# ESG-Valued Portfolio Optimization and Dynamic Asset Pricing

Davide Lauria  Department of Mathematics & Statistics, Texas Tech University,
Lubbock TX 79409-1042, U.S.A<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author<EMAIL_ADDRESS>W. Brent Lindquist  Department
of Mathematics & Statistics, Texas Tech University, Lubbock TX 79409-1042,
U.S.A<EMAIL_ADDRESS><EMAIL_ADDRESS>Stefan Mittnik  Department
of Statistics, Ludwig Maximilians Universität München, 80799 München, DE. and
Scalable Capital, 80538 München, DE<EMAIL_ADDRESS>Svetlozar T.
Rachev  Department of Mathematics & Statistics, Texas Tech University, Lubbock
TX 79409-1042, U.S.A<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract ESG ratings provide a quantitative measure for socially responsible
investment. We present a unified framework for incorporating numeric ESG
ratings into dynamic pricing theory. Specifically, we introduce an ESG-valued
return that is a linearly constrained transformation of financial return and
ESG score. This leads to a more complex portfolio optimization problem in a
space governed by reward, risk and ESG score. The framework preserves the
traditional risk aversion parameter and introduces an ESG affinity parameter.
We apply this framework to develop ESG-valued: portfolio optimization; capital
market line; risk measures; option pricing; and the computation of shadow
riskless rates.

Keywords ESG scores, optimal portfolio, efficient frontier, option pricing,
riskless rates

##  1 Introduction

Socially responsible investing (SRI) is a broad approach advocating investment
in activities and companies that produce positive impact on the environment
and society. The definition of SRI is quite general; its practices span a
heterogeneous system of policies and scopes [27, 73]. The heterogeneity within
the SRI movement and forces driving differing SRI outlooks have been studied
on four levels: terminological, definitional, strategic and practical [43,
66]. Within the vast realm of SRI, a more focused approach in investment is
emerging based upon ESG ratings, wherein each company is separately evaluated
in three categories: environmental, social and governance. Evaluation is
performed on the basis of factors within each category. The total ESG score of
a company is then determined by a combination of three category scores,
providing a “sustainability ordering” between firms.

ESG investing is experiencing fast growth as institutional investors, such as
mutual and pension funds, now provide SRI-related financial products. (See for
instance the recent report from the Organization for Economic Co-operation and
Development on ESG investing [18], and references therein.) Different
approaches have been taken by asset managers to integrate ESG factors into
investment decisions [3, 71]. These approaches generally fall into three
strategy groups. The first strategy, an exclusionary approach, “filters out
certain companies based upon products or certain corporate behavior” [13]. The
second strategy, the inclusionary approach, “involves adjusting the weights of
an investment in a firm according to whether its behavior is more or less
socially responsible” (ibid). There is a degree of subjectivity in both
approaches. In the third, empirical-based strategy, asset managers seek for
more exposure in companies with higher ESG scores, thus avoiding the
subjectivity of the exclusionary and inclusionary approaches through the use
of an empirical measure that does not exclude participation in less virtuous
entities. According to the work of Berry at al. [13], most socially
responsible financial products focus on the exclusionary approach. In
contrast, they found that investors prefer to reward firms who display overall
positive social policies, rather than exclude the less responsible through
negative screening. A similar conclusion was reached by Amel-Zadeh and
Serafeim [3]. These results could explain the slow reception of SRI, and drive
new approaches to implement the third class of strategies.

Research has highlighted two main reasons behind the push for ESG products:
ethical beliefs and improved financial performance [41]. Investigation of the
connection between stakeholder preferences (ethical beliefs) and responsible
investing includes the work of Bénabou and Tirole [11], Krüger [51], Lins et
al. [53], Liang and Renneboog [52], and Starks at al. [68]. Krüger
investigated how stock markets react to positive and negative events connected
with corporate social responsibility, finding a strong reaction to negative
events. Starks at al. found that long-term investors are more attracted toward
higher rated ESG companies, “tilting their portfolios towards firms with high-
ESG profiles”.

The literature has been more focused on the performance of ESG investing
compared to conventional return-centric investing. An extensive literature
review of over 2000 articles [33] concludes that 90% of the studies “find a
nonnegative ESG [to] corporate financial performance relation” and that this
relationship appears to be stable over time. Brooks and Oikonomou [19] obtain
similar conclusions after reviewing a large set of studies focusing on the
connection between ESG disclosure and performance and their effects on firm
value. They found “a positive and statistically significant but economically
modest link” between corporate social performance and firm financial
perfomance. They also found a strong negative relationship between financial
risk, whether systematic or idiosyncratic, and corporate social performance.
Bello [10] performed a comparison between socially responsible and traditional
mutual funds to attempt to quantify the impact of ethical screening on
portfolio diversification and overall performance. He reported that the two
classes of mutual funds do not differ statistically in the characteristics of
assets they hold, in diversification, or in overall performance. In constrast,
a similar study [1] on European mutual funds reports a superior “efficiency”
of funds that focus on high ESG rated securities. Giese et al. [39] studied
ESG causality on (as opposed to “correlation with”) financial performance by
considering transmission channels of systematic and idiosyncratic risk within
a standard discounted cash flow model, through which ESG ratings data
(specifically MSCI ESG ratings data) impact valuation and performance. An
interesting outcome of their study relates to the intensity and longevity of
ESG signals (ESG rating changes) compared with factors such as momentum or
volatility.

A component of the literature has focused on the effect of specific components
of the three ESG categories. In the area of governance, Pan et al. [61] found
a negative relationship between returns and high pay ratios. They claim that,
in 2018, inequality-averse investors decreased the relative amount invested in
stocks with a high pay ratio. Krueger at al. [50] surveyed institutional
investors on whether, and how, they integrate climate risk into their
financial decisions. The majority of those interviewed consider climate change
to have relevant implication for financial performance and have consequently
taken action in their practices to accommodate such risk. Görgen et al. [40]
have investigated the impact of carbon risk on equity prices of “brown” and
“green” firms. Their research uncovered two opposing effects: while brown
firms display higher average returns, companies that become less green show a
drop in returns.

Other relationships between ESG ratings and the financial system have been
studied. Examples include the effect of ESG scores on credit ratings [7, 32]
and on the fixed income market [44, 45, 46]. The analysis of ESG factors
associated with a fixed income security, especially one issued by a
government, involves dealing with complex methodological and technical
implications [44]. The literature identifies two main approaches to
translating ESG ratings into the fixed income market. The first is to issue
products specifically designed to fund projects having positive environmental
or social impact. Examples of securities issued with the specific purpose of
financing projects related to environmental and climate change objectives are
green and blue bonds111 The first green bond was issued by the European
Investment Bank in 2007. https://www.eib.org/en/investor-
relations/cab/index.htm [29, 31]. Ehlers et al. [31] have documented lower
credit spreads222 The spread of yield at issuance over the yield curve of an
appropriate treasury security of the same maturity at issuance date. on
borrowing green bonds compared to that of a traditional counterpart issued by
the same institutions. Polbennikov at al. [63] found that ESG corporate bond
portfolios have experienced a modest positive incremental return. The second
approach has a broader view. It consists of assigning the ESG rating of the
company to their fixed income products, imposing a tighter connection between
the global business activity of the issuer and the bond rating [2, 60]. A
related issue is the ESG valuation of general fixed income securities backed
by governments or municipalities [6].

A major concern of ESG ratings is the variability of scores among different
rating agencies, which raises the issues of reliability and potential bias
selection; see for instance [12, 15, 21, 25, 38], and references therein. Berg
et al. [12] analyzed ESG scores from six large providers. They decomposed the
spread among scores into three main sources:

  * i)

scope (56%), defined by the range of company attributes that the score intends
to measure;

  * ii)

measurement (38%), the methodology used to compute a particular attribute; and

  * iii)

weights (6%), how the attribute values are combined to form a final score;

and were able to attribute the percentage weight (quoted above) to each
source. The main source of variability found among ESG scoring systems was the
selection of attributes (the scope). The authors warn of possible bias effects
involved in the assignment of ESG scores as well as in research attempting to
study the link between ESG scores and financial performance. For example, a
“rater effect” was discovered under which “a rater’s overall view of a firm
influences the measurement of specific categories”. Bilio et al. [15] found
substantially low agreement in the assignment of ESG scores by four rating
agencies. They investigated the impact of this variability on financial
performance by comparing the Jensen-alpha for two equi-weighted portfolios:
one formed by those companies that were rated by all four ESG rating agencies
and the other composed of companies that were rated by none of the agencies.
They found no evidence for a difference in financial performance between these
two portfolios.

The construction of optimal portfolios is one corner-stone of modern finance
theory. The traditional approach is based on two measures: financial reward
and financial risk. The reward measure is usually represented by the expected
value of future portfolio returns. Several measures have been proposed to
quantify the risk related to the unpredictable component of the variability of
portfolio returns; the most common of these are: the standard deviation [56],
the absolute deviation [49], the value at risk (VaR) [54], and the conditional
value at risk (CVaR) [65]. A common optimization approach consists of
minimizing the reward-risk trade-off on the basis of a statistical model for
future asset returns, see for instance [48] and [55]. The idea of
incorporating ESG information or more general SRI aspects in this optimization
problem was first pursued in the framework of negative screening. An example
is the portfolio strategy approach proposed by Geczy and Guerard [36], which
combined negative screening with a factor model and mean-variance
optimization.

A more complex set of approaches have added ESG information directly to the
portfolio selection and/or optimization problem. Bilbao at al. [14] introduced
a two step optimization model. In the first step, an efficient frontier is
computed using expected value of wealth as the reward measure and either mean-
variance or CVaR as the risk measure. An investor social behavior satisfaction
index is then computed on the efficient frontier. In the second step, the
portfolio showing the best financial and social behavior is selected. The
approach taken by Hirschberger et al. [42] and Utz et al. [70] involves
portfolio selection in which sustainability is modeled, after risk and return,
as a third selection criterion. Gasser at al. [35] have incorporated an ESG
score as a linear term in the classical mean-variance optimization. Chen at
al. [23] proposed a three step approach: the first step applies data
envelopment analysis to the scores in each of the three ESG categories to
obtain a more informative total score; the second step creates a restricted
investment universe using financial attributes and the ESG scores obtained
from the first step; the third step applies standard mean-variance
optimization to the restricted portfolio. Pedersen et al. [62] integrated the
relative ESG score of each asset into the objective function and studied the
differential impact of environmental and governance scores on portfolio
performance. Schmidt [67] added the portfolio ESG value to the objective
function. Asset weights in a long-only portfolio were then optimized in terms
of return, mean variance and ESG value. In Cesarone et al. [20], the ESG
dimension is added as a hard constraint under mean-variance optimization with
long-only positions solved numerically under data-driven scenarios. Their
analysis is conducted on four investment universes: DJIA, Euro Stocxx 50,
FTSE100, NASDAQ100 and S&P500. They find that optimal portfolios with highest
ESG constraints gave better results only within the DJIA and S&P500 universes.

The discussion above underscores the fact that SRI investing and ESG ratings
are “here to stay”, and that societal pressures will require that the
optimization of financial returns be considered within a larger SRI framework.
We note that ESG scores are an “SRI measure” that, to first-order
approximation, is independent of the traditional return and risk-measures used
in portfolio optimization.333 Literature reviews such as those done by Friede
et al. [33] and Brooks and Oikonomou [19] have revealed less-than-conclusive
results regarding the strength of correlation between ESG scores and financial
performance. It is our goal, in this and further work, to produce a
consistent, ESG-based, asset pricing framework upon which pricing,
optimization, and risk management practices can be based. Our objective is to
add the ESG score as a third dimension, while preserving the main machinery
developed in dynamic asset pricing theory (see for instance [30]). As a
result, we will be able to find optimal reward-risk-ESG asset allocations, but
also be able to hedge the ESG-valued risk by pricing any contingent claims
traded in financial markets. Our work has commonality with approaches
discussed above in that we propose a linearly constrained combination of asset
returns and ESG scores to produce ESG-valued returns. Our work then builds
consistently upon this ESG-valued framework, extending its development into
principle components of modern finance theory.

In Section 2 we present this linearly constrained transformation; in Section 3
we develop its application to portfolio optimization, and analyze empirically
the ESG-valued efficient frontier using a portfolio of stocks from the Dow
Jones Industrial Average. In Section 4 we consider ESG-valued measures
appropriate for evaluating the performance of ESG-valued optimizations. In
Section 5, we extend the results by developing an ESG-valued risk-free rate to
enable the identification of tangent portfolios to the ESG-valued efficient
frontier, thus formulating an ESG-valued capital asset pricing model (CAPM).
The extension to an ESG-valued model for valuing European contingent claims is
covered in Section 6. A numerical study is also presented. In Section 7 we
investigate the effect of the ESG-adjustment on the computation of a riskless
rate in markets having no riskless asset; the so-called shadow riskless rate
as defined by Rachev et al. [64]. Final discussion is presented in Section 8.

##  2 ESG-Valued Returns

We define the ESG-valued return as an affine combination of the ESG score for
any asset and its return. Let $\text{ESG}_{i,t}$ be the ESG score for asset
$i$ on day $t$, and $\varsigma_{i,t}$ denote the value of $\text{ESG}_{i,t}$
appropriately normalized to the range $[-1,1]$.444 This normalization is
discussed in Section 2.1. Since ESG scores are usually provided as positive
values, the normalization to the range $[-1,1]$ produces a positive (negative)
shift in $\varsigma_{i,t}$ for those assets with ESG scores above (below) some
determined “average” value of the original scale. For any $\lambda\in[0,1]$,
we then define the ESG-valued return

$\zeta_{i,t}(\lambda)=\lambda\frac{\varsigma_{i,t}}{c}+\left(1-\lambda\right)r_{i,t}.$
(1)

The constant $c\in R$ is needed to insure that the ranges of
$\varsigma_{i,t}/c$ and $r_{i,t}$ are comparable. The value of $c$ will depend
on the return time period; for the daily returns considered here, we used the
value $c=255$ to approximate the number of trading days in a year. Note that
$\lambda$ is chosen by the investor; for our purposes, we consider it to be
independent of $i$ and $t$. The ESG-valued return defined in (1) assigns more
weight to the ESG score of each asset as $\lambda\uparrow 1$. In the limit
$\lambda=1$, the investor abandons any concern regarding the financial risk-
reward trade-off and concentrates solely on the ESG risk-reward impact on the
portfolio. Conversely, when $\lambda=0$, the investor abandons any interest in
ESG.

To avoid confusion, the phrases “ESG-valued return” and “adjusted return” will
always refer to $\zeta_{i,t}(\lambda)$. The word “return” used alone will
refer to the financial return $r_{i,t}$.

###  2.1 Scaling of ESG Values

To use the ESG-valued return (1), the investor has to determine a scoring
system that reflects desired ESG criteria. As discussed in the Introduction,
preference might, for example, be given to considering environmental
implications over social aspects; the choice may be driven by ethical values
or by consideration of profit maximization.555 Some literature has focused on
the higher returns provided by portfolios based on certain ESG criteria.
Pedersen at al. [62] found that governance factors are more linked to
financial performance than those related to environmental impact. Choice of
criteria will direct choice of scoring methodology and perhaps scoring agency.

Given a user-determined decision on the scoring system, we first consider the
issue related to the time scale $\Delta t$ embedded in (1). Currently, ESG
scores are provided on an annual basis,666 Although some providers distribute
monthly scores, these data are usually expensive and are not provided on a
regular basis. whereas the time scale (hence, the return data) required in (1)
may vary from monthly to intra-day. Proceding with our choice of $\Delta t=1$
day, we discuss the issue of assigning values to $\text{ESG}_{i,t}$ on a daily
basis. A straightforward approach is to utilize the last published ESG score
for each date $\tau\in\left\\{t-T+1,\dots,t\right\\}$. While this takes
advantage of the most recent ESG information, the ESG score for asset $i$ will
remain constant over the period of a year. Under this approach,
$\varsigma_{i,t}$ in (1) is the normalized value of
$\text{ESG}_{i,t_{\text{y}}}$ for asset $i$ at the last yearly issue date,
$t_{\text{y}}$, of new ESG values. A clear drawback to this approach is the
introduction of a (potentially large) jump discontinuity in the value of
$\varsigma_{i,t}$ as the ESG year changes, with a consequent discontinuity in
the value of the ESG-valued return. With ESG scores only available at yearly
time points, a fill-in procedure to produce a smoothly varying daily ESG score
would require both interpolation and extrapolation methods. An alternate
approach would be to model the ESG score as a random variable. This approach,
which should be more appropriate as the ESG score is affected by uncertainty,
is challenging to implement given the current, relatively low frequency of ESG
data, especially when compared to that of financial returns.

Taking the above considerations into account, we invoke the philosophy of
using the most current data available and adopt the following procedure in
this paper. In most financial computations (e.g. portfolio optimization,
option pricing), a “look-back” window of $T$ days is required in order to
compute statistics summarizing the random behavior of the assets under
consideration. We utililze the most current ESG value available for each stock
during the look-back period, regardless of the size of $T$. For example, to
obtain optimized portfolio weights for 06/06/2019 using a window of 510 days
(5/25/2017 through 6/5/2019), the most recent ESG scores available from
Refinitiv were from 12/31/2018. These ESG scores were used in (1) to compute
adjusted returns for each day in the look-back period.

We consider next the issue of normalizing agency-provided ESG scores to the
interval $[-1,1]$. While a linear normalization appears immediate and obvious,
this is based upon inherent assumptions of linearity within the agency’s
methodology. To illustrate this issue quantitatively we consider the ESG
scores from Refinitiv for 29 of the stocks in the Dow Jones Industrial Average
(DJIA). These are shown in Table A.1 for the years 2013 through 2020.
Refinitiv ESG scores are based on a $[0,100]$ scale.777 Environmental, Social
and Governance Scores from Refinitiv; Feb. 2021.
https://www.refinitiv.com/content/dam/marketing/en_us/documents/methodology/refinitiv-
esg-scores-methodology.pdf We applied the obvious linear transformation to
normalize these to $[-1,1]$. Box-whisker summaries of the resultant
transformed scores are shown, by year, in Fig. 1. Very few of the stocks have
scaled ESG values that are negative, implying very few (and none for 2018
through 2020) stocks have a Refinitiv score below 50. The ESG scores of the
lower rated companies improved fairly rapidly over the period 2013 through
2017. Adjusting the ESG normalization to reflect a yearly change in the median
ESG value (which might, for example, be interpreted as the appropriate
percentile for separating “good” from “bad” ESG ratings) would introduce
hidden variability into (1). Addressing this issue in further depth remains
outside of the intended scope of this paper; therefore we proceed with the
simple linear transformation to normalize the provided Refinitiv scores from
$[0,100]$ to $[-1,1]$.

Figure 1: Box-whisker summaries of normalized Refinitiv data for the DJIA data
listed in Table A.1.

##  3 ESG-Valued Portfolio Optimization

We apply the ESG-valued return model to portfolio optimization. We consider a
quite general framework; at the end of day888 We develop our formalism for
daily returns. Reformulation for other time intervals is straightforward,
assuming appropriate time-varying values for the ESG scores can be found. $t$
an investor has to decide how to reallocate the value $W_{t}$ of a portfolio
among a universe ${\cal I}$ of $I$ assets by selecting the fraction (the
weight) $\theta_{i}$ to be invested in asset $i=1,\dots,I$. The investor
observes the series of $T$ previous asset returns $\\{r_{i,\tau};\
\tau=t-T+1,\dots,t;\ i=1,\dots,I\\}$ and, based upon some discrete-state,
parametric or non-parametric, statistical model, obtains a series of $S$ one-
period-ahead returns $\\{\hat{r}^{s}_{i,t+1};\ s=1,\dots,S;\ i=1,\dots,I\\}$.
Under these assumptions, given the vector of portfolio weights
$\theta=\left[\theta_{1},\dots,\theta_{I}\right]^{\prime}$, the simulated
return of the portfolio between $t$ and $t+1$ is defined by the set of $S$
scenarios

$\hat{R}_{t+1}=\left\\{\hat{R}^{s}_{t+1}=\sum_{i=1}^{I}\theta_{i}\hat{r}^{s}_{i,t+1};\
s=1,\dots,S\right\\}.$ (2)

The traditional approach is to assign weights by optimizing the trade-off
between the financial risk and financial reward of the portfolio. The reward
is usually measured by the expected value of the future return
$\mathbb{E}[\hat{R}_{t+1}]$, whereas there are many choices for the risk
measure $\mathbb{V}[\hat{R}_{t+1}]$.999 Artzner at al. [4] proposed an
axiomatic approach to define the minimum set of propertie that
$\mathbb{V}[\cdot]$ should satisfy to be a coherent measure of risk. Frittelli
at al. [34] have specified requirements to define a convex risk measure.  Once
the risk measure $\mathbb{V}[\cdot]$ has been selected, the optimal portfolio
can be obtained from the following minimization problem,

$\min_{\theta}\left\\{-\alpha\mathbb{E}[\hat{R}_{t+1}]+(1-\alpha)\mathbb{V}[\hat{R}_{t+1}]\right\\},$
(3)

subject to the constraints

$\displaystyle\sum_{i=1}^{I}\theta_{i}=1,$ (4)
$\displaystyle\sum_{i=1}^{I}|\theta_{i}-\bar{\theta}_{i}|\leq\gamma,$ (5)
$\displaystyle\theta_{i}\geq 0;\ i=1,\dots,I,$ (6)

for some parameter $\alpha\in[0,1]$ that represents the risk-aversion attitude
of the investor. Constraint (5) forces the portfolio turnover to be less than
the pre-specified parameter $\gamma$; here $\bar{\theta}_{i},i=1,\dots,I$ are
the portfolio weights at time $t$. Constraint (6) avoids short-selling and can
be relaxed when the investor wishes to enter into such positions.

###  3.1 ESG Efficient Frontier

The ensemble of ESG-valued return scenarios is

$\hat{Z}_{t+1}=\left\\{\hat{Z}^{s}_{t+1}=\sum_{i=1}^{I}\theta_{i}\hat{\zeta}^{s}_{i,t+1};\
s=1,\dots,S\right\\},$ (7)

where

$\left\\{\hat{\zeta}^{s}_{i,t+1}(\lambda)\right\\}_{s=1}^{S}=\lambda\frac{\varsigma_{i,t}}{c}+\left(1-\lambda\right)\left\\{\hat{r}^{s}_{i,t+1}\right\\}_{s=1}^{S}.$
(8)

We consider the portfolio optimization problem (3)-(6) when $\hat{R}_{t+1}$ is
replaced by $\hat{Z}_{t+1}$. We illustrate ESG-valued optimization by
application to a specific portfolio consisting of 29 of the 30 stocks101010
Due to its shorter span of existence, stock from DOW Inc is not included in
the portfolio. comprising the DJIA. We use the ESG scores in Table A.1
provided by Refinitiv for each asset over the period 2013-2020. We employ the
scaling methodology discussed in Section 2.1 for daily ESG scores.

To generate the ensemble of values
$\left\\{\hat{r}^{s}_{i,t+1};s=1,\dots,S\right\\}$ in (8), we applied an
ARMA($p_{i},q_{i}$)-GARCH(1,1) model to the set of historical returns
$\\{r_{i,\tau};\tau=t-T+1,\dots,t;i=1,\dots,I\\}$. We employed the R package
rugarch [37] to perform the ARMA-GARCH fits. The function autoarfima was used
to fit the ARMA parameters $p_{i}$ and $q_{i}$ over the ranges $0\leq
p_{i}\leq 2$, $0\leq q_{i}\leq 2$ using the Bayesian information criterion.
Separate fits were performed for each stock. For each asset, a GARCH(1,1) fit
was then performed on the residuals of the best-fit ARMA model using the
function ugarchfit and the assumption that the innovations are normal inverse
Gaussian (NIG) [8, 9]. In the case in which ugarchfit failed to converge, the
autocorrelation
$\mathbb{E}[r_{i,\tau_{1}}^{2},r_{i,\tau_{2}}^{2}],\tau_{1}\neq\tau_{2}$ was
computed to check for persistence. If no persistence was found, the GARCH(1,1)
component was dropped in preference of a pure ARMA model.111111 Persistence
was found to be both stock and time dependent; even for the same stock, the
presence of persistence could change over time. The $I\times T$ set of
innovations resulting from the ensemble of ARMA($p_{i},q_{i}$)-GARCH(1,1) fits
was then fit to a multidimensional NIG model using the R package ghyp [72].
From the parameterized, multidimensional NIG model, $I\times S$ innovations
were randomly generated. The generated innovations for asset $i$ were then fed
into the ARMA($p_{i},q_{i}$)-GARCH(1,1) fit to generate $S$ sample returns for
day $t+1$ for asset $i$. The choice of the NIG distribution family for the
innovations was motivated by its flexibility in modeling potentially different
heavy-tailed behavior for each assset, while preserving asset return processes
as semimartingales, a property that is necessary to perform option pricing.
Since NIG distributions are closed under affine transformations, the ESG-
valued returns (8) remain within the same distribution class as the unadjusted
returns, again preserving the semimartingale property.

Two choices were used for the risk metric $\mathbb{V}[\hat{R}_{t+1}]$ in (3),
mean-variance (MV) and mean-CVaRβ (mCVaRβ) with $\beta=0.95$ and $0.99$.
Details on solving the minimization problem (3)-(6) using each of these
measures are provided in Appendix B. Let
$\theta^{*}=[\theta^{*}_{1},\dots,\theta^{*}_{I}]$ denote the vector of
optimal weights for any of these solutions.121212 For brevity we omit labeling
the $\lambda$ and $\alpha$ dependence of $\theta^{*}$. We can then define the
optimal portfolio return and ESG-valued return,

$\displaystyle\hat{R}^{*}_{t+1}(\lambda)$
$\displaystyle=\sum_{i=1}^{I}\theta_{i}^{*}r_{i,t+1},$ (9)
$\displaystyle\hat{Z}^{*}_{t+1}(\lambda)$
$\displaystyle=\sum_{i=1}^{I}\theta_{i}^{*}\zeta_{i,t+1}(\lambda),$ (10)

where $r_{i,t+1}$ are the realized asset returns on the close of trading day
$t+1$, from which the ESG-valued returns $\zeta_{i,t+1}(\lambda)$ are
computed. The portfolio price is computed in the usual manner,

$P_{t}(\lambda)=P_{0}e^{R_{\text{cum}}(t;\lambda)};\quad
R_{\text{cum}}(t;\lambda)=\sum_{\tau=1}^{t}\hat{R}^{*}_{\tau}(\lambda).$ (11)

The ESG-valued numeraire for the portfolio is similarly defined as

$P^{(Z)}_{t}(\lambda)=P^{(Z)}_{0}e^{Z_{\text{cum}}(t;\lambda)};\quad
Z_{\text{cum}}(t;\lambda)=\sum_{\tau=1}^{t}\hat{Z}^{*}_{\tau}(\lambda).$ (12)

We assign the initial adjusted value of this numeraire to be $P^{(Z)}_{0}=1$.
For simplicity we refer to $P^{(Z)}_{t}(\lambda)$ as the ESG-valued price. For
$\lambda=0$, the portfolio price (11) time series is just a
$P_{0}/P^{(Z)}_{0}$ scaled version of the ESG-valued price (12). Any
significant differences only occur for $\lambda>0$. The normalized ESG score
for the optimized portfolio is

$\varsigma^{*}_{t+1}=\sum_{i=1}^{I}\theta_{i}^{*}\varsigma_{i,t+1}.$ (13)

As we have invoked a linear normalization
$\varsigma_{i,t}=\mathcal{L}(\text{ESG}_{i})$, this is equivalent to

$\text{ESG}^{*}_{t+1}=\sum_{i=1}^{I}\theta_{i}^{*}\text{ESG}_{i,t+1}.$ (14)

We illustrate the optimization procedure by solving for optimal weights for
the date 12/30/2019 for our illustrative portfolio. We use a historical look-
back period of $T=510$ days and generate $S=10,000$ simulated returns. Three
optimizations were obtained, using MV, mCVaR0.95 and mCVaR0.99 risk measures.
Optimizations were performed for four choices of
$\lambda=\left\\{0,0.25,0.50,0.75\right\\}$. For each $\lambda$, the efficient
frontier was computed using the sequence of values $\alpha\in$ {0, 0.01, 0.02,
…, 0.99}.131313 For this portfolio, there is essentially no change in the
optimized solutions over the range $\alpha\in[0.9,0.99]$. The efficient
frontier can be examined either in the three-dimensional space
$(\mathbb{V}[\hat{Z}^{*}],\mathbb{E}[\hat{Z}^{*}],\text{ESG}^{*})$ or in
$(\mathbb{V}[\hat{R}^{*}],\mathbb{E}[\hat{R}^{*}],\text{ESG}^{*})$. The
efficient frontiers are plotted in
$(\mathbb{V}[\hat{Z}^{*}_{t+1}],\mathbb{E}[\hat{Z}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space in Fig. 2.

a  b  c  d  e  f

Figure 2: ESG-valued return efficient frontiers (top) plotted in
$(\mathbb{V}[\hat{Z}^{*}_{t+1}],\mathbb{E}[\hat{Z}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space and (bottom) projected onto the
$(\mathbb{V}[\hat{Z}^{*}_{t+1}],\mathbb{E}[\hat{Z}^{*}_{t+1}])$ plane. Left to
right: a,d) MV, b,e) mCVaR0.95 and c,f) mCVaR0.99 optimizations. Points on
each curve represent equally spaced values of $\alpha\in$ {0, 0.01, 0.02, …,
0.99}.

At the higher $\lambda$ values, both $\text{ESG}^{*}$ and
$\mathbb{E}[\hat{Z}^{*}]$ increase rapidly for low-to-moderate values of the
parameter $\alpha$. This is especially noticeable for the MV portfolios when
$\lambda=0.5$ and $0.75$ (Fig. 2(a)). Increasing the value of $\lambda$
decreases the value of the ESG-valued risk measure $\mathbb{V}[\hat{Z}^{*}]$
and increases the value of the $\mathbb{E}[\hat{Z}^{*}]$. The increase of
$\mathbb{E}[\hat{Z}^{*}]$ with $\lambda$ follows from (8); increasing
$\lambda$ puts more weight on optimizing the ESG score of the portfolio. The
observed increase of $\text{ESG}^{*}$ with $\lambda$ follows from the same
reasoning. As we hold ESG scores constant throughout a year’s time, increasing
$\lambda$ puts more weight on $\varsigma_{i,t}$ resulting in the observed
decrease of $\mathbb{V}[\hat{Z}^{*}]$.

Fig. 3 investigates the results in
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space, using the traditional $\mathbb{V}[\hat{R}]$ and $\mathbb{E}[\hat{R}]$
axes.

a  b  c  d  e  f

Figure 3: ESG-valued return efficient frontiers (top) plotted in
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space and (bottom) projected onto the
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}])$ plane. Left to
right: a,d) MV, b,e) mCVaR0.95 and c,f) mCVaR0.99 optimizations. Points on
each curve in (a)-(c) represent equally spaced values of $\alpha\in$ {0, 0.01,
0.02, …, 0.99}.

The $\lambda=0$ curve represents the traditional efficient frontier under
which ESG scores are not considered. For this portfolio and date, on the
$\lambda=0$ frontier $\text{ESG}^{*}_{t+1}$ increases with increasing values
of $\alpha$; the choice of more risky portfolios giving rise to a higher ESG
score. This is not a general result as it depends on the distribution of
returns estimated at the particular date.141414 In the supplementary material
to this paper we show the $\lambda=0$ efficient frontier computed for a
different date in which the portfolio ESG score initially increases with
$\alpha$, but then decreases as $\alpha$ and risk continue to increase. For
the higher $\lambda$ values, $\mathbb{E}[\hat{R}^{*}]$ also increases rapidly
for low-to-moderate values of the parameter $\alpha$; again most noticeably
for the MV portfolios when $\lambda=0.5$ and $0.75$ (Fig. 3(a)). There is a
nonlinear decrease in both $\mathbb{E}[\hat{R}^{*}_{t+1}]$ and
$\mathbb{V}[\hat{R}^{*}_{t+1}]$ as $\lambda$ increases (for any fixed value of
$\alpha$), which is seen most clearly in the projection on the
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}])$ plane. The
most significant decrease in these values occurs when $\lambda$ increases from
0.5 to 0.75. The behavior of this non-linear decrease changes with $\alpha$.
When viewed in the full
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space, it is clear that this shift (at fixed $\alpha$) in the efficient
frontier shift is toward larger $\text{ESG}^{*}_{t+1}$ values. Noticeable on
the $\lambda=0.75$ efficient frontiers is a “kink” in its convex shape. For
mCVaR0.99 optimization, the $\lambda=0.75$ efficient frontier becomes less
defined due to sparsity of data at the 99’th percentile.

Since new ESG scores arrived on 12/31/2019, the choice of the optimization
date of 12/30/2019 enabled a test of the stability of the optimization results
by comparing the use of in-sample (released on 12/31/2018) versus out-of-
sample (released on 12/31/2019) ESG scores. Fig. 4 plots the relative
difference151515 The relative difference in the portfolio ESG score is defined
in the usual manner as (ESG(out-of-sample) $-$ ESG(in-sample))/ESG(in-sample).
between the ESG score of the optimal portfolio computed using individual asset
ESG scores released on 12/31/2019 compared with the computations in Figs. 2
and 3 computed using the ESG data released on 12/31/2018.

a MV b mCVaR0.95 c mCVaR0.99

Figure 4: Relative difference between the ESG score of the 12/30/2019 MV and
MCVaR optimized portfolios computed using ESG scores released on 12/31/2019
(out-of-sample) compared with the computations obtained using ESG scores
released on 12/31/2018 (in-sample). Note the truncated range of the $\alpha$
axis for the MV optimized results.

In general, over all three risk measures, the relative difference does not
exceed 5% percent, indicating a quite stable ESG portfolio value (at least for
this date and these securities). The relative difference becomes zero as
$\alpha$ increases, approaching zero for smaller values of $\alpha$ as
$\lambda$ increases. For the MV optimized portfolios, the relative ESG
difference rapidly disappears over the range $0\leq\alpha\leq 0.1$, The
sensitivity of the MV optimal solution to ESG score dramatically decreases for
values $\alpha>0.1$ as the optimized portfolios become extremely concentrated
in very few securities. The relative difference persists longest under
mCVaR${0.99}$ optimization, staying roughly constant until $\alpha>0.6$.

In Fig. 5 we depict the optimal portfolio weight composition along the
efficient frontiers (i.e. as a function of $\alpha$) of the mCVaR0.99
optimizations for each value of $\lambda$ considered.

Figure 5: Variation of the weight composition of the mCVaR0.99 optimal
portfolios along each efficient frontier (as a function $\alpha$). Each panel
corresponds to a different choice of $\lambda$. Assets with lower ESG scores
are depicted with warmer colors. The legend identifies each stock and the
stock’s ESG score as assigned by Refinitiv on 12/31/2018.

For a given choice of $\lambda$, the diversity of the portfolio decreases as
$\alpha$ increases; at high $\alpha$ the optimizer focuses on those assets
with highest expected return. For a fixed value of $\alpha$, an increase in
$\lambda$ leads to a portfolio composition that decreases the weight of lower
ESG-scoring stocks. The effect is most noticeable for $\lambda=0.75$. Similar
results (not shown) were obtained for the MV and mCVaR0.95 optimizations.

##  4 ESG-Valued Performance Measures

A common approach to evaluate the performance of an investment strategy is to
use one or more reward-risk ratios (RRRs) to quantify the trade-off between
expected reward and the risk associated with the strategy. There is an
extensive list of such ratios, though not all of them posses the properties of
a coherent risk measure; see Cheridito and Kromer [24] for a complete overview
of RRRs and an analysis of their properties. For instance, the extensively
used Sharpe ratio does not satisfy the monotonicity property. The development
of ESG-valued RRRs involves deep questions concerning the desired properties
that the measures should possess; for example, the coherent risk measure
properties developed by Cherdito and Kromer may comprise a necessary, but not
sufficient, set of properties for ESG-valued RRRs. The scope of this question
is outside of this paper. We illustrate here a “straightforward” strategy for
developing ESG-valued RRRs.

For any choice of an RRR, one strategy for developing an ESG-valued
counterpart involves replacing the portfolio return $\hat{R}$ in the
definition of the RRR with the ESG-valued return $\hat{Z}$. For illustration,
we consider the stable tail adjusted return (STAR) ratio [58],

$\text{STAR}_{\beta}(\hat{R})=\frac{\mathbb{E}[(\hat{R}-r_{f,t})^{+}]}{\text{ETL}_{\beta}[(\hat{R}-r_{f,t})^{+}]},$
(15)

where $r_{f,t}$ is a risk-free rate appropriate for day $t$. The ESG-valued
STAR ratio is then

$\text{STAR}_{\beta}(\hat{Z})=\frac{\mathbb{E}[(\hat{Z}-r_{f,t})^{+}]}{\text{ETL}_{\beta}[(\hat{Z}-r_{f,t})^{+}]}.$
(16)

The ESG-valued STAR ratio satisfies all four properties of a coherent risk
measure. As the STAR ratio will vary along the efficient frontier (i.e. with
$\alpha$) and from frontier to frontier (i.e. with $\lambda$), in Fig. 6 we
summarize the behavior of the STAR ratio as the surface STAR($\lambda,\alpha$)
for the mCVaR0.99 optimization.

Figure 6: The ESG-valued STAR ratio computed as a function of the selected
$\alpha$ and $\lambda$ values for the mCVaR0.99 optimization using equations
(16) and (17).

(To compute the STAR ratio, we used the 10-year U.S. treasury yield rate to
determine daily values for $r_{f,t}$. Computation of the expectation and ETL
in the STAR ratio is performed over the ensemble of ESG-valued returns
generated for day $t+1$ based upon the information available at day $t$.) As
only four values for $\lambda$ were computed while the $\alpha$ grid is much
finer, the smoothly interpolated surface appears coarser in the $\lambda$
direction. The surface shows a generally convex relationship between the ESG-
valued STAR ratio and $\lambda$ and $\alpha$, although variation with $\alpha$
is smaller. The “kink” that appears in the $\lambda=0.75$ efficient frontier
produces the non-convexity observed in the surface.

Another strategy would be to replace $r_{f,t}$ with an ESG-valued risk-free
rate, such as that developed in equation (18) for defining tangent portfolios.
Fig. 6 also shows the surface STAR($\lambda,\alpha$) for the mCVaR0.99
optimization using the formulation

$\text{STAR}_{\beta}(\hat{Z})=\frac{\mathbb{E}[(\hat{Z}-\zeta_{f,t}(\lambda))^{+}]}{\text{ETL}_{\beta}[(\hat{Z}-\zeta_{f,t}(\lambda))^{+}]}.$
(17)

with $\zeta_{f,t}(\lambda)$ defined in (18). Because the risk-free rate
$r_{f,t}$ has no $\lambda$ dependence, as $\lambda$ increases the behavior of
the STAR ratio is effectively that of
$\mathbb{E}[\hat{Z}]/\text{ETL}_{\beta}[\hat{Z}]$ which increases with
$\lambda$. In contrast, the risk-free rate $\zeta_{f,t}$ has strong $\lambda$
dependence, especially so since this risk-free rate is assigned an ESG score
of 100, larger that that of any assset in the optimized portfolio and,
consequently, larger than that of the portfolio. As a result, for large
$\lambda$ the $\zeta_{f,t}$ term becomes the dominating factor in the
numerator and denominator of (17). Apparently this effect exists even for
small, positive values of $\lambda$, resulting in a monotonic decrease in the
STAR ratio as $\lambda$ increases. In contrast to the $r_{f,t}$ surface, which
shows no strong $\alpha$ dependence, as $\lambda$ increases, the $\alpha$
dependence of the $\zeta_{f,t}$ surface becomes more pronouced and the STAR
ratio decreases with a decrease in $\alpha$ (more risk-averse optimization
producing smaller STAR ratios). The STAR ratio becomes negative under these
conditions.

###  4.1 Performance Over Time

We now investigate the ESG-optimized investment results over time using the
same DJIA-derived portfolio of stocks as in Section 3.1. We consider a daily
investment horizon spanning the period 01/03/2017 through 12/30/2020. Using
the procedure outlined in Section 3.1 involving 1-day ahead forecasting based
upon two years of previous returns, for each day $t$ we computed an efficient
frontier using the sequence of values $\alpha\in[0,0.001,0.002,...,0.9]$ for
each value of $\lambda\in[0,0.25,0.5,0.75]$, obtaining a set of optimal asset
weights for each $\alpha,\lambda,t$ combination. We employed only the MV and
mCVaR0.99 optimizations to examine central and tail behavior. Transaction
costs were included in the evaluation by assuming a cost of 2 basis points for
each buy and each sell order executed in rebalancing the portfolio. We imposed
a daily turnover constraint of 0.4% ($\gamma=0.004$ in (5)) to approximate
reasonable business practice. The results are compared to a benchmark
consisting of the equally weighted, buy-and-hold portfolio (EWBH) of the same
29 DJIA stocks.161616 With reference to equations (9) and (10), the EWBH
portfolio is defined as follows. $I=29$; the stock for Dow Inc. remains
excluded. No scenario sets are generated; no optimization is performed. At
$t=0$, $\theta_{i}^{*}=1/29$. For $t>0$, the number of shares of each stock
remains fixed at $n_{i,t}=n_{i,0}$; thus
$\theta_{i,t+1}^{*}=n_{i,0}P_{i}(t)/\sum_{j=1}^{I}n_{j,0}P_{j}(t)$, where
$P_{i}(t)$ is the price of the asset at the close of trading day $t$.

The efficient frontier portfolios were evaluated using standard performance
measures: total return (Tot Rtn), annualized return (Ann Rtn), average
turnover (AvgTO), expected tail loss (ETL95)171717 Also known as conditional
value-at-risk or expected shortfall.  and expected tail return (ETR95), both
at 95% significance levels, and maximum drawdown (MDD). ETR95 is the reward
counterpart of ETL95, measuring the average value of those returns above the
0.95 quantile. MDD is generally defined as the maximum drop of portfolio
returns over all time intervals that can be formed as a partition of the
investment period [22]. As performance measures, we also considered the
average ESG score of a portfolio over the time period, as well as its standard
deviation. Table A.2 presents time-averaged values for these summary
performance measures for select ($\lambda,\alpha$) efficient frontier
portfolios for both mCVaR0.99 and MV optimizations.

The total and annualized returns of the optimized portfolios are generally
superior to that of the benchmark for values of $\alpha\geq 0.7$ for mCVaR0.99
and $\alpha\geq 0.1$ for MV optimizations. For fixed $\lambda\geq 0.5$, as
$\alpha$ increases, AvgTO tends to fall below the daily constraint value of
0.4%. For fixed $\alpha>0$, AvgTO decreases with $\lambda$. This effect is of
particular importance under MV optimization, which suffers from instability of
solutions and high turnover rates which can downgrade financial performance
[26, 47, 59]. For mCVaR0.99, except for the largest values of $\lambda$ and
$\alpha$, the magnitudes of ETL95 and ETR95 are less than that of the EWBH
benchmark. For MV, these magnitudes generally exceed that of the benchmark. In
the opimized portfolios, ETL95 always exceeds ETR95 in magnitude, with the
suggestion that, for MV optimization, at large values of $\lambda$ the
magnitude difference decreases as $\alpha$ increases. All optimal portfolios
have lower MDD than the benchmark. For optimization under mCVaR0.99, the MDD
decreases with $\lambda$ for each fixed value of $\alpha$. Average ESG scores
for the optimized portfolios virtually always exceed that of the benchmark.
Under mCVaR0.99 optimization the standard deviation of the ESG score is
generally smaller than that of the benchmark; for MV optimization the reverse
is true.

a Price  b ESG Score c Price  d ESG Score e Price  f ESG Score

Figure 7: (a)-(d) Time series over the period 01/03/2017 through 12/30/2020 of
price and ESG score for $\alpha=0.7$ efficient frontier portfolios obtained
from the mCVaR0.99 optimization with (a)-(b) no and (c)-(d) 0.4% daily
turnover restriction. Plots (e)-(f) are for the tangent portfolios with 0.4%
daily turnover restriction. In each, a comparison is made against the equi-
weighted porfolio EWBH.

The time series for both the price (assuming an initial investment of $10,000)
and ESG score for the $\alpha=0.7$ point of the
$\lambda\in\\{0,0.25,0.5,0.75\\}$ efficient frontiers of the mCVaR0.99
optimized portfolios are shown in Fig. 7. The values for the benchmark
portfolio, EWBH, are also shown. As previously noted, the price and ESG time
series were computed with a daily turnover constraint of 0.4%. For discussion
purposes, these time series were also computed under optimization with no
turnover constraint. The unconstrained time series are also plotted in Fig. 7.
With no turnover constraints, the ESG optimized ($\lambda>0$) portfolios
outperform the benchmark, both in cumulative price and portfolio ESG value.
With a realistic daily turnover constraint of 0.4% to control transaction
costs, the EWBH benchmark outperforms the $\alpha=0.7$ optimized portfolios
until the Covid-19 pandemic. Post pandemic, the $\lambda=0.75,\alpha=0.7$
portfolio exhibits strong recovery while the $\lambda=0.5,\alpha=0.7$
portfolio displays recovery competitive with that of the benchmark. Each
turnover-unconstrained portfolio rapidly establishes a separate ESG level
(around which it displays variance). Under the turnover constraint, the
portfolio ESG scores climb to their separate levels more slowly, but with much
less variance. The noticeable discontinuity in portfolio ESG values on
01/01/2020 arises from rather large changes in the ESG ratings of a number of
these companies in the 12/31/2019 data release compared to that of 12/31/2018.
(See Table A.1.)

We computed moment values for the distribution of the returns for the same
($\lambda,\alpha$) selection of efficient frontier portfolios as in Table A.2.
Specifically we consider the mean, median, standard deviation (Std), skewness
(Skew), and excess kurtosis (ExKurt). Table A.3 summarizes these values for
both optimizations. For each fixed value of $\lambda$, mean, median returns
and the Std increase with $\alpha$, much more so under MV optimization.
Skewness and ExKurt decrease as $\alpha$ increases.

We also computed the portfolios’ performance relative to five popular RRRs:
the Sharpe (SR), Sortino, STAR, Rachev and Gini ratios [24].181818 Here we
define the Sharpe, Sortino and STAR ratios using only the portfolio returns
and not the excess returns relative to a benchmark. These values are
summarized in Table A.4. The optimized portfolios generally produce superior
RRR values compared to the benchmark over all ranges of $\lambda$ and
$\alpha$. For $\lambda=0.5$ and $0.75$, the RRRs for MV generally outperform
those for mCVaR0.99.

When $\alpha=0$, the choice of $\lambda$ does not significantly alter the
performance of the optimal portfolios as the investor is only concerned with
the risk of the investment, which is unaffected by the ESG score.

##  5 ESG-Valued Tangent Portfolios

The efficient frontiers in the
$(\mathbb{V}[\hat{Z}^{*}],\mathbb{E}[\hat{Z}^{*}])$ plane (see Fig. 2) can be
used to identify the ESG-valued tangent portfolio. This approach will
naturally lead to an ESG reformulation of the capital asset pricing model
(CAPM), the security market line (SML), and the two-fund separation theorem
[69]. Determining the tangent portfolio for each ESG-valued efficient frontier
in this space requires identification of an appropriate risk-free rate process
$r_{f,t}$.

The appropriate risk-free rate to be used should also be ESG-valued as in (1).
This raises the issue of assigning an ESG score to a fixed-income security, in
particular to any government bond. As a first attempt to address this, we
proceeded as follows. We used the yield on the 10-year U.S. treasury bond
$r_{f,t}$ (appropriate scaled to daily rates) and assigned the government bond
a maximum ESG score of 100.191919 Another possibility is to use an ESG score
for the appropriate country for a government bond. Some agencies are beginning
to provide such scores, though not necessarily in the same scale used for
companies. The motivation behind this choice is that the risk-free rate is not
among the set of choices available to the investor; any other ESG value for
the risk-free rate will be more arbitrary than simply assigning the maximum
value. From (1), the appropriate ESG-valued riskless rate is then

$\zeta_{f,t}(\lambda)=\lambda\frac{1}{c}+\left(1-\lambda\right)r_{f,t}.$ (18)

The ESG-valued tangent portfolio for date $t$ will be uniquely determine by
the straight line from the point $(0,\zeta_{f,t}(\lambda))$ which is tangent
to the $\lambda$-specified ESG-valued efficient frontier.

Figure 8: (a)-(d) ESG-valued portfolio values, $P^{(Z)}_{t}(\lambda)$,
obtained for the tangent portfolios under mCVaR0.99 optimization, compared
with the corresponding $P^{(Z)}_{t}(\lambda)$ time series for the EWBH
strategy and the ESG-valued DJIA index. (e) Difference between the
$P^{(Z)}_{t}(\lambda)$ values for the tangent portfolios and the ESG-valued
DJIA index.

Using (18), the ESG-valued tangent portfolio can be identified on any
efficient frontier parameterized by ($\lambda,\alpha,t$).202020 It is critical
to emphasize that the tangent portfolios are identified using the efficient
frontiers computed in
$(\mathbb{V}[\hat{Z}^{*}],\mathbb{E}[\hat{Z}^{*}],\text{ESG}^{*})$ space and
not in $(\mathbb{V}[\hat{R}^{*}],\mathbb{E}[\hat{R}^{*}],\text{ESG}^{*})$
space. The optimization is performed in the former space, not the latter. As
can be seen by comparing Figs. 2 and 3, convexity of the efficient frontiers
can only be guaranteed in the former space.  These ESG-valued tangent
portfolios were identified for the (0.4% daily turnover constrained)
portfolios computed in Section 4.1. Fig. 7 shows the price performance
(obtained from the returns (9)) as well as the ESG scores for the tangent
portfolios computed with mCVaR0.99 optimization. For $\lambda>0$, the price
performance of these turnover constrained tangent portfolios is competitive
with that obtained by the turnover unconstrained, $\alpha=0.7$ efficient
frontier portfolios (compare Figs. 7 (a) and (e)). Allowing for the initial
transient behavior due to the turnover constraint, for $\lambda>0$ the ESG
scores for the turnover constrained tangent portolios are superior to those
for the turnover unconstrained, $\alpha=0.7$ efficient frontier portfolios
(compare Figs. 7 (b) and (f)).

The $P^{(Z)}_{t}(\lambda)$ time series are displayed in Fig. 8 for these
mCVaR0.99 optimized tangent portfolios. $P^{(Z)}_{t}(\lambda)$ time-series are
also shown for the EWBH benchmark as well as for an ESG-valued DJIA
index.212121 With reference to equations (9) and (10), the DJIA index is
defined as follows. $I=29$. No scenario sets are generated;
$\theta_{i}^{*}=w_{i}^{\text{DJ}}/(1-w_{\text{Dow}})$ where
$w_{i}^{\text{DJ}}$ is the DJIA published weight for component $i$ and
$w_{\text{Dow}}$ is the published weight for Dow Inc., which is excluded from
our benchmark index. We utilized the published DJIA weights applicable to
30/12/2020 in composing our index.  Note the close agreement between the EWBH
and ESG-valued index values; both portfolios are comprised of the same stocks,
with different weights applied. The differences between ESG-valued tangent
portfolio and the ESG-valued index values are also plotted in Fig. 8. As noted
earlier, the $\lambda=0$, ESG-valued plots are just scaled versions of the
corresponding price plots. As $\lambda$ increases, the differences between the
optimized portfolio ESG-valued time series and those of the benchmarks (Fig. 8
(e)) grow more rapidly than the corresponding differences between optimized
portfolio and benchmark price series (not shown, but can be inferred from Fig.
7).

Table A.5 provides the performance, moment, and RRR measures for these four
tangent portfolios. At the higher values of $\lambda$, the tangent portfolios
outperform the benchmark. Compared to the benchmark, total (and hence
annualized) return almost doubles for the tangent portfolio as $\lambda$
increases. With the exception of one Gini value, the value for every RRR
considered is better than the benchmark as $\lambda$ increases. Trivially the
tangent portfolio average ESG score improves with $\lambda$. (The standard
deviation of ESG decreases because more weight is given to the ESG score which
remains constant over each year.) The average turnover considerably decreases
as $\lambda$ increases.

##  6 ESG-Valued Option Valuation

We consider the effect of ESG-valued returns on option pricing. Paralleling
equations (11) and (12), we shall refer to the valuations given to options
using ESG-valued returns as “ESG-valued option values” or simply “option
values” and reserve the phrase “option price” to refer to the valuation of
options based on traditional financial returns. When $\lambda=0$, ESG-valued
option values become equal to option prices. For a given date $t$, using the
discrete methodology outlined in Appendix C, we computed European contingency
claim option values for a sequence of expiration dates $t+T$,
$T\in[T_{\text{min}},T_{\text{max}}]$ and strike values
$K\in[K_{\text{min}},K_{\text{max}}]$ using an ESG-valued tangent portfolio as
the underlying. We chose the date $t=12/30/2019$ corresponding to the
efficient frontiers discussed in Sections 3.1 and 5. Option values were
computed for each tangent portfolio. The same methodology was employed to
obtain option values for the same date using the ESG-valued DJIA index as the
underlying. We are interested in the difference between option values based
upon the adjusted DJIA index and the tangent portfolios.

Call and put option values were expressed as functions of $T$ and “ESG-
moneyness” $M=K/P^{Z}_{t}$, where $P^{Z}_{t}$ is the ESG-valued price of the
relevant portfolio on day $t$. We considered ESG-moneyness values in the range
$M\in[0.5,1.5]$ and maturity times $T\in[15,252]$ days. Comparisons of the
call and put value surfaces based on the ESG-valued DJIA index and on the
mCVaR0.99 optimized tangent portfolio for selected values of $\lambda$ are
given in Fig. 9.

a $\lambda=0$ b $\lambda=0.25$ c $\lambda=0.5$ d $\lambda=0.75$ e Call f Put

Figure 9:  (a) to (d): (top) call and (bottom) put option values for select
choices of $\lambda$. The orange surfaces corresponds to option values for the
ESG-valued DJIA index; blue surfaces represents option values for the
mCVaR0.99 tangent portfolios. The difference in (e) call and (f) put option
values between the ESG-valued DJIA index and the mCVaR0.99 tangent portfolios.

The respective call and put option value differences,
$\Phi_{c}^{(\text{mCVaR}_{0.99})}(T,K,\lambda)-\Phi_{c}^{(\text{DJIA})}(T,K,\lambda)$
and
$\Phi_{p}^{(\text{mCVaR}_{0.99})}(T,K,\lambda)-\Phi_{p}^{(\text{DJIA})}(T,K,\lambda)$,
between the tangent portfolios and the DJIA index are also given in Fig. 9.

When $\lambda=0$, call and put values (and hence, call and put option prices)
written on the index and the mCVaR0.99 tangent portfolio are practically
equal. The difference increases with $\lambda$, especially as strike values
move further into-the-money. By construction, ESG-valued tangent portfolios
have a higher ESG score, which in turn implies a positive shift in their
aggregate value, creating higher call and put values. The effect of a positive
$\lambda$ on call and put values is different as a function of time to
maturity; at constant in-the-money values of $K$, call values increase with
$T$ while put values decrease.

a $\lambda=0$ b $\lambda=0.25$ c $\lambda=0.5$ d $\lambda=0.75$

Figure 10:  Implied volatility surfaces computed from the (orange surfaces)
ESG-valued DJIA option values and (blue surfaces) the tangent portfolios
corresponding to the selected choices for $\lambda$.

Fig. 10 compares implied volatility (IV) surfaces, computed by fitting to the
Black-Scholes model, for call options based on the DJIA index and the
mCVaR0.99-optimized tangent portfolios. For $\lambda=0$ and in-the-money $K$
values, the IV surface of the DJIA index is above that of the tangent
portfolio for short maturity times. For longer maturity dates it drops below.
As $\lambda$ increases, the two IV surfaces move upwards (overall volatility
increases); the difference between the surfaces decreases; and the volatility
“smirk” becomes steeper.

##  7 ESG-Valued Shadow Riskless Rate

The existence of a risk-free rate in the economy is a common and implicit
assumption of modern financial theory. However, the situation in which the
risk-free rate is not available for a market has been also considered. The
seminal work is that of Black [16], who proposed a CAPM model without assuming
the existence of a riskless rate. Black later introduced a “shadow real rate”
as an option on fixed-income derivatives [17]. More recently, a different
approach has been introduced by Rachev at al. [64], in which a shadow riskless
rate is defined as a perpetual option on the set of risky securities selected
as the investment universe. Rachev at al. computed analytic formulas for the
shadow riskless rate under different models of the stochastic process driving
the market uncertainty. The simplest case, where diffusion is driven by
Brownian motions, is summarized in Appendix D.

We analyze the effect of ESG-valued returns on the shadow riskless rate (SRR)
derived in (D.3) of Appendix D for an investment universe defined by the 29
stocks comprising our ESG-valued DJIA index. For each trading date in the
period from 01/03/2017 through 12/31/2020, using two-year moving windows we
estimated the vector $[\mu_{i}]$ of mean values of, and the variance-
covariance matrix $\Sigma$ for, the 29 stock returns. The asset (row) entries
in $\Sigma$ were sorted in order of decreasing variance value. The elements
$\sigma_{i,j};\ i=1,\dots,N;\ j=1,\dots,N-1$ required in (D.3) were obtained
from the upper triangular matrix $L^{\prime}$ obtained from the Cholesky
decomposition $\Sigma=LL^{\prime}$. To eliminate one dimension and obtain 28
Brownian motions, the last column of $L^{\prime}$ was replaced with a linear
combination of its last two columns.

a  b

Figure 11:  Time series from 01/03/2017 through 12/31/2020 for select values
of $\lambda$ of (a) the SRR computed from ESG-valued returns on a universe of
29 DFJIA assets and (b) the information ratio of the SRR.

The SRR computations were performed on the DJIA index for the values
$\lambda\in(0,0.25,0.5,0.75)$.222222 We perform the SRR computations on the
DJIA index rather than the optimized portfolios because the Dow Index is
conventially recognized as an indicator of the general health of the US stock
market (though there are valid arguments against this).  The time series for
the four SRRs are presented in Fig. 11. The SRR levels show separation in
rough proportion to the value of $\lambda$; however this proportion is
affected by each year-end readjustment of ESG values. The separation between
SRR curves appears to be roughly constant for the calendar years 2017 and
2019. Over 2018 the separations between the curves narrow. The Covid 19 crash
in 2020 initially removes differences between SRR curves; after which they
again begin to separate.

Figure 12:  The curves $\mu_{\text{IR}}(\lambda)$,
$\sigma_{\text{IR}}(\lambda)$ and the parametric curve
$(\mu_{\text{IR}}(\lambda),\sigma_{\text{IR}}(\lambda))$. The dashed line is a
linear fit.

We define the standard deviation of the price deflator process $\pi$ by
$\sigma_{\pi}=\sqrt{\sum_{j=1}^{N-1}\sigma_{\pi,j}^{2}}$. This enables the
definition of the SRR information ratio,
IR${}_{\text{SRR}}=\mu_{\pi}/\sigma_{\pi}$. The four time series for
IR${}_{\text{SRR}}$ are also plotted in Fig. 11. For each time series, the
average $\mu_{\text{IR}}(\lambda)$ and standard deviation
$\sigma_{\text{IR}}(\lambda)$ values of IR${}_{\text{SRR}}$ can be computed.
The behaviors of $\mu_{\text{IR}}(\lambda)$ and $\sigma_{\text{IR}}(\lambda)$
for $\lambda\in${ 0,0.01,0.02,…,0.89,0.9} are presented in Fig. 12. The value
of $\mu_{\text{IR}}(\lambda)$ intially grows with $\lambda$, reaching a
maximum time-averaged IR${}_{\text{SRR}}$ value at $\lambda=0.31$, after which
the averaged SRR information ratio decreases. The standard deviation
$\sigma_{\text{IR}}(\lambda)$ decreases with $\lambda$; for $\lambda>0.2$, the
decrease is linear. These results quantify the nonlinear impact of $\lambda$
on the shadow riskless rate.

##  8 Discussion

In response to environmental changes and investor, societal, and governmental
pressure, ESG ratings are assuming an important role in financial markets.
Thus motivated, we have introduced a consistent ESG-valued framework for the
inclusion of ESG ratings into dynamic pricing theory. There are two
fundamental parameters in the framework; in addition to the traditional
financial risk aversion parameter ($\alpha\in[0,1]$), we introduce the ESG
affinity parameter $\lambda\in[0,1]$. When $\lambda=0$, our pricing framework
reduces to traditional financial-risk versus financial-reward valuation. As
$\lambda$ increases, the framework places increased emphasis on the impact of
ESG ratings on valuation. Thus our results are presented in terms of
comparisons of valuations for $\lambda>0$ compared with $\lambda=0$. In this
work we have applied the ESG-valued framework to portfolio optimization, risk
measures, option pricing, and the concept of shadow riskless rates. Although
the ESG-valued return (1) is linearly dependent on ESG scores through
$\lambda$, other non-linearities in portfolio and option valuation ultimately
result in non-linear dependence on $\lambda$.

The philosophy incorporated in our framework is that socially responsible
investing places a value on an investment portfolio that is competitive with
(or “additional to”) financial return. Individual investor emphasis on that
value is adjusted through the ESG affinity parameter. As a consequence, our
framework produces valuations (e.g. for portfolios, options, riskless rates)
in terms of an ESG-valued numeraire which reduces to a standard financial
price when $\lambda=0$. Standardization of such a numeraire will become
necessary if ESG-valued valuation is to be adopted. This would require the
development of market indices expressed in terms of a standard ESG (or SRI)
numeraire.

If adopted, a framework such as that developed here will require consideration
of a myriad of issues, some of which are being pursued as described in the
Introduction; others arise from considerations encountered while implementing
the framework. In our view, one of the most important limitations is the
deterministic nature of the ESG rankings employed here; we have ignored the
variability of agency-provided ESG scores. As noted in the introduction, the
definition of methodologies, and the consequent creation of databases related
to ESG scores has experienced huge growth in the last ten years, resulting in
a wide set of options in the choice of rating agency. Although it is not our
intention to further address that issue in this paper, as it has been
extensively explored in literature cited, we provide some insight on its
impact on our ESG-valued valuation framework by considering scores from two
different providers. In the supplement to this paper, we have repeated the
computational results presented in this manuscript using ESG scores provided
by RobecoSAM.

Despite the general disagreement of ESG scores between these two providers, we
have observed that optimal portfolios performances, and hence efficient
frontiers, on the DJIA index present relative stable results between the two
scoring systems. This in turn implies stable results for the option pricing
example in Section 6. However, optimized portfolio compositions can
considerably change when computed with RobecoSAM scores. This in turn is
reflected on the shadow riskless rate implicit in the market, even if computed
under Gaussian hypotheses, exhibiting wide variation in the SRR depending on
the score system used. This result confirms the need for a stronger effort
toward a convergence of methodologies to asses ESG scores.

Given the current and foreseable variation in ESG rating systems and
methodologies, our framework accommodates (and in fact argues in favor of) ESG
ratings as random variables whose stochastic properties reflect these
variations. Evaluation of these properties will become more definitive once
ESG ratings are updated on time scales shorter than the calendar year.
Ideally, company-specific ESG ratings would change on a time scale intrinsic
with the company’s activity.

Our work reveals that adoption of an ESG-valued valuation framework will
require resolution of critical research issues. (i) Our model requires that
ESG-ratings be rescaled to be comparable to financial returns, which requires
scaling to a bounded interval $[\varsigma^{-},\varsigma^{+}]$ with
$\varsigma^{-}<0$, $\varsigma^{+}>0$. This requires the identification of a
threshold ESG value $\text{ESG}_{0}$ satisfying $\varsigma(\text{ESG}_{0})=0$
which effectively separates “good” ESG ratings from “bad” ESG ratings. (ii)
Traditional financial-reward : financial-risk metrics must be modified to
incorporate ESG ratings, thus becoming ESG-valued-reward : ESG-valued-risk
metrics. This raises the research question of “coherence”, the necessary set
of properties that such adjusted metrics should satisfy. (iii) As excess
return is often measured relative to a government set risk-free rate, the
concept of applying ESG ratings to governments and government-issued bonds
becomes a required research focus. (iv) The option pricing model explored here
is based upon a discretization of a continuum model which captures limited
information (specifically, only the standard deviation of the financial
returns of the underlying). Thus another research avenue is the application of
this ESG-valued framework to option valuation models that are capable of
capturing more of the microstructure of the underlying asset price (and, when
it becomes available, that of its ESG ratings). (v) The methodology used here
for the computation of the shadow riskless rate requires identification of the
$N\times N-1$ standard deviation parameters $\sigma_{i,j}$ (equation (D.1) of
Appendix D) from the $N\times N$ elements of the variance-covariance matrix of
asset returns. There is currently no theory governing this identification; the
Cholesky decomposition approach used here is ad hoc.

##  Appendix A Tables

Ticker | 12/31 | 12/31 | 12/31 | 12/30 | 12/29 | 12/31 | 12/31 | 12/31
---|---|---|---|---|---|---|---|---
| 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020
MMM | 88 | 90 | 86 | 88 | 88 | 87 | 89 | 92
AXP | 50 | 51 | 51 | 68 | 73 | 73 | 78 | 78
AMGN | 67 | 65 | 68 | 66 | 75 | 72 | 74 | 74
AAPL | 61 | 57 | 54 | 62 | 69 | 70 | 67 | 73
BA | 59 | 73 | 66 | 69 | 80 | 78 | 80 | 83
CAT | 75 | 69 | 75 | 73 | 65 | 69 | 69 | 72
CVX | 73 | 73 | 80 | 77 | 88 | 84 | 79 | 84
CSCO | 83 | 85 | 85 | 87 | 87 | 88 | 88 | 89
KO | 77 | 72 | 73 | 77 | 75 | 70 | 65 | 76
GS | 63 | 67 | 72 | 62 | 70 | 73 | 86 | 86
HD | 63 | 78 | 70 | 81 | 83 | 74 | 72 | 73
HON | 68 | 53 | 63 | 67 | 68 | 74 | 75 | 83
IBM | 79 | 73 | 81 | 76 | 77 | 80 | 71 | 73
INTC | 86 | 91 | 91 | 90 | 87 | 88 | 88 | 88
JNJ | 91 | 92 | 93 | 91 | 90 | 89 | 88 | 89
JPM | 71 | 71 | 78 | 81 | 83 | 77 | 82 | 82
MCD | 77 | 70 | 64 | 69 | 71 | 75 | 70 | 78
MRK | 78 | 71 | 72 | 80 | 81 | 77 | 80 | 82
MSFT | 92 | 92 | 93 | 91 | 90 | 93 | 93 | 93
NKE | 68 | 65 | 75 | 65 | 70 | 71 | 72 | 74
PG | 60 | 59 | 72 | 68 | 66 | 65 | 71 | 73
CRM | 46 | 41 | 46 | 56 | 63 | 72 | 62 | 67
TRV | 42 | 49 | 43 | 36 | 44 | 63 | 70 | 66
UNH | 62 | 64 | 67 | 82 | 78 | 80 | 72 | 71
VZ | 72 | 69 | 60 | 57 | 66 | 68 | 75 | 76
V | 21 | 51 | 52 | 71 | 72 | 71 | 53 | 54
WBA | 39 | 53 | 60 | 71 | 72 | 74 | 81 | 94
WMT | 78 | 78 | 79 | 78 | 77 | 74 | 81 | 85
DIS | 61 | 62 | 64 | 67 | 65 | 76 | 78 | 71

Table A.1: Refinitiv ESG scores for the DJIA Index components.

Model | Tot. Ret | Ann. Ret | AvgTO | ETL95 | ETR95 | MDD | ESG*
---|---|---|---|---|---|---|---
|  | (%) | (%) | (%) | (%) | (%) | (%) | avg | std
EWBH | 90.14 | 17.68 | 0.00 | -3.41 | 2.86 | 33.00 | 72.72 | 2.97
$\lambda$ | $\alpha$ |  mCVaR0.99 |  |  |
0.00 | 0.0 | 70.75 | 14.53 | 0.40 | -2.74 | 2.41 | 28.47 | 71.97 | 2.05
| 0.3 | 71.53 | 14.66 | 0.40 | -2.74 | 2.40 | 28.36 | 71.99 | 2.06
| 0.5 | 71.85 | 14.71 | 0.40 | -2.75 | 2.41 | 28.36 | 72.02 | 2.01
| 0.7 | 73.88 | 15.05 | 0.40 | -2.76 | 2.44 | 27.62 | 72.33 | 2.17
| 0.9 | 82.60 | 16.49 | 0.40 | -2.90 | 2.55 | 28.23 | 72.61 | 2.93
0.25 | 0.0 | 71.17 | 14.60 | 0.40 | -2.73 | 2.40 | 28.20 | 72.55 | 1.78
| 0.3 | 70.80 | 14.53 | 0.40 | -2.73 | 2.41 | 28.05 | 72.86 | 1.71
| 0.5 | 71.60 | 14.67 | 0.40 | -2.73 | 2.41 | 27.66 | 73.17 | 1.68
| 0.7 | 73.35 | 14.96 | 0.40 | -2.75 | 2.44 | 26.94 | 74.83 | 1.77
| 0.9 | 92.13 | 18.00 | 0.39 | -2.96 | 2.65 | 25.71 | 78.59 | 2.53
0.50 | 0.0 | 69.77 | 14.36 | 0.40 | -2.73 | 2.40 | 28.01 | 73.78 | 1.56
| 0.3 | 69.62 | 14.33 | 0.40 | -2.73 | 2.41 | 27.89 | 74.30 | 1.57
| 0.5 | 84.62 | 16.81 | 0.39 | -2.86 | 2.57 | 25.23 | 80.36 | 2.72
| 0.7 | 93.51 | 18.22 | 0.38 | -2.97 | 2.68 | 24.78 | 82.13 | 3.18
| 0.9 | 112.02 | 20.98 | 0.36 | -3.20 | 2.94 | 24.37 | 84.51 | 4.14
0.75 | 0.0 | 68.57 | 14.15 | 0.40 | -2.74 | 2.41 | 27.43 | 76.50 | 1.95
| 0.3 | 72.50 | 14.82 | 0.40 | -2.79 | 2.46 | 26.67 | 78.85 | 2.38
| 0.5 | 80.31 | 16.12 | 0.39 | -2.84 | 2.53 | 25.69 | 80.89 | 2.87
| 0.7 | 112.57 | 21.06 | 0.34 | -3.26 | 2.98 | 25.39 | 86.65 | 4.97
| 0.9 | 143.49 | 25.30 | 0.33 | -3.61 | 3.35 | 27.40 | 88.19 | 5.66
$\lambda$ | $\alpha$ | MV |  |  |
0.00 | 0.0 | 76.82 | 15.54 | 0.40 | -2.84 | 2.46 | 29.17 | 71.93 | 2.59
| 0.05 | 84.53 | 16.80 | 0.40 | -3.06 | 2.61 | 30.62 | 72.38 | 2.80
| 0.10 | 90.68 | 17.77 | 0.38 | -3.33 | 2.80 | 32.39 | 73.49 | 3.22
| 0.15 | 90.64 | 17.77 | 0.39 | -3.55 | 2.96 | 35.00 | 74.17 | 3.46
| 0.20 | 88.69 | 17.46 | 0.40 | -3.73 | 3.08 | 37.27 | 74.30 | 3.63
0.25 | 0.0 | 85.13 | 16.90 | 0.40 | -2.95 | 2.52 | 30.21 | 72.01 | 2.81
| 0.05 | 84.74 | 16.83 | 0.40 | -3.21 | 2.74 | 30.96 | 77.46 | 3.49
| 0.10 | 102.24 | 19.54 | 0.40 | -3.51 | 3.08 | 30.64 | 81.10 | 4.29
| 0.15 | 109.34 | 20.60 | 0.40 | -3.74 | 3.31 | 31.39 | 82.49 | 4.72
| 0.20 | 114.60 | 21.36 | 0.40 | -3.87 | 3.43 | 32.07 | 83.14 | 5.01
0.50 | 0.0 | 92.87 | 18.12 | 0.40 | -3.14 | 2.63 | 31.68 | 72.00 | 2.78
| 0.05 | 93.35 | 18.19 | 0.40 | -3.42 | 3.01 | 29.52 | 81.58 | 4.96
| 0.10 | 144.99 | 25.50 | 0.35 | -3.76 | 3.54 | 27.01 | 86.22 | 5.80
| 0.15 | 170.92 | 28.74 | 0.32 | -3.95 | 3.75 | 27.47 | 87.59 | 5.94
| 0.20 | 180.31 | 29.86 | 0.30 | -4.05 | 3.84 | 27.72 | 88.01 | 5.99
0.75 | 0.0 | 100.63 | 19.30 | 0.40 | -3.42 | 2.87 | 33.14 | 72.14 | 2.97
| 0.05 | 101.12 | 19.38 | 0.39 | -3.57 | 3.20 | 28.20 | 83.86 | 5.66
| 0.10 | 166.27 | 28.18 | 0.31 | -3.88 | 3.67 | 27.43 | 87.69 | 6.04
| 0.15 | 191.69 | 31.17 | 0.27 | -4.05 | 3.87 | 27.53 | 88.51 | 6.04
| 0.20 | 202.70 | 32.41 | 0.26 | -4.15 | 3.96 | 28.03 | 88.76 | 6.11

Table A.2: Performance measure values for the period 01/03/2017 through
12/30/2020 for select efficient frontier portfolios optimized under mCVaR0.99
and MV. Note the difference between mCVaR0.99 and MV in the range of $\alpha$
provided.

Model | Mean | Median | Std | Skew | ExKurt | Model | Mean | Median | Std | Skew | ExKurt
---|---|---|---|---|---|---|---|---|---|---|---
|  | $\times 1^{-4}$ | $\times 1^{-4}$ | $\times 1^{-2}$ |  |  |  | $\times 1^{-4}$ | $\times 1^{-4}$ | $\times 1^{-2}$ |  |
EWBH | 6.4 | 11.1 | 1.3 | -1.02 | 21.9 |  |  |  |  |  |
$\lambda$ | $\alpha$ |  mCVaR0.99 | $\alpha$ | MV
0.00 | 0.0 | 5.3 | 8.3 | 1.1 | -0.92 | 22.1 | 0.00 | 5.7 | 9.3 | 1.1 | -0.93 | 20.4
| 0.3 | 5.4 | 8.0 | 1.1 | -0.92 | 22.2 | 0.05 | 6.1 | 10.7 | 1.2 | -1.08 | 22.9
| 0.5 | 5.4 | 8.5 | 1.1 | -0.92 | 22.3 | 0.10 | 6.4 | 11.1 | 1.3 | -1.17 | 23.1
| 0.7 | 5.5 | 9.6 | 1.1 | -0.90 | 22.4 | 0.15 | 6.4 | 11.2 | 1.4 | -1.32 | 23.9
| 0.9 | 6.0 | 9.9 | 1.2 | -0.97 | 22.7 | 0.20 | 6.3 | 11.5 | 1.5 | -1.40 | 24.1
0.25 | 0.0 | 5.3 | 8.7 | 1.1 | -0.88 | 21.8 | 0.00 | 6.1 | 10.1 | 1.1 | -1.02 | 21.0
| 0.3 | 5.3 | 8.6 | 1.1 | -0.89 | 21.8 | 0.05 | 6.1 | 10.6 | 1.3 | -1.05 | 23.2
| 0.5 | 5.4 | 8.5 | 1.1 | -0.88 | 21.8 | 0.10 | 7.0 | 12.3 | 1.4 | -0.91 | 20.8
| 0.7 | 5.5 | 10.2 | 1.1 | -0.82 | 22.1 | 0.15 | 7.4 | 12.5 | 1.5 | -0.90 | 20.2
| 0.9 | 6.5 | 10.6 | 1.2 | -0.68 | 20.4 | 0.20 | 7.6 | 12.0 | 1.5 | -0.88 | 19.4
0.50 | 0.0 | 5.3 | 9.6 | 1.1 | -0.83 | 21.2 | 0.0 | 6.5 | 11.7 | 1.2 | -1.08 | 22.3
| 0.3 | 5.3 | 9.5 | 1.1 | -0.83 | 21.4 | 0.05 | 6.6 | 10.6 | 1.4 | -0.74 | 22.0
| 0.5 | 6.1 | 11.1 | 1.1 | -0.51 | 19.0 | 0.10 | 8.9 | 13.4 | 1.5 | -0.43 | 17.5
| 0.7 | 6.6 | 11.2 | 1.2 | -0.43 | 18.7 | 0.15 | 9.9 | 12.6 | 1.6 | -0.46 | 16.2
| 0.9 | 7.5 | 11.0 | 1.3 | -0.34 | 17.3 | 0.20 | 10.3 | 11.7 | 1.6 | -0.49 | 16.0
0.75 | 0.0 | 5.2 | 9.4 | 1.1 | -0.70 | 19.8 | 0.0 | 6.9 | 12.3 | 1.3 | -1.11 | 23.3
| 0.3 | 5.4 | 10.4 | 1.1 | -0.60 | 19.0 | 0.05 | 7.0 | 12.2 | 1.4 | -0.52 | 20.1
| 0.5 | 5.9 | 10.8 | 1.1 | -0.47 | 18.3 | 0.10 | 9.7 | 13.3 | 1.6 | -0.43 | 16.4
| 0.7 | 7.5 | 11.5 | 1.3 | -0.23 | 15.8 | 0.15 | 10.7 | 11.8 | 1.6 | -0.48 | 15.8
| 0.9 | 8.9 | 12.8 | 1.4 | -0.36 | 16.0 | 0.20 | 11.0 | 11.8 | 1.7 | -0.52 | 16.0

Table A.3:  Moment values for the return distributions obtained for the period
01/03/2017 through 12/30/2020 for select efficient frontier portfolios
optimized under mCVaR0.99 and MV. Note the difference between mCVaR0.99 and MV
in the range of $\alpha$ provided.

Model | SR | Sortino | STAR | Rachev | Gini | Model | SR | Sortino | STAR | Rachev | Gini
---|---|---|---|---|---|---|---|---|---|---|---
|  | (%) | (%) | (%) | (%) | (%) |  | (%) | (%) | (%) | (%) | (%)
EWBH | 4.86 | 6.59 | 1.88 | 83.86 | 12.26 |  |  |  |  |  |
$\lambda$ | $\alpha$ |  mCVaR0.99 | $\alpha$ | MV
0.00 | 0.0 | 4.87 | 6.69 | 1.94 | 87.79 | 12.42 | 0.0 | 5.09 | 6.96 | 2.00 | 86.56 | 12.97
| 0.3 | 4.91 | 6.75 | 1.96 | 87.84 | 12.55 | 0.05 | 5.09 | 6.92 | 1.99 | 85.10 | 13.03
| 0.5 | 4.92 | 6.76 | 1.96 | 87.76 | 12.55 | 0.10 | 4.92 | 6.66 | 1.93 | 83.94 | 12.35
| 0.7 | 4.98 | 6.85 | 2.00 | 88.42 | 12.66 | 0.15 | 4.62 | 6.21 | 1.81 | 83.29 | 11.47
| 0.9 | 5.16 | 7.08 | 2.07 | 88.03 | 13.11 | 0.20 | 4.34 | 5.81 | 1.69 | 82.50 | 10.68
0.25 | 0.0 | 4.91 | 6.76 | 1.96 | 87.99 | 12.53 | 0.0 | 5.34 | 7.28 | 2.08 | 85.36 | 13.70
| 0.3 | 4.89 | 6.73 | 1.95 | 88.13 | 12.46 | 0.05 | 4.85 | 6.60 | 1.90 | 85.36 | 12.30
| 0.5 | 4.93 | 6.79 | 1.97 | 88.32 | 12.56 | 0.10 | 5.02 | 6.89 | 2.00 | 87.79 | 12.40
| 0.7 | 4.95 | 6.84 | 1.99 | 88.64 | 12.61 | 0.15 | 4.93 | 6.78 | 1.97 | 88.48 | 12.05
| 0.9 | 5.45 | 7.59 | 2.20 | 89.56 | 13.77 | 0.20 | 4.93 | 6.78 | 1.96 | 88.78 | 11.96
0.50 | 0.0 | 4.85 | 6.67 | 1.93 | 87.92 | 12.33 | 0.0 | 5.38 | 7.31 | 2.08 | 83.94 | 13.87
| 0.3 | 4.83 | 6.65 | 1.92 | 88.02 | 12.27 | 0.05 | 4.82 | 6.66 | 1.92 | 88.23 | 12.03
| 0.5 | 5.31 | 7.43 | 2.13 | 89.90 | 13.41 | 0.10 | 5.84 | 8.26 | 2.37 | 94.17 | 14.38
| 0.7 | 5.49 | 7.72 | 2.21 | 90.18 | 13.81 | 0.15 | 6.18 | 8.76 | 2.51 | 94.89 | 15.14
| 0.9 | 5.80 | 8.21 | 2.34 | 91.87 | 14.46 | 0.20 | 6.24 | 8.84 | 2.53 | 94.94 | 15.26
0.75 | 0.0 | 4.78 | 6.60 | 1.90 | 88.23 | 12.05 | 0.0 | 5.22 | 7.08 | 2.03 | 83.98 | 13.32
| 0.3 | 4.90 | 6.80 | 1.95 | 88.27 | 12.36 | 0.05 | 4.87 | 6.80 | 1.95 | 89.64 | 11.97
| 0.5 | 5.19 | 7.24 | 2.07 | 89.17 | 13.07 | 0.10 | 6.20 | 8.80 | 2.51 | 94.65 | 15.28
| 0.7 | 5.77 | 8.17 | 2.30 | 91.53 | 14.39 | 0.15 | 6.48 | 9.20 | 2.63 | 95.40 | 15.94
| 0.9 | 6.13 | 8.69 | 2.45 | 92.86 | 15.27 | 0.20 | 6.54 | 9.28 | 2.66 | 95.63 | 16.10

Table A.4:  RRR values obtained for the period 01/03/2017 through 12/30/2020
for select efficient frontier portfolios optimized under mCVaR0.99 and MV.
Note the difference between mCVaR0.99 and MV in the range of $\alpha$
provided.

Model | Tot. Ret | Ann. Ret | AvgTO | ETL95 | ETR95 | MDD | ESG* |  |
---|---|---|---|---|---|---|---|---|---
| (%) | (%) | (%) | (%) | (%) | (%) | avg | std |  |
EWBH | 90.14 | 17.69 | 0.00 | -3.41 | 2.86 | 32.87 | 72.72 | 2.97 |  |
$\lambda$ |  mCVaR0.99 |  |
0.00 | 95.01 | 18.45 | 0.40 | -3.48 | 2.92 | 34.06 | 74.04 | 3.49 |  |
0.25 | 114.91 | 21.40 | 0.35 | -3.80 | 3.38 | 31.09 | 82.98 | 4.67 |  |
0.50 | 129.63 | 23.46 | 0.33 | -3.60 | 3.36 | 26.86 | 84.58 | 4.69 |  |
0.75 | 176.66 | 29.43 | 0.23 | -3.97 | 3.78 | 27.49 | 88.08 | 5.83 |  |
$\lambda$ | MV |  |
0.00 | 93.80 | 18.26 | 0.42 | -3.76 | 3.13 | 37.23 | 74.64 | 3.76 |  |
0.25 | 118.44 | 21.90 | 0.33 | -4.10 | 3.60 | 34.23 | 83.64 | 5.22 |  |
0.50 | 154.95 | 26.77 | 0.25 | -4.12 | 3.83 | 30.23 | 86.09 | 5.62 |  |
0.75 | 207.56 | 32.95 | 0.16 | -4.21 | 4.03 | 28.14 | 88.51 | 6.06 |  |
Model | Mean | Median | Std | Skew | ExKurt | Mean | Median | Std | Skew | ExKurt
| $\times 10^{-4}$ | $\times 10^{-4}$ | $\times 10^{-2}$ |  | $\times 10^{-4}$ | $\times 10^{-4}$ | $\times 10^{-2}$ |
EWBH | 6.4 | 11.1 | 1.3 | -1.02 | 21.9 |
$\lambda$ |  mCVaR0.99 | MV
0.00 | 6.7 | 11.4 | 1.4 | -1.28 | 23.9 | 6.6 | 11.6 | 1.5 | -1.36 | 23.7
0.25 | 7.6 | 11.9 | 1.5 | -0.86 | 19.1 | 7.8 | 11.7 | 1.6 | -0.92 | 18.7
0.50 | 8.3 | 12.6 | 1.4 | -0.50 | 16.8 | 9.3 | 12.9 | 1.7 | -0.64 | 16.7
0.75 | 10.1 | 11.7 | 1.6 | -0.45 | 15.8 | 11.2 | 13.0 | 1.7 | -0.51 | 15.1
Model | SR | Sortino | STAR | Rachev | Gini | SR | Sortino | STAR | Rachev | Gini
| (%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | (%) | (%)
EWBH | 4.86 | 6.59 | 1.88 | 83.86 | 12.26 |  |  |  |  |
$\lambda$ |  mCVaR0.99 | MV
0.00 | 4.87 | 6.57 | 1.91 | 83.82 | 12.19 | 4.48 | 6.00 | 1.75 | 83.05 | 11.02
0.25 | 5.02 | 6.92 | 2.00 | 88.87 | 12.18 | 4.79 | 6.57 | 1.90 | 87.84 | 11.51
0.50 | 5.69 | 8.01 | 2.30 | 93.20 | 13.92 | 5.61 | 7.85 | 2.26 | 92.94 | 13.57
0.75 | 6.29 | 8.93 | 2.55 | 95.31 | 15.44 | 6.52 | 9.25 | 2.66 | 95.70 | 15.88

Table A.5: Performance measures, return moment values and RRR results obtained
for the tangent portfolios under mCVaR0.99 and MV optimization during the
period 01/03/2017 to 12/30/2020.

##  Appendix B MV and mCVaRβ ESG-Valued Optimization

Extension of MV and mCVaRβ optimization to ESG-valued returns is
straightforward. Consider a universe of $I$ assets and the corresponding
series of simulated ESG-valued returns $\\{\hat{\zeta}^{s}_{i,t+1}(\lambda)$;
$s=1,\dots,S$; $i=1,\dots,I\\}$. For MV optimization, let
$\mu=[\mu_{1},\dots,\mu_{I}]$ (with transpose denoted by $\mu^{\prime}$)
denote the vector of the sample means and $\Sigma$ the corresponding variance-
covariance matrix computed from the adjusted returns. Let $\theta_{i}\in[0,1]$
denote the portfolio weight232323 Without loss of generality, we consider
long-only portfolio optimization. for asset $i$. The ESG-valued MV
optimization is obtained as the solution of the quadratic programming problem
[56, 57]

$\min_{\theta}\left\\{-\alpha\theta^{\prime}\mu+\left(1-\alpha\right)\theta^{\prime}\Sigma
w\right\\},$ (B.1)

subject to the constraints

$\displaystyle\theta_{i}\geq 0,\text{ for }i=1,\dots,I,$ (B.2)
$\displaystyle\sum_{i=1}^{I}\theta_{i}=1,$ (B.3)

for a particular choice of $\alpha$ and $\lambda\in[0,1]$, where $\alpha$ is
the risk-aversion parameter.

The ESG-valued mCVaRβ solution is obtained from the minimization [65]

$\min_{\theta,\xi}\left\\{-\alpha\theta^{\prime}\mu+(1-\alpha)\left[\xi+S^{-1}\left(1-\beta\right)^{-1}\sum_{s=1}^{S}\left[-x_{s}-\xi\right]^{+}\right]\right\\},$
(B.4)

where

$x_{s}=\sum_{i=1}^{I}\hat{\zeta}^{s}_{i,t+1}(\lambda)\theta_{i},\text{ for
}s=1,\dots,S,$ (B.5)

subject to

$\displaystyle\theta_{i}\geq 0,\text{ for }i=1,\dots,I,$ (B.6)
$\displaystyle\sum_{i=1}^{I}\theta_{i}=1,$ (B.7)
$\displaystyle\xi\in\mathbb{R},$ (B.8)

for a particular choice of $\alpha$ and $\lambda\in[0,1]$. We use 0.95 and
0.99 as the values for the significance coefficient $\beta$. Equations
(B.4)-(B.8) can be reduced to a linear problem [65].

##  Appendix C ESG-Valued Option Valuation: Discrete Model

Consider the problem of pricing a European contingent claim on the underlying
securuity $\mathbb{S}$ on day $t$ having strike value $K$ and maturity date
$t+T$. We utilize the following discrete option pricing approach. Following
the procedure in Section 3.1, we fit an ARMA($p,q$)-GARCH(1,1) model to a two-
year window of historical returns of $\mathbb{S}$, assuming that the
innovations are distributed according to an NIG distribution. From the NIG
fit, we randomly generate a set of of $T=T_{\text{min}},\dots,T_{\text{max}}$
innovations which, using the fitted ARMA-GARCH parameters, is conveted into a
return trajectory $\\{\hat{r}_{t+T};\
T=T_{\text{min}},\dots,T_{\text{max}}\\}$. We generate an ensemble of
$S=20{,}000$ such return trajectories, $\left\\{\hat{r}^{s}_{t+T};\
T=T_{\text{min}},\dots,T_{\text{max}};\ s=1,\dots,S\right\\}$. Using the ESG
score for day $t$, from (1) each trajectory can then be converted to a
trajectory of ESG-valued returns $\left\\{\hat{\zeta}^{s}_{t+T};\
T=T_{\text{min}},\dots,T_{\text{max}};\ s=1,\dots,S\right\\}$ and consequently
a set of ESG-valued price trajectories $\left\\{\hat{P}^{(Z,s)}_{t+T};\
T=T_{\text{min}},\dots,T_{\text{max}};\ s=1,\dots,S\right\\}$.242424 As noted
in Section 6, we used $T_{\text{min}}=15$, $T_{\text{max}}=252$.

We compute an ESG-valued price for call and put options in the standard way;
the contract value is given by the discounted expected value of the contract
pay-off, conditioned to a filtration ${\cal F}_{t}$, where the conditional
expectation is with respect to a risk-neutral measure $\mathbb{Q}$ [28], which
discounts expected values at the ESG-valued risk-free rate
$\zeta_{f,t}(\lambda)$ defined in (18),

$P^{(Z)}_{t}(\lambda)=\mathbb{E}^{\mathbb{Q}}\left[P^{(Z)}_{T}(\lambda)e^{-\zeta_{f,t}(\lambda)T}|{\cal
F}_{t}\right].$ (C.1)

Given the risk neutral probability measure $\mathbb{Q}$, European call (c) and
put (p) ESG-valued option values on the underlying can be computed as

$\displaystyle\Phi_{c}(t,T,K,\lambda)$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[\max\left(\hat{P}^{(Z)}_{T}(\lambda)-K,0\right)e^{-\zeta_{f,t}(\lambda)T}|{\cal
F}_{t}\right],$ (C.2) $\displaystyle\Phi_{p}(t,T,K,\lambda)$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[\max\left(K-\hat{P}^{Z}_{T}(\lambda),0\right)e^{-\zeta_{f,t}(\lambda)T}|{\cal
F}_{t}\right].$

In the discrete setting of this paper,

$\mathbb{E}^{\mathbb{Q}}\left[P^{(Z)}_{T}(\lambda)e^{-\zeta_{f,t}(\lambda)T}|{\cal
F}_{t}\right]=\sum_{s=1}^{S}q_{s}P^{(Z,s)}_{T}(\lambda)e^{-\zeta_{f,t}(\lambda)T},$

and the option valuation problem is reduced to identifying $\mathbb{Q}$
through the coefficients $q_{s}$. We solve for these coefficients by
minimizing the Kullback-Leibler divergence between $\mathbb{Q}$ and the
natural world probability $\mathbb{P}$ [5]. This involves solving the convex
problem

$\min_{\\{q_{s}\\}}\sum_{s=1}^{S}q_{s}\ln(q_{s}/p_{s}),$ (C.3)

subject to the constraints

$\displaystyle\sum_{s=1}^{S}q_{s}=1,$ (C.4)
$\displaystyle\sum_{s=1}^{S}q_{s}P^{(Z,s)}_{T}(\lambda)e^{-\zeta_{f,t}(\lambda)T}=P^{(Z)}_{t},$
(C.5) $\displaystyle 0<\ q_{s}<1;\quad s=1,\dots,S.$ (C.6)

In (C.3), $p_{s}$ is the probability of trajectory $s$ under $\mathbb{P}$. As
each trajectory is equally likely, $p_{s}=1/S$. Option values are then
computed from the discrete version of (C.2)

$\displaystyle\Phi_{c}\left(t,T,K\right)$
$\displaystyle=\sum_{s=1}^{S}q_{s}\left[\max\left(\hat{P}^{(Z,s)}_{T}-K,0\right)e^{-\zeta_{f,t}(\lambda)T}\right],$
(C.7) $\displaystyle\Phi_{p}\left(t,T,K\right)$
$\displaystyle=\sum_{s=1}^{S}q_{s}\left[\max\left(K-\hat{P}^{(Z,s)}_{T},0\right)e^{-\zeta_{f,t}(\lambda)T}\right].$

##  Appendix D Shadow Riskless Rate

Consider a set of $N$ risky securities whose prices $P_{i,t}$; $i=1,\dots,N$
are driven by $N-1$ Brownian motions252525 More complex pricing models have
been considered [64]; here we employ the basic Brownian motion model.
$[dB_{1,t},\dots,dB_{N-1,t}]$. Thus

$dP_{i,t}=P_{i,t}\mu_{i}dt+P_{i,t}\sum_{j=1}^{N-1}\sigma_{i,j}dB_{j,t},\quad
i=1,\dots,N.$ (D.1)

To eliminate arbitrage opportunities in the market defined by (D.1), we search
for the unique state-price deflator

$d\pi_{t}=\pi_{t}\mu_{\pi}dt+\pi_{t}\sum_{j=1}^{N-1}\sigma_{\pi,j}dB_{j,t},$
(D.2)

such that the discounted prices $P_{i,t}\pi_{t}$, $i=1,\dots,N$ are
martingales. It is easy to show262626 This follows directly from Itô’s
formula. that this requirement is equivalent to solving the linear system

$\begin{bmatrix}-1&-\sigma_{1,1}&\dots&-\sigma_{1,N-1}\\\
-1&-\sigma_{2,1}&\dots&-\sigma_{2,N-1}\\\ \vdots&\vdots&\dots&\vdots\\\
-1&-\sigma_{N,1}&\dots&-\sigma_{N,N-1}\\\
\end{bmatrix}\begin{bmatrix}\mu_{\pi}\\\ \sigma_{\pi,1}\\\ \vdots\\\
\sigma_{\pi,N-1}\\\ \end{bmatrix}=\begin{bmatrix}\mu_{1}\\\ \mu_{2}\\\
\vdots\\\ \mu_{N}\\\ \end{bmatrix},$ (D.3)

for $[\mu_{\pi},\sigma_{\pi,1},\dots,\sigma_{\pi,N-1}]$. Following the
treatment in Duffie [30, Chapter 6D], the risk premium for each security is

$\mu_{i}-\hat{r}=-\sum_{j=1}^{N-1}\sigma_{i,j}\sigma_{\pi,j},\text{ for
}i=1,\dots,N,$ (D.4)

where $\hat{r}=-\mu_{\pi}$ is the short-term, shadow riskless rate (SRR)
observed in the market.

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## Supplement to ESG-Valued Portfolio Optimization and Dynamic Asset Pricing

This appendix contains supplementary material to the article ESG Optimizing
Portfolios. Here we provide further analyses on the Refinitiv data.
Additionally we replicate all analyses in the manuscript by considering ESG
scores provided by the rating agency, RobecoSAM; enabling preliminary results
on the impact of the choice of the ESG score provider.

###  S.1 RobecoSAM ESG Data: Comparison with Refinitiv

The ESG ratings produced by RobecoSam are released at the end of a calendar
year and are available for application to the years 2017 through 2021. The
ratings for 29 of the companies in the the Dow Jones Industrial Average (DJIA)
used in our analyses are summarized in Table S.1.

Ticker | 12/30 | 12/29 | 12/31 | 12/31 | 12/31 | Ticker | 12/30 | 12/29 | 12/31 | 12/31 | 12/31
---|---|---|---|---|---|---|---|---|---|---|---
| 2016 | 2017 | 2018 | 2019 | 2020 |  | 2016 | 2017 | 2018 | 2019 | 2020
MMM | 93 | 84 | 88 | 87 | 91 | JPM | 35 | 41 | 38 | 43 | 58
AXP | 47 | 72 | 69 | 61 | 68 | MCD | 69 | 64 | 60 | 53 | 76
AMGN | 82 | 88 | 89 | 87 | 95 | MRK | 61 | 53 | 57 | 65 | 73
AAPL | 30 | 36 | 18 | 36 | 25 | MSFT | 97 | 93 | 91 | 92 | 96
BA | 42 | 36 | 32 | 32 | 78 | NKE | 77 | 76 | 71 | 78 | 80
CAT | 81 | 91 | 88 | 88 | 94 | PG | 50 | 64 | 58 | 55 | 64
CVX | 60 | 57 | 62 | 53 | 60 | CRM | 65 | 83 | 85 | 83 | 91
CSCO | 89 | 88 | 100 | 100 | 100 | TRV | 55 | 63 | 58 | 41 | 50
KO | 41 | 41 | 39 | 39 | 64 | UNH | 98 | 95 | 100 | 95 | 96
GS | 75 | 77 | 70 | 66 | 71 | VZ | 33 | 23 | 19 | 20 | 49
HD | 56 | 68 | 75 | 68 | 62 | V | 61 | 72 | 80 | 76 | 82
HON | 29 | 44 | 31 | 27 | 48 | WBA | 13 | 47 | 55 | 50 | 81
IBM | 69 | 69 | 66 | 56 | 69 | WMT | 49 | 44 | 48 | 52 | 68
INTC | 73 | 84 | 70 | 44 | 86 | DIS | 52 | 55 | 56 | 54 | 73
JNJ | 59 | 49 | 71 | 83 | 84 |  |

Table S.1: Robeco ESG scores for the DJIA Index components.

Box-whisker summaries of the resultant normalized scores $\varsigma_{i,t}$ are
shown, by year, in Fig. S.1.

Figure S.1: Box-whisker summaries of $[-1,1]$ normalized RobecoSAM data for
the DJIA data listed in Table S.1.

Compared to the Refinitiv data (manuscript, Fig. 1), for this set of companies
the yearly RobecoSAM data has smaller mean value, but larger spread. Thus
while the Refinitiv data reveals lower outliers, the Robeco data does not.
There are also instances (CSCO and UNH) where RobecoSAM assigns a “perfect”
ESG score.

Figure S.2: Normalized ESG ratings from Refinitive and RobecoSAM for 2016 and
2020.

| Kendall Tau |  | Spearman Rho
---|---|---|---
| Ref16 | Reb16 | Ref20 | Reb20 |  | Ref16 | Reb16 | Ref20 | Reb20
Ref16 | 1 | 0.22 | 0.49 |  |  | 1 | 0.35 | 0.64 |
|  | (0.095) |  ($10^{-4}$) |  |  |  | (0.067) |  ($2\cdot 10^{-4})$ |
Reb16 |  | 1 |  |  $-$0.08 |  |  | 1 |  |  $-$0.11
|  |  |  | (0.55) |  |  |  |  | (0.58)
Ref20 |  |  | 1 | 0.15 |  |  |  | 1 | 0.17
|  |  |  | (0.26) |  |  |  |  | (0.37)

Table S.2: Rank correlation measures between Refinitiv and RobecoSAM ESG
ratings released in 2016 and 2020. The numbers in parenthesis represent
p-values. Ref16 = Refinitiv, 2016; etc.

Fig. S.2 presents a comparison of the normalized $\varsigma_{i,t}$ ESG ratings
from Refinitiv and RobecoSAM released at the end of 2016 and 2020. In both
plots the companies are ordered from lowest-to-highest Refinitiv ESG rating.
The plots show both the change in company rankings with time for the same
rating agency as well as the different rankings assigned by Refinitiv and
RobecoSAM in the same year. For each of these two years, the ESG rankings of
these 29 companies were computed separately for the Refinitive and RobecoSAM
data. Table S.2 shows the results of the Kendall Tau and Spearman Rho tests on
the correlation between the rankings for the same year by the two rating
agencies; as well as the correlation between the rankings over the two
different years by the same agency. The p-values indicate that, even at the 5%
significance level, there is no correlation between the Refinitiv and
RobecoSAM rankings for the same year. There is a strong correlation between
Refinitiv rankings for 2016 and 2020. There is however no correlation between
the RobecoSAM rankings for 2016 and 2020.

###  S.2 Efficient Frontiers

In Section 2.2, efficient frontiers were computed for the MV, mCVaR0.95 and
mCVaR0.99 optimizations for the date 12/30/2019 using the ESG ratings from
Refinitiv. The efficient frontiers displayed (manscript Figs. 2 and 3) a
positive, monotonic, dependence of the optimal portfolio’s ESG score with
increasing value of the risk paramenter $\alpha$, even for the ESG affinity
parameter value $\lambda=0$. We provide here an alternate example, obtained by
computing the efficient frontiers for the date 03/30/2020, again using the ESG
ratings provided by Refinitiv. These efficient frontiers are plotted in
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space and projected onto the
$(\text{ESG}^{*}_{t+1},\mathbb{E}[\hat{R}^{*}_{t+1}])$ and
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}])$ planes in Fig.
S.3.

a  b  c  d  e  f  g  h  i

Figure S.3: Refinitiv-based ESG-adjusted return efficient frontiers for
03/30/2020 plotted in (top)
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space; projected onto the (middle
)$(\text{ESG}^{*}_{t+1},\mathbb{E}[\hat{R}^{*}_{t+1}])$ and (bottom)
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}])$ planes. From
left to right the optimizations are: a,d,g) MV, b,e,h) MCVaR0.95 and c,f,i)
MCVaR0.99.

The date 03/16/2020 represents the low point of the drop in the Dow Jones
Industrial average due to the Covid 19 pandemic, with the average closing at
$19,170. By 03/30/2020, the index had regained somewhat to $21,053. However,
it was not until 11/08/2020 that the DJI average surpassed the value of
$29,398 it held just prior to the pandemic crash. For 03/30/2020, only the MV
optimization was capable of projecting positive expected returns; the tail
risk CVaR optimizations project negative returns. For the optimizations on
12/30/2019, the efficient frontiers converged to the same point as
$\alpha\uparrow 1$ for all four values of $\lambda$. For the 03/30/2020
optimization, convergence only occurs for the three smaller values of
$\lambda$ under MV optimization. For the mCVaR optimizations, for each value
of $\lambda$, the efficient frontiers approach different values as
$\alpha\uparrow 1$. The projections on the
$(\text{ESG}^{*}_{t+1},\mathbb{E}[\hat{R}^{*}_{t+1}])$ indicate radically
different behaviors of the portfolio ESG values along the efficient frontiers.
Under the MV optimization, for $\lambda=0$, the ESG value falls and then
rises. For $\lambda=0.25,0.5$, the ESG value rises and then falls. For
$\lambda=0.75$, the ESG value monotonically increases with $\alpha$. For the
mCVaR optimizations, the ESG value decreases as $\alpha$ increases for the
$\lambda=0,0.25$ efficient frontiers; for the other two values of $\lambda$ it
increases. For the mCVaR optimizations with $\lambda=0.75$, the change in
$\mathbb{E}[\hat{R}^{*}_{t+1}]$ with respect to $\alpha$ is non-monotonic.
This is also seen in the MV optimization with $\lambda=0.5$ and to a small
extent for $\lambda=0.75$.

a  b  c  d  e  f

Figure S.4: RobecoSAM-based ESG-adjusted return efficient frontiers for
12/30/2019 (top) plotted in
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}],\text{ESG}^{*}_{t+1})$
space and (bottom) projected onto the
$(\mathbb{V}[\hat{R}^{*}_{t+1}],\mathbb{E}[\hat{R}^{*}_{t+1}])$ plane. From
left to right the optimizations are: a,d) MV, b,e) MCVaR0.95 and c,f)
MCVaR0.99.

In Fig. S.4, we show efficient frontiers computed for the same date 12/30/2019
as in the manuscript, but now using the RobecoSAM scores. (Compare with
manuscript Fig. 3.) The $\lambda=0.25$ RobecoSAM efficient frontiers display
behavior comparable to that of the $\lambda=0.25$ and $0.5$ Refinitiv
efficient frontiers. The $\lambda=0.5$ RobecoSAM efficient frontiers display
behavior comparable to that of the $\lambda=0.75$ Refinitiv efficient
frontiers. However the $\lambda=0.75$ RobecoSAM efficient frontiers display
very different behaviors compared to those for Refinitiv. In particular, for
MV optimization the $\lambda=0.75$ RobecoSAM efficient frontier shows
decreasing expected return at hign mean standard deviation values.

Figure S.5: Variation of the weight composition along each efficient frontier
(as a function $\alpha$) for the MCVaR0.99 optimal portfolios with
$\lambda=0,0.25,0.5,0.75$. Assets with lower ESG scores are depicted with
warmer colors.

The lack of correlation between Refinitiv and RobecoSAM ESG ratings will
impact optimal portfolio composition. For each choice of $\lambda$, the
compositions, as a function of $\alpha$, of the mCVaR0.99 optimal portfolios
of Fig. S.4 are shown in Fig. S.5. Thirteen assets are represented in the
mCVaR0.99 efficient frontier portfolios in Fig. 5 of the manuscript while 18
are represented in Fig. S.5. Twelve of the assets are represented in both the
Refinitiv and RobecoSAM based optimizations. Refinitiv ESG scores for the 13
represented assets range from 65 to 83, while the RobecoSAM scores for the 18
represented assets range from 18 to 100.

###  S.3 Performance Measures

The performance measures Tot Rtn, Ann Rtn, AvgTO, ETL95, ETR95, and MDD, as
well as the average and standard deviation of the portfolio ESG scores, over
the time period 01/03/2017 through 12/30/2020 for select values of $\alpha$ on
the RobecoSAM-based mCVaR0.99 and MV optimized efficient frontiers are
presented in Table S.3 of Section S.7. Except for the ESG scores, the
performance measures for the $\lambda=0$ efficient frontiers will be identical
whether using Refinitiv or RobecoSAM data. For $\lambda>0$, we note the same
general trends in the performance measures as the value of $\lambda$ increases
as were observed in the Refinitiv data. However, the performance measures
obtained using the two different ESG data sets will differ, reflecting the
time-average of the types of differences displayed for the date 12/30/2019 in
Fig. 2 of the manuscript for Refinitiv data compared with Fig. S.4 for
RobecoSAM data. For given values of $\lambda$ and $\alpha$, for mCVaR0.99
optimization the return performance is generally better for RobecoSAM data,
while for MV optimization the return performance is better for Refinitiv data.
The Refinitiv data based AvgTO, MDD and ESG scores are generally better under
both optimizations. Under mCVaR0.99 optimization, the ETL/ETR values are
better for Refinitiv data; while Robesco data produces better values under MV
optimization. There are also $\lambda$-dependant differences. For example,
while the Avg ESG score is higher by as much as 15 to 16 points for Refinitv-
based optimization (compared with RovbecoSAM) when $\lambda=0$, this
difference decreases as $\lambda$ increases. When $\lambda=0.75$ the
Refinitive-based optimizations produce a higher Avg ESG score by as little as
0.25 (mCVaR) to 4.5 (MV optimization). The other pronounced $\lambda$
dependence is in the total (and annual) returns under MV optimization. The
difference in total return provided by Refinitiv-based data compared to
RobecoSAM-based data grows from 8.5 for $\lambda=0.25$ to 42.0 for
$\lambda=0.75$.

Values for the EWBH benchmark are also given. Relative to EWBH, the Tot. Ret,
Ann. Ret, and avg ESG score of the optimized portfolios outperform that of the
benchmark as $\lambda$ increases. Under MCVaR0.99 optimization, the MDD
remains better than that of EWBH for all values of $\lambda$, while the MDD
for MV optimzation is better once $\lambda\geq 0.25$. Only for the smaller
values of $\lambda$ are the ETL/ETR values for the optimized portfolios better
than those for EWBH.

Moments of the Robeco-SAM based return distributions for the period 01/03/2017
through 12/30/2020 are summarized in Table S.4 of Section S.7. Comparing with
manuscript Table A.3 for Refinitiv based data, mCVaR0.99 Mean, Std, and ExKurt
values are smaller and Skew values larger for Refinitiv data; Median values
are roughly compariable. Under MV optimization, except for ExKurt, the moment
values are generally smaller for RobecsoSAM data. Relative to EWBH, Mean and
ExKurt values of the optimized portfolios become better as $\lambda$
increases. For mCVaR0.99 optimization, Skew values are always better than
those for EWBH; for MV optimization, Skew values improve beyond EWBH as
$\lambda$ increases. Std values for the optimized portfolios begin to exceed
those for EWBH as $\lambda$ increases.

Risk-reward ratio values for the RobecoSAM based optimizations are summarized
in Table S.5 of Section S.7. Comparing with manuscript Table A.4 for Refinitiv
based data, we find that better RRR values are essentially split between
Refinitiv and RobecoSAM data based optimizations. The strongest exception is
that Rachev ratio values are more frequently higher under MV optimization for
the Refinitiv-based data. All RRR values for the optimized portfolios almost
always exceed those for the EWBH benchmark.

###  S.4 Tangent Portfolios: Capital Market Lines

The $P_{t}^{(Z)}(\lambda)$ time series computed from the ESG-adjusted tangent
portfolios under mCVaR0.99 optimization are shown in Fig. S.6 for
$\lambda\in\left\\{0,0.25,0.5,0.75\right\\}$.

Figure S.6: (a)-(d) RobecoSAM-based ESG-adjusted portfolio values,
$P_{t}^{(Z)}(\lambda)$, obtained for tangent portfolios under mCVaR0.99
optimization, compared with the corresponding $P_{t}^{(Z)}(\lambda)$ time
series for the EWBH strategy and the ESG-adjusted DJIA index. (e) Differences
between the $P_{t}^{(Z)}(\lambda)$ values for the tangent portfolios and the
DJIA index.

Also plotted are the corresponding time series for the EWBH and DJIA index
benchmarks. All ESG-adjusted values are now computed using the RobecoSAM data.
Comparing with the results in Fig. 8 of the manscript for Refinitiv data, we
confirm that the corresponding time series must be identical when $\lambda=0$.
For $\lambda>0$, we see a greater difference between the EWBH and DJIA index
benchmarks under RobecoSAM data. In addition, the optimized portfolio values
perform slightly better under the RobecoSAM data than under the Refinitiv
data.

Table S.6 summarizes the performance measures, return distribution moments and
RRR values for the EWBH benchmark and the tangent portfolios optimized under
mCVaR0.99 and MV. This summary should be compared to that in manuscript Table
A.5 for the Refinitiv based computations. In Section S.3 we have discussed the
differences between between RobecoSAM and Refinitiv based values for the
performance measures, return distribution moments and RRRs for efficient
frontier portfolios characterized by the same values of $\lambda$ and
$\alpha$. The tangent portfolios on the corresponding efficient frontiers will
have the same value of $\lambda$ but different $\alpha$ value (in general).
Thus for $\lambda>0$, we expect some differences between the results in Tables
S.6 and A.5 and those observed in Section S.3. Under both optimizations, Tot.
Ret. is better under Refinitiv data at the higher $\lambda$ values, while
RobescoSAM Tot. Ret. is better at lower $\lambda$ values. The same is true for
MDD. AvgTO is better for Refinitiv data, whle avg ESG is better for RobecoSAM.
There are no discernible trends in the ETL/ETR results, although they are
generally better for RobecoSAM data.

At the larger values of $\lambda$, Refinitiv based results are generally
better for the return Mean, Median, Skew and ExKurt moments. At
$\lambda=0.75$, RRR values for Refinitive tangent portfolios outperform thosse
for RobecoSAM. Refinitiv tangent portfolios for $\lambda=0.75$ perform well as
they were heavily concentrated in Microsoft stock; the security with the best
return performance over the investment period.

Fig. S.7 compares daily portfolio ESG scores (manuscript equation (14))
computed under mCVaR0.99 optimization with $\lambda=\\{0.5,0.75\\}$ under
Refinitiv and RobecoSAM.

Figure S.7:  ESG values obtained by mCVaR0.99 optimized tangent portfolios and
the EWBH benchmark under Refinitiv and RobecoSAM data.

Also plotted is the ESG score for the EWBH benchmark. Although the portfolio
ESG scores for RobecoSAM start from a lower value in 2017, they rise more
rapidly; by 2018, RobecoSAM based portfollio ESG scores are larger than those
based upon Refinitiv data. ESG values change more noticeably under Refinitiv
data than under RobecoSAM data as $\lambda$ grows from 0.5 to 0.75. The effect
produced by the yearly adjustment of RobecoSAM and Refinitiv ESG scores
produces noticeable discontinuities in the ESG values. Interestingly, for 2019
there was a noticeable downtrend in the ESG scoring by both agencies. The
effect of the large change in RobecoSAM ESG scores from 2019 to 2020 is most
apparent in the EWBH benchmark results.

###  S.5 Option Pricing

Option prices were computed for 12/30/2019 using either the RobecoSAM ESG-
adjusted tangent portfolios or the DJIA index as the underlying asset. The
call and put price surfaces for these tangent portfolios and the DJIA index
are given in Fig. S.8.

a $\lambda=0$ b $\lambda=0.25$ c $\lambda=0.5$ d $\lambda=0.75$ e Call f Put

Figure S.8:  Figures (a) to (d): (top) call and (bottom) put option prices for
select choices of $\lambda$. The orange surfaces corresponds to option prices
for the ESG-adjusted DJIA index; blue surfaces represents option prices for
the tangent portfolios. Figures (e) and (f). The difference in call and put
option prices between the ESG-adjusted DJIA index and the MCV tangent
portfolios.

Comparing the RobecoSAM results with the Refinitiv results in Fig. 9 of the
manuscript, most noticeable is the size of the difference between the tangent
portfolio and DJIA index for both call and put surfaces for each value of
$\lambda$, with the differences being larger for Refinitiv-based option
pricing.

Implied volatilities computed from the RobescoSAM based option prices are
presented in Fig. S.9.

a $\lambda=0$ b $\lambda=0.25$ c $\lambda=0.5$ d $\lambda=0.75$

Figure S.9:  Implied volatility surfaces computed from the (orange surfaces)
ESG-adjusted DJIA option prices and (blue surfaces) the tangent portfolios
corresponding to the selected choices for $\lambda$.

The results are qualitatively similar to those for Refinitiv data (manuscript
Fig. 10).

###  S.6 Shadow Riskless Rate

The realized shadow riskless rate (SRR) time series computed using RobecoSAM
data are shown in Fig. (S.10) and are to be compared to the Refinitiv based
results in Fig. 11 of the manuscript.

a  b

Figure S.10:  Time series from 01/03/2017 through 12/31/2020 for select values
of $\lambda$ of (a) the SRR computed from ESG-adjusted returns on a universe
of 29 DFJIA assets for select values, and (b) the information ratio of the
SRR.

For $\lambda>0$, the change in rating agencies dramatically affected the sign
of the shadow rate. Under the RobecoSAM ESG data, non-zero values of $\lambda$
drive the shadow riskless rate negative, with a decreasing (more negative)
value as $\lambda$ increases. The reason for this behavior is not yet
understood. Similar to the results noted for SRR based on Refinitiv data
(manuscript Fig. 11), the separation between SRR values, while showing some
variation with time, appears roughly linearly proportional to $\lambda$. As a
result of the negative sign for the SRR, the information ratio for RobecoSAM
based data is negative. However, similar to the behavior under Refinitiv based
data, as $\lambda$ increases, the magnitude of the information ratio tends to
decrease.

Figure S.11:  The curves $\mu_{\text{IR}}(\lambda)$,
$\sigma_{\text{IR}}(\lambda)$ and the parametric curve
$(\mu_{\text{IR}}(\lambda),\sigma_{\text{IR}}(\lambda))$. The dashed line is a
linear fit.

The time averaged values $\mu_{\text{IR}}(\lambda)$ and
$\sigma_{\text{IR}}(\lambda)$ for the RobecoSAM-based SRR information ratio
are shown in Fig. S.11. Interesting differences and similarities emerge from
the comparison with manuscript Fig. 12 for the Refinitiv data. Again the
dependence of $\mu_{\text{IR}}(\lambda)$ on $\lambda$ is “parabolic”, but with
a local minimum (at $\lambda=0.25$) rather than a local maximum. Except when
$\lambda=0$, the time averaged $\mu_{\text{IR}}(\lambda)$ is negative over the
range $\lambda\in[0.01,0.90]$. The behavior of the standard deviation
$\sigma_{\text{IR}}(\lambda)$ for RobecoSAM based data is qualitatively
similar to that for Refinitiv but now with a local minimum developed at
$\lambda=0.11$, prior to following a linear trend for $\lambda>0.4$. This
linear trend in $\sigma_{\text{IR}}(\lambda)$ for RobecoSAM based data is
virtually identical to that developed for the Refinitiv based data.

###  S.7 Tables

Model | Tot. Ret | Ann. Ret | AvgTO | ETL95 | ETR95 | MDD | ESG*
---|---|---|---|---|---|---|---
|  | (%) | (%) | (%) | (%) | (%) | (%) | avg | std
EWBH | 90.14 | 17.68 | 0.00 | -3.41 | 2.86 | 33.00 | 72.72 | 2.97
$\lambda$ | $\alpha$ |  MCVaR0.99
0.00 | 0.0 | 70.75 | 14.53 | 0.4 | -2.74 | 2.41 | 28.47 | 54.94 | 3.69
| 0.3 | 71.53 | 14.66 | 0.4 | -2.74 | 2.40 | 28.36 | 54.94 | 3.59
| 0.5 | 71.85 | 14.71 | 0.4 | -2.75 | 2.41 | 28.36 | 55.20 | 3.74
| 0.7 | 73.88 | 15.05 | 0.4 | -2.76 | 2.44 | 27.62 | 56.32 | 3.57
| 0.9 | 82.60 | 16.49 | 0.4 | -2.90 | 2.55 | 28.23 | 58.22 | 2.88
0.25 | 0.0 | 70.57 | 14.49 | 0.4 | -2.77 | 2.43 | 28.94 | 56.98 | 3.36
| 0.3 | 72.55 | 14.83 | 0.4 | -2.77 | 2.43 | 28.55 | 58.07 | 3.52
| 0.5 | 74.24 | 15.11 | 0.4 | -2.80 | 2.45 | 28.61 | 59.48 | 3.59
| 0.7 | 86.38 | 17.10 | 0.4 | -2.95 | 2.60 | 28.11 | 67.40 | 4.16
| 0.9 | 107.58 | 20.34 | 0.4 | -3.31 | 2.97 | 28.94 | 76.69 | 4.99
0.50 | 0.0 | 75.82 | 15.38 | 0.4 | -2.82 | 2.47 | 28.88 | 61.48 | 3.68
| 0.3 | 78.75 | 15.86 | 0.4 | -2.88 | 2.51 | 28.67 | 64.54 | 3.98
| 0.5 | 84.48 | 16.79 | 0.4 | -2.95 | 2.58 | 28.71 | 67.95 | 4.50
| 0.7 | 100.75 | 19.32 | 0.4 | -3.38 | 3.08 | 29.12 | 80.85 | 6.13
| 0.9 | 122.16 | 22.43 | 0.37 | -3.70 | 3.52 | 29.00 | 87.87 | 8.50
0.75 | 0.0 | 88.55 | 17.44 | 0.4 | -3.03 | 2.67 | 28.83 | 71.63 | 4.57
| 0.3 | 91.38 | 17.88 | 0.4 | -3.19 | 2.85 | 29.00 | 77.06 | 5.21
| 0.5 | 95.75 | 18.56 | 0.4 | -3.35 | 3.09 | 28.67 | 82.20 | 6.34
| 0.7 | 110.92 | 20.82 | 0.37 | -3.65 | 3.51 | 29.73 | 88.94 | 8.87
| 0.9 | 115.23 | 21.45 | 0.34 | -3.82 | 3.66 | 31.38 | 89.93 | 9.25
$\lambda$ | $\alpha$ | MV
0.00 | 0.0 | 76.82 | 15.54 | 0.4 | -2.84 | 2.46 | 29.17 | 57.21 | 3.19
| 0.05 | 84.53 | 16.80 | 0.40 | -3.06 | 2.61 | 30.62 | 57.24 | 3.28
| 0.10 | 90.68 | 17.77 | 0.38 | -3.33 | 2.80 | 32.39 | 58.01 | 2.73
| 0.15 | 90.64 | 17.77 | 0.39 | -3.55 | 2.96 | 35.00 | 58.24 | 2.68
| 0.20 | 88.69 | 17.46 | 0.40 | -3.73 | 3.08 | 37.27 | 58.78 | 2.09
0.25 | 0.0 | 82.99 | 16.55 | 0.4 | -2.94 | 2.52 | 30.14 | 58.30 | 2.67
| 0.05 | 85.00 | 16.87 | 0.4 | -3.08 | 2.65 | 30.05 | 65.25 | 2.14
| 0.10 | 89.48 | 17.59 | 0.4 | -3.18 | 2.74 | 30.12 | 68.38 | 3.33
| 0.15 | 95.56 | 18.53 | 0.4 | -3.27 | 2.83 | 30.23 | 72.19 | 4.58
| 0.20 | 100.63 | 19.30 | 0.4 | -3.35 | 2.92 | 30.30 | 74.27 | 5.43
0.50 | 0.0 | 92.55 | 18.07 | 0.4 | -3.13 | 2.63 | 31.79 | 60.41 | 2.02
| 0.05 | 98.67 | 19.01 | 0.4 | -3.34 | 2.90 | 30.34 | 74.25 | 6.29
| 0.10 | 105.55 | 20.04 | 0.4 | -3.48 | 3.05 | 30.60 | 78.52 | 7.63
| 0.15 | 109.45 | 20.61 | 0.4 | -3.56 | 3.16 | 30.20 | 81.19 | 8.24
| 0.20 | 115.15 | 21.43 | 0.4 | -3.64 | 3.31 | 29.53 | 83.66 | 8.82
0.75 | 0.0 | 100.69 | 19.31 | 0.32 | -3.42 | 2.87 | 33.07 | 64.37 | 2.38
| 0.05 | 107.80 | 20.37 | 0.32 | -3.55 | 3.13 | 30.39 | 79.30 | 8.76
| 0.10 | 110.05 | 20.70 | 0.32 | -3.61 | 3.22 | 30.02 | 82.42 | 8.93
| 0.15 | 114.46 | 21.34 | 0.33 | -3.72 | 3.43 | 29.90 | 85.88 | 9.46
| 0.20 | 119.54 | 22.06 | 0.33 | -3.76 | 3.51 | 30.16 | 86.57 | 9.59

Table S.3:  Performance measure values for the period 01/03/2017 through
12/30/2020 for select efficient frontier portfolios optimized under MCVaR0.99
and MV using RobecoSAM data. Note the difference between MCVaR0.99 and MV in
the range of $\alpha$ considered.

Model | Mean | Median | Std | Skew | ExKurt | Model | Mean | Median | Std | Skew | ExKurt
---|---|---|---|---|---|---|---|---|---|---|---
|  | $\times 1^{-4}$ | $\times 1^{-4}$ | $\times 1^{-2}$ |  |  |  | $\times 1^{-4}$ | $\times 1^{-4}$ | $\times 1^{-2}$ |  |
EWBH | 6.4 | 11.1 | 1.3 | -1.02 | 21.9 |  |  |  |  |  |
$\lambda$ | $\alpha$ |  mCVaR0.99 | $\alpha$ | MV
0.00 | 0.0 | 5.3 | 8.3 | 1.1 | -0.92 | 22.1 | 0.00 | 5.7 | 9.3 | 1.1 | -0.93 | 20.4
| 0.3 | 5.4 | 8.0 | 1.1 | -0.92 | 22.2 | 0.05 | 6.1 | 10.7 | 1.2 | -1.08 | 22.9
| 0.5 | 5.4 | 8.5 | 1.1 | -0.92 | 22.3 | 0.10 | 6.4 | 11.1 | 1.3 | -1.17 | 23.1
| 0.7 | 5.5 | 9.6 | 1.1 | -0.90 | 22.4 | 0.15 | 6.4 | 11.2 | 1.4 | -1.32 | 23.9
| 0.9 | 6.0 | 9.9 | 1.2 | -0.97 | 22.7 | 0.20 | 6.3 | 11.5 | 1.5 | -1.40 | 24.1
0.25 | 0.0 | 5.3 | 8.6 | 1.1 | -0.93 | 22.9 | 0.00 | 6.0 | 9.9 | 1.1 | -1.01 | 21.0
| 0.3 | 5.4 | 8.9 | 1.1 | -0.91 | 22.9 | 0.05 | 6.1 | 10.1 | 1.2 | -0.98 | 22.8
| 0.5 | 5.5 | 8.8 | 1.1 | -0.91 | 23.5 | 0.10 | 6.4 | 11.0 | 1.2 | -0.95 | 23.2
| 0.7 | 6.2 | 9.8 | 1.2 | -0.89 | 24.1 | 0.15 | 6.7 | 10.6 | 1.3 | -0.90 | 23.0
| 0.9 | 7.3 | 10.6 | 1.3 | -0.96 | 23.5 | 0.20 | 6.9 | 10.2 | 1.3 | -0.88 | 22.4
0.50 | 0.0 | 5.6 | 9.0 | 1.1 | -0.86 | 23.5 | 0.00 | 6.5 | 11.6 | 1.2 | -1.09 | 22.1
| 0.3 | 5.8 | 9.8 | 1.1 | -0.81 | 23.3 | 0.05 | 6.8 | 11.6 | 1.3 | -0.86 | 22.3
| 0.5 | 6.1 | 9.9 | 1.2 | -0.81 | 23.5 | 0.10 | 7.2 | 10.2 | 1.4 | -0.82 | 21.2
| 0.7 | 6.9 | 10.4 | 1.4 | -0.78 | 22.0 | 0.15 | 7.4 | 9.7 | 1.4 | -0.83 | 20.5
| 0.9 | 7.9 | 10.2 | 1.5 | -0.61 | 18.7 | 0.20 | 7.6 | 9.8 | 1.5 | -0.75 | 19.7
0.75 | 0.0 | 6.3 | 9.8 | 1.2 | -0.73 | 23.0 | 0.00 | 6.9 | 12.2 | 1.3 | -1.11 | 23.3
| 0.3 | 6.5 | 10.4 | 1.3 | -0.66 | 22.0 | 0.05 | 7.3 | 10.8 | 1.4 | -0.82 | 20.7
| 0.5 | 6.7 | 10.8 | 1.4 | -0.58 | 19.8 | 0.10 | 7.4 | 9.7 | 1.4 | -0.76 | 19.1
| 0.7 | 7.4 | 10.2 | 1.5 | -0.54 | 17.8 | 0.15 | 7.6 | 9.6 | 1.5 | -0.72 | 19.2
| 0.9 | 7.6 | 11.2 | 1.6 | -0.70 | 19.0 | 0.20 | 7.8 | 10.4 | 1.5 | -0.72 | 19.4

Table S.4:  Moment values for the return distributions obtained for the period
01/03/2017 through 12/30/2020 for select efficient frontier portfolios
optimized under MCVaR0.99 and MV.

Model | SR | Sortino | STAR | Rachev | Gini | Model | SR | Sortino | STAR | Rachev | Gini
---|---|---|---|---|---|---|---|---|---|---|---
|  | (%) | (%) | (%) | (%) | (%) |  | (%) | (%) | (%) | (%) | (%)
EWBH | 4.86 | 6.59 | 1.88 | 83.86 | 12.26 |  |  |  |  |  |
$\lambda$ | $\alpha$ |  mCVaR0.99 | $\alpha$ | MV
0.00 | 0.0 | 4.87 | 6.69 | 1.94 | 87.79 | 12.40 | 0.00 | 5.09 | 6.96 | 2.00 | 86.56 | 12.97
| 0.3 | 4.91 | 6.75 | 1.96 | 87.84 | 12.55 | 0.05 | 5.09 | 6.92 | 1.99 | 85.10 | 13.03
| 0.5 | 4.92 | 6.76 | 1.96 | 87.76 | 12.55 | 0.10 | 4.92 | 6.66 | 1.93 | 83.94 | 12.35
| 0.7 | 4.98 | 6.85 | 2.00 | 88.42 | 12.66 | 0.15 | 4.62 | 6.21 | 1.81 | 83.29 | 11.47
| 0.9 | 5.16 | 7.08 | 2.07 | 88.03 | 13.11 | 0.20 | 4.34 | 5.81 | 1.69 | 82.50 | 10.68
0.25 | 0.0 | 4.80 | 6.59 | 1.92 | 87.54 | 12.29 | 0.00 | 5.25 | 7.15 | 2.05 | 85.57 | 13.45
| 0.3 | 4.90 | 6.73 | 1.96 | 87.57 | 12.55 | 0.05 | 5.07 | 6.92 | 1.99 | 86.10 | 12.99