# ESG-coherent risk measures for sustainable investing

Gabriele Torri Corresponding author<EMAIL_ADDRESS>Department of
Management, University of Bergamo, Bergamo 24127, Italy  Rosella Giacometti
Department of Management, University of Bergamo, Bergamo 24127, Italy  Darinka
Dentcheva  School of Engineering and Science, Stevens Institute of Technology,
Hoboken, NJ 07030, USA  Svetlozar T. Rachev  Department of Mathematics and
Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA  W. Brent
Lindquist  Department of Mathematics and Statistics, Texas Tech University,
Lubbock, TX 79409-1042, USA

(August 28, 2024)

###### Abstract

The growing interest in sustainable investing calls for an axiomatic approach
to measures of risk and reward that focus not only on financial returns, but
also on measures of environmental and social sustainability, i.e.
environmental, social, and governance (ESG) scores. We propose definitions for
ESG-coherent risk measures and ESG reward–risk ratios based on functions of
bivariate random variables that are applied to financial returns and ESG
scores, extending the traditional univariate measures to the ESG case. We
provide examples and present an empirical analysis in which the ESG-coherent
risk measures and ESG reward–risk ratios are used to rank stocks.

##  1 Introduction

ESG investing refers to the integration of environmental, social, and
governance considerations into the asset allocation process. It is one of the
most significant trends in the asset management industry, due to an increased
focus on sustainability and to the increasing availability of information
related to non-financial impacts. Governments and transnational organizations
are acknowledging the importance of ESG by promoting specific legislation.
Under general voluntary commitments, such as the UN Principles for Responsible
Investment (UN-PRI) initiative launched in 2006, regulators are beginning to
require financial institutions to comply with specific standards.

ESG investing encompasses a broad array of approaches, and its market
practices are very heterogeneous, with different terminologies, definitions,
and strategies. These practices vary due to the cultural and ideological
diversity of investors (Sandberg et al.,, 2009; Widyawati,, 2020). According
to Amel-Zadeh and Serafeim, (2018), the most significant motivation for
incorporating ESG factors is related to financial performance, as
sustainability factors are perceived as relevant to investment returns. That
is, investors believe that ESG data can be used to identify potential risks
and opportunities, and that such information is not yet fully incorporated
into market prices. Hence, ESG information should help investors to control
risk better and improve their financial performance. In line with Schanzenbach
and Sitkoff, (2020) we employ the expression risk-return ESG to refer to
investment strategies that use ESG factors to improve returns while lessening
risk. Academic evidence on the role of ESG in enhancing performance is
inconclusive. The meta-analysis conducted by Revelli and Viviani, (2015) shows
that sustainable and responsible investing (SRI) is neither a weakness nor a
strength compared with conventional investing. Similar conclusions have been
drawn by Hornuf et al., (2022). From a theoretical point of view, risk-return
ESG does not pose any specific challenge, as ESG is treated as any other
information (e.g. balance sheet data, macroeconomic indicators, sentiment
analysis, etc.) and is integrated in the investment process without affecting
the main goal of improving the risk-adjusted performance.111 We note that, if
ESG information is already incorporated in market prices, any strategy that
restricts investment based on ESG criteria would result in sub-optimal
allocation in terms of monetary performance. Moreover we underline that risk-
return ESG strategies are not necessarily more sustainable than traditional
ones. Indeed, an investor may implement a contrarian-ESG approach, investing
in less sustainable assets if they are expected to outperform the market.

A second motivation that guides ESG strategies is the desire to improve the
sustainable profile of the portfolios for ethical reasons or to improve the
investors’ green image (Amel-Zadeh and Serafeim,, 2018).222 Almel-Zadeh and
Serafim reported additional motivations that push asset managers to propose
ESG products, including growing client demand, the effectiveness of ESG
investing in bringing about change in companies, and formal client mandates.
For our purposes, these drivers can be attributed to either material or
ethical motivations. Sustainability then becomes part of the investment goals,
alongside monetary performance and the riskiness of the position. Schanzenbach
and Sitkoff, (2020) refer to investment strategies that incorporate ESG
screenings for moral or ethical reasons as collateral benefit ESG, as they aim
to provide benefits to a third party, rather than to improve risk-adjusted
returns. We refer to investors who include sustainability considerations for
ethical reasons as ESG-oriented investors, to distinguish them from risk-
return investors who care exclusively about the financial risks and returns of
a position.

The employment of ethical motivations challenges many of the traditional
assumptions adopted in finance theory, as ESG-oriented investors consider
monetary risks and rewards in their investment process as well as a set of
other determinants that cannot be reduced to the financial performance alone.
To show how the ethical motivation breaks common investing principles,
consider two stocks with the same reward–risk profile, but with different ESG
scores. They may be equivalent for a risk-return investor but not for an ESG-
oriented investor, as one of the two companies may be more environmentally
sustainable or may adopt stricter human right policies. An ESG-oriented
investor may decide to deliberately worsen their risk-adjusted performance to
comply with non-negotiable principles, for instance by excluding certain
sectors or companies, thus reducing the diversification of their portfolio.333
It is common practice for sustainable investment funds to exclude tobacco,
weapons, fossil fuels, and other controversial sectors. The divestment
campaign aimed at South Africa’s apartheid regime in the 1980s was one of the
key turning points in the history of responsible investing (Schanzenbach and
Sitkoff,, 2020).

The dichotomy between ethical and financial motivations of ESG strategies is
also connected to country-specific legal frameworks. As mentioned, ESG
investing is now high on the agenda of regulators as a consequence of interest
from market participants and pressure from initiatives such as UN-PRI. Due to
the highly regulated nature of the asset management industry, the growth of
ESG investing is tightly related to national and international regulatory
frameworks, and views on this topic differ substantially across jurisdictions.
In particular, in the United States the debate focuses on investments subject
to the Employee Retirement Income Security Act of 1974 (ERISA): Schanzenbach
and Sitkoff, (2020) argue that, in general under American trust fiduciary law,
the use of ESG is only permissible to improve financial performance (i.e.
risk-return ESG), as the trustee of a fund must act in the sole interest of
the beneficiaries, and the inclusion of ethical considerations in the
investment process may breach this principle (collateral benefit ESG may still
be possible under direct authorization by the beneficiary or for charity
endowments under specific conditions). Other jurisdictions more extensively
support ESG investing in all its forms. As an example, in Europe the
Commission Delegated Regulation (EU) 2021/1253 of 21 April 2021444https://eur-
lex.europa.eu/legal-content/en/TXT/?uri=CELEX:32021R1253 amends the Markets in
Financial Instruments Directive II (MiFID II) and imposes on financial
advisors and portfolio managers the requirement to carry out an assessment of
the sustainability preferences of their clients (and potential clients) in
order to offer products that are in line with client preferences.

It is clear that the growing interest and surrounding controversy related to
ESG investing requires the development of a solid theoretical framework for
supporting the decision making process of ESG-oriented investors.

Currently, the most common ESG portfolio strategies involve negative
screenings (i.e., the exclusion of specific sectors or companies) or a pre-
selection of best-in-class companies according to ESG scores. It is only
recently that more formal techniques for the construction of a portfolio
suitable for an ESG-oriented investor have been discussed in the literature.
One approach to combining financial and sustainability goals is the
introduction of optimal asset allocations that include ESG constraints. This
stream of the literature extends the classical reward–risk analysis by adding
a third dimension: the sustainability, represented by ESG scores. An investor
would then be able to assess a portfolio in three dimensions: return, risk,
and ESG. Such an approach has been pursued by, among others, Utz et al.,
(2015) and Cesarone et al., (2022) who studied the efficient frontier of non-
dominated portfolios in the reward–risk-ESG space. This approach extends the
traditional optimal portfolio literature that started with Markowitz, (1952)
and which aims to find portfolios with the highest expected return for a given
level of risk, where risk is measured using the portfolio variance
(Markowitz,, 1952), a coherent risk measure such as the average value at risk
(AVaR)555 The AVaR is also known in the literature as Conditional Value at
Risk (CVaR) or Expected Shortfall (ES) (Rockafellar et al.,, 2000), or an
asymmetric deviation measure (Giacometti et al.,, 2021).666 An alternative
approach for including ESG in the frontier analysis was proposed by Pedersen
et al., (2021), who identified a two-dimensional ESG efficient frontier by
considering the Sharpe ratio and the ESG score. One drawback of this modeling
framework is that it fails to take into account the stochasticity of ESG
scores: it either uses only the expected values of the scores or assumes that
ESG scores can be treated as constants for the duration of the investment
period.

Here, we aim to address this limitation and, more generally, introduce a way
to measure the risk associated with an ESG-oriented position. We propose an
axiomatic approach based on the idea that risk comes from two drivers:
monetary performance (i.e., the financial returns of a position) and
sustainability (represented by the ESG score of the company).777For
convenience, in the rest of this paper, we refer to these two dimensions as
the monetary and ESG components of risk. These quantities are random and not
necessarily independent, and we can represent them as a bivariate random
variable. Hence, it is natural to refer to the rich literature on multivariate
risk measures (Jouini et al.,, 2004; Hamel,, 2009; Wei and Hu,, 2014; Ekeland
et al.,, 2012). Such measures have been developed to study portfolios of non-
perfectly fungible assets (e.g., assets valued in multiple currencies) or
assets that are difficult to price. We propose using a bivariate risk measure
to deal with a single asset that can be evaluated over two dimensions: the
monetary returns and the sustainability (represented by ESG scores). We then
define ESG-coherent risk measures as an extension of the coherent risk
measures introduced by Artzner et al., (1999).

Since different investors may have different attitudes towards ESG, the
proposed measures are parametrized using a value $\lambda\in[0,1]$ to
explicitly take into account the subjective trade-off between sustainability
and financial performance: when $\lambda=0$ an investor cares exclusively
about financial risk, when $\lambda=1$ the investor cares only about ESG. In
addition to ESG-coherent risk measures, we define ESG-coherent reward measures
and ESG reward–risk ratios.

Section 2 discusses our interpretation of ESG scores and outlines how we
propose to use them. Section 3 introduces ESG-coherent risk measures and
provides examples, while Section 4 introduces ESG reward–risk ratios. Section
5 presents an empirical example using real data. Section 6 highlights some
conclusions and perspectives for future research.

##  2 Measuring ESG and financial performance

The first step in defining ESG risk measures is to clarify how we characterize
and measure the sustainability.

In general, following an approach common in the literature, we use the ESG
scores of a company as a proxy for sustainability (see, e.g., Pedersen et
al.,, 2021; Utz et al.,, 2015; Cesarone et al.,, 2022). Here, we clarify how
to treat such a variable from both practical and theoretical points of view,
establishing the basis for measuring ESG scores in a way that allows us to
consistently use them alongside monetary values for risk measurement. In
practice, to measure risk, we define a bivariate random variable
$X=[r\;\text{ESG}]^{\prime}$, where the first component measures the monetary
risk and the second measures the ESG risk. We emphasize that, at this stage,
we keep the discussion on an abstract level, without discussing the
composition and estimation of ESG scores. This allows us not to be limited by
the current state-of-the-art market ESG scores, which are still far from being
standardized and comparable across data providers (see, e.g., Berg et al.,,
2019; Billio et al.,, 2021).

Several approaches appear in the literature for modeling the monetary
component. Artzner et al., (1999) define coherent risk measures for a random
variable that represents net worth by following the principle that “bygones
are bygones,” meaning that future net worth is the only thing that matters. In
an alternative approach, the random variable used to compute monetary risk
represents the return of a financial position. This last approach, which is
often used in practical applications for the measurement of the risk of equity
portfolios, introduces some differences in the interpretation of the axioms
(see Rachev et al.,, 2011, Chapter 6).888Alternatively, some authors prefer to
define risk measures computed on a variable that represents losses, thus
assuming that lower values of the variable are preferred by an investor. Such
an assumption does not significantly alter the analysis: it simply changes
some signs in the definition of risk (see, e.g., Rockafellar and Uryasev,,
2013).

Similarly, we need to establish the quantity measured by the random variable
ESG and the sign convention used. It is critical to identify the nature of the
ESG variable; in particular, one must consider whether it is a stock variable
(measured at a specific point in time) or a flow variable (measured over an
interval of time, as some sort of sustainability return). Based on the way
they are computed, we argue that ESG scores belong to the second category:
indeed, they represent the current level of sustainability of the production
and commercial practices of a company, which directly affects the current
impact of that company on the world.999 The scores are typically computed as
functions of several indicators related to the production methods, the supply
chain management, the industry in which the company operates, the transparency
of its governance, the presence of specific policies on human rights
violations, etc. We can think of ESG as a broad measure of externalities
(positive or negative) that are generated over time. In this sense, the ESG
score of a company is the rate at which it accumulates non-monetary
“satisfaction” for the investor. The total non-monetary satisfaction for an
investor is proportional to the holding time of the investment.

Our approach is to regard ESG as a sustainability flow. This quantity does not
necessarily need to be expressed in monetary terms, and the value of this
component is different for each investor.101010A monetary market price of
sustainability may exist, but each investor may assign a subjective value to
it. Assuming that the sustainability flow is proportional to the initial
investment, we can rescale the ESG rating at a time $t$ to obtain
$\text{esg}_{t}$, which represents the instantaneous sustainability flow of an
asset, and it can be modeled as a stochastic process. For a given time
horizon, the satisfaction of the investor depends on the amount of
sustainability accumulated over time, which, for the interval of time from $0$
to $T$, is defined by

$\text{ESG}_{T}:=\int_{0}^{T}\text{esg}_{s}ds.$ (1)

Under this approach, ESG scores currently available from the market are
interpreted as the accumulated sustainability over one year, and could be used
to calibrate the process $\text{esg}_{t}$. In the simple case of a constant
instantaneous flow esg, we have $\text{ESG}_{T}=T\cdot\text{esg}$.

The stochasticity of the process $\text{esg}_{t}$ can be traced to two
sources: the first is the uncertainty of the measurement of its value, and the
second is the uncertainty in the evolution of the sustainability policies
within the company, which is driven by the choices made by the management, by
market conditions, and by the regulatory framework. We also expect a
correlation structure between ESG scores and financial returns due to the
presence of common driving factors related, for instance, to sector-wide or
country-wide dynamics.

With regard to a sign convention, it is natural to express ESG scores in such
a way that higher scores are preferable. We use this convention in the rest of
this paper. For consistency, we also assume a preference for higher values for
the monetary component: that is, we consider a preference for a gain in future
net worth.

This interpretation of ESG as a sustainability flow pairs well with the use of
returns, rather than the final net worth, to model the monetary component. As
with returns, the accumulated sustainability for the unit investment is
expressed as a number which depends on the length of the studied period of
time.

Unless stated otherwise, in the rest of this paper we equate the monetary
component of the investment with the rate of return $r_{T}$ and the
sustainability component with the accumulated sustainability $\text{ESG}_{T}$.
Dropping the time dependence from the notation to improve readability, the
bivariate random variable used to compute the ESG-coherent risk measures is
therefore

$X=\begin{bmatrix}r\\\ \text{ESG}\end{bmatrix}.$ (2)

With reference to the empirical implementation of this framework, we note that
the standardization of ESG scores across data providers is currently a major
issue, and the procedures used to estimate such scores are under continuous
development. In particular, there is a dearth of forward-looking information
regarding the sustainability of a company. ESG scores are typically based on
the current or past business practices of the firm, while measures of the ESG
risk (and reward) should focus on the sustainability of the firm over, and
beyond, any investment horizon. Our procedure is general enough to be adapted
to improved estimators of company sustainability, when they are developed.
Further research should focus on the modeling and estimation of
$\text{esg}_{t}$ and the computation of the ESG score.

##  3 ESG-coherent risk measures

We begin with the axioms that define a coherent risk measure (Artzner et al.,,
1999). These axioms make it possible to identify measures with desirable
properties, assisting both investors and regulators. Consider a convex set
$\mathcal{X}\subseteq\mathcal{L}_{p}(\Omega,\mathcal{F},\mathbb{P})$ of real-
valued random variables $X$ which are defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, have finite $p$-moments ($p\geq 1$), and
are indistinguishable up to a set of $P$-measure zero. We assume that the
random variables represent the returns or the payoff of an asset. The
functional $\rho(X):\mathcal{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is a
coherent risk measure if it satisfies the following properties:

  * (SUB)

sub-additivity: if $X_{1},X_{2}\in\mathcal{X}$, then
$\rho(X_{1}+X_{2})\leq\rho(X_{1})+\rho(X_{2})$;

  * (PH)

positive homogeneity: if $\alpha\in\mathbb{R}_{+}$ and $X\in\mathcal{X}$, then
$\rho(\alpha X)=\alpha\rho(X)$;

  * (TI)

translation invariance: if $a$ is deterministic, $\rho(X+a)=\rho(X)-a$;

  * (MO)

monotonicity: if $X_{1},X_{2}\in\mathcal{X}$ and $X_{1}\leq X_{2}$ a.s., then
$\rho(X_{1})\geq\rho(X_{2})$.

Examples of coherent risk measures are AVaR and expectile; but VaR and
standard deviation are not coherent risk measures.111111 Alternative
axiomizations of risk measures, such as convex risk measures (Föllmer and
Schied,, 2002) and regular risk measures (Rockafellar and Uryasev,, 2013) have
been proposed since the work of (Artzner et al.,, 1999).

The definition of a coherent risk measure is based upon a univariate random
variable, and on the idea that the risk is a function only of the returns of
the asset (consistent with the assumption that an investor is only interested
in the monetary outcome of their position). In some contexts, the monetary
outcome does not fully characterize the risk. Examples include: portfolios in
two countries having floating exchange rates whose payoffs in the different
currencies are not perfectly substitutable (the Siegel paradox, see Black,,
1989), the fact that various maturities for interest rate products are not
perfect substitutes (failure of the pure expectation hypothesis), and cases in
which it is difficult to attribute a monetary equivalent to various dimensions
of risk, such as environmental or health risks. The same principle can be used
for the ESG-oriented investor, whose risk depends on both the monetary return
of a financial position and its sustainability, represented in our analysis by
its ESG score. Such an approach allows an investor to deal with trade-offs
between different goals that, although they are very different in nature, have
to be taken into account in the investment process.

###  3.1 ESG-coherent risk measures — axiomatic definition

We introduce bivariate risk measures and apply them to the financial
performance and sustainability of a position.121212 As discussed in Section 2,
we use periodic returns for the monetary component and ESG scores (which
represent the accumulated sustainability of the investment) for the
sustainability component. The definition could be extended to alternative
specifications that can be represented using two random variables with a joint
distribution such that an investor has a preference for both higher $r$ and
higher ESG values (e.g., using the final net worth and the accumulated
sustainability multiplied by the initial value of the position). Consider a
convex set $\mathcal{X}_{2}$ of random vectors $X=[r\;ESG]^{\prime}$, defined
on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with values in
$\mathbb{R}^{2}$. We use the short-hand notation
$\mathcal{X}_{2}=\mathcal{L}_{p}(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^{2})$
for the space of random vectors with two components and finite $p$-th moments,
which are indistinguishable on sets of $P$-measure zero. Here
$p\in[1,\infty]$. An ESG risk measure is then a functional of the form
$\boldsymbol{\rho}(X):\mathcal{X}_{2}\rightarrow\mathbb{R}\cup\\{+\infty\\}$.
As in the univariate case, it is possible to axiomatically characterize a set
of measures that have desirable properties. These axioms are a bivariate
generalization of SUB, PH, TI, and MO:

  * (SUB-M)

sub-additivity: if $X_{1},X_{2}\in\mathcal{X}_{2}$, then
$\boldsymbol{\rho}(X_{1}+X_{2})\leq\boldsymbol{\rho}(X_{1})+\boldsymbol{\rho}(X_{2})$;

  * (PH-M)

positive homogeneity: if $\beta\in\mathbb{R}_{+}$ and $X\in\mathcal{X}_{2}$,
then $\boldsymbol{\rho}(\beta X)=\beta\boldsymbol{\rho}(X)$;

  * (TI-M)

translation invariance: if $y\in\mathbb{R}^{2}$,
$\boldsymbol{\rho}(X+y)=\boldsymbol{\rho}(X)+\boldsymbol{\rho}(y)$;

  * (MO-M)

monotonicity: if $X_{1},X_{2}\in\mathcal{X}_{2}$ and ($r_{1}\leq
r_{2}\wedge\text{ESG}_{1}\leq\text{ESG}_{2}$) a.s., then
$\boldsymbol{\rho}(X_{1})\geq\boldsymbol{\rho}(X_{2})$.

We note the following.

  * •

A multivariate generalization of the first two axioms is straightforward.

  * •

The identification of the value of $\boldsymbol{\rho}(y)$, where $y$ is a
constant vector, appearing in axiom TI-M, is discussed in subsection 3.2. It
is related to the risk of an ESG safe asset (SA), defined as a position having
constant values for both monetary and ESG components.

  * •

In line with the literature, we impose a monotonicity condition for which
zeroth-order stochastic dominance implies the a.s. ordering of risks.
Alternative approaches may be based on first- and second-order stochastic
dominance; we leave such an analysis for future studies and maintain the most
general (weakest) condition. We emphasize that the ordering induced by the
rule in MO-M is partial.

This set of axioms is consistent with the work of Rüschendorf, (2006), Wei and
Hu, (2014), and Chen and Hu, (2020), which define risk for multivariate
vectors using scalar-valued functions. A key difference is that we maintain a
more general formulation of the translation invariance (TI-M) axiom, as we aim
to further parametrize the function for ESG-coherent risk measures.

Some authors have proposed set-valued risk measures (Jouini et al.,, 2004;
Hamel,, 2009; Hamel et al.,, 2013). Such proposals start from the idea that,
“risk is the amount of cash that needs to be added to a position to make it
acceptable” (Artzner et al.,, 1999). Extending the reasoning to a multivariate
setting with multiple assets, it is possible to make a position acceptable by
adding any of several “safe” portfolios with deterministic payoff, each
characterized by a different combination of assets with deterministic payoff.
In such a framework, the risk measure is given by a combination of all safe
portfolios that make the position acceptable. In the context ESG investing,
where each asset is evaluated on the basis of two dimensions (returns and
sustainability), we can imagine a market with multiple ESG safe assets with
different return and ESG. The advantage of this approach is to provide a more
complete assessment of the risk in relation to the multiple drivers of risk,
and has nicer mathematical properties, but at the cost of greater complexity
and the inability to directly rank positions. Our approach to compute a scalar
risk based on a bivariate random variable can be seen as a special case of the
set-valued risk measures, in which we consider only one specific ESG safe
asset.

To better characterize the risk for an ESG-oriented investor, we introduce a
fifth axiom, that takes into account the treatment of measures computed on
constant vectors, allowing thus to characterize the risk of an ESG safe asset.
Consider a constant vector $y\in\mathbb{R}^{2}$, and a scalar
$\lambda\in[0,1]$, then

  * (LH-M)

lambda homogeneity: if $y=\left[\begin{matrix}r_{f}\\\
r_{\text{ESG}}\end{matrix}\right]\in\mathbb{R}^{2}$, then
$\boldsymbol{\rho}(y)=-\big{(}(1-\lambda)r_{f}+\lambda r_{\text{ESG}}\big{)}$.

Axiom LH-M specifies the form of $\boldsymbol{\rho}(y)$ used in TI-M and
explicitly highlights the trade-off between the monetary and sustainability
values using the parameter $\lambda$, which represents an investor preference
for the relative weighing between the monetary and ESG components of risk.
This specification ensures that
$\boldsymbol{\rho}(\boldsymbol{1})=-1,\quad\forall\lambda\in[0,1]$, where
$\boldsymbol{1}$ is the vector $[1,1]^{\prime}$. To make the dependency on
$\lambda$ of a risk measure clear, we introduce $\lambda$ as a subscript:
$\boldsymbol{\rho}_{\lambda}(X)$, and we finally provide the following
definition:

######  Definition 1 (ESG-coherent risk measure).

Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$, a parameter
$\lambda\in[0,1]$, and $X=\begin{bmatrix}r\\\ ESG\end{bmatrix}$ belonging to a
set of bivariate random variables $\mathcal{X}_{2}$ where $r$ measures the
monetary returns of a position or portfolio and ESG measures its
sustainability. We define an ESG-coherent risk measure as any functional
$\boldsymbol{\rho}_{\lambda}(X):\mathcal{X}_{2}\rightarrow\mathbb{R}\cup\\{+\infty\\}$
that satisfies the five axioms from SUB-M through LH-M.

The class of ESG-coherent risk measures extends coherent measures to the
multivariate setting and provides a way to control the trade-off between the
two sources of risk. This trade-off depends on the preferences of the
individual investor expressed by $\lambda$. Axioms SUB-M and PH-M guarantee
that an ESG-coherent risk measure is convex. This allows an investor to
diversify between monetary risk and ESG risk, as highlighted by the following
remarks.

######  Remark.

Given $X=[r,\;ESG]^{\prime}\in\mathcal{X}_{2}$ and an ESG risk measure
$\boldsymbol{\rho}_{\lambda}(X)$, for an investor with a given $\lambda$ the
pure monetary risk and pure ESG risk are defined by
$\boldsymbol{\rho}_{\lambda}([r_{X},\;0]^{\prime})$ and
$\boldsymbol{\rho}_{\lambda}([0,\;\text{ESG}_{X}]^{\prime})$, respectively.

######  Remark (Convexity of an ESG-coherent risk measure).

If $\boldsymbol{\rho}_{\lambda}(X)$ is an ESG-coherent risk measure, from
SUB-M and PH-M we observe that

$\boldsymbol{\rho}_{\lambda}(X)\leq\boldsymbol{\rho}_{\lambda}([r_{X},\;0]^{\prime})+\boldsymbol{\rho}_{\lambda}([0,\;\text{ESG}_{X}]^{\prime}).$
(3)

That is, the risk of a position is always less than or equal to the sum of the
pure ESG risk and the pure monetary risk (the investor diversifies between the
ESG risk and monetary risk).

We note that the definition of an ESG-coherent risk measure remains agnostic
concerning the measurement of either the financial performance or the ESG
score; the former can be measured in terms of the final value, profit and
loss, or periodic returns (Artzner et al.,, 1999), and the ESG component can
be computed according to multiple methodologies and aggregated over time
following several approaches. The only requirement is that the investor must
have a preference for both higher financial gain and higher ESG scores (hence,
the monetary part must be expressed in terms such that gains are positive).

In an analogous manner, we can define ESG-coherent reward measures that extend
the work of Rachev et al., (2008) (see Appendix A).

####  3.1.1 Dual representation

It is well known that coherent risk measures have a dual representation $-$
the supremum of a certain expected value over a risk envelope (Ruszczyński and
Shapiro,, 2006; Rockafellar,, 2007; Ang et al.,, 2018). For ESG-coherent risk
measures, the dual representation is provided in Proposition 1.

######  Proposition 1.

Given $X=[r,\;ESG]^{\prime}\in\mathcal{X}_{2}$ and an ESG-coherent risk
measure $\boldsymbol{\rho}_{\lambda}(X)$ that satisfies axioms SUB-M through
LH-M, the dual representation of the risk measure is

$\displaystyle\boldsymbol{\rho}_{\lambda}(X)$
$\displaystyle=\sup_{\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}}\left\\{-\int_{\Omega}\left[\zeta_{1}(\omega)r(\omega)+\zeta_{2}(\omega)\text{ESG}(\omega)\right]P(d\omega)\right\\},$
(4)

where $\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ contains non-negative
functions
$(\zeta_{1},\zeta_{2})\in\mathcal{L}_{q}(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}^{2})$
whose expected value is $[1-\lambda,\quad\lambda]^{\prime}$. Furthermore,
$\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ is equal to the convex
subdifferential of $\boldsymbol{\rho}_{\lambda}([0,0]^{\prime})$.

The proof of Proposition 1 is provided in Appendix B. Using a more compact
notation, (4) can be written

$\boldsymbol{\rho}_{\lambda}(X)=\sup_{\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}}-\mathbb{E}[\zeta_{1}r+\zeta_{2}\text{ESG}].$

Proposition 2 addresses the marginal ESG-coherent risk measure when
$\lambda=1$ or $\lambda=0$.

######  Proposition 2.

If $\boldsymbol{\rho}_{\lambda}$ is an ESG-coherent risk measure, then

$\boldsymbol{\rho}_{0}([r,\;\text{ESG}]^{\prime})=\boldsymbol{\rho}_{0}([r,\;0]^{\prime}),$
$\boldsymbol{\rho}_{1}([r,\;\text{ESG}]^{\prime})=\boldsymbol{\rho}_{1}([0,\;\text{ESG}]^{\prime}).$

###### Proof.

For $\lambda=0$, we know from the dual representation that $\zeta_{2}=0$ a.s.,
since it has zero expected value and is non-negative. Hence,

$\boldsymbol{\rho}_{0}([r,\;\text{ESG}]^{\prime})=\sup_{\zeta\in\mathcal{A}_{0}}-\int_{\Omega}\zeta_{1}(\omega)r(\omega)P(d\omega)=\boldsymbol{\rho}_{0}([r,\;0]^{\prime}).$

Analogously, for $\lambda=1$ we have

$\boldsymbol{\rho}_{1}([r,\;\text{ESG}]^{\prime})=\sup_{\zeta\in\mathcal{A}_{1}}-\int_{\Omega}\zeta_{2}(\omega)\text{ESG}(\omega)P(d\omega)=\boldsymbol{\rho}_{1}([0,\;\text{ESG}]^{\prime}).$

∎

Proposition 2 states that the risk for an investor with $\lambda=0$ is not
affected by the ESG score of the asset, while the risk for an investor with
$\lambda=1$ is not affected by the monetary returns.

###  3.2 Hedging risk by investing in ESG safe assets

To provide a more complete interpretation of ESG-coherent risk measures, it is
useful to study how ESG-oriented investors can hedge a risky position by
investing in an ESG safe asset. In a traditional univariate framework, a safe
asset is an asset whose payoff is deterministic;131313 Note that a position
can have a risk equal to zero and not be a safe asset. Similarly, an asset
with a deterministic payoff (i.e., a safe asset) can have a risk that is
different from zero. We can understand this point better by considering a
position with a return $r$ and risk $\rho(r)=m$. If $\rho(\cdot)$ is a
coherent risk measure, by axiom TI we have $\rho(r+m)=0$. That is, a position
with a return $r^{\prime}=r+m$ has zero risk (but its returns are not
necessarily constant). its return is a constant $r_{f}$. If we define the risk
on a random variable that represents the returns, axioms TI and PO provide
that the risk of a portfolio composed of the safe asset and a risky position
$r$ is

$\rho((1-w)r+wr_{f})=(1-w)\rho(r)-wr_{f},$ (5)

where $w\in[0,1]$ is the weight of the risky position in the portfolio. More
formally, we address the problem of an investor who is willing to reduce the
risk of a position to an acceptable level $\kappa$ by creating a portfolio
consisting of the risky position, and of the smallest possible amount of the
safe asset. The motivation is, for instance, to satisfy requirements imposed
by regulators or by the institutional mandate. Formally the problem is

$\displaystyle w^{*}=\arg\min_{w}$ $\displaystyle(w)$ s.t.
$\displaystyle\rho((1-w)X+wr_{f})\leq\kappa$ $\displaystyle 0\leq w\leq 1.$
(6)

We assume that $-r_{f}<\kappa\leq\rho(X)$. Since the risk of a portfolio is an
affine function of $w$ as shown in (6), the solution is:

$w^{*}=\dfrac{\rho(X)-\kappa}{\rho(X)+r_{f}}.$ (7)

That is, the risky position can be hedged by constructing a portfolio that
contains the safe asset having weight $w^{*}$.141414 If the risk measure were
defined in terms of final net worth rather than returns, to hedge a risky
position with risk $m$, it would be necessary to add a cash position. For a
broader discussion of the interpretation of the axioms expressed in terms of
returns rather than the final net worth, see Rachev et al., (2011, Chapter 6).

For ESG-oriented investing, an ESG safe asset is a position having constant
values for both monetary and ESG components. We can postulate the existence of
several types of such ESG safe assets, distinguished by different combinations
of constant ESG and monetary return.

Consider first the case in which only one type of ESG safe asset is available
in the market:

$\text{SA}:=\begin{bmatrix}r_{f}\\\ r_{\text{ESG}}\end{bmatrix}.$

We know that by axiom LH-M, for an ESG-coherent risk measure, the risk of this
ESG safe asset is
$\boldsymbol{\rho}_{\lambda}(\text{SA})=-((1-\lambda)r_{f}+\lambda
r_{\text{ESG}})$. The problem of hedging the risk of an asset $X$ is analogous
to the univariate case: an investor with a given $\lambda$ wants to construct
a portfolio with ESG-risk smaller or equal than $\kappa$ by adding to $X$ the
smallest amount of the ESG safe asset SA (i.e. minimizing its weight in the
portfolio):

$\displaystyle w_{\lambda}^{*}=\arg\min_{w}$ $\displaystyle(w)$ s.t.
$\displaystyle\boldsymbol{\rho}_{\lambda}((1-w)X+w\text{SA})\leq\kappa$
$\displaystyle 0\leq w\leq 1.$ (8)

We assume again that $-(1-\lambda)r_{f}-\lambda
r_{\text{ESG}}<\kappa\leq\rho(X).$ The solution is

$w_{\lambda}^{*}=\dfrac{\boldsymbol{\rho}_{\lambda}(X)-\kappa}{(1-\lambda)r_{f}+\lambda
r_{\text{ESG}}+\boldsymbol{\rho}_{\lambda}(X)}.$ (9)

$w_{\lambda}^{*}$ is unique and its value varies with the investor preference
$\lambda$. In practice, such an ESG safe asset could be achieved by the
investor making a guaranteed loan to an institution (either a for-profit
company, a government, or a non-profit institution) that has a positive and
stable environmental or social impact, which generates an interest $r_{f}$ for
the investor.

To provide further intuitions, we discuss three special cases characterized by
specific ESG safe assets. With the exception of case 3, we assume that the ESG
safe assets have non-negative monetary return and ESG.

  1. 1.

The only ESG safe asset is a pure monetary safe asset described by

$\text{SA}_{\text{CASH}}:=\begin{bmatrix}r_{f}\\\ 0\end{bmatrix},$

where $r_{f}\geq 0$. An example of this is a risk-free, zero-coupon bond
issued by a governmental institution not associated with a specific ESG
profile.151515 Rating agencies are starting to compute ESG scores for
countries as well, although the criteria are different from those used to
calculate companies’ scores. The identification of pure monetary and pure ESG
safe assets will be a significant challenge for practitioners and scholars. An
ESG investor can hedge the risk of a position $X$ by constructing the
portfolio $(1-w_{\lambda}^{*})X+w_{\lambda}^{*}\text{SA}_{\text{CASH}}$ with

$w^{*}_{\lambda}=\dfrac{\boldsymbol{\rho}_{\lambda}(X)-\kappa}{(1-\lambda)r_{f}+\boldsymbol{\rho}_{\lambda}(X)}.$

We note that if $\lambda=1$ (i.e., an investor cares exclusively about the ESG
component) $\boldsymbol{\rho}_{1}(\text{SA}_{\text{CASH}})=0$.

  2. 2.

The only ESG safe asset is a pure ESG safe asset described by

$\text{SA}_{\text{ESG}}:=\begin{bmatrix}0\\\ r_{\text{ESG}}\end{bmatrix}.$

The analysis of this case is symmetrical to that for the pure monetary safe
asset.

  3. 3.

There exists an ESG safe asset having a monetary return of –100%. We consider
the special case of an ESG safe asset described by

$\text{SA}_{CHARITY}:=\begin{bmatrix}-1\\\ r_{\text{ESG}}\end{bmatrix}.$

An example of this is a donation to a non-profit organization that has a
positive and constant ESG score. Such an asset produces a monetary return of
$-100\%$, and clearly it is not a relevant investment opportunity for a risk-
return investor with $\lambda=0$.161616Assuming the absence of tax benefits
associated to the donation, or other forms of monetary gains. On the contrary,
for an ESG-oriented investor with $\lambda>0$ it could be rational to invest
in such asset. The measured risk of such an asset is
$\boldsymbol{\rho}(\text{SA})=(1-\lambda)-\lambda r_{\text{ESG}}$; That is,
for an investor with a given $\lambda$ the risk is negative if
$\lambda>\tilde{\lambda}=1/(1+r_{\text{ESG}})$. For a
“$\lambda<\tilde{\lambda}$” investor, such an asset provides no opportunity to
hedge a risky position $X$; however for a “$\lambda>\tilde{\lambda}$”
investor, such an ESG safe asset provides a meaningful hedging tool through
the donation of a wealth fraction:

$w^{*}_{\lambda}=\dfrac{\boldsymbol{\rho}_{\lambda}(X)-\kappa}{\lambda-1+\lambda
r_{\text{ESG}}+\boldsymbol{\rho}_{\lambda}(X)}$

to a project with a positive environmental or social impact.

We can also study the case with multiple ESG safe assets available in the
market. If the ESG safe assets $\text{SA}_{i},\;i=1,\dots,n$, are available,
an investor can hedge a risky position $X$ by creating a portfolio
$w_{0}X+w_{1}\text{SA}_{1}+w_{2}\text{SA}_{2}+\dots+w_{n}\text{SA}_{n}$ with
$\sum_{i=0}^{n}w_{i}=1$, so that the risk is equal to $\kappa$. We can
formulate an optimization problem analogous to 8, with the difference that the
objective function to maximize is the weight of the risky asset $w_{0}$ (that
is equivalent to minimizing the sum of the portfolio weights invested in ESG
safe assets). Formally:

$\displaystyle\max_{w_{0},w_{1},w_{2},\dots,w_{n}}$ $\displaystyle(w_{0})$
s.t.
$\displaystyle\boldsymbol{\rho}_{\lambda}(w_{0}X+w_{1}\text{SA}_{1}+w_{2}\text{SA}_{2}+\dots+w_{n}\text{SA}_{n}){\leq\kappa,}$
$\displaystyle\sum_{i=1}^{n+1}w_{i}=1,\quad 0\leq w_{i}\leq 1,\quad
i=1,\dots,n+1.$ (10)

The constraint set defines a convex feasible subset of $\mathbb{R}^{n}$ for
the investment allocations, $w_{i}$. The solution is not a diversified
portfolio; only one ESG safe asset is selected. This follows from the fact
that, due to LH-M and TI-M, the risk of a sum of ESG safe assets is the sum of
their risk:
$\boldsymbol{\rho}(\delta\text{SA}_{i}+(1-\delta)\text{SA}_{j})=\delta\boldsymbol{\rho}(\text{SA}_{i})+(1-\delta)\boldsymbol{\rho}(\text{SA}_{j}),\;\delta\in[0,1]$.
For any couples of ESG safe assets $\text{SA}_{i}$, $\text{SA}_{j}$ the risk
of their convex combination is always greater or equal than
$\min(\boldsymbol{\rho}(\text{SA}_{i}),\boldsymbol{\rho}(\text{SA}_{j})$.
Hence to minimize the risk it is always convenient to choose the ESG safe
asset with the smallest risk.171717 An exception is when two or more ESG safe
assets with exactly the same risk are available. In such a case, they are
indistinguishable to the investor in terms of risk, and either can be chosen.
Once the ESG safe asset with the lowest risk is identified, the problem then
is the same as (8) where only one ESG safe asset was available. Note however
that the choice of ESG safe asset is different for each investor, as the
optimization is dependence on the investor’s value for $\lambda$.

######  Remark.

In general, the choice of the ESG safe asset is not influenced by the
characteristics of the risky position, it is only influenced by the
availability and price of ESG safe assets and the $\lambda$ preference of the
investor.

###  3.3 Examples of ESG-coherent risk measures

After discussing ESG-coherent risk measures in general, we present two
approaches for extending univariate risk measures to a bivariate setting. In
particular, starting from a univariate coherent risk measure $\rho(r)$, we
identify the ESG-coherent risk measures $\text{ESG-}\boldsymbol{\rho}(X)$, and
$\text{ESG-}\boldsymbol{\rho}^{l}(X)$. The measures encompass
$\lambda\in[0,1]$ as a parameter, thus they are families of bivariate risk
measures suitable for investors having differing ESG “inclinations”. In the
following subsection we apply these approaches to the well-known coherent risk
measure, average value at risk (AVaR), resulting in ESG-coherent risk
measures. In the subsequent subsection we apply these approaches to two non-
coherent risk measures, variance and volatility, producing ESG extensions that
are not ESG-coherent.

Our first approach to generalizing a univariate risk measure $\rho(r)$
utilizes a linear combination of $r$ and ESG. For $X=\left[\begin{matrix}r\\\
\text{ESG}\end{matrix}\right]$

$\displaystyle\text{ESG-}\boldsymbol{\rho}_{\lambda}\left(X\right)$
$\displaystyle:=\rho\big{(}(1-\lambda)r+\lambda\text{ESG}\big{)}.$ (11)

######  Proposition 3.

If $\rho(\cdot)$ is a (univariate) coherent risk measure, then
$\text{ESG-}\boldsymbol{\rho}_{\lambda}(\cdot)$ is an ESG-coherent risk
measure.

###### Proof.

Since the right-hand-side of (11) involves a direct application of
$\rho(\cdot)$, the extended function
$\text{ESG-}\boldsymbol{\rho}_{\lambda}(\cdot)$ inherits the properties SUB-M
and PH-M. It is straightforward to show that axiom MO of the univariate
function implies MO-M. Consider two vectors $X_{1}$,
$X_{2}\in\mathcal{X}_{2}$. If $(r_{1}\leq
r_{2}\wedge\text{ESG}_{1}\leq\text{ESG}_{2})$ a.s., then from the monotonicity
of $\rho(\cdot)$, we have

$\text{ESG-}\boldsymbol{\rho}_{\lambda}(X_{1})=\rho((1-\lambda)r_{1}+\lambda\text{ESG}_{1})\geq\rho((1-\lambda)r_{2}+\lambda\text{ESG}_{2})=\text{ESG-}\boldsymbol{\rho}_{\lambda}(X_{2}).$

Finally, using the translation invariance of $\rho(\cdot)$, we can prove TI-M
and LH-M. ∎

We note the following properties of
$\text{ESG-}\boldsymbol{\rho}_{\lambda}(\cdot)$.

  * •

If $\rho(r)$ is convex, then
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$ is a convex
function of $\lambda$.

  * •

If $\rho(r)$ is a coherent risk measure and if the ESG scores are constant,
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$ is an affine
transformation of $\rho(r)$ computed on returns only.

  * •

For $\lambda=0$ and $\lambda=1$,
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$ is equal to
the univariate $\rho(r)$ computed on returns alone or $\rho(\text{ESG})$
computed on ESG scores alone, respectively.

  * •

Suppose that $r$ and ESG have the same marginal distributions. Then
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$ is at its
maximum for perfect positive co-monotonicity between $r$ and ESG, and it is at
its minimum for perfect negative co-monotonicity.

Our second approach utilizes a linear combination of univariate risk measures,

$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}\left(\left[\begin{matrix}r\\\
\text{ESG}\end{matrix}\right]\right):=(1-\lambda)\rho(r)+\lambda\rho(\text{ESG}).$
(12)

It is straightforward to show that
$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}(\cdot)$ is an ESG-coherent risk
measure if $\rho(\cdot)$ is coherent because axioms SUB-M, PH-M, TI-M, MO-M,
and LH-M follow from the respective univariate axioms.

We note the following properties of
$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}(\cdot)$.

  * •

The measure
$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}([r,\text{ESG}]^{\prime})$ is more
conservative than
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$ as it is
linear in $\lambda$ and, hence, always greater than or equal to
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$.

  * •

$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}([r,\text{ESG}]^{\prime})$ is
equivalent to
$\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$ for the case
of perfect co-monotonicity between $r$ and ESG (i.e. when no diversification
between the two is possible). In this sense, we can consider it a worst-case
measure of $\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$.

  * •

For the limiting cases $\lambda=0$ and $\lambda=1$, we have
$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}([r,\text{ESG}]^{\prime})=\text{ESG-}\boldsymbol{\rho}_{\lambda}([r,\text{ESG}]^{\prime})$.

  * •

For $\lambda=0$ or $\lambda=1$,
$\text{ESG-}\boldsymbol{\rho}^{l}_{\lambda}([r,\text{ESG}]^{\prime})$
corresponds to the univariate risk measure $\rho(X)$ computed on just the
monetary part or just the ESG part, respectively.

####  3.3.1 ESG-AVaR

We demonstrate the two approaches presented in equations (11) and (12) to
develop ESG-coherent risk measures based on AVaR. Given a random bivariate
vector
$X=[r,\;\text{ESG}]^{\prime}\in\mathcal{X}_{2}=\mathcal{L}_{1}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$,
the first measure, given by (11), is181818 See Ogryczak and Ruszczynski,
(2002); Rockafellar and Uryasev, (2002) for the extremal representation.

$\displaystyle\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}\left(\left[\begin{matrix}r\\\
\text{ESG}\end{matrix}\right]\right)$
$\displaystyle:=\text{AVaR}_{\tau}\left((1-\lambda)r+\lambda\text{ESG}\right)$
$\displaystyle=\inf_{\beta\in\mathbb{R}}\left\\{\frac{1}{1-\tau}\mathbb{E}\left[\left(\beta-((1-\lambda)r+\lambda\text{ESG})\right)^{+}\right]-\beta\right\\},$
(13)

where $(a)^{+}$ denotes max$(a,0)$. As discussed above,
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(\cdot)$ is ESG-coherent. It is
similar to the multivariate expected shortfall presented by Ekeland et al.,
(2012) (although their measure lacks the parametrization using $\lambda$).
Since $\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(\cdot)$ is computed on
univariate data (i.e., as a linear combination of $r$ and ESG), numerical
applications using $\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(\cdot)$ do
not present any particular challenge; it is possible to fully utilize existing
procedures developed for AVaR for risk estimation, portfolio optimization, and
risk management. (See, e.g. Shapiro et al.,, 2021.)

The dual representation of
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(\cdot)$ is

$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}\left(\left[\begin{matrix}r\\\
\text{ESG}\end{matrix}\right]\right)=\sup_{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(X)}}\left(-\mathbb{E}[r\zeta_{1}+\text{ESG}\zeta_{2}]\right),$
(14)

where

$\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}}=\left\\{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2}):\;[\zeta_{1},\;\zeta_{2}]^{\prime}=\xi[1-\lambda,\;\lambda]^{\prime};\;0\leq\xi\leq\frac{1}{1-\tau}\;\text{a.s.
}\mathbb{E}[\xi]=1\right\\}.$ (15)

The derivation of this dual representation is given in Appendix C.

The second measure, given by (12), is

$\displaystyle\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}\left(\left[\begin{matrix}r\\\
\text{ESG}\end{matrix}\right]\right)$
$\displaystyle:=(1-\lambda)\text{AVaR}_{\tau}(r)+\lambda\text{AVaR}_{\tau}(\text{ESG}).$
(16)

$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(\cdot)$ is also an ESG-
coherent. It can be viewed as the limit of
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(\cdot)$ in the case of
comonotonicity; i.e., an asset for which it is not possible to diversify
between the monetary and ESG components as they are comonotone. From an
economic perspective,
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(\cdot)$ is significant for
investors who consider the worst-case scenario in terms of the dependency
structure.

The dual representation of
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(X)$ is

$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}\left(\left[\begin{matrix}r\\\
\text{ESG}\end{matrix}\right]\right)=\sup_{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(X)}}-\mathbb{E}[r\zeta_{1}+\text{ESG}\zeta_{2}],$
(17)

where

$\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}}=\left\\{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2}):\mathbb{E}[\zeta_{1}]=1-\lambda;\mathbb{E}[\zeta_{2}]=\lambda;\zeta_{1},\zeta_{2}\geq
0;\zeta_{1}\leq\frac{1-\lambda}{1-\tau};\zeta_{2}\leq\frac{\lambda}{1-\tau}\right\\}.$
(18)

The derivation of the dual representation of
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(\cdot)$ is also given in
Appendix C.

####  3.3.2 Non-ESG-coherent measure examples

It is well known that the standard deviation $\sigma$ (the volatility) and the
variance $\mathbb{V}$ are not coherent risk measures. We consider the
application of (11) and (12) to $\sigma$ and $\mathbb{V}$ and show that, in
all cases, the result is an ESG measure that does not satisfy the ESG-
coherency axioms.

Given a vector
$X=[r,\;\text{ESG}]^{\prime}\in\mathcal{X}_{2}=\mathcal{L}_{2}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$,
from (11) the ESG variance and ESG volatility are

$\displaystyle\text{ESG-}\mathbb{V}_{\lambda}(X)$
$\displaystyle:=\mathbb{V}[(1-\lambda)r+\lambda\text{ESG}],$ (19)
$\displaystyle\text{ESG-}\sigma_{\lambda}(X)$
$\displaystyle:=\sqrt{\text{ESG-}\mathbb{V}_{\lambda}(X)}.$ (20)

$\text{ESG-}\mathbb{V}_{\lambda}(\cdot)$ is not ESG-coherent, as it does not
satisfy PH-M, MO-M, SUB-M, and LH-M. $\text{ESG-}\sigma_{\lambda}(\cdot)$ is
not ESG-coherent, as it does not satisfy MO-M and LH-M.

Using (12), the corresponding risk measures are

$\displaystyle\text{ESG-}\mathbb{V}^{l}_{\lambda}(X)$
$\displaystyle:=(1-\lambda)\mathbb{V}[r]+\lambda\mathbb{V}[\text{ESG}],$ (21)
$\displaystyle\text{ESG-}\sigma^{l}_{\lambda}(X)$
$\displaystyle:=\sqrt{\text{ESG-}\mathbb{V}^{l}_{\lambda}(X)}.$ (22)

The former does not satisfy PH-M, MO-M, SUB-M, and LH-M, and the latter does
not satisfy MO-M and LH-M.

A summary of the properties of the examples considered in sections 3.3.1 and
3.3.2 is given in Table 1.

Table 1: Summary of the properties satisfied by ESG risk measures. Risk measure | SUB-M | PH-M | TI-M | MO-M | LH-M
---|---|---|---|---|---
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$\text{ESG-}\mathbb{V}_{\lambda}$ |  |  | $\checkmark$ |  |
$\text{ESG-}\sigma_{\lambda}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |
$\text{ESG-}\mathbb{V}^{l}_{\lambda}$ |  |  | $\checkmark$ |  |
$\text{ESG-}\sigma^{l}_{\lambda}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  |

##  4 ESG reward–risk ratios

It is natural to extend the ESG framework to reward–risk ratios (RRRs), used
to measure risk-adjusted performance of an investment. Following Cheridito and
Kromer, (2013), a reward–risk ratio $\alpha(r)$ in a univariate setting is

$\alpha(r):=\dfrac{\theta(r)^{+}}{\rho(r)^{+}},$ (23)

where $\theta(r):\mathcal{X}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$ and
$\rho(r):\mathcal{X}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ are risk and
reward measures, respectively. Cheridito and Kromer, (2013) identified four
conditions desirable for RRRs:

  * (MO-R)

monotonicity: if $r_{1},r_{2}\in\mathcal{X}$ and $r_{1}\leq r_{2}$ a.s., then
$\alpha(r_{1})\geq\alpha(r_{2})$;

  * (QC-R)

quasi-concavity: if $r_{1},r_{2}\in\mathcal{X}$ and $\delta\in[0,1]$, then
$\alpha(\delta r_{1}+(1-\delta)r_{2})\geq\min(\alpha(r_{1}),\alpha(r_{2}))$;

  * (SI-R)

scale invariance: if $r\in\mathcal{X}$ and $\delta>0$ s.t. $\delta
r\in\mathcal{X}$, then $\alpha(\delta r)=\alpha(r)$;

  * (DB-R)

distribution-based: $\alpha(r)$ only depends on the distribution of $r$ under
$\mathbb{P}$.

Following the approach used for risk and reward measures, we introduce ESG
reward–risk ratios (ESG-RRRs). Let $X=[r,\;\text{ESG}]^{\prime}$. We define an
ESG-RRR
$\boldsymbol{\alpha}_{\lambda}:\mathcal{X}_{2}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$
by

$\boldsymbol{\alpha}_{\lambda}(X):=\dfrac{\boldsymbol{\theta}_{\lambda}(X)^{+}}{\boldsymbol{\rho}_{\lambda}(X)^{+}},$
(24)

where $\boldsymbol{\theta}_{\lambda}(X)$ and $\boldsymbol{\rho}_{\lambda}(X)$
are ESG reward and risk measures as defined in Section 3 and Appendix A.191919
In principle, it is possible for the risk and reward measures to have
different values of $\lambda$. We consider the case of values of lambda for
both the numerator and the denominator for conciseness. The extension of the
Cheridito-Kromer conditions for ESG-RRRs are:

  * (MO-RM)

monotonicity: if $X_{1},X_{2}\in\mathcal{X}_{2}$ and ($r_{1}\leq
r_{2}\wedge\text{ESG}_{1}\leq\text{ESG}_{2}$) a.s., then
$\boldsymbol{\alpha}_{\lambda}(X_{1})\leq\boldsymbol{\alpha}_{\lambda}(X_{2})$;

  * (QC-RM)

quasi-concavity: if $X_{1},X_{2}\in\mathcal{X}_{2}$ and $\delta\in[0,1]$, then
$\boldsymbol{\alpha}_{\lambda}(\delta
X_{1}+(1-\delta)X_{2})\geq\min(\boldsymbol{\alpha}_{\lambda}(X_{1}),\boldsymbol{\alpha}_{\lambda}(X_{2}))$;

  * (SI-RM)

scale invariance: if $X\in\mathcal{X}_{2}$ and $\delta>0$ s.t. $\delta
X\in\mathcal{X}_{2}$, then $\boldsymbol{\alpha}_{\lambda}(\delta
X)=\boldsymbol{\alpha}_{\lambda}(X)$;

  * (DB-RM)

distribution-based: $\boldsymbol{\alpha}_{\lambda}(X)$ only depends on the
distribution of $X$ under $\mathbb{P}$.

Verification of conditions QC-RM and SI-RM is done on the domain of
$\boldsymbol{\alpha}_{\lambda}(X)$. Verification of DB-RM depends on the
bivariate distribution of $X\in\mathcal{X}_{2}$ and not on the univariate
distribution of the returns. As in the case of the risk measure axiom MO-M,
verification of MO-RM requires the use of partial ordering.

######  Proposition 4.

The ESG-RRR (24), where $\boldsymbol{\theta}_{\lambda}(X)$ is an ESG-coherent
risk measure and $\boldsymbol{\rho}_{\lambda}(X)$ is an ESG-coherent reward
measure (Appendix A) satisfies conditions MO-RM, QC-RM and SI-RM.

###### Proof.

MO-RM follows from the monotonicity of $\boldsymbol{\rho}_{\lambda}(X)$ and
$\boldsymbol{\theta}_{\lambda}(X)$. QC-RM follows from the convexity of
$\boldsymbol{\rho}_{\lambda}(X)$ and the concavity of
$\boldsymbol{\theta}_{\lambda}(X)$. SI-RM follows from the corresponding
properties of $\boldsymbol{\rho}_{\lambda}(X)$ and
$\boldsymbol{\theta}_{\lambda}(X)$. ∎

######  Remark.

In general, risk and reward measures may not depend on the distribution of
returns and ESG under a single probability measure, as in the case of robust
reward–risk ratios, which take into account the fact that agents do not know
with certainty the distribution of the random variables (Cheridito and
Kromer,, 2013).

###  4.1 Examples of ESG reward–risk ratios

We present six examples of ESG reward–risk ratios derived from RRRs commonly
used in the literature. The ratios are obtained by generalizing the univariate
reward–risk ratios using the approach described by (11). Note that, as in the
case of risk measures, the definition of a reward–risk ratio can be based on
several alternative specifications of the random variable
$X=[r,\;\text{ESG}]^{\prime}$. In particular, the ratios can be computed using
the rate of returns, the excess returns over a risk-free rate, the final
wealth, profit and/or losses, etc. The same logic applies to the ESG
component. Here, we only provide hints concerning which approach to use in
practice: the choice depends on the specific needs of the practitioner or
regulator who uses these measures.

  * •

ESG Sharpe ratio. The Sharpe ratio is the ratio between the excess return of
an asset and its standard deviation over a period of time. The ESG Sharpe
ratio is

$\text{ESG-}\text{SR}_{\lambda}(X):=\dfrac{\text{ESG-}\mu_{\lambda}(X)-\text{SA}}{\text{ESG-}\sigma_{\lambda}(X)}.$
(25)

where

$\text{ESG-}\mu_{\lambda}(X):=(1-\lambda)\mathbb{E}[r]+\lambda\mathbb{E}[\text{ESG}],$
(26)

the ESG standard deviation $\text{ESG-}\sigma_{\lambda}(X)$ is given by (20),
and $\text{SA}=[r_{f},\;r_{\text{ESG}}]^{\prime}$ is the vector of return and
ESG of an ESG safe asset. $\text{ESG-}\mu_{\lambda}$ is an ESG-coherent reward
measure, while $\text{ESG-}\sigma_{\lambda}$ is not an ESG-coherent risk
measure. The ESG Sharpe ratio satisfies QC-RM, SI-RM, and DB-RM, but it does
not satisfy condition MO-RM.202020 The SI-RM property applies if
$\text{SA}=[0,\;0]^{\prime}$ or if $X$ is intended to represent a vector of
excess returns over the risk-free rate.

  * •

ESG Rachev ratio. The univariate Rachev ratio is defined as (Biglova et al.,,
2004)

$\text{RR}_{\beta,\gamma}(r):=\dfrac{\text{AVaR}_{\beta}(-r)}{\text{AVaR}_{\gamma}(r)}.$
(27)

We generalize to an ESG-RR by replacing AVaR with ESG-AVaR. Let
$X\in\mathcal{X}_{2}$. Then,

$\text{ESG-}\text{RR}_{\beta,\gamma,\lambda}(X):=\dfrac{\text{ESG-}\text{AVaR}_{\lambda,\beta}(-X)}{\text{ESG-}\text{AVaR}_{\lambda,\gamma}(X)}.$
(28)

Note that ESG-RR satisfies MO-RM, SI-RM and DB-RM but fails to satisfy QC-RM
since the numerator is not concave.

  * •

ESG STAR ratio. When $\beta=0$, the ESG Rachev ratio becomes an ESG
generalization (ESG-STARR) of the stable tail-adjusted return ratio,

$\text{ESG-}\text{STARR}_{\alpha,\lambda}(X):=\dfrac{\text{ESG-}\mu_{\lambda}(X)}{\text{ESG-}\text{AVaR}_{\lambda,\alpha}(X)}.$
(29)

As a special case of the ESG Rachev ratio, ESG-STARR satisfies conditions MO-
RM, SI-RM and DB-RM; as the numerator is linear, it also satisfies QC-RM.

  * •

ESG Sortino–Satchell ratio. The univariate Sortino–Satchell ratio (Sortino and
Satchell,, 2001) is defined as $\mathbb{E}[r]^{+}/||r^{-}||_{p}$, where
$||r||_{p}=\left(\int_{-\infty}^{\infty}|x|^{p}f_{r}(x)dx\right)^{1/p}$ and
$r\sim f_{r}(x)$. We extend this measure to the bivariate ESG setting by212121
The definition provided here assumes a target return of 0, similarly to
Cheridito and Kromer, (2013).

$\text{ESG-}\text{SS}_{\lambda}(X)=\dfrac{\left(\text{ESG-}\mu_{\lambda}(X)\right)^{+}}{||Y^{-}||_{p}},$
(30)

where $Y=(1-\lambda)r+\lambda\text{ESG}$ and $Y\sim f(y)$. This measure
satisfies all four ESG-RRR conditions.

The proposed formulation assumes a required rate of return and ESG target of
zero. We can introduce a non-zero target by subtracting a bivariate vector
(e.g. the return of an ESG safe asset
$\text{SA}=[r_{f},\;r_{\text{ESG}}]^{\prime}$) from the numerator before
applying the positive operator and using
$\tilde{Y}=(1-\lambda)(r-r_{f})+\lambda(\text{ESG}-r_{\text{ESG}})$ in the
denominator in place of $Y$. In such a case, this measure no longer satisfies
SI-RM.

  * •

ESG Omega ratio. Defining $Y=(1-\lambda)r+\lambda\text{ESG}$, where $F(y)$ is
the cumulative distribution function of $Y$, the ESG version of the Omega
ratio (Keating and Shadwick,, 2002) is

$\text{ESG-}\Omega_{\lambda}(X)=\dfrac{\int_{\tau}^{\infty}[1-F(y)]dy^{+}}{\int_{-\infty}^{\tau}F(y)dy}.$
(31)

Equivalently (see Farinelli and Tibiletti,, 2008),

$\text{ESG-}\Omega_{\lambda}(X)=\dfrac{\mathbb{E}[(Y-\tau)^{+}]}{\mathbb{E}[(Y-\tau)^{-}]}.$
(32)

This measure satisfies MO-RM and DB-RM. It does not satisfy QC-RM, and SI-RM
is only satisfied if $\tau=0$.

  * •

ESG Farinelli–Tibiletti ratio. This ratio aims to take into account the
asymmetry in the return distribution; it generalizes the Omega ratio. By
defining $Y=(1-\lambda)r+\lambda\text{ESG}$, the ESG version of the ratio can
be defined as

$\text{ESG-}\text{FTR}_{\lambda,m,n,p,q}(X)=\dfrac{||(Y-m)^{+}||_{p}}{||(n-Y)^{+}||_{q}},$
(33)

where $m,n\in\mathbb{R}$ and $p,q>0$. This ratio satisfies MO-RM and DB-RM. It
also satisfies SI-RM if $n=m=0$. It does not satisfy QC-RM (see the example in
Cheridito and Kromer,, 2013, Section 4.3).

Table 2 summarizes the properties of these six ESG-RRRs.

Table 2: Summary of the properties of ESG reward–risk ratios.  Ratio | MO-RM | QC-RM | SI-RM | DB-RM
---|---|---|---|---
ESG Sharpe |  | $\checkmark$ | $\checkmark$ | $\checkmark$
ESG Rachev | $\checkmark$ |  | $\checkmark$ | $\checkmark$
ESG STAR | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
ESG Sortino–Satchell | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
ESG Omega | $\checkmark$ |  | $\checkmark$ | $\checkmark$
ESG Farinelli–Tibiletti | $\checkmark$ |  | $\checkmark$ | $\checkmark$

##  5 Empirical analysis

We present an empirical exercise in which we estimate a set of ESG risk
measures and ESG reward–risk ratios. We use such measures to rank equity
assets from the Dow-Jones Industrial Average (DJIA) index, assessing the role
of lambda for different measures.

First, we describe the dataset. We consider 28 constituents of the Dow-Jones
Industrial Average index in the period from January 3, 2006 to October 29,
2021 (almost 15 years). To match the frequencies of the two datasets, we make
the assumption that the ESG scores are constant within each year. For this
period, daily returns and yearly ESG scores are available. The annual score is
normalized between -1 and 1. Hence, for each day, we obtain daily ESG values
by dividing the annual value by the number of trading days in a year (252)
(this is consistent with assuming a constant sustainability flow for the
entire year, as discussed in Section 2). As shown in Figure 1, the series of
ESG scores show an increasing trend over time, as well as a reduction in the
cross-sectional variability, with more homogeneous scores across companies in
recent years. The financial time series show the well-known properties of
equity time series, such as limited serial correlations, and volatility
clustering for the returns. In the analysis we consider the historical time
series and we do not address the modeling of the data. Concerning the relation
between the ESG score and the financial performance, the data does not
highlight any clear relation. Panels C and D plot the average ESG score across
the entire period in relation to the average returns and the return
volatility, respectively. Higher ESG scores seem to correlate slightly with
lower returns, but the relation does not appear to be strong (correlation:
$-0.16$). We see a slightly positive correlation between ESG scores and
volatility (correlation: $0.09$).

Since the characteristics of the time series evolve over time, we focus the
analysis on a narrower period of four years: October 30, 2017 to October 29,
2021. Table 3 reports the names of the companies, the properties of the
distribution of daily returns, and the statistics of the annual ESG scores for
the period from October 30, 2017 to October 29, 2021. The dataset presents a
wide cross-sectional variability in terms of the average return and risk, as
well as in terms of the average ESG scores, despite the fact that the data
showed a certain degree of convergence over the last few years, with almost no
negative scores appearing in the sample after 2017. Since the ESG data are
annual and are assumed to be constant within each year, they show a very small
volatility compared to the returns. This is partially a consequence of the
different sampling frequency of the dataset (annual vs. daily), but it also
reflects the fact that the changes in the sustainability profile of a company
happen at a much slower pace than the shifts seen in the monetary value of a
stock. The scales of the average returns and average ESG scores are broadly
comparable.

Figure 1: Panel A: Evolution of the ESG score of the constituents of the index. The thicker blue line shows the evolution of the average return. Panel B: Evolution of prices for the stocks. The thick blue line is the value of a daily rebalanced equally weighted portfolio of all the stocks (to improve readability, the price has been normalized to 1 at the beginning of the considered time period, and the upper limit of the $Y$ axis has been set to 20, cutting off some stocks that showed extremely high growth in the observed period). Panel C: Scatterplot of average daily returns and the average ESG score for each asset. Panel D: Scatterplot of the daily standard deviation of returns and the average ESG score for each asset. Name | GICS Sector Name | Avg. return | St. dev. returns | Avg. ESG | St. dev. ESG
---|---|---|---|---|---
3M Co. | Industrials | $-0.031$ | $0.271$ | $0.775$ | $0.002$
American Express Co. | Financials | $0.218$ | $0.374$ | $0.502$ | $0.004$
Amgen Inc. | Health Care | $0.077$ | $0.266$ | $0.466$ | $0.002$
Apple Inc. | IT | $0.379$ | $0.326$ | $0.386$ | $0.003$
Boeing Co. | Industrials | $0.077$ | $0.512$ | $0.594$ | $0.003$
Caterpillar Inc. | Industrials | $0.153$ | $0.332$ | $0.375$ | $0.003$
Chevron Corp. | Energy | $0.068$ | $0.362$ | $0.668$ | $0.004$
Cisco Systems Inc. | IT | $0.164$ | $0.290$ | $0.759$ | $0.001$
Coca-Cola Co. | Consumer Staples | $0.075$ | $0.219$ | $0.430$ | $0.006$
Goldman Sachs Group Inc. | Financials | $0.191$ | $0.336$ | $0.552$ | $0.010$
Home Depot Inc. | Consumer Discretionary | $0.239$ | $0.278$ | $0.515$ | $0.005$
Honeywell International Inc. | Industrials | $0.150$ | $0.276$ | $0.452$ | $0.004$
IBM Corp. | IT | $-0.011$ | $0.284$ | $0.506$ | $0.005$
Intel Corp. | IT | $0.097$ | $0.378$ | $0.758$ | $0.001$
Johnson & Johnson | Health Care | $0.058$ | $0.215$ | $0.779$ | $0.002$
JPMorgan Chase & Co. | Financials | $0.181$ | $0.326$ | $0.615$ | $0.003$
McDonald’s Corp. | Consumer Discretionary | $0.130$ | $0.249$ | $0.430$ | $0.002$
Merck & Co. Inc. | Health Care | $0.143$ | $0.234$ | $0.595$ | $0.002$
Microsoft Corp. | IT | $0.388$ | $0.297$ | $0.846$ | $0.001$
Nike Inc. | Consumer Discretionary | $0.320$ | $0.305$ | $0.423$ | $0.002$
Procter & Gamble Co. | Consumer Staples | $0.148$ | $0.218$ | $0.377$ | $0.004$
Salesforce Inc. | IT | $0.337$ | $0.360$ | $0.313$ | $0.006$
Travelers Companies Inc. | Financials | $0.095$ | $0.293$ | $0.195$ | $0.014$
UnitedHealth Group Inc. | Health Care | $0.241$ | $0.309$ | $0.509$ | $0.005$
Verizon Communications Inc. | Communication Services | $0.039$ | $0.195$ | $0.410$ | $0.006$
WBA Inc. | Consumer Staples | $-0.021$ | $0.340$ | $0.586$ | $0.011$
Walmart Inc. | Consumer Staples | $0.158$ | $0.231$ | $0.578$ | $0.005$
Walt Disney Co. | Communication Services | $0.184$ | $0.313$ | $0.451$ | $0.006$
Table 3: Annualized mean and standard deviation of daily returns and ESG
scores.

Figure 2: Visual representation of the ranking of the assets in the DJIA index
based on $\text{ESG-}\text{AVaR}_{\lambda,\tau}$ ($\tau$ equal to 0.95 and
0.99), $\text{ESG-}\mu_{\lambda}$, the ESG Rachev ratio ($\tau$ equal to 0.95
and 0.99), the standard deviation, $\text{ESG-}\text{STARR}_{\alpha,\lambda}$
(the STAR ratio with p = 2), and the Sharpe ratio. Lambda values are arranged
on the X axis from 0 (only the monetary component considered, left) to 1 (only
the ESG component considered, right), and companies are color-coded by sector.

For each of the stocks, we compute a set of ESG risk measures and ESG
reward–risk ratios, considering a grid of different values of $\lambda$. We
then use the measures to rank the stocks in the dataset. In particular, we
compute $\text{ESG-}\text{AVaR}_{\lambda,\tau}(X)$ (with $\tau$ equal to 0.95
and 0.99), the ESG standard deviation ($\text{ESG-}\sigma_{\lambda}$), the
ESG-mean ($\text{ESG-}\mu_{\lambda}$), the ESG Rachev ratio (with $\tau$ equal
to 0.95 and 0.99), the ESG STAR ratio (with $\tau$ equal to 0.95 and 0.99),
and the ESG Sharpe ratio. Figure 2 displays the ordering of the assets: each
company is color-coded according to the industrial sector in which it
operates, and the companies in the upper part of the plot are those with the
highest value for the indicator. On the horizontal axis, we vary the
investor’s $\lambda$. Tables 4 and 5in Appendix D report the names of the
companies in the top and bottom five positions for a selected sample of
indicators and values of $\lambda$.222222 For brevity we report only the
measures for the 95th quantile. The complete rankings and the results for the
99th quantile are available upon request.

We make the following observations:232323 We emphasize that the range of
$\lambda$ that we considered is quite extreme, and realistically an investor
would choose a small value of $\lambda$ to generate only minor variations
compared to traditional market portfolios focused only on monetary
considerations.

  * •

ESG-AVaR tends to give similar rankings for similar levels of $\lambda$ up to
0.5; then, the ranking changes significantly and converges to a significantly
different ranking for $\lambda=1$ compared to $\lambda=0$. We can explain this
behavior by taking into consideration that the ESG scores are characterized by
a very small variability over time; hence, they tend to behave mostly as
shifts of the distribution of returns, and the variability of the monetary
returns dominates the determination of ESG-AVaR (and thus the ranking) even
for relatively large values of $\lambda$. For values of $\lambda$ close to 1,
the ESG component becomes dominant in the determination of the ranking. The
results are similar for $\tau=0.95$ and $\tau=0.99$; the main difference is
that for the highest value of $\tau$, the monetary component remains dominant
for higher values of $\lambda$.

  * •

The ranking according to the Rachev ratio is also stable for
$\lambda\lessapprox 0.5$. This is related to the fact that this ratio is
computed as the ratio of the AVaRs for the top and bottom tails of the
distribution.

  * •

The ranking according to the STAR ratio shows significant changes when
$\lambda$ changes. This can be attributed in large part to the fact that the
numerator of the ratio ($\text{ESG-}\mu_{\lambda}$) changes significantly with
$\lambda$.

  * •

The ranking according to the standard deviation is almost identical for all
values of $\lambda$, except for values very close to 1. This is because, as
stated previously, the variability of ESG scores is minimal, and the ESG
standard deviation is almost exclusively driven by monetary returns. Unlike
the AVaR, the standard deviation is not influenced by parallel shifts of the
distribution; hence, increasing the value of $\lambda$ does not cause any
changes in the ranking.

  * •

$\text{ESG-}\mu_{\lambda}$, the ESG Sharpe ratio, and the ESG Sortino-Satchell
ratio rankings change significantly for different values of $\lambda$. These
changes are driven by the significant differences in the rankings of the
average returns and average sustainability, which is also suggested by Panel C
in Figure 1.

Overall, the very low volatility of the ESG scores in this analysis greatly
affects the results, and the ESG score can almost be treated as constant
compared to the returns. This could be related to the lower sampling frequency
of the ESG data compared to the returns (annual vs daily), but it is likely to
be a property of the data due to a greater stability in the level of
sustainability of a company, which is influenced by the characteristics of the
productive processes and stable firm-level characteristics; in contrast, stock
prices have high uncertainty and volatility because they are forward-looking
and reflect the market expectations of the future profitability of a company.
Further empirical research and an increase in the quality of the ESG data are
required for more extensive results, but the framework proposed here is
flexible enough to be used to develop empirical analyses both to answer
theoretical questions and to implement viable investment strategies suitable
for ESG-oriented investors.

##  6 Conclusions

Individuals and institutions are increasingly aware of the non-monetary impact
of their investments, and they are willing to structure their portfolios so
that not only are the monetary risk and gains optimal but also the
environmental, social, and governance implications are taken into account.

This shift in investors’ goals challenges the traditional financial
literature, requiring the development of new analytical tools to describe the
behavior of ESG-oriented investors. Our work contributes to the literature by
introducing ESG-coherent risk measures, a framework for the measurement of
risk for investors with both monetary and ESG goals that provides an axiomatic
definition in which risk is measured as a function of a bivariate random
variable. The investor in our approach is still rational but follows a multi-
dimensional evaluation that considers not only the monetary part but also the
sustainability.

This framework extends the traditional coherent risk measures approach of
Artzner et al., (1999). The measures we provide can be used in several
contexts, and due to the introduction of a parameter $\lambda$, they can be
adapted to individuals with different relative preferences for the monetary
and ESG components of risk. We also provide the dual representation, present
several examples that generalize univariate risk measures, and introduce
related ESG reward–risk ratios.

We stress that the goal of the approach proposed is not to integrate ESG
scores for improving monetary risk-adjusted performances, but to take into
account the ethical preference of an investor for sustainable assets. This
challenges the traditional assumptions of monetary profit-maximizing and risk-
minimizing agents.

This paper is only an initial step in the development of a new financial
theory that will be capable of describing the behavior of ESG-oriented
investors. Future work will study optimal asset allocations, utility theory,
and asset pricing for ESG-oriented investors.

## Acknowledgments

Project funded under the National Recovery and Resilience Plan (NRRP), Mission
4 Component 2 Investment 1.3 - Call for tender No. 341 of 15/03/2022 of
Italian Ministry of University and Research funded by the European Union –
NextGenerationEU. Award Number: PE_00000018, Concession Decree No. 1558 of
11/10/2022 adopted by the Italian Ministry of University and Research, CUP
F83C22001720001, GROWING RESILIENT INCLUSIVE AND SUSTAINABLE - GRINS

##  Appendix A ESG-coherent reward measures

Similarly to how we defined ESG-coherent risk measures, we define ESG-coherent
reward measures. As discussed by Rachev et al., (2008), in the univariate
case, a reward measure can be defined as any functional defined on the space
of the random variables of interest, that is, it should be isotonic with
market preferences. Still, it is useful to formalize axiomatically the
characteristics of reward measures. We now extend to the multivariate setting
the axioms defined by Rachev et al., (2008), and we add the axiom of lambda
homogeneity (LH-M+). Considering a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and a set of bivariate random variables
$\mathcal{X}_{2}$, we can define an ESG-adjusted reward measure
$\boldsymbol{\theta}_{\lambda}(X):\mathcal{X}_{2}\rightarrow\mathbb{R}\cup\\{\pm\infty\\}$,
where $\lambda\in[0,1]$ is then ESG-coherent if the following axioms are
satisfied:

  * (SUP-M+)

Super-additivity: if $X_{1},X_{2}\in\mathcal{X}_{2}$, then
$\boldsymbol{\theta}_{\lambda}(X_{1}+X_{2})\geq\boldsymbol{\theta}_{\lambda}(X_{1})+\boldsymbol{\theta}_{\lambda}(X_{2})$;

  * (PH-M+)

Positive Homogeneity: if $\beta\geq 0$ and $X\in\mathcal{X}_{2}$, then
$\boldsymbol{\theta}_{\lambda}(\beta
X)=\beta\boldsymbol{\theta}_{\lambda}(X)$;

  * (TI-M+)

Translation Invariance: if $y\in\mathbb{R}^{2}$,
$\boldsymbol{\theta}_{\lambda}(X+a)=\boldsymbol{\theta}_{\lambda}(X)+\boldsymbol{\theta}_{\lambda}(y)$;

  * (MO-M+)

Monotonicity: if $X_{1},X_{2}\in\mathcal{X}_{2}$ and ($r_{1}\leq
r_{2}\wedge\text{ESG}_{1}\leq\text{ESG}_{2}$) a.s., then
$\boldsymbol{\theta}_{\lambda}(X_{1})\leq\boldsymbol{\theta}_{\lambda}(X_{2})$;

  * (LH-M+)

Lambda Homogeneity: if $y=\begin{bmatrix}y_{1}\\\
y_{2}\end{bmatrix}\in\mathbb{R}^{2}$, then
$\boldsymbol{\theta}_{\lambda}(y)=(1-\lambda)y_{1}+\lambda y_{2}$.

##  Appendix B Proof of dual representation of ESG-coherent risk measures

For convenience define $Z=-X$ where $X$ is the bivariate vector of monetary
returns and ESG. Consider the space
$\mathcal{Z}=\mathcal{L}_{p}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$,
$p\in[1,\infty)$. Its dual space $\mathcal{Z}^{*}$ is isomorphic to
$\mathcal{L}_{q}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$, where $q\in(1,\infty]$
is such that $1/p+1/q=1$ with $q=\infty$ for $p=1$.

Assume that
$\boldsymbol{\rho}_{\lambda}:\mathcal{Z}\to\mathbb{R}\cup\\{\infty\\}$ is a
proper convex lower-semicontinuous function. Note that
$\boldsymbol{\rho}_{\lambda}$ is convex when SUB-M and PH-M hold.

Consider the bilinear form $\langle\cdot,\cdot\rangle$ on the product
$\mathcal{Z}\times\mathcal{Z}^{*}$, which is defined as follows for
$Z\in\mathcal{Z}$ and $\zeta\in\mathcal{Z}^{*}$:

$\langle\zeta,Z\rangle=\int_{\Omega}\zeta_{1}(\omega)Z_{1}(\omega)+\zeta_{2}(\omega)Z_{2}(\omega)\rangle
dP(\omega).$

This form provides the corresponding continuous linear functionals on
$\mathcal{Z}$ and $\mathcal{Z}^{*}$ when equipped with appropriate topologies.
For each fixed $\zeta\in\mathcal{Z}^{*}$, the mapping
$Z\mapsto\langle\zeta,Z\rangle$ is a continuous linear functional on
$\mathcal{Z}$, equipped with the norm topology. For $p\in(0,1),$ all
continuous linear functional on $\mathcal{Z}^{*}$ have this form. For $p=1$,
we equip $Z^{*}$ with the weak∗ topology to have the representation by the bi-
linear form valid. The Fenchel conjugate function
$\boldsymbol{\rho}_{\lambda}^{*}:\mathcal{Z}^{*}\to\overline{\mathbb{R}}$ of
the risk measure $\boldsymbol{\rho}_{\lambda}$ is defined as

$\boldsymbol{\rho}_{\lambda}^{*}(\zeta)=\sup_{Z\in\mathcal{Z}}\big{\\{}\langle\zeta,Z\rangle-\boldsymbol{\rho}_{\lambda}(Z)\big{\\}},$

and the conjugate of $\boldsymbol{\rho}_{\lambda}^{*}$ (the bi-conjugate
function) is defined as

$\boldsymbol{\rho}_{\lambda}^{**}(Z)=\sup_{\zeta\in\mathcal{Z}^{*}}\big{\\{}\langle\zeta,Z\rangle-\boldsymbol{\rho}_{\lambda}^{*}(\zeta)\big{\\}}.$

$\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ denotes the domain of
$\boldsymbol{\rho}_{\lambda}^{*}.$ The Fenchel-Moreau Theorem, which is valid
in paired spaces, implies that
$\boldsymbol{\rho}_{\lambda}^{**}(Z)=\boldsymbol{\rho}_{\lambda}(Z)$ whenever
$\boldsymbol{\rho}_{\lambda}$ is proper, convex, and lower-semicontinuous.

Claim: The monotonicity axiom holds iff for all
$\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$, $\zeta\geq 0$ a.s.

Assume that the opposite is true. This means that there exists a set
$\Delta\in\mathcal{F}$ with $P(\Delta)>0$ such that for $\omega\in\Delta$, we
have $\zeta_{i}(\omega)<0$ for $i=1$ or $i=2$. We define
$\bar{Z}_{i}=\mathbbm{1}_{\Delta\cap{\zeta_{i}<0}}$, where $\mathbbm{1}_{B}$
is the indicator function of the event $B$. Take any $Z$ with support in
$\Delta$ such that $\boldsymbol{\rho}_{\lambda}(Z)$ is finite and and define
$Z_{t}:=Z-t\bar{Z}$. Then for $t\geq 0$, we have that $Z_{t}\leq Z$
componentwise, and
$\boldsymbol{\rho}_{\lambda}(Z_{t})\leq\boldsymbol{\rho}_{\lambda}(Z)$ by
monotonicity. Consequently,

$\boldsymbol{\rho}_{\lambda}^{*}(\zeta)\geq\sup_{t\in\mathbb{R}_{+}}\big{\\{}\langle\zeta,Z_{t}\rangle-\boldsymbol{\rho}_{\lambda}(Z_{t})\big{\\}}\geq\sup_{t\in\mathbb{R}_{+}}\big{\\{}\langle\zeta,Z\rangle-t\langle\zeta,\bar{Z}\rangle-\boldsymbol{\rho}_{\lambda}(Z)\big{\\}}.$

On the right-hand side, $\langle\zeta,\bar{Z}\rangle<0$ on $\Delta$ and zero
otherwise, while the other terms under the supremum are fixed. Hence, the
supremum is infinite and
$\zeta\not\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}.$

Conversely, suppose that every
$\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ is nonnegative. Then for
every $\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ and $Z\geq
Z^{\prime}$ componentwise, we have

$\langle\zeta,Z^{\prime}\rangle=\int_{\Omega}\zeta_{1}(\omega)Z_{1}^{\prime}(\omega)+\zeta_{2}(\omega)Z_{2}^{\prime}(\omega)dP(\omega)\leq\int_{\Omega}\zeta_{1}(\omega)Z_{1}(\omega)+\zeta_{2}(\omega)Z_{2}(\omega)dP(\omega)=\langle\zeta,Z\rangle.$

Consequently

$\boldsymbol{\rho}_{\lambda}(Z)=\sup_{\zeta\in\mathcal{Z}^{*}}\big{\\{}\langle\zeta,Z\rangle-\boldsymbol{\rho}_{\lambda}^{*}(\zeta)\big{\\}}\geq\sup_{\zeta\in\mathcal{Z}^{*}}\big{\\{}\langle\zeta,Z^{\prime}\rangle-\boldsymbol{\rho}_{\lambda}^{*}(\zeta)\big{\\}}=\boldsymbol{\rho}_{\lambda}(Z^{\prime}).$

Claim: The positive homogeneity axiom holds iff $\boldsymbol{\rho}_{\lambda}$
is the support function of $\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$.

Suppose that $\boldsymbol{\rho}_{\lambda}(tZ)=t\boldsymbol{\rho}_{\lambda}(Z)$
for all $Z\in\mathcal{Z}$. For all $t>0$ and for all $Z\in\mathcal{Z}$

$\boldsymbol{\rho}_{\lambda}^{*}(\zeta)=\sup_{Z\in\mathcal{Z}}\big{\\{}\langle\zeta,Z\rangle-\boldsymbol{\rho}_{\lambda}(Z)\big{\\}}\geq\langle\zeta,tZ\rangle-\boldsymbol{\rho}_{\lambda}(tZ)$

Thus for all $t>0$

$\boldsymbol{\rho}_{\lambda}^{*}(\zeta)=\sup_{Z\in\mathcal{Z}}\big{\\{}\langle\zeta,Z\rangle-\boldsymbol{\rho}_{\lambda}(Z)\big{\\}}\geq\sup_{Z\in\mathcal{Z}}\big{\\{}\langle\zeta,tZ\rangle-t\boldsymbol{\rho}_{\lambda}(Z)\big{\\}}=t\boldsymbol{\rho}_{\lambda}^{*}(\zeta).$

Hence, if $\boldsymbol{\rho}_{\lambda}^{*}(\zeta)$ is finite, then
$\boldsymbol{\rho}_{\lambda}^{*}(\zeta)=0$ as claimed. Furthermore,

$\boldsymbol{\rho}_{\lambda}(0)=\sup_{\zeta\in\mathcal{Z}^{*}}\big{\\{}\langle\zeta,0\rangle-\boldsymbol{\rho}_{\lambda}^{*}(\zeta)\big{\\}}=0.$

For the converse, if
$\boldsymbol{\rho}_{\lambda}(Z)=\sup_{\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}}\langle\zeta,Z\rangle$,
then $\boldsymbol{\rho}_{\lambda}$ is positively homogeneous as a support
function of a convex set. Hence when the positive homegeneity property holds,
the conjugate function is the indicator function of convex analysis of the set
$\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}.$

Claim: The translation axiom
$\boldsymbol{\rho}_{\lambda}(Z+a)=\boldsymbol{\rho}_{\lambda}(Z)+\boldsymbol{\rho}_{\lambda}(a)$
holds if and only if $\int_{\Omega}\zeta(\omega)P(d\omega)=\mu$, where $\mu$
is a constant vector and $\boldsymbol{\rho}_{\lambda}(a)=\langle\mu,a\rangle$.
We write $\mu_{\zeta}=\int_{\Omega}\zeta(\omega)P(d\omega)$ and fix an
arbitrary constant $a$. We infer

$\boldsymbol{\rho}_{\lambda}^{*}(\zeta)=\sup_{Z\in\mathcal{Z}}\big{\\{}\langle\zeta,Z+a\rangle-\boldsymbol{\rho}_{\lambda}(Z+a)\big{\\}}=\sup_{Z\in\mathcal{Z}}\Big{\\{}\int_{\Omega}\langle
Z+a,\zeta(\omega)\rangle
P(d\omega)-\boldsymbol{\rho}_{\lambda}(Z)-\boldsymbol{\rho}_{\lambda}(a)\Big{\\}}\\\
=\boldsymbol{\rho}_{\lambda}^{*}(\zeta)-\boldsymbol{\rho}_{\lambda}(a)+\langle
a,\mu_{\zeta}\rangle.$

Hence, $\boldsymbol{\rho}_{\lambda}(a)=\langle a,\mu_{\zeta}\rangle$ for all
$\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ and for all constant
vectors $a$. This is possible only if

$\mu_{\zeta}=\mu\quad\forall\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}.$

For the converse, consider

$\boldsymbol{\rho}_{\lambda}(Z+a)=\sup_{\zeta\in\mathcal{Z}^{*}}\big{\\{}\langle\zeta,Z+a\rangle-\boldsymbol{\rho}_{\lambda}^{*}(\zeta)\big{\\}}=\sup_{\zeta\in\mathcal{Z}^{*}}\Big{\\{}\int_{\Omega}\langle
Z,\zeta(\omega)\rangle
P(d\omega)-\boldsymbol{\rho}_{\lambda}^{*}(\zeta)+\int_{\Omega}\langle
a,\zeta(\omega)\rangle P(d\omega)\Big{\\}}\\\
=\boldsymbol{\rho}_{\lambda}(Z)+\langle a,\mu\rangle.$

Claim: If additionally LH-M holds, then
$\mu=\big{[}(1-\lambda),\lambda\big{]}^{\prime}$ and, hence, for all
$\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$, we have

$\int_{\Omega}\zeta_{1}(\omega)+\zeta_{2}(\omega)P(d\omega)=1.$

In summary, when axioms SUB-M through LH-M are satisfied, then the dual
representation of the risk measure is

$\displaystyle\boldsymbol{\rho}_{\lambda}(X)$
$\displaystyle=\sup_{\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}}-\int_{\Omega}\zeta_{1}(\omega)r(\omega)+\zeta_{2}(\omega)\text{ESG}(\omega)P(d\omega)$
(34) $\displaystyle\boldsymbol{\rho}_{\lambda}(X)$
$\displaystyle=-\inf_{\zeta\in\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}}\int_{\Omega}\zeta_{1}(\omega)r(\omega)+\zeta_{2}(\omega)\text{ESG}(\omega)P(d\omega)$

where $\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ contains non-negative
functions $(\zeta_{1}(\omega),\zeta_{2}(\omega))$ on $\mathbb{R}^{2}$ whose
expected value is $\big{[}(1-\lambda),\quad\lambda\big{]}^{\prime}$.
Furthermore, $\mathcal{A}_{\boldsymbol{\rho}_{\lambda}}$ is equal to the
convex subdifferential of $\boldsymbol{\rho}_{\lambda}([0,0]^{\prime})$. Note
that in (34) we adjusted the signs since we express the risk measure in terms
of $X$ and not $Z=-X$:

##  Appendix C Proofs of dual representation for
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}$ and
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}$

Let
$X=[r,\;\text{ESG}]^{\prime}\in\mathcal{X}_{2}=\mathcal{L}_{1}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$
be a bivariate random variable associated with an asset, and for convenience,
define $Z:=[Z_{1},\;Z_{2}]^{\prime}=-X$ (i.e., the corresponding vector with
inverted signs).

Dual representation for $\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}$

Here, we prove that $\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(X)$ has the
following dual representation:

$\displaystyle\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}(X)=\sup_{[\zeta_{1}\;\zeta_{2}]^{\prime}\in\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}}}\mathbb{E}[Z_{1}\zeta_{1}+Z_{2}\zeta_{2}],\quad\text{with}$
$\displaystyle\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}}=\left\\{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2}):\;[\zeta_{1},\;\zeta_{2}]^{\prime}=\xi[1-\lambda,\;\lambda]^{\prime};\;0\leq\xi\leq\frac{1}{1-\tau}\;\text{a.s.
}\mathbb{E}[\xi]=1\right\\}.$

We assume
$\mathcal{X}_{2}=\mathcal{L}_{1}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$, which
entails that the paired space is
$\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$.

$\displaystyle\boldsymbol{\rho}(Z)$
$\displaystyle=\min_{\beta\in\mathbb{R}}\Big{\\{}\frac{1}{1-\tau}\mathbb{E}\big{[}\big{(}\beta-((1-\lambda)r+\lambda\text{ESG})\big{)}_{+}\big{]}-\beta\Big{\\}},$
$\displaystyle=\min_{\beta\in\mathbb{R}}\Big{\\{}\beta+\frac{1}{1-\tau}\mathbb{E}\big{[}\big{(}(1-\lambda)Z_{1}+\lambda
Z_{2}-\beta\big{)}_{+}\big{]}\Big{\\}},\quad\tau\in(0,1).$

Using the rules of subdifferential calculus and Strassen’s theorem (Strassen,
(1965)), we get

$\partial\mathbb{E}\big{[}\big{(}(1-\lambda)Z_{1}+\lambda
Z_{2}-\beta\big{)}_{+}\big{)}_{+}\big{]}=[1-\lambda,\;\lambda]^{\prime}\xi,\quad\text{where
}\xi(\omega)=\begin{cases}1&\text{ if }(1-\lambda)Z_{1}+\lambda Z_{2}>\beta\\\
0&\text{ if }(1-\lambda)Z_{1}+\lambda Z_{2}<\beta\\\ [0,1]&\text{ if
}(1-\lambda)Z_{1}+\lambda Z_{2}=\beta.\end{cases}$

Note that $\xi\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P)$ and $\xi\geq 0$.
We define $\zeta\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2})$
by setting $\zeta=[1-\lambda,\;\lambda]^{\prime}\xi$ for any measurable
selection
$\xi\in\partial\mathbb{E}\big{[}\big{(}(1-\lambda)r+\lambda\text{ESG})\big{)}_{+}\big{]}.$
Let $\tilde{\mathcal{A}}$ be the set containing all such elements $\zeta$.
Evidently, we have $\boldsymbol{0}\leq\zeta\leq[1-\lambda,\;\lambda]^{\prime}$
a.s. for all. Using the subgradient inequality at $\bar{Z}=(\beta,\beta)$, we
obtain for all $Z$

$\mathbb{E}\big{[}\big{(}((1-\lambda)Z_{1}+\lambda
Z_{2})-\beta\big{)}_{+}\big{]}\geq\langle\zeta,Z-\beta(1,1)\rangle=\langle\zeta,Z\rangle-\beta\mathbb{E}[\xi].$

On the other hand, for any $\zeta=[1-\lambda,\;\lambda]^{\prime}\xi$

$\langle\zeta,Z\rangle=\langle[1-\lambda,\;\lambda]^{\prime}\xi,Z-\beta(1,1)\rangle+\beta\langle[1-\lambda,\;\lambda]^{\prime}\xi,(1,1)\rangle\\\
=\langle\xi,((1-\lambda)Z_{1}+\lambda
Z_{2}-\beta)\rangle+\beta\mathbb{E}[\xi]\leq\mathbb{E}\big{[}\big{(}(1-\lambda)Z_{1}+\lambda
Z_{2}-\beta)_{+}\big{]}+\beta\mathbb{E}[\xi].$

Hence, we can represent

$\mathbb{E}\big{[}\big{(}(1-\lambda)Z_{1}+\lambda
Z_{2}-\beta)_{+}\big{]}+=\max_{\zeta\tilde{\mathcal{A}}}\Big{(}\langle\zeta,Z\rangle-\beta\mathbb{E}[\xi]\Big{)}.$

Now, we can express the risk measure as follows:

$\displaystyle\boldsymbol{\rho}(Z)$
$\displaystyle=\min_{\beta\in\mathbb{R}}\bigg{\\{}\beta+\frac{1}{1-\tau}\max_{\zeta\in\tilde{\mathcal{A}}}\Big{(}\langle\zeta,Z\rangle-\beta\mathbb{E}[\xi]\Big{)}\bigg{\\}}$
$\displaystyle=\max_{\zeta\in\tilde{\mathcal{A}}}\inf_{\beta\in\mathbb{R}}\bigg{\\{}\beta+\frac{1}{1-\tau}\langle\zeta,Z\rangle-\frac{\beta}{1-\tau}\mathbb{E}[\xi]\bigg{\\}}=\max_{\zeta\in\tilde{\mathcal{A}}}\inf_{\beta\in\mathbb{R}}\bigg{\\{}\beta(1-\frac{1}{1-\tau}\mathbb{E}[\xi])+\frac{1}{1-\tau}\langle\zeta,Z\rangle\bigg{\\}}.$

The exchange of the “$\min$” and “$\max$” operations is possible, because the
function in braces is bilinear in $(\beta,\zeta)$, and the set
$\tilde{\mathcal{A}}$ is compact. The inner minimization with respect to
$\beta$ yields $-\infty$, unless $\mathbb{E}[\xi]=1-\tau$. Consequently,

$\boldsymbol{\rho}(Z)=\max_{{\zeta=\xi[1-\lambda,\lambda]^{\prime}\in\mathcal{A}}\atop{\mathbb{E}[\xi]=1-\tau}}\frac{1}{1-\tau}\langle
Z,\zeta\rangle.$

Setting $\zeta^{\prime}=\zeta/(1-\tau)$, we obtain the support set

$\partial\boldsymbol{\rho}(0)=\bigg{\\{}\zeta\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2}):\zeta=\frac{\xi}{1-\tau}[1-\lambda,\lambda]^{\prime};\xi\geq
0,\ \mathbb{E}[\xi]=1-\tau,\ \|\xi\|_{\infty}\leq 1\bigg{\\}}.$

Dual representation for $\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}$

We now prove that the dual representation of
$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(X)$ is defined as follows:

$\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(X)=\sup_{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}}}\mathbb{E}[Z_{1}\zeta_{1}+Z_{2}\zeta_{2}],$
(35)

with $Z:=[Z_{1},\;Z_{2}]^{\prime}=-X$ and

$\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}}=\left\\{[\zeta_{1},\;\zeta_{2}]^{\prime}\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P;\mathbb{R}^{2}):\mathbb{E}[\zeta_{1}]=1-\lambda;\mathbb{E}[\zeta_{2}]=\lambda;\zeta_{1},\zeta_{2}\geq
0;\zeta_{1}\leq\frac{1-\lambda}{1-\tau};\zeta_{2}\leq\frac{\lambda}{1-\tau}\right\\}.$
(36)

We use the known representation of Average Value at Risk for scalar random
variables (cf. Shapiro et al.,, 2021, Example 6.19 eq. 6.76). Denote the dual
set in that representation by $\mathcal{A}^{\prime}$, i.e.,

$\mathcal{A}^{\prime}=\left\\{\xi\in\mathcal{L}_{\infty}(\Omega,\mathcal{F},P):\;\mathbb{E}[\xi]=1;\;0\leq\xi\leq\frac{1}{1-\tau}\;\text{a.s.}\right\\},\\\
$

Hence

$\displaystyle\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(X)$
$\displaystyle=(1-\lambda)\sup_{\xi\in\mathcal{A}^{\prime}}\mathbb{E}[\xi,Z_{1}]+\lambda\sup_{\xi\in\mathcal{A}^{\prime}}\mathbb{E}[\xi,Z_{2}]$
$\displaystyle=\sup_{\xi\in\mathcal{A}^{\prime}}\mathbb{E}[(1-\lambda)\xi,Z_{1}]+\sup_{\xi\in\mathcal{A}^{\prime}}\mathbb{E}[\lambda\xi,Z_{2}]$
$\displaystyle=\sup_{[\xi_{1},\xi_{2}]^{\prime}\in\mathcal{A}^{\prime}\times\mathcal{A}^{\prime}}\Big{(}\mathbb{E}[(1-\lambda)\xi_{1},Z_{1}]+\mathbb{E}[\lambda\xi_{2},Z_{2}]\Big{)}.$
(37)

Now, we define the set
$\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}}=(1-\lambda)\mathcal{A}^{\prime}\times\lambda\mathcal{A}^{\prime}$
and continue the last chain of equations as follows:

$\displaystyle\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}(X)$
$\displaystyle=\sup_{[\xi_{1},\xi_{2}]^{\prime}\in\mathcal{A}^{\prime}\times\mathcal{A}^{\prime}}\mathbb{E}[(1-\lambda)\xi,Z_{1}]+\mathbb{E}[\lambda\xi,Z_{2}]=\sup_{\zeta\in\mathcal{A}_{\text{ESG}\text{-}\text{AVaR}_{\lambda,\tau}^{l}}}\langle\zeta,Z\rangle.$

This concludes the proof.

##  Appendix D Ranking of companies based on ESG risk measures, reward
measures, and RRRs.

ranking by $\text{ESG-}\text{AVaR}_{\lambda,0.95\%}$
---
rank |  $\lambda=0$ |  $\lambda=0.25$ |  $\lambda=0.5$ |  $\lambda=0.75$ |  $\lambda=1$
1 |  Boeing Co. |  Boeing Co. |  Boeing Co. |  Boeing Co. |  Travelers Companies Inc.
2 |  Intel Corp. |  Intel Corp. |  Intel Corp. |  Intel Corp. |  Salesforce Inc.
3 |  WBA Inc. |  WBA Inc. |  WBA Inc. |  Salesforce Inc. |  Verizon Comm. Inc.
4 |  American Express Co. |  American Express Co. |  American Express Co. |  WBA Inc |  Apple Inc.
5 |  Chevron Corp. |  Salesforce Inc. |  Salesforce Inc. |  American Express Co. |  Goldman Sachs Group Inc.
24 |  Coca-Cola Co. |  Johnson & Johnson |  Johnson & Johnson |  Procter & Gamble Co. |  3M Co.
25 |  Merck & Co. Inc. |  Merck & Co. Inc. |  Merck & Co. Inc. |  Merck & Co. Inc. |  Intel Corp.
26 |  Procter & Gamble Co. |  Procter & Gamble Co. |  Procter & Gamble Co. |  Johnson & Johnson |  Cisco Systems Inc.
27 |  Walmart Inc. |  Walmart Inc. |  Walmart Inc. |  Walmart Inc. |  Johnson & Johnson
28 |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Microsoft Corp.
ranking by ESG-standard deviation
rank |  $\lambda=0$ |  $\lambda=0.25$ |  $\lambda=0.5$ |  $\lambda=0.75$ |  $\lambda=1$
1 |  Boeing Co. |  Boeing Co. |  Boeing Co. |  Boeing Co. |  Travelers Companies Inc.
2 |  Intel Corp. |  Intel Corp. |  Intel Corp. |  Intel Corp. |  WBA Inc.
3 |  American Express Co. |  American Express Co. |  American Express Co. |  American Express Co. |  Goldman Sachs Group Inc.
4 |  Chevron Corp. |  Chevron Corp. |  Chevron Corp. |  Chevron Corp. |  Verizon Comm. Inc.
5 |  Salesforce Inc. |  Salesforce Inc. |  Salesforce Inc. |  Salesforce Inc. |  Walt Disney C.o
24 |  Walmart Inc. |  Walmart Inc. |  Walmart Inc. |  Walmart Inc. |  3M Co.
25 |  Coca-Cola Co. |  Coca-Cola Co. |  Coca-Cola Co. |  Coca-Cola Co. |  Johnson & Johnson
26 |  Procter & Gamble Co. |  Procter & Gamble Co. |  Procter & Gamble Co. |  Procter & Gamble Co. |  Microsoft Corp.
27 |  Johnson & Johnson |  Johnson & Johnson |  Johnson & Johnson |  Johnson & Johnson |  Intel Corp.
28 |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Cisco Systems Inc.
ranking by ESG-mean
rank |  $\lambda=0$ |  $\lambda=0.25$ |  $\lambda=0.5$ |  $\lambda=0.75$ |  $\lambda=1$
1 |  Microsoft Corp. |  Microsoft Corp. |  Microsoft Corp. |  Microsoft Corp. |  Microsoft Corp.
2 |  Apple Inc. |  Apple Inc. |  Cisco Systems Inc. |  Cisco Systems Inc. |  Johnson & Johnson
3 |  Salesforce Inc. |  Nike Inc. |  Intel Corp. |  Johnson & Johnson |  3M Co.
4 |  Nike Inc. |  Salesforce Inc. |  Johnson & Johnson |  Intel Corp. |  Cisco Systems Inc.
5 |  UnitedHealth Group Inc. |  UnitedHealth Group Inc. |  Apple Inc. |  3M Co. |  Intel Corp.
24 |  Johnson & Johnson |  3M Co. |  Procter & Gamble Co. |  Salesforce Inc. |  Apple Inc.
25 |  Verizon Comm. Inc. |  Travelers Companies Inc. |  Coca-Cola Co. |  Caterpillar Inc. |  Procter & Gamble Co.
26 |  IBM Corp |  Verizon Comm. Inc. |  IBM Corp. |  Procter & Gamble Co. |  Caterpillar Inc.
27 |  WBA Inc. |  WBA Inc |  Verizon Comm. Inc. |  Verizon Comm. Inc. |  Salesforce Inc.
28 |  3M Co. |  IBM Corp. |  Travelers Companies Inc. |  Travelers Companies Inc. |  Travelers Companies Inc.
Table 4: Top and bottom ranking according to several ESG risk and reward
measures ($\text{ESG-}\text{AVaR}_{\lambda,0.95\%}$, ESG-standard deviation,
ESG-mean) for investors with different $\lambda$. ranking by ESG Sharpe ratio
---
rank |  $\lambda=0$ |  $\lambda=0.25$ |  $\lambda=0.5$ |  $\lambda=0.75$ |  $\lambda=1$
1 |  Microsoft Corp. |  Microsoft Corp. |  Microsoft Corp. |  Johnson & Johnson |  Cisco Systems Inc.
2 |  Apple Inc. |  Apple Inc. |  Johnson & Johnson |  Microsoft Corp. |  Intel Corp.
3 |  Nike Inc. |  Nike Inc. |  Walmart Inc. |  Cisco Systems Inc. |  Microsoft Corp.
4 |  Salesforce Inc. |  Home Depot Inc. |  Cisco Systems Inc. |  3M Co. |  Johnson & Johnson
5 |  Home Depot Inc. |  Walmart Inc. |  Merck & Co. Inc. |  Merck & Co. Inc. |  3M Co.
24 |  Verizon Comm. Inc. |  3M Co. |  IBM Corp. |  American Express Co. |  Verizon Comm. Inc.
25 |  Boeing Co. |  Travelers Companies Inc. |  Caterpillar Inc. |  Caterpillar Inc. |  Goldman Sachs Group Inc.
26 |  IBM Corp |  Boeing Co. |  WBA Inc. |  Salesforce Inc. |  Salesforce Inc.
27 |  WBA Inc. |  IBM Corp. |  Boeing Co. |  Boeing Co. |  WBA Inc.
28 |  3M Co. |  WBA Inc. |  Travelers Companies Inc. |  Travelers Companies Inc |  Travelers Companies Inc.
ranking by $\text{ESG-}\text{STARR}_{\lambda,0.95\%}$
rank |  $\lambda=0$ |  $\lambda=0.25$ |  $\lambda=0.5$ |  $\lambda=0.75$ |  $\lambda=1$
1 |  3M Co. |  WBA Inc. |  Travelers Companies Inc. |  Travelers Companies Inc. |  Verizon Comm. Inc.
2 |  WBA Inc. |  IBM Corp |  Boeing Co. |  Boeing Co. |  Salesforce Inc.
3 |  IBM Corp. |  Boeing Co. |  WBA Inc. |  Salesforce Inc. |  Goldman Sachs Group Inc.
4 |  Boeing Co. |  Travelers Companies Inc. |  IBM Corp |  Caterpillar Inc. |  Apple Inc.
5 |  Chevron Corp. |  3M Co. |  Caterpillar Inc. |  IBM Corp |  Walt Disney Co.
24 |  Home Depot Inc. |  Home Depot Inc. |  Cisco Systems Inc. |  Cisco Systems Inc. |  Microsoft Corp.
25 |  Salesforce Inc. |  Apple Inc. |  Merck & Co. Inc. |  Merck & Co. Inc. |  Johnson & Johnson
26 |  Nike Inc. |  Walmart Inc. |  Johnson & Johnson |  Walmart Inc. |  Intel Corp.
27 |  Apple Inc. |  Nike Inc. |  Walmart Inc. |  Microsoft Corp. |  Cisco Systems Inc.
28 |  Microsoft Corp. |  Microsoft Corp. |  Microsoft Corp. |  Johnson & Johnson |  Travelers Companies Inc.
ranking by $\text{ESG-}\text{RR}_{\lambda,0.95\%}$
rank |  $\lambda=0$ |  $\lambda=0.25$ |  $\lambda=0.5$ |  $\lambda=0.75$ |  $\lambda=1$
1 |  Walt Disney Co. |  Walmart Inc. |  Walmart Inc. |  Walmart Inc. |  Travelers Companies Inc.
2 |  Walmart Inc. |  Walt Disney Co. |  Walt Disney Co. |  Microsoft Corp. |  Cisco Systems Inc.
3 |  Boeing Co. |  Boeing Co. |  UnitedHealth Group Inc. |  Walt Disney Co. |  Intel Corp.
4 |  Nike Inc. |  Nike Inc. |  Nike Inc. |  Merck & Co. Inc. |  Microsoft Corp.
5 |  UnitedHealth Group Inc. |  UnitedHealth Group Inc. |  Boeing Co. |  JPMorgan Chase & Co. |  Johnson & Johnson
24 |  Home Depot Inc. |  Home Depot Inc. |  Caterpillar Inc. |  Salesforce Inc. |  Walt Disney Co.
25 |  Travelers Companies Inc. |  Travelers Companies Inc. |  Coca-Cola Co. |  Coca-Cola Co. |  WBA Inc.
26 |  IBM Corp. |  Johnson & Johnson |  3M Co. |  Caterpillar Inc. |  Goldman Sachs Group Inc.
27 |  Johnson & Johnson |  IBM Corp. |  Travelers Companies Inc. |  IBM Corp |  Verizon Comm. Inc.
28 |  3M Co. |  3M Co. |  IBM Corp. |  Travelers Companies Inc. |  Salesforce Inc.
Table 5: Top and bottom ranking according to several ESG-RRRs (ESG Sharpe
ratio, $\text{ESG-}\text{STARR}_{\lambda,0.95\%}$,
$\text{ESG-}\text{RR}_{\lambda,0.95\%}$) for investors with different
$\lambda$.

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