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WORKING PAPER · NO. 2020-24
Sustainable Investing in Equilibrium
Lubos Pastor, Robert F. Stambaugh, and Lucian A. Taylor
FEBRUARY 2020
Sustainable Investing in Equilibrium
Lˇubosˇ Pa′stor
Robert F. Stambaugh
Lucian A. Taylor*
February 14, 2020
Abstract
We present a model of investing based on environmental, social, and gover-
nance (ESG) criteria. In equilibrium, green assets have negative CAPM alphas,
whereas brown assets have positive alphas. Green assets’ negative alphas stem
from investors’ preference for green holdings and from green stocks’ ability to
hedge climate risk. Green assets can nevertheless outperform brown ones during
good performance of the ESG factor, which captures shifts in customers’ tastes
for green products and investors’ tastes for green holdings. The latter tastes pro-
duce positive social impact by making firms greener and shifting real investment
from brown to green firms. The ESG investment industry is at its largest, and
the alphas of ESG-motivated investors are at their lowest, when there is large
dispersion in investors’ ESG preferences.
JEL classifications: G11, G12
Keywords: sustainable investing, socially responsible investing, ESG, social impact
*Pa′stor is at the University of Chicago Booth School of Business. Stambaugh and Taylor are at the
Wharton School of the University of Pennsylvania. Pa′stor and Stambaugh are also at the NBER. Pa′stor is
additionally at the National Bank of Slovakia and the CEPR. The views in this paper are the responsibility of
the authors, not the institutions they are affiliated with. We are grateful for comments from Rui Albuquerque,
Malcolm Baker, George Constantinides, Alex Edmans, Gene Fama, Sam Hartzmark, John Heaton, Ralph
Koijen, Yrjo Koskinen, Raghu Rajan, Jeff Wurgler, Josef Zechner, and seminar audiences at the University
of Chicago, WU Vienna, and the National Bank of Slovakia.
1. Introduction
Sustainable investing is an investment approach that considers not only financial but also
environmental, social and governance (ESG) objectives. This approach initially gained pop-
ularity by imposing negative screens under the umbrella of socially responsible investing
(SRI), but its scope has expanded significantly in recent years. Assets managed with an eye
on sustainability have grown to tens of trillions of dollars, and they seem poised to grow
further.1 Given the rapid growth of ESG-driven investing, it seems important to understand
its effects on asset prices and corporate investment.
We analyze both financial and real effects of sustainable investing through the lens of a
general equilibrium model. The model features many heterogeneous firms and many het-
erogeneous agents, yet it is highly tractable, yielding simple and intuitive expressions for
the quantities of interest. The model illuminates the key channels through which agents’
preferences for sustainability can move asset prices, tilt portfolio holdings, determine the size
of the ESG investment industry, and cause real impact on society.
In the model, firms differ in the sustainability of their activities. “Green” firms generate
positive externalities for society, “brown” firms impose negative externalities, and there are
different shades of green and brown. Agents differ in their preferences for sustainability,
or “ESG preferences,” which have multiple dimensions. First, agents derive utility from
holdings of green firms and disutility from holdings of brown firms. Second, agents care
about firms’ aggregate social impact. In a model extension, agents additionally care about
climate risk. Naturally, agents also care about financial wealth.
We show that agents’ tastes for green holdings affect asset prices. The greener the firm,
the lower is its cost of capital in equilibrium. Greener assets have lower CAPM alphas. Green
assets have negative alphas, whereas brown assets have positive alphas. Consequently, agents
with stronger ESG preferences, whose portfolios tilt more toward green assets and away from
brown assets, earn lower expected returns. Yet such agents are not unhappy because they
derive utility from their holdings.
The model implies three-fund separation, whereby each agent holds the market portfolio,
the risk-free asset, and an “ESG portfolio,” which is largely long green assets and short
brown assets. Agents with stronger-than-average tastes for green holdings go long the ESG
1According to the 2018 Global Sustainable Investment Review, sustainable investing assets exceeded $30
trillion globally at the start of 2018, a 34% increase in two years. As of November 2019, more than 2,600
organizations have become signatories to the United Nations Principles of Responsible Investment (PRI),
with more than 500 new signatories in 2018/2019, according to the 2019 Annual Report of the PRI.
1
portfolio, agents with weaker tastes go short, and agents with average tastes hold the market
portfolio. If there is no dispersion in ESG tastes, all agents simply hold the market. Even if all
agents derive a large amount of utility from green holdings, they end up holding the market
if their ESG tastes are equally strong, because asset prices adjust to reflect those tastes.
In this equal-taste case, the ESG investment industry does not exist: despite the strong
demand for green holdings, a market index fund is all that is needed to satisfy investors.
For the ESG industry to exist, some dispersion in ESG tastes is necessary. The larger the
dispersion, the larger the industry.
We illustrate the economic significance of the above effects by calibrating a case in which
there are two types of investors, those sharing equal concerns about ESG and those not con-
cerned at all. The key free parameter is ?, the maximum certain return an ESG-concerned
investor is willing to forego in exchange for investing in her desired portfolio instead of the
market. The negative alpha such investors earn is greatest, and the ESG industry is largest,
when dispersion in ESG tastes is greatest (here meaning ESG investors constitute half of
total wealth). That worst-case alpha is substantially smaller than ?, however, because equi-
librium prices adjust to ESG demands, thereby pushing the market portfolio toward the
portfolio desired by ESG investors. For example, when ESG investors have a ? of 4%, their
worst-case alpha is only ?2%. This difference between ? and alpha provides ESG investors
with an “investor surplus”: In order to hold their desired portfolio, they sacrifice less return
than they are willing to. The same price-adjustment mechanism lessens the impact of ESG
investing on the ESG industry’s size, which we measure as the aggregate ESG dollar tilt
away from the market portfolio. For example, if the ESG industry reaches 33% of the stock
market’s value when ? is 1%, then doubling the strength of ESG concerns (raising ? to 2%)
increases that maximum industry size by less than half, to 46% of the market’s value.
Our model implies that sustainable investing leads to positive social impact. We define
social impact as the product of a firm’s ESG characteristic and the firm’s operating capital.
We show that agents’ tastes for green holdings increase firms’ social impact, through two
channels. First, firms choose to become greener, because greener firms have higher market
values. Second, real investment shifts from brown to green firms, due to shifts in firms’
cost of capital (up for brown firms, down for green firms). We obtain positive aggregate
social impact even if agents have no direct preference for it, shareholders do not engage with
management, and managers simply maximize market value.
We introduce an “ESG factor” that captures unexpected changes in ESG concerns. These
concerns can change in two ways: customers may shift their demands for goods of green
providers, or investors may change their appreciation for green holdings. The ESG factor
2
affects the relative performance of green and brown assets, both ex post and ex ante. Ex
post, the factor’s positive realizations boost green assets while hurting brown ones. If ESG
concerns strengthen unexpectedly, green assets can outperform brown ones despite having
lower expected returns. Ex ante, green and brown assets have opposite ESG factor exposures,
which push these assets’ market betas in opposite directions.
Finally, we extend the model by allowing climate to enter investors’ utility. Expected
returns then depend not only on market betas and investors’ tastes, but also on climate
betas, which measure firms’ exposures to climate shocks. Evidence suggests that brown
assets have higher climate betas than green assets (e.g., Choi, Gao, and Jiang, 2019, and
Engle et al., 2019), which pushes up brown assets’ expected returns in our model. The idea
is that investors dislike unexpected deteriorations in the climate. If the climate worsens
unexpectedly, brown assets lose value relative to green assets (e.g., due to new government
regulation that penalizes brown firms). Because brown firms lose value in states of the world
investors dislike, they are riskier, so they must offer higher expected returns. Greener stocks
thus have lower CAPM alphas not only because of investors’ tastes for green holdings, but
also because of their ability to better hedge climate risk.
The climate factor is likely to be related to the ESG factor because climate shocks are
likely to affect both customers’ tastes for green products and investors’ tastes for green
holdings. In the special case where climate shocks are the only reason behind shifts in
agents’ tastes, the two factors coincide. A two-factor asset pricing model then prices all
stocks, with the factors being the market portfolio and the ESG factor. In general, though,
a multi-factor approach may be necessary to capture the risk associated with ESG investing
because agents’ tastes may shift also for reasons unrelated to climate shocks.
Our theoretical treatment of climate risk is related to recent empirical work on the im-
plications of such risk for asset prices. Hong, Li, and Xu (2019) analyze the response of food
stock prices to climate risks. Bolton and Kacperczyk (2019) argue that investors demand
compensation for exposure to carbon risk in the form of higher returns on carbon-intensive
firms. Ilhan, Sautner, and Vilkov (2019) show that firms with higher carbon emissions ex-
hibit more tail risk and more variance risk. Engle et al. (2019) develop a procedure to
dynamically hedge climate risk by constructing mimicking portfolios that hedge innovations
in climate news series obtained by textual analysis of news sources. Bansal, Ochoa, and Kiku
(2016) model climate change as a long-run risk factor. Krueger, Sautner, and Starks (2019)
find that institutional investors consider climate risks to be important investment risks.
Besides climate risk, studies have identified multiple other aspects of ESG-related risk.
3
Hoepner et al. (2018) find that ESG engagement reduces firms’ downside risk, as well as
their exposures to a downside-risk factor. Luo and Balvers (2017) find a premium for boycott
risk. We complement these studies with a theoretical contribution. We construct an ESG
risk factor that is driven by unexpected shifts in ESG concerns of firms’ customers and
market investors. We show that green and brown assets have opposite exposures to this
factor, and we link these exposures to market betas.
Prior studies report that green assets underperform brown assets, in various contexts.
Hong and Kacperczyk (2009) find that “sin” stocks (i.e., stocks of public firms producing
alcohol, tobacco, and gaming, which we would classify as brown) outperform non-sin stocks.
They argue that social norms lead investors to demand compensation for holding sin stocks.
Barber, Morse, and Yasuda (2018) find that venture capital funds that aim not only for
financial return but also for social impact earn lower returns than other funds. They argue
that investors derive nonpecuniary utility from investing in dual-objective funds. Baker et
al. (2018) and Zerbib (2019) find that green bonds tend to be priced at a premium, offering
lower yields than traditional bonds. Both studies argue that the premium is driven by
investors’ environmental concerns. Similarly, Chava (2014) and El Ghoul et al. (2011) find
that greener firms have a lower implied cost of capital. All of these results are consistent
with the effects of tastes for green holdings on the cost of capital in our model.
Some studies find the opposite result, based on different definitions of green and brown.
Firms perform better if they are better-governed, judging by employee satisfaction (Edmans,
2011) or strong shareholder rights (Gompers, Ishii, and Metrick, 2003), or if they have higher
ESG ratings in the 1992–2004 period (Kempf and Osthoff, 2007). These results are consistent
with our model if ESG concerns strengthened unexpectedly over the sample period.
Our model is related to prior theoretical studies of sustainable investing. Heinkel, Kraus,
and Zechner (2001) build an equilibrium model in which exclusionary ethical investing affects
firm investment. They consider two types of investors, one of which refuses to hold shares
in polluting firms. The resulting reduction in risk sharing increases the cost of capital
of polluting firms, depressing their investment. Albuquerque, Koskinen, and Zhang (2019)
construct a model in which a firm’s socially responsible investments increase customer loyalty,
giving the firm more pricing power. This power makes the firm less risky and thus more
valuable. Unlike these models, ours features neither a lack of risk sharing nor pricing power;
instead, the main force is investors’ tastes for holding green assets.
Tastes for holding assets can affect prices, as emphasized by Fama and French (2007).
Baker et al. (2018) build a model featuring two types of investors with mean-variance
4
preferences, where one type also has tastes for green assets. Their model predicts that green
assets have lower expected returns and more concentrated ownership, and they find support
for these predictions in the universe of green bonds. Pedersen, Fitzgibbons, and Pomorski
(2019) consider the same two types of mean-variance investors, but also add a third type
that is unaware of firms’ ESG scores. This lack of awareness is costly if firms’ ESG scores
predict their profits. The authors show that stocks with higher ESG scores can have either
higher or lower expected returns, depending on the wealth of the third type of investors.
They obtain four-fund separation and derive the ESG-efficient frontier characterizing the
tradeoff between the ESG score and the Sharpe ratio.
While these studies share some modeling features with ours, we offer novel insights that
do not appear in those studies. We show that the size of the ESG investment industry, as well
as investors’ alphas, crucially depend on the dispersion in investors’ ESG tastes. We relate
market betas to assets’ exposures to an ESG risk factor. We show that this factor, along
with the market, prices assets in a two-factor model. We also show that positive realizations
of this factor, which result from shifts in customers’ and investors’ tastes, can result in green
assets outperforming brown. We have a continuum of investors with multiple dimensions of
ESG preferences. Including climate in those preferences, for example, results in the pricing
of climate risk. Finally, we show that ESG investing has positive social impact.
Positive social impact also emerges from the model of Oehmke and Opp (2020), but
through a different channel. Key ingredients to generating impact in their model are fi-
nancing constraints and coordination among agents. Our model does not include those
ingredients, but it produces social impact nevertheless, through tastes for green holdings.
To emphasize these tastes, we do not model shareholder engagement with management,
which is another channel through which ESG investing can potentially increase market value
(e.g., Dimson, Karakas, and Li, 2015). In our model, value-maximizing managers make
their firms greener voluntarily, without pressure from shareholders, because greener firms
command higher market values.2
Our assumption that some investors derive nonpecuniary benefits from green holdings
has a fair amount of empirical support in the mutual fund literature. Mutual fund flows
respond to ESG-salient information, such as Morningstar sustainability ratings (Hartzmark
and Sussman, 2019) and environmental disasters (Bialkowski and Starks, 2016). Flows to SRI
mutual funds are less volatile than flows to non-SRI funds (Bollen, 2007) and less responsive
2Theoretical work on sustainable investing also includes Friedman and Heinle (2016), Gollier and Pouget
(2014), and Luo and Balvers (2017). Empirical work includes Geczy, Stambaugh, and Levin (2005), Hong and
Kostovetsky (2012), and Cheng, Hong, and Shue (2016), among others. For surveys of the early literature,
see Bauer, Koedijk, and Otten (2005) and Renneboog, ter Horst, and Zhang (2008).
5
to negative past performance (Renneboog, ter Horst, and Zhang, 2011). Investors in SRI
funds also indicate willingness to forgo financial performance to accommodate their social
preferences (Riedl and Smeets, 2017).
This paper is organized as follows. Section 2 presents our baseline model. Section 3
explores the model’s quantitative implications. Section 4 extends the baseline model by
allowing ESG concerns to shift over time, giving rise to the ESG factor. Section 5 extends the
baseline model by letting agents care about the climate, showing that climate risk commands
a risk premium. Section 6 discusses social impact. Section 7 concludes.
2. Model
The model considers a single period, from time 0 to time 1, in which there are N firms,
n = 1, . . . , N . Let r?n denote the return on firm n’s shares in excess of the riskless rate, rf ,
and let r? be the N × 1 vector whose nth element is r?n. We assume r? is normally distributed:
r? = μ+ ? , (1)
where μ contains equilibrium expected excess returns and ? ～ N(0,Σ). In addition to finan-
cial payoffs, firms produce social impact. Each firm n has an observable “ESG characteristic”
gn, which can be positive (for “green” firms) or negative (for “brown” firms). Firms with
gn > 0 have positive social impact, meaning they generate positive externalities (e.g., clean-
ing up the environment). Firms with gn < 0 have negative social impact, meaning they
generate negative externalities (e.g., polluting the environment). In Section 6, we model
firms’ social impact in greater detail.
There is a continuum of agents who trade firms’ shares and the riskless asset. The
riskless asset is in zero net supply, whereas each firm’s stock is in positive net supply. Let
Xi denote an N × 1 vector whose nth element is the fraction of agent i’s wealth invested in
stock n. Agent i’s wealth at time 1 is W?1i = W0i (1 + rf +X
′
i r?), where W0i is the agent’s
initial wealth. Besides liking wealth, agents also derive utility from holding green stocks and
disutility from holding brown stocks.3 Each agent i has exponential (CARA) utility
V (W?1i, Xi) = ?e?AiW?1i?b′iXi , (2)
where Ai is the agent’s absolute risk aversion and bi is an N × 1 vector of nonpecuniary
benefits that the agent derives from her stock holdings. Holding the riskless asset brings no
3We frame the discussion in terms of green and brown stocks, but our main ideas apply more broadly to
any set of green and brown assets, such as bonds and private equity investments.
6
such benefit. The benefit vector has agent-specific and firm-specific components:
bi = dig , (3)
where g is an N×1 vector whose nth element is gn and di ≥ 0 is a scalar measuring agent i’s
“ESG taste.” Agents with higher values of di have stronger tastes for the ESG characteristics
of their holdings. In addition to having ESG tastes, agents care about firms’ aggregate social
impact, but that component of preferences does not affect agents’ portfolio choice or asset
prices. Therefore, we postpone the discussion of that component until Section 6.3.
2.1. Expected Returns
Due to their infinitesimal size, agents take asset prices (and thus also the return distribution)
as given when choosing their optimal portfolios, Xi, at time 0. To derive the first-order con-
dition for Xi, we compute the expectation of agent i’s utility in equation (2) and differentiate
it with respect to Xi. As we show in the Appendix, agent i’s portfolio weights are
Xi =
1
ai
Σ?1
(
μ+
1
ai
bi
)
, (4)
where ai ≡ AiW0i is agent i’s relative risk aversion. For tractability, we assume that ai = a
for all agents. We define wi to be the ratio of agent i’s initial wealth to total initial wealth:
wi ≡ W0i/W0, where W0 = ∫iW0idi. The market-clearing condition requires that x, the
N × 1 vector of weights in the market portfolio, satisfies
x =
∫
i
wiXi di
=
1
a
Σ?1μ+
dˉ
a2
Σ?1g , (5)
where dˉ ≡ ∫iwididi ≥ 0 is the wealth-weighted mean of ESG tastes di across agents. Note
that dˉ > 0 unless the mass of agents who care about ESG is zero. Solving for μ gives
μ = aΣx? dˉ
a
g . (6)
Premultiplying by x′ gives the market equity premium, μM = x′μ:
μM = aσ
2
M ?
dˉ
a
x′g , (7)
where σ2M = x
′Σx is the variance of the market return. In general, the equity premium
depends on the average of ESG tastes, dˉ, through x′g, which is the overall “greenness” of the
market portfolio. If the market is net green (x′g > 0) then stronger ESG tastes (i.e., larger
7
dˉ) reduce the equity premium. If the market is net brown (x′g < 0), stronger ESG tastes
increase the premium. For simplicity, we make the natural assumption that the market
portfolio is ESG-neutral,
x′g = 0 , (8)
so that the equity premium is independent of agents’ ESG tastes. In this case, equation (7)
implies a = μM/σ
2
M . Combining this with equation (6) and noting that the vector of market
betas is β = (1/σ2M)Σx, we obtain our first proposition.
Proposition 1. Expected excess returns in equilibrium are given by
μ = μMβ ? dˉ
a
g . (9)
We see that expected excess returns deviate from their CAPM values, μMβ, due to ESG
tastes for holding green stocks.
Corollary 1. If dˉ > 0, the expected return on stock n is decreasing in gn.
As long as the mass of agents who care about sustainability is nonzero, dˉ is positive, and
expected returns are decreasing in stocks’ ESG characteristics. Because the alpha of stock
n is defined as αn ≡ μn ? μMβn, equation (9) yields the following corollary.
Corollary 2. The alpha of stock n is given by
αn = ? dˉ
a
gn . (10)
If dˉ > 0, green stocks have negative alphas, and brown stocks have positive alphas. Moreover,
greener stocks have lower alphas.
As long as some agents care about sustainability, equation (10) implies that the CAPM
alphas of stocks with gn > 0 are negative, the alphas of stocks with gn < 0 are positive, and
αn is decreasing with gn. Furthermore, the negative relation between αn and gn is stronger
when risk aversion, a, is lower and when the average ESG taste, dˉ, is higher.
Proposition 2. The mean and variance of the excess return on agent i’s portfolio are
E(r?i) = μM ? δi
(
dˉ
a3
g′Σ?1g
)
(11)
Var(r?i) = σ
2
M + δ
2
i
(
1
a4
g′Σ?1g
)
, (12)
where δi ≡ di ? dˉ.
8
Both equations are derived in the Appendix. Agents with δi > 0 earn below-market
expected returns because their portfolios tilt toward stocks with negative alphas. In contrast,
agents with δi < 0 earn above-market returns because they tilt toward positive-alpha stocks.
All agents with δi 6= 0 hold portfolios more volatile than the market.
Corollary 3. If dˉ > 0 and g 6= 0, agents with larger δi earn lower expected returns.
Under the conditions of this corollary, the term in parentheses in equation (11) is strictly
positive. Therefore, agents with stronger ESG tastes (i.e., larger δi) earn lower expected
returns. The conditions are not satisfied if no agents care about ESG (dˉ = 0) or if all firms
are ESG-neutral (g = 0); in that case, E(r?i) is independent of δi because all agents hold the
market. The effect of δi on E(r?i) is stronger when the average ESG taste is stronger (i.e.,
when dˉ is larger), when risk aversion a is smaller, and when g′Σ?1g is larger.
The low expected returns earned by ESG-sensitive agents do not imply these agents are
unhappy. As we show in the Appendix, agent i’s expected utility in equilibrium is given by
E
{
V (W?1i)
}
= Vˉ e?
δ2
i
2a2
g′Σ?1g , (13)
where Vˉ is the expected utility if the agent has δi = 0. We see that expected utility is
increasing in δ2i (note from equation (2) that Vˉ < 0), so it is larger for agents with larger
absolute values of δi. The more an agent’s ESG taste di deviates from the average in either
direction, the more ESG preferences contribute to the agent’s utility. High-δi investors
derive utility from their holdings of green stocks, while low-δi investors derive utility from
the positive alphas of brown stocks.
2.2. Portfolio Tilts
Substituting for μ from equation (9) into equation (4), we obtain an agent’s portfolio weights:
Proposition 3. Agent i’s equilibrium portfolio weights are given by
Xi = x+
δi
a2
(
Σ?1g
)
. (14)
Proposition 3 implies three-fund separation as each agent’s portfolio can be implemented
with three assets: the riskless asset, the market portfolio x, and an “ESG portfolio” whose
weights are proportional to Σ?1g. Agents with δi > 0 go long the ESG portfolio; agents with
δi < 0 short the portfolio. Agent i’s portfolio departs from the market portfolio due to the
second term in equation (14), which we refer to as agent i’s “ESG tilt.”
9
The ESG tilt is zero for agents whose ESG taste is average, in that di = dˉ. Such agents
hold the market portfolio. Interestingly, agents with di = 0 hold a portfolio that departs
from the market in the direction away from ESG. It is suboptimal for an investor to say “I
don’t care about ESG, so I’m just going to hold the market.” Investors who do not care about
ESG must tilt away from ESG, otherwise they are not optimizing. The market portfolio is
optimal for investors who care about ESG to an average extent, but not for those who do
not care about ESG at all.
Corollary 4. If there is no dispersion in ESG tastes across agents then all agents hold the
market portfolio.
No dispersion in ESG tastes implies di = dˉ, and so zero ESG tilt, for all i. Interestingly,
even if all agents derive a large amount of utility from green holdings, they end up holding
the market if their tastes are equally strong. The reason is that stock prices adjust to reflect
those tastes, making the market everybody’s optimal choice. Some dispersion in ESG tastes
is necessary for an ESG investment industry to exist.
If the covariance matrix Σ is diagonal, meaning all risk is idiosyncratic, then the ESG
portfolio weights are positive for green stocks (whose gn > 0), negative for brown stocks
(whose gn < 0), and lower for stocks with more volatile returns. A similar result obtains
when Σ has a simple one-factor structure, allowing systematic risk:
Σ = σ2ιι′ + η2IN , (15)
where ι is an N × 1 vector of ones and IN is an identity matrix, because in that case4
Σ?1g =
1
η2
??g ? gˉ
η2
Nσ2
+ 1
ι
?? , (16)
where gˉ = ι′g/N is the mean gn across firms. As N gets large, the ESG portfolio goes
long stocks that are greener than average (gn > gˉ) and short stocks that are browner than
average (gn < gˉ). The ESG portfolio’s positions are smaller when idiosyncratic risk η
2 is
higher, because tilting toward the ESG portfolio exposes investors to more idiosyncratic risk.
In general, the ESG tilt depends also on the covariances among stocks. If a stock is
positively correlated with a greener stock, the former stock may be shorted by agents who
want to hold the greener stock and hedge their risk exposure to it. In principle, even a green
stock could be shorted if it is sufficiently correlated with a stock that is even greener.
4With Σ given in equation (15), we have Σ?1 = 1η2
(
IN ? 1η2/σ2+N ιι′
)
.
10
The ESG tilt is smaller when risk aversion, a, is higher, because the tilt exposes agents to
additional risk (see equation (12)). Holding ESG preferences (δi) constant, those preferences
are reflected less strongly in agents’ portfolios if agents’ risk aversion is higher.
3. Quantitative Implications
To explore the model’s quantitative implications, we consider a special case with two types
of agents: ESG investors, for whom di = d > 0, and non-ESG investors, for whom di = 0.
ESG investors thus consume nonpecuniary benefits dg, whereas non-ESG investors consume
no benefits (see equation (3)). Let λ denote the fraction of total wealth belonging to ESG
investors, so that 1? λ is the corresponding fraction for non-ESG investors.
3.1 Expected Returns and Portfolio Tilts
In this setting, dˉ = λd, so from equation (9) the vector of expected excess returns becomes
μ = μMβ ? λd
a
g . (17)
As λ increases, expected returns on green stocks decrease, whereas expected returns on brown
stocks increase. In this comparative static sense, growing interest in ESG increasingly pushes
stock prices in the direction of their ESG characteristics.
The portfolio weights for each type of investor follow directly from equation (14), with
δi = (1? λ)d for an ESG investor and δi = ?λd for a non-ESG investor:
Xesg = x+ (1? λ) d
a2
Σ?1g (18)
Xnon = x? λ d
a2
Σ?1g . (19)
Both ESG tilts depend on λ in an interesting way. As λ→ 0, all investing is non-ESG, and
all capital is invested in the market portfolio x because Xnon → x. As λ → 1, all investing
is ESG, and again, all capital is invested in the market because Xesg → x. In other words,
whether λ → 0 or λ → 1, all portfolios converge to the market portfolio. When λ → 0,
everybody holds the market because there are no ESG investors. When λ → 1, everybody
holds the market because ESG preferences are fully embedded in market prices.
From equation (11), the difference in expected excess returns earned by the two types of
investors is
E(r?esg)? E(r?non) = ?λd
2
a3
g′Σ?1g . (20)
11
An ESG investor thus earns a lower expected return than a non-ESG investor. The
performance gap is larger when there is a greater presence of ESG investors (i.e., when λ is
larger). In this comparative static sense, growth in ESG investing deepens the performance
gap. The gap is also larger when the two types of investors are further apart in their ESG
tastes (i.e., when d is larger), when risk aversion a is smaller, and when g′Σ?1g is larger.
3.2. Parameter Specifications
We further simplify our setting by assuming that Σ has the simple one-factor structure given
in equation (15) and setting β = ι, x = (1/N)ι, ι′g = 0 (i.e., x′g = 0), and g′g = 1. With
these assumptions, as the number of assets (N) grows large, the mean and variance of market
returns, the certainty-equivalent return of ESG investors, and other aggregate quantities of
interest do not depend on N , as will be evident below.
This simple setting has five free parameters: λ, a, σ, η, and d. We present results over
the entire (0, 1) range of λ. We specify a and σ so that the return on the market portfolio
has a mean of μM = 0.08 and a standard deviation of σM = 0.20, corresponding roughly to
annual empirical estimates. To translate these values of μM and σM into implied values for
a and σ, first note that the above assumptions imply the variance of the market, x′Σx, is
σ2M =
1
N2
ι′
(
σ2ιι′ + η2IN
)
ι = σ2 +
η2
N
, (21)
so we set σ2 = σ2M , taking the limit as N grows large. Next, recall from equation (7) that
a = μM/σ
2
M . We set η
2 = (0.7/0.3)σ2, so that the common market factor explains 30% of
the variance of each individual stock’s return.
The remaining free parameter, d, reflects the strength of ESG tastes. We calibrate this
parameter by choosing ?, the maximum rate of return that an ESG investor is willing to
sacrifice, for certain, in order to invest in her desired portfolio rather than in the market
portfolio. The sacrifice is greatest when there are no other ESG investors, i.e., when λ ≈ 0,
because that is when the ESG investor’s portfolio most differs from the market portfolio.
Specifically, we define ? ≡ r?esg?r?M , where r?esg is the ESG investor’s certainty equivalent
excess return when investing in the optimal ESG portfolio, and r?M is the same investor’s
corresponding certainty equivalent if forced to hold the market portfolio instead. Both
certainty equivalents are computed for λ = 0. For a given ?, the corresponding value of d is
d =
√
2?a3η2 . (22)
12
We derive this equation, along with the expressions for r?esg and r
?
M , in the Appendix. In the
following analysis, we consider four values of ?: 1, 2, 3, and 4% per year.
3.3. Correlation Between the ESG Return and the Market Return
The correlation between the return on an ESG investor’s portfolio and the return on the
market portfolio is derived in the Appendix:
ρ (r?esg, r?M) =
σ√
σ2 + 2?
a
(1? λ)2
. (23)
Figure 1 plots the value of ρ (r?esg, r?M) as λ goes from 0 to 1. The correlation takes its lowest
value at λ = 0. For ? = 0.01, that value is nearly 0.9, whereas for ? = 0.04, it is just over
0.7. As ? increases, indicating that ESG investors feel increasingly strongly about ESG,
those investors’ portfolios become increasingly different from the market portfolio in terms
of ρ (r?esg, r?M), and this effect is strongest when λ = 0. However, as λ approaches 1, so does
ρ (r?esg, r?M). When ESG investors hold an increasingly large fraction of wealth, market prices
adjust to their preferences, and all portfolios converge to the market portfolio.
3.4. ESG versus Non-ESG Expected Portfolio Returns
The difference in expected excess returns on the portfolios of the two investor types is
E{r?esg} ? E{r?non} = ?2λ? , (24)
as shown in the Appendix. Figure 2 plots this difference as λ goes from 0 to 1. The difference
is zero at λ = 0, but it declines linearly as λ increases. At λ = 1, ESG tastes are fully reflected
in prices, and the difference is at its largest in magnitude. In that scenario, the difference is
-2% when ? = 0.01, but it is ?8% when ? = 0.04. ESG investors thus earn significantly
lower returns than non-ESG investors when the former account for a large fraction of wealth
(i.e., λ is large) and their ESG tastes are strong (i.e., ? is large).
The certainty equivalent returns of the two types, r?esg for ESG investors and r
?
non for non-
ESG investors, are both increasing in ?, but r?esg decreases with λ whereas r
?
non increases
with λ, as we show in the Appendix. As λ increases, stock prices are affected more by ESG
investors’ tastes, so these investors must pay more for the green stocks they desire. The
resulting drop in r?esg need not imply, however, that an ESG investor is made less happy by
an increased presence of ESG investors. With the latter, there is also greater social impact
13
of ESG investing, as we discuss in Section 6. The additional utility that the ESG investor
derives from the greater social impact, as in equation (57), can exceed the drop in utility
corresponding to the lower r?esg. Non-ESG investors, on the other hand, do prefer to be lonely
in their ESG tastes. A non-ESG investor is happiest when all other investors are ESG (λ = 1),
because that scenario maximizes deviations of prices from pecuniary fundamentals, which
the non-ESG investor exploits to her advantage. This investor’s preference for loneliness in
ESG tastes is even stronger if she derives utility from social impact because that impact is
maximized when λ = 1.
3.5. Alphas and the Investor Surplus
The alphas of the ESG and non-ESG investors’ portfolios are derived in the Appendix:
αesg = ?2λ(1? λ)? (25)
αnon = 2λ
2? . (26)
Panel A of Figure 3 plots αesg as λ goes from 0 to 1. ESG investors earn zero alpha at
both extremes of λ. Their portfolio differs most from the market portfolio when λ = 0, but
all stocks have zero alphas in that scenario, because there is no impact of ESG investors on
prices. At the other extreme, when λ = 1, many stocks have non-zero alphas, due to the
price impacts of ESG investors, but ESG investors hold the market, so again they earn zero
alpha. Otherwise, ESG investors earn negative alpha, which is greatest in magnitude when
λ = 0.5. At that peak, αesg = ?0.5% when ? = 0.01, but αesg = ?2% when ? = 0.04.
Interestingly, these worst-case alphas are substantially smaller than the corresponding
?’s. For example, when ESG investors are willing to give up 2% certain return to hold
their portfolio rather than the market (i.e., ? = 0.02), their worst-case alpha is only ?1%.
The reason is that equilibrium stock prices adjust to ESG demands. These demands push
the market portfolio toward the portfolio desired by ESG investors, thereby bringing those
investors’ negative alphas closer to zero. Through this adjustment of market prices, ESG
investors earn an “investor surplus” in that they do not have to give up as much return as
they are willing to in order to hold their desired portfolio.
The magnitude of this investor surplus is easy to read off Panel B of Figure 3, which
plots αesg as a function of ?. For any given value of λ, investor surplus is the difference
between the corresponding solid line and the dashed line, which has a slope of ?1. The
surplus increases with ? because the stronger the ESG investors feel about greenness, the
14
more they move market prices. The relation between the surplus and λ is richer. Formally,
investor surplus I ≡ αesg + ? follows quickly from equation (25):
I = ?[1? 2λ(1? λ)] . (27)
Because 0 ≤ λ ≤ 1, the value in brackets is always between 0.5 and 1, so I is always between
?/2 and ?. It reaches its smallest value of ?/2 when λ = 0.5 and its largest value of ?
when λ = 0 or 1. For example, when ? = 0.02, I ranges from 1% to 2% depending on λ.
Figure 4 plots αnon as a function of λ and ?. Like ESG investors, non-ESG investors
earn zero alpha when λ = 0 or ? = 0. However, αnon increases in both λ or ?. This alpha
can be as large as 8% when λ = 1 and ? = 0.04. A non-ESG investor earns the highest
alpha when all other investors are ESG (i.e., λ = 1) and when those investors’ ESG tastes
are strong (i.e., ? is large) because the price impact of ESG is then particularly large. By
going long brown stocks, whose alphas are positive and large, and short green stocks, whose
alphas are negative and large, the non-ESG investor earns a large positive alpha.
Given the simplifying assumption that all assets have unit betas, the differences between
the alphas plotted in Figures 3 and 4 are equal to the differences in expected returns plotted
in Figure 2. Specifically, from equations (24) through (26), αesg?αnon = E{r?esg}?E{r?non}.
3.6. Size of the ESG Investment Industry
We define the size of the ESG investment industry by the aggregate amount of ESG-driven
investment that deviates from the market portfolio, divided by the stock market’s total value.
In general, this aggregate ESG tilt is given by
T =
∫
i:di>0
wiTi di , (28)
where
Ti =
1
2
ι′|Xi ? x| . (29)
The aggregate ESG tilt, T , is a wealth-weighted average of agent-specific tilts, Ti, across
all agents who care at least to some extent about ESG (i.e., di > 0). Each Ti is one half
of the sum of the absolute values of the N elements of agent i’s ESG tilt, |Xi ? x|. We
compute absolute values of portfolio tilts because ESG-motivated investors both over- and
under-weight stocks relative to the market. We divide by two because we do not want
to double-count: for each dollar that an agent moves into a green stock, she must move
a dollar out of another stock. The value of Ti is formally equivalent to agent i’s active
15
share (Cremers and Petajisto, 2009), with the market portfolio as the benchmark, but its
interpretation is different: instead of measuring the activeness of the agent’s portfolio, Ti
measures the portfolio’s ESG-induced tilt away from the market.
With two types of agents, the expression for T simplifies to
T =
1
2
λι′|Xesg ? x| = λ(1? λ)
√
?
2a
ι′|g| , (30)
as we show in the Appendix. The aggregate tilt depends on the absolute values of the
elements of g. To evaluate ι′|g| in this quantitative exercise, we further assume that the
elements of g are normally distributed across stocks, in addition to the previous assumptions
that these elements have a mean of zero and a variance of 1/N (recall x′g = (1/N)ι′g = 0
and g′g = 1). Then ι′|g| = NE(|gn|) =
√
2N/pi. Therefore,
T = λ(1? λ)
√
?N
api
. (31)
We set the number of assets here to N = 100. That number is considerably smaller than
the actual number of stocks in the U.S. market, but recall that we assume equal market
weights across stocks. We reduce N as a concession to the fact that the actual distribution
of firm size in the U.S. market is quite disperse. Another reason to choose a small N is
that we do not impose any investment constraints. As investors go long and short, the
sum of the absolute values of their short positions increases with N , without bounds. In
reality, however, investors often face short-sale or margin constraints that would prevent this
from happening. Choosing a smaller N helps offset the effect of a growing number of short
positions on T . Given the arbitrariness inherent in the choice of N , we are more interested
in the dependence of T on λ and ? than in the magnitude of T per se. The overall level of
T depends on N , but the patterns with respect to λ and ? do not.
Figure 5 plots T for different values of λ and ?. In Panel A, λ goes from 0 to 1. At
both λ = 0 and λ = 1, we have T = 0 because all investors hold the market portfolio. The
maximum value of T in equation (31) always occurs at λ = 0.5, the maximum of λ(1?λ). In
Panel B, ? goes from 0 to 0.04. Larger values of ? produce larger values of T . This relation
between ? and T is concave (see also equation (31)). For example, the ESG industry peaks
at 46% of the stock market’s value when ? = 0.02, but doubling the strength of ESG tastes
(raising ? to 0.04) increases that maximum industry size by less than half, to 65% of the
market’s value. We see that the price impact of ESG tastes weakens their impact on the size
of the ESG investment industry.
16
4. The ESG Factor
In this section, we extend our model from Section 2 to show how firms’ ESG characteristics
can emerge as sensitivities to a risk factor—the ESG factor. The strength of ESG concerns
can change over time, both for investors in firms’ shares and for the customers who buy the
firms’ goods and services. If ESG concerns strengthen, customers may shift their demands
for goods and services to greener providers (the “customer” channel), and investors may
derive more utility from holding the stocks of greener firms (the “investor” channel). Both
channels contribute to the ESG factor’s risk in our framework.
To model the investor channel, we assume that the average ESG taste dˉ shifts unpre-
dictably from time 0 to time 1. To model the customer channel, we need to model firm
profits. Let u?n denote the financial payoff (profit in our one-period setting) that firm n
produces at time 1, for each dollar invested in the firm’s stock at time 0. We assume a
two-factor structure for the N × 1 vector of these payoffs:
u?? E0{u?} = z?hh+ z?gg + ζ? , (32)
where E0{ } denotes expectation as of time 0, the random variables z?h and z?g have zero
means, and ζ? is a mean-zero vector that is uncorrelated with z?h, z?g, and dˉ and has a
diagonal covariance matrix, Λ. The shock z?h can be viewed as a macro output factor, with
the elements of h being firms’ sensitivities to that pervasive shock. The shock z?g represents
the effect on firms’ payoffs of unanticipated shifts in customers’ demands. These shifts can
result not only from changes in consumers’ tastes but also from revisions of government
policy. For example, pro-environmental regulations may subsidize green products, leading
to more customer demand, or handicap brown products, leading to less demand. A positive
z?g shock increases the payoffs of green firms but hurts those of brown firms.
To assess how shifts in ESG tastes affect asset prices, we need to price stocks not only
at time 0, as we have done so far, but also at time 1, after the preference shift in dˉ occurs.
To make this possible in our simple framework, we split time 1 into two times, 1? and 1+,
that are close to each other. We calculate prices p1 as of time 1
?, by which time ESG tastes
have shifted and all risk associated with u? has been realized. Stockholders receive u? at time
1+. During the instant between times 1? and 1+, these payoffs are riskless. For economy of
notation, we assume the risk-free rate rf = 0.
There are two generations of agents, Gen-0 and Gen-1. Gen-0 agents live from time
0 to time 1?; Gen-1 agents live from time 1? to 1+. Gen-1 agents have identical tastes
of di = dˉ1, a condition that gives them finite utility, given the absence of both risk and
17
position constraints during their lifespan. Neither a nor g change across generations. At
time 1?, Gen-0 agents sell stocks to Gen-1 agents at prices p1, which depend on Gen-1
ESG tastes dˉ1 and the financial payoff u?. This simple setting maintains single-period payoff
uncertainty while also allowing risk stemming from shifts in ESG tastes to enter via both
channels described earlier.
4.1. Two Channels
Given that the payoff u?n is known at the time when the price p1,n is computed, p1,n is equal
to u?n discounted at the expected return implied by equation (9) with βn set to zero:
p1,n =
u?n
1? gn
a
dˉ1
≈ u?n + gn
a
dˉ1 . (33)
The approximation above holds well for typical discount rates, which are not too far from
zero.5 Representing it as an equality for all assets gives
p1 = u?+
1
a
dˉ1g , (34)
which is the vector of payoffs to Gen-0 agents. Its expected value at time 0 equals
E0{p1} = E0{u?}+ 1
a
E0{dˉ1}g . (35)
Note that p1?E0{p1} equals the vector of unexpected returns for Gen-0 agents, because u?n
is the firm’s payoff per dollar invested in its stock at time 0. From equations (32) through
(35), these unexpected returns, ? = r? ? E0{r?}, are given by
? = z?hh+ f?gg + ζ? (36)
= Bf? + ζ? , (37)
where B = [h g], f? = [z?h f?g]
′, with E0{f?} = 0, and f?g denotes the “ESG factor” given by
f?g = z?g +
1
a
[
dˉ1 ? E0{dˉ1}
]
. (38)
The two components of f?g correspond to the two ESG risk channels discussed earlier: z?g
represents the customer channel while the other term represents the investor channel. While
the customer channel follows closely from the structure assumed in equation (32), the investor
channel emerges from the equilibrium dependence of stock prices on dˉ.
The elements of f?gg in equation (36) drive a wedge between expected and realized returns
for ESG-motivated agents in Gen-0. We thus have the following proposition.
5For arbitrary rates ρ1 ≈ 0 and ρ2 ≈ 0, we have 1+ρ11?ρ2 =
(1+ρ1)(1+ρ2)
1?ρ22
≈ (1 + ρ1)(1 + ρ2) ≈ 1 + ρ1 + ρ2,
neglecting ρ22 and ρ1ρ2. Setting u?n = 1 + ρ1 and
gn
a dˉ1 = ρ2 gives the approximation in equation (33).
18
Proposition 4. Green (brown) stocks perform better (worse) than expected if ESG concerns
strengthen unexpectedly via either the customer channel or the investor channel.
As noted earlier, green stocks generally have lower expected returns than brown stocks.
If f?g is positive, however, such an outcome boosts the realized performance of green stocks
while hurting that of brown stocks. If one computes average returns over a sample period
when ESG concerns consistently strengthened more than investors expected, so that the
average of f?g over that period is strongly positive, then green stocks could outperform brown
stocks, contrary to what is expected.
4.2. The ESG Factor’s Effects on Market Betas
Besides affecting stock returns ex post, the ESG factor also affects returns ex ante by influ-
encing market betas. As we show in the Appendix, the vector of market betas is
β = hβh + gβg +
1
σ2M
Λx , (39)
where βh ≡ Cov{?M , z?h}/σ2M , βg ≡ Cov{?M , f?g}/σ2M , and ?M ≡ x′? is the unexpected market
return. In words, a stock’s market beta depends on the stock’s loading on the macro factor
(hn) times that factor’s loading on the market (βh), plus the stock’s loading on the ESG factor
(gn) times that factor’s loading on the market (βg), plus a term reflecting idiosyncratic risk.
Substituting from equation (36) into ?M ≡ x′?, we immediately obtain
βg = (x
′h)Cov{z?h, f?g}/σ2M + (x′g)Var(f?g)/σ2M . (40)
The overall stock market surely loads positively on the macro factor, z?h, meaning x
′h > 0.
Also, recall from equation (8) that x′g = 0, so the second term in equation (40) drops out.
Proposition 5. If Cov{z?h, f?g} > 0 then ESG factor risk raises the market betas of green
stocks but lowers the betas of brown stocks. These effects are reversed if Cov{z?h, f?g} < 0.
If Cov{z?h, f?g} > 0 then βg > 0 in equation (40), which then increases β’s of green stocks,
and reduces β’s of brown stocks, through equation (39). If Cov{z?h, f?g} < 0 then βg < 0,
and the effect on the β’s is the opposite. The sign of Cov{z?h, f?g}, the covariance between
the macro factor and the ESG factor, is unclear. On the one hand, a positive covariance is
supported by the evidence of Bansal, Wu, and Yaron (2018) that green stocks outperform
brown stocks in good times but underperform in bad times. Those authors argue that green
stocks are similar to luxury goods in that they are in higher demand when the economy
19
does well and thus financial concerns matter less. On the other hand, the covariance could
be negative if z?h and f?g have opposite exposures to climate risk. Adverse climate shocks
are likely to be accompanied by positive realizations of f?g, as we discuss in Section 5. If
such shocks are large enough to reduce aggregate output then they are also accompanied by
negative realizations of z?h. This common dependence on climate shocks then makes z?h and
f?g negatively correlated. Albuquerque, Koskinen, and Zhang (2019) find empirically that
greener firms, as measured by high corporate social responsibility scores, have lower market
betas. Their evidence is consistent with a negative correlation between z?h and f?g.
Finally, if we relax the assumption from equation (8) that x′g = 0, the role of ESG factor
risk depends on the overall greenness of the market portfolio. If the market is net green,
so that x′g > 0, then the second term in equation (40) is positive, further increasing the
covariance between the market return and the ESG factor. As the economy becomes greener,
x′g rises, pushing up βg in equation (40). The greenifying of the economy thus makes green
stocks increasingly exposed to the market, and brown stocks decreasingly so.
4.3. Two-Factor Pricing
Under the above setting, the ESG factor, with its mean shifted to a non-zero value, also
produces near-zero alphas in a two-factor model in which the market return is the other
factor. This result obtains when the market portfolio is neither green nor brown (x′g = 0)
and is also well diversified, in that xn ≈ 0 for all n (a large-N scenario). Those conditions
imply that excess returns are closely approximated by the regression relation
r? = θr?M + g(f?g + μg) + ν? , (41)
with E{ν?|r?M , f?g} = 0, θ = (1/x′h)h, and
μg = μMβg ? dˉ/a , (42)
as we show in the Appendix. We thus obtain the following proposition.
Proposition 6. Each stock has zero alpha with respect to a two-factor model with the market
factor and the ESG factor, with stock n’s loading on the ESG factor equal to gn.
The premium associated with the ESG factor, μg in equation (42), has two components.
The first component, μMβg, is a part of the market risk premium. The second component,
?dˉ/a, is not a risk premium; it is characteristic-based. Recall from equation (10) that
?gdˉ/a is a vector of stocks’ CAPM alphas. The variation in f?g represents shifts in tastes;
20
it represents common risk, but this risk is not priced, except for the part that comoves with
the market. Only market risk, captured by β, is priced in Section 4. The ESG factor affects
expected returns only through its effect on β, which we discuss earlier in Section 4.2.
The reason why ESG factor risk is not priced in this section, above and beyond its co-
movement with the market, is that its variation is unrelated to agents’ utility function in
equation (2). However, ESG factor risk is priced if the utility function is modified so that
agents care about the realizations of f?g. In the following section, we modify the utility func-
tion by adding a concern about the global climate. We show that climate risk is priced, and
we draw parallels between climate shocks and ESG factor realizations. If agents care about
climate shocks and those shocks are correlated with f?g—a scenario we consider plausible—
then we should observe the ESG factor’s risk being priced in any empirical setting that does
not fully control for climate shocks.
5. Climate Risk
Sustainable investing is motivated in part by concerns about global climate change (part
of “E” in ESG). Many experts expect climate change to impair quality of life, essentially
lowering utility of the typical individual beyond what is captured by climate’s effect on
wealth. Unanticipated climate changes present investors with an additional source of risk.
This section extends our model from Section 2 to include such risk.
Let C? denote climate at time 1, which is unknown at time 0. We modify the utility
function for individual i in equation (2) to include C? as follows:
V (W?1i, Xi, C?) = ?e?AiW?1i?b′iXi?ciC? , (43)
where ci ≥ 0, so that agents dislike low realizations of C?. Define cˉ ≡ ∫iwicidi, the wealth-
weighted mean of climate sensitivity across agents. We assume C? is normally distributed,
and without loss of generality we set E{C?} = 0 and Var{C?} = 1. Besides replacing equation
(2) by equation (43), we maintain all other assumptions from Section 2.
5.1. Expected Returns
Climate risk has a transparent effect on equilibrium stock returns.
Proposition 7. Expected excess returns in equilibrium are given by
μ = μMβ ? dˉ
a
g + cˉ
(
1? ρ2MC
)
ψ , (44)
21
where ψ is the N×1 vector of “climate betas,” that is slope coefficients on C? in a multivariate
regression of ? on both ?M and C?, and ρMC is the correlation between ?M and C?.
Expected returns depend on climate betas, ψ, which represent firms’ exposures to non-
market climate risk. To understand the regression defining ψ, recall that ? is an N×1 vector
of unexpected stock returns from equation (1) and ?M is the unexpected market return. A
firm’s climate beta is its loading on C? after controlling for the market return.
Compared to equation (9), expected excess returns contain an additional component
given by the last term on the right-hand side of equation (44). Stock n’s climate beta, ψn,
enters expected return positively. Thus, a stock with a negative ψn, which provides investors
with a climate-risk hedge, has a lower expected return than it would in the absence of climate
risk. Vice versa, a stock with a positive ψn, which performs particularly poorly when the
climate worsens unexpectedly, has a higher expected return.
Climate betas, ψn, are likely to be related to ESG characteristics, gn, as we argue next.
5.2. Green Stocks as Climate Hedges
Green stocks seem more likely than brown stocks to hedge climate risk. This hedging asym-
metry can be motivated through both channels described in Section 4. First, consider the
customer channel. Unexpected worsening of the climate can heighten consumers’ climate
concerns, prompting greater demands for goods and services of greener providers. These
demands can arise not only from consumers’ choices but also from government regulation.
Negative climate shocks can lead governments to adopt regulations that favor green providers
or penalize brown ones. Half of the institutional investors surveyed by Krueger, Sautner,
and Starks (2019) state that climate risks related to regulation have already started to ma-
terialize. Second, consider the investor channel. Unexpected worsening of the climate can
strengthen investors’ preference for green holdings (i.e., increase dˉ). For example, Choi, Gao,
and Jiang (2019) show that retail investors sell carbon-intensive firms in extremely warm
months, consistent with dˉ rising in such months. Climate shocks are thus likely to correlate
negatively with both components of the ESG factor, f?g, in equation (38). Green stocks,
which have positive exposures to f?g, are likely to have negative exposures to C?. In other
words, the correlation between gn and ψn across firms is likely to be negative.
Evidence also suggests that greener stocks are better climate hedges. Empirical studies
find that returns on green (brown) stocks have positive (negative) correlations with adverse
climate shocks. For example, Choi, Gao, and Jiang (2019) show that green firms, as measured
22
by low carbon emissions, outperform brown firms during months with abnormally warm
weather, which the authors argue alert investors to climate change. Similarly, Engle et al.
(2019) report that green firms, as measured by high E-Scores from Sustainalytics, outperform
brown firms in periods with negative climate news. Both studies thus show that a high-minus-
low gn stock portfolio is a good hedge against climate risk, indicating that gn is negatively
correlated with ψn across firms.
In the special case where this negative correlation is perfect, so that
ψn = ?ξgn , (45)
where ξ > 0 is a constant, equation (44) simplifies to
μ = μMβ ?
[
dˉ
a
+ cˉ
(
1? ρ2MC
)
ξ
]
g . (46)
Stock n’s CAPM alpha is then given by
αn = ?
[
dˉ
a
+ cˉ
(
1? ρ2MC
)
ξ
]
gn . (47)
Both terms inside the brackets are positive, so the negative relation between αn and gn is
stronger than in Corollary 2. Greener stocks now have lower CAPM alphas not only because
of investors’ tastes for green holdings, but also because of greener stocks’ ability to better
hedge climate risk. Climate risk thus represents another reason to expect green stocks to
underperform brown ones over the long run. For the same reason, green stocks have a lower
cost of capital than brown stocks relative to the CAPM.
5.3. Climate-Hedging Portfolio
Finally, climate risk has interesting implications for investors’ asset holdings.
Proposition 8. Agent i’s equilibrium portfolio weights are given by
Xi = x+
δi
a2
(
Σ?1g
)
? γi
a
(
Σ?1σC
)
, (48)
where γi ≡ ci ? cˉ and σC is an N × 1 vector of covariances between ?n and C?.
Proposition 8 implies four-fund separation. The first three funds are the same as in
Proposition 3; the fourth one is a climate-hedging portfolio whose weights are proportional
to Σ?1σC . Agents with γi > 0, whose climate sensitivity is above average, short the hedging
portfolio, whereas agents with γi < 0 go long.
23
The climate-hedging portfolio, Σ?1σC , is a natural mimicking portfolio for C?. To see this,
note that the N elements of Σ?1σC are the slope coefficients from the multiple regression
of C? on ?. Therefore, the return on the hedging portfolio has the highest correlation with C?
among all portfolios of the N stocks. Investors in our model hold this maximum-correlation
portfolio, to various degrees determined by their γi, to hedge climate risk.
The climate-hedging portfolio is likely to favor green stocks over brown. The reason is
that green stocks are generally better climate hedges than brown stocks, as discussed earlier
in Section 5.2. Yet the climate-hedging portfolio is not necessarily simply long green stocks
and short brown ones. Even if σC were perfectly correlated with g across firms, σC in
equation (48) is pre-multiplied by Σ?1, which in general makes the climate-hedging portfolio
weights imperfectly aligned with stocks’ ESG characteristics.
5.4. Climate Risk versus ESG Risk
In this section, we relate the climate variable, C?, from Section 5, to the ESG factor, f?g, from
Section 4. At first sight, the two variables are not comparable because Sections 4 and 5 are
different extensions of the baseline model in Section 2. To make C? and f?g comparable, we
embed them in the same modeling framework. Specifically, we rederive the results in Section
4 after replacing the utility function in equation (2) by that in equation (43) and maintaining
all other assumptions of Section 4. We continue to assume that firm payoffs are given by
the two-factor structure in equation (32), so that the effect of C? on firm payoffs is spanned
by z?h and z?g. In this modified setting, the derivation in Section 4.1 remains unchanged,
except that the last term in equation (44), cˉ (1? ρ2MC)ψ, appears on the right-hand sides
of equations (33) through (35). For simplicity, we assume that this term, which represents
the premium for climate risk, is constant over time. Therefore, this term subtracts out in
equation (36) and the ESG factor, f?g, remains the same as in equation (38). Therefore, C?
from equation (43) and f?g from equation (38) are directly comparable, after all.
The two factors, C? and f?g, are likely to be negatively correlated. Recall that f?g from
equation (38) has two components reflecting the customer and investor channels. Both
components are likely to be negatively correlated with C?, as discussed in Section 5.2. When
the climate worsens unexpectedly, consumers may shift their demands toward green products
and away from brown products, either of their own volition or prompted by new government
regulations subsidizing green products or taxing, possibly even prohibiting, brown products.
In addition, worse climate can elevate investors’ tastes for green holdings, for example, as a
result of stronger public pressure on institutional investors to divest from brown assets.
24
If climate shocks are the only reason behind shifts in customers’ and investors’ tastes,
C? and f?g are perfectly negatively correlated. In this special case, the two factors coincide.
Without loss of generality, we scale them so that f?g = ?C?. Stocks’ loadings on the single
factor are then given by the ESG characteristics: ψ = ?g. Moreover, stocks are priced by
only two factors: the market portfolio and the ESG factor. As we show in the Appendix,
the two-factor asset pricing formula in equation (41) holds also in this setting, except that
the ESG factor’s premium in that equation is redefined to include an extra term:
μg = μMβg ? dˉ/a? cˉ
(
1? ρ2MC
)
. (49)
The last term in μg, which is absent from equation (42), reflects compensation for climate
risk. This compensation is negative because greener firms are better hedges against this risk.
The ESG factor’s premium thus has two risk-based components, μMβg for market risk and
?cˉ (1? ρ2MC) for climate risk, and one taste-based component, ?dˉ/a.
This special case is appealing because it can serve as motivation for empiricists to use
a single factor to capture the risk associated with ESG investing. Moreover, it shows that
firms’ loadings on the single factor are simply proportional to firms’ ESG characteristics, gn.
However, in general, a multi-factor approach may be required because the tastes of customers
and investors may shift also for reasons unrelated to climate news, such as environmental
activism and political pressure. Whether a single factor is sufficient to capture all ESG-
related risk is an interesting question for future empirical research.
6. Social Impact
Does sustainable investing produce real social impact? We explore this question by adding
firms’ choices of investment and ESG characteristics to the baseline model from Section 2.
We define the social impact of firm n as
Sn ≡ gnKn , (50)
where Kn is the firm’s operating capital. Social impact captures the firm’s total amount
of externalities, which depends on both the nature of the firm’s operations (gn) and their
scale (Kn). We consider two scenarios. In Section 6.1, we let the firm’s manager choose Kn
while taking gn as given. In Section 6.2, we allow the manager to choose both Kn and gn.
Throughout, the manager maximizes the firm’s market value at time 0.
The extra assumptions we make here change none of the previous sections’ predictions.
Since investors are atomistic, they still take asset prices and firms’ ESG characteristics as
25
given, even though firms now choose those characteristics. Firms’ choices of Kn and gn affect
their market values, which are consistent with the expected returns derived in Section 2.
6.1. Green Firms Invest More, Brown Firms Less
The firm is initially endowed with operating capital K0,n > 0. The firm’s manager chooses
how much additional capital ?Kn to buy, while taking the firm’s ESG characteristic, gn, as
given. The firm’s capital investment produces time-0 cash flow of ??Kn? κn2 (?Kn)2, where
κn > 0 controls capital-adjustment costs. The firm uses capital to produce expected gross
cash flow at time 1 equal to ΠnKn, where Πn is a positive quantity denoting one plus the
firm’s gross profitability.
The optimal amount of additional capital is derived in the Appendix:
?Kn(d) =
1
κn
?? Πn
1 + rf + μMβn ? dagn
? 1
?? . (51)
This value is increasing in gn, indicating that greener firms invest more, ceteris paribus.
For any firm n, agents’ ESG tastes induce social impact equal to the difference between
the firm’s actual social impact and its hypothetical impact if agents did not care about ESG:
Sn(d)? Sn(0) = gn
(
?Kn(d)??Kn (0)
)
. (52)
We prove the following proposition in the Appendix.
Proposition 9. Firm n’s ESG-induced social impact is positive:
Sn(d)? Sn(0) = d g
2
n Πn
aκn
(
1 + rf + μMβn ? dagn
)
(1 + rf + μMβn)
> 0 , (53)
as long as dˉ > 0 and gn 6= 0. Moreover, this impact is increasing in d, decreasing in a,
increasing in Πn, decreasing in κn, and decreasing in βn.
The intuition behind this result builds on equation (51), which shows that ESG tastes
lead green firms to invest more and brown firms to invest less. ESG tastes reduce the cost of
capital for green firms, which increases the NPV of those firms’ projects and hence also those
firms’ investment. And vice versa, ESG tastes increase brown firms’ cost of capital, reducing
their investment. As a result, ESG tastes tilt investment from brown to green firms, which
increases the social impact for both types of firms.
26
The comparative statics are also intuitive. Social impact is larger when ESG tastes are
stronger (i.e., when dˉ is larger) because stronger tastes move asset prices more. The impact
is also larger when risk aversion is weaker (i.e., a is smaller) because less risk-averse agents
tilt their portfolios more to accommodate their tastes, again resulting in larger price effects
(see Propositions 1 and 3). The impact is larger when capital is less costly to adjust (i.e.,
when κn is smaller) because more investment reallocation takes place. The impact is also
larger when firms are more productive (i.e., when Πn is larger) because a given change in
the cost of capital has a larger effect on investment. Finally, the impact is larger for firms
with smaller market betas because such firms have a lower cost of capital to begin with, so
the ESG-induced change in their cost of capital is relatively larger.
In our model, investors’ ESG tastes tilt real investment from brown to green firms because
those tastes generate alphas, which affect the cost of capital, which in turn affects investment.
There is considerable empirical support for this mechanism. Baker and Wurgler (2012) survey
papers that find a negative relation between corporate investment and alpha. Most of these
papers interpret alpha as mispricing, whereas our paper’s ESG-induced alphas do not reflect
mispricing. We expect ESG-induced alphas to have an especially strong effect on investment.
Whereas mispricing is transient, firms’ ESG traits are highly persistent, which makes ESG-
induced alphas highly persistent. Van Binsbergen and Opp (2019) show that when alphas
are more persistent, they have stronger effects on investment.
6.2. Firms Become Greener
We now extend the framework from Section 6.1 by allowing firm n’s manager to choose
not only Kn but also gn. The firm is initially endowed with an ESG characteristic g0,n.
The manager chooses both ?Kn and ?gn, the change in the firm’s ESG characteristic. For
example, a coal power producer can increase its gn by installing scrubbers. Adjusting gn is
costly: it reduces the firm’s time-1 cash flow by a fraction ωn
2
(?gn)
2, where ωn > 0 controls
ESG-adjustment costs.
We prove the following proposition in the Appendix.
Proposition 10. Firm n’s value-maximizing choices of ESG adjustment and investment are
?gn
(
d
)
≈ d
aωn
(54)
?Kn
(
d
)
=
1
κn
?? Πn
(
1? ωn
2
(?gn(d))
2
)
1 + rf + μMβn ? dagn(d)
? 1
?? , (55)
where gn(d) = g0,n + ?gn(d), and the approximation uses log(1 + x) ≈ x for small x.
27
Both choices are intuitive. When d > 0, investors value ESG characteristics, so all
firms choose to become greener (i.e, ?gn > 0) to increase their market value. This effect
is especially strong when risk aversion a is low because ESG characteristics then have large
effects on market values. Firms also adjust gn by more when doing so is less costly.
As in Section 6.1, ESG tastes lead green firms to invest more and brown firms to invest
less. The denominator in equation (55) shows that ESG tastes reduce the cost of capital for
green firms, which increases their projects’ NPV and hence investment. And vice versa, ESG
tastes increase brown firms’ cost of capital, reducing their projects’ NPV and investment. In
addition, ESG tastes affect expected cash flows in the numerator of equation (55). Stronger
ESG tastes induce all firms, green and brown, to adjust their gn by more, which reduces
their expected cash flows, and hence also their investment.
Agents’ ESG tastes now increase social impact not only by tilting investment from brown
to green firms, as before, but also by making firms greener:
Sn(d)? Sn (0) = gn,0
(
?Kn(d)??Kn (0)
)
+Kn(d) ?gn(d) . (56)
The first term reflects the investment effect analogous to equation (52). As discussed pre-
viously, when firms cannot change their gn’s, ?Kn(d) ? ?Kn(0) is positive for green firms
and negative for brown firms, making this term positive for both types of firms. When firms
can change their gn’s, the first term in equation (56) is still generally positive. The second
term reflects firms’ capital becoming greener. This term is also positive since ?gn(d) > 0,
as implied by equation (54).
Figure 6 plots the ESG-induced social impact across firms with different initial ESG
characteristics. We see that all firms have positive social impact. The two colored regions
indicate the two sources of social impact from equation (56). The second source, from firms
becoming greener, is roughly equal across firms (top green region). The first source, from
tilting investment toward green firms, is zero for an ESG-neutral firm, but it is large for very
green or very brown firms, which experience the largest shifts in investment (bottom blue
region). Due to this non-monotonicity, the overall social impact induced by ESG-motivated
investors is largest for firms with extreme ESG characteristics, but it is strictly positive even
for ESG-neutral firms.
The aggregate social impact induced by ESG investors, denoted S(d)? S(0), is the sum
of Sn(d)? Sn(0) across firms n. This sum can be computed off the curve in Figure 6. Since
this curve is convex in g0,n, S(d) ? S(0) is greater when there is more dispersion in ESG
characteristics across firms. A larger dispersion in g0,n deepens the cost-of-capital differentials
between green and brown firms, leading to larger investment differentials. With green firms
28
investing more and brown firms investing less, aggregate social impact increases.
Figure 7 illustrates how aggregate social impact varies with the strength of ESG prefer-
ences. We assume firms differ only in their initial ESG characteristics gn,0, which we assume
are uniformly distributed with mean zero. The figure shows that S(d) ? S(0) increases
as ESG preferences strengthen, which is intuitive. We also see that both sources of social
impact from equation (56) grow larger as ESG preferences strengthen. These results hold
whether ESG preferences strengthen because more investors are ESG (Panel A), or because
ESG investors have stronger ESG tastes (Panel B).
We have made the standard assumption that managers maximize the firm’s market value.
This assumption makes sense if, for example, managers wish to maximize the value of their
stock-based compensation. Alternatively, a manager could maximize shareholder welfare,
which depends not just on market value but also on the firm’s ESG characteristics (e.g., Hart
and Zingales, 2017). Such behavior could result from shareholders engaging actively with the
firm, so that managers run the firm as shareholders desire (e.g., Dyck et al., 2019), or from
shareholders appointing managers whose preferences match their own. Our model arguably
provides a lower bound on social impact. Extending the model so that managers additionally
care about their firms’ ESG characteristics should produce ?gn values (and hence social
impact) even larger than we currently predict. Put differently, we show that ESG-motivated
investors generate social impact even without direct engagement by shareholders, and even
if managers do not care directly about firms’ ESG characteristics. Even a “selfish” manager
who cares only about market value behaves in a way that increases social impact.
6.3. Preferences for Aggregate Social Impact
As noted in Section 2, agents care not only about their holdings but also about firms’
aggregate social impact, S =
∑N
n=1 Sn. We assume each agent i’s utility is increasing in S:
U(W?1i, Xi, S) = V (W?1i, Xi) + hi(S) , (57)
where h′i(S) > 0 and V is the original utility function from equation (2). (The additive
specification is not needed; our results are identical if S enters utility multiplicatively.)
Proposition 11. If agents derive utility also from aggregate social impact (equation (57)),
all of our results in Propositions 1 through 10 and Corollaries 1 through 4 continue to hold.
The inclusion of S in the utility function does not affect any of our prior results. The
reason is that agents are infinitesimally small. Small agents take stock prices, and hence S,
29
as given when choosing their portfolios. Therefore, agents’ preference for S does not affect
their portfolio choice. When an agent tilts toward green stocks, she generates a positive
externality on other agents via the hi(S) term in their utility. Being small, though, she
does not internalize this effect. As the preference for S does not affect portfolio choice, it
does not affect equilibrium asset prices, real investment, or S. In the model of Oehmke
and Opp (2020), agents’ preference for social impact does lead to impact because agents are
assumed to coordinate. In our model, agents cannot coordinate. Social impact is caused by
the inclusion of Xi, not S, in the utility function in equation (57).
7. Conclusion
We analyze both financial and real effects of sustainable investing in a highly tractable general
equilibrium model. The model produces a number of empirical implications regarding asset
prices, portfolio holdings, the size of the ESG investment industry, climate risk, and the
social impact of sustainable investing. We summarize those implications below.
First, ESG preferences move asset prices. Stocks of greener firms have lower ex ante
CAPM alphas, especially when risk aversion is low and the average ESG preference is strong.
Green stocks have negative alphas, whereas brown stocks have positive alphas. Green stocks’
negative alphas stem from two sources: investors’ tastes for green holdings and such stocks’
ability to hedge climate risk. Green and brown stocks have opposite exposures to an ESG
risk factor, which captures unexpected changes in ESG concerns of customers and investors.
If either kind of ESG concerns strengthen unexpectedly over a given period of time, green
stocks can outperform brown stocks over that period, despite having lower alphas. Under
plausible assumptions, stocks are priced by a two-factor asset pricing model, where the
factors are the market portfolio and the ESG factor. In general, though, multiple factors
may be required to fully capture asset exposures to ESG and climate risks.
Second, portfolio holdings exhibit three-fund separation. Investors with stronger-than-
average ESG tastes hold portfolios that have a green tilt away from the market portfolio,
whereas investors with weaker-than-average ESG tastes have a brown tilt. These tilts are
larger when risk aversion is lower. Investors with stronger ESG tastes earn lower expected
returns, especially when risk aversion is low and the average ESG taste is high. In the
model extension that adds climate risk, we obtain four-fund separation, with the fourth
fund representing a climate-hedging portfolio with a green tilt.
Third, the size of the ESG investment industry—the aggregate dollar amount of ESG-
30
driven investment that deviates from the market portfolio, scaled by total market value— is
increasing in the heterogeneity of investors’ ESG preferences. If there is no dispersion, there
is no ESG industry because everyone holds the market.
Finally, sustainable investing generates positive social impact, in two ways. First, it leads
firms to become greener. Second, it induces more real investment by green firms and less
investment by brown firms.
While the model’s predictions for alphas have been examined empirically by prior studies,
most of its other predictions remain untested, presenting opportunities for future empirical
work. One challenge is that our model aims to describe the world of the present and the
future, but not necessarily the world of the past. Although the “sin” aspects of investing
have been recognized for decades, the emphasis on ESG criteria is a recent phenomenon.
How the model fits in various time periods is a question for empirical work.
31
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.7
0.75
0.8
0.85
0.9
0.95
1
C
or
re
la
tio
n(
ES
G,
ma
rke
t)
= 0.01
= 0.02
= 0.03
= 0.04
Figure 1. Correlation of ESG investor’s portfolio return with the market return.
The figure plots the correlation between the returns on the ESG investor’s portfolio and the
market portfolio. Results are plotted against λ, the fraction of wealth belonging to ESG in-
vestors, and for different values of ?, the maximum certain return an ESG investor would
sacrifice to invest in the ESG portfolio instead of the market portfolio.
32
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
E
(r)
E
SG

-

E
(r)
N
O
N
= 0.01
= 0.02
= 0.03
= 0.04
Figure 2. ESG versus non-ESG expected portfolio return. The figure plots the expected
excess return on the portfolio of ESG investors minus the corresponding value for non-ESG
investors. Results are plotted against λ, the fraction of wealth belonging to ESG investors, and
for different values of ?, the maximum certain return an ESG investor would sacrifice to invest
in the ESG portfolio instead of the market portfolio.
33
Panel A. The Role of λ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.02
-0.018
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
ES
G
= 0.01
= 0.02
= 0.03
= 0.04
Panel B. The Role of ?
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
ES
G
= 0.1 or 0.9
= 0.2 or 0.8
= 0.3 or 0.7
= 0.5
Figure 3. Alphas of ESG Investors. This figure plots the alpha for the portfolio held by
ESG investors as a function of λ, the fraction of wealth belonging to ESG investors, and ?, the
maximum certain return an ESG investor would sacrifice to invest in the ESG portfolio instead
of the market portfolio. Panel A plots the ESG alpha as a function of λ for four different values
of ?; Panel B flips the roles of λ and ?. The dashed line in Panel B has a slope of ?1. The
differences between the solid lines and the dashed line represent investor surplus.
34
Panel A. The Role of λ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
NO
N
= 0.01
= 0.02
= 0.03
= 0.04
Panel B. The Role of ?
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
NO
N
= 1
= 0.75
= 0.5
= 0.25
Figure 4. Alphas of Non-ESG Investors. This figure plots the alpha for the portfolio held
by non-ESG investors as a function of λ, the fraction of wealth belonging to ESG investors, and
?, the maximum certain return an ESG investor would sacrifice to invest in the ESG portfolio
instead of the market portfolio. Panel A plots the ESG alpha as a function of λ for four different
values of ?; Panel B flips the roles of λ and ?.
35
Panel A. The Role of λ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ag
gre
ga
te
ES
G
tilt
= 0.04
= 0.03
= 0.02
= 0.01
Panel B. The Role of ?
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Ag
gre
ga
te
ES
G
tilt
= 0.5
= 0.3 or 0.7
= 0.2 or 0.8
= 0.1 or 0.9
Figure 5. Size of the ESG Industry. The figure plots the aggregate dollar size of ESG
investors’ deviations from the market portfolio (the ESG “tilt”), expressed as a fraction of the
market’s total capitalization. In Panel A, results are plotted against λ, the fraction of wealth
belonging to ESG investors, and for different values of ?, the maximum certain return an ESG
investor would sacrifice to invest in the ESG portfolio instead of the market portfolio. In Panel
B, results are plotted against ? and for different values of λ.
36
Figure 6. Firm-Level Social Impact. This figure plots Sn(d) ? Sn(0), the social impact
induced by ESG-motivated investors, for different firms n. The horizontal axis indicates the
firm’s initial ESG characteristic, gn,0. The two regions indicate the components of Sn(d)?Sn(0)
from equation (56). This figure uses the same parameters as the previous figures, with λ = 0.5
and ? = 0.02, as well as rf = 0.02, K0,n = 1, Πn = 1.2, ωn = 0.5, and κn = 1. These parameter
values produce d = 0.0864, ?gn(d) = 0.0864, ?Kn(0) = 0.0909, and ?Kn(d) ranging from
0.0228 to 0.1726.
37
Panel A. The Role of λ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
Firms become greener
Green firms invest more, brown less
Panel B. The Role of ?
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Firms become greener
Green firms invest more, brown less
Figure 7. Aggregate Social Impact. The figure plots S(d) ? S(0), the aggregate social
impact induced by ESG-motivated investors. We assume the firms’ initial ESG characteristics
g0,n are uniformly distributed in [?
√
3,
√
3]. (These endpoints maintain g′0ι = 0 and g
′
0g0 = 1.)
The two colored regions indicate the components of Sn(d)?Sn(0) from equation (56), aggregated
across firms. In Panel A, results are plotted against λ, the fraction of wealth belonging to ESG
investors, assuming ? = 0.02. In Panel B, results are plotted against ?, the maximum certain
return an ESG investor would sacrifice to invest in the ESG portfolio instead of the market
portfolio, assuming λ = 0.5. All remaining parameter values are the same as in Figure 6.
38
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41
Appendix. Proofs and Derivations
Derivation of Equation (4):
To compute agent i’s expected utility, we rely on equation (2), the relation W?1i = W0i(1+
rf +X
′
i r?), and the fact that r? is normally distributed, r? ～ N(μ,Σ):
E
{
V (W?1i, Xi)
}
= E
{
?e?AiW?1i?b′iXi
}
= E
{
?e?Ai[W0i(1+rf+X′i r?)]?b′iXi
}
= ?e?ai(1+rf )E
{
e
?aiX′i[r?+ 1ai bi]
}
= ?e?ai(1+rf )e?aiX′i[E(r?)+ 1ai bi]+ 12a2iX′iV ar(r?)Xi
= ?e?ai(1+rf )e?aiX′i[μ+ 1ai bi]+ 12a2iX′iΣXi (A1)
where ai ≡ AiW0i is agent i’s relative risk aversion. Agents take μ and Σ as given. Differen-
tiating with respect to Xi, we obtain the first-order condition
?ai[μ+ 1
ai
bi] +
1
2
a2i (2ΣXi) = 0 , (A2)
from which we obtain agent i’s portfolio weights
Xi =
1
ai
Σ?1
(
μ+
1
ai
bi
)
. (A3)
Derivation of Equation (5):
The nth element of agent i’s portfolio weight vector, Xi, is given by
Xi,n =
W0i,n
W0i
(A4)
where W0i,n is the dollar amount invested by agent i in stock n. Let W0,n ≡ ∫iW0i,ndi
denote the total amount invested in stock n by all agents. Then the nth element of the
market-weight vector, x, is given by
xn =
W0,n
W0
=
1
W0
∫
i
W0i,ndi =
1
W0
∫
i
W0iXi,ndi =
∫
i
W0i
W0
Xi,ndi =
∫
i
wiXi,ndi (A5)
Note that
∑N
n=1 xn = 1 because
∑N
n=1W0,n = W0, which follows from the riskless asset being
in zero net supply. Plugging in for Xi from equation (A3) and imposing ai = a, we have
x =
∫
i
wiXidi
=
∫
i
wi
[
1
a
Σ?1
(
μ+
1
a
bi
)]
]di
=
1
a
Σ?1μ
(∫
i
widi
)
+
1
a2
Σ?1g
(∫
i
wididi
)
=
1
a
Σ?1μ+
dˉ
a2
Σ?1g . (A6)
42
Derivation of Equation (11):
Agent i’s expected excess return is given by E(r?i) = X
′
iμ. We take μ from equation (9)
and express Xi in terms of x by subtracting equation (5) from equation (4). Recalling the
assumption x′g = 0 from equation (8), we obtain agent i’s expected excess return as
E(r?i) = X
′
iμ
=
[
x′ +
δi
a2
g′Σ?1
] [
μMβ ? dˉ
a
g
]
=
[
x′ +
δi
a2
g′Σ?1
] [
μM
σ2M
Σx? dˉ
a
g
]
= μM ? d
a
x′g +
δiμM
a2σ2M
g′x? δid
a3
g′Σ?1g
= μM ? δidˉ
a3
g′Σ?1g. (A7)
Derivation of Equation (12):
Recall that agent i’s excess portfolio return is r?i = X
′
i r?, where r? ～ N(μ,Σ). Therefore,
Var(r?i) = X
′
i ΣXi
=
[
x′ +
δi
a2
g′Σ?1
]
Σ
[
x+
δi
a2
Σ?1g
]
= x′Σx+
δi
a2
g′Σ?1Σx+ x′Σ
δi
a2
Σ?1g +
δ2i
a4
g′Σ?1ΣΣ?1g
= x′Σx+
δi
a2
g′x+
δi
a2
x′g +
δ2i
a4
g′Σ?1g .
Recognizing that x′Σx = σ2M and x
′g = 0, we have
Var(r?i) = σ
2
M +
δ2i
a4
g′Σ?1g ,
which is equation (12). We see that Var(r?i) > σ
2
M as long as δi 6= 0.
Derivation of Equation (13):
The second exponent in agent i’s expected utility in equation (A1) contains the terms
?aX ′iμ, ?X ′ibi, and (a2/2)X ′iΣXi. The first of these is simply minus a times the expression
in equation (A7). The second is given by
?X ′ibi = ?
[
x′ +
δi
a2
g′Σ?1
]
[dig]
= ?diδi
a2
g′Σ?1g, (A8)
43
and the third is given by
a2
2
X ′iΣXi =
a2
2
[
x′ +
δi
a2
g′Σ?1
]
Σ
[
x+
δi
a2
Σ?1g
]
=
a2
2
σ2M +
δ2i
2a2
g′Σ?1g, (A9)
recalling x′g = 0 in both cases. Adding the three terms then gives
?aX ′iμ?X ′ibi + (a2/2)X ′iΣXi = ?aμM +
δidˉ
a2
g′Σ?1g ? diδi
a2
g′Σ?1g +
a2
2
σ2M +
δ2i
2a2
g′Σ?1g
= ?aμM + a
2
2
σ2M +
1
a2
(
δid? diδi + 1
2
δ2i
)
g′Σ?1g
= ?a
(
μM ? a
2
σ2M
)
? δ
2
i
2a2
g′Σ?1g. (A10)
Substituting this exponent into equation (A1) gives
E
{
V (W?1i, Xi)
}
= ?e?a(1+rf )e?a(μM?a2σ2M)?
δ2
i
2a2
g′Σ?1g
=
[
?e?a(1+rf )e?a(μM?a2σ2M)
]
e?
δ2
i
2a2
g′Σ?1g
= Vˉ e?
δ2
i
2a2
g′Σ?1g, (A11)
noting that the bracketed term is, Vˉ , the agent’s expected utility if δi = 0.
Derivation of Equation (22):
To implement the approach for calibrating d, we first note that under the assumptions
given, the vector of the ESG investor’s portfolio weights in equation (18) becomes
Xesg =
1
N
ι+ (1? λ) d
a2
[
σ2ιι′ + η2IN
]?1
g
=
1
N
ι+ (1? λ) d
a2
[
1
η2
(
IN ? 1
η2/σ2 +N
ιι′
)]
g
=
1
N
ι+ (1? λ) d
a2η2
g, (A12)
and the variance of the ESG investor’s portfolio return, for large N , is
X ′esgΣXesg =
[
1
N
ι′ + (1? λ) d
a2η2
g′
] [
σ2ιι′ + η2IN
] [ 1
N
ι+ (1? λ) d
a2η2
g
]
= σ2 + (1? λ)2 d
2
a4η2
. (A13)
With expected utility as given by equation (A1), an ESG investor’s certainty equivalent
excess return from holding the ESG portfolio is then, for large N ,
r?esg = X
′
esg(μ+
d
a
g)? a
2
X ′esgΣXesg
44
= X ′esg(μMβ ?
λd
a
g +
d
a
g)? a
2
X ′esgΣXesg
=
[
1
N
ι′ + (1? λ) d
a2η2
g′
] [
aσ2M ι? λ
d
a
g +
d
a
g
]
? a
2
[
σ2M + (1? λ)2
d2
a4η2
]
=
1
2
[
aσ2M +
(1? λ)2d2
a3η2
]
. (A14)
If the ESG investor is instead constrained to hold the market portfolio, the resulting certainty
equivalent excess return is given by
r?M = x
′μ? a
2
x′Σx
= μM ? a
2
σ2M
=
aσ2M
2
. (A15)
The ESG investor’s certainty-equivalent gain from investing as desired, versus investing in
the market, is therefore
r?esg ? r?M =
(1? λ)2d2
2a3η2
. (A16)
when N is large. As noted, this difference in certainty equivalents is largest when λ = 0.
Solving for d with ? ≡ r?esg ? r?M then gives equation (22).
Derivation of the Certainty Equivalent Excess Return of a Non-ESG Investor:
The non-ESG investor’s portfolio weights in equation (19) are
Xnon =
1
N
ι? λ d
a2
[
σ2ιι′ + η2IN
]?1
g
=
1
N
ι? λ d
a2
[
1
η2
(
IN ? 1
η2/σ2 +N
ιι′
)]
g
=
1
N
ι? λ d
a2η2
g, (A17)
and the variance of the non-ESG investor’s portfolio return, for large N , is
X ′nonΣXnon =
[
1
N
ι′ ? λ d
a2η2
g′
] [
σ2ιι′ + η2IN
] [ 1
N
ι? λ d
a2η2
g
]
= σ2 + λ2
d2
a4η2
. (A18)
A non-ESG investor’s certainty equivalent excess return from holding the non-ESG portfolio
is then, for large N ,
r?non = X
′
nonμ?
a
2
X ′nonΣXnon
= X ′non(μMβ ?
λd
a
g)? a
2
X ′nonΣXnon
45
=[
1
N
ι′ ? λ d
a2η2
g′
] [
aσ2M ι? λ
d
a
g
]
? a
2
[
σ2M + λ
2 d
2
a4η2
]
=
1
2
[
aσ2M +
λ2d2
a3η2
]
. (A19)
Derivation of Equation (23):
The correlation between the ESG investor’s return and the market return is equal to
ρ (r?esg, r?M) =
Cov
(
X ′esg ?, x
′?
)
√
Var
(
X ′esg ?
)√
Var (x′?)
=
X ′esgΣx√
X ′esgΣXesg
√
x′Σx
. (A20)
From equations (15) and (A12), recalling that x = (1/N)ι and ι′g = 0, we obtain
X ′esgΣx =
[
1
N
ι′ + (1? λ) d
a2η2
g′
] [
σ2ιι′ + η2IN
] [ 1
N
ι
]
= σ2, (A21)
for large N . Substituting from equations (A13) and (A21) into equation (A20), recalling
x′Σx = σ2 and equation (22), gives equation (23).
Derivation of Equation (24):
Applying equation (20) gives
E{r?esg} ? E{r?non} = ?λd
2
a3
g′Σ?1g
= ?λd
2
a3
g′
[
σ2ιι′ + η2IN
]?1
g
= ?λd
2
a3
g′
[
1
η2
(
IN ? 1
η2/σ2
ιι′
)]
g
= ? λd
2
a3η2
. (A22)
Plugging in for d2 from equation (22), we obtain equation (24).
Derivations of Equations (25) and (26):
Let α denote the N × 1 vector of alphas given by equation (10). Taking Xesg from
equation (A12), the alpha of the ESG investor is given by
αesg = X
′
esgα
=
[
1
N
ι′ + (1? λ) d
a2η2
g′
] [
?λd
a
g
]
= ?λ(1? λ) d
2
a3η2
. (A23)
46
By using Xnon from equation (A17), we obtain the alpha of the non-ESG investor:
αnon = X
′
nonα
=
[
1
N
ι′ ? λ d
a2η2
g′
] [
?λd
a
g
]
= λ2
d2
a3η2
. (A24)
Plugging in for d2 from equation (22), we obtain equations (25) and (26).
Derivation of Equation (30): Using equation (A12) and x = (1/N)ι,
T =
1
2
λι′|Xesg ? x|
=
1
2
λι′|(1? λ) d
a2η2
g|
=
1
2
λ(1? λ) d
a2η2
ι′|g|. (A25)
Plugging in for d from equation (22), we obtain equation (30).
Derivation of Equations (39) and (40):
From equation (37), the return covariance matrix is Σ = BCov{f? , f? ′}B′+ Λ. The N × 1
vector of market betas can therefore be written as
β =
1
σ2M
Σx =
1
σ2M
{
BCov{f? , f? ′}B′ + Λ
}
x =
1
σ2M
{
BCov{f? , f? ′B′}+ Λ
}
x
=
1
σ2M
{
BCov{f? , ?′}+ Λ
}
x =
1
σ2M
{
BCov{f? , ?′x}+ Λx
}
=
1
σ2M
{
BCov{f? , ?M}+ Λx
}
=
1
σ2M
{
BCov{
(
z?h
f?g
)
, ?M}+ Λx
}
=
1
σ2M
{
[h g]
(
Cov{z?h, ?M}
Cov{f?g, ?M}
)
+ Λx
}
= h
Cov{z?h, ?M}
σ2M
+ g
Cov{f?g, ?M}
σ2M
+
1
σ2M
Λx = hβh + gβg +
1
σ2M
Λx , (A26)
which is equation (39). Substituting from equation (36) into ?M ≡ x′? yields
?M = (x
′h)z?h + (x′g)f?g + x′ζ? , (A27)
which immediately implies equation (40).
Derivation of Equations (41) and (42):
Combining equations (9) and (36) gives
r? = μ+ ?
= βμM ? g dˉ
a
+ hz?h + gf?g + ζ? . (A28)
47
With x′β = 1 and x′g = 0, premultiplying the above by x′ gives the excess market return as
r?M = μM + (x
′h)z?h + x′ζ? , (A29)
implying
z?h =
(
r?M ? μM ? x′ζ?
)
/x′h. (A30)
Substituting into equation (A28) and then using equation (39) gives
r? = βμM ? g dˉ
a
+ h
(
r?M ? μM ? x′ζ?
)
/x′h+ gf?g + ζ?
=
(
hβh + gβg +
1
σ2M
Λx
)
μM ? g dˉ
a
+ h
[
r?M ? μM ? x′ζ?
]
/x′h+ gf?g + ζ?
= h
(
βhμM +
[
r?M ? μM ? x′ζ?
]
/x′h
)
+ g
(
f?g + βgμM ? dˉ
a
)
+
μM
σ2M
Λx+ ζ?
= θr?M + g
(
f?g + βgμM ? dˉ
a
)
+ ν?, (A31)
where θ = (1/x′h)h, and
ν? = hμM
(
βh ? 1
x′h
)
+
μM
σ2M
Λx? h
(
x′ζ?
x′h
)
+ ζ? . (A32)
Equation (A31) provides the desired relation, but it remains to show that E{ν?|r?M , f?g} ≈ 0.
Recall that βh ≡ Cov{?M , z?h}/σ2M . Substituting for ?M from equation (A27) and recalling
x′g = 0, we obtain
βh =
(x′h)var(z?h)
(x′h)2var(z?h) + var(x′ζ?)
. (A33)
If the market is well diversified with N large, such that xn ≈ 0, then var(x′ζ?) = x′Λx ≈ 0,
and thus βh ≈ 1/x′h, thereby making the first term in equation (A32) approximately zero.
The second term in that equation is also approximately zero if xn ≈ 0, as the nth ele-
ment of that vector is
(
μMVar(ζ?n)/σ
2
M
)
xn. The third and fourth terms in equation (A32)
have zero means, so we have E{ν?} ≈ 0. Because Cov{ζ? , f?g} = 0, it remains to show that
Cov{ζ? , r?M} ≈ 0. That result follows from equation (A29), which implies that the nth ele-
ment of Cov{ζ? , r?M} equals Var(ζ?n)xn, approximately zero if xn ≈ 0.
Derivation of Equation (44):
Modifying the earlier derivation of equation (4), we obtain
E
{
V (W?1i, Xi, C?)
}
= ?e?ai(1+rf )E
{
e
?aiX′i[r?+ 1ai bi]?ciC?
}
= ?e?ai(1+rf )e?aiX′i[E(r?)+ 1ai bi]+ 12a2iX′iV ar(?)Xi+aiciX′iCov(?,C?)+ 12 c2i V ar(C?)
= ?e?ai(1+rf )e?aiX′i[μ+ 1ai bi]+ 12a2iX′iΣXi+aiciX′iσC+ 12 c2i σ2C , (A34)
48
where σC ≡ Cov(?, C?). Differentiating with respect to Xi gives the first-order condition
?ai[μ+ 1
ai
bi] + a
2
iΣXi + aiciσC = 0 (A35)
from which we obtain agent i’s portfolio weights
Xi =
1
ai
Σ?1
(
μ+
1
ai
bi ? ciσC
)
. (A36)
Again imposing the market-clearing condition and ai = a gives
x =
∫
i
wiXi di
=
1
a
Σ?1μ+
dˉ
a2
Σ?1g ? cˉ
a
Σ?1σC , (A37)
which implies
μ = aΣx? dˉ
a
g + cˉσC . (A38)
Premultiplying by x′, again imposing the assumption x′g = 0, gives
μM = aσ
2
M + cˉσMC , (A39)
where σMC ≡ Cov(?M , C?) = x′σC . Solving equation (A39) for a and substituting into the
first term on the right-hand side of equation (A38) gives
μ =
μM ? cˉσMC
σ2M
Σx? dˉ
a
g + cˉσC
= (μM ? cˉσMC)β ? dˉ
a
g + cˉσC
= μMβ ? dˉ
a
g + cˉ
(
σC ? σMC
σ2M
σM
)
, (A40)
noting β = (1/σ2M)σM = (1/σ
2
M)Σx. To see that the third term on the right-hand side of
equation (A40) is the same as that in equation (44), first observe that in the multivariate
regression of ? on ?M and C?, the N × 2 matrix of slope coefficients is given by
[σM σC ]
[
σ2M σMC
σMC σ
2
C
]?1
=
1
σ2Mσ
2
C ? σ2MC
[
σ2CσM ? σMCσC σ2MσC ? σMCσM
]
,
so the second column is given by
ψ =
σ2MσC ? σMCσM
σ2Mσ
2
C ? σ2MC
. (A41)
Using equation (A41), we can rewrite the third term on the right-hand side of equation (A40)
as
cˉ
(
σC ? σMC
σ2M
σM
)
= cˉ
σ2Mσ
2
C ? σ2MC
σ2M
ψ
= cˉ
(
1? ρ2MC
)
ψ , (A42)
49
recalling that σC = 1.
Derivation of Equation (48):
Substituting for μ from equation (A40) into equation (A36) and setting ai = a, we obtain
Xi =
1
a
Σ?1
(
μ+
1
a
bi ? ciσC
)
=
1
a
Σ?1
(
μMβ ? dˉ
a
g + cˉ
(
σC ? σMC
σ2M
σM
)
+
1
a
bi ? ciσC
)
=
μM
a
Σ?1β ? 1
a
Σ?1cˉ
σMC
σ2M
σM +
1
a
Σ?1
(
di
a
g ? dˉ
a
g
)
? 1
a
Σ?1 (ci ? cˉ)σC
=
μM
a
Σ?1β ? 1
a
Σ?1 (cˉσMC)
σM
σ2M
+
1
a
Σ?1
δi
a
g ? ci ? cˉ
a
Σ?1σC (A43)
Noting from equation (A39) that cˉσMC = μM ? aσ2M , and that β = σMσ2M =
Σx
σ2M
, we have
Xi =
μM
a
Σ?1β ? 1
a
Σ?1
(
μM ? aσ2M
)
β +
δi
a2
Σ?1g ? ci ? cˉ
a
Σ?1σC
= σ2MΣ
?1β +
δi
a2
Σ?1g ? ci ? cˉ
a
Σ?1σC
= x+
δi
a2
Σ?1g ? ci ? cˉ
a
Σ?1σC , (A44)
which is equation (48).
Derivation of Equation (49):
This derivation follows exactly the same steps as our prior derivation of equations (41)
and (42), except that instead of using μ from equation (9) we use μ from equation (44). As
a result, we obtain a counterpart of equation (A31) with an extra term:
r? = θr?M + g
(
f?g + βgμM ? dˉ
a
)
+ cˉ
(
1? ρ2MC
)
[ψ ? θ(x′ψ)] + ν? , (A45)
where θ and ν? are the same as before. In the special case to which equation (49) pertains,
ψ = ?g. To see this, note from equation (A45) that g is the slope on f?g from the multiple
regression of stock returns on market returns and f?g, whereas ψ is defined in the context
of equation (44) as the slope on C? in the multiple regression of stock returns on market
returns and C?. In this special case, f?g = ?C?, which implies ψ = ?g. It then follows that
x′ψ = x′g = 0, and equation (A45) simplifies to
r? = θr?M + g
(
f?g + βgμM ? dˉ
a
? cˉ
(
1? ρ2MC
))
+ ν? , (A46)
which immediately delivers equation (49).
50
Derivation of equation (53):
The firm’s value at time 0 is
υn = ??Kn ? κn
2
(?Kn)
2 +
Πn (K0,n + ?Kn)
1 + rf + μMβn ? dagn
. (A47)
The manager maximizes υn by choosing ?Kn. The first-order condition yields
?Kn(d) =
1
κn
?? Πn
1 + rf + μMβn ? dagn
? 1
?? . (A48)
Substituting into equation (52) produces
Sn(d)? Sn(0) = gn 1
κn
?? Πn
1 + rf + μMβn ? dagn
? Πn
1 + rf + μMβn
?? (A49)
= gn
Πn
κn
?? dagn
(1 + rf + μMβn ? dagn)(1 + rf + μMβn)
?? , (A50)
which produces equation (53). Comparative statics for Πn, βn, and κn follow immediately
from equation (53). For the comparative statics for d and a, we define d? ≡ d/a and compute
?
?d?
(
Sn(d)? Sn(0)
)
=
g2nΠn
κn(1 + rf + μMβn)
[
(1 + rf + μMβn ? d?gn) + d?gn
(1 + rf + μMβn ? d?gn)2
]
(A51)
=
g2nΠn
κn
[
1
(1 + rf + μMβn ? d?gn)2
]
, (A52)
which is positive if gn 6= 0. Since Sn(d)?Sn(0) increases in d?, it increases in d and decreases
in a.
Derivation of equations (54) and (55):
The firm’s value at time 0 is now
υn = ??Kn ? κn
2
(?Kn)
2 +
Πn (K0,n + ?Kn)
(
1? ωn
2
(?gn)
2
)
1 + rf + μMβn ? da(gn,0 + ?gn)
. (A53)
The manager maximizes υn by choosing ?gn and ?Kn. The choice of ?gn depends only on
the third term of equation (A53), and we can maximize its log. Using the approximation
that log(1 + x) ≈ x and ignoring terms without ?gn, the choice of ?gn simplifies to
max
?gn
?ωn
2
(?gn)
2 +
d
a
?gn . (A54)
The first-order condition delivers equation (54). Without taking logs, the first-order condi-
tion for ?Kn is
?1? κn?Kn +
Πn
(
1? ωn
2
(?gn)
2
)
1 + rf + μMβn ? dagn
= 0 , (A55)
which delivers equation (55).
51