4. Pricing Assets With Higher Moments Evidence From The

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Int. Fin. Markets, Inst. and Money 20 (2010) 5167

Contents lists available at ScienceDirect

Journal of International Financial

Markets, Institutions & Moneyj ou r na l ho m e pa ge : w w w . e l s e v i e r . c o m / l o c a t e / i n t f i n

Pricing assets with higher moments: Evidence from the

Australian and us stock markets

Phuong Doan a, Chien-Ting Lin b,, Ralf Zurbruegg b

a Business School, University of RMIT, Melbourne, VIC 3000, Australiab Business School, University of Adelaide, Adelaide, SA 5005, Australia

a r t i c l e i n f o

Article history:

Received 10 January 2009

Accepted 12 October 2009

Available online 22 October 2009

JEL classification:

G11

G12

Keywords:

Asset pricing

Co-skewness

Co-kurtosis

Fama and French 3 factors

Australian stock market

a b s t r a c t

This paper investigatesthe importance of highermoments of return

distributions in capturing the variation of average stock returns for

companies listed in the leading S&P US and Australian indices. We

find that Australian stocks are more negatively skewed but less

leptokurtic than US stocks. As a result, we find that co-skewness

plays a more important role in explaining Australian returns whileco-kurtosis is consistently influential for US stock returns. We pos-

tulate that the differences in results are related to the underlying

firm characteristics of the companies in the two indices, where

principally the Australian firms are noticeably smaller than their

US counterparts and concentrated in a smaller number industry

sectors. This implies that for many smaller exchanges around the

world higher moment characteristics displayed by the US market

maynot be applicable. We also show our results are robust to partly

explaining average stock returns in the presence of size, value, and

momentum effects.

2009 Elsevier B.V. All rights reserved.

1. Introduction

It has long been well documented that stock returns do not follow a normal distribution. For exam-

ple, Mandelbrot (1963) and Mandelbrot and Taylor (1967) show that stock returns exhibit excess

kurtosis, also commonly referred to as fat tail distributions. Fama (1965) finds that large stock returns

tend to be followed by stock returns of similar magnitude but in the opposite direction. This can lead

to the volatility clustering effect that is related to how information arrives and is received by the mar-

ket (see Campell and Hentschel (1992)). This clustering in return volatility has raised a fundamental

Corresponding author. Tel.: +61 8 8303 6461; fax: +61 8 8303 7243.

E-mail address: [email protected] (C.-T. Lin).

1042-4431/$ see front matter 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.intfin.2009.10.002
http://www.sciencedirect.com/science/journal/10424431http://www.elsevier.com/locate/intfinmailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_12/dx.doi.org/10.1016/j.intfin.2009.10.002http://localhost/var/www/apps/conversion/tmp/scratch_12/dx.doi.org/10.1016/j.intfin.2009.10.002mailto:[email protected]://www.elsevier.com/locate/intfinhttp://www.sciencedirect.com/science/journal/10424431


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52 P. Doan et al. / Int. Fin. Markets, Inst. and Money 20 (2010) 5167

question on whether a mean and variance asset pricing model using only the first two moments of

the return distribution is adequate in capturing variation in average stock returns. Subsequent volu-

minous empirical tests on Sharpes CAPM (1964) have largely rejected the validity of the model which

assumes that an investors utility function is quadratic and that the co-movement with the market

return is the only important factor in pricing stocks (see Campbell et al. (1995) for a comprehensive

review).

Given that the empirical stock return distribution is observed to be asymmetric and leptokurtic, a

natural extension of the elegant but oversimplified two-moment asset pricing model is to incorporate

the co-skewness (third moment) and co-kurtosis (fourth moment) factors. An investor whose utility is

non-quadratic and is described by non-increasing absolute risk aversion may prefer positive skewness

and less kurtosis in the return distribution. Stocks of negative co-skewness and of larger co-kurtosis

with the market should therefore be related to higher risk premia. Therefore, movement of higher

co-moments unfavourable to the investors risk preferences requires compensation in the form of

additional returns. This particular approach of characterizing stock pricing behaviour not only can be

intuitively appealing but may also improve the explanatory power of a model on the expected stock

returns.

In this paper, we examine the importance of co-skewness and co-kurtosis for average stock returns,along with the well documented Fama and French (1993) 3 common risk factors (namely firm size,

book-to-market equity (BV/MV), and market returns) and the Jegadeesh and Titman (1993) momen-

tum effect. In particular, we test the presence of higher co-moment effects in the Australian stock

market and compare them with those in the US market. Our interest in the behaviour of Australian

stocks rests with the glaring absence of any direct studies on the pricing of higher co-moments in the

current Australian literature despite some evidence of skewness and kurtosis in the stock return dis-

tribution. For instance, Beedles (1986) and Alles and Spowart (1995) find that Australian stocks exhibit

significant skewness. Furthermore, Bird and Gallagher (2002) and Brands and Gallagher (2004) docu-

ment that Australian mutual funds are characterized by a leptokurtic distribution. In particular, they

noticed that portfolio returns of larger funds had more negative skewness and larger kurtosis relative

to smaller mutual funds. Although they suggest that the non-normal distribution may have implica-tions for diversification benefits, they did not pursue the analysis to directly measure this through

these higher moments.

Even for studies on the US, direct examination of higher moments is usually quite limited, and

approaches to examining it can be varied. Fang and Lai (1997) examine the importance of co-skewness

and co-kurtosis within the four-moment CAPM framework. Dittmar (2002) tests the four moment

factors with non-linear pricing kernels to improve the pricing kernels ability to describe the cross-

section of returns. His methodology is linked to the nonparametric models ofBansal and Viswanathan

(1993) and Chapman (1997) in which the pricing kernel is non-linear in the market return. On the

other hand, Kan and Zhou (2003) and Ando and Hodoshima (2006) examine the robustness of the

asymptotic covariance matrix of least square errors (LSE) of alphas and betas in a linear asset pricing

model when the joint distribution of the factors and error terms may not be normal or conditionallyhomoskedastic. In contrast, our approach is more consistent with the spirit of Ross APT (1976) or

Mertons ICAPM (1973) in which additional factors such as size, BV/MV, and momentum may also

capture variation in average stock returns. Our approach can therefore be viewed as a more direct test

on the presence of higher co-moments.

We draw a comparison of return behaviour between stocks listed as part of the Australian S&P

ASX 300 index and the US S&P 500 to highlight the potential different roles that co-skewness and

co-kurtosis perform in each market. Since an average Australian firm tends to be smaller and less

volatile than in other developed markets, negative skewness could be more dominant than kurtosis

in pricing stocks. On the other hand, an average US firm is larger but more volatile (also shown in

the descriptive statistics in Tables 1 and 2) such that its variance risk or kurtosis could be a more

influential factor. Despite this casual observation on the different stock market characteristics, nostudies to our knowledge have addressed the potential differential pricing effect of co-skewness and

co-kurtosis. Most studies, especially those in the US, rather focus simply on the skewness of the return

distribution, when kurtosis could be equally or more important. Earlier works including Arditti (1967),

Kraus and Litzenberger (1976), Friend and Westerfield (1980), Lim (1989), Harvey and Siddique (1999,


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Table 1

Summary statistics of the returns of 25 US portfolios formed by size and BV/MV: January 1992July 2007.

Size BV/MV Mean SD Unconditional

skewness

Excess unconditional

kurtosis

No

Ja

Large Low Portfolio 11 0.00027 0.01193 0.55992 7.05873 86

2 Portfolio 12 0.00005 0.01076 0.03064 2.81503 13

3 Portfolio 13 0.00005 0.01115 0.21915 5.51658 51

4 Portfolio 14 0.00003 0.01213 0.21107 5.52309 51

High Portfolio 15 0.00013 0.01593 0.21826 4.01917 27

2 Low Portfolio 21 0.00019 0.01103 0.04318 4.08743 28

2 Portfolio 22 0.00034 0.00920 0.07564 3.28607 18

3 Portfolio 23 0.00026 0.01060 0.01666 4.92388 40

4 Portfolio 24 0.00025 0.01168 0.03459 5.16342 44

High Portfolio 25 0.00044 0.01191 0.04082 3.03562 15

3 Low Portfolio 31 0.00042 0.00941 0.14778 3.93310 26

2 Portfolio 32 0.00046 0.00963 0.03205 4.58425 35

3 Portfolio 33 0.00037 0.00979 0.10882 3.77506 244 Portfolio 34 0.00038 0.01137 0.10549 3.36970 19

High Portfolio 35 0.00056 0.01074 0.04084 7.05465 83

4 Low Portfolio 41 0.00027 0.01047 0.04773 5.97045 60

2 Portfolio 42 0.00042 0.01001 0.21201 6.50191 71

3 Portfolio 43 0.00040 0.01024 0.19771 3.86752 25

4 Portfolio 44 0.00045 0.01039 0.01058 3.95522 26

High Portfolio 45 0.00046 0.01109 0.02718 4.59737 35

Small Low Portfolio 51 0.00069 0.01381 0.01686 6.95935 81

2 Portfolio 52 0.00044 0.01244 0.01017 3.38753 19

3 Portfolio 53 0.00051 0.01097 0.05701 3.74465 23

4 Portfolio 54 0.00045 0.01024 0.12154 4.44389 33

High Portfolio 55 0.00048 0.01131 0.09959 5.63357 53

Index 0.000106 0.004235 0.124432 4.543455 34

The sample consists of all stocks listed at any point in time during the sample period that were part of the S&P 500. Each portfolio comp

the intersection of 5 size and 5 BV/MV groups. Portfolio 11 contains large-cap and low BV/MV stocks while portfolio 55 contains small

are based on the direct method. The daily returns of each portfolio are the value-weighted returns of stocks in the portfolio. Unconditio

fourth moment of the daily returns. The JarqueBera normality test is a test of whether the stock returns are normally distributed.


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Table 2

Summary statistics of 25 Australian portfolios formed by size and book-to-market value: January 2001July 2007.

Size BV/MV Mean SD Unconditional

skewness

Excess unconditional

kurtosis

No

Jar

Large Low Portfolio 11 0.001078 0.015451 0.4141 28.01132 552 Portfolio 12 0.000641 0.011673 0.89338 8.17112 4

3 Portfolio 13 0.00097 0.013862 0.14476 5.442801 2

4 Portfolio 14 0.000314 0.013093 1.37125 22.56568 36

High Portfolio 15 0.000682 0.014213 0.61091 5.383741 2

2 Low Portfolio 21 0.0000503 0.009885 1.1804 11.55367 9

2 Portfolio 22 0.000167 0.008852 0.54744 3.587539

3 Portfolio 23 0.000248 0.010429 0.47329 3.539279

4 Portfolio 24 0.00003 0.011205 0.7352 4.796208 1

High Portfolio 25 0.000373 0.010618 0.72905 5.276479 2

3 Low Portfolio 31 0.000369 0.008598 0.26806 3.737123 1

2 Portfolio 32 0.000331 0.009714 0.53622 6.153249 2

3 Portfolio 33 0.00045 0.009091 0.40085 3.661214

4 Portfolio 34 0.00082 0.009157 0.1679 2.891133

High Portfolio 35 0.000275 0.012259 0.55098 3.001116

4 Low Portfolio 41 0.000477 0.007567 0.04241 2.27273

2 Portfolio 42 0.000363 0.007203 0.12133 2.1406

3 Portfolio 43 0.000313 0.008484 0.57527 4.228345 1

4 Portfolio 44 0.000315 0.010089 2.64937 41.52602 124

High Portfolio 45 0.000828 0.008785 0.40689 3.543715

Small Low Portfolio 51 0.0005 0.009136 0.507958 6.132786 2

2 Portfolio 52 0.000445 0.00817 1.5602 16.25734 19

3 Portfolio 53 0.000599 0.00726 0.43624 2.173984

4 Portfolio 54 0.000637 0.007411 0.41203 2.219143

High Portfolio 55 0.000404 0.008728 0.35203 1.771397

Index 0.000154 0.002988 0.488384 3.058635

Thesample consists of all stockslisted at any point in time during the sampleperiod that were part of the S&P ASX300. Each portfolio com

the intersection of 5 size and 5 BV/MV groups. Portfolio 11 contains large-cap and low BV/MV stocks while portfolio 55 contains small

are based on the direct method. The daily returns of each portfolio are the value-weighted returns of stocks in the portfolio. Unconditio

fourth moment of the daily returns. The JarqueBera normality test is a test of whether the stock returns are normally distributed.


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P. Doan et al. / Int. Fin. Markets, Inst. and Money 20 (2010) 5167 55

2000) and Smith (2007) have examined the return distribution that only includes skewness. Our study

intends to fill the gap in the literature by addressing the relative importance of higher co-moments

in markets of different characteristics. Examining these two different markets also provides some

robustness checks on the importance of each pricing factor.

We find strong evidence for the higher co-moment factors in the US stocks and co-skewness in

Australian stocks. Consistent with the investor preference theory discussed earlier, average stock

returns are negatively related to co-skewness but positively related to co-kurtosis. These two factors

remain robust when we regress them along with excess market returns, size, BV/MV, and momentum

in our data. Our results therefore suggest that both co-skewness and co-kurtosis explain part of the

return variation that is not captured by these other well known factors. Our findings do not support

Chung et al. (2006) who argue that Fama and French factors are proxies for the pricing of higher order

co-moments, but are more consistent with Smith (2007) who finds that adding co-skewness to the

Fama and French 3 factor model improves the explanatory power of the model.

Our analysis also shows that although co-skewness is important in both the Australian and US

markets, its influence varies in degree. The co-skewness effect is stronger for the Australian stocks

compared to the co-kurtosis effect for US stocks. Theimportance of the co-skewness effect can perhaps

be partly explained by the positive relationship between size and skewness which is found to be morepronounced in Australia than in the US. Since the average size of the sampled Australian firm is smaller

than the US firm,1 it follows that Australian stocks may appear to be more sensitive to downside risk

given that the return distribution is more negatively skewed. Co-skewness may therefore play a larger

role in the Australian market.

On the other hand, the return distribution in the US appears to be more leptokurtic as the stock

returns tend to be more volatile. Subsequently, the significance of the co-kurtosis effectis more notice-

able in the US data. The source of the larger return volatility in the US market may be traced to the

specific characteristics of the US firms. Contrary to Australian firms which are more closely related to

primary industries in commodity and mining,2 US firms (at least from our sample of S&P 500 firms) are

more represented by technology related firms or high growth firms. Their returns therefore display

more extreme values at both tails, leading to co-kurtosis being more influential for US stocks.The remainder of the paper is as follows. Section 2 describes the data and our methodology while

Section 3 presents the empirical results. Section 4 concludes the study.

2. Data and methodology

2.1. Data

Our sample consists of all stocks in the Australian S&P ASX 300 and the US S&P 500 indices. The

advantage of the data is that they are a good proxy for the Australian and the US market portfolios,

plus are derived using similar weighting methodology from the S&P. They include the leading compa-

nies, by market capitalization, across different industries for each country. Another advantage is thatthese relatively large stocks mitigate the nonsynchronous trading problem that tends to be associated

with smaller firms. Scholes and Williams (1977) and Dimson (1979) show that small stocks where

infrequent trading may be common can cause positive serial correlation in stock returns. 3 However,

since larger stocks tend to exhibit less skewness and kurtosis, the selection of the firms in our sample

may bias against us finding the presence of higher co-moments.

All the stock and index daily returns are obtained from Datastream. The sample period for the

Australian data starts from inception of the S&P ASX 300 series in January 2001July 2007, yielding

approximately 510,000 firm-year observations. Since the US data are available from an earlier date, the

1

At the end of 2007 the average market capitalization of S&P500 and ASX300 stocks were US$ 5.9 billion and US$ 3.3 billion,respectively.2 Also, the ASX 300 is not well diversified across industries, relative to the S&P 500, with over 51% of stocks in the Australian

index (as of the end of 2007) being classified as either a financial or resource stock.3 We chose to utilise daily data for our study as Kirchler and Huber (2007) argue that skewness and kurtosis become more

prominent when higher frequency data are examined.


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sample period starts from January 1992 to July 2007 for a total of 2.03 million firm-year observations.

To directly compare the results between the two markets, we also run the analyses over the same

sampled period from January 2001 to July 2007 on both sets of data. Since the outcomes are essentially

the same from those over the full sampled period, we did not tabulate them in the paper. For proxies

of the risk-free rates, we use the 90-day bank bill and the 30-day Treasury bill rates of Australia and

the US, respectively.

2.2. Portfolio formation and measurements

To form portfolios based on size and BV/MV, we followthemethodology ofFama and French (1993).

Firms of S&P ASX 300 and S&P 500 for each year are ranked according to their market capitalization

at the beginning of the year and are divided into five quintiles with about equal number of stocks

in each quintile. We then take the difference between returns of the biggest and smallest portfolios

(SMB) to mimic the risk factor in returns relating to firm size. The stocks are further ranked by BV/MV

independently and sorted into fiveportfolios. TheHML factor is then estimatedby takingthe difference

in returns between the highest and lowest BV/MV ratios. The 25 portfolios are subsequently formedby the intersection of five size and five BV/MV quintiles as shown in Table 1. We repeat the process

each year to rebalance the portfolios and to estimate the size and value factors from the ASX 300 and

S&P 500 stocks.

For co-skewness (or co-kurtosis) factors, the co-skewness (or co-kurtosis) of each stock is first

calculated according to the following equations;

Co-skewness:

iM =E[{ri E(ri)}{rM E(rM)}

2]

E{ri E(ri)}

2E[{rM E(rM)}]2

=Cov(ri, r

2)

SD(ri)Var(rM)(1)

Co-kurtosis:

iM =E[{ri E(ri)}

2{rM E(rM)}

2]

E{ri E(ri)}2E[{rM E(rM)}]

2=

Cov(r2, r2)

Var(ri)Var(rM)(2)

where ri and rM are the returns of stock i and the market returns respectively, and E(ri) and E(rM)

are the expected returns of stock i and the expected market returns, respectively. The stocks are then

ranked based on their co-skewness (or co-kurtosis) and are then sorted into five quintile portfolios

with an approximate equal number of stocks. Therefore, quintile 1 contains the highest co-skewness

(or co-kurtosis) and quintile 5 the lowest. The difference of the return of the highest co-skewness

(co-kurtosis) portfolio minus the return of lowest co-skewness (co-kurtosis) portfolio captures thereturn premium that is related to co-skewness (co-kurtosis).

Finally, we follow the methodology of Jegadeesh and Titman (1993) to estimate the momentum

factor. Stocks are first sorted into five quintiles in descending order on the basis of their past daily

returns. Based on these rankings, we form an equal-weighted portfolio within each quintile. The first

quintile portfolio containing stocks with the highest returns is the winners portfolio and the bottom

portfolio is the losers. The difference in returns between the winner portfolio and the loser portfolio

is attributed to the return premium for the momentum strategy.

3. Empirical analysis

3.1. Summary statistics

We first present the summary statistics of the daily returns of the 25 portfolios of the US and

Australian stocks in Tables 1 and 2, respectively. In addition to the mean and standard deviation of the


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returns, the reported unconditional skewness and excess kurtosis are calculated as follows:

Skewness =1

T 1

Tt=1

Rp R

R

3(3)

Excess kurtosis =1

T 1

Tt=1

RP R

R

4 3 (4)

where Rp is the daily return of a portfolio, R is the standard deviation of the portfolio returns, and Tisthe number of observations. The excess kurtosis is obtained by subtracting the unconditional kurtosis

from (3), the unconditional kurtosis of a normal distribution. For co-skewness and co-kurtosis in our

later regression analysis, we estimate them according to Eqs. (3) and (4).

Consistent with earlier studies, Table 1 shows that smaller portfolios tend to outperform larger

portfolios in the US even after we control for BV/MV. This size effect is most apparent on the returns of

the largest portfolio in which 4 out of 5 sub-portfolios sorted by BV/MV earn less average returns than

the S&P 500 index. The remaining sub-portfolios within each portfolio however earn higher averagereturns than the index. Controlling for the size effect, we find that the value effect is less apparent. The

increase in average returns from lowest to highest BV/MV sub-portfolios in each size portfolio is less

than monotonic. On the standard deviation of the sub-portfolios, we also fail to detect any systematic

patterns across either size or BV/MV.

As expected, the average returnof each portfolio is asymmetric and leptokurtic. Of the 25 portfolios,

14 and 11 of these portfolios exhibit negative and positive skewness, respectively. Size appears to be

positively related to skewness.4 Table 1 shows that 4 out of the 5 smallest portfolios according to size

tend to exhibit negative skewness. Overall, the average unconditional skewness is low for the US data,

ranging from 0.56 to 0.22. The return distribution however exhibits heavy tails systematically where

the unconditional excess kurtosis estimates range from 2.82 to 7.06. In other words, the kurtosis of the

sampled portfolios has largely exceeded the kurtosis of 3 for the normal distribution. Based on theseobservations, if co-kurtosis is priced in the US, our subsequent analysis should find that the fourth-

moment will be more influential on the average stock returns. The JarqueBera tests of standard

normality, which measure the difference of skewness and kurtosis of the series with those of the

normal distribution, also show that every sub-portfolio is significantly non-normal at the 1% level.

For the Australian data, Table2 shows that small portfolios earn on average higher returns than large

portfolios. In particular, only the largest sub-portfolios based on size have negative average returns.

We also fail to find a monotonic relationship between average returns and BV/MV after controlling for

size. There also appears to be little correlation between size and BV/MV with standard deviation. The

stockreturnbehaviourforthefirsttwomomentsthusfarissimilartothatoftheUStabulatedin Table1.

When we estimate unconditional skewness and co-kurtosis for the Australian data, we find that the

stock returns are more asymmetric but less leptokurtic. The average skewness for the ASX 300 indexis 0.4884 compared to 0.1244 for the S&P 500. Furthermore, 24 out of the 25 sub-portfolio return

distributions are negatively skewed. The degree of the negative skewness for the sub-portfolios is also

larger than that found in the US data. For example, the negative skewness in 9 of the 24 sub-portfolios

is larger than the largest negatively skewed portfolio in the US sample. In contrast, Australian stock

returns appear to be less leptokurtic than those in the US The excess kurtosis for the Australian and

US markets are 3.059 and 4.543, respectively. The excess kurtosis however is more dispersed in the

Australian sub-portfolios where it varies from 1.77 to 41.52, compared to the range of 2.82 and 7.06

in the US stocks. Before we run the regression analyses of average portfolio returns on the higher

co-moments and other controlled factors, we examine the correlations between these independent

variables in both markets. On the US stocks shown in Table 3, the correlations across different variables

are generally low. They range from

0.54 to 0.68 where the majority of the correlations fall within

0.2to 0.2. In particular, the correlations between the two high co-moments and SMB and HML are weak,

4 We present the correlations in Table 3 which shows that size is positively correlated with skewness in both markets.


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Table 3

The correlations between explanatory variables.

RM Rf Co-skewness Co-kurtosis SMB HML Momentum

US

RM Rf

1

Co-skewness 0.0288 1

Co-kurtosis 0.5430 0.1119 1

SMB 0.1631 0.0693 0.0528 1

HML 0.1634 0.0475 0.2635 0.6837 1

Momentum 0.0941 0.0767 0.4220 0.1781 0.0697 1

Australia

RM Rf 1

Co-skewness 0.5334 1

Co-kurtosis 0.3115 0.4416 1

SMB 0.2144 0.3799 0.0681 1

HML 0.1984 0.2434 0.2937 0.2082 1

Momentum 0.1223 0.4740 0.1176 0.5926 0.1690 1

This table reports the correlations between excess market returns, RM Rf, co-skewness, co-kurtosis, size (SMB), value (HML),and momentum for the US S&P 500 stocks and the S&P ASX 300 Australian stocks.

between 0.26 and 0.07, which implies that co-skewness and co-kurtosis are not proxies for SMB and

HML. It is also interesting to note that the highest correlation of 0.68 is found between SMB and HML.5

Similarly, the correlations are also low across the independent variables in the Australian market.

Not surprisingly, excess market returns tend to correlate more with other factors but not to the extent

that it creates a multicollinearity issue in our regression analyses. The positive correlations between

co-skewness and co-kurtosis contradict those in the US market, although they remain very low. It

suggeststhatthere maynot be a systematicrelationshipamong these independentfactors. Momentum

however tends to be consistently negatively correlated with both co-skewness and co-kurtosis in both

markets. Measured as the returns of winner portfolios minus the returns of loser portfolios, a largermomentum effect is perhaps related to larger negative skewness of the loser portfolio than that of the

winner portfolio (see Harvey and Siddique (2000)).

3.2. The co-skewness and co-kurtosis effect

In our regression analyses, we first examine the sensitivity of excess portfolio returns to co-

skewness and co-kurtosis alone. We therefore regress the excess daily returns of the 25 sub-portfolios

formed by size and BV/MV on the two higher co-moments only according to the equation below:

Rp,t Rf,t = + 1CoSt + 2CoKt + et (5)

where Rp,tis the return of the portfolio at timet, Rf,tis therisk-free rate at time t,CoStis the co-skewnessfactor at time t, and CoKt is the co-kurtosis at time t.

As shown in Table 4, we find that co-skewness is notconsistently related to the US portfolio returns.

The significance of co-skewness is only found in 7 out of 25 sub-portfolios of which they tend to be

small in firm size and high in BV/MV. It suggests that co-skewness captures limited variation in returns

that is not explained by the size and BV/MV effects. We suspect that the smaller dispersion of skewness

in the US sub-portfolios shown in Table 1 may not co-vary well with the variability of returns in the

time-series regressions. For co-kurtosis, we find that it is influential in 24 out of the 25 US portfolios.

Its economic significance is also quite apparent as a 1% increase in co-kurtosis is related to an increase

in average returns of between 0.3% and 0.7%

Unlike the US portfolios, we find that both co-moments are important in the Australian market

although the explanatory power of co-skewness seems to be more robust for average portfolio returns.

5 We estimate the variance inflation factor (VIF) for each independent factor to examine if the correlationsamong them could

bias the results in our regression analysis. We find that the highest VIF coefficient obtained is 3.13 relating to co-skewness.

Multicollinearity among the variables is not usually considered a problem unless a VIF exceeds 5.


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Co-skewness in 24 of the 25 sub-portfolios is significant compared to 17 sub-portfolios for the co-

kurtosis. Furthermore, 4 out of the 17 sub-portfolios have negative signs (rather than positive signs)

for thestock returns. Therefore, while the returns of themajority of the sub-portfolios can be explained

by co-kurtosis, these are not as consistent as with co-skewness. One possible explanation for the

differences in the importance of the high co-moments in the two markets may be related to their firm

characteristics. An average firm in the US sample tends to be larger than its counterpart in Australia.

If smaller firms tend to associate with negative skewness, then co-skewness should play a larger role

in asset pricing. Comparing unconditional skewness across portfolios in Tables 1 and 2 does show

that the Australian portfolios, on average, exhibit more negative skewness than the US portfolios. The

larger daily return volatility of 0.004% among the US stocks compared to 0.0001% of the Australian

stocks however indicates that excess kurtosis is on average higher for the US stocks (4.54 in the US

vs. 3.06 in Australia). With a larger number of high growth stocks, such as those in the computer

related industries represented in the S&P 500 data, it may explain why our US sampled stocks exhibit

higher excess kurtosis. It follows that co-kurtosis may be more influential on the average returns for

the US stocks. On the contrary, the sampled Australian firms tend to be concentrated in the mining

and resource sectors. These firms tend to be more mature and their corresponding volatilities are also

lower. As a result, co-kurtosis plays a lesser role than those found for the average US stock.

3.3. Multivariate regression analysis and robustness checks

If higher co-moments do explain average returns, then they must also remain important in the

presence of other well known factors. We therefore add the excess market returns to our regressions

below:

Rp,t Rf,t = + (Rm,t Rf,t) + 1CoSt + 2CoKt + et (6)

where Rm,t is the return of the market index at time t, and the other variables are defined earlier in Eq.

(5).

Eq. (6) allows us to test whether the covariance of the market return volatility with the portfolioreturns (co-skewness) and that with the portfolio volatility (co-kurtosis) captures variation in average

stock returns in addition to the covariance of the market return and portfolio return. Hence, in this

test, we examine if the second moment of the market index is just as important as its first moment in

explaining portfolio returns and volatility. Table 5 shows that at least one of the higher co-moments

remains significant in both markets when we add the excess market returns. In fact, co-skewness and

co-kurtosis in the US returns explain 15 and 17 of the 25 sub-portfolio returns, respectively. Overall,

24 out of the 25 sub-portfolios have at least one higher co-moment statistically significant at the 5%

level. Co-skewness in the Australian portfolios remains an important factor where 18 out of 25 sub-

portfolios are statistically significant in explaining average portfolio returns. However, the results for

co-kurtosis aremixed and of those sub-portfolios that canbe explained by a significant fourth moment,

it is unclear how it affects the returns as the direction associated with the moment is inconsistent andswitches between being positive and negative.

Next we regress the average portfolio returns in each sample by size, BV/MV, and momentum

effect. That is:

Rp,t Rf,t = + (Rm,t Rf,t) + s SMB + h HML+ 3CoSt + 4CoKt + M+ et (7)

where SMB is small minus big, HML is high minus low, and Mis themomentum by takingthe difference

in the returns of the winner and loser portfolios. The remaining variables are defined in the earlier

equations.

Table 6 shows that thenegative relationshipbetween co-skewness and the average returns, and the

positive relationships between co-kurtosis and the average returns in the US portfolios have changed

little from earlier analyses, despite us adding size, value, and momentum effects. More specifically, co-skewness and co-kurtosis are significant in 19 and 21 of the 25 sub-portfolios, respectively. It suggests

that not only the higher co-moments are important in pricing assets but they also capture different

elements of risk that other established factors fail to explain. Results in Table 7 for the Australian port-

folios are again consistent with our earlier findings. The significance of co-skewness remains robust


11/17


12/17

Table 5 (Continued )

BV/MV Size US portfolios Australian portfolio

Intercept RM Rf Co-

skewness

Co-kurtosis Adj. R2 Intercept RM

4 Portfolio

44

0.00030

(2.96)*

1.65434

(58.60)**

0.07770

(5.50)**

0.18726

(15.27)**

0.62 0.00006

(0.31)

2.00

(26.

Small Portfolio

54

0.00270

(2.87)*

1.83673

(70.55)**

0.06077

(4.67)**

0.10072

(0.11)

0.67 0.00031

(3.19*)*

2.08

(54.

5 Large Portfolio

15

0.00002

(0.05)

1.53906

(25.85)**

0.01703

(0.57)

0.30495

(11.79)**

0.29 0.00015

(0.48)

0.11

(0.92 Portfolio

25

0.00031

(2.26)*

1.58506

(41.58)**

0.00200

(0.10)

0.23974

(14.49)**

0.48 0.00069

(3.38)**

0.94

(11.

3 Portfolio

35

0.00042

(3.35)**

1.51777

(42.94)**

0.01200

(0.67)

0.12655

(8.24)**

0.45 0.00041

(1.72)

1.48

(16.

4 Portfolio

45

0.00035

(3.14)**

1.51027

(48.60)**

0.09642

(6.21)**

0.31892

(23.62)**

0.60 0.00063

(3.93)**

1.67

(26.

Small Portfolio

55

0.00035

(3.44)**

1.65781

(57.77)**

0.14911

(10.39)**

0.32394

(25.99)**

0.67 0.000174

(1.20)

1.97

(35.

This table reports the regression results of the excess portfolios returns on market excess returns, co-skewness and co-kurtosis over the

denote t-statistics at the 5% and 1% level, respectively.


13/17


14/17


15/17


Table 8

Adjusted R2 for regressions of 25 US portfolios and 25 Australian portfolios formed by size and BV/MV.

Low BV/MV 2 3 4 High BV/MV

US stocks

Size

Big 0.78 0.58 0.60 0.43 0.43

2 0.61 0.56 0.59 0.56 0.51

3 0.53 0.54 0.56 0.59 0.47

4 0.44 0.52 0.60 0.62 0.65

Small 0.35 0.51 0.64 0.68 0.79

Australian stocks

Size

Big 0.66 0.45 0.30 0.30 0.53

2 0.32 0.24 0.29 0.37 0.74

3 0.36 0.37 0.42 0.38 0.45

4 0.36 0.42 0.43 0.41 0.49

Small 0.35 0.51 0.64 0.72 0.63

This table reports the adjusted R2 of the regression results reported in Tables 6 and 7.

as it captures variations of 21 out of the 25 sub-portfolio returns. The co-kurtosis effect, however,

continues to be weak where it explains only 7 out of 25 sub-portfolios with the correct positive sign.

In summary, the picture remains similar throughout our analyses where we find co-kurtosis is espe-

cially important for the US stock returns, while co-skewness plays a more important role for Australian

stocks. Although not tabulated in the paper, we also conducted further checks on our results by incor-

porating a GARCH (1,1) effect in the time-series regressions to adjust for conditional heteroscedasticity.

However, the results remain consistent to what we previously illustrated.

Our overall findings are consistent with Arditti (1967), Scott and Horvath (1980), Fang and Lai

(1997), and Galagedera et al. (2002) who argue that investors have a negative preference for even

moments (i.e. variance and kurtosis) and a positive for odd moments (i.e. return and skewness).Furthermore, Smith (2007) finds that by adding co-skewness to the Fama and French 3 factor model,

the model performs better than either the 3 factor or three-moment models alone. Table 8 shows the

adjusted R2 for the regressions in Tables 6 and 7. Similar to Smith (2007), adding Fama and French

3 factors and momentum to higher co-moments increases the explanatory power of the model. The

adjusted R2 for the 25 sub-portfolios range from 0.24 to 0.79. Our findings, however, do not find

support Chung et al. (2006) who suggest that the Fama and French 3 factors are proxies for the higher

co-moments.

3.4. Fama and Macbeth regressions

To investigate if co-skewness and co-kurtosis are priced and command significant risk premiumsin the Australian and US stock markets, we run cross-sectional regressions following the Fama and

MacBeth (1973) methodology. First, we estimate the sensitivity of a firms excess return to the risk

premium related to each risk factor for each day:

Rp,t Rf,t = p,k,t + p,k,t RPk,t + ep,t t = 59, , 0 (8)

where Rp,t Rf,t is portfoliops excess return for day t, Rk,t is the risk premium related to the kth factor

which includes co-skewness, co-kurtosis, size, BV/ME, market, and momentum. The regression is run

across each of these six factors so that at the end of each day portfoliop possesses a vector of sensitivity

estimates to these factors. This estimation process is repeated daily for the sample period.

Next, we run the following cross-sectional regression at the end of each day to examine if co-

skewness and co-kurtosis are priced or have predictive power over portfolio returns:

Rp,t+1 = k,t+1 +

Kk=1

k,t+1 p,k,t + ep,t+1 p = 1, , N (9)


16/17


Table 9

FamaMacbeth regression estimates.

Variables US Australia

Co-skewness 0.0001

(0.38)

0.0122**

(2.80)

Co-kurtosis 0.0019*

(2.10)

0.0007

(0.12)

SMB 0.0005

(1.86)

0.0172**

(2.72)

HML 0.0003

(1.08)

0.0025

(1.24)

Market 0.0003

(1.30)

0.0008

(0.52)

Momentum 0.0024**

(3.40)

0.0181

(1.78)

Thistable presentsthe results of FamaMacbeth regressions for co-skewness, co-kurtosis, SMB,

HML, market, and momentum. At the end of each day, the following cross-sectional regres-

sion is run, Rp,t+1 = k,t+1 +

Kk=1

k,t+1 p,k,t + ep,t+1, p = 1, , Nwhere Rp,t+1 is the portfolio

return at t, p,k,t is the portfolio beta for the kth factor at time t, and ep,t+1 is the error term at

time t. The test statistic for k,t+1 is calculated as tk = (k 0/rk ) where k =1T

Tj=1

k,j . * and

** denote t-statistics at the 5% and the 1% level, respectively.

Under the null hypothesis, test statistics for Ak,t+1 can be obtained as follows:

tk =k 0

rk(10)

where k =1

T

Tj=1

k,j

Table 9 reports the results of cross-sectional regressions according to the Fama and MacBeth (1973)

methodology. Consistent with the results of time-series regressions reported earlier, we find that co-

kurtosis and co-skewness aresignificantly related to expected portfolio returns in the US and Australia,

respectively, in the presence of size, book-to-market, market, and momentum factors. US stocks which

are characterized by larger variance and less skewness can be better explained by the variance risk or

kurtosis of the return distribution. On the other hand, Australian stocks which tend to be smaller andmore negatively skewed are more correlated with the co-skewness factor. As a result, at least one of

the higher co-moment factors is priced for the stocks. In sum, risk related to higher co-moment does

not appear to be proxied by the well documented four common risk factors.

4. Conclusions

This paper suggests that co-skewness and co-kurtosis are important in pricing stocks. The degree of

the importance, however, depends on the firm characteristics of the stocks and the risk preference of

investors. As Australian stocks tend to be more skewed and less leptokurtic, we find that co-skewness

plays a more significant role in explaining average stock returns. For US stocks, co-kurtosis is more

influential as returns exhibit higher excess kurtosis. We believe the differences in results betweenthe US and Australian markets come from the fact that the Australian stocks are relatively small to

begin with, when compared to their counterparts in the US. The implication being that for many

medium to small sized exchanges, co-skewness may be a more relevant factor than co-kurtosis. Our

results are also robust to a number of model specifications. Although size, BV/MV, and momentum are


17/17


correlated with co-skewness and co-kurtosis, the importance of co-skewness and co-kurtosis remain

largely unchanged in their presence. It implies that the higher co-moments capture parts of variation

in average stock returns that are not explained by their effects.

Adding co-skewness and co-kurtosis also improves the explanatory power of the Carhart (1997)

four-factor model that includes market, size, BV/MV, and the momentum factors. This is despite the

fact that we believe our sample is bias against us finding higher co-moment effects, as we focus on

only analyzing the larger capitalized firms in the US and Australia. If our study included smaller cap

stocks we would expect even further evidence of the presence of higher moments. We therefore

believe that our findings do support the need to incorporate higher co-moments into asset pricing

models and developing a theoretical asset pricing model that addresses the non-normality of the

return distribution should remain important for future research.

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