Multivariate stochastic dominance applied to sector-based portfolio selection

Kouaissah,, Noureddine; Ortobelli Lozza,, Sergio

doi:10.1093/imaman/dpaa004

Multivariate stochastic dominance applied to sector-based portfolio selection

Kouaissah,, Noureddine;Ortobelli Lozza,, Sergio 2021-01-30 00:00:00 Abstract In this study, we investigate whether sector-weighted portfolios based on alternative parametric assumptions are consistent with multivariate stochastic dominance (MSD) conditions for a class of non-satiable risk-averse investors. Focusing specifically on stable symmetric and Student’s t distributions, we propose and motivate an MSD rule to determine a partial order among sectors, based on a comparison between (i) location, (ii) dispersion parameters and (iii) either stability indices (for stable symmetric distributions) or degrees of freedom (for Student’s t distributions). The proposed MSD rule is applied to the US equity market to evaluate whether and how the derived stochastic dominance conditions are relevant to investors’ decisions. The empirical study confirms that the proposed MSD rule is effective and that the tail behaviour of returns is relevant to the optimization of portfolios for non-satiable investors. 1. Introduction The introduction of multivariate stochastic dominance (MSD) rules among different market activity sectors is fundamental to understanding the market trend. Moreover, equity portfolios constructed based on financial benchmarks associated with specific economic sectors are emerging as an efficient way to take advantage of business sector cyclicality. In this study, we develop and apply a parametric MSD framework to sector-based portfolio selection problems. We focus on a very general class of investors, assumed to be risk-averse and non-satiable (Levy, 1992; Müller & Stoyan, 2002), and on given parametric assumptions of the assets’ return distributions. We use dominance among market sectors to express a MSD of a given order between portfolios, including stocks belonging to specific economic sectors. Essentially, this work aims to evaluate the impact of dominance among market sectors on optimal investors’ choices and understand whether optimal portfolio choices imply some manner of orderings between chosen and non-chosen assets (market sectors). Specifically, it examines the extent to which the dominance observed among some market sectors persists into the composition of an optimal portfolio. It endeavours to answer the following questions: (i) if non-Gaussian elliptical distributions (say, symmetric stable or Student’s t) are assumed, which multivariate rule should be used with respect to other parameters (index of stability and degree of freedom)? (ii) Does dominance among market sectors exist? (iii) What impact do distributional assumptions have on investors’ choices? Research on operationalizing stochastic dominance rules for parametric families of distributions goes back to Ali (1975) and many related studies (see, e.g., Kroll & Levy, 1986; Levy, 2006). Differently from these studies, we examine stochastic dominance rules for parametric multivariate families of distributions. In particular, we first propose a parametric distributional comparison, after which we test its efficiency using a non-parametric approach. Our main purpose is to identify the best sectors in which to invest and help investors and fund managers to construct their equity portfolios. Thus, we focus on the issue of ranking different market sectors from a non-satiable risk-averse investor’s point of view, a decision under uncertainty that can be addressed using different approaches, such as utility theory (see, Ingersoll, 1987) or game theory (see, Moretti et al., 2011). Furthermore, from a portfolio theory perspective, individual preferences can be summarized by maximum gain and minimum risk, and, in this setting, several reward-risk performance measures have been proposed in the literature to account for the characteristics of different asset returns (see, e.g., Szegö, 1994, Rachev et al. (2008a), Ortobelli et al., 2019. Unlike these approaches, this work relies on the theory of stochastic orderings. In a univariate context, stochastic dominance rules establish a partial order in the space of distribution functions by quantifying the concept of one random variable being ‘preferable’ to another. Because the decision problem here concerns multivariate random elements (i.e., market sectors), the purpose is to apply stochastic dominance rules that are simple and applicable to a multivariate framework. On the one hand, a large body of literature has presented and examined the implications of stochastic dominance in portfolio theory (see, among others, Bruni et al., 2012; Levy, 2006; Osuna, 2012; Post & Kopa, 2017; Roman et al., 2006). For instance, Roman et al. (2013) evaluated the effectiveness of enhanced indexation models based on second-order stochastic dominance (SSD). Later, Bruni et al. (2017) proposed a new type of approximate stochastic dominance rule for enhanced indexation. On the other hand, numerous theoretical and empirical studies have been developed to test portfolio efficiency based on stochastic dominance (see, e.g., Kopa & Post, 2015; Kuosmanen, 2004; Post, 2003; Post & Potì, 2017; Scaillet & Topaloglou, 2010). However, most existing multivariate statistical tests for stochastic efficiency (see, e.g., Post & Kopa, 2017) are not suitable to examine dominance among market sectors, primarily because these tests require a comparison between vectors of the same dimension and consider an order among the marginals, which is not realistic among market sectors. Recall that each market sector has a different number of assets from which some components may cover different periods. In a recent contribution, Arvanitis et al. (2019) introduced a model-free concept of stochastic spanning along with a test procedure for implementation based on sub-sampling and linear programming. Spanning evaluates and determines whether including new securities or relaxing investment constraints enhances the investment opportunity set for a general class of all risk-averse investors. According to behavioural finance literature (see, e.g., Levy & Levy (2002), Müller & Stoyan, 2002; Ortobelli et al., 2017a,b), we want to rank different market sectors from the point of view of different non-satiable risk-averse investors. Thus, the research question in this work aligns with contemporaneous research on MSD analysis in portfolio theory (see, Arvanitis et al. (2019), Anyfantaki et al., 2018). In contrast to Arvanitis et al. (2019), this work compares choice sets of portfolios that do not have common portfolios. In this sense, the non-parametric test proposed by Arvanitis et al. cannot be used to compare different sectors, but it can still be used to test if the dominant sectors contain (span) the market-efficient set. Moreover, the stochastic dominance relations must consider return anomalies (e.g., heavy tails and skewness). In financial literature, it is well-known that asset returns present heavy tails and skewness. In particular, empirical research has established some stylized facts about asset returns: (a) clustering of volatility, (b) skewness and (c) fat tails (see, e.g., Kim et al., 2011). Different models have been proposed and experimentally tested to explain these empirical facts; for a survey of criticism, see Mandelbrot (1963), Fama (1970) and Rachev & Mittnik (2000). Stable Paretian and Student’s t distributions (both heavy-tailed) can explain the observed excess kurtosis of financial returns and represent an improvement over the Gaussian assumption (see, e.g., Blattberg & Gonedes, 1974). Conceptually, stable Paretian distributions are a natural generalization of the Gaussian distribution. In the non-Gaussian case, a stable distribution exhibits fatter tails than a normal distribution and is more peaked around its centre (i.e., leptokurtic). For portfolio analysis, the most critical feature of these distributions is that their variances (and higher moments) are not finite. Given that stable distributions depend on location and scale parameters, they are also identified by a parameter which specifies the shape of the distribution in terms of skewness and, more importantly, one that determines the asymptotic behaviour of the tails. Evidence provided by several authors (see, e.g., Ortobelli et al., 2016a) suggests that one could explicitly account for the observed ‘fat tails’ using the symmetric-stable distribution. This study considers another family of symmetric distributions that also accounts for the heavy tails of return distributions. This alternative is the Student’s t distribution, which is empirically and theoretically supported by several studies (see, e.g., Fergusson & Platen, 2006 and literature therein). Therefore, because of the importance of stable Paretian and Student’s t distributions for approximating random financial variables, we propose stochastic dominance rules that are aimed at ranking either Student’s t or sub-Gaussian random vectors. In so doing, we derive criteria that select the best market sector either to invest in or to consider for portfolio diversification purposes. In the empirical comparison, we examine the ex-ante and ex-post dominance between stock sectors, assuming that the returns are either Student’s t-distributed or in the domain of attraction of a stable sub-Gaussian law. Empirical evidence shows that the proposed methodology can assist decision makers in determining the optimal portfolio by considering the impact of stochastic dominance among market sectors. In particular, we show that the dominance observed among market sectors can explain the percentage concentration of invested stocks. Specifically, the dominant sectors mostly contain a higher fraction of the optimal portfolio we invested in. The rest of the paper is organized as follows: in Section 2, we first define a simple and applicable MSD rule based on the mean-variance approach. Then, we propose a definition of a multivariate stochastic ordering aimed at ranking different market sectors whose vectors of returns are either in the domain of attraction of a stable sub-Gaussian law or are Student’s t-distributed. In Section 3, we present an empirical comparison to evaluate possible dominance relations between several market sectors of the S&P 500. Finally, Section 4 summarizes our conclusions. 2. Stochastic dominance between market sectors This section introduces multivariate stochastic orderings among market sectors. Our primary goal is to compare market sectors from investors’ points of views and examine the impact of the observed dominance of the optimal choices. Thus, we introduce multivariate ordering rules that are especially useful for ranking different market sectors. Recall that, when we compare two random variables, |$X$| and |$Y$|⁠, with respect to a given univariate order of preferences, represented by the symbol |$\succ $|⁠, we say that |$X$| dominates |$Y$| with respect to |$\succ $| (namely |$X\succ Y$|⁠) if suitable conditions are satisfied. In general, these conditions involve the distribution functions of |$X$| and |$Y$|⁠, say |$F_X$| and |$F_Y$| (see, e.g., Levy, 1992; Lodwick, 1989, Castellano & Cerqueti (2018)). Classical stochastic orders are reviewed in the following definition. Definition 2.1 – First-order stochastic dominance: we say that |$X$| dominates |$Y$| with respect to the first-order stochastic dominance (in symbols, |$X$| FSD |$Y$|⁠) if and only if |$F_X (t)\leqslant F_Y (t)$| for any |$t$| that belongs to |$\mathbb{R}$| or, equivalently, |$X$| FSD |$Y$| if and only if |$E(g(X))\geqslant E(g(Y))$| for any increasing function |$g$|⁠. – SSD (increasing concave order): we say that |$X$| dominates |$Y$| with respect to the SSD order (in symbols, |$X$| SSD |$Y$| or |$X\geqslant _{icv}Y$|⁠) if and only if $$\begin{equation*}\displaystyle{\int_{-\infty}^{t}F_X (z) \, \textrm{d}z\leqslant \int_{-\infty}^{t}F_Y (z) \, \textrm{d}z}, \, \forall \, t\in \mathbb{R}\end{equation*}$$ or, equivalently, |$X\geqslant _{icv}Y$| if and only if |$E(g(X))\geqslant E(g(Y))$| for any increasing and concave function |$g$|⁠, i.e., any non-satiable risk-averse investor prefers |$X$| to |$Y$|⁠. Recall that investors are risk averse if they have concave utility function (see Ingersoll, 1987). In general, the extension of a given order of preferences |$\succ $| to the multivariate case is not trivial because in some practical cases it could be challenging to satisfy the conditions of MSD. In this context, although the natural generalizations of FSD and SSD can be found, for instance in Shaked & Shanthikumar (1994) and Müller & Stoyan (2002), their dominance rules are far from being applicable, except in some specific cases. Indeed, multivariate stochastic orders consider the dependence structure of random vectors and cannot be based merely on component-wise comparisons of the marginal distributions. It is very rare to find a multivariate ordering rule that is based on comparisons between functionals. One idea born from the definition proposed by Ortobelli et al. (2016a) is shown in the following definition. Suppose that there are two sectors, |$A$| and |$B$|⁠, composed, respectively, of |$n$| and |$s$| assets. Let the vectors of non-negative allocations among risky assets of sectors |$A$| and |$B$|⁠, respectively, be denoted by |$x=[x_1,x_2,\ldots ,x_n ]^{\prime}$| and |$y=[y_1,y_2,\ldots ,y_s ]^{\prime}$|⁠, whose sum is equal to 1 (i.e., |$\sum _{i=1}^n x_i=\sum _{j=1}^s y_j=1$|⁠). Moreover, assume that no short sales are allowed (i.e., |$x_i\geqslant 0$| and |$y_j\geqslant 0$|⁠, |$\forall \,\, i=1,\ldots ,n$| and |$j=1,\ldots ,s$|⁠). Definition 2.2 We say that a sector |$A$| with |$n$| assets strongly dominates sector |$B$| with |$s\leqslant n$| assets with respect to a multivariate preference ordering |$\succ $| (namely, FSD and SSD) if for any vector of returns |$Y_B$| of |$B$|⁠, there exists a vector |$X_A$| of sector |$A$| such that |$X_A\succ Y_B$|⁠. Similarly, we say that a sector |$A$| with |$n$| assets weakly dominates another sector |$B$| with |$s$| assets with respect to a multivariate preference ordering |$\succ $| if for any given portfolio of sector |$B$| with return |$y^{\prime}Y_B$|⁠, there exists a portfolio of sector |$A$| with return |$x^{\prime} X_A$| such that |$x^{\prime} X_A \succ y^{\prime} Y_B$|⁠. Generally, the stronger dominance implies the weak one, as clarified in the following example. Example 2.1 Suppose that the returns of sectors |$A$| and |$B$| are jointly Gaussian distributed. Suppose also that the two sectors have the same number of assets |$n$|⁠, vector averages |$\mu _A$| and |$\mu _B$|⁠, and variance-covariance matrices |$Q_A$| and |$Q_B$|⁠, such that |$\mu _A \geqslant \mu _B$| and |$(Q_A-Q_B)$| is negative semi-definite. Then, sector |$A$| strongly dominates sector |$B$| with respect to the increasing concave multivariate (ICVM) order. Moreover, under these assumptions, sector |$A$| weakly dominates sector |$B$| with respect to the concave order because |$x^{\prime}\mu _A \geqslant x^{\prime}\mu _B$| and |$x^{\prime}Q_{A}x\leqslant x^{\prime}Q_{B}x$| for any vector |$x\geqslant 0$|⁠. Note that this weak dominance between Gaussian distributed vectors is also known in the literature as the increasing positive linear concave multivariate order (see Müller & Stoyan, 2002). The increasing positive linear concave multivariate ordering defined in Example 2.1 is strictly related to the mean-variance rule, which is widely used in finance to solve the portfolio optimization problem. According to this principle, investors aim at minimizing variance (risk) for some fixed level of the mean (reward). In mean-variance framework, we call the efficient frontier (EF) the set of portfolios with greater mean and lower variance. However, we can extend this concept with respect to any given investor preference as follows. Definition 2.3 We call the EF with respect to a given preference |$\succ $| the set of portfolios |$x^{\prime}r$| (⁠|$\sum _{i=1}^n x=1$|⁠, |$x_i\geqslant 0$|⁠) that are not dominated with respect to the preference |$\succ $|⁠, i.e., |$\textrm{EF}_{\succ }=\lbrace x\in \mathbb{R}^n: \sum _{i=1}^n x_i=1, \, x_i\geqslant 0, \, \nexists \, y\in \mathbb{R}^n: \sum _{i=1}^n y_i=1, \, y_i\geqslant 0: y^{\prime}r\succ x^{\prime}r\rbrace $|⁠. Typically, in MSD literature, the comparison between vectors of the same dimension with respect to some orderings is used (see, e.g., Kopa & Post, 2015; Post & Kopa, 2017). However, this typical comparison among vectors is generally stronger than the one introduced in Definition 2.2 because in this definition we do not take into account an order among the marginals. Let us clarify this concept. Example 2.2 Suppose the sector |$A$| is composed of two Gaussian independent distributed random variables |$X_1\sim N(1,2)$| and |$X_2\sim N(2,1)$|⁠. Similarly, sector |$B$| is composed of two Gaussian independent distributed random variables |$Y_1\sim N(2,2)$| and |$Y_2\sim N(1,3)$|⁠. Observe that vector |$X=[X_1,X_2]^{\prime}$| (with |$\mu _X=[1,2]^{\prime}$| and variance-covariance matrix |$Q_X$|⁠) does not dominate vector |$Y=[Y_1,Y_2]^{\prime}$| (with |$\mu _Y=[2,1]^{\prime}$| and variance-covariance matrix |$Q_Y$|⁠) with respect to the ICVM order. However, sector |$A$| dominates sector |$B$| with respect to the ICVM order because the |$\tilde{X}=\left [ X_2,X_1\right ]$| ICVM dominates vector |$Y=\left [ Y_1,Y_2\right ]$| according to Example 2.1. Example 2.2 clarifies an important difference between sector dominance and vector dominance. Moreover, Examples 2.1 and 2.2 are based on the mean-variance approach (see Markowitz, 1952) justified by assuming Gaussian distributed returns. However, we argue that a proper dominance rule for ranking market sectors should depend on the most reliable models. In particular, in financial literature, the mean-variance approach has been extended to a mean-risk approach, where used risk measures |$\sigma $| are consistent with the choices of risk-averse investors (i.e., if |$E(X)=E(Y)$| and |$X$| SSD |$Y$|⁠, then |$\sigma _X\leqslant \sigma _Y$|⁠) such as the conditional value-at-risk |$CVaR_{\gamma }(X)=-\frac{1}{\gamma }\int _0^{\gamma }F_X^{-1}(q) \,\textrm{d}q$| (see Artzner et al., 1999), the mean absolute deviation |$MAD_{X}=E(|X-E(X)|)$| (see Konno & Yamazaki, 1991) and many others (see Szegö, 1994). In the following sections, we propose a distributional approach to the dominance problem that is based either on stable Paretian (sub-Gaussian) or Student’s t distributional assumptions. Recall that this class of elliptically symmetric distributions does not capture the skewness of stock portfolio returns. In response to these challenges, Ortobelli et al. (2016b) define a new stochastic order, weaker than the SSD, and prove that it holds if some conditions related to the skewness parameters are verified. Furthermore, in the context of portfolio choices, it is well known that diversification tends to introduce negative skewness (see, e.g., Simkowitz & Beedles (1978)). Thus, a natural extension of this research would be an MSD rule that also considers skewness and its relation to portfolio diversification. 2.1 Stable Paretian sub-Gaussian distribution Let |$r_{t+1}=[r_{1,t+1},\ldots ,r_{n,t+1}]^{\prime}$| denote the vector of gross returns on date |$t+1$|⁠. We define the i-th gross return between time |$t$| and time |$t+1$| as |$r_{i,t}=\frac{P_{t+1,i}}{P_{t,i}}$| and the i-th log return as |$\ln (r_{i,t})$|⁠, where |$P_{t,i}$| is the price of the i-th asset at time |$t$|⁠. In empirical finance literature, it is well known that asset returns follow heavy-tailed distributions. In this framework, it can be assumed that the returns are in the domain of attraction of a stable law such that $$\begin{equation} Pr(|r_{i}|>t)\approx t^{-\alpha_{i}}L_{i}(t)\,\,\,\textrm{as}\,\,t\rightarrow\infty, \end{equation}$$(2.1) where |$0<\alpha _{i}<2$| and |$L_{i}(t)$| is a slowly varying function at infinity, i.e., $$\begin{equation} \lim_{t\rightarrow\infty}\frac{L_{i}(ct)}{L_{i}(t)}\rightarrow1\,\,\,\,\,\,\,\,\,\,\forall \,c>0, \end{equation}$$(2.2) (see Rachev & Mittnik, 2000 and the references therein). The tail condition implies that any gross return |$r_{i}$| follows an |$\alpha $|-stable distribution |$S_{\alpha }(\sigma ,\beta ,\mu )$|⁠, where |$\alpha \in (0,2]$| is the so-called stability index (which specifies the asymptotic behaviour of the tails), |$\sigma>0$| is the dispersion parameter, |$\beta \in \lbrack -1,1]$| is the skewness parameter and |$\mu \in \mathbb{R}$| is the location parameter. It can typically be assumed that the vector of returns |$r$| is |$\alpha $|-stable sub-Gaussian distributed. Recall that an |$\alpha $|-stable sub-Gaussian distribution is an elliptical distribution that is symmetric around the mean (because |$\beta =0$|⁠). Its characteristic function has the following form: $$\begin{equation*} \varPhi_r(u)=E(\mathrm{exp}(iu^{\prime}r))=\mathrm{exp}(-(u^{\prime}Qu)^{\frac{\alpha}{2}}+iu^{\prime}\mu),\end{equation*}$$ where |$Q=[q_{ij}]$| is a positive definite dispersion matrix and |$\mu $| is the mean vector (when |$\alpha>1$|⁠). Sub-Gaussian returns are elliptically distributed for a fixed value of |$\alpha $|⁠; when |$\alpha =2$|⁠, we obtain the Gaussian distribution. Therefore, according to Ortobelli et al. (2016a), it is important to define a suitable order of preferences to deal with distributions having different |$\alpha $| values. In the following, we determine a ranking criterion to compare sub-Gaussian distributions according to SSD. It has been proven that SSD can be verified by comparing the values of the stability, dispersion and location parameters. Thus, the following theorem (see Ortobelli et al., 2016a) can be recalled: Theorem 2.1 Let |$X_1\sim S_{\alpha _1}(\sigma _1,0,\mu _1)$|⁠, and |$X_2\sim S_{\alpha _2}(\sigma _2,0,\mu _2)$|⁠. Suppose that |$\alpha _1>\alpha _2>1$|⁠, and |$\sigma _1\leqslant \sigma _2$|⁠. If |$\mu _1\geqslant \mu _2$|⁠, then |$X_1$| SSD |$X_2$|⁠. The results of Theorem 2.1 can be extended to a multivariate setting, generalizing the multivariate mean-dispersion approach presented in Example 2.1 by considering the asymptotic behaviour of the tail distributions. In particular, we obtain the following corollary. Corollary 2.1 Let |$A$| and |$B$| be two sectors with |$n$| and |$m$| assets, respectively. Assume that the returns |$r_A$| of sector |$A$| are jointly |$\alpha _A$|-stable sub-Gaussian distributed with vector mean |$\mu _A$| and dispersion matrix |$Q_A$| and the returns |$r_B$| of sector |$B$| are jointly |$\alpha _B$|-stable sub-Gaussian distributed with vector mean |$\mu _B$| and dispersion matrix |$Q_B$|⁠. Assume that |$\alpha _A\geqslant \alpha _B>1$| and for any portfolio of returns |$y^{\prime}r_A$| of sector |$A$|⁠, there exists a portfolio of returns |$x^{\prime}r_B$| of sector |$B$| such that |$y^{\prime}Q_Ay \leqslant x^{\prime}Q_Bx$| (and at least one inequality holds strictly, i.e., either |$y^{\prime}Q_Ay < x^{\prime}Q_Bx$| or |$\alpha _A>\alpha _B$| ). Then, |$y^{\prime}\mu _A \geqslant x^{\prime}\mu _B$| implies |$y^{\prime}r_A$| SSD |$ x^{\prime}r_B$| (i.e., sector |$A$| weakly SSD dominates sector |$B$|⁠). Proof. See Appendix A. The dominance obtained by stable sub-Gaussian distribution could be stronger than other mean-risk approaches, as proved in the following corollary. Corollary 2.2 Let |$X_1\sim S_{\alpha _1}(\sigma _1,0,\mu _1)$| and |$X_2\sim S_{\alpha _2}(\sigma _2,0,\mu _2)$|⁠. If |$\alpha _1\geqslant \alpha _2\geqslant 1$|⁠, |$\mu _1\geqslant \mu _2$| and |$MAD_{X_1}\leqslant \frac{\varGamma (1-\frac{1}{\alpha _1})}{\varGamma (1-\frac{1}{\alpha _2})}MAD_{X_2}$|⁠, then |$X_1$| SSD |$X_2$|⁠. Proof. See Appendix A. Observe that when |$X$| and |$Y$| are two random variables with the same mean and different MAD (⁠|$MAD_{X}\neq MAD_{Y}$|⁠), then |$X$| SSD |$Y$| implies that |$MAD_{X}< MAD_{Y}$|⁠. However, we cannot say the converse. Thus, Corollary 2.2 gives conditions to invert the relationship between stable symmetric distributions. This study explores another family of symmetric elliptical distributions that also deal with the observed fat tails of asset returns. We propose ranking criteria that aim to compare Student’s t distributions according to SSD. Specifically, we prove that in the case of Student’s t distributions, the degree of freedom is crucial for establishing multivariate stochastic ordering. 2.2 Student’s t distribution Several studies have used Student’s t distributions to model asset log returns (see, e.g., Blattberg & Gonedes, 1974; Fergusson & Platen, 2006; Markowitz & Usmen, 1996; Praetz, 1972). The following equation gives the Student’s t density function: $$\begin{equation} f_{X}(x)=\frac{1}{\sigma\sqrt{\nu\pi}}\frac{\varGamma((\nu+1)/2)}{\varGamma(\nu/2)}\left(1+\frac{(\frac{x-\mu}{\sigma})^2}{\nu}\right)^{-\frac{\nu+1}{2}}, \end{equation}$$(2.3) where |$\varGamma (z)=\int _0^{\infty }x^{z-1}e^{-x}\,\textrm{d}x$| is the gamma function, |$\mu $| is the location parameter, |$\sigma $| is the scale parameter and |$\nu $| is a positive parameter called the degree of freedom. The Student’s t distribution has the following properties: (i) all moments of order |$m<\nu $| are finite (i.e., the mean exists for |$\nu>1$|⁠, and the variance is finite only for |$\nu>2$|⁠); (ii) when |$\nu =1$|⁠, the Student’s t density function is the Cauchy density or ‘Lorentzian’ function; (iii) it has fatter tails than the density function of a normal distribution; and (iv) for values |$\nu \rightarrow \infty $|⁠, it converges asymptotically to the normal distribution. For our purposes, the most important parameter of this distribution is the degree of freedom, denoted by |$\nu $| (especially when |$\nu>2$|⁠). While similarities between the Student’s t and an |$\alpha $|-stable sub-Gaussian distribution exist, these two distributions have very different theoretical and empirical implications (see, e.g., Blattberg & Gonedes, 1974). On the one hand, the stability property distinguishes the stable and Student’s t distributions; the sum of identically and independently distributed (i.i.d) |$\alpha $|-stable distributions is still |$\alpha $|-stable distributed, while the sum of i.i.d Student’s t distributions converges to a Gaussian distribution for |$\nu>2$| and to |$\nu $|-stable distribution for |$\nu <2$|⁠. On the other hand, the Student’s t model allows the use of well-known density functions while the stable density functions are known in only a few cases (e.g., |$\alpha =\frac{1}{2}$|⁠, |$\alpha =1$| and |$\alpha =2$|⁠). Consequently, the likelihood function of the Student’s t model can be expressed in closed form, and maximum likelihood (ML) estimates for all parameters of the model can be easily obtained. For the stable distribution, ML estimates are generally obtained by inverting the Fourier transform (see, e.g., Rachev & Mittnik, 2000). Student’s t distribution belongs to the elliptically distributed family (see, e.g., Schoutens (2003) and Samorodnitsky & Taqqu, 1994). It is therefore important to define a suitable order of preferences to deal with distributions for different values of |$\nu $|⁠. The following determines a ranking criterion that aims to compare univariate Student’s t distributions according to SSD. The fact that SSD can be verified by comparing the degree of freedom, dispersion and location parameters is shown below. Theorem 2.2 Let |$X_1\sim T_{\nu _1}(\sigma _1,\mu _1)$| and |$X_2\sim T_{\nu _2}(\sigma _2,\mu _2)$|⁠. Suppose |$\nu _1\geqslant \nu _2$| and |$\sigma _1\leqslant \sigma _2$| with at least one strict inequality. If |$\mu _1\geqslant \mu _2$|⁠, then |$X_1$| SSD |$X_2$|⁠. Proof. See Appendix A. The results of Theorem 2.2 can be extended to a multivariate setting that considers the asymptotic behaviour of the tail distributions. This extension yields the following MSD among market sectors. Corollary 2.3 Let |$A$| and |$B$| be two sectors with |$n$| and |$m$| assets, respectively. Assume that the returns |$r_A$| of sector |$A$| are jointly Student’s t distributed with vector mean |$\mu _A$| and dispersion matrix |$Q_A$| and the returns |$r_B$| of sector |$B$| are jointly Student’s t distributed with vector mean |$\mu _B$| and dispersion matrix |$Q_B$|⁠. Assume that |$\nu _A\geqslant \nu _B\geqslant 2$| and for any portfolio of returns |$y^{\prime}r_A$| of sector |$A$|⁠, there exists a portfolio of returns |$x^{\prime}r_B$| of sector |$B$| such that |$y^{\prime}Q_Ay \leqslant x^{\prime}Q_Bx$| (and at least one inequality holds strictly, i.e., either |$y^{\prime}Q_Ay < x^{\prime}Q_Bx$| or |$\nu _A>\nu _B$| ). Then, |$y^{\prime}\mu _A \geqslant x^{\prime}\mu _B$| implies |$y^{\prime}r_A$| SSD |$ x^{\prime}r_B$| (i.e., sector |$A$| weakly SSD dominates sector |$B$|⁠). Proof. See Appendix A. Observe that the scale parameter |$\sigma $| does not correspond to a standard deviation (SD) of the normal distribution. Similar to the stable distribution, the dominance obtained under the Student’t assumption implies different mean-variance ordering, according to the following Corollary. Corollary 2.4 Let |$X_1\sim T_{\nu _1}(\sigma _1,\mu _1)$| and |$X_2\sim T_{\nu _2}(\sigma _2,\mu _2)$|⁠. If |$\nu _1\geqslant \nu _2\geqslant 2$|⁠, |$\mu _1\geqslant \mu _2$| and |$\textrm{Std}_1\leqslant \sqrt{\frac{\nu _1(\nu _2-2)}{\nu _2(\nu _1-2)}}\textrm{Std}_2$|⁠, then |$X_1$| SSD |$X_2$|⁠. Proof. See Appendix A. Observe that when |$X$| and |$Y$| are two random variables with the same mean and different finite variances, then |$X$| SSD |$Y$| implies |$var(X)< var(Y)$|⁠. Moreover, we cannot state the converse, except under the conditions of Corollary 2.4. 3. Empirical analysis This section applies the multivariate weak dominance rules to compare S&P 500 sectors empirically. The results of these methods are compared with those obtained by the weak concave multivariate order described in Example 2.1 and Corollaries 2.1 and 2.3. Empirically, it is well known that the conditions for the strong form of stochastic dominance are not easily verified in practice. Therefore, we employ the multivariate weak dominance rule under the assumption that the vector of returns for each sector is either Gaussian, Student’s t or |$\alpha $|-stable sub-Gaussian distributed. In general, the asset returns are typically heavy-tailed (see, e.g., Rachev & Mittnik, 2000); thus, determining the tail probabilities is key. The general aim is to examine to what extent the dominance observed among market sectors impacts the optimal choices of a non-satiable risk-averse investor under different distributional hypotheses of asset returns. Specifically, we apply the stochastic dominance rules stated in the previous section to empirically compare optimal choices of the components of the S&P 500 index. Doing so, we also verify the validity of some classical elliptical distributional assumptions. The goal of this empirical section is twofold. First, it verifies if sectors not chosen by non-satiable risk-averse investors are really dominated (in the SSD sense) by other sectors. Second, it evaluates if one among three elliptical distributional assumptions is more suitable than the others. Both objectives are important to better address investors’ choices. This section is divided into three subsections to address these objectives. The first subsection examines the statistical characteristics of the returns for each sector individually and all sectors collectively. The second subsection verifies the proposed SSD dominance rules during the period 2004–2017, when short sales were not allowed. In particular, Subsection 3.2 examines the so-called mean-dispersion EF under different distributional hypotheses (Gaussian, stable sub-Gaussian and Student’s t) and computes the SSD EFs. Then, it compares the EFs to determine whether the conditions for the weak ICVM hold. Next, it uses a non-parametric approach to test the efficiency of the frontiers obtained under three different distributional assumptions and, using the test proposed by Arvanitis et al. (2019), it also evaluates if the dominant sectors contain a market-efficient set. Finally, the third subsection examines whether an active investor can use these findings to achieve economically and statistically significant out-of-sample performance compared with the S&P 500 index. 3.1 Data set description We consider daily returns of the S&P 500 specific economic sectors for the period from December 2004 to January 2017, for a total of 3037 observations. A specific dynamic data set is obtained by considering the S&P 500 sector components available from the Thomson Reuters DataStream database (431 assets). In particular, according to the Global Industry Classification Standard, we distinguish the following specific economic sectors: (1) information technology (IT), (2) financials (FI), (3) health care (HC), (4) consumer discretionary (CD), (5) industrials (IN), (6) consumer staples (CS), (7) energy (EN), (8) utilities (UT), (9) real estate (RE) and (10) materials (MA). We omit the telecommunication services sector because it has very few assets and some of them are not available over the entire period. In practice, for both stable and Student’s t distributions, we estimate the unknown parameters using the ML method. There are two main approaches to the problem of ML estimation for stable Paretian cases. Modern ML estimation techniques for stable distributions utilize either (i) the fast Fourier transform method to approximate the stable density function (see, e.g., Rachev & Mittnik, 2000) or (ii) the direct integration method (see Nolan, 2001). Both approaches are comparable in terms of efficiency, and variances in performance result from different approximation algorithms. Our analysis uses the first approach. As observed by Kring et al. (2009), the estimation of |$q_{ij}$| (where |$i\neq j$|⁠) can be seen as a difference between squared scale parameters of returns |$(r_{i}\pm r_{j})/2$|⁠, i.e., |$q_{ij}=\sigma ^2_{(r_i+r_j)/2}-\sigma ^2_{(r_i-r_j)/2}$|⁠. Moreover, it should be stressed that the common stability index is estimated using the sample mean of the stability parameters of |$10,000$| random portfolios (i.e., |$\alpha =\frac{1}{10,000}\sum _{i=1}^{10,000}\alpha _i$|⁠), as suggested by Ortobelli et al. (2004). Similarly, we obtain the common degrees of freedom (df) using the sample mean of |$10,000$| random portfolios. Table 1 reports the average ML estimates of the stable Paretian and Student’s t parameters and summary statistics for the average return on assets for each sector individually and for all sectors collectively: mean, SD, skewness and kurtosis. Table 1 Average statistics for the log returns of different sectors and ML estimates of the stable Paretian and Student’s t parameters Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Open in new tab Table 1 Average statistics for the log returns of different sectors and ML estimates of the stable Paretian and Student’s t parameters Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Open in new tab According to Simkowitz & Beedles (1978), log returns (in Table 1) present a slight negative skewness (as the asymmetry parameter |$\beta $| and skewness are generally negative close to zero). Moreover, the parameter estimates suggest the presence of a quite remarkable tail-weightiness (the kurtosis exceeds three, while the stability parameter |$\alpha $| is less than two). The FI sector returns present, on average, the lowest mean |$\mu $| and |$alpha$| and the highest SD and kurtosis. Computing the Jarque–Bera statistic with a 95% confidence level to test whether asset returns follow a normal distribution shows that the Gaussian hypothesis is always rejected. On average, the Kolmogorov–Smirnov test suggests rejecting approximately |$17.12\%$| of the Student’s t distributional hypothesis and |$13.44\%$| of the stable Paretian hypothesis. Thus, it is reasonable to assume from the obtained results that the asset returns of each sector are jointly elliptical distributed because we observe strong evidence of tail weightiness and there is little evidence of skewness. 3.2 SSD sector dominance Generally, there are three distinct steps to verify SSD dominance rules among sectors under the different distributional assumptions. The first step fits the mean-dispersion EF by considering a unique index of stability for the stable distributional assumption and a unique value for the df of the multivariate Student’s t distribution. The second step identifies the existence of SSD dominance between sectors whose components have been chosen and sectors whose assets are not chosen among the assets of the fitted EFs of the first step. Clearly, sector dominance always occurs under the Gaussian assumption, based only on the mean-variance Markowitz rule. However, for the stable and Student’s t distributional assumptions, different indexes of stability and df for the chosen and non-chosen sectors must also be considered to verify existing SSD sector dominance. Thus, it first values a common index of stability (df) of the non-chosen sectors (obtained in the first step) and then fits the mean-dispersion SSD EF of these assets. Second, it values a common index of stability (df) of the sectors chosen in the first step. If the common index of stability (df) of the chosen sectors is not larger than the index of stability (df) of the non-chosen sectors, a common index of stability (df) of the chosen sectors is computed using only those sectors whose index of stability (df) is estimated to be greater than the common index of stability (df) of non-chosen sectors. Then, the mean-dispersion SSD EF of the sectors chosen in the first step is fit. Then, according to the dominance rules of Corollaries 2.1 and 2.3, weak sector dominance is verified. In particular, it verifies if, for any portfolio belonging to the fitted EF of non-chosen sectors, there exists a portfolio of the fitted EF of the chosen sectors in which SSD dominates. Notably, sector dominance (according to the dominance rules of Corollaries 2.1 and 2.3) is always present in some of the chosen sectors with respect to unchosen ones, even under elliptical non-Gaussian distributional assumptions (see Table 3). The last step tests if, for any efficient portfolio of the non-chosen sector, there exists a portfolio of the chosen sectors that SSD dominates according to a non-parametric approach (see, Davidson & Jean-Yves, 2000). This third step should reveal when the observed sector dominance can be guaranteed to hold in reality. Thus, in some sense, this last step also serves to validate the distributional assumption. All three steps have been applied every month (e.g., 20 trading days) from 1 December 2007 until the end of January 2017. Thus, according to the first and second steps, the mean-dispersion SSD EFs are estimated using the assets that were active during the last three years (750 daily historical observations).1 Therefore, at every recalibration time, the EF is fitted by solving the optimization problem for 100 levels of the mean as follows: $$\begin{equation} \begin{array} [c]{c} \min_{x}x^{\prime}Qx\\ \text{s.t.}\\ \sum_{i=1}^{n}x_{i}=1;\,x^{\prime}\mu=m\\ x_{i}\geqslant0;\ \ i=1,\ldots,n, \end{array} \end{equation}$$(3.1) where |$x^{\prime }Qx$| is the risk measure (variance or dispersion measure) associated with portfolio |$x^{\prime }r$| and |$n$| is the number of all assets, which are grouped by sector. The asset returns of sectors that are unchosen in the SSD mean-dispersion EF are identified. Subsequently, whether an MSD between chosen and unchosen sectors exists is verified. Then, a non-parametric approach is used to test the efficiency of the parametric dominance (see, e.g., Davidson & Jean-Yves, 2000). It is assumed that a sector |$A$| parametrically dominates a sector |$B$| to give a meaningful intuition of a non-parametric test. Then, whether for any optimal portfolio of sector B (fitted among 100 levels of the mean), there exists a portfolio of sector |$A$| that non-parametrically dominates those of sector |$B$| (based on their empirical distributions) is verified. Finally, the non-parametric test proposed by Arvanitis et al. (2019) is used to evaluate when the dominant sectors contain the market-efficient set. The results of this empirical analysis are reported in Tables 2 and 3 and Figures 1–3. Table 2 reports the mean of the percentage invested in each sector over different periods (e.g., subprime crisis, EU credit crisis and post-crisis period) using either the Gaussian, Student’s t or stable distributional assumption. Table 2 The mean of the percentage invested in each sector over different periods considering three distributional assumptions From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 Open in new tab Table 2 The mean of the percentage invested in each sector over different periods considering three distributional assumptions From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 Open in new tab Table 3 Weak SSD sector dominance . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| Open in new tab Table 3 Weak SSD sector dominance . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| Open in new tab From Table 2, we observe that |$\bullet $| The percentage invested in each sector differs slightly from one distributional hypothesis to another. The IT, HC, CD and CS sectors have higher weights than their FI, IN, EN, UT, RE and MA counterparts over the entire period examined for each of the Gaussian, Student’s t or stable distribution hypotheses. These results can be explained by the effects of financial crises on these sectors. |$\bullet $| The IT sector has a higher weight post-crisis than during other periods of market distress (subprime and European credit crisis), which is not the case for the CD sector, which shows losses of significant weights after the European credit crisis. |$\bullet $| The HC and CS sectors obtain higher weights during the subprime crisis and lower weights throughout the European credit crisis (even though they have the highest percentage). |$\bullet $| After suffering from the subprime mortgage crisis, which was triggered by a substantial decline in home prices after the collapse of a housing bubble, the RE sector gains more weight throughout the European credit crisis. Example of mean-variance SSD dominance (June 2010). Fig. 1. Open in new tabDownload slide Fig. 1. Example of mean-variance SSD dominance (June 2010). Open in new tabDownload slide Example of alpha-mean-dispersion SSD dominance (January 2008). Fig. 2. Open in new tabDownload slide Fig. 2. Example of alpha-mean-dispersion SSD dominance (January 2008). Open in new tabDownload slide Example of the degree of freedom-mean-dispersion SSD dominance (January 2009). Fig. 3. Open in new tabDownload slide Fig. 3. Example of the degree of freedom-mean-dispersion SSD dominance (January 2009). Open in new tabDownload slide These results highlight the significant differences between the Gaussian and the other two distributions. In particular, the weights obtained from Student’s t distribution are generally located between those of the stable and Gaussian hypotheses. This could be explained by the previously stressed fact (see Section 2); for higher |$\nu $|⁠, the Student’s t random variable will tend to a normal distribution, while for lower values of |$\nu $|⁠, it presents heavy tails as the stable distribution. The differences among these distributions have already been examined by several studies (see, e.g., Ortobelli et al., 2004), which showed the superiority of heavy-tailed distributions in portfolio theory (see Rachev & Mittnik, 2000). The empirical investigation also shows that the percentage invested in each sector appears to be unstable over time. This observation emphasizes the importance of regularly recalibrating the portfolio. Table 3 reports the number of times required to achieve zero investment, number of times each sector is parametrically dominated by other sectors in the SSD sense, non-parametric SSD dominance, single (ratio between non-parametric and parametric SSD for each sector individually) percentage of tested SSD dominance and the percentages of times we obtain that the non-parametric SSD sector dominance is able to contain all of the SSD efficient set, according to the test proposed by Arvanitis et al. (2019). From Table 3, we observe that |$\bullet $| The number of times (in months) that zero percent invested is high for the FI, IN, EN and RE sectors; this fact is also confirmed by the number of times these sectors are dominated by the remaining sectors. |$\bullet $| Under the three assumptions, the number of times that parametric SSD verifies and zero investment happens are equal, which is not the case for the non-parametric evaluation. |$\bullet $| The difference between parametric and non-parametric SSD is more evident under the Gaussian hypothesis than using the stable and Student’s t distributions. |$\bullet $| Single percentages of tested SSD dominance are higher for stable and Student’s t distributions than for the Gaussian assumption. |$\bullet $| The percentage of times that non-parametric SSD dominance is able to contain all the SSD efficient set according to the test proposed by Arvanitis et al. (2019) is sufficiently high for all three distributional approaches (always higher than 50%). In particular, the stable and Student’s t distributions seem more capable of capturing dominance in the analysed cases. Notably, the number of times a given sector is dominated by the rest of the sectors is critically dependent on distributional assumptions. For instance, the FI sector is dominated 36 times when the stable sub-Gaussian assumed, while it is dominated 33 and 29 times for the Student’s t and Gaussian distributions, respectively. In contrast, the RE sector is dominated 22 and 13 times when the Gaussian and Student’s t distributions are respectively considered, and it is never dominated under the stable sub-Gaussian hypothesis. Comparing the overall number of dominances from the three different rules (i.e., mean-variance, df-mean-dispersion and alpha-mean-dispersion) results in 164, 151 and 141 SSD dominances, respectively. These results show significant differences between the three approaches; thus, distributional assumptions greatly affect outcomes. Moreover, according to the test proposed by Arvanitis et al., it could be deduced that a sector analysis is sufficiently able to determine those sectors that weakly SSD dominate all markets (i.e., they contain the SSD efficient set), in particular, when the stable and Student’s t distributional assumptions are considered. Thus, the proposed analysis can also be used to reduce the dimensionality of the portfolio problem. Interestingly, Tables 2 and 3 show the extent to which the dominance observed among some sectors impacts the composition of an optimal portfolio. While dominated sectors generally have lower percentages of optimal portfolio weights, dominant sectors earn the highest percentages of optimal portfolio weights. These results confirm the fact that the dominance observed among some sectors persists into optimal portfolio composition. This observation is of great practical importance because it allows investors to optimize their portfolios by considering only a few dominant sectors; it could also be significant in large-scale portfolio selection problems. We deduce from this ex-ante analysis that the best performing sectors are IT, HC, CD and CS. This result is also confirmed by Figures 2–3, which report three examples of the mean-variance, alpha-mean-dispersion and degree of freedom-mean-dispersion dominance. From Figures 2–3, it is worth noting that the dominated sectors have lower values for |$\alpha $| and |$\nu $| than the dominant sectors when we use stable and Student’t distributions, respectively. For instance, Figure 2 shows the case when the FI, IN and MA sectors are weakly dominated by the other sectors (such as IT, HC, CD and CS) in terms of the ICVM order. 3.3 An ex-post comparison among ‘efficient portfolios’ This section compares the ex-post wealth of portfolios optimised under various distributional hypotheses. The empirical analysis aims to determine whether an active investor can use these portfolio models to outperform the S&P 500 index and two other benchmarks: one based on the uniform strategy (investing the same percentage of wealth in each asset) and one based on a sector momentum strategy (which rebalances into the sectors that performed best in the previous year). Accordingly, starting from 1 December 2007, the mean-dispersion EFs under the three distributional assumptions for the asset returns are fitted monthly (every 20 trading days). In particular, 100 optimal portfolios obtained by minimizing the variance or dispersion measures for a fixed mean (according to problem (3.1)) and varying the mean from the mean of the global minimum dispersion portfolio are considered. It is assumed that each risk-averse investor maintains the same ex-ante level of mean returns over time when the three types of EFs are approximated. Thus, at each recalibration time (every 20 trading days), first a common index of stability (in the stable case) and a common value for the df (in the Student’s t case) are estimated and, as in Section 3.2, the ML estimator for all of the parameter estimates is used. Then, the EFs for the three distributional assumptions are fitted. Finally, out-of-sample wealth obtained starting with an initial wealth equal to one and using the solutions of optimal portfolios for each fixed level of mean are computed. For each distributional assumption, the wealth evolution of 100 portfolio strategies (one corresponding to each level of the mean) are obtained, and a total of 2289 ex-post daily portfolio observations are made. The same method is used to obtain the ex-post wealth of the uniform strategy and the sector momentum strategy. Table 4 Number of the portfolio strategies that SSD dominate the benchmarks or other portfolio strategies . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable Open in new tab Table 4 Number of the portfolio strategies that SSD dominate the benchmarks or other portfolio strategies . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable Open in new tab Table 5 Statistics on the benchmarks and best strategies . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 Open in new tab Table 5 Statistics on the benchmarks and best strategies . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 Open in new tab Once the ex-post strategies are obtained, the non-parametric test (see Davidson & Jean-Yves, 2000) is used to evaluate whether SSD dominance applies among the returns of these ex-post strategies and the market benchmarks. We also evaluate some statistics of the ex-post returns obtained with the benchmark strategies and the three distributional approaches. In particular, we evaluate the risk measures |$VaR_{5\%}$| and |$CVaR_{5\%}$| (where |$VaR_q(x^{\prime}r)=-F_{x^{\prime}r}^{-1}(q)=-inf\lbrace v|P(x^{\prime}r \leqslant v>q)\rbrace $| and |$CVaR_{\gamma }(x^{\prime}r)=\frac{1}{\gamma }\int _0^{\gamma }VaR_{q}(x^{\prime}r) \,\textrm{d}q$|⁠), the risk-reward performance measures the Sharpe ratio |$SR(x^{\prime}r)=E(x^{\prime}r-r_f)/\sigma _{x^{\prime}r}$| and Sortino ratio |$ \textrm{SoR}(x^{\prime}r)=E(x^{\prime}r-r_f)/(E((y-x^{\prime}r)_{+}^{2}))^{1/2}$| (where |$r_f$| is the risk-free return—here considered null—and |$\sigma _{x^{\prime}r}$| is the portfolio standrad deviaition; the function |$(v)_{+}^{2}=( \textrm{max}(v,0))^2$| and |$y$| is the minimum acceptable return level for the investors, which is null in our analysis, i.e., |$y=0$|⁠, (for further details, see Sharpe, 1994 and Sortino & Price, 1994)), and the certainty equivalent gross returns (C.E.) for the power utility functions |$U(x^{\prime}r)=((1+ x^{\prime}r)^{1-\rho }-1)/(1-\rho ))$| with relative risk aversion (RRA) |$\rho =2,4,8$|⁠. We evaluate these statistics for the benchmarks and average the statistics of the best 16, 21 and 24 strategies for the Gaussian, Student’s t and stable approaches where SSD dominates the S&P 500 index (according to Table 4). The results of this ex-post analysis are reported in Tables 4 and 5. Table 4 reports the number of portfolio strategies (out of a possible 100) of each distributional approach where SSD dominates the benchmark strategies or other portfolio strategies. Table 5 reports the statistics of the benchmarks and of the best Gaussian, Student’s t and stable strategies. From this comparison, we can deduce that |$\bullet $| Benchmark strategies (S&P 500 index, uniform strategy and sector momentum strategy) are not SSD comparable by themselves, even if there exist deep differences between them. All benchmark strategies present risk measures VaR and CVaR higher than distributional strategies. However, the risk-reward performance and certainty equivalent values of the Uniform strategy and sector momentum strategy are always higher than those for the Gaussian approach and slightly lower than those for the other distributional approaches. For the Gaussian, Student’s t and stable approaches, respectively, 16, 21 and 24 strategies SSD dominate the S&P 500 index, while the number of distributional strategies that SSD dominate the uniform strategy and the sector momentum strategy is much smaller. Moreover, according to Pflug et al. (2012) and DeMiguel et al. (2009), the uniform strategy performs very well because 13, 9 and 8 respective strategies of the Gaussian, Student’s t and stable approaches are actually dominated by the uniform strategy. Similarly, the sector momentum strategy SSD dominates 9, 6 and 8 of the Gaussian, Student’s t and stable strategies, respectively, confirming that sector outperformance is often persistent. |$\bullet $| The stable and the Student’s t approaches perform better than the Gaussian approach in terms of SSD dominance risk measures, risk-reward performance and certainty equivalent values, even in the ex-post analysis. Thus, we confirm that the distributional hypotheses seriously influence the outcomes. The fact that normal distribution may yield misleading results has already been documented in several studies (see, e.g., Rachev & Mittnik, 2000). With respect to the Student’s t assumption, our empirical findings show that it may yield significant improvements from the classical mean-variance approach, especially if compared with the alpha-mean-dispersion criterion. Therefore, it can be asserted that the Student’s t distribution may be more reliable and the stable sub-Gaussian distribution may present more appropriate dominance rules. |$\bullet $| All of these distributional approaches (Gaussian, Student’s t and stable) present several portfolio strategies that outperform the S&P 500 index and some strategies that outperform the uniform strategy and the sector momentum strategy. This result emphasizes the managerial relevance and the impact of sector dominance on optimal investors’ choices in the US stock market. Overall, these results confirm and strongly support the assertion that weak MSD-based methodologies can effectively compare and rank different market sectors. Interestingly, the observed dominance of the optimal choices has the power to explain the percentage concentration of invested stocks. Finally, the significant differences observed between the three approaches suggest that portfolio selection models must account for the asymptotic (tail) behaviour of returns. 4. Conclusions This study introduces a methodology for comparing portfolios homogeneous with respect to market sectors relying on MSD principles. In this context, a possible application aimed at ranking S&P 500 sectors (namely, IT, FI, HC, CD, IN, CS, EN, UT, RE and MA) is proposed. In practice, the proposed dominance rules can be used by non-satiable risk-averse investors to identify the best market sectors in which to invest. The results show that this method yields remarkably different outcomes from the more traditional mean-variance approach. The non-parametric tests suggest that both stable symmetric and Student’s t distributions are preferable to the Gaussian approach for identifying MSD between sectors. The primary contribution of the empirical comparison is to examine the impact of the distributional assumptions on asset allocation decisions. In particular, the significant differences observed between the mean-variance and the stable Paretian approaches suggest that portfolio selection models must consider the asymptotic (tail) behaviour of returns. Therefore, this paper proposes another family of heavy-tailed distributions (Student’s t) and shows that SSD can be verified by comparing the values of the degree of freedom, dispersion and location parameters. It also discusses the implications of various distributional hypotheses on optimal portfolio choices and highlights the notable relationship between stochastic dominance among market sectors and the fraction of optimal portfolios. Finally, it emphasizes the managerial relevance of our results with an ex-post analysis that compare the different distributional approaches with three alternative benchmarks. 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Let |$Z_i=\frac{X_i-\mu _i}{\sigma _i}$| (for |$i=1,2$|⁠) be such that |$Z_1\sim T_{\nu _1}(1,0)$| and |$Z_2\sim T_{\nu _2}(1,0)$|⁠. Analytical and empirical studies ensure that the distribution functions of |$Z_1$| and |$Z_2$| have a single crossing point. Conceptually, the distribution function can be written as follows: $$\begin{equation} F_{\nu}(z)=\frac{1}{2}\left(1+\textrm{sgn}(z)\left(1-I_{\left(\frac{\nu}{z^2+\nu}\right)}\left(\frac{\nu}{2},\frac{1}{2}\right)\right)\right), \end{equation}$$(A.1) where |$I_{(x)}(a,b)$| is the regularized Beta function (also known as the incomplete Beta function ratio) (see, e.g., Shaw (2006)). We can distinguish three particular cases: |$z=0$|⁠, we have |$F_{\nu }(0)=\frac{1}{2}$|⁠, |$\forall \,\, \nu>0$|⁠; |$\forall \,\, z>0$|⁠, we have |$F_{\nu }(z)=1-\frac{1}{2}I_{\left (\frac{\nu }{z^2+\nu }\right )}\left (\frac{\nu }{2},\frac{1}{2}\right )$|⁠; and |$\forall \,\, z<0$|⁠, we have |$F_{\nu }(z)=\frac{1}{2}I_{\left (\frac{\nu }{z^2+\nu }\right )}\left (\frac{\nu }{2},\frac{1}{2}\right )$|⁠. From the properties of the regularized Beta function (see, e.g., Johnson et al., 1995 and literature therein) and from the fact that |$0<\frac{\nu }{z^2+\nu }<1$| for any |$\forall \,\, z\neq 0$| and |$\nu>0$|⁠, it clearly follows that |$F_{\nu _1}(z)<F_{\nu _2}(z)$|⁠, |$\forall \,\, z<0$|⁠, and |$\nu _1>\nu _2$|⁠; |$F_{\nu _1}(z)>F_{\nu _2}(z)$|⁠, |$\forall \,\, z>0$|⁠, and |$\nu _1>\nu _2$|⁠; |$F_{\nu _1}(0)=F_{\nu _2}(0)=\frac{1}{2}$|⁠, |$\forall \,\, \nu>0$|⁠. Thus, the distribution functions cross only once, and |$F_{\nu _1}(z)$| is below |$F_{\nu _2}(z)$| to the left crossing point. Consequently (see, e.g., Hanoch & Levy (1969) and Rachev et al. (2008b)), it follows that |$Z_1$| SSD |$Z_2$| (which is equivalent to |$X_1$| SSD |$X_2$| as proved by Ortobelli & Rachev, 2001). This completes the proof of Theorem 2.2. Proof Corollary 2.3. Since any linear combination |$x\in \mathbb{R}^n$| of Student’s t vector |$r$| (⁠|$\mu \geqslant 2$|⁠) with location (mean) |$\mu $| and dispersion |$Q$| is still a Student’s t distribution with parameter |$ x^{\prime}\mu $| and dispersion parameter |$ \sigma = \sqrt{x^{\prime}Qx}$|⁠, then it follows the thesis as a consequence of Theorem 2.2. Proof Corollary 2.4. Since |$\forall $| |$\nu \geqslant 2$|⁠, we have |$\textrm{Std}=\sigma \sqrt{\frac{\nu }{\nu -2}}$|⁠, we get the results of Corollary 2.4. © The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press http://www.deepdyve.com/lp/oxford-university-press/multivariate-stochastic-dominance-applied-to-sector-based-portfolio-Awt0UYff3H

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Robert Blattberg, Nicholas Gonedes (1974)
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The Journal of Business, 50
Sebastian Kring, S. Rachev, Markus Höchstötter, F. Fabozzi (2009)
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Publisher: Oxford University Press
Copyright: © The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
ISSN: 1471-678X
eISSN: 1471-6798
DOI: 10.1093/imaman/dpaa004
Publisher site: See Article on Publisher Site

Abstract

Abstract In this study, we investigate whether sector-weighted portfolios based on alternative parametric assumptions are consistent with multivariate stochastic dominance (MSD) conditions for a class of non-satiable risk-averse investors. Focusing specifically on stable symmetric and Student’s t distributions, we propose and motivate an MSD rule to determine a partial order among sectors, based on a comparison between (i) location, (ii) dispersion parameters and (iii) either stability indices (for stable symmetric distributions) or degrees of freedom (for Student’s t distributions). The proposed MSD rule is applied to the US equity market to evaluate whether and how the derived stochastic dominance conditions are relevant to investors’ decisions. The empirical study confirms that the proposed MSD rule is effective and that the tail behaviour of returns is relevant to the optimization of portfolios for non-satiable investors. 1. Introduction The introduction of multivariate stochastic dominance (MSD) rules among different market activity sectors is fundamental to understanding the market trend. Moreover, equity portfolios constructed based on financial benchmarks associated with specific economic sectors are emerging as an efficient way to take advantage of business sector cyclicality. In this study, we develop and apply a parametric MSD framework to sector-based portfolio selection problems. We focus on a very general class of investors, assumed to be risk-averse and non-satiable (Levy, 1992; Müller & Stoyan, 2002), and on given parametric assumptions of the assets’ return distributions. We use dominance among market sectors to express a MSD of a given order between portfolios, including stocks belonging to specific economic sectors. Essentially, this work aims to evaluate the impact of dominance among market sectors on optimal investors’ choices and understand whether optimal portfolio choices imply some manner of orderings between chosen and non-chosen assets (market sectors). Specifically, it examines the extent to which the dominance observed among some market sectors persists into the composition of an optimal portfolio. It endeavours to answer the following questions: (i) if non-Gaussian elliptical distributions (say, symmetric stable or Student’s t) are assumed, which multivariate rule should be used with respect to other parameters (index of stability and degree of freedom)? (ii) Does dominance among market sectors exist? (iii) What impact do distributional assumptions have on investors’ choices? Research on operationalizing stochastic dominance rules for parametric families of distributions goes back to Ali (1975) and many related studies (see, e.g., Kroll & Levy, 1986; Levy, 2006). Differently from these studies, we examine stochastic dominance rules for parametric multivariate families of distributions. In particular, we first propose a parametric distributional comparison, after which we test its efficiency using a non-parametric approach. Our main purpose is to identify the best sectors in which to invest and help investors and fund managers to construct their equity portfolios. Thus, we focus on the issue of ranking different market sectors from a non-satiable risk-averse investor’s point of view, a decision under uncertainty that can be addressed using different approaches, such as utility theory (see, Ingersoll, 1987) or game theory (see, Moretti et al., 2011). Furthermore, from a portfolio theory perspective, individual preferences can be summarized by maximum gain and minimum risk, and, in this setting, several reward-risk performance measures have been proposed in the literature to account for the characteristics of different asset returns (see, e.g., Szegö, 1994, Rachev et al. (2008a), Ortobelli et al., 2019. Unlike these approaches, this work relies on the theory of stochastic orderings. In a univariate context, stochastic dominance rules establish a partial order in the space of distribution functions by quantifying the concept of one random variable being ‘preferable’ to another. Because the decision problem here concerns multivariate random elements (i.e., market sectors), the purpose is to apply stochastic dominance rules that are simple and applicable to a multivariate framework. On the one hand, a large body of literature has presented and examined the implications of stochastic dominance in portfolio theory (see, among others, Bruni et al., 2012; Levy, 2006; Osuna, 2012; Post & Kopa, 2017; Roman et al., 2006). For instance, Roman et al. (2013) evaluated the effectiveness of enhanced indexation models based on second-order stochastic dominance (SSD). Later, Bruni et al. (2017) proposed a new type of approximate stochastic dominance rule for enhanced indexation. On the other hand, numerous theoretical and empirical studies have been developed to test portfolio efficiency based on stochastic dominance (see, e.g., Kopa & Post, 2015; Kuosmanen, 2004; Post, 2003; Post & Potì, 2017; Scaillet & Topaloglou, 2010). However, most existing multivariate statistical tests for stochastic efficiency (see, e.g., Post & Kopa, 2017) are not suitable to examine dominance among market sectors, primarily because these tests require a comparison between vectors of the same dimension and consider an order among the marginals, which is not realistic among market sectors. Recall that each market sector has a different number of assets from which some components may cover different periods. In a recent contribution, Arvanitis et al. (2019) introduced a model-free concept of stochastic spanning along with a test procedure for implementation based on sub-sampling and linear programming. Spanning evaluates and determines whether including new securities or relaxing investment constraints enhances the investment opportunity set for a general class of all risk-averse investors. According to behavioural finance literature (see, e.g., Levy & Levy (2002), Müller & Stoyan, 2002; Ortobelli et al., 2017a,b), we want to rank different market sectors from the point of view of different non-satiable risk-averse investors. Thus, the research question in this work aligns with contemporaneous research on MSD analysis in portfolio theory (see, Arvanitis et al. (2019), Anyfantaki et al., 2018). In contrast to Arvanitis et al. (2019), this work compares choice sets of portfolios that do not have common portfolios. In this sense, the non-parametric test proposed by Arvanitis et al. cannot be used to compare different sectors, but it can still be used to test if the dominant sectors contain (span) the market-efficient set. Moreover, the stochastic dominance relations must consider return anomalies (e.g., heavy tails and skewness). In financial literature, it is well-known that asset returns present heavy tails and skewness. In particular, empirical research has established some stylized facts about asset returns: (a) clustering of volatility, (b) skewness and (c) fat tails (see, e.g., Kim et al., 2011). Different models have been proposed and experimentally tested to explain these empirical facts; for a survey of criticism, see Mandelbrot (1963), Fama (1970) and Rachev & Mittnik (2000). Stable Paretian and Student’s t distributions (both heavy-tailed) can explain the observed excess kurtosis of financial returns and represent an improvement over the Gaussian assumption (see, e.g., Blattberg & Gonedes, 1974). Conceptually, stable Paretian distributions are a natural generalization of the Gaussian distribution. In the non-Gaussian case, a stable distribution exhibits fatter tails than a normal distribution and is more peaked around its centre (i.e., leptokurtic). For portfolio analysis, the most critical feature of these distributions is that their variances (and higher moments) are not finite. Given that stable distributions depend on location and scale parameters, they are also identified by a parameter which specifies the shape of the distribution in terms of skewness and, more importantly, one that determines the asymptotic behaviour of the tails. Evidence provided by several authors (see, e.g., Ortobelli et al., 2016a) suggests that one could explicitly account for the observed ‘fat tails’ using the symmetric-stable distribution. This study considers another family of symmetric distributions that also accounts for the heavy tails of return distributions. This alternative is the Student’s t distribution, which is empirically and theoretically supported by several studies (see, e.g., Fergusson & Platen, 2006 and literature therein). Therefore, because of the importance of stable Paretian and Student’s t distributions for approximating random financial variables, we propose stochastic dominance rules that are aimed at ranking either Student’s t or sub-Gaussian random vectors. In so doing, we derive criteria that select the best market sector either to invest in or to consider for portfolio diversification purposes. In the empirical comparison, we examine the ex-ante and ex-post dominance between stock sectors, assuming that the returns are either Student’s t-distributed or in the domain of attraction of a stable sub-Gaussian law. Empirical evidence shows that the proposed methodology can assist decision makers in determining the optimal portfolio by considering the impact of stochastic dominance among market sectors. In particular, we show that the dominance observed among market sectors can explain the percentage concentration of invested stocks. Specifically, the dominant sectors mostly contain a higher fraction of the optimal portfolio we invested in. The rest of the paper is organized as follows: in Section 2, we first define a simple and applicable MSD rule based on the mean-variance approach. Then, we propose a definition of a multivariate stochastic ordering aimed at ranking different market sectors whose vectors of returns are either in the domain of attraction of a stable sub-Gaussian law or are Student’s t-distributed. In Section 3, we present an empirical comparison to evaluate possible dominance relations between several market sectors of the S&P 500. Finally, Section 4 summarizes our conclusions. 2. Stochastic dominance between market sectors This section introduces multivariate stochastic orderings among market sectors. Our primary goal is to compare market sectors from investors’ points of views and examine the impact of the observed dominance of the optimal choices. Thus, we introduce multivariate ordering rules that are especially useful for ranking different market sectors. Recall that, when we compare two random variables, |$X$| and |$Y$|⁠, with respect to a given univariate order of preferences, represented by the symbol |$\succ $|⁠, we say that |$X$| dominates |$Y$| with respect to |$\succ $| (namely |$X\succ Y$|⁠) if suitable conditions are satisfied. In general, these conditions involve the distribution functions of |$X$| and |$Y$|⁠, say |$F_X$| and |$F_Y$| (see, e.g., Levy, 1992; Lodwick, 1989, Castellano & Cerqueti (2018)). Classical stochastic orders are reviewed in the following definition. Definition 2.1 – First-order stochastic dominance: we say that |$X$| dominates |$Y$| with respect to the first-order stochastic dominance (in symbols, |$X$| FSD |$Y$|⁠) if and only if |$F_X (t)\leqslant F_Y (t)$| for any |$t$| that belongs to |$\mathbb{R}$| or, equivalently, |$X$| FSD |$Y$| if and only if |$E(g(X))\geqslant E(g(Y))$| for any increasing function |$g$|⁠. – SSD (increasing concave order): we say that |$X$| dominates |$Y$| with respect to the SSD order (in symbols, |$X$| SSD |$Y$| or |$X\geqslant _{icv}Y$|⁠) if and only if $$\begin{equation*}\displaystyle{\int_{-\infty}^{t}F_X (z) \, \textrm{d}z\leqslant \int_{-\infty}^{t}F_Y (z) \, \textrm{d}z}, \, \forall \, t\in \mathbb{R}\end{equation*}$$ or, equivalently, |$X\geqslant _{icv}Y$| if and only if |$E(g(X))\geqslant E(g(Y))$| for any increasing and concave function |$g$|⁠, i.e., any non-satiable risk-averse investor prefers |$X$| to |$Y$|⁠. Recall that investors are risk averse if they have concave utility function (see Ingersoll, 1987). In general, the extension of a given order of preferences |$\succ $| to the multivariate case is not trivial because in some practical cases it could be challenging to satisfy the conditions of MSD. In this context, although the natural generalizations of FSD and SSD can be found, for instance in Shaked & Shanthikumar (1994) and Müller & Stoyan (2002), their dominance rules are far from being applicable, except in some specific cases. Indeed, multivariate stochastic orders consider the dependence structure of random vectors and cannot be based merely on component-wise comparisons of the marginal distributions. It is very rare to find a multivariate ordering rule that is based on comparisons between functionals. One idea born from the definition proposed by Ortobelli et al. (2016a) is shown in the following definition. Suppose that there are two sectors, |$A$| and |$B$|⁠, composed, respectively, of |$n$| and |$s$| assets. Let the vectors of non-negative allocations among risky assets of sectors |$A$| and |$B$|⁠, respectively, be denoted by |$x=[x_1,x_2,\ldots ,x_n ]^{\prime}$| and |$y=[y_1,y_2,\ldots ,y_s ]^{\prime}$|⁠, whose sum is equal to 1 (i.e., |$\sum _{i=1}^n x_i=\sum _{j=1}^s y_j=1$|⁠). Moreover, assume that no short sales are allowed (i.e., |$x_i\geqslant 0$| and |$y_j\geqslant 0$|⁠, |$\forall \,\, i=1,\ldots ,n$| and |$j=1,\ldots ,s$|⁠). Definition 2.2 We say that a sector |$A$| with |$n$| assets strongly dominates sector |$B$| with |$s\leqslant n$| assets with respect to a multivariate preference ordering |$\succ $| (namely, FSD and SSD) if for any vector of returns |$Y_B$| of |$B$|⁠, there exists a vector |$X_A$| of sector |$A$| such that |$X_A\succ Y_B$|⁠. Similarly, we say that a sector |$A$| with |$n$| assets weakly dominates another sector |$B$| with |$s$| assets with respect to a multivariate preference ordering |$\succ $| if for any given portfolio of sector |$B$| with return |$y^{\prime}Y_B$|⁠, there exists a portfolio of sector |$A$| with return |$x^{\prime} X_A$| such that |$x^{\prime} X_A \succ y^{\prime} Y_B$|⁠. Generally, the stronger dominance implies the weak one, as clarified in the following example. Example 2.1 Suppose that the returns of sectors |$A$| and |$B$| are jointly Gaussian distributed. Suppose also that the two sectors have the same number of assets |$n$|⁠, vector averages |$\mu _A$| and |$\mu _B$|⁠, and variance-covariance matrices |$Q_A$| and |$Q_B$|⁠, such that |$\mu _A \geqslant \mu _B$| and |$(Q_A-Q_B)$| is negative semi-definite. Then, sector |$A$| strongly dominates sector |$B$| with respect to the increasing concave multivariate (ICVM) order. Moreover, under these assumptions, sector |$A$| weakly dominates sector |$B$| with respect to the concave order because |$x^{\prime}\mu _A \geqslant x^{\prime}\mu _B$| and |$x^{\prime}Q_{A}x\leqslant x^{\prime}Q_{B}x$| for any vector |$x\geqslant 0$|⁠. Note that this weak dominance between Gaussian distributed vectors is also known in the literature as the increasing positive linear concave multivariate order (see Müller & Stoyan, 2002). The increasing positive linear concave multivariate ordering defined in Example 2.1 is strictly related to the mean-variance rule, which is widely used in finance to solve the portfolio optimization problem. According to this principle, investors aim at minimizing variance (risk) for some fixed level of the mean (reward). In mean-variance framework, we call the efficient frontier (EF) the set of portfolios with greater mean and lower variance. However, we can extend this concept with respect to any given investor preference as follows. Definition 2.3 We call the EF with respect to a given preference |$\succ $| the set of portfolios |$x^{\prime}r$| (⁠|$\sum _{i=1}^n x=1$|⁠, |$x_i\geqslant 0$|⁠) that are not dominated with respect to the preference |$\succ $|⁠, i.e., |$\textrm{EF}_{\succ }=\lbrace x\in \mathbb{R}^n: \sum _{i=1}^n x_i=1, \, x_i\geqslant 0, \, \nexists \, y\in \mathbb{R}^n: \sum _{i=1}^n y_i=1, \, y_i\geqslant 0: y^{\prime}r\succ x^{\prime}r\rbrace $|⁠. Typically, in MSD literature, the comparison between vectors of the same dimension with respect to some orderings is used (see, e.g., Kopa & Post, 2015; Post & Kopa, 2017). However, this typical comparison among vectors is generally stronger than the one introduced in Definition 2.2 because in this definition we do not take into account an order among the marginals. Let us clarify this concept. Example 2.2 Suppose the sector |$A$| is composed of two Gaussian independent distributed random variables |$X_1\sim N(1,2)$| and |$X_2\sim N(2,1)$|⁠. Similarly, sector |$B$| is composed of two Gaussian independent distributed random variables |$Y_1\sim N(2,2)$| and |$Y_2\sim N(1,3)$|⁠. Observe that vector |$X=[X_1,X_2]^{\prime}$| (with |$\mu _X=[1,2]^{\prime}$| and variance-covariance matrix |$Q_X$|⁠) does not dominate vector |$Y=[Y_1,Y_2]^{\prime}$| (with |$\mu _Y=[2,1]^{\prime}$| and variance-covariance matrix |$Q_Y$|⁠) with respect to the ICVM order. However, sector |$A$| dominates sector |$B$| with respect to the ICVM order because the |$\tilde{X}=\left [ X_2,X_1\right ]$| ICVM dominates vector |$Y=\left [ Y_1,Y_2\right ]$| according to Example 2.1. Example 2.2 clarifies an important difference between sector dominance and vector dominance. Moreover, Examples 2.1 and 2.2 are based on the mean-variance approach (see Markowitz, 1952) justified by assuming Gaussian distributed returns. However, we argue that a proper dominance rule for ranking market sectors should depend on the most reliable models. In particular, in financial literature, the mean-variance approach has been extended to a mean-risk approach, where used risk measures |$\sigma $| are consistent with the choices of risk-averse investors (i.e., if |$E(X)=E(Y)$| and |$X$| SSD |$Y$|⁠, then |$\sigma _X\leqslant \sigma _Y$|⁠) such as the conditional value-at-risk |$CVaR_{\gamma }(X)=-\frac{1}{\gamma }\int _0^{\gamma }F_X^{-1}(q) \,\textrm{d}q$| (see Artzner et al., 1999), the mean absolute deviation |$MAD_{X}=E(|X-E(X)|)$| (see Konno & Yamazaki, 1991) and many others (see Szegö, 1994). In the following sections, we propose a distributional approach to the dominance problem that is based either on stable Paretian (sub-Gaussian) or Student’s t distributional assumptions. Recall that this class of elliptically symmetric distributions does not capture the skewness of stock portfolio returns. In response to these challenges, Ortobelli et al. (2016b) define a new stochastic order, weaker than the SSD, and prove that it holds if some conditions related to the skewness parameters are verified. Furthermore, in the context of portfolio choices, it is well known that diversification tends to introduce negative skewness (see, e.g., Simkowitz & Beedles (1978)). Thus, a natural extension of this research would be an MSD rule that also considers skewness and its relation to portfolio diversification. 2.1 Stable Paretian sub-Gaussian distribution Let |$r_{t+1}=[r_{1,t+1},\ldots ,r_{n,t+1}]^{\prime}$| denote the vector of gross returns on date |$t+1$|⁠. We define the i-th gross return between time |$t$| and time |$t+1$| as |$r_{i,t}=\frac{P_{t+1,i}}{P_{t,i}}$| and the i-th log return as |$\ln (r_{i,t})$|⁠, where |$P_{t,i}$| is the price of the i-th asset at time |$t$|⁠. In empirical finance literature, it is well known that asset returns follow heavy-tailed distributions. In this framework, it can be assumed that the returns are in the domain of attraction of a stable law such that $$\begin{equation} Pr(|r_{i}|>t)\approx t^{-\alpha_{i}}L_{i}(t)\,\,\,\textrm{as}\,\,t\rightarrow\infty, \end{equation}$$(2.1) where |$0<\alpha _{i}<2$| and |$L_{i}(t)$| is a slowly varying function at infinity, i.e., $$\begin{equation} \lim_{t\rightarrow\infty}\frac{L_{i}(ct)}{L_{i}(t)}\rightarrow1\,\,\,\,\,\,\,\,\,\,\forall \,c>0, \end{equation}$$(2.2) (see Rachev & Mittnik, 2000 and the references therein). The tail condition implies that any gross return |$r_{i}$| follows an |$\alpha $|-stable distribution |$S_{\alpha }(\sigma ,\beta ,\mu )$|⁠, where |$\alpha \in (0,2]$| is the so-called stability index (which specifies the asymptotic behaviour of the tails), |$\sigma>0$| is the dispersion parameter, |$\beta \in \lbrack -1,1]$| is the skewness parameter and |$\mu \in \mathbb{R}$| is the location parameter. It can typically be assumed that the vector of returns |$r$| is |$\alpha $|-stable sub-Gaussian distributed. Recall that an |$\alpha $|-stable sub-Gaussian distribution is an elliptical distribution that is symmetric around the mean (because |$\beta =0$|⁠). Its characteristic function has the following form: $$\begin{equation*} \varPhi_r(u)=E(\mathrm{exp}(iu^{\prime}r))=\mathrm{exp}(-(u^{\prime}Qu)^{\frac{\alpha}{2}}+iu^{\prime}\mu),\end{equation*}$$ where |$Q=[q_{ij}]$| is a positive definite dispersion matrix and |$\mu $| is the mean vector (when |$\alpha>1$|⁠). Sub-Gaussian returns are elliptically distributed for a fixed value of |$\alpha $|⁠; when |$\alpha =2$|⁠, we obtain the Gaussian distribution. Therefore, according to Ortobelli et al. (2016a), it is important to define a suitable order of preferences to deal with distributions having different |$\alpha $| values. In the following, we determine a ranking criterion to compare sub-Gaussian distributions according to SSD. It has been proven that SSD can be verified by comparing the values of the stability, dispersion and location parameters. Thus, the following theorem (see Ortobelli et al., 2016a) can be recalled: Theorem 2.1 Let |$X_1\sim S_{\alpha _1}(\sigma _1,0,\mu _1)$|⁠, and |$X_2\sim S_{\alpha _2}(\sigma _2,0,\mu _2)$|⁠. Suppose that |$\alpha _1>\alpha _2>1$|⁠, and |$\sigma _1\leqslant \sigma _2$|⁠. If |$\mu _1\geqslant \mu _2$|⁠, then |$X_1$| SSD |$X_2$|⁠. The results of Theorem 2.1 can be extended to a multivariate setting, generalizing the multivariate mean-dispersion approach presented in Example 2.1 by considering the asymptotic behaviour of the tail distributions. In particular, we obtain the following corollary. Corollary 2.1 Let |$A$| and |$B$| be two sectors with |$n$| and |$m$| assets, respectively. Assume that the returns |$r_A$| of sector |$A$| are jointly |$\alpha _A$|-stable sub-Gaussian distributed with vector mean |$\mu _A$| and dispersion matrix |$Q_A$| and the returns |$r_B$| of sector |$B$| are jointly |$\alpha _B$|-stable sub-Gaussian distributed with vector mean |$\mu _B$| and dispersion matrix |$Q_B$|⁠. Assume that |$\alpha _A\geqslant \alpha _B>1$| and for any portfolio of returns |$y^{\prime}r_A$| of sector |$A$|⁠, there exists a portfolio of returns |$x^{\prime}r_B$| of sector |$B$| such that |$y^{\prime}Q_Ay \leqslant x^{\prime}Q_Bx$| (and at least one inequality holds strictly, i.e., either |$y^{\prime}Q_Ay < x^{\prime}Q_Bx$| or |$\alpha _A>\alpha _B$| ). Then, |$y^{\prime}\mu _A \geqslant x^{\prime}\mu _B$| implies |$y^{\prime}r_A$| SSD |$ x^{\prime}r_B$| (i.e., sector |$A$| weakly SSD dominates sector |$B$|⁠). Proof. See Appendix A. The dominance obtained by stable sub-Gaussian distribution could be stronger than other mean-risk approaches, as proved in the following corollary. Corollary 2.2 Let |$X_1\sim S_{\alpha _1}(\sigma _1,0,\mu _1)$| and |$X_2\sim S_{\alpha _2}(\sigma _2,0,\mu _2)$|⁠. If |$\alpha _1\geqslant \alpha _2\geqslant 1$|⁠, |$\mu _1\geqslant \mu _2$| and |$MAD_{X_1}\leqslant \frac{\varGamma (1-\frac{1}{\alpha _1})}{\varGamma (1-\frac{1}{\alpha _2})}MAD_{X_2}$|⁠, then |$X_1$| SSD |$X_2$|⁠. Proof. See Appendix A. Observe that when |$X$| and |$Y$| are two random variables with the same mean and different MAD (⁠|$MAD_{X}\neq MAD_{Y}$|⁠), then |$X$| SSD |$Y$| implies that |$MAD_{X}< MAD_{Y}$|⁠. However, we cannot say the converse. Thus, Corollary 2.2 gives conditions to invert the relationship between stable symmetric distributions. This study explores another family of symmetric elliptical distributions that also deal with the observed fat tails of asset returns. We propose ranking criteria that aim to compare Student’s t distributions according to SSD. Specifically, we prove that in the case of Student’s t distributions, the degree of freedom is crucial for establishing multivariate stochastic ordering. 2.2 Student’s t distribution Several studies have used Student’s t distributions to model asset log returns (see, e.g., Blattberg & Gonedes, 1974; Fergusson & Platen, 2006; Markowitz & Usmen, 1996; Praetz, 1972). The following equation gives the Student’s t density function: $$\begin{equation} f_{X}(x)=\frac{1}{\sigma\sqrt{\nu\pi}}\frac{\varGamma((\nu+1)/2)}{\varGamma(\nu/2)}\left(1+\frac{(\frac{x-\mu}{\sigma})^2}{\nu}\right)^{-\frac{\nu+1}{2}}, \end{equation}$$(2.3) where |$\varGamma (z)=\int _0^{\infty }x^{z-1}e^{-x}\,\textrm{d}x$| is the gamma function, |$\mu $| is the location parameter, |$\sigma $| is the scale parameter and |$\nu $| is a positive parameter called the degree of freedom. The Student’s t distribution has the following properties: (i) all moments of order |$m<\nu $| are finite (i.e., the mean exists for |$\nu>1$|⁠, and the variance is finite only for |$\nu>2$|⁠); (ii) when |$\nu =1$|⁠, the Student’s t density function is the Cauchy density or ‘Lorentzian’ function; (iii) it has fatter tails than the density function of a normal distribution; and (iv) for values |$\nu \rightarrow \infty $|⁠, it converges asymptotically to the normal distribution. For our purposes, the most important parameter of this distribution is the degree of freedom, denoted by |$\nu $| (especially when |$\nu>2$|⁠). While similarities between the Student’s t and an |$\alpha $|-stable sub-Gaussian distribution exist, these two distributions have very different theoretical and empirical implications (see, e.g., Blattberg & Gonedes, 1974). On the one hand, the stability property distinguishes the stable and Student’s t distributions; the sum of identically and independently distributed (i.i.d) |$\alpha $|-stable distributions is still |$\alpha $|-stable distributed, while the sum of i.i.d Student’s t distributions converges to a Gaussian distribution for |$\nu>2$| and to |$\nu $|-stable distribution for |$\nu <2$|⁠. On the other hand, the Student’s t model allows the use of well-known density functions while the stable density functions are known in only a few cases (e.g., |$\alpha =\frac{1}{2}$|⁠, |$\alpha =1$| and |$\alpha =2$|⁠). Consequently, the likelihood function of the Student’s t model can be expressed in closed form, and maximum likelihood (ML) estimates for all parameters of the model can be easily obtained. For the stable distribution, ML estimates are generally obtained by inverting the Fourier transform (see, e.g., Rachev & Mittnik, 2000). Student’s t distribution belongs to the elliptically distributed family (see, e.g., Schoutens (2003) and Samorodnitsky & Taqqu, 1994). It is therefore important to define a suitable order of preferences to deal with distributions for different values of |$\nu $|⁠. The following determines a ranking criterion that aims to compare univariate Student’s t distributions according to SSD. The fact that SSD can be verified by comparing the degree of freedom, dispersion and location parameters is shown below. Theorem 2.2 Let |$X_1\sim T_{\nu _1}(\sigma _1,\mu _1)$| and |$X_2\sim T_{\nu _2}(\sigma _2,\mu _2)$|⁠. Suppose |$\nu _1\geqslant \nu _2$| and |$\sigma _1\leqslant \sigma _2$| with at least one strict inequality. If |$\mu _1\geqslant \mu _2$|⁠, then |$X_1$| SSD |$X_2$|⁠. Proof. See Appendix A. The results of Theorem 2.2 can be extended to a multivariate setting that considers the asymptotic behaviour of the tail distributions. This extension yields the following MSD among market sectors. Corollary 2.3 Let |$A$| and |$B$| be two sectors with |$n$| and |$m$| assets, respectively. Assume that the returns |$r_A$| of sector |$A$| are jointly Student’s t distributed with vector mean |$\mu _A$| and dispersion matrix |$Q_A$| and the returns |$r_B$| of sector |$B$| are jointly Student’s t distributed with vector mean |$\mu _B$| and dispersion matrix |$Q_B$|⁠. Assume that |$\nu _A\geqslant \nu _B\geqslant 2$| and for any portfolio of returns |$y^{\prime}r_A$| of sector |$A$|⁠, there exists a portfolio of returns |$x^{\prime}r_B$| of sector |$B$| such that |$y^{\prime}Q_Ay \leqslant x^{\prime}Q_Bx$| (and at least one inequality holds strictly, i.e., either |$y^{\prime}Q_Ay < x^{\prime}Q_Bx$| or |$\nu _A>\nu _B$| ). Then, |$y^{\prime}\mu _A \geqslant x^{\prime}\mu _B$| implies |$y^{\prime}r_A$| SSD |$ x^{\prime}r_B$| (i.e., sector |$A$| weakly SSD dominates sector |$B$|⁠). Proof. See Appendix A. Observe that the scale parameter |$\sigma $| does not correspond to a standard deviation (SD) of the normal distribution. Similar to the stable distribution, the dominance obtained under the Student’t assumption implies different mean-variance ordering, according to the following Corollary. Corollary 2.4 Let |$X_1\sim T_{\nu _1}(\sigma _1,\mu _1)$| and |$X_2\sim T_{\nu _2}(\sigma _2,\mu _2)$|⁠. If |$\nu _1\geqslant \nu _2\geqslant 2$|⁠, |$\mu _1\geqslant \mu _2$| and |$\textrm{Std}_1\leqslant \sqrt{\frac{\nu _1(\nu _2-2)}{\nu _2(\nu _1-2)}}\textrm{Std}_2$|⁠, then |$X_1$| SSD |$X_2$|⁠. Proof. See Appendix A. Observe that when |$X$| and |$Y$| are two random variables with the same mean and different finite variances, then |$X$| SSD |$Y$| implies |$var(X)< var(Y)$|⁠. Moreover, we cannot state the converse, except under the conditions of Corollary 2.4. 3. Empirical analysis This section applies the multivariate weak dominance rules to compare S&P 500 sectors empirically. The results of these methods are compared with those obtained by the weak concave multivariate order described in Example 2.1 and Corollaries 2.1 and 2.3. Empirically, it is well known that the conditions for the strong form of stochastic dominance are not easily verified in practice. Therefore, we employ the multivariate weak dominance rule under the assumption that the vector of returns for each sector is either Gaussian, Student’s t or |$\alpha $|-stable sub-Gaussian distributed. In general, the asset returns are typically heavy-tailed (see, e.g., Rachev & Mittnik, 2000); thus, determining the tail probabilities is key. The general aim is to examine to what extent the dominance observed among market sectors impacts the optimal choices of a non-satiable risk-averse investor under different distributional hypotheses of asset returns. Specifically, we apply the stochastic dominance rules stated in the previous section to empirically compare optimal choices of the components of the S&P 500 index. Doing so, we also verify the validity of some classical elliptical distributional assumptions. The goal of this empirical section is twofold. First, it verifies if sectors not chosen by non-satiable risk-averse investors are really dominated (in the SSD sense) by other sectors. Second, it evaluates if one among three elliptical distributional assumptions is more suitable than the others. Both objectives are important to better address investors’ choices. This section is divided into three subsections to address these objectives. The first subsection examines the statistical characteristics of the returns for each sector individually and all sectors collectively. The second subsection verifies the proposed SSD dominance rules during the period 2004–2017, when short sales were not allowed. In particular, Subsection 3.2 examines the so-called mean-dispersion EF under different distributional hypotheses (Gaussian, stable sub-Gaussian and Student’s t) and computes the SSD EFs. Then, it compares the EFs to determine whether the conditions for the weak ICVM hold. Next, it uses a non-parametric approach to test the efficiency of the frontiers obtained under three different distributional assumptions and, using the test proposed by Arvanitis et al. (2019), it also evaluates if the dominant sectors contain a market-efficient set. Finally, the third subsection examines whether an active investor can use these findings to achieve economically and statistically significant out-of-sample performance compared with the S&P 500 index. 3.1 Data set description We consider daily returns of the S&P 500 specific economic sectors for the period from December 2004 to January 2017, for a total of 3037 observations. A specific dynamic data set is obtained by considering the S&P 500 sector components available from the Thomson Reuters DataStream database (431 assets). In particular, according to the Global Industry Classification Standard, we distinguish the following specific economic sectors: (1) information technology (IT), (2) financials (FI), (3) health care (HC), (4) consumer discretionary (CD), (5) industrials (IN), (6) consumer staples (CS), (7) energy (EN), (8) utilities (UT), (9) real estate (RE) and (10) materials (MA). We omit the telecommunication services sector because it has very few assets and some of them are not available over the entire period. In practice, for both stable and Student’s t distributions, we estimate the unknown parameters using the ML method. There are two main approaches to the problem of ML estimation for stable Paretian cases. Modern ML estimation techniques for stable distributions utilize either (i) the fast Fourier transform method to approximate the stable density function (see, e.g., Rachev & Mittnik, 2000) or (ii) the direct integration method (see Nolan, 2001). Both approaches are comparable in terms of efficiency, and variances in performance result from different approximation algorithms. Our analysis uses the first approach. As observed by Kring et al. (2009), the estimation of |$q_{ij}$| (where |$i\neq j$|⁠) can be seen as a difference between squared scale parameters of returns |$(r_{i}\pm r_{j})/2$|⁠, i.e., |$q_{ij}=\sigma ^2_{(r_i+r_j)/2}-\sigma ^2_{(r_i-r_j)/2}$|⁠. Moreover, it should be stressed that the common stability index is estimated using the sample mean of the stability parameters of |$10,000$| random portfolios (i.e., |$\alpha =\frac{1}{10,000}\sum _{i=1}^{10,000}\alpha _i$|⁠), as suggested by Ortobelli et al. (2004). Similarly, we obtain the common degrees of freedom (df) using the sample mean of |$10,000$| random portfolios. Table 1 reports the average ML estimates of the stable Paretian and Student’s t parameters and summary statistics for the average return on assets for each sector individually and for all sectors collectively: mean, SD, skewness and kurtosis. Table 1 Average statistics for the log returns of different sectors and ML estimates of the stable Paretian and Student’s t parameters Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Open in new tab Table 1 Average statistics for the log returns of different sectors and ML estimates of the stable Paretian and Student’s t parameters Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Sector . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . All sectors . Assets . 58 . 57 . 55 . 65 . 55 . 31 . 30 . 27 . 31 . 22 . 431 . Gaussian distribution hypothesis Mean 0.038|$\%$| 0.009|$\%$| 0.052|$\%$| 0.046|$\%$| 0.038|$\%$| 0.046|$\%$| 0.022|$\%$| 0.029|$\%$| 0.035|$\%$| 0.034|$\%$| 0.036|$\%$| St. dev. 2.262|$\%$| 2.830|$\%$| 1.958|$\%$| 2.323|$\%$| 2.093|$\%$| 1.501|$\%$| 2.624|$\%$| 1.455|$\%$| 2.658|$\%$| 2.208|$\%$| 2.232|$\%$| Skew. |$-0.1201$| |$-0.2891$| |$-0.2203$| |$-0.0276$| |$-0.2780$| |$-0.0752$| |$-0.2368$| |$-0.0433$| |$-0.1789$| |$-0.2102$| |$-0.1703$| Kurt. 11.362 12.935 12.634 9.964 9.857 11.815 9.036 8.788 10.657 9.023 10.869 J–B test 1 1 1 1 1 1 1 1 1 1 1 Student’s t distribution Df. |$\nu $| 3.6212 3.0991 3.6271 3.6297 3.6878 3.6548 3.9681 4.5523 3.1941 3.5916 3.6153 Sigma |$\sigma $| 1.499 |$\%$| 1.461|$\%$| 1.272|$\%$| 1.532|$\%$| 1.399|$\%$| 0.975|$\%$| 1.785|$\%$| 0.988|$\%$| 1.433|$\%$| 1.444|$\%$| 1.404|$\%$| Mean |$\mu $| 0.050|$\%$| 0.034|$\%$| 0.061|$\%$| 0.039|$\%$| 0.065|$\%$| 0.056|$\%$| 0.080|$\%$| 0.056|$\%$| 0.068|$\%$| 0.063|$\%$| 0.054|$\%$| K–S test 0.19 0.21 0.14 0.16 0.14 0.20 0.18 0.13 0.17 0.21 0.1712 Stable distribution hypothesis Alpha |$\alpha $| 1.664 1.526 1.661 1.653 1.651 1.665 1.681 1.708 1.548 1.639 1.636 Beta |$\beta $| 0.0240 |$-0.0302$| 0.0300 0.0653 |$-0.0214$| |$-0.0121$| |$-0.0414$| |$-0.1438$| |$-0.0756$| |$-0.0038$| |$-0.0082$| Sigma |$\sigma $| 1.206|$\%$| 2.326|$\%$| 1.023|$\%$| 1.233|$\%$| 1.122|$\%$| 0.784|$\%$| 1.423|$\%$| 0.787|$\%$| 1.176|$\%$| 1.162|$\%$| 1.278 |$\%$| Mean |$\mu $| 0.065 |$\%$| |$-0.259$||$\%$| 0.081|$\%$| 0.078|$\%$| 0.056|$\%$| 0.057|$\%$| 0.048|$\%$| 0.020|$\%$| 0.026|$\%$| 0.053|$\%$| 0.017|$\%$| K–S test 0.17 0.18 0.08 0.13 0.09 0.18 0.21 0.12 0.08 0.11 0.1344 Open in new tab According to Simkowitz & Beedles (1978), log returns (in Table 1) present a slight negative skewness (as the asymmetry parameter |$\beta $| and skewness are generally negative close to zero). Moreover, the parameter estimates suggest the presence of a quite remarkable tail-weightiness (the kurtosis exceeds three, while the stability parameter |$\alpha $| is less than two). The FI sector returns present, on average, the lowest mean |$\mu $| and |$alpha$| and the highest SD and kurtosis. Computing the Jarque–Bera statistic with a 95% confidence level to test whether asset returns follow a normal distribution shows that the Gaussian hypothesis is always rejected. On average, the Kolmogorov–Smirnov test suggests rejecting approximately |$17.12\%$| of the Student’s t distributional hypothesis and |$13.44\%$| of the stable Paretian hypothesis. Thus, it is reasonable to assume from the obtained results that the asset returns of each sector are jointly elliptical distributed because we observe strong evidence of tail weightiness and there is little evidence of skewness. 3.2 SSD sector dominance Generally, there are three distinct steps to verify SSD dominance rules among sectors under the different distributional assumptions. The first step fits the mean-dispersion EF by considering a unique index of stability for the stable distributional assumption and a unique value for the df of the multivariate Student’s t distribution. The second step identifies the existence of SSD dominance between sectors whose components have been chosen and sectors whose assets are not chosen among the assets of the fitted EFs of the first step. Clearly, sector dominance always occurs under the Gaussian assumption, based only on the mean-variance Markowitz rule. However, for the stable and Student’s t distributional assumptions, different indexes of stability and df for the chosen and non-chosen sectors must also be considered to verify existing SSD sector dominance. Thus, it first values a common index of stability (df) of the non-chosen sectors (obtained in the first step) and then fits the mean-dispersion SSD EF of these assets. Second, it values a common index of stability (df) of the sectors chosen in the first step. If the common index of stability (df) of the chosen sectors is not larger than the index of stability (df) of the non-chosen sectors, a common index of stability (df) of the chosen sectors is computed using only those sectors whose index of stability (df) is estimated to be greater than the common index of stability (df) of non-chosen sectors. Then, the mean-dispersion SSD EF of the sectors chosen in the first step is fit. Then, according to the dominance rules of Corollaries 2.1 and 2.3, weak sector dominance is verified. In particular, it verifies if, for any portfolio belonging to the fitted EF of non-chosen sectors, there exists a portfolio of the fitted EF of the chosen sectors in which SSD dominates. Notably, sector dominance (according to the dominance rules of Corollaries 2.1 and 2.3) is always present in some of the chosen sectors with respect to unchosen ones, even under elliptical non-Gaussian distributional assumptions (see Table 3). The last step tests if, for any efficient portfolio of the non-chosen sector, there exists a portfolio of the chosen sectors that SSD dominates according to a non-parametric approach (see, Davidson & Jean-Yves, 2000). This third step should reveal when the observed sector dominance can be guaranteed to hold in reality. Thus, in some sense, this last step also serves to validate the distributional assumption. All three steps have been applied every month (e.g., 20 trading days) from 1 December 2007 until the end of January 2017. Thus, according to the first and second steps, the mean-dispersion SSD EFs are estimated using the assets that were active during the last three years (750 daily historical observations).1 Therefore, at every recalibration time, the EF is fitted by solving the optimization problem for 100 levels of the mean as follows: $$\begin{equation} \begin{array} [c]{c} \min_{x}x^{\prime}Qx\\ \text{s.t.}\\ \sum_{i=1}^{n}x_{i}=1;\,x^{\prime}\mu=m\\ x_{i}\geqslant0;\ \ i=1,\ldots,n, \end{array} \end{equation}$$(3.1) where |$x^{\prime }Qx$| is the risk measure (variance or dispersion measure) associated with portfolio |$x^{\prime }r$| and |$n$| is the number of all assets, which are grouped by sector. The asset returns of sectors that are unchosen in the SSD mean-dispersion EF are identified. Subsequently, whether an MSD between chosen and unchosen sectors exists is verified. Then, a non-parametric approach is used to test the efficiency of the parametric dominance (see, e.g., Davidson & Jean-Yves, 2000). It is assumed that a sector |$A$| parametrically dominates a sector |$B$| to give a meaningful intuition of a non-parametric test. Then, whether for any optimal portfolio of sector B (fitted among 100 levels of the mean), there exists a portfolio of sector |$A$| that non-parametrically dominates those of sector |$B$| (based on their empirical distributions) is verified. Finally, the non-parametric test proposed by Arvanitis et al. (2019) is used to evaluate when the dominant sectors contain the market-efficient set. The results of this empirical analysis are reported in Tables 2 and 3 and Figures 1–3. Table 2 reports the mean of the percentage invested in each sector over different periods (e.g., subprime crisis, EU credit crisis and post-crisis period) using either the Gaussian, Student’s t or stable distributional assumption. Table 2 The mean of the percentage invested in each sector over different periods considering three distributional assumptions From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 Open in new tab Table 2 The mean of the percentage invested in each sector over different periods considering three distributional assumptions From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 From . To . Period . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis December 2007 January 2017 All periods 0.1227 0.0080 0.2794 0.1891 0.0430 0.2286 0.0031 0.0364 0.0905 0.0082 December 2008 September 2009 Subprime crisis 0.0675 0.0350 0.3917 0.1603 0.0182 0.3161 0.0000 0.0086 0.0000 0.0090 September 2009 January 2013 EU credit crisis 0.1207 0.0060 0.2455 0.2776 0.0000 0.1808 0.0000 0.0200 0.1491 0.0086 January 2013 January 2017 Post crisis 0.1333 0.0051 0.2887 0.1215 0.0822 0.2534 0.0062 0.0544 0.0575 0.0076 Student’s t distribution hypothesis December 2007 January 2017 All periods 0.1202 0.0063 0.2591 0.1383 0.0342 0.2243 0.0061 0.0413 0.1042 0.0093 December 2007 September 2009 Subprime crisis 0.0431 0.0135 0.4346 0.1134 0.0282 0.2790 0.0031 0.0137 0.0129 0.0315 September 2009 January 2013 EU credit crisis 0.0986 0.0093 0.2531 0.2412 0.0000 0.1793 0.0009 0.0319 0.1736 0.0121 January 2013 January 2017 Post crisis 0.1405 0.0063 0.2185 0.1137 0.0625 0.2539 0.0091 0.0545 0.0786 0.0067 Stable distribution hypothesis December 2007 January 2017 All periods 0.1100 0.0054 0.2561 0.1367 0.0298 0.2235 0.0080 0.0489 0.1737 0.0150 December 2007 September 2009 Subprime crisis 0.0238 0.0054 0.4765 0.0747 0.0346 0.2591 0.0078 0.0201 0.0398 0.0613 September 2009 January 2013 EU credit crisis 0.0659 0.0005 0.2700 0.1825 0.0001 0.1756 0.0013 0.0376 0.2547 0.0188 January 2013 January 2017 Post crisis 0.1601 0.0094 0.2089 0.1094 0.0533 0.2567 0.0134 0.0630 0.1295 0.0042 Open in new tab Table 3 Weak SSD sector dominance . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| Open in new tab Table 3 Weak SSD sector dominance . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| . IT . FI . HC . CD . IN . CS . EN . UT . RE . MA . Gaussian distribution hypothesis Zero invest. SSD 1 29 0 0 42 0 69 0 22 1 # SSD param. 1 29 0 0 42 0 69 0 22 1 # non–param SSD 0 18 0 0 26 0 42 0 14 0 Percentage of tested SSD dom. 0|$\%$| 62|$\%$| – – 61|$\%$| – 60|$\%$| – 64|$\%$| 0|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 54.9 |$\%$| Student’s t distribution hypothesis Zero invest. SSD 1 33 0 0 43 0 60 0 13 1 # SSD param. 1 33 0 0 43 0 60 0 13 1 # non–param. SSD 0 33 0 0 36 0 54 0 12 1 Percentage of tested SSD dom. – 100|$\%$| – – 84|$\%$| – 90|$\%$| – 92|$\%$| 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 80.6 |$\%$| Stable distribution hypothesis Zero invest.SSD 3 36 0 0 46 0 54 0 0 2 # SSD param. 3 36 0 0 46 0 54 0 0 2 # non–param. SSD 0 36 0 0 42 0 51 0 0 2 Percentage of tested SSD dom. – 100|$\%$| – – 91|$\%$| – 94|$\%$| – – 100|$\%$| |$\%$| of times dominant sectors contain the SSD efficient set (Arvanitis et al.’s test) 84 |$\%$| Open in new tab From Table 2, we observe that |$\bullet $| The percentage invested in each sector differs slightly from one distributional hypothesis to another. The IT, HC, CD and CS sectors have higher weights than their FI, IN, EN, UT, RE and MA counterparts over the entire period examined for each of the Gaussian, Student’s t or stable distribution hypotheses. These results can be explained by the effects of financial crises on these sectors. |$\bullet $| The IT sector has a higher weight post-crisis than during other periods of market distress (subprime and European credit crisis), which is not the case for the CD sector, which shows losses of significant weights after the European credit crisis. |$\bullet $| The HC and CS sectors obtain higher weights during the subprime crisis and lower weights throughout the European credit crisis (even though they have the highest percentage). |$\bullet $| After suffering from the subprime mortgage crisis, which was triggered by a substantial decline in home prices after the collapse of a housing bubble, the RE sector gains more weight throughout the European credit crisis. Example of mean-variance SSD dominance (June 2010). Fig. 1. Open in new tabDownload slide Fig. 1. Example of mean-variance SSD dominance (June 2010). Open in new tabDownload slide Example of alpha-mean-dispersion SSD dominance (January 2008). Fig. 2. Open in new tabDownload slide Fig. 2. Example of alpha-mean-dispersion SSD dominance (January 2008). Open in new tabDownload slide Example of the degree of freedom-mean-dispersion SSD dominance (January 2009). Fig. 3. Open in new tabDownload slide Fig. 3. Example of the degree of freedom-mean-dispersion SSD dominance (January 2009). Open in new tabDownload slide These results highlight the significant differences between the Gaussian and the other two distributions. In particular, the weights obtained from Student’s t distribution are generally located between those of the stable and Gaussian hypotheses. This could be explained by the previously stressed fact (see Section 2); for higher |$\nu $|⁠, the Student’s t random variable will tend to a normal distribution, while for lower values of |$\nu $|⁠, it presents heavy tails as the stable distribution. The differences among these distributions have already been examined by several studies (see, e.g., Ortobelli et al., 2004), which showed the superiority of heavy-tailed distributions in portfolio theory (see Rachev & Mittnik, 2000). The empirical investigation also shows that the percentage invested in each sector appears to be unstable over time. This observation emphasizes the importance of regularly recalibrating the portfolio. Table 3 reports the number of times required to achieve zero investment, number of times each sector is parametrically dominated by other sectors in the SSD sense, non-parametric SSD dominance, single (ratio between non-parametric and parametric SSD for each sector individually) percentage of tested SSD dominance and the percentages of times we obtain that the non-parametric SSD sector dominance is able to contain all of the SSD efficient set, according to the test proposed by Arvanitis et al. (2019). From Table 3, we observe that |$\bullet $| The number of times (in months) that zero percent invested is high for the FI, IN, EN and RE sectors; this fact is also confirmed by the number of times these sectors are dominated by the remaining sectors. |$\bullet $| Under the three assumptions, the number of times that parametric SSD verifies and zero investment happens are equal, which is not the case for the non-parametric evaluation. |$\bullet $| The difference between parametric and non-parametric SSD is more evident under the Gaussian hypothesis than using the stable and Student’s t distributions. |$\bullet $| Single percentages of tested SSD dominance are higher for stable and Student’s t distributions than for the Gaussian assumption. |$\bullet $| The percentage of times that non-parametric SSD dominance is able to contain all the SSD efficient set according to the test proposed by Arvanitis et al. (2019) is sufficiently high for all three distributional approaches (always higher than 50%). In particular, the stable and Student’s t distributions seem more capable of capturing dominance in the analysed cases. Notably, the number of times a given sector is dominated by the rest of the sectors is critically dependent on distributional assumptions. For instance, the FI sector is dominated 36 times when the stable sub-Gaussian assumed, while it is dominated 33 and 29 times for the Student’s t and Gaussian distributions, respectively. In contrast, the RE sector is dominated 22 and 13 times when the Gaussian and Student’s t distributions are respectively considered, and it is never dominated under the stable sub-Gaussian hypothesis. Comparing the overall number of dominances from the three different rules (i.e., mean-variance, df-mean-dispersion and alpha-mean-dispersion) results in 164, 151 and 141 SSD dominances, respectively. These results show significant differences between the three approaches; thus, distributional assumptions greatly affect outcomes. Moreover, according to the test proposed by Arvanitis et al., it could be deduced that a sector analysis is sufficiently able to determine those sectors that weakly SSD dominate all markets (i.e., they contain the SSD efficient set), in particular, when the stable and Student’s t distributional assumptions are considered. Thus, the proposed analysis can also be used to reduce the dimensionality of the portfolio problem. Interestingly, Tables 2 and 3 show the extent to which the dominance observed among some sectors impacts the composition of an optimal portfolio. While dominated sectors generally have lower percentages of optimal portfolio weights, dominant sectors earn the highest percentages of optimal portfolio weights. These results confirm the fact that the dominance observed among some sectors persists into optimal portfolio composition. This observation is of great practical importance because it allows investors to optimize their portfolios by considering only a few dominant sectors; it could also be significant in large-scale portfolio selection problems. We deduce from this ex-ante analysis that the best performing sectors are IT, HC, CD and CS. This result is also confirmed by Figures 2–3, which report three examples of the mean-variance, alpha-mean-dispersion and degree of freedom-mean-dispersion dominance. From Figures 2–3, it is worth noting that the dominated sectors have lower values for |$\alpha $| and |$\nu $| than the dominant sectors when we use stable and Student’t distributions, respectively. For instance, Figure 2 shows the case when the FI, IN and MA sectors are weakly dominated by the other sectors (such as IT, HC, CD and CS) in terms of the ICVM order. 3.3 An ex-post comparison among ‘efficient portfolios’ This section compares the ex-post wealth of portfolios optimised under various distributional hypotheses. The empirical analysis aims to determine whether an active investor can use these portfolio models to outperform the S&P 500 index and two other benchmarks: one based on the uniform strategy (investing the same percentage of wealth in each asset) and one based on a sector momentum strategy (which rebalances into the sectors that performed best in the previous year). Accordingly, starting from 1 December 2007, the mean-dispersion EFs under the three distributional assumptions for the asset returns are fitted monthly (every 20 trading days). In particular, 100 optimal portfolios obtained by minimizing the variance or dispersion measures for a fixed mean (according to problem (3.1)) and varying the mean from the mean of the global minimum dispersion portfolio are considered. It is assumed that each risk-averse investor maintains the same ex-ante level of mean returns over time when the three types of EFs are approximated. Thus, at each recalibration time (every 20 trading days), first a common index of stability (in the stable case) and a common value for the df (in the Student’s t case) are estimated and, as in Section 3.2, the ML estimator for all of the parameter estimates is used. Then, the EFs for the three distributional assumptions are fitted. Finally, out-of-sample wealth obtained starting with an initial wealth equal to one and using the solutions of optimal portfolios for each fixed level of mean are computed. For each distributional assumption, the wealth evolution of 100 portfolio strategies (one corresponding to each level of the mean) are obtained, and a total of 2289 ex-post daily portfolio observations are made. The same method is used to obtain the ex-post wealth of the uniform strategy and the sector momentum strategy. Table 4 Number of the portfolio strategies that SSD dominate the benchmarks or other portfolio strategies . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable Open in new tab Table 4 Number of the portfolio strategies that SSD dominate the benchmarks or other portfolio strategies . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable . Gaussian . Student . Stable . S&P 500 . Uniform . Momentum . Gaussian 2 |$\succ $| SSD 7 |$\succ $| SSD 6 |$\succ $| SSD 16 |$\succ $| SSD 5 |$\succ $| SSD 6 |$\succ $| SSD Student 15 |$\succ $| SSD 1 |$\succ $| SSD 8 |$\succ $| SSD 21 |$\succ $| SSD 9 |$\succ $| SSD 10 |$\succ $| SSD Stable 17 |$\succ $| SSD 11 |$\succ $| SSD 1 |$\succ $| SSD 24 |$\succ $| SSD 8 |$\succ $| SSD 11 |$\succ $| SSD S&P 500 2 |$\succ $| SSD 1 |$\succ $| SSD 1 |$\succ $| SSD Not comparable Not comparable Not comparable Uniform 13 |$\succ $| SSD 9 |$\succ $| SSD 8 |$\succ $| SSD Not comparable Not comparable Not comparable Momentum 9 |$\succ $| SSD 6 |$\succ $| SSD 6 |$\succ $| SSD Not comparable Not comparable Not comparable Open in new tab Table 5 Statistics on the benchmarks and best strategies . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 Open in new tab Table 5 Statistics on the benchmarks and best strategies . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 . S&P 500 . Uniform . Momentum . Best Gaussian . Best Student . Best Stable . VaR 5% 0.02061 0.02255 0.02349 0.02048 0.02035 0.02041 CVaR 5% 0.03391 0.03767 0.04107 0.03378 0.03340 0.03367 Sharpe 0.01421 0.03200 0.02935 0.02373 0.03782 0.04173 Sortino 0.01323 0.03004 0.02678 0.02222 0.03448 0.03903 C.E. RRA=2 1.00010 1.00036 1.00035 1.00022 1.00049 1.00051 C.E. RRA=4 0.99991 1.00014 1.00006 1.00004 1.00020 1.00028 C.E. RRA=8 0.99954 0.99968 0.99946 0.99967 0.99960 0.99982 Open in new tab Once the ex-post strategies are obtained, the non-parametric test (see Davidson & Jean-Yves, 2000) is used to evaluate whether SSD dominance applies among the returns of these ex-post strategies and the market benchmarks. We also evaluate some statistics of the ex-post returns obtained with the benchmark strategies and the three distributional approaches. In particular, we evaluate the risk measures |$VaR_{5\%}$| and |$CVaR_{5\%}$| (where |$VaR_q(x^{\prime}r)=-F_{x^{\prime}r}^{-1}(q)=-inf\lbrace v|P(x^{\prime}r \leqslant v>q)\rbrace $| and |$CVaR_{\gamma }(x^{\prime}r)=\frac{1}{\gamma }\int _0^{\gamma }VaR_{q}(x^{\prime}r) \,\textrm{d}q$|⁠), the risk-reward performance measures the Sharpe ratio |$SR(x^{\prime}r)=E(x^{\prime}r-r_f)/\sigma _{x^{\prime}r}$| and Sortino ratio |$ \textrm{SoR}(x^{\prime}r)=E(x^{\prime}r-r_f)/(E((y-x^{\prime}r)_{+}^{2}))^{1/2}$| (where |$r_f$| is the risk-free return—here considered null—and |$\sigma _{x^{\prime}r}$| is the portfolio standrad deviaition; the function |$(v)_{+}^{2}=( \textrm{max}(v,0))^2$| and |$y$| is the minimum acceptable return level for the investors, which is null in our analysis, i.e., |$y=0$|⁠, (for further details, see Sharpe, 1994 and Sortino & Price, 1994)), and the certainty equivalent gross returns (C.E.) for the power utility functions |$U(x^{\prime}r)=((1+ x^{\prime}r)^{1-\rho }-1)/(1-\rho ))$| with relative risk aversion (RRA) |$\rho =2,4,8$|⁠. We evaluate these statistics for the benchmarks and average the statistics of the best 16, 21 and 24 strategies for the Gaussian, Student’s t and stable approaches where SSD dominates the S&P 500 index (according to Table 4). The results of this ex-post analysis are reported in Tables 4 and 5. Table 4 reports the number of portfolio strategies (out of a possible 100) of each distributional approach where SSD dominates the benchmark strategies or other portfolio strategies. Table 5 reports the statistics of the benchmarks and of the best Gaussian, Student’s t and stable strategies. From this comparison, we can deduce that |$\bullet $| Benchmark strategies (S&P 500 index, uniform strategy and sector momentum strategy) are not SSD comparable by themselves, even if there exist deep differences between them. All benchmark strategies present risk measures VaR and CVaR higher than distributional strategies. However, the risk-reward performance and certainty equivalent values of the Uniform strategy and sector momentum strategy are always higher than those for the Gaussian approach and slightly lower than those for the other distributional approaches. For the Gaussian, Student’s t and stable approaches, respectively, 16, 21 and 24 strategies SSD dominate the S&P 500 index, while the number of distributional strategies that SSD dominate the uniform strategy and the sector momentum strategy is much smaller. Moreover, according to Pflug et al. (2012) and DeMiguel et al. (2009), the uniform strategy performs very well because 13, 9 and 8 respective strategies of the Gaussian, Student’s t and stable approaches are actually dominated by the uniform strategy. Similarly, the sector momentum strategy SSD dominates 9, 6 and 8 of the Gaussian, Student’s t and stable strategies, respectively, confirming that sector outperformance is often persistent. |$\bullet $| The stable and the Student’s t approaches perform better than the Gaussian approach in terms of SSD dominance risk measures, risk-reward performance and certainty equivalent values, even in the ex-post analysis. Thus, we confirm that the distributional hypotheses seriously influence the outcomes. The fact that normal distribution may yield misleading results has already been documented in several studies (see, e.g., Rachev & Mittnik, 2000). With respect to the Student’s t assumption, our empirical findings show that it may yield significant improvements from the classical mean-variance approach, especially if compared with the alpha-mean-dispersion criterion. Therefore, it can be asserted that the Student’s t distribution may be more reliable and the stable sub-Gaussian distribution may present more appropriate dominance rules. |$\bullet $| All of these distributional approaches (Gaussian, Student’s t and stable) present several portfolio strategies that outperform the S&P 500 index and some strategies that outperform the uniform strategy and the sector momentum strategy. This result emphasizes the managerial relevance and the impact of sector dominance on optimal investors’ choices in the US stock market. Overall, these results confirm and strongly support the assertion that weak MSD-based methodologies can effectively compare and rank different market sectors. Interestingly, the observed dominance of the optimal choices has the power to explain the percentage concentration of invested stocks. Finally, the significant differences observed between the three approaches suggest that portfolio selection models must account for the asymptotic (tail) behaviour of returns. 4. Conclusions This study introduces a methodology for comparing portfolios homogeneous with respect to market sectors relying on MSD principles. In this context, a possible application aimed at ranking S&P 500 sectors (namely, IT, FI, HC, CD, IN, CS, EN, UT, RE and MA) is proposed. In practice, the proposed dominance rules can be used by non-satiable risk-averse investors to identify the best market sectors in which to invest. The results show that this method yields remarkably different outcomes from the more traditional mean-variance approach. The non-parametric tests suggest that both stable symmetric and Student’s t distributions are preferable to the Gaussian approach for identifying MSD between sectors. The primary contribution of the empirical comparison is to examine the impact of the distributional assumptions on asset allocation decisions. In particular, the significant differences observed between the mean-variance and the stable Paretian approaches suggest that portfolio selection models must consider the asymptotic (tail) behaviour of returns. Therefore, this paper proposes another family of heavy-tailed distributions (Student’s t) and shows that SSD can be verified by comparing the values of the degree of freedom, dispersion and location parameters. It also discusses the implications of various distributional hypotheses on optimal portfolio choices and highlights the notable relationship between stochastic dominance among market sectors and the fraction of optimal portfolios. Finally, it emphasizes the managerial relevance of our results with an ex-post analysis that compare the different distributional approaches with three alternative benchmarks. 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Invest. , 3 , 59 – 65 . Google Scholar Crossref Search ADS WorldCat Stubbs , R. A. & Vance , P. ( 2006 ) Computing return estimation error matrices for robust optimization . Axioma White Paper . Szegö , G. ( 1994 ) Risk Measures for the 21st Century . Chichester : Wiley . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Appendix. A Proof Corollary 2.1. Since any linear combination |$x\in \mathbb{R}^n$| of |$\alpha $|-stable sub-Gaussian vector |$r$| (⁠|$\alpha \geqslant 1$|⁠) with location (mean) |$\mu $| and dispersion |$Q$| is still an |$\alpha $|-stable sub-Gaussian distributed with parameter |$ x^{\prime}\mu $| and dispersion parameter |$ \sigma = \sqrt{x^{\prime}Qx}$|⁠, then it follows the thesis as a consequence of Theorem 2.1. Proof Corollary 2.2. Since |$\forall $| |$\alpha>1$|⁠, we have that |$\sigma =\frac{\pi }{2\varGamma (1-\frac{1}{\alpha })}MAD_{X}$|⁠, where |$MAD_{X}=E(|X-E(X)|)$|⁠; then, the thesis of Corollary 2.2. holds. Proof Theorem 2.2. Let |$Z_i=\frac{X_i-\mu _i}{\sigma _i}$| (for |$i=1,2$|⁠) be such that |$Z_1\sim T_{\nu _1}(1,0)$| and |$Z_2\sim T_{\nu _2}(1,0)$|⁠. Analytical and empirical studies ensure that the distribution functions of |$Z_1$| and |$Z_2$| have a single crossing point. Conceptually, the distribution function can be written as follows: $$\begin{equation} F_{\nu}(z)=\frac{1}{2}\left(1+\textrm{sgn}(z)\left(1-I_{\left(\frac{\nu}{z^2+\nu}\right)}\left(\frac{\nu}{2},\frac{1}{2}\right)\right)\right), \end{equation}$$(A.1) where |$I_{(x)}(a,b)$| is the regularized Beta function (also known as the incomplete Beta function ratio) (see, e.g., Shaw (2006)). We can distinguish three particular cases: |$z=0$|⁠, we have |$F_{\nu }(0)=\frac{1}{2}$|⁠, |$\forall \,\, \nu>0$|⁠; |$\forall \,\, z>0$|⁠, we have |$F_{\nu }(z)=1-\frac{1}{2}I_{\left (\frac{\nu }{z^2+\nu }\right )}\left (\frac{\nu }{2},\frac{1}{2}\right )$|⁠; and |$\forall \,\, z<0$|⁠, we have |$F_{\nu }(z)=\frac{1}{2}I_{\left (\frac{\nu }{z^2+\nu }\right )}\left (\frac{\nu }{2},\frac{1}{2}\right )$|⁠. From the properties of the regularized Beta function (see, e.g., Johnson et al., 1995 and literature therein) and from the fact that |$0<\frac{\nu }{z^2+\nu }<1$| for any |$\forall \,\, z\neq 0$| and |$\nu>0$|⁠, it clearly follows that |$F_{\nu _1}(z)<F_{\nu _2}(z)$|⁠, |$\forall \,\, z<0$|⁠, and |$\nu _1>\nu _2$|⁠; |$F_{\nu _1}(z)>F_{\nu _2}(z)$|⁠, |$\forall \,\, z>0$|⁠, and |$\nu _1>\nu _2$|⁠; |$F_{\nu _1}(0)=F_{\nu _2}(0)=\frac{1}{2}$|⁠, |$\forall \,\, \nu>0$|⁠. Thus, the distribution functions cross only once, and |$F_{\nu _1}(z)$| is below |$F_{\nu _2}(z)$| to the left crossing point. Consequently (see, e.g., Hanoch & Levy (1969) and Rachev et al. (2008b)), it follows that |$Z_1$| SSD |$Z_2$| (which is equivalent to |$X_1$| SSD |$X_2$| as proved by Ortobelli & Rachev, 2001). This completes the proof of Theorem 2.2. Proof Corollary 2.3. Since any linear combination |$x\in \mathbb{R}^n$| of Student’s t vector |$r$| (⁠|$\mu \geqslant 2$|⁠) with location (mean) |$\mu $| and dispersion |$Q$| is still a Student’s t distribution with parameter |$ x^{\prime}\mu $| and dispersion parameter |$ \sigma = \sqrt{x^{\prime}Qx}$|⁠, then it follows the thesis as a consequence of Theorem 2.2. Proof Corollary 2.4. Since |$\forall $| |$\nu \geqslant 2$|⁠, we have |$\textrm{Std}=\sigma \sqrt{\frac{\nu }{\nu -2}}$|⁠, we get the results of Corollary 2.4. © The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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Multivariate stochastic dominance applied to sector-based portfolio selection

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Multivariate stochastic dominance applied to sector-based portfolio selection

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