Higher order moment risk in efficient futures portfolios

Journal of Economics and Business 65 (2013) 33– 54

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Journal of Economics and Business

Higher order moment risk in efficient futures portfolios

Leyuan Youa,∗, Duong Nguyenb

a Texas State University-San Marcos, 601 University Dr., San Marcos, TX 78666, USAb University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, USA

a r t i c l e i n f o

Article history:Received 16 September 2011Received in revised form 14 September2012Accepted 15 September 2012

JEL classification:G150

Keywords:Higher momentsValue-at-riskFour-moment value-at-riskDiversificationFutures

a b s t r a c t

This study examined whether mean–variance (M–V) frameworkhelps to efficiently diversify away tail risk due to skewness, kurtosis,and other higher moments. We found that M–V efficient portfo-lios do not have significant higher moment risk, because the riskmeasured by the two-moment value-at-risk (VaR) was not signif-icantly different from risk measured by the higher moment VaRs.This result was not caused by the large number of assets includedin the portfolio. With only nine assets in the M–V portfolio, about85% of the diversifiable loss measured by higher moment VaR wasdiversified away. Furthermore, with less than nine assets in M–Vefficient portfolios, the M–V technique diversified away the highermoment risk even more efficiently than did volatility.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Half a century ago, Markowitz’s classic portfolio theory quantified the investor’s risk as the standarddeviation of the return distribution and simplified the portfolio selection process into optimization in amean–variance (M–V) framework. The Markowitz portfolio theory assumed an underlying normal dis-tribution/quadratic utility function for asset’s returns. If the return was normal, the first two moments(i.e., mean and variance) alone were sufficient to explain the distribution. As a result, investors shouldnot care about higher order moments. However, it is well documented in the literature that finan-cial asset’s returns are not normally distributed (Aggarwal & Aggarwal, 1993; Arditti, 1967; Fama,1965; Gorton & Rouwenhorst, 2006). This implies that investors may consider higher order moments

∗ Corresponding author. Tel.: +1 512 245 3215; fax: +1 512 245 3089.E-mail address: [email protected] (L. You).

0148-6195/$ – see front matter © 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jeconbus.2012.09.003

dx.doi.org/10.1016/j.jeconbus.2012.09.003

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mailto:[email protected]

dx.doi.org/10.1016/j.jeconbus.2012.09.003

34 L. You, D. Nguyen / Journal of Economics and Business 65 (2013) 33– 54

in their investment decision. Quite a number of studies have pointed out the importance of consid-ering higher order moment, such as skewness and kurtosis, in asset pricing (Dittmar, 2002; Nguyen& Puri, 2009; Rubinstein, 1973; Scott & Horvath, 1980). Consequently, some researchers have triedto optimize portfolios in a three-moment or four-moment framework (Jondeau & Rockinger, 2006;Prakash, Chang, & Pactwa, 2003). However, such techniques also come with serious issues and resultin lowered portfolio returns. Moreover, some researchers have suggested the use of further higherorder moments beyond the first four moments so that higher order moments can specify the returndistribution more precisely in taking into account the investors’ concern for extreme outcomes ontheir investment or the tail end of the return distribution (see Chung, Johnson, & Schill, 2006; Nguyen& Puri, 2009; Rubinstein, 1973). However, another strand of literature (Aggarwal & Aggarwal, 1993;DeFusco, Karels, & Muralidhar, 1996; Simkowitz & Beedles, 1978; Sun & Yan, 2003) suggests thatinstead of considering complex higher order moments of the distribution, investors might choose thesimple naïve diversification, because it reduces non-normality to some degree.

Given the popularity of the M–V framework and findings regarding the naïve portfolio, the followingquestions are of important interest: Does the M–V optimization also help to diversify away the riskdue to higher moments, such as skewness and kurtosis? Mean–variance optimization assumes thequadratic utility function for investors, who are not concerned with higher moment risk in theory; itaims to reduce risk measured by standard deviation. But, the mean–variance optimization techniquemay also lead to reduction in higher moment risk, which provides added benefit for its users, who, inreality, are also concerned with higher moment risk. If mean–variance technique can reduce highermoment risk, then to what degree can it diversify away the higher moment risk? In another context,if the investors hold efficient portfolios based on M–V optimization, should they still be concernedwith higher moment risk? How many assets should be included in a portfolio to optimize the highermoment risk?

This study aims to answer the above-mentioned questions and provides more understanding onthe efficient frontier portfolios resulting from M–V optimization. The second purpose of this paperis to examine the diversification benefit of including various financial and commodity assets in aportfolio. More specifically, this study compares efficient portfolios of a wide range of futures assetswith a traditional portfolio to examine the diversification benefit of a wide range of assets. Subse-quently, it employs the two-moment value-at-risk (VaR2) and the higher moment value-at-risk (VaR)measures to evaluate the M–V efficient portfolios and compares the results. VaR2 calculates the poten-tial loss using the return distribution’s mean and standard deviation, whereas the three-momentVaR (VaR3) utilizes the mean, standard deviation, and skewness. Furthermore, the four-moment VaR(VaR4) uses the mean, standard deviation, skewness, and kurtosis, and finally, the five-moment VaR(VaR5) includes a fifth moment risk to calculate the potential loss. Hence, the difference between VaR2and the higher moment VaR is the potential risk due to higher moments. For example, the differencebetween VaR2 and VaR3 is the potential loss/risk due to skewness, whereas the difference betweenVaR2 and VaR4 is the potential risk due to skewness and kurtosis. Comparing the results of VaR2 andhigher moment VaR helps in understanding whether significant higher moment risk still persist in effi-cient portfolios. Subsequently, the analysis is further extended to include simulations to determinethe effect of portfolio size on higher moment risk diversification in M–V efficient portfolios. Futurescontracts on financial and commodity assets are employed in this study, instead of cash assets, fortheir advantages over cash assets.1

This study observed that it is beneficial to diversify among different types of futures assets. Moreimportantly, traditional VaR2 and the higher moment VaR results were found to be very similarfor efficient portfolios. Therefore, the M–V technique is sufficient to control the higher momentrisk as long as investors hold well-diversified M–V portfolios. Simulation results indicated that the

1 The advantage of futures over cash assets are (1) futures have lower transaction cost because they are zero-investmentvehicles and do not need the large cash investment; (2) futures react to new information more quickly than do individualstocks, and hence futures do not have stale price problems; (3) the bid-ask bounce for futures are typically limited to one tick,whereas cash stock prices can be significantly affected by a relatively large bid-ask bounce at the close; and (4) futures do nothave the liquidity risk issues that can exist when buying a large block of stocks outside the United States. Hence, a portfoliomanager could diversify internationally by purchasing U.S. stocks and stock index futures from other countries.

L. You, D. Nguyen / Journal of Economics and Business 65 (2013) 33– 54 35

above-mentioned finding is not due to large number of assets in the efficient portfolios. With onlynine futures included in the portfolio, 85% of the diversifiable risk measured by higher moment VaRdisappeared. Furthermore, with less than nine assets in the M–V efficient portfolios, its potential lossmeasured by higher moment VaR decreased even more rapidly than that of VaR, as the portfolio sizeincreased. The M–V technique diversified away the higher moment risk more efficiently than didvolatility.

This paper is structured as follows: Section 2 examines issues related to futures diversification anddifferent risk measures. Section 3 presents the data, and the Section 4 describes the method. Section5 analyzes the results and Section 6 concludes the paper.

2. Futures diversification and risk measures

This section examines issues related to futures diversification studies, findings regarding individualassets and portfolios’ return distributions, and traditional VaR2 and higher moment VaRs.

Previous studies on futures diversification have typically focused on adding a commodity futuresindex, such as Goldman Sachs commodity index (GSCI), to a traditional portfolio of stocks and bonds.For example, Satyanarayan and Varangis (1996), Anson (1998), and Garrett and Taylor (2001) con-cluded that adding a commodity index futures to a portfolio of stocks and bonds can improve theportfolio’s return or reduce its risk.2 However, employing only commodity index futures generates abias, especially when the index is heavily weighted in one type of futures. For example, GSCI futuresindex is overweighed on energy. In addition, only using an index obscures the benefits of using sepa-rate categories and individual futures contracts. Therefore, in this paper, we have employed individualfutures contract to construct M–V efficient portfolios. Furthermore, earlier studies have also ignoredthe effect of skewness, kurtosis, and other higher moments, although studies have shown that returnsof futures and most of other financial assets are not normally distributed. Hence, this study has alsoexamined the higher moments risk in efficient portfolios.

It is well documented in the literature that most of financial assets exhibit non-normality, for exam-ple, excessive skewness and leptokurtosis (Aggarwal & Aggarwal, 1993; Arditti, 1967; Fama, 1965;Gorton & Rouwenhorst, 2006). Excessive skewness implies asymmetric return payoff, and leptokurto-sis (fat tail) means larger price movements, which will result in more potential loss than that predictedby normal distribution. This excessive potential loss is considered as the higher moment risk or tail risk.In the M–V framework, investors are assumed to care only about two moments–mean and variance–forportfolio returns. In general, however, investors may care about higher moments–skewness, kurtosis,and so on. For example, Scott and Horvath (1980) show that investors should have a negative prefer-ence for even moments and a positive preference for odd ones. Rubinstein (1973) demonstrates thata risk measure, when distributions are non-normal, requires not only measuring covariance with themarket but also the higher-order co-moments (co-skewness, co-kurtosis, and so on) with the market.Chung et al. (2006) test Rubinstein’s assertion and find that while each co-moment individually isunable to explain returns, a set of co-moments taken together can do so. They find that a set of sys-tematic co-moments of the third through the tenth orders substantially lowers the level of significanceof the Fama-French factors (SMB and HML). Therefore, they conclude that the SMB and HML factorsare simply proxies for higher-order systematic co-moments. Further justification and explanation ofthe fifth higher moment is given in Appendix A.

Various studies (Jondeau & Rockinger, 2006; Prakash et al., 2003) have also tried to optimizeportfolios in a three- or four-moment framework to consider higher moments and concluded withsignificant changes in resultant optimal portfolios from those in a two-moment framework. However,such techniques are hand-in-hand with serious problems, such as estimation errors and computationcomplexities. In addition, they may not result in better portfolios than the M–V efficient portfolios.

2 Studies argue that the economic rationale for including commodity futures in a portfolio is their ability to hedge againstchanges in inflation. Lummer and Siegel (1993) and Gorton and Rouwenhorst (2006) found that the GSCI total return is negativelycorrelated with stocks and bonds, and positively correlated with inflation over the long run. In addition, Jensen, Johnson, andMercer (2000) showed that the GSCI increases the optimal portfolio return for a given risk during restrictive monetary policy.


For example, Bergh and Rensburg (2008) found that the mean–variance–skewness–kurtosis (MVSK)-efficient portfolios have lower returns than do M–V portfolios.

Some studies, however, point out that deviation from normality decreases as the number of assetsin a naïve portfolio increases. For example, Simkowitz and Beedles (1978), Aggarwal and Aggarwal(1993), and Cromwell, Taylor, and Yoder (2000) concluded that increasing the number of assets in anaïve portfolio reduces the portfolios’ standard deviation and excessive kurtosis. At the same time, theyalso recognized that increasing the portfolio size induces undesirable negative skewness. However,they neither considered optimal portfolios nor higher moment risk in those portfolios.

Many risk measures have been developed during the last several decades, and the traditional VaR(VaR2), which measures the potential loss at a confidence level, has become an industry standardin measuring risk, and it is widely used by portfolio managers, treasurers, and regulators. Yet, VaR2ignores the higher moments because it only includes return and standard deviation in its calculation.3

As returns are not normal, theoretically, to calculate VaR, mean and variance alone are not sufficient,and higher moments should be considered. Consequently, VaR2 is extended to VaR4, which measuresthe potential loss of an investment, given a set of return, standard deviation, skewness, and kurtosisvalues (Favre & Galeano, 2002). Some papers have claimed the superiority of VaR4 over VaR2, suchas the study by Liang and Park (2007). In our study, in addition to VaR2 and VaR4, we also extendedVaR2 to VaR3 and VaR5 to consider the effect of different moments on portfolios risk. These differentmoment VaRs were found to separate out the risk due to mean and standard deviation from thatdue to skewness, kurtosis, and other higher moments. If risk due to mean and variance (VaR2) ofM–V portfolios is not significantly different from higher moment risk, M–V technique is consideredsufficient to diversify away higher moment risk.

3. Data

We obtain weekly returns for the 39 most actively traded futures contracts for the period1994–2003 from the Commodity Research Bureau (CRB). These futures contracts include all major con-tracts traded in the U.S. markets as determined by the volume of the trade. The sample data includestwo commodity indices (GSCI and CRB), five stock indices, four interest rates, seven currencies, and 21commodity futures. The commodity futures group includes three metals, four energies, and 14 agricul-tural contracts. A naïve portfolio is generated from individual futures contracts by equally weightingreturns from each of the futures represented in this study. The data is divided into ten annual periodsto examine if the results were consistent over time.

4. Methodology

4.1. M–V optimal portfolio

The efficient portfolios in this study are generated by Markowitz M–V technique. The M–V approachoptimizes the risk-return relationship via quadratic programming. Let us consider that a portfolio Phas n assets. Let

XP = [x1, x2, . . . . . . . . . xn] (1)

be a 1 × n vector of portfolio weights, where xi is the proportion of a particular asset relative to theentire portfolio. The return and variance of the portfolio are expressed as

RP =∑n

i=1xiRi (2)

3 It also has been criticized for its inability to capture the potential loss beyond the confidence level, and it also lacks coherence(Artzner, Delbaen, Eber, & Heath, 1997; Artzner, Delbaen, Eber, & Heath, 1999). For this reason, we supplement our VaR4 resultswith expected shortfall (ES), which is coherent and also captures the tail risk. Expected shortfall is calculated as the worst 5%returns for each portfolio.


�2P =

n∑i=1

n∑j=1

xixj�ij (3)

where RP and Ri are the return on portfolio and asset i, respectively, �2P is the variance of the portfolio,

and �ij is the covariance between asset i and j. The efficient set is generated by finding optimal risk-return portfolios over all possible return choices.

4.2. VaRs

Different techniques have been developed to manage the downside risk and potential loss. Amongthose newly developed downside tail risk measures, VaR is the most popular, largely because of itsadoption by international bank regulators. It measures the maximum potential loss over an investmenthorizon at a certain confidence level. Mathematically, the VaR at a 100(1 − ˛)% confidence level isexpressed as

VaR˛(x) = sup{x|p[X ≥ x] > ˛} (4)

For example, a 95% VaR ( ̨ = 5%) using weekly holding period indicates that, on average, the maximumloss that occurs over a week should only break the limit five in every 100 cases. If a portfolio’s 95%VaR is calculated as $2 with a $100 investment in that portfolio using weekly prices, then we are 95%confident that, on average, the maximum loss on that portfolio during a week is $2.

Traditional VaR is calculated by employing only the first two moments: mean and variance. There-fore, VaR2 does not consider higher moments, and it cannot capture the higher moment risk. Favreand Galeano (2002) use Cornish–Fisher expansion to extend VaR2 to VaR4 to explicitly incorporateskewness and kurtosis and claim that VaR4 does not depend on any distribution assumptions. In thispaper, we further advance their efforts and expand the traditional VaR into higher moment VaRs thattake into account two, three, four, and five moments. The moments are given in Appendix B. By com-paring different moment VaRs, we were able to examine the effect of each moment (the third moment,skewness; the fourth moment, kurtosis; and the fifth moment) on VaR.

VaRs of various moments are provided in the following equations:

VAR = � + ˝(˛) × � (5)

For VaR2:

˝(˛) = z(˛) (6)

For VaR3:

˝(˛) = z(˛) + 16

[z(˛)2 − 1] (7)

For VaR4:

˝(˛) = z(˛) + 16

(z(˛)2 − 1)S + 124

(z(˛)3 − 3z(˛))K + 136

(2z(˛)3 − 5z(˛))S2 (8)

For VaR5:

˝(˛) = z(˛) + 16

(z(˛)2 − 1)S + 124

(z(˛)3 − 3z(˛))K + 136

(2z(˛)3 − 5z(˛))S2 + 1120

(z(˛)4

− 6z(˛)2 + 3)F − 124

(z(˛)4 − 5z(˛)2 + 2)S × K + 1324

(12z(˛)4 − 53z(˛)2 + 17)S3 (9)

where �, �, and S are the first three moments of portfolio P, K is the excess kurtosis, F isthe fifth moment, and z(˛) is the critical value from normal distribution for probability (1 − ˛);


Table 1Basic statistics of the futures contracts.

Security number Mean (%) Geometric mean (%) SD (%) Skewness Kurtosis 5th moment

Naïve portfolio 0.09 0.09 0.86 −0.20 0.35 −0.98GSCI 0.08 0.05 2.52 −0.60 2.78 −10.49CRB index 0.07 0.06 1.15 −0.19 0.56 −0.57S&P 500 0.14 0.11 2.35 −0.35 1.61 −3.74Dow Jones 0.06 0.02 2.70 −0.27 0.93 0.59NADAQ 100 0.09 −0.04 5.05 −0.60 3.34 −8.13S&P 400 0.12 0.08 2.46 −0.54 2.59 −8.94Nikkei −0.05 −0.09 2.85 0.01 1.23 0.85T-bond 30-year 0.07 0.06 1.42 −0.45 0.50 0.12T-note 10-year 0.04 0.03 0.98 −0.41 0.92 −2.07T-note 5-year 0.04 0.04 0.62 −0.38 1.23 −2.05T-note 2-year 0.02 0.02 0.29 −0.17 2.47 −1.46U.S. dollar −0.01 −0.02 1.09 −0.09 0.55 0.21British pound 0.07 0.06 1.06 −0.17 0.85 0.48Japanese yen −0.02 −0.03 1.70 1.68 12.11 98.67Swiss franc 0.00 −0.01 1.52 0.23 1.11 0.19Australian dollar 0.05 0.04 1.39 −0.26 1.53 −3.42Canadian dollar −0.01 −0.01 0.75 0.10 0.70 −1.05Mexican peso 0.16 0.16 1.28 −0.39 5.13 6.34Copper 0.31 0.27 2.85 −0.14 1.17 −4.61Gold 0.03 0.02 1.65 0.36 3.91 8.76Silver 0.14 0.10 2.83 0.13 2.57 2.16Crude oil 0.40 0.29 4.60 −0.66 2.13 −6.53Heating oil 0.32 0.21 4.78 −0.22 2.89 −0.36Natural gas 0.26 0.02 6.93 −0.05 0.96 −1.43Unleaded gasoline NY 0.56 0.45 4.72 −0.57 1.97 −5.73Cocoa 0.30 0.23 3.89 0.21 2.86 −4.27Coffee “C” 0.23 0.09 5.40 0.29 1.93 3.30Corn 0.00 −0.05 2.96 0.47 1.37 0.87Cotton −0.04 −0.09 3.29 −0.17 3.05 −11.94Feeder cattle −0.02 −0.03 1.75 −0.03 2.15 0.80Live cattle 0.09 0.06 2.16 0.18 0.64 0.53Lean hog −0.12 −0.19 3.66 −0.42 3.20 −8.78Orange juice −0.12 −0.20 3.97 0.59 4.05 13.89Soybeans 0.24 0.20 2.77 0.17 0.98 0.85Soybean meal 0.26 0.21 3.24 0.61 1.68 2.39Soybean oil −0.08 −0.12 2.84 0.32 0.72 1.66Sugar #11 0.13 0.03 4.23 −1.49 8.76 −54.48Wheat −0.19 −0.24 3.32 0.61 1.99 3.04KC wheat 0.12 0.07 3.06 0.56 1.57 2.96

for example, z(˛) = −2.326, −1.960, and −1.645 for 1%, 2.5%, and 5%, respectively.4 In this study, allVaRs are calculated at the 95% and 99% confidence levels for a portfolio with $100 investment.

5. Empirical results

5.1. Basic statistics

Table 1 provides basic statistics for the 39 futures contracts and the equally weighted naïve port-folio over the period 1994–2003. Consistent with previous studies on the distribution of futures, thisstudy showed that stock index futures and commodity futures have high returns and standard devi-ations, whereas currency and interest rate futures have low returns and standard deviations. Stockindex futures returns were found to be negatively skewed, whereas commodities futures had mixed

4 Favre and Galeano (2002) used negative standard deviations to make the results positive. We followed a similar approachin our study.


Table 2Normality tests of the futures contracts.

Security number Shapiro–Wilk Kolmogorov–Smirnov Cramer–von Mises Anderson–Darling

Naïve portfolio 0.9951* 0.0315 0.0656 0.4285GSCI 0.9708*** 0.0450*** 0.2748*** 1.6888***

CRB index 0.9935** 0.04635*** 0.1387** 0.9254**

S&P 500 0.9830*** 0.0380** 0.2256*** 1.5320***

Dow Jones 0.9730*** 0.1454*** 0.5883*** 2.9633***

NADAQ 100 0.9688*** 0.0519*** 0.2197*** 1.3375***

S&P 400 0.9745*** 0.0481*** 0.2806*** 1.8820***

Nikkei 0.9902*** 0.0306 0.0663 0.5371T-bond 30-year 0.9875*** 0.0464*** 0.2780*** 1.6959***

T-note 10-year 0.9885*** 0.0488*** 0.1672*** 1.0165***

T-note 5-year 0.9862*** 0.0471*** 0.2105*** 1.4266***

T-note 2-year 0.9697*** 0.0477*** 0.4198*** 2.8988***

U.S. dollar 0.9953 0.0253 0.0633 0.5034British pound 0.9911*** 0.0529*** 0.2638*** 1.4903***

Japanese yen 0.9196*** 0.0593*** 0.5684*** 3.4505***

Swiss franc 0.9897*** 0.0353* 0.1180** 0.8014***

Australian dollar 0.9861*** 0.0412** 0.1675*** 1.1062***

Canadian dollar 0.9925*** 0.0531*** 0.1607** 1.0212**

Mexican peso 0.9363*** 0.0821*** 0.7088*** 4.4639***

Copper 0.9892*** 0.0354* 0.1288** 0.7971**

Gold 0.9469*** 0.0754*** 1.0802*** 6.1533***

Silver 0.9717*** 0.0586*** 0.5502*** 3.1379***

Crude oil 0.9733*** 0.0600*** 0.3247*** 1.9090***

Heating oil 0.9702*** 0.0549*** 0.3535*** 2.2396***

Natural gas 0.9916*** 0.0375* 0.1225** 0.7996**

Unleaded gasoline NY 0.9772*** 0.0385** 0.2021*** 1.5217***

Cocoa 0.9671*** 0.0719*** 0.6934*** 3.7877***

Coffee “C” 0.9796*** 0.0508*** 0.3921*** 2.2926***

Corn 0.9759*** 0.0636*** 0.5393*** 3.2523***

Cotton 0.9702*** 0.0546*** 0.4314*** 2.6713***

Feeder cattle 0.9764*** 0.0566*** 0.4708*** 2.8702***

Live cattle 0.9934** 0.0362* 0.1124* 0.7989**

Lean hog 0.9646*** 0.0621*** 0.5561*** 3.3283***

Orange juice 0.9565*** 0.0695*** 0.6824*** 4.1112***

Soybeans 0.9893*** 0.0409** 0.1520** 1.1425***

Soybean meal 0.9740*** 0.056*** 0.4164*** 2.5929***

Soybean oil 0.9909*** 0.0428*** 0.1719*** 1.0459***

Sugar #11 0.921*** 0.0730*** 0.7237*** 4.3038***

Wheat 0.9724*** 0.0438*** 0.2517*** 1.9457***

KC wheat 0.9788*** 0.0463*** 0.2886*** 1.9081***

Any test results indicating normality are in bold.* Significance at 10% level.

** Significance at 5% level.*** Significance at 1% level.

skewness values. Furthermore, all of the futures had excessive kurtosis. The naïvely constructed port-folio possessed an average return and low risk when compared with individual futures contracts,indicating potential diversification benefit by employing different individual futures. Table 2 pro-vides various normality tests, namely, Shapiro–Wilk, Kolmogorov–Smirnov, Cramer–von Mises, andAnderson–Darling, for all futures contracts in our sample. In almost all the cases, normality wasstrongly rejected. As returns did not follow normal distribution, the first two moments alone werenot sufficient to characterize the return distribution. Hence, higher order moments were considered.The normality test results, together with the existence of higher moments and heterogeneous charac-teristics of different types of futures, proved these contracts to be ideal assets to test the various riskmeasures.


Table 3Average correlations within and among different groups of futures.

Indices Stock index Interest rate Currency Metal Energy Agricultural

Indices 0.74Stock index 0.17 0.61Interest rate 0.04 0.03 0.58Currency 0.08 0.04 0.04 0.04Metal 0.23 0.02 0.00 0.07 0.37Energy 0.55 0.03 0.03 0.02 0.06 0.58Agricultural 0.21 0.05 −0.02 0.01 0.04 0.02 0.13

Average 0.28 0.14 0.11 0.04 0.11 0.18 0.06

5.2. Correlation

Due to limited space, this study does not report pair-wise correlations between individual futurescontract, and only correlations between different categories (groups) of futures are reported. Table 3provides the within- and between-group average correlations over ten annual periods from 1994 to2003. The within-group correlation is calculated by averaging pair-wise correlations among all thefutures in each group over the ten annual periods. Correlations among each group are obtained in asimilar way. All of the within-group correlations are high, except for the currency and agriculturalfutures. The SIFs within-group correlation value of 0.61 is the highest among all within-group correla-tions. This finding is consistent with previous findings that equity markets are more integrated in therecent past.5 When compared with within-group correlations, correlations between different groups(excluding the index futures because they are heavily influenced by some futures) are exceptionallylow, ranging from −0.02 to 0.06. Previous research documented negative correlations between com-modity futures and equities; however, we do not find such a correlation in our study. The commodityfutures correlate minimally with SIFs, with correlation values of 0.2, 0.3, and 0.5 for energy, metal, andagriculture, respectively. Overall, these correlation results indicate that diversifying among differentgroups of assets provides better potential diversification benefit than does diversifying within onegroup of futures.

5.3. Efficient portfolios

In this section, Markowitz M–V efficient portfolios of different assets are examined using the VaR2and higher moment VaRs.

5.3.1. Basic statistics and M–V frontierTable 4 provides the first five moments and the Sharpe ratios of M–V portfolios. These efficient

portfolios generally have better risk-return characteristics (higher Sharpe ratio) than those of individ-ual futures. This is also confirmed in the efficient frontiers. Figs. 1 and 2 illustrate the efficient frontiersof M–V portfolios along with individual assets for years 1994 and 2003. It can be observed that theefficient frontier dominates individual assets as well as the equally weighted naïve portfolio and abenchmark portfolio of stock index futures and interest rate futures (which is similar to a traditionalportfolio of bonds and stocks). This result holds true for every year from 1994 to 2003.6 Furthermore,efficient portfolios have smaller skewness and kurtosis values than do individual assets, indicatingthe advantages of an efficient portfolio over individual futures, besides standard deviation and returncharacteristics.

Table 4 also presents various VaRs that take into account mean, variance, skewness, kurtosis, andthe fifth moment at a 95% confidence level for a $100 investment in efficient portfolios for each annual

5 Bertoneche (1979) found that correlations between markets were below 0.20 in the 1970s. By the 1980s, Odier and Solnik(1993) observed many correlations to be around 0.50, which is a substantial increase. Similarly, Shawky, Kuenzel, and Mikhail(1997) determined the correlations to be above 0.50 in most of the pair-wise combinations for data from the early 1990s.

6 The results are not presented here but are available upon request.

L. You,

D.

Nguyen

/ Journal

of Econom

ics and

Business 65 (2013) 33– 54

41

Table 4The first five moments and VaRs of the Markowitz-efficient portfolios using a 95% confidence level.

Portfolio no.

1 2 3 4 5 6 8 12 17 21 25

1994Mean (%) 1.139 1.107 0.912 0.892 0.820 0.771 0.734 0.340 0.133 0.010 0.004SD (%) 3.146 2.942 1.920 1.816 1.617 1.501 1.368 0.447 0.175 0.013 0.010Skewness −0.280 −0.314 −0.500 −0.518 −0.577 −0.605 −0.592 −0.374 −0.287 −0.383 −0.456Kurtosis 0.146 0.196 −0.202 −0.273 −0.256 −0.255 −0.314 0.073 −0.026 0.227 0.4815th moment −0.019 −0.082 1.542 1.923 2.209 2.359 2.408 −0.179 −0.125 0.179 1.115VaR2 4.037 3.732 2.247 2.096 1.840 1.689 1.516 0.396 0.155 0.137 0.136VaR3 4.288 3.995 2.521 2.363 2.103 1.956 1.747 0.444 0.169 0.152 0.150VaR4 4.274 3.978 2.519 2.364 2.105 1.954 1.746 0.442 0.169 0.151 0.149VaR5 4.269 3.966 2.547 2.403 2.129 1.971 1.764 0.432 0.166 0.151 0.155Mean (%) 0.940 0.863 0.773 0.755 0.639 0.627 0.603 0.510 0.280 0.092 0.029

1995SD (%) 3.360 2.349 1.596 1.476 0.870 0.832 0.764 0.568 0.300 0.129 0.099Skewness 1.041 0.263 0.051 0.050 0.014 0.001 0.009 0.153 0.311 0.494 −0.097Kurtosis 2.919 0.911 −0.329 −0.373 −0.098 −0.053 −0.023 0.024 −0.178 0.092 −0.1195th moment 6.466 0.146 −0.438 −0.325 −0.903 −0.828 −0.709 −0.535 −0.588 0.107 0.269VaR2 4.588 3.001 1.852 1.673 0.792 0.741 0.654 0.424 0.213 0.120 0.134VaR3 3.592 2.826 1.828 1.652 0.789 0.741 0.652 0.399 0.186 0.102 0.136VaR4 3.326 2.780 1.839 1.663 0.791 0.742 0.652 0.399 0.187 0.101 0.136VaR5 3.942 2.713 1.809 1.645 0.752 0.708 0.626 0.384 0.184 0.106 0.138Mean (%) 1.496 1.404 1.175 1.154 0.809 0.806 0.624 0.362 0.068 0.044 −0.013

1996SD (%) 4.739 3.042 1.928 1.898 1.080 1.056 0.755 0.395 0.137 0.107 0.080Skewness 0.290 0.197 0.064 0.067 0.057 0.062 0.185 −0.129 −0.401 −0.611 −0.456Kurtosis −0.034 −0.348 0.250 0.261 0.259 0.240 0.125 −0.270 1.624 1.768 0.6535th moment −0.341 0.308 1.829 1.825 0.976 0.995 1.386 1.264 0.234 −1.165 −0.734VaR2 6.299 3.600 1.997 1.969 0.968 0.932 0.618 0.289 0.158 0.132 0.145VaR3 5.908 3.430 1.962 1.933 0.950 0.913 0.579 0.303 0.173 0.151 0.155VaR4 5.904 3.449 1.952 1.923 0.944 0.908 0.576 0.305 0.169 0.146 0.154VaR5 5.873 3.540 2.121 2.088 0.994 0.957 0.626 0.327 0.183 0.152 0.153Mean (%) 1.768 1.551 1.057 0.912 0.763 0.701 0.584 0.377 0.147 0.078 0.038

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Portfolio no.

1 2 3 4 5 6 8 12 17 21 25

1997SD (%) 8.139 6.410 3.039 2.270 1.650 1.472 1.098 0.595 0.202 0.139 0.115Skewness −0.444 −0.423 −0.615 −0.612 −0.605 −0.551 −0.486 −0.220 −0.352 0.002 0.222Kurtosis 0.079 −0.074 0.342 0.343 0.025 −0.029 0.006 0.080 0.778 0.236 −0.1535th moment 0.866 1.203 0.694 1.066 1.902 1.889 1.748 1.605 −1.701 −2.115 −1.151VaR2 11.625 8.995 3.942 2.822 1.952 1.721 1.222 0.602 0.187 0.151 0.152VaR3 12.651 9.766 4.474 3.217 2.236 1.951 1.373 0.639 0.207 0.151 0.145VaR4 12.608 9.754 4.431 3.185 2.223 1.944 1.368 0.637 0.203 0.150 0.145VaR5 12.757 9.929 4.399 3.206 2.254 1.991 1.419 0.684 0.193 0.136 0.140Mean (%) −0.134 1.144 0.940 0.498 0.435 0.397 0.348 0.078 0.045 0.046 0.035

1998SD (%) 3.927 3.878 2.862 1.206 1.015 0.912 0.865 0.282 0.176 0.137 0.118Skewness 0.179 −0.570 −0.306 0.429 0.484 0.448 0.437 0.435 0.002 0.001 0.001Kurtosis 1.321 −0.124 −0.174 0.502 0.650 0.717 0.734 1.086 0.194 0.072 0.1195th moment 1.711 2.226 1.431 1.270 1.026 0.496 0.615 2.801 0.876 0.477 0.490VaR2 6.594 5.237 3.768 1.487 1.234 1.103 1.075 0.386 2.290 −0.016 −1.515VaR3 6.393 5.866 4.017 1.340 1.094 0.987 0.967 0.351 0.244 0.180 0.160VaR4 6.286 5.852 4.022 1.323 1.077 0.970 0.952 0.344 0.234 0.177 0.156VaR5 6.462 5.975 4.168 1.387 1.113 0.970 0.955 0.368 0.231 0.176 0.154Mean (%) 1.600 1.467 1.357 1.339 1.270 1.197 1.015 0.697 0.290 0.110 0.044

1999SD (%) 4.604 3.160 2.642 2.580 2.359 2.181 1.680 1.023 0.403 0.162 0.089Skewness −0.983 −0.391 −0.145 −0.103 0.049 0.182 0.443 −0.038 0.110 0.122 −0.133Kurtosis 1.445 0.384 0.034 0.029 −0.179 −0.275 −0.433 1.016 0.105 −0.940 −0.9365th moment −2.127 −0.658 −1.814 −2.015 −2.117 −2.069 −2.225 −3.212 −2.209 −0.543 0.847VaR2 5.973 3.730 2.989 2.905 2.612 2.391 1.748 0.986 0.374 0.157 0.103VaR3 7.260 4.081 3.098 2.981 2.579 2.278 1.536 0.998 0.361 0.151 0.107VaR4 7.042 4.048 3.095 2.979 2.587 2.289 1.545 0.977 0.360 0.154 0.108VaR5 6.172 3.962 2.858 2.723 2.345 2.090 1.468 0.821 0.316 0.153 0.110

2000Mean (%) 2.758 2.628 2.542 2.299 1.637 1.530 1.070 0.487 0.212 0.144 0.102SD (%) 5.933 5.361 5.093 4.297 2.620 2.383 1.513 0.541 0.192 0.125 0.105Skewness 0.015 −0.138 −0.160 −0.221 −0.266 −0.232 0.013 0.419 0.166 0.056 0.111Kurtosis 0.610 0.043 −0.168 −0.305 0.025 0.012 −0.153 0.601 0.469 0.230 0.1685th moment 1.735 0.717 0.454 0.217 −0.540 −0.643 −0.657 −0.838 −1.656 −1.586 −1.153VaR2 7.002 6.192 5.836 4.770 2.672 2.391 1.420 0.403 0.104 0.061 0.070VaR3 6.977 6.402 6.068 5.040 2.870 2.548 1.414 0.338 0.095 0.059 0.067VaR4 6.904 6.395 6.082 5.063 2.865 2.545 1.419 0.330 0.093 0.058 0.067VaR5 7.402 6.585 6.165 5.042 2.781 2.460 1.370 0.298 0.075 0.048 0.060

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Table 4 (Continued)

Portfolio no.

1 2 3 4 5 6 8 12 17 21 25

2001Mean (%) 1.326 1.275 0.696 0.669 0.477 0.456 0.375 0.162 0.101 0.070 0.033SD (%) 5.227 4.986 2.410 2.206 1.497 1.418 0.962 0.313 0.202 0.176 0.162Skewness 0.425 0.417 0.388 0.366 0.175 0.150 0.050 0.037 0.355 0.488 0.593Kurtosis −0.231 −0.215 0.060 0.057 −0.185 −0.218 −0.332 −0.401 0.089 0.791 1.4875th moment −1.189 −1.123 −0.914 −0.936 −0.980 −0.993 −0.858 −1.107 −0.593 0.497 2.055VaR2 7.272 6.928 3.269 2.960 1.987 1.877 1.208 0.353 0.231 0.220 0.234VaR3 6.640 6.336 3.003 2.731 1.912 1.816 1.194 0.350 0.211 0.196 0.206VaR4 6.647 6.341 2.993 2.723 1.917 1.822 1.201 0.352 0.210 0.192 0.200VaR5 6.572 6.271 2.925 2.651 1.856 1.762 1.163 0.336 0.206 0.192 0.204

2002Mean (%) 1.251 1.174 1.080 1.021 0.961 0.914 0.807 0.590 0.420 0.206 0.150SD (%) 4.935 3.775 2.984 2.446 2.086 1.888 1.560 0.926 0.576 0.311 0.189Skewness −1.699 −1.108 −0.877 −0.504 −0.183 0.004 −0.120 −0.503 −0.741 −0.638 −0.494Kurtosis 6.740 2.563 1.477 0.338 −0.378 −0.527 −0.377 −0.256 0.034 0.074 0.2315th moment −25.655 −7.189 −3.789 −1.051 0.107 −0.284 0.210 1.928 2.812 1.640 0.302VaR2 6.868 5.036 3.829 3.002 2.471 2.191 1.759 0.933 0.527 0.305 0.161VaR3 9.252 6.225 4.573 3.353 2.580 2.189 1.812 1.065 0.648 0.361 0.188VaR4 8.313 5.943 4.441 3.325 2.594 2.209 1.823 1.066 0.642 0.358 0.186VaR5 3.483 4.681 3.854 3.161 2.576 2.183 1.826 1.092 0.642 0.358 0.185

2003Mean (%) 1.000 0.984 0.932 0.912 0.894 0.892 0.792 0.709 0.558 0.298 0.098SD (%) 3.650 3.467 2.254 1.962 1.799 1.779 1.138 0.931 0.709 0.371 0.140Skewness 0.293 0.333 0.520 0.705 0.735 0.751 0.716 0.273 0.497 0.395 0.184Kurtosis −0.415 −0.398 0.248 0.935 1.269 1.338 1.543 0.343 0.178 −0.179 −0.3905th moment −1.197 −1.315 −0.004 1.001 2.159 2.250 4.552 1.249 −0.586 −1.001 −0.592VaR2 5.004 4.719 2.776 2.316 2.065 2.034 1.080 0.823 0.609 0.312 0.132VaR3 4.700 4.391 2.443 1.923 1.689 1.654 0.848 0.750 0.509 0.271 0.125VaR4 4.725 4.412 2.420 1.867 1.625 1.588 0.802 0.742 0.503 0.271 0.126VaR5 4.619 4.313 2.480 1.979 1.773 1.736 0.984 0.791 0.502 0.265 0.124

The portfolios are presented in descending order of risk from left to right. Portfolios with similar amounts of risk are not presented because of space limitation.


1994

-1.0

-0.5

0.0

0.5

1.0

0.0 1.0 2.0 3.0 4.0 5.0

Standard Deviation

Return%Optimal

Portfolio

Tangent

Benchmark

Naïve

Portfolio

Stock Index

Interest

Rate

Currency

Metal

Energy

Agricultural

Fig. 1. M–V Markowitz portfolio for 1994.

2003

-1.5

-1.0

-0.5

0.0

0.5

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Standard Deviation

Return% Optimal

Portfolio

Benchmark

Tangent

Naïve

Portfolio

Stock Index

Interest Rate

Currency

Metal

Energy

Agricultural

Fig. 2. M–V Markowitz portfolios for 2003.

period. The VaR2 calculates the potential loss due to return and standard deviation, and the highermoment VaRs provide potential loss due to return, standard deviation, skewness, kurtosis, and thefifth moment. Our argument is that if skewness, kurtosis, and higher moments are important, then weshould observe significant difference between higher moment VaRs and VaR2. The difference betweenVaR2 and higher moment VaRs is the risk incurred by higher moments.

In Table 4, M–V portfolios are ranked in the order of descending risk (standard deviation), andportfolios of similar risk are removed due to space consideration. An eyeball test shows that VaR2values are not much different from higher moment VaRs. This is confirmed by the t-test and non-parametric Wilcoxon test, shown in Table 5. As shown in Table 5, the difference between VaR2 andhigher moment VaRs (VaR3, VaR4, and VaR5) is obtained using the 95% confidence level. It can beobserved that all t-statistics and Wilcoxon statistics for all the annual periods are not significant at10% level or better, which shows that higher moment VaRs are not statistically different from VaR2.The last column provides the sum of the difference between VaR2 and higher moment VaRs over theten years in our sample period. It shows that the accumulated difference between VaR2 and VaR3over the ten-year period is 0.25, and this number is reduced to 0.186 for the accumulated differencebetween VaR2 and VaR4, and it further decreases to 0.056 for the accumulated difference betweenVaR2 and VaR5. This shows that if we only consider skewness, then the potential loss due to skewnessis larger than the potential loss due to skewness and kurtosis. The more moments are included in thecalculation of VaR loss, the less difference remains between VaR2 and higher moment VaR. Thus, thedifference between M–V efficient portfolios and mean–variance–skewness (MVS)-efficient portfolio is

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Table 5Difference between VaR2 and higher moment VaRs for M–V efficient portfolios using a 95% confidence level.

Year Sum

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

VaR3 − VaR2Average 0.096 0.057 −0.020 0.122 −0.011 0.037 0.018 −0.082 0.154 −0.121 0.250t-Test 0.31 0.22 −0.05 0.19 −0.03 0.10 0.02 −0.17 0.43 −0.42Wilcoxon test 729 761 649 1127 1011 975 942 853 1726 1167

VaR4 − VaR2Average 0.0947 0.068 −0.022 0.116 −0.020 0.028 0.016 −0.083 0.122 −0.133 0.186t-Test 0.31 0.27 −0.06 0.18 −0.05 0.07 0.03 −0.17 0.35 −0.46Wilcoxon test 731 761 658 1124 1016 977 944 852 1727 1167

VaR5 − VaR2Average 0.100 0.057 0.010 0.137 0.001 −0.071 0.020 −0.103 −0.039 −0.106 0.056t-Test 0.32 0.22 0.02 0.21 0.00 −0.20 0.04 −0.22 −0.14 −0.37Wilcoxon test 720 764 640 1150 1025 989 977 857 1730 1160

None of the difference is significant at 10% level or below in both t-test and non-parametric Wilcoxon test.


-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 1 2 3 4 5 6 7

VAR4 Var2ES Individual futuresNaive

Fig. 3. The return-loss graph for M–V optimal portfolios in 1994.

-0.01

0

0.01

0.02

0 5 10 15 20

ES VaR4 VaR2

Individual futuresNaive


-0.02

-0.01

0

0.01

0.02

0.03

0 5 10 15

ES VaR2 Individual Naïve VaR4


more significant than that between M–V and mean–variance–skewness–kurtosis (MVSK) portfolios.The implication of these results is that the significant difference between the optimal portfolios ina three-moment framework and those in a two-moment framework reported in previous studies(Chunhachinda, Dandapani, Hamid, & Prakash, 1997; Lai, 1991; Prakash et al., 2003) may not indicatethat the three-moment framework is superior to a two-moment framework and that diversificationin a three-moment world may not be a complete analysis. In general, Table 5 indicates that the riskdue to higher moments, including skewness, kurtosis, and the fifth moment, is not significant in M–VV-efficient portfolios. As a robustness check, we also computed the VaR2 and higher moment VaRusing the 99% confidence level and examined the difference between them. The results are reportedin Tables 6 and 7. As can be seen from these tables, results with a 99% confidence level are very similarto those reported earlier with a 95% confidence level. The t-statistics and Wilcoxon statistics indicatethat there is no statistically significant difference between VaR2 and higher moment VaRs even witha 99% confidence level.

The VaR2 and higher moment VaR of efficient portfolios are plotted against the portfolio returns.As VaR3, VaR4, and VaR5 are very similar and as most of the literature has focused on skewnessand kurtosis, we used VaR4 to represent higher moment VaR. Because of limited space, only figuresfor every other three years are presented. Figs. 3–6 illustrate efficient portfolios using return against


Table 6VaRs of the Markowitz-efficient portfolios using a 99% confidence level.

Portfolio no.

1 2 3 4 5 6 8 12 17 21 25

1994VaR2 6.179 5.737 3.555 3.333 2.941 2.721 2.448 0.701 0.274 0.227 0.210VaR3 6.828 6.415 4.262 4.025 3.626 3.388 3.044 0.824 0.311 0.264 0.246VaR4 6.843 6.441 3.990 3.726 3.328 3.092 2.763 0.809 0.304 0.264 0.250VaR5 6.765 6.337 3.838 3.576 3.129 2.875 2.587 0.788 0.301 0.256 0.238

1995VaR2 6.876 4.601 2.938 2.679 1.385 1.308 1.175 0.811 0.418 0.209 0.201VaR3 4.303 4.148 2.879 2.625 1.376 1.307 1.169 0.747 0.349 0.162 0.208VaR4 5.222 4.587 2.754 2.494 1.356 1.297 1.165 0.745 0.326 0.152 0.205VaR5 10.072 4.720 2.749 2.489 1.355 1.296 1.165 0.747 0.330 0.166 0.205

1996VaR2 9.526 5.672 3.310 3.262 1.703 1.651 1.133 0.443 0.251 0.205 0.200VaR3 8.515 5.232 3.220 3.169 1.658 1.603 1.030 0.484 0.292 0.253 0.227VaR4 8.328 4.940 3.330 3.281 1.722 1.660 1.042 0.480 0.336 0.282 0.233VaR5 8.409 4.924 3.341 3.293 1.726 1.665 1.051 0.479 0.313 0.242 0.222

1997VaR2 17.168 13.361 6.012 4.368 3.076 2.723 1.969 1.007 0.324 0.246 0.231VaR3 19.821 15.353 7.387 5.389 3.810 3.319 2.362 1.103 0.377 0.246 0.212VaR4 19.370 14.811 7.197 5.251 3.592 3.142 2.266 1.104 0.404 0.254 0.206VaR5 18.767 14.475 6.526 4.758 3.303 2.955 2.168 1.098 0.387 0.253 0.206

1998VaR2 9.268 7.878 5.717 2.309 1.925 1.724 1.664 0.578 0.364 0.274 0.240VaR3 8.750 9.504 6.361 1.928 1.564 1.424 1.386 0.488 0.339 0.267 0.230VaR4 9.914 8.917 6.144 1.987 1.629 1.507 1.472 0.539 0.372 0.282 0.243VaR5 10.106 8.407 6.112 2.110 1.778 1.625 1.579 0.583 0.379 0.283 0.244

1999VaR2 9.109 5.882 4.788 4.663 4.219 3.877 2.892 1.684 0.649 0.267 0.164VaR3 12.436 6.790 5.070 4.858 4.134 3.584 2.345 1.712 0.616 0.253 0.173VaR4 12.317 6.892 5.070 4.865 4.033 3.417 2.051 1.955 0.624 0.216 0.153VaR5 7.679 6.655 5.054 4.853 4.021 3.401 2.105 1.943 0.624 0.213 0.155

2000VaR2 11.043 9.843 9.305 7.697 4.456 4.014 2.451 0.771 0.235 0.146 0.142VaR3 10.978 10.386 9.903 8.394 4.968 4.421 2.436 0.605 0.211 0.141 0.133VaR4 11.823 10.401 9.654 8.010 4.914 4.379 2.382 0.645 0.231 0.147 0.137VaR5 11.849 10.391 9.665 8.026 4.869 4.352 2.380 0.700 0.233 0.147 0.137

2001VaR2 10.832 10.323 4.911 4.463 3.007 2.843 1.863 0.566 0.369 0.340 0.344VaR3 9.199 8.794 4.223 3.869 2.814 2.686 1.828 0.557 0.316 0.277 0.274VaR4 8.561 8.217 4.120 3.787 2.732 2.601 1.752 0.528 0.311 0.294 0.309VaR5 8.780 8.417 4.238 3.878 2.727 2.595 1.748 0.527 0.319 0.322 0.361

2002VaR2 10.229 7.607 5.861 4.668 3.892 3.477 2.821 1.564 0.628 0.516 0.290VaR3 16.391 10.681 7.785 5.575 4.173 3.470 2.958 1.906 0.828 0.662 0.359VaR4 18.806 11.198 7.951 5.534 3.962 3.238 2.812 1.762 0.758 0.620 0.352VaR5 −10.521 5.195 5.654 5.208 3.978 3.236 2.823 1.693 0.669 0.554 0.330

2003VaR2 7.490 7.081 4.311 3.652 3.290 3.246 1.855 1.457 1.092 0.565 0.227VaR3 7.186 6.753 3.978 3.259 2.914 2.866 1.623 1.384 0.992 0.523 0.220VaR4 6.233 5.766 3.351 2.698 2.485 2.442 1.447 1.318 0.797 0.420 0.194VaR5 6.221 5.779 3.654 3.473 3.354 3.362 2.007 1.350 0.876 0.433 0.193

The portfolios are presented in descending order of risk from left to right. Portfolios with similar amounts of risk are notpresented because of space limitation.

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Table 7Difference between VaR2 and higher moment VaRs for M–V-efficient portfolios using a 99% confidence level.

Year Sum

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

VaR3 − VaR2Average 0.249 −0.148 −0.050 0.315 −0.028 0.095 0.049 −0.212 0.399 −0.122 0.547t-Test 0.51 0.40 −0.09 0.32 −0.05 0.15 0.06 −0.30 0.67 −0.28Wilcoxon test 715 775 639 1137 1024 969 943 877 1711 1151

VaR4 − VaR2Average 0.172 −0.115 −0.043 0.254 0.016 0.091 0.061 −0.273 0.419 −0.323 0.259t-Test 0.36 0.30 −0.08 0.26 0.03 0.15 0.07 −0.40 0.66 −0.78Wilcoxon test 726 783 628 1141 977 1001 943 878 1733 1198

VaR5 − VaR2Average 0.114 0.071 −0.050 0.160 0.037 −0.066 0.068 −0.229 −0.520 −0.164 −0.579t-Test 0.24 0.15 −0.09 0.17 0.06 −0.12 0.08 −0.34 −1.04 −0.38Wilcoxon test 738 777 645 1143 963 999 931 816 1792 1171

None of the difference is significant at 10% level or below in both t-test and non-parametric Wilcoxon test.


-0.02

-0.01

0

0.01

0.02

0.03

0 5 10 15

ES VaR2 Individual Naïve VaR4


VaR2 and VaR4 for the years 1994, 1997, 2000, and 2003 to form different return-loss sets.7 Individualasset’s returns are plotted against their VaR4. It can be noted that the return-VaR4 frontier of efficientportfolios dominate those of individual assets for every year, indicating that when the potential loss iscalculated using VaR4, which explicitly considers the skewness and kurtosis values, efficient portfoliosstill perform better than do individual assets.

On comparing VaR2 with VaR4, one can find that these two frontiers lie very close to each other,indicating that if we consider the risk associated with higher moments, the potential loss does notvary much from VaR2. In fact, in 1994 and 1997, it can be noted that the VaR4 lies northwest to theVaR2, implying that for every level of return, the loss measured by VaR4 is actually slightly smallerthan for VaR2. Furthermore, after taking into account the skewness and kurtosis, the potential loss isnot increased but is reduced in these two cases. In short, the VaR4 results are very similar to those ofVaR2 when evaluating the M–V efficient portfolios, because the risk due to skewness and kurtosis inefficient portfolios is not significant. These results hold true for all the annual periods in our sampleperiod.

The significant reduction in higher moment risk and the similarity between VaR2 and VaR4 impliesthat the root cause may be the normality of M–V portfolios. So, we test the normality of these M–Vportfolios. Table 8 presents p-values for Shapiro–Wilk normality tests for the top 20 portfolios eachyear.8 At the 95% confidence level, 23 of 200 M–V portfolios reported in Table 8 are not normallydistributed, which means 88.5% of M–V portfolios in our sample are normal.

Therefore, the following can be concluded: Diversifying in an M–V framework also effectivelyreduces the higher moment/tail risk. This is because M–V portfolios are mostly normal. More compli-cated techniques that take into account skewness, kurtosis, and higher moment values, which althoughresults in portfolios different from M–V efficient ones, may not add much value as long as investorshold well-diversified efficient portfolios. This result is consistent over our sample period.

This study confirmed the claim by Jondeau and Rockinger (2006) that the M–V framework providesa good approximation of the expected utility maximization under moderate non-normality. If investorshold well-diversified M–V efficient portfolios, then VaR2 should be sufficient to estimate the potentialloss, and the effects of skewness, kurtosis, and the fifth moment on the potential loss of efficientportfolios would be marginal.

5.4. VaR, higher moment VaR, and number of assets in portfolio

One may argue that the above-mentioned results are due to large numbers of assets includedin M–V portfolios. Aggarwal and Aggarwal (1993) pointed out that with 25 securities in the naïveportfolio, the percentage of positive-skewed portfolio return distributions is reduced to zero; however,they also found that the percentage of negative-skewed portfolio return distributions had increasedsignificantly.

7 Expected shortfall (ES), which remedies some of issues with VaR2 and VaR4, is also plotted along with VaRs. The ES resultsare very similar to VaR2 and VaR4, which strengthens our conclusion that the M–V technique also reduces tail risk.

8 Portfolios after the top 20 tend to have extremely similar returns and risk characteristics.


Table 8Normality tests for mean–variance-efficient portfolios.

Portfolio no. 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

1 0.89 0.01 0.90 0.36 0.51 0.01 0.48 0.52 0.00 0.242 0.88 0.33 0.78 0.39 0.09 0.59 0.93 0.53 0.01 0.243 0.07 0.98 0.69 0.10 0.66 0.46 0.95 0.66 0.03 0.154 0.05 0.98 0.71 0.05 0.69 0.37 0.68 0.70 0.22 0.075 0.04 0.29 0.98 0.11 0.60 0.27 0.69 0.82 0.53 0.036 0.04 0.35 0.97 0.14 0.73 0.30 0.74 0.81 0.46 0.037 0.05 0.72 0.97 0.13 0.73 0.25 0.84 0.87 0.52 0.048 0.05 0.72 0.65 0.15 0.72 0.11 0.83 0.86 0.54 0.039 0.06 0.77 0.61 0.27 0.47 0.39 0.82 0.87 0.51 0.13

10 0.40 0.81 0.72 0.29 0.48 0.38 0.80 0.77 0.28 0.5511 0.48 0.85 0.74 0.29 0.47 0.34 0.51 0.76 0.27 0.6812 0.34 0.88 0.85 0.12 0.26 0.33 0.49 0.63 0.12 0.8313 0.35 0.94 0.86 0.15 0.28 0.33 0.49 0.62 0.02 0.8914 0.63 0.97 0.86 0.14 0.29 0.41 0.62 0.86 0.02 0.9015 0.63 0.99 0.13 0.12 0.39 0.41 0.65 0.86 0.01 0.6416 0.66 0.92 0.12 0.11 0.63 0.41 0.61 0.80 0.01 0.5117 0.64 0.90 0.11 0.35 0.64 0.40 0.42 0.56 0.02 0.4518 0.58 0.78 0.08 0.33 0.89 0.41 0.37 0.49 0.02 0.4519 0.56 0.24 0.08 0.33 0.96 0.51 0.33 0.43 0.02 0.4220 0.64 0.21 0.07 0.28 0.96 0.54 0.35 0.42 0.03 0.44

This table provides p-values for Shapiro–Wilk normality tests for mean–variance optimal portfolios. Other normality testsprovide similar results. Italic and bold values indicate non-normality at 5% significance level.

Table 9VaR2 and VaR4 over different asset portfolios.

No. of assets in the portfolio VaR2 VaR4 Difference (%)

1 5.70 6.15 7.882 4.33 4.59 5.853 3.72 3.89 4.584 3.34 3.48 4.355 3.08 3.20 4.006 2.92 3.03 3.777 2.75 2.85 3.628 2.62 2.71 3.339 2.54 2.62 3.29

10 2.47 2.56 3.45

Therefore, in this section, random combinations of different assets are carried out to determinethe relationship between the number of assets in a M–V optimal portfolio and diversifiable loss. Thishelps to determine the effect of portfolio size on diversifiable two-moment and higher moment riskand discover the optimal number of assets in an M–V portfolio to diversify away higher moment risk.In our study, for every year, k (k is not greater than 109) futures contracts were randomly chosen fromthe 39 futures pool to form 1000 or fewer k-asset portfolios, and their VaR2 and VaR4 were calculatedand then averaged across the portfolios. These VaR2 and VaR4 were then averaged over ten annualperiods to obtain the final results. Fig. 7A and B illustrates VaR2 and VaR4 against k, the number ofassets in the portfolio, respectively. Table 9 reports VaR2 and VaR4 for each k. Furthermore, Fig. 7Aand B shows that when the number of assets in the portfolio increases, both VaR2 and VaR4 decreaseat a decreasing rate as expected. However, it must be noted that VaR4 decreases slightly more rapidlythan does VaR2. With one asset in the portfolio, on average, VaR4 is 7.88% higher than VaR2. However,with nine assets, VaR4 is 3.29% higher than is VaR2, and then it levels off. Hence, it can be concludedthat when a portfolio has less than nine assets, the M–V technique diversifies the higher momentrisk more efficiently than does volatility. Moreover, as the gap between VaR2 and VaR4 narrows, the

9 Simulations with more than ten assets take a very long time to run and cannot be handled by a PC.


2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

1 2 3 4 5 6 7 8 9 10

VaR

2

No. of Assets

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

1 2 3 4 5 6 7 8 9 10

VaR

4

No. of Assets

A

B

Fig. 7. (A) The relationship between the number of assets in a portfolio and VaR2. (B) The relationship between the number ofassets in a portfolio and VaR4.

higher moment risk diminishes. With nine assets in a portfolio, about 80–90% of the diversifiable lossmeasured by VaR4 is diversified away. The risk due to higher moments, such as skewness and kurtosis,is effectively reduced in these portfolios. Thus, this study concludes that if an investor is diversifyingin an M–V framework, with nine assets, he/she can achieve effective risk reduction due to highermoments.

6. Conclusion

By recognizing that assets’ returns are not normally distributed, many studies have tried to optimizeportfolios in a three- or four-moment framework and have reported different portfolios than M–V ones.Some studies have shown that naïve diversification diversifies away part of the higher moment riskas naïve portfolios deviate less from normality, when compared with individual assets. In this paper,we aimed to analyze the risk due to higher moments (skewness, kurtosis, and the fifth moment) inthe M–V efficient portfolios and tried to determine whether the M–V technique is sufficient to controlthe higher moment risk or whether portfolio managers should adopt more complicated techniques tocontrol higher moment risk. In addition, this paper also examined whether combining different futuresprovides diversification benefits on a return-standard deviation basis and when higher moments areconsidered.

By employing a wide range of futures assets, this study has reported that futures assets exhibitvarious degrees of skewness and kurtosis, and normality tests showed that individual futures are notnormally distributed. Furthermore, correlations between different types of futures were low, whichsupport diversification among them. The M–V efficient portfolios on a risk-return basis dominateindividual assets, an equally weighted naïve portfolio, and a traditional portfolio for every year over thesample period. Even when we considered the higher moments, the domination of the M–V portfoliosover individual assets was still evident.

Furthermore, efficient portfolios also had smaller skewness and kurtosis values when comparedwith individual futures. In addition, the higher moment VaRs were not significantly different fromVaR2 for these M–V efficient portfolios. When the efficient portfolio returns were plotted againstVaR2 and VaR4, the two return-loss sets remained very close to each other and were not statistically


different for all the annual periods. Normality tests show that M–V portfolios are mostly normal. Thisreduction in tail risk is not due to large number of assets included in this study. We observed thatwhen as many as nine assets were included in the portfolio, the majority of the diversifiable loss couldbe diversified away and the higher moment risk was effectively reduced for efficient portfolios. Withless than nine assets in the optimal portfolio, the M–V technique diversified away the higher momentrisk more effectively than did volatility.

However, we would also like to point out that our conclusion is drawn from ex-post results, someof the mis-specified inputs to M–V framework might cause the resulting portfolio to exhibit greatertail risk than a framework that optimizes for higher moments. Only if the portfolio manager canestimate returns and risk accurately, then perhaps are they better off employing the M–V frameworkthan using other complicated techniques, such as optimizing in a three- or four-moment framework,especially considering issues and costs come with those techniques and that the resultant three- orfour-moment-efficient portfolios have lower returns than does the M–V-efficient portfolio.

Appendix A.

One might argue that most return distributions can be reasonably characterized by the first fourmoments; therefore, resorting to further higher moments does not seem appealing. However, weargue that there is no reason to stop with the fourth moment. Rubinstein (1973) and Chung et al.(2006) point out that a set of higher moments is a measure of the likelihood of extreme outcomes,a matter of great importance to risk-averse investors. Variance, skewness, and kurtosis might givesome information about the tail of the outcome distribution. However, they fall short of specifyingthe tail precisely. Only a set of higher moments can fully specify the tail of the distribution. A precisespecification of the right tail via higher moments is required to measure contingent returns in events,such as lotteries, out-of-the-money options, and extreme geopolitical shocks that could possibly dryup the market liquidity. Risk-averse investors, therefore, must concern themselves with the accuratedescription of the tail.

To further justify the importance of higher moments beyond the fourth moment, consider thefollowing lottery example: an investor attempts to optimize a portfolio made of two independentassets: A and B. The two assets have the following payoffs:

A payoff: B payoff:$1 loss (999 times in 1000) $1 gain (999 times in 1000)$999 gain (1 time in 1000) $999 loss (1 time in 1000)

From the above-mentioned distribution, any rational risk-averse investor would choose A because ofsmall downside risk and large upside gain. Therefore, if asset returns are independent, the optimalportfolio should consist of 100% A. However, it might not be the case when we consider this lottery inthe mean–variance–skewness–kurtosis framework.

We obtained the following results after computing the A and B central moments:

• A and B have the same mean (zero) and variance (999);• skewness is 31.57 for A and −31.57 for B; and• kurtosis is 998 for both A and B.

Based on the mean–variance framework, the optimal portfolio should be 50% A and 50% B becausethis is the minimum variance portfolio (both assets have the same mean and variance). Based on thefour-moment framework, the optimal portfolio is still 50% A and 50% B, because it produces minimumvariance and kurtosis. The conclusion, based only on the mean–variance–skewness–kurtosis, mightnot reflect the small downside risk and the probability to earn large upside return (extreme event). Asa result, we should consider the total return distribution to solve the problem, and hence the higherorder moments should be considered.


Appendix B.

Higher moment VaRs are calculated as follows:

VAR = � + ˝(˛) × � (B1)

Using Cornish–Fisher expansion to expand ˝(˛), using the first five cumulants,

˝(˛) = z(˛) + 16

(z(˛)2 − 1)�3 + 124

(z(˛)3 − 3z(˛))�4 + 136

(2z(˛)3 − 5z(˛))�23 + 1

120(z(˛)4

− 6z(˛)2 + 3)�5 − 124

(z(˛)4 − 5z(˛)2 + 2)�3 × �4 + 1324

(12z(˛)4 − 53z(˛)2 + 17)�33

(B2)

where z(˛) is the critical value from normal distribution for probability (1 − ˛); for example,z(˛) = −2.326, −1.960, and −1.645 for 1%, 2.5%, and 5%, respectively.

�3, �4, and �5 are the third, fourth, and fifth cumulant, respectively. They are calculated as follows:

�3 = E[(X − �)3]�3

= skewness = S (B3)

�4 = E[(X − �)4]�4

− 3 = excess kurtosis = K (B4)

�5 = E[(X − �)5]�5

− 10E[(X − �)3]

�3× E[(X − �)2]

�2= fifth moment = F (B5)

where � = E(X) is the mean, �2 = E(X − �)2 is the variance, and � is the standard deviation. S is theskewness, K is the excess kurtosis, and F is the fifth moment.

The above-mentioned definitions are for a population. In our study, we used a sample, and thereforesome adjustments were necessary. In particular,

skewness = S = n

(n − 1)(n − 2)

∑(Xi − X̄

�

)3

(B6)

excess kurtosis = K = n(n + 1)(n − 1)(n − 2)(n − 3)

∑(Xi − X̄

�

)4

− 3(n − 1)2

(n − 2)(n − 3)(B7)

Fifth moment = F = n(n + 1)(n + 2)(n − 1)(n − 2)(n − 3)(n − 4)

∑(Xi − X̄

�

)5

− 10n

(n − 1)(n − 2)

∑(Xi − X̄

�

)3

× (n − 1)2

(n − 2)(n − 3)(B8)

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Higher order moment risk in efficient futures portfolios