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
Motivations In the financial literature, it is well known that asset returns are not normally ditributed, see Mandelbrot (1963), Fama (1965), and Rachev and Mittnik (2000). This paper focuses on the issue of ranking different financial sectors from the point 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 that can be simple and applicable to a multivariate framework, relying on the theory of stochastic orderings.
Stochastic dominance between sectors In the univariate case, we say that X dominates Y with respect to a given univariate order of preferences and we write X Y when appropriate conditions are satisfied. Generally, these conditions involve the distribution functions of X and Y, say F X and F Y. An especially useful ordering is the second order (or increasing concave) stochastic dominance (SSD): we say that X SSD Y if and only if t F X (z) F Y (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 the multivariate case, especially since in some practical cases it could be very difficult to satisfy the conditions of multivariate dominance. In this regard, the natural generalizations of the first- and second-degree stochastic orderings can be 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 = x 1, x 2,, x n and y = y 1, y 2,, y s 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 B with s assets with respect to a multivariate preference ordering if, for any vector of returns Y B composed of t u = min s, n assets from sector B, there exists a vector X A of sector A such that X A Y B. Similarly, we say that a sector A with n assets weakly dominates another sector B with s assets with respect to the multivariate preference ordering if for any given portfolio of sector B with return y Y B, there exists a portfolio of the sector A with return x X A such that x X A y Y B.
Example Suppose that the returns of sectors A and B are jointly elliptically distributed with finite variance. Suppose also that the two sectors have the same number of assets n, vector of averages μ A and μ B, and variancecovariance matrices Q A and Q B such that μ A μ B and Q A Q B is negative semi-definite. Then, sector A strongly dominates sector B with respect to the increasing concave multivariate order (see Muller and Stoyan (2002)). Moreover, under these assumptions sector A weakly dominates sector B with respect to the concave order, because x μ A x μ B and x Q A x x Q B x for any vector x 0. Note that this weak dominance between elliptically distributed vectors is also known in the literature as the increasing positive linear concave multivariate order (see Muller and Stoyan (2002)).
Robust portfolio optimization approach The increasing positive linear concave multivariate ordering defined in Example 1 is strictly related to the mean variance rule, which is widely used in finance to solve 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 = μ 1, μ 2,, μ n is normally distributed. Then, given an estimate of expected returns μ and a covariance matrix of the estimates of expected returns, it is assumed that the true expected returns lie inside the confidence region: μ μ 1 μ μ κ 2 (1) with probability 100η per cent, where κ 2 = χ n 2 1 η and χ n 2 is the percentile of the chi-squared distribution with n degrees of freedom.
Robust portfolio approach Under these assumptions, the classical mean-variance problem can be modelled with a robust approach. In particular, any optimal robust portfolio is solution of the following optimization problem: min x x Qx n x i = 1; x i 0 (2) i=1 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 classical portfolio optimization problem based on the mean-variance approach, when κ = 0.
Stable distributions Stable distributions are described by their characteristic function, for deeper discussion see among others Rachev and Mittnik (2000). The characteristic function (and thus the density function) of a stable distribution 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~S α σ, β, μ. A sub-gaussian distribution is a special case of an α-stable distribution, obtained by setting β = 0.
Asymptotic multivariate dominance Following Ortobelli et al. (2016), we determine a ranking criteria aiming to compare sub-gaussian distributions according to SSD. In particular, it has been found that SSD can be verified by comparing the values of the stability, dispersion and location parameters. Theorem 1 (Ortobelli et al. (2016)). Let X 1 ~S α1 σ 1, 0, μ 1 and X 2 ~S α2 σ 2, 0, μ 2. Suppose α 1 > α 2 > 1, σ 1 σ 2 and μ 1 μ 2. Then X 1 SSD X 2. The results of Theorem 1 can be extended to a multivariate context, generalizing the multivariate mean variance approach described in Example 1, by taking into account the asymptotic behaviour of the tail distributions. This yields the asymptotic multivariate dominance between financial sectors.
Empirical analysis We apply the asymptotic multivariate weak dominance rule in order to compare empirically the SP 500 sectors. We compare the results of this method with those obtained by the weak concave multivariate order defined in Example 1. First of all, we examine the statistical characteristics of the returns of each sector. 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 mean x μ A, and finally we compare the efficient frontiers determining whether the condition for the weak asymptotic increasing concave multivariate holds. Moreover, we present a similar analysis, assuming that the returns of each sector are normally distributed and we compare the mean variance efficient frontiers, as suggested in Example 1.
Empirical analysis We consider the returns of the stock sectors of the S&P 500, through the period 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 from January 1, 2005 until January 1, 2017. Every month (20 trading days) we estimate the mean variance efficient frontiers and the stable mean dispersion efficient frontiers of the sectors, by using the assets which were active during the last 4 years (1000 daily historical observations). Therefore, every month, we fit the efficient frontier solving the optimization problem for 40 levels of the mean.
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 Jan. 2013 Dec. 2008 Dec. 2008 Sept. 2009 Jan. 2013 Sept. 2009 Jan. 2013 Jan. 2017 Jan. 2017 Sept. 2009 Jan. 2013 Jan. 2017 Subprime crisis 0.067 0.035 0.391 0.160 0.018 0.316 0.000 0.008 0.000 0.009 EU credit crisis 0.120 0.006 0.245 0.277 0.000 0.180 0.000 0.020 0.149 0.008 post crisis 0.133 0.005 0.288 0.121 0.082 0.253 0.006 0.054 0.057 0.007 Stabe distribution hypothesis all period 0.110 0.005 0.256 0.136 0.029 0.223 0.008 0.048 0.173 0.015 Subprime crisis 0.023 0.005 0.476 0.074 0.034 0.259 0.007 0.020 0.039 0.061 EU credit crisis 0.065 0.000 0.270 0.182 0.000 0.175 0.001 0.037 0.254 0.018 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 all 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 rejected 1 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. times # dominated 3 36 0 0 46 0 54 0 0 2 n.c. 0 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
Conclusion We have introduced a methodology to compare different financial sectors based on asymptotic multivariate stochastic dominance. From a practical point of view, the proposed dominance rules can be used by non-satiable risk averse investors in order to identify the best financial market to invest in. The primary contribution of the empirical comparison presented in this paper is the analysis of the impact of the distributional assumptions on asset allocation decisions. Further research could involve theoretical and empirical studies: 1. A natural extension of this research would be a multivariate stochastic dominance rule that also takes skewness into account. 2. The use of nonparametric approaches
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