Evidence from International Stock Markets

Evidence from world(prenominal) Stock MarketsPortfolio Selection with Four Moments Evidence from International Stock Markets nonwithstanding the world(prenominal) variegation suggested by several researchers (e.g. Grulbel, 1968 charge and Sarnat, 1970 Solnik, 1974) and the increased consolidation of capital markets, the planetary house deflect has not decreased (Thomas et. al., 2004 and Coeurdacier and Rey, 2013) and there is no complete explanation of this develop. Further more(prenominal), there atomic t exclusivelyy 18 the fastgrowing concerns of investor for extreme pre tends1 and the investors p bushelence toward bizarre moments (e.g. mean and lopsidedness) and an aversion toward even moments (e.g. part and kurtosis) considered by numerous studies (e.g. Levy, 1969 Arditti, 1967 and 1971 Jurczenko and Maillet, 2006). agree to these reasons, this paper propose to investigate whether the internalisation of investor preferences in the high(prenominal) moments into the outside(a) addition apportionment paradox keep help apologise the kinfolk bias puzzle. The get will allow investor preferences to depend not solo the offset dickens moments (i.e. mean and variance) but also on the high moments, such(prenominal) as lopsidedness and kurtosis, by apply the polynomial conclusion programming (PGP) admittance and then gene identify the three-dimensional high-octane marge.The master(prenominal) objective of the proposed weigh is to investigate whether the internalization of lopsidedness and kurtosis into the foreign gillyflower portfolio survival causes these issues The changes in the creationion of optimum portfolios, the patterns of relationships mingled with moments, and the less diversification comp bed to the mean-variance model.Since several researchers (e.g. Grulbel, 1968 Levy and Sarnat, 1970 Solnik, 1974) suggest that coronation in a portfolio of equities across foreign markets abide enceinte diversification oppor tunities, then investors should rebalance there portfolio international from domesticated toward foreign equities. However, US investors continue to hold equity portfolios that ar big(p)ly henpecked by domestic summations. Thomas et. al. (2004) reported that by the end of 2003 US investors held only 14 percent of their equity portfolios in foreign mental strains. Furthermore, Coeurdacier and Rey (2013) also reported that in 2007, US investors hold more than 80 percent of domestic equities.Many explanations bind been recommended in the literary works to explain this piazza bias puzzle include direct barriers such as capital controls and transaction costs (e.g. Stulz, 1981 Black, 1990 Chaieb and Errunza, 2007), and indirect barriers such as information costs and high estimation un authoritativety for foreign than domestic equities (e.g. Brennan and Cao, 1997Guidolin, 2005 Ahearne et. al., 2004). Nevertheless, several studies (e.g. Karolyi and Stulz, 1996 Lewis, 1999) suggest s that these explanations be weakened since the direct costs to international investment have come down signifi lavatorytly everyplacetime and the financial globalization by electronic trading increases modifys of information and decreases uncertainty across markets.Since the neo portfolio possible action of Markowitz (1952) indicates how risk-averse investors can construct optimum portfolios ground upon mean-variance trade-off, there ar numerous studies on portfolio alternative in the framework of the first two moments of the recall scatterings. However, as many researchers (e.g., Kendall and Hill, 1953Mandelbrot, 1963a and 1963b Fama, 1965) discovered that the presence of significant lopsidedness and special kurtosis in asset take place distributions, there is a great concern that higher(prenominal)moments than the variance should be accounted in portfolio endurance.The motivation for the generalization to higher moments arises from the theoretical work of Levy (19 69) provided the cubic gain program program bit depending on the first three moments. Later, the empirical works of Arditti (1967 and 1971) documented the investors preference for corroboratory lopsidedness and aversion negative lopsidedness in return distributions of various(prenominal) stocks and mutual funds, respectively. Even Markowitz (1959) himself also supports this aspect by suggesting that a mean-semi-variance trade-off 2, which gives priority to avoiding downside risk, would be superior to the original mean-variance advancement.While the enormousness of the first three moments was recognized, there were some arguments on the incorporation of higher moments than the trey into the analysis. First, Arditti (1967) suggested that most of the information about any probability distribution is contained in its first three moments. Later, Levy (1969) argued that even the higher moments ar or so work ons of the first moments, but not that they ar small in magnitude.se veral(prenominal) authors (Levy, 1969 Samuelson, 1970 Rubinstein, 1973) also recommend that in general the higher moments than the variance cannot be neglected, except when at least one of the following conditions must be trueAll the higher moments beyond the first are zero.The derivatives of utility usage are zero for the higher moments beyond the second.The distributions of asset returns are normal or the utility enjoyments are quadratic.However, ample yard (e.g., Kendall and Hill, 1953 Mandelbrot, 1963a and 1963b Fama, 1965) presented not only the higher moments beyond the first and their derivatives of the utility function are not zero, but also the asset returns are not normally distributed. Furthermore, several researchers (Tobin, 1958 Pratt, 1964 Samuelson, 1970 Levy and Sarnat, 1972) indicate that the assumption of quadratic utility function is appropriate only when return distributions are compact. Therefore, the higher moments of return distributions, such as skewness, a re relevant to the investors decision on portfolio plectrum and cannot be ignored.In the field of portfolio theory with higher moments, Samuelson (1970) was the first author who recommends the importance of higher moments than the second for portfolio analysis. He extracts that when the investment decision restrict to the bounded time horizon, the use of mean-variance analysis deceases insufficient and the higher moments than the variance become more relevant in portfolio selection. Therefore, he developed three-moment model based on the cubic utility function which express by Levy (1969)3. chase Samuelson(1970), turn of events of studies (e.g. Jean, 1971, 1972 and 1973 Ingersoll, 1975 and Schweser, 1978) explained the importance of skewness in security returns, derived the risk premium as functions of the first three moments, and generated the three-dimensional efficacious frontier with a risk-free asset.Later, Diacogiannis (1994) proposed the multi-moment portfolio optimiz ation programme by minimizing variance at any given train of anticipate return and skewness. Consequently, Athayde and Flores (1997) developed portfolio theory taking the higher moments than the variance into consideration in a utility maximizing context. The expressions in this paper greatly simplified the numerical solutions of the multi-moment portfolio optimum asset allocation problems4.23Levy (1969) localises the cubic utility function as U(x) which has the form U(x) = ax + bx + cx , where x is a random variable and a,b,c are co efficients. This function is concave in a certain range but convex in an opposite.Jurczenko, E. and Maillet, B. (2006) Multi-Moment Asset assignation and determine Models, Wiley Finance, p. xxii.Different approaches have been developed to incorporate the indivi treble preferences for higher- hallow moments into portfolio optimization. These approaches can be divided into two of import groups, the original and treble approaches.The multiple appr oach starts from a specification of the higher-moment utility function by using the Taylors series expansion to link between the utility function and the moments of the return distribution. Then, the dual approaches will determine the optimal portfolio via its parameters reflecting preferences for the moments of asset return distribution. Harvey et. al. (2004) uses this approach to construct the set of the three-moment efficient frontier by using two sets of returns3. The results guide that as the investors preference in skewness increases, there are fast change points in the expect utility that lead to dramatically modifications in the allocation of the optimal portfolio. Jondeau and Rockinger (2003 and 2006) and Guidolin and Timmermann (2008) extend the dual approach in portfolio selection from three- to quartette-moment framework.A shortcoming of this dual approach is that the Taylor series expansion whitethorn converge to the anticipate utility under restrictive conditions. That is for some utility functions (e.g. the exponential function), the expansion converges for all possible levels of return, whereas for some types of utility function (e.g. the logarithm-power function), the convergence of Taylor series expansion to the evaluate utility is ensured only over a restricted range6. Furthermore, since Taylor series expansion have an infinite number of terms, then using a finite number of terms creates the brusqueness error.To circumvent these problems, the primal approach parameters that used to weight the moment deviations are not relate precisely to the utility function. Tayi and Leonard (1988) introduced the Polynomial Goal schedule (PGP), which is a primal approach to solve the goal in portfolio optimization by trade-off between competing and conflicting objectives. Later, Lai (1991) is the first researcher who proposed this order to solve the multiple objectives determining the set of the mean-variance-skewness efficient portfolios. He illustr ated the three-moment portfolio selection with three objectives, which are maximizing both the expected return and the skewness, and minimizing the variance of asset returns.Follows Lai (1991) who uses a sample of five stocks and a risk-free asset, Chunhachinda et. al. (1997) and Prakash et. al. (2003) fancys three-moment portfolio selection by using international stock indices.Regarding the under-diversification, many studies (e.g. Simkowitz and Beedles, 1978 Mitton and Vorkink, 2004 and Briec et. al., 2007) suggested that incorporation of the higher moments in the investors objective functions can explain portfolio under-diversification. Home bias puzzle is one of the under-diversification. It is a tendency to invest in a large proportion in domestic securities, even there are dominance gains from diversification of investment portfolios across national markets. Guidolin and Timmermann (2008)4 indicate that home bias in US can be explained by incorporate the higher moments (i.e. skewness and kurtosis) with distinct bull and bear regimes in the investors objective functions.several(prenominal) researchers use the primal and the dual approaches to examine theinternational portfolio selection. Jondeau and Rockinger (2003 and 2006) and Guidolin and Timmermann (2008) applied the dual approaches using a higher-order Taylor expansion of the utility function. They provide the empirical evidence that under large departure from newton of the return distribution, the higher-moment optimization is more efficient than the mean-variance framework. Chunhachinda et. al. (1997) and Prakash (2003) applied the Polynomial Goal Programming (PGP), which is a primal approach, to determine the optimal portfolios of international stock indices. Their results indicated that the incorporation of skewness into the portfolio selection problem causes a major change in the allocation of the optimal portfolio and the trade-off between expected return and skewness of the efficient portf olio. appurtenance 1 presents methodology and selective information of the previous papers that study international portfolio selection with higher moments.In the proposed study, I will extend PGP approach to the mean-variance-skewnesskurtosis framework and investigate the international asset allocation problem that whether the incorporation of investor preferences in the higher moments of stock return distributions returns can help explain the home bias puzzle.Since previous research (e.g. Levy, 1972 Singleton and Wingerder, 1986) points out that the estimated honours of the moments of the asset return distribution sensitive to the prime(prenominal)s of an investment horizon, I will examine daily, each week, and monthly data sets in the study5.The sample data will harp of daily, weekly, and monthly rates of return of five international indices for all acquirable data from January 1975 to December 2016. These five indices cover the stock markets in the main geographical areas , namely the United States, theUnited Kingdom, Japan, the Pacific region (excluding Japan), and europium (excluding United Kingdom)6. Moreover, the study also use three-month US Treasury invoice rates as the existence of the risk-free asset in order that the investor is not restricted to invest only in risky assets.The data source of these indices is the Morgan Stanley Capital International Index (MSCI) who reports these international price indices as converted into US dollar at the spot foreign exchange rate. The MSCI stock price indices and T-bill rates are available in Datastream.The methodology proposed in the study consists of two parts. First, the rate of return distribution of to each one international index will be adjudicateed for normality by using the Shapiro-Wilk test. Then, the PGP approach will be utilized to determine the optimal portfolio in the quaternionmoment framework.4.1 Testing for normality of return distributionAt the rise of the empirical work, I will test the normality of return distributions of international stock indices and the US T-bill rates. This test provides the foundation for examine the portfolio selection problem in the mean-variance-skewness-kurtosis framework.Although several methods are developed, there is an ample evidence that the ShapiroWilk is the outflank choice for evaluating normality of data under various specifications of the probability distribution. Shapiro et. al. (1968) provide an empirical sampling study of the sensitivities of nine normality-testing procedures and concluded that among those procedures, the Shapiro-Wilk statistic is a largely superior measure of non-normality. More recently,Razali and Wah (2011) compared the power of four statistical tests of normality via Monte Carlo simulation of sample data generated from various alternation distributions. Their results support that Shapiro-Wilk test is the most powerful normality test for all types of the distributions and sample sizes.The Shapi ro-Wilk statistic is defined aswhere is the i th order statistic (rate of returns), . / is the sample mean, are the expected values of the order statistics of self-sufficingand identically distributed random variables sampled from the standard normal, andV is the covariance intercellular substance of those order statistics.Note that the values of are provided in Shapiro-Wilk (1965) get across based on the order i.The Shapiro-Wilk tests the null possible action of normalityH0 The macrocosm is normally distributed.H1 The population is not normally distributed. If the p-value is less than the significant level (i.e. 1%, 5%, or 10%), then the null hypothesis of normal distribution is rejected. Thus, there is statistical evidence that the sample return distribution does not came from a normally distributed population. On the other hand, if the p-value is great than the chosen alpha level, then the null hypothesis that the return distribution came from a normally distributed popul ation cannot be rejected.4.2 Solving for the multi-objective portfolio problemFollowing Lai (1991) and Chunhachinda et. al. (1997), the multi-objective portfolio selection with higher momentscan be examined based on the following assumptionsInvestors are risk-averse individuals who maximize the expected utility of their end-ofperiod wealth.There are n + 1 assets and the (n + 1)th asset is the risk-free asset.All assets are marketable, perfectly divisible, and have express liability.The borrowing and lending rates are equal to the rate of return r on the risk-free asset.The capital market is perfect, there are no taxes and transaction costs.Unlimited short sales of all assets with rich use of the proceeds are allowed.The mean, variance, skewness, and kurtosis of the rate of return on asset are assumed to exist for all risky assets for 1,2, . Then, I define the variables in the analysis as= ,, , be the transpose of portfolio component , where is the part of wealth invested in the th risky asset,= ,, , be the transpose of whose mean denoted by ,= the rate of return on the th risky asset,= the rate of return on the risk-free asset,= a (n x 1) vector of expected excess rates of return,= the expectation operator,= the (n x 1) vector of ones,= the variance-covariance (n x n) matrix of ,= the skewness-coskewness (n x n2) matrix of ,= the kurtosis-cokurtosis (n x n3) matrix of .Then, the mean, the variance, the skewness, and the kurtosis of the portfolio returns can be defined as7,, -,8Kurtosis = = - - .Note that because of certain symmetries, only ((n+1)*n)/2 elements of the skewnesscoskewness matrix and ((n+2)*(n+1)*n)/6 elements of the kurtosis-cokurtosis matrix must be computed. The components of the variance-covariance matrix, the skewness-coskewness matrix, and the kurtosis-cokurtosis matrix can be computed as follows , , , , , .Therefore, the optimal solution is to select a portfolio component . The portfolio selection can be determined by solving the foll owing multiple objectives, which are maximizing the expected return and the skewness while minimizing the variance and the kurtosis,,-, = - -. checkmate to1.Since the percentage invested in each asset is the main concern of the portfolio decision, Lai (1991) suggests that the portfolio choice can be rescaled and restricted on the unit variance quadruplet (i.e. 1 ). Under the condition of unit variance, the portfolio selection problem with skewness and kurtosis (P1) can be formulated as follows ,-,(P1) = - - ,subject to 1 ,1 .Usually, the solution of the problem (P1) does not satisfy three objectives (,, ) simultaneously. As a result, the above multi-objective problem (P1) involves a two-step procedure. First, a set of non-dominated solutions independent of investors preferences is developed. Then, the next step can be accomplished by incorporating investors preferences for objectives into the formula of a polynomial goal programming (PGP). Consequently, portfolio selection by w elcome the multiple objectives that is the solution of PGP can be achieved.In PGP the objective function ( ) does not contain a portfolio component , it contains deviational variables ( , , ) which show deviations between goals and what can be achieved, given a set of constrains. Therefore, the objective function ( ) is minimization of the deviation variables ( , , ) to determine the portfolio component. Moreover, if the goals are at the same priority level, the deviations from the goals ( , , ) are non-negative variables.Given an investors preferences among mean, skewness, and kurtosis ( , , ), a PGP model can be expressed as .subject to - ,-- ,(P2) - - = - ,1 ,1 ,,, 0 .where- = the extreme value of objective when they are optimized individually, then- 1 , - 1 ,and - 1 ,= the non-negative variables which represent the deviation of and -,= the non-negative parameters representing the investors subjective degree of preferences between objectives,The combinations of represent diver gent preferences of the mean, the skewness, and the kurtosis of a portfolio return. For example, the higher , the more important the mean (skewness or kurtosis) of the portfolio return is to the investor. Thus, the efficient portfolios are the solutions of problem (P2) for various combinations of preferences .The expected results provided in this section refer to two parts of methodology, the normality test and the international portfolio optimization in four-moment framework.5.1 The expected results of the normality testMany researches examine the international stock indices and found that most of the stock return distributions exhibit skewness and their excess kurtosis are far from zero. For instance, in the work of Chunhachinda et. al. (1997), the Shapiro-Wilk statistics indicate 5 markets and 11 markets reject the null hypothesis of normal distribution at ten percent significant level, for weekly and monthly data, respectively. Prakash et. al. (2003) use the Jarque-Bera test to outpouring the normality of each international stock index, their results indicate that for 17 markets for weekly returns and 10 markets for monthly returns reject the null hypothesis of normal distribution five percent significant level.Therefore, I expected that the Shapiro-Wilk tests in the proposed study will be significant and reject the null hypothesis of normality. In other words, the return distributions of international stock markets during the period under study are expected to be non-normal.5.2 The expected results of the multi-objective portfolio selection5.2.1 The changes in the allocation of optimal portfoliosChunhachinda et. al. (1997) and Prakash et. al. (2003) both indicated that the incorporation of skewness into the portfolio selection problem causes a major change in the allocation of the optimal portfolio. However, their definitions of a major change are different. Chunhachinda et. al. (1997) found that there is a modification in the allocation when they compar e between the mean-variance and the mean-variance-skewness efficient portfolios. However, both types of portfolios are dominated by the investment components of only four markets9. On the other hand, Prakash (2003) results show that the structural weights of the mean-variance and the mean-variance-skewness optimal portfolios are dominated by different markets.Therefore, I expected that when I compare between of the mean-variance efficient portfolios, the three-moment efficient portfolios, and the mean-variance efficient portfolios, the percentage invested in each asset will be different in magnitude and ranking.5.2.2 The trade-off between expected return and skewnessMost of the studies of international portfolio selection with higher moments (e.g. Chunhachinda et. al., 1997 Prakash et. al., 2003 Jondeau and Rockinger, 2003 and 2006) reported that the mean-variance efficient portfolios have the higher expected return while the three-moment efficient portfolios have great skewness. T hus, they indicated that after incorporation of skewness into portfolio selection problem, the investor will trade the expected return of the portfolio for the skewness. More recently, Davies et. al. (2005) applied PGP to determine the set of the four-moment efficient funds of hedge funds and found not only the trade-off between the mean and the skewness, but also the trade-off between the variance and the kurtosis.Thus, I expected to discover the trade-off between the expected return and the skewness and the trade-off between the variance and the kurtosis. In addition, I will also investigate other relationships between the moments of return distribution and report them in both numerical and graphical ways.5.2.3 The less diversification compared to the mean-variance model. To investigate whether the incorporation of higher moments than the second (i.e. skewness and kurtosis) can help explain the home bias puzzle, I will examine the hypothesisH0 ZMV ZMVSK.H1 ZMV ZMVSK.where ZMV an d ZMVSK are the number of nonzero weights of the mean-variance efficient portfolios and the four-moment efficient portfolios, respectively.If the number of nonzero weights of the mean-variance efficient portfolios (ZMV) is greater than the number of nonzero weights of the four-moment efficient portfolios (ZMVSK), then I will rejected the null hypothesis. This implies that the incorporation of the higher moments into the portfolio decision can help explain the home bias puzzle.However, the results from the publications are mixed. On one hand, several researchers (e.g. Prakash et. al., 2003 Briec et. al., 2007 Guidolin and Timmermann, 2008) provided the evidence that the incorporation of skewness into the portfolio selection causes the less diversification in the efficient portfolio. On the other hand, the results of some studies (e.g. Chunhachinda et. al., 1997 Jondeau and Rockinger, 2003 and 2006) found that when compare with the mean-variance efficient portfolios, the diversificat ion of the higher-moment efficient portfolios attend to be same or even became more diversify.I expected the results to show that the four-moment efficient portfolio is less diversified than the mean-variance one. In other words, the incorporation of the skewness and the kurtosis into the international portfolio selection can help explain the home bias.1 Jurczenko, E. and Maillet, B. (2006) Multi-Moment Asset Allocation and Pricing Models, Wiley Finance, p. xxii.2 Semi-variance is a measure of the dispersion of all observations that fall below the average or target value of a data set.3 The first set consists of four stocks and the second set consists of four equity indices, two commodities, and a risk-free asset. 6Jurczenko, E. Maillet, B., and Merlin, P. (2006) Multi-Moment Asset Allocation and Pricing Models, Wiley Finance, p. 52.4 Guidolin and Timmermann (2008) analyze the portfolio selection problem by using the dual approach.5 Chunhachinda et. al. (1997) and Prakash et. al. ( 2003) studied the portfolio selection across national stock markets by using two data sets, weekly and monthly data.6 Guidolin and Timmermann (2008) reported that these markets represent about 97% of the world equity market capitalization.7 I use the derivations of skewness and kurtosis as provided in the textbook Multi-Moment Asset Allocation and Pricing Models of Jurczenko and Maillet (2006) to translate the expectation operators into the matrix terms.8 Let A be an (n-p) matrix and B an (m-q) matrix. The (mn-pq) matrix A-B is called the of matrix A and matrix B9 The four markets are Hong Kong, Netherlands, Singapore, and Switzerland. These markets have high rankings of the coefficient of variation under the sample period.