Performance of Risk Measures in Portfolio Construction on Central and South-East European Emerging Markets

doi:10.4236/ajor.2011.14027

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American Journal of Oper ations Research, 2011, 1, 236-242

doi:10.4236/ajor.2011.14027 Published Online December 2011 (http://www.SciRP.org/journal/ajor)

Performance of Risk Measures in Portfolio Construction

on Central and South-East Eur opean Emerging Markets

Jelena Vidovic

Department of Fi na nce , University Centre for Professional Studies, Split, Croatia

E-mail: [email protected]

Received May 24, 2011; revised June 30, 2011; accepted July 8, 2011

Abstract

Aim of this paper is to characterize different risk measures in portfolio construction on seven Central and

South-East European stock markets; Slovenia, Croatia, Hungary, Poland, Chez Republic, Romania and Tur-

key. Selected countries are members of EU, except Croatia and Turkey which have candidate status. Em-

pirical part of this paper consists of three stages; at first descriptive statistics on stock returns was performed,

afterwards different risk measures were employed in portfolio construction and in the last part, portfolios

were tested in the out-of-sample period. Results indicate presence of extreme kurtosis and skewness in stock

return series. Resulting portfolios incorporate stocks with extremely high kurtosis and stocks with negative

skewness. Portfolio construction based only on risk and return results in major exposure to extreme returns

and unsatisfactory portfolio out-of-sample results.

Keywords: Alternative Risk Measures, Central and South-East European Emerging Markets, Portfolio,

Skewness, Kurtosis

1. Introduction

Transition economies in Central and South-Eastern

Europe (CSEE) represent very attractive investment area

for foreign investors. In the past few years these stock

markets witnessed tremendous growth both in number of

listed securities as well as in market capitalization. In

2007 stock market indices grew tremendously. In 2007

value of stock market index on Zagreb Stock Exchange

rose 63.20%, Slovene stock market index rose 71.0%,

while German stock market index rose 22.3%. Risk and

illiquidity of stocks are the main problem on these mar-

kets especially in the recent period when the global

economy was struck by financial crisis. Aim of this pa-

per is to define characteristics of stock returns on CSEE

equity markets. Portfolios of stocks will be formed on

these markets using five risk measures: variance, semiva-

riance, lower partial moment when target return is equal

to 0, Mean Absolute Deviation and Conditional Value at

Risk. Out-of-sample analysis of formed portfolios char-

acterizes selected risk measures and their performance in

presence of kurtosis and skewness.

Paper is organized as follows: Section 2 presents a

brief review of previous researches. Risk measures

employed in forming portfolios are presented in Sec-

tion 3. Section 4 discusses data and methodology. Sec-

tion 5 presents results of empirical analysis. Results

include portfolio composition and their out-of-sample

performance. In Section 6 main conclusions were dr-

awn.

2. Previous Researches

The analysis of emerging capital markets has increased

substantially in the recent years, however many studies

had failed to take into account the characteristics of

emerging markets in their analysis. The underlying as-

sumption of the standard mean-variance model is that

stock returns must be normally distributed, however this

is in direct contradiction with the empirical evidence

concerning the distribution of emerging markets returns.

[1-5] concluded that Central and South-Eastern European

equity markets have fat tails indicating presence of many

extreme observations. According to [3] who examined

the use of downside risk measures in construction of an

optimal portfolio, the use of downside risk measures re-

sults in significant improvement in the out-of-sample

performance of those portfolios. [6] examined benefits of

diversification into three leading Central European eq-

uity markets using lower partial moment in the presence

237

J. VIDOVIC

of nonnormality of returns on those markets. Their study

shows that investors could benefit diversifying into Cen-

tral European equity markets. These results are supported

by the relatively low short term correlations as well as

the lack of cointegration between these markets and de-

veloped equity markets, their prospects for future eco-

nomic growth and positive impact associated with their

recent accession to the EU. Similar conclusion was

brought in [7] where EU accession was highlighted as

the key contributor to the reduction of risk on these mar-

kets. [8] indicates that South-Eastern Europe emerging

markets are loosely related in periods of normal eco-

nomic activity while in conditions of economic recession

they exhibit strong interrelationship. [9] concluded that

return distribution appears to be leptokurtic for all Euro

stock markets. Investors are more exposed to the risk

since the distribution of returns has a greater exposure to

outlier events and bias to the downside.

3. Risk Measures

Many researches had their idea of the “best” risk meas-

ure to be applied in the portfolio selection; mean lower

partial moment [10-12], Mean-absolute deviation (MAD)

[13] and Conditional Value-at-Risk (CVaR) [14]. This

study involves empirical analysis of most important risk

measures proposed and compared in literature in con-

struction of optimal portfolios on selected markets.

Variance which is by its definition measure of disper-

sion considers the positive and negative deviations from

the mean as potential risk.

Variance of security is defined as:



iii

ExE x



 (1)

where 2



-variance of security , i

is random return

on security , and is expectation operator.

In the case of variance, over-performance relative to

the mean is penalized just as much as under-perfor-

mance. In order to overcome this anomaly [10] pro-

posed semivariance as risk measure. The natural exten-

sion of the semivariance [11,12] is the lower partial

moment risk.

Semivariance is a statistical measure equal to sum of

square deviations from the mean, taking into account

only observations below the mean:

 



iii

Exx Ex









, (2)

Semivariance = (3)

()







Two alternative downside risk measures are examined

in this paper. The first uses the mean as target rate, the

mean semivariance, and the second uses a target rate of

zero. These two risk measures are denoted as LPMM and

LPM0. Both measures compute risk using only returns

below the mean return or alternatively below a target

return. In the presence of negative skewness in a return

series downside returns will occur in larger magnitudes

than upside returns, the opposite is true in the presence of

positive skewness.

The absolute deviation of random variable is expected

absolute value of difference between the random variable

and its mean. [13] proved that minimizing Mean Abso-

lute Deviation (MAD) is similar to minimizing variance

if stock returns are multivariate normally distributed.

Mean absolute deviation can be calculated using follow-

ing expression:



MAD ii

Ex Ex









 (4)

The Conditional Value-at-Risk (CVaR) can be ex-

plained trough Value-at-Risk (VaR) [15]. Following

equation defines conditional expectation in the lower tail

of the distribution of returns and is equal to average of

returns beyond VaR at level







CVaRxEx xVaRx





 



(5)

Unlike Value-at-Risk, the CVaR is coherent risk

measure [16,17].

4. Data and Methodology

Analysis includes seven stock markets form CSEE re-

gion, five countries are members of EU: Poland, Czech

Republic, Hungary and Romania, while two; Croatia and

Turkey have candidate status. On every market ten

stocks [18,19] from correspondent stock index were se-

lected. Data series consists of 500 daily closing prices for

each security in time period from November 2007 until

the end of October 2009. Stocks were selected according

to following criteria: stock was listed before 2007 and all

stock prices were available in period from November

2007 until October 2009, selected stocks are members of

national stock index which includes best stocks in the

country irrelevant which sector they belong. Stocks

which have the biggest share in construction of national

index have advantage.

Series of daily logarithmic returns for each stock were

calculated. In order to determine whether the stock re-

turns follow the normal distribution in this paper are

presented results of descriptive statistics (mean, standard

deviation, skewness and kurtosis) and normality tests.

Normality tests conducted in this paper are Shapiro

Wilk (W test) and Kolmogorov-Smirnov (K-S) D test

[20].

J. VIDOVIC

238

Using five risk measures for each stock market portfo-

lios were estimated. Estimated portfolios were analyzed

in the out-of-sample period from November 2008 until

October 2009. For each estimated portfolio Cumulative

Abnormal Returns (CAR-s) against German stock index

DAX [6] were calculated. Standard event study method-

ology was used, abnormal daily returns of each portfolio

against DAX index were calculated:

Romania, Chez Republic and Croatia. As expected the

standard deviation indicates high level of risk in the

CSEE markets. According to results of normality tests

almost all 80 observed CSEE stocks do not pass nor-

mality test. Descriptive statistics indicates that all st-

ocks have negative returns what is expected due pres-

ence of crisis, kurtosis is always greater than zero and

statistically significant indicating fat tails and presence

of many extreme observations. According to data from

Table 2 and Table 5, stocks from Turkey and Poland

have lowest kurtosis coefficients which do not exceed 2,

while values of kurtosis coefficients from Table 1, Ta-

ble 3, Table 4, Table 6 and Table 7 show that all

stocks from Croatia, Romania, Slovenia, Chez Republic

and Hungary have very high kurtosis coefficients.

Similar conclusion can be drawn by observing results of

normality tests; only two stocks from Poland pass W

test and only one stock from Turkey passes KS normal-

ity test. Nonnormality, extreme returns and high kurto-

sis are rather rule than exception when observing stock

returns in emerging CSEE markets. Correlation coeffi-

cients of all stocks are very high and positive indicating

existence of crisis on capital markets giving little space

for diversification. This situation is characteristic for all

capital markets in the region.

itit mt

Rrr (6)

where it is daily return () for portfolio and is

the appropriate benchmark return.

rtimt

The CAR from the beginning of the first day until the

last day of trading is the summation of abnormal returns.

CAR AR



t

(7)

CARs were estimated assuming benchmark was the

portfolios normal return.

5. Empirical Results

Selected markets could be divided in two groups depen-

dently on risk level measured by standard deviation;

countries with lower level of risk are: Poland, Turkey,

Hungary and Slovenia while higher risks can be found in

Table 1. Results of descriptive statistics for stocks from Croatia and their portfolio weights.

Descriptive statistics ADGR ATPL HT DLKVIGHZERNTULPL PODR INGRKRAS

SKEWNESS 0.317* –0.218 0.425* 1.043* 1.131*–0.107 –0.701* –0.074 0.803*0.216

KURTOSIS 5.064* 5.982* 6.462* 10.44* 8.604*2.047*13.053* 3.451* 9.753*2.69*

Resulting portfolios

MV, LPMM, LPM0, CVaR, MAD0.00% 0.00% 69.16%0.00% 0.00%0.00%0.00% 30.84% 0.00%0.00%

*Denotes statistical significance at 5% level.

Table 2. Results of descriptive statistics for stocks fr om Poland and their portfolio weights.

Descriptive statistics TPSA PKN SOBK GTC PGN KGHM PKOBP PBG BZWBAPE

SKEWNESS 0.244 –0.107 0.658* 0.1520.007–0.553* –0.116 –0.077 0.0780.017

KURTOSIS 1.335* 1.358* 10.034*1.752*0.775*7.307* 1.319* 0.188 1.681*0.831*

Resulting portfolios

MV 57.22% 0.00% 38.59%0.00%4.19%0.00% 0.00% 0.00% 0.00%0.00%

LPMM 55.72% 0.00% 41.22%0.00%3.06%0.00% 0.00% 0.00% 0.00%0.00%

LPM0 55.30% 0.00% 41.94%0.00%2.75%0.00% 0.00% 0.00% 0.00%0.00%

CVaR 56.65% 0.00% 39.60%0.00%3.76%0.00% 0.00% 0.00% 0.00%0.00%

MAD 51.63% 0.00% 48.37%0.00%0.00%0.00% 0.00% 0.00% 0.00%0.00%

*Denotes statistical significance at 5% level.

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J. VIDOVIC

Table 3. Results of descriptive statistics for stocks fr om Romania and their portfolio weights.

Descriptive statistics SNPP BRDX ROMPAZOMTSEL BATR ALRO ATBE SCDBBCCA

SKEWNESS –0.571*–0.228 –0.475*0.242 0.255 –11.15* –0.467*–0.563* 0.195 –0.414*

KURTOSIS 5.129* 3.797* 5.625* 1.341* 3.565* 153.136* 3.742* 4.94* 4.244* 5.979*

Resulting portfolios

MV 0.00% 0.00% 0.00% 39.18%0.00% 0.00% 22.26%0.00% 0.00% 38.55%

LPMM 0.00% 0.00% 0.00% 39.65%0.00% 0.00% 24.28%0.00% 0.00% 36.07%

LPM0 0.00% 0.00% 0.00% 39.62%0.00% 0.00% 24.15%0.00% 0.00% 36.24%

CVaR 0.00% 0.00% 0.00% 40.06%0.00% 0.00% 26.07%0.00% 0.00% 33.87%

MAD 0.00% 0.00% 0.00% 34.03%0.00% 0.00% 0.00% 0.00% 0.00% 65.97%

*Denotes statistical significance at 5% level.

Table 4. Results of descriptive statistics for stocks fr om Slovenia and their portfolio weights.

Descriptive statistics TLSG SAVA MELR HDOG PILR ARPO LKPG GORE KRKG PETG

SKEWNESS 0.006 0.189 0.168 –0.16 –0.026 0.413* 0.154 –0.41* –0.321* 0.528*

KURTOSIS 2.79* 3.311* 8.792* 6.189* 3.363* 4.785* 2.827* 1.637* 2.514* 3.528*

Resulting portfolios

MV 0.77% 0.00% 15.33% 49.22% 15.71% 0.00% 0.00% 0.00% 18.97% 0.00%

LPMM 1.59% 0.00% 12.36% 50.19% 16.72% 0.00% 0.00% 0.00% 19.14% 0.00%

LPM0 2.55% 0.00% 12.06% 51.20% 16.40% 0.00% 0.00% 0.00% 17.79% 0.00%

CVaR 0.00% 0.00% 28.69% 47.30% 2.16% 0.00% 0.00% 0.00% 21.85% 0.00%

MAD 25.39% 0.00% 0.00% 74.61% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

*Denotes statistical significance at 5% level.

Table 5. Results of descriptive statistics for stocks fr om Turkey and their portfolio weights.

Descriptive statistics TCELL GARAN HALKTUPRISAEFESISCTRYKBNKERGEL SAHOLAKBNK

SKEWNESS 0.012 0.484* 0.32* –0.259 –0.1260.316*–0.079 0.376* 0.379* 0.479*

KURTOSIS 2.165* 1.337* 2.514*1.285* 1.324*2.308*2.104* 2.010* 1.620* 2.921*

Resulting portfolios

MV 16.40% 0.00% 0.00%37.89% 36.81%0.00%0.00% 6.33% 2.57% 0.00%

LPMM 15.61% 0.00% 0.00%35.94% 37.82%0.00%0.00% 7.63% 3.00% 0.00%

LPM0 15.30% 0.00% 0.00%35.47% 38.28%0.00%0.00% 7.32% 3.63% 0.00%

CVaR 25.55% 0.00% 0.00%6.35% 44.52%2.32% 1.31% 18.02% 1.93% 0.00%

MAD 0.00% 0.00% 0.00%69.30% 30.70%0.00%0.00% 0.00% 0.00% 0.00%

*Denotes statistical significance at 5% level.

Observed stock returns have asymmetric distribution.

Skewness coefficients are quite different between coun-

tries generally, and between stocks within every stock

market. Generally, skewness coefficients are statistically

significant for major part of stocks but the sign of skew-

ness coefficients is quite different. According to Tab le 3

and Table 6 most stocks from Romania and Chez Re-

public have statistically significant negative skewness.

Table 1 and Table 5 show that most stocks from Croa-

tia and Turkey have positive skewness coefficient. Ac-

cording to Table 2, Table 4 and Table 7 most stocks

from Slovenia, Poland and Hungary do not have statisti-

cally significant skewness coefficient.

Using five different risk measures portfolios on every

market were formed and their CARs were calculated.

esults indicate that application of standard MV model

J. VIDOVIC

240

Table 6. Results of descriptive statistics for stoc ks fr om Che z Republic and their portfolio weights.

Descriptive statistics CEZ BKOMUNPEPGSNTABKCETVECMP SPTT ORCOAAAA

SKEWNESS –0.116 –0.614*–0.103 –0.713*0.224 –1.314*–3.066* 0.252 0.357*–0.303

KURTOSIS 7.073* 4.996* 7.483*6.184*5.753*10.762*36.486* 12.543* 8.274*2.35*

Resulting portfolios

MV, LPMM, LPM0, CVaR, 0.00% 0.00% 0.00%0.00%26.67%0.00%0.00% 73.33% 0.00% 0.00%

MAD 0.00% 0.00% 0.00%0.00%0.00%0.00%0.00% 99.89% 0.00%0.11%

*Denotes statistical significance at 5% level.

Table 7. Results of descriptive statistics for stocks fr om Hungary and their portfolio weights.

Descriptive statistics GDRB FHBK FOTH EMAS EGIS OTPB PANP RABA MTEL MOLB

SKEWNESS –0.788* 0.868* 1.05* –0.363* –0.26 0.218 0.015 –0.493* 0.061 –0.027

KURTOSIS 6.432* 6.178* 10.694* 4.062* 7.818* 6.445* 3.433* 7.443* 6.406* 7.033*

Resulting portfolios

MV 53.41% 0.00% 0.00% 19.65% 0.00% 0.00% 0.00% 0.00% 26.94% 0.00%

LPMM 50.65% 0.00% 0.00% 16.66% 0.00% 0.00% 0.00% 0.00% 32.69% 0.00%

LPM0 50.83% 0.00% 0.00% 16.84% 0.00% 0.00% 0.00% 0.00% 32.33% 0.00%

CVaR 51.02% 0.00% 0.00% 17.05% 0.00% 0.00% 0.00% 0.00% 31.93% 0.00%

MAD 35.35% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 64.65% 0.00%

*Denotes statistical significance at 5% level.

is questionable when CSEE emerging markets are ob-

served. In order to overcome the nonnormality problem

downside risk measures, LPMM and LPM0 were intro-

duced. According to results from Tables 1 to 7, MV,

LPMM and LMP0 portfolios have similar composition

and similar stock weights and consequentially they have

similar results in the out-of-sample period. These con-

clusions are compatible with [6]. MAD portfolios from

all 7 markets share common characteristic; MAD portfo-

lios are composed from smaller number of assets than

other portfolios. CVaR portfolios are very hard to char-

acterize because their composition is quite different than

composition of other portfolios and their out-of-sample

performance is different than performance of other port-

folios. If MAD portfolios and CVaR portfolios are com-

pared, CVaR portfolios are more stable while MAD

portfolios have faster drops. These results are in accor-

dance with results of previous researches [21,22]. This

can be proved by examining volatility of CVaR and

MAD portfolios in the out-of-sample period measured by

standard deviation of portfolio returns in Table 8. Gen-

erally, comparison of CARs in the out-of-sample period

does not give answer on the best risk measure. Accord-

ing to results from Table 8, application of standard MV

model on these markets is quite questionable. Better re-

sults are possible what can be seen in case of Turkey,

Slovenia, Romania and Poland were MAD or CVaR

portfolios over-perform MV, LPMM and LPM0 portfo-

lios. These results do not have continuity. CVaR portfo-

lio over-performs in case of Romania and gives good

results in case of Slovenia. MAD portfolios have better

results in case of Poland, Slovenia and Turkey but these

portfolio returns are always accompanied by higher vola-

tility in the out-of-sample period. In case of Hungary

best result is achieved by application of standard MV

model.

6. Conclusions

In this paper behavior of risk measures in situation of

nonnormality and their impact on portfolio composition

were investigated. Employed statistical methods affirmed

presence of nonormality and extreme kurtosis accompa-

nied with skewness which is statistically significant for

major part of stocks. The use of downside risk measures

does not give significant improvement in the portfolio

performance in the out-of-sample period. Employed risk

measures are not able to recognize excess kurtosis and

skewness in stock returns allowing highly risky securities

to enter portfolio. CARs were calculated in order to fol-

low the performance of resulting portfolios in the out-of-

sample period. According to results, CVaR portfolios

have slightly more stable returns in the out-of-sample

period while MAD portfolios have highest volatility.

LPMM, LPM0 and variance portfolios have similar

omposition, volatility and out-of-sample results. Natural c

241

J. VIDOVIC

Table 8. Performance of resulting portfolios in the out-of-sample period.

Risk measure Portfolios expected

return (%) Expected monthly

return (%) Standard deviation of

portfolio return CAR at the end of

period (%)

Croatia

MV, LPMM, LPM0, CVaR, MAD –0.0021 0.0005 2.2235 –0.5130

Poland

MV –0.1698 –0.1242 2.2089 –26.9950

LPMM –0.1686 –0.1218 2.2119 –26.7923

LPM0 –0.1683 –0.1211 2.2131 –26.7692

CVaR –0.1693 –0.1233 2.2098 –27.9404

MAD –0.1655 –0.1152 2.2297 –26.3223

Romania

MV –0.1230 –0.1480 3.0093 –24.4724

LPMM –0.1182 –0.1458 3.0307 –23.5257

LPM0 –0.1185 –0.1459 3.0292 –23.5895

CVaR –0.1140 –0.1438 3.0531 –22.6873

MAD –0.1755 –0.1731 3.0527 –34.9204

Slovenia

MV 0.0041 0.1809 2.9196 0.3880

LPMM 0.0026 0.1781 2.9464 0.2446

LPM0 0.0033 0.1801 2.9698 0.3111

CVaR 0.0335 0.2319 2.8635 3.1472

MAD 0.0448 0.2642 3.5960 4.2149

Turkey

MV 0.1276 0.1307 2.8488 31.5103

LPMM 0.1262 0.1291 2.8428 31.1751

LPM0 0.1262 0.1289 2.8409 31.1710

CVaR 0.1079 0.1098 2.7865 26.6409

MAD 0.1463 0.1504 3.1612 36.1460

Chez Republic

MV, LPMM, LPM, CVaR –0.0451 –0.7725 2.2258 –9.8411

MAD –0.0777 –0.0200 2.5121 –16.9433

Hungary

MV 0.1322 0.1511 2.2899 28.2907

LPMM 0.1292 0.1469 2.2857 27.6593

LPM0 0.1294 0.1472 2.2858 27.6985

CVaR 0.1296 0.1475 2.2859 27.7422

MAD 0.1128 0.1238 2.4001 24.1482

extension of this paper would be presentation of risk

measure which should take into account information on

skewness and kurtosis of stock returns.

7. References

[1] G. Bekaert and R. H. Campbell, “Research in Emerging

J. VIDOVIC

242

Markets Finance: Looking to the Future,” Emerging Mar-

kets Review, Vol. 3, No. 4, 2002, pp. 429-448.

doi:10.1016/S1566-0141(02)00045-6

[2] G. Bekaert and C. Harvey, “Emerging Markets Finance,”

Journal of Empirical Finance, Vol. 10, No. 1-2, 2003, pp.

3-56. doi:10.1016/S0927-5398(02)00054-3

[3] S. Stevenson, “Emerging Markets, Downside Risk and

the Asset Allocation Decision,” Emerging Markets Re-

view, Vol. 2, No. 1, 2001, pp. 50-66.

doi:10.1016/S1566-0141(00)00019-4

[4] R. Susmel, “Extreme Observations and Diversification in

Latin America Emerging Equity Markets,” Journal of In-

ternational Money and Finance, Vol. 20, No. 7, 2001, pp.

971-986. doi:10.1016/S0261-5606(01)00014-6

[5] J. Vidović and Z. Aljinović, “Research on Stock Returns

in Central and South-East European Transitional Econo-

mies,” Proceedings KOI 10th International Symposium

on Operational Research in Slovenia, Nova Gorica, 23 -

25 September 2009, pp. 237-246.

[6] C. G. Gilmore, G. M. McManus and A. Tezel, “Portfolio

Allocations and the Emerging Equity Markets of Central

Europe,” Journal of Multinational Financial Manage-

ment, Vol. 15, No. 3, 2005, pp. 287-300.

doi:10.1016/j.mulfin.2004.12.001

[7] C. A. J. Middleton, S. G. M. Fifield and D. M. Power,

“Investment in Central and Eastern European Equities,”

Studies in Economics and Finance, Vol. 24, No. 1, 2007,

pp. 13-31. doi:10.1108/10867370710737364

[8] T. Gklezakou and J. Mylonakis, “Interdependence of the

Developing Stock Markets, before and during the Eco-

nomic Crisis: The Case of South Europe,” Journal of

Money, Investment and Banking, Vol. 11, 2009, pp. 70-

78.

[9] J. Bley, “European Stock Market Integration: Fact or

Fiction?” Journal of International Financial Markets, In-

stitutions and Money, Vol. 19, No. 5, 2009, pp. 759-776.

[10] H. M. Markowitz, “Portfolio Selection: Efficient Diversi-

fication of Investments,” John Wiley & Sons, New York,

1959.

[11] P. C. Fishburn, “Mean-Risk Analysis with Risk Associ-

ated with Below-Target Returns,” American Economic

Review, Vol. 67, No. 2, 1977, pp. 116-126.

[12] V. S. Bawa and E. B. Lindenberg, “Capital Market Equi-

librium in a Mean-Lower Partial Moments Framework,”

Journal of Financial Economics, Vol. 5, No. 2, 1977, pp.

198-200. doi:10.1016/0304-405X(77)90017-4

[13] H. Konno and H. Yamazaki, “Mean-Absolute Deviation

Portfolio Optimization Model and Its Application to To-

kyo Stock Market,” Management Science, Vol. 37, No. 5,

1991, pp. 519-531. doi:10.1287/mnsc.37.5.519

[14] S. Uryasev, “Conditional Value-at-Risk (CVaR): Algo-

rithms and Applications,” Working Paper, University of

Florida, 2002. http://www.ise.ufl.edu/uryasev

[15] A. Alexandre, M. Houkari and J.-P. Laurent, “Spectral

Risk Measures and Portfolio Selection,” Journal of Bank-

ing and Finance, Vol. 32, No. 9, 2008, pp. 1870-1882.

doi:10.1016/j.jbankfin.2007.12.032

[16] C. Acerbi and D. Tasche, “On the Coherence of Expected

Shortfall,” Journal of Banking and Finance, Vol. 26, No.

7, 2002, pp. 1487-1503.

doi:10.1016/S0378-4266(02)00283-2

[17] R. T. Rockafellar and S. Uraysev, “Optimization of Con-

ditional Value-at-Risk,” Journal of Risk, Vol. 2, No. 3,

2000, pp. 21-41.

[18] G. Y. N. Tang, “How Efficient Is Naive Portfolio Diver-

sification? An Educational Note,” Omega—The Interna-

tional Journal of Management Science, Vol. 32, No. 2,

2004, pp. 155-160.

[19] L. R. Irala and P. Patil, “Portfolio Size and Diversifica-

tion,” SMCS Journal of Indian Management, Vol. 4, No.

1, 2007, pp. 1-6.

[20] S. S. Shapiro, M. B. Wilk and H. J. Chen, “A Compara-

tive Study of Various Tests of Normality,” Journal of the

American Statistical Association, Vol. 63, No. 324, 1968,

pp. 1343-1372. doi:10.2307/2285889

[21] Y. Simaan, “Estimation Risk in Portfolio Selection: The

Mean Variance Model versus the Mean Absolute Devia-

tion Model,” Management Science, Vol. 43, No. 10, 1997,

pp. 1437-1446. doi:10.1287/mnsc.43.10.1437

[22] E. Angelelli, R. Mansini and G. M. Speranza, “A Com-

parison of MAD and CVaR Model with Real Features,”

Journal of Banking & Finance, Vol. 32, No. 7, 2008, pp.

1188-1197. doi:10.1016/j.jbankfin.2006.07.015