The Market Measure of Carbon Risk and its Impact on the Minimum Variance Portfolio
Théo Roncalli, Théo Le Guenedal, Frédéric Lepetit, Thierry Roncalli, Takaya Sekine
TThe Market Measure of Carbon Risk andits Impact on the Minimum Variance Portfolio ∗ Th´eo RoncalliMaster in Bioinformatics & BiostatisticsUniversity Paris-Saclay, Paris [email protected]
Th´eo Le Guenedal, Fr´ed´eric Lepetit, Thierry Roncalli, Takaya SekineQuantitative ResearchAmundi Asset Management, France [email protected]
January 2021
Abstract
Like ESG investing, climate change is an important concern for asset managersand owners, and a new challenge for portfolio construction. Until now, investors havemainly measured carbon risk using fundamental approaches, such as with carbon in-tensity metrics. Nevertheless, it has not been proven that asset prices are directlyimpacted by these fundamental-based measures. In this paper, we focus on anotherapproach, which consists in measuring the sensitivity of stock prices with respect to acarbon risk factor. In our opinion, carbon betas are market-based measures that arecomplementary to carbon intensities or fundamental-based measures when managinginvestment portfolios, because carbon betas may be viewed as an extension or forward-looking measure of the current carbon footprint. In particular, we show how this newmetric can be used to build minimum variance strategies and how they impact theirportfolio construction.
Keywords:
Carbon, climate change, risk factor, carbon beta, carbon intensity, minimumvariance portfolio.
JEL classification:
C61, G11.
Key findings:
1. Measuring carbon risk is different if we consider a fundamental-based approach byusing carbon intensity metrics or a market-based approach by using carbon betas.2. Managing relative carbon risk implies to overweight green firms, whereas managingabsolute carbon risk implies having zero exposure to the carbon risk factor. The firstapproach is an active management bet, while the second case is an immunizationinvestment strategy.3. Both specific and systematic carbon risks are important when building a minimumvariance portfolio and justify combining fundamental and market approaches of carbonrisk. ∗ The authors are grateful to Martin Nerlinger from the University of Augsburg, who provided themwith the time series of the BMG risk factor (see https://carima-project.de/en/downloads for more detailsabout this carbon risk factor). They would also like to thank Melchior Dechelette, Lauren Stagnol andBruno Taillardat for their helpful comments. a r X i v : . [ q -f i n . P M ] J a n INTRODUCTION
According to Mark Carney (2019), climate change is one of the big current challenges facedby the financial sector with the goal to accelerate the transition to a low carbon economy.It concerns all the financial institutions: central banks, commercial banks, insurance com-panies, asset managers, asset owners, etc. Among the several underlying topics, the riskmanagement of climate change will be one of the pillars of the future regulation in order toensure financial sector resilience to a tail risk. Since risk management must concern bothphysical and transition risks (Carney, 2015), incorporating climate change when managingbanks’ credit portfolios is not obvious. The question is how climate change impacts thedefault probability of issuers. The same issue occurs when we consider stock and bondportfolios of asset managers and owners. Indeed, we have to understand how asset pricesreact to climate change. This is why we have to develop new risk metrics in order to assessthe relationship between climate change and asset returns. However, we face data collectionissues when we consider this broad subject. Therefore, we focus here on carbon risk sinceit is the main contributor to climate change and we have more comprehensive and robustdata on carbon metrics at the issuer level.The general approach to managing an investment portfolio’s carbon risk is to reduce orcontrol the portfolio’s carbon footprint, for instance by considering CO2 or CO2e emissions.This approach assumes that the carbon risk will materialize and that having a portfoliowith a lower exposure to CO2 emissions will help to avoid some future losses. The mainassumption of this approach is then to postulate that firms that currently have high carbonfootprints will be penalized in the future in comparison with firms that currently have lowcarbon footprints. In this paper, we use an alternative approach. We define carbon risk froma financial point of view, and we consider that the carbon risk of equities corresponds to themarket risk priced in by the stock market. This carbon financial risk can be decomposedinto a common (or systematic) risk factor and a specific (or idiosyncratic) risk factor. Sinceidentifying the specific risk is impossible, we focus on the common risk factor that drivesthe carbon risk. The objective is then to build a market-based risk measure to manage thecarbon risk in investment portfolios. This is exactly the framework proposed by G¨orgen etal. (2019) in their seminal paper.In this framework, the carbon financial risk of a stock corresponds to its price sensitivityto the carbon risk factor. This carbon beta is a market-based relative risk and may be viewedas an extension or forward-looking measure of the carbon footprint, where the objective isto be more exposed to green firms than to brown ones. In this case, this is equivalent topromoting stocks with a negative carbon beta over stocks with a positive carbon beta. Thisapproach of relative carbon risk differs from the approach of absolute carbon risk, which ismeasured at the stock level by the absolute value of the carbon beta, because absolute carbonrisk considers that both large positive and negative carbon beta values incur a financial riskthat must be reduced. This is an agnostic or neutral method, contrary to the first methodwhich is more related to investors’ moral values or convictions.Since the 2008 Global Financial Crisis, institutional investors have widely used minimumvariance (MV) strategies to reduce their equity investments’ market risk. While the originalidea of these strategies was to reduce the portfolio’s volatility, today the goal of minimumvariance strategies is to manage the largest financial unrewarded risks and not just volatilityrisk. This is why sophisticated MV programs also include idiosyncratic valuation risk,reputational risk, etc. In this context, incorporating climate risk into minimum varianceportfolios is natural. Therefore, we propose a two-factor model that is particularly adaptedto this investment strategy and show that the solution depends on whether we would liketo manage relative or absolute carbon risk. This implies that we consider transition risks, not physical rirks. THE MARKET MEASURE OF CARBON RISK
To manage a portfolio’s carbon risk, carbon risk needs to be measured at the companylevel. There are different ways to measure this risk, including the fundamental and mar-ket approaches. In this paper, we 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 compare thisseminal approach with a simplified approach, which consists in using direct metrics such ascarbon intensity. Once carbon betas are computed, we can analyze the carbon risk of eachcompany priced in by the stock market and compare it with the carbon intensity, which isthe most used fundamental-based measure of carbon risk. We also discuss the differencebetween relative and absolute carbon risk.
Measuring a company’s carbon risk using the carbon beta of its stock price was first proposedby G¨orgen et al. (2019). In what follows, we summarize their approach and test alternativeapproaches. Moreover, we suggest using the Kalman filter in order to estimate the dynamiccarbon beta of stock prices.
The Carima approach
The goal of the carbon risk management (Carima) project, de-veloped by G¨orgen et al. (2019), is to develop “ a quantitative tool in order to assess theopportunities of profits and the risks of losses that occur from the transition process ”. TheCarima approach combines a market-based approach and a fundamental approach. Indeed,the carbon risk of a firm or a portfolio is measured by considering the dynamics of stockprices which are partly determined by climate policies and transition processes towards agreen economy. Nevertheless, a prior fundamental approach is important to quantify car-bon risk. In practical terms, the fundamental approach consists in defining a carbon riskscore for each stock in an investment universe using a set of objective measures, whereas themarket approach consists in building a brown minus green or BMG carbon risk factor, andcomputing the risk sensitivity of stock prices with respect to this BMG factor. Therefore,the carbon factor is derived from climate change-relevant information from numerous firms.In the Carima approach, the BMG factor is developed using a large amount of climate-relevant information provided by different databases. In the following, we detail the method-ology used by the Carima project to construct the BMG factor. Two steps are required todevelop this new common risk factor: (1) the development of a scoring system to determineif a firm is green, neutral or brown and (2) the construction of a mimicking factor portfoliofor carbon risk which has a long exposure to brown firms and a short exposure to greenfirms. The first step consists in defining a brown green score (BGS) using a fundamentalapproach 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. Each variable is transformedinto a dummy derived with respect to the median, meaning that 1 corresponds to a brownvalue and 0 corresponds to a green value. Then, G¨orgen et al. (2019) classified the variablesinto three different dimensions that may affect the stock value of a firm in the event ofunexpected shifts towards a low carbon economy: (1) value chain, (2) public perception and(3) adaptability. The value chain dimension mainly deals with current emissions while theadaptability dimension reflects potential future emissions determined in particular by emis-sion reduction targets and environmental R&D spending. Then, three scores are created3nd correspond to the average of all variables contained in each dimension: the value chain
VC, the public perception
PP and the non-adaptability
NA. It follows that each score hasa range between 0 and 1. G¨orgen et al. (2019) proposed defining the brown green score(BGS) using the following equation:BGS i ( t ) = 23 (0 . · VC i ( t ) + 0 . · PP i ( t )) + NA i ( t )3 (0 . · VC i ( t ) + 0 . · PP i ( t )) (1)The higher the BGS value, the browner the firm. The second step consists in constructing abrown minus green (BMG) risk factor. Here the Carima project considers an average BGSfor each stock that corresponds to the mean value of the BGS over the period in question,from 2010 to 2016. The construction of the BMG factor follows the methodology of Famaand French (1992), which consists in splitting the stocks into six portfolios: small green(SG), small neutral (SN), small brown (SB), big green (BG), big neutral (BN) and bigbrown (BB). Then, the return of the BMG factor is defined as follows: R bmg ( t ) = 12 ( R SB ( t ) + R BB ( t )) −
12 ( R SG ( t ) + R BG ( t )) (2)where the returns of each portfolio are value-weighted. Alternative approaches
Since the Carima approach is based on 55 variables from 4ESG databases, it may be complicated for investors and academics to reproduce the BMGfactor of G¨orgen et al. (2019). This is why Roncalli et al. (2020) proposed several proxiesthat may be easily computed. They used the same approach to build the BMG factor, butreplaced the brown green score by simple scoring systems using a single variable. Amongthe different tested factors , Roncalli et al. (2020) showed that the Carima BMG factor ishighly correlated to two BMG factors based on (1) the carbon intensity derived on the threescopes (Trucost dataset) and (2) the MSCI carbon emissions exposure score (MSCI, 2020).In Exhibit 1, we report the cumulative performance of these two factors and the Carimafactor. We observe that the three factors are very similar and highly correlated. On average,we observe that brown firms slightly outperformed green firms from 2010 to 2012. Then,the cumulative return fell by almost 35% because of the unexpected path in the transitionprocess towards a low carbon economy. From 2016 to the end of the study period, brownfirms created a slight excess performance. Overall, the best-in-class green stocks outperformthe worst-in-class green stocks over the study period with an annual return of 2 .
52% for theCarima factor, 3 .
09% for the carbon intensity factor and 4 .
01% for the factor built with thecarbon emissions exposure score.
Estimation of the carbon beta
G¨orgen et al. (2019) and Roncalli et al. (2020) testedseveral models to estimate the carbon beta by considering different sets of risk factors thatinclude market, size, value and momentum risk factors. While the first authors used astatic approach by assuming that the carbon beta is constant over the period, the secondauthors proposed a dynamic approach by assuming that the betas are time-varying. Thisis more realistic since carbon betas may evolve with the introduction of a climate-relatedpolicy, a firm’s environmental controversies, a change in the firm’s environmental strategy,an increased incorporation of carbon risk into portfolio strategies, etc. In what follows, weconsider the dynamic approach with a two-factor model. Let R i ( t ) be the monthly returnof stock i at time t . We assume that: R i ( t ) = α i ( t ) + β mkt ,i ( t ) R mkt ( t ) + β bmg ,i ( t ) R bmg ( t ) + ε i ( t ) (3) To build these factors, they have considered the stocks that were present in the MSCI World indexduring the 2010-2018 period. where R mkt ( t ) is the return of the market risk factor, R bmg ( t ) is the return of the BMGfactor and ε i ( t ) ∼ N (cid:0) , ˜ σ i (cid:1) is a white noise. The alpha component and the beta sensitivitiesfollow a random walk: α i ( t ) = α i ( t −
1) + η alpha ,i ( t ) β mkt ,i ( t ) = β mkt ,i ( t −
1) + η mkt ,i ( t ) β bmg ,i ( t ) = β bmg ,i ( t −
1) + η bmg ,i ( t ) (4)where η alpha ,i ( t ) ∼ N (cid:16) , σ ,i (cid:17) , η mkt ,i ( t ) ∼ N (cid:16) , σ ,i (cid:17) and η bmg ,i ( t ) ∼ N (cid:16) , σ ,i (cid:17) are three independent white noise processes.In the sequel of the paper, we use the Carima factor to estimate the carbon beta. Forthe market factor, we use the time series provided by Kenneth French on his website. Weestimate α i ( t ), β mkt ,i ( t ) and β bmg ,i ( t ) for the stocks that belong to the MSCI World indexbetween January 2010 and December 2018 using the Kalman filter (Fabozzi and Francis,1978). Moreover, we scale the Carima risk factor so that it has the same volatility as themarket risk factor over the entire period, implying that the magnitude of the carbon beta β bmg ,i ( t ) may be understandable and comparable to the magnitude of the market beta β mkt ,i ( t ).The average carbon beta of a stock is equal to 0 .
05, which is close to zero, whereas themonthly variation of the carbon beta has a standard deviation of 6 . .
02 and 5 . More precisely, 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. whereas the other sectors have a neutral or negative carbon beta. The resultsdiffer slightly from those obtained by G¨orgen et al. (2019) and Roncalli et al. (2020), whoprovided a sector analysis by considering a constant carbon beta over the period 2010–2018.Exhibit 2: Box plots of the dynamic carbon betas at the end of 2018 -0.5 0 0.5 1 1.5 2EnergyMaterialsReal EstateIndustrialsUtilitiesCommunication ServicesConsumer DiscretionaryFinancialsConsumer StaplesInformation TechnologyHealth Care 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 In Exhibit 3, we report β bmg , R ( t ) for several MSCI universes at the end of each year:World (WD), North America (NA), EMU, Europe-ex-EMU (EU) and Japan (JP). Whateverthe study period, the carbon beta β bmg , R ( t ) is positive in North America, which impliesthat American stocks are negatively influenced by an acceleration in the transition processtowards a green economy. The average carbon beta is always negative in the Eurozone.Overall, the Eurozone has always a lower average carbon beta than the world as a whole,whereas the opposite is true for North America. Nevertheless, the negative sensitivity ofEuropean equity returns has dramatically decreased since 2010 and the BMG betas aregetting closer for North America and the Eurozone. In the previous paragraph, the relative carbon risk of a stock i at time t is measured by itscarbon beta value: RCR i ( t ) = β bmg ,i ( t )A majority of investors will prefer stocks with a negative carbon beta over stocks witha positive carbon beta. However, an investment portfolio with a negative carbon beta is This is in line with Bouchet and Le Guenedal (2020) who demonstrated that credit risks are morematerial in the energy and materials sectors. Therefore, the market perceives these sectors as the entrypoints for systemic financial carbon risks.
ACR i ( t ) = | β bmg ,i ( t ) | Exhibit 4 presents the sector analysis of the absolute carbon risk at the end of December2018. From this point of view, utilities is the least exposed sector to absolute carbon risk,whereas the energy and materials are the sectors that are the most exposed.Exhibit 4: Box plots of the absolute carbon risk at the end of 2018
ACR i ( t ) is also a pricing magnitude measure of the carbon risk. Let us consider aninvestment universe with two stocks. We assume that β bmg , (1) = 0 . β bmg , (1) = − . .
5. One year later, we obtain β bmg , (2) = 1 and β bmg , (2) = −
1. In this case, therelative carbon risk of the investment universe has not changed and is always equal to zero.7owever, its absolute carbon risk has increased and is now equal to 1. It is obvious that thecarbon risk is priced in more in the second period than in the first period.We have reported the absolute carbon risk by region in Exhibit 5. We notice that thecarbon risk was priced in more in 2011 and 2012, because of the pricing magnitude in theEurozone. In this region, the absolute carbon risk has dramatically decreased from 50%in 2011 to 27% in 2018. More globally, we observe a convergence between the differentdeveloped regions. One exception is Japan, where the absolute carbon risk is 50% lowerthan in Europe and North America.Exhibit 5: Absolute carbon risk by region (end of year)Year WD NA EMU EU JP2010 0.35 0.32 0.50 0.35 0.302011 0.34 0.32 0.51 0.32 0.312012 0.28 0.24 0.40 0.24 0.272013 0.28 0.26 0.31 0.24 0.302014 0.27 0.26 0.30 0.24 0.262015 0.27 0.29 0.27 0.27 0.202016 0.29 0.31 0.30 0.30 0.202017 0.27 0.29 0.30 0.28 0.202018 0.28 0.29 0.27 0.29 0.20
ESG rating agencies have developed many fundamental measures and scores to assess afirm’s carbon risk. For instance, the most well-known is the carbon intensity CI i ( t ), whichinvolves scopes 1, 2 and 3. In this paper, a firm’s carbon risk corresponds to the carbonbeta priced in by the financial market. It is not obvious that there is a strong relationshipbetween fundamental and market measures, because we may observe wide discrepanciesbetween the market perception of the carbon risk and the carbon intensity of the firm. Forinstance, the linear correlation between CI i ( t ) and β bmg ,i ( t ) is equal to 17 .
4% at the endof December 2018. If we consider the BMG factor built directly with the carbon intensity(Exhibit 1), the correlation increases but remains relatively low since it is equal to 21 . CI i ( t ) and β bmg ,i ( t ) is then morecomplex as seen in Exhibit 6.This result is easily understandable because the stock market incorporates dimensionsother than carbon intensity to price in the carbon risk. From a fundamental point ofview, if the carbon intensity of two firms is equal to 100, they present the same carbon risk.Nevertheless, we know that their risks depend on other factors and parameters. For instance,it is difficult to compare two firms with the same carbon intensity if they belong to twodifferent sectors or countries. The trajectory of the carbon intensity is also another importantfactor. For instance, the risk is not the same if one firm has dramatically decreased itscarbon intensity in recent years. Moreover, the adaptability issue, the capacity to transformits business with investments in green R&D or its financial resources to absorb transitioncosts (Bouchet and Le Guenedal, 2020) are other important parameters that impact themarket perception of the firm’s carbon risk. Therefore, carbon intensity is less appropriateto describe financial risks than carbon beta. In other words, the carbon beta is an integratedmeasure of the different fundamental factors affecting a firm’s carbon risk.8xhibit 6: Scatter plot of CI i ( t ) and β bmg ,i ( t ) at the end of 2018
10 25 50 100 250 500 1000 5000-1.5-1-0.500.511.5
In Exhibit 7, we report the correlation between CI i ( t ) and β bmg ,i ( t ) at the end ofDecember 2018. We notice that it is higher in the Eurozone than in other regions. Inparticular, the correlation in Japan is very low (less than 5%). Moreover, we observe that italso differs with respect to the sector. For instance, the financial sector presents the lowestcorrelation value, certainly because the carbon risk of financial institutions is less connectedto their greenhouse gas emissions than their (green and brown) investments and financingprograms.Exhibit 7: Correlation in % between CI i ( t ) and β bmg ,i ( t ) at the end of 2018Sector WD NA EMU EU JPFinancials 18.2 20.1 29.2 20.9 -1.5Energy 18.2 17.8 31.8 24.5 3.8Materials 20.3 24.8 37.2 28.0 5.4Information Technology 20.4 21.0 34.2 26.1 3.2Health Care 20.9 21.3 34.5 26.3 4.2Consumer Staples 21.5 22.3 35.1 26.4 4.3Communication Services 21.6 22.2 32.2 24.7 6.6Consumer Discretionary 22.3 23.1 37.8 25.8 2.6Real Estate 22.4 22.5 34.3 26.1 6.1Industrials 23.6 23.8 38.7 31.6 8.1Utilities 26.6 29.8 26.5 26.1 8.4All sectors 21.4 22.3 33.8 26.2 4.69 INCORPORATING CARBON RISK INTO MINI-MUM VARIANCE PORTFOLIOS
There is an increasing appetite from fund managers of minimum variance portfolios to takeinto account carbon risk because of two main reasons. First, it is a financial and regulationrisk that may negatively impact stock returns. The second reason is that it is highly soughtafter by institutional investors. In what follows, we show how to incorporate carbon risk intothese strategies. In particular, we provide an analytical formula which is useful to understandthe impact of carbon betas on the minimum variance portfolio and the covariance matrix ofstock returns. We also discuss the different practical implementations of minimum varianceportfolios when we consider market and fundamental measures of carbon risk.
In this paragraph, we extend the famous formula of the minimum variance portfolio whenwe complement the market risk factor with the BMG factor. Then, we illustrate how theminimum variance portfolio selects stocks in the presence of carbon risk.
Extension of the one-factor GMV formula
We consider the global minimum variance(GMV) portfolio, which corresponds to this optimization program: x (cid:63) = arg min 12 x (cid:62) Σ x (5)s.t. (cid:62) n x = 1where x is the vector of portfolio weights and Σ is the covariance matrix of stock returns.In the capital asset pricing model, we recall that: R i ( t ) = α i + β mkt ,i R mkt ( t ) + ε i ( t ) (6)where R i ( t ) is the return of asset i , R mkt ( t ) is the return of the market factor, ε i ( t ) ∼N (cid:0) , ˜ σ i (cid:1) is the idiosyncratic risk and ˜ σ i is the idiosyncratic volatility. Clarke et al. (2011)and Scherer (2011) showed that: x (cid:63)i = σ ( x (cid:63) )˜ σ i (cid:18) − β mkt ,i β (cid:63) mkt (cid:19) (7)where β (cid:63) mkt is a threshold and σ ( x (cid:63) ) is the variance of the GMV portfolio. Therefore, wenote that the minimum variance portfolio is exposed to stocks with low specific volatility˜ σ i and low beta β mkt ,i . More precisely, if asset i has a market beta β mkt ,i smaller than thethreshold β (cid:63) mkt , the weight of this asset is positive: x (cid:63)i >
0. If β mkt ,i > β (cid:63) mkt , then x (cid:63)i < β (cid:63) mkt is anotherthreshold. In this case, if β mkt ,i < β (cid:63) mkt , x (cid:63)i > 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 ) (8)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.Roncalli et al. (2020) showed 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) (9)10here β (cid:63) mkt and β (cid:63) bmg are two threshold values. In the case of long-only portfolios, we obtaina similar formula: x (cid:63)i = σ ( x (cid:63) )˜ σ i max (cid:32) − β mkt ,i β (cid:63) mkt − β bmg ,i β (cid:63) bmg ; 0 (cid:33) (10)but with other values of the thresholds β (cid:63) mkt and β (cid:63) bmg . Interpretation of these results
Contrary to the single-factor model, the impact ofsensitivities is more complex in the two-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 . Wededuce that the ratio β mkt ,i β (cid:63) mkt is an increasing function of β mkt ,i , but the ratio β bmg ,i β (cid:63) bmg maybe an increasing or a decreasing function of β bmg ,i . The GMV portfolio will then alwaysprefer stocks with low market betas, but not necessarily stocks with low carbon betas. Forinstance, it may prefer stocks with high carbon betas if β (cid:63) bmg is negative.In the long-only case, a stock is selected if it satisfies the following inequality: β mkt ,i β (cid:63) mkt + β bmg ,i β (cid:63) bmg ≤ β mkt ,i and β bmg ,i . Nevertheless,Roncalli et al. (2020) showed that the long-only MV portfolio tends to prefer stocks withlow absolute carbon risk.We recall that the volatility of stock i is equal to σ i = β ,i σ + β ,i σ +˜ σ i , whereas the covariance between stocks i and j is equal to σ i,j = β mkt ,i β mkt ,j σ + β bmg ,i β bmg ,j σ . Therefore, choosing stocks with low volatilities implies considering stockswith low values of β ,i . In a similar way, removing stocks with high positive correlationsimplies removing stocks with high values of β bmg ,i β bmg ,j . This explains that the MV portfoliowill prefer stocks with low values of | β bmg ,i | . We now apply the previous framework to the MSCI World index at December 2018, andillustrate the difference between absolute and relative carbon risk when we consider theminimum variance portfolio. Moreover, we compare these market-based approaches withimplementations of minimum variance portfolios that use fundamental carbon risk metrics.
Impact of carbon risk
In Exhibit 8, we indicate the stocks that make up the MV portfoliowith respect to their beta values β mkt ,i and β bmg ,i . We find that the most important axis isthe market beta. Indeed, the market risk of a stock determines whether the stock is includedin the MV portfolio or not whereas the carbon risk adjusts the weights of the asset. As wecan see, the portfolio overweights assets whose market and carbon sensitivities are both closeto zero. This solution is satisfactory if the original motivation is to reduce the portfolio’sabsolute carbon risk, but it is not satisfactory if the objective is to manage the portfolio’srelative carbon risk. Considering relative carbon risk
In order to circumvent the previous drawback, wecan directly add a constraint in the optimization program: β bmg ( x ) = n (cid:88) i =1 x i × β bmg ,i ≤ β +bmg (11) Moreover, it generally takes a high absolute value. -0.5 -0.25 0 0.25 0.5 0.75 1-1.5-1-0.500.511.522.5
Exhibit 9: Weights of the constrained MV portfolio ( β +bmg = − . -0.5 -0.25 0 0.25 0.5 0.75 1-1.5-1-0.500.511.522.5 β bmg ( x ) is the carbon beta of portfolio x and β +bmg is the maximum tolerance of theinvestor with respect to the relative carbon risk. We consider the previous example. If wewould like to impose a carbon sensitivity of lower than − .
25, we obtain results given inExhibit 9. The comparison with Exhibit 8 shows that the MV portfolio tends to select stockswith both a low market sensitivity and a negative carbon beta. Moreover, large weights areassociated with large negative values of β bmg ,i on average. Managing both market and fundamental risk measures
The previous method isnot the standard approach when managing carbon risk in investment portfolios. Indeed, theasset management industry generally considers constraints on carbon intensity measures.Following Andersson et al. (2016), we can impose individual constraints on the differentstocks: x i = 0 if CI i ≤ CI + (12)or we can use a global constraint: WACI ( x ) = n (cid:88) i =1 x i × CI i ≤ WACI + (13)where CI i is the carbon intensity of stock i and WACI ( x ) is the weighted average carbonintensity of portfolio x . CI + and WACI + are the individual and portfolio thresholds thatare accepted by investors.We may wonder whether managing the fundamental measure of carbon risk is equivalentto managing the market measure of carbon risk . A preliminary answer has been providedpreviously since we have found that the correlation between β bmg ,i and CI i is less than30% on average. In Exhibit 10, we compute the minimum variance portfolio by consideringseveral threshold values of β +bmg . We notice that using a lower value of β +bmg reduces the valueof WACI ( x ), but WACI ( x ) remains very high because some issuers have a low commoncarbon risk, but a high idiosyncratic carbon risk. We have also reported the number ofstocks N ( x ) in the MV portfolio. As expected, it decreases when we impose a strongerconstraint. Exhibit 11 is a variant of Exhibit 10 by considering a constraint WACI + on theportfolio’s carbon intensity instead of a constraint β +bmg on the portfolio’s carbon beta. Here,the impact on the portfolio’s carbon beta is low when we strengthen the constraint. Indeed,the portfolio’s carbon beta β bmg ( x ) is equal to 1.43% when we target a carbon intensity of500, whereas it drops to 1.33% when the constraint on the carbon intensity is set to 50.Exhibit 10: Minimum variance portfolios with a relative carbon beta constraint β +bmg β bmg ( x ) WACI ( x ) N ( x )1.43% 538 105-10.00% -10.00% 501 100-20.00% -20.00% 422 89-40.00% -40.00% 289 70These results show that the two optimization problems give two different solutions interms of carbon risk. Therefore, it makes sense to combine the approaches by imposing twoconstraints: (cid:26) WACI ( x ) ≤ WACI + β bmg ( x ) ≤ β +bmg (14) In what follows, we also impose that CI + = 4 000. WACI + WACI ( x ) β bmg ( x ) N ( x )500 500 1.43% 105250 250 1.37% 103100 100 1.36% 9850 50 1.33% 82Exhibit 12: Minimum variance portfolios with carbon beta and intensity constraints WACI + WACI ( x ) β bmg ( x ) N ( x ) WO ( x )500 430 -20.00% 111 74.65%250 250 -20.00% 86 75.26%100 100 -20.00% 79 74.87%50 50 -20.00% 74 74.99%Moreover, the threshold β +bmg allows us to reduce the common carbon risk, but not theidiosyncratic carbon risk. The WACI constraint circumvents this problem. Exhibit 12presents the results for several values of WACI + when β +bmg is equal to − WACI + = 500 and β +bmg = − WO ( x ) is equalto 75% on average. This means that 25% of the minimum variance portfolio allocation ischanged when we add the market carbon constraint β +bmg = − This paper considers the seminal approach of G¨orgen et al. (2019) to measuring carbon risk.While many asset managers and owners use carbon intensity, we focus on the carbon betawhich is priced in by the market. The carbon beta is estimated using a two-step approach.First, we build a brown-minus-green risk factor. Second, we perform a Kalman filtering inorder to obtain the time-varying carbon beta. By considering this dynamic framework, wehighlight several stylized facts. We show that this market measure is very different froma traditional fundamental measure of the carbon risk. The main reason is that carbonintensity is not the only dimension that is priced in by the market.Another important result is the difference between relative and absolute carbon risk.Investors that are sensitive to relative carbon risk prefer stocks with a negative carbon betaover stocks with a positive carbon beta, whereas investors that are sensitive to absolutecarbon risk prefer stocks with a carbon beta close to zero. Managing relative carbon riskimplies having a negative exposure to the carbon risk factor, whereas managing absolutecarbon risk implies having zero exposure to the carbon risk factor. The first case is an activemanagement bet since the performance may be negative if brown stocks outperform greenstocks. Nevertheless, this approach reduces exposure to firms that face a threat of environ-mental regulation (Maxwell et al., 2000). The second case is an immunization investmentstrategy against carbon risk. However, this hedging strategy is not widely implemented byinstitutional and passive investors because of their moral values and convictions, and theygenerally prefer to implement relative carbon risk strategies.Introducing carbon risk into a minimum variance portfolio is a hot topic among as-14et managers and owners. Indeed, the goal of a minimum variance portfolio is to build alow volatility strategy on the equity market. This is achieved by considering a strong riskmanagement approach on several dimensions. Originally, the strategy only focused on theportfolio’s volatility. Since the 2008 Global Financial Crisis, it has included other risk di-mensions that can burst the equity market such as credit risk and valuation risk. Climaterisk has become another important dimension, especially because minimum variance strate-gies are massively implemented by ESG institutional investors. In this context, the questionof carbon metrics is important. In this paper, we show that managing the carbon intensityof minimum variance portfolios has little impact on their carbon beta. The opposite is nottrue, but the effect of managing the carbon beta on carbon intensity is limited. This is whywe propose combining the market and fundamental approaches to carbon risk. Anotherissue concerns the choice of the market carbon risk measure. We show that the optimizationprogram of a minimum variance portfolio naturally considers absolute carbon risk. However,relative carbon risk can also be an option if the investor’s goal is not to hedge the carbonrisk, but to be a green investor.
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