#  New approximate stochastic dominance approaches for Enhanced Indexation
models ⋆

Francesco Cesarone1, Justo Puerto2
1 _Roma Tre University - Department of Business Economics_
<EMAIL_ADDRESS>
2 _Universidad de Sevilla - Instituto Universitario de Investigación
Matemática (IMUS)_
<EMAIL_ADDRESS>

###### Abstract

In this paper, we discuss portfolio selection strategies for Enhanced
Indexation (EI), which are based on stochastic dominance relations. The goal
is to select portfolios that stochastically dominate a given benchmark but
that, at the same time, must generate some excess return with respect to a
benchmark index. To achieve this goal, we propose a new methodology that
selects portfolios using the ordered weighted average (OWA) operator, which
generalizes previous approaches based on minimax selection rules and still
leads to solving linear programming models. We also introduce a new type of
approximate stochastic dominance rule and show that it implies the almost
Second-order Stochastic Dominance (SSD) criterion proposed by Lizyayev and
Ruszczyński (2012). We prove that our EI model based on OWA selects portfolios
that dominate a given benchmark through this new form of stochastic dominance
criterion. We test the performance of the obtained portfolios in an extensive
empirical analysis based on real-world datasets. The computational results
show that our proposed approach outperforms several SSD-based strategies
widely used in the literature, as well as the global minimum variance
portfolio.

Keywords: Portfolio selection; Stochastic dominance; Portfolio optimization;
Expected shortfall; Ordered weighted average operator.

⋆ This paper is dedicated to the memory of Moisés Rodríguez-Madrena

##  1 Introduction

It is well-known among academics and practitioners that active portfolio
management is unlikely to generate extra returns over their benchmarks (Bufalo
et al., 2022). For this reason, passive asset management has attracted great
interest in recent decades, offering returns close to benchmarks at a lower
cost. Index tracking (IT) is a popular problem for dealing with the
construction of passive management portfolios and has received large attention
in the literature. IT basically consists of a constrained optimization
problem, in which one minimizes the distance between a benchmark and the
tracking portfolio by typically using a fixed number of assets less than those
available in the investment universe. Comprehensive surveys on this problem
can be found in Beasley et al. (2003), Canakgoz and Beasley (2009), Guastaroba
and Speranza (2012), Bruni et al. (2012, 2013, 2015, 2017), Scozzari et al.
(2013), Cesarone et al. (2019), and, more recently, in Shu et al. (2020),
Sant’Anna et al. (2022).

A step towards a more sophisticated strategy, which has attracted increasing
attention over the past decade, is the Enhanced Indexation (EI) problem, where
the aim is not only to track a market index but simultaneously to generate an
excess return over the benchmark. An extensive review of the literature on EI
is provided in Canakgoz and Beasley (2009), Guastaroba et al. (2016), and more
recently in Guastaroba et al. (2020), Beraldi and Bruni (2022).

This paper focuses on portfolio selection strategies for EI based on
Stochastic Dominance (SD) relations. Indeed, one relatively recent and
promising approach for Enhanced Indexation consists of selecting portfolios
that stochastically dominate a given benchmark. SD provides a (partial) order
in the space of random returns, has connections with Expected Utility Theory,
and takes into account the full information regarding the distribution of the
portfolio random return (see Whitmore and Findlay, 1978; Levy, 1992; Shaked
and Shanthikumar, 1994; Müller and Stoyan, 2002; Levy, 2015). Roman et al.
(2013) develop a Second-order Stochastic Dominance (SSD) approach, which
consists of a multi-objective optimization problem to construct portfolios
whose returns dominate those of a benchmark (see also Roman et al., 2006;
Fábián et al., 2011a). Among the infinite Pareto-optimal solutions, the
authors scalarize their multi-objective model as a Minimax problem, which is
linear and is solved efficiently by cutting plane techniques. Several SSD-
based approaches are tested in Hodder et al. (2015), where the authors
implement the exact SSD methods developed by Kuosmanen (2004) and Kopa and
Post (2015).

Alternative portfolio selection approaches for EI, based on relaxing SD rules,
i.e., using some forms of approximate SD, can be found in Bruni et al. (2012),
Bruni et al. (2015), Bruni et al. (2017), and Sharma et al. (2017). In fact,
as discussed by Leshno and Levy (2002), in some economic situations, the
relaxation of SD conditions could provide advantages w.r.t. the exact ones.
Lizyayev and Ruszczyński (2012) propose different relaxations of SD for both
First and Second-order rules, named $\varepsilon$-Almost-FSD
($\varepsilon$-AFSD) and $\varepsilon$-Almost-SSD ($\varepsilon$-ASSD),
respectively. They also describe the optimization models for potential
applications of $\varepsilon$-AFSD and $\varepsilon$-ASSD to portfolio
selection, but without providing empirical tests on real datasets.
Furthermore, as discussed in the conclusions of Post and Kopa (2017), “the
current literature offers little guidance for the specification of “epsilon”,
or the admissible violation area.” We point out that our approach does not
require to explicitly specify “epsilon” since it aims to obtain solutions that
satisfy the SD conditions with minimum violations.

Further developments and generalizations on Almost Stochastic Dominance (ASD)
conditions can be found in Levy et al. (2010), Tzeng et al. (2013), Post and
Kopa (2013), Guo et al. (2014), Denuit et al. (2014), and Tsetlin et al.
(2015). However, there appear to be no available applications of ASD to
portfolio selection.

According to our literature review, the reader may realize that there is room
for improvement in the portfolio selection methods based on SD rules. Our goal
is to elaborate in this direction by proposing a new methodology and to shed
some light on this field, showing some of their connections. Our contributions
can be summarized as follows. First, we address the multi-objective
representation of the EI problem proposed by Roman et al. (2013) by an
alternative approach based on the Ordered Weighted Average (OWA) operator
(Yager, 1988). Its advantage is that it allows us to consider a set of the
worst realized objectives in the problem, whereas the Minimax operator, used
in Roman et al. (2013), pays attention only to the maximum of them.
Furthermore, the convenience of this type of scalarization is that it allows
highlighting a broader family of Pareto-optimal portfolios, improving the
properties of the Minimax ones. The application of the OWA operator for
scalarizing multi-objective optimization problems has been widely studied in
the literature (see, e.g., Fernández et al., 2014, 2017). Indeed, OWA is a
flexible operator that generalizes several criteria, including the Minimax one
(see, e.g., Ogryczak, 2000; Nickel and Puerto, 2006). As a consequence, we can
provide a more general method for highlighting a novel subset of Pareto-
optimal solutions of the multi-objective problem in Roman et al. (2013).
Second, we propose a new methodology to select a portfolio that dominates a
given benchmark index in terms of SD. This methodology uses the OWA
representation developed in Marín et al. (2020) and leads to solving linear
programming problems. Furthermore, it is more robust and contains as special
cases some known SD-based approaches to asset allocation (see Lizyayev and
Ruszczyński, 2012; Roman et al., 2013). Third, with the purpose of providing a
common framework to understand and explain approximate dominance rules, we
introduce a new rule that we call Cumulative Second-order
$\mathcal{E}$-Stochastic Dominance (CS$\mathcal{E}$SD), and we show that it
implies the family of $\varepsilon$-ASSD rules proposed by Lizyayev and
Ruszczyński (2012). Exploiting some results in multi-criteria optimization
(Ehrgott, 2005), we prove that the optimal solution of our EI model based on
OWA selects portfolios that dominate the benchmark index in terms of
CS$\mathcal{E}$SD. Fourth, to test and validate the performance of the
portfolios obtained by our proposed strategies, we provide extensive empirical
analysis based on the FTSE100, NASDAQ100, SP500, and Fama & French 49 Industry
portfolios datasets, comparing their out-of-sample behavior with that of the
portfolios constructed by several SD-based approaches proposed in the
literature and with that of the long-only, long-short, 1-norm, and 2-norm
minimum variance portfolios.

The remainder of the paper is organized as follows. In Section 2, we recall
the most commonly used exact stochastic dominance rules and present new
approximate stochastic dominance criteria. In Section 3, we introduce the
multi-objective optimization model of Roman et al. (2013), and then we provide
a general method for finding a class of Pareto-optimal solutions, which SSD-
dominate or almost SSD-dominate a given benchmark index. In Section 4, we
discuss some theoretical features of our proposed approach, providing some
statistical and economic interpretations and some illustrative examples.
Section 5 presents the results of an extensive empirical analysis based on
real-world data, comparing the out-of-sample performance of our proposed
approach with other concurrent strategies. Conclusions and further research
projects are presented in Section 6.

##  2 Preliminary concepts on some exact and approximate SD criteria

One of the reasons that has made the use of stochastic dominance criteria for
portfolio selection particularly attractive is that these criteria are able to
provide a partial order in the space of random variables by avoiding the
specification of a particular utility function, which represents the
preferences of an investor. Indeed, from Expected Utility Theory (EUT, see,
e.g., Morgenstern and Von Neumann, 1953), given a utility function, a
decision-maker prefers a random variable to another if it provides a larger
value in terms of its expected utility, i.e., it determines a greater
satisfaction. However, the specification of a utility function is a strictly
subjective issue, although it is a key element of EUT.

For the sake of readability, we recall here the most commonly used exact
stochastic dominance criteria. Let $A$ and $B$ be two random variables with
cumulative distribution functions $F_{A}(\alpha)=\mathbf{Pr}(A\leq\alpha)$ and
$F_{B}(\alpha)=\mathbf{Pr}(B\leq\alpha)$ for $\alpha\in\mathbb{R}$.

######  Definition 1 (Zero-order Stochastic Dominance - ZSD).

$A$ is preferred to $B$ w.r.t. ZSD iff

$F_{A-B}(0)=\mathbf{Pr}(A-B\leq 0)=0$ (1)

The ZSD relation represents behavior of a decision maker who prefers a random
variable over another only when the first gives better outcomes than the
second in (almost) all states of the world.

######  Definition 2 (First-order Stochastic Dominance - FSD).

$A$ is preferred to $B$ w.r.t. FSD iff

$F_{A}(\alpha)\leq F_{B}(\alpha)\ \ \ \ \forall\alpha\in\mathbb{R}.$ (2)

######  Definition 3 (Second-order Stochastic Dominance - SSD).

$A$ is preferred to $B$ w.r.t. SSD iff

$\int_{-\infty}^{\alpha}F_{A}(\tau)d\tau\
\leq\int_{-\infty}^{\alpha}F_{B}(\tau)d\tau\ \ \ \
\forall\alpha\in\mathbb{R}.$ (3)

These two last definitions require that each inequality is strict in at least
one case.

As discussed, e.g., in Levy (1992), these relations can be linked to EUT in
terms of different classes of utility functions. Indeed, $A$ is preferred to
$B$ w.r.t. FSD if and only if $E[u(A)]\geq E[u(B)]$ for all non-decreasing
utility functions $u$; $A$ is preferred to $B$ w.r.t. SSD if and only if the
same holds for all non-decreasing and concave utility functions. The SSD
relation is less demanding than that of ZSD and of FSD. More precisely, given
an SD criterion of order $v$, when increasing the order, the $v+1$-SD
condition becomes less restrictive. Furthermore, $v$-SD implies $(v+1)$-SD,
while the opposite is not necessarily true (see, e.g., Levy, 2015). In this
context, Post and Kopa (2017) remark that Third-order Stochastic Dominance
(TSD), although very appealing from a theoretical perspective, is highly
demanding to test. Therefore, these authors propose a method of selecting
investment portfolios that dominate a given benchmark in terms of TSD, which
is less demanding. More precisely, their strategy is based on a refinement of
the “SuperConvex” TSD (SCTSD) approach of Bawa et al. (1985) that leads to a
portfolio selection model formulated as a convex quadratic constrained
programming. As stated by the authors, SCTSD can be seen as an approximation
of TSD although in any case results in a difficult quadratic program. For the
reasons mentioned above, we have restricted ourselves from considering TSD in
our experiments. We want to remain within the framework of LP models, avoiding
the more complex quadratic or even superconvex mathematical programs. We also
observe that fractional-order SD rules extending the classical integer-order
SD have recently been proposed in the literature (see Müller et al., 2017;
Huang et al., 2020). However, there currently are no portfolio selection
applications in this framework.

In this paper, we essentially focus on the SSD condition and its relaxations,
characterizing our approach as an LP. An example of an approximate (relaxed)
version of the SSD criterion is given in Lizyayev and Ruszczyński (2012).

######  Definition 4 (Almost Second-order Stochastic Dominance - ASSD).

Given a tolerance $\vartheta>0,\,A$ is preferred to $B$ w.r.t. ASSD if

$\int_{-\infty}^{\alpha}\left(F_{A}(\tau)-F_{B}(\tau)\right)d\tau\leq\vartheta\quad\forall\alpha\in\left[\alpha^{\prime},\alpha^{\prime\prime}\right],$

where $\left[\alpha^{\prime},\alpha^{\prime\prime}\right]$ is the combined
range of outcomes of $A$ and $B$.

Another example of relaxation of SD conditions can be found in Bruni et al.
(2017), where the authors introduce the Zero-order $\varepsilon$-Stochastic
Dominance ($\mathrm{Z}\varepsilon\mathrm{SD}$) relation, which is a relaxation
(approximation) of the standard ZSD (see Definition 1). Although in their work
only the ZSD relaxation is addressed, this idea could also be extended to
higher-order stochastic dominance rules.

######  Remark 1 ($\mathrm{Z}\varepsilon\mathrm{SD}$ vs. ASSD).

We observe that the extension of the $\mathrm{Z}\varepsilon\mathrm{SD}$
approach to SSD, i.e., the Second-order $\varepsilon$-Stochastic Dominance
($\mathrm{S}\varepsilon\mathrm{SD}$), is equivalent to ASSD, by simply
considering $\varepsilon=\vartheta$.

Based on Definitions 3 and 4, the SSD and ASSD relations can be expressed by
the following equivalent criteria.

  * •

$E[u(A)]-E[u(B)]\geq-\tau$ holds for all non-decreasing and concave utility
functions $u$;

  * •

$E[[\eta-A]_{+}]-E[[\eta-B]_{+}]\leq\tau$ holds for all $\eta\in\mathbb{R}$;

  * •

$Tail_{\beta}(A)-Tail_{\beta}(B)\geq-\tau$ holds for all $\beta\in(0,1]$,
where $Tail_{\beta}(A)$ is the unconditional expectation of the worst
$\beta$100% outcomes of $A$.

Clearly, these conditions are equivalent to SSD for $\tau=0$ and to ASSD for
$\tau>0$.

In the remaining part of the paper, we assume that the asset returns are
discrete random variables defined on a discrete state space, and that there
are $T$ states of nature, each with probability $\pi_{t}$ with $t=1,...,T$. We
use a look-back approach, where the realizations of the discrete random
returns correspond to the historical scenarios, and the investment decision is
performed on an in-sample window of $T$ historical outcomes, each with
probability $\pi_{t}=1/T$, as is typically done in portfolio optimization
(see, e.g., Carleo et al., 2017; Cesarone, 2020; Bellini et al., 2021, and
references therein).

######  Remark 2 (Equally likely states and Shannon entropy).

We observe that given a discrete random variable with probability distribution
$\pi_{t}$ (where $t=1,...,T$) the associated Shannon entropy value
$S(\pi)=-\sum_{t=1}^{T}\pi_{t}\log_{2}(\pi_{t})$ is a measure for the degree
of uncertainty or ignorance about the probability distribution. Its value is
zero when the probability of an event is 1 (i.e., $\pi_{t}=1$ for a
$t=1,...,T$) or a certainty, while it is maximum when all the states of nature
are equally likely ($\pi_{t}=1/T$), namely the case of maximum uncertainty
(see, e.g., Myung et al., 1996). As for instance in Post and Kopa (2017), our
experiments consider the case of maximum ignorance about the probability
distribution $\pi_{t}$.

We denote by $R^{I}$ the return of a given benchmark index and by $R$ the
return of a portfolio of assets. Furthermore, we indicate by $(R_{(j)})_{1\leq
j\leq T}$ the ordered realizations such that $R_{(1)}\leq\ldots\leq R_{(T)}$,
and, reversing the order, by $(R^{(j)})_{1\leq j\leq T}$ the ordered
realizations such that $R^{(1)}\geq\ldots\geq R^{(T)}$. Let us assume w.l.o.g.
that the scenarios of the portfolio return $R_{t}$ and the index $R^{I}_{t}$
occur with probabilities $\pi_{t}$ for $t=1,...,T$.

In a discrete world, Definitions 2 and 3 can be then expressed as follows

  * •

$R$ FSD-dominates $R^{I}$ iff
$\displaystyle\sum_{R_{t}\leq\alpha}\pi_{t}\leq\sum_{R^{I}_{t}\leq\alpha}\pi_{t},\;\forall\alpha\in\mathbb{R}.$

  * •

$R$ SSD-dominates $R^{I}$ iff
$\eta\sum_{R_{t}\leq\eta}\pi_{t}-\sum_{R_{t}\leq\eta}R_{t}\pi_{t}\leq\eta\sum_{R_{t}^{I}\leq\eta}\pi_{t}-\sum_{R_{t}^{I}\leq\eta}R_{t}^{I}\pi_{t},\
\forall\eta\in\mathbb{R}.$

Note that for a given threshold $\beta=\sum_{R_{t}\leq\eta}\pi_{t}$, we have
that $Tail_{\beta}(R)=\displaystyle\sum_{R_{t}\leq\eta}\pi_{t}R_{t}$. Then,
since

$CVaR_{\beta}(R)=-\frac{1}{\beta}\sum_{R_{t}\leq\eta}\pi_{t}R_{t},$ (4)

we have that $Tail_{\beta}(R)=-\displaystyle\beta CVaR_{\beta}(R)$. Thus,
requiring $Tail_{\beta}(R)\geq Tail_{\beta}(R_{I})$ is equivalent to consider
$CVaR_{\beta}(R)\leq CVaR_{\beta}(R_{I})$ for all $\beta\in(0,1]$.

In the particularly interesting case, that is usual in portfolio optimization
(see also Remark 2), where $\pi_{t}=1/T$ for all $t$, the above criteria
simplify resulting in:

  * •

$R$ FSD-dominates $R_{I}$ iff $R_{(j)}\geq R^{I}_{(j)}$ for all
$j=1,\ldots,T$.

  * •

$R$ SSD-dominates $R_{I}$ iff
$\sum_{t=1}^{j}R_{(t)}\geq\sum_{t=1}^{j}R^{I}_{(t)}$ for all $j=1,\ldots,T$.

Also in this case,
$Tail_{\frac{j}{T}}(R)=\displaystyle\frac{1}{T}\sum_{t=1}^{j}R_{(t)}$. Next,
since

$CVaR_{\frac{j}{T}}(R)=-\frac{1}{j}\sum_{t=1}^{j}R_{(t)},$ (5)

we have that
$Tail_{\frac{j}{T}}(R)=-\displaystyle\frac{j}{T}CVaR_{\frac{j}{T}}(R)$. Thus,
requiring $Tail_{\frac{j}{T}}(R)\geq Tail_{\frac{j}{T}}(R_{I})$ is equivalent
to consider $CVaR_{\frac{j}{T}}(R)\leq CVaR_{\frac{j}{T}}(R_{I})$ for all
$j=1,\ldots,T$.

When discrete random variables are considered, we can introduce a _cumulative_
version of $\mathrm{S}\varepsilon\mathrm{SD}$, named here
$\mathrm{CS}\varepsilon\mathrm{SD}$, similar to the
$\mathrm{CZ}\varepsilon\mathrm{SD}$ relation developed in Bruni et al. (2017).
This last approximate stochastic dominance relation was thought as a criterion
for bounding the cumulative losses for Enhanced Indexation purposes and, among
other properties, Bruni et al. (2017) show that
$\mathrm{CZ}\varepsilon\mathrm{SD}$ implies Z$\varepsilon$SD. Therefore,
similarly to how $\mathrm{Z}\varepsilon\mathrm{SD}$ can be extended to
$\mathrm{S}\varepsilon\mathrm{SD}$, $\mathrm{CZ}\varepsilon\mathrm{SD}$ can
also be generalized to SSD. This generalization leads to the following
definition and the fact that this new criterion implies
$\mathrm{S}\varepsilon\mathrm{SD}$, or, equivalently, ASSD (see Section 4,
Proposition 2, point 1).

######  Definition 5 (Cumulative Second-order $\varepsilon$-Stochastic
Dominance - $CS\varepsilon SD$).

Given a tolerance $\varepsilon>0$, $A$ is preferred to $B$ w.r.t.
$CS\varepsilon SD$ if

$\sum_{t\in\mathcal{S}}CVaR_{\frac{t}{T}}(A)-\sum_{t\in\mathcal{S}}CVaR_{\frac{t}{T}}(B)\leq\varepsilon\quad\forall\mathcal{S}\subseteq\mathcal{T},$
(6)

where $\mathcal{T}=\\{1,\ldots,T\\}$.

Definition 5 requires to check an exponential number of conditions to verify
$CS\varepsilon SD$. However, the next result shows that the condition can be
verified in polynomial time.

For this purpose, we need to introduce some additional notation. Let us denote
by $\sigma$ the permutation that provides the sorting of
$\left(CVaR_{\frac{t}{T}}(A)-CVaR_{\frac{t}{T}}(B)\right)_{1\leq t\leq T}$ as
$CVaR_{\frac{\sigma(1)}{T}}(A)-CVaR_{\frac{\sigma(1)}{T}}(B)\geq\ldots\geq
CVaR_{\frac{\sigma(T)}{T}}(A)-CVaR_{\frac{\sigma(T)}{T}}(B)$. Next, for the
ease of presentation, and whenever the framework is clear, we denote

$\sum_{j=1}^{t}\left(CVaR_{\frac{\cdot}{T}}(A)-CVaR_{\frac{\cdot}{T}}(B)\right)^{(j)}:=\sum_{j=1}^{t}CVaR_{\frac{\sigma(j)}{T}}(A)-CVaR_{\frac{\sigma(j)}{T}}(B)$

######  Proposition 1.

Given a tolerance $\varepsilon>0,A$ is preferred to $B$ w.r.t. $CS\varepsilon
SD$ iff

$\sum_{j=1}^{t}\left(CVaR_{\frac{\cdot}{T}}(A)-CVaR_{\frac{\cdot}{T}}(B)\right)^{(j)}\leq\varepsilon\quad\forall
t\in\mathcal{T}.$

###### Proof.

We observe that Conditions (6) are equivalent to require

$\max_{\mathcal{S}\subseteq\mathcal{T}:|\mathcal{S}|=t}\left\\{\sum_{t\in\mathcal{S}}CVaR_{\frac{t}{T}}(A)-\sum_{t\in\mathcal{S}}CVaR_{\frac{t}{T}}(B)\right\\}\leq\varepsilon\quad\forall
t\in\mathcal{T}.$

Furthermore, the maximum in the left-hand-side is attained on the $t$ largest
values, so that

$\max_{\mathcal{S}\subseteq\mathcal{T}:|\mathcal{S}|=t}\left\\{\sum_{t\in\mathcal{S}}CVaR_{\frac{t}{T}}(A)-\sum_{t\in\mathcal{S}}CVaR_{\frac{t}{T}}(B)\right\\}=\sum_{j=1}^{t}\left(CVaR_{\frac{\cdot}{T}}(A)-CVaR_{\frac{\cdot}{T}}(B)\right)^{(j)}.$

This proves the result. ∎

We point out that in Proposition 1 we have a single threshold $\varepsilon$,
$\forall t$. However, it can be generalized so that we can impose different
tolerances $\varepsilon_{t}$ for each $t\in\mathcal{T}$, as described in the
following definition.

######  Definition 6 (Cumulative Second-order $\mathcal{E}$-Stochastic
Dominance - $CS\mathcal{E}SD$).

Given a tolerance vector
$\mathcal{E}=(\varepsilon_{1},\ldots,\varepsilon_{T})\in\mathbb{R}^{T}_{>0}$,
$A$ is preferred to $B$ w.r.t. $CS\mathcal{E}SD$ if

$\sum_{j=1}^{t}\left(CVaR_{\frac{\cdot}{T}}(A)-CVaR_{\frac{\cdot}{T}}(B)\right)^{(j)}\leq\varepsilon_{t}\quad\forall
t\in\mathcal{T}.$

In the next section, we introduce the Enhanced Indexation model of Roman et
al. (2013), which properly generalized is closely connected to
$CS\mathcal{E}SD$, as shown in Section 4.1.

##  3 Enhanced Indexation based on stochastic dominance

Here, we first report the multi-objective approach of Roman et al. (2013) for
finding portfolios that SSD-dominate or almost SSD-dominate a given benchmark
index. Then, we provide a general method for finding a novel subset of Pareto-
optimal solutions of such a multi-objective model. More precisely, we
generalize the existing minimax selection rule for the multi-objective
approach of Roman et al. (2013), which, as shown in Section 3.1, is strictly
linked to the ASSD conditions of Lizyayev and Ruszczyński (2012) (see
Definition 4, Remarks 1 and 3). Thus still keeping the model as an LP, in
Section 3.2 we propose more flexible choices of Pareto-optimal solutions of
such a multi-objective model, which, as discussed in Section 4, are related to
the $CS\mathcal{E}SD$ rules and as we shall see, imply those of Lizyayev and
Ruszczyński (2012).

We consider a setting where $n$ assets are available in the investment
universe with rates of return described by the random variables
$R_{1},\ldots,R_{n}$. A portfolio is identified with a vector
$x=(x_{1},\ldots,x_{n})\in\mathbb{R}_{+}^{n}$, where $x_{k}$ denotes the
fraction of capital invested in the $k^{th}$ asset. Hence, the (linear)
portfolio return is given by $R(x)=\sum_{k=1}^{n}x_{k}R_{k}$.

As mentioned above, Roman et al. (2013) propose the following multi-objective
model whose Pareto-optimal solutions SSD-dominate or almost SSD-dominate a
given benchmark index (see also Roman et al., 2006; Fábián et al., 2011b, a;
Valle et al., 2017; Cesarone et al., 2022a)

$\begin{array}[]{rl}\displaystyle\min_{x}&\left[(CVaR_{\frac{1}{T}}(R(x))-CVaR_{\frac{1}{T}}(R_{I})),\ldots,(CVaR_{\frac{T}{T}}(R(x))-CVaR_{\frac{T}{T}}(R_{I}))\right]\\\
\mbox{s.t.}&x\in C\end{array}$ (7)

where $C$ is a polyhedron. Alternatively, they find the SSD efficient
portfolios by solving a slightly different multi-objective optimization
problem based on centered Tails

$\begin{array}[]{rl}\displaystyle\max_{x}&\left[(Tail_{\frac{1}{T}}(R(x))-Tail_{\frac{1}{T}}(R_{I})),\ldots,(Tail_{\frac{T}{T}}(R(x))-Tail_{\frac{T}{T}}(R_{I}))\right]\\\
\mbox{s.t.}&x\in C\end{array}$ (8)

In the following, we will focus on Problem (7) and then briefly discuss how
the results obtained from this problem are also valid for Problem (8).

###  3.1 Minimax scalarization

Among the infinite Pareto-optimal points belonging to the efficient
$T$-dimensional hypersurface generated by Problem (7), Roman et al. (2013)
consider the one obtained by solving the following Minimax problem

$\begin{array}[]{rl}\displaystyle\min_{x}&\displaystyle\max_{1\leq j\leq
T}(CVaR_{\frac{j}{T}}(R(x))-CVaR_{\frac{j}{T}}(R_{I}))\\\ \mbox{s.t.}&x\in
C\end{array}$ (9)

This model can be expressed as an LP problem, using the CVaR reformulation of
Rockafellar and Uryasev (2000, 2002). However, due to the high number of
variables and constraints (more than $T^{2}$), Problem (9) is solved by
implementing cutting plane techniques, as proposed by Fábián et al. (2011a).

We note that the above problem is equivalent to

$\begin{array}[]{rl}\displaystyle\min_{x}&\displaystyle\left(CVaR_{\frac{\cdot}{T}}(R(x))-CVaR_{\frac{\cdot}{T}}(R_{I})\right)^{(1)}\\\
\mbox{s.t.}&x\in C\end{array}$ (10)

######  Remark 3 (Minimax scalarization vs. ASSD conditions).

From Problem (10), one can directly observe the link between Model (9) in the
paper by Roman et al. (2013), $SSD$ (Definition 3) and ASSD (Definition 4)
conditions. Indeed, denoting by $x^{\star}$ the optimal solution of Problem
(10), if the corresponding optimal value function
$\left(CVaR_{\frac{\cdot}{T}}(R(x^{\star}))-CVaR_{\frac{\cdot}{T}}(R_{I})\right)^{(1)}$
is nonpositive (positive), then the corresponding optimal portfolio SSD-
dominates (ASSD-dominates) the benchmark index.

###  3.2 Ordered weighted average scalarization

To address the multi-objective optimization problems (7) and (8), we propose
here an approach based on the Ordered Weighted Average (OWA) operator. As
shown below, using the OWA scalarization, we are able to provide a more
general methodology to highlight a new subset of Pareto-optimal solutions of
such multi-objective problems based on centered CVaRs (see Section 3.2.1) and
centered Tails (see Section 3.2.2).

####  3.2.1 Maximizing OWA of the centered CVaRs

An alternative scalarization of Problem (7) consists of the following simple
idea: rather than minimizing the maximum centered CVaR, namely Model (9), one
can investigate the performance of the Pareto-optimal portfolios obtained by
minimizing the sum of the $k$ maximal CVaRs for a fixed $k=1,\ldots,T$. It is
well-known that this form of scalarization always results in Pareto-optimal
solutions since the ordered median function is an isotone function provided
than the lambda weights are non-negative (see Lemma 1.1 in Nickel and Puerto,
2006). Obviously, when $k=1$, one recovers the Minimax approach of Roman et
al. (2013). More specifically, we deal with Problem (7) by scalarizing the $T$
goals in the following way. To simplify the presentation, we introduce some
notation. Let us denote by $g_{t}(x):=CVaR_{t/T}(R(x))-CVaR_{t/T}(R_{I})$ with
$t\in\mathcal{T}$, and following our notation on Section 2, by
$(g^{(j)}(x))_{1\leq t\leq T}$ the elements of a vector such that
$g^{(1)}(x)\geq g^{(2)}(x)\geq\ldots\geq g^{(T)}(x)$. We then consider the
following problem

$\begin{array}[]{rl}\displaystyle\min_{x}&\displaystyle\sum_{j=1}^{T}\lambda^{j}g^{(j)}(x)\\\
\mbox{s.t.}&x\in C\end{array}$ (11)

where $(\lambda^{j})_{1\leq j\leq T}$ is a vector such that
$\lambda^{1}\geq\lambda^{2}\geq\ldots\geq\lambda^{T}\geq 0$. We can observe
that our approach is less conservative (i.e., risk-averse) than that of Roman
et al. (2013), since Problem (11) takes into account all differences between
CVaRs at different confidence levels and not just the worst one. However, our
approach still models conservative behavior, since larger weights are assigned
to the worst differences.

Thus, our goal is to obtain a tractable representation for the problem that
minimizes the sum of an ordered weighted average (OWA) of the centered CVaRs,
$(CVaR_{\cdot/T}(R(x))-CVaR_{\cdot/T}(R_{I}))_{1\leq j\leq T}^{(j)}$, with
weights $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{T}\geq 0$. Note that
if $\lambda=(1,0,\ldots,0)$ we obtain the Minimax portfolio (as in Roman et
al., 2013), whereas if
$\lambda=(1,1,\stackrel{{\scriptstyle(k)}}{{\ldots}},1,0,\ldots,0)$ we get the
portfolio minimizing the sum of the $k$-worse centered CVaRs.

To achieve a more tractable formulation of Problem (11), we denote by
$f_{t}(x)=-g_{t}(x)=CVaR_{t/T}(R_{I})-CVaR_{t/T}(R(x))$ $\forall
t\in\mathcal{T}$, and by $(f_{(j)}(x))_{1\leq t\leq T}$ the elements of a
vector such that $f_{(1)}(x)\leq f_{(2)}(x)\leq\ldots\leq f_{(T)}(x)$. Hence,
$f_{(j)}(x)=-g^{(j)}(x)$ $\forall j\in\mathcal{T}$. We then obtain the
following maximization problem which is equivalent to (11)

$\begin{array}[]{rl}\displaystyle\max_{x}&\displaystyle\sum_{j=1}^{T}\lambda^{j}f_{(j)}(x)\\\
\mbox{s.t.}&x\in C\end{array}$ (12)

where, again, $(\lambda^{j})_{1\leq j\leq T}$ is a vector such that
$\lambda^{1}\geq\lambda^{2}\geq\ldots\geq\lambda^{T}\geq 0$. Furthermore,
using the approach described in Nickel and Puerto (2006) (Theorem 1.3, pag.
13) and Marín et al. (2020), we can write

$\displaystyle\sum_{j=1}^{T}\lambda^{j}f_{(j)}(x)=$
$\displaystyle\displaystyle\min_{z}\;\sum_{t=1}^{T}\sum_{j=1}^{T}\lambda^{j}f_{t}(x)z_{jt}$
(13a) $\displaystyle\mbox{s.t.}\hskip
2.84544pt\sum_{j=1}^{T}z_{jt}=1,\;t=1,\ldots,T,$ (13b) $\displaystyle\hskip
17.07182pt\sum_{t=1}^{T}z_{jt}=1,\;j=1,\ldots,T,$ (13c) $\displaystyle\hskip
17.07182ptz_{jt}\in\\{0,1\\},$ (13d)

Since the above problem is an assignment problem, and the constraint matrix
(13b)-(13c) is totally unimodular, we can relax the integrality constraints
(13d) into $z_{jt}\geq 0$, thus obtaining an LP. Then, Problem
(13a)-(13b)-(13c) with $z_{jt}\geq 0$ has an exact dual so that

$\displaystyle\sum_{j=1}^{T}\lambda^{j}f_{(j)}(x)=$
$\displaystyle\displaystyle\max_{u,v}\;\sum_{t=1}^{T}u_{t}+\sum_{j=1}^{T}v_{j}$
(14) $\displaystyle\mbox{s.t.}\hskip
2.84544ptu_{t}+v_{j}\leq\lambda^{j}f_{t}(x),\;\forall j,t=1,\ldots,T,$
$\displaystyle\hskip 17.07182ptu_{t},v_{j}\mbox{ free}\qquad\forall
j,t=1,\ldots,T$

Therefore, if we wish to maximize the OWA of the centered CVaRs, then we can
proceed by solving the following problem

$\begin{array}[]{llll}\displaystyle\max_{x}&\sum_{j=1}^{T}\lambda^{j}f_{(j)}(x)=&\displaystyle\max_{x,u,v}&\sum_{t=1}^{T}u_{t}+\sum_{j=1}^{T}v_{j}\\\
&&\mbox{s.t.}&u_{t}+v_{j}\leq\lambda^{j}(CVaR_{t/T}(R_{I})-CVaR_{t/T}(R(x))),\;\forall
j,t\\\ &&&u_{t},v_{j}\mbox{ free},\;x\in C\end{array}$ (15)

where $f_{(j)}(x)=(CVaR_{\cdot/T}(R_{I})-CVaR_{\cdot/T}(R(x)))_{(j)}$ and $C$
is a polyhedron. This representation still needs some further refinement to
become linear since the CVaR function requires an appropriate expansion.
Knowing that for a fixed portfolio $x\in C$

$\displaystyle CVaR_{t/T}(R(x)):=$
$\displaystyle\min_{\zeta\in\mathbb{R},d}\;\zeta+\frac{T}{t}\sum_{j=1}^{T}\pi_{j}d_{j}$
$\displaystyle\mbox{s.t.}\hskip
2.84544ptd_{j}\geq-\zeta-\sum_{\ell=1}^{n}r_{\ell j}x_{\ell},\quad
j=1,\ldots,T,$ $\displaystyle\hskip 17.07182ptd_{j}\geq 0,\;\forall j\,,$

we can express $f_{t}(x)=CVaR_{t/T}(R_{I})-CVaR_{t/T}(R(x))$ using the
following linear program

$\begin{array}[]{lll}f_{t}(x):=&\displaystyle\max_{\zeta\in\mathbb{R},d}\;CVaR_{t/T}(R_{I})-\zeta-\frac{T}{t}\sum_{j=1}^{T}\pi_{j}d_{j}&\\\
&\mbox{s.t.}\hskip 2.84544ptd_{j}\geq-\zeta-\sum_{\ell=1}^{n}r_{\ell
j}x_{\ell},\quad j=1,\ldots,T,\\\ &\hskip 17.07182ptd_{j}\geq 0,\;\forall
j.\end{array}$ (16)

Now, the largest feasible region for the inequalities
$u_{t}+v_{j}+\lambda^{j}(\zeta_{t}+\frac{1}{t}\sum_{k=1}^{T}d_{tk})\leq\lambda^{j}CVaR_{t/T}(R_{I}),\;\forall
j,t$ subject to the constraints of the above problem, is attained when
$\lambda^{j}(\zeta_{t}+\frac{1}{t}\sum_{k=1}^{T}d_{tk})$ is minimized. Thus,
we obtain an alternative valid formulation for (15) substituting
$CVaR_{t/T}(R(x))$ by $\lambda^{j}(\zeta_{t}+\frac{1}{t}\sum_{k=1}^{T}d_{tk})$
plus the constraints in (16). Hence, Problem (15) becomes equivalent to the
following one:

$\begin{array}[]{ll}\displaystyle\max_{x,u,v,d,\zeta}&\displaystyle\sum_{t=1}^{T}u_{t}+\sum_{j=1}^{T}v_{j}\\\
\quad\mbox{s.t.}&\displaystyle
u_{t}+v_{j}\leq\lambda^{j}(CVaR_{t/T}(R_{I})-\zeta_{t}-\frac{1}{t}\sum_{k=1}^{T}d_{tk}),\;\forall
j,t\\\ &\displaystyle d_{tk}\geq-\zeta_{t}-\sum_{\ell=1}^{n}r_{\ell
k}x_{\ell},\;t,k=1,\ldots,T,\\\ &\zeta_{t},u_{t},v_{j}\mbox{ free, }d_{tk}\geq
0,\\\ &x\in C\,,\end{array}$ (17)

where, again, $C$ is a polyhedron. This last formulation is a Linear Program
(LP) with $n+T+T+T^{2}+T$ (namely, $x,u,v,d,\zeta$) variables, and $2T^{2}$
constraints plus those defining the polyhedron $C$. This LP problem returns
the portfolio $x$ that maximizes the ordered weighted average (OWA) of the
centered CVaRs. The reader may think that these figures are large, but still,
the model solves as a linear program, which simplifies a lot due to nowadays
efficiency of current solvers.

####  3.2.2 Maximizing OWA of the centered Tails

In this section, we briefly discuss how to extend the results obtained from
the multi-objective optimization problem based on centered CVaRs for Problem
(8), based on centered Tails, which is recalled here for the sake of
readability:

$\begin{array}[]{rl}\displaystyle\max_{x}&\left[(Tail_{\frac{1}{T}}(R(x))-Tail_{\frac{1}{T}}(R_{I})),\ldots,(Tail_{\frac{T}{T}}(R(x))-Tail_{\frac{T}{T}}(R_{I}))\right]\\\
\mbox{s.t.}&x\in C\,.\end{array}$ (18)

Roman et al. (2013) scalarize Model (18) considering the following Maximin
problem

$\begin{array}[]{rl}\displaystyle\max_{x}&\displaystyle\min_{1\leq j\leq
T}(Tail_{\frac{j}{T}}(R(x))-Tail_{\frac{j}{T}}(R_{I}))\\\ \mbox{s.t.}&x\in
C\end{array}$ (19)

Note that Problems (9) and (19) lead to different solutions (see Remark 1 in
Roman et al., 2013). Following the same procedure described in Section 3.2.1,
we can denote by $h_{t}(x):=Tail_{t/T}(R(x))-Tail_{t/T}(R_{I})$ with
$t\in\mathcal{T}$ and by $(h_{(j)}(x))_{1\leq t\leq T}$ the elements of a
vector such that $h_{(1)}(x)\leq h_{(2)}(x)\leq\ldots\leq h_{(T)}(x)$.
Therefore, we consider the following problem

$\begin{array}[]{rl}\displaystyle\max_{x}&\displaystyle\sum_{j=1}^{T}\lambda^{j}h_{(j)}(x)\\\
\mbox{s.t.}&x\in C\end{array}$ (20)

where $(\lambda^{j})_{1\leq j\leq T}$ is a vector such that
$\lambda^{1}\geq\lambda^{2}\geq\ldots\geq\lambda^{T}\geq 0$.

Following the same arguments that lead to Problem (15) and knowing that for a
fixed portfolio $x\in C$, we can express $h_{t}(x)$ as the following linear
program

$\displaystyle h_{t}(x):=$
$\displaystyle\max_{\zeta\in\mathbb{R},d}\;-\frac{t}{T}\zeta-\sum_{j=1}^{T}\pi_{j}d_{j}-Tail_{t/T}(R_{I})$
(21) $\displaystyle\mbox{s.t.}\hskip
2.84544ptd_{j}\geq-\zeta-\sum_{\ell=1}^{n}r_{\ell j}x_{\ell},\quad
j=1,\ldots,T,$ $\displaystyle\hskip 17.07182ptd_{j}\geq 0,\;\forall j\,,$

we can reformulate Problem (20) as

$\begin{array}[]{rl}\displaystyle\max_{x,u,v,d,\zeta}&\displaystyle\sum_{t=1}^{T}u_{t}+\sum_{j=1}^{T}v_{j}\\\
\mbox{s.t.}&\displaystyle
u_{t}+v_{j}\leq\lambda_{j}(-\frac{t}{T}\zeta_{t}-\frac{1}{T}\sum_{k=1}^{T}d_{tk}-Tail_{t/T}(R_{I})),\;\forall
j,t\\\ &\displaystyle d_{tk}\geq-\zeta_{t}-\sum_{\ell=1}^{n}r_{\ell
k}x_{\ell},\;t,k=1,\ldots,T,\\\ &\zeta_{t},u_{t},v_{j}\mbox{ free, }d_{tk}\geq
0,\\\ &x\in C.\end{array}$ (22)

Again, the above problem is an LP, and its solution returns the portfolio $x$
that maximizes the ordered weighted average (OWA) of the centered tails.

##  4 Theoretical Analysis

This section discusses some theoretical features of the OWA maximization of
centered CVaRs approach and the CS$\mathcal{E}$SD relations. More precisely,
Section 4.1 shows the theoretical link between Model (7) scalarized by the OWA
operator and the CS$\mathcal{E}$SD rule. In Section 4.2, we provide some
statistical motivations for using the OWA operator. Furthermore, we discuss
the economic interpretation of the CS$\mathcal{E}$SD concept, emphasizing its
main advantages. Finally, Section 4.3 shows some illustrative examples about
the $S\varepsilon SD$ (or, equivalently, ASSD), $CS\varepsilon SD$, and
$CS\mathcal{E}SD$ relations in three practical cases.

###  4.1 OWA maximization and CS$\mathcal{E}$SD

We provide here the theoretical relationship between Model (12), which
consists of maximizing the OWA of the centered CVaRs, and the
CS$\mathcal{E}$SD criterion, described in Definition 6.

Let $g^{(t)}(x):=\left(CVaR_{t/T}(R(x))-CVaR_{t/T}(R_{I})\right)^{(t)}$ and
$G^{(t)}(x):=\sum_{j=1}^{t}g^{(j)}(x)$ for each $t\in\mathcal{T}$. Note that
the efficient portfolios, whose returns dominate the return of a given
benchmark index $R_{I}$ w.r.t. CS$\mathcal{E}$SD, are those obtained by
solving the following problems

$\begin{array}[]{rl}\displaystyle\min_{x}&\displaystyle G^{(j)}(x)\\\
\mbox{s.t.}&G^{(t)}(x)\leq\varepsilon_{t}\quad t=1,\ldots,T\quad t\neq j\\\
&x\in C\end{array}$ (23)

for each $j\in\mathcal{T}$. In the following we show that Problem (23) is
equivalent to Problem (12). For the sake of readability, we recall below some
well-known concepts in multi-criteria optimization. Consider the following
multi-objective optimization problem

$\min_{x\in\mathcal{X}}\left(h_{1}(x),h_{2}(x),\ldots h_{p}(x)\right)\,.$ (24)

A standard technique to scalarize Problem (24) is the $\varepsilon$-constraint
method, namely

$\begin{array}[]{rl}\displaystyle\min_{x}&\displaystyle h_{j}(x)\\\
\mbox{s.t.}&h_{k}(x)\leq\varepsilon_{k}\quad k=1,\ldots,p\quad k\neq j\\\
&x\in\mathcal{X}\end{array}$ (25)

An alternative is the following weighted sum problem

$\min_{x\in\mathcal{X}}\sum_{k=1}^{p}\gamma_{k}h_{k}(x)$ (26)

We first recall a theorem that links these two methods.

######  Theorem 7 (Theorem 4.6 of Ehrgott (2005)).

  1. 1.

Suppose $\hat{x}$ is an optimal solution of (26). If $\gamma_{j}>0$ there
exists $\hat{\varepsilon}$ such that $\hat{x}$ is an optimal solution of (25),
too.

  2. 2.

Suppose $\mathcal{X}$ is a convex set and
$h_{k}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ are convex functions. If $\hat{x}$
is an optimal solution of (25) for some $j$, there exists
$\hat{\gamma}\in\mathbb{R}_{\geq}^{p}$ such that $\hat{x}$ is optimal for
$\min_{x\in\mathcal{X}}\sum_{k=1}^{p}\hat{\gamma}_{k}h_{k}(x)$.

The following result is instrumental to get the announced equivalence between
models.

######  Lemma 8.

Assume that $C$ is a convex set. Then, $G^{(t)}$ is a convex function for all
$t\in\mathcal{T}$.

###### Proof.

Let $t\in\mathcal{T}$. For $x,y\in C$ and $\alpha\in[0,1]$ we have that

$G^{(t)}(\alpha x+(1-\alpha)y)=\sum_{j=1}^{t}g^{(j)}(\alpha x+(1-\alpha)y).$

Let $k_{1},\ldots,k_{t}$ be indices such that

$\sum_{j=1}^{t}g^{(j)}(\alpha x+(1-\alpha)y)=\sum_{j=1}^{t}g_{k_{j}}(\alpha
x+(1-\alpha)y).$

Thus, since CVaR function is convex,

$\sum_{j=1}^{t}g_{k_{j}}(\alpha
x+(1-\alpha)y)\leq\alpha\sum_{j=1}^{t}g_{k_{j}}(x)+(1-\alpha)\sum_{j=1}^{t}g_{k_{j}}(y).$

Finally, note that

$\alpha\sum_{j=1}^{t}g_{k_{j}}(x)+(1-\alpha)\sum_{j=1}^{t}g_{k_{j}}(y)\leq\alpha\sum_{j=1}^{t}g^{(j)}(x)+(1-\alpha)\sum_{j=1}^{t}g^{(j)}(y)=\alpha
G^{(t)}(x)+(1-\alpha)G^{(t)}(y).$

∎

Exploiting Theorem 7 and Lemma 8, we now provide the theoretical link between
Problems (23) and (11).

######  Theorem 9.

  1. 1.

Suppose $\hat{x}$ is an optimal solution of Problem (11). If
$\lambda^{j}-\lambda^{j+1}>0$, then there exists
$\underline{\varepsilon}\in\mathbb{R}_{\geq}^{T}$ such that $\hat{x}$ is an
optimal solution of Problem (23), too.

  2. 2.

If $\hat{x}$ is an optimal solution of Problem (23) for some $j$, there exists
$\hat{\lambda}\in\mathbb{R}_{\geq}^{T}$ with
$\hat{\lambda}_{1}\geq\ldots\geq\hat{\lambda}_{T}$ such that $\hat{x}$ is
optimal for Problem (11).

###### Proof.

  1. 1.

We note that the objective function of Problem (11) can be rewritten as
follows

$\displaystyle\sum_{j=1}^{T}\lambda^{j}g^{(j)}(x)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{T}\sum_{t=j}^{T}\left(\lambda^{t}-\lambda^{t+1}\right)g^{(j)}(x)\quad\mbox{or,
equivalently,}$ (27) $\displaystyle=$
$\displaystyle\sum_{j=1}^{T}\sum_{t=1}^{j}\left(\lambda^{j}-\lambda^{j+1}\right)g^{(t)}(x)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{T}\left(\lambda^{j}-\lambda^{j+1}\right)G^{(j)}(x)$

where $G^{(j)}:=\sum_{t=1}^{j}g^{(t)}$ for each $j\in\mathcal{T}$ and we
assume that $\lambda^{T+1}=0$. Thus, we can reformulate Problem (11) as

$\begin{array}[]{rl}\displaystyle\min_{x}&\displaystyle\sum_{j=1}^{T}\left(\lambda^{j}-\lambda^{j+1}\right)G^{(j)}(x)\\\
\mbox{s.t.}&x\in C\,.\end{array}$ (29)

Knowing that, by construction,
$\lambda^{1}\geq\lambda^{2}\geq\ldots\geq\lambda^{T}\geq 0$ and by hypothesis
of item 1, $\lambda^{j}-\lambda^{j+1}>0$ for some $j\in\mathcal{T}$, the
result follows by Theorem 7.

  2. 2.

Assuming that $\hat{x}$ is an optimal solution of Problem (23) and using point
2 of Theorem 7, there must exist some
$\hat{\gamma}=\left(\hat{\gamma}^{1},\hat{\gamma}^{2},\ldots,\hat{\gamma}^{T}\right)\in\mathbb{R}_{\geq}^{T}$
such that

$\sum_{t=1}^{T}\hat{\gamma}^{t}G^{(t)}(x)\geq\sum_{t=1}^{T}\hat{\gamma}^{t}G^{(t)}(\hat{x})$

for all $x\in C$. Equivalently,

$\sum_{t=1}^{T}\hat{\gamma}^{t}\sum_{j=1}^{t}g^{(j)}(x)\geq\sum_{t=1}^{T}\hat{\gamma}^{t}\sum_{j=1}^{t}g^{(j)}(\hat{x})$

Exploiting relations (27) and (1) and denoting by
$\hat{\gamma}^{t}=\hat{\lambda}^{t}-\hat{\lambda}^{t+1}$ $\forall
t\in\mathcal{T}$, but with $\hat{\lambda}^{T+1}=0$, we have

$\sum_{t=1}^{T}\sum_{j=t}^{T}\left(\hat{\lambda}^{j}-\hat{\lambda}^{j+1}\right)g^{(t)}(x)\geq\sum_{t=1}^{T}\sum_{j=t}^{T}\left(\hat{\lambda}^{j}-\hat{\lambda}^{j+1}\right)g^{(t)}(\hat{x})\,,$

namely,

$\sum_{t=1}^{T}\hat{\lambda}^{t}g^{(t)}(x)\geq\sum_{t=1}^{T}\hat{\lambda}^{t}g^{(t)}(\hat{x})\,,$

for all $x\in C$, which completes the proof.

∎

###  4.2 Statistical and economic interpretations of CS$\mathcal{E}$SD

In the previous section, we have shown (see Theorem 9) the equivalence between
the maximization of the OWA of the centered CVaRs, and the CS$\mathcal{E}$SD
rule.

Next, we note that the OWA of the centered CVaRs, for
$\lambda=(1,\stackrel{{\scriptstyle t}}{{\ldots}},1,0,\ldots,0)$ can be
interpreted as the $t/T$-CVaR of the centered CVaRs. Therefore, from a
statistical viewpoint, according to Bertsimas and Sim (2004), it is more
robust to maximize the $t/T$-CVaR of the centered CVaRs than the minimax model
($1/T$-CVaR) proposed by Roman et al. (2013). It has been well-known since the
early ’70s that minimax models provide robust solutions against uncertain
data, although they may be too conservative. An alternative is to consider the
minimization of $t$-sums. In this regard, the reader may note that our
proposal, namely the minimization of the $t$ largest losses of a sequence,
provides more robust solutions than considering only the largest loss ($t=1$).
The interested reader is referred to the papers by Bertsimas and Sim (2004),
Puerto et al. (2017), and the references therein for further insights on
$k$-sum optimization.

On the other hand, based on the equivalence provided by Theorem 9, we can also
give a insightful interpretation to the CS$\mathcal{E}$SD rule. Indeed, any
CS$\mathcal{E}$SD portfolio comes from an optimal solution of an OWA solution
of centered CVaRs for particular choices of lambda weights. In other words,
the CS$\mathcal{E}$SD portfolios optimize weighted sums of centered CVaRs;
each choice of the lambda parameters reflects the importance given by the
decision-maker to the ordinal sequence of centered CVaRs.

Finally, we discuss on the advantages of the $CS\varepsilon SD$ concept. We
prove that it is a refinement of the concept of ASSD in Lizyayev and
Ruszczyński (2012) and, moreover, that the expected utility of a
$CS\varepsilon SD$ portfolio does not exceed the expected utility of the
dominating one by more than $\varepsilon$ for a large class of reasonable
utility functions.

######  Proposition 2.

Let $A$ and $B$ be two random variables.

  1. 1.

If $A$ $CS\varepsilon SD$-dominates $B$ then $A$ ASSD-dominates $B$
($CS\varepsilon SD\Rightarrow$ ASSD).

  2. 2.

If A $CS\varepsilon SD$-dominates B, then $E[u(A)]+\varepsilon\geq E[u(B)]$
for any nondecreasing concave utility function $u$ with first derivative
$u^{\prime}\leq 1$.

###### Proof.

From Proposition 1 it follows that $CS\varepsilon SD$ implies ASSD, or
equivalently $S\varepsilon SD$, which requires
$\left(CVaR_{\frac{\cdot}{T}}(A)-CVaR_{\frac{\cdot}{T}}(B)\right)^{(1)}\leq\varepsilon$.
This proves assertion (1.).

Next, to prove assertion (2.) we use Theorem 2 in Lizyayev and Ruszczyński
(2012) and the result in assertion (1.). ∎

###  4.3 Illustrative examples for $S\varepsilon SD$, $CS\varepsilon SD$ and
$CS\mathcal{E}SD$

To support intuition, in Fig. 1, we give some illustrative examples of how the
$S\varepsilon SD$ (and hence the Minimax scalarization), $CS\varepsilon SD$
and $CS\mathcal{E}SD$ (and hence the OWA scalarization) conditions work in
three practical situations covering all possible real-world cases. More
precisely, we consider two discrete random variables $A$ and $B$ represented
by five scenarios $t=1,...,5$. Denoting by
$g^{(t)}=\left(CVaR_{\cdot/T}(A)-CVaR_{\cdot/T}(B)\right)^{(t)}$ and
$G^{(t)}:=\sum_{j=1}^{t}g^{(j)}$, on the left side of Fig. 1 we show a bar
graph of $g^{(t)}$ as a function of $t$, while on the right side we report
$G^{(t)}$ as a function of $t$. In Case 1, $g^{(t)}>0$ $\forall t$, therefore
all the SSD inequalities are violated; in Case 2, $g^{(t)}>0$ for $t=1,2,3$
and $g^{(t)}<0$ for $t=4,5$; in Case 3, $g^{(t)}<0$ $\forall t$, namely $A$
SSD-dominates $B$.

In simple terms, the Minimax scalarization of Roman et al. (2013) consists in
fixing a single threshold “epsilon” (see the blue dashed line on the left side
of Fig. 1) and minimizing it; therefore, we can only focus on the worst
violation $g^{(1)}$. On the other hand, using the OWA scalarization with an
appropriate choice of lambda weights, we can potentially consider one or more
thresholds "epsilon" (see the red dashed lines on the right side of Fig. 1),
limiting not only the worst violation $G^{(1)}=g^{(1)}$ but also the remaining
cumulative sums of the violations.

Figure 1: Illustrative examples of $S\varepsilon SD$ (left side),
$CS\varepsilon SD$ and $CS\mathcal{E}SD$ (right side).

##  5 Empirical Analysis

We present here a detailed empirical analysis where the performance of
efficient portfolios obtained by the OWA maximization of centered CVaRs (or
Tails), are compared with those achieved by different SD-based strategies
proposed in the literature for Enhanced Indexation. In addition, we also
consider the long-only, long-short, 1-norm, and 2-norm minimum variance
portfolios. Summing up, we thus compare the portfolio selection models listed
below.

  * •

1-OWA-CVaR$\,\equiv\,$ RomanCVaR: Model (17) with $\lambda=(1,0,0,\ldots,0)$,
which is equivalent to Model (9) (Roman et al., 2013).

  * •

$k$-OWA-CVaR-$\beta$: Model (17) with
$\lambda=(1,1,\stackrel{{\scriptstyle(k)}}{{\ldots}},1,0,\ldots,0)$, where $k$
rounds $\beta T$ to the nearest integer, and the values of $\beta$ considered
are $5\%,10\%,25\%,50\%,75\%,100\%$.

  * •

$k$-OWA-CumCVaR-$\beta$: Model (17) with
$\lambda=(k,k-1,k-2,\ldots,2,1,0,\ldots,0)$, where $k$ rounds $\beta T$ to the
nearest integer, and the values of $\beta$ considered are
$5\%,10\%,25\%,50\%,75\%,100\%$.

  * •

1-OWA-Tail$\,\equiv\,$ RomanTail: Model (22) with $\lambda=(1,0,0,\ldots,0)$,
which is equivalent to Model (19) (Roman et al., 2013).

  * •

$k$-OWA-Tail-$\beta$: Model (22) with
$\lambda=(1,1,\stackrel{{\scriptstyle(k)}}{{\ldots}},1,0,\ldots,0)$, where $k$
rounds $\beta T$ to the nearest integer, and the values of $\beta$ considered
are $5\%,10\%,25\%,50\%,75\%,100\%$.

  * •

$k$-OWA-CumTail-$\beta$: Model (22) with
$\lambda=(k,k-1,k-2,\ldots,2,1,0,\ldots,0)$, where $k$ rounds $\beta T$ to the
nearest integer, and the values of $\beta$ considered are
$5\%,10\%,25\%,50\%,75\%,100\%$.

  * •

KP2011Min: Model (10) of Kopa and Post (2015) with

$w_{s}=\left\\{\begin{array}[]{ll}0.001,&\mbox{for $s=2,\ldots,T$}\\\
1-\sum_{i=2}^{T}w_{i},&\mbox{for $s=1$}\end{array}\right.$

(see also Hodder et al., 2015).

  * •

MinV: long-only minimum variance portfolio (Markowitz, 1952, 1959),
$x_{{MinV}}^{*}=\displaystyle\operatorname*{arg\,min}_{x\geq 0}x^{T}\Sigma x$,
where $\Sigma$ is the sample covariance matrix obtained from the returns of
the assets belonging to an investment universe.

  * •

MinVsh: long-short minimum variance portfolio,
$x_{{MinV\\!sh}}^{*}=\displaystyle\operatorname*{arg\,min}_{x\in\mathbb{R}}x^{T}\Sigma
x$.

  * •

MinVN1: 1-norm constrained minimum variance portfolio,
$x_{{MinV\\!N1}}^{*}=\displaystyle\operatorname*{arg\,min}_{x\in\mathbb{R}}x^{T}\Sigma
x+\delta_{1}\sum_{k=1}^{n}|x_{k}|$ (see, e.g. Dhingra et al., 2023).

  * •

MinVN2: 2-norm constrained minimum variance portfolio,
$x_{{MinV\\!N2}}^{*}=\displaystyle\operatorname*{arg\,min}_{x\in\mathbb{R}}x^{T}\Sigma
x+\delta_{2}\sum_{k=1}^{n}x^{2}_{k}$ (see, e.g. Coqueret, 2015; Dhingra et
al., 2023).

The models have been implemented in MATLAB R2019b, and they make calls to
XPRESS solver version 8.9 for solving the LP problems. All experiments were
run on a computer DellT5500 with an Intel(R) Xeon(R) processor with a CPU
X5690 at 3.75 GHz and 48 GB of RAM.

###  5.1 Datasets and Methodology

In this section, we provide some details about the four real-world datasets on
which we test all the portfolio selection strategies described in the previous
section:

  1. 1.

FTSE100 (Financial Times Stock Exchange, UK), containing 80 assets and 3715
observations (Oct 2006-Dec 2020);

  2. 2.

NASDAQ 100 (National Association of Securities Dealers Automated Quotation,
USA), containing 54 assets and 3715 observations (Oct 2006-Dec 2020);

  3. 3.

SP500 (Standard & Poor’s, USA), containing 336 assets and 3715 observations
(Oct 2006-Dec 2020);

  4. 4.

FF49 (Fama & French 49 Industry portfolios, USA), containing 49 portfolios
considered as assets and using the subsample where all the returns of the 49
industries are available, namely from July 1969 to June 2023.

The first three datasets are publicly available on the website
https://www.francescocesarone.com/data-sets and consist of daily prices,
adjusted for dividends and stock splits and obtained from Refinitiv (see, for
more details, Cesarone et al., 2022b). For the last dataset, we use daily
returns downloaded from
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Research.
In this case, we consider as a benchmark the Equally-Weighted portfolio.

Our aim is to examine the performance of all models described above, using a
rolling time windows scheme of evaluation. More precisely, we consider an in-
sample time window of 6 months (i.e., 125 financial days). We then evaluate
the portfolio out-of-sample performance over the next month (i.e., 20
financial days). Next, we shift the in-sample window by 1 month to cover the
out-of-sample period, recalculate the optimal portfolio relative to the new
in-sample window, and repeat the operation.

For each portfolio strategy, we calculate some standard performance measures
recalled below (see, e.g., Cesarone et al., 2015, 2016, 2020; Bruni et al.,
2017; Cesarone and Colucci, 2017).

  * •

The average $\hat{\mu}^{out}$ (ExpRet) and the standard deviation
$\hat{\sigma}^{out}$ (Vol) of the out-of-sample portfolio returns.

  * •

The Sharpe ratio defined as $\hat{\mu}^{out}/\hat{\sigma}^{out}$ (Sharpe,
1966, 1994).

  * •

The Maximum Drawdown (MDD, see, e.g., Chekhlov et al., 2005, and references
therein). Denoting by $R_{\tau}^{out}$ the out-of-sample portfolio returns for
each portfolio strategy, we first consider the cumulative out-of-sample
portfolio returns, which correspond to the values of wealth after $\tau$
periods $W_{\tau}=W_{\tau-1}(1+R_{\tau}^{out})$, where
$\tau=M+1,\ldots,\bar{T}$, $M=125$, $M$ is the length of the in-sample window,
$\bar{T}$ is the number of observations available for a single dataset, and
the initial wealth $W_{M}=1$. We define the drawdowns as

$dd_{\tau}=\frac{W_{\tau}-\max_{M+1\leq s\leq\tau}(W_{s})}{\max_{M+1\leq
s\leq\tau}(W_{s})}.$

Note that the drawdowns $dd_{\tau}$ are obviously negative. The Maximum
Drawdown $Mdd$ corresponds to the worst drawdown over the entire out-of-sample
period, namely $Mdd=\min_{M+1\leq\tau\leq T}(dd_{\tau})$.

  * •

The Ulcer Index, which is defined as the square root of the mean of the
squared percentage drawdowns $dd_{\tau}$ with $\tau=M+1,\ldots,T$

$UI=\sqrt{\frac{\sum_{\tau=M+1}^{T}dd_{\tau}^{2}}{T-M}}.$ (30)

It evaluates the depth and the duration of drawdowns in prices over the out-
of-sample period (see, e.g., Martin and McCann, 1989).

  * •

The Sortino ratio (Sortino et al., 2001), defined as the ratio between the
average of $R^{out}$ and its downside deviation

$\frac{E[R^{out}-r_{f}]}{\sigma(\min\\{R^{out}-r_{f},\,0\\})}\,.$

  * •

The Rachev ratio with confidence levels of $5\%$ (Biglova et al., 2004),
defined as the ratio between the average of the best $5\%$ values of $R^{out}$
(with the opposite sign) and that of the worst $5\%$ values of $R^{out}$

$\frac{C\\!V\\!a\\!R_{5\%}(-R^{out})}{C\\!V\\!a\\!R_{5\%}(R^{out})}\,.$

  * •

The Turnover (see, e.g., DeMiguel et al., 2009) evaluates the amount of
trading required to perform in practice the portfolio strategy, and is defined
as

$Turnover=\frac{1}{Q}\sum_{q=1}^{Q}\sum_{k=1}^{n}\mid x_{q,k}-x_{q-1,k}\mid,$

where $Q$ represents the number of rebalances, $x_{q,k}$ is the portfolio
weight of asset $k$ after rebalancing, and $x_{q-1,k}$ is the portfolio weight
before rebalancing at time $q$. Lower turnover values indicate better
portfolio performance. We point out that this definition of portfolio turnover
is a proxy of the effective one, since it evaluates only the amount of trading
generated by the models at each rebalance, without considering the trades due
to changes in asset prices between one rebalance and the next. Thus, by
definition, the turnover of the EW portfolio is zero.

  * •

The Jensen’s Alpha (AlphaJ, Jensen, 1968), defined as the intercept of the
line given by the linear regression of $R^{out}$ on $R_{I}^{out}$, namely
$\alpha=E[R^{out}]-\beta E[R_{I}^{out}]$, where
$\beta=Cov(R^{out},R_{I}^{out})/\sigma^{2}(R_{I}^{out})$.

  * •

The Value-at-Risk of the out-of-sample portfolio returns $R_{\tau}^{out}$ with
a confidence level equal to 1% (VaR1).

  * •

The Appraisal ratio (Treynor and Black, 1973), defined as the Jensen’s Alpha
divided by the standard deviation of the residual return $\epsilon_{r}$,
namely:

$\frac{\alpha}{\sigma(\epsilon_{r})}\,.$

where $\epsilon_{r}=R^{out}-\alpha-\beta E[R_{I}^{out}]$. The larger is its
value, the better is the portfolio performance. This measure is particularly
used by practitioners for evaluating the asset selection ability of a model.
It grasps the idea that a portfolio manager to generate $\alpha$ has to
deviate from the benchmark, i.e., take residual risk $\sigma(\epsilon_{r})$.

  * •

The Information ratio (Info, Goodwin, 1998), defined as the expected value of
the difference between the out-of-sample portfolio return and that of the
benchmark index divided by the standard deviation of such difference, namely

$\frac{E[R^{out}-R_{I}^{out}]}{\sigma(R^{out}-R_{I}^{out})}\,.$

The larger is its value, the better is the portfolio performance. This measure
is equivalent to the Appraisal ratio with $\beta=1$.

  * •

The Omega ratio, defined as
$\displaystyle\frac{E[\max(0,R^{out}-\eta)]}{E[\min(0,R^{out}-\eta)]}\,,$
where $\eta$ is fixed to 0.

  * •

The average number of selected assets (ave #) and running time in seconds.

  * •

The Return On Investment (ROI), namely the time-by-time return generated by
each portfolio strategy over a specified time horizon $\Delta H$. More
precisely, ROI is defined as follows

$ROI_{\tau}=\frac{W_{\tau}-W_{\tau-\Delta H}}{W_{\tau-\Delta
H}}\qquad\tau=M+\Delta H+1,\ldots,\bar{T}$ (31)

where $W_{\tau}=W_{\tau-\Delta H}\prod_{t=\tau-\Delta
H+1}^{\tau}(1+R_{t}^{out})$ is the portfolio wealth generated by investing at
the beginning of the time horizon the amount of capital $W_{\tau-\Delta H}$.

We also test the statistical significance of our results. To this end, the
significance of the difference between the Sharpe ratios of the selected
portfolios and that of the benchmark is tested both using the bootstrapping
and HAC methodologies as presented in Ledoit and Wolf (2008). Moreover, to
test the statistical significance of the difference between the variances of
the selected portfolios and that of the benchmark, we use the bootstrapping
method as proposed in Ledoit and Wolf (2011). For both the bootstrap tests, we
choose $M=3000$ bootstrap repetitions and block sizes $b=1,10$. Furthermore,
we adopt a $t$-statistic as in Goodwin (1998) to test the statistical
significance of the positivity of the Information ratio.

Tables 1, 2, 3, and 4 report the $p$-values in percentage

  * •

for the Sharpe ratio differences in the columns
Sharpe$\shortrightarrow$p-v/$b=1$, Sharpe$\shortrightarrow$p-v/$b=10$
(bootstrap test), and Sharpe$\shortrightarrow$p-v/HAC (HAC test);

  * •

for the variance differences in the columns Vol$\shortrightarrow$p-v/$b=1$ and
Vol$\shortrightarrow$p-v/$b=10$ (bootstrap test);

  * •

for the positivity of the Information ratio in the column
InfoRatio$\shortrightarrow$p-v ($t$-test).

Note that for FTSE100, NASDAQ100, and SP500 the benchmark is represented by
the Market index, while for FF49 by the Equally-Weighted portfolio.

###  5.2 Computational results

In this section, we report the out-of-sample results obtained by the OWA
maximization of centered CVaRs approach, that are compared with those obtained
by the Roman strategy with CVaRs and by the long-only, long-short, 1-norm, and
2-norm minimum variance portfolios.

We point out that for the 1-norm constrained minimum variance portfolio we set
$\delta_{1}=10^{-5}$, while for the 2-norm constrained minimum variance
portfolio $\delta_{2}=10^{-3}$. Furthermore, in the case of SP500, since the
solution of the standard minimum variance model might not be unique, we solve
the 2-norm minimum variance model regularizing the covariance matrix with
$\delta_{2}=10^{-6}$.

Appendix A provides some additional computational results that compare the OWA
maximization of centered Tails approach with that of Kopa and Post, which can
be interpreted by means of Tails of portfolio returns, and, again, the MinV
portfolio.

To simplify the reading of the results reported in the following tables, for
each dataset, we show with different colors the rank of the performance
results of the proposed models. More specifically, for each column
corresponding to a specific performance measure the colors span from deep-
green to deep-red, where deep-green represents the best value while deep-red
the worst one.

In Tables 1, 2, 3, and 4 we provide the out-of-sample performance results
obtained by the SD-based strategies using CVaRs, that are compared with the
long-only, long-short, 1-norm, 2-norm minimum variance portfolios and the
Market Index on FTSE100, NASDAQ100, SP500, and FF49, respectively. We can
observe that the $k$-OWA-CVaR-$\beta$ and $k$-OWA-CCVaR-$\beta$ portfolios
with $\beta=5\%,10\%$ typically show the best performance in terms of ExpRet,
Sharpe, Sortino, Rachev, AlphaJ, Info, Appraisal, Omega. We find that on
FTSE100 and FF49, the Sharpe and Information ratios obtained with the $k$-OWA-
CVaR-$\beta$ and $k$-OWA-CCVaR-$\beta$ portfolios ($\beta=5\%,10\%$) are
statistically different from the benchmark, while statistically significant
differences are not realized in the other investment universes. These
strategies also seem to provide slightly better performance than the Roman-
CVaR portfolio. On the other hand, the best values in terms of risk, namely
Vol, MDD, Ulcer, and VaR1, are shown by the $k$-OWA-CVaR-$\beta$ and $k$-OWA-
CCVaR-$\beta$ portfolios with $\beta=75\%,100\%$, and by the MinV, MinVN1, and
MinVN2 portfolios. We observe that on all the datasets analyzed, the variances
obtained with the $k$-OWA-CVaR-$\beta$ and $k$-OWA-CCVaR-$\beta$ portfolios
($\beta=75\%,100\%$), and with the MinV, MinVN1, and MinVN2 portfolios are
statistically different from the benchmark. Note that by increasing the values
of $\beta$ for the $k$-OWA-CVaR-$\beta$ and $k$-OWA-CCVaR-$\beta$ portfolios,
an investor can mitigate risk similarly to the minimum variance portfolios.
The best performance in terms of Turnover are typically achieved by the MinV
and MinVN2 portfolios and by the $k$-OWA-CVaR-$\beta$ and $k$-OWA-
CCVaR-$\beta$ portfolios with $\beta=75\%,100\%$.

Table 1: Out-of-sample performance results for FTSE100

Approach | ExpRet | Vol | Sharpe | MDD | Ulcer | Sortino | Rachev5 | Turn | AlphaJ | InfoRatio | ApprRatio | VaR1 | Omega
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | value | p-v/b=1 | p-v/b=10 | value | p-v/b=1 | p-v/b=10 | p-v/HAC |  |  |  |  |  |  | value | p-v |  |  |
Roman-CVaR | 0.0490% | 1.3023% | 22.26% | 17.39% | 0.0377 | 1.63% | 1.63% | 1.37% | -0.4736 | 0.1382 | 0.0549 | 0.9897 | 1.0020 | 0.0428% | 0.0418 | 0.62% | 0.0461 | 0.0357 | 1.1214
k-OWA-CVaR-5 | 0.0537% | 1.2828% | 32.69% | 27.32% | 0.0419 | 0.40% | 0.70% | 0.49% | -0.4521 | 0.1262 | 0.0612 | 0.9829 | 0.9786 | 0.0475% | 0.0470 | 0.25% | 0.0520 | 0.0351 | 1.1357
k-OWA-CVaR-10 | 0.0541% | 1.2622% | 47.62% | 46.08% | 0.0428 | 0.50% | 0.50% | 0.31% | -0.4448 | 0.1242 | 0.0631 | 0.9884 | 0.9728 | 0.0480% | 0.0480 | 0.21% | 0.0535 | 0.0342 | 1.1392
k-OWA-CVaR-25 | 0.0531% | 1.1948% | 72.51% | 73.04% | 0.0445 | 0.10% | 0.43% | 0.14% | -0.4497 | 0.1203 | 0.0647 | 0.9821 | 0.9084 | 0.0471% | 0.0509 | 0.12% | 0.0579 | 0.0329 | 1.1436
k-OWA-CVaR-50 | 0.0470% | 1.0698% | 0.03% | 0.03% | 0.0439 | 0.13% | 0.23% | 0.11% | -0.4326 | 0.1136 | 0.0621 | 0.9541 | 0.8198 | 0.0413% | 0.0497 | 0.15% | 0.0610 | 0.0298 | 1.1378
k-OWA-CVaR-75 | 0.0385% | 1.0037% | 0.03% | 0.03% | 0.0384 | 0.47% | 0.43% | 0.42% | -0.4295 | 0.1196 | 0.0539 | 0.9458 | 0.7601 | 0.0331% | 0.0404 | 0.78% | 0.0536 | 0.0279 | 1.1187
k-OWA-CVaR-100 | 0.0384% | 0.9591% | 0.03% | 0.03% | 0.0400 | 0.40% | 0.37% | 0.37% | -0.4040 | 0.1060 | 0.0561 | 0.9401 | 0.7093 | 0.0333% | 0.0397 | 0.87% | 0.0555 | 0.0263 | 1.1235
k-OWA-CCVaR-5 | 0.0538% | 1.2853% | 30.09% | 30.49% | 0.0419 | 0.50% | 0.50% | 0.50% | -0.4573 | 0.1285 | 0.0613 | 0.9919 | 0.9828 | 0.0476% | 0.0472 | 0.24% | 0.0521 | 0.0349 | 1.1356
k-OWA-CCVaR-10 | 0.0545% | 1.2753% | 36.75% | 33.52% | 0.0427 | 0.47% | 0.40% | 0.36% | -0.4469 | 0.1244 | 0.0626 | 0.9825 | 0.9768 | 0.0484% | 0.0479 | 0.21% | 0.0532 | 0.0351 | 1.1385
k-OWA-CCVaR-25 | 0.0541% | 1.2351% | 69.51% | 68.44% | 0.0438 | 0.23% | 0.40% | 0.23% | -0.4518 | 0.1261 | 0.0642 | 0.9867 | 0.9401 | 0.0480% | 0.0494 | 0.16% | 0.0555 | 0.0340 | 1.1420
k-OWA-CCVaR-50 | 0.0529% | 1.1448% | 7.53% | 6.43% | 0.0462 | 0.17% | 0.10% | 0.08% | -0.4349 | 0.1117 | 0.0666 | 0.9757 | 0.8708 | 0.0470% | 0.0535 | 0.07% | 0.0622 | 0.0315 | 1.1476
k-OWA-CCVaR-75 | 0.0467% | 1.0705% | 0.03% | 0.03% | 0.0436 | 0.10% | 0.20% | 0.13% | -0.4348 | 0.1154 | 0.0617 | 0.9542 | 0.8111 | 0.0410% | 0.0492 | 0.16% | 0.0604 | 0.0303 | 1.1368
k-OWA-CCVaR-100 | 0.0416% | 1.0116% | 0.03% | 0.03% | 0.0411 | 0.27% | 0.40% | 0.23% | -0.4286 | 0.1175 | 0.0578 | 0.9477 | 0.7606 | 0.0362% | 0.0439 | 0.44% | 0.0573 | 0.0281 | 1.1275
MinV | 0.0362% | 0.9254% | 0.03% | 0.03% | 0.0391 | 0.13% | 0.33% | 0.13% | -0.3802 | 0.0909 | 0.0545 | 0.9577 | 0.5836 | 0.0310% | 0.0407 | 0.75% | 0.0595 | 0.0271 | 1.1215
MinVsh | 0.0266% | 1.2719% | 13.53% | 14.20% | 0.0209 | 48.32% | 49.05% | 47.75% | -0.4923 | 0.1848 | 0.0289 | 0.9422 | 5.1897 | 0.0241% | 0.0123 | 23.16% | 0.0197 | 0.0352 | 1.0603
MinVN1 | 0.0166% | 0.9100% | 0.03% | 0.03% | 0.0183 | 49.08% | 48.52% | 47.85% | -0.4815 | 0.1895 | 0.0249 | 0.9055 | 1.2402 | 0.0134% | 0.0077 | 32.25% | 0.0172 | 0.0263 | 1.0547
MinVN2 | 0.0195% | 0.8738% | 0.03% | 0.03% | 0.0223 | 14.46% | 14.43% | 14.08% | -0.4312 | 0.1335 | 0.0301 | 0.8793 | 0.2878 | 0.0147% | 0.0153 | 17.94% | 0.0282 | 0.0253 | 1.0666
Index | 0.0083% | 1.2117% |  |  | 0.0069 |  |  |  | -0.4783 | 0.1549 | 0.0095 | 0.9257 |  |  |  |  |  | 0.0345 | 1.0210

Table 2: Out-of-sample performance results for NASDAQ100

Approach | ExpRet | Vol | Sharpe | MDD | Ulcer | Sortino | Rachev5 | Turn | AlphaJ | InfoRatio | ApprRatio | VaR1 | Omega
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Value | p-v/b=1 | p-v/b=10 | value | p-v/b=1 | p-v/b=10 | p-v/HAC |  |  |  |  |  |  | value | p-v |  |  |
Roman-CVaR | 0.0770% | 1.4215% | 61.01% | 63.48% | 0.0541 | 44.05% | 44.05% | 43.66% | -0.4041 | 0.1135 | 0.0780 | 0.9674 | 0.9220 | 0.0238% | 0.0136 | 20.76% | 0.0286 | 0.0407 | 1.1733
k-OWA-CVaR-5 | 0.0771% | 1.4016% | 73.18% | 73.44% | 0.0550 | 38.19% | 36.49% | 37.46% | -0.4123 | 0.1114 | 0.0795 | 0.9750 | 0.9166 | 0.0242% | 0.0141 | 19.91% | 0.0299 | 0.0393 | 1.1759
k-OWA-CVaR-10 | 0.0721% | 1.3743% | 9.46% | 10.13% | 0.0524 | 51.88% | 52.85% | 51.81% | -0.3992 | 0.1155 | 0.0756 | 0.9679 | 0.9196 | 0.0201% | 0.0084 | 30.81% | 0.0255 | 0.0386 | 1.1672
k-OWA-CVaR-25 | 0.0634% | 1.3221% | 0.03% | 0.03% | 0.0479 | 86.34% | 85.87% | 85.85% | -0.3791 | 0.1119 | 0.0685 | 0.9541 | 0.8848 | 0.0131% | -0.0021 | 54.96% | 0.0175 | 0.0381 | 1.1538
k-OWA-CVaR-50 | 0.0532% | 1.1964% | 0.03% | 0.03% | 0.0445 | 86.50% | 87.17% | 86.43% | -0.3865 | 0.1141 | 0.0631 | 0.9322 | 0.7893 | 0.0076% | -0.0149 | 81.33% | 0.0113 | 0.0348 | 1.1436
k-OWA-CVaR-75 | 0.0509% | 1.1073% | 0.03% | 0.03% | 0.0459 | 98.33% | 98.20% | 98.19% | -0.3558 | 0.0945 | 0.0652 | 0.9310 | 0.6941 | 0.0091% | -0.0175 | 85.20% | 0.0142 | 0.0315 | 1.1493
k-OWA-CVaR-100 | 0.0504% | 1.0427% | 0.03% | 0.03% | 0.0484 | 84.34% | 84.11% | 83.32% | -0.3495 | 0.0836 | 0.0690 | 0.9538 | 0.6033 | 0.0120% | -0.0172 | 84.80% | 0.0191 | 0.0301 | 1.1574
k-OWA-CCVaR-5 | 0.0766% | 1.4083% | 96.57% | 96.70% | 0.0544 | 40.85% | 38.69% | 41.01% | -0.4092 | 0.1102 | 0.0786 | 0.9743 | 0.9178 | 0.0236% | 0.0135 | 20.99% | 0.0290 | 0.0396 | 1.1739
k-OWA-CCVaR-10 | 0.0716% | 1.3912% | 40.45% | 41.05% | 0.0514 | 57.28% | 60.61% | 59.67% | -0.4106 | 0.1192 | 0.0740 | 0.9649 | 0.9259 | 0.0191% | 0.0077 | 32.35% | 0.0238 | 0.0390 | 1.1639
k-OWA-CCVaR-25 | 0.0703% | 1.3520% | 0.63% | 0.47% | 0.0520 | 56.51% | 56.58% | 55.55% | -0.3757 | 0.1113 | 0.0748 | 0.9586 | 0.8920 | 0.0192% | 0.0063 | 35.37% | 0.0247 | 0.0388 | 1.1664
k-OWA-CCVaR-50 | 0.0577% | 1.2852% | 0.03% | 0.03% | 0.0449 | 89.30% | 90.10% | 89.56% | -0.3936 | 0.1208 | 0.0636 | 0.9397 | 0.8634 | 0.0087% | -0.0092 | 70.87% | 0.0120 | 0.0384 | 1.1440
k-OWA-CCVaR-75 | 0.0539% | 1.1974% | 0.03% | 0.03% | 0.0450 | 91.24% | 91.60% | 91.09% | -0.3818 | 0.1114 | 0.0640 | 0.9351 | 0.7834 | 0.0085% | -0.0138 | 79.59% | 0.0125 | 0.0351 | 1.1458
k-OWA-CCVaR-100 | 0.0521% | 1.1206% | 0.03% | 0.03% | 0.0465 | 97.63% | 98.00% | 97.72% | -0.3598 | 0.0954 | 0.0660 | 0.9341 | 0.7000 | 0.0099% | -0.0159 | 82.92% | 0.0153 | 0.0321 | 1.1510
MinV | 0.0465% | 0.9999% | 0.03% | 0.03% | 0.0465 | 97.87% | 97.90% | 97.57% | -0.3485 | 0.0865 | 0.0661 | 0.9357 | 0.4399 | 0.0099% | -0.0214 | 90.00% | 0.0162 | 0.0286 | 1.1537
MinVsh | 0.0293% | 1.1254% | 0.03% | 0.03% | 0.0260 | 24.83% | 24.36% | 24.12% | -0.4100 | 0.1767 | 0.0365 | 0.9553 | 2.4534 | 0.0070% | -0.0260 | 94.00% | 0.0069 | 0.0307 | 1.0801
MinVN1 | 0.0372% | 0.9681% | 0.03% | 0.03% | 0.0385 | 63.78% | 61.55% | 60.89% | -0.3618 | 0.1140 | 0.0546 | 0.9410 | 1.0237 | 0.0113% | -0.0241 | 92.50% | 0.0143 | 0.0271 | 1.1232
MinVN2 | 0.0506% | 1.0264% | 0.03% | 0.03% | 0.0493 | 71.11% | 72.88% | 71.64% | -0.3795 | 0.1090 | 0.0697 | 0.9108 | 0.2424 | 0.0096% | -0.0197 | 88.10% | 0.0187 | 0.0297 | 1.1630
Index | 0.0651% | 1.4093% |  |  | 0.0462 |  |  |  | -0.5371 | 0.1339 | 0.0654 | 0.9481 |  |  |  |  |  | 0.0407 | 1.1523

Table 3: Out-of-sample performance results for SP500

Approach | ExpRet | Vol | Sharpe | MDD | Ulcer | Sortino | Rachev5 | Turn | AlphaJ | InfoRatio | ApprRatio | VaR1 | Omega
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | value | p-v/b=1 | p-v/b=10 | value | p-v/b=1 | p-v/b=10 | p-v/HAC |  |  |  |  |  |  | value | p-v |  |  |
Roman-CVaR | 0.0420% | 1.2895% | 79.91% | 81.24% | 0.0326 | 61.15% | 60.05% | 60.91% | -0.4693 | 0.1429 | 0.0457 | 0.9223 | 1.1162 | 0.0142% | 0.0081 | 31.37% | 0.0181 | 0.0384 | 1.1011
k-OWA-CVaR-5 | 0.0436% | 1.2707% | 32.59% | 32.76% | 0.0343 | 50.35% | 50.95% | 49.53% | -0.4837 | 0.1483 | 0.0482 | 0.9214 | 1.0794 | 0.0162% | 0.0102 | 27.16% | 0.0210 | 0.0385 | 1.1069
k-OWA-CVaR-10 | 0.0415% | 1.2505% | 6.70% | 7.70% | 0.0332 | 56.61% | 56.41% | 56.58% | -0.5044 | 0.1636 | 0.0467 | 0.9252 | 1.0823 | 0.0147% | 0.0076 | 32.43% | 0.0192 | 0.0380 | 1.1030
k-OWA-CVaR-25 | 0.0378% | 1.1674% | 0.03% | 0.03% | 0.0323 | 62.51% | 63.05% | 61.13% | -0.4963 | 0.1744 | 0.0454 | 0.9162 | 1.0596 | 0.0126% | 0.0032 | 42.52% | 0.0177 | 0.0337 | 1.0995
k-OWA-CVaR-50 | 0.0335% | 1.0643% | 0.03% | 0.03% | 0.0315 | 68.18% | 68.74% | 67.09% | -0.4913 | 0.1886 | 0.0438 | 0.8816 | 1.0046 | 0.0106% | -0.0022 | 55.16% | 0.0163 | 0.0315 | 1.0984
k-OWA-CVaR-75 | 0.0304% | 0.9740% | 0.03% | 0.03% | 0.0312 | 70.21% | 70.61% | 68.98% | -0.4783 | 0.1732 | 0.0433 | 0.8806 | 0.9382 | 0.0092% | -0.0062 | 64.47% | 0.0158 | 0.0287 | 1.0987
k-OWA-CVaR-100 | 0.0312% | 0.9165% | 0.03% | 0.03% | 0.0341 | 51.95% | 51.15% | 50.57% | -0.4976 | 0.1599 | 0.0475 | 0.8905 | 0.8700 | 0.0116% | -0.0050 | 61.65% | 0.0206 | 0.0275 | 1.1101
k-OWA-CCVaR-5 | 0.0428% | 1.2754% | 42.75% | 43.85% | 0.0336 | 54.65% | 55.88% | 54.67% | -0.4756 | 0.1444 | 0.0471 | 0.9156 | 1.0895 | 0.0154% | 0.0091 | 29.24% | 0.0197 | 0.0382 | 1.1045
k-OWA-CCVaR-10 | 0.0433% | 1.2593% | 15.19% | 16.59% | 0.0344 | 49.58% | 49.35% | 49.57% | -0.4857 | 0.1494 | 0.0483 | 0.9229 | 1.0782 | 0.0162% | 0.0097 | 28.01% | 0.0210 | 0.0372 | 1.1069
k-OWA-CCVaR-25 | 0.0399% | 1.2137% | 0.20% | 0.53% | 0.0328 | 58.41% | 58.01% | 58.32% | -0.5004 | 0.1584 | 0.0462 | 0.9314 | 1.0711 | 0.0138% | 0.0057 | 36.69% | 0.0185 | 0.0345 | 1.1012
k-OWA-CCVaR-50 | 0.0378% | 1.1338% | 0.03% | 0.03% | 0.0334 | 55.98% | 55.61% | 54.61% | -0.5000 | 0.1722 | 0.0467 | 0.9047 | 1.0367 | 0.0135% | 0.0032 | 42.31% | 0.0194 | 0.0328 | 1.1031
k-OWA-CCVaR-75 | 0.0342% | 1.0587% | 0.03% | 0.03% | 0.0323 | 62.18% | 60.11% | 61.17% | -0.4908 | 0.1848 | 0.0450 | 0.8822 | 0.9932 | 0.0114% | -0.0013 | 53.07% | 0.0177 | 0.0314 | 1.1008
k-OWA-CCVaR-100 | 0.0307% | 0.9829% | 0.03% | 0.03% | 0.0312 | 69.61% | 70.44% | 69.24% | -0.4924 | 0.1813 | 0.0433 | 0.8771 | 0.9414 | 0.0095% | -0.0057 | 63.42% | 0.0159 | 0.0296 | 1.0983
MinV | 0.0279% | 0.8816% | 0.03% | 0.03% | 0.0317 | 65.04% | 65.81% | 64.39% | -0.4578 | 0.1412 | 0.0440 | 0.8897 | 0.6970 | 0.0083% | -0.0095 | 71.52% | 0.0164 | 0.0253 | 1.1060
MinVsh | 0.0237% | 0.8700% | 0.03% | 0.03% | 0.0272 | 99.80% | 99.67% | 99.76% | -0.4850 | 0.2241 | 0.0381 | 0.9321 | 3.4491 | 0.0129% | -0.0097 | 71.99% | 0.0166 | 0.0262 | 1.0887
MinVN1 | 0.0188% | 0.8396% | 0.03% | 0.03% | 0.0223 | 75.14% | 74.08% | 75.73% | -0.4776 | 0.2018 | 0.0307 | 0.9090 | 1.6692 | 0.0065% | -0.0149 | 81.42% | 0.0092 | 0.0245 | 1.0726
MinVN2 | 0.0242% | 0.8134% | 0.03% | 0.03% | 0.0297 | 85.57% | 86.07% | 85.12% | -0.4167 | 0.1660 | 0.0408 | 0.8807 | 0.6760 | 0.0087% | -0.0119 | 76.11% | 0.0150 | 0.0246 | 1.0995
Index | 0.0352% | 1.2960% |  |  | 0.0272 |  |  |  | -0.5678 | 0.1653 | 0.0378 | 0.9099 |  |  |  |  |  | 0.0393 | 1.0926

Table 4: Out-of-sample performance results for FF49

Approach | ExpRet | Vol | Sharpe | Sortino | Rachev5 | Turn | AlphaJ | InfoRatio | ApprRatio | VaR1 | Omega
---|---|---|---|---|---|---|---|---|---|---|---
|  | value | p-v/b=1 | p-v/b=10 | value | p-v/b=1 | p-v/b=10 | p-v/HAC |  |  |  |  | value | p-v |  |  |
Roman-CVaR | 10.7464% | 95.4540% | 0.03% | 0.03% | 0.1126 | 0.03% | 0.03% | 0.00% | 0.1626 | 0.9610 | 0.8274 | 4.8309% | 0.0540 | 0.00% | 0.0890 | 2.6588 | 1.3926
k-OWA-CVaR-5 | 10.6469% | 94.3017% | 0.03% | 0.03% | 0.1129 | 0.03% | 0.03% | 0.00% | 0.1627 | 0.9568 | 0.8016 | 4.7910% | 0.0527 | 0.00% | 0.0897 | 2.6147 | 1.3944
k-OWA-CVaR-10 | 10.4907% | 92.9753% | 0.03% | 0.03% | 0.1128 | 0.03% | 0.03% | 0.00% | 0.1622 | 0.9505 | 0.7800 | 4.6959% | 0.0507 | 0.00% | 0.0899 | 2.5955 | 1.3950
k-OWA-CVaR-25 | 10.2073% | 89.1384% | 0.03% | 0.03% | 0.1145 | 0.03% | 0.03% | 0.00% | 0.1640 | 0.9397 | 0.7301 | 4.6138% | 0.0469 | 0.00% | 0.0935 | 2.5450 | 1.4059
k-OWA-CVaR-50 | 9.3912% | 82.3421% | 0.03% | 0.03% | 0.1141 | 0.03% | 0.03% | 0.00% | 0.1621 | 0.9216 | 0.6318 | 4.1242% | 0.0340 | 0.00% | 0.0948 | 2.2656 | 1.4098
k-OWA-CVaR-75 | 8.6147% | 76.5745% | 0.03% | 0.03% | 0.1125 | 0.03% | 0.03% | 0.00% | 0.1593 | 0.9223 | 0.5465 | 3.6686% | 0.0196 | 1.15% | 0.0931 | 2.1367 | 1.4083
k-OWA-CVaR-100 | 7.8655% | 72.1837% | 0.03% | 0.03% | 0.1090 | 0.03% | 0.03% | 0.00% | 0.1550 | 0.9499 | 0.4568 | 3.2263% | 0.0052 | 27.24% | 0.0857 | 2.0144 | 1.3954
k-OWA-CCVaR-5 | 10.6470% | 94.6393% | 0.03% | 0.03% | 0.1125 | 0.03% | 0.03% | 0.00% | 0.1621 | 0.9572 | 0.8106 | 4.7676% | 0.0527 | 0.00% | 0.0890 | 2.6202 | 1.3926
k-OWA-CCVaR-10 | 10.5938% | 93.7354% | 0.03% | 0.03% | 0.1130 | 0.03% | 0.03% | 0.00% | 0.1628 | 0.9565 | 0.7938 | 4.7636% | 0.0521 | 0.00% | 0.0901 | 2.5965 | 1.3951
k-OWA-CCVaR-25 | 10.4592% | 91.6365% | 0.03% | 0.03% | 0.1141 | 0.03% | 0.03% | 0.00% | 0.1641 | 0.9501 | 0.7591 | 4.7480% | 0.0503 | 0.00% | 0.0922 | 2.5895 | 1.4025
k-OWA-CCVaR-50 | 9.9291% | 86.6683% | 0.03% | 0.03% | 0.1146 | 0.03% | 0.03% | 0.00% | 0.1637 | 0.9317 | 0.6843 | 4.4557% | 0.0427 | 0.00% | 0.0943 | 2.4592 | 1.4081
k-OWA-CCVaR-75 | 9.3683% | 82.1244% | 0.03% | 0.03% | 0.1141 | 0.03% | 0.03% | 0.00% | 0.1622 | 0.9215 | 0.6270 | 4.1232% | 0.0334 | 0.01% | 0.0947 | 2.2680 | 1.4099
k-OWA-CCVaR-100 | 8.7489% | 77.5644% | 0.03% | 0.03% | 0.1128 | 0.03% | 0.03% | 0.00% | 0.1599 | 0.9222 | 0.5628 | 3.7591% | 0.0220 | 0.53% | 0.0931 | 2.1458 | 1.4083
MinV | 6.2402% | 68.1950% | 0.03% | 0.03% | 0.0915 | 0.27% | 0.57% | 1.17% | 0.1294 | 0.9556 | 0.2986 | 1.7604% | -0.0253 | 99.84% | 0.0527 | 1.8493 | 1.3299
MinVsh | 5.9300% | 62.6256% | 0.03% | 0.03% | 0.0947 | 8.90% | 8.86% | 11.69% | 0.1372 | 1.0241 | 2.3053 | 4.0419% | -0.0174 | 97.84% | 0.0704 | 1.6200 | 1.3315
MinVN1 | 5.9299% | 62.6202% | 0.03% | 0.03% | 0.0947 | 8.90% | 9.96% | 11.68% | 0.1372 | 1.0241 | 2.3039 | 4.0416% | -0.0174 | 97.84% | 0.0704 | 1.6198 | 1.3315
MinVN2 | 5.9399% | 62.4297% | 0.03% | 0.03% | 0.0951 | 8.26% | 8.26% | 10.77% | 0.1378 | 1.0232 | 2.2554 | 4.0402% | -0.0174 | 97.81% | 0.0707 | 1.6143 | 1.3339
Index - EW | 7.5825% | 100.6446% |  |  | 0.0753 |  |  |  | 0.1062 | 0.9500 |  |  |  |  |  | 2.8491 | 1.2568

In Tables 5, 6, 7 we report some statistics of ROI based on a 3-years time
horizon, computed using Expression (31) with $\Delta H=750$ days, for all the
portfolio strategies analyzed. Again, for the FTSE100, NASDAQ100, and SP500
daily datasets, we can observe that the most profitable portfolio strategies
appear to be the $k$-OWA-CVaR-$\beta$ and $k$-OWA-CCVaR-$\beta$ portfolios
with $\beta=5\%,10\%$, along with the Roman-CVaR approach, in terms of mean,
25%, 50%, 75%, and 95% percentile of ROI.

Table 5: Summary statistics of ROI based on a 3-years time horizon for FTSE100

Approach | ExpRet | Vol | 5%-perc | 25%-perc | 50%-perc | 75%-perc | 95%-perc
---|---|---|---|---|---|---|---
Roman-CVaR | 65% | 45% | -21% | 20% | 80% | 99% | 123%
k-OWA-CVaR-5 | 69% | 45% | -13% | 25% | 81% | 104% | 124%
k-OWA-CVaR-10 | 68% | 45% | -10% | 21% | 81% | 100% | 130%
k-OWA-CVaR-25 | 64% | 41% | -7% | 21% | 77% | 96% | 116%
k-OWA-CVaR-50 | 53% | 33% | -5% | 20% | 64% | 78% | 95%
k-OWA-CVaR-75 | 43% | 28% | -7% | 14% | 53% | 63% | 76%
k-OWA-CVaR-100 | 43% | 27% | -3% | 17% | 50% | 63% | 80%
k-OWA-CCVaR-5 | 69% | 45% | -14% | 28% | 82% | 105% | 125%
k-OWA-CCVaR-10 | 69% | 45% | -12% | 23% | 82% | 106% | 126%
k-OWA-CCVaR-25 | 67% | 44% | -8% | 20% | 81% | 101% | 124%
k-OWA-CCVaR-50 | 61% | 38% | -3% | 22% | 74% | 90% | 108%
k-OWA-CCVaR-75 | 53% | 33% | -5% | 20% | 63% | 77% | 97%
k-OWA-CCVaR-100 | 47% | 30% | -5% | 16% | 57% | 69% | 85%
MinV | 37% | 19% | 3% | 25% | 41% | 50% | 63%
MinVsh | 33% | 24% | -15% | 24% | 39% | 49% | 63%
MinVN1 | 29% | 28% | -27% | 17% | 38% | 48% | 62%
MinVN2 | 25% | 18% | -12% | 16% | 30% | 38% | 49%
Index | 8% | 15% | -19% | 1% | 10% | 17% | 31%

Table 6: Summary statistics of ROI based on a 3-years time horizon for
NASDAQ100

Approach | ExpRet | Vol | 5%-perc | 25%-perc | 50%-perc | 75%-perc | 95%-perc
---|---|---|---|---|---|---|---
Roman-CVaR | 79% | 30% | 12% | 61% | 86% | 102% | 117%
k-OWA-CVaR-5 | 79% | 27% | 14% | 66% | 84% | 98% | 114%
k-OWA-CVaR-10 | 74% | 29% | 14% | 58% | 78% | 96% | 112%
k-OWA-CVaR-25 | 62% | 28% | 6% | 45% | 64% | 83% | 101%
k-OWA-CVaR-50 | 52% | 27% | -4% | 37% | 55% | 72% | 91%
k-OWA-CVaR-75 | 50% | 21% | 3% | 36% | 54% | 67% | 76%
k-OWA-CVaR-100 | 46% | 17% | 7% | 36% | 49% | 59% | 69%
k-OWA-CCVaR-5 | 78% | 28% | 15% | 65% | 83% | 98% | 116%
k-OWA-CCVaR-10 | 73% | 29% | 10% | 58% | 78% | 95% | 111%
k-OWA-CCVaR-25 | 70% | 28% | 10% | 53% | 72% | 92% | 109%
k-OWA-CCVaR-50 | 56% | 29% | -2% | 38% | 58% | 79% | 99%
k-OWA-CCVaR-75 | 53% | 27% | -3% | 38% | 56% | 74% | 92%
k-OWA-CCVaR-100 | 51% | 22% | 3% | 37% | 55% | 68% | 78%
MinV | 41% | 19% | 0% | 30% | 44% | 55% | 70%
MinVsh | 42% | 46% | -25% | 3% | 41% | 80% | 120%
MinVN1 | 42% | 30% | -14% | 18% | 41% | 69% | 85%
MinVN2 | 52% | 26% | -13% | 49% | 58% | 67% | 82%
Index | 54% | 23% | 2% | 45% | 55% | 68% | 88%

Table 7: Summary statistics of ROI based on a 3-years time horizon for SP500

Approach | ExpRet | Vol | 5%-perc | 25%-perc | 50%-perc | 75%-perc | 95%-perc
---|---|---|---|---|---|---|---
Roman-CVaR | 37% | 24% | -13% | 28% | 40% | 48% | 77%
k-OWA-CVaR-5 | 40% | 24% | -15% | 33% | 43% | 53% | 76%
k-OWA-CVaR-10 | 38% | 24% | -19% | 33% | 42% | 50% | 72%
k-OWA-CVaR-25 | 34% | 23% | -18% | 29% | 39% | 46% | 65%
k-OWA-CVaR-50 | 31% | 24% | -24% | 25% | 36% | 47% | 57%
k-OWA-CVaR-75 | 27% | 21% | -24% | 24% | 32% | 42% | 50%
k-OWA-CVaR-100 | 30% | 21% | -20% | 27% | 36% | 43% | 51%
k-OWA-CCVaR-5 | 39% | 24% | -14% | 32% | 42% | 51% | 75%
k-OWA-CCVaR-10 | 39% | 24% | -15% | 34% | 43% | 52% | 75%
k-OWA-CCVaR-25 | 36% | 23% | -17% | 30% | 40% | 47% | 69%
k-OWA-CCVaR-50 | 34% | 23% | -18% | 30% | 38% | 46% | 66%
k-OWA-CCVaR-75 | 31% | 24% | -24% | 26% | 37% | 46% | 57%
k-OWA-CCVaR-100 | 28% | 22% | -26% | 25% | 34% | 43% | 51%
MinV | 26% | 20% | -16% | 18% | 27% | 43% | 53%
MinVsh | 33% | 31% | -38% | 28% | 44% | 53% | 62%
MinVN1 | 22% | 25% | -34% | 16% | 26% | 40% | 52%
MinVN2 | 28% | 24% | -30% | 24% | 35% | 43% | 51%
Index | 30% | 22% | -22% | 24% | 32% | 42% | 60%

##  6 Conclusions

This paper has analyzed portfolio selection strategies for EI that are based
on SD relations. We have introduced a new type of approximate stochastic
dominance rule, named $CS\mathcal{E}SD$, and we have proved that it implies
the almost SSD proposed by Lizyayev and Ruszczyński (2012). We have proposed a
new methodology that select portfolios using the OWA operator which
generalizes previous approaches based on minimax selection rules, and still
leads to solving linear programming models. Next, we have proved that our
enhanced indexation model based on OWA selects portfolios that dominate the
benchmarks in terms of this new form of stochastic dominance rule, and it
highlights encouraging out-of-sample performances with respect to a benchmark
index. Finally, we have tested and validated the performance of our new
proposal reporting an extensive empirical analysis based on three real-world
datasets belonging to major stock markets across the world (FTSE100, NASDAQ100
and SP500) and on the Fama and French 49 industry portfolios.

This paper has also opened new avenues of research into the EI field. In a
follow-up paper, we propose to extend the analysis to portfolio selection
strategies based on approximate stochastic dominance conditions by considering
an appropriate reshaping of the original benchmark index, as suggested by
Valle et al. (2017) and Cesarone et al. (2022a).

## References

  * Bawa et al. (1985) Bawa, V. S., Bodurtha Jr, J. N., Rao, M., and Suri, H. L. (1985).  On determination of stochastic dominance optimal sets.  The Journal of Finance, 40(2):417–431.
  * Beasley et al. (2003) Beasley, J. E., Meade, N., and Chang, T.-J. (2003).  An evolutionary heuristic for the index tracking problem.  European Journal of Operational Research, 148(3):621–643.
  * Bellini et al. (2021) Bellini, F., Cesarone, F., Colombo, C., and Tardella, F. (2021).  Risk parity with expectiles.  European Journal of Operational Research, 291(3):1149–1163.
  * Beraldi and Bruni (2022) Beraldi, P. and Bruni, M. E. (2022).  Enhanced indexation via chance constraints.  Operational Research - An International Journal, 22:1553–1573.
  * Bertsimas and Sim (2004) Bertsimas, D. and Sim, M. (2004).  The price of robustness.  Operations Research, 52(1):35–53.
  * Biglova et al. (2004) Biglova, A., Ortobelli, S., Rachev, S. T., and Stoyanov, S. (2004).  Different approaches to risk estimation in portfolio theory.  The Journal of Portfolio Management, 31(1):103–112.
  * Bruni et al. (2015) Bruni, R., Cesarone, F., Scozzari, A., and Tardella, F. (2015).  A linear risk-return model for enhanced indexation in portfolio optimization.  OR Spectrum, 37(3):735–759.
  * Bruni et al. (2017) Bruni, R., Cesarone, F., Scozzari, A., and Tardella, F. (2017).  On exact and approximate stochastic dominance strategies for portfolio selection.  European Journal of Operational Research, 259(1):322–329.
  * Bruni et al. (2012) Bruni, R., Cesarone, F., Scozzari, A., Tardella, F., et al. (2012).  A new stochastic dominance approach to enhanced index tracking problems.  Economics Bulletin, 32(4):3460–3470.
  * Bruni et al. (2013) Bruni, R., Cesarone, F., Scozzari, A., Tardella, F., et al. (2013).  No arbitrage and a linear portfolio selection model.  Economics Bulletin, 33(2):1247–1258.
  * Bufalo et al. (2022) Bufalo, D., Bufalo, M., Cesarone, F., and Orlando, G. (2022).  Straightening skewed markets with an index tracking optimizationless portfolio.  arXiv preprint arXiv:2203.13766.
  * Canakgoz and Beasley (2009) Canakgoz, N. A. and Beasley, J. E. (2009).  Mixed-integer programming approaches for index tracking and enhanced indexation.  European Journal of Operational Research, 196(1):384–399.
  * Carleo et al. (2017) Carleo, A., Cesarone, F., Gheno, A., and Ricci, J. M. (2017).  Approximating exact expected utility via portfolio efficient frontiers.  Decisions in Economics and Finance, 40(1):115–143.
  * Cesarone (2020) Cesarone, F. (2020).  Computational Finance: MATLAB® Oriented Modeling.  Routledge.
  * Cesarone et al. (2022a) Cesarone, F., Cesetti, R., Orlando, G., Martino, M. L., and Ricci, J. M. (2022a).  Comparing ssd-efficient portfolios with a skewed reference distribution.  Mathematics, 11(1):50.
  * Cesarone and Colucci (2017) Cesarone, F. and Colucci, S. (2017).  Minimum risk versus capital and risk diversification strategies for portfolio construction.  Journal of the Operational Research Society, pages 1–18.
  * Cesarone et al. (2019) Cesarone, F., Lampariello, L., and Sagratella, S. (2019).  A risk-gain dominance maximization approach to enhanced index tracking.  Finance Research Letters, 29:231–238.
  * Cesarone et al. (2022b) Cesarone, F., Martino, M. L., and Carleo, A. (2022b).  Does esg impact really enhance portfolio profitability?  Sustainability, 14:2050.
  * Cesarone et al. (2016) Cesarone, F., Moretti, J., Tardella, F., et al. (2016).  Optimally chosen small portfolios are better than large ones.  Economics Bulletin, 36(4):1876–1891.
  * Cesarone et al. (2015) Cesarone, F., Scozzari, A., and Tardella, F. (2015).  Linear vs. quadratic portfolio selection models with hard real-world constraints.  Computational Management Science, 12(3):345–370.
  * Cesarone et al. (2020) Cesarone, F., Scozzari, A., and Tardella, F. (2020).  An optimization–diversification approach to portfolio selection.  Journal of Global Optimization, 76(2):245–265.
  * Chekhlov et al. (2005) Chekhlov, A., Uryasev, S., and Zabarankin, M. (2005).  Drawdown measure in portfolio optimization.  International Journal of Theoretical and Applied Finance, 8(01):13–58.
  * Coqueret (2015) Coqueret, G. (2015).  Diversified minimum-variance portfolios.  Annals of Finance, 11(2):221–241.
  * DeMiguel et al. (2009) DeMiguel, V., Garlappi, L., and Uppal, R. (2009).  Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?  Review of Financial Studies, 22(5):1915–1953.
  * Denuit et al. (2014) Denuit, M. M., Huang, R. J., Tzeng, L. Y., and Wang, C. W. (2014).  Almost marginal conditional stochastic dominance.  Journal of Banking & Finance, 41:57–66.
  * Dhingra et al. (2023) Dhingra, V., Gupta, S. K., and Sharma, A. (2023).  Norm constrained minimum variance portfolios with short selling.  Computational Management Science, 20(1):6.
  * Ehrgott (2005) Ehrgott, M. (2005).  Multicriteria optimization, volume 491.  Springer Science & Business Media.
  * Fábián et al. (2011a) Fábián, C. I., Mitra, G., and Roman, D. (2011a).  Processing second-order stochastic dominance models using cutting-plane representations.  Mathematical Programming, 130(1):33–57.
  * Fábián et al. (2011b) Fábián, C. I., Mitra, G., Roman, D., and Zverovich, V. (2011b).  An enhanced model for portfolio choice with ssd criteria: a constructive approach.  Quantitative Finance, 11(10):1525–1534.
  * Fernández et al. (2014) Fernández, E., Pozo, M. A., and Puerto, J. (2014).  Ordered weighted average combinatorial optimization: Formulations and their properties.  Discrete Applied Mathematics, 169:97–118.
  * Fernández et al. (2017) Fernández, E., Pozo, M. A., Puerto, J., and Scozzari, A. (2017).  Ordered weighted average optimization in multiobjective spanning tree problem.  European Journal of Operational Research, 260(3):886–903.
  * Goodwin (1998) Goodwin, T. H. (1998).  The information ratio.  Financial Analysts Journal, 54(4):34–43.
  * Guastaroba et al. (2016) Guastaroba, G., Mansini, R., Ogryczak, W., and Speranza, M. G. (2016).  Linear programming models based on omega ratio for the enhanced index tracking problem.  European Journal of Operational Research, 251(3):938–956.
  * Guastaroba et al. (2020) Guastaroba, G., Mansini, R., Ogryczak, W., and Speranza, M. G. (2020).  Enhanced index tracking with cvar-based ratio measures.  Annals of Operations Research, 292(2):883–931.
  * Guastaroba and Speranza (2012) Guastaroba, G. and Speranza, M. G. (2012).  Kernel search: An application to the index tracking problem.  European Journal of Operational Research, 217(1):54–68.
  * Guo et al. (2014) Guo, X., Post, T., Wong, W.-K., and Zhu, L. (2014).  Moment conditions for almost stochastic dominance.  Economics Letters, 124(2):163–167.
  * Hodder et al. (2015) Hodder, J. E., Jackwerth, J. C., and Kolokolova, O. (2015).  Improved portfolio choice using second-order stochastic dominance.  Review of Finance, 19(4):1623–1647.
  * Huang et al. (2020) Huang, R. J., Tzeng, L. Y., and Zhao, L. (2020).  Fractional degree stochastic dominance.  Management Science, 66(10):4630–4647.
  * Jensen (1968) Jensen, M. C. (1968).  The performance of mutual funds in the period 1945-1964.  The Journal of Finance, 23(2):389–416.
  * Kopa and Post (2015) Kopa, M. and Post, T. (2015).  A general test for ssd portfolio efficiency.  OR Spectrum, 37(3):703–734.
  * Kuosmanen (2004) Kuosmanen, T. (2004).  Efficient diversification according to stochastic dominance criteria.  Management Science, 50(10):1390–1406.
  * Ledoit and Wolf (2008) Ledoit, O. and Wolf, M. (2008).  Robust performance hypothesis testing with the Sharpe ratio.  Journal of Empirical Finance, 15(5):850–859.
  * Ledoit and Wolf (2011) Ledoit, O. and Wolf, M. (2011).  Robust performances hypothesis testing with the variance.  Wilmott, 2011(55):86–89.
  * Leshno and Levy (2002) Leshno, M. and Levy, H. (2002).  Preferred by “all” and preferred by “most” decision makers: Almost stochastic dominance.  Management Science, 48(8):1074–1085.
  * Levy (1992) Levy, H. (1992).  Stochastic dominance and expected utility: Survey and analysis.  Management Science, 38(4):555–593.
  * Levy (2015) Levy, H. (2015).  Stochastic dominance: Investment decision making under uncertainty.  Springer.
  * Levy et al. (2010) Levy, H., Leshno, M., and Leibovitch, B. (2010).  Economically relevant preferences for all observed epsilon.  Annals of Operations Research, 176(1):153–178.
  * Lizyayev and Ruszczyński (2012) Lizyayev, A. and Ruszczyński, A. (2012).  Tractable almost stochastic dominance.  European Journal of Operational Research, 218(2):448–455.
  * Marín et al. (2020) Marín, A., Ponce, D., and Puerto, J. (2020).  A fresh view on the discrete ordered median problem based on partial monotonicity.  European Journal of Operational Research, 286(3):839–848.
  * Markowitz (1952) Markowitz, H. (1952).  Portfolio selection.  The Journal of Finance, 7(1):77–91.
  * Markowitz (1959) Markowitz, H. (1959).  Portfolio selection: Efficient diversification of investments.  Cowles Foundation for Research in Economics at Yale University, Monograph 16. John Wiley & Sons Inc., New York.
  * Martin and McCann (1989) Martin, P. G. and McCann, B. B. (1989).  The Investor’s Guide to Fidelity Funds.  Wiley.
  * Morgenstern and Von Neumann (1953) Morgenstern, O. and Von Neumann, J. (1953).  Theory of games and economic behavior.  Princeton University Press.
  * Müller et al. (2017) Müller, A., Scarsini, M., Tsetlin, I., and Winkler, R. L. (2017).  Between first-and second-order stochastic dominance.  Management Science, 63(9):2933–2947.
  * Müller and Stoyan (2002) Müller, A. and Stoyan, D. (2002).  Comparison methods for stochastic models and risks, volume 389.  Wiley.
  * Myung et al. (1996) Myung, I. J., Ramamoorti, S., and Bailey Jr, A. D. (1996).  Maximum entropy aggregation of expert predictions.  Management Science, 42(10):1420–1436.
  * Nickel and Puerto (2006) Nickel, S. and Puerto, J. (2006).  Location theory: a unified approach.  Springer Science & Business Media.
  * Ogryczak (2000) Ogryczak, W. (2000).  Multiple criteria linear programming model for portfolio selection.  Annals of Operations Research, 97(1):143–162.
  * Post and Kopa (2013) Post, T. and Kopa, M. (2013).  General linear formulations of stochastic dominance criteria.  European Journal of Operational Research, 230(2):321–332.
  * Post and Kopa (2017) Post, T. and Kopa, M. (2017).  Portfolio choice based on third-degree stochastic dominance.  Management Science, 63(10):3381–3392.
  * Puerto et al. (2017) Puerto, J., Rodríguez-Chía, A., and Tamir, A. (2017).  Revisiting k-sum optimization.  Mathematical Programming, 165(2):579–604.
  * Rockafellar and Uryasev (2000) Rockafellar, R. and Uryasev, S. (2000).  Optimization of Conditional Value-at-Risk.  Journal of Risk, 2(3):21–42.
  * Rockafellar and Uryasev (2002) Rockafellar, R. T. and Uryasev, S. (2002).  Conditional Value-at-Risk for General Distributions.  Journal of Banking & Finance, 26:1443–1471.
  * Roman et al. (2006) Roman, D., Darby-Dowman, K., and Mitra, G. (2006).  Portfolio construction based on stochastic dominance and target return distributions.  Mathematical Programming, 108(2):541–569.
  * Roman et al. (2013) Roman, D., Mitra, G., and Zverovich, V. (2013).  Enhanced indexation based on second-order stochastic dominance.  European Journal of Operational Research, 228(1):273–281.
  * Sant’Anna et al. (2022) Sant’Anna, L. R., Righi, M. B., Müller, F. M., and Guedes, P. C. (2022).  Risk measure index tracking model.  International Review of Economics & Finance, 80:361–383.
  * Scozzari et al. (2013) Scozzari, A., Tardella, F., Paterlini, S., and Krink, T. (2013).  Exact and heuristic approaches for the index tracking problem with ucits constraints.  Annals of Operations Research, 205(1):235–250.
  * Shaked and Shanthikumar (1994) Shaked, M. and Shanthikumar, J. G. (1994).  Stochastic orders and their applications.  Academic press.
  * Sharma et al. (2017) Sharma, A., Agrawal, S., and Mehra, A. (2017).  Enhanced indexing for risk averse investors using relaxed second order stochastic dominance.  Optimization and Engineering, 18(2):407–442.
  * Sharpe (1966) Sharpe, W. F. (1966).  Mutual fund performance.  The Journal of Business, 39(1):119–138.
  * Sharpe (1994) Sharpe, W. F. (1994).  The sharpe ratio.  Journal of portfolio management, 21(1):49–58.
  * Shu et al. (2020) Shu, L., Shi, F., and Tian, G. (2020).  High-dimensional index tracking based on the adaptive elastic net.  Quantitative Finance, 20(9):1513–1530.
  * Sortino et al. (2001) Sortino, F. A., Satchell, S., and Sortino, F. (2001).  Managing downside risk in financial markets.  Butterworth-Heinemann.
  * Treynor and Black (1973) Treynor, J. L. and Black, F. (1973).  How to use security analysis to improve portfolio selection.  The Journal of Business, 46(1):66–86.
  * Tsetlin et al. (2015) Tsetlin, I., Winkler, R. L., Huang, R. J., and Tzeng, L. Y. (2015).  Generalized almost stochastic dominance.  Operations Research, 63(2):363–377.
  * Tzeng et al. (2013) Tzeng, L. Y., Huang, R. J., and Shih, P.-T. (2013).  Revisiting almost second-degree stochastic dominance.  Management Science, 59(5):1250–1254.
  * Valle et al. (2017) Valle, C. A., Roman, D., and Mitra, G. (2017).  Novel approaches for portfolio construction using second order stochastic dominance.  Computational Management Science, 14(2):257–280.
  * Whitmore and Findlay (1978) Whitmore, G. A. and Findlay, M. C. (1978).  Stochastic dominance: an approach to decision-making under risk.  Lexington Books.
  * Yager (1988) Yager, R. R. (1988).  On ordered weighted averaging aggregation operators in multicriteria decisionmaking.  IEEE Transactions on systems, Man, and Cybernetics, 18(1):183–190.

##  Appendix A SD-based strategies using Tails

For the sake of completeness, we report here additional tables containing the
computational results obtained by the OWA maximization of centered Tails
approach along with the RomanTail and KP2011Min strategies, described in
Section 5. These portfolios are also compared with the MinV portfolio and the
Market Index. More precisely, in Tables 8, 9, and 10, we show the out-of-
sample performance results obtained on the FTSE100, NASDAQ100, and SP500
datasets, respectively. From these empirical findings we can observe three
main aspects. On the one hand, as also highlighted by Roman et al. (2013), the
portfolios obtained with the SD models using centered CVaRs, except for
FTSE100, lead to better performance. On the other hand, we can notice that the
$k$-OWA-Tail-$\beta$ and $k$-OWA-CTail-$\beta$ strategies with low $\beta$
typically continue to show the best performance in terms of profitability.
Finally, the risk mitigation feature of our approach, obtained by increasing
the $\beta$ values, is no longer found in the “Tails” case.

Table 8: Out-of-sample performance results of the SD-based strategies using
Tails compared with the MinV portfolio and the Market Index on the FTSE100
daily dataset

Approach | ExpRet | Vol | Sharpe | MDD | Ulcer | Sortino | Rachev | AlphaJ | VaR1 | Omega | ave # | time
---|---|---|---|---|---|---|---|---|---|---|---|---
Index | 0.0083% | 1.2117% | 0.0069 | -0.4783 | 0.1549 | 0.0095 | 0.9257 |  | 0.0345 | 1.0210 |  |
Roman-Tail | 0.0541% | 1.0800% | 0.0501 | -0.5032 | 0.1140 | 0.0742 | 1.0280 | 0.00049 | 0.0293 | 1.1630 | 10.5 | 30.0
k-OWA-Tail-5 | 0.0520% | 1.0080% | 0.0516 | -0.4956 | 0.1232 | 0.0736 | 0.9736 | 0.00047 | 0.0282 | 1.1611 | 11.5 | 24.4
k-OWA-Tail-10 | 0.0472% | 0.9732% | 0.0485 | -0.4436 | 0.1085 | 0.0686 | 0.9628 | 0.00042 | 0.0270 | 1.1497 | 12.6 | 32.4
k-OWA-Tail-25 | 0.0409% | 0.9774% | 0.0418 | -0.4246 | 0.1113 | 0.0589 | 0.9467 | 0.00036 | 0.0268 | 1.1291 | 13.6 | 36.9
k-OWA-Tail-50 | 0.0395% | 0.9882% | 0.0400 | -0.4049 | 0.1095 | 0.0564 | 0.9552 | 0.00034 | 0.0267 | 1.1232 | 14.7 | 47.7
k-OWA-Tail-75 | 0.0407% | 0.9959% | 0.0408 | -0.4120 | 0.1114 | 0.0576 | 0.9519 | 0.00035 | 0.0277 | 1.1265 | 15.0 | 61.7
k-OWA-Tail-100 | 0.0419% | 1.0138% | 0.0413 | -0.4320 | 0.1190 | 0.0580 | 0.9475 | 0.00036 | 0.0282 | 1.1279 | 14.3 | 69.7
k-OWA-CTail-5 | 0.0555% | 1.0285% | 0.0540 | -0.4881 | 0.1178 | 0.0774 | 0.9867 | 0.00050 | 0.0285 | 1.1706 | 11.2 | 17.9
k-OWA-CTail-10 | 0.0504% | 0.9893% | 0.0509 | -0.4570 | 0.1180 | 0.0725 | 0.9646 | 0.00045 | 0.0270 | 1.1582 | 12.1 | 17.4
k-OWA-CTail-25 | 0.0412% | 0.9649% | 0.0427 | -0.4354 | 0.1111 | 0.0600 | 0.9443 | 0.00036 | 0.0264 | 1.1314 | 13.4 | 17.1
k-OWA-CTail-50 | 0.0398% | 0.9770% | 0.0407 | -0.3995 | 0.1059 | 0.0573 | 0.9483 | 0.00035 | 0.0264 | 1.1253 | 14.2 | 34.2
k-OWA-CTail-75 | 0.0397% | 0.9837% | 0.0403 | -0.4034 | 0.1089 | 0.0569 | 0.9526 | 0.00034 | 0.0265 | 1.1243 | 14.6 | 78.9
k-OWA-CTail-100 | 0.0403% | 0.9911% | 0.0407 | -0.4091 | 0.1115 | 0.0574 | 0.9525 | 0.00035 | 0.0273 | 1.1257 | 14.8 | 132.7
MinV | 0.0362% | 0.9254% | 0.0391 | -0.3802 | 0.0909 | 0.0545 | 0.9577 | 0.00031 | 0.0271 | 1.1215 | 17.9 | 0.0
KP2011Min | 0.0494% | 0.0102 | 0.0486 | -0.4199 | 0.1069 | 0.0695 | 0.9551 | 0.00044 | 0.0289 | 1.1541 | 13.1 | 51.4

Table 9: Out-of-sample performance results of the SD-based strategies using
Tails compared with the MinV portfolio and the Market Index on the NASDAQ100
daily dataset

Approach | ExpRet | Vol | Sharpe | MDD | Ulcer | Sortino | Rachev5 | AlphaJ | VaR1 | Omega | ave # | time
---|---|---|---|---|---|---|---|---|---|---|---|---
Index | 0.0651% | 1.4093% | 0.0462 | -0.5371 | 0.1339 | 0.0654 | 0.9481 |  | 0.0407 | 1.1523 |  |
Roman-Tail | 0.0522% | 1.1618% | 0.0450 | -0.4293 | 0.1081 | 0.0649 | 0.9742 | 0.00009 | 0.0329 | 1.1443 | 7.9 | 15.9
k-OWA-Tail-5 | 0.0529% | 1.1439% | 0.0463 | -0.4231 | 0.1094 | 0.0666 | 0.9722 | 0.00010 | 0.0314 | 1.1474 | 8.3 | 16.8
k-OWA-Tail-10 | 0.0606% | 1.1446% | 0.0529 | -0.3657 | 0.0824 | 0.0760 | 0.9710 | 0.00018 | 0.0312 | 1.1706 | 9.0 | 22.2
k-OWA-Tail-25 | 0.0554% | 1.1547% | 0.0480 | -0.3667 | 0.0957 | 0.0686 | 0.9584 | 0.00013 | 0.0326 | 1.1553 | 9.8 | 27.6
k-OWA-Tail-50 | 0.0540% | 1.1368% | 0.0475 | -0.3643 | 0.1002 | 0.0677 | 0.9484 | 0.00012 | 0.0320 | 1.1542 | 11.1 | 36.8
k-OWA-Tail-75 | 0.0525% | 1.1219% | 0.0468 | -0.3643 | 0.0994 | 0.0666 | 0.9383 | 0.00010 | 0.0322 | 1.1519 | 11.7 | 49.3
k-OWA-Tail-100 | 0.0512% | 1.1251% | 0.0455 | -0.3585 | 0.0956 | 0.0647 | 0.9349 | 0.00009 | 0.0321 | 1.1479 | 11.6 | 43.4
k-OWA-CTail-5 | 0.0533% | 1.1433% | 0.0466 | -0.4185 | 0.1079 | 0.0670 | 0.9694 | 0.00011 | 0.0322 | 1.1479 | 8.2 | 12.7
k-OWA-CTail-10 | 0.0571% | 1.1406% | 0.0500 | -0.3935 | 0.0961 | 0.0720 | 0.9718 | 0.00015 | 0.0312 | 1.1600 | 8.7 | 13.2
k-OWA-CTail-25 | 0.0545% | 1.1453% | 0.0476 | -0.3784 | 0.0975 | 0.0681 | 0.9625 | 0.00012 | 0.0326 | 1.1536 | 9.4 | 15.2
k-OWA-CTail-50 | 0.0533% | 1.1446% | 0.0466 | -0.3695 | 0.1012 | 0.0665 | 0.9528 | 0.00011 | 0.0317 | 1.1509 | 10.4 | 31.6
k-OWA-CTail-75 | 0.0542% | 1.1343% | 0.0477 | -0.3664 | 0.0990 | 0.0681 | 0.9487 | 0.00012 | 0.0318 | 1.1552 | 11.1 | 57.8
k-OWA-CTail-100 | 0.0534% | 1.1269% | 0.0474 | -0.3648 | 0.0989 | 0.0674 | 0.9435 | 0.00011 | 0.0323 | 1.1539 | 11.5 | 97.3
MinV | 0.0465% | 0.9999% | 0.0465 | -0.3486 | 0.0865 | 0.0661 | 0.9357 | 0.00010 | 0.0286 | 1.1537 | 13.0 | 0.0
KP2011Min | 0.0442% | 1.0830% | 0.0408 | -0.3807 | 0.1000 | 0.0577 | 0.9245 | 0.00004 | 0.0320 | 1.1310 | 10.6 | 47.5

Table 10: Out-of-sample performance results of the SD-based strategies using
Tails compared with the MinV portfolio and the Market Index on the SP500 daily
dataset

Approach | ExpRet | Vol | Sharpe | MDD | Ulcer | Sortino | Rachev5 | AlphaJ | VaR1 | Omega | ave # | time
---|---|---|---|---|---|---|---|---|---|---|---|---
Index | 0.0352% | 1.2960% | 0.0272 | -0.5678 | 0.1653 | 0.0378 | 0.9099 |  | 0.0393 | 1.0926 |  |
Roman-Tail | 0.0312% | 0.9921% | 0.0314 | -0.5120 | 0.1950 | 0.0435 | 0.9018 | 0.00010 | 0.0288 | 1.1015 | 13.8 | 70.3
k-OWA-Tail-5 | 0.0306% | 0.9754% | 0.0313 | -0.5016 | 0.1520 | 0.0432 | 0.8960 | 0.00010 | 0.0294 | 1.0997 | 14.3 | 37.4
k-OWA-Tail-10 | 0.0322% | 0.9535% | 0.0338 | -0.4853 | 0.1353 | 0.0468 | 0.8819 | 0.00012 | 0.0269 | 1.1065 | 15.0 | 49.4
k-OWA-Tail-25 | 0.0362% | 0.9500% | 0.0381 | -0.4748 | 0.1266 | 0.0532 | 0.8983 | 0.00016 | 0.0299 | 1.1208 | 16.8 | 56.3
k-OWA-Tail-50 | 0.0318% | 0.9488% | 0.0336 | -0.4964 | 0.1706 | 0.0468 | 0.8931 | 0.00012 | 0.0295 | 1.1058 | 18.4 | 64.4
k-OWA-Tail-75 | 0.0304% | 0.9602% | 0.0316 | -0.4916 | 0.1763 | 0.0440 | 0.8836 | 0.00010 | 0.0296 | 1.0998 | 19.1 | 77.3
k-OWA-Tail-100 | 0.0304% | 0.9866% | 0.0308 | -0.4935 | 0.1833 | 0.0428 | 0.8745 | 0.00009 | 0.0295 | 1.0970 | 18.5 | 90.7
k-OWA-CTail-5 | 0.0304% | 0.9902% | 0.0307 | -0.5104 | 0.1669 | 0.0424 | 0.8992 | 0.00009 | 0.0300 | 1.0982 | 14.2 | 27.5
k-OWA-CTail-10 | 0.0308% | 0.9623% | 0.0320 | -0.5185 | 0.1638 | 0.0442 | 0.8813 | 0.00011 | 0.0288 | 1.1010 | 14.6 | 24.8
k-OWA-CTail-25 | 0.0317% | 0.9484% | 0.0334 | -0.4864 | 0.1287 | 0.0465 | 0.8961 | 0.00012 | 0.0284 | 1.1051 | 16.0 | 23.9
k-OWA-CTail-50 | 0.0333% | 0.9415% | 0.0354 | -0.4946 | 0.1537 | 0.0494 | 0.8960 | 0.00013 | 0.0295 | 1.1119 | 17.8 | 46.7
k-OWA-CTail-75 | 0.0317% | 0.9445% | 0.0335 | -0.4968 | 0.1681 | 0.0468 | 0.8925 | 0.00011 | 0.0289 | 1.1058 | 18.3 | 86.9
k-OWA-CTail-100 | 0.0313% | 0.9531% | 0.0329 | -0.4952 | 0.1740 | 0.0458 | 0.8884 | 0.00011 | 0.0291 | 1.1039 | 18.9 | 170.2
MinV | 0.0279% | 0.8816% | 0.0317 | -0.4578 | 0.1412 | 0.0440 | 0.8897 | 0.00008 | 0.0253 | 1.1060 | 23.1 | 0.1
KP2011Min | 0.0326% | 0.9715% | 0.0336 | -0.5227 | 0.1748 | 0.0467 | 0.8873 | 0.00012 | 0.0309 | 1.1090 | 17.0 | 69.1

Finally, Tables 11, 12, 13 report some statistics of the 3-years time horizon
ROI obtained by all the analyzed portfolio strategies on the FTSE100,
NASDAQ100, and SP500 daily datasets, respectively. Again, we can note that the
most profitable portfolios are those obtained by the $k$-OWA-Tail-$\beta$ and
$k$-OWA-CTail-$\beta$ models with low $\beta$, although in general, the SD
models with Tails are less performing w.r.t. those formulated with CVaRs.

Table 11: Summary statistics of ROI based on a 3-years time horizon for
FTSE100

Approach | ExpRet | Vol | 5%-perc | 25%-perc | 50%-perc | 75%-perc | 95%-perc
---|---|---|---|---|---|---|---
Index | 8% | 15% | -19% | 1% | 10% | 17% | 31%
Roman-Tail | 60% | 36% | 2% | 33% | 54% | 88% | 124%
k-OWA-Tail-5 | 59% | 35% | 0% | 35% | 51% | 86% | 120%
k-OWA-Tail-10 | 49% | 28% | 2% | 26% | 48% | 69% | 97%
k-OWA-Tail-25 | 44% | 29% | -3% | 16% | 52% | 64% | 83%
k-OWA-Tail-50 | 42% | 27% | -5% | 15% | 52% | 62% | 74%
k-OWA-Tail-75 | 44% | 28% | -5% | 16% | 54% | 65% | 79%
k-OWA-Tail-100 | 47% | 30% | -6% | 16% | 58% | 70% | 85%
k-OWA-CTail-5 | 61% | 38% | 1% | 35% | 53% | 89% | 128%
k-OWA-CTail-10 | 56% | 33% | 1% | 33% | 48% | 81% | 115%
k-OWA-CTail-25 | 44% | 29% | -3% | 17% | 47% | 65% | 88%
k-OWA-CTail-50 | 42% | 27% | -4% | 16% | 51% | 62% | 78%
k-OWA-CTail-75 | 42% | 27% | -5% | 15% | 52% | 62% | 75%
k-OWA-CTail-100 | 44% | 28% | -5% | 16% | 53% | 64% | 77%
MinV | 37% | 19% | 3% | 25% | 41% | 50% | 63%
KP2011Min | 57% | 35% | 1% | 28% | 55% | 85% | 114%

Table 12: Summary statistics of ROI based on a 3-years time horizon for
NASDAQ100

Approach | ExpRet | Vol | 5%-perc | 25%-perc | 50%-perc | 75%-perc | 95%-perc
---|---|---|---|---|---|---|---
Index | 54% | 23% | 2% | 45% | 55% | 68% | 88%
Roman-Tail | 52% | 24% | 1% | 40% | 48% | 65% | 95%
k-OWA-Tail-5 | 52% | 21% | 0% | 43% | 54% | 65% | 84%
k-OWA-Tail-10 | 59% | 21% | 10% | 49% | 61% | 72% | 91%
k-OWA-Tail-25 | 51% | 19% | 5% | 42% | 54% | 63% | 79%
k-OWA-Tail-50 | 51% | 21% | 3% | 37% | 58% | 66% | 76%
k-OWA-Tail-75 | 50% | 21% | 3% | 37% | 56% | 66% | 75%
k-OWA-Tail-100 | 50% | 21% | 3% | 37% | 54% | 67% | 77%
k-OWA-CTail-5 | 55% | 24% | 2% | 44% | 54% | 69% | 95%
k-OWA-CTail-10 | 55% | 22% | 4% | 46% | 57% | 68% | 90%
k-OWA-CTail-25 | 51% | 20% | 2% | 43% | 54% | 63% | 80%
k-OWA-CTail-50 | 50% | 20% | 1% | 39% | 55% | 64% | 75%
k-OWA-CTail-75 | 51% | 20% | 4% | 39% | 57% | 66% | 75%
k-OWA-CTail-100 | 51% | 21% | 3% | 38% | 57% | 66% | 75%
MinV | 41% | 19% | 0% | 30% | 44% | 55% | 70%
KP2011Min | 43% | 21% | -3% | 29% | 46% | 60% | 71%

Table 13: Summary statistics of ROI based on a 3-years time horizon for SP500

Approach | ExpRet | Vol | 5%-perc | 25%-perc | 50%-perc | 75%-perc | 95%-perc
---|---|---|---|---|---|---|---
Index | 30% | 22% | -22% | 24% | 32% | 42% | 60%
Roman-Tail | 28% | 24% | -26% | 22% | 34% | 44% | 56%
k-OWA-Tail-5 | 29% | 21% | -17% | 23% | 34% | 42% | 53%
k-OWA-Tail-10 | 29% | 19% | -13% | 25% | 33% | 40% | 55%
k-OWA-Tail-25 | 32% | 18% | -7% | 28% | 36% | 42% | 56%
k-OWA-Tail-50 | 29% | 21% | -22% | 27% | 34% | 44% | 51%
k-OWA-Tail-75 | 27% | 22% | -24% | 23% | 32% | 42% | 50%
k-OWA-Tail-100 | 28% | 23% | -26% | 25% | 33% | 43% | 51%
k-OWA-CTail-5 | 27% | 21% | -20% | 21% | 34% | 41% | 51%
k-OWA-CTail-10 | 28% | 20% | -19% | 25% | 33% | 41% | 52%
k-OWA-CTail-25 | 28% | 17% | -10% | 24% | 31% | 39% | 54%
k-OWA-CTail-50 | 31% | 20% | -17% | 28% | 37% | 42% | 52%
k-OWA-CTail-75 | 29% | 21% | -22% | 26% | 34% | 43% | 50%
k-OWA-CTail-100 | 29% | 21% | -24% | 25% | 34% | 43% | 50%
MinV | 26% | 20% | -16% | 18% | 27% | 43% | 53%
KP2011Min | 31% | 22% | -22% | 28% | 38% | 44% | 52%