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A Sustainable Capital Asset Pricing Model

(S-CAPM): Evidence from Environmental

Integration and Sin Stock Exclusion*

Olivier David Zerbib

EDHEC Business School, Nice, France

Abstract

This article shows how sustainable investing—through the joint practice of exclu- sionary screening and environmental, social, and governance (ESG) integration— affects asset returns. I develop an asset pricing model with partial segmentation and heterogeneous preferences. I characterize two exclusion premia generalizing Merton’s (1987) premium on neglected stocks and a taste premium that clarifies the relationship between ESG and financial performance. Focusing on US stocks, I esti- mate the model by applying it to sin stocks as excluded assets and using the hold- ings of green funds to proxy for environmental integration. The average annual ex- clusion effect is 2.79% for the period 1999–2019. Although the annual taste effect ranges from –1.12% to þ 0.14% across industries for 2007–19, the taste effect spread between the top and bottom terciles of companies within each industry can exceed 2% per year. Finally, I estimate and explain the dynamics of these premia.

Keywords: Sustainable finance, Asset pricing, ESG, Sin stocks

* I am grateful to Marcin Kacperczyk (the editor) and two anonymous referees for the insightful com-

ments and suggestions, which significantly improved the article. I also sincerely thank Rob Bauer,

Milo Bianchi, Claire Bonello, Marco Ceccarelli, Julio Crego, Patricia Crifo, Joost Driessen, Esther

Eiling, Caroline Flammer, Olivier Guéant, James Guo, Ulrich Hege, Ying Jiao, Sonia Jimenez Garces,

Frank de Jong, Nabil Kazi-Tani, Peter Kondor, Felix Kübler, Augustin Landier, Dong Lou, Valéry

Lucas-Leclin, Sophie Moinas, Lionel Melin, Martin Oehmke, Sébastien Pouget, Kevin Ratsimiveh,

Christian Robert, Bert Scholtens, Paul Smeets, Dimitri Vayanos, Michela Verardo, Alexander

Wagner, workshop participants at the London School of Economics, Tilburg University, University

of Zurich–SFI, Toulouse School of Economics, CREST, Paris Dauphine University, University of Lille,

University of Orléans, I Care, and ISFA, Boston University, University of Toronto—Rotman School of

Management, RSM Erasmus Rotterdam, EDHEC, ESSEC, ESCP, ESADE, Aix-Marseille School of

Economics, University of Luxembourg, and the Climate Economics Chair, and conference partici-

pants at GRASFI (2020), Paris December Meeting (2020), Geneva Summit on Sustainable Finance

(2020) for their valuable comments and suggestions. This article was written when I was at Tilburg

University and Boston University, and previously circulated under the title “A Sustainable Capital

Asset Pricing Model (S-CAPM): Evidence from green investing and sin stock exclusion.”

VC The Author(s) 2022. Published by Oxford University Press on behalf of the European Finance Association.

All rights reserved. For permissions, please email: [email protected]

Review of Finance, 2022, 1345–1388

https://doi.org/10.1093/rof/rfac045

Advance Access Publication Date: 21 July 2022

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JEL classification: G12, G11

Received December 29, 2021; accepted June 30, 2022 by Editor Marcin Kacperczyk.

1. Introduction

Sustainable investing, which accounts for more than one-quarter of the total assets under

management (AUM) in the USA (US SIF, 2018) and more than half of those in Europe

(GSIA, 2018), is usually based on the inclusion of several criteria related to environmental,

social, or governance (ESG) issues in investment decisions.1 As of December 2019, of the

453 mutual funds investing in the USA and classified as green by Bloomberg,2 57% of them

were also classified as “socially responsible.” This substantial proportion highlights the fact

that environmental and social criteria are often jointly considered by sustainable investors.

The two most widely used sustainable investment practices are exclusionary screening

and ESG integration (GSIA, 2018). On the one hand, exclusionary screening involves the

exclusion of certain assets from the range of eligible investments, usually the most socially

controversial assets, such as the stocks of companies in the tobacco, alcohol, gambling, and

weapons industries, also referred to as “sin stocks.” On the other hand, ESG integration

involves factoring ESG criteria into investment decisions. Specifically, when sustainable

investors focus on the environment, which, according to a recent survey by Macquarie

(2021), is sustainable investors’ primary concern and is perceived as the greatest risk on a

global scale (World Economic Forum, 2021), they overweight (underweight) assets with

the highest (lowest) environmental score. The exclusion of assets based on social criteria

and the integration of environmental criteria (referred to as environmental integration),

which are often jointly implemented by sustainable investors, can create major supply and

demand imbalances, thereby influencing market prices. This study develops a theoretical

framework and provides empirical evidence on how these two sustainable investing practi-

ces—separately and jointly—affect asset returns.

To reflect the dual practice of exclusion and integration by sustainable investors, I de-

velop an asset pricing model with partial segmentation and heterogeneous preferences.

Specifically, I propose a single-period equilibrium model populated by two investor groups:

regular investors that invest freely in all available assets and have mean–variance preferen-

ces and sustainable investors that exclude certain assets and adjust their mean–variance

preferences by internalizing a private cost of externalities for the assets in which they invest.

For example, sustainable investors would exclude sin stocks, while assets with a high cost

of externalities can be thought of as the assets of companies with high environmental

1 Sustainable investing is also referred to as socially responsible investing, responsible investing,

and ethical investing. In the European Parliament legislative resolution of April 18, 2019

(COM(2018)0354—C8-0208/2018–2018/0179(COD)), sustainable investments are defined as

“investments in economic activities that contribute to environmental or social objectives as well

[sic] their combination, provided that the invested companies follow good governance practices

and the precautionary principle of ‘do no significant harm’ is ensured, i.e. that neither the environ-

mental nor the social objective is significantly harmed.” In the USA, the AUM in sustainable inves-

ting amounted to USD 12 trillion in 2018 and increased by 38% from 2016 to 2018 (US SIF, 2018).

2 Green funds are classified under the attributes “environmentally friendly,” “clean energy,” and

“climate change.”

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footprint (or brown) and those with a low or negative cost of externalities can be viewed as

those of companies with low environmental footprints (or green).

I propose a unified pricing formula for all assets in the market; namely, the assets

excluded by sustainable investors (hereinafter, excluded assets) and the assets in which they

invest by over- or underweighting them (hereinafter, investable assets). Two types of pre-

mia are induced by sustainable investors: a taste premium and two exclusion premia.

The taste premium materializes through three effects. First, on investable asset returns,

the taste premium is induced by sustainable investors’ tastes for assets owing to the cost of

externalities that they internalize. Consistent with Pastor, Stambaugh, and Taylor (2021b),

this premium increases with the cost of externalities and the wealth share of sustainable

investors. Second, as a consequence, the market risk premium is also adjusted by the aver-

age taste premium. Third, the taste premium arises by commonality on excluded asset

returns: in equilibrium, regular investors overweight investable assets that have the highest

cost of externalities to provide liquidity to sustainable investors who take the opposite pos-

ition. Therefore, in order to diversify their allocation, regular investors most highly value

the excluded assets that are the least correlated with the investable assets having a high cost

of externalities. In other words, when applied to sin stock exclusion and environmental in-

tegration, the taste premium on a sin stock is all the higher, as the asset is positively corre-

lated with the brownest investable assets.

Two exclusion premia affect the excluded asset returns. The exclusion premia result

from a reduction in the investor base and are related to Errunza and Losq’s (1985) super

risk premium and Roon’s (2005) local segmentation premium. I show that one of the two

exclusion premia is a generalized form of the premium on neglected stocks characterized by

Merton (1987). Both exclusion premia are structured similarly and reflect the dual hedging

effect of regular and sustainable investors. Specifically, regular investors, who are com-

pelled to hold the excluded market portfolio, most highly value the assets that are the least

correlated with this portfolio. Simultaneously, sustainable investors, who seek to replicate

the hedging portfolio built from investable assets that are most closely correlated with the

excluded assets, most highly value the assets that are positively correlated with this hedging

portfolio. In practice, the exclusion premia increase when the excluded assets increasingly

behave like a separate group from the investable assets. The exclusion effect is the sum of

the two exclusion premia. Although the exclusion effect on asset returns is, on average,

positive, as empirically assessed by Hong and Kacperczyk (2009) and Chava (2014), I show

that this effect can be negative for an individual excluded asset; for example, when it is

negatively correlated with the other excluded assets. Finally, a cross-effect of one of the two

exclusion premia also drives the investable asset returns.

I empirically validate the theoretical predictions by estimating the model using US stocks

in the Center for Research in Security Prices (CRSP) database from December 1999 to

December 2019 for excluded stocks, and from December 2007 to December 2019 for in-

vestable stocks due to data constraints. More precisely, I use sin stocks to constitute the

assets excluded by sustainable investors, and I apply the environmental integration proced-

ure for investable assets by proxying sustainable investors’ tastes for the stocks of green

firms.

Beyond the econometric specification issue, there are three main reasons for the mixed

results in the empirical literature on the link between environmental and financial perform-

ances. First, the identification of a company’s environmental performance through a par-

ticular environmental metric is a weak proxy for the average tastes of sustainable investors

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for green firms: the various metrics used to assess the environmental impacts of assets lack

a common definition, show low commensurability (Gibson et al., 2020; Berg, Koëlbel, and

Rigobon, 2022), and are updated with a low frequency, typically on an annual basis.

Second, these studies fail to capture the increase in the proportion of green investors over

time. Third, realized returns, used as the dependent variable, are both driven by the taste ef-

fect and the unexpected shifts in investors’ tastes (Pastor, Stambaugh, and Taylor, 2021b).

Indeed, even if a company’s environmental footprint remains unchanged, sustainable

investors’ tastes are dynamically adjusted in response to technological changes, the institu-

tional and socio-political environment, climate policies, reputational risks, and investors’

awareness of environmental issues, thereby affecting realized returns (Bolton and

Kacperczyk, 2022). Hence, the absence of control for unexpected shifts in tastes while using

realized returns as a proxy for expected returns induces a critical omitted variable bias. For

example, if sustainable investors’ tastes for green companies unexpectedly increase, green

assets may outperform brown assets while the former have a lower taste premium than the

latter.

Therefore, I construct a proxy for the tastes of green investors that allows me to address

the three issues raised. First, to circumvent the use of environmental metrics, this agnostic

ex post proxy reflects green investors’ private costs of environmental externalities. I identify

453 green funds worldwide with investments in US equities as of December 2019 and use

the FactSet data to determine their holding history on a quarterly basis. For a given stock

and on a given date, the approximated cost of externalities is the relative difference between

the weight of the stock in the market portfolio and its weight in the US allocation of the

green funds. The higher the cost, the more a stock is underweighted by the green funds on

that date, and vice versa when it is negative. Second, the proxy captures the share of green

investors’ wealth through the proportion of US stocks managed by green funds relative to

the market value of the investment universe. Third, I control for the unexpected shifts in

green investors’ tastes by using the variation of this proxy over time.

For investable stocks, the taste premium is significant from 2007 onward, irrespective of

whether it is estimated by constructing industry-sorted or industry-size double-sorted port-

folios. The taste premium remains significant after controlling for the unexpected shifts in

tastes, as well as for the small-minus-big (SMB), high-minus-low (HML; Fama and French,

1993), and momentum (MOM; Carhart, 1997) factors. At the industry level, the taste ef-

fect ranges from �1.12% to þ0.14%. Indeed, environmental integration significantly con-

tributes toward modifying the expected returns of the industries most impacted by the

ecological transition. For example, on average, during the period 2007–19, green investors

induced additional annual returns of 0.50% for the petroleum and natural gas industry

when compared with the electrical equipment industry. This taste effect has steadily

increased over time, reaching 1.23% between 2013 and 2019. However, many industries

are highly heterogeneous and include companies with different environmental footprints,

such as utilities or electrical equipment. Thus, I perform an intra-industry analysis by

repeating the estimation on portfolios doubly sorted by industry and carbon emissions, and

I estimate the taste effect differential between the tercile of the most carbon-intensive com-

panies (top 33%) and that of low-emitting companies (bottom 33%). For example, the an-

nual taste effect differential reaches 2.46% for utilities and 0.68% for electrical equipment

companies. This differential is high for industries that are exposed to the ecological transi-

tion and have high intra-industry heterogeneity. Conversely, this differential is close to zero

for coal companies, reflecting the very similar treatment by sustainable investors of all coal

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companies, which are substantially underweighted, irrespective of their carbon emissions.

Finally, I also find weak evidence supporting the cross-effect of sin stock exclusion on in-

vestable stock returns.

Regarding sin stocks, I find that both exclusion premia and the taste premium are sig-

nificant and remain so when the SMB, HML, and MOM factors are included. The ordinary

least squares (OLS) adjusted-R2 and generalized least squares (GLS) R2 of the estimated

model are higher than those obtained under Carhart’s (1997) four-factor model. The an-

nual average exclusion effect amounts to 2.79% for the period from December 1999 to

December 2019. I also show that the exclusion effect increased sharply during the 2007–08

crisis because the covariances between the sin stocks increased faster than those of the sin

stocks with the other assets. In addition, consistent with the theory, the exclusion effect is

negative for thirty-three out of the seventy-seven sin stocks analyzed.

1.1 Related Literature

The results of this study contribute to two literature strands on asset pricing. First, they

clarify the relationship between the environmental and financial performances of assets by

building on the heterogeneous preferences and disagreement literature.3 The empirical evi-

dence regarding the effects of ESG integration on asset returns is mixed, as several studies

point to the existence of a negative relationship between ESG performance and stock

returns,4 while others argue in favor of a positive effect.5 Pedersen, Fitzgibbons, and

Pomorski (2021) and Pastor, Stalbaugh, and Taylor (2021b) provide theoretical contribu-

tions on how ESG integration by sustainable investors affects asset returns.6 Pedersen,

Fitzgibbons, and Pomorski (2021) show that when the market is populated by ESG-

motivated, ESG-aware, and ESG-unaware investors, the optimal allocation satisfies a four-

fund separation and is characterized by an ESG-efficient frontier. The authors derive an

asset pricing equation in cases where all investors are ESG-motivated or ESG-unaware.

Pastor, Stalbaugh, and Taylor (2021b) show that green assets have negative alphas, brown

assets have positive alphas, and the alphas of ESG-motivated investors are at their lowest

when investors’ ESG tastes are largely dispersed. Extending the conceptual framework laid

out by Fama and French (2007), I contribute to this literature strand in two ways. First,

from a theoretical viewpoint, when sustainable investors jointly practice ESG integration

and exclusionary screening, I show that sustainable investors’ tastes (i) affect the market

premium and (ii) induce a taste premium on the expected returns of the assets they exclude,

in addition to the taste premium on investable assets characterized by Pastor, Stalbaugh,

and Taylor (2021b). Second, from an empirical viewpoint, this is the first paper (i) in which

the asset pricing specification is estimated using a microfounded proxy for sustainable

investors’ revealed tastes for green companies, (ii) accounting for the increase in green

3 A vast literature has examined the effects of heterogeneous preferences, disagreement, and differ-

ences of opinion on asset returns and prices, including Fama and French (2007); Bhamra and Uppal

(2014); Baker, Hollifield, and Osambela (2016); and Atmaz and Basak (2018).

4 See Renneboog, Ter Horst, and Zhang (2008) and Barber, Morse, and Yasuda (2019). Moreover,

Chava (2014) shows that the same effect applies to the expected returns. Bolton and Kacperczyk

(2021) and Hsu, Li, and Tsou (2019) show that companies emitting the most greenhouse gases earn

higher stock returns than companies emitting the lowest levels.

5 See Edmans (2011) and Krüger (2015).

6 Both papers focus on ESG integration and do not address exclusionary screening.

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investing, and (iii) unexpected shifts in green investors’ tastes. Recent independent papers

also use proxies for investors’ beliefs in climate-related financial risks (e.g., Sautner et al.,

2021, from earnings call discussions) and investors’ shifts in climate concerns (Ardia et al.,

2021; Pastor, Stalbaugh, and Taylor, 2021a). In addition, analyzing carbon-transition risk

in a global sample, Bolton and Kacperczyk (2022) estimate the impact of technological,

socio-economic, regulatory, and investor awareness changes on asset returns.

The results of this study also contribute to the literature on exclusionary screening by

bridging the gap with market segmentation. From a theoretical viewpoint, this study extends

the analysis of Heinkel, Kraus, and Zechner (2001) by characterizing the risk factors associ-

ated with exclusionary screening. I show that the exclusion effect results from the sum of two

exclusion premia, which are related to the premia identified by Errunza and Losq (1985) in

the case of excluded assets and by de Jong and de Roon (2005) as an indirect effect on invest-

able assets. I show that both premia apply to all assets in the market and thus, I identify the

cross-effect of exclusion on investable stock returns. Moreover, I demonstrate that one of the

two exclusion premia is a generalized form of Merton’s (1987) premium on neglected stocks.

Compared with Merton (1987), this study emphasizes the importance of considering non-

independent returns because the exclusion effect is driven by covariances between assets.

From an empirical viewpoint, the magnitude of the average annual exclusion effect for sin

stocks is close to the 2.5% obtained by Hong and Kacperczyk (2009) and is substantially

lower than the 16% found by Luo and Balvers (2017). In addition, beyond the average effect,

the individual exclusion effect is negative for several sin stocks. Finally, Berk and van

Binsbergen (2021) give an approximation of the effect of exclusionary screening on expected

returns. Calibrated on the FTSE USA 4 Good index compared with the FTSE USA, they find

a small effect in the period 2015–20. I show that while the average exclusion effect on sin

stock returns was indeed small in this period, it was large during the 2008–09 crisis because

the intra-group dynamic strengthened. That is, the covariances between sin stocks increased

more than the covariances of sin stocks with non-sin stocks.

The remainder of this article is structured as follows. Section 2 presents the equilibrium

equations of the model and characterizes the resulting premia. Section 3 describes the iden-

tification method used in the empirical analysis when the model is applied to sin stocks

regarded as excluded assets, and to environmental integration for characterizing investors’

tastes for investable assets. Sections 4 and 5 present the empirical results on investable and

excluded stock returns, respectively. Section 6 concludes the article. The Appendix A con-

tains the main proofs, and the Online Appendix provides additional proofs and details

about the empirical analysis.

2. Asset Pricing with Partial Segmentation and Heterogeneous Preferences

To reflect sustainable investors’ dual practice of excluding certain assets (e.g., sin stocks)

and over- or underweighting other assets (e.g., through their preferences for green company

stocks), I develop an asset pricing model with partial segmentation and heterogeneous pref-

erences among investors. I show how the expected excess returns deviate from those pre-

dicted by the capital asset pricing model (CAPM) and identify two types of premia that

occur in equilibrium: a taste premium and two exclusion premia. I also show that exclusion

and taste premia have cross-effects on investable and excluded assets.

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2.1 Motivating Examples

Even among different types of investors, certain common features emerge from their sustain-

able investment policies. For example, AXA IM (asset manager of the insurer AXA), BNP

Paribas AM (asset manager of the bank BNP Paribas), and the US Conference of Catholic

Bishops (USCCB) have sustainable investment guidelines that involve both the integration of

environmental issues and the exclusion of “sin stocks.”7 Specifically, in addition to factoring

environmental footprints into their sustainable investment strategies, these asset managers

and asset owners exclude stocks from the tobacco, gambling (USCCB), and unconventional

weapons industries. Regarding the weapon industry, USCCB excludes companies that pro-

duce biological and chemical weapons, landmines, nuclear weapons, weapons of mass de-

struction; AXA IM excludes producers of white phosphorus weapons; and BNP Paribas AM

excludes manufacturers of controversial weapons. These strategies are not isolated, and the

USCCB states that “Many dioceses, eparchies, and religious communities have also been seek-

ing to apply these guidelines through their own policies on corporate responsibility. We hope

that they are helpful to others who wish to be both ethical and responsible to the common

good in the investments they make.” Consistent with the aggregate practice of sustainable

investing as well as these sustainable investment guidelines, I develop a model in which sus-

tainable investors practice both exclusion and integration.

2.2 Model Setup and Assumptions

The economy is populated by two investor groups: regular and sustainable investors.

Regular investors can invest in the whole market and have mean–variance preferences,

while sustainable investors can only invest in part of the market and, in addition to their

mean–variance preferences, have tastes for the assets in which they invest. Sustainable

investors can be thought of as an aggregate ESG fund that both (i) have an exclusionary

policy based on social criteria, for example, by excluding sin stocks and (ii) practice envir-

onmental integration by overweighting the greenest stocks and underweighting the brown-

est stocks. Notably, this simple setup does not lose generality compared with a model

comprising several sustainable investors that practice either exclusion, integration, or

both.8 Formally, the model is based on the following assumptions.

Assumption 1 (Single-Period Model). Agents operate in a single-period model from time t

to tþ 1. They receive an endowment at time t, have no other source of income, trade at

time t, and derive utility from their wealth at time tþ 1.

Assumption 2 (Gaussian Returns). The market is composed of nI þ nX risky assets,

I1; . . . ; InI ;X1; . . . ;XnX

, whose returns are normally distributed, and one risk-free asset.

Assumption 3 (Partial Segmentation). Regular investors invest freely in all assets in the mar-

ket. Sustainable investors restrict their risky asset allocation to the sub-market of investable

7 The sustainable investment guidelines of AXA IM are available here: https://www.axa-im.com/

sites/corporate/files/2021-09/axa-im-ESG-Standards-Policy-EN-sept-21.pdf; those of BNP, here:

https://group.bnpparibas/uploads/file/2021_eu_sustainable_finance_disclosure_bnp_paribas_asset

_management_english.pdf; and those of the USCCB, here: https://www.usccb.org/resources/

Socially%20Responsible%20Investment%20Guidelines%202021%20(003).pdf.

8 In the Online Appendix, I derive the results in a more general framework with several sustainable

investors having different exclusionary and integration practices.

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assets, which is composed of assets I1; . . . ; InI , and exclude the sub-market of excluded

assets, which is composed of assets X1; . . . ;XnX (e.g., the sin stocks). The proportion of the

excluded assets’ market value is denoted by q 2 ½0; 1�. The wealth shares of sustainable and

regular investors are p and 1� p, respectively.

Assumption 4 (Heterogeneous Preferences). Investors have mean–variance preferences, and

their relative risk aversion is denoted by c. However, contrary to regular investors, sustain-

able investors have specific tastes for the assets in which they invest; for example, they favor

the greenest companies’ stocks. Therefore, they subtract a deterministic private cost of

externalities, ck, from the expected returns on each investable asset k 2 fI1; . . . ; InI g in their

mean–variance optimization program.9 C ¼ ðcI1 ; . . . ; cInI

Þ0 is the vector of stacked costs for

investable assets I1; . . . ; InI , where the prime symbol stands for the transposition operator.

The cost of externalities of the value-weighted portfolio of investable assets is denoted by

cmI (Figure 1).

Assumption 5 (Perfect Market). The market is perfect and frictionless.

Assumption 6 (Free Lending and Borrowing). Investors can lend and borrow freely, without

any constraint, at the same exogenous interest rate.

The specific assumptions adopted in this model are those of a partially segmented market

(Assumption 3) in which investors have heterogeneous preferences (Assumption 4). I do not

consider the partial segmentation assumption as a limiting case of the heterogeneous prefer-

ences assumption with no-short-sales constraint because exclusionary screening and inte-

gration correspond to distinct practices applied to different types of assets. Indeed,

exclusionary practices are often used to exclude the most controversial assets (e.g., sin

stocks), while integration is used to modulate a portfolio’s exposure to a specific issue (e.g.,

companies’ carbon footprints). Consequently, as emphasized by Bolton and Kacperczyk

(2022), exclusionary screening generates an extensive margin adjustment on the cost of cap-

ital, while integration induces an intensive margin adjustment because sustainable investors

require higher compensation for holding the assets they dislike.

By characterizing sustainable investors’ practices through both exclusion and environ-

mental integration, the developed model subsumes two types of previous models. On the

one hand, when the cost of externalities is zero (i.e., focusing on Assumption 3), the present

framework is reduced to that of segmentation models, such as the I-CAPM (Errunza and

Losq, 1985; de Jong and de Roon, 2005), and that used by Luo and Balvers (2017), who

analyze the effects of excluding a specific set of assets. The assumptions of the present

model generalize those of Merton’s (1987) model since I do not impose any particular spe-

cification on asset returns, and these are not independent.10

On the other hand, when the market is not segmented (i.e., focusing on Assumption 4),

the present model is reduced to a model of differences of opinions, in which sustainable

9 As detailed in the Appendix, regular investors have an exponential utility, while sustainable invest-

ors adjust their exponential utility by internalizing a deterministic private cost of externalities, as

in Pastor, Stalbaugh, and Taylor (2021b).

10 However, it should be noted that Merton allows each stock to be neglected by a different number

of investors, while in the present model, all excluded stocks are excluded by the same proportion

of total wealth, p.

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investors adjust their expected returns on each available asset by internalizing a private cost

of externalities. The setup is related to that of Acharya and Pedersen (2005): the cost of il-

liquidity is replaced here by a deterministic cost of externalities, which is internalized only

by a fraction of the investors. Unlike the illiquidity cost, which fluctuates daily, the cost of

environmental externalities varies with high inertia and does not necessarily need to be

modeled as a stochastic factor. The internalization of the cost of externalities—which is

modeled here as a linear adjustment of the expected excess return—is consistent with other

theoretical studies on ESG investing (Gollier and Pouget, 2014; Pastor, Stambaugh, and

Taylor, 2021b; Pedersen, Fitzgibbons, and Pomorski, 2021). Notably, the cost of external-

ities can have a negative value and reflect the internalization of positive externalities by sus-

tainable investors. This occurs for companies whose assets may benefit from enhanced

returns in the future, for example, the greenest companies in a given industry.

2.3 Premia Induced by Sustainable Investing

Subscripts I and X are used here as generic indices, denoting the vectors of nI investable

assets and nX excluded assets, respectively. To simplify the notation, the time subscripts are

omitted, and all returns, r, are considered in excess of the risk-free rate. Therefore, the ex-

cess return on any asset k in the market is denoted by rk. The vectors of excess returns on

assets I ¼ ðI1; . . . ; InI Þ and X ¼ ðX1; . . . ;XnX

Þ are denoted by rI and rX, respectively. I refer

to the value-weighted portfolios of investable assets and of excluded assets as the investable

market and excluded market portfolios, respectively. The excess returns on the investable

market, excluded market, and market are denoted by rmI ; rmX

, and rm, respectively. I use r

to denote the standard deviation of the excess returns on an asset and q for the correlation

coefficient (multiple correlation coefficient) between the excess returns on two assets (be-

tween one asset and a vector of assets). Let bkmI be the slope coefficient of the regression of

the excess returns on asset k 2 fI1; . . . InI ;X1; . . . ;XnX

g on the excess returns on the invest-

able market, mI, and a constant. Let BkI ¼ ðbkI1 ; . . . ; bkInI

Þ be the row vector of the slope

coefficients in a multiple regression of asset k’s excess returns on the excess returns on the

investable assets, rI1 ; . . . ; rInI

, and a constant. Covðrk; rmX jrIÞ and Covðrk; rmX

jrmI Þ refer to

Figure 1. Graphical overview of the financial setup. This graph depicts the two types of investors

involved (sustainable and regular investors), their scope of eligible assets, and the tastes of sustain-

able investors through the private costs of externalities, ðcIk Þk2f1;...;nI g, they internalize.

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the conditional covariances between rk and rmX , given the vector of returns rI and return

rmI , respectively.

Proposition 1 (S-CAPM).

1. The expected excess return on any asset k 2 fI1; . . . InI ;X1; . . . ;XnX

g is

EðrkÞ ¼ bkmI

� EðrmI

Þ � pcmI

� þ pBkIC|fflfflffl{zfflfflffl}

Taste premium

þ c p

1� p qCovðrk; rmX

jrIÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Exclusion-asset premium

þ cqCovðrk; rmX jrmI Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Exclusion-market premium

:

(1)

2. Particularly,

(i) the expected excess return on any investable asset Ik (k 2 f1; . . . ;nIg) is

EðrIk Þ ¼ bIkmI

� EðrmI

Þ � pcmI

� þ pcIk|{z}

Taste premium

þ cqCovðrIk ; rmX jrmI Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Exclusion-market premium

; (2)

(ii) the expected excess return on any excluded asset Xk (k 2 f1; . . . ; nXg) is

EðrXk Þ ¼ bkmI

� EðrmI

Þ � pcmI

� þ pBXkIC|fflfflfflffl{zfflfflfflffl}

Taste premium

þ c p

1� p qCovðrXk

; rmX jrIÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Exclusion-asset premium

þ cqCovðrXk ; rmX jrmI Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Exclusion-market premium

:

(3)

Proposition 1 shows that sustainable investors’ exclusion and integration practices in-

volve two types of additional premia in equilibrium: two exclusion premia11—the exclu-

sion-asset and exclusion-market premia—and a taste premium. The presence of the

exclusion-market premium on investable asset returns and the taste premium on excluded

asset returns reflects the cross effects of exclusion and integration practices. Compared with

the previous papers on partially segmented markets (Errunza and Losq, 1985; de Jong and

de Roon, 2005), I show that equilibrium returns can be expressed in a unified form for all

assets in the market [Equation (1)]. As in de Jong and de Roon (2005) and Eiling (2013),

the expected excess returns are expressed with respect to those on the investable market,

which is the largest investment universe accessible to all investors in a partially segmented

market. However, the expected return on the investable market is lowered by the taste pre-

mium on this market, pcmI .

Three limiting cases can be considered. First, when sustainable investors do not exclude

assets but have different tastes for investable assets from regular investors, the exclusion

premia disappear because q¼ 0, and only the taste premium remains. In addition, the in-

vestable market, mI, and the market, m, coincide. Denoting the beta of asset k with respect

to the market by bkm and the average cost of externalities in the market by cm, the expected

excess return on asset k is

EðrkÞ ¼ bkm

� EðrmÞ � pcm

� þ pck: (4)

11 The exclusion premia are not random variables but scalars because, for a multivariate normal dis-

tribution, the conditional covariance does not depend on the given values (see Lemma 1 in the

Appendix).

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This equilibrium equation is the same as the one in Pastor, Stambaugh, and Taylor

(2021b), except that the authors deliberately assume cm ¼ 0 for simplicity. Specifically,

when the economy is only populated by sustainable investors (p¼1), the equilibrium equa-

tion reduces to Acharya and Pedersen’s (2005) liquidity-adjusted CAPM with a determinis-

tic illiquidity cost.

Second, when sustainable investors only practice exclusion and have similar tastes to

those of regular investors (8k 2 f1; . . . ; nIg; cIk ¼ 0), the taste premium vanishes, and only

the exclusion premia remain. Equation (2) reduces to the I-CAPM equilibrium equation for

investable assets, as in de Jong and de Roon (2005)12:

EðrIk Þ ¼ bIkmI

EðrmI Þ þ cqCovðrIk

; rmX jrmI Þ: (5)

Equation (3) is also related to de Jong and de Roon (2005), who express the equilibrium

equation for excluded assets’ expected excess returns with respect to the vector of invest-

able assets’ expected returns, EðrIÞ. I extend their result to express the expected excess

returns on excluded assets with respect to those on the investable market, EðrmI Þ, as

EðrXk Þ ¼ bXkmI

EðrmI Þ þ c

p

1� p qCovðrXk

; rmX jrIÞ þ cqCovðrXk

; rmX jrmI Þ: (6)

Finally, in the absence of sustainable investors (p¼ 0), there are no longer any excluded

assets (q¼ 0; mI and m coincide), and the model boils down to the CAPM.

2.3. a. Taste premium

The taste premium induced by sustainable investors’ tastes arises in equilibrium for the in-

vestable asset Ik (pBIkIC ¼ pcIk ), and by commonality for the excluded asset Xk (pBXkIC).

Applied primarily to investable assets, this premium is proportional to the cost of exter-

nalities: the higher the cost of externalities, the higher the premium to incentivize sustain-

able investors to acquire the considered asset, and vice versa when the cost of externalities

is low. This finding aligns with the literature on the differences of opinions (e.g., Jouini and

Napp, 2007; Atmaz and Basak, 2018), in which the assets’ expected returns increase (de-

crease) when a group of investors is pessimistic (optimistic), and the finding of Pastor,

Stambaugh, and Taylor (2021b), who show that brown (green) assets have positive (nega-

tive) alphas. The taste premium also increases with the proportion of sustainable investors,

p, as shown by Fama and French (2007) and Gollier and Pouget (2014).

In addition, in a partially segmented market, the taste premium also arises on excluded

asset returns by commonality. This premium is an indirect effect, which is explained as fol-

lows. In equilibrium with market clearing, regular investors overweight the assets with the

highest cost of externalities to provide liquidity to sustainable investors who take the op-

posite position.13 Consequently, to diversify their allocation, regular investors value most

12 The local segmentation premium in de Jong and de Roon (2005) can be expressed as a conditional

covariance between asset returns (see Lemma 1 in the Appendix).

13 The weights ws;I and wr ;I in equilibrium follow directly from the first-order conditions of sustain-

able investors’ optimization program (first row of System [3] in the Appendix) and the market

clearing condition. This effect is similar to the one in De Angelis, Tankov, and Zerbib (2022). It is

also consistent with disagreement models in which some investors have an optimistic view on the

market while others have a pessimistic one (Osambela, 2015; Atmaz and Basak, 2018). In such a

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highly the excluded assets that are the most negatively correlated with the investable assets

having a high cost of externalities (i.e., the assets Xk, k 2 f1; . . . ; nXg, for which BXkIC is

negative). Conversely, they require a higher expected return to hold the excluded assets that

are the most positively correlated with the investable assets having a high cost of external-

ities (i.e., the assets Xk, k 2 f1; . . . ;nXg, for which BXkIC is positive) to compensate for

their lesser diversification.

Finally, by internalizing externalities on investable assets, sustainable investors simul-

taneously adjust their total exposure to the investable market and impact the market pre-

mium through cmI . When they internalize a positive global cost of externalities (cmI

> 0),

they underweight the investable market, and the market premium is negatively adjusted.

The opposite effect applies when the global cost of externalities is negative. Therefore,

focusing on asset Ik, the total taste effect caused by sustainable investors’ tastes is a relative

effect:

Taste effect for investable asset Ik ¼ pcIk|{z} Taste premium

� bIkmI pcmI|fflfflfflfflfflffl{zfflfflfflfflfflffl}

Market effect

:

Consequently, although the weighted average cost of externalities on the investable mar-

ket, cmI , is not necessarily zero, the weighted average taste effect is zero.14

2.3. b. Exclusion premia

Two exclusion premia arise in equilibrium on excluded assets’ expected excess returns: the

exclusion-asset premium, c p 1�p qCovðrXk

; rmX jrIÞ, and exclusion-market premium,

cqCovðrXk ; rmX jrmI Þ. As a cross effect, the exclusion-market premium, cqCovðrIk

; rmX jrmI Þ,

also arises in equilibrium on investable assets’ expected excess returns, while the exclusion-

asset premium is zero.

The exclusion-asset premium is the super risk premium, as characterized by Errunza

and Losq (1985) for excluded assets in partially segmented markets.15 The exclusion-

market premium is the local segmentation premium that de Jong and de Roon (2005) iden-

tify for investable assets.16

As outlined in Corollary 1, the exclusion premia are induced by the joint hedging effect

of regular investors compelled to hold excluded assets and sustainable investors who cannot

hold them.

case, the risk is transferred from the pessimists to the optimists, who increase their holdings of

the assets under consideration.

14 The weighted average taste effect on the investable market is pcmI � bmI mI

pcmI ¼ 0.

15 Using different levels of risk aversion, and denoting regular investors’ risk aversion by cr and the

global risk aversion by c, the exclusion-asset premium is cr

1�p � c � �

qCovðrk ; rmX jrIÞ. Errunza and

Losq (1985) use absolute risk aversions, while relative risk aversions are used in the present

model.

16 I show that both exclusion premia apply to all assets in the market; indeed,

c p 1�p qCovðrIk

; rmX jrIÞ ¼ 0. However, when the expected returns on investable assets, EðrIk

Þ, are

expressed with respect to the expected market returns, EðrmÞ, the exclusion-asset premium is

not zero (see the proof of Proposition 2).

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Corollary 1 (Breakdown of the Exclusion Premia).

The exclusion premia can be expressed as the difference between a regular investor effect

and a sustainable investor effect:

c p

1� p qCovðrk; rmX

jrIÞ ¼ c p

1� p qCovðrk; rmX

Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Regular investor effect

� c p

1� p qCov

� EðrkjrIÞ;EðrmX

jrIÞ �

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Sustainable investor effect

; (7)

cqCovðrk; rmX jrmI Þ ¼ cqCovðrk; rmX

Þ|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Regular investor effect

� cqCovðEðrkjrmI Þ;EðrmX

jrmI ÞÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Sustainable investor effect

: (8)

The former effect is induced by regular investors’ need for diversification: because they

are compelled to hold the excluded market portfolio, they value most highly the assets that

are the most negatively correlated with this portfolio. The latter effect is related to the hedg-

ing need of sustainable investors, who cannot hold excluded assets. As the second-best solu-

tion, they seek to purchase from regular investors the hedging portfolios most positively

correlated with the excluded market and built from investable assets, with returns of

EðrmX jrIÞ, and from the investable market portfolio, with returns of EðrmX

jrmI Þ. As a result,

sustainable investors value most highly the hedging portfolios of asset k if they are highly

correlated with the hedging portfolios of the excluded market.

Notably, when the joint dynamics of excluded assets strengthen and diverge from those

of investable assets, the exclusion premia increase as the regular investor effects increase

and the sustainable investor effects decrease.

The exclusion-asset premium is a generalized form of Merton’s (1987) premium on neglected

stocks. As proven in detail in the Online Appendix, Proposition 2 characterizes this by expressing

the expected excess returns on excluded assets as a function of the market returns, rm.

Proposition 2 (A Generalized Form of Merton’s (1987) Premium on Neglected Stocks).

Let ~bXkm ¼ CovðrXk

;rmI Þ

Covðrm ;rmI Þ . When the expected excess returns on Xk are expressed with respect

to those on the market portfolio, the exclusion-asset premium is

c p

1� p qCovðrXk

� ~bXkmqrmX ; rmX jrIÞ; (9)

and is a generalized form of Merton’s (1987) premium on neglected stocks.

The generalized form of Merton’s (1987) premium on neglected stocks is equal to

c p 1�p qCovðrXk

; rmX jrIÞ, which is adjusted by factor �c p

1�p ~bXkmq2

VarðrmX jrIÞ to express the

expected excess returns on excluded assets with respect to those on the market.

Hong and Kacperczyk (2009) and Chava (2014) empirically show that sin stocks have

higher expected returns than otherwise comparable stocks. Although this finding is, on

average, true, it is not always true for individual stocks (see Proposition 3).

Proposition 3 (Sign of the Exclusion Premia).

i. The exclusion premia on an excluded asset are not necessarily positive.

ii. The exclusion premia on the excluded market portfolio are always positive or zero and

equal to

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cqVar rmXð Þ p

1� p 1� qmXI

� � þ 1� qmXmI

� �� � : (10)

When an excluded asset is sufficiently negatively correlated with the excluded market, the

exclusion premia are likely to be negative.17 In this case, regular investors are strongly incen-

tivized to diversify their risk exposure by purchasing the excluded asset. However, although

the exclusion effect on individual assets is not necessarily positive (Proposition 3 [i]), the

value-weighted average exclusion effect is always positive or zero (Proposition 3 [ii]).

3. Empirical Analysis Applied to Sin Stock Exclusion and Green Investing: The Identification Strategy

I estimate the proposed model by (i) treating sin stocks as excluded assets and (ii) proxying

sustainable investors’ tastes for green assets using green fund holdings. In this section, I de-

scribe the data used, the proxy developed for approximating sustainable investors’ tastes,

and the identification method.

3.1 Data and Proxy Design

3.1. a. Sin stocks as excluded assets

Although the practice of exclusionary screening has previously targeted other objectives, such as

the boycott of the South African state during the apartheid regime (Teoh, Ivo, and Paul, 1999),

it is now mainly applied to sin stocks. However, there is no consensus on the scope of sin indus-

tries to be excluded. Luo and Balvers (2017) provide a summary of the sin industries analyzed in

the existing literature. The tobacco, alcohol, and gaming industries are always regarded as sin

industries. Several authors include the defense industry, but Hong and Kacperczyk (2009) ex-

clude it from US data, noting that not all US investors regard it as a controversial industry. Some

studies also include the pornography and coal industries as sin stocks. I carry out an analysis on

the exclusion of US sin stocks and follow Hong and Kacperczyk (2009) by focusing on the tri-

umvirate of sins, consisting of the tobacco, alcohol, and gaming industries. I check the validity

of the results by performing a robustness test including the defense industry.

I start from all the common stocks (share type codes 10 and 11) listed on the New York

Stock Exchange (NYSE), American Stock Exchange (AMEX), and National Association of

Securities Dealers Automated Quotations exchange (NASDAQ; exchange codes 1, 2, and

3) in the CRSP database. I use the Standard Industrial Classification (SIC) to identify forty-

eight different industries. The alcohol (SIC 4), tobacco (SIC 5), and defense (SIC 26) indus-

tries are directly identifiable from this classification. Since the classification does not distin-

guish gaming companies from those in the hotel and entertainment industries, in line with

Hong and Kacperczyk (2009), I define a 49th industrial category consisting of gaming

based on the North American Industry Classification System (NAICS). Gaming companies

have the following NAICS codes: 7132, 71312, 713210, 71329, 713290, 72112, and

721120. Therefore, out of the forty-nine industries, I focus on the three sin industries of al-

cohol, tobacco, and gaming, which accounted for seventy-seven stocks in the period from

December 31, 1999 to December 31, 2019. Over this period, the number of companies

decreased and the market capitalization of all sin companies increased (Table I).

17 Specifically, when the correlation of an excluded asset with the excluded market is lower than

that of their replicating portfolios using investable assets, the exclusion premia are negative.

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3.1. b. Sustainable investors’ tastes for green firms

Because of sustainable investors’ major interest in environmental issues (see, e.g.,

Macquarie, 2021), I apply their ESG integration preferences to their tastes for green

firms.18 Many empirical studies have investigated the effects of a company’s environmental

performance on its stocks’ excess returns. Yet, the results differ significantly for at least

three main reasons. First, this heterogeneity lies in the fact that the identification of a com-

pany’s environmental performance through a particular environmental metric is a weak

proxy for sustainable investors’ tastes for green firms. Indeed, several dozens of environ-

mental impact metrics are offered by various data providers, covering a wide range of

themes, methods, and analytical scopes. These metrics lack a common definition and di-

verge significantly (Berg, Koëlbel, and Rigobon, 2022).19 For instance, Gibson et al. (2020)

show that the average correlation between the environmental impact metrics of six major

ESG data providers was 42.9% for the period 2013–17. Each available metric reflects spe-

cific information and the average taste of all sustainable investors for green firms can hardly

be captured by a single metric. Moreover, these metrics are generally only available on an

annual basis. Second, the empirical studies fail to capture the increase in the proportion of

green investors and thus, the growing impact of their tastes over time. Third, by using real-

ized returns as proxy for expected returns, these papers omit to control for the effect of the

Table I. Profile of the sin industries

This table reports the number of firms and the total market capitalization corresponding to the

alcohol, tobacco, gaming, and defense industries in the period from December 31, 1999 to

December 31, 2019.

Number of firms Average market

capitalization ($ billion)

Alcohol Tobacco Gaming Defense Alcohol Tobacco Gaming Defense

December 1999–

December 2004

25 7 14 24 2.8 18.4 3 1.6

December 2004–

December 2009

15 9 12 31 3.5 24 4.7 3.2

December 2009–

December 2014

15 9 11 21 2.1 36.8 4.9 4.4

December 2014–

December 2019

13 10 10 9 6 50.3 13.6 7.4

18 I use “tastes for green firms” and “green tastes” interchangeably to refer to the tastes of green

funds proxied by their asset holdings as described in this section.

19 These metrics cover different environmental themes, such as greenhouse gas emissions, air qual-

ity, water management, waste treatment, impact on biodiversity, and thematic and global environ-

mental ratings (e.g., KLD ratings). Even for greenhouse gas emissions, various metrics are

available: carbon intensity, two-degree alignment, avoided emissions, green share, and emission

scores, among others. Additionally, data providers often have their own calculation methods and

analysis scopes. The calculation is further complicated by inconsistencies the data reported by

companies, as well as by the differences in the treatment of data gaps and the benchmarking

options chosen by data providers.

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unexpected shifts in sustainable investors’ tastes on realized returns (Pastor, Stalbaugh, and

Taylor, 2021b). Indeed, investors’ tastes for green assets are intrinsically dynamic because

they are continuously changing as a result of changes in the socio-economic and climate

environments (Bolton and Kacperczyk, 2022). Hence, for example, if the proportion of

green investors or their tastes for green companies unexpectedly increase, green assets may

outperform brown assets while the former have a lower taste premium than the latter.

Consequently, I construct a proxy for the green tastes of sustainable investors that

allows me to address the three issues raised. I circumvent the first two issues by approxi-

mating the shifts in tastes of sustainable investors from a qualitative and quantitative view-

point: I approximate both the cost of environmental externalities defined in the model,

ðcIk Þk2f1;...;nIg, and sustainable investors’ wealth share, p, by using green fund holdings. Such

a proxy for the taste premium allows me to address the third issue by constructing a proxy

for the unexpected shifts in sustainable investors’ tastes (see Section 4.4).

3.1.b.1 Proxy for the cost of environmental externalities. In Proposition 4, we give a first-

order approximation of the cost of externalities for investable asset Ik.

Proposition 4 (Proxy for the Cost of Externalities).

Let us denote sustainable investors’ optimal weight of Ik by w�s;Ik and the market weight of

Ik by wm;Ik . Let us assume that (i) sustainable investors do not account for the correlations

among asset returns when internalizing the cost of externalities of asset Ik, (ii) the share of

sustainable investors’ wealth, p, is small, and (iii) the taste premium, pcIk , is small com-

pared with the expected return, EðrIk Þ. The cost of environmental externalities, cIk

, is

approximated as

cIk ’

wm;Ik �w�s;Ik

wm;Ik

EðrIk Þ: (11)

By providing a micro-foundation of the form of the cost of externalities, Proposition 4 is

intended to allow the construction of a reasonable and intuitive proxy. Under assumptions

(i)–(iii), the cost of externalities of asset Ik has the form of a relative difference between the

weight of this asset in the benchmark and its weight in the sustainable investors’ portfolio.

In other words, the cost of externalities is positive when green funds underweight asset Ik

relative to the benchmark (e.g., in the case of a brown asset), and it is negative when they

overweight asset Ik (e.g., in the case of a green asset).

Assumption (i), which relaxes the dependency structures between assets, aims at not giv-

ing too much weight to (a) the structure of the model which, by nature, simplifies the reality

(two groups of investors, one of which practices exclusion and integration) and (b) the ap-

plication case of the model (exclusion of sin stocks and integration of environmental

issues). Without assumption (i), we would impose a specific dependency structure between

assets, omitting all other structures that are not modeled. Assumption (ii) applied only to

investors practicing environmental integration in the period 2007–19 is realistic since the

total AUM by sustainable investors in the USA reached 25% in 2018. Finally, we validate

hypothesis (iii) ex post: on all assets, for the period from December 2007 to December

2019, the median, the 95% percentile, and the 99% percentile of the ratio of the absolute

value of the estimated taste premium to the absolute value of the realized return are 0.35%,

4.1%, and 10.7%, respectively.

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Therefore, I exclude the expected return, EðrIk Þ, in the approximation of Proposition 4 to

avoid endogeneity bias, and I define the proxy for the cost of externalities of asset Ik, ~cIk , as

~cIk ¼

wm;Ik �w�s;Ik

wm;Ik

: (12)

I compute the microfounded proxy, ~cIk , using the holding history of all the listed green

funds investing in US equities. Specifically, among all funds listed by Bloomberg on

December 2019, I select the 453 funds whose asset management mandate includes environ-

mental guidelines (“environmentally friendly,” “climate change,” and “clean energy”), of

which the investment asset classes are defined as “equity,” “mixed allocation,” and

“alternative,”20 with the geographical investment scope including the USA.21 I retrieve the

entire asset holding history of each of these funds on a quarterly basis (March, June,

September, and December) via the data provider FactSet. The number of green funds

exceeded 100 in 2010 and reached 200 in 2018. I aggregate the holdings of all green funds

on a quarterly basis and focus on the US stock investment universe in CRSP (referred to as

the US allocation). Given the large number of investable stocks and to mitigate the noise

caused by outliers, I perform the analysis on portfolios. To estimate the taste premium

across industries, I construct industry-sorted portfolios (Section 4.1.a). I extend the analysis

by constructing portfolios doubly sorted by industry and carbon emissions to estimate the

taste premium based on the environmental footprints of the companies within each industry

(Section 4.1.b). To illustrate the construction of ~cIk , let us consider the case with industry-

sorted portfolios. The investable market consists of forty-six industries corresponding to

the forty-nine industries from which the three sin industries have been removed. For every

quarter t, I calculate the weight of each industry Ik in the US allocation of the aggregate

green fund to estimate w�s;Ik at date t. I estimate wm;Ik

as the weight of industry Ik in the in-

vestment universe. I construct proxy ~cIk by substituting the estimates of w�s;Ik

and wm;Ik in

Equation (12). I then extend the value of the proxy over the next 2 months of the year in

which no holding data are available.

This agnostic factor serves as a proxy for the sustainable investors’ revealed green tastes

by comparing green funds’ asset allocations with the asset weights in the investment uni-

verse. It offers the dual advantage of covering a large share of the assets in the market (46%

of the stocks at the end of 2019) and being constructed from a minimal fraction of the

AUM (green funds’ AUM accounted for only 0.12% of the market capitalization of the in-

vestment universe at the end of 2019).22 Therefore, by using proxy ~cIk , I implicitly assume

that all sustainable investors have fairly similar green tastes to those revealed by the aggre-

gated 453 green funds, and I test this assumption by estimating the asset pricing model.23

20 The last two categories include diversified funds that also invest in equities.

21 The geographical areas selected are “global,” “international,” “multi,” “North American region,”

“Organization for Economic Co-operation and Development countries,” and “the USA” (see the

Online Appendix).

22 The AUMs of the 453 green funds account for only 0.12% of the total market capitalization of the

investment universe for two main reasons: most green investments are made through the propri-

etary funds of institutional investors (pension funds, life insurers, etc.) rather than via open-ended

funds; not all green funds worldwide are necessarily listed in Bloomberg and FactSet.

23 Given that the list of green funds is not historically available, I acknowledge that the proposed

proxy may introduce survivorship bias. However, given the massive and steady increase in green

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In line with the gradual development of green investing during the 2000s and concomi-

tantly with the enforcement of the US Securities and Exchange Commission’s (SEC’s)

February 2004 amendment requiring US funds to disclose their holdings on a quarterly

basis, the number of green funds reporting their holdings exceeded fifty as of 2007.

Therefore, to construct a sufficiently robust proxy for the taste premium, I start the analysis

from December 2007. Table II summarizes the proxy for the cost of environmental exter-

nalities and the excess returns for the various investable industries in descending order of

average cost, ~cIk , for the period from December 2007 to December 2019.

This ranking shows that the industries least held by green funds include fossil energies

(coal, petroleum, and natural gas), highly polluting manufacturing industries (defense, and

printing and publishing), polluting transportation (aircraft and shipping containers), and min-

ing (non-metallic and industrial mining, and precious metals). However, to be able to over-

weight the least polluting companies, green investors not only underweight the most polluting

companies, but also some of the companies with the largest market capitalizations.

Particularly, they substantially underweight the largest companies in the investment universe,

which belong to the entertainment (e.g., Time Warner and Walt Disney), retail (e.g.,

Walmart), communication (e.g., Verizon and CBS), banking (e.g., JP Morgan, Wells Fargo,

and Citigroup), and insurance (e.g., Berkshire Hathaway, United Health, and AIG) industries.

This is the reason that these specific industries are also at the top of the ranking in Table II.

Therefore, when estimating the effective impact of green investing on asset returns, the under-

weighting of companies with very large capitalizations that we observe on green fund hold-

ings should be taken into account. Naturally, the use of environmental ratings or carbon

footprints as proxy for green investors’ tastes does not allow capturing this effect.

3.1.b.2 Proxy for the proportion of sustainable investors’ wealth. To capture the shifts in

tastes from a quantitative viewpoint, I construct a proxy for the proportion of sustainable

investors’ wealth, p. I estimate the proportion of managed assets following environmental

guidelines as the market value of the US stocks in the 453 green funds divided by the market

value of the investment universe at each considered date. The proxy is denoted by ~p and

defined as

~pt ¼ Market value of US stocks in green fund holdings in t

Total market capitalization of US stocks in t : (13)

From December 2007 to December 2019, ~p increased from 0.02% to 0.12% (see the

Online Appendix).

3.2 Econometric Specifications

I carry out the estimations based on the equations in Proposition 1 being applied to sin

stocks for excluded assets and green funds’ tastes—through ~cIk and ~p—to reflect sustain-

able investors’ preferences.24 I assume that the cost of externalities is proportional to its

investments, the net creation of green funds can be assumed to be positive over the period. Thus,

the number of closed green funds should be limited compared with the number of green funds still

in operation. Additionally, it can be assumed that the average tastes of the closed funds do not

differ significantly from the average tastes of the funds still in operation.

24 The estimations were coded using the software R and the scripts are available at the following

URL: https://drive.google.com/drive/folders/1SbK0DEpyibMIKfw9bl7uTUhtLV8fY6PA?usp=sharing.

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Table II. Descriptive statistics of the investable industries

This table reports the descriptive statistics for the proxy for the cost of environmental external-

ities, ~c , and the monthly returns in excess of the 1-month T-Bill for the period from December

31, 2007 to December 31, 2019, in each of the forty-six investable industries (i.e., the forty-nine

SIC industries from which the alcohol, tobacco, and gaming industries have been excluded).

The construction of the proxy for the cost of environmental externalities is described in Section

3.1.b. In this table, the industries are ranked in descending order of the average proxy ~c .

Environmental cost proxy Returns

Industry name Mean Median St.

Dev.

Min. Max. Mean Median St.

Dev.

Min. Max.

Defense 0.87 0.83 0.08 0.72 0.96 0.021 0.018 0.011 �0.001 0.039

Aircraft 0.69 0.72 0.09 0.47 0.80 0.018 0.018 0.004 0.004 0.028

Precious metals 0.66 0.61 0.08 0.52 0.75 0.008 0.015 0.018 �0.026 0.043

Printing and

publishing

0.58 0.58 0.05 0.43 0.66 0.017 0.017 0.009 0.000 0.039

Non-metallic and

industrial metal

mining

0.54 0.63 0.18 0.17 0.86 0.013 0.012 0.009 �0.007 0.038

Coal 0.52 0.53 0.25 0.32 0.99 �0.002 �0.006 0.018 �0.041 0.039

Agriculture 0.50 0.40 0.61 �1.58 1.00 0.017 0.018 0.011 �0.006 0.036

Entertainment 0.41 0.38 0.18 0.15 0.64 0.025 0.024 0.006 0.010 0.035

Personal services 0.38 0.38 0.04 0.29 0.46 0.016 0.017 0.005 0.004 0.025

Petroleum and natural

gas

0.36 0.33 0.08 0.27 0.58 0.008 0.008 0.006 �0.005 0.023

Candy and soda 0.36 0.32 0.10 0.28 0.57 0.010 0.010 0.003 0.005 0.018

Communication 0.32 0.27 0.09 0.24 0.49 0.014 0.013 0.005 0.005 0.025

Trading 0.32 0.30 0.09 0.22 0.50 0.014 0.014 0.005 0.002 0.026

Retail 0.29 0.28 0.11 0.15 0.47 0.015 0.015 0.005 0.006 0.024

Banking 0.27 0.27 0.07 0.19 0.44 0.012 0.012 0.005 �0.002 0.026

Pharmaceutical

products

0.23 0.22 0.03 0.19 0.29 0.017 0.017 0.006 0.007 0.029

Insurance 0.22 0.18 0.20 0.04 0.57 0.015 0.014 0.004 0.005 0.025

Meals 0.19 0.18 0.09 0.10 0.41 0.017 0.016 0.004 0.010 0.032

Shipbuilding and rail-

road equipment

0.19 0.10 1.12 �2.28 0.92 0.014 0.014 0.007 0.000 0.032

Chemicals 0.16 0.21 0.12 �0.26 0.25 0.015 0.015 0.005 0.007 0.033

Real estate 0.14 0.11 0.22 �0.13 0.50 0.017 0.017 0.009 0.003 0.044

Clothes apparel 0.13 0.24 0.21 �0.10 0.50 0.018 0.020 0.008 0.004 0.038

Transportation 0.11 0.15 0.17 �0.18 0.43 0.016 0.016 0.004 0.010 0.029

Recreation 0.10 0.09 0.18 �0.11 0.57 0.014 0.014 0.006 0.003 0.031

Steel works 0.08 0.06 0.49 �0.54 0.74 0.012 0.011 0.004 0.005 0.028

Business services 0.05 0.05 0.07 �0.01 0.23 0.019 0.019 0.003 0.011 0.029

Computers 0.02 0.05 0.14 �0.25 0.17 0.018 0.016 0.005 0.010 0.035

Automobiles and

trucks

�0.05 �0.02 0.07 �0.16 0.05 0.016 0.013 0.010 0.003 0.050

Shipping containers �0.08 0.30 0.52 �1.13 0.64 0.013 0.013 0.004 0.005 0.026

Consumer goods �0.10 �0.02 0.14 �0.38 0.09 0.010 0.009 0.004 0.003 0.021

(continued)

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proxy: cIk ¼ jc~cIk

and C ¼ jc ~C (jc 2 Rþ) for investable stock Ik and the vector of invest-

able stocks, I, respectively. Similarly, I assume that the share of sustainable investors’

wealth is proportional to its proxy: p ¼ jp ~p (jp 2 Rþ).

3.2. a. Investable asset specification

For each investable asset Ik (k 2 f1; . . . ;nIg), Equation (2) is written as

EðrIk Þ ¼ ðEðrmI

Þ � pcmI ÞbIkmI

þ jpjc ~p~cIk þ cqCovðrIk

; rmX jrmI Þ: (14)

The three independent variables are the beta coefficient, bIkmI , the proxy for the taste

factor, ~p~cIk , and the exclusion-market factor, qCovðrIk

; rmX jrmI Þ.

3.2. b. Excluded asset specification

For each excluded asset Xk (k 2 f1; . . . ; nXg), Equation (3) is written as

EðrXk Þ¼ � EðrmI

Þ�pcmI

� bXkmI

þjpjc ~pBXkI ~Cþc

p

1�p qCovðrXk

;rmX jrIÞþcqCovðrXk

;rmX jrmI Þ:

(15)

The four independent variables of the estimation are the beta coefficient, bXkmI , the

proxy for the taste factor, ~pBXkI ~C, the exclusion-asset factor, qCovðrXk

; rmX jrIÞ, and the

Table II. Continued

Environmental cost proxy Returns

Industry name Mean Median St.

Dev.

Min. Max. Mean Median St.

Dev.

Min. Max.

Rubber and plastic

products

�0.18 �0.12 0.54 �1.61 0.39 0.018 0.018 0.008 0.004 0.046

Healthcare �0.22 �0.19 0.14 �0.39 0.04 0.014 0.015 0.006 0.002 0.026

Food products �0.23 �0.21 0.10 �0.41 �0.05 0.014 0.015 0.005 0.003 0.021

Medical equipment �0.26 �0.27 0.09 �0.46 �0.15 0.017 0.018 0.004 0.006 0.026

Fabricated products �0.33 0.11 1.05 �3.44 0.66 0.014 0.016 0.010 �0.005 0.034

Chips �0.40 �0.40 0.14 �0.73 �0.22 0.017 0.017 0.004 0.008 0.027

Textiles �0.54 �0.69 0.64 �1.88 0.61 0.021 0.021 0.007 0.010 0.046

Wholesale �0.57 �0.59 0.13 �0.71 �0.25 0.016 0.016 0.005 0.008 0.029

Utilities �0.59 �0.50 0.28 �1.12 �0.27 0.010 0.010 0.003 0.001 0.018

Business supplies �0.77 �0.62 0.42 �1.44 0.16 0.015 0.015 0.006 0.005 0.037

Machinery �0.83 �0.77 0.37 �1.81 �0.40 0.012 0.010 0.006 0.002 0.036

Construction

materials

�2.17 �1.97 0.63 �3.54 �1.45 0.018 0.017 0.005 0.008 0.038

Construction �2.33 �2.95 1.44 �4.36 �0.44 0.016 0.015 0.005 0.005 0.027

Electrical equipment �2.58 �2.43 0.43 �3.51 �2.06 0.013 0.013 0.005 0.003 0.030

Measuring and con-

trol equipment

�2.63 �2.57 0.28 �3.85 �2.29 0.019 0.018 0.004 0.012 0.031

Other �6.62 �6.56 2.40 �11.93 �3.48 0.012 0.012 0.002 0.005 0.018

Investable market

portfolio mI

�0.02 �0.02 0.00 �0.02 �0.01 0.015 0.015 0.003 0.009 0.027

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exclusion-market factor, qCovðrXk ; rmX jrmI Þ. As shown in the correlation matrix reported in

the Online Appendix, the correlations between all factors are low.

4. Stock Returns with Tastes for Green Firms

In this section, I empirically assess the effect of sustainable investors’ green tastes and that

of their exclusion of sin stocks on investable stock excess returns. The taste premium sig-

nificantly impacts excess returns. I find weak evidence supporting the effect of sin stock ex-

clusion on investable stock returns.

4.1 Main Estimation

I estimate the following three models. (i) The S-CAPM, corresponding to Equation (14):

EðrIk Þ ¼ aþ dmktbIkmI

þ dtaste ~p~cIk þ dex:mktqCovðrIk

; rmX jrmI Þ; (16)

(ii) the four-factor S-CAPM (denoted as 4F S-CAPM), corresponding to the S-CAPM

specification to which the SMB, HML (Fama and French, 1993), and MOM (Carhart,

1997) betas are added25; and (iii) for benchmarking purposes, the four-factor model

(denoted as 4F model), corresponding to the CAPM specification with respect to the invest-

able market returns to which the SMB, HML, and MOM betas are added.

I perform a two-stage cross-sectional regression (Fama and MacBeth, 1973) with

Newey and West (1987) standard errors to account for heteroskedasticity and serial correl-

ation. Investable assets account for 5,660 stocks in the period from December 2007 to

December 2019, and the estimations are carried out on stock portfolios using value-

weighted returns. All returns are in excess of the 1-month Treasury Bill (T-bill) rate. In the

first pass, I compute the dependent and independent variables over a 3-year rolling period

at monthly intervals. The betas are estimated as univariate betas. Specifically,

qCovðrI; rmX jrmI Þ ¼ CovðrI; rXjrmI

ÞqX, where qX is the vector of weights of the excluded

assets in the market and CovðrI; rXjrmI Þ is computed as a Schur complement from stacked

excess returns (see Lemma 1 in the Appendix). qCovðrIk ; rmX jrmI Þ is the kth entry of vector

qCovðrI; rmX jrmI Þ.26 In the second pass, for each month, I run the cross-sectional regressions

of the nI dependent variables on a constant and the independent variables. The estimated

loadings are equal to the average over the number of cross-sectional regressions. To evalu-

ate and compare the models, I report the OLS adjusted R2 of the cross-sectional regressions.

As suggested by Kandel and Stambaugh (1995) and Lewellen, Nagel, and Shanken (2010),

I also report the GLS R2 as an alternative measure of model fit because it is determined by

the factor’s proximity to the minimum-variance boundary.

The detailed descriptive statistics of the dependent and independent variables and the

correlation matrix are given in the Online Appendix. The mean of the proxy for the taste

factor, ~p ~C, is �2� 10�4, and its median is 10�5. The proxy reaches a maximum of 10�3,

and the minimum is �7� 10�3.

25 The three factors are downloaded from Kenneth French’s website: https://mba.tuck.dartmouth.

edu/pages/faculty/ken.french/data_library.html.

26 I estimate the inverse of the investable asset covariance matrix assuming that returns follow a

one-factor model (Ledoit and Wolf, 2003).

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4.1. a. Inter-industry taste effect

Table III reports the estimates of the three specifications using industry-sorted portfolios

for the period from December 31, 2007 to December 31, 2019. Consistent with the model

predictions, the taste premium is significant (t-statistic of 2.07) and its loading is positive

(d̂taste ¼ 0:17). When the SMB, HML, and MOM factors are included, this premium

becomes highly significant (t-statistic of 4.55) and the loading increases to 0.49. The annual

average market effect is �d̂taste ~p~cmI ¼ 0:25 basis point (bp).27 Therefore, the market effect

is negligible, and the taste effect is almost exclusively driven by the taste premium.

Although the exclusion-market premium is positive and significant when considered in-

dividually, it is not significant in the S-CAPM specification. There are at least two possible

non-exclusive reasons for this low significance: either sustainable investors have not (yet)

sufficiently priced this second-order effect, or it is priced but difficult to identify because of

the small number of sin stocks (seventy-seven) in the excluded asset market, which covary

chaotically with the forty-six investable industry portfolios whose total asset scope is 5,660

stocks.

For each industry, Table IV provides the average annual taste effect estimates using the

S-CAPM. Compared with the industry ranking in Table II, which is based on proxy ~cIk ,

Table IV provides a ranking according to the taste effect, d̂taste ~p~cIk þ d̂taste ~p~cmI

bIkmI , which

includes the market effect, d̂taste ~p~cmI bIkmI

. The rankings slightly differ because bIkmI is not

sorted as ~cIk .

The taste effect ranges from �1.12% to þ0.14% for the different industries.

Specifically, the return differential between industries differently impacted by the ecological

transition is substantial. For example, green investors induce additional annual returns of

0.50% for the petroleum and natural gas industry compared with the electrical equipment

industry.

4.1. b. Intra-industry taste effect

Although some industries can fairly be identified as brown (e.g., coal, aircraft, petroleum,

and natural gas), most of them include companies with very different environmental foot-

prints. For example, the utilities industry contains carbon-intensive companies (e.g., gas

utilities) and low-emitting companies (e.g., renewable energy utilities). Thus, following

Bolton and Kacperczyk (2021, 2022), who emphasize the more significant impact on asset

returns of carbon emissions compared with carbon intensities, I use the total yearly emis-

sions in tons of CO2 equivalent per firm provided by S&P-Trucost to identify the climate

footprint of each firm. I check that the size of the firm does not significantly change the esti-

mates by controlling for the SMB factor. I carry out the analysis focusing on two cases: (i)

the emissions of scopes 1 and 2, namely, the considered firm’s direct emissions related to its

activity (scope 1) and indirect emissions from the generation of purchased energy (scope 2)

and (ii) the sum of the emissions of scopes 1–3, that is, the previous emissions to which are

added those of the rest of the upstream value chain, via suppliers, and downstream value

chain, via customers. I divide each industry into three terciles (see, e.g., In, Park, and

Monk, 2019) to build 138 (¼ 46�3) portfolios doubly sorted by industry and tercile of

emissions, and I repeat the estimation using these portfolios.

27 The proxies for the value-weighted average cost of externalities and the taste factor of the invest-

able market, ~cmI and ~p ~cmI

, are –55 and –0.12 bps, respectively, over the period.

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For both sets of portfolios sorted by scopes 1 and 2 and scopes 1–3, the taste premium is

positive and significant (Table V). It is even more significant when all three scopes are cov-

ered. I then estimate the difference in taste effect per industry between the tercile of brown

companies (top 33% of carbon emissions) and that of green companies (bottom 33% of

carbon emissions); the results are reported in Table VI. Most industries have a positive taste

effect differential. These industries are mainly those that are most impacted by the ecologic-

al transition and have high intra-industry heterogeneity. Taking the example of the

Table III. Cross-sectional regressions for investable stock industry-sorted portfolios with tastes

for green firms

This table presents the estimates of the S-CAPM on the value-weighted monthly returns in ex-

cess of the 1-month T-Bill for forty-six investable stock industry-sorted portfolios for the period

from December 31, 2007 to December 31, 2019. The specification of the S-CAPM is as follows:

EðrIk Þ ¼ aþ dmktbIk mI þ dtaste ~p ~c Ik þ dex:mktqCovðrIk ; rmX

jrmI Þ, where rIk is the value-weighted ex-

cess return on industry portfolio Ik (k ¼ 1; . . . ;nI ), bIk mI is the slope of an OLS regression of rIk on

rmI ; ~p is the proxy for the proportion of sustainable investors’ wealth; ~c Ik is the proxy for the

cost of environmental externalities of industry Ik; q is the proportion of the excluded assets’

market value in the market, and CovðrIk ; rmX jrmI Þ is the covariance of the excess return on port-

folio Ik with that of the excluded market, the excess returns on the investable market being

given. This specification is compared with two other specifications: (i) the 4F S-CAPM, which is

the S-CAPM to which the betas of the Fama and French (1993) size and value factors and the

Carhart (1997) MOM factor are added, and (ii) the 4F model, which is the CAPM with respect to

the investable market returns to which the betas of the Fama and French (1993) size and value

factors and the Carhart (1997) MOM factor are added: EðrIk Þ ¼ aþ dmktbIk mI þ dSMBbIk SMBþ

dHMLbIk HMLþ dMOMbIk MOM. These specifications are estimated using the Fama and MacBeth

(1973) procedure. First, the variables are estimated portfolio-by-portfolio in a 3-year rolling win-

dow at monthly intervals. In the second pass, a cross-sectional regression is performed month-

by-month on all the portfolios. The estimated parameter is the average value of the estimates

obtained on the 109 months during the period. t-values, estimated following Newey and West

(1987) with three lags, are reported between parentheses. The last column reports the average

OLS adjusted-R2 and the GLS R2 on the row underneath. The 95% confidence intervals are

shown in brackets.

a dmkt dtaste dex:mkt dSMB dHML dMOM Adj. OLS/GLS R2

Estimate 0.0143 0.0004 0.05 [0.03, 0.07]

t-value (13) (0.44) 0.07 [0.05, 0.09]

Estimate 0.0149 0.174 �0.02 [�0.02, �0.01]

t-value (24.16) (2.2) 0.01 [0, 0.01]

Estimate 0.0149 119.2 0.06 [0.04, 0.08]

t-value (26.22) (2.15) 0.08 [0.06, 0.1]

Estimate 0.0144 0.0004 0.1922 0.03 [0.02, 0.05]

t-value (12.95) (0.44) (2.55) 0.08 [0.06, 0.1]

Estimate 0.0137 0.0012 0.1737 56.1 0.08 [0.06, 0.11]

t-value (10.51) (1.13) (2.07) (0.77) 0.14 [0.12, 0.17]

Estimate 0.0148 0.0024 0.491 �105.7 0.0001 0.0005 0.000 0.22 [0.19, 0.26]

t-value (14.54) (2.71) (4.55) (�1.94) (0.36) (2.26) (0.09) 0.33 [0.3, 0.36]

Estimate 0.0139 0.0028 0.000 0.0004 0.000 0.23 [0.19, 0.27]

t-value (14.81) (3.14) (0.14) (2.14) (0.15) 0.3 [0.26, 0.33]

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estimation using scopes 1 and 2 carbon emissions, the electrical equipment and utilities

industries have an annual taste effect differential of 2.46% and 0.68%, respectively.

Notably, the electrical equipment industry has the lowest average taste effect (see Table IV;

�0.44% per year, apart from the special case of the “Other” industry). Conversely, the

coal industry has one of the highest average taste effects (0.12% per year) but almost no

within-industry taste effect differential, reflecting the very similar treatment by sustainable

investors of all coal companies, which are substantially underweighted, regardless of their

carbon emissions.

The taste premium associated with sustainable investors’ underweighting of certain

assets can have a substantial effect both at the industry and firm levels. Therefore, environ-

mental integration can be a valuable tool for sustainable investors willing to have an impact

on companies’ practices by raising their cost of capital. Their effect will be all the greater

the higher their proportion of wealth.

Table IV. Annual green taste effect estimates by industry

For all forty-six investable SIC industries, this table reports the estimates of the annual taste ef-

fect d̂taste ~p ~c Ik þ d̂taste ~p ~c mI bIk mI

, which is the sum of the taste premium and the market effect.

The market effect, d̂taste ~p ~c mI bIk mI

, accounts for only 0.25 bps in the total taste effect. The indus-

tries are ranked in descending order of their taste effect.

Industry name Taste

effect (%)

Industry name Taste

effect (%)

Defense 0.14 Transportation 0.02

Aircraft 0.12 Business services 0.01

Coal 0.12 Computers 0.01

Printing and publishing 0.1 Automobiles and trucks 0

Precious metals 0.1 Shipping containers 0

Non-metallic and industrial

metal mining

0.09 Consumer goods �0.02

Agriculture 0.07 Fabricated products �0.02

Entertainment 0.07 Healthcare �0.03

Personal services 0.07 Food products �0.04

Candy and soda 0.06 Medical equipment �0.04

Petroleum and natural gas 0.06 Rubber and plastic products �0.05

Communication 0.06 Textiles �0.05

Trading 0.06 Chips �0.06

Retail 0.05 Shipbuilding and railroad equipment �0.07

Banking 0.05 Wholesale �0.09

Pharmaceutical products 0.04 Utilities �0.1

Meals 0.04 Business supplies �0.1

Insurance 0.04 Machinery �0.13

Clothes apparel 0.03 Construction materials �0.37

Chemicals 0.03 Construction �0.37

Steel works 0.03 Measuring and control equipment �0.43

Real estate 0.03 Electrical equipment �0.44

Recreation 0.02 Other �1.12

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Table V. Cross-sectional regressions for investable stock industry-carbon emissions double-

sorted portfolios with tastes for green firms

This table presents the estimates of the S-CAPM on the value-weighted monthly returns in ex-

cess of the 1-month T-Bill for 138 investable stock industry-carbon emissions double-sorted

portfolios for the period from December 31, 2007 to December 31, 2019. For each industry, we

build three portfolios that correspond to the first, second, and third terciles ranked by carbon

emissions. In Panel A, we focus on scopes 1 and 2 emissions, while in Panel B, we use scopes

1–3 emissions. The specification of the S-CAPM is as follows: EðrIk Þ ¼ aþ dmktbIk mI þ dtaste ~p ~c Ikþ

dex:mktqCovðrIk ; rmX jrmI Þ, where rIk is the value-weighted excess return on industry portfolio Ik

(k ¼ 1; . . . ;nI ), bIk mI is the slope of an OLS regression of rIk on rmI

; ~p is the proxy for the propor-

tion of sustainable investors’ wealth; ~c Ik is the proxy for the cost of environmental externalities

of industry Ik; q is the proportion of the excluded assets’ market value in the market, and

CovðrIk ; rmX jrmI Þ is the covariance of the excess return on portfolio Ik with that of the excluded

market, the excess returns on the investable market being given. This specification is compared

with the 4F S-CAPM, which is the S-CAPM to which the betas of the Fama and French (1993)

size and value factors and the Carhart (1997) MOM factor are added. These specifications are

estimated using the Fama and MacBeth (1973) procedure. First, the variables are estimated

portfolio-by-portfolio in a 3-year rolling window at monthly intervals. In the second pass, a

cross-sectional regression is performed month-by-month on all the portfolios. The estimated

parameter is the average value of the estimates obtained on the 109 months during the period.

t-Values, estimated following Newey and West (1987) with three lags, are reported between

parentheses. The last column reports the average OLS adjusted R2 and the GLS R2 on the row

underneath. The 95% confidence intervals are shown in brackets.

a dmkt dtaste dex:mkt dSMB dHML dMOM Adj. OLS/GLS R2

Panel A: Double-sorted industry-carbon emissions (Scopes 1þ 2) portfolios

Estimate 0.0134 0.0005 0.05 [0.04, 0.07]

t-value (10) (0.35) 0.06 [0.05, 0.08]

Estimate 0.0141 0.1945 0 [0, 0]

t-value (19.19) (1.62) 0.01 [0, 0.01]

Estimate 0.014 62.5 0.03 [0.02, 0.04]

t-value (21) (1.55) 0.04 [0.03, 0.05]

Estimate 0.0135 0.0005 0.2519 0.05 [0.04, 0.07]

t-value (9.81) (0.33) (2.03) 0.07 [0.05, 0.08]

Estimate 0.0136 0.0004 0.1956 36.9 0.07 [0.05, 0.09]

t-value (10.1) (0.31) (1.54) (0.93) 0.09 [0.07, 0.11]

Estimate 0.0126 0.0021 0.1437 �123.9 0.0001 �0.0002 �0.0001 0.17 [0.14, 0.19]

t-value (10.93) (1.74) (1.61) (�2.42) (0.9) (�1.18) (�2.71) 0.2 [0.18, 0.23]

Panel B: Double-sorted industry-carbon emissions (Scope 1þ 2þ 3) portfolios

Estimate 0.0129 0.0009 0.05 [0.04, 0.07]

t-value (10.47) (0.71) 0.06 [0.05, 0.08]

Estimate 0.0143 0.2765 0 [0, 0]

t-value (19.35) (2.37) 0.01 [0, 0.01]

Estimate 0.014 77 0.04 [0.03, 0.06]

t-value (21.11) (1.95) 0.05 [0.04, 0.07]

Estimate 0.0131 0.0009 0.3222 0.06 [0.04, 0.07]

t-value (10.26) (0.69) (2.63) 0.07 [0.05, 0.09]

Estimate 0.0134 0.0007 0.2696 54.3 0.08 [0.06, 0.1]

t-value (11.12) (0.56) (2.16) (1.38) 0.1 [0.08, 0.12]

Estimate 0.0123 0.0024 0.185 �79.7 0.0001 �0.0002 �0.0001 0.19 [0.16, 0.22]

t-value (11.67) (2.03) (2.13) (�1.49) (0.5) (�0.97) (�1.17) 0.23 [0.2, 0.25]

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Table VI. Annual taste effect spread by industry between the 33% greenest companies and the

33% brownest companies

For all forty-six investable SIC industries, this table reports the estimates of the annual spread

between the taste effect of the tercile of the brownest companies and that of the greenest com-

panies. The estimation is made for both scopes 1 and 2 emissions and scopes 1–3 emissions.

Annual taste effect (%)

Industry name Industry-carbon emissions

(Scope 1þ 2)

Industry-carbon emissions

(Scope 1þ 2þ 3)

Diff. in

brown versus

green tercile

Brown

tercile

Green

tercile

Diff. in

brown versus

green tercile

Brown

tercile

Green

tercile

Electrical equipment 2.46 �0.23 �2.69 3.2 �0.42 �3.62

Other 1.59 �0.91 �2.5 2.23 �1.2 �3.43

Machinery 1.15 �0.02 �1.17 1.4 �0.11 �1.51

Construction materials 0.93 �0.3 �1.23 1 �0.47 �1.47

Utilities 0.68 0.02 �0.66 1.06 0.02 �1.04

Textiles 0.62 0.12 �0.5 0.74 0.16 �0.58

Shipbuilding and railroad equipment 0.56 0.06 �0.5 0.71 0.02 �0.69

Computers 0.4 0.05 �0.35 0.97 0.07 �0.9

Healthcare 0.29 0.03 �0.26 0.46 0.07 �0.39

Aircraft 0.24 0.16 �0.08 0.35 0.23 �0.12

Wholesale 0.2 �0.01 �0.21 0.47 �0.01 �0.48

Non-metal. and indus. metal mining 0.19 0.15 �0.04 0.31 0.21 �0.1

Chips 0.19 �0.01 �0.2 0.22 0.01 �0.21

Automobiles and trucks 0.18 0.08 �0.1 0.29 0.12 �0.17

Food products 0.15 �0.03 �0.18 0.2 �0.05 �0.25

Transportation 0.15 0.05 �0.1 0.14 0.07 �0.07

Fabricated products 0.14 0.08 �0.06 0.16 0.07 �0.09

Business supplies 0.12 0.07 �0.05 0.02 0.09 0.07

Steel works 0.08 0.06 �0.02 0.24 0.14 �0.1

Defense 0.06 0.18 0.12 0.08 0.24 0.16

Personal services 0.06 0.09 0.03 0.04 0.13 0.09

Agriculture 0.05 0.14 0.09 0.1 0.19 0.09

Precious metals 0.05 0.13 0.08 0.07 0.18 0.11

Chemicals 0.03 0.03 0 0.06 0.04 �0.02

Coal 0.03 0.14 0.11 0.04 0.19 0.15

Business services 0.02 0.03 0.01 0.02 0.04 0.02

Petroleum and natural gas 0.01 0.08 0.07 0.02 0.11 0.09

Medical equipment 0.01 �0.02 �0.03 0.03 �0.03 �0.06

Retail 0 0.07 0.07 �0.03 0.09 0.12

Insurance 0 0.05 0.05 �0.03 0.06 0.09

Trading 0 0.06 0.06 �0.03 0.08 0.11

Rubber and plastic products �0.02 0.03 0.05 0.03 0.03 0

Printing and publishing �0.03 0.09 0.12 �0.08 0.13 0.21

Cand and soda �0.03 0.07 0.1 �0.06 0.09 0.15

Entertainment �0.03 0.08 0.11 �0.06 0.11 0.17

(continued)

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4.2 Alternative Estimations

I perform alternative estimations, the results of which are available in the Online Appendix.

First, the estimate of the taste premium is robust to a first-pass regression using a 5-year

rolling window, and its significance increases. Second, when using equally weighted

returns, the taste premium is not significant, but the exclusion-market premium becomes

significant and positive, as predicted by the model. Third, I repeat the estimation using a set

of 230 (¼ 46� 5) industry-size portfolios doubly sorted by industry and market capitaliza-

tion quintiles. The taste premium is significant and consistent with the estimation using in-

dustry portfolios. Finally, the estimated taste premium is significant and consistent with

that of the main estimation when using only the proxy for the cost of externality, ~c, as the

taste factor.

4.3 Reverse Causality Bias

The first concern is the risk of reverse causality bias through proxy ~c. In other words, is

dtaste significant because the return on industry Ik affects the relative weight differential be-

tween the market and sustainable investors’ asset allocation in this industry, wm;Ik

�w� s;Ik

wm;Ik

?

Since the industry weights of green investors and those of the market vary slowly over time,

I repeat the regression using proxy ~c lagged by 3 years to ensure that the returns estimated

in the first pass of the Fama–MacBeth regression do not affect the proxy retroactively.

The taste premium is highly significant (t-statistics of 3.09) and positive (d̂taste ¼ 0:47).

The estimate is robust to the inclusion of the SMB, HML, and MOM factors. Although the

loading is higher than that of the main model, this estimation supports the significant effect

of the taste premium on investable asset returns. The results are reported in the Online

Appendix.

Table VI. Continued

Annual taste effect (%)

Industry name Industry-carbon emissions

(Scope 1þ 2)

Industry-carbon emissions

(Scope 1þ 2þ 3)

Diff. in

brown versus

green tercile

Brown

tercile

Green

tercile

Diff. in

brown versus

green tercile

Brown

tercile

Green

tercile

Communication �0.04 0.06 0.1 �0.06 0.09 0.15

Measuring and control equip. �0.04 �0.53 �0.49 �0.11 �0.77 �0.66

Real estate �0.05 0 0.05 �0.08 �0.01 0.07

Consumer goods �0.06 0 0.06 �0.02 0 0.02

Pharmaceutical products �0.07 0.05 0.12 �0.07 0.07 0.14

Clothes apparel �0.08 0 0.08 �0.18 �0.01 0.17

Banking �0.09 0.05 0.14 �0.12 0.07 0.19

Meals �0.11 0.04 0.15 �0.17 0.03 0.2

Recreation �0.12 �0.03 0.09 �0.21 �0.05 0.16

Shipping containers �0.13 �0.07 0.06 �0.25 �0.15 0.1

Construction �0.38 �0.34 0.04 �0.46 �0.51 �0.05

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4.4 Unexpected Shifts in Tastes

As Bolton and Kacperczyk (2022) point out, the taste premium is intrinsically dynamic

even if the environmental footprint of the considered company remains unchanged: unex-

pected changes in technologies, institutional and socio-political environment, climate poli-

cies, reputation, and investor awareness push sustainable investors to adjust their

environmental tastes constantly, that is, the costs of externalities that they internalize.

However, as emphasized by Pastor, Stambaugh, and Taylor (2021b), the adjustment of the

costs of externalities has an impact on realized returns in the opposite direction of the effect

on expected returns. For example, when the tastes for green companies increase over a

period, a green asset can have a negative taste premium and yet outperform brown assets.

Consequently, omitting to control for the unexpected changes in tastes when using realized

returns as proxy for expected returns induces a critical omitted variable bias. The failure to

account for the shifts in green investors’ tastes due to unexpected environmental, societal,

and economic changes may, therefore, partly explain why the results of the empirical analy-

ses on the link between ESG and financial performance are mixed. Pastor, Stambaugh, and

Taylor (2021b) suggest using the in- and out-flows of ESG-tilted funds as proxy for this ef-

fect. The analysis of green fund holdings thus offers a dual advantage: (i) constructing a

proxy for the unexpected shifts in green investors’ tastes at a monthly frequency that is (ii)

homogeneous with the proxy for the taste premium. Therefore, I define the proxy for the

unexpected shifts in green investors’ tastes for asset Ik between t�1 and t as the variation of

the taste factor between these two dates:

D~pt~cIk ;t ¼ ~pt~cIk ;t � ~pt�1~cIk ;t�1: (17)

An increase (or decrease) in the taste factor should lead to a decrease (or increase) in the

short-term returns. Indeed, when sustainable investors’ tastes for firm Ik decrease (~cIk

increases; hence, D~p~cIk increases), realized returns, rIk

, should decrease; conversely, when

sustainable investors’ tastes for firm Ik increase (~cIk decreases; hence, D~p~cIk

decreases), real-

ized returns, rIk , should increase. A similar reasoning applies to ~p. Therefore, I perform a

robustness check on the following augmented specification, and I expect the loading of

D~p~cIk to be negative:

EðrIk Þ ¼ aþ dmktbIkmI

þ dtaste ~p~cIk þ duD~p~cIk

þ dex:mktqCovðrIk ; rmX jrmI Þ: (18)

Table VII, Panel A, reports the estimates for all industries. Although the taste premium

is not significant in the augmented S-CAPM, it becomes significant when controlling for the

SMB, HML, and MOM factors (hereinafter referred to as the augmented 4F S-CAPM). Its

loading is in line with that estimated in the main specification. However, two industries

have experienced massive divestments by green investors since 2012: the relative weights of

the coal and construction industries in the portfolios of green investors relative to the mar-

ket weights, ~c, have dropped from �0.48 to �0.93 and from 3.3 to 0.43, respectively, from

December 2012 to December 2019. Therefore, I repeat the estimation by removing these

outliers. Panel B presents the estimates for all industries except coal. The taste premium is

significant in the absence of the exclusion-market premium and remains significant for the

augmented 4F S-CAPM. The estimates are in line with those of the main estimation. Panel

C presents the estimates for all industries except coal and construction. The taste premium

is highly significant for the augmented S-CAPM and the augmented 4F S-CAPM. The load-

ing is twice as high for the augmented S-CAPM than for the S-CAPM but is similar for the

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Table VII. Cross-sectional regressions for investable stock industry-sorted portfolios with tastes

for green firms and unexpected shifts in tastes

This table presents the estimates of the augmented S-CAPM with unexpected shifts in tastes on

the value-weighted monthly returns in excess of the 1-month T-Bill for forty-six investable stock

industry-sorted portfolios for the period from December 31, 2007 to December 31, 2019. Panels

A–C present the estimates on all industries, all industries without the coal industry, and all

industries without the coal and construction industries, respectively. The specification is written

as follows: EðrIk Þ ¼ aþ dmktbIk mI þ dtaste ~p ~c Ik þ duD~p ~c Ik þ dex:mktqCovðrIk ; rmX

jrmI Þ, where rIk is the

value-weighted excess return on industry portfolio Ik (k ¼ 1; . . . ;nI ); bIk mI is the slope of an OLS

regression of rIk on rmI ; ~p is the proxy for the proportion of sustainable investors’ wealth; ~c Ik is

the proxy for the cost of environmental externalities of industry Ik; D~p ~c Ik is the proxy for the un-

expected shifts in tastes; q is the proportion of the excluded assets’ market value in the market,

and CovðrIk ; rmX jrmI Þ is the covariance of the excess return on portfolio Ik with that of the

excluded market, the excess returns on the investable market being given. This specification is

compared with the augmented 4F S-CAPM, which is the augmented S-CAPM to which the betas

of the Fama and French (1993) size and value factors and the Carhart (1997) MOM factor are

added. These specifications are estimated using the Fama and MacBeth (1973) procedure. First,

the variables are estimated portfolio-by-portfolio in a 3-year rolling window at monthly inter-

vals. In the second pass, a cross-sectional regression is performed month-by-month on all the

portfolios. The estimated parameter is the average value of the estimates obtained on the

109 months during the period. t-Values, estimated following Newey and West (1987) with three

lags, are reported between parentheses. The last column reports the average OLS adjusted R2

and the GLS R2 on the row underneath. The 95% confidence intervals are shown in brackets.

a dmkt dtaste du dex:mkt dSMB dHML dMOM Adj. OLS/GLS R2

Panel A: All industries

Estimate 0.0145 0.0003 �0.1562 �18.5 0.03 [0.01, 0.05]

t-value (12.94) (0.31) (�1.05) (�2.22) 0.1 [0.08, 0.11]

Estimate 0.014 0.001 �0.1977 �14.9 46.3 0.08 [0.06, 0.11]

t-value (10.67) (0.96) (�1.44) (�1.78) (0.62) 0.16 [0.14, 0.18]

Estimate 0.015 0.0022 0.2496 �9.3 �113.6 0.0001 0.0004 0.000 0.22 [0.18, 0.26]

t-value (14.91) (2.43) (1.69) (�1.27) (�2.01) (0.39) (2.1) (�0.17) 0.34 [0.31, 0.37]

Panel B: All industries without the coal industry (SIC 29)

Estimate 0.0136 0.0015 0.1879 �8.8 0.02 [0, 0.05]

t-value (16.68) (1.84) (1.66) (�1.32) 0.09 [0.07, 0.11]

Estimate 0.0132 0.0021 0.0983 �8.3 82.1 0.03 [0.01, 0.06]

t-value (18.39) (2.53) (0.89) (�1.19) (1.57) 0.12 [0.1, 0.14]

Estimate 0.014 0.002 0.2704 �8.7 15.9 0.0002 0.0001 0.0002 0.13 [0.09, 0.16]

t-value (19.46) (2.13) (1.87) (�1.27) (0.3) (1.96) (0.62) (2.09) 0.27 [0.24, 0.29]

Panel C: All industries without the coal (SIC 29) and construction (SIC 18) industries

Estimate 0.0137 0.0014 0.3642 �13.2 0.03 [0, 0.05]

t-value (16.35) (1.68) (3.08) (�1.94) 0.09 [0.07, 0.11]

Estimate 0.0132 0.002 0.2947 �12.7 80.4 0.03 [0.01, 0.06]

t-value (17.64) (2.42) (2.39) (�1.77) (1.54) 0.12 [0.1, 0.15]

Estimate 0.0141 0.0019 0.546 �12.7 9.8 0.0003 0.0001 0.0002 0.13 [0.1, 0.16]

t-value (18.83) (1.9) (3.06) (�1.68) (0.19) (2.08) (0.61) (2.13) 0.27 [0.24, 0.3]

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augmented 4F S-CAPM and the 4F S-CAPM. In addition, the premium for the unexpected

shifts in tastes becomes significant and, as expected, its loading is negative. Finally, under

the augmented S-CAPM, when the coal or the coal and construction industries are

removed, the exclusion-market premium is weakly significant and positive, as predicted by

the model.

4.5 Taste Effect over Time

I analyze the dynamics of the taste premium by repeating the estimation over several sub-

periods. Given the violent effect induced by the divestment from the coal industry in the

period 2012–19 and the short periods over which these estimations are carried out, the lat-

ter are performed on all industries except coal in this subsection.

First, I repeat the estimation over three consecutive sub-periods within the period 2007–

19. The significance of the taste premium increases over time, reaching a t-statistic of 7.27

between 2013 and 2019. In addition, although the average taste premium is stable over

time, the difference in taste premium between the brown and green industries increases.

This spread between the petroleum and natural gas industry and the electrical equipment

industry increased from 50 bps in the period 2007–13 to 1.23% in the period 2013–19.28

The detailed tables are available in the Online Appendix. Second, I repeat the estimation

over 3-year rolling periods for the second pass. The dynamics depicted in Figure 2 show the

steady increase in the taste effect spread between the petroleum and natural gas and elec-

trical equipment industries.

5. Sin Stock Returns

I perform an empirical analysis to assess the effect of sustainable investors’ exclusion of sin

stocks and the indirect effect of their green tastes on sin stocks’ excess returns. I show that

the exclusion premia significantly impact the excess returns. I also find evidence supporting

the cross-effect of green tastes on sin stocks’ excess returns.

5.1 Main Estimation

I estimate the following three models. (i) The S-CAPM, corresponding to Equation (15):

EðrXk Þ ¼ aþ dmktbXkmI

þ dtaste ~pBXkI ~Cþ dex:assetqCovðrXk

; rmX jrIÞþ dex:mktqCovðrXk

; rmX jrmI Þ;

(19)

(ii) the four-factor S-CAPM (denoted as 4F S-CAPM), corresponding to the S-CAPM

specification to which the SMB, HML, and MOM betas are added; and (iii) for benchmark-

ing purposes, the four-factor model (denoted as 4F model), corresponding to the CAPM

with respect to the investable market returns to which the SMB, HML, and MOM betas

are added.

In the same way as for investable assets, I estimate the models using a two-pass regres-

sion on seventy-seven single sin stocks in the period from December 1999 to December

2019, for an annual average number of forty-one stocks.29 Given the substantial noise that

28 The taste effect is higher when the coal industry is removed compared with the entire period in

the main estimation.

29 In the robustness check that includes the defense industry, and I work with ninety-eight sin

stocks, giving an annual mean number of fifty-one stocks.

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occurs when performing regressions on a small number of single stocks, especially when

several of them have extreme return variations, I trim the returns at the 3% level, which

corresponds, on average, to removing the highest outlier and lowest outlier in each cross-

sectional regression. As a robustness check, I also perform the estimation on winsorized

returns.

Table VIII reports the estimates of the three specifications for sin stocks using industry-

sorted portfolios of investable assets. The OLS adjusted R2 of 14% is higher under the

S-CAPM than under the 4F model (11%). In addition, the estimation of the exclusion pre-

mia supports the model predictions. First, the loadings of the exclusion-asset and exclusion-

market factors are positive (d̂ex:asset ¼ 91:5 and d̂ex:index ¼ 79:5, respectively) and significant

(t-statistics of 3.75 and 2.42, respectively). The estimates are robust to the inclusion of the

SMB, HML, and MOM factors. As shown in the Online Appendix, albeit slightly lower,

the estimates are robust to winsorizing the returns. Second, the taste premium is positive

(d̂taste ¼ 1:8) and significant (t-statistics of 1.62).

I estimate the exclusion effect as follows. For each sin stock, I calculate the individual

exclusion effect as the average over time of the sum of the estimated exclusion-asset and

exclusion-market premia. The exclusion effect is the average over all sin stocks of the indi-

vidual exclusion effects. For the period from December 1999 to December 2019, the exclu-

sion effect is 2.79% per year. This effect is of a similar magnitude as the one estimated on

US sin stocks by Hong and Kacperczyk (2009) for the period 1965–2006 (2.5%). However,

it is substantially lower than the annual 16% effect estimated by Luo and Balvers (2017)

for the period 1999–2012 and based on the same modeling framework (in the absence of

green tastes). Additionally, consistent with Proposition 3, I find that the exclusion effect is,

on average, positive, but it is negative for thirty-three out of seventy-seven sin stocks

(Figure 3).

Calculated similarly, the average taste premium on sin stocks’ excess returns is 1.1%

per year, which corresponds to the compensation required by regular investors to hold sin

stocks due to their correlation with the brownest investable stocks. The taste premium

amounts to 28% (¼ 1:1%=½1:1%þ 2:79%�) of the total effect induced on sin stocks’ cost of

capital by sustainable investors practicing exclusion and environmental integration.

Figure 2. Evolution of the taste effect. This figure shows the evolution of the taste effect for the invest-

able market, the petroleum and natural gas industry, and the electrical equipment industry in the

period from December 2007 to December 2019. The first and second passes are both estimated over

3-year rolling periods.

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Table VIII. Cross-sectional regressions on sin stocks’ excess returns

This table provides the estimates obtained with the S-CAPM on the value-weighted monthly

returns in excess of the 1-month T-Bill for seventy-seven sin stocks for the period from

December 31, 1999 to December 31, 2019. The specification is as follows: EðrXk Þ ¼ aþ

dmktbXk mI þ dtaste ~pBXk I

~C þ dex:assetqCovðrXk ; rmX

jrIÞ þ dex:mktqCovðrXk ; rmX

jrmI Þ, where rXk

is the

value-weighted excess return on stock Xk (k ¼ 1; . . . ;nX ), and bXk mI is the slope of an OLS re-

gression of rXk on rmI

; ~pBXk I ~C is the proxy for the taste factor and ~p is the proxy for the propor-

tion of sustainable investors’ wealth; q is the proportion of the excluded assets’ market value in

the market, and CovðrXk ; rmX

jrI Þ (and CovðrXk ; rmX

jrmI Þ) are the covariances of the excess returns

on stock Xk with those on the excluded market, the excess returns on the investable market

(and the vector of investable assets, respectively) being given. The investable assets are ana-

lyzed using forty-six industry-sorted portfolios. The S-CAPM specification is compared with two

other specifications: (i) the 4F S-CAPM, which is the S-CAPM to which the betas of the Fama

and French (1993) size and value factors and the Carhart (1997) MOM factor have been added

and (ii) the 4F model, which is the CAPM with respect to the investable market to which the

betas of the Fama and French (1993) size and value factors and the Carhart (1997) MOM factor

have been added: EðrXk Þ ¼ aþ dmktbXk mI

þ dSMBbXk SMB þ dHMLbXk HML þ dMOMbXk MOM. These spec-

ifications are estimated using the Fama and MacBeth (1973) procedure. First, the variables are

estimated, stock-by-stock, in a 3-year rolling window, at monthly intervals. In the second pass,

a cross-sectional regression is performed on a monthly basis on all the stocks. The returns are

trimmed at the 3% level, which corresponds, on average, to removing the highest outlier and

lowest outlier in each cross-sectional regression. The estimated parameter is the average value

of the estimates obtained on all months during the period of interest. t-Values, estimated fol-

lowing Newey and West (1987) with three lags, are reported between parentheses. The last col-

umn reports the average OLS adjusted R2 and the GLS R2 on the row underneath. The 95%

confidence intervals are in brackets.

a dmkt dtaste dex:asset dex:mkt dSMB dHML dMOM Adj. OLS/GLS R2

Estimate 0.0119 0.0017 0.03 [0.02, 0.04]

t-value (9.53) (1.79) 0.04 [0.03, 0.05]

Estimate 0.0135 0.5733 0.03 [0.02, 0.04]

t-value (9.99) (0.61) 0.07 [0.06, 0.09]

Estimate 0.0129 32.6 0.05 [0.03, 0.06]

t-value (9.23) (1.89) 0.06 [0.05, 0.07]

Estimate 0.0118 73 0.08 [0.06, 0.1]

t-value (9.13) (2.61) 0.09 [0.08, 0.11]

Estimate 0.012 79.8 70.4 0.1 [0.08, 0.12]

t-value (8.88) (3.76) (2.3) 0.15 [0.13, 0.17]

Estimate 0.0118 �0.0003 98.1 84.7 0.12 [0.1, 0.14]

t-value (8.91) (�0.28) (4.1) (2.47) 0.19 [0.16, 0.21]

Estimate 0.0124 �0.0013 1.8 91.5 79.5 0.14 [0.12, 0.16]

t-value (9.43) (�0.98) (1.62) (3.75) (2.42) 0.24 [0.22, 0.26]

Estimate 0.0124 0.0000 1.7 107.2 72.5 �0.0001 �0.0002 0.0004 0.23 [0.2, 0.25]

t-value (9.63) (�0.01) (1.45) (3.81) (1.89) (�0.6) (�1.01) (2.41) 0.38 [0.36, 0.4]

Estimate 0.0124 0.0008 0.0000 �0.0002 0.0004 0.11 [0.09, 0.13]

t-value (10) (0.65) (�0.27) (�1.42) (2.26) 0.19 [0.17, 0.21]

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5.2 Alternative Estimations

I perform additional analyses presented in this subsection and detailed in the Online

Appendix. In all robustness tests, the S-CAPM has higher OLS adjusted R2 and GLS R2

than those of the 4F model. I repeat the estimation in three alternative cases: (i) using a

5-year rolling window for the first pass, (ii) using equally weighted returns, and (iii) includ-

ing the defense industry among sin industries. In all three cases, both exclusion premia are

significant and the exclusion effect is of a similar magnitude to that in the main estimation.

5.3 Dynamics of Sustainable Investors’ Wealth in the Exclusion-Asset Premium

Unlike the taste factor (~pBXkI ~C) that takes into account the proxy for the proportion of sus-

tainable investors’ wealth, the exclusion-asset factor (qCovðrXk ; rmX jrIÞ) does not incorpor-

ate it. Yet, the exclusion-asset premium (c p 1�p qCovðrXk

; rmX jrIÞ) includes p. Therefore,

I repeat the estimation using proxy ~p in the following specification:

EðrXk Þ ¼ aþ dmktbXkmI

þ dtaste ~pBXkI ~C

þ dex:asset

~p

1� ~p qCovðrXk

; rmX jrIÞ þ dex:mktqCovðrXk

; rmX jrmI Þ:

(20)

As expected, under the S-CAPM and the 4F S-CAPM, the estimates of both exclusion

factors are significant and positive (see the Online Appendix). In addition, under the S-

CAPM, the average annual exclusion effect is equal to 2.80%, in line with the one esti-

mated using the main specification.

5.4 Exclusion Effect over Time

I estimate the S-CAPM over four consecutive periods within the 1999–2019 timeframe. In

each period, at least one of the two exclusion factors is significant (see the Online

Appendix). In addition, to highlight the dynamics of the exclusion effect, I repeat the

second-pass estimation using a 3-year rolling window from 2002 to 2019 (blue line on

Figure 4).30 The average exclusion effect rose sharply and was high during the 2007–08

Figure 3. Distribution of the annual exclusion effect. This figure shows the distribution of the annual

exclusion effect, d̂ex:assetqCovt ðrX ; rmX jrI Þ þ d̂ex:mktqCovt ðrX ; rmX

jrmI Þ, over all sin stocks estimated in

the period from December 31, 1999 to December 31, 2019.

30 The second pass starts in 2002 because the variables are computed using a 3-year rolling window

in the first pass.

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crisis as shown in Figure 4. Note that since the first pass of the estimation spans 3 years, the

premia estimated in the second pass smoothen the effect of the crisis on the figure: the effect

starts to materialize in 2008 (as the second pass uses the first pass 2005–08) and vanishes in

2012 (as the second pass uses the first pass 2009–12). This spike in the exclusion effect is

explained by the fact that during the 2007–08 crisis, the covariances of each sin stock and

the portfolio of sin stocks (regular investor effect in Corollary 1) increased faster than the

covariances of the replicating portfolios (using non-sin stocks) of each sin stock and the

portfolio of sin stocks (sustainable investor effect in Corollary 1). For an intuitive interpret-

ation, the sustainable investor effect can be related to the correlation of the sin stocks with

the portfolio of non-sin stocks. Therefore, the discrepancy between these two effects can be

understood as sin stocks behaving increasingly like a homogeneous and separate group

from other stocks. Consequently, the increase in the gap between these two effects during

the crisis led to an increase in the exclusion premia and hence, the exclusion effect. In the

Online Appendix, I show how these two effects varied throughout the whole period using a

sample of sin stocks.

This result suggests that even in the presence of a limited number of sustainable invest-

ors and when averaged over all excluded assets, the effect of exclusionary screening on the

targeted companies’ cost of capital can be quite pronounced. Therefore, an opportune im-

pact investing strategy would be to increase exclusionary screenings when the targeted

assets have dynamics that diverge from all other assets.

5.5 Discussion: “Exit” versus “Voice” for Impact

Exclusionary screening and shareholder engagement are often compared as two opposing

approaches to impact investing: while the former involves divesting from companies to in-

crease their cost of capital and incentivize them to reform, the latter requires investing in

companies to push them to improve their practices as an active shareholder.

Through a theoretical model, Broccardo, Hart, and Zingales (2020) study the relative

effectiveness of exclusionary screening (“exit”) and shareholder engagement (“voice”) in

promoting socially desirable outcomes in companies. They show that, for a sufficiently

Figure 4. Evolution of the exclusion effect. This figure shows the evolution of the exclusion effect,

d̂ex:assetqCovðrX ; rmX jrI Þ þ d̂ex:mktqCovðrX ; rmX

jrmI Þ, estimated using a rolling S-CAPM, in the period

from December 1999 to December 2019. The first and second passes are both estimated over 3-year

rolling periods.

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large number of sustainable investors, engagement is more effective than exclusion, notably

because investors’ individual incentives are aligned with social incentives. Indeed, they

point out that the marginal impact of divestment is limited, especially when there are

enough regular investors to buy the asset under consideration. Berk and van Binsbergen

(2021) reach a similar conclusion by showing that the impact of exclusion on the cost of

capital “can be closely approximated by a simple formula.” Calibrating this formula on the

FTSE USA and FTSE USA 4 Good indices for the period from December 2015 to December

2020, they find that the effect on the cost of capital is negligible—in the order of a few bps

depending on the assumptions chosen.

In this article, I show that the effect of exclusionary screening on the cost of capital is

not necessarily negligible. The conclusion differs from that of Berk and van Binsbergen

(2021) for four main reasons. First, I show that the exclusion effect for a given asset is the

sum of two conditional covariances, generalizing the premium on neglected stocks

(Merton, 1987), while Berk and van Binsbergen (2021) find an approximation of the exclu-

sion effect on the cost of capital. However, in the particular case where the excluded assets

increasingly behave like a separate group from the other assets, the exclusion effect

increases as in the approximation found by Berk and van Binsbergen (2021).31 Second, I

focus on sin stocks, whereas Berk and van Binsbergen (2021) analyze the broader scope of

the stocks that are included in the FTSE USA but not in the FTSE USA 4 Good. Third, and

most importantly, I carry out a dynamic empirical analysis on sin stocks from 1999 onward

and show that although the average exclusion effect was small in the period 2015–20, as

shown by Berk and van Binsbergen (2021), it was high during the 2007–08 crisis. Fourth,

Berk and van Binsbergen (2021) estimate an average effect on an aggregate basis by com-

paring two indices and using their correlation, while here, I estimate the exclusion effect

stock by stock, using the covariance matrix structure. Although negative for several stocks,

the exclusion effect is positive and large for other stocks, in some cases above 10% annually

(Figure 3).

However, exclusionary screening and shareholder engagement are not necessarily con-

flicting practices, and implementing them in concert may increase their efficiency. For ex-

ample, the California State Teachers’ Retirement System (CalSTRS), the largest teachers’

retirement fund in the USA, managing approximately USD 320 billion as of January 2022,

has a sustainable investment management process that involves both engagement and ex-

clusion. The process is broken down into three stages (CalSTRS, 2017). When a company

in the portfolio violates CalSTRS’ ESG policy, (1) “CalSTRS will actively engage, in a con-

structive manner, corporate management whose actions are inconsistent with this policy.”

(2) “After all reasonable efforts have been made to constructively engage corporate man-

agement [. . .] and the corporate remedies are insufficient or nonresponsive, CalSTRS will

inform [their] active investment managers that, to the extent suitable alternate investments

are available [. . .], the managers will invest in these alternatives until the CalSTRS policy

violations cease.” (3) “Upon remedy of the policy violation, CalSTRS will inform the active

investment managers and passive managers that the securities can be purchased [. . .].”

31 Rewritten with this papers’ notation, Berk and van Binsbergen (2021) approximate the exclusion

effect by EðrmÞ p 1�p qð1� q2Þ, where q is the correlation between the excluded portfolio and the

non-excluded portfolio.

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6. Conclusion

In this article, I develop an asset pricing model with partial segmentation and heterogeneous pref-

erences to describe the effects of exclusionary screening and integration practices by sustainable

investors on expected asset returns. By estimating this model for sin stock exclusion and green

investing, I show that the exclusion and taste premia significantly affect asset returns. I also find

evidence for the cross effects of exclusion and tastes between investable and excluded stocks.

The findings suggest that the impact of sustainable investors on a company’s cost of cap-

ital can be substantial in many cases. Therefore, without contradicting the implementation

of shareholder engagement policies, exclusionary screening and ESG integration can be ef-

fective tools for contributing to the ecological transition.

The conclusions of the model presented in this article remain valid in a more general case.

The Online Appendix presents the derivation of the expected excess returns on investable

assets in the case of several sustainable investors with different tastes and exclusion scopes.

Future empirical research could build on this study and that of Broccardo, Hart, and

Zingales (2020) by disentangling the impacts of engagement and investment screening on

companies’ practices. In addition, impact investing is more efficient when sustainable invest-

ors account for the investments of all market players in their investment decision (Oehmke

and Opp, 2020; Green and Roth, 2021) or when markets are subject to search frictions

(Landier and Lovo, 2020). Another avenue for future research is to estimate the impact bene-

fit when sustainable investors overweight poorly funded companies that are inclined to be-

come greener in their portfolios rather than already well-funded green companies.

Data Availability

The data underlying this article were provided by CRSP, Compustat, FactSet, Bloomberg,

and S&P-Trucost under licence. Data will be shared on request to the corresponding author

with permission of CRSP, Compustat, FactSet, Bloomberg, or S&P-Trucost.

Supplementary Material

Supplementary data are available at Review of Finance online.

Funding

I gratefully acknowledge the financial support from the AXA Research Fund through the research

initiative entitled Climate risks in institutional asset allocation.

Conflict of Interest

None declared.

Appendix A: Derivation of the S-CAPM and Main Proofs

Problem Setup

We model regular investors and sustainable investors on an aggregate basis: one generic

regular investor (referred to using subscript r) and one generic sustainable investor (referred

to using subscript s).

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Heterogeneous preferences. The two groups of investors maximize at time t the expected

utility of their terminal wealth at time tþ1. We denote by ca j the absolute risk aversion of

investors j (j 2 fr; sg) and by Wj;t and Wj;tþ1 their wealth on t and tþ 1, respectively.

However, investors have heterogeneous preferences. On the one hand, regular investors

have an exponential utility. They select the optimal vector of weights of risky assets, wj,

corresponding to the solution of the following optimization problem:

max wr

E UrðWr;tþ1Þ � �

¼ max wr

E 1� e�ca r Wr;tþ1ð Þ:

On the other hand, sustainable investors have specific tastes for assets; they adjust their

exponential utility by internalizing a deterministic private cost of externalities as in Pastor,

Stambaugh, and Taylor (2021b). We denote by CW the vector of private costs of external-

ities that sustainable investors internalize in their utility function; CW has the same unit as a

wealth. Sustainable investors’ utility decreases when the cost of externalities increases; they

select the optimal vector of weights of risky assets, ws, corresponding to the solution of the

following optimization problem:

max ws

E UsðWs;tþ1Þ � �

¼ max ws

E 1� e�ca s Ws;tþ1þw0sC

W � �

:

In Pastor, Stambaugh, and Taylor (2021b), investors internalize nonpecuniary benefits,

which positively impact their utility. In the present article, sustainable investors internalize

costs of externalities, which negatively impact their utility.

Partially segmented market. Investors can invest in a risk-free asset, the return on which is

denoted by rf, and in risky assets. Sustainable investors can only invest in investable risky

assets, the returns on which are denoted by the nI � 1 vector RI, while regular investors can

invest in investable and excluded risky assets, the returns on which are denoted by the ðnI þ nXÞ � 1 vector R ¼ RI RX

� �0 . We assume that risky asset returns are normally distributed.

Mean–variance problems. Without loss of generality, we assume that investors have the

same relative risk aversion, c ¼Wj;tca j (j 2 fr; sg). We denote by C ¼ 1

c CW the vector of pri-

vate costs of environmental externalities per unit of relative risk aversion; C has the same

unit as a return. We now work with vector C and refer to its entries as the private costs of

environmental externalities (without referring to the normalization by the risk aversion). C

is a nI � 1 vector that applies to investable assets, which are the only ones that sustainable

investors can trade. We denote by r ¼ R� rf 1nIþnX ; rI ¼ RI � rf 1nI

, and rX ¼ RX � rf 1nX

the vectors of excess returns on all assets, investable assets, and excluded assets, respective-

ly, where 1n is the vector of ones of length n 2 N �.

The weights of regular investors in investable and excluded assets are denoted by wr;I

and wr;X, respectively; the weights of sustainable investors in investable assets are denoted

by ws;I. All weights add up to one, including the weight of the risk-free asset. Since the

wealth in tþ1 is normally distributed and CW is deterministic, sustainable investors’

expected utility writes

EðUsðWs;tþ1ÞÞ ¼ 1� E e �ca

s Ws;t 1þw0 s;I

RIþ 1�w0 s;I

1nIð Þrfð Þþw0 s;I

CW � �

¼ 1� e�c 1þrfð Þe�cw0 s;I

EðrIÞ�Cð Þþc2

2 w0 s;I VarðrIÞws;I :

:

Similarly, regular investors’ expected utility is

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EðUrðWr;tþ1ÞÞ ¼ 1� e�c 1þrfð Þe �c

wr;I

wr;X

� � 0EðrÞþc2

2

wr;I

wr;X

� � 0VarðrÞ

wr;I

wr;X

� � :

Let us also denote the vectors lI ¼ EtðrIÞ; lX ¼ EtðrXÞ and the matrices RXX ¼ VartðrXÞ; RII ¼ VartðrIÞ; RXI ¼ CovtðrX; rIÞ; RIX ¼ CovtðrI; rXÞ. Therefore:

– Regular investors choose their optimal asset allocation by solving the following

problem:

max wr;I ;wr;Xð Þ wr;I

wr;X

� �0 lI

lX

� � � c

2

wr;I

wr;X

� �0 RII RIX

RXI RXX

� � wr;I

wr;X

� � : (A.1)

– Sustainable investors choose their optimal asset allocation by solving the following

problem:

maxws;I w0s;IðlI � CÞ � c

2 w0s;IRIIws;I: (A.2)

First-order conditions. Denoting the inverse of the risk aversion by k ¼ 1 c, regular investors

and sustainable investors therefore solve the following first-order conditions:

k lI � Cð Þ ¼ RIIws;I

k lI

lX

� � ¼ RII RIX

RXI RXX

� � wr;I

wr;X

� � :

8>>< >>: (A.3)

Proof of Proposition 1: S-CAPM

Lemma 1 (Preliminary Results).

The covariance column vector between the vector of excess returns on investable assets, rI,

and the excess returns on the investable market, rmI , is denoted by rImI

; rmII refers to the

covariance line vector between rmI and rI. rXmI

and rmIX are defined similarly.

We denote by qX the weight vector of the excluded assets’ market values as a fraction of

the market value of the investment universe and q 2 ½0; 1� the share of the excluded mar-

ket’s value as a fraction of the market value of the investment universe.

Assuming that the returns are normally distributed, rmI is non-zero and RII is nonsingu-

lar, we have the following equalities:

1. (i) RXX � 1 r2

mI

rXmI rmIX ¼ VartðrXjrmI

Þ,

(ii) RIX � 1 r2

mI

rImI rmIX ¼ CovtðrI; rXjrmI

Þ,

(iii) RXX � RXIR �1 II RIX ¼ VartðrXjrIÞ,

(iv) rXmX � RXIR

�1 II RImX

¼ CovtðrX; rmX jrIÞ.

2. CovtðrI; rXjrmI ÞqX ¼ qCovtðrI; rmX

jrmI Þ.

Proof: See the Online Appendix. h

From here on, the time subscripts will be omitted to simplify the notations.

Derivation of the expected excess returns on I. Multiplying the first rows of System (3) by

the wealth of investors s and r, respectively, we have

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k Ws þWrð ÞlI � kWsC ¼ RII Wsws;I þWrwr;Ið Þ þ RIX Wrwr;Xð Þ: (A.4)

Dividing by the total wealth W, and noting that Ws

W ¼ p and Wr

W ¼ 1� p, we obtain

klI ¼ RII Wsws;I þWrwr;I

W

� � þ RIX

Wrwr;X

W

� � þ kpC: (A.5)

Denoting by DI and DX the column vectors equal to the total demand for stocks I and

X, respectively, we have Wsws;I þWrwr;I ¼ DI and Wrwr;X ¼ DX. Consequently,

klI ¼ RII DI

W þ RIX

DX

W þ kpC: (A.6)

In equilibrium, the total demand of assets is equal to the total supply in the entire mar-

ket (S). The same holds for the markets of investable (SI) and excluded (SX) assets: W¼ S,

DI ¼ SI, and DX ¼ SX. The ðnX � 1Þ weight vectors of the excluded assets’ values as a frac-

tion of the market value are denoted by qX ¼ SX

S . Therefore,

klI ¼ RII SI

S þ RIXqX þ kpC: (A.7)

We denote by q the proportion of the excluded market’s value as a fraction of the mar-

ket value of the investment universe. The share of the investable market’s value is 1� q. Let

us denote by wI the vector of market values of stocks ðIkÞk2f1;...;nIg as a fraction of the invest-

able market’s value. Therefore, we have SI

S ¼ ð1� qÞwI, and Equation (A.7) rewrites

klI ¼ 1� qð ÞRIIwI þ RIXqX þ kpC: (A.8)

Multiplying by w0I, we obtain

kw0IlI ¼ 1� qð Þw0IRIIwI þw0IRIXqX þ kpw0IC: (A.9)

Since w0IlI ¼ lmI is the expected excess return on the investable market, and denoting

cmI ¼ w0IC and the row vector of covariances rmIX ¼ w0IRIX,

klmI ¼ 1� qð Þr2

mI þ rmIXqX þ kpcmI

: (A.10)

Therefore, assuming r2 mI 6¼ 0,

1� qð Þ ¼ 1

r2 mI

klmI � rmIXqX � kpcmI

� � : (A.11)

Substituting Equation (A.11) into Equation (A.8) and noting that the column vector of

covariances is rImI ¼ RIIwI, we obtain

lI ¼ lmI � pcmIð Þ

1

r2 mI

rImI þ pCþ c RIX �

1

r2 mI

rImI rmIX

! qX: (A.12)

Denoting by bImI ¼ 1

r2 mI

rImI the vector of slope of the regression of the excess returns on

the investable assets, rI, on the excess returns on the investable market, rmI , and a constant,

and from Lemma 1, we rewrite the above equation as follows using vector notations:

EðrIÞ ¼ EðrmI Þ � pcmI

� � bImI þ pCþ cqCovðrI; rmX

jrmI Þ: (A.13)

Derivation of the expected excess returns on X. Assuming that RII is nonsingular, the first

row of System (3) yields

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wr;I ¼ R�1 II klI � RIXwr;Xð Þ: (A.14)

Substituting wr;I into the second row of System (3), we have

klX ¼ kRXIR �1 II lI � RXIR

�1 II RIXwr;X þ RXXwr;X: (A.15)

Multiplying by Wr

W , we obtain

k Wr

W lX ¼ k

Wr

W RXIR

�1 II lI �

Wr

W RXIR

�1 II RIXwr;X þ

Wr

W RXXwr;X: (A.16)

Since in equilibrium W¼ S, and knowing that ð1� pÞ ¼ Wr

W and wr;X Wr

S ¼ qX, we have

lX ¼ RXIR �1 II lI þ

c 1� p

RXX � RXIR �1 II RIX

� � qX: (A.17)

Substituting lI into the previous equation, and since rImI ¼ RIIwI,

lX ¼ lmI � pcmIð Þ

1

r2 mI

RXIR �1 II RIIwI þ pRXIR

�1 II C

þ c RXIR �1 II RIX �

1

r2 mI

RXIR �1 II RIIwIrmIX

! qX þ

c 1� p

RXX � RXIR �1 II RIX

� � qX:

(A.18)

By adding and subtracting cRXXqX to the previous equation,

lX ¼ lmI � pcmIð Þ

1

r2 mI

RXIR �1 II RIIwI þ pRXIR

�1 II C

þ c RXIR �1 II RIX � RXX

� � qX þ c RXX �

1

r2 mI

RXIR �1 II RIIwIrmIX

! qX

þ c 1� p

RXX � RXIR �1 II RIX

� � qX:

(A.19)

We denote bXmI ¼ 1

r2 mI

RXIwI and BXI ¼ RXIR �1 II . Noting that c

1�p� c ¼ c p 1�p and from

Lemma 1, the previous equation is simplified as follows using vector notations:

EðrXÞ ¼ EðrmI Þ � pcmI

� � bXmI

þ pBXICþ c p

1� p qCovðrX; rmX

jrIÞ þ cqCovðrX; rmX jrmI Þ:

(A.20)

Derivation of the general pricing formula. For any investable asset Ik,

CovðrIk ; rmX jrIÞ ¼ rIkmX

� rIkIR �1 II rImX

¼ rIkmX � rIkmX

¼ 0; (A.21)

and

BIkIC ¼ rIkIR �1 II C ¼ cIk

: (A.22)

Therefore, for any asset k 2 fI1; . . . ; InI ;X1; . . . ;XnX

g,

EðrkÞ¼bkmI EðrmI

Þ�pcmI

� � þpBkICþc

p

1�p qCovðrk;rmX

jrIÞþcqCovðrk;rmX jrmI Þ: (A.23)

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Proof of Corollary 1: Expression of the Exclusion Premia as the Difference between a

Regular Investor Effect and a Sustainable Investor Effect

(i) From the law of total covariance, we express the expectation of the conditional covari-

ance as a difference between two covariances:

EðCovðrk; rmX jrIÞÞ ¼ Covðrk; rmX

Þ � CovðEðrkjrIÞ;EðrmX jrIÞÞ: (A.24)

Since the conditional covariance of multivariate normal distributions is independent of

the conditioning variable (see Lemma 1), EðCovðrk; rmX jrIÞÞ ¼ Covðrk; rmX

jrIÞ. By multiply-

ing the previous equation by c p 1�p q, we obtain the expected result.

(ii) The proof is analogous for the exclusion-market premium.

Proof of Proposition 3: Sign of the Exclusion Premia

(i) Let us focus on the exclusion-asset premium. Since c; q � 0, and p 2 ½0; 1�; c p 1�p q is

positive.

As shown in Lemma 1, the conditional covariance is equal to

qCovðrX; rmX jrIÞ ¼ RXX � RXIR

�1 II RIX

� � qX: (A.25)

When there is at least one excluded asset, that is, q> 0 and qX 6¼ 0nX , denoting by wX ¼

1 q qX > 0 the vector of weights of assets X in the excluded market, we express the covariance

matrix as the product of a Schur complement by a strictly positive vector of weights:

CovðrX; rmX jrIÞ ¼ RXX � RXIR

�1 II RIX

� � 1

q qX ¼ RXX � RXIR

�1 II RIX

� � wX: (A.26)

However, RII is positive-definite (because it is nonsingular positive semidefinite) and

with RII RIX

RXI RXX

� � being positive semidefinite, Schur complement RXX � RXIR

�1 II RIX

� � is

positive semidefinite. Therefore, the exclusion-asset effects for assets X are the elements of

the vector being the product of a semidefinite positive matrix by a strictly positive vector of

weights. Consequently, not all elements of this vector are necessarily positive.

The same applies to the exclusion-market premium.

(ii) The expected excess return of the excluded market EðrmX Þ is obtained by multiplying

the vector of excluded assets’ expected excess returns EðrXÞ by their weight in the excluded

market w0X:

EðrmX Þ ¼ ðEðrmI

Þ � pcmI Þw0XbXmI

þ pw0XBXIC

þc p

1� p qw0XCovðrX; rmX

jrIÞ þ cqw0XCovðrX; rmX jrmI Þ : (A.27)

Since the covariance and the conditional covariance are bilinear, we have

EðrmX Þ ¼ bmXmI

ðEðrmI Þ � pcmI

Þ þ pBmXICþ c p

1� p qVarðrmX

jrIÞ þ cqVarðrmX jrmI Þ:

(A.28)

Let qmXmI be the correlation coefficient between the excess returns on the excluded mar-

ket, mX, and those on the investable market, mI, and qmXI be the multiple correlation coeffi-

cient between the excess returns on the excluded market, mX, and those on the vector of

investable assets’ excess returns, I. Since VarðrmX jrIÞ ¼ VarðrmX

Þ 1� qmXI

� � and

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VarðrmX jrmI Þ ¼ VarðrmX

Þ 1� qmXmI

� � (because the returns are Gaussian), the positivity of

the exclusion premia follows.

Proof of Proposition 4: Cost of Externalities

Let w�r;I and w�r;X be regular investors’ optimal weight vector of investable and excluded

assets, respectively; w�s;I is sustainable investors’ weight vector of investable assets.

Intuition of the proof. By substituting the first-order condition of sustainable investors into

the first-order condition of regular investors via risk aversion c ¼ 1 k (using system of

Equation (3)), the cost of externalities of asset Ik; k 2 f1; . . . ; nIg, is

cIk ¼

CovðrIk ; r0IÞðw�r;I �w�s;IÞ þ CovðrIk

; r0XÞw�r;X CovðrIk

; r0IÞw�r;I þ CovðrIk ; r0XÞw�r;X

EðrIk Þ: (A.29)

Proof: Let us focus on asset Ik. We assume that asset returns are independent (assump-

tion (i)). From System (3),

w�r;Ik ¼ k

EðrIk Þ

VarðrIk Þ ; w�s;Ik

¼ k EðrIk

Þ � cIk

VarðrIk Þ : (A.30)

But, the market weight of Ik is

wm;Ik ¼ ð1� pÞk EðrIk

Þ VarðrIk

Þ þ pk EðrIk

Þ � cIk

VarðrIk Þ ¼ k

EðrIk Þ

VarðrIk Þ � pk

cIk

VarðrIk Þ :

Therefore,

wm;Ik �w�s;Ik

wm;Ik

EðrIk Þ ¼

k EðrIk

Þ VarðrIk

Þ � pk cIk

VarðrIk Þ � k

EðrIk Þ�cIk

VarðrIk Þ

k EðrIk

Þ VarðrIk

Þ � pk cIk

VarðrIk Þ

EðrIk Þ: (A.31)

Simplifying the above expression,

wm;Ik �w�s;Ik

wm;Ik

EðrIk Þ ¼ cIk

� pcIk

1� pcIk

EðrIk Þ : (A.32)

Using the first-order expansion 1

1� pcIk EðrIk

Þ

’ 1þ pcIk

EðrIk Þ, when

pcIk

EðrIk Þ is small (assumption (iii)),

wm;Ik �w�s;Ik

wm;Ik

EðrIk Þ ’ 1� p 1� ð1� pÞcIk

EðrIk Þ

! ! cIk : (A.33)

When p is small (assumption (ii)), wm;Ik

�w� s;Ik

wm;Ik

EðrIk Þ ’ cIk

:

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