of firm i. The authors formulate the Markowitz portfolio optimization problem by taking into account the stranded asset (assets where returns may fall below expectation, due to climate regulation) risk. For this purpose, let A denote the fair value of the company’s assets and $\alpha$ the standing ratio (the explicit form for $\alpha$ is available in the paper). Let ${\displaystyle {\overrightarrow{w}}_{n\times 1}}$, ${\displaystyle {\overrightarrow{\mu }}_{n\times 1}}$ and ${\displaystyle {\mathrm{\Sigma }}_{n\times n}}$ denote the weighted vector, the expected return vector, and the covariance matrix, respectively. Further, if ${\displaystyle {\overrightarrow{1}}_{n\times 1}}$ denotes the unit vector and ${\mu }^{\ast }$ is the target return of the green portfolio, then the sustainable portfolio management problem can be presented as,

$$\begin{array}{c}\hfill \underset{\overrightarrow{w}}{min}{\overrightarrow{w}}^{\top }\mathrm{\Sigma }\overrightarrow{w},\end{array}$$

such that,

$$\begin{array}{c}\hfill {\overrightarrow{w}}^{\top }\overrightarrow{\mu }={\mu }^{\ast },{\overrightarrow{w}}^{\top }\overrightarrow{1}=1\text{and}A\overrightarrow{w}\le a,\end{array}$$ 18

with the last inequality representing an appropriate constraint.

The next natural progression to the Markowitz approach in sustainable portfolio optimization is the consideration of the efficient frontier, which was addressed in a recent article on the ESG framework, the motivation for which stems from the fact that there are very few formal guidelines on the incorporation of ESG by portfolio managers. In this context, the authors (Pedersen et al. 2021) seek to reconcile the costs, with the benefits of investing in ESG instruments. Accordingly, a portfolio optimization approach for ESG, where the Markowitz setup meets the goals of sustainable finance is set up. The problem is the choice of a portfolio of n risky assets and one risk-free asset. Let $\overrightarrow{r}={\left(r_1,,,r_2,,,\cdots ,,,r_n\right)}^{\top }$ and $\overrightarrow{s}={\left(s_1,,,s_2,,,\cdots ,,,s_n\right)}^{\top }$ be the return vector of the n risky assets, and the corresponding ESG score vector, with $r_f$ being the risk-free rate. The investors are divided into exhaustive categories of Unaware/Uninterested (U) in ESG scores, Aware(A) and using ESG scores, and motivated (M) using ESG scores. While the investment solutions for the categories of U and A can be determined in the traditional Markowitz setup, the M segment of investors is the focus of the work. Accordingly, we consider, for the investor M, the portfolio with the weight vector $\overrightarrow{w}={\left(w_1,,,w_2,,,\cdots ,,,w_n\right)}^{\top }$ for the risky assets. Using the standard mean-variance approach, the tangency portfolio is the one that maximizes the Sharpe Ratio, while considering the ESG score level. The ESG Shape Ratio frontier, with an ESG score of $\tilde{\overrightarrow{s}}$ is defined as,

$$\begin{array}{c}\hfill \underset{\overrightarrow{w}}{max}\left(\frac{{\overrightarrow{w}}^{\top }\overrightarrow{\mu }}{\sqrt{{\overrightarrow{w}}^{\top }\mathrm{\Sigma }\overrightarrow{w}}}\right),\end{array}$$ 19

such that,

$$\begin{array}{c}\hfill {\overrightarrow{w}}^{\top }\overrightarrow{1}=1\text{and}{\overrightarrow{w}}^{\top }\overrightarrow{S}=\tilde{\overrightarrow{s}}.\end{array}$$ 20

In a recent preprint (Roncalli et al. 2021), the (post-2008 crisis) resorting to minimum variance strategies, to achieve a reduction of market risk on equity investment, is emphasized. The renewed interest in this approach has, in turn, led to the consideration of idiosyncratic valuation risk and reputational risk (among others). The quantification of the carbon risk measurement is based on the Carima approach, wherein the BGS is defined as a function of the value chain and public perception, in the case of each firm, with higher values of BGS indicating the firm is browner. Also considered is the Brown Minus Green (BMG) factor, which follows the methodology of Fama and French. One drawback of the Carima approach is the complexity, resulting from the 55 variables and 4 ESG databases being used in the exercise, and accordingly, several proxies to this were proposed. Another metric (or indicator) considered, was the carbon beta, whose estimation is based on several risk factors, and through the usages of both static and dynamic approaches. The CAPM-based formula presented in Roncalli et al. (2021) for returns of the i-th asset is given by,

$$\begin{array}{c}\hfill r_i(t)={\alpha }_i+{\beta }_{M,i}{r}_M(t)+{\beta }_{BMG,i}{r}_{\mathit{BMG}}(t)+{ϵ}_i(t),\end{array}$$ 21

where ${\alpha }_i$, ${\beta }_{M,i}$, $r_{M,i}$, ${\beta }_{BMG,i}$, $r_{BMG,i}$ and ${ϵ}_i$ are alpha, beta, returns, carbon-beta, carbon-returns, and idiosyncratic risk, respectively. With these returns, the global minimum variance portfolio is given by,

$$\begin{array}{c}\hfill min{\overrightarrow{w}}^{\top }\mathrm{\Sigma }\overrightarrow{w},\text{such that}{\overrightarrow{w}}^{\top }\overrightarrow{1}=1.\end{array}$$ 22

Assuming the independence of $r_M(t)$, $r_{\mathit{BMG}}(t)$ and ${ϵ}_i(t)$, Roncalli et al. (2021) obtained the minimum m variance portfolio as,

$$\begin{array}{c}\hfill w_i^{\ast }=\frac{{\sigma }^2\left({\overrightarrow{w}}^{\ast }\right)}{{\tilde{\sigma }}_i^2}\left(1,-,\frac{B_{M,i}}{B_M^{\ast }},-,\frac{B_{BMG,i}}{B_{\mathit{BMG}}^{\ast }}\right),\end{array}$$ 23

where ${\sigma }^2({\overrightarrow{w}}^{\ast }),{\tilde{\sigma }}_i^2,{\beta }_M^{\ast }$ and ${\beta }_{\mathit{BMG}}^{\ast }$ denote the minimum variance, the variance of idiosyncratic risk, the threshold of market beta and threshold of carbon beta, respectively.

Sustainable investment strategies taking into account ESG criteria in equilibrium are studied in a recent publication (Pástor et al. 2021). The model allows for a spectrum of different shades of green and brown firms. The three-fund separation setup accommodates investor hedging in a risk-free asset, the market portfolio, and an ESG portfolio (whose constituent assets are chosen by virtue of the greenness of the asset). The model is considered for a single period, in the case of a portfolio of n assets. Let $g_i,i=1,2,\cdots ,n$ be the observable “ESG” characteristics, which are positive (negative) in the case of green (brown) firms (equivalently, clean impact and polluting impact, respectively, on the environment). As an illustrative case, the authors choose an exponential utility, in the case of the investors. Let $\overrightarrow{g}={(g_1,\cdots ,g_n)}^{\top }$ and the market portfolio $\overrightarrow{w_M}={(w_{M_1},\cdots ,w_{M_n})}^{\top }$. Accordingly, ${\displaystyle {\overrightarrow{w_M}}^{\top }\overrightarrow{g}>0(<0)}$ signifies a net green (brown) market. However, for brevity, it is assumed that the market portfolio is ESG neutral i.e., ${\overrightarrow{w_M}}^{\top }g=0$. Let the vector of market return and market betas be denoted by $\overrightarrow{{\mu }_M}$ and $\overrightarrow{{\beta }_M}$, respectively. The corresponding vectors in case of the ESG portfolios are denoted by $\overrightarrow{{\mu }_g}$ and $\overrightarrow{{\beta }_g}$, respectively. Then it is established that the expected excess return (in a state equilibrium) is given by,

$$\begin{array}{c}\hfill \mu ={\overrightarrow{{\mu }_m}}^{\top }\overrightarrow{{\beta }_m}+{\overrightarrow{{\mu }_g}}^{\top }\overrightarrow{{\beta }_g},\end{array}$$ 24

which follows from the regression model,

$$\begin{array}{c}\hfill \tilde{r}={\tilde{r}}_m{\tilde{\beta }}_m+{\tilde{r}}_g{\tilde{\beta }}_g+\widetilde{ϵ},\end{array}$$ 25

where for the idiosyncratic risk,

$$\begin{array}{c}\hfill E(\widetilde{ϵ}|{\tilde{r}}_m,{\tilde{\beta }}_m,{\tilde{r}}_g,{\tilde{\beta }}_g)=0.\end{array}$$ 26

Now, this two-factor model can be rewritten as,

$$\begin{array}{c}\hfill \tilde{r}={\tilde{r}}_{\mu }{\tilde{\beta }}_m+\tilde{g}{\tilde{f}}_g+\widetilde{ϵ},\end{array}$$ 27

where ${\tilde{f}}_g$ is the excess return emanating from the ESG portfolio.

In another recent paper, the authors (Roncalli et al. 2020) developed the BMG risk factor for carbon risk, and its estimation of the carbon beta, based on a multi-factor model, followed by the management of carbon risk for investment portfolios. The formulation of the BMG factor, hinges on climate-pertinent data sets, through a scoring system-based declassification of firms into green, neutral or brown, followed by a portfolio setup (with long and short positions in brown and green firms, respectively). The BGS is taken as a function of Value Chain, Public Perception, and Non-Adaptability. The return of the BMG $r_{\mathit{BMG}}$, introduced earlier, is defined in terms of (up to six) combinations of small and big firms, with green, neutral, and brown classifications. A major desirable aspect of the BMG factor is the fact that BGS is determined, based on multiple data sets, thereby reducing biases. The most generalized model considered is the five-factor (including BMG) model, as follows,

$$\begin{array}{cc}\hfill r_i(t)=& {\alpha }_i+{\beta }_{M,i}{r}_M(t)+{\beta }_{BMG,i}{r}_{\mathit{BMG}}(t)\hfill \\ \hfill & +{\beta }_{SMB,i}{r}_{\mathit{SMB}}(t)+{\beta }_{HML,i}{r}_{\mathit{HML}}(t)+{\beta }_{WML,i}{r}_{\mathit{WML}}(t)+{ϵ}_i(t),\hfill \end{array}$$ 28

where SMB, HML, and WML represent the size, value, and momentum factors, respectively. For a simpler model involving only $r_M$ and $r_{\mathit{BMG}}$ being time-dependent, $({\alpha }_i(t),{\beta }_{M,i}(t),{\beta }_{BMG,i}(t))$ is assumed to follow a random walk process given by,

$$\begin{array}{c}\hfill ({\alpha }_i(t),{\beta }_{M,i}(t),{\beta }_{BMG,i}(t))=({\alpha }_i(t-1),{\beta }_{M,i}(t-1),{\beta }_{BMG,i}(t-1))+\overrightarrow{\xi }(t),\end{array}$$ 29

where $\overrightarrow{\xi }(t)=({\xi }_{i,1}(t),{\xi }_{i,2}(t),{\xi }_{i,2}(t))$ follows a multivariate normal distribution with a zero mean vector.

Discussion

Regardless of whether one believes in climate change or not, it is futile to deny that there is a growing consensus around the world that accelerated climate change and its consequences may significantly affect human lives and livelihoods for the worse. Since markets are often driven by sentiment, the interest of academics and investors in climate-associated risks is hardly surprising. In this paper, we have reviewed 41 papers in total and described the mathematical models that are used in answering questions related to the market pricing of carbon risk, and portfolio optimization techniques to hedge against it. At this juncture it is important to note that there are a few caveats to our results, for instance, this is intended to be a review of the literature on methods, and therefore a detailed review and analysis of results including geographical bias and the interaction with policy and regulation is beyond the scope of this work. Furthermore, stock prices are not the only indicators of financial risks in markets. Several studies that deal with climate risks in other financial instruments such as bonds (Baker et al. 2018) or options (Ilhan et al. 2021), could have contributed overall to the general understanding of climate risks in markets, but have not been reviewed as our study is focused on stock returns. However, considering stock returns alone, our review yields interesting results nonetheless.

We find a disparity in the two sets of literature that we have surveyed. While the literature on risk measures and estimates is vast, yet still growing, the literature on hedging climate risk is scarcer, and the area continues to be nascent more than seven years after the Paris agreement. However, our findings in this review need not be representative of the state of things in the industry. Since our review predominantly comprises academic work, we cannot comment on unpublished strategies to hedge climate risk that are used by industry experts. Furthermore, most of the papers reviewed on risk measures were models designed for empirical analysis. The literature on carbon premium is so vast because it is an empirical problem that needs to be studied by different methods over different data sets in different countries and different markets. The hedging problem is more theoretical, and since the transition away from fossil fuels is still impending, it is difficult to test the strategies empirically, which may cause difficulties in developing and verifying strategies.

In the context of risk measures, despite the literature being vast, the number of models is not as large, with a handful of models showing up repeatedly. This need not necessarily be a problem, and it is entirely possible that existing models adequately describe the data. But the models themselves often do not agree with one another and different data sets sometimes yield different results even with the usage of similar models. There is little consensus on the existence of carbon premia. Among the better-cited works, some authors (Oestreich and Tsiakas 2015; Bolton and Kacperczyk 2021a) find a significant premium, but others (Görgen et al. 2020) find no significant (weakly negative) premium, and some researchers (Pastor et al. 2022) find a significant negative premium. In our sample, 33.33% of the studies that conducted some empirical analysis found a positive carbon premium, while 25.92 % found a negative carbon premium. 40.74% either found mixed results or no premium, neither positive nor negative. This demonstrates that the carbon premium itself is sensitive to both data and the methods used and the results available today need not be robust. Figure 1 shows the mosaic plot for FFE and LRM models. Both models have an approximately equal fraction of positive estimates of the carbon premium but FFE models also find negative premia and mixed results while LRM studies do not. This may also be because our sample of studies includes only 6 LRM studies but 20 FFE studies.

Fig. 1.

Mosaic Plot of Models and Results for Empirical Estimation: We observe that close to a third of both studies using Fama-French Extended (FFE) models and studies using Linear Risk Models (LRM) estimate positive carbon premia. However, an equal number of FFE studies find negative premia but no LRM study finds a negative premium. In fact, two-thirds of LRM studies find no premium

Moreover, very few existing studies use methods that check for causality. In the context of the existence of a carbon premium, it is very important to establish that excess emissions causally impact stock returns, either for the positive or the negative, and the literature would benefit from works that include rigorous econometric techniques to check for causality. Furthermore, there is a significant geographical bias in the existing literature with very few empirical studies in the Global South. Most of the literature is still restricted to The European Union and The United States of America, and a few studies focus on China or other parts of Asia. The models have thus not been tested on the emerging markets of Asia, Africa, and Latin America, and thus there is still considerable scope in empirical research on the carbon premium around the World.

The need to develop appropriate causal models and reach a consensus on the existence of a carbon premium or an equity greenium is also particularly important from the point of view of the regulator or the policymaker. When policy is made to control carbon emissions, such as the introduction of carbon taxes, the primary motive behind such regulation is to reduce carbon emissions of firms by making it expensive and inefficient to do business which harms the environment, thus reducing investments into existing and future businesses of the kind. The existence of an equity greenium is evidence for the policymaker that investor interest in polluting stocks is reducing. On the other hand, no premium implies that the policy made little difference to investor behavior while dealing with polluting stocks. The existence of the carbon premium is somewhere in between, where investors are still willing to invest in polluting firms, as long as the firms compensate with some level of higher returns. Thus policymakers, by assessing the existence and nature of the premium in the market, can gauge the effectiveness of their policies and stay with them or introduce new policies if need be.

Coming to portfolio approaches, Markowitz’s mean-variance analysis and the Capital Asset Pricing Model have been the fundamental pillars of modern portfolio theory, which has stood the test of time, for the last seven decades. Accordingly, upon identification of climate change as a growing risk, for the financial markets, it was natural that climate risk would be examined in the paradigm of modern portfolio theory, by building on the two aforesaid traditional theoretical frameworks. In our review, we find that some studies incorporate these frameworks directly (Andersson et al. 2016), with almost no methodological difference to the core optimization problem, while others depart (Roston 2021) from the traditional framework to incorporate climate risk.

Hedging climate risk predominantly has two parts, namely, choosing an appropriate framework and choosing suitable risk-return quantifiers. For the former, by design, our review is restricted to only mean-variance optimization and CAPM approaches. Within these restrictions, we find that the authors typically choose three direct measures for the risk quantifiers, namely, carbon emissions or intensity, ESG scores, and BMG scores. Only one paper used Carbon Beta as a risk factor. Carbon emissions and carbon intensity are absolute measures and deal with one facet of the transition risk. Since all climate risk cannot be broken down into carbon emissions (Randalls 2011), therefore it may be useful to work with more composite measures and indices. This is where the choice of ESG and BGM scores come in. In general, these scores can potentially be more comprehensive in accounting for non-emission-related transition risks. However, even these scores suffer a similar drawback as with absolute emissions or emissions intensity, which is that the models assume that the long-term carbon-related risk is a linear function of these quantities. While such an assumption is simple and parsimonious, the existence of more compound measures such as the carbon premium (Bolton and Kacperczyk 2021a), the equity greenium (Pastor et al. 2022), or the single-event transition risk (Nag et al. 2023) allows the incorporation of more direct risk factors which are supposed to be the market’s quantifier of risk.

This points to a new direction of research in climate finance which bridges the gaps between the development of risk measures and formulating hedging strategies. With an evergrowing consensus on impending climate change, the coming few years could potentially see some serious policies from governments around the world in the direction of climate transition. It is, therefore, in the best interests of investors and citizens of the world that the global economy divests from fossil fuels. Therefore the development of suitable risk measures and formulating strategies based on those measures are going to be instrumental in mapping the route to the transition of the economy so that investors can transition away from fossil fuels without making significant losses. It is also entirely possible that new research in the area will show that investors could make more profits by transitioning and thus bring about a new paradigm in finance and asset management.

Another change that could bring about an improvement in the quality and robustness of studies is the better availability of ESG data and disclosures. A significant fraction of financial data, including environmental data, especially in an organized format is behind various paywalls, which restricts research and a better understanding of market dynamics. Furthermore, the literature on disclosure points to other problems such as an association with disclosure volume and quality (Saha et al. 2019) or the possibility of ESG risk being manipulated by institutional investors which could increase the risk (Farag et al. 2022). And the free access to data would improve the transparency of firm performance and lend considerable help to the burgeoning literature on understanding climate risk and in general the interaction between climate change, regulation, and markets.

Funding

No funding was received for this study, by any of the authors.

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Human participants or animals

This article does not contain any studies with human participants or animals performed by any of the authors.