Journal of Risk
ISSN:
14651211 (print)
17552842 (online)
Editorinchief: Farid AitSahlia
Genetic algorithmbased portfolio optimization with higher moments in global stock markets
Need to know
 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.
Abstract
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 thirtythree 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 multiobjective 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.
Introduction
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 nonGaussian 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 nonGaussian 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 thirtythree 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, multiobjective 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, multiobjective 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 outofsample 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.
2 Data and methodology
2.1 Data
Monthly values of thirtythree 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 (thirtythree 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 riskaverse individuals, who maximize the expected utility of their endofperiod wealth.
 •
There are $n$ risky assets, and investors do not have access to any riskfree 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 ${R}_{\mathrm{p}}$ 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
$M({R}_{\mathrm{p}})$  $=E({X}^{\prime}(R\overline{R})),$  
$V({R}_{\mathrm{p}})$  $=E{({X}^{\prime}(R\overline{R}))}^{2},$  
$S({R}_{\mathrm{p}})$  $=E{({X}^{\prime}(R\overline{R}))}^{3},$  
$K({R}_{\mathrm{p}})$  $=E{({X}^{\prime}(R\overline{R}))}^{4}.$ 
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:
$$\mathrm{minimize}f(x)=\lambda (\overline{r})+(1\lambda ){\left(\sum _{i=1}^{N}\frac{{({r}_{p,i}\overline{r})}^{2}}{N}\right)}^{1/2},$$  
$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}{X}^{\prime}I& =1,\hfill \\ \hfill {r}_{p,i}& =\sum {x}_{i}{r}_{j},\hfill \\ \hfill \overline{r}& =\sum _{i=1}^{N}\frac{{r}_{p,i}}{N}.\hfill \end{array}$$  (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:
$$\mathrm{minimize}f(x)={\lambda}_{1}(\overline{r})+{\lambda}_{2}{\left(\sum _{i=1}^{N}\frac{{({r}_{i}\overline{r})}^{2}}{N}\right)}^{1/2}{\lambda}_{3}\left(\frac{{\sum}_{i=1}^{N}({({r}_{i}\overline{r})}^{3}/N)}{{({({r}_{i}\overline{r})}^{2}/N)}^{3/2}}\right),$$  
$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}{X}^{\prime}I& =1,\hfill \\ \hfill {r}_{p,i}& =\sum {x}_{i}{r}_{j},\hfill \\ \hfill \overline{r}& =\sum _{i=1}^{N}\frac{{r}_{p,i}}{N},\hfill \\ \hfill \sum _{i=1}^{N}{\lambda}_{i}& =1.\hfill \end{array}$$  (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:
$$\begin{array}{cc}\hfill \mathrm{minimize}f(x)& ={\lambda}_{1}(\overline{r})+{\lambda}_{2}{\left(\sum _{i=1}^{N}\frac{{({r}_{i}\overline{r})}^{2}}{N}\right)}^{1/2}\hfill \\ & +{\lambda}_{3}\left(\frac{{\sum}_{i=1}^{N}({({r}_{i}\overline{r})}^{3}/N)}{{({({r}_{i}\overline{r})}^{2}/N)}^{3/2}}\right)+{\lambda}_{4}\left(\frac{{\sum}_{i=1}^{N}({({r}_{i}\overline{r})}^{4}/N)}{{({({r}_{i}\overline{r})}^{2}/N)}^{4/2}}\right),\hfill \end{array}$$  
$$\begin{array}{cc}\hfill \mathrm{such}\mathrm{that}{X}^{\prime}I& =1,\hfill \\ \hfill {r}_{p,i}& =\sum {x}_{i}{r}_{j},\hfill \\ \hfill \overline{r}& =\sum _{i=1}^{N}\frac{{r}_{p,i}}{N},\hfill \\ \hfill \sum _{i=1}^{N}{\lambda}_{i}& =1.\hfill \end{array}$$  (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 multiobjective 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; Qifa 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 realworld 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 networkbased 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 wellknown 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 multiobjective optimization problems, as it generates the entire set of Paretooptimal 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 bestknown 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:
 (i)
it scores each member of the current population by computing their fitness value;
 (ii)
it selects parents based on their fitness;
 (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;
 (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
 (v)
it replaces the current population with their children to form the next generation.
 (i)
 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 $({\mathrm{P}}_{1,0})$ 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 $({\mathrm{P}}_{1,1})$ 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 outofsample 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 outofsample 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.
Mean  SD  Skew  Excess  

Index  Country  (%)  (%)  (%)  kurtosis 
HANGSENG  Hong Kong  0.26  0.43  $$0.641  1.186${}^{\prime}$ 
STI  Singapore  0.23  0.36  $$1.078  3.965${}^{\prime}$ 
ASX 200  Australia  0.27  0.14  $$1.020  1.270${}^{\prime}$ 
BEL  Belgium  $$0.07  0.28  $$1.408  3.809${}^{\prime}$ 
CAC  France  $$0.27  0.30  $$0.638  0.699 
DAX  Germany  0.08  0.46  $$0.950  2.768${}^{\prime}$ 
ATEX  Greece  $$1.34  0.97  $$0.510  1.328${}^{\prime}$ 
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${}^{\prime}$ 
AEX  Netherlands  $$0.36  0.38  $$1.040  2.302${}^{\prime}$ 
ATX  Austria  0.51  0.43  $$1.554  5.293${}^{\prime}$ 
ISEQ  Ireland  $$0.22  0.39  $$0.876  1.503${}^{\prime}$ 
NZX 50  New Zealand  0.57  0.13  $$0.697  0.812 
OMX COPENHAGEN  Denmark  0.44  0.34  $$0.768  1.787${}^{\prime}$ 
OMX HELSIKNI  Finland  $$0.22  0.43  $$0.126  1.223${}^{\prime}$ 
SSE 50  China  0.28  0.66  $$0.525  1.523${}^{\prime}$ 
SENSEX  India  0.85  0.56  $$0.475  1.188${}^{\prime}$ 
LQ 45  Indonesia  1.11  0.64  $$1.138  4.813${}^{\prime}$ 
FTSE MALAYSIA  Malaysia  0.38  0.21  $$0.521  1.096${}^{\prime}$ 
PSEI  Philippines  0.69  0.41  $$0.757  1.953${}^{\prime}$ 
IPC  Mexico  1.22  0.34  $$0.593  0.771 
MERVAL  Argentina  1.06  1.09  $$0.176  3.697${}^{\prime}$ 
IBOVESPA  Brazil  0.86  0.57  $$0.528  0.777 
BUX  Hungary  0.42  0.53  $$0.812  2.297${}^{\prime}$ 
IPSA  Chile  0.75  0.53  $$1.686  6.388${}^{\prime}$ 
ISE100  Turkey  1.05  1.31  $$0.227  1.859${}^{\prime}$ 
MADEX  Morocco  0.13  0.16  $$1.033  3.539${}^{\prime}$ 
RTS  Russia  1.42  1.22  $$0.753  1.871${}^{\prime}$ 
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.
(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  4.87 
SENSEX  3.71  SET  16.86  WIG 20  22.22 
RTS  8.18  WIG 20  9.19  SSE 50  3.69 
IPC  0.80  RTS  1.88  MADEX  11.88 
MERVAL  0.10  PSEI  5.79  SET  24.74 
ISE 100  0.72  BUX  1.46  IBOVESPA  4.88 
BUX  0.22  LQ 45  7.99  LQ 45  14.56 
PSEI  0.49  MERVAL  6.52  FTSE MALAYSIA  6.15 
FTSE MALAYSIA  0.56  FTSE MALAYSIA  2.45  SENSEX  3.93 
LQ 45  0.72  IBOVESPA  0.20  MERVAL  2.99 
(b) Developed markets (May–September 2007)  
MV model  MVS model  MVSK model  
Weight  Weight  Weight  
Index  (%)  Index  (%)  Index  (%) 
HANG SENG  32.63  HANG SENG  1.97  ISEQ  70.16 
ATX  32.57  STI  0.77  OMX COP.  9.17 
ATHEX  13.21  ASX 200  5.04  STI  5.78 
OMX HELSIKNI  17.09  BEL  2.30  FTSE MIB  4.05 
STI  0.46  CAC  2.57  ATX  2.20 
NZX 50  1.68  DAX  0.55  AEX  4.23 
ISEQ  0.90  ATHEX  35.20  CAC  0.67 
ASX 200  0.97  FTSE MIB  0.22  NASDAQ  2.26 
OMX COP.  0.37  OMX 30  46.58  ATHEX  0.93 
FTSE MIB  0.23  SMI  4.80  SMI  0.48 
(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  8.29  ISEQ  5.36  SSE 50  15.06 
SENSEX  3.30  BEL  1.02  ISE 100  2.09 
IPC  2.31  WIG 20  21.93  LQ 45  0.59 
ATHEX  0.41  OMX 30  7.69  IPC  1.12 
IBOVESPA  0.38  NZX 50  1.29  WIG 20  23.23 
PSEI  0.99  LQ 45  5.42  TSEC 50  6.10 
ASX 200  0.41  ASX 200  1.06  DAX  6.06 
OMX HELSIKNI  0.30  SENSEX  0.59  NASDAQ  4.07 
ATX  0.29  OMX HELSIKNI  4.96  BEL  4.28 
3.2 Insample portfolios: risk–return analysis
Four portfolios were constructed annually for each panel using the three different models (MV, MVS and MVSK). The insample 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 outofsample 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 outofsample 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.
(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 outofsample period
Realized returns were computed for each portfolio across the outofsample 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 outofsample 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 outofsample 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.
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  8.43*  0.52  4.76*  3.84  1.24  3.91  2.55  1.53  3.63  2.27  1.60* 
2004  1.20  8.79*  0.14  1.49  5.84  0.26  2.55*  2.64  0.97*  1.19  2.65  0.45 
2005  2.11*  7.91*  0.27  0.60  5.81  0.10  1.65  2.86  0.58*  2.08  3.54  0.58* 
2006  2.68  4.00*  0.67  1.29  3.42  0.38  3.61*  3.02  1.20*  1.99  3.05  0.65 
2007  0.70  9.67*  0.07  3.13*  7.49  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  4.89  0.79  3.36  4.56  0.74  4.05  3.73  1.08*  4.61*  4.94*  0.93 
2010  0.86  4.29  0.20  1.48*  2.64  0.56*  0.68  6.16*  0.11  1.47  3.34  0.44 
2011  $$1.02  4.14  $$0.25  $$1.27  5.73*  $$0.22  1.45*  4.35  0.33*  $$0.70  4.37  $$0.16 
2012  1.52  3.10  0.49  1.34  4.52*  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.
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  5.32*  0.46  2.97  3.04  0.98*  2.40  3.76  0.64 
2004  3.22  2.82  1.14*  3.54*  3.14*  1.13*  1.15  2.91  0.39  1.12  1.92  0.58 
2005  3.87*  4.74  0.82*  2.24  4.96*  0.45  1.70  3.22  0.53  1.85  2.91  0.64 
2006  1.09  4.65*  0.23  1.61  3.45  0.47  2.41*  3.86  0.62*  1.37  2.80  0.49 
2007  $$1.37  6.36*  $$0.22  2.57*  2.67  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  6.42*  0.19  2.77*  5.32  0.52*  2.37  6.09  0.39 
2010  2.63*  4.49  0.59*  2.11  4.74*  0.45  0.53  3.51  0.15  0.80  4.39  0.18 
2011  $$0.38  5.20  $$0.07  $$2.28  7.11*  $$0.32  1.91*  3.60  0.53*  $$1.30  4.92  $$0.26 
2012  0.54  4.13  0.13  3.03*  4.20*  0.72*  1.50  3.53  0.42  1.01  3.66  0.28 
Outofsample 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.
Outofsample 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.
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  7.55*  0.37  4.41*  3.12  1.41  3.68  2.27  1.62*  2.96  2.68  1.10 
2004  2.04  7.61*  0.27  2.93*  4.31  0.68  2.64  2.61  1.01*  1.15  1.91  0.61 
2005  3.33  7.58*  0.44  2.11  3.84  0.55  5.34*  3.19  1.67*  1.96  3.08  0.64 
2006  1.74  6.01*  0.29  1.86  3.38  0.55  2.17*  3.06  0.71*  1.65  2.66  0.62 
2007  0.22  9.02*  0.02  1.06*  6.39  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*  3.71  1.49*  1.84  5.39  0.34  4.70  6.09*  0.77  3.39  5.35  0.63 
2010  0.85  4.11  0.21  1.91*  5.72*  0.33*  1.10  3.32  0.33*  1.11  3.79  0.29 
2011  0.50  3.49  0.14  1.89*  4.28  0.44  1.39  2.90  0.48*  $$1.03  4.53*  $$0.23 
2012  2.23  2.98  0.75  1.63  4.82*  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.
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 outofsample 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.
We also note that the volatility of the outofsample 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, valueatrisk (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.
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 thirtythree 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.
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