Measuring and Managing Carbon Risk in Investment Portfolios
Théo Roncalli, Théo Le Guenedal, Frédéric Lepetit, Thierry Roncalli, Takaya Sekine
MMeasuring and Managing Carbon Risk inInvestment Portfolios ∗ Th´eo RoncalliQuantitative ResearchAmundi Asset Management, Paris [email protected]
Th´eo Le GuenedalQuantitative ResearchAmundi Asset Management, Paris [email protected]
Fr´ed´eric LepetitQuantitative ResearchAmundi Asset Management, Paris [email protected]
Thierry RoncalliQuantitative ResearchAmundi Asset Management, Paris [email protected]
Takaya SekineQuantitative ResearchAmundi Asset Management, Paris [email protected]
August 2020
Abstract
This article studies the impact of carbon risk on stock pricing. To address this, weconsider the seminal approach of G¨orgen et al. (2019), who proposed estimating thecarbon financial risk of equities by their carbon beta. To achieve this, the primary taskis to develop a brown-minus-green (or BMG) risk factor, similar to Fama and French(1992). Secondly, we must estimate the carbon beta using a multi-factor model. WhileG¨orgen et al. (2019) considered that the carbon beta is constant, we propose a time-varying estimation model to assess the dynamics of the carbon risk. Moreover, we testseveral specifications of the BMG factor to understand which climate change-relateddimensions are priced in by the stock market. In the second part of the article, wefocus on the carbon risk management of investment portfolios. First, we analyze howcarbon risk impacts the construction of a minimum variance portfolio. As the goal ofthis portfolio is to reduce unrewarded financial risks of an investment, incorporatingthe carbon risk into this approach fulfils this objective. Second, we propose a newframework for building enhanced index portfolios with a lower exposure to carbon riskthan capitalization-weighted stock indices. Finally, we explore how carbon sensitivitiescan improve the robustness of factor investing portfolios.
Keywords:
Carbon, climate change, risk factor, Kalman filter, minimum variance portfolio,enhanced index portfolio, factor investing.
JEL classification:
C61, G11. ∗ We are grateful to Martin Nerlinger from the University of Augsburg, who provided us with the timeseries of the BMG risk factor (see https://carima-project.de/en/downloads for more details about thiscarbon risk factor). We would also like to thank Melchior Dechelette and Bruno Taillardat for their helpfulcomments. a r X i v : . [ q -f i n . P M ] A ug easuring and Managing Carbon Risk in Investment Portfolios The general approach to managing the carbon risk of an investment portfolio is to reduceor control the portfolio’s carbon footprint, for instance by considering CO emissions. Thisapproach supposes that the carbon risk will materialize and having a portfolio with a lowerexposure to CO emissions will help to avoid some future losses. The main assumption ofthis approach is then to postulate that firms currently with high carbon footprints will bepenalized in the future in comparison with firms currently with low carbon footprints.In this article, we use an alternative approach. We define carbon risk from a financialpoint of view, and we consider that the carbon risk of equities corresponds to the market riskpriced in by the stock market. This carbon financial risk can be decomposed into a common(or systematic) risk factor and a specific (or idiosyncratic) risk factor. Since identifying thespecific risk is impossible, we focus on the common risk factor that drives the carbon risk.The objective is then to build a market-based carbon risk measure to manage this marketrisk in investment portfolios. This is exactly the framework proposed by G¨orgen et al. (2019)in their seminal paper.G¨orgen et al. (2019) proposed extending the Fama-French-Carhart model by including abrown-minus-green (or BMG) risk factor. Using the sorted portfolios technique popularizedby Fama and French (1992), they build a factor-mimicking portfolio based on a scoringmodel and more than fifty carbon risk variables. They then defined the carbon financial riskof a stock using its price sensitivity to the BMG factor or its carbon beta. In this paper, weexplore the original approach of G¨orgen et al. (2019) and estimate a time-varying model inorder to analyze the dynamics of the carbon risk. Moreover, we make the distinction betweenrelative and absolute carbon risk. Relative carbon risk may be viewed as an extension orforward-looking measure of the carbon footprint, where the objective is to be more exposedto green firms than to brown firms. In this case, this is equivalent to promoting stocks with anegative carbon beta over stocks with a positive carbon beta. Absolute carbon risk considersthat both large positive and negative carbon beta values incur a financial risk that must bereduced. This is an agnostic or neutral method, contrary to the first method which is morerelated to investors’ moral values. In this paper, an important issue concerns the climatechange-related dimensions that are priced in by the financial market. According to Delmas et al. (2013), the concept of environmental performance encompasses several dimensionsbut there is no consensus on universally accepted environmental performance indicators.To address this issue, we compare pricing models with different criteria: current carbonfootprint, carbon management, climate change score and environmental risk.The carbon beta of a stock can be interpreted as its carbon-related systematic risk.Therefore, it contains financial information that is extremely useful from a trading pointof view. In particular, it can be used to improve the construction of a minimum varianceportfolio, the main goal of which is to avoid unrewarded risks. It can also be used in theinvestment scope of enhanced indexing or factor investing. For these different illustrations,we develop an analytical framework to better understand the impact of carbon betas.This paper is organized as follows. Section Two presents the seminal approach of G¨orgen et al. (2019), and reviews the pricing impact of the carbon risk factor. Besides the staticanalysis, we also consider a dynamic approach where the carbon beta is estimated usingthe Kalman filter. Then we test the contribution of the different climate-change relateddimensions. Section Three is dedicated to investment portfolio management considering theinformation deduced from the carbon beta values. First, we focus on the minimum varianceportfolio, before extending the analytical results to enhanced index portfolios and explaininghow carbon betas can also be used in a factor investing framework. Finally, Section Fouroffers some concluding remarks. 2easuring and Managing Carbon Risk in Investment Portfolios To manage a portfolio’s carbon risk, it is important to measure carbon risk at the companylevel. There are different ways to measure this risk, including the fundamental and marketapproaches. In this paper, we will favor the second approach because it provides a betterassessment of the impact of climate-related transition risks on each company’s stock price.Moreover, the market-based approach allows us to mitigate the issue of a lack of climatechange-relevant information. In what follows, we present this latest approach by using themimicking portfolio for carbon risk developed by G¨orgen et al. (2019). We also discuss thedifferent climate change-relevant dimensions to determine which dimensions are priced in bythe market.
The goal of the carbon risk management (Carima) project, developed by G¨orgen et al. (2019), is to develop ‘ a quantitative tool in order to assess the opportunities of profits andthe risks of losses that occur from the transition process ’. The Carima approach combines amarket-based approach and a fundamental approach. Indeed, the carbon risk of a firm or aportfolio is measured by considering the dynamics of stock prices which are partly determinedby climate policies and transition processes towards a green economy. Nevertheless, a priorfundamental approach is important to quantify carbon risk. In a practical manner, thefundamental approach consists in defining a carbon risk score for each stock of a universeusing a set of objective measures, whereas the market approach consists in building a brownminus green or BMG carbon risk factor, and computing the risk sensitivity of stock priceswith respect to this BMG factor. Therefore, the carbon factor is derived from climatechange-relevant information from numerous firms.
The development of the BMG factor is based on a large amount of climate-relevant informa-tion provided by different databases. In the following, we report the methodology used bythe Carima project to construct the BMG factor and thereby obtain a deeper understand-ing of the results . Two steps are required to develop this new common risk factor: (1) thedevelopment of a scoring system to determine if a firm is green, neutral or brown and (2)the construction of a mimicking factor portfolio for carbon risk which has a long exposureto brown firms and a short exposure to green firms.The first step consists in defining a brown green score (BGS) with a fundamental ap-proach to assess the carbon risk of different firms. This scoring system uses four ESGdatabases over the period from 2010 to 2016: Thomson Reuters ESG, MSCI ESG Ratings,Sustainalytics ESG ratings and the Carbon Disclosure Project (CDP) climate change ques-tionnaire. Overall, 55 carbon risk proxy variables are retained . Then, G¨orgen et al. (2019)classified the variables into three different dimensions that may affect the stock value of afirm in the event of unexpected shifts towards a low carbon economy:1. Value chain (impact of a climate policy or a cap and trade system on the differentactivities of a firm: inbound logistics and supplier chain, manufacturing production,sales, etc.); A more exhaustive presentation is available in the Carima manual, which can be downloaded at thefollowing address: https://carima-project.de/downloads . The governance and social variables of a traditional ESG analysis or even certain environmental variablessuch as waste recycling, water consumption or toxic emissions have been deleted. value chain
VC, the publicperception
PP and the non-adaptability
NA. It follows that each score has a range between 0and 1. G¨orgen et al. (2019) proposed defining the brown green score (BGS) by the followingequation:BGS i ( t ) = 23 (cid:0) . · VC i ( t ) + 0 . · PP i ( t ) (cid:1) + NA i ( t )3 (cid:0) . · VC i ( t ) + 0 . · PP i ( t ) (cid:1) (1)The higher the BGS value, the browner the firm. The value chain and public perception axesdirectly influence stock prices in the case of unexpected changes in the transition process.However, G¨orgen et al. (2019) considered that the impact of the value chain score is moreimportant than the impact of the public perception score. The adaptability axis influencesthe equity value in a different way. Indeed, it mitigates the upward or downward impacts ofthe two other axes. The less adaptable a firm is, the greater the impact of an unexpectedacceleration in the transition process. In total, almost 1 650 firms are retained thanks tosufficient data covered.The second step consists in constructing a BMG carbon risk factor. Here the Carimaproject considers an average BGS for each stock that corresponds to the mean value of theBGS over the period in question, from 2010 to 2016. The construction of the BMG factorfollows the methodology of Fama and French (1992, 1993), which consists in splitting thestocks into six portfolios: Green Neutral BrownSmall SG SN SBBig BG BN BBwhere the classification is based on the terciles of the aggregating BGS and the medianmarket capitalization. Then, the return of the BMG factor is defined as follows: R bmg ( t ) = 12 (cid:0) R SB ( t ) + R BB ( t ) (cid:1) − (cid:0) R SG ( t ) + R BG ( t ) (cid:1) (2)where the returns of each portfolio is value-weighted by market capitalization. The BMGfactor can then be integrated as a new common risk factor into a multi-factor model. Somestatistical details are reported in Table 9 on page 51, whereas the historical cumulativeperformance of the BMG factor is showed in Figure 19 on page 52. According to the factordeveloped for the Carima project, brown firms slightly outperform green firms from 2010to the end of 2012. During the next three years, the cumulative return fell by almost 35%because of the unexpected path in the transition process towards a low carbon economy.From 2016 to the end of the study period, brown firms created a slight excess performance.Overall, the best-in-class green stocks outperform the worst-in-class green stocks over thestudy period with an annual return of 2 . Many advantages can be attributed to the BMG factor. Some biases in the construction ofESG databases are offset since the BGS scores are derived from several databases. Moreover,the tests performed by G¨orgen et al. (2019) showed that there are no significant country-specific or sector-specific effects . Even though the BMG factor has many benefits, it can besubject to some disadvantages, starting with the treatment of variables. The transformationof continuous and discrete variables into a dummy variable with respect to the median valuefixes the problem of extreme values, but does not differentiate between values based on theirdistance from the median. Besides, the most important problem is that no rebalancing takesplace. Some tests performed by G¨orgen et al. (2019) showed that less than five percent offirms shifted between the green, neutral and brown portfolios during the study period butsuch a decision presents some consistency problems in the long-run. For instance, the resultsobtained by the average BGS score for the 2010-2016 period have been generalized for thefollowing two years.Another limit involves the size-specific effects in the BMG factor. Table 1 reports thecorrelation matrix of common risk factors during the sample period. While the value (HML)and momentum (WML) factors are not significantly correlated to the size (SMB) factor, theBMG factor is influenced by size characteristics. Mitigating this problem can be difficultsince the carbon risk factor has been derived from the methodology of Fama and French(1992). The most plausible explanation of this correlation is that among the studied firms,the green firms have the largest market capitalizations as we can see in Figure 28 on page 56.In this case, when big firms outperform small firms, both the SMB and BMG factor returnsdecrease. Furthermore, preventing the BMG factor from capturing size-specific effect is animportant but difficult matter to solve.Table 1: Correlation matrix of factor returns (in %)Factor MKT SMB HML WML BMGMKT 100 . ∗∗∗ SMB 1 .
41 100 . ∗∗∗ HML 11 . − .
93 100 . ∗∗∗ WML − .
59 3 . − . ∗∗∗ . ∗∗∗ BMG 5 .
33 20 . ∗∗ . ∗∗∗ − . ∗∗ . ∗∗∗ Source : G¨orgen et al. (2019).
Selecting numerous variables allows us to avoid some important dependencies on a vari-able and incorporate a lot of climate change-relevant information. Nevertheless, we havedouble counting problems. For instance, the carbon emissions score and the climate changetheme score in the MSCI ESG Ratings are both taken into account when developing theBGS score but the carbon emissions score is integrated into the climate change theme score.Moreover, some variables in the public perception dimension are not exclusive to the car-bon risk dimension, such as the ESG score developed by Sustainalytics ESG ratings or theIndustry-adjusted Overall score developed by MSCI ESG Ratings.
The first carbon risk objective is to assess the relevance of the BMG factor during thestudy period. To do this, we follow the analysis of G¨orgen et al. (2019), but our analysis is In the following, we will find some sector-specific effects for short periods. et al. (2019) considereda universe of 39 500 stocks, whereas we only consider the stocks that were present in theMSCI World index during the 2010-2018 period. As a result, our investment universe hasless than 2 000 stocks, but we think that a restricted universe makes more sense than a verylarge universe. Indeed, the computation of a market beta is already difficult for some smalland micro stocks because of OTC pricing and low trading activity. Therefore, calculating acarbon beta is even more difficult for such equities.It may be worthwhile to compare different common factor models to measure the infor-mation gain related to the carbon risk factor. The first studied model is the CAPM modelintroduced by Sharpe (1964) which is defined by: R i ( t ) = α i + β mkt ,i R mkt ( t ) + ε i ( t ) (3)where R i ( t ) is the return of asset i , α i is the alpha of the asset i , R mkt ( t ) is the return ofthe market factor, β mkt ,i is the systematic risk (or the market beta) of stock i and ε i ( t )is the idiosyncratic risk. We may also consider that the risk is multi-dimensional with themodel developed by Fama and French (1992): R i ( t ) = α i + β mkt ,i R mkt ( t ) + β smb ,i R smb ( t ) + β hml ,i R hml ( t ) + ε i ( t ) (4)where R smb ( t ) is the return of the size (or small minus big) factor, β smb ,i is the SMBsensitivity (or the size beta) of stock i , R hml ( t ) is the return of the value (or high minus low)factor and β hml ,i is the HML sensitivity (or the value beta) of stock i . Nevertheless, thesetwo models do not include the carbon risk. Furthermore, we also consider the MKT+BMGmodel: R i ( t ) = α i + β mkt ,i R mkt ( t ) + β bmg ,i R bmg ( t ) + ε i ( t ) (5)and the extended Fama-French (FF+BMG) model: R i ( t ) = α i + β mkt ,i R mkt ( t ) + β smb ,i R smb ( t ) + β hml ,i R hml ( t ) + β bmg ,i R bmg ( t ) + ε i ( t ) (6)where R bmg ( t ) is the return of the carbon risk factor and β bmg ,i is the BMG sensitivityof stock i . Another well-known model is the four-factor model (4F) developed by Carhart(1997). This model corresponds to the following equation: R i ( t ) = α i + β mkt ,i R mkt ( t ) + β smb ,i R smb ( t ) + β hml ,i R hml ( t ) + β wml ,i R wml ( t ) + ε i ( t ) (7)where R wml ( t ) is the return of the momentum (or winners minus losers) factor and β wml ,i is the WML sensitivity of stock i . Again, we may include the carbon risk factor to obtain afive-factor (4F+BMG) model: R i ( t ) = α i + β mkt ,i R mkt ( t ) + β smb ,i R smb ( t ) + β hml ,i R hml ( t ) + β wml ,i R wml ( t ) + β bmg ,i R bmg ( t ) + ε i ( t ) (8)In minimum variance or enhanced index portfolios, we assume that the factor returns areuncorrelated.Risk factor model estimates were performed on single stocks during the 2010-2018 pe-riod . In Table 2, we have reported a comparison between the common factor models andtheir nested models by computing the average difference of the adjusted R and the pro-portion of stocks for which the Fisher test is significant at 10%, 5% and 1%. According to In this article, we only consider the stocks that were in the MSCI World index for at least three yearsduring the 2010-2018 period. Moreover, we do not consider the returns for the period during which the stockis outside the index. R F -testdifference 10% 5% 1%CAPM vs FF 1.74 34 . . . . . . . . . . . .
84F vs 4F+BMG 1.76 23 . . . obtained by G¨orgen et al. (2019),meaning that the carbon factor plays a key role in the variation of stock returns.Figure 1 reports the sector analysis of the carbon beta ˆ β bmg ,i estimated with Model(5). The box plots provide the median, the quartiles and the 5% and 95% quantiles of thecarbon beta. The energy, materials, real estate and, to a lesser extent, industrial sectorsare negatively impacted by an unexpected acceleration in the transition process towardsa green economy, certainly because these four sectors are responsible for a large part ofgreenhouse gas emissions (GHG). Indeed, the energy and the materials sectors have a largescope 1 mainly because of oil and gas drilling and refining for the former and the extractionand processing of raw materials for the latter . Overall, the energy sector is the mostsensitive to an unexpected acceleration in the transition process but the carbon beta rangeis widest for the materials sector, which indicates a high heterogenous risk for this sector.The latter is mostly influenced by the growth in material demand per capita. In terms ofthe industrial sector, construction and transport are responsible for much of global finalenergy consumption, which leads to a high carbon risk for this sector. In the real estatesector, the firms have a large scope 2 and energy efficiency can be improved in many cases.If we consider a long-run investment, a transition process that reduces climate change canprotect households from physical risks like climate hazards. Nevertheless, a short-run visionsupposes that a climate policy negatively impacts households that over-consume. Therefore,real estate investment trusts are highly sensitive to climate-related policies. One surprisingresult involves utility firms which do not have a substantial positive carbon beta whereastheir scope 1 is on average the largest of any sector. This overall neutral carbon sensitivityfor utilities is explained by their carbon emissions management and efforts to reduce carbonexposure. Indeed, Le Guenedal et al. (2020) have shown that utilities – power generationaccording to the Sectoral Decarbonization Approach (SDA) – have been aggressive in theirinflexion of carbon intensity trajectories.If we consider the sectors positively impacted by an unexpected shift towards a greeneconomy, these primarily include health care, information technology and consumer sta- See Table IA.2 in G¨orgen et al. (2019). Sector taxonomy is based on the Global Industry Classification Standard (GICS). Moreover, scope 3 of the basic materials sector is very large. -1 0 1 2 3 4EnergyMaterialsReal EstateIndustrialsUtilitiesCommunication ServicesConsumer DiscretionaryConsumer StaplesFinancialsInformation TechnologyHealth Care ples because of their low GHG emissions. Financials are also part of this group, but theinterpretation of the carbon risk differs. Indeed, the carbon risk of financial institutionsis less connected to their GHG emissions than their investments and financing programs.The greener a financial institution’s investment, the lower its carbon beta. The low valueof the median beta implies that financial firms integrate carbon risk into their investmentstrategies or that financials are not significantly disadvantaged by the relative carbon risk.
Remark 1.
These results are coherent, but slightly different from those obtained by G¨orgen et al. (2019). Certainly, this difference comes from the investment universe, which is moreliquid in our case. For instance, we obtain less high median carbon betas – except for theenergy and materials sectors – since our universe of stocks includes only the world’s biggestfirms.
We have also reported in Figure 20 on page 52 the box plots for four other investmentuniverses: Eurozone, Europe ex EMU, North America and Japan. The energy sector re-mains the most negatively impacted by an unexpected acceleration in the transition processregardless of the region under review. The integration of carbon risk in the Eurozone issubstantial, especially in the financial sector where green investments are widely taken intoaccount. The inclusion of this carbon risk is also significant in Europe ex EMU and Japan,whereas it is very mixed in North America. Concerning the real estate sector, BMG risk ishighly integrated in Europe ex EMU and slightly in North America because of their largeexposure to climate risks , while the integration of carbon risk is different in Japan becausereal estate investment trusts are more short-sighted despite the vulnerability of this sector . Almost all the companies with a negative carbon beta are English, while the one remaining with apositive carbon beta is Swiss. Real estate in United States is especially exposed to rising sea levels and hurricanes. Real estate in Japan is exposed to the physical risk of typhoons.
In this section, we suppose that the risks are time-varying. For instance, carbon risk mayevolve with the introduction of a climate-related policy, a firm’s environmental controversy, achange in the firm’s environmental strategy, a greater integration of carbon risk into portfoliostrategies, etc. Therefore, we use the following dynamic common factor model : R i ( t ) = R ( t ) (cid:62) β i ( t ) + ε i ( t ) (9)where R ( t ) = (cid:0) , R mkt ( t ) , R bmg ( t ) (cid:1) is the vector of factor returns, β i ( t ) is the vector offactor betas : β i ( t ) = α i ( t ) β mkt ,i ( t ) β bmg ,i ( t ) (10)and ε i ( t ) is a white noise. We assume that the state vector β i ( t ) follows a random walkprocess: β i ( t ) = β i ( t −
1) + η i ( t ) (11)where η i ( t ) ∼ N (cid:0) , Σ β,i (cid:1) is the white noise vector and Σ β,i is the covariance matrix ofthe white noise. Several specifications of Σ β,i may be used , but we assume that Σ β,i is adiagonal matrix in the following. As previously, the time-varying risk factor model is used onsingle stocks during the 2010-2018 period . Below, we provide the average of two forecasterror criteria between the OLS model and the SSM model:Model OLS SSMMAE 4 .
95% 4 . .
45% 6 . .
95% in the OLS model while it is equal to 4 .
63% inthe SSM model. Overall, the SSM model reduces the mean absolute error value of the lastobservation date by 12 .
17% with respect to the OLS model.In Table 3, we have reported the proportion of firms for which the t -student test of theestimation of the covariance matrix Σ β,i is significant at 10%, 5% and 1% confidence levels.We notice that the coefficients of the covariance matrix are significant for a substantialnumber of firms implying that between 10% and 15% of stocks present time-varying marketand carbon risks.In Figure 2, we have reported the variation of the average carbon beta by region .Whatever the study period, the carbon beta β bmg , R ( t ) is positive in North America, whichimplies that American stocks are negatively influenced by an acceleration in the transition The beta estimates are based on the state space model (SSM) and the Kalman filter algorithm describedin Appendix A.1 on page 41. In this model, we only consider the dynamics of market and carbon risks. We have also performed thesame analysis with the 4F+BMG model, but the results are noisier. For instance, we can assume that (cid:0) Σ β,i (cid:1) , is equal to zero, implying that the alpha coefficient α i ( t ) isconstant. As previously, we only consider the stocks that were in the MSCI World index for at least three yearsduring the 2010-2018 period and we take into account only the returns for the period during which the stockis in the MSCI World index. The average carbon beta β bmg , R ( t ) for the region R at time t is calculated as follows: β bmg , R ( t ) = (cid:80) i ∈R β bmg ,i ( t )card R α .
97 4 .
10 0 . β mkt .
95 10 .
22 3 . β bmg .
00 5 .
90 1 . . Overall, the Eurozone has always a lower average carbon beta than the worldas a whole, whereas the opposite is true for North America. Nevertheless, the sensitivityof European equity returns to carbon risk dramatically increases and the BMG betas aregetting closer for North America and the Eurozone. In Europe ex EMU, the BMG betais higher than in the Eurozone but their trends are very similar. Regarding the Japanesefirms, the trend has tracked the world as a whole since 2013 but with a lower carbon beta.We notice that the carbon risk is not driven by climate agreements in the short run. Forinstance, the 2030 climate and energy framework, which includes EU-wide targets and policyobjectives for the period from 2021 to 2030 does not influence the average European carbonbeta in 2014 certainly because of the lack of binding commitments. Another example is the2015 Paris Climate Agreement, which does not include fiscal pressure mechanisms. Becauseof the differences between expectations and constraints, the Paris Climate Agreement hasnot been followed by a significant increase in the carbon beta and has been concomitantwith the outperformance of brown stocks one quarter later . In February 2016, the globalincrease of the carbon beta is related to a sector-specific effect. Indeed, the materials sectorhas largely outperformed because of a substantial increase in gold, silver and zinc priceswhereas the market index has decreased. Furthermore, some firms in the materials sectorwere not driven by the market for a short period, implying a sharp increase in the carbonbeta. This explains that the carbon beta returns to its long-term trend some months later. Remark 2.
In terms of the results obtained, one issue concerns the increase in the averagecarbon beta in Europe, and we wonder if this is due to a geographical area effect or a carbonsensitivity specific effect – the firms with the most negative carbon betas may be less influ-enced by the BMG factor over time. To answer this question, we use the method of sortingportfolios, which has been popularized by Fama and French (1992). Every month, we rankthe stocks with respect to their carbon beta, and form five quintile portfolios. Portfolio Q corresponds to the lowest carbon beta stocks while Portfolio Q corresponds to the highest carbon beta stocks. The stocks are equally weighted in each portfolio and the portfoliosare rebalanced every month. In Figure 21 on page 53, we have reported the average carbonbeta of the five sorted portfolios for each region. Since European stocks are the most posi-tively impacted by an unexpected change in the transition process towards a green economy,we have to compare similar portfolios between the regions. For instance, Portfolio Q in theEurozone and Portfolio Q in North America started with an average carbon beta aroundminus one. In the Eurozone, the increase in the carbon beta for Portfolio Q is much higherthan the increase of Portfolio Q in North America. The Europe ex EMU region is anotherexample. Portfolio Q in this region is comparable with Portfolio Q in North America sincetheir average carbon betas started at a similar level. Nevertheless, the average carbon betaof Portfolio Q in Europe ex EMU increases while it decreases for Portfolio Q in NorthAmerica. We can deduce that the increase of the carbon beta in Europe is not due to asensitivity effect for the stocks that are the most negatively sensitive to the BMG factor, butto a geographical effect. In Japan, it is also negative most of the time. We recall that the performance of the brown minus green portfolio is given in Figure 19 on page 52. β bmg , R ( t ) by region Figure 3: Dynamics of the average absolute carbon risk | β | bmg , R ( t ) by region | β | bmg , R ( t ) foreach region R . The higher the value of | β | bmg , R ( t ), the greater the impact (positive ornegative) of carbon risk on stock returns. Curiously, we notice that the integration of carbonrisk in the financial market decreases over time. In particular, there is a substantial decreasein 2012 and then a stabilization of the global average absolute carbon beta . The Eurozonewas the region with the highest sensitivity to carbon risk but this decreased sharply by almost44% between 2010 and 2018. However, we observe all regions converging except Japan .The convergence of absolute sensitivities between large geographical regions indicates thatinvestors see carbon risk as a global issue.We may also be interested in carbon risk trends by sector. In the static analysis, werecall that the energy, materials and real estate sectors were the most negatively impactedby an unexpected acceleration in the transition process, whereas the opposite is true for thehealth care, information technology and consumer staples sectors. Figures 4 and 5 providethe trends in the median carbon beta β bmg , S ( t ) for the sector S at time t , which is definedas follows: β bmg , S ( t ) = median i ∈S β bmg ,i ( t )We distinguish four categories. The first one concerns high positively sensitive sectors tothe carbon factor. This includes only the energy sector. The stock price of the latteris increasingly negatively influenced by the movements of the carbon factor. The secondcategory includes the materials and real estate sectors for which the positive sensitivity ofstock price to carbon factor is much more moderate. The third category includes the sectorswith a neutral or a low negative sensitivity to the carbon factor. This category is madeup of the industrials, utilities, communication services, consumer discretionary, consumerstaples, financials and information technology sectors. The last category, including only thehealth care sector, concerns a moderate negative sensitivity to the carbon factor. However,this sector is getting closer and closer to the carbon risk-neutral category over time, eventhough we continue to observe a gap.Among the second category, we observe that the materials and real estate sectors startedwith a similar median carbon beta. However, the spread has been increasing between thetwo sectors since 2016, because the median carbon risk is stable in the case of the real estatesector whereas it is increasing for the materials sector. This gap may be persistent in thelong run, implying that the materials sector may be increasingly affected by carbon risk.Concerning the third category, we observe that the industrials sector was mostly negativelyinfluenced by the BMG factor, but it has become a carbon risk-neutral sector . Overall,sector differentiation is more important than geographical breakdown for investors sincemarket-based carbon risks converge both in absolute and relative values at the geographicallevel. It is calculated as follows: | β | bmg , R ( t ) = (cid:80) i ∈R (cid:12)(cid:12) β bmg ,i ( t ) (cid:12)(cid:12) card R As we have seen in Table 9 on page 51, the volatility of the BMG factor is lower than the volatility ofthe MKT factor. A variation of the carbon beta can not be interpreted in an ordinary scale. The decrease of | β | bmg , R ( t ) in March 2012 is not due to a climate-related policy but to green stocksfar outperforming as we can see in Figure 19 on page 52. At the same time, the European market declinedwhile the American market increased. Therefore, the carbon beta considerably increased for green Europeanstocks, which was driven mostly by the European market’s return rather than their carbon return. In asimilar way, the carbon beta decreased for brown American stocks, which was driven mostly by the Americanmarket’s return rather than their carbon return. Carbon risk pricing in Japan is around 25% lower than globally. This increase in the median carbon beta may push the industrial sector to the second category in thefuture. β bmg , S ( t ) by sector Figure 5: Dynamics of the median carbon risk β bmg , S ( t ) for the energy sector Remark 3.
We may also consider the absolute average carbon beta for each sector. Resultsare reported in Figure 22 on page 53. In this case, we distinguish two main categories. Thefirst one corresponds to high carbon pricing. This category includes the energy and materialssectors. The second one includes the sectors with a low (either upward or downward) carbonsensitivity to stock prices.
Figure 6: Density of the carbon risk first differenceThe advantage of this dynamic analysis is to assume that common risks are time-varying.In Figure 6, we have reported the density of the monthly variations β bmg ,i ( t ) − β bmg ,i ( t − . The extreme changesare more explained by regional or sector-related effects. This confirms that β bmg ,i ( t ) ismore a low-frequency systematic measure than a high-frequency idiosyncratic measure ofthe carbon risk. The brown minus green or BMG factor developed by G¨orgen et al. (2019) is an aggregationof numerous variables and we wonder what climate change-related dimensions are most This is for example the case of some famous controversial events, e.g. Volkswagen, Bayer, etc. What-ever the variation of the carbon factor, the firm’s stock return decreases in the case of an environmentalcontroversy. In this section, the dimensions do not correspond to the dimensions previously introduced (value chain,public perception and adaptability). When we refer to climate-related dimensions, it concerns any variablesinvolved in climate change.
Source : MSCI (2020).
In what follows, we present some risk factors built on different climate-related dimensions(Figure 7). These factors have long exposure to worst-in-class green stocks and short expo-sure to best-in-class green stocks. To obtain results that are comparable with the carbon riskfactor developed for the Carima project, we use the methodology of Fama and French (1992,1993). Nevertheless, the returns of the four portfolios (SG, BG, SB and BB) are equallyweighted , the portfolio weights are rebalanced every month and the stock universe is the We have also derived the risk factors with a capitalization-weighted scheme. The results do not change et al. (2019), their carbon risk is more influenced bytheir investments than by their carbon emissions. We retain this sector because excludingfinancial firms would result in lower statistical significance of the factors. Moreover, wethink that financial firms are exposed to some climate-related dimensions. Unless otherwisespecified, all the variables come from the MSCI ESG Ratings dataset.
The first comparison concerns the factors built on the exposure to carbon pricing and reg-ulatory caps: (1) the carbon intensity derived on the three scopes and (2) the carbonemissions exposure score based on the carbon-intensive business activities and the currentor potential future carbon regulations (MSCI, 2020). In Figure 8, we have reported thecumulative performance of these two factors and the carbon factor developed by G¨orgen et al. (2019). We observe that the three factors are very similar. For instance, we have astrong linear correlation greater than 90% between the carbon intensity and carbon emis-sions exposure factors. Because of the greater similarity with the Carima factor, we wonderif the carbon intensity risk measure is the only carbon dimension priced in by the market .Table 10 on page 51 provides the correlation between the carbon risk factors and the refer-ence factors. We have a close correlation between the two current carbon factors and theCarima factor, but not so much (58% and 64%). In Table 11 on page 51, we have reported acomparison between multi-factor models and their nested models. There is a slightly largerfactor exposure for the two current carbon risk factors with respect to the Carima risk fac-tor. Hence, the carbon exposure is a dimension widely taken into account by the marketto determine changes in stock prices, but we cannot deduce that this dimension is the onlydimension priced in because our methodology and our universe of stocks to build factors aredifferent from the Carima project. Remark 4.
We have also built the factors with the capitalization-weighted scheme and ex-cluding financial firms. In this case, the carbon intensity factor has a smaller factor exposurethan the Carima factor because of a strong correlation with the market, value and momen-tum factors. However, this decrease in explanatory power is mainly due to the exclusion offinancials. Indeed, this sector has little exposure to the potential risk of increased costs linkedto carbon pricing or cap and trade systems. We have seen previously in Figure 1 on page8 that the financial sector has a negative relative carbon risk. Even though carbon risk isconstructed differently for this sector, excluding it means losing information. Concerning theCW carbon emissions exposure factor, the results are halfway between the two EW factorsand the Carima factor. Nevertheless, the better results of explanatory power for this CWfactor in comparison with the Carima factor are too low to conclude that the carbon exposureis the only carbon dimension priced in by the market. much. We have a correlation of around 95% between EW and CW factors. Nevertheless, we obtain a betterexplanatory power for the multi-factor regression models when we consider an equally-weighted scheme. This factor has already been proposed by In et al. (2017) on a universe of American stocks. The three scopes are available in the Trucost dataset. Assets are selected every month with a reportinglag of one year. It is obvious that numerous other carbon variables explain the fluctuations in stock prices but some (orall) of them increase the explanatory power of a multi-factor model because they are correlated with thecarbon intensity.
While prior literature (Semenova and Hassel, 2015) distinguishes between environmentalperformance (or management) and environmental risk (or exposure), we focus on the samedistinction but on the carbon dimension. In what follows, we will provide an answer towhether carbon emissions exposure or carbon management contributes more to variationsin stock prices. The carbon emissions exposure mainly involves current carbon emissionsand factors inherent to the firm’s business whereas the carbon emissions management isabout future potential carbon emissions and measures the efforts to reduce this exposure.If we consider the carbon axes used by the Carima project, the first one corresponds tothe value chain axis and the second one to the adaptability axis. In Table 11 on page 51,we remark that both carbon exposure and carbon management significantly increase theexplanatory power of the common factor models but this variation is greater with the firstfactor. With respect to the other carbon risk factors, the carbon management factor has alower correlation with the Carima factor as we can see in Table 10 on page 51. Moreover,the close correlation of this factor with the Carhart risk factors is a problem in the case ofminimum variance and enhanced index portfolios where the risk factors are supposed to beuncorrelated. Furthermore, this carbon factor is not an alternative option to the referencecarbon factor.Despite the issues with the carbon management factor, abandoning this dimension wouldbe unfortunate. The carbon emissions factor built on the carbon emissions score, which isderived from both the carbon emissions management and exposure scores, allows us toovercome this problem. The higher the exposure score and the lower the managementscore, the lower the carbon emissions score. The latter assesses the capacity of a firm tohandle increasing carbon costs. As we can see in Table 11 on page 51, aggregating the twodimensions leads to an increase in explanatory power, which is significant for almost 19%of the stocks at the threshold of 5%. Moreover, as seen in Table 10 on page 51, the carbon17easuring and Managing Carbon Risk in Investment Portfoliosemissions dimension is the most closely correlated to the Carima risk factor.
Remark 5.
The median sector carbon beta coefficients have almost the same ranks for thecarbon exposure and the aggregated carbon factors. Nonetheless, we have very different re-sults with the carbon management factor. In the case of carbon management, the carbonbetas are very high for the health care sector because of the lack of environmental perfor-mances. For the utilities and materials sector, the carbon beta has significantly decreased –the utilities sector has the lower average carbon beta. These results are not surprising sincethe more environmentally responsible firms face greater environmental challenges (Delmasand Blass, 2010; Rahman and Post, 2012). We confirm again that the exposure factor is bet-ter than the management factor since the aggregated carbon factor is associated to a higheradjusted R coefficient while it is very related to the carbon exposure dimension. Let us consider now the three main climate-related dimensions: environment, climate changeand carbon emissions. The carbon dimension is nested into the climate dimension whichis itself nested into the environmental dimension (see Figure 7 on page 15). The last twofactors are based respectively on the environmental pillar score and the climate changetheme score available in the MSCI ESG Ratings dataset. The environmental pillar includesthe climate change dimension but also environmental opportunities, waste and recycling,and natural capital. The climate change scope includes carbon emissions, environmentalrisk financing, climate change vulnerability of insurance companies and the product carbonfootprint (MSCI, 2020). The three new factors are very similar with a correlation betweenthem around 75% and 80%. In Table 11 on page 51, we notice that the mimicking factorportfolio for environmental risk is more closely correlated with the factor built by G¨orgen etal. (2019) than the climate change factor, certainly because the Carima risk factor doesn’tjust incorporate carbon emissions variables. One issue with the environmental factor is thatit is significantly negatively correlated with the market factor .In Figure 24 on page 54, we have reported the sector analysis of the carbon beta ˆ β bmg ,i estimated with the MKT+BMG model. We notice that the carbon emissions and environ-mental factors are very similar while there are some differences with the climate changefactor. For this latter factor, the median and quantiles are associated with small relativecarbon risk β bmg ,i for most sectors. These small carbon risks are mainly offset by the finan-cial sector’s higher carbon risk. While the financial sector has the lower median carbon betawith the carbon emissions factor, it ranks eighth with the climate change factor because ittakes into account the vulnerability of insurance companies to insured individuals’ physicalrisks and the integration of the environmental component into banks’ or asset managers’business models. We also notice that the carbon beta of the consumer staples and discre-tionary sectors have overall increased because the climate change factor takes into accountthe product carbon footprint. Numerous climate-related dimensions can be used to measure carbon risk. Among thestudied factors, some of them are more appropriate for assessing stock price fluctuations.Figure 9 provides the dynamics of the average absolute carbon risk | β | bmg ,i ( t ) for someclimate change-related dimensions and the Carima factor. In order to have comparable In a minimum variance portfolio where the average relative carbon risk is negative, a bear market mayimply a higher loss since the best-in-class green stocks underperform the other stocks. | β | bmg ,i ( t ) carbon betas, each carbon factor BMG j has been standardized so that its monthly volatilityis equal to 1%: ˜ R bmg ,j ( t ) = R bmg ,j ( t )100 × σ bmg ,j ( t )where ˜ R bmg ,j ( t ) is the return of the standardized carbon factor and σ bmg ,j ( t ) is the con-ditional volatility of the factor BMG j estimated with a GARCH(1,1) model. In this case,the factors’ volatilities are the same at any time t . Overall, the trends of absolute car-bon risk in the environment, climate change and carbon emissions factors are very similar.Therefore, we think that the carbon emissions dimension is highly priced in, whereas addingother environmental variables does not significantly increase the average absolute carbonbeta . We can observe that the Carima factor is less priced in than the three main climatechange-related factors. One reason might be the different methodologies between the fac-tors. Nevertheless, our results do not change much with a capitalization-weighted scheme orby excluding financial firms. Therefore, we may wonder whether including a large range ofenvironmental variables is more informative. The carbon intensity dimension is less pricedin by the stock market at the beginning of the study period in comparison with the threemain climate change-related dimensions, but this gap has been eliminated since 2017.According to Table 11 on page 51, carbon emissions is the better climate change-relatedfactor to explain stock price fluctuations, followed by the carbon emissions exposure andcarbon intensity factors. By splitting the period into two equal subperiods , we obtain the Our results do not mean that the other environmental dimensions are barely priced in. For a widerange of environmental variables, the associated average absolute betas are often high but not so much asthe carbon emissions dimension. By taking into account several variables into one factor like the climatechange and environmental factors do, there is some collinearity between variables and it is difficult, from astatistical point of view, to determine which dimension is predominant in stock price variations. The first period starts at the beginning of 2010 and ends in mid-2014 and the second period starts inmid-2014 and ends at the end of 2018. R difference for the CAPM model against the MKT+BMG model:1 st period 2 nd period2010-2014 2014-2018Carima 1.16 2.21Carbon intensity 1.43 2.53Carbon emissions 2.18 2.39Climate change 1.98 1.83Environment 1.35 2.17The use of the BMG factors to explain stock price fluctuations is not consistent over time.According to these results, the carbon intensity dimension is currently the most importantclimate change-related axis in the two-factor model. Concerning the environment and cli-mate change dimensions, the results and their correlations with the market factor lead tothese factors being abandoned when it comes to managing carbon risk in investment portfo-lios. Overall, the carbon intensity and carbon emissions dimensions are the more interestingalternative factors to the Carima factor. Remark 6.
Even though the climate change-related dimensions are less priced in by thestock market over time, their integration – except for the climate change dimension – in aCAPM model significantly increases the explanatory power during the second period. This isbecause the CAPM model has an average adjusted R equal to . during the 1 st periodand . during the 2 nd period. Moreover, if we compute a single carbon factor model,the adjusted R is lower during the 2 nd period. Therefore, our results are coherent with theobservation that the carbon risk is less priced in today than before. In what follows, we consider how to manage carbon risk in an investment portfolio. Threemethods are used: the minimum variance strategy, the enhanced index portfolio and factorinvesting. For each method, we use an easily understood example and then we apply themethod to the MSCI World index. In this last case, we take into account the dynamic betasof the MKT+BMG model estimated by the Kalman filter, implying that we use the betacoefficients and the weights of the MSCI World index at the end of December 2018.In this section, it is important to keep in mind the distinction between absolute andrelative carbon risks. In the first case, the underlying idea is to have a neutral exposureto the BMG factor. In other words, we search the closest carbon exposure to zero. Inthe second case, the objective is to have a negative exposure to carbon risk. These twoapproaches lead us to consider different objective functions or constraints of the portfoliooptimization program.
We consider the global minimum variance (GMV) portfolio, which corresponds to this op-timization program: x (cid:63) = arg min 12 x (cid:62) Σ x (12)s.t. (cid:62) n x = 120easuring and Managing Carbon Risk in Investment Portfolioswhere x is the vector of portfolio weights and Σ is the covariance matrix of stock returns.The solution is given by the well-known formula: x (cid:63) = Σ − n (cid:62) n Σ − n (13)Problem (12) can be extended by considering other constraints: x (cid:63) = arg min 12 x (cid:62) Σ x (14)s.t. (cid:40) (cid:62) x = 1 x ∈ ΩFor instance, the most famous formulation is the long-only optimization problem whereΩ = [0 , n (Jagannathan and Ma, 2003). In the capital asset pricing model, we recall that: R i ( t ) = α i + β mkt ,i R mkt ( t ) + ε i ( t ) (15)where R i ( t ) is the return of asset i , R mkt ( t ) is the return of the market factor and ε i ( t ) isthe idiosyncratic risk. It follows that the covariance matrix Σ can be decomposed as:Σ = β mkt β (cid:62) mkt σ + D where β mkt = (cid:0) β mkt , , . . . , β mkt ,n (cid:1) is the vector of betas, σ is the variance of the marketportfolio and D = diag (cid:0) ˜ σ , . . . , ˜ σ n (cid:1) is the diagonal matrix of specific variances. Using theSherman-Morrison-Woodbury formula , we deduce that the inverse of the covariance matrixis: Σ − = D − − σ σ ϕ (cid:16) ˜ β mkt , β mkt (cid:17) ˜ β mkt ˜ β (cid:62) mkt where ˜ β mkt ,i = β mkt ,i / ˜ σ i and ϕ (cid:16) ˜ β mkt , β mkt (cid:17) = ˜ β (cid:62) mkt β mkt . Solution (13) becomes: x (cid:63) = σ ( x (cid:63) ) D − n − σ σ ϕ (cid:16) ˜ β mkt , β mkt (cid:17) ˜ β mkt ˜ β (cid:62) mkt n Using this new expression, Scherer (2011) showed that: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:32) − β mkt ,i β (cid:63) mkt (cid:33) (16)where: β (cid:63) mkt = 1 + σ ϕ (cid:16) ˜ β mkt , β mkt (cid:17) σ ˜ β (cid:62) mkt n (17) This is provided in Appendix A.2 on page 42. The expression of Σ − is obtained with A = D and u = v = σ mkt β mkt . i has a beta β mkt ,i smaller than β (cid:63) mkt , the weight of this asset is positive ( x (cid:63)i > β mkt ,i > β (cid:63) mkt , then x (cid:63)i <
0. Clarke etal. (2011) extended Formula (16) to the long-only case with the threshold β (cid:63) mkt defined asfollows: β (cid:63) mkt = 1 + σ (cid:80) β mkt ,i <β (cid:63) mkt ˜ β mkt ,i β mkt ,i σ (cid:80) β mkt ,i <β (cid:63) mkt ˜ β mkt ,i (18)In this case, if β mkt ,i > β (cid:63) mkt , x (cid:63)i = 0. We consider an extension of the CAPM by including the BMG risk factor: R i ( t ) = α i + β mkt ,i R mkt ( t ) + β bmg ,i R bmg ( t ) + ε i ( t ) (19)where R bmg ( t ) is the return of the BMG factor and β bmg ,i is the BMG sensitivity (or thecarbon beta) of stock i . Moreover, we assume that R mkt ( t ) and R bmg ( t ) are uncorrelated.It follows that the expression of the covariance matrix becomes:Σ = β mkt β (cid:62) mkt σ + β bmg β (cid:62) bmg σ + D In Appendix A.3 on page 43, we show that the GMV portfolio is defined as: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:32) − β mkt ,i β (cid:63) mkt − β bmg ,i β (cid:63) bmg (cid:33) (20)where β (cid:63) mkt and β (cid:63) bmg are two threshold values given by Equations (43) and (44) on page 45.In the case of long-only portfolios, we obtain a similar formula: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:32) − β mkt ,i β (cid:63) mkt − β bmg ,i β (cid:63) bmg (cid:33) if β mkt ,i β (cid:63) mkt + β bmg ,i β (cid:63) bmg ≤
10 otherwise (21)but the expressions of the thresholds β (cid:63) mkt and β (cid:63) bmg are different from those obtained inthe GMV case.Contrary to the single-factor model, the impact of sensitivities is more complex in thetwo-factor model. Indeed, we know that ¯ β mkt ≈ β bmg ≈
0. It follows that β (cid:63) mkt is positive, but β (cid:63) bmg may be positive or negative. We deduce that the ratio β mkt ,i β (cid:63) mkt is anincreasing function of β mkt ,i . Therefore, the MV portfolio selects assets that present a lowMKT beta value. For the BMG factor, the impact of β bmg ,i is more complex. Let us firstcompute the volatility of the asset i . We have: σ i = β ,i σ + β ,i σ + ˜ σ i Selecting low volatility assets is then equivalent to considering assets with a low absolutevalue (cid:12)(cid:12) β bmg ,i (cid:12)(cid:12) . If we calculate the correlation between assets i and j , we obtain: ρ i,j = β mkt ,i β mkt ,j σ + β bmg ,i β bmg ,j σ σ i σ j They are given by Equations (47) and (48) on page 47. β mkt ,i β mkt ,j is generally positive, whereas the crossproduct β bmg ,i β bmg ,j is positive or negative. In this context, diversifying a portfolio consistsin selecting assets with low values of β mkt ,i β mkt ,j . We then observe consistency betweenlow volatility and low correlated assets if we consider market beta contributions. In termsof BMG sensitivities, diversifying a portfolio consists in selecting assets with high negativevalues of β bmg ,i β bmg ,j . Therefore, we have to choose assets, with high absolute values (cid:12)(cid:12) β bmg ,i β bmg ,j (cid:12)(cid:12) , and opposite signs of β bmg ,i and β bmg ,j . We do not have consistency betweenlow volatility and low correlated assets if we consider BMG contributions. This explains thatthe ratio β bmg ,i β (cid:63) bmg may be an increasing or decreasing function of β bmg ,i . The MV portfoliowill then overweight assets with a negative value of β bmg ,i only if β (cid:63) bmg is positive. Otherwise,the MV portfolio may prefer assets with a positive sensitivity to the BMG factor. Remark 7.
Let us denote by x (cid:63) (cid:0) β mkt , β bmg (cid:1) the minimum variance portfolio that dependson the parameters β mkt and β bmg . We have the following properties: x (cid:63) (cid:0) β mkt , − β bmg (cid:1) = x (cid:63) (cid:0) β mkt , β bmg (cid:1) β (cid:63) mkt (cid:0) β mkt , − β bmg (cid:1) = β (cid:63) mkt (cid:0) β mkt , β bmg (cid:1) β (cid:63) bmg (cid:0) β mkt , − β bmg (cid:1) = − β (cid:63) bmg (cid:0) β mkt , β bmg (cid:1) Changing the BMG sensitivities by their opposite values does not change the solution . We consider an example given in Roncalli (2013, Example 24, page 168). The investmentuniverse is made up of five assets. Their market beta is respectively equal to 0 .
9, 0 .
8, 1 .
2, 0 . . β (cid:63) mkt = 1 . β (cid:63) mkt = 0 . β mkt ,i β bmg ,i CAPM MKT+BMGGMV MV GMV MV1 0 . − .
50 147 .
33 0 .
00 166 .
55 33 .
542 0 .
80 0 .
70 24 .
67 9 .
45 21 .
37 1 .
463 1 .
20 0 . − .
19 0 . − .
80 0 .
004 0 .
70 0 .
90 74 .
20 90 .
55 65 .
06 64 .
995 1 . − . − .
01 0 . − .
18 0 . − .
5, 0 .
7, 0 .
2, 0 . − .
3, whereas the volatility of the BMGfactor is set to 10%. In the case of the GMV, the thresholds are equal to β (cid:63) mkt = 1 . This result holds for both the GMV portfolio and the long-only MV portfolio. β (cid:63) bmg = 19 . β (cid:63) mkt = 0 . β (cid:63) bmg = 9 . β mkt , β (cid:63) mkt = 0 . . > β mkt , β (cid:63) mkt + β bmg , β (cid:63) bmg = 0 . . − . . < β mkt ,i Parameter set β bmg ,i GMV MV β bmg ,i GMV MV1 0 . − .
50 105 .
46 0 .
00 1 .
50 105 .
46 0 .
002 0 . − .
50 27 .
88 19 .
48 0 .
50 27 .
88 19 .
483 1 .
20 3 .
00 40 .
19 13 . − .
00 40 .
19 13 .
614 0 . − .
20 76 .
77 66 .
91 1 .
20 76 .
77 66 .
915 1 . − . − .
30 0 .
00 0 . − .
30 0 . − . − .
5, 3 . − . − .
9. For the long/short MV portfolio, we obtain β (cid:63) mkt = 1 . β (cid:63) bmg = − . β (cid:63) mkt = 0 . β (cid:63) bmg = − . β (cid:63) mkt , but thevalue of β (cid:63) bmg is different. Indeed, we obtain β (cid:63) bmg = +19 . β (cid:63) bmg = +9 . in Section 2.1.4 on page 9. By computingthe long-only MV portfolio, we obtain β (cid:63) mkt = 0 . β (cid:63) bmg = 5 . β mkt ,i and β bmg ,i . We verify that the most important axis is the MKT beta. Indeed, themarket risk of a stock determines whether the stock is included in the MV portfolio ornot whereas the carbon risk adjusts the weights of the asset. As we can see, the portfoliooverweights assets whose MKT and BMG sensitivities are both close to zero. This solutionis satisfactory if the original motivation is to reduce the portfolio’s absolute carbon risk, butit is not satisfactory if the objective is to manage the portfolio’s relative carbon risk. We have reported the scatter plot of MKT and BMG sensitivities in Figure 23 on page 54. We observea low positive correlation between β mkt ,i and β bmg ,i . -0.5 -0.25 0 0.25 0.5 0.75 1-2024 In order to circumvent the previous drawback, we can directly add a BMG constraint in theoptimization program: x (cid:63) = arg min 12 x (cid:62) Σ x (22)s.t. (cid:62) n x = 1 β (cid:62) bmg x ≤ β +bmg x ≥ n where β +bmg is the maximum tolerance of the investor with respect to the relative BMG risk.In this case, the values of β bmg ,i influence both the covariance matrix and the optimizationproblem. In this case, it is not possible to obtain an analytical solution. Nevertheless,we can always analyze the solution using the framework developed by Jagannathan andMa (2003). Introducing the BMG constraint is equivalent to applying a shrinkage of thecovariance matrix : ˜Σ = Σ + λ bmg (cid:16) β bmg (cid:62) n + n β (cid:62) bmg (cid:17) (23)where λ bmg ≥ β (cid:62) bmg x ≤ β +bmg . We deduce that the shrinkage matrix is equal to:˜Σ = σ β mkt β (cid:62) mkt − (cid:32) λ bmg σ mkt σ bmg (cid:33) + σ (cid:16) ˙ β bmg ˙ β (cid:62) bmg (cid:17) + D The analysis of Jagannathan and Ma (2003) only includes bound constraints. The extension to otherlinear constraints can be found in Roncalli (2013). β bmg = β bmg + λ bmg σ n . By imposing that the MV portfolio has a carbon beta lowerthan β +bmg , we implicitly introduce two effects:1. first, we shift the BMG sensitivities by a positive scalar λ bmg σ ;2. second, we reduce the MKT covariance matrix by a uniform parallel shift, because ofthe term λ bmg σ mkt σ bmg .Therefore, the BMG constraint β (cid:62) bmg x ≤ β +bmg can be interpreted as an active view.In order to illustrate the BMG constraint, we consider the previous examples and imposethat the BMG sensitivity of the MV portfolio cannot be positive. Results are reported inTable 6. We notice that the invariance of the BMG sign is broken. For instance, we do notobtain the same solution between parameter sets .Table 6: Composition of the constrained MV portfolio ( β +bmg = 0)Asset β mkt ,i Parameter set β bmg ,i MV β bmg ,i MV β bmg ,i MV1 0 . − .
50 64 . − .
50 0 .
00 1 .
50 0 .
002 0 .
80 0 .
70 0 . − .
50 19 .
48 0 .
50 16 .
113 1 .
20 0 .
20 0 .
00 3 .
00 13 . − .
00 25 .
894 0 .
70 0 .
90 35 . − .
20 66 .
91 1 .
20 58 .
005 1 . − .
30 0 . − .
90 0 .
00 0 .
90 0 . λ bmg
65 bps 0 56 bpsWe consider again the MSCI World index universe at December 2018. Since asset man-agers are mostly interested in long-only portfolios, we only report the long-only MV portfolio.If we would like to impose that the BMG sensitivity is lower than − .
5, we obtain resultsgiven in Figure 11. The comparison with the previous results (Figure 10) shows that theMV portfolio tends to select assets with both a low MKT beta and a negative BMG sensitiv-ity. Moreover, large weights are associated with large negative values of β bmg ,i on average.These results can be explained because the Lagrange coefficient λ bmg is equal to 29 bps. Ofcourse, the magnitude of the shrinkage depends on the value of β +bmg . The lower the BMGconstraint, the higher the Lagrange coefficient. For instance, we report the relationshipbetween β +bmg and λ bmg in Figure 12. This trade-off is not free since it will also impact thevolatility of the MV portfolio.For some time now, an important preoccupation of asset managers and asset ownersis about the GHG emissions associated with their investment portfolios. There are manyportfolio carbon footprint metrics but most of them are not consistent over time becauseof the equity ownership approach. To overcome this issue, we use the weighted averagecarbon intensity (or WACI) recommended by the Task Force on Climate-related FinancialDisclosures (TCFD, 2017): WACI ( x ) = n (cid:88) i =1 x i · CI i We recall that this is not the case of the unconstrained MV portfolio, which better promotes weak BMGsensitivities. β +bmg = − . -0.5 -0.25 0 0.25 0.5 0.75 1-2024 Figure 12: Relationship between β +bmg and λ bmg -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 10204060 -0.5 -0.4 -0.3 -0.2 -0.1 08008509009501000105011001150 where CI i is the carbon intensity of issuer i . We define the carbon intensity CI as theissuer’s direct and first-tier indirect GHG emissions divided by the revenue. This measureis expressed in tons CO e per million dollars in revenue. In Figure 13, we have reported therelationship between WACI and β +bmg of the MV portfolio. We remark that the lower therelative carbon risk threshold, the lower the portfolio exposure to carbon-intensive compa-nies. Nevertheless, the carbon intensities related to these different portfolios are very highin comparison with the equally-weighted portfolio whose WACI is approximatively equal to315. The reason is that carbon footprint metrics cannot be necessarily interpreted as riskmetrics. Some firms in the energy sector whose stock prices follow a singular pathway haveboth non-significant market and carbon betas, whereas they have a high carbon intensity.These unusual firms, whose carbon risk is captured in the idiosyncratic risk, contribute todramatically increasing the WACI of the MV portfolio. In order to circumvent this issue,we can add a constraint to the MV problem: x (cid:63) = arg min 12 x (cid:62) Σ x (24)s.t. (cid:62) n x = 1 β (cid:62) bmg x ≤ β +bmg x i = 0 if CI i > CI + x ≥ n where CI + is a maximum carbon intensity threshold. We have reported in Figure 25 on page55 the relationship between WACI and β +bmg of the MV portfolio when the carbon intensitythreshold CI + is equal to the WACI of the EW portfolio. In this case, we considerably The direct emissions correspond to the scope 1 emissions and the first-tier indirect emissions correspondto the GHG emissions of the firm’s direct suppliers (scope 2 emissions + some upstream scope 3 emissions). β +bmg . Even ifwe impose β +bmg = 0 and CI + = 2 000, we obtain a WACI around 225, which is very low incomparison with the MV portfolio without a carbon intensity threshold. Figure 26 on page55 provides the tradeoff between the portfolio volatility σ ( x ), the carbon risk threshold β +bmg and the carbon intensity threshold CI + . This latter constraint has no substantial impact onportfolio volatility. Therefore, it is possible to reduce the weighted average carbon intensitywithout substantially increasing volatility. Remark 8.
In order to highlight the difference between a market measure of carbon riskand a fundamental measure of carbon risk, we have reported in Figure 27 on page 56 therelationship between CI i and β bmg ,i . On average, the linear correlation is equal to . for the Carima factor. It slightly increases if we consider the carbon intensity factor, but itremains lower than . Enhanced index portfolios may be obtained by considering the portfolio optimization methodin the presence of a benchmark (Roncalli, 2013). For that, we define b = ( b , . . . , b n ) and x = ( x , . . . , x n ) as the asset weights in the benchmark and the portfolio. The tracking errorbetween the active portfolio x and its benchmark b is the difference between the portfolio’sreturn and the benchmark’s return: R (cid:0) x | b (cid:1) = R ( x ) − R ( b )= ( x − b ) (cid:62) R where R = ( R , . . . , R n ) is the vector of asset returns. The volatility of the tracking error R (cid:0) x | b (cid:1) corresponds to the standard deviation of R ( x ) − R ( b ): σ (cid:0) x | b (cid:1) = (cid:113) ( x − b ) (cid:62) Σ ( x − b )The optimization problem of enhanced index portfolios consists in replacing the portfolio’svolatility with the portfolio’s tracking error volatility in a minimum variance framework andimposing long-only weights: x (cid:63) = arg min 12 ( x − b ) (cid:62) Σ ( x − b ) (25)s.t. (cid:62) n x = 1 x ≥ n x ∈ ΩIf no other constraint is added (Ω = R n ), the optimal solution x (cid:63) is the benchmark b .But this framework only makes sense if we impose a second objective using the restriction x ∈ Ω. For instance, we can impose that the optimal portfolio has a carbon beta less thana threshold as in the case of the minimum variance problem:Ω = (cid:110) x ∈ R n : β (cid:62) bmg x ≤ β +bmg (cid:111) (26)Another approach consists in excluding the first m stocks that present the largest carbonrisk beta: Ω = (cid:110) x ∈ R n : x i = 0 if β bmg ,i ≥ β ( m,n )bmg (cid:111) (27)29easuring and Managing Carbon Risk in Investment Portfolioswhere β ( m,n )bmg = β bmg ,n − m +1: n is the ( n − m + 1) -th order statistic of (cid:0) β bmg , , . . . , β bmg ,n (cid:1) .These two approaches are similar to the ones proposed by Andersson et al. (2016). Thedifference comes from the fact that we use a market measure to estimate the carbon risk,whereas Andersson et al. (2016) measured the carbon risk directly using the carbon intensity.The two methods have their own advantages and drawbacks. It is obvious that the methodof Andersson et al. (2016) is more objective than our method, because it directly uses thecarbon footprint of the issuer. However, it is difficult to know if the stock price is sensitive tothis carbon footprint measure, especially since there are several carbon intensity measures .We understand that the carbon footprint is an ecological environment risk for planet Earth.But it is less obvious that it corresponds to the carbon market risk which is priced in by thestock market at the security level. If this were the case, it would mean that two corporatefirms with the same carbon footprint present the same carbon beta, regardless of the firms’other characteristics. Our method is less objective since the carbon risk is estimated throughthe dynamics of stock prices, and also depends on the methodology to build the carbon riskfactor. Nevertheless, it is more relevant from a financial point of view, because we considerthe carbon risk directly priced in by the stock market. In a sense, the first method has alonger-term horizon, whereas the second method is short-term by construction. Remark 9.
We can replace the absolute threshold β +bmg with a relative threshold. Indeed,imposing a reduction of the carbon risk with respect to the benchmark, e.g. β (cid:62) bmg ( x − b ) ≤− ∆ bmg , is equivalent to using an absolute threshold, e.g. β +bmg = β (cid:62) bmg b − ∆ bmg where ∆ bmg corresponds to the relative or absolute difference between the benchmark’s carbon risk andthe threshold value of relative carbon risk. The mathematical analysis of the optimization problem (25) with the constraint (26) isgiven in Appendix A.4 on page 47. We show that ∆ i = x (cid:63)i − b i is a decreasing function of thescaled BMG sensitivity ˘ β bmg ,i , which is equal to (cid:0) Σ − β bmg (cid:1) i . To illustrate this property, weconsider the example used on page 23 (parameter set β +bmg = 0. Results are reported in Table 7. Weverify that underweights and overweights depend on the sign of ˘ β bmg ,i . If ˘ β bmg ,i is negative,∆ i is positive and the asset is overweighted with respect to the benchmark. Otherwise, theasset is underweighted if ˘ β bmg ,i is positive.Table 7: Enhanced index portfolioAsset b i x (cid:63)i ∆ i β bmg ,i ˘ β bmg ,i .
00% 36 .
77 16 . − . − .
382 20 .
00% 17 . − .
88% 0 . .
223 20 .
00% 11 . − .
39% 0 . .
464 20 .
00% 12 . − .
97% 0 . .
105 20 .
00% 22 .
48 2 . − . − . The previous example gives the impression that underweights and overweights can also bepredicted thanks to the BMG sensitivity β bmg ,i . In this example, β bmg ,i and ˘ β bmg ,i have We generally distinguish scope 1, 2 and 3 carbon emissions. According to the Greenhouse Gas Protocol(2013), scope 1 corresponds to all direct emissions from the firm’s activities, scope 2 includes indirectemissions from electricity purchased and used by the firm, whereas scope 3 measures all other indirectemissions from the firm’s activities. β bmg ,i is less relevant than ˘ β bmg ,i ,we apply the previous framework to the MSCI World index universe. We consider that thebenchmark is the EW portfolio, and we impose that the BMG sensitivity is less than zero – β +bmg = 0. In Figures 14 and 15, we have reported the relationships between β bmg ,i , ˘ β bmg ,i and ∆ i = x (cid:63)i − b i . We verify that ˘ β bmg ,i is a better statistic than β bmg ,i for estimating theweighting direction. Indeed, the relationship between β bmg ,i and ∆ i = x (cid:63)i − b i is noisierthan the relationship between ˘ β bmg ,i and ∆ i = x (cid:63)i − b i . Moreover, in this last case, therelationship is almost linear.In what follows, we always consider that the benchmark is the capitalization-weighted(CW) portfolio. We have again reported the relationships between β bmg ,i , ˘ β bmg ,i and ∆ i = x (cid:63)i − b i in Figures 29 and 30 on page 57. In this case, we impose a relative carbon risk ofthe portfolio less than − . β +bmg = − .
3. We can notice that some assets are not in linewith the linear relationship between ˘ β bmg ,i and ∆ i = x (cid:63)i − b i . Since we consider a long-onlyportfolio, these assets are excluded in the optimized portfolio. The slope of the relationshipbetween ˘ β bmg ,i and ∆ i is steeper in the current case than in the case where the benchmarkis the EW portfolio for two reasons. The first one is that we have set a higher ∆ bmg andthe second one is that a steeper slope allows us to offset the assets whose weights x (cid:63)i havealready reached the value of zero. Remark 10.
Since the relationship between β bmg ,i and ∆ i = x (cid:63)i − b i is not a monotonicallydecreasing function, the order-statistic optimization problem defined by the restriction (27)is not equivalent to the max-threshold optimization problem defined by the inequality (26). Table 8: Regional composition of the portfolio (in %)Region b R x (cid:63) R ∆ R Eurozone 10 .
89 12 .
77 1 . .
83 10 . − . .
16 64 . − . .
70 8 .
90 0 . .
41 2 . − . R are respectively equalto b R = (cid:80) i ∈R b i and x (cid:63) R = (cid:80) i ∈R x (cid:63)i , the long/short regional exposure ∆ R of the optimizedportfolio with respect to the benchmark is equal to ∆ R = x (cid:63) R − b R . The long/short expo-sure for the Eurozone and Japan is positive while it is negative for Europe ex EMU andNorth America. We notice that these results are consistent with the regional trend analysisprovided by Figure 2 on page 11. Indeed, the higher a region’s relative carbon risk, thelower the long/short exposure to a region. Long/short exposure to the Eurozone is highin absolute and relative values. Indeed, the optimized portfolio has a long exposure to theEurozone of almost 17 .
2% higher than the benchmark. Nevertheless, the exposure to Eu-ropean ex EMU and North American stocks has not changed significantly with respect tothe benchmark. This result is quite surprising for North America since the average relativecarbon risk β bmg , R ( t ) was high in December 2018. For the rest of the world, long/short We impose a smaller threshold value of carbon risk when we consider the CW portfolio as the benchmarkbecause the relative carbon risk is equal to 0 . − . β +bmg = 0 in the case where the benchmark is the CW portfolio,the optimized portfolio x (cid:63) is then equal to the benchmark b . β bmg ,i and ∆ i = x (cid:63)i − b i for the EW benchmark -2 -1 1 2 3 4 5-6-4-2246 Figure 15: Relationship between ˘ β bmg ,i and ∆ i = x (cid:63)i − b i for the EW benchmark -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1-6-4-2246 .
5% with respect to the benchmark. Weobtain such a result because of the very high average relative carbon risk β bmg , R ( t ) of theother regions which is around 0 .
75 at the end of December 2018.We may also be interested in the sector composition of the optimized portfolio . Figure16 provides the portfolio’s long/short exposure ∆ S with respect to the benchmark, whichis defined as ∆ S = (cid:80) i ∈S (cid:0) x (cid:63)i − b i (cid:1) where S is the sector. In this case, the difference (bothin absolute and relative terms) between the CW and optimized portfolios is high. Theenergy sector is the most impacted. Indeed, the weight of the energy sector is 6 .
00% inthe benchmark while it is equal to 4 .
02% in the optimized portfolio which represents adecrease of 33%. The materials sector closely follows the energy sector with a decrease of itsexposure from 4 .
58% to 3 . .
42% and 14 . .
95% to 16 .
51% for the information technology sector and 13 .
32% to 14 .
34% forthe health care sector. The weight for the other sectors has changed slightly. Overall, theresults are consistent with the results obtained in Figures 1, 4 and 5. In comparison withthe benchmark, the optimized portfolio is more exposed to the sectors which are positivelyimpacted by an unexpected acceleration in the transition process towards a green economy.Figure 16: Long/short sector exposure ∆ R of the portfolio (in %) E n e r g y M a t e r i a l s R e a l E s t a t e I n d u s t r i a l s U t ili t i e s C o m m u n i c a t i o n S e r v i c e s C o n s u m e r D i s c r e t i o n a r y C o n s u m e r S t a p l e s F i n a n c i a l s I n f o r m a t i o n T e c h n o l o g y H e a l t h C a r e -2-1012 In what follows, we again consider that the benchmark is the CW portfolio. Figure 17provides the relationships between the difference between the benchmark’s carbon risk andthe portfolio’s carbon risk ∆ bmg , the tracking error σ (cid:0) x | b (cid:1) , the active share AS (cid:0) x | b (cid:1) ,the number of excluding stocks N (cid:0) x | b (cid:1) and the weighted average carbon intensity WACIfor the max-threshold optimization problem. We notice that the relationship between ∆ bmg We recall that β +bmg = − . σ (cid:0) x | b (cid:1) is linear. Indeed, we can demonstrate that : σ (cid:0) x (cid:63) | b (cid:1) ≈ c ∆ bmg By decreasing the relative carbon risk of the portfolio by 0 .
1, the tracking error increases byalmost 65 bps whatever the initial value of the portfolio’s carbon risk. In the current opti-mization problem, the active share remains relatively low for any value of ∆ bmg . Moreover,we verify that the higher the ∆ bmg , the lower the WACI.Figure 17: Solution of the max-threshold optimization problem
Figure 31 on page 58 provides the solution to the order-statistic optimization problemdefined by the constraint (27). In this case, we solve a standard tracking error problemby having zero exposure to the first m stocks with higher carbon betas. We can noticethat for a same level of relative carbon risk tolerance, the tracking error and the activeshare are always larger in the case of the order-statistic problem in comparison with themax-threshold problem. By excluding the assets with the higher relative carbon risk, theoptimization problem excludes itself the assets with a low relative carbon risk in order tohave a carbon exposure similar to the benchmark. Hence, we must exclude a large numberof stocks to move from ∆ bmg = 0 to ∆ bmg = 0 .
1. This implies both high active share andtracking error. For instance, the tracking error is almost twice that of the max-thresholdoptimization problem when ∆ bmg is equal to 0 . (cid:110) x ∈ R n : x i = 0 if b i β bmg ,i ≥ (cid:0) b (cid:12) β bmg (cid:1) ( m,n ) (cid:111) (28)where (cid:0) b (cid:12) β bmg (cid:1) ( m,n ) = (cid:0) b (cid:12) β bmg (cid:1) n − m +1: n is the ( n − m + 1) -th order statistic of thevector (cid:0) b β bmg , , . . . , b n β bmg ,n (cid:1) . In this case, we exclude the assets with both high weight The semi-formal proof is given in Appendix A.4.3 on page 50.
For the minimum variance portfolio, we have seen that managing the relative or absolutecarbon risk leads to two different portfolio optimization programs. In the previous section,we considered the case where the fund manager would like to reduce exposure to the browneststocks. In order to decrease the BMG sensitivity, the optimized portfolio increases itsexposure to the greenest stocks. This is normal if the investor’s moral values are to fightclimate risk. However, this implies taking a bet particularly in the short run. Indeed, if theinvestor chooses a negative value of β +bmg (e.g. β +bmg = − (cid:26) x ∈ R n : (cid:12)(cid:12)(cid:12) β (cid:62) bmg x (cid:12)(cid:12)(cid:12) ≤ | β | +bmg (cid:27) where | β | +bmg is the maximum sensitivity to absolute carbon risk. This is equivalent toimposing the following inequality constraints : (cid:0) β bmg − β bmg (cid:1) (cid:62) x ≤ (cid:32) | β | +bmg | β | +bmg (cid:33) Again, we obtain a QP problem, which is easy to solve. The special case | β | +bmg = 0corresponds to the neutral exposure to the absolute carbon risk . From the viewpoint ofpassive management, imposing that β (cid:62) bmg x = 0 can be justified because the objective ofpassive management is to implement no active bets. Remark 11.
While we have two specific optimization programs in the case of the minimumvariance portfolios, the boundary between relative and absolute carbon risk is not obviouswhen we consider tracking error optimization problems. Indeed, imposing a neutral absolutecarbon risk is equivalent to using an inequality constraint on the relative carbon risk . Bennani et al. (2018) and Drei et al. (2019) discussed the relationships between ESG in-vesting and factor investing. Among the different results, they showed that ESG may be Indeed, we have: (cid:12)(cid:12)(cid:12) β (cid:62) bmg x (cid:12)(cid:12)(cid:12) ≤ | β | +bmg ⇔ − | β | +bmg ≤ β (cid:62) bmg x ≤ + | β | +bmg ⇔ (cid:40) β (cid:62) bmg x ≤ | β | +bmg | β | +bmg ≥ − β (cid:62) bmg x In this case, the inequality constraints are replaced by the equality constraint β (cid:62) bmg x = 0, and theoptimization program remains a QP problem. If β (cid:62) bmg b > β (cid:62) bmg b < β (cid:62) bmg x ≤ β (cid:62) bmg x ≥ factor zoo ’ (Cochrane, 2011).Indeed, we think that the ESG risk factor does make more sense than the carbon risk factorin a factor investing framework, because it represents a broader investment type. Moreover,ESG investing is today a big investment topic of institutional investors, whereas carbon riskis embedded in the Environment pillar of ESG. Carbon risk is more a risk managementsubject than an investing approach. This is why we speak about ESG investing, but notabout carbon investing.Nevertheless, as shown previously, carbon risk is a financial risk and is interesting tomanage. Since equity factor investing is benchmarked against capitalization-weighted in-dices, we can use the framework developed in Section 3.2 on page 29. Instead of using abottom-up approach to define a carbon risk factor, it is easier to implement an overlay or atop-down approach that controls the relative or absolute carbon risk of the factor investingportfolio with respect to its benchmark. This paper studies the methodology proposed by G¨orgen et al. (2019) for measuring carbonrisk in investment portfolios. We confirm the results of these authors, that showed thatcarbon risk is priced in at the stock level and is relevant in a cross-sectional multi-factoranalysis. By considering a dynamic framework, we highlight several stylized facts. First, wenotice that carbon risk was priced in more at the beginning of the 2010s than it is today.Nevertheless, this is mainly due to the Eurozone. Second, we observe a convergence ofabsolute carbon risk pricing among the different regions, except in Japan. If we focus onrelative carbon risk, we confirm another transatlantic divide that we generally observe inESG investing (Bennani et al. , 2018; Drei et al. , 2019). On average, European stocks havea negative carbon beta, whereas it is positive for North America. We also observe somedifferences between sectors. For instance, there is clearly a difference in the dynamics of thecarbon beta between the materials and energy sectors, and the other sectors.Because the brown-minus-green (BMG) factor developed by G¨orgen et al. (2019) is basedon more than 50 proxy variables, investors may have some difficulty understanding which riskdimension is priced in by the market. For each stock, these authors calculate a brown greenscore, which is based on three dimensions: value chain, public perception and adaptability.Each dimension is the result of mixing several sub-dimensions. In this article, we havepreferred to consider very basic dimensions based on the Trucost and MSCI databases. Wehave focused our analysis on carbon intensity data provided by Trucost, and data on carbonemissions exposure, carbon management, the climate change score and the environmentpillar provided by MSCI. All these dimensions have an explanatory power that is in line36easuring and Managing Carbon Risk in Investment Portfolioswith the Carima factor of G¨orgen et al. (2019). However, at first order, we find that carbonintensity and carbon emissions exposure are the best alternative approaches to the Carimafactor.ESG rating agencies and other NGOs have developed many fundamental measures andscores to assess a firm’s carbon risk. Carbon beta is not a new measure. However, theapproach of G¨orgen et al. (2019) is definitively original since it is a market-based measureand not a fundamental-based measure. The carbon beta of a stock corresponds to thecarbon risk of the stock priced in by the financial market. Therefore, we may observe widediscrepancies between the market perception of the carbon risk, and for instance the directvalue of the carbon intensity. Our analysis shows that they are weakly correlated (less than30%). We can draw a parallel with the value risk factor. Indeed, the value beta of a stockmay be related to its book-to-price, but they are two different measures. And the financialmarket may consider that a stock with a high book-to-price is not necessarily a value stock,but a growth stock.The carbon beta therefore constitutes useful information for managing carbon risk ininvestment portfolios. In this article, we have mainly studied two investment strategies:the minimum variance portfolio and the enhanced index portfolio. Our results highlightthe difference between managing absolute carbon risk and relative carbon risk. Managingabsolute carbon risk implies having zero exposure to the BMG factor, whereas managingrelative carbon risk implies having negative exposure to the BMG factor. In the first case,the objective is to propose an immunization-hedging investment strategy against carbonrisk. In the second case, we explicitly take an active management bet by overweightinggreen stocks and underweighting brown stocks. This second approach is certainly the mostfrequently observed, even in passive management, because of investors’ moral values. Weshow that the two approaches led us to consider different objective functions or constraintsof the portfolio optimization program, implying that we obtain very different solutions.Another finding is that managing market-based carbon risk and fundamental-based carbonrisk does not give the same solution, even though we observe similar properties betweenthe optimized portfolios. This is particularly true in the case of enhanced index portfolios.Finally, we discuss whether the BMG factor can be considered as a new factor or not,alongside traditional factors (size, value, momentum, etc.). Our conviction is that carbonrisk is a risk management subject, and not an investment style such as ESG investing. This iswhy we consider that carbon risk is more appropriate for better defining a minimum varianceportfolio than improving the diversification of a factor investing portfolio. Nevertheless, allour results confirm the initial findings of G¨orgen et al. (2019). Investors must be aware thatcarbon risk is priced in by the stock market. This is why they must measure and managethis risk, especially when it is too high or when it is incompatible with the fiduciary dutiesof their investment portfolios. By considering for instance a direct measure of carbon risk such as carbon intensity.
References
Andersson , M.,
Bolton , P., and
Samama , F. (2016), Hedging Climate Risk,
FinancialAnalysts Journal , 72(3), pp. 13-32.
Baldauf , M.,
Garlappi , L., and
Yannelis , C. (2020), Does Climate Change Affect RealEstate Prices? Only if you Believe in it,
Review of Financial Studies , 33(3), pp. 1256-1295.
Batista , M. (2008), A Note on A Generalization of Sherman-Morrison-Woodbury Formula, arXiv , 0807.3860.
Batista , M., and
Karawia , A.R.A.I. (2009), The Use of the Sherman-Morrison-WoodburyFormula to Solve Cyclic Block Tri-diagonal and Cyclic Block Penta-diagonal Linear Sys-tems of Equations,
Applied Mathematics and Computation , 210(2), pp. 558-563.
Bennani , L.,
Le Guenedal , T.,
Lepetit , F., Ly , L., Mortier , V.,
Roncalli , T. and
Sekine , T. (2018), How ESG Investing Has Impacted the Asset Pricing in the EquityMarket,
Amundi Discussion Paper , 36, . Bernardini , E.,
Di Giampaolo , J.,
Faiella , I., and
Poli , R. (2019), The Impact ofCarbon Risk on Stock Returns: Evidence from the European Electric Utilities,
Journalof Sustainable Finance & Investment , forthcoming, pp. 1-26.
Bolton , P., and
Kacperczyk , M.T. (2019), Do Investors Care about Carbon Risk?,
SSRN , . Bolton , P., and
Kacperczyk , M.T. (2020), Carbon Premium around the World,
SSRN , . Campiglio , E.,
Monnin , P., and von Jagow , A. (2019), Climate Risks in Financial Assets,
Council on Economic Policies , Discussion Note , 2019/2.
Choi , D.,
Gao , Z., and
Jiang , W. (2020), Attention to Global Warming,
Review of Finan-cial Studies , 33(3), pp. 1112-1145.
Carhart , M.M. (1997), On Persistence in Mutual Fund Performance,
Journal of Finance ,52(1), pp. 57-82.
Cochrane , J.H. (2011), Presidential Address: Discount Rates,
Journal of Finance , 66(4),pp. 1047-1108.
Clarke , R.G., de Silva
H., and
Thorley , S. (2011), Minimum Variance Portfolio Com-position,
Journal of Portfolio Management , 37(2), pp. 31-45.
Delmas , M.A., and
Blass , V.D. (2010), Measuring Corporate Environmental Performance:the Trade-offs of Sustainability Ratings,
Business Strategy and the Environment , 19(4),pp. 245-260.
Delmas , M.A.,
Etzion , D., and
Nairn-Birch , N. (2013), Triangulating EnvironmentalPerformance: What do Corporate Social Responsibility Ratings really capture?,
Academyof Management Perspectives , 27(3), pp. 255-267.
Drei , A.,
Le Guenedal , T.,
Lepetit , F.,
Mortier , V.,
Roncalli , T. and
Sekine ,T. (2019), ESG Investing in Recent Years: New Insights from Old Challenges,
AmundiDiscussion Paper , 42, .38easuring and Managing Carbon Risk in Investment Portfolios
Engle , R.F.,
Giglio , S.,
Kelly , B.,
Lee , H., and
Stroebel , J. (2020), Hedging ClimateChange News,
Review of Financial Studies , 33(3), pp. 1184-1216.
Fama , E.F., and
French , K.R. (1992), The Cross-Section of Expected Stock Returns,
Journal of Finance , 47(2), pp. 427-465.
Fama , E.F., and
French , K.R. (1993), Common Risk Factors in the Returns on Stocksand Bonds,
Journal of Financial Economics , 33(1), pp. 3-56.Greenhouse Gas Protocol (2013),
Technical Guidance for Calculating Scope 3 Emmissions,Supplement to the Corporate Value Chain (Scope 3), Accounting & Reporting Standard ,in partnership with the Carbon Trust.
Golub , G.H., and
Van Loan , C.F. (2013),
Matrix Computations , Fourth edition, JohnsHopkins University Press.
G¨orgen , M.,
Jacob , A.,
Nerlinger , M.,
Riordan , R.,
Rohleder , M., and
Wilkens ,M. (2019), Carbon Risk,
SSRN , . In , S.Y., Park , K.Y., and
Monk , A. (2017), Is ‘Being Green’ Rewarded in the Market?An Empirical Investigation of Decarbonization Risk and Stock Returns,
InternationalAssociation for Energy Economics , Singapore Issue, pp. 46-48.
Jagannathan , R., and Ma , T. (2003), Risk Reduction in Large Portfolios: Why Imposingthe Wrong Constraints Helps, Journal of Finance , 58(4), pp. 1651-1684.
Kim , Y.B., An , H.T., and Kim , J.D. (2015), The Effect of Carbon Risk on The Cost ofEquity Capital,
Journal of Cleaner Production , 93, pp. 279-287.
Krueger , P.,
Sautner , Z., and
Starks , L.T. (2020), The Importance of Climate Risksfor Institutional Investors,
Review of Financial Studies , 33(3), pp. 1067-1111.
Le Guenedal , T.,
Girault , J.,
Jouanneau , M.,
Lepetit , F., and
Sekine , T. (2020),Trajectory Monitoring in Portfolio Management and Issuer intentionality Scoring,
AmundiWorking Paper , 97, . Maillard , S.,
Roncalli , T. and
Te¨ıletche , J. (2010), The Properties of EquallyWeighted Risk Contribution Portfolios,
Journal of Portfolio Management , 36(4), pp. 60-70.MSCI (2020), MSCI ESG Ratings Methodology, MSCI ESG Research, April 2020.
Rahman , N., and
Post , C. (2012), Measurement Issues in Environmental Corporate SocialResponsibility (ECSR): Toward a Transparent, Reliable, and Construct Valid Instrument,
Journal of Business Ethics , 105(3), pp. 307-319.
Randers , J. (2012), Greenhouse Gas Emissions per Unit of Value Added (“GEVA”) – ACorporate Guide to Voluntary Climate Action,
Energy Policy , 48, pp. 46-55.
Roncalli , T. (2013),
Introduction to Risk Parity and Budgeting , Chapman & Hall/CRCFinancial Mathematics Series.
Roncalli , T. (2017), Alternative Risk Premia: What Do We Know?, in Jurczenko, E.(Ed.),
Factor Investing and Alternative Risk Premia , ISTE Press – Elsevier.
Roncalli , T. (2020a),
Handbook of Financial Risk Management , Chapman & Hall/CRCFinancial Mathematics Series. 39easuring and Managing Carbon Risk in Investment Portfolios
Roncalli , T. (2020b), ESG & Factor Investing: A New Stage Has Been Reached,
AmundiViewpoint . Scherer , B. (2011), A Note on the Returns from Minimum Variance Investing,
Journal ofEmpirical Finance , 18(4), pp. 652-660.
Semenova , N., and
Hassel , L.G. (2015), On the Validity of Environmental PerformanceMetrics,
Journal of Business Ethics , 132(2), pp. 249-258.
Sharpe , W.F. (1964), Capital Asset Prices: A Theory of Market Equilibrium under Con-ditions of Risk,
Journal of Finance , 19(3), pp. 425-442.Task Force on Climate-related Financial Disclosures (2017), Recommendations of the TaskFornce on Climate-related Financial Disclosures, Final Report, June 2017.40easuring and Managing Carbon Risk in Investment Portfolios
AppendixA Mathematical results
A.1 Time-varying estimation with Kalman filter
The time-varying risk factor model can be written as a state space model: (cid:40) y ( t ) = x ( t ) (cid:62) β ( t ) + ε ( t ) β ( t ) = β ( t −
1) + η ( t ) (29)where ε ( t ) ∼ N (cid:0) , σ ε (cid:1) , η ( t ) ∼ N (cid:0) K +1 , Σ β (cid:1) and K is the number of risk factors. Forinstance, in the case of the MKT+BMG model, y ( t ) corresponds to the asset return R i ( t ), x ( t ) is a 3 × R mkt ( t ) and R bmg ( t ) and: β ( t ) = α i ( t ) β mkt ,i ( t ) β bmg ,i ( t ) (30)It follows that the variable y t is observable, but this is not the case for the state vector β ( t ). The Kalman filter is a statistical tool to estimate the distribution function of β ( t ).Let β (0) ∼ N ( β , P ) be the initial position of the state vector. We note ˆ β (cid:0) t | t − (cid:1) = E (cid:2) β ( t ) | F ( t − (cid:3) and ˆ β (cid:0) t | t (cid:1) = E (cid:2) β ( t ) | F ( t ) (cid:3) as the optimal estimators of β ( t ) giventhe available information until time t − t . P (cid:0) t | t − (cid:1) and P (cid:0) t | t (cid:1) are the covariancematrices associated with ˆ β (cid:0) t | t − (cid:1) and ˆ β (cid:0) t | t (cid:1) . Since the estimate of y ( t ) is equal toˆ y (cid:0) t | t − (cid:1) = x ( t ) (cid:62) ˆ β (cid:0) t | t − (cid:1) , we can compute the variance F ( t ) of the innovation process v ( t ) = y ( t ) − ˆ y (cid:0) t | t − (cid:1) . These different quantities can be calculated thanks to the Kalmanfilter, which consists in the following recursive algorithm (Roncalli, 2020a, page 654): ˆ β (cid:0) t | t − (cid:1) = ˆ β (cid:0) t − | t − (cid:1) P (cid:0) t | t − (cid:1) = P (cid:0) t − | t − (cid:1) + Σ β v ( t ) = y ( t ) − x ( t ) (cid:62) ˆ β (cid:0) t | t − (cid:1) F ( t ) = x ( t ) (cid:62) P (cid:0) t | t − (cid:1) x ( t ) + σ ε ˆ β (cid:0) t | t (cid:1) = ˆ β (cid:0) t | t − (cid:1) + (cid:32) P (cid:0) t | t − (cid:1) F ( t ) (cid:33) x ( t ) v ( t ) P (cid:0) t | t (cid:1) = I K +1 − (cid:32) P (cid:0) t | t − (cid:1) F ( t ) (cid:33) x ( t ) x ( t ) (cid:62) P (cid:0) t | t − (cid:1) (31)In this model, the parameters σ u and Σ β are unknown and can be estimated by the methodof maximum likelihood. Since v ( t ) ∼ N (cid:0) , F ( t ) (cid:1) , the log-likelihood function is equal to: (cid:96) ( θ ) = − T π ) − T (cid:88) t =1 (cid:32) ln F ( t ) + v ( t ) F ( t ) (cid:33) (32)where θ = (cid:0) σ , Σ (cid:1) . Maximizing the log-likelihood function requires specifying the initialconditions β and P , which are not necessarily known. In this case, we use the linearregression y ( t ) = x ( t ) (cid:62) β + ε ( t ), and the OLS estimates ˆ β ols and ˆ σ ε (cid:0) X (cid:62) X (cid:1) − to initialize β and P . The algorithm is initialized with values ˆ β (cid:0) | (cid:1) = β and P (cid:0) | (cid:1) = P . A.2 Sherman-Morrison-Woodbury formula
Suppose u and v are two n × A is an invertible n × n matrix. We can showthat (Golub and Van Loan, 2013): (cid:16) A + uv (cid:62) (cid:17) − = A − −
11 + v (cid:62) A − u A − uv (cid:62) A − (33)Batista (2008) and Batista and Karawia (2009) extended the SMW formula when the outerproduct is a sum: A + m (cid:88) k =1 u k v (cid:62) k − = A − − A − U S − V (cid:62) A − (34)where U = (cid:0) u · · · u m (cid:1) and V = (cid:0) v · · · v m (cid:1) are two n × m matrices, and S = I + T and T = (cid:0) T i,j (cid:1) are two m × m matrices where T i,j = v (cid:62) i A − u j .In the case m = 2, the SMW formula becomes: (cid:16) A + u v (cid:62) + u v (cid:62) (cid:17) − = A − − A − U S − V (cid:62) A − (35)where: S = (cid:32) v (cid:62) A − u v (cid:62) A − u v (cid:62) A − u v (cid:62) A − u (cid:33) Since we have: S − = 1 | S | (cid:32) v (cid:62) A − u − v (cid:62) A − u − v (cid:62) A − u v (cid:62) A − u (cid:33) where | S | = (cid:16) v (cid:62) A − u (cid:17) (cid:16) v (cid:62) A − u (cid:17) − v (cid:62) A − u v (cid:62) A − u = 1 + v (cid:62) A − u + v (cid:62) A − u + v (cid:62) A − u v (cid:62) A − u − v (cid:62) A − u v (cid:62) A − u We deduce that: | S | · U S − V (cid:62) = (cid:0) u u (cid:1) (cid:32) v (cid:62) A − u − v (cid:62) A − u − v (cid:62) A − u v (cid:62) A − u (cid:33) (cid:32) v (cid:62) v (cid:62) (cid:33) = u v (cid:62) + u v (cid:62) A − u v (cid:62) − u v (cid:62) A − u v (cid:62) − u v (cid:62) A − u v (cid:62) + u v (cid:62) + u v (cid:62) A − u v (cid:62) If A is a diagonal matrix, we can simplify the previous SMW formula (35). Indeed, wehave: | S | = 1 + n (cid:88) s =1 u ,s v ,s + u ,s v ,s a s,s + n (cid:88) s =1 u ,s v ,s a s,s n (cid:88) s =1 u ,s v ,s a s,s − n (cid:88) s =1 u ,s v ,s a s,s n (cid:88) s =1 u ,s v ,s a s,s (36)42easuring and Managing Carbon Risk in Investment Portfoliosand : | S | · U S − V (cid:62) = n (cid:88) s =1 u ,s v ,s a s,s u v (cid:62) + n (cid:88) s =1 u ,s v ,s a s,s u v (cid:62) − n (cid:88) s =1 u ,s v ,s a s,s u v (cid:62) − n (cid:88) s =1 u ,s v ,s a s,s u v (cid:62) (37)If we assume that u is uncorrelated to v , u is uncorrelated to v , u and v are centeredaround 0, we obtain: | S | ≈ n (cid:88) s =1 u ,s v ,s + u ,s v ,s a s,s + n (cid:88) s =1 u ,s v ,s a s,s n (cid:88) s =1 u ,s v ,s a s,s (38)and: | S | · U S − V (cid:62) ≈ n (cid:88) s =1 u ,s v ,s a s,s u v (cid:62) + n (cid:88) s =1 u ,s v ,s a s,s u v (cid:62) (39) A.3 Minimum variance portfolio in the MKT+BMG model
Remark 12.
In what follows, we use the notations β i and γ i instead of β mkt ,i and β bmg ,i to simplify the notations. We have: R i ( t ) = α i + β i R mkt ( t ) + γ i R bmg ( t ) + ε i ( t )It follows that the covariance matrix is:Σ = ββ (cid:62) σ + γγ (cid:62) σ + D where β = ( β , . . . , β n ) is the vector of MKT betas, γ = ( γ , . . . , γ n ) is the vector of BMGbetas, σ is the variance of the market portfolio, σ is the variance of the BMG factorand D = diag (cid:0) ˜ σ , . . . , ˜ σ n (cid:1) is the diagonal matrix of specific variances. We recall that theGMV portfolio is equal to: x (cid:63) = Σ − n (cid:62) n Σ − n = σ ( x (cid:63) ) · Σ − n (40)because we have: σ ( x (cid:63) ) = x (cid:63) (cid:62) Σ x (cid:63) = (cid:62) n Σ − (cid:62) n Σ − n Σ Σ − n (cid:62) n Σ − n = 1 (cid:62) n Σ − n Let A = (cid:0) a i,j (cid:1) and C = (cid:0) c i,j (cid:1) be two n × n matrices and b a vector of dimension n . We recall that: (cid:0) A diag ( b ) C (cid:1) i,j = n (cid:88) s =1 a i,s b s c s,j We deduce that: (cid:16) a a (cid:62) diag ( b ) c c (cid:62) (cid:17) i,j = a ,i c ,j n (cid:88) s =1 a ,s b s c ,s where a , a , c and a are n × A.3.1 General formula
We use the generalized Sherman-Morrison-Woodbury with A = D , u = v = σ mkt β and u = v = σ bmg γ . It follows that the inverse of the covariance matrix is equal to:Σ − = D − − D − U S − V (cid:62) D − where U = V = (cid:0) σ mkt β σ bmg γ (cid:1) and: S = (cid:32) σ β (cid:62) D − β σ mkt σ bmg β (cid:62) D − γσ mkt σ bmg β (cid:62) D − γ σ γ (cid:62) D − γ (cid:33) We notice that: n (cid:88) s =1 u ,s v ,s a s,s = σ n (cid:88) s =1 β s ˜ σ s = σ ϕ (cid:16) ˜ β, β (cid:17) n (cid:88) s =1 u ,s v ,s a s,s = σ n (cid:88) s =1 γ s ˜ σ s = σ ϕ (˜ γ, γ ) n (cid:88) s =1 u ,s v ,s a s,s = n (cid:88) s =1 u ,s v ,s a s,s = σ mkt σ bmg n (cid:88) s =1 β s γ s ˜ σ s = σ mkt σ bmg ϕ (cid:16) ˜ β, γ (cid:17) where ˜ β and ˜ γ are the standardized vectors of β and γ by the idiosyncratic variances and ϕ ( x, y ) = x ◦ y is the outer product. Using Equation (36), we obtain: | S | = 1 + σ ϕ (cid:16) ˜ β, β (cid:17) + σ ϕ (˜ γ, γ ) + σ σ (cid:18) ϕ (cid:16) ˜ β, β (cid:17) ϕ (˜ γ, γ ) − ϕ (cid:16) ˜ β, γ (cid:17)(cid:19) Equation (37) becomes: | S | · U S − V (cid:62) = σ (cid:16) σ ϕ (˜ γ, γ ) (cid:17) ββ (cid:62) + σ (cid:18) σ ϕ (cid:16) ˜ β, β (cid:17)(cid:19) γγ (cid:62) − σ σ ϕ (cid:16) ˜ β, γ (cid:17) (cid:16) γβ (cid:62) + βγ (cid:62) (cid:17) Finally, the inverse of the covariance matrix has the following expression:Σ − = D − − M − where: M − = ω ˜ β ˜ β (cid:62) + ω ˜ γ ˜ γ (cid:62) − ω (cid:16) ˜ γ ˜ β (cid:62) + ˜ β ˜ γ (cid:62) (cid:17) and: ω = | S | ω = ω − · σ (cid:16) σ ϕ (˜ γ, γ ) (cid:17) ω = ω − · σ (cid:18) σ ϕ (cid:16) ˜ β, β (cid:17)(cid:19) ω = ω − · σ σ ϕ (cid:16) ˜ β, γ (cid:17) Therefore, the solution (40) becomes: x (cid:63) = σ ( x (cid:63) ) (cid:16) D − n − M − n (cid:17) x (cid:63) = σ ( x (cid:63) ) (cid:16) ξ − ˜ ω ˜ β − ˜ ω ˜ γ (cid:17) (41)where ξ = (cid:16) ˜ σ − , . . . , ˜ σ − n (cid:17) , ˜ ω = ω ˜ β (cid:62) n − ω ˜ γ (cid:62) n and ˜ ω = ω ˜ γ (cid:62) n − ω ˜ β (cid:62) n . Weconclude that: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:18) − β i β (cid:63) − γ i γ (cid:63) (cid:19) (42)where β (cid:63) = ˜ ω − and γ (cid:63) = ˜ ω − . Remark 13.
If we develop the expression of β (cid:63) and γ (cid:63) , we obtain: β (cid:63) = 1 + σ ϕ (cid:16) ˜ β, β (cid:17) + σ ϕ (˜ γ, γ ) + σ σ (cid:18) ϕ (cid:16) ˜ β, β (cid:17) ϕ (˜ γ, γ ) − ϕ (cid:16) ˜ β, γ (cid:17)(cid:19) σ (cid:32) ˜ β (cid:62) n + σ (cid:18) ϕ (˜ γ, γ ) ˜ β (cid:62) n − ϕ (cid:16) ˜ β, γ (cid:17) ˜ γ (cid:62) n (cid:19)(cid:33) (43) and: γ (cid:63) = 1 + σ ϕ (cid:16) ˜ β, β (cid:17) + σ ϕ (˜ γ, γ ) + σ σ (cid:18) ϕ (cid:16) ˜ β, β (cid:17) ϕ (˜ γ, γ ) − ϕ (cid:16) ˜ β, γ (cid:17)(cid:19) σ (cid:32) ˜ γ (cid:62) n + σ (cid:18) ϕ (cid:16) ˜ β, β (cid:17) ˜ γ (cid:62) n − ϕ (cid:16) ˜ β, γ (cid:17) ˜ β (cid:62) n (cid:19)(cid:33) (44) A.3.2 Special cases
Let us assume that stock returns are not sensitive to the BMG risk factor, i.e. γ i = 0. Itfollows that: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:18) − β i β (cid:63) (cid:19) and: β (cid:63) = 1 + σ ϕ (cid:16) ˜ β, β (cid:17) σ ˜ β (cid:62) n We retrieve Equations (16) and (17) that have been formulated by Scherer (2011).On average, we observe that the sensitivities β i are distributed around 1 .
0, whereas thesensitivities γ i are distributed around 0 .
0. Then, we can assume the hypothesis H that (cid:80) ns =1 w s γ s is equal to zero when w is a vector that is not correlated to γ . Under H , wededuce that: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:18) − β i β (cid:63) − γ i γ (cid:63) (cid:19) and: β (cid:63) = 1 + σ ϕ (cid:16) ˜ β, β (cid:17) + σ ϕ (˜ γ, γ ) + σ σ ϕ (cid:16) ˜ β, β (cid:17) ϕ (˜ γ, γ ) σ (cid:16) σ ϕ (˜ γ, γ ) (cid:17) ˜ β (cid:62) n and: γ (cid:63) = 1 + σ ϕ (cid:16) ˜ β, β (cid:17) + σ ϕ (˜ γ, γ ) + σ σ ϕ (cid:16) ˜ β, β (cid:17) ϕ (˜ γ, γ ) σ (cid:18) σ ϕ (cid:16) ˜ β, β (cid:17)(cid:19) ˜ γ (cid:62) n A.3.3 Extension to the long-only minimum variance portfolio
The extension to the long-only case follows the semi-formal proof formulated by Clarke etal. (2011). We note I u = { , . . . , n } as the investment universe. We consider the long-onlyminimum variance (MV) portfolio, which corresponds to the optimization program: x (cid:63) = arg min 12 x (cid:62) Σ x (45)s.t. (cid:40) (cid:62) n x = 1 x ≥ n The associated Lagrange function is then equal to: L ( x ; λ , λ ) = 12 x (cid:62) Σ x − λ (cid:16) (cid:62) n x − (cid:17) − λ (cid:62) ( x − n )The first-order condition is: ∂ L ( x ; λ , λ ) ∂ x = Σ x − λ n − λ = n whereas the Kuhn-Tucker conditions are min ( λ i , x i ) = 0 for i = 1 , . . . , n . The optimalsolution is given by: x (cid:63) = Σ − ( λ n + λ )We deduce that : x mv = x gmv + δ + where: x gmv = Σ − n (cid:62) n Σ − n and: δ + = (cid:32) I n − Σ − n (cid:62) n (cid:62) n Σ − n (cid:33) Σ − λ Therefore, imposing the long-only constraint is equivalent to adding a correction term δ + to the GMV portfolio x (cid:63)gmv in order to satisfy the constraint x i ≥ I (cid:48) u ⊂ I u as this constrained investment universe. If we restrict the analysis to the set I (cid:48) u , the GMV portfolio is equal to: ˜ x gmv = ˜Σ − n (cid:48) (cid:62) n (cid:48) ˜Σ − n (cid:48) where ˜Σ is the submatrix of Σ corresponding to the restricted universe and n (cid:48) = card I (cid:48) u isthe number of assets that belong to the long-only MV portfolio. By construction, we havethe equivalence between long-only MV and restricted GMV portfolios: x mv ≡ ˜ x gmv Because we have: (cid:62) n x = 1 ⇔ λ (cid:62) n Σ − n + (cid:62) n Σ − λ = 1 ⇔ λ = 1 − (cid:62) n Σ − λ (cid:62) n Σ − n x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:18) − β i β (cid:63) − γ i γ (cid:63) (cid:19) if β i β (cid:63) + γ i γ (cid:63) ≤
10 otherwise (46)However, contrary to the GMV case, the threshold values are endogenous and not exogenous: β (cid:63) = 1 + (cid:80) i ∈I (cid:48) u (cid:16) σ ˜ β i β i + σ ˜ γ i γ i (cid:17) + σ σ (cid:18)(cid:80) i ∈I (cid:48) u ˜ β i β i (cid:80) i ∈I (cid:48) u ˜ γ i γ i − (cid:16)(cid:80) i ∈I (cid:48) u ˜ β i γ i (cid:17) (cid:19) σ (cid:18)(cid:80) i ∈I (cid:48) u ˜ β i + σ (cid:16)(cid:80) i ∈I (cid:48) u ˜ γ i γ i (cid:80) i ∈I (cid:48) u ˜ β i − (cid:80) i ∈I (cid:48) u ˜ β i γ i (cid:80) i ∈I (cid:48) u ˜ γ i (cid:17)(cid:19) (47)and: γ (cid:63) = 1 + (cid:80) i ∈I (cid:48) u (cid:16) σ ˜ β i β i + σ ˜ γ i γ i (cid:17) + σ σ (cid:18)(cid:80) i ∈I (cid:48) u ˜ β i β i (cid:80) i ∈I (cid:48) u ˜ γ i γ i − (cid:16)(cid:80) i ∈I (cid:48) u ˜ β i γ i (cid:17) (cid:19) σ (cid:18)(cid:80) i ∈I (cid:48) u ˜ γ i + σ (cid:16)(cid:80) i ∈I (cid:48) u ˜ β i β i (cid:80) i ∈I (cid:48) u ˜ γ i − (cid:80) i ∈I (cid:48) u ˜ β i γ i (cid:80) i ∈I (cid:48) u ˜ β i (cid:17)(cid:19) (48)Indeed, we first need to determine I (cid:48) u in order to calculate β (cid:63) and γ (cid:63) . A.4 Analysis of the tracking error optimization problem
A.4.1 General formulation of the optimization program
We consider the following optimization program: x (cid:63) = arg min 12 ( x − b ) (cid:62) Σ ( x − b ) (49)s.t. (cid:62) n x = 1The associated Lagrange function is then equal to: L ( x ; λ ) = 12 x (cid:62) Σ x − x (cid:62) Σ b − λ (cid:16) (cid:62) n x − (cid:17) = 12 x (cid:62) Σ x − x (cid:62) (Σ b + λ n ) + λ We deduce that the first-order condition is: ∂ L ( x ; λ ) ∂ x = Σ x − (Σ b + λ n ) = n Since we have x = Σ − (Σ b + λ n ) and (cid:62) n x = 1, it follows that: (cid:62) n x = 1 ⇔ (cid:62) n Σ − (Σ b + λ n ) = 1 ⇔ λ (cid:16) (cid:62) n Σ − n (cid:17) = 1 ⇔ λ = 0and we obtain the trivial solution x (cid:63) = b . We notice that this solution remains valid if weintroduce long-only constraints x ≥ n because the Kuhn-Tucker conditions min ( λ i , x i ) = 0are already satisfied. 47easuring and Managing Carbon Risk in Investment PortfoliosWe now consider the optimization program: x (cid:63) = arg min 12 ( x − b ) (cid:62) Σ ( x − b ) (50)s.t. (cid:62) n x = 1 x ≥ n β (cid:62) bmg x ≤ β +bmg The associated Lagrange function is then equal to: L (cid:0) x ; λ , λ, λ bmg (cid:1) = 12 x (cid:62) Σ x − x (cid:62) Σ b − λ (cid:16) (cid:62) n x − (cid:17) − λ (cid:62) x + λ bmg (cid:16) β (cid:62) bmg x − β +bmg (cid:17) = 12 x (cid:62) Σ x − x (cid:62) (cid:0) Σ b + λ n + λ − λ bmg β bmg (cid:1) + λ − λ bmg β +bmg where λ ≥ (cid:62) n x = 1, λ ≥ n isthe vector of Lagrange multipliers of the bound constraints x ≥ n and λ bmg ≥ β (cid:62) bmg x ≤ β +bmg . We deduce that the first-order condition is: ∂ L (cid:0) x ; λ , λ, λ bmg (cid:1) ∂ x = Σ x − (cid:0) Σ b + λ n + λ − λ bmg β bmg (cid:1) = n implying that: x (cid:63) = Σ − (cid:0) Σ b + λ n + λ − λ bmg β bmg (cid:1) = b + λ Σ − n + Σ − λ − λ bmg Σ − β bmg = x (cid:63) ( b ) + x (cid:63) (cid:0) β bmg (cid:1) (51)where x (cid:63) ( b ) = b + λ Σ − n and x (cid:63) (cid:0) β bmg (cid:1) = Σ − λ − λ bmg Σ − β bmg . We notice that Portfolio x (cid:63) ( b ) only depends on the benchmark b , the covariance matrix Σ and the Lagrange multiplier λ . It does not depend on the BMG sensitivities, and may be considered as a constant term.Moreover, we can set λ = 0, because the constraint (cid:62) n x = 1 has no impact on the solution x (cid:63) , but only scales the optimal portfolio such as the weights sum up to 100%. In this case,we have x (cid:63) ( b ) = b and x (cid:63) (cid:0) β bmg (cid:1) is the long/short portfolio x (cid:63) − b that depends on severalparameters: ∆ = x (cid:63) − b ≈ Σ − λ − λ bmg Σ − β bmg (52)At first sight, these parameters are the covariance matrix Σ, the vector λ of Lagrangecoefficients, the Lagrange coefficient λ bmg and the vector β bmg of carbon risk sensitivities.In fact, this interpretation of Equation (52) is misleading. The two important quantities arethe scaled BMG beta vector ˘ β bmg = Σ − β bmg and the threshold value β +bmg , since λ and λ bmg are endogenous . A.4.2 Special cases
Let us assume that Σ = diag (cid:0) σ , . . . , σ n (cid:1) is a diagonal matrix. We have∆ i = λ i σ i − λ bmg β bmg ,i σ i (53) The vector λ and the scalar λ bmg are related to these two quantities. They do not add more degrees offreedom to the optimization problem. i is a decreasing function of β bmg ,i . Moreover, the Kuhn-Tucker conditionsimplies the following property: x i > ⇒ λ i = 0 ⇒ (cid:26) ∆ i > β bmg ,i < i < β bmg ,i > σσ (cid:62) (cid:12) R where R = C n ( ρ ).Maillard et al. (2010) showed that Σ − = Γ (cid:12) R − where Γ i,j = σ i σ j and: R − = ρ n (cid:62) n − (cid:0) ( n − ρ + 1 (cid:1) I n ( n − ρ − ( n − ρ − − = ρ Γ (cid:12) (cid:0) n (cid:62) n (cid:1) − (cid:0) ( n − ρ + 1 (cid:1) Γ (cid:12) I n ( n − ρ − ( n − ρ − ρξ Γ − (cid:0) ( n − ρ + 1 (cid:1) ξ ˜Γwhere ˜Γ = diag (Γ) is the diagonal matrix with ˜Γ i,i = Γ i,i , and: ξ = 1( n − ρ − ( n − ρ − u ∈ R n be a vector. We deduce that: v = Σ − u = ρξ Γ u − (cid:0) ( n − ρ + 1 (cid:1) ξ ˜ u where ˜ u is a vector with ˜ u i = σ − i u i . Finally, we obtain: v i = ρξ n (cid:88) j =1 Γ i,j u j − (cid:0) ( n − ρ + 1 (cid:1) ξ ˜ u i = nρξσ i ¯ s − (cid:0) ( n − ρ + 1 (cid:1) ξσ i s i (55)where s i = σ − i u i and ¯ s = n − (cid:80) nj =1 s j . If n is very large and we assume that ρ >
0, wehave: ξ ≈ nρ ( ρ − v i ≈ − ρ (cid:18) s i − ¯ sσ i (cid:19) (56)We only consider this case in order to simplify the expression of ∆ i , which is otherwisecomplex and does not help to interpret the impact of the BMG constraint. We obtain:∆ i ≈ − ρ ) σ i (cid:32) λ i σ i − λσ (cid:33) + λ bmg (1 − ρ ) σ i (cid:32) β bmg σ − β bmg ,i σ i (cid:33) = ∆ i + ∆ i (57)We notice that the overweight or underweight of an asset will depend on the relative positionof the statistic σ − i β bmg ,i with respect to its mean. If it is below the mean, the second term∆ i is positive. Generally, we would observe an overweight of asset i .49easuring and Managing Carbon Risk in Investment PortfoliosIn the general case, we have:∆ i = (cid:16) Σ − λ (cid:17) i − λ bmg ˘ β bmg ,i (58)If we omit the impact of the lower bound, ∆ i is positive if ˘ β bmg ,i is negative. A.4.3 Approximation of the tracking error volatility
We recall that the first-order condition is Σ ( x (cid:63) − b ) = λ n + λ − λ bmg β bmg where λ , λ and λ bmg are the Lagrange coefficients associated with the constraints (cid:62) n x = 1, x ≥ n and β (cid:62) bmg x ≤ β +bmg . We deduce that: σ (cid:0) x (cid:63) | b (cid:1) = ( x (cid:63) − b ) (cid:62) Σ ( x (cid:63) − b )= ( x (cid:63) − b ) (cid:62) (cid:0) λ n + λ − λ bmg β bmg (cid:1) = ( x (cid:63) − b ) (cid:62) λ − λ bmg ( x (cid:63) − b ) (cid:62) β bmg = ( x (cid:63) − b ) (cid:62) λ + λ bmg ∆ bmg ≈ c ∆ The last result is obtained because we have ( x (cid:63) − b ) (cid:62) λ ≈ λ bmg is almost a linear function of ∆ bmg for reasonable values of ∆ bmg .For instance, we report in Figure 18 the relationship between ∆ bmg and λ bmg when thebenchmark corresponds to the CW index. This explains that the tracking error volatility isapproximatively a linear function of ∆ bmg .Figure 18: Relationship between ∆ bmg and λ bmg See Equation (51) on page 48.
B Additional results
Table 9: Statistics of factor returns
Factor µ σ γ γ MDD BM WM
MKT 7 .
63% 13 . − .
34 3 . − .
4% 10 . − . − .
10% 4 .
65% 0 .
14 2 . − .
6% 3 . − . − .
82% 5 .
92% 0 .
25 2 . − .
0% 4 . − . .
74% 8 . − .
14 2 . − .
1% 6 . − . − .
52% 6 . − .
06 3 . − .
3% 4 . − . − .
09% 5 .
13% 0 .
14 3 . − .
1% 4 . − . − .
01% 5 .
47% 0 .
36 3 . − .
7% 5 . − . .
15% 3 .
71% 0 .
20 2 . − .
7% 3 . − . .
05% 4 .
68% 0 .
31 4 . − .
8% 5 . − . − .
78% 4 .
56% 0 .
25 2 . − .
8% 3 . − . − .
26% 4 .
63% 0 .
48 3 . − .
8% 4 . − . The statistics µ and σ are the annualized return and volatility, γ and γ are the skewness andkurtosis coefficients, MDD is the observed maximum drawdown, BM and WM are the best andworst monthly returns. BMG corresponds to the Carima factor developed by G¨orgen et al. (2019). Table 10: Correlation matrix of factor returns (in %)
Factor MKT SMB HML WML BMGCarbon intensity − .
46 13 .
71 8 . − .
04 58 . ∗∗∗ Carbon emissions exp. − .
71 14 .
95 4 . − .
03 64 . ∗∗∗ Carbon emissions mgmt. − . ∗ . ∗∗ − . ∗∗ . ∗∗ . ∗∗∗ Carbon emissions 1 .
22 25 . ∗∗∗ − .
23 5 .
15 72 . ∗∗∗ Climate change − .
02 16 . ∗ .
43 2 .
07 61 . ∗∗∗ Environment − . ∗∗∗ . ∗∗ − .
33 3 .
70 68 . ∗∗∗ BMG corresponds to the Carima factor developed by G¨orgen et al. (2019).
Table 11: Comparison of cross-section regressions (in %)
Adjusted R F -test Adjusted R F -testdifference 10% 5% 1% difference 10% 5% 1%CAPM vs MKT+BMG FF vs FF+BMGCarima 1.74 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Source : G¨orgen et al. (2019).
Figure 20: Box plots of the carbon sensitivities by sector and region -2 -1 0 1 2 31015602055502530404535
Eurozone -2 -1 0 1 2 31015602055502530404535
Europe ex EMU -2 0 2 41015602055502530404535
North America -2 -1 0 1 2 31015602055502530404535
Japan
The sorted GICS classification is: energy (10), materials (15), industrials (20), consumer discre-tionary (25), consumer staples (30), health care (35), financials (40), information technology (45),communication services (50), utilities (55) and real estate (60). The box plots provide the median,the quartiles and the 10% and 90% quantiles of the carbon beta.
Eurozone
Europe ex EMU
North America
Japan
Figure 22: Dynamics of the average absolute carbon risk | β | bmg , S ( t ) by sector -0.5 0 0.5 1 1.5 2 2.5 3-4-3-2-10123456 Figure 24: Box plots of the carbon sensitivities by sector and factor -2 0 2 41015202530354045505560
Carbon emissions -2 0 2 41015202530354045505560
Climate change -2 0 2 41015202530354045505560
Environment
The sorted GICS classification is: energy (10), materials (15), industrials (20), consumer discre-tionary (25), consumer staples (30), health care (35), financials (40), information technology (45),communication services (50), utilities (55) and real estate (60). The box plots provide the median,the quartiles and the 10% and 90% quantiles of the carbon beta. CI + = 315) -0.5 -0.4 -0.3 -0.2 -0.1 0859095100 Figure 26: Volatility of the constrained MV portfolios -0.5 -0.4 -0.3 -0.2 -0.1 0250350450550
Based on the MKT+BMG model, the monthly volatility of the capitalisation-weighted MSCI Worldindex is 13 .
10 25 50100 500 1500 5000-5-4-3-2-101234510 25 50100 500 1500 5000-5-4-3-2-1012345 10 25 50100 500 1500 5000-5-4-3-2-1012345
Figure 28: Scatter plot of CW weights and BMG sensitivities
50 100 150 200-4-202468 β bmg ,i and ∆ i = x (cid:63)i − b i for the CW benchmark -2 -1 1 2 3 4 5-6-4-2246 Figure 30: Relationship between ˘ β bmg ,i and ∆ i = x (cid:63)i − b i for the CW benchmark -1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1-6-4-2246 Figure 32: Solution of the alternative order-statistic optimization problem0 0.1 0.2 0.3 0.4 0.50100200300400 0 0.1 0.2 0.3 0.4 0.502550751000 0.1 0.2 0.3 0.4 0.5050010001500 0 0.1 0.2 0.3 0.4 0.560120180240300360