FINANCE E L S E V I E R Journal of Banking & Finance 21 (1997) 143-167
Portfolio selection and skewness" Evidence from intemational stock markets
Pomchai Chunhachinda a, Krishnan Dandapani b, Shahid Hamid b Arun J. Prakash b,*
a Department of Finance and Banking, Thammasat University, Bangkok, 10200, Thailand b Department of Finance, BA 208, Florida International University, Miami, FL 33199, USA
Received 15 October 1995; accepted 16 May 1996
Abstract
This paper finds that the returns of the world's 14 major stock markets are not normally distributed, and that the correlation matrix of these stock markets was stable during the January 1988-December 1993 time period. Polynomial goal programming, in which investor preferences for skewness can be incorporated, is utilized to determine the optimal portfolio consisting of the choices of 14 international stock indexes. The empirical findings suggest that the incorporation of skewness into an investor's portfolio decision causes a major change in the construction of the optimal portfolio. The evidence also indicate that investors trade expected return of the portfolio for skewness.
JEL classification: G11; G15
Keywords: Multi-objective portfolio selection; Skewness; International portfolio diversification; Inter- temporal stability test
I. Introduct ion
Since the seminal works o f Markowi tz (1952), Sharpe (1964), and Lintner
(1965), numerous studies on portfol io select ion and per formance measures have
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been based upon only the first two moments of return distributions. However, there is a controversy over the issue of whether higher moments should be accounted for in portfolio selection. Many researchers (e.g., Arditti, 1967, 1971; Samuelson, 1970; Rubinstein, 1973) argue that the higher moments cannot be neglected unless there is reason to believe that the asset returns are normally distributed and the utility function is quadratic, or that the higher moments are irrelevant to the investor's decision.
In fact, there is ample empirical evidence (e.g., Fama, 1965; Arditti, 1971; Simkowitz and Beedles, 1978) indicating that individual security and portfolio returns are not normally distributed. Several studies including Arditti (1967, 1971), Jean (1971, 1973), and Levy and Sarnat (1972) also demonstrate that skewness is an important factor in explaining the security and portfolio retums. Samuelson (1970) shows that the higher moment is relevant to the investor's decision on portfolio selection. Moreover, Hanoch and Levy (1970) and Levy and Sarnat (1972) point out that the quadratic utility function is subject to some serious drawbacks. Other studies (e.g., Bierwag, 1974; Borch, 1974; Levy, 1974) also question the adequacy of using quadratic approximation for the utility function in practical applications.
Therefore, this development is based upon the earlier argument that the higher moments of return distributions are relevant to the investor's decision and cannot be neglected. The objective of this study is threefold. First, the return distributions of 14 international stock markets are tested for normality using the Wilk-Shapiro test. Second, the inter-temporal stability of these stock markets are investigated using the Sen and Puri test, which accounts for non-normal return distributions. Our effort extends the Meric and Meric (1989) study on the stability of interna- tional stock markets by incorporating skewness and offers a different methodology to test the relevant hypothesis of stability without making any of the distributional assumptions which were prevalent in the prior studies. Finally, portfolio selection with skewness is empirically applied to the sample of international stock markets. This was complicated by the fact that, in the presence of skewness, selecting a portfolio is a trade-off between competing and conflicting objectives, i.e., the investor tries to maximize expected return and skewness, while simultaneously minimizing variance. To solve this multi-objective portfolio problem, this study extends the work of Lai (1991) by utilizing polynomial goal programming, which incorporates investor preferences for skewness.
The remainder of this paper is organized as follows. Section 2 summarizes the contributions of prior research. Section 3 discusses the empirical applications of polynomial goal programming and describes the methodology for multi-objective portfolio selection with skewness. The specification of data is presented in Section 4. Sections 5 and 6 discuss the tests and the results for the normality of return distributions and the stability of stock markets. The empirical results for the portfolios of international stocks are presented in Section 7. The last section provides concluding remarks.
P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 145
2. Prior contributions on higher-moment portfolio selection
Several researchers (e.g., Arditti, 1967; Samuelson, 1970; Rubinstein, 1973) argue that in order to ignore the third and higher moments, at least one of the following conditions must be true: 1. The distribution has negligible variation. Thus, any moments beyond the first
are zero. 2. The derivatives of the applicable utility function are zero for the third and
higher moments. 3. The distribution of asset returns are normal, or the investors' utility functions
are quadratic. Tsiang (1972, 1974) supports the use of quadratic approximation for the utility
functions in practical problems, assuming that the risk taken by the investor is small compared to his total wealth. However, others (e.g., Bierwag, 1974; Botch, 1974; Levy, 1974) remain skeptical of its application. Hanoch and Levy (1970) point out that the quadratic utility function implies increasing absolute risk aversion (which is contrary to the normal assumption of decreasing absolute risk aversion). Levy and Samat (1972) also point out that the assumption of a quadratic utility function is appropriate only for relatively low returns, which precludes its use for many types of investments.
Regarding the return distributions of an individual security and a portfolio, a substantial number of studies (e.g., Fama, 1965; Arditti, 1971; Sirnkowitz and Beedles, 1978; Singleton and Wingender, 1986) provide evidence indicating that these distributions are not symmetrical. Furthermore, the important role of skew- ness in explaining security returns is demonstrated by Arditti (1967, 1971), Jean (1971, 1973), and Levy and Sarnat (1972). Consequently, more attention has been directed to the existence and the importance of skewness in the portfolio decision.
Samuelson (1970) shows that when the investment decision is restricted to a finite time interval, the use of mean-variance approximation becomes inadequate, and the higher moments become more relevant to portfolio choice. 1 Jean (1971, 1973) extends the portfolio analysis to three or more parameters, and derives the risk premium for higher moments similar to that for the two moments case. Ingersoll (1975) proposes normative security pricing models by extending the CAPM to incorporate the effect of skewness on security pricing. Empirical tests by Kraus and Litzenberger (1976) show that the three moment CAPM fits the return distribution better than the two moment CAPM. Simkowitz and Beedles (1978) examine the behavior of skewness of portfolio returns at different degrees of diversification, and report that increasing diversification results in a progressive loss of portfolio skewness.
l i t should be noted, however, that Levy and Markowitz (1979) and Markowitz (1991) have suggested that for relatively small deviations in rates of return the mean-variance may approximately maximize expected utility even if distributions are not normal.
146 P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167
Although numerous researchers have demonstrated the existence of a risk premium for skewness, only a few studies to date have contributed to the construction of a portfolio with skewness. These studies, which are more relevant to this paper, are Arditti and Levy (1975) (hereafter, AL), Kumar et al. (1978) (hereafter, KPE), and Lai (1991). AL suggest a nonlinear mathematical program to solve the mean-variance-skewness efficient set in a multi-period framework. They assume that the rates of return are identically and independently distributed with zero skewness in each period. Unfortunately, this assumption can be mislead- ing; for instance, even though the third moment in a single period appears to be insignificant, the third moment in a multi-period is not necessarily insignificant. 2 Lai (1991) points out that the optimal portfolio derived by omitting the third central moment of each period in a multi-period model could be an inefficient portfolio. Moreover, AL also ignore the effect of co-skewness (curvilinear interac- tion) on the portfolio selection.
KPE propose linear goal programming to solve the goal conflicts inherent in the portfolio selection of dual-purpose funds (hereafter, DPF). Since, DPF issue income and capital shares in an equal amount, its capital structure has the possibility of a serious conflict of objectives. This conflict occurs because DPF attempt to maximize dividend income to income shareholders and capital apprecia- tion to capital shareholders in one portfolio. Therefore, the conceptual framework proposed by KPE intends to maximize the returns to both groups of shareholders without any single preference or bias to one or the other group.
Lai (1991) applies polynomial goal programming (PGP), introduced by Tayi and Leonard (1988), to solve the portfolio selection with skewness. The optimal portfolio is chosen from a sample of five domestic stocks. In the presence of skewness, selecting a portfolio is a trade-off between competing and conflicting objectives. On the one hand, the investors try to maximize the expected return and the skewness, and on the other hand, they try to minimize the variance of the portfolio returns. To solve this problem, therefore, portfolio selection must depend on an investor's subjective judgements and relative preferences on objectives. Lai (1991) demonstrates that investor's preferences can be incorporated into polyno- mial goal programming, from which portfolio selection with skewness is deter- mined. The important features of PGP are: (1) the existence of an optimal solution, (2) the flexibility of incorporating investor's preferences, and (3) the relative simplicity of computational requirements.
In the next section, we briefly summarize the work of Lai (1991) and then empirically apply the methodology to construct portfolios chosen from among 14
2 In equation (7) of Arditti and Levy (1975), the expansion of [(1 +/zi) 3 +3(1 + p~i)o -2 +/x31] n includes the terms of nCi(1 + ~i)3(n-i)l.~ i31, for all i = 1 ..... n, where nCi is the number of combina- tions of n taken i at a time. Even though J/~3i is insignificant, nCi(l + ~i)3(n-i)l.~i31 may be significant and may not be omitted (see Lai, 1991, endnote 3]
P. Chunhachinda et al. / Journal o f Banking & Finance 21 (1997) 143-167 147
international stock market indexes based on the data from January 1988 to December 1993.
3. Empirical applications of multi-objective portfolio selection
In this section, we briefly describe the Lai (1991) multi-objective portfolio selection model to incorporate the skewness of the return distributions, assuming that the moments higher than the second are behaviorally justified. 3 The outline of the assumptions necessary to obtain the portfolio selection are provided. These assumptions are generally accepted, and their justification can be found in the literature. Following Lai (1991), the ideal portfolio selection with higher moments can be formulated based on the following assumptions: 1. Investors are risk-averse individuals who maximize the expected utility of their
end-of-period wealth. 2. There are n + 1 assets and the (n + 1)th asset is the risk-free asset. 3. Al l assets are marketable, perfectly divisible, and have limited liability. 4. The borrowing and lending rates are equal to the rate of return r on the
risk-free asset. 5. The capital market is perfect. There are no taxes and transaction costs. 6. Unlimited short sales of all assets with full use of the proceeds are allowed.
The mean, variance, and higher moments of the rate of return R i on asset i are assumed to exist for all risky assets i ( i = 1,2 . . . . . n). The n × n var iance-covar i - ance matrix V of asset rates of return is positive definite. 4 And, let
X T
/~i =
E --
( X l , X 2 . . . . . Xn) be the transpose of portfolio component X, where x i is the percentage of wealth invested in the ith risky asset.
(/~1,/~2 . . . . . Rn) be the transpose o f / ~ whose mean is denoted by R,
the rate of return on the ith asset, the mean rate of return on the risk-free asset,
a n X 1 vector of expected excess rates of return, the expectation operator.
3 Kraus and Litzenberger (1976) ignore the moments higher than three when deriving the skewness preference model. They argue that any moments beyond the third are not behaviorally justified.
4 To maintain the consistency and readability, the notations in this section are borrowed from Lai (1991).
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Then, the mean, the variance, and the skewness of the portfolio returns can be 5 defined as:
Mean = X V ( R - ~) , (1)
Variance = X T V X , withV as the variance-covariance matrix of R, (2)
Skewness = E[ x T ( / ~ -- R) ] 3 (3)
The optimal solution is to select a portfolio component X such that its multiple objectives are optimized, i.e., maximize the expected rate of return, minimize the variance and maximize the skewness. Thus, in the context of skewness, the portfolio selection can be formulated by solving the following multiple objective programming problem: 6
Maximize 01 = X X ( R - ~),
Minimize 0 2 = X x V X ,
M a x i m i z e O3 = E[ 3 ,
subject to: X T I = 1.
Since the relative percentage invested in each asset is the main concern of the portfolio decision, Lai (1991) suggests that the portfolio choice X can be rescaled and restricted on the unit variance space, i.e., {X] X ' r V X = 1}. Under the condition of unit variance, the portfolio selection with skewness (P1) can be formulated as follows:
Maximize 01 = X x ( R - ? ) ,
(P1) Maximize O3 = E[ X T ( R - R) ] 3'
subject to: X ' r l = 1,
x ' r v x = 1.
Usually, any single solution of the problem (P1) does not satisfy both objec- tives (01 and 0 3) simultaneously. As a result, the above multi-objective problem involves a two-step procedure. First, a set of non-dominated solutions independent
5 Throughout this study, skewness is defined as the third central moment. Some studies use the statistical definition of skewness which is defined as /x~ //z32 .
6 Lal (1991) points out that an alternative approach of portfolio selection with higher moments is to apply Taylor's expansion of investor utility around the mean of the portfolio returns. However, the utility approach has to specify an investor's exact utility function which is generally unknown or too complicated to be applied, whereas the polynomial goal programming approach requires only the investor's preference for mean, variance, and the skewness of portfolio returns.
P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 149
of investor 's preferences is developed. Then, given the above set of solutions, the investor selects the most preferable solution. The next step can be accomplished by incorporating investor 's preferences for objectives (or the marginal rate of substitution between the objectives) into the construction of a polynomial goal programming problem. 7 Consequently, portfolio selection with higher moments is
a solution of PGP, and multiple objectives can be achieved. In PGP, the objective function does not contain choice variables. Normally, it
contains deviational variables, which represent deviations between goals and what can be achieved, given a set of constraints. Therefore, the objective function causes deviational variables to determine the values of choice variables. As a result, the objective of PGP is to minimize the sum of the deviational variables. If the goals are at the same priority level, the relative amount of deviation from the
goal is always positive. Given an investor 's preferences among mean and skewness (Pl, P3), a PGP
model can be formulated as:
'Minimize O = ( d l ) P ' + (d3) p3 (a)
subject to: x T ( R - r ) + d l = Ol*, (b)
(P2) E [ X T ( R - R ) ] 3 + d 3 = O ; , (c)
x T / = 1, (d)
x T v x = 1, (e)
dl , d 3 ~ O,
where Oi* = the extreme values of objective O i when they are optimized individually rather than simultaneously, d i = nonnegative variables which represent the deviation of 0 i from Oi*, pi = nonnegative parameters representing the investor 's subjective degree of preferences (or trade-off) between objectives. 8 The marginal rate of substitution (MRS) between objectives (say, 0 i and Oj) is
expressed as
MRSij = [ ( O 0 / O d i ) / ( O 0 / O d j ) ] = ( Pi /Pj)[ d ~ i - ' / d p j - 1].
Lai (1991) notes that the MRSij can be used to measure the relative desirability
7 PGP is much more flexible than the linear programming. It allows a simultaneous solution of a system of multiple objectives rather than of a single objective. The objective function of goal programming may consist of heterogeneous units of measure such as dollars and yen, rather than one type of unit.
s Different combinations of Pi represent different preferences of the mean, the variance, and the skewness of a portfolio return. For example, the higher the Pt (P3), the more important the mean (skewness) of the portfolio return is to the investor.
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of forgoing the objective i in order to gain the objective j. Since the MRS is the negative of the slope of the indifference curve, the corresponding indifference curve can be approximated from the polynomial objective function by varying the degree of preference between objectives (pi). As pointed out by Tayi and Leonard (1988), individual preferences between bundles of commodities and their compar- isons can be used to develop consistent indifference curves. In addition, the polynomial objective function with deviational variables (d i) also helps in devel- oping a local approximation to the investor's underlying utility function.
Since an indifference curve covers the entire objective space, the optimal portfolio is the one that has an indifference curve tangent to the frontier of non-dominated points. Thus, a non-dominated portfolio is efficient in the presence of moments higher than the second. The efficient portfolios are the solutions of problem (P2) for various combinations of preferences Pr Jean (1973) shows that investors with different preferences will not necessarily choose the same combina- tion of risky securities, even if they have homogeneous expectations about the distribution of security returns. Thus, the risk premium for each investor will not be identical if they have different preferences.
Since we have fourteen international stock markets, the various expressions in (P2) can be computed as follows:
14 x T ( R - r ) = EXj(Rj - -? ' ) , (4)
j=l
14 14 [ 14 14 1 E [ x T ( R - R ) ] 3 = j=IEX?S3i "-1- 3i~=llj~=lXi2XjSii j+ j=lEXiXTSiJj'] (5)
iv~j
14 X~I = ~_,Xj,
j=l
14 14 14 x w x = Ex? + E Ex,
i=1 i=l j=l i4:j
To be able to solve the PGP problem, computed. For i, j = 1, 2 . . . . . 14:
1 ~v
1 N
7"= -~ E r t , t=l
(6)
(7)
the following set of statistics are
(8)
(9)
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1 N - - 2
o'i2 = -~t~=l( e i t - - e i ) , (10)
I N
O'ij~ ~ t ~ = l ( R i t - g i ) ( e j t - Rj) , ( l l )
1 N - - 3
El(R, ,- R 3 , (12)
1 N - - 2
Siij = -~t~=l( e i t - e i ) ( e j t - e j ) , (13)
1 N - - 2
Sijj : ~t~=l( Rit - e i ) ( ejt - ej) , (14)
where o-ij measures the covariances. Sii j and Sij j measure the co-skewness (curvilinear interactions) which occurs in the joint distribution of R i and Rj. 9
The solution to (P2) can be obtained using the generalized reduced gradient method, which is found to be more robust, particularly for non-linear problems with equality and inequality constraints. Currently, the gradient method has been made available in different forms of software packages. However, in this study, we utilize the Generalized Reduced Gradient Software (GRG2), which has been found by Lasdon et al. (1978) to be more robust and efficient for larger-sized problems. Then, the solution X for various combinations of Pl and p~ is obtained. This solution is the set of mean-va r i ance - skewness efficient portfolios. For Pl = 1 and P3 = 0, the solution of problem (P2) is just a special case of the mean-var iance efficient portfolio. The portfolio weights X of the mean-var iance efficient portfolio are also obtained, and compared to those of the m e a n - var iance-skewness efficient portfolio.
4 . T h e d a t a
The sample data consists of weekly, as well as monthly, rates for return of 14 international stock market indexes from January 1988 to December 1993. 10 The data used to construct the return series which includes the stock indexes and the
9 Jean (1971) points out that the co-skewness (Sii j, SO j) is related to the third moment (S 3) in the same way that the covariance (~rij) is related to the second moment (o-/2). The signs and sizes of the co-skewness will vary as the type and degree of curvilinear relationship between the two securities changes.
l0 As pointed out by Levy (1972), Fogler and Radcliffe (1974), and Singleton and Wingender (1986) the assumed investment holding periods should be arbitrarily chosen so that the empirical results can be compared, since the measurement of higher moments could be sample sensitive.
152 P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167
exchange rates of each country was collected from Barron 's . 11 To minimize any intra-country risk, only well-diversified indexes are chosen to represent the portfolio of each country. The index prices are converted to dollar prices using their respective exchange rates. Holding period returns for each country index ( j ) are calculated as follows:
I j tXj t - I j t - l Xj t -1 Rj, = (15) •,_lxj,_l
where I j t= the stock market index in country j at time t, Sit = the country j exchange rate at time t, expressed in US dollars, R jr = the exchange-rate-adjusted rate of return from investment in country j ' s stock at time t. The weekly and monthly risk-free rates of return are obtained from the
three-month US Treasury bill rates reported in the Federal Reserve Bulletin. The average weekly and monthly risk-free rates during the period under study are 0.1093 and 0.4738 percents, respectively.
5. Testing for normality of return distributions
The methodology employed in this study can be summarized in three parts. First, the return distribution of each international index is tested for normality. Second, the inter-temporal stabilities of the international stock markets are investi- gated. Finally, with different combinations of an investor's preferences for objec- tives, optimal portfolio selections with skewness are determined.
We begin the empirical work by testing the normality of return distributions of the 14 international stock indexes. This test will set the stage for the empirical work in the next section. If the test results support the non-normality of return distributions and the evidence also supports the skewed returns, we will construct an optimal portfolio with skewness. The Wilk-Shapiro test (W-test) is selected as the methodology, since it is found to be the best for testing normality under various alternative specifications of the probability distribution. 12
Usually, the population from which the samples are drawn are infinite. As such, the population parameters are never known. Under these circumstances, Karels and Prakash (1987) point out that the W-test provides the best 'omnibus ' indicator
11 The stock indexes are provided by Morgan Stanley Capital International Perspective, Geneva, and are adjusted for dividends.
12 Shapiro et al. (1968) compare the sensitivities of nine statistical procedures for evaluating the normality of a complete sample. They find that among those procedures, the W-test provides a generally superior measure of non-normality.
P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 153
of non-normality against various specifications of altemative hypotheses. Further- more, since the W-test is scale and origin invariant, one can avoid the problems associated with the estimation of population parameters based on a given sample. Specifically, the W-test tests the hypothesis:
H0: The parent population is normally distributed.
Hi: The parent population is not normally distributed.
To determine whether to reject the null hypothesis of normality. It is necessary to examine the probability associated with the W-statistic. If this probability is less than some specified level (such as 0.05), it means that the null hypothesis cannot be supported at the five percent level of significance.
5.1. The results o f normality tests
For preliminary analysis, Table 1 lists the means and the variances of the rates of return of the 14 international stock markets. A look at the first column reveals that Hong Kong has the highest means for both weekly and monthly rates of return, followed by Singapore. Respectively, Japan and Italy provide the lowest means for weekly and monthly investment horizons. Specifically, Italy is the only market whose mean return is negative. In column two, the evidence indicates that the variability of weekly returns for the Netherlands is the lowest, whereas that of Hong Kong is the highest. For monthly rates of return, France has the highest variance of returns, while the US has the lowest.
Table 1 also provides the values of skewness and kurtosis for each of the indexes' rates of return. The evidence indicates that, for weekly rates of return, most of the international stock markets exhibit positive skewness (except for France, Germany, and Switzerland). For monthly rates of return, Italy, the Netherlands, and Switzerland also show negative skewness. Interestingly, the monthly skewness and kurtosis of France appear to be the highest, whereas their weekly counterparts are relatively low. Similar evidence also appears in the Singapore market which has the highest weekly skewness and kurtosis, but relatively low monthly skewness and kurtosis. The incompatibility between weekly and monthly higher co-moments may be attributed to the intervalling effects which also appear in the Chunhachinda et al. (1994) study on the higher-moment performance measure of the international stock markets.
The results of the test for normality of return distributions using the W-test are also provided in the last column of Table 1. For weekly rates of return, the probability associated with the W-statistic indicates that the null hypothesis of a normal distribution for five markets cannot be supported at the ten percent level of
154 P. Chunhachinda et al. / Journal o f Banking & Finance 21 (1997) 143-167
[..
.=~
0 . ~ "~ o
0 .=
.=.
r~
I I I
d c5 c5,..~,.~ ¢ , i o c5 ~ , - z , . . ~ , - z o" o" I
a~
~d
v
~ t"q 0 t",l t ' q ~."~ 0 C . t ' q "~1" ¢',1 ¢¢~ ~ ¢ q c5 o c5 o c5 c5 o o o c 5 o c5 o o ~ ~. ~
P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 155
significance, i.e., five of the 14 distributions exhibit significant skewness. These five markets include Hong Kong, Italy, Japan, the Netherlands, and Singapore.
For monthly rates of return, there are only three markets (the Netherlands, Sweden, and the UK) for which the results support the null hypothesis of a normal distribution at the ten percent level of significance. In other words, 11 of the 14 international markets exhibit significant skewness of return distributions. This empirical finding is consistent with previous studies (e.g., Arditti, 1971; Fogler and Radcliffe, 1974) which find that the longer the assumed holding periods, the more the return distributions exhibit skewness.
Informally, the kurtosis of a data set can also be examined to provide a check of normality, as the kurtosis of a normal distribution is equal to zero. The information provided in the fourth column of Table 1 indicates that the kurtosis of the 14 international stock markets are far from zero. Hence, at this step of the analysis, we can safely assume from the empirical findings that the return distributions of international stock markets during the period under study are not normally distributed. Therefore, this assumption becomes a valid argument used to develop the empirical works in the next section.
6. Testing for inter-temporal stability of international stock markets
The stability of international stock markets is important in helping investors realize the potential gains from international diversification. If the observed structural relationships are stable over time, the investors can use the ex post patterns of co-movement to proxy the ex ante co-movements. The issue of inter-temporal stability has been investigated by many studies (e.g., Meric and Meric, 1989; Philippatos et al., 1983; Maldonado and Saunders, 1981; Watson, 1980; Panton et al., 1976). These studies contributed significantly in addressing the issue of intertemporal stability of the returns of various stock markets. From the perspective of this paper, Meric and Meric (1989) requires a special mention. In their work, Meric and Meric used 1973-1987 data and upgraded the results of prior studies. We use the period from January 1988 to December 1993 and, in light of the non-normality of stock returns, employ a distribution-free statistical technique to test for intertemporal stability. Thus, our effort not only extends the Meric and Meric (1989) study, it also offers a different methodology to test relevant hypothesis without making any of the distributional assumptions which were prevalent in other studies. These studies have reported contradictory results which may be attributed to their employment of different methodologies, sample countries, and sample periods. To date, no such study takes the non-normality of return distributions into consideration.
Therefore, this paper investigates the inter-temporal stability of international stock markets using the Parhizgari and Prakash (1989) algorithm for the Sen and
156 P. Chunhachinda et al. / Journal o f Banking & Finance 21 (1997) 1 4 3 - 1 6 7
Table 2 Results of tests of the equality of correlation matrices
Sen and Puri L-stat a
A. 1.5-year sub-periods Jan. 1988 to June 1989, and July 1989 to Dec. 1990 July 1989 to Dec. 1990, and Jan. 1991 to June 1992 Jan. 1991 to June 1992, and July 1992 to Dec. 1993
B. Two-year sub-periods Jan. 1988 to Dec. 1989, and Jan. 1990 to Dec. 1991 Jan. 1990 to Dec. 1991, and Jan. 1992 to Dec. 1993
C. Three-yearsub-periods Jan. 1988 to Dec. 1990, and Jan. 1991 to Dec. 1993
1.4419 1.4321 1.6178
1.0773 1.2628
0.9033
a For large samples, L-statistics will be distributed as Chi-squared distributions. For 105 degrees of freedom, the Chi-squared statistic (critical value) at the five percent level of significance is approxi- mately 130. If the values of L-statistic exceed the critical value, this leads to the rejection of the null hypothesis that the correlation matrices are the same.
Puri (1968) me thodo logy which assumes mul t ivar ia te non-normal populat ions. 13
This me thodo logy was or iginal ly des igned to test the equal i ty o f dispers ion
matr ices wi thout assuming the ident i ty o f locat ion vectors. However , for our
purpose, Parhizgar i and Prakash ' s or iginal a lgor i thm was amended to test the
equal i ty o f correlat ion matrices. Thus, we test the hypotheses
H0: The corre la t ion matr ix o f sub-per iod one is the same as that o f sub-per iod
two.
H i : The correlat ion matr ix o f sub-per iod one is not the same as that o f
sub-per iod two.
The rates o f return o f nat ional s tock indexes are used to calculate corre la t ion
matrices. The 1 9 8 8 - 1 9 9 3 per iod is d iv ided into four 1.5-year, three two-year , and
two three-year sub-periods. Then, the correlat ion matr ices for each consecu t ive
sub-periods are calculated and tested. Fo r this study, the L-statistic o f the Sen and
Puri test wil l be distr ibuted as a chi -squared distr ibution with 105 [ = (2 - 1)14(14
+ 1 ) / 2 ] degrees o f f reedom. At this degrees o f f reedom, the crit ical va lue for the
chi -squared statistic at the f ive percent level is approximate ly 130.
13 If X (k) - i X ( k ) ~:(k)) ( a = 1 . . . . . nk; k = 1 . . . . . c) are nk independently and identically dis- - - \ l a , " " " ~ - - p c t l ,
tributed p-variate random vectors from c multivariate populations with the kth population's correlation matrix ¢(g). The L statistics test the hypothesis, H0:~(k) = ~ V k against HI: ~(k) :~ ¢ for at least one pair.
P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 157
6.1. The results of inter-temporal stability tests
Table 2 presents the L-statistics of the Sen and Puri test obtained by comparing the correlation matrices of the consecutive 1.5-year, two-year, and three-year time periods. The results indicate that the L-statistics of all pairs of correlation matrices in all sub-periods are lower than the critical value. Therefore, the null hypothesis that the correlation matrices of the consecutive periods are the same cannot be rejected. In other words, there exists inter-temporal stability in the relationships among the 14 international stock markets.
Interestingly, the evidence also indicates that the L-statistics of the longer sub-periods are lower than those of the shorter sub-periods. Specifically, the L-statistics of three-year sub-periods are the lowest. This implies that, when measured for longer periods of time, the correlation matrices seem to be more stable. This evidence is consistent with the Meric and Meric (1989) study, which finds that the longer the time period, the more the ex post patterns of co-movement will be better proxies for the ex ante co-movements. Philippatos et al. (1983) also note that the longer investment horizons yield patterns of stationarity, and may also legitimately represent holding periods of international financial investors.
Thus, the important implication of this finding to the portfolio decision is that the portfolio efficient frontier may be considered relatively stable over time. This will increase the accuracy of using the past patterns of co-movement to forecast the future co-movements of the stock markets, i.e., enhancing the chance for the investors to realize the potential gains from international diversification.
7. Empirical results of the multi-objective portfolio selection model
Tables 3 and 4 provide the mean values of the variances and the covariances for the weekly and the monthly rates of return, respectively. The values appearing in bold face represent the variances of the 14 international stock markets, and all other values represent the covariances. As can be seen, all covariances are relatively low compared to the variances, for both the weekly as well as the monthly rates of return. This evidence implies that after forming a portfolio, a substantial amount of unsystematic risk is diversified.
Tables 5 and 6 also provide the mean values of skewness (diagonal) and co-skewness (off-diagonal) for the weekly and the monthly rates of return, respectively. For each assumed investment horizon, there are 182 co-skewness values (curvilinear interactions). The sizes and signs of the coskewness will vary depending upon the degree of curvilinear relationship between the two markets. Interestingly, there are 84 negative curvilinear interactions for the weekly invest- ment horizon, as opposed to 54 for the monthly counterpart. In order to obtain the mean-variance-skewness efficient portfolios, the mean values of variances, co- variances, skewness, and co-skewness from Tables 3 -6 will be used to construct a
158 P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167
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P, Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 159
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160 P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167
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P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 161
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162 P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167
Table 7 Coefficient of variation rankings of international stock markets
* CV represents the coefficient of variation = ( tr /2) .
polynomial goal programming for both the weekly and the monthly investment horizons.
Table 7 lists the means, the standard deviations, and the coefficients of variation (CV) of the stock markets ' rates of return. The ranking of CV (the last column of the table) may provide some information, although preliminary, with regard to the potential candidacy for inclusion in the optimal portfolio. A look at the ranking of CV reveals that Hong Kong ranks at the top of the list (providing the least risk per unit of return), whereas Japan and Italy rank at the bottom for both the weekly and the monthly rates of return. This evidence contradicts the Chunhachinda et al. (1994) study which finds that Japan ranks at the top during period 1987 to 1989. The poor ranking of Japan in this study may be attributed to the economic recession during the subsequent period (1990-1993).
Using the information in Tables 3 - 6 , portfolio selection allowing no short sales is determined for both the weekly as well as the monthly investment horizons. First, the values of O 1 and 03* have to be computed separately. To obtain O~*, equation (b) in (P2) is set as the objective function, and then maximized subject to constraints (d) and (e). Similarly, 03* can be obtained by maximizing equation (c) subject to constraints (d) and (e). Then, substitute the obtained values of O~* and 03* back into equations (b) and (c). Next, minimize the objective function (a) subject to constraints (b), (c), (d) and (e), and the portfolio solution is obtained. These optimization processes are then repeated for both weekly and monthly portfolio selection.
Table 8 presents the optimal portfolio solution when short sales are not allowed, i.e., the portfolio weight x i can only take on posit ive values. The results
P. Chunhachinda et aL/ Journal of Banking & Finance 21 (1997) 143-167
Table 8
The opt imal portfol io selection a l lowing no short sales
163
p l = l , p 3 = 0 a p 1 = l , p 3 = 1 b P l = l , P 3 = 2 b p 1 = 2 , p 3 = l b
Week Month Week Month Week Month Week Month
Austra l ia . . . . . . .
Be lg ium . . . . . . . .
C a n a d a . . . . . . . . France 4.35 5.44 2.26 3.02 1.32 1.91 1.32 1.90
Ge rmany 7.65 6.13 21.13 8.77 11.59 - 11.58 -
Hong Kong 22.51 35.46 21.70 32.83 15.67 29.33 15.66 29.34
a This combina t ion represents the m e a n - v a r i a n c e efficient portfolio.
b This combina t ion represents the m e a n - v a r i a n c e - s k e w n e s s efficient portfolio.
c Skewness represents the third central co -moments o f the opt imal portfolio return.
show that different combinations of Pl and P3 result in different portfolio compositions. Clearly, the evidence demonstrates that the incorporation of skew- ness into an investor's portfolio decision causes a major change in the construction of their optimal portfolio.
Interestingly, Australia, Belgium, Canada, Italy and Japan are not included in the optimal portfolios, except under Pl = l, P3 = 2 , and Pl = 2, P3 = 1 for the monthly portfolio. The exclusion may be due to relatively poor performances of those stock markets, as can be seen in Table 7 by the rank of coefficients of variation (ranging from l0 to 14).
For Pl = 1, and P3 = 0 (the mean-variance efficient portfolio), the Netherlands (with the lowest variance) and Hong Kong (with the highest mean return) have dominant components of 25.70 and 22.51 percent, respectively, in the optimal portfolio for the weekly investment horizon. For the monthly investment horizon, Hong Kong has the largest component of 35.46 percent, followed by Singapore with a component of 27.47 percent in the optimal portfolio. However, Sweden and the U.K. are excluded from the monthly optimal portfolio.
For Pl = 1 and P3 = 1 (the mean-variance-skewness efficient portfolio), again, the Netherlands is the most dominant component in the weekly portfolio, followed by Hong Kong and Germany. For the monthly investment horizon,
164 P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167
Singapore has the largest component of 33.21 percent, and Hong Kong has the second largest component of 32.83 percent in the optimal portfolio.
For the other two mean-variance-skewness efficient portfolios (pl = 1, Ps = 2, and Pl = 2, P3 ---- 1) the compositions of the portfolio appear to be very similar. For the weekly investment horizon, the optimal portfolios are dominated by the components of the Netherlands and Switzerland markets. For the monthly invest- ment horizon, however, Hong Kong has the largest component of approximately 29 percent in both optimal portfolios.
Generally, both the mean-variance (P3 = 0 ) and the mean-variance-skewness (P3 v~ 0) efficient portfolios seem to be dominated by the investment components of only four markets, including Hong Kong, the Netherlands, Singapore, and Switzerland. These markets seem to have a high ranking of the coefficient of variation, and have either relatively high mean return, low variance, or high skewness.
As reported in Table 8, the mean-variance efficient portfolios have the highest expected return of 0.334 and 1.620 percent among various portfolios, for both weekly and monthly investment horizons. This evidence is consistent with the notion that the expected return of the mean-variance efficient portfolio must dominate any other portfolios given the same level of variance. On the other hand, all skewness of the mean-variance-skewness efficient portfolios (Ps ~ O) are
greater than those of the mean-variance efficient portfolios (ps = 0). This implies that the investor will trade the expected return of the portfolio for the skewness. However, the optimal portfolios obtained from PGP may not retain their optimal status for long due to the changing of the correlation structure. If this is the case, then the portfolio must be revised periodically by rerunning the PGP model.
We also obtained the optimal portfolios after allowing for short sales of various combinations of Pl and P3. As expected, we found that efficient frontiers of no-short-sales portfolios lie within the frontiers of short-sales portfolios. Thus, with short sales allowed, investors will be better off, as they are able to achieve a higher level of satisfaction. 14
8. Conclusions
In this study, the return distributions of 14 international stock markets are tested for normality using the Wilk-Shapiro W-test. The evidence indicates that 5 of the 14 weekly return distributions, and 11 of the 14 monthly return distributions exhibit significant skewness. This finding sets the stage for multi-objective portfolio selection (with skewness), and confirms our argument that higher mo- ments cannot be neglected in the portfolio selection.
14 For brevity, we do not report the empirical results for short sales portfolios. Interested readers may obtain the results from the author of correspondence.
P. Chunhachinda et al. / Journal of Banking & Finance 21 (1997) 143-167 165
The Sen and Purl test, which accounts for non-normality of return distributions, is applied to test the inter-temporal stability of 14 international stock markets. The empirical findings suggests that there is strong evidence to support inter-temporal stability in international stock market movements during the period under study. Moreover, when measured for a longer period of time, the correlation structure seems to be more stable, thus representing a better proxy for future relationships. This empirical evidence is in conformity with the Meric and Meric (1989) study. Thus, the important implication for portfolio decisions is that the investor can use the past patterns of co-movements as a proxy for the future co-movements of international stock markets.
The portfolio selection with skewness is determined using the Lai (1991) polynomial goal programming in which investor preferences among objectives can be easily incorporated. The empirical evidence suggests that the incorporation of skewness into the investor's portfolio decision causes a major change in the construction of the optimal portfolio. Under both the short-sales and the no-short- sales cases, the mean-variance and mean-variance-skewness efficient portfolios appear to be dominated by the investment components of only four markets. These four markets have high rankings of coefficients of variation, and have either a relatively high mean return, low variance, or high skewness. The empirical evidence also indicates that the expected returns of the mean-variance efficient portfolios are the highest among the various portfolios, for both the weekly as well as the monthly investment horizons. On the other hand, the skewness values of the mean-variance-skewness efficient portfolios are found to be higher than those of the mean-variance efficient portfolios. This evidence implies that an investor will trade the expected return of a portfolio for skewness. The evidence also indicates that by allowing short sales, investors are better off as they attain higher expected return and skewness simultaneously.
Acknowledgements
We are indebted to our colleagues at the FIU Finance Workshop for helpful suggestions. We are also grateful to two anonymous referees for their helpful comments. All of the remaining errors are, of course, the authors alone.
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