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Investor preferences and portfolio selection: isdiversification an appropriate strategy?C. James Hueng a & Ruey Yau ba Department of Economics , Western Michigan University , Kalamazoo, MI 49008, U.S.Ab Department of Economics , National Central University , Taoyuan, Taiwan 32001,R.O.C.Published online: 18 Feb 2007.

To cite this article: C. James Hueng & Ruey Yau (2006) Investor preferences and portfolio selection: is diversification anappropriate strategy?, Quantitative Finance, 6:3, 255-271, DOI: 10.1080/14697680600680134

To link to this article: http://dx.doi.org/10.1080/14697680600680134

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Quantitative Finance, Vol. 6, No. 3, June 2006, 255–271

Investor preferences and portfolio selection:

is diversification an appropriate strategy?

C. JAMES HUENG*y and RUEY YAUz

yDepartment of Economics, Western Michigan University, Kalamazoo, MI 49008, U.S.A.zDepartment of Economics, National Central University, Taoyuan, Taiwan 32001, R.O.C.

(Received 6 July 2004; in final form 24 February 2006)

This paper analyzes the relationship between diversification and several distributionalcharacteristics that have risk implications for stock returns. We develop a flexible three-parameter distribution to model the stock returns. Using data on the current 30 DJIA stocks,we show that an investor’s strategy on diversification depends on the measures of risk forparticular concerns. For example, investors who desire to increase positive skewness wouldhold a less diversified portfolio, while those who care more about extreme losses would hold amore diversified portfolio. Experimenting with a more general pool of stocks yields the sameconclusions.

Keywords: Diversification; Asymmetric Generalized t distribution; Skewness

1. Introduction

Research conducted in the mean-variance framework haslong shown that diversification offers the benefit of redu-cing unsystematic risk (e.g. Sharpe 1964, Lintner 1965,Evans and Archer 1968, Fielitz 1974). However, empiricalevidence has revealed that investors do not tend to holdfully diversified portfolios. For example, Goetzmann andKumar (2002) examine more than 40,000 equity invest-ment accounts over a 6-year period from 1991 to 1996and find that the average investor has four stocks inher/his portfolio, while less than 5% of investors holdmore than 10 stocks in their portfolios. Furthermore,more than 25% of portfolios contain only one stock,and more than 50% contain fewer than three. Given thebenefits typically associated with holding a diversifiedportfolio, the apparent lack of diversification amongmost investors is somewhat puzzling.

The perception that investors do not weigh downsiderisk equally with upside potential provides an explanationto this phenomenon. In this case, the variance is no longeran appropriate measure for risk. Kraus and Litzenberger(1976) build a three-factor CAPM framework by expli-citly incorporating skewness in investors’ preference.Malevergne and Sornette (2005) derive a modified

efficient frontier where higher moments (up to the eighthorder) replace variance as the measure of risk. Theydemonstrate how the traditional portfolio optimizationframework can be improved. Cooley (1977) conducts anexperiment to test the perception of risk on the part ofinstitutional investors and finds that, among 56 institu-tional investors who were asked to rate distributionsaccording to perceived risk, at least 29 associated theasymmetry of return distributions with risk. In particular,the investors associated increases in risk with increases innegative skewness, indicating a preference for positiveskewness.§ Harvey and Siddique (2000) and Premaratneand Tay (2002) explore the role of skewness in assetreturns. If an asset contributes positive skewness to adiversified portfolio, then that asset will be valuable andhas a higher price (or lower expected return).

Investors who prefer positive skewness would seekto construct portfolios that have this characteristic.However, this complicates the portfolio selection deci-sion, because a desire on the part of investors to obtainpositive skewness may not be compatible with the familiarmethod of constructing a diversified portfolio in order toreduce risk. As shown in Simkowitz and Beedles (1978),even though portfolio variance decreases as diversifica-tion occurs, skewness may either increase or decrease

*Corresponding author. Email: James.Hueng@wmich.edu§Arrow (1974) theoretically shows that risk-averse investors with non-increasing risk aversion prefer positively skewedinvestment positions.

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online # 2006 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/14697680600680134

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with diversification. Based on computations of the rawskewness, they find that skewness is rapidly reducedby diversification for a pool of 549 common stocks.Aggarwal and Aggarwal (1993) and Cromwell et al.(2000) also record similar results.

Aside from variance and skewness, there exist otherrisk measurements that are of particular concern to spe-cific investors. For example, recent research on Value atRisk (VaR) indicates that investors may be concernedstrongly about the likely maximum loss, i.e. the left tailof the distribution of a portfolio’s expected returns.Of particular interest to us is the relationship betweendiversification and the other information contained inthe distribution beyond variance and skewness. Forexample, investors may have a stronger preference for alower expected extreme-loss than for a higher skewness.As such, the relationship between expected extreme lossesand diversification adds another dimension to the portfo-lio selection process. Indeed, Gaivoronski and Pflug(2004) conclude that investors who are concerned withVaR will not achieve comparable results using a portfolioselection methodology that relies on another risk measuresuch as variance; rather, VaR is a very different measurethat deserves its own place in the portfolio selectionprocess.

To achieve our stated goals, we propose a parametricmodel to estimate the population distribution of stockreturns. It is well known that stock returns are notnormally distributed. Therefore, to model the highermoments in stock returns, we develop the ‘AsymmetricGeneralized t (AGT)’ distribution, an asymmetric versionof McDonald and Newey’s (1988) generalized t (GT)distribution. Although this is a simple three-parameterdistribution, it is very flexible in that it permitsvery diverse levels of higher moments.y This distributionnests several popular distributions often seen in theliterature, including the normal and the Student’s tdistributions. This general distribution allows us toestimate the population distribution of the returns andprovides us with information on a wide range ofdistributional characteristics that have risk implications.

We focus on the current 30 stocks in the Dow JonesIndustrial Average (DJIA) for both computational tract-ability and its desired nature of a well-diversified portfo-lio. The results show that first, consistent with theprevious studies using sample moments, our resultsshow a trade-off between low variance and high skewness:diversification reduces the portfolio variance, but at thesame time also reduces the skewness. Second, expectedextreme losses become smaller when the portfolio sizeincreases. As such, an investor’s strategy of diversificationdepends on the measures of risk for particular concerns.

For example, investors who want to increase positiveskewness would hold a less diversified portfolio, whilethose who care more about extreme losses would holda more diversified portfolio.

The next section outlines the model and the propertiesof the AGT distribution. Section 3 presents the empiricalresults by using the DJIA component stocks. Section 4experiments with a more general pool of stocks to testthe robustness of our conclusions. Section 5 offers theconclusion.

2. The model and methodology

The conditional mean of the stock returns is modeled asa simple AR(m) process in an attempt to estimate thezero-mean, serially uncorrelated residuals:

yt ¼ �þXmi¼1

�iyt�i þ "t, ð1Þ

where yt represents the daily stock returns and lag lengthm is chosen by the Ljung-Box Q tests as the minimum lagthat renders serially uncorrelated residuals (at the 5%significance level) up to 30 lags from an ordinaryleast squares (OLS) regression. Note that the OLSregression is used only to determine the number of laggedreturns to be included in the conditional mean. Thecoefficients in the conditional mean equation will bejointly estimated with the conditional variance equationusing the full information maximum-likelihood (FIML)estimation.

The conditional variance, denoted as ht � Eð"2t jt� 1Þ,follows a GARCH process. It is well-documented in thefinance literature that stock returns have asymmetriceffects on predictable volatilities; see, among others,Glosten et al. (1993) and Bekaert and Wu (2000).Therefore, we use an asymmetric GARCH modelproposed by Glosten, Jaganathan and Runkle (the GJRmodel), which is claimed to be the best parametric modelamong a wide range of predictable volatility modelsexperimented by Engle and Ng (1993):

ht ¼ �þ � � ht�1 þ � � "2t�1 þ � � Iþt�1 � "2t�1, ð2Þ

where Iþt�1 ¼ 1 if "t�1 >0 and Iþt�1 ¼ 0 otherwise.The relationship between the return residual and

its conditional variance can be specified as "t ¼ffiffiffiffiht

pvt,

where vt is a zero-mean and unit-variance random vari-able. The specification of the distribution of the stochasticprocess {vt} determines the distribution of yt. The mostcommonly used distribution for vt is the standard normal

yOther flexible alternatives include the exponential generalized beta of the second kind (EGB2), transformations of normallydistributed variables discussed by Johnson (1949), and a family of modified Weibull distributions proposed by Malevergne andSornette (2004). Wang et al. (2001) apply the EGB2 to GARCH models. The AGT distribution proposed in this paper is at leastequally as flexible as the EGB2 distribution. Whereas the EGB2 distribution imposes limited ranges on higher moments, the AGTdistribution has no such limits. The modified Weibull distribution is characterized by only two parameters, but it does not nest theStudent’s t distribution, the most often used statistical distribution to capture the fat-tail behaviors in asset returns.

256 C. J. Hueng and R. Yau

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distribution, which is symmetric and has a kurtosiscoefficient of three. While the GARCH specificationmakes some allowance for unconditional excess kurtosis,according to Bollerslev et al. (1992), it is still unableto adequately model the fat-tailed properties of thestock returns. In response to the levels of kurtosisfound in stock return data, Bollerslev (1987) combinesa t-distribution and a GARCH (1,1) model. Thet-distribution has fatter tails than are found in the normaldistribution.

McDonald and Newey (1988) propose the Generalizedt (GT) distribution, which is a more generaldistribution that can accommodate both leptokurtosis(thicker tailed than the normal distribution) andplatykurtosis (thinner tailed than the normal distribu-tion). The probability density function (pdf) of the GTdistribution is:

fGTðxt;!, p, qÞ ¼p

2!q1=pBðð1=pÞ, qÞ 1þ ðjxtjp=q!pÞÞ

qþð1=pÞ,�

where B(�) is the beta function. The parameters p and qare positive and p� q> n in order for the nth momentto exist. The mean is zero and the variance is !2q2=p�Bð3=p, q� 2=pÞ=Bð1=p, qÞ. The kurtosis is ½Bð1=p, qÞ=Bð3=p,q� 2=pÞ�2½Bð5=p,q� 4=pÞ=Bð1=p,qÞ� with p� q>4.Special cases of the GT include the power exponentialor Box–Tiao (BT) (q ! 1), Student’s t ( p¼ 2), normal( p¼ 2 and q ! 1), and Laplace ( p¼ 1 and q ! 1).The apparent advantage of the GT distribution overthe t-distribution is its flexibility in modeling highermoments.y

While the GT distribution is more flexible than thepopular normal and student’s t distributions in model-ing the fourth moment, it is still symmetric and unableto model the skewness in stock returns. A distributionwith the ability to capture all of the first fourmoments could provide more flexibility for modelingreturns. To obtain such a distribution, we develop anasymmetric version of the generalized t distribution,where we define:

hðztÞ ¼

fGTjztj

1þ r

� �for zt � 0,

fGTjztj

1� r

� �for zt < 0,

8>>><>>>:

where �1<r<1.z This specification is a proper pdf andallows different rates of descent for zt>0 and zt<0, andtherefore, it allows for skewness. Next, we scale zt bydefining xt¼ zt/A, where A is a positive scaling constantfor simplifying the notation.

The density function of the AGT distribution can bewritten as:

fAGTðxtÞ ¼ hðztÞ �@zt@xt

¼ hðA � xtÞ � A

¼

fGTA � jxtj

1þ r

� �� A for xt � 0,

fGTA � jxtj

1� r

� �� A for xt < 0:

8>>><>>>:

Specifically, let A ¼ð2!q1=pBð1=p, qÞÞ

p , and the pdf becomes:

fAGTðxt; p, q, rÞ

¼

1þ2Bðð1=pÞ, qÞ � jxtj

pð1þ rÞ

� �p� ��q�ð1=pÞ

for xt � 0,

1þ2Bðð1=pÞ, qÞ � jxtj

pð1� rÞ

� �p� ��q�ð1=pÞ

for xt < 0:

8>>><>>>:

As is apparent, the AGT distribution nests the GT, BT,Laplace, t, and normal distributions. (See the Appendixfor more details on this distribution.) As in the GT dis-tribution, p and q control the height and tails of thedensity. The additional parameter r controls the rate ofdescent of the density around x¼ 0. Specifically, whenr>0, the mode of the density is to the left of zero andthe distribution skews to the right, and vice versa whenr<0. When r¼ 0, the distribution is symmetric. Figure 1plots this density for several different parameterizations.

Let vt¼ (xt��)/�. The standard AGT distributionwith zero mean and unit variance has the followingprobability density function:

fSAGTðvt; p, q, rÞ

¼

� 1þ2Bðð1=pÞ, qÞ � j� � vt þ �j

pð1þ rÞ

� �p� ��q�ð1=pÞ

for � � vt þ � � 0,

� 1þ2Bðð1=pÞ, qÞ � j� � vt þ �j

pð1� rÞ

� �p� ��q�ð1=pÞ

for � � vt þ � < 0:

8>>>>>>>><>>>>>>>>:

ð3Þ

The conditional mean equation (1) and the GJRmodel (2) with the conditional distribution implied byequation (3) are jointly estimated by the full informationmaximum-likelihood (FIML) method. When estimatingthe model, we restrict p� q>2 for the variance toexist.§ The likelihood function is obtained as:

gAGTð"tjhtÞ ¼ fSAGTðvtÞ �@vt@"t

¼ fSAGT"tffiffiffiffiht

p !

�1ffiffiffiffiht

p :

yThe GT distribution does not restrict the level of kurtosis. In addition, it can be shown that the square of the skewness is less thanone plus the kurtosis [(SK)2<KUþ 1]. Therefore, a wider range for kurtosis also allows a wider range for skewness.zHansen (1994) uses the same technique to develop an asymmetric t-distribution. In a different framework, Theodossiou (1998) alsodevelops a skewed version of the GT distribution, but his distribution has four parameters and a more complicated pdf.§We use a logistic transformation � ¼ Lþ ððU� LÞ=1þ expð�!ÞÞ to set constraints on the parameters. With this transformation,even if the parameter ! is allowed to vary over the entire real line, � will be constrained to lie in the region [L,U]. Specifically, p and qare restricted to be between 0 and 50 and r is between �1 and 1.

Investor preferences and portfolio selection 257

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3. Empirical results

3.1. Data and model estimation

We focus on the current 30 components of the DJIA forboth computational tractability and the prominence ofthe index in the investment arena.y The DJIA is verywidely quoted in investment news and is among themost scrutinized indicators of U.S. stock market per-formance. It includes a wide variety of industries andis composed of blue chip stocks that are typicallyindustry leaders.z Many investors choose to invest in

these stocks in order to achieve diversification and long-term growth. We collect daily return data from the CRSPdatabase from January 1990 to December 2002, for atotal of 3280 observations for each stock.

Panel A of table 1 reports the sample mean andstandard deviation of individual daily stock returns(in percentage). The average daily return ranges from0.038% to 0.143%. The daily standard deviations aregenerally high, ranging from 1.457% to 2.915%. PanelB reports the estimated values of several key parametersin the model. In the conditional variance equation, asexpected, the estimated �s and �s are statistically signifi-cant at the 5% level and show that the volatilities of thereturns are highly persistent for all 30 stocks. In addition,the estimated GJR parameters (�s) are all negative andmost of them are statistically significant, showing thatthe return shocks have asymmetric effects on predictablevariance. Negative shocks tend to cause more volatilitiesthan do positive shocks.

Most of the estimated distributional parameters aresignificantly different from zero at the 5% level. Of parti-cular interest is the estimate of r. Recall that r governs theasymmetry of the distribution. When r>0, the distribu-tion is positively skewed. When r<0, the distribution isnegatively skewed. All the 30 estimated rs are positive,and 22 of them are significantly different from zero atthe 5% level.

To test the fit of the model, we conduct Newey’s (1985)GMM specification test by using orthogonality condi-tions implied by correct specifications. Correct specifica-tions require that the standard AGT-distributed vt bei.i.d. and have zero mean and unit variance. Therefore,we test the moment conditions E(vt)¼ 0 and E(2t )¼ 1,as well as the serial correlation in vt at lags one throughfour. Panel C of table 2 reports the t-statistics for the sixselected orthogonality conditions, along with thechi-square statistic for the joint test. The chi-square testshows that our model fits very well, with all but one(stock code JPM) of the statistics being statisticallyinsignificant at the conventional significance level.Considered individually, almost all of the six orthogon-ality conditions are insignificantly different from zero atthe 5% level, with only four exceptions.

3.2. Diversification and risk measures

To investigate how the behaviors of portfolios’ distribu-tional characteristics vary with increasing diversification,we construct portfolios containing different numbers ofstocks in the following manner. First, a stock is randomlyselected from the pool of the 30 DJIA stocks and thereturns from this one-stock portfolio are used to estimate

yNote that we are using the 30 stocks that are currently the components of the Dow Jones Index. Some of them were not in the Indexin the earlier part of the sample. As of the date this paper is written, the latest change of the components in the DJIA was on 8 April2004. Table 1 includes a list of the stock symbols.zThe industries represented by the DJIA include materials, electronics, food/beverages/tobacco, financial services, aviation/aerospace, heavy equipment, chemicals, petroleum, automobiles, retail, computer hardware/software/services, pharmaceuticals,household supplies, telecommunications, and entertainment.

−4 −3 −2 1 0 1 2 3 40.00

(a)

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50P

DF

−3 −1 0 1(b)

2 3 40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

PD

F

−2−4

Figure 1. Asymmetric Generalized Student t Density (a) forp¼ 2, q¼ 100, and: ___ r¼ 0 - - - r¼ 0.5 � � � r¼�0.5; (b) forr¼ 0, and: ___ p¼ 2 q¼ 3 (t distribution), - - - p¼ 2 andq¼ 100 (Normal), � � � p¼ 1 q¼ 100 (Laplace).

258 C. J. Hueng and R. Yau

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Table

1.Estim

atedcoefficients

andGMM

specificationtest

statistics.

(A)Sample

statistics

(B)Estim

atedcoeffi

cientsa

(C)GMM

specificationtests

t-statisticsb

2

Stock

Mean

Std

dev

��

�p

qr

(1)

(2)

(3)

(4)

(5)

(6)

Statisticc

AA

0.058

2.100

0.943*

0.071*

�0.043

2.054*

3.443*

0.132*

0.011

�0.175

0.810

0.645

0.697

�2.549*

8.510

AIG

0.074

1.747

0.933*

0.078*

�0.039*

2.028*

3.784*

0.090*

�0.149

�0.210

0.752

0.293

0.438

0.434

1.246

AXP

0.072

2.256

0.924*

0.098*

�0.062*

2.408*

2.378*

0.076*

�0.234

�0.281

0.454

0.686

�1.948

�1.239

6.916

BA

0.043

2.039

0.928*

0.075*

�0.025

2.312*

1.885*

0.052*

�0.126

�0.066

0.728

0.781

1.142

�0.534

2.643

C0.117

2.334

0.923*

0.091*

�0.077*

2.257*

2.658*

0.065*

�0.003

0.283

0.919

0.557

�0.013

�1.266

2.662

CAT

0.065

2.058

0.969*

0.043*

�0.029*

2.124*

2.359*

0.126*

�0.622

�0.632

0.276

�0.410

�1.415

�0.479

4.716

DD

0.052

1.854

0.966*

0.040*

�0.021*

2.075*

3.334*

0.099*

�0.403

�0.125

0.528

�0.001

�1.887

0.311

3.700

DIS

0.041

2.072

0.971*

0.040*

�0.031*

2.409*

1.955*

0.089*

�0.494

0.024

0.566

�1.786

�1.933

0.345

7.822

GE

0.070

1.749

0.942*

0.086*

�0.068*

2.349*

3.092*

0.054

�0.535

�0.158

0.487

�0.123

0.137

�0.002

0.600

GM

0.038

2.068

0.917*

0.086*

�0.053*

1.898*

5.224*

0.118*

�0.245

�0.236

0.825

�2.106*

�1.463

0.347

7.581

HD

0.107

2.314

0.927*

0.094*

�0.073*

2.315*

2.609*

0.055*

�0.649

0.268

0.827

0.160

�0.230

0.432

1.204

HON

0.064

2.226

0.866*

0.162*

�0.112*

2.249*

1.968*

0.047

0.237

0.410

0.690

�1.259

�1.258

�1.308

5.616

HPQ

0.082

2.752

0.972*

0.040*

�0.028*

2.446*

1.729*

0.060*

�0.389

�0.009

0.610

0.526

�0.319

�1.028

1.783

IBM

0.067

2.126

0.927*

0.101*

�0.083*

2.551*

1.474*

0.071*

�0.061

�0.868

0.558

�1.156

0.285

1.220

3.576

INTC

0.125

2.915

0.950*

0.058*

�0.038

3.007*

1.372*

0.047

�1.191

�0.637

0.046

0.128

�0.751

0.017

2.839

JNJ

0.081

1.647

0.893*

0.110*

�0.088*

2.345*

2.816*

0.052*

�0.443

0.059

0.499

0.170

�0.589

�0.620

1.099

JPM

0.072

2.446

0.932*

0.114*

�0.100*

2.001*

3.912*

0.068*

0.123

0.174

3.625*

�0.297

�1.405

�0.088

13.980*

KO

0.066

1.692

0.934*

0.083*

�0.066*

2.381*

2.363*

0.078*

�0.448

�0.108

1.299

0.013

�0.536

0.188

2.157

MCD

0.038

1.758

0.968*

0.032*

�0.014

2.332*

2.336*

0.098*

�0.580

�0.131

0.635

0.261

�1.458

�1.104

3.881

MMM

0.059

1.543

0.980*

0.021

�0.004

1.885*

2.864*

0.063*

�0.302

�0.048

1.486

0.278

0.199

�0.633

2.998

MO

0.063

2.064

0.812*

0.163*

�0.040

2.564*

1.319*

0.028

�0.983

0.362

0.780

1.049

�0.130

1.428

4.419

MRK

0.070

1.781

0.938*

0.074*

�0.062*

2.211*

2.894*

0.064*

�0.759

0.005

0.514

0.363

�0.411

0.398

1.095

MSFT

0.143

2.393

0.870*

0.132*

�0.087*

2.303*

2.688*

0.093*

�0.231

�0.061

0.858

0.720

�1.588

�0.452

4.082

PFE

0.098

1.968

0.922*

0.084*

�0.046*

2.284*

3.224*

0.051

�0.424

�0.107

0.955

�0.425

�2.092*

�1.948

8.630

PG

0.072

1.731

0.916*

0.079*

�0.047

2.351*

2.246*

0.048

�0.467

0.601

0.071

0.118

0.402

�0.639

0.850

SBC

0.047

1.842

0.917*

0.099*

�0.052*

1.779*

7.564*

0.056*

�0.119

0.027

0.576

0.747

�0.572

0.443

1.531

UTX

0.073

1.874

0.907*

0.121*

�0.085*

1.963*

3.523*

0.048

�0.178

0.513

0.931

0.552

�0.136

�0.397

1.619

VZ

0.043

1.799

0.944*

0.074*

�0.045*

2.049*

3.886*

0.069*

�0.067

0.055

0.306

0.114

�0.052

�0.447

0.292

WMT

0.090

2.053

0.944*

0.071*

�0.042*

2.150*

3.681*

0.073*

�0.052

�0.138

0.447

�0.442

0.037

0.213

0.411

XOM

0.056

1.457

0.934*

0.073*

�0.034*

2.073*

4.350*

0.047

�0.031

0.077

1.154

0.008

�0.157

0.204

1.570

Theasterisk*indicatessignificance

atthe5%

level.

(a)Theparametersare

forequations(2)and

(3).

Allcomputationswereperform

edusingtheGAUSS

MAXLIK

module.Theestimated

standard

errors

werecalculated

with

therobust

standard

errors

correspondingto

resultssummarizedin

Greene(2003,p.520).

(b)Thenullhypotheses

are

(1)E(v

t)¼0,(2)Eðv

2 t�1Þ¼

0,(3)E(v

t�v t�1)¼0,(4)E(v

t�v t�2)¼0,(5)E(v

t�v t�3)¼0,and(6)E(v

t�v t�4)¼0,wherev tisthestandardized

residualfrom

theAR-G

ARCH-A

GTmodel.

(c)Thisisajointtestforthenullhypothesisthatthemodeliscorrectlyspecified

basedonthemomentconditions.Ithasa2(6)distribution.Thecovariance

matrix

oftheorthogonality

conditionsiscalculatedusing

theconsistentestimatorproposedbyNew

eyandWest(1987,1994)withaBartlettkernel.

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the model. Another stock randomly chosen from theremaining 29 stocks is then added to form a two-stockportfolio. The arithmetic average of the returns of thesetwo stocks is used to estimate the model. We keep addingone stock at a time to form 30 portfolios with sizes ran-ging from one to 30, where the portfolio with size n isa subset of the portfolio with size nþ 1.

Several distributional characteristics with risk impli-cations are next calculated from the estimations: theunconditional variance, the skewness parameter r, andthe 5% and 95% quantiles.y Since we do not assume theexistence of higher moments beyond the variance, theusual measures of skewness and kurtosis may not exist.However, our parametric model provides all the informa-tion needed for the distribution. The parameter rmeasuresthe asymmetry of the distribution directly, and the 5% and

95% quantiles show the thickness of the tails. These riskmeasures are plotted against the portfolio sizes. We callthese curves the ‘diversification structures of risk’. Bydoing this we are able to observe how a risk measure varieswith the portfolio size (the degree of diversification). Forexample, if the plot is downward-sloping, then the riskmeasure is decreasing with diversification.

The whole process is repeated 100 times to yield 100simulated samples. Therefore, there are 100 simulatedcurves for each risk measure. Note that the results forthe 30-stock portfolio are identical across these 100simulations. These curves are plotted in figures 2(a)–(d),which show a similar pattern: the absolute value of theslope is very big when the portfolio size is small, quicklydeclines as the portfolio size is approaching 10, and staysalmost unchanged afterward.

yThere are two approaches to calculate the distributional characteristics of risks for a portfolio’s returns. The first method is toapply estimation schemes to the portfolio returns directly. This is the method employed in this paper and in Campbell et al. (2001).The second approach is to compute the distributional characteristics for the portfolio returns from a multivariate model, in whichthe dependence structure among individual stock returns needs to be identified beforehand. Examples of this are Guidolin andTimmermann (2006) and Malevergne and Sornette (2004).

Table 2. Estimation results for equation (4): Ri,n¼ aiþ bi � (1/exp(n))þ ci � (n/exp(n))þ ei,n; DJIA stocks.

Risk measure Variance Skewness 5% quantile 95% quantile 1% VaR 5% VaR(A) Estimation results

Avg R2 0.880 0.758 0.805 0.857 0.683 0.651

Average of estimated coefficients and their t-statistics (in parentheses)

�̂a 1.587 (116.4) �0.019 (�19.27) �1.975 (�225.5) 2.085 (243.5) 3.311 (246.1) 2.085 (224.2)�̂b 6.308 (9.917) �0.033 (�1.900) �1.860 (�8.706) 2.147 (9.422) 1.042 (2.110) 0.230 (0.726)

�̂c 1.460 (3.891) 0.273 (16.78) �1.179 (�6.541) 1.525 (7.688) 3.017 (6.379) 2.340 (7.456)

(B) Diversifiable and diversified risk

Average risk measure for n ¼ 1 ðin percentagesÞ : �̂aþ

�̂bþ �̂c

expð1Þ

!

4.445 0.069 �3.093 3.436 4.805 3.030

Diversifiable risk:

�̂bþ �̂c

expð1Þ

2.858 0.088 �1.118 1.351 1.493 0.945

n Diversified risk:

�̂bþ �̂c

expð1Þ�

�̂b �

1

expðnÞþ �̂c �

n

expðnÞ

� �( )

2 1.609 0.019 �0.547 0.648 0.536 0.2813 2.326 0.049 �0.849 1.016 0.991 0.5844 2.635 0.069 �0.997 1.200 1.253 0.7705 2.766 0.079 �1.066 1.285 1.385 0.8656 2.820 0.084 �1.096 1.323 1.446 0.9107 2.843 0.086 �1.109 1.339 1.473 0.9308 2.852 0.087 �1.114 1.346 1.485 0.9399 2.855 0.088 �1.116 1.349 1.490 0.94310 2.857 0.088 �1.117 1.350 1.492 0.94411 2.857 0.088 �1.118 1.351 1.493 0.94512 2.857 0.088 �1.118 1.351 1.493 0.94513 2.858 0.088 �1.118 1.351 1.493 0.94514–30 2.858 0.088 �1.118 1.351 1.493 0.945

Note: All risk measures, except for skewness, are in percentage.

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The risk measures apparently are all non-linearlyassociated with portfolio sizes. Therefore, to model thediversification structure of risk, we propose the followingspecification:

Ri, n ¼ ai þ bi �1

expðnÞþ ci �

n

expðnÞþ ei, n, ð4Þ

where R is one of the four risk measures, i¼ 1, . . . , 100 isthe sample index, and n¼ 1, . . . , 30 is the portfolio size.yAccording to this specification, the risk measure has anasymptote a. The second term 1/exp(n) decays rapidly as nincreases. The third term n/exp(n) also decays rapidly,but not as fast as the second term. Hence, the speed

of decay depends on the relative magnitudes of theestimated b and c. This specification allows monotonic,humped, or S shapes, depending on the values of band c. According to this structure, the ‘diversifiable’risk is (bþ c)/exp(1), and (bþ c)/exp(1)� [b(1/exp(n))þc(n/exp(n))] shows the diversified risk by holding n stocks.

In the first four columns of Panel (A) in table 2, wereport (for each risk measure) the average R-squared andestimated coefficients over 100 samples. The model fitsquite well as the average R-squareds are high and almostall the average coefficients, with only one exception,are statistically significant at the conventional level.Figures 3(a)–(d) plot the estimated diversification struc-ture of risk, along with its upper and lower one-standard

yThis specification has been used in the literature on yield curves. See, for example, Nelson and Siegel (1987).

(a)

(c) (d)

(b)

Figure 2. Risk measures against portfolio sizes (100 samples) (DJIA).

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deviation bounds, for each risk measure using theaverages of the estimated coefficients. It can be seenthat, first of all, there is a reduction tendency in returnvariance and return skewness as the portfolio sizeincreases. This is consistent with the findings in previousstudies using sample raw moments. Based on the fact thatinvestors prefer lower return variance and more positiveskewness, there exists a trade-off between lower varianceand higher skewness when the portfolio is diversified.Second, both tails in the distribution are thicker whenthe portfolio is less diversified, as shown in the resultsfor the 5% and 95% quantiles.

The first four columns of Panel (B) in table 2 report thediversifiable risk and the diversified risk by diversifyingthe portfolio with n stocks. The first, third, and fourthcolumns show that over 90% of the diversifiable risk canbe diversified (the benefit of diversification) by holdingfive stocks in the portfolio. The second column shows

that 90% of the diversifiable skewness is reduced (the

cost of diversification) by holding five stocks. The gains

or losses from diversification all vanish very quickly and

are ignorable when the portfolio size is greater than 10.

3.3. Value at risk analysis

The goal of a quality risk system is to generate validforecast distributions to enhance executive decision-making. Hence, most risk systems require models togenerate valid ex-ante estimates of the forecast distribu-tion, especially the left tail of the distribution. Recentresearch on Value-at-Risk (VaR) indicates that investorsmay be strongly concerned about the likely maximumloss, i.e. the left tail of the distribution of a portfolio’sexpected returns. The evaluation of such large losses is atopic of increasing importance to investors operating in

(a) (b)

(c) (d)

Figure 3. Estimated diversification structure of risk (DJIA).

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today’s tumultuous market environment. One majoradvantage of the proposed AGT distribution in thispaper is its flexibility in estimating this risk measure. Ofparticular interest to us is the relationship between thismaximum loss concern and the diversification strategy.

The estimation results from the previous subsection areused to calculate the 1% and 5% conditional VaRs fora one-day horizon.y Figures 4(a)–(b) plot the estimateddiversification structure of risk along with its upper andlower one-standard deviation bounds. The last twocolumns of table 2 report the regression results basedon equation (4). It is shown that the expected extremelosses tend to be smaller as the portfolio size increases.The average expected maximum 1% and 5% daily lossesare 4.805% and 3.030%, respectively, for a single-stockportfolio. The diversifiable losses are 1.493% and0.945%, respectively. The average gains of diversifyingthe portfolio to a two-stock fund are 0.536% and0.281%, respectively. More than 90% of the diversifiablelosses are gone with a five-stock portfolio. The reductionin the extreme losses fades away as the portfolio size goesbeyond five.

4. Alternative data for robustness check

4.1. Market pool

Even though the DJIA stocks are the most popularstocks held by investors and provide a well-diversifiedportfolio, one may argue that they still do not providea representative picture of the market. To test the

robustness of our conclusions, we replicate the empiricalanalysis for a more general set of stocks. This data setconsists of all NYSE and AMEX firms that have com-plete return data (no missing value) during the sampleperiod (January 1990 to December 2002). We follow theconventional method and exclude stocks that do not havea CRSP share code of 10 or 11, i.e. we only includeordinary common shares and exclude REITs, closed-endfunds, primes, and scores. There are a total of 457 stocksin this pool.

The random selection methods used before areapplied to this new pool of data. We extend the maximumportfolio size to 50 to see whether our conclusions changewhen the portfolios include more than 30 stocks.The simulation results analogous to those in figures 2–4are plotted in figures 5–7. Table 3 shows that the resultsare analogous to those in table 2. As expected, thesimulations from the market pool yield a wider range ofeach risk measure. However, the qualitative results areconsistent with those from the DJIA data set. First, theabsolute value of the slope is very big when the portfoliosize is small, quickly declines as the portfolio size isapproaching 10, and stays almost unchanged afterward,even beyond the portfolio size of 30. Our model of diver-sification structure of risk still fits quite well.

Second, variance and skewness decline as the portfoliosize increases, i.e. there exists a trade-off between lowervariance and higher skewness when the portfolio is diver-sified. In addition, both tails in the distribution are thickerwhen the portfolio is less diversified, and the expectedextreme losses (VaR) tend to be smaller as the portfoliosize increases.

yWe also use weekly data to estimate the 1% and 5% conditional VaRs at the weekly horizon. All qualitative patterns observedbased on the daily data are well preserved in the weekly data. Specifically, the average expected maximum 1% and 5% weekly lossesare 10.455% and 6.692%, respectively, for a single-stock portfolio. The diversifiable losses are 3.393% and 2.019%, respectively.More than 92% of the diversifiable losses are gone with a five-stock portfolio.

(a) (b)

Figure 4. (a) Estimated diversification structure of risk (1-day horizon 1% VaR) (DJIA); (b) estimated diversification structure ofrisk (1-day horizon 5% VaR) (DJIA).

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Finally, 90% of the diversifiable risk can bediversified (the benefit of diversification) by holding fiveto six stocks, and 90% of the diversifiable skewness isreduced (the cost of diversification) by holding six stocksin the portfolio. The gains or losses from diversificationall vanish very quickly and are ignorable when theportfolio size is greater than 10. Therefore, extendingthe sample to the market pool does not change ourconclusions.

4.2. Characteristics-based diversification

Investors can achieve diversification by investing in theDJIA stocks, because the DJIA includes a wide variety ofindustries. However, these stocks are typically industryleaders and have high market capitalizations and tradingvolumes. Studies such as Campbell et al. (2001) suggest

that returns on small firms are more volatile and there-fore, to reduce the higher risk (measured by volatility) ofsmall firms, it may be optimal to hold 20–30 stocks in aportfolio. This suggestion clearly is not supported by oursimulations of industry-wise diversification. However, itwould be interesting to see whether diversification strate-gies based on stock characteristics such as firm sizes andliquidities would yield results different from those withindustry-wise diversification. This section experimentswith these two alternative strategies to test the robustnessof our conclusions.

We first rank those 457 stocks used in the previoussection according to their average sizes (market values)over the sample period. The seven stocks with the smallestsizes are dropped and the rest are put into 30 size-basedgroups, ranging from the smallest-size group to thebiggest-size group, with each group including 15 stocksof similar sizes. We then construct portfolios containing

(a) (b)

(c) (d)

Figure 5. Risk measures against portfolio sizes (100 samples) (market pool).

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different numbers of stocks in the following manner.

First, a stock is randomly selected from the smallest-size

group and the returns from this one-stock portfolio are

used to estimate the model.y Next, a stock is randomly

selected from the second-to-the-smallest-size group and

added to form a two-stock portfolio. The arithmetic aver-

age of the returns of these two stocks is used to estimate

the model. We keep adding one stock at a time from

a bigger size group to form 30 portfolios with portfolio

sizes ranging from one to 30, where the portfolio with

size n is a subset of the portfolio with size nþ 1.

The thirtieth stock is randomly selected from the

biggest-size group. Therefore, diversification is achieved

by adding stocks from bigger size groups. The whole

process is again repeated 100 times to yield 100 simulated

samples, and the diversification structure of risk model is

estimated using these 100 samples. Figures 8 (a)–(f) plot

the estimated diversification structure of risk for the six

risk measures.The second diversification strategy we experiment with

in this section is based on the liquidity of the stocks,

which is measured by the turnover ratios (trading volume

yWe select stocks from size-ordered groups (from the smallest to the biggest), because we want to see the diversification effect whenbigger-size stocks are added to the portfolio, in an attempt to compare our results with those from studies such as Campbell et al.(2001). A more general approach is to randomly select a size-group, randomly select another size-group from the other 29 groups,and so on. We also experiment with this more general approach and obtain similar results. The same sampling approach is alsoapplied to the liquidity-based strategy experimented upon later. The results from both experiments do not change our conclusionsand are available from the authors upon request.

(a) (b)

(c) (d)

Figure 6. Estimated diversification structure of risk (market pool).

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Table 3. Estimation results for equation (4): Ri,n¼ aiþ bi � (1/exp(n))þ ci � (n/exp(n))þ ei,n; market pool.

Risk measure Variance Skewness 5% quantile 95% quantile 1% VaR 5% VaR(A) Estimation results

Avg R2 0.879 0.601 0.818 0.843 0.676 0.647Average of estimated coefficients and their t-statistics (in parentheses)

�̂a 1.142 (70.427) �0.050 (�32.268) �1.687 (�139.910) 1.731 (151.558) 2.863 (106.910) 1.801 (105.085)�̂b 9.963 (5.973) �0.374 (�13.888) �1.828 (�4.556) 1.811 (3.962) �1.962 (�1.824) �1.719 (�2.613)

�̂c 3.797 (4.607) 0.686 (25.250) �3.168 (�9.981) 4.104 (11.626) 8.838 (9.418) 5.797 (10.036)

(B) Diversifiable and diversified risk

Average risk measure for n ¼ 1 ðin percentagesÞ : �̂aþ

�̂bþ �̂c

expð1Þ

!

6.204 0.065 �3.525 3.907 5.393 3.302

Diversifiable risk:

�̂bþ �̂c

expð1Þ

5.062 0.115 �1.838 2.176 2.530 1.500

n Diversified risk:

�̂bþ �̂c

expð1Þ�

�̂b �

1

expðnÞþ �̂c �

n

expðnÞ

� �( )

2 2.686 �0.020 �0.733 0.820 0.403 0.1643 3.999 0.031 �1.274 1.473 1.307 0.7204 4.601 0.071 �1.572 1.842 1.918 1.1075 4.867 0.094 �1.719 2.026 2.245 1.3176 4.981 0.105 �1.786 2.111 2.403 1.4187 5.029 0.111 �1.816 2.148 2.475 1.4658 5.049 0.113 �1.829 2.165 2.507 1.4859 5.057 0.114 �1.834 2.171 2.520 1.49410 5.060 0.114 �1.837 2.174 2.526 1.49811 5.061 0.115 �1.837 2.175 2.528 1.49912 5.062 0.115 �1.838 2.176 2.529 1.50013–50 5.062 0.115 �1.838 2.176 2.530 1.500

Note: All risk measures, except for skewness, are in percentage.

(a) (b)

Figure 7. (a) Estimated diversification structure of risk (1-day horizon 1% VaR) (market pool); (b) estimated diversificationstructure of risk (1-day horizon 5% VaR) (market pool).

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(a) (b)

(c) (d)

(e) (f)

Figure 8. Estimated diversification structure of risk (size-based diversification).

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(b)(a)

(d)(c)

(f)(e)

Figure 9. Estimated diversification structure of risk (liquidity-based diversification).

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divided by shares outstanding). We rank those 457 stocksaccording to their average turnover ratios over the sampleperiod and conduct simulations similar to what wehave done in the size-based diversification strategy.Figures 9 (a)–(f) plots the estimated diversificationstructure of risk for those six risk measures. Figures 8and 9 show that our conclusions are not changed byusing different diversification strategies based on stockcharacteristics: variance, skewness, kurtosis, and extremelosses all decrease with increasing diversification, andthe gains or losses from diversification all vanish veryquickly and are ignorable when the portfolio size isgreater than 10.

5. Discussion and conclusion

This paper addresses additional dimensions in theanalysis of portfolio diversification and risk. In additionto the conventional measures of risk, namely variance andskewness, we propose a parametric model to estimate thewhole distribution of asset returns and investigate therelationships between diversification and other distribu-tional characteristics that have risk implications.

Our results indicate that variance, kurtosis, andextreme losses decrease with increasing diversification.As such, the goal of an investor who wants to decreasevariance and extreme losses would be to hold a more-diversified portfolio. However, portfolio skewness alsodecreases with increasing diversification. Therefore, aninvestor who desires higher skewness would choose tohold a less-diversified portfolio. These results indicatethat an investor’s strategy of diversification depends onwhich risk measure is the main concern.

Our results also shed some light on the empirical puzzlethat investors do not tend to hold fully diversified portfo-lios. A possible explanation is that most investors do notweigh downside risk equally with upside potential andprefer more positively skewed returns over low-returnvariance and extreme losses. Furthermore, taking intoaccount the transaction cost and information costembedded in managing a more diversified portfolio,investors are likely not to hold more than 10 stocks intheir portfolios since the benefit of diversification beyond10 stocks is limited. The cost outweighs the benefit ofa highly diversified portfolio.

Possible extensions of this paper are considered. First,the literature on the risk–return tradeoffs using a three-moment CAPM indicates that stocks which decrease theskewness of a portfolio should have higher expectedreturns, that is, stocks with lower systematic skewness(i.e. coskewness with the market portfolio) outperformstocks with higher systematic skewness (see, for example,Kraus and Litzenberger (1976) and Harvey andSiddique (2000)). In other words, investors prefer higher

systematic skewness. Idiosyncratic skewness, on the otherhand, does not affect expected returns. Therefore, itwould be interesting to decompose the skewness measureinto systematic skewness and idiosyncratic skewness andsee whether the former decreases as the portfolio sizeincreases, i.e. diversification decreases coskewness.yUnfortunately, we are unable to do this decompositionusing the current parametric model, because it wouldrequire a bivariate AGT distribution to jointly modelthe individual stock return and the market return,which is beyond the scope of this paper. We leave thistask to future studies.

Another possible extension of the paper is to analyzethe actual holdings of stocks by investors in the market,rather than using the simulation method proposed in thispaper. This would provide us with evidence from the realworld, rather than from counterfactual simulations. Forexample, a unique data set used by Barber and Odean(2000) and Mitton and Vorkink (2006), which consistsof the investments of 78,000 households from January1991 to December 1996, would serve this purpose. Sincethis data set is not publicly available, we leave thisextension to researchers who have access to it.

Technical appendix: specifics of the asymmetric

generalized t distribution

The generalized t (GT) distribution in McDonald andNewey (1988) has the following pdf:

fGTðxt;!, p, qÞ ¼p

2!q1=pB ð1=pÞ, qð Þð1þ ðjxtjp=q!pÞÞ

qþð1=pÞ,

where B(�) is the beta function. To transform thissymmetric distribution to an asymmetric one, we define:

hðztÞ ¼

fGTjztj

1þ r

� �for zt � 0,

fGTjztj

1� r

� �for zt < 0,

8>>><>>>:

where �1<r<1. This specification allows different ratesof descent for zt� 0 and zt<0. Next, we scale zt by defin-ing xt¼ zt/A, where A is a scaling constant in order tosimplify the notation. The density function of the asym-metric generalized t (AGT) distribution can be written as:

fAGTðxtÞ ¼ hðztÞ �@zt@xt

¼ hðA � xtÞ � A

¼

fGTA � jxtj

1þ r

� �� A for xt � 0,

fGTA � jxtj

1� r

� �� A for xt < 0:

8>>><>>>:

y In a more recent strand of the literature, Barberis and Huang (2005) and Kumar (2005) argue that models with cumulativeprospect-theoretic preferences imply that idiosyncratic skewness should be priced as well. In this case, the relationship betweendiversification and idiosyncratic skewness is also an interesting topic.

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Specifically,

Let A¼(2!q1/pB((1/p),q))/p; as such, the pdf becomes:

fAGTðxt; p, q, rÞ

¼

f1 ¼ 1þ 2B ð1=pÞ, qð Þ �jxtjpð1þrÞ

pn o�q�ð1=pÞ

for xt � 0,

f2 ¼ 1þ 2B ð1=pÞ, qð Þ �jxtjpð1�rÞ

pn o�q�ð1=pÞ

for xt < 0:

8><>:

The nth raw moment of xt is:

Mn ¼

Z 1

�1

xnfAGTðxÞdx

¼

Z 1

0

xnf1ðxÞdxþ

Z 0

�1

xnf2ðxÞdx

¼

Z 1

0

xnf1ðxÞdxþ ð�1ÞnZ 1

0

xnf2ðxÞdx:

Using the formula

Z 1

0

xnð1þk �xpÞ�mdx¼Bnþ1

p,m�

nþ1

p

� �p �kðnþ1Þ=p �1

and letting

k1 ¼2B ð1=pÞ, qð Þ

pð1þ rÞ

� �p

, k2 ¼2B ð1=pÞ, qð Þ

pð1� rÞ

� �p

,

and m¼ qþ (1/p), the nth raw moment becomes:

Mn ¼ Bnþ 1

p,m�

nþ 1

p

� �p � k

ðnþ1Þ=p1

�1

þ ð�1ÞnBnþ 1

p,m�

nþ 1

p

� �p � k

ðnþ1Þ=p2

�1

¼ ð1þ rÞnþ1pnBnþ 1

p,q�

n

p

� �2B

1

p,q

� �� ��n�1

þ ð�1Þnð1� rÞnþ1pnBnþ 1

p,q�

n

p

� �2B

1

p,q

� �� ��n�1

¼ ð�1Þnð1� rÞnþ1þ ð1þ rÞnþ1

� �pnB

nþ 1

p,q�

n

p

� �

� 2B1

p,q

� �� ��n�1

:

Therefore, the mean is

� � M1 ¼ 4rp � B2

p, q�

1

p

� �2B

1

p, q

� �� ��2

;

the variance is �2� M2 � �2; the skewness (SK) is:

Eðx� �Þ3

�3¼

E x3 � 3�x2 þ 3�2x� �3�

�3

¼M3 � 3�ð�2

þ �2Þ þ 3�2�� �3

�3

¼M3 � 3��2

� �3

�3;

and the kurtosis (KU) is

Eðx� �Þ4

�4¼

Eðx4 � 4�x3 þ 6�2x2 � 4�3xþ �4Þ

�4

¼M4 � 4�M3 þ 6�2M2 � 4�3�þ �4

�4

¼M4 � 4�ðM3 � 3��2

� �3Þ � 6�2�2

� �4

�4

¼M4 � 4� � SK � �3

� 6�2�2� �4

�4:

Note that

Prðx � 0Þ ¼

Z 1

0

f1ðxÞdx ¼

Z 1

0

x0f1ðxÞdx

¼ ð1þ rÞB1

p, q

� �2B

1

p, q

� �� ��1

¼1þ r

2:

Similarly,

Prðx < 0Þ ¼

Z 0

�1

f2ðxÞdx ¼1� r

2,

which shows the asymmetry of the distribution whenr 6¼ 0. Furthermore, Pr(x� 0)þPr(x<0)¼ 1 confirmsthat fAGT is a proper pdf.

Acknowledgements

We are grateful to the editor, two anonymous referees,and the participants in the 2003 Conference ofHigh-Frequency Financial Data in Taipei for helpfulcomments. Naturally, all remaining errors are ours. Yauacknowledges the research support from the NationalScience Council of the Republic of China (NSC93-2415-H-008-007).

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fAGTðxt;A,!, p, q, rÞ ¼

p � A

2!q1=pB ð1=pÞ, qð Þð1þ ðA � jxtj=ð1þ rÞÞpð1=q!pÞÞqþð1=pÞ

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p � A

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for xt < 0:

8>>><>>>:

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