Journal of Risk

Risk.net

Genetic algorithm-based portfolio optimization with higher moments in global stock markets

Saranya Kshatriya and Krishna Prasanna

  • The distributional characteristics of the stock market returns are analysed.
  • The importance of higher moments in emerging as well as in developed markets are investigated
  • Higher moment’s model outperforms the traditional mean variance model especially in emerging markets.

Markowitz’s mean–variance portfolio model is widely used in the field of investment management. The changing dynamics of markets have resulted in higher uncertainties surrounding returns. Returns have often been found to be skewed and extreme events observed to be frequent. These characteristics are measured by skewness and kurtosis, which need to be accommodated in the definition of risk. They should also be included in the portfolio optimization process. The purpose of this paper is to investigate the impact of including higher moments in the estimation of risk in the process of international portfolio diversification. Our study is based on a sample of thirty-three globally traded stock market indexes, including emerging as well as developed markets, for the period between 2000 and 2012. Our inclusion of skewness and kurtosis makes portfolio optimization a nonlinear, nonconvex and multi-objective problem; this has been solved with the use of a genetic algorithm. Empirical results demonstrate that the higher moments model outperforms the traditional mean–variance model across the time period. The results of this study may be useful to fund managers, portfolio managers and investors, aiding them in understanding the behavior of the stock market and in selecting an optimal portfolio model among various alternative portfolio models.

1 Introduction

Markowitz’s celebrated mean–variance theory (1952, 1959) focused mainly on the first two moments of returns, namely, the expected return and the variance. Of late, market dynamics have changed considerably, and prior studies have also noted that assets exhibit non-Gaussian behavior (Mandelbrot 1963; Fama 1965; Peiró 1999; Harvey and Siddique 1999; Premaratne and Bera 2000). Hence, the distributional properties of an asset cannot be completely captured by its return and variance.

Stock markets are often volatile due to extreme events such as financial crises, and this results in excess kurtosis. Markets also often move in a single direction, exhibiting either a bullish or a bearish pattern. This unidirectional movement indicates the presence of skewness in the return distribution. In all such cases, the variance alone would not be sufficient to capture the risk of an asset accurately (Lai 1991; Samuelson 1970). Given these volatility patterns, the traditional mean–variance theory would lead to suboptimal portfolios. Hence, it is essential to incorporate the non-Gaussian behavior of markets in optimal asset allocation decisions. Consequently, a more generic model that captures these risk dynamics needs to be considered in portfolio selection and allocation procedures.

The existing literature stands divided on the significance of higher moments. Some state they should not be neglected unless there is reason to believe the asset returns are normally distributed and the utility function is quadratic, while others state they are irrelevant to the investor’s decision (Samuelson 1970; Arditti 1971; Scott and Horvath 1980; Lai 1991; Konno and Suzuki 1995; Chunhachinda et al 1997; Rubinstein 2002; Prakash et al 2003; Lai et al 2006).

In addition, one set of studies indicates that higher moments need to be incorporated in investment decisions (Lai 1991; de Athayde and Flôres Jr 2004; Jondeau and Rockinger 2006; Lai et al 2006; Pindoriya et al 2010; Vermorken et al 2012), while another indicates that mean–variance portfolios are more efficient than the portfolios constructed using higher moments (Konno and Suzuki 1995; Chunhachinda et al 1997; Prakash et al 2003; Maillet and Merlin 2009). Given the context, this paper investigates the controversy over the issue of considering higher moments in portfolio selection.

This paper investigates the distributional characteristics of stock market returns and analyzes the significance of higher moments. In particular, it examines the impact of including higher moments in the risk estimation process for international asset allocation. The sample data comprises thirty-three global stock market indexes, which include emerging as well as developed markets, over the period 2000–2012. The portfolios are constructed based on three alternate models: the mean–variance (MV) model, the mean–variance–skewness (MVS) model and the mean–variance–skewness–kurtosis (MVSK) model.

A genetic algorithm has been used to solve this nonlinear, multi-objective optimization; it allocates optimal weights to the stocks in the portfolio. For robustness, the higher moments model is evaluated based on the realized performance of the portfolios.

The empirical results indicate that the MVSK and MVS models outperform the MV model when emerging markets, both individually and in combination with developed markets, are considered. However, there was no clear evidence of any one model outperforming the others in the case of developed markets. This research provides empirical evidence about the relevance of higher moments in international portfolio decisions.

The contributions of this study are manifold. First, this paper identifies the role of higher moments in emerging and developed markets, and provides insights into their market behavior that may be relevant to international investors. Emerging markets are found to be more sensitive to higher moments compared with developed markets.

Second, unlike prior studies – which have adopted a polynomial goal programming (PGP) model, that is, a local optimizer – this paper adopts a global optimization technique to solve the nonconvex, nonlinear, multi-objective function. Chang et al (2009) and Li et al (2010) have used genetic programming to solve the MVS model. However, to our knowledge, the present paper is the first to use genetic programming in the context of solving an optimization problem that includes four moments simultaneously.

Third, unlike most of the past studies that have evaluated the significance of higher moments using expected returns, we evaluated the use of both expected returns and realized returns in investigating consistency in the out-of-sample performance of the higher moments model.

Fourth, the study considered data from 2000 to 2012. This period is characterized by the occurrence of several extreme events, such as the global financial crisis of 2007–08, the European crisis, the Korean conflict and others. Thus, our sample length allows us to analyze the effects of economic cycles on asset price behavior.

Fifth, both emerging and developed markets were considered in the study. This allows us to investigate the relevance of higher moments in emerging markets in comparison with developed markets. Thus, this paper contributes to the academic literature in the broad area of portfolio management, while also providing insights into portfolio diversification decisions that may be useful to fund managers, portfolio managers and investors seeking to invest in global markets.

The remainder of this paper is organized as follows. Section 2 describes the data and methodology used in constructing the portfolio using MV, MVS and MVSK models. Section 3 presents our empirical results and findings, and Section 4 reports our conclusions.

2 Data and methodology

2.1 Data

Monthly values of thirty-three stock market indexes (including fifteen emerging and eighteen developed markets) are considered in this study. The sample period spans from January 2000 to December 2012. Our market classification is based on the Standard & Poor’s (S&P) emerging and developed markets list as of May 31, 2012. The required index data is sourced from the Bloomberg database.

In order to check the validity of higher moments, the sample data is clustered into three panels. The first panel consists of fifteen emerging market indexes, the second panel consists of eighteen developed market indexes and the third consists of both emerging and developed market indexes (thirty-three indexes in total).

2.2 Model assumptions

We assume that an investor chooses their portfolio from n risky stocks. The weightages of the stocks in the portfolio are allocated in such a way that the investor’s expected utility of wealth gets maximized. Following Lai (1991) and Kemalbay et al (2011), we also assume the following.

  • Investors are risk-averse individuals, who maximize the expected utility of their end-of-period wealth.

  • There are n risky assets, and investors do not have access to any risk-free asset. This implies that the portfolio weights must sum to one.

  • All assets are marketable and perfectly divisible.

  • The capital market is perfect; there are no taxes or transaction costs.

  • Short selling is not allowed, implying that portfolio weights must be positive.

2.3 Alternate portfolio models

The three alternative portfolio models considered in this study are MV, MVS and MVSK. Higher moments contribute to portfolio performance in the following way. Positive skewness indicates that smaller negatives (rather than higher negative returns) are more favorable from an investor’s perspective; therefore, skewness must be maximized so as to avoid negative skewness. Minimizing kurtosis infers that investors do not want extreme events to happen. Investors do not prefer the occurrence of extreme negative events, as kurtosis includes both positive and negative sides. Minimizing kurtosis will thus protect investors from extreme losses as well as extreme gains.

Let Rp be the return from the portfolio, X be the weight vector, R be the vector of individual returns, V be the variance, S be the skewness and K be the kurtosis. The first four moments of the portfolio are calculated as

  M(Rp) =E(X(R-R¯)),  
  V(Rp) =E(X(R-R¯))2,  
  S(Rp) =E(X(R-R¯))3,  
  K(Rp) =E(X(R-R¯))4.  

The preference for moments of a distribution is positive for every odd central moment and negative for every even central moment (Scott and Horvath 1980).

2.3.1 The MV model

The objectives of the MV model are to (i) maximize the mean return and (ii) minimize the standard deviation of the portfolio:

  minimizef(x)=-λ(r¯)+(1-λ)(i=1N(rp,i-r¯)2N)1/2,  
  suchthatXI=1,rp,i=xirj,r¯=i=1Nrp,iN.   (2.1)

2.3.2 The MVS model

The three objectives of the MVS model are to (i) maximize the mean return, (ii) minimize the standard deviation and (iii) maximize the skewness of the portfolio:

  minimizef(x)=-λ1(r¯)+λ2(i=1N(ri-r¯)2N)1/2-λ3(i=1N((ri-r¯)3/N)((ri-r¯)2/N)3/2),  
  suchthatXI=1,rp,i=xirj,r¯=i=1Nrp,iN,i=1Nλi=1.   (2.2)

2.3.3 The MVSK model

The conflicting objectives of the MVSK model are to (i) maximize the mean and skewness, while (ii) minimizing the standard deviation and kurtosis:

  minimizef(x)=-λ1(r¯)+λ2(i=1N(ri-r¯)2N)1/2+λ3(i=1N((ri-r¯)3/N)((ri-r¯)2/N)3/2)+λ4(i=1N((ri-r¯)4/N)((ri-r¯)2/N)4/2),  
  suchthatXI=1,rp,i=xirj,r¯=i=1Nrp,iN,i=1Nλi=1.   (2.3)

2.4 Genetic algorithm

In the presence of higher order moments (skewness and kurtosis), the problem of portfolio selection turns into a nonconvex, nonlinear optimization problem, characterized by multiple conflicting and competing objectives. A more generic way of solving multi-objective programming is to combine the multiple objectives into a single objective function. Different optimization techniques have been employed to solve the above objective function.

The technique most frequently used in solving the aforementioned problem is that of PGP (see Lai et al 2006; Qi-fa et al 2007; Mhiri and Prigent 2010; Kemalbay et al 2011; Prakash et al 2003; Wang and Xia 2012; Chunhachinda et al 1997). However, a much debated weakness of PGP is that its solutions are not Pareto efficient. This violates a fundamental concept of decision theory that no rational decision maker will knowingly choose a solution that is not Pareto efficient.

Certain studies have used a Bayesian decision framework for the process of optimization. However, a traditional Bayesian framework will experience computational difficulties when solving real-world scenarios that have uncertainties (Harvey et al 2010).

A number of other studies have used heuristic techniques such as table search and simulated annealing (SA) (see, for example, Fernández and Gómez 2007). The table search and SA are more problem dependent and have a smaller probability of generating a global optimum. Heuristics that are based on greedy search algorithms lead to a local optimal solution.

Maringer and Parpas (2007) applied stochastic algorithms such as differential evolution (DE) and stochastic differential equations (SDEs) to solve the mean–variance–skewness–kurtosis framework. Their results indicated that DE and SDEs require sophisticated initial values for the optimization process and would otherwise lead to a suboptimal solution.

Yu et al (2008) proposed an integrated radial basis function (RBF) neural network-based mean–variance–skewness optimization model for portfolio selection. Based on gradient descent, the RBF either converges to a local minimum or consumes more time in the process of finding the optimal gradient.

To overcome the above limitations, the present study has used the well-known global optimization technique of genetic programming, following Chang et al (2009) and Li et al (2010). While these authors used a genetic algorithm (GA) for modeling skewness, mean and variance, we use the same method to model all four moments of the return distribution.

GA is less problem dependent and handles a large variety of future uncertainties. In addition, it is noted for being computationally efficient compared with the other optimization techniques used in prior studies. Further, GA is more suitable for solving multi-objective optimization problems, as it generates the entire set of Pareto-optimal solutions unlike other traditional mathematical programming techniques.

2.4.1 Formulation of the genetic algorithm

Initiated by Holland (1992), GA is based on Darwin’s “survival of the fittest” principle. Developed as the best-known evolutionary technique (Goldberg and Holland 1988; Mitchell 1996), GA has attracted much attention for its role in solving portfolio optimization problems. Arnone et al (1993) proposed GA for the unconstrained portfolio optimization problem with downside risk associated with the portfolio. Kyong et al (2005) used GA for index fund management to support portfolio optimization. Lin and Liu (2008) proposed GA for portfolio selection problems with minimum transaction lots.

GA starts by randomly generating an initial population that consists of a constant number of chromosomes. With respect to our problem, each chromosome represents the weight of an individual index. The fitness of the chromosomes is evaluated on the basis of their evaluation functions. For the present study, the evaluation function is the objective function described above. The basic steps in GA are shown as follows.

Step 1.

The algorithm begins by creating a random initial population.

Step 2.

The algorithm creates a sequence of new populations. At each step, the algorithm uses the individuals in the current generation to create the next population. To create the new population, the algorithm performs the following steps:

  1. (i)

    it scores each member of the current population by computing their fitness value;

  2. (ii)

    it selects parents based on their fitness;

  3. (iii)

    it selects individuals in the current population that have lower fitness and designates them as elite; these elite individuals are passed on to the next population;

  4. (iv)

    it produces children from the parents, either by making random changes to a single parent (mutation) or by combining the vector entries of a pair of parents (crossover); and

  5. (v)

    it replaces the current population with their children to form the next generation.

Step 3.

If the termination condition is satisfied then stop; otherwise, go back to Step 2.

A detailed description of our proposed GA for a portfolio optimization problem based on the above GA is presented below.

This study used a population size of one hundred. The population consists of random initial solutions to the optimization function. Then, the fitness of each solution in the population is evaluated using (2.1) for the MV model, (2.2) for the MVS model and (2.3) for the MVSK model. Based on this evaluation, the new population (set of solutions) is generated using mutations and crossovers. This process repeats till the weighted average change in the fitness function value over the generations is less than the tolerance level of 10-6.

2.5 Portfolio construction

Portfolio construction includes two processes: (i) selection and (ii) allocation. For both asset selection and allocation, GA is employed. The selection criteria for the MV model is based on the mean and variance of the index; for the MVS model it is mean, variance and skewness, while for the MVSK model it is mean, variance, skewness and kurtosis.

The first sample portfolio (P1,0) was constructed by taking the monthly returns of all fifteen emerging stock market indexes between January 2000 and December 2002. The sample period for the second portfolio (P1,1) spans from April 2000 to March 2003. Likewise, all forty portfolios were constructed progressively using data from the emerging markets on a quarterly rolling basis up to December 2012. For each of the three models examined – MV, MVS and MVSK – forty portfolios were constructed, iterating 120 portfolios. These results are contained in panel (a). The same procedure has been followed for both panel (b) (consisting of developed stock indexes) and panel (c) (consisting of both emerging and developed stock indexes). Altogether, 360 portfolios were constructed.

Since the portfolios were constructed on a rolling basis, the holding or out-of-sample period for each portfolio is the following three months. For the first constructed portfolio covering the period between January 2000 and December 2002, the holding period is comprised of January, February and March 2003. Realized returns for each of the constructed portfolios were computed over the holding period; 120 out-of-sample monthly returns for each model across the three panels have been computed.

3 Empirical results and findings

Descriptive statistics of eighteen developed and fifteen emerging stock market indexes are presented in Table 1. Among the developed market indexes, the NZX 50 (New Zealand) had the highest average monthly return of 0.57%, while the ATEX (Greece) had the lowest at -1.34%. ATEX was found to exhibit high volatility, while NZX 50 had the lowest. Negative skewness was exhibited by most of the indexes. Significant excess kurtosis was exhibited by very few indexes.

Among the emerging stock market indexes, RTS (Russia) had the highest average monthly return of 1.42%, while TSEC 50 (Taiwan) had the lowest average monthly return of -1.34%. It is also to be noted from Table 1 that all emerging market indexes except TSEC 50 had a positive average monthly return. MADEX (Morocco) had the lowest monthly variance of 0.16%, while ISE 100 (Turkey) exhibited the highest variance of 1.31%. Most of the indexes exhibited negative skewness, and eleven out of fifteen indexes had very significant excess kurtosis.

Table 1: Descriptive statistics of the developed and emerging market indexes. [The mean, variance, skewness and excess kurtosis of the developed and emerging market indexes considered in our study. Monthly returns are used to compute these statistics. Sample data spans the period 2000–2012. “Mean” values indicate the average monthly return of each index; “SD” indicates the average monthly standard deviation; “Skew” and “Excess kurtosis” indicate the average monthly skewness and excess kurtosis, respectively; and indicates significant excess kurtosis (that is, excess kurtosis>1).]
    Mean SD Skew Excess
Index Country (%) (%) (%) kurtosis
HANGSENG Hong Kong 0.26 0.43 -0.641 1.186
STI Singapore 0.23 0.36 -1.078 3.965
ASX 200 Australia 0.27 0.14 -1.020 1.270
BEL Belgium -0.07 0.28 -1.408 3.809
CAC France -0.27 0.30 -0.638 0.699
DAX Germany 0.08 0.46 -0.950 2.768
ATEX Greece -1.34 0.97 -0.510 1.328
FTSE MIB Italy -0.58 0.39 -0.393 0.854
OMX 30 Sweden -0.05 0.38 -0.435 0.952
SMI Switzerland -0.01 0.17 -0.731 0.739
FTSE 100 UK -0.03 0.18 -0.717 0.748
NASDAQ USA -0.09 0.67 -0.719 1.499
AEX Netherlands -0.36 0.38 -1.040 2.302
ATX Austria 0.51 0.43 -1.554 5.293
ISEQ Ireland -0.22 0.39 -0.876 1.503
NZX 50 New Zealand 0.57 0.13 -0.697 0.812
OMX COPENHAGEN Denmark 0.44 0.34 -0.768 1.787
OMX HELSIKNI Finland -0.22 0.43 -0.126 1.223
SSE 50 China 0.28 0.66 -0.525 1.523
SENSEX India 0.85 0.56 -0.475 1.188
LQ 45 Indonesia 1.11 0.64 -1.138 4.813
FTSE MALAYSIA Malaysia 0.38 0.21 -0.521 1.096
PSEI Philippines 0.69 0.41 -0.757 1.953
IPC Mexico 1.22 0.34 -0.593 0.771
MERVAL Argentina 1.06 1.09 -0.176 3.697
IBOVESPA Brazil 0.86 0.57 -0.528 0.777
BUX Hungary 0.42 0.53 -0.812 2.297
IPSA Chile 0.75 0.53 -1.686 6.388
ISE100 Turkey 1.05 1.31 -0.227 1.859
MADEX Morocco 0.13 0.16 -1.033 3.539
RTS Russia 1.42 1.22 -0.753 1.871
TSEC 50 Taiwan -0.14 0.53 -0.070 0.762
WIG 20 Poland 0.24 0.56 -0.191 0.771

Emerging market indexes are characterized by positive returns and lower levels of negative skewness compared with developed markets, both of which are positive features in investment decision making. Also, the asset optimization process maximizes both returns and skewness. From an investor’s perspective, positive skewness is better than negative skewness, since small positive losses are better than higher negative losses. A prospective investor might also expect excess kurtosis to be minimal so as to avoid extreme negative returns.

3.1 Optimal weights of global indexes

The ten best indexes in each of the three panels were selected based on the respective criteria of the MV, MVS and MVSK models for each period. The weights of the selected indexes were then allocated using the GA. Table 2 presents the optimal weights of the portfolio under each model over a period spanning May to September 2007. The optimal weights for the other periods are not reported in the interests of brevity.

China’s stock market index was given the highest weightage of 85% during the May–September 2007 period under the MV model. Under the MVS model, however, its weightage is halved, and under MVSK model it is almost 4%. We note that the weightages for the same index differ across the three models. As can be seen in Table 2, the change in investment pattern across the portfolios with the differing models clearly shows that the incorporation of skewness and kurtosis into an investor’s portfolio decision making has a major impact on the construction of the optimal portfolio. Similar observations were also reported by Kemalbay et al (2011) and Mhiri and Prigent (2010).

The asset allocation with the MV model is mainly based on mean and variance. The return can be high due to excess kurtosis or skewness. When higher moments are considered as risk with the MVS and MVSK models, those assets with excess kurtosis and skewness will get lower weightage compared with their weights under the MV model. Thus, our empirical findings endorse the view that the incorporation of skewness and kurtosis into portfolio decisions does indeed cause a major change in the resultant optimal portfolio.

Table 2: Optimal investment pattern under MV, MVS and MVSK models. [The weights generated by the optimizer for each selected index in the portfolios constructed using the MV, MVS and MVSK models. Panel (a) consists of fifteen emerging market indexes. Panel (b) consists of eighteen developed market indexes. Panel (c) consists of thirty-three emerging and developed market indexes.]
           
(a) Emerging markets (May–September 2007)
MV model MVS model MVSK model
     
  Weight   Weight   Weight
Index (%) Index (%) Index (%)
SSE 50 84.55 SSE 50 47.65 RTS 04.87
SENSEX 03.71 SET 16.86 WIG 20 22.22
RTS 08.18 WIG 20 09.19 SSE 50 03.69
IPC 00.80 RTS 01.88 MADEX 11.88
MERVAL 00.10 PSEI 05.79 SET 24.74
ISE 100 00.72 BUX 01.46 IBOVESPA 04.88
BUX 00.22 LQ 45 07.99 LQ 45 14.56
PSEI 00.49 MERVAL 06.52 FTSE MALAYSIA 06.15
FTSE MALAYSIA 00.56 FTSE MALAYSIA 02.45 SENSEX 03.93
LQ 45 00.72 IBOVESPA 00.20 MERVAL 02.99
           
(b) Developed markets (May–September 2007)
MV model MVS model MVSK model
     
  Weight   Weight   Weight
Index (%) Index (%) Index (%)
HANG SENG 32.63 HANG SENG 01.97 ISEQ 70.16
ATX 32.57 STI 00.77 OMX COP. 09.17
ATHEX 13.21 ASX 200 05.04 STI 05.78
OMX HELSIKNI 17.09 BEL 02.30 FTSE MIB 04.05
STI 00.46 CAC 02.57 ATX 02.20
NZX 50 01.68 DAX 00.55 AEX 04.23
ISEQ 00.90 ATHEX 35.20 CAC 00.67
ASX 200 00.97 FTSE MIB 00.22 NASDAQ 02.26
OMX COP. 00.37 OMX 30 46.58 ATHEX 00.93
FTSE MIB 00.23 SMI 04.80 SMI 00.48
           
(c) Emerging as well as developed markets (May–September 2007)
MV model MVS model MVSK model
     
  Weight   Weight   Weight
Index (%) Index (%) Index (%)
SSE 50 83.41 SSE 50 50.74 RTS 37.29
RTS 08.29 ISEQ 05.36 SSE 50 15.06
SENSEX 03.30 BEL 01.02 ISE 100 02.09
IPC 02.31 WIG 20 021.93 LQ 45 00.59
ATHEX 00.41 OMX 30 07.69 IPC 01.12
IBOVESPA 00.38 NZX 50 01.29 WIG 20 23.23
PSEI 00.99 LQ 45 05.42 TSEC 50 06.10
ASX 200 00.41 ASX 200 01.06 DAX 06.06
OMX HELSIKNI 00.30 SENSEX 00.59 NASDAQ 04.07
ATX 00.29 OMX HELSIKNI 04.96 BEL 04.28

3.2 In-sample portfolios: risk–return analysis

Four portfolios were constructed annually for each panel using the three different models (MV, MVS and MVSK). The in-sample period for each portfolio comprises monthly returns for the three previous years. This process was repeated every quarter on a rolling basis. The expected returns of the constructed portfolios for that year were used to compute the average monthly return and standard deviation for that year. Table 3 presents the average monthly returns and standard deviations of the portfolios across the different models and panels.

In the case of the emerging markets, returns with the MVSK model were higher in eight out of ten years. Both MV and MVSK models have high standard deviation in four out of ten years. However, the MV model did not provide the highest returns in any of the years considered. Since the returns with the MVS and MVSK models were better than with the MV model, one might have expected the realized or out-of-sample returns would be higher with a model that includes higher moments, ie, the MVS or the MVSK.

Yet, in the developed markets context, the MV model had the highest return in all the years compared with the MVS and MVSK models. Standard deviation for the MVSK model was high in most of the years. It is evident from Table 3 that, in developed markets, the performance of the MV model is better than either the MVS or the MVSK model. This implies that the out-of-sample returns (realized returns) of the MV model should be better than those of either the MVS or the MVSK model.

When both emerging and developed markets were considered together, the returns of the MVSK model were higher than those generated by the MVS or the MV model. Standard deviation was also found to be higher for most years. Hence, we expect the realized returns for both MVSK and MVS models to be higher.

Table 3: Risk–return profile of the portfolios over the in-sample period (all values are percentages). [The average monthly returns and standard deviation (SD) in four portfolios, based on data pertaining to each particular year. The highest values for return and SD in each year, across all alternative models, are highlighted with asterisks.]
             
(a) Emerging markets
  MV model MVS model MVSK model
       
Year Return SD Return SD Return SD
2003 -0.280 2.920 -0.410 3.560 0.125* 4.481*
2004 -0.650 4.902* -0.300 4.100 1.016* 4.380
2005 -0.720 4.985* -0.220 3.410 1.940* 3.330
2006 0.080 3.936* 0.890 2.900 1.282* 2.570
2007 0.260 3.832* 0.380 2.200 1.552* 2.810
2008 -0.080 3.530 0.230 4.021* 1.094* 3.530
2009 -0.850 5.020 -0.199* 6.773* -0.250 6.580
2010 -0.470 5.110 -0.350 6.280 0.020* 6.712*
2011 -0.160 5.450 0.220 7.120 0.407* 7.644*
2012 0.000 4.560 0.657* 4.520 0.550 5.123*
             
(b) Developed markets
  MV model MVS model MVSK model
       
Year Return SD Return SD Return SD
2003 0.54* 3.83 -2.33 7.67* -1.55 6.20
2004 1.37* 3.98 -0.86 7.03* -0.32 6.69
2005 2.50* 3.77 0.96 6.84* 1.20 4.33
2006 3.13* 4.06 1.71 3.71 0.05 5.23*
2007 2.46* 3.96 -0.18 4.92* 1.53 3.22
2008 1.13* 5.72 0.18 5.22 0.26 4.91
2009 0.07* 7.43* -1.24 6.28 -1.32 5.33
2010 -0.11* 8.01* -1.09 7.67 -1.22 6.06
2011 0.55* 6.32 -0.89 7.78* -0.38 6.64
2012 1.66* 5.31 -1.64 7.81* 0.71 4.40
             
(c) Both developed and emerging markets
  MV model MVS model MVSK model
       
Year Return SD Return SD Return SD
2003 -0.56 2.53 -0.33* 2.38 -0.37 5.83*
2004 -0.85 4.16 -0.50 3.31 1.16* 4.98*
2005 -0.64 4.46* -0.16 3.43 1.45* 3.17
2006 0.15 3.32* 0.48 2.10 1.42* 2.76
2007 0.23 4.07* 1.03 3.31 1.06* 2.57
2008 -0.21 3.60 0.48 4.75* 0.49* 3.34
2009 -0.62 4.49 -0.75 6.14 -0.36* 6.39*
2010 -0.48* 4.23 -0.51 4.35 -1.38 5.65*
2011 -0.14* 4.09 -1.61 7.90* -0.79 6.66
2012 -0.12 3.28 0.08 3.94 0.50* 4.29*

3.3 Analysis of holding period returns during the out-of-sample period

Realized returns were computed for each portfolio across the out-of-sample period. Optimal weights from the GA of each model were applied to the historical index values to obtain the holding period realized returns. Our study also compared the out-of-sample portfolio returns of the three competing models with the global minimum variance portfolio. The realized returns of the global minimum variance model were obtained by assigning equal weights to all the indexes in the portfolio. The return and risk characteristics of all portfolios across all three panels are reported in Tables 4, 5 and 6 (pertaining to panels (a), (b) and (c), respectively).

For the emerging markets, average monthly realized return and standard deviation during the out-of-sample period are presented in Table 4. The MVS and MVSK models exhibited higher returns than the MV model in all years, with the exception of 2005. The MVS model had the highest returns in the years 2003, 2007 and 2010, while the MVSK model had the highest returns in each of the remaining six years. We also note that the MVSK model had the lowest standard deviation in most years except 2010. The MVS model had the highest standard deviation in 2008, 2011 and 2012, while the MV model had the highest standard deviation in the remaining six years.

Table 4: Realized returns and risk characteristics of emerging markets. [Fifteen emerging market indexes were considered under panel (a). Using the weights generated by the optimizer holding period, monthly return and standard deviation were computed for the subsequent three months. This table presents the average monthly returns and standard deviation for each year. For each model, the return/risk ratio was also computed and has been presented. Asterisks (*) indicate the highest returns, standard deviation and return/risk ratio values for each year across all alternate models.]
  MV model MVS model MVSK model Global minimum model
         
  Return SD Return/ Return SD Return/ Return SD Return/ Return SD Return/
Year (%) (%) risk (%) (%) (%) risk (%) (%) risk (%) (%) risk
2003 4.37 08.43* 0.52 4.76* 03.84 1.24 3.91 2.55 1.53 3.63 2.27 1.60*
2004 1.20 08.79* 0.14 1.49 05.84 0.26 2.55* 2.64 0.97* 1.19 2.65 0.45
2005 2.11* 07.91* 0.27 0.60 05.81 0.10 1.65 2.86 0.58* 2.08 3.54 0.58*
2006 2.68 04.00* 0.67 1.29 03.42 0.38 3.61* 3.02 1.20* 1.99 3.05 0.65
2007 0.70 09.67* 0.07 3.13* 07.49 0.42 3.02 1.36 2.21* 1.19 3.85 0.31
2008 -2.08 11.11 -0.19 -5.35 12.78* -0.42 -0.57* 3.42 -0.17* -5.44 7.58 -0.72
2009 3.87 04.89 0.79 3.36 04.56 0.74 4.05 3.73 1.08* 4.61* 4.94* 0.93
2010 0.86 04.29 0.20 1.48* 02.64 0.56* 0.68 6.16* 0.11 1.47 3.34 0.44
2011 -1.02 04.14 -0.25 -1.27 05.73* -0.22 1.45* 4.35 0.33* -0.70 4.37 -0.16
2012 1.52 03.10 0.49 1.34 04.52* 0.30 1.74* 2.59 0.67* 0.97 2.74 0.35

The performance of a portfolio should always be computed as a combination of risk and return; hence, the return/risk ratios were computed for our study. The return/risk ratio obtained with the MVSK model was the highest in all years except 2010. This indicates that, except in 2010, the MVSK model provided the higher returns with lower standard deviation in all years. The MVS model generated mixed results. In 2010, its performance was better than that of the MV and MVSK models. During 2010, significant negative skewness dominated the kurtosis. Although the MVSK model accommodates skewness well, the equal preference given to the kurtosis in this model reduced the efficacy of its performance in 2010.

Even across the global minimum variance portfolios, the MVSK model portfolios exhibited better performances. No superiority of the MVS model over the global minimum variance model was witnessed.

Table 5: Realized return and risk characteristics of developed markets. [Eighteen developed market indexes were considered under panel (b). Using the weights generated by the optimizer holding period, monthly return and standard deviation were computed for the subsequent three-month period. The table presents the average monthly returns and standard deviation in each year. For each model, the return/risk ratio was also computed and has been presented. Asterisk (*) indicates the highest return, standard deviation and return/risk ratio in each year across the alternative models.]
  MV model MVS model MVSK model Global minimum model
         
  Return SD Return/ Return SD Return/ Return SD Return/ Return SD Return/
Year (%) (%) risk (%) (%) (%) risk (%) (%) risk (%) (%) risk
2003 3.25* 3.70 0.88 2.43 05.32* 0.46 2.97 3.04 0.98* 2.40 3.76 0.64
2004 3.22 2.82 1.14* 3.54* 03.14* 1.13* 1.15 2.91 0.39 1.12 1.92 0.58
2005 3.87* 4.74 0.82* 2.24 04.96* 0.45 1.70 3.22 0.53 1.85 2.91 0.64
2006 1.09 4.65* 0.23 1.61 03.45 0.47 2.41* 3.86 0.62* 1.37 2.80 0.49
2007 -1.37 6.36* -0.22 2.57* 02.67 0.96* -1.01 3.96 -0.25 -0.89 4.64 -0.19
2008 -4.69* 9.18 -0.51* -7.70 10.33* -0.75 -6.85 9.44 -0.73 -4.70 7.22 -0.65
2009 1.54 5.96 0.26 1.22 06.42* 0.19 2.77* 5.32 0.52* 2.37 6.09 0.39
2010 2.63* 4.49 0.59* 2.11 04.74* 0.45 0.53 3.51 0.15 0.80 4.39 0.18
2011 -0.38 5.20 -0.07 -2.28 07.11* -0.32 1.91* 3.60 0.53* -1.30 4.92 -0.26
2012 0.54 4.13 0.13 3.03* 04.20* 0.72* 1.50 3.53 0.42 1.01 3.66 0.28

Out-of-sample results for the developed markets are reported in Table 5. It can be observed in Table 5 that, in the context of developed markets, no model consistently outperformed the others. The MV model had the highest returns for four years, the MVS model had the highest returns for three years and the MVSK model had the highest returns for the remaining three years. The portfolios under the MVS model proved highly volatile compared with the other models. In the years 2003, 2006, 2009 and 2011, MVSK outperformed the other models. In 2004, 2005, 2008 and 2010, the MV model was the top performer, while the MVS model outperformed the other models for the remaining years. The performances of the global minimum variance portfolios and the MVSK model portfolios are almost similar. The same can be said for the MVS model. These results clearly show that, in developed markets, no single model outperforms the others.

Out-of-sample results for portfolios comprising both emerging and developed markets are presented in Table 6. The MV model had the highest return in 2009 only, while the MVS model had higher returns for five years (2003, 2004, 2007, 2010 and 2011) and the MVSK model garnered the highest in the remaining four years (2005, 2006, 2008 and 2012). Portfolios under the MV and MVSK models exhibited high volatility in all years, except for 2009. The standard deviation for the MVSK model is noted to be highest in 2009. The return/risk ratio was higher for the MVSK model in all years, except for 2008 and 2009. The return/risk ratio is highest for the MVS model in 2008 and for the MV model in 2009.

Table 6: Realized return and risk characteristics of global markets. [Thirty-three emerging and developed market indexes were considered under panel (c). Using the weights generated by the optimizer holding period, monthly returns and standard deviation values were computed for the subsequent three-month period. The table presents the average monthly return and standard deviation for each year. For each model, the return/risk ratio was also computed and has been presented. Asterisk (*) indicates the highest returns, standard deviation and highest return/risk ratio for each year across the alternative models.]
  MV model MVS model MVSK model Global minimum model
         
  Return SD Return/ Return SD Return/ Return SD Return/ Return SD Return/
Year (%) (%) risk (%) (%) (%) risk (%) (%) risk (%) (%) risk
2003 2.80 07.55* 0.37 4.41* 03.12 1.41 3.68 2.27 1.62* 2.96 2.68 1.10
2004 2.04 07.61* 0.27 2.93* 04.31 0.68 2.64 2.61 1.01* 1.15 1.91 0.61
2005 3.33 07.58* 0.44 2.11 03.84 0.55 5.34* 3.19 1.67* 1.96 3.08 0.64
2006 1.74 06.01* 0.29 1.86 03.38 0.55 2.17* 3.06 0.71* 1.65 2.66 0.62
2007 0.22 09.02* 0.02 1.06* 06.39 0.17 0.98 5.16 0.19* 0.06 4.22 0.01
2008 -9.73 12.51 -0.78 -2.86 14.99* -0.19* -2.57* 8.12 -0.32 -5.03 7.20 -0.70
2009 5.52* 03.71 1.49* 1.84 05.39 0.34 4.70 6.09* 0.77 3.39 5.35 0.63
2010 0.85 04.11 0.21 1.91* 05.72* 0.33* 1.10 3.32 0.33* 1.11 3.79 0.29
2011 0.50 03.49 0.14 1.89* 04.28 0.44 1.39 2.90 0.48* -1.03 4.53* -0.23
2012 2.23 02.98 0.75 1.63 04.82* 0.34 3.11* 2.50 1.24* 0.99 3.16 0.31

We note that the MVS model exhibited a superior performance to the MV model in all years except 2009 and 2012. During the year of the financial crisis (2008) both the MVS and MVSK models performed better than the MV model. Compared with the global minimum variance portfolios, the MVSK model portfolios had high return/risk ratios during the whole sample period. We also witnessed the superiority of the MVS model over the global minimum variance model. These findings indicate that the higher moments model helps to constrain the effects of extreme events, thus limiting huge losses. Global fund managers and investors would benefit from including skewness and kurtosis in their asset allocation processes.

The study analyzed whether there is a significant difference between the performance of the MVSK model and that of the other models. A t test has been conducted over the return/risk ratios. The null hypothesis of the t test is that the difference between the return/risk ratios of the two series is zero. The results of our t test are presented in Table 7. The test analyzes whether the risk/return ratios generated by the MVSK/MVS model differ from the MV model/global minimum portfolio’s return/risk ratio. It is observed from Table 7 that return/risk ratios for the MVSK model differs significantly from those generated by MV, MVS and the global minimum variance portfolio in the case of panels (a) and (c). However, the return/risk ratios of the MVSK, MVS, MV and global minimum portfolios exhibit no significant differences in the case of developed markets.

Table 7: Robustness check using t test. [The t test results based on out-of-sample data. The null hypothesis of the test is that the difference between the two return/risk ratio series is zero. * indicates significance at 1%, ** indicates significance at 5% and *** indicates significance at 10%.]
          Emerging and
  Emerging markets Developed markets developed markets
       
  ? -statistic ? -value ? -statistic ? -value ? -statistic ? -value
MVSK_MV 2.836** 0.020 -0.068 0.947 2.505** 0.034
MVSK_MVS 2.888** 0.018 -0.314 0.761 2.387** 0.041
MVSK_Global 2.119*** 0.063 1.176 0.270 3.957* 0.003
MVS_MV 0.633 0.542 0.324 0.753 0.766 0.463
MVS_Global -1.435 0.185 1.205 0.259 1.454 0.180

Figures 1, 2 and 3 present the realized returns of the three alternate models over the out-of-sample period. Figure 1 refers to emerging markets, Figure 2 refers to developed markets and Figure 3 refers to all the markets. The x-axes represent time and the y-axes represent returns.

It can be observed from the figures that during the crisis period (2008), the MVS model had the highest negative returns compared with the other two models in the case of Figure 1. A similar observation holds true in the case of Figure 3. However, in the case of Figure 2, we note that the magnitude of the negative returns is almost the same for all three models. The MVSK model had the least negative returns during the crisis across all figures and panels, except in the case of developed markets.

Realized returns of emerging market portfolios.
Figure 1: Realized returns of emerging market portfolios.
Realized returns of developed market portfolios.
Figure 2: Realized returns of developed market portfolios.
Realized returns of the combined panel portfolios.
Figure 3: Realized returns of the combined panel portfolios.

We also note that the volatility of the out-of-sample period returns is high in MV and MVS models in the case of emerging markets as well as emerging and developed markets combined. In developed markets, the volatility is mostly similar across all models. Figures 1 and 3 depict a greater consistency in the returns generated by the MVSK model than observed with the other models, indicating a reduction in extreme losses or gains.

The overall summary statics of the realized returns are presented in Table 8. This table reports annualized monthly returns, annualized standard deviations, value-at-risk (VaR) at a 95% confidence level and the return/risk ratios of the realized returns across all models and panels. We note that the MVSK model has the highest annualized return in panels (a) and (c), and the MV model outperforms all others in panel (b). The standard deviation of the MVSK model is relatively low compared with the other models in all three panels. The VaR, which measures the highest probable loss at a 95% confidence level, also indicates that the risk pertaining to the MVSK model is less. A similar finding is noted from the resulting return/risk ratio values. Thus, the overall results provide evidence in support of the MVSK model over the MV model.

Table 8: Summary of the risk–return profile of realized returns. [Returns values indicate the annualized monthly return of the portfolio; SD indicates the annualized monthly standard deviation (monthly standard deviation × square root of 12). The return/risk ratio was computed as a ratio of the annualized returns to annualized standard deviations during the period between 2003 and 2012. Panel (a) represents fifteen emerging market indexes. Panel (b) includes eighteen developed market indexes. Panel (c) considers all thirty-three indexes from both emerging and developed markets. Asterisks (*) indicate the highest return, highest standard deviation and highest return/risk ratio across the alternative models.]
         
Panel (a)
  Annualized Annualized VaR at Return to
Model return SD 95% risk ratio
MV model 0.17 0.25 -0.240 0.69
MVS model 0.13 0.23 -0.250 0.57
MVSK model 0.27* 0.13* 0.057* 2.10*
         
Panel (b)
  Annualized Annualized VaR at Return to
Model return SD 95% risk ratio
MV model 0.12* 0.20 -0.21* 0.58*
MVS model 0.11 0.22 -0.26 0.48
MVSK model 0.08 0.18* -0.21 0.47
         
Panel (c)
  Annualized Annualized VaR at Return to
Model return SD 95% risk ratio
MV model 0.11 0.27 -0.33 0.42
MVS model 0.20 0.23 -0.17 0.89
MVSK model 0.27* 0.16* 0.01* 1.67*

Overall, the empirical results of this study are broadly in consensus with Chunhachinda et al (1997) and Prakash et al (2003). Clearly, the inclusion of higher moments improved the international diversification benefits, especially while considering portfolios in emerging markets. The results also indicate that higher moments play a significant role in portfolio optimization. Hence, it is implied that higher moments need to be considered in the asset selection and allocation processes. The results of this study also show that there are good opportunities for investment in emerging markets.

4 Conclusion

The purpose of this paper was to investigate the impact of including higher moments in the portfolio optimization process. Changing market dynamics necessitates the consideration of more generic models that incorporate skewness and kurtosis in portfolio construction. Empirical results indicate that investment decisions should be based not only on the variance but also on other risk properties. The MVSK optimization model that includes all four moments of the return distribution (that is, the mean, standard deviation, skewness and kurtosis) performed better in the process of selection and optimization of thirty-three global indexes across emerging and developed markets, providing higher returns, lower standard deviations and higher return/risk ratios. Our results prove that the incorporation of skewness and kurtosis improved the portfolio return performance. The higher risks considered in the optimization process shed light on the return–risk of the asset prices. Thus, our study supports the inclusion of higher moments in the portfolio selection and optimization processes.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

Acknowledgements

We thank the anonymous reviewer, whose comments substantially improved the paper.

References

  • Arditti, F. D. (1971). Another look at mutual fund performance. Journal of Financial and Quantitative Analysis 6, 909–912 (https://doi.org/10.2307/2329910).
  • Arnone, S., Loraschi, A., and Tettamanzi, A. (1993). A genetic approach to portfolio selection. Neural Network World 6, 597–604. URL: http://bit.ly/2pKJwj7.
  • Chang, T.-J., Yang, S.-C., and Chang, K.-J. (2009). Portfolio optimization problems in different risk measures using genetic algorithm. Expert Systems with Applications 36(7), 10529–10537 (https://doi.org/10.1016/j.eswa.2009.02.062).
  • Chunhachinda, P., Dandapani, K., Hamid, S., and Prakash, A. J. (1997). Portfolio selection and skewness: evidence from international stock markets. Journal of Banking & Finance 21(2), 143–167 (https://doi.org/10.1016/S0378-4266(96)00032-5).
  • de Athayde, G. M., and Flôres, R. G., Jr. (2004). Finding a maximum skewness portfolio: a general solution to three-moments portfolio choice. Journal of Economic Dynamics and Control 28(7), 1335–1352 (https://doi.org/10.1016/S0165-1889(02)00084-2).
  • Fama, E. F. (1965). The behaviour of stock market prices. Journal of Business 38, 34–105. URL: http://bit.ly/2FFjy7t.
  • Fernández, A., and Gómez, S. (2007). Portfolio selection using neural networks. Computers & Operations Research 34, 1177–1191 (https://doi.org/10.1016/j.cor.2005.06.017).
  • Goldberg, D. E., and Holland, J. H. (1988). Genetic algorithms and machine learning. Machine Learning 3(2), 95–99 (https://doi.org/10.1023/A:1022602019183).
  • Harvey, C. R., and Siddique, A. (1999). Autoregressive conditional skewness. Journal of Financial and Quantitative Analysis 34(04), 465–487 (https://doi.org/10.2307/2676230).
  • Harvey, C. R., Liechty, J. C., Liechty, M. W., and Muller, P. (2010). Portfolio selection with higher moments. Quantitative Finance 10(5), 469–485 (https://doi.org/10.1080/14697681003756877).
  • Holland, J. H. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. Massachusetts Institute of Technology Press. URL: http://bit.ly/2F0nfYf.
  • Jondeau, E., and Rockinger, M. (2006). Optimal portfolio allocation under higher moments. European Financial Management 12(1), 29–55 (https://doi.org/10.1111/j.1354-7798.2006.00309.x).
  • Joro, T., and Na, P. (2006). Portfolio performance evaluation in a mean–variance–skewness framework. European Journal of Operational Research 175, 446–461 (https://doi.org/10.1016/j.ejor.2005.05.006).
  • Jurczenko, E., Maillet, B. B., and Merlin, P. (2005). Hedge funds portfolio selection with higher-order moments: a non-parametric mean–variance–skewness–kurtosis efficient frontier. Working Paper, Social Science Research Network. URL: http://bit.ly/2FE7JOR.
  • Kane, A. (1982). Skewness preference and portfolio choice. Journal of Financial and Quantitative Analysis 17, 15–25 (https://doi.org/10.2307/2330926).
  • Kemalbay, G., Özkut, C. M., and Franko, C. (2011). Portfolio selection with higher moments: a polynomial goal programming approach to ISE-30 index. Ekonometri ve İstatistik e-Dergisi 13, 41–61. URL: http://bit.ly/2F4CGu8.
  • Konno, H., and Suzuki, K. (1995). A mean–variance–skewness portfolio optimization model. Journal of the Operations Research Society of Japan 38(2), 173–187 (https://doi.org/10.15807/jorsj.38.173).
  • Konno, H., and Yamazaki, H. (1991). Mean–absolute deviation portfolio optimization model and its application to Tokyo stock market. Management Science 37(5), 519–531. URL: http://bit.ly/2ovZMnd.
  • Konno, H., Shirakawa, H., and Yamazaki, H. (1993). A mean–absolute deviation–skewness portfolio optimization model. Annals of Operations Research 45(1), 205–220 (https://doi.org/10.1007/BF02282050).
  • Kyong, J. O., Tae, Y. K., and Sungky, M. (2005). Using genetic algorithm to support portfolio optimization for index fund management. Expert Systems with Applications 28, 371–379 (https://doi.org/10.1016/j.eswa.2004.10.014).
  • Lai, K. K., Yu, L., and Wang, S. (2006). Mean–variance–skewness–kurtosis based portfolio optimization. Computer and Computational Sciences 2, 292–297 (https://doi.org/10.1109/IMSCCS.2006.239).
  • Lai, T. Y. (1991). Portfolio selection with skewness: a multiple-objective approach. Review of Quantitative Finance and Accounting 1, 293–305 (https://doi.org/10.1007/BF02408382).
  • Levy, H., and Markowitz, H. (1979). Approximating expected utility by a function of mean and variance. American Economic Review 69(3), 308–317. URL: http://bit.ly/2F0S1QM.
  • Li, D., and Ng, W. L. (2000). Optimal dynamic portfolio selection: multiperiod mean–variance formulation. Mathematical Finance 10, 387–406 (https://doi.org/10.1111/1467-9965.00100).
  • Li, X., Qin, Z., and Kar, S. (2010). Mean–variance–skewness model for portfolio selection with fuzzy returns. European Journal of Operational Research 202(1), 239–247 (https://doi.org/10.1016/j.ejor.2009.05.003).
  • Lin, C. C., and Liu, Y. T. (2008). Genetic algorithms for portfolio selection problems with minimum transaction lots. European Journal of Operational Research 185, 393–404 (https://doi.org/10.1016/j.ejor.2006.12.024).
  • Maillet, B. M., and Merlin, P. M. (2009). Robust higher-order moments and efficient portfolio selection. Working Paper, Social Science Research Network. URL: http://bit.ly/2GS5Tce3.
  • Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business 36, 394–419 (https://doi.org/10.1007/978-1-4757-2763-0_14).
  • Maringer, D., and Parpas, P. (2007). Global optimization of higher order moments in portfolio selection. Journal of Global Optimization 43(2–3), 219–230 (https://doi.org/10.1007/s10898-007-9224-3).
  • Markowitz, H. (1952). Portfolio selection. Journal of Finance 7, 77–91 (https://doi.org/10.1111/j.1540-6261.1952.tb01525.x).
  • Markowitz, H. (1959). Portfolio Selection Efficient Diversification of Investments. Wiley. URL: http://bit.ly/2Cu4jiE.
  • Mhiri, M., and Prigent, J.-L. (2010). International portfolio optimization with higher moments. International Journal of Economics and Finance 2(5), 157–169 (https://doi.org/10.5539/ijef.v2n5p157).
  • Mitchell, M. (1996). An Introduction to Genetic Algorithms. MIT Press. URL: http://bit.ly/2t14Sg5.
  • Peiró, A. (1999). Skewness in financial returns. Journal of Banking & Finance 23(6), 847–862 (https://doi.org/10.1016/S0378-4266(98)00119-8).
  • Pindoriya, N. M., Singh, S. N., and Singh, S. K. (2010). Multi-objective mean–variance–skewness model for generation portfolio allocation in electricity markets. Electric Power Systems Research 80(10), 1314–1321 (https://doi.org/10.1016/j.epsr.2010.05.006).
  • Prakash, A. J., Chang, C. H., and Pactwa, T. E. (2003). Selecting a portfolio with skewness: recent evidence from US, European, and Latin American equity markets. Journal of Banking & Finance 27(7), 1375–1390 (https://doi.org/10.1016/S0378-4266(02)00261-3).
  • Premaratne, G., and Bera, A. K. (2000). Modeling asymmetry and excess kurtosis in stock return data. Working Paper 00–123, Illinois Research & Reference. URL: http://bit.ly/2FdL6D4.
  • Qi-fa, X., Cui-xia, J., and Pu, K. (2007). Dynamic portfolio selection with higher moments risk based on polynomial goal programming. In International Conference on Management Science and Engineering, 2007, pp. 2152–2157. Institute of Electrical and Electronics Engineers, Piscataway, NJ.
  • Rubinstein, M. (2002). Markowitz’s “portfolio selection”: a fifty-year retrospective. Journal of Finance 57(3), 1041–1045. URL: http://bit.ly/2Cr1sXT.
  • Samuelson, P. (1970). The fundamental approximation of theorem of portfolio analysis in terms of means, variances and higher moments. Review of Economic Studies 37, 537–542. URL: http://bit.ly/2HUkwx0.
  • Scott, R. C., and Horvath, P. A. (1980). On the direction of preference for moments of higher order than the variance. Journal of Finance 35, 915–919 (https://doi.org/10.2307/2327209).
  • Vermorken, M. A., Medda, F. R., and Schröder, T. (2012). The diversification delta: a higher-moment measure for portfolio diversification. Journal of Portfolio Management 39(1), 67–74 (https://doi.org/10.3905/jpm.2012.39.1.067).
  • Wang, S., and Xia, Y. (2002). Portfolio Selection and Asset Pricing. Lecture Notes in Economics and Mathematical Systems. Springer (https://doi.org/10.1007/978-3-642-55934-1).
  • Yu, L., Wang, S., and Lai, K. K. (2008). Neural network based mean–variance–skewness model for portfolio selection. Computers & Operations Research 35, 34–46 (https://doi.org/10.1016/j.cor.2006.02.012).

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