Noureddine Kouaissah, Sergio Ortobelli, Tomas Tichy

University of Bergamo, Italy and

VŠB-Technical University of Ostrava, Czech Republic

CMS Bergamo, 05/2017

Agenda

Motivations

Stochastic dominance between sectors

Empirical analysis

Conclusion and future research

In the financial literature, it is well known that asset returns are not normallyditributed, see Mandelbrot (1963), Fama (1965), and Rachev and Mittnik(2000).

This paper focuses on the issue of ranking different financial sectors from thepoint of view of different non-satiable investors.

As our decision problem is concerned with multivariate random elements (i.e.financial sectors), the aim of this paper is to introduce a dominance rule thatcan be simple and applicable to a multivariate framework, relying on thetheory of stochastic orderings.

Motivations

In the univariate case, we say that 𝑋 dominates 𝑌 with respect to a givenunivariate order of preferences ≻ and we write 𝑋 ≻ 𝑌 when appropriateconditions are satisfied. Generally, these conditions involve the distributionfunctions of 𝑋 and 𝑌, say 𝐹𝑋 and 𝐹𝑌. An especially useful ordering is the secondorder (or increasing concave) stochastic dominance (SSD): we say that 𝑋 SSD 𝑌if and only if

−∞𝑡

𝐹𝑋(𝑧) − 𝐹𝑌(𝑧) 𝑑𝑧 ≤ 0 for any 𝑡 in ℝ

see, among, others Levy (1992).

Generally, it is not trivial to extend any given order of preferences ≻ to themultivariate case, especially since in some practical cases it could be verydifficult to satisfy the conditions of multivariate dominance. In this regard, thenatural generalizations of the first- and second-degree stochastic orderings canbe found, for instance, in Muller and Stoyan (2002).

Stochastic dominance between sectors

Suppose that there are two sectors 𝐴 and 𝐵, composed, respectively, of

𝑛 and 𝑠 assets. Denote by 𝑥 = 𝑥1, 𝑥2, … , 𝑥𝑛′ and 𝑦 = 𝑦1, 𝑦2, … , 𝑦𝑠

′ the

vectors which contain the percentages of investments in the risky assets of

sectors 𝐴 and 𝐵, respectively. Assume that no short sales are allowed.

Definition 1. We say that a sector 𝐴 with 𝑛 assets strongly dominates sector 𝐵with 𝑠 assets with respect to a multivariate preference ordering ≻ if, for anyvector of returns 𝑌𝐵 composed of 𝑡 ≤ 𝑢 = 𝑚𝑖𝑛 𝑠, 𝑛 assets from sector 𝐵, thereexists a vector 𝑋𝐴 of sector 𝐴 such that 𝑋𝐴 ≻ 𝑌𝐵. Similarly, we say that a sector 𝐴with 𝑛 assets weakly dominates another sector 𝐵 with 𝑠 assets with respect tothe multivariate preference ordering ≻ if for any given portfolio of sector 𝐵 withreturn 𝑦′𝑌𝐵, there exists a portfolio of the sector 𝐴 with return 𝑥′𝑋𝐴 such that𝑥′𝑋𝐴 ≻ 𝑦′𝑌𝐵.

Stochastic dominance between sectors

Suppose that the returns of sectors 𝐴 and 𝐵 are jointly ellipticallydistributed with finite variance. Suppose also that the two sectors have thesame number of assets 𝑛, vector of averages 𝜇𝐴 and 𝜇𝐵, and variance-covariance matrices 𝒬𝐴 and 𝒬𝐵 such that 𝜇𝐴 ≥ 𝜇𝐵 and 𝒬𝐴 − 𝒬𝐵 is negativesemi-definite.

Then, sector 𝐴 strongly dominates sector 𝐵 with respect to the increasingconcave multivariate order (see Muller and Stoyan (2002)). Moreover,under these assumptions sector 𝐴 weakly dominates sector 𝐵 with respectto the concave order, because 𝑥′𝜇𝐴 ≥ 𝑥′𝜇𝐵 and 𝑥′𝒬𝐴𝑥 ≤ 𝑥′𝒬𝐵𝑥 for anyvector 𝑥 ≥ 0.

Note that this weak dominance between elliptically distributed vectors isalso known in the literature as the increasing positive linear concavemultivariate order (see Muller and Stoyan (2002)).

Example

The increasing positive linear concave multivariate ordering defined in Example1 is strictly related to the mean–variance rule, which is widely used in finance tosolve the portfolio optimization problem.

Robust portfolio optimization

According to Ceria and Stubbs (2006) (see also Fabozzi et al. (2007)),assume that the vector of expected returns 𝜇 = 𝜇1, 𝜇2, … , 𝜇𝑛 is normallydistributed. Then, given an estimate of expected returns 𝜇 and a covariancematrix of the estimates of expected returns ∑, it is assumed that the trueexpected returns lie inside the confidence region:

𝜇 − 𝜇 ′∑−1 𝜇 − 𝜇 ≤ 𝜅2 (1)

with probability 100𝜂 per cent, where 𝜅2 = 𝜒𝑛2 1 − 𝜂 and 𝜒𝑛

2 is the percentileof the chi-squared distribution with 𝑛 degrees of freedom.

Robust portfolio optimization approach

Under these assumptions, the classical mean-variance problem can be modelledwith a robust approach. In particular, any optimal robust portfolio is solution ofthe following optimization problem:

min𝑥 𝑥′𝒬𝑥

∑𝑖=1𝑛 𝑥𝑖 = 1; 𝑥𝑖 ≥ 0 (2)

𝜌 𝑥 = 𝑚

where 𝜌 𝑥 = 𝑥′ 𝜇 − 𝜅 𝑥′∑𝑥 is the corrected mean and represents an

alternative reward measure. Clearly, this formulation generalizes the classicalportfolio optimization problem based on the mean-variance approach,when 𝜅 = 0.

Robust portfolio approach

Stable distributions are described by their characteristic function, fordeeper discussion see among others Rachev and Mittnik (2000).

The characteristic function (and thus the density function) of a stabledistribution is described by four parameters:

• 𝛼: the index of stability or the shape parameter, 𝛼 ∈ 0,2• 𝛽: the skewness parameter, 𝛽 ∈ −1,1 ;• 𝜎: the scale parameter, 𝜎 ∈ 0,+∞ ;• 𝜇: the location parameter, 𝜇 ∈ −∞,+∞ .

When a random variable 𝑋 follows the 𝛼-stable distribution characterized by those parameters, then we denote 𝑋~𝑆𝛼 𝜎, 𝛽, 𝜇 .

A sub-Gaussian distribution is a special case of an 𝛼-stable distribution, obtained by setting 𝛽 = 0.

Stable distributions

Following Ortobelli et al. (2016), we determine a ranking criteria aiming tocompare sub-Gaussian distributions according to SSD. In particular, it has beenfound that SSD can be verified by comparing the values of the stability, dispersionand location parameters.

Theorem 1 (Ortobelli et al. (2016)). Let 𝑋1~𝑆𝛼1 𝜎1, 0, 𝜇1 and 𝑋2~𝑆𝛼2 𝜎2, 0, 𝜇2 .

Suppose 𝛼1 > 𝛼2 > 1, 𝜎1 ≤ 𝜎2 and 𝜇1 ≥ 𝜇2. Then 𝑋1SSD 𝑋2.

The results of Theorem 1 can be extended to a multivariate context, generalizingthe multivariate mean–variance approach described in Example 1, by taking intoaccount the asymptotic behaviour of the tail distributions. This yields theasymptotic multivariate dominance between financial sectors.

Asymptotic multivariate dominance

We apply the asymptotic multivariate weak dominance rule in order tocompare empirically the SP 500 sectors. We compare the results of thismethod with those obtained by the weak concave multivariate order definedin Example 1.

First of all, we examine the statistical characteristics of the returns of eachsector. Then, we verify the dominance rules proposed over the decade 2005–2017.

Then, we determine the so-called alpha–mean–dispersion efficient frontier,computing the portfolio with minimum dispersion 𝑥′𝒬𝑥 for any fixed mean𝑥′𝜇𝐴, and finally we compare the efficient frontiers determining whether thecondition for the weak asymptotic increasing concave multivariate holds.

Moreover, we present a similar analysis, assuming that the returns of eachsector are normally distributed and we compare the mean–variance efficientfrontiers, as suggested in Example 1.

Empirical analysis

We consider the returns of the stock sectors of the S&P 500, through theperiod January 2005 till January 2017, namely:

1. Information Technology (IF)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)10. Materials (MA)

Empirical analysis

We compare the efficient frontiers of the SPA 500 sectors starting fromJanuary 1, 2005 until January 1, 2017.

Every month (20 trading days) we estimate the mean– variance efficientfrontiers and the stable–mean–dispersion efficient frontiers of the sectors, byusing the assets which were active during the last 4 years (1000 dailyhistorical observations).

Therefore, every month, we fit the efficient frontier solving the optimizationproblem for 40 levels of the mean.

Empirical analysis

Table 1: Average of weights invested in each sector over different periods

From To MA 1000 IT FI HC CD IN CS EN UT RE MA

Dec. 2008 Jan.

2017 all period 0.122 0.008 0.279 0.189 0.043 0.228 0.003 0.036 0.095 0.008

Dec. 2008 Sept.

2009

Subprime

crisis 0.067 0.035 0.391 0.160 0.018 0.316 0.000 0.008 0.000 0.009

Sept.

2009

Jan.

2013

EU credit

crisis 0.120 0.006 0.245 0.277 0.000 0.180 0.000 0.020 0.149 0.008

Jan. 2013 Jan.

2017

post

crisis 0.133 0.005 0.288 0.121 0.082 0.253 0.006 0.054 0.057 0.007

Stabe distribution hypothesisDec. 2008 Jan.

2017

all period

0.110 0.005 0.256 0.136 0.029 0.223 0.008 0.048 0.173 0.015

Dec. 2008 Sept.

2009

Subprime

crisis 0.023 0.005 0.476 0.074 0.034 0.259 0.007 0.020 0.039 0.061

Sept.

2009

Jan.

2013

EU credit

crisis 0.065 0.000 0.270 0.182 0.000 0.175 0.001 0.037 0.254 0.018

Jan. 2013 Jan.

2017

post

crisis 0.160 0.009 0.208 0.109 0.053 0.256 0.013 0.063 0.129 0.004

Table 1: Statistics for the assets returns, McCulloch’s stable paramters estimation

IT FI HC CD IN CS EN UT RE MA allMean % 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

rejected1 1 1 1 1 1 1 1 1 1 1

0% invest.

times 1 29 0 0 42 0 69 0 22 1 n.c.

Stabe distribution hypothesis Alpha 𝛼 1.565 1.444 1.575 1.550 1.533 1.585 1.603 1.630 1.449 1.527 1.542

Beta 𝛽 0.0088 -0.0257 0.0296 00472 -0.0280 -0.0143 -0.0495 -0.0912 -0.0563 -0.0119 -0.0097

Sigma 𝜎 % 1.156 1.160 0.986 1.176 1.066 0.760 1.380 0.762 1.132 1.108 1.085

Delta 𝛿 % 0.0519 0.0280 0.0633 0.0744 0.0354 0.0477 0.0304 0.0079 0.1063 0.0503 0.0507

0% invest.

times3 36 0 0 46 0 54 0 0 2 n.c.

#

dominated0 36 0 0 28 0 11 0 0 2 n.c.

Figure 1: Example of Mean-Variance dominance

Figure 2: Example of alpha-Mean-Dispersion dominance

We have introduced a methodology to compare different financial sectorsbased on asymptotic multivariate stochastic dominance.

From a practical point of view, the proposed dominance rules can be usedby non-satiable risk averse investors in order to identify the best financialmarket to invest in.

The primary contribution of the empirical comparison presented in thispaper is the analysis of the impact of the distributional assumptions onasset allocation decisions.

Further research could involve theoretical and empirical studies:

1. A natural extension of this research would be a multivariate stochasticdominance rule that also takes skewness into account.

2. The use of nonparametric approaches

Conclusion

1. Chopra VK, Ziemba W T (1993). The effect of errors in means, variances, and covariances on optimal

portfolio choice. J Portf Manag 19(2):6–11

2. Fabozzi FJ, Kolm PN, Pachamanova DA, Focardi SM (2007) Robust portfolio optimization. The Journal of

Portfolio Management 33(3):40–48

3. Fama EF (1965). Portfolio analysis in a stable Paretian market. Manag Sci 11:404–419

4. Mandelbrot BB (1963). The variation of certain speculative prices. J Bus 26:394–419

5. Ortobelli S, Lando T, Petronio F, Tichy T (2016) Asymptotic Multivariate Dominance: A financial

application. Metholo Comput Appl Probab 18:1097-1115

6. Rachev S, Mittnik S (2000) Stable Paretian models in finance. Wiley, Chichester

References