research article realized jump risk and equity return in...

Research ArticleRealized Jump Risk and Equity Return in China

Guojin Chen1 Xiaoqun Liu2 Peilin Hsieh1 and Xiangqin Zhao2

1 School of Economics and TheWang Yanan Institute for Studies in Economics Xiamen University Xiamen Fujian 361005 China2 School of Economics Xiamen University Xiamen Fujian 361005 China

Correspondence should be addressed to Peilin Hsieh peilinhgmailcom

Received 2 March 2014 Revised 10 May 2014 Accepted 11 May 2014 Published 23 June 2014

Academic Editor Fenghua Wen

Copyright copy 2014 Guojin Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We utilize the realized jump components to explore a new jump (including nonsystematic jump and systematic jump) risk factormodel After estimating daily realized jumps from high-frequency transaction data of the Chinese A-share stocks we calculatemonthly jump size monthly jump standard deviation and monthly jump arrival rate and then use those monthly jump factors toexplain the return of the following month Our empirical results show that the jump tail risk can explain the equity return For thelarge capital-size stocks large cap stock portfolios and index one-month lagged jump risk factor significantly explains the assetreturn variation Our results remain the same even when we add the size and value factors in the robustness tests

1 Introduction

Jump is a source which attributes to the fat tail of a returndistribution If a market is incomplete and thus investorscannot form amarket portfolio to diversify the nonsystematicrisk away firm-specific and industry-level risk may have theinfluence over return premiumand affect asset pricesmakingthe classical capital asset pricing model (CAPM) unable toexplain the returns as perfectly as the theory says Inspired bythis idea this paper considers the jump the tail risk and agood supplemental factor to explain the asset returns

After CAPM model was proposed how well this modelperforms has been always challenged Fama and French [1 2]demonstrate that the CAPMmodel is unable to explain cross-sectional stock returns well and find that value premium(HML) and size (SMB) factors outperform themarketmodelAs a result they proposed the well-known Fama-Frenchthree-factor model which subsequently was extended tothe models with liquidity and momentum factors FurtherHarvey and Siddique [3] and Wen and Yang [4] arguethat conditional coskewness which could capture similarinformation contained in the size and book-to-market ratiocomplements market beta to explain the return Merton [5]also supplements a theoretical argument tomarket model AsCAPMmodel does not price idiosyncratic risk (IV) he claimsthat incomplete information which hinders the investors

to diversify their portfolios leads to the positive relationbetween idiosyncratic risk and expected stock return

According to the studies the empirical results of cross-sectional relation between the idiosyncratic volatility andexpected return are mixed Ang et al [6] show thathigh idiosyncratic volatility in one lagged month predictsabysmally low average returns in the next month SimilarlyGuo and Robert [7] propose that in addition to stock marketvolatility the aggregate idiosyncratic volatility (IV) could bethe source of risk that determines the equity return In theirpreposition IV may be considered as a proxy for variance ofthe risk factors of a multiple-factor or intertemporal capitalasset pricing model (ICAPM) Guo and Robert [8] furtherused the average idiosyncratic volatility (AV-IV) as proxy forinvestment opportunity cost on G7 countries and concludedthat IV contributes as well as the book-to-market factorto explaining the cross-sectional stock returns In additionto empirical works Campbell et al [9] also provide a verynoticeable argument for the declining explanatory powerof traditional market model In their paper they find theincrease of firm-level volatility level and market volatilitylevel making correlations between the stock returns lowerand diversification difficult Therefore seeking firm-specificor aggregate level risk factors becomes amore important issuefor asset pricing

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 721635 13 pageshttpdxdoiorg1011552014721635

2 Discrete Dynamics in Nature and Society

Recently several studies have presented the empiricalevidence in favor of jump risk effects on financial assetprices by using nonparametric method and high-frequencydatabase Since the realized volatility (RV) the realized rangevolatility (RRV) and realized jump volatility (RJV) wereproposed those nonparametric measures have been appliedextensively even for the researches in volatility parametricmodels for example GARCH-RV and the heterogeneousautoregressive with realized volatility (HAR-RV) modelThose nonparametric measures not only can be used todepict the volatility dynamic nature such as clusteringlong memory asymmetric leverage effect and the price-volume relation in financial market microstructure theory[10ndash18] but also can be considered as the risk indicationsTherefore investigating those risksrsquo influences on asset pric-ing is of great interest Adrian and Rosenberg [19] explorea new three-factor pricing model including the long-runvolatility component the short-run volatility componentand the market return and they conclude that their modeloutperforms the Fama-French model through reducing thepricing biased errors Apparently volatility can be a risk factorwhich accounts the return premium Following Adrian andRosenberg [19] Kelly and Jiang [20] develop a tail risk factormodel Both models are based on the structural model bytheoretical derivationwith explicit economicalmeanings Fortail risk model by Kelly the bivariate calibration of tail riskfactors withmacrovariables can be based on the long-run riskmodel proposed byBansal andYaron [21] and the disaster riskmodel proposed by Rietz [22] and Barro [23]

Though there are a large number of researches seekingdifferent risk factors for the returns all of them focus on thereturns of index bond market and credit spread Moreovermost of those works are using high-frequency data in theUnited States Inspired by those works we propose a jump-based model to examine the relations between the realizedjump components and equity returns at China market Wewill begin with an illustration of the nonparametric jumpestimation method After daily realized jump volatility isestimated from 5-minute high-frequency trading data wethen calculate monthly jump size jump standard deviationand jump arrival intensity Finally these jump componentsare used to investigate their predicting and explaining powerover one-month-ahead equity return

The remaining parts of this paper are organized as followsSection 2 introduces the identification method for the high-frequency realized jump and presents the jump risk factormodel and Section 3 describes our sample data and itsdescriptive statistics Empirical results are demonstrated inSection 4 and Section 5 is robustness tests analysis Finallythe conclusion is in Section 6

2 Preliminaries and Theories

21 Realized Jump Risk While jumps are known to bevery crucial in the asset pricing [32] estimation of jumpcomponents by parametricmodel has been questioned for thestability over different sample time periods With the avail-ability of high-frequency data nonparametric estimation

method has been developed rapidly Andersen and Bollerslev[33] Barndorff-Nielsen and Shephard [34 35] and Meddahi[36] have presented the use of the so-called realized variancemeasures by utilizing the information in the intraday data formeasuring and forecasting volatilities Barndorff-Nielsen andShephard [37 38] developed a series of the seminal work onbipower variation measure which is then used to divide theRV into continuous diffusion volatility and jump volatility(see Andersen et al [39] and Huang and Tauchen [40])Under the reasonable presumption that jumps on financialmarkets are usually rare and large we follow Huang andTauchen [40] to assume that there is at most one jump perday and that the jump size dominates the daily return when ajump occurs These assumptions allow us to extract the dailyrealized jumps and further to explicitly calculate the monthlyjump intensity size and standard deviation Tauchen andZhou [29] have demonstrated that jump parameters can beprecisely estimated and that the statistic inference is reliable

Compared to parametric models in which jump isestimated by the maximum likelihood MCMC or GMMmethods nonparametric estimation has merit of convenientestimation without assuming specifying underlying driftdiffusion and jump functions The assumption of one jumpper day fits to the compound Poisson jump process ([41]also utilizes the Poisson jump process to describe rare andlarge return jumps which are presumably the responses to thearrivals of important news) and it should be pointed out thatbipower variation also works for the infinite activity jumpsdespite the fact that we focus only on the case of rare and largejumps The following presents the details of jump estimationand detection

Let 119901119905= log(119875

119905) denote the time 119905 logarithmic price of

the asset and assume that it evolves in continuous time as ajump-diffusion process

119889119901119905= 120583119905119889119905 + 120590

119905119889119882119905+ 119869119905119889119902119905 (1)

where 119889119902119905is a Poisson jump process with intensity 120582

119869and

119869119905is the corresponding log jump size distribution following

normal (120583119869 120590119869) Consider

RV119905=

119872

sum

119895=1

1199032

119905119895997888rarr int

119905

119905minus1

1205902

119904119889119904 + int

119905

119905minus1

1198692

119904119889119902119904 (2)

where 119872 is the total number of trades during time 119905 and119905 + 1 and 119895 is the indication for each trade In the realisticfinancial markets the price volatility of financial asset isnot continuous but contains jumps due to the influencearoused by information shock on market Barndorff-Nielsenand Shephard [37 38] proposed two general measures forthe quadratic variation process-realized variance and realizedbipower variation this is presented in the following

BV119905equiv120587

2

119872

119872 minus 1

119872

sum

119895=2

10038161003816100381610038161003816119903119905119895

10038161003816100381610038161003816

10038161003816100381610038161003816119903119905119895minus1

10038161003816100381610038161003816997888rarr int

119905

119905minus1

1205902

119904119889119904 (3)

where radic1205872 = Ε(Π119905) Π119905is a standardized normal distribu-

tion random variable and 119872(119872 minus 1) is the amendment to

Discrete Dynamics in Nature and Society 3

sample size According to Barndorff-Nielsen and Shephardthe difference between RV

119905and BV

119905is just the consistent

estimator of the discrete jump variation when119872 rarr infin thatis

RV119905minus BV119905

119872rarrinfin

997888997888997888997888997888997888rarr RJV119905 (4)

A variety of jump detection techniques are proposed andstudied by Barndorff-Nielsen and Shephard [37] Huang andTauchen [40] and Andersen et al [14] In fact in the processof calculating the discrete jump variation the existence ofdifferent intraday sampling frequency may lead to some kindof calculation errors Here we adopt the ratio statistic favoredby their findings

119885119905=

(RV119905minus BV119905) RV119905

radic(12058724 + 120587 minus 5) (1119872)max (1TP119905BV2119905)

(5)

where TP119905is the tripower quarticity that Barndorff-Nielsen

and Shephard [37] define as

TP119905=

119872

119872 minus 2sdot

119872

4[Γ (76) Γ (12)]3

sdot

119872

sum

119894=3

10038161003816100381610038161199031199051198941003816100381610038161003816431003816100381610038161003816119903119905119894minus1

1003816100381610038161003816431003816100381610038161003816119903119905119894minus2

100381610038161003816100381643

(6)

The test statistic has an asymptotical normal distributionUnder the significance level 1 minus 120572 we can get the estimateof the discrete jump variation

RJV119905= 119868 (119885

119905gt Φminus1

120572)radic[RV

119905minus BV119905] (7)

whereΦ is the cumulative distribution function of a standardnormal and 119868(119885

119905gt Φminus1

120572) is the resulting indicator function

on whether there is a jump during the day 119868(119885119905gt Φminus1

120572)

equals 1 when a jump is detected at day 119905 and 0 otherwiseIn the process of actual operation we need to choose anappropriate120572 and Tauchen and Zhou [29] propose that whenjump contributions are 10 and 80 the significance levelshould be 099 and 0999 respectively

With the above test of119885119905statistic and the related bipower

variation theory we can get the estimator of RJV119905and then

calculate the monthly jump size (Size RJVmonth) monthlyjump size mean (Mean RJVmonth) monthly jump arrival rate(Arr RJVmonth) and monthly jump size standard deviation(Std RJVmonth)The jump components are defined as follows

Size RJVmonth = sum Size RJVday (8)

Mean RJVmonth =Size RJVmonth119873 RJV days

(9)

Arr RJVmonth =119873 RJV days

days (10)

Std RJVmonth = [sum((Size RJVday minusMean RJVday)2

)]12

(11)

where119873 RJV days is the total number of days when realizedjump occurs and ldquodaysrdquo denotes the trading days in a monthWe follow the previous studies of China stock market to setthe confidence level 120572 at 095 in this paper Our results showthat among 5 different capital-size portfolios monthly jumpsize for the largest-cap portfolio is 8 and it is 9 for thesmallest-cap portfolio The jump size mean is significantlylarger than 0 for all portfolios Standard deviation is about032sim036 and jump arrival rate is about 19 for allportfolios

22 JumpComponents Risk FactorModel under the IncompleteMarket There are a fewmethods formeasuring time-varyingtail risk First Kelly and Jiang [20] devise a panel approachto estimate economy-wide conditional tail risk by usingcommon fluctuation of the stocks The framework is basedon the long-run risk literature by Bansal and Yaron [21] andtime-varying rare disaster model by Gabaix [24] as well as byWachter [25] Secondly Bollerslev et al [26] examine howthe variance risk premium (VRP) implied in index optionprices relates to the equity premium As VRP is an ex-antemeasure that represents investorsrsquo expectation for future riskthe realized jump is ex-post measure for tail risk

Owing to the trading constraints including short-selling constraint liquidity issue and budget constraint theinvestors are not able to formmarket portfolio effectively andcannot diversify the nonsystematic risk away As a result therisk of the individual stock may need to be priced and therisk information could be contained in the historical returncharacteristics (eg skewness and kurtosis) This paperconsiders the jump the resource of stock risk and should affectthe asset prices We thus do an extensive investigation on therelation between jump risk and equity returns Our empiricalworks cover the index returns return of portfolios and stockreturns

Guo and Robert [8] argue that it is the omitted variablesproblem that results in the failure of the CAPM and theyderive the equation about the effect of the idiosyncraticvolatility on risk premium Their argument is analogous toour reasoning on the tail risk of individual stock and can beexpressed in following expression

119864119905(119903119894119905+1

) = 120574119872Cov119905(119903119894119905+1

119903119872119905+1

) + 120574119867Cov119905(119903119894119905+1

119903119867119905+1

)

= 120574119872

Cov119905(119903119894119905+1

119903119872119905+1

)

Var119905(119903119872119905+1

)Var119905(119903119872119905+1

)

+ 120574119867

Cov119905(119903119894119905+1

119903119867119905+1

)

Var119905(119903119867119905+1

)Var119905(119903119867119905+1

)

= 120574119872120573119894119872119905

Var119905(119903119872119905+1

) + 120574119867120573119894119867119905

Var119905(119903119867119905+1

)

(12)

It says that the expected return on any asset is a function ofconditional variances of stock market returns 119903

119872119905+1 and the

risk factor 119903119867119905+1

is omitted in the CAPMUnder some moderate conditions average idiosyncratic

volatility (IV119905) is the proxy for volatility of 119903

119867119905+1 and MV

119905is


the proxy of aggregatemarket volatility at time 119905+1 thereforewe can write

119903119894119905+1

= 120572119894119905+ 120574119872120573119894119872

MV119905+ 120574119867120573119894119867

IV119905+ 120577119894119905+1

(13)

For simplicity we assume constant betas in (13) as Bollerslevet al [27] do Suppose that the intercept is zero in (13) theloading on stock market volatility is equal to the market betascaled by the price of market risk 120574

119872 Similarly the loading

on idiosyncratic volatility is equal to the beta on the omittedrisk factor scaled by its risk price 120574

119867 Therefore we can use

(13) to explain the cross-sectional stock returns even thoughwe donot observe the risk factor 119903

119867119905+1Thismethod provides

a direct link between time series and cross-sectional stockreturn predictability If the value premium is an omitted riskfactor as argued by Fama and French [28] we can expectthat its volatility should have predictive power for the stockreturns similar to that of average idiosyncratic volatility

Jump volatility risk shares similar characteristics regard-ing what generates IV For example Bollerslev et al [27]provide a new framework for estimating the systematic andidiosyncratic jump tail risks in financial asset pricing In theirpaper the opinion dispersion is subsequently related to thejump and it should be pointed out that their concept isalso relevant to incomplete market hypothesisTherefore thetheoretical argument over idiosyncratic volatility and jumpvolatility risk on equity returns is based on incompletemarketfoundation

For an individual stock the jump could result fromnonsystematic information as well as systematic informationAccording to the theory the index jump carries the systematicinformation which also leads to the jump of individualstocksTherefore when investigating the nonsystematic jumpeffect on the returns we also do both regressions withmarket jumps as well as without market jumps separatelyIn Section 4 we examine the nonsystematic and systematicjump effects on the returns of index portfolios and stocksOur regression models for the stock return and portfolio arethe following

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905 + 120577

119872119898+1

(14)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

(15)

As for index return the estimation model is given as follows

119903119875119898+1

= 120572119875+ 120573119875size119875Size RJV

119898+ 120573119875std119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119875119898+1

(16)

For each individual stock Size RJV Std RJV and Arr RJVdenote monthly jump size jump standard deviation andjump arrival rate respectively 119875Size RJV 119875Std RJV and119875Arr RJV represent monthly market index jump size index

jump standard deviation and index jump arrival rate119898 is theindication ofmonth 119894 and119901 are notations for individual stock(portfolio) and index (market portfolio) In the robustnesstests we run the regression models incorporating Fama-French factors and regression models with nonlinear jumpcomponents Those models are given as follows

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120574

119894sizeSize RJV119898lowast Arr RJV

119898

+ 120574119894stdStd RJV

119898lowast Arr RJV

119898+ 120577119894119905+1

(17)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898

+ 120574119894size ArrSize RJV


119898

+ 120574119894std ArrStd RJV


119898

+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(18)119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Std119875Std RJV

119898+ 120573119875Arr119875Arr RJV119898

+ 120574SMBSMB119898+ 120574HMLHML

119898+ 120577119894119898+1

(19)

When (17) expands (14) to include the nonlinear jumpmulti-ple terms Size RJV

119898lowastArr RJV

119898 Std RJV

119898lowastArr RJV

119898 into

repressors (18) extends (17) so we can test nonlinearity effectof jump components under the control of the index jumpcomponents Finally (19) is used to investigate the nonlinearjump effect on return with control of Fama-French factors

3 Estimation of Jump Components

31 Data and Summary Statistics Intraday high-frequencytrading contains noises On the one hand low samplingfrequency may fail to depict the actual volatility informationon that day On the other hand high sampling frequencymay lead to the problem of micronoise which may affectthe results As suggested by the former literature we usefive-minute high-frequency data for return calculation Tominimize the noise we then divide 200 randomly selectedstocks into 5 portfolios based on the stock market valueand then use the equally weighted return for realized jumpestimationThe data comes from the CSMARhigh-frequencydatabase and sampling period begins from January 4 2007and ends on October 31 2013 There are 82 months and49 transactions per day (including one overnight tradingdata and 48 intraday trading data) The individual stocksrsquomonthly returns and the index returns are directly from theCSMAR financial database We calculate all monthly jumpcomponents from high-frequency transaction and run the


00

02

04

06

08

10

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

10 20 30 40 50 60 70 80

00

02

04

06

08

10

10 20 30 40 50 60 70 80

Arr

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80

Std

Size Mean

Figure 1 Time series characteristic of index jump components

Table 1 Descriptive statistics for index jump components

Index 000002 Size Mean Arr StdMean 00864 00124 02608 00019Median 00437 00096 02247 00013Maximum 09120 00570 08947 00118Minimum 00000 00000 00000 00000Std dev 01361 00096 01990 00021Skewness 36540 21444 12761 22811Kurtosis 194318 90465 45444 92647Jarque-Bera 110499 18776 3040 20521Probability 00000 00000 00000 00000

regression on monthly return over monthly jump compo-nents

For market-level (systematic) realized jump estimationour sample data is high-frequency return of Shanghai com-posite index Table 1 demonstrates descriptive statistics ofindex jump components and Figure 1 shows the time seriesof index jump components

In comparison with the study of Tauchen and Zhou[29] on SampP the A-share index jump size mean is 124while it is 65 for SampP index The jump arrival rates are2608 for Shanghai index and 133 for SampP index standarddeviations are 019 for Shanghai index and 0525 forSampP index The difference is mainly due to the differentsignificant level selected for jump filtration Our paper sets

120572 equal 095 allowing smaller jumps and increasing thejump arrival rate Therefore we have a larger sample size ofrealized jumps and larger sample size leads to lower jump sizevariance Additionally as self-evident in Figure 1 the jumpcomponents are apparently time varying

Table 2 shows the jump statistics of the largest-cap andsecond largest-cap portfolios each of which contains 40stocks And Figure 2 demonstrates time series of jump com-ponents for size 1 portfolio

Compared with jump components of index return themagnitude ofmonthly jump sizemean is larger for portfoliosand the arrival rate tends to be lower The way we sort theportfolios by the size of capitalization is inspired not onlyby traditional cross-sectional return researches but also byBollerslev et al works in [27 30] in which they show thattheir model works better for large cap stocks or for stockstraded actively (descriptive statistics for other portfolios arelisted in the appendix) While the statistics of portfolio withsmaller size stocks are shown in the appendix we find veryinteresting pattern over jump components of 5 portfolios Asseen in Table 3 and Figure 3 monthly jump size jump meanjump intensity and jump size variance tend to be lower forlarger capitalization stocksTherefore jump componentsmaycapture the similar information of the size factor

32 Jump Components of the Index and of the Individual StockIn Figure 4 we compare jump components of individualstock the largest-cap portfolio and index over our sample


000

005

010

015

020

025

030

10 20 30 40 50 60 70 80001

002

003

004

005

10 20 30 40 50 60 70 80

00

01

02

03

04

05

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

Arr Std

Size Mean

Figure 2 Time series characteristic of individual-based portfolio

0080

0084

0088

0092

0096

1 2 3 4 50019

0020

0021

0022

0023

0024

0188

0190

0192

0194

0196

0198

0200

00031

00032

00033

00034

00035

00036

00037

1 2 3 4 5

1 2 3 4 51 2 3 4 5

Arr Std

Size Mean

Figure 3 Trend of jump components for portfolios


Table 2 Descriptive statistics of jump components for size 1 and size 2 portfolios

SZ1 Size Mean Arr Std SZ2 Size Mean Arr StdMean 00803 00199 01899 00032 Mean 00887 00225 01895 00035Median 00617 00187 01713 00023 Median 00656 00202 01707 00029Maximum 02813 00442 04134 00106 Maximum 03011 00488 04264 00097Minimum 00304 00108 00870 00013 Minimum 00429 00119 01065 00014Std dev 00553 00077 00786 00019 Std Dev 00596 00082 00744 00018Skewness 20477 12664 17262 15554 Skewness 20234 11964 19767 13557Kurtosis 62403 43168 53520 53717 Kurtosis 60927 41452 61131 45256Jarque-Bera 93181 27843 59623 52283 Jarque-Bera 88634 24043 86513 33071Probability 00000 00000 00000 00000 Probability 00000 00000 00000 00000

Table 3 Descriptive statistics of jump components for portfolios

Portfolio mean Size Mean Arr StdSZ1 00803 00199 01899 00032SZ2 00887 00225 01895 00035SZ3 00918 00230 01918 00035SZ4 00939 00229 01982 00036SZ5 00933 00230 01947 00037Observations 82 82 82 82

000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13

Mean RJV sizeSZ1Mean size

Pmean RJV size

Figure 4 Jump size mean for a stock a size 1 portfolio and an A-share index

period The stock is one randomly picked stock in size 1portfolio and we can find the jump comovements betweenthe stock portfolio and index In this quick glance themonthly jump size mean is the largest for the individualstock and the lowest for the indexThe figure strongly impliesthat index jump components could also be a proxy for thesystematic risk

4 Forecasting One-Month-AheadEquity Returns

Realize that jump components have been applied for pre-dicting and explaining the return of market in the USAand China However for the return of stock and portfoliono empirical work has been done by using the realizedjump measure The existing researches for stock market inChina mainly have been focusing on either macroeconomic

Table 4 Forecasting one-month-ahead index returns using realizedjump risk factor model

Variables 119862 119875Size 119875Arr 119875Std Adj1198772

Coefficients 0027 0278lowastlowast minus0055 minus18695lowastlowastlowast 0080Note The numbers in the table are coefficients and ldquolowastrdquo ldquolowastlowastrdquo ldquolowastlowastlowastrdquorepresents 10 5 and 1 significance level same for the following tables

Table 5 Forecasting one-month-ahead individual stock returnsusing realized jump volatility-based factor model

Stock 119862 Size RJV Arr RJV Std RJV Adj1198772

600362 0115lowastlowastlowast 0977lowast minus0760lowastlowast minus9494 0047600100 0044 0533 minus0357 minus3494lowast 0010600859 0037 0602lowast minus0206 minus14241lowastlowast 0039

variables or Fama-French multifactor models Our paperthus makes the contribution of applying jump componentsto study the equity return and exploring their power topredict the return By aggregating the daily realized jumpinto monthly jump size we then use the monthly realizedjump components as risk factors to explain one-month-ahead equity return Meanwhile we also consider indexjump components systematic risks and add them as controlvariables in our tests

41 Forecasting One-Month-Ahead Index Return For themarket-level realized jump our sample is the return ofShanghai composite A-share index (000002) Because thecalculations of monthly jump size and monthly jump sizemean are highly related we use monthly jump size as oneof the explanatory variables Table 4 shows the results ofregression In linear model the monthly jump size (119875size)


Table 6 Forecasting one-month-ahead individual stock portfolio returns using realized jump volatility-based factormodel withmarket-leveljump components

Stock 119862 Size RJV Arr RJV Std RJV 119875size 119875Std 119875Arr Adj1198772

600362 minus004 6024lowastlowast minus002 minus12957lowastlowast 5045 minus0042 minus33608lowastlowast 0113600100 0072lowastlowast 0654lowast minus039 minus3818lowast 0305 minus0108 minus15798 0015600859 0057lowast 0890lowastlowast minus0251 minus15176lowastlowastlowast 0135 minus0014 minus22315lowastlowastlowast 0104

Table 7 Forecasting one-month-ahead individual-based portfolio returns using realized jump volatility-based factor model

Size 119862 Size RJV Std RJV Arr RJV Adj1198772

1 0073lowastlowast 0725 minus13634lowast minus0379 00112 0120lowastlowast 1390lowastlowast minus20060 minus0823lowastlowast 00243 0107lowastlowast 1132lowast minus26017lowast minus0539 00164 0099 1117 minus19607 minus0588lowast mdash5 0079 0848 minus11404 minus0517 mdash

and monthly jump standard deviation (119875std) are significantwhile the coefficients are positive for the jump size andnegative for jump standard deviation It is quite intuitive thatthe larger jump size implies higher tail risk such that investorsrequire higher return However the negative sign for jumpstandard deviation is not easy to comprehend

While Guo and Robert [7 8] also find that the averageidiosyncratic volatility has negative coefficients on futureindex return they claim that the average idiosyncraticvolatility may capture the information of opportunity costWhen higher opportunity lowers the return the increaseof idiosyncratic volatility decreases the returns We thusconjecture that the jump standard deviation may also catchthe information of opportunity cost

42 Forecasting One-Month-Ahead Stock Return FollowingBollerslev et al [27 30] using top 40 large cap stocks or stockstraded actively for empirical analysis we also pick up largecap stocks among our randomly selected stocks for predictingpower tests For each stock we first regress the individualstockrsquos return on its realized jump components and then addmarket components into explanatory variables for advancedinvestigation Tables 5 and 6 demonstrate the results of tworegressions for 3 large cap stocks The results of more stockslisted in our largest-cap portfolio are shown in the appendixAccording to the results for stock we find that jump size iscommonly a tail risk factor in explaining the future returnand at least one jump component is a significant factor inregression

Moreover the coefficient signs for jump size and jumpstandard deviation are consistent with the results of indexreturn From the general equilibrium perspective if the tailrisk cannot be diversified away higher jump size implieshigher tail risk and requires higher return for compensationAs for negative relation between return and jump standarddeviation we think that the jump standard deviation ishighly negatively correlated with book-to-market ratio forthe success or failure of a project leading to the jump hasstronger impact on stocks with lower book-to-market ratioHence higher jump size deviation implies lower book-to-market ratio While Gou and Robert [7 8] empirically show

the negative correlation between idiosyncratic volatility andbook-to-market ratio we are collecting more accountinginformation for another further work Table 6 shows theresults with the control of market jump components TheAdj1198772 increases after we include index jump componentsinto the regression model

43 Forecasting One-Month-Ahead Portfolio Return Forportfolio return analysis we apply the convention of cross-section return research which divides the total sample into5 portfolios based on the stock market capital size Usingthe portfolio return helps us to reduce the trading noise ofthe individual stock As a few researches apply realized jumpto study the index return we supplement the first empiricalstudy of realized jump effect on the portfolio and the stockreturn Similar to our empirical work on stocks we firstestimate portfolio return over the portfolio jump componentsand show the results in Table 7 Then we include the marketindex jump components as the control variables and list theresults in Table 8

According to Table 7 the jump components work betterto explain the future return for larger-cap portfolios includ-ing size 1 to size 3 For smaller size portfolios (size 4 and size5) the jump components perform marginally in predictingreturn The Adj1198772 is also very low for small cap portfolioThose results are consistent with the paper by Bollerslev et al[27 30] in which they show that their model performs betterfor large cap stocks In addition the signs of the coefficientsagree with results of index and stocks Our conclusion doesnot alter if we add market-level jump components intothe regression model however the explaining power of themodel increases greatly With the control of market jumpcomponents all of jump components including monthlyjump size (Size RJV) jump standard deviation (Std RJV)and arrival rate (Arr RJV) become significant for size 2portfolios Finally we find that jump arrival rate also hassignificantly negative impact on return for large cap portfolioHowever the role of jump arrival rate should not be overem-phasized because size has correlations with jump arrival rateowing to jump detection statistics design In this paper weset 120572 equal to 095 leading to jump size mean ranging from


Table 8 Forecasting one-month-ahead individual-based portfolio returns using realized jump volatility-based factor model with market-level jump components

Size 119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Adj1198772

1 0083lowastlowast 1237lowast minus0581lowast minus9260 minus0170 0033 0033 00472 0139lowastlowast 2022lowastlowast minus1092lowastlowast minus16304lowastlowast minus0171 0042lowastlowast 0042lowastlowast 01083 0099lowast 1184 minus0597 minus12683 0246 minus0036lowast minus0036lowast 00404 0088 1649lowast minus0728 minus7980 minus0272 minus0009 minus0009 00575 0088 1389 minus0620 minus7020 0112 minus0086 minus0086 0015

Table 9 Forecasting one-month-ahead individual returns using realized jump volatility-based factor model with FF factors

119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std SMB HML Adj1198772

600362 0150lowastlowastlowast 1280lowastlowast minus0806lowastlowast minus11864 0484lowast minus0089 minus35944lowastlowastlowast minus0049 minus0671 0103600100 0067lowast 0643lowast minus0382 minus3764lowast 0300 minus0110 minus14427 0258 0027 0015600859 0047 0882lowast minus0260 minus15119lowastlowast 0118 minus0011 minus19235lowastlowast 0597lowast 0133 0123

Table 10 Forecasting one-month-ahead individual returns using realized jump volatility-based factor model with cross-term

119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

600362 0167lowastlowastlowast 1842lowastlowast minus0837lowastlowast minus40387lowast 0513lowast minus0112 minus40058lowastlowast 129012 minus2333 0115600100 0087lowastlowast minus0185 minus031 3484 0256 minus0113 minus8573 minus51175 2044 0018600859 0047 0761 minus0249 minus2221 0138 minus0009 minus22127lowastlowastlowast minus69190 1124 0096

Table 11 Forecasting one-month-ahead index returns using realized jump volatility-based factor model with cross-term of its jumpcomponents

Index 119862 119875size 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

0015 0691 minus0042 minus8972 minus56573 minus0162 0058000002 119862 119875size 119875Arr 119875Std Adj1198772

0027 0278lowastlowast minus0055 minus18695lowastlowastlowast 0080

1 to 3 and jump arrival rate ranging from 15 to 25 Ifthe confidence level 120572 increases to 099 the stricter criterianaturally discriminate against smaller jumps As a result outdata contains only larger jumps leading to lower arrival rateThis paper follows the previous study using the ease criteriato filter the jump Compared with Tauchen and Zhou [29]documenting 6 jump size mean our jump size mean is 2and is more practical for nature of jump dynamics

In summary of 41 42 and 43 the monthly jumpsize (Size RJV) is a significant risk factor which positivelyinfluences one-month-ahead return while the jump sizestandard deviation (Std RJV) has negative impact on thereturn After including market jump components we derivethe same conclusion andfind the increasedAdj1198772 Overall asshown in the appendix even for the individual stock at leastone jump component is a significant factor for return with orwithout the control of market jump components

5 Robustness Tests

In this section we proceed with the robustness tests fromtwo perspectives First because many empirical works useFama-French factor models to explain cross-sectionalreturns this paper thus compares jump risk factors togetherwith size and book-to-market ratio for stock return Secondly

we consider jump componentsrsquo nonlinear effect on returntherefore the multiples of jump risk factors are added toregression models for advanced tests If those multiple termsare significant and greatly improve the predicting powerthen the effect of jump risk factors on return is probablynonlinear

Table 9 shows the jump risk factors regression withcontrol of size (SMB) and book-to-market ratio (HML)for stocks The jump size and standard deviation remainsignificant andAdj1198772 does not increase Additionally Fama-French factors are not significant In other words the one-month-ahead return is influenced by the jump effect ratherthan FF factors

Table 10 demonstrates the results of regressions whichincorporate themultiples of jump risk factorsThoughAdj1198772improves slightly the multiple terms are not significant

Table 11 shows the robustness results for index jump effecton the returns As seen not only nonlinear terms Std lowast Arrand Size lowast Arr are insignificant but also Adj1198772 drops

Table 12 shows the robustness results for 5 portfoliosAfter adding control variables of multiple terms all jumpcomponents became insignificant for size 1 portfolio WhileAdj1198772 does not increase the nonlinear terms are not signifi-cant


Table 12 Forecasting one-month-ahead individual-based portfolio returns using realized jump volatility-based factormodel with cross-termof its jump components

Size 119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

1 0091 1139 minus0612 minus10041 minus0146 0028 minus15147 5862 0194 00202 0146 4829lowast minus1439lowast minus74502 minus0734 0156 minus27168lowastlowast 295313 minus9148 01033 0058 2062 minus0491 minus18227 0070 0000 minus23763lowast 8667 minus2071 00174 0111 3156 minus0946 minus45107 minus0560 0032 minus23857lowast 146138 minus4150 00405 0099 2762 minus0756 minus40191 minus0021 minus0054 minus28734lowastlowast 145713 minus4528 0042Size 1 to size 5 portfolios are arranged in the order of capitalization Size 1 is the largest-cap portfolio while size 5 is the smallest-cap portfolio The numbers inthe table are coefficients and ldquolowastrdquo ldquolowastlowastrdquo and ldquolowastlowastlowastrdquo represent 10 5 and 1 significance levels

Table 13 Descriptive statistics of jump components for size 3 to size 5 portfolios

SZ3 Size Mean Arr Std SZ4 Size Mean Arr Std SZ5 Size Mean Arr StdMean 00918 00230 01918 00035 Mean 00939 00229 01982 00036 Mean 00933 00230 01947 00037Median 00690 00212 01756 00030 Median 00719 00200 01778 00030 Median 00712 00210 01727 00033Maximum 03286 00508 04283 00090 Maximum 03002 00505 04330 00089 Maximum 02663 00499 04324 00095Minimum 00326 00131 00855 00012 Minimum 00447 00140 01032 00018 Minimum 00386 00147 01085 00011Std dev 00604 00081 00779 00016 Std dev 00568 00079 00759 00016 Std dev 00552 00077 00762 00016Skewness 20464 13540 16768 13756 Skewness 19224 15546 17292 15800 Skewness 16370 15211 15507 14567Kurtosis 66092 46886 53955 45527 Kurtosis 58266 51466 54538 51632 Kurtosis 48022 50022 47581 51850Jarque-Bera 10174 34799 58033 34099 Jarque-Bera 77807 48773 61438 50105 Jarque-Bera 47720 45318 43423 45311Probability 00000 00000 00000 00000 Probability 00000 00000 00000 00000 Probability 00000 00000 00000 00000

Observations 82 82 82 82

Table 14 Descriptive statistics for individual stock average jumpcomponents

600362 Size Mean Arr StdMean 0106062 0023454 0216076 0003758Median 0074818 0023504 0195238 0002885Maximum 0528641 0063170 0733333 0018655Minimum 0000000 0000000 0000000 0000000Std dev 0090047 0011790 0135851 0003497Skewness 1740065 0680288 0938441 1552614Kurtosis 7637096 3866075 4460514 6301100Jarque-Bera 1148477 8887622 1932393 7017733Probability 0000000 0011751 0000064 0000000

In short after controlling size and value factors we findthat the realized jump risk can explain and predict the equityreturn Therefore jump is an important risk factor neededto be considered for asset pricing and risk management Wealso find that nonlinearmodels do not work better than linearmodels [31]

6 Conclusion

For risk-aversion investors jump risk which leads to returndistribution with fat tail can greatly affect investorsrsquo percep-tion about how risky a company is or how it captures the riskinformation As long as nonsystematic risk of the individualstock cannot be diversified away the firm-level and industry-level risk should be priced into the return This paper applies

the realized jump considers a measure of tail risk and useshigh-frequency data to estimate jump components includingjump size mean jump size standard deviation and jumpintensity We argue that trading constraints make investorsunable to diversify nonsystematic risk away hence the tailrisk should be priced and jump should be able to explain theasset return

Our data contain high-frequency trades of 200 randomlyselected stocks which are the composite stocks for Shanghaicomposite index Our empirical works are divided into 3parts including index return portfolio returns and stockreturns Under the control of Fama-French factors ourresults show that jump factors can explain one-month-aheadreturn for index high cap portfolio and high cap stocksMoreover we also try to use nonlinear combinations of jumpcomponents to explain and predict the returns However noevidence supports that the nonlinear models are better thanlinear models

In our research the monthly jump size and jump sizestandard deviation are generally significant factors for equityreturnThe positive coefficient for jump size is quite intuitiveas it is directly linked to the tail risk However the negativecoefficient for jump size standard deviation is for the firsttime documented in the literature We claim that the jumpstandard deviation is positively related to opportunity costwhich decreases the returns as similar argument was madefor the idiosyncratic volatility Obviously further theoreticalmodels are needed after we find that jump components canexplain the return We are also looking forward to the theorymodel delivering the insights about how jump componentsare prices and looking forward to seeing more empirical


Table 15

119862 Size RJV Arr RJV Std RJV Adj1198772

600066 0039 minus0162 0069 minus4371 0051119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Adj1198772

600066 0056lowast 0145 minus0058 minus5543lowast 0165 0017 minus17954lowast 0069119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

600066 0028 0601 minus0024 5626 0147 0051 minus19204lowastlowast minus68284 minus0303 0062119862 Mean RJV Arr RJV Std RJV Adj1198772

600276 0003 minus1517 0138lowast 4749lowast 0045119862 Mean RJV Arr RJV Std RJV 119875mean 119875Arr 119875Std Adj1198772

600276 0009 minus1856 0122 4901lowast 4206lowast minus0117lowast minus939 0064119862 Mean RJV Arr RJV Std RJV 119875mean 119875Arr 119875Std Std lowast Arr Mean lowast Arr Adj1198772

600276 0025 minus1484 003 minus3703 4598lowastlowast minus0121lowast minus10859 42980lowastlowast minus2751 0148119862 Size RJV Arr RJV Std RJV Adj1198772

600377 0115lowastlowastlowast 0977lowast minus0760lowastlowast minus9494 0047119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Adj1198772

600377 minus004 6024lowastlowast minus002 minus12957lowastlowast 5045 minus0042 minus33608lowastlowast 0113119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

600377 0055 minus0711 minus0268 22229lowast 0247lowast minus0043 minus16372lowastlowast minus141272lowast 5712lowast 0115119862 Mean RJV Arr RJV Std RJV Adj1198772

600079 minus0015 5303lowastlowast minus0181 minus11173lowast 0073119862 Mean RJV Arr RJV Std RJV 119875mean 119875Arr 119875Std Adj1198772

600079 minus004 6024lowastlowast minus002 minus12957lowastlowast 5045 minus0042 minus33608lowastlowast 0112119862 Mean RJV Arr RJV Std RJV 119875mean 119875Arr 119875Std Std lowast Arr Mean lowast Arr Adj1198772

600079 minus0116lowast 9884lowastlowast 0504lowast minus2751 486 minus0063 minus27503lowast minus34662 minus27183 0146119862 Size RJV Arr RJV Std RJV Adj1198772

601006 minus0004 0177 0110 minus9650lowast 0052119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Adj1198772

601006 0015 0358 0067 minus7748lowast 0188 minus0071 minus12002 0053119862 Size RJV Arr RJV Std RJV 119875mean 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

601006 0031 0002 0020 minus14181 0192 minus0082 minus9975 40557 0393 0044119862 Size RJV Arr RJV Std RJV Adj1198772

600748 0049 minus0004 minus0178 0717 mdash119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Adj1198772

600748 0111lowastlowast 0536 minus0187 minus2107 0642lowastlowast minus0253 minus47805lowastlowastlowast 0101119862 Size RJV Arr RJV Std RJV 119875size 119875Arr 119875Std Std lowast Arr Size lowast Arr Adj1198772

600748 0119lowast minus0019 minus0226 9983 0608lowastlowast minus0225 minus47241lowastlowastlowast minus66184 2291 0087

researches on what information is contained in the jumpcomponents

Appendix

Tables 13 14 and 15 show the descriptive statistics ofindividual-based portfolio 3 to portfolio 5 and individualstock respectively Figure 5 shows the time series of individ-ual stockrsquos jump components

We forecast one-month-ahead stock return using thefollowing

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean

Figure 5 Time series characteristic of individual stock

+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was partly supported by National Science Founda-tion of China (Grant no 71071132)

References

[1] E F Fama and K R French ldquoThe cross-section of expectedstock returnsrdquo The Journal of Financial vol 47 no 2 pp 427ndash465 1992

[2] E F Fama and K R French ldquoCommon risk factors in thereturns on stocks and bondsrdquo Journal of Financial Economicsvol 33 no 1 pp 3ndash56 1993

[3] C R Harvey and A Siddique ldquoConditional skewness in assetpricing testsrdquoThe Journal of Finance vol 6 no 3 pp 1263ndash12952000

[4] F Wen and X Yang ldquoSkewness of return distribution andcoefficient of risk premiumrdquo Journal of Systems Science andComplexity vol 22 no 3 pp 360ndash371 2009

[5] R C Merton ldquoA simple model of capital market equilibriumwith incomplete informationrdquo The Journal of Finance vol 42no 3 pp 483ndash510 1987

[6] A Ang R Hodrick Y Xing and X Zhang ldquoHigh idiosyncraticvolatility and low returns international and further US evi-dencerdquo Working Paper Columbia Univerity 2006

[7] H Guo and S Robert ldquoIdiosyncratic volatility stock marketvolatility and expected stock returnsrdquo Journal of Business ampEconomic Statistics vol 24 no 1 pp 43ndash56 2006

[8] H Guo and S Robert ldquoAverage idiosyncratic volatility in G7countriesrdquo The Review of Financial Studies vol 21 no 3 pp1259ndash1296 2008

[9] J Y Campbell M Lettau B G Malkiel and Y Xu ldquoHave indi-vidual stocks become more volatile An empirical explorationof idiosyncratic riskrdquo The Journal of Finance vol 56 no 1 pp1ndash43 2001

[10] T G Andersen T Bollerslev F X Diebold and P Labys ldquoThedistribution of realized exchange rate volatilityrdquo Journal of theAmerican Statistical Association vol 96 no 453 pp 42ndash552001

[11] T G Andersen T Bollerslev F Diebold and H EbensldquoThe distribution of realized stock return volatilityrdquo Journal ofFinancial Economics vol 61 no 1 pp 43ndash76 2001

[12] S J Koopman B Jungbacker and E Hol ldquoForecasting dailyvariability of the SampP 100 stock index using historical realisedand implied volatility measurementsrdquo Journal of EmpiricalFinance vol 12 no 3 pp 445ndash475 2005

[13] F Corsi ldquoA simple approximate long-memorymodel of realizedvolatilityrdquo Journal of Financial Econometrics vol 7 no 2 pp174ndash196 2009


[14] T G Andersen T Bollerslev and F X Diebold ldquoRoughing itup including jump components in themeasurementmodelingand forecasting of return volatilityrdquo The Review of Economicsand Statistics vol 89 no 4 pp 701ndash720 2007

[15] T G Andersen T Bollerslev and X Huang ldquoA reduced formframework formodeling volatility of speculative prices based onrealized variation measuresrdquo Journal of Econometrics vol 160no 1 pp 176ndash189 2011

[16] F Wen and Z Liu ldquoA copula-based correlation measure andits application in chinese stock marketrdquo International Journalof Information Technology amp Decision Making vol 8 no 4 pp787ndash801 2009

[17] C Huang X Gong X Chen and F Wen ldquoMeasuring andforecasting volatility in Chinese stock market using HAR-CJ-M modelrdquo Abstract and Applied Analysis vol 2013 Article ID143194 13 pages 2013

[18] C Huang C Peng X Chen and F Wen ldquoDynamics analysisof a class of delayed economic modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 962738 12 pages 2013

[19] T Adrian and J Rosenberg ldquoStock returns and volatilitypricing the short-run and long-run components ofmarket riskrdquoThe Journal of Finance vol 63 no 6 pp 2997ndash3030 2008

[20] B Kelly and H Jiang ldquoTail Risk and Asset Pricesrdquo WorkingPaper no 19375 The National Bureau of Economic Research2013

[21] R Bansal and A Yaron ldquoRisks for the long run a potentialresolution of asset pricing puzzlesrdquoThe Journal of Finance vol59 no 4 pp 1481ndash1509 2004

[22] T A Rietz ldquoThe equity risk premium a solutionrdquo Journal ofMonetary Economics vol 22 no 1 pp 117ndash131 1988

[23] R Barro ldquoRare disasters and asset markets in the twentiethcenturyrdquoQuarterly Journal of Economics vol 121 no 3 pp 823ndash866 2006

[24] X Gabaix ldquoVariable rare disasters an exactly solved frame-work for ten puzzles in macro-financerdquo Quarterly Journal ofEconomics vol 127 no 2 pp 645ndash700 2012

[25] J A Wachter ldquoCan time-varying risk of rare disasters explainaggregate stock market volatilityrdquo The Journal of Finance vol63 no 3 pp 987ndash1035 2013

[26] T Bollerslev M Gibson and H Zhou ldquoDynamic estimation ofvolatility risk premia and investor risk aversion from option-implied and realized volatilitiesrdquo Journal of Econometrics vol160 no 1 pp 235ndash245 2011

[27] T Bollerslev V Todorov and S Z Li ldquoJump tails extremedependencies and the distribution of stock returnsrdquo Journal ofEconometrics vol 172 no 2 pp 307ndash324 2013

[28] E Fama and K French ldquoMultifactor explanations of assetpricing anomaliesrdquoThe Journal of Finance vol 51 no 1 pp 55ndash84 1996

[29] G Tauchen and H Zhou ldquoRealized jumps on financial marketsand predicting credit spreadsrdquo Journal of Econometrics vol 160no 1 pp 102ndash118 2011

[30] T Bollerslev T H Law and G Tauchen ldquoRisk jumps anddiversificationrdquo Journal of Econometrics vol 144 no 1 pp 234ndash256 2008

[31] G Qin C Huang Y Xie and F Wen ldquoAsymptotic behaviorfor third-order quasi-linear differential equationsrdquo Advances inDifferential Equations vol 2013 article 305 pp 1ndash8 2013

[32] A R Jarrow and E R Rosenfeld ldquoJump risks and the intertem-poral captial asset pricing modelrdquo The Journal of Business vol57 no 3 pp 337ndash351 1984

[33] T G Andersen and T Bollerslev ldquoAnswering the skepticsyes standard volatility models do provide accurate forecastsrdquoInternational Economic Review vol 39 no 4 pp 885ndash905 1998

[34] O Barndorff-Nielsen and N Shephard ldquoEconometric analysisof realized volatility and its use in estimating stochastic volatilitymodelsrdquo Journal of the Royal Statistical Society B StatisticalMethodology vol 64 no 2 pp 253ndash280 2002

[35] O Barndorff-Nielsen and N Shephard ldquoEstimating quadraticvariation using realized variancerdquo Journal of Applied Economet-rics vol 17 no 5 pp 457ndash477 2002

[36] N Meddahi ldquoA theoretical comparison between integrated andrealized volatilityrdquo Journal of Applied Econometrics vol 17 no5 pp 479ndash508 2002

[37] O E Barndorff-Nielsen and N Shephard ldquoPower and bipowervariation with stochastic volatility and jumpsrdquo Journal of Finan-cial Econometrics vol 2 no 1 pp 1ndash37 2004

[38] O E Barndorff-Nielsen and N Shephard ldquoEconometrics oftesting for jumps in financial economics using bipower varia-tionrdquo Journal of Financial Econometrics vol 4 no 1 pp 1ndash302006

[39] T G Andersen T Bollerslev and F Diebold ldquoSome like itsmooth and some like it rough untangling continuous andjump components in measuring modeling and forecastingasset return volatilityrdquo Working Paper Department of Eco-nomics University of Pennsylvania 2003

[40] X Huang and G Tauchen ldquoThe relative contribution of jumpsto total price variancerdquo Journal of Financial Econometrics vol 3no 4 pp 456ndash499 2005

[41] R Merton ldquoOption pricing when underlying stock returns arediscontinuousrdquo Journal of Financial Economics vol 3 no 1-2pp 125ndash144 1976

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Recently several studies have presented the empiricalevidence in favor of jump risk effects on financial assetprices by using nonparametric method and high-frequencydatabase Since the realized volatility (RV) the realized rangevolatility (RRV) and realized jump volatility (RJV) wereproposed those nonparametric measures have been appliedextensively even for the researches in volatility parametricmodels for example GARCH-RV and the heterogeneousautoregressive with realized volatility (HAR-RV) modelThose nonparametric measures not only can be used todepict the volatility dynamic nature such as clusteringlong memory asymmetric leverage effect and the price-volume relation in financial market microstructure theory[10ndash18] but also can be considered as the risk indicationsTherefore investigating those risksrsquo influences on asset pric-ing is of great interest Adrian and Rosenberg [19] explorea new three-factor pricing model including the long-runvolatility component the short-run volatility componentand the market return and they conclude that their modeloutperforms the Fama-French model through reducing thepricing biased errors Apparently volatility can be a risk factorwhich accounts the return premium Following Adrian andRosenberg [19] Kelly and Jiang [20] develop a tail risk factormodel Both models are based on the structural model bytheoretical derivationwith explicit economicalmeanings Fortail risk model by Kelly the bivariate calibration of tail riskfactors withmacrovariables can be based on the long-run riskmodel proposed byBansal andYaron [21] and the disaster riskmodel proposed by Rietz [22] and Barro [23]

Though there are a large number of researches seekingdifferent risk factors for the returns all of them focus on thereturns of index bond market and credit spread Moreovermost of those works are using high-frequency data in theUnited States Inspired by those works we propose a jump-based model to examine the relations between the realizedjump components and equity returns at China market Wewill begin with an illustration of the nonparametric jumpestimation method After daily realized jump volatility isestimated from 5-minute high-frequency trading data wethen calculate monthly jump size jump standard deviationand jump arrival intensity Finally these jump componentsare used to investigate their predicting and explaining powerover one-month-ahead equity return

The remaining parts of this paper are organized as followsSection 2 introduces the identification method for the high-frequency realized jump and presents the jump risk factormodel and Section 3 describes our sample data and itsdescriptive statistics Empirical results are demonstrated inSection 4 and Section 5 is robustness tests analysis Finallythe conclusion is in Section 6

2 Preliminaries and Theories

21 Realized Jump Risk While jumps are known to bevery crucial in the asset pricing [32] estimation of jumpcomponents by parametricmodel has been questioned for thestability over different sample time periods With the avail-ability of high-frequency data nonparametric estimation

method has been developed rapidly Andersen and Bollerslev[33] Barndorff-Nielsen and Shephard [34 35] and Meddahi[36] have presented the use of the so-called realized variancemeasures by utilizing the information in the intraday data formeasuring and forecasting volatilities Barndorff-Nielsen andShephard [37 38] developed a series of the seminal work onbipower variation measure which is then used to divide theRV into continuous diffusion volatility and jump volatility(see Andersen et al [39] and Huang and Tauchen [40])Under the reasonable presumption that jumps on financialmarkets are usually rare and large we follow Huang andTauchen [40] to assume that there is at most one jump perday and that the jump size dominates the daily return when ajump occurs These assumptions allow us to extract the dailyrealized jumps and further to explicitly calculate the monthlyjump intensity size and standard deviation Tauchen andZhou [29] have demonstrated that jump parameters can beprecisely estimated and that the statistic inference is reliable

Compared to parametric models in which jump isestimated by the maximum likelihood MCMC or GMMmethods nonparametric estimation has merit of convenientestimation without assuming specifying underlying driftdiffusion and jump functions The assumption of one jumpper day fits to the compound Poisson jump process ([41]also utilizes the Poisson jump process to describe rare andlarge return jumps which are presumably the responses to thearrivals of important news) and it should be pointed out thatbipower variation also works for the infinite activity jumpsdespite the fact that we focus only on the case of rare and largejumps The following presents the details of jump estimationand detection

Let 119901119905= log(119875

119905) denote the time 119905 logarithmic price of

the asset and assume that it evolves in continuous time as ajump-diffusion process

119889119901119905= 120583119905119889119905 + 120590

119905119889119882119905+ 119869119905119889119902119905 (1)

where 119889119902119905is a Poisson jump process with intensity 120582

119869and

119869119905is the corresponding log jump size distribution following

normal (120583119869 120590119869) Consider

RV119905=

119872

sum

119895=1

1199032

119905119895997888rarr int

119905

119905minus1

1205902

119904119889119904 + int

119905

119905minus1

1198692

119904119889119902119904 (2)

where 119872 is the total number of trades during time 119905 and119905 + 1 and 119895 is the indication for each trade In the realisticfinancial markets the price volatility of financial asset isnot continuous but contains jumps due to the influencearoused by information shock on market Barndorff-Nielsenand Shephard [37 38] proposed two general measures forthe quadratic variation process-realized variance and realizedbipower variation this is presented in the following

BV119905equiv120587

2

119872

119872 minus 1

119872

sum

119895=2

10038161003816100381610038161003816119903119905119895

10038161003816100381610038161003816

10038161003816100381610038161003816119903119905119895minus1

10038161003816100381610038161003816997888rarr int

119905

119905minus1

1205902

119904119889119904 (3)

where radic1205872 = Ε(Π119905) Π119905is a standardized normal distribu-

tion random variable and 119872(119872 minus 1) is the amendment to



119905and BV




119872rarrinfin

997888997888997888997888997888997888rarr RJV119905 (4)


119885119905=

(RV119905minus BV119905) RV119905


(5)



TP119905=

119872

119872 minus 2sdot

119872

4[Γ (76) Γ (12)]3

sdot

119872

sum

119894=3

10038161003816100381610038161199031199051198941003816100381610038161003816431003816100381610038161003816119903119905119894minus1

1003816100381610038161003816431003816100381610038161003816119903119905119894minus2

100381610038161003816100381643

(6)


RJV119905= 119868 (119885

119905gt Φminus1

120572)radic[RV

119905minus BV119905] (7)


119905gt Φminus1



120572)







(9)


days (10)


)]12

(11)





119864119905(119903119894119905+1

) = 120574119872Cov119905(119903119894119905+1

119903119872119905+1

) + 120574119867Cov119905(119903119894119905+1

119903119867119905+1

)

= 120574119872

Cov119905(119903119894119905+1

119903119872119905+1

)

Var119905(119903119872119905+1

)Var119905(119903119872119905+1

)

+ 120574119867

Cov119905(119903119894119905+1

119903119867119905+1

)

Var119905(119903119867119905+1

)Var119905(119903119867119905+1

)

= 120574119872120573119894119872119905

Var119905(119903119872119905+1

) + 120574119867120573119894119867119905

Var119905(119903119867119905+1

)

(12)


119872119905+1 and the

risk factor 119903119867119905+1



119867119905+1 and MV

119905is



119903119894119905+1

= 120572119894119905+ 120574119872120573119894119872

MV119905+ 120574119867120573119894119867

IV119905+ 120577119894119905+1

(13)










119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905 + 120577

119872119898+1

(14)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

(15)


119903119875119898+1

= 120572119875+ 120573119875size119875Size RJV

119898+ 120573119875std119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119875119898+1

(16)



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120574


119898

+ 120574119894stdStd RJV


119898+ 120577119894119905+1

(17)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898

+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(18)119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Std119875Std RJV

119898+ 120573119875Arr119875Arr RJV119898

+ 120574SMBSMB119898+ 120574HMLHML

119898+ 120577119894119898+1

(19)


119898lowastArr RJV

119898 Std RJV

119898lowastArr RJV

119898 into





00

02

04

06

08

10

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

10 20 30 40 50 60 70 80

00

02

04

06

08

10

10 20 30 40 50 60 70 80

Arr

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80

Std

Size Mean












000

005

010

015

020

025

030

10 20 30 40 50 60 70 80001

002

003

004

005

10 20 30 40 50 60 70 80

00

01

02

03

04

05

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

Arr Std

Size Mean


0080

0084

0088

0092

0096

1 2 3 4 50019

0020

0021

0022

0023

0024

0188

0190

0192

0194

0196

0198

0200

00031

00032

00033

00034

00035

00036

00037

1 2 3 4 5

1 2 3 4 51 2 3 4 5

Arr Std

Size Mean







000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13


Pmean RJV size











































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905and BV




119872rarrinfin

997888997888997888997888997888997888rarr RJV119905 (4)


119885119905=

(RV119905minus BV119905) RV119905


(5)



TP119905=

119872

119872 minus 2sdot

119872

4[Γ (76) Γ (12)]3

sdot

119872

sum

119894=3

10038161003816100381610038161199031199051198941003816100381610038161003816431003816100381610038161003816119903119905119894minus1

1003816100381610038161003816431003816100381610038161003816119903119905119894minus2

100381610038161003816100381643

(6)


RJV119905= 119868 (119885

119905gt Φminus1

120572)radic[RV

119905minus BV119905] (7)


119905gt Φminus1



120572)







(9)


days (10)


)]12

(11)





119864119905(119903119894119905+1

) = 120574119872Cov119905(119903119894119905+1

119903119872119905+1

) + 120574119867Cov119905(119903119894119905+1

119903119867119905+1

)

= 120574119872

Cov119905(119903119894119905+1

119903119872119905+1

)

Var119905(119903119872119905+1

)Var119905(119903119872119905+1

)

+ 120574119867

Cov119905(119903119894119905+1

119903119867119905+1

)

Var119905(119903119867119905+1

)Var119905(119903119867119905+1

)

= 120574119872120573119894119872119905

Var119905(119903119872119905+1

) + 120574119867120573119894119867119905

Var119905(119903119867119905+1

)

(12)


119872119905+1 and the

risk factor 119903119867119905+1



119867119905+1 and MV

119905is



119903119894119905+1

= 120572119894119905+ 120574119872120573119894119872

MV119905+ 120574119867120573119894119867

IV119905+ 120577119894119905+1

(13)










119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905 + 120577

119872119898+1

(14)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

(15)


119903119875119898+1

= 120572119875+ 120573119875size119875Size RJV

119898+ 120573119875std119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119875119898+1

(16)



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120574


119898

+ 120574119894stdStd RJV


119898+ 120577119894119905+1

(17)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898

+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(18)119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Std119875Std RJV

119898+ 120573119875Arr119875Arr RJV119898

+ 120574SMBSMB119898+ 120574HMLHML

119898+ 120577119894119898+1

(19)


119898lowastArr RJV

119898 Std RJV

119898lowastArr RJV

119898 into





00

02

04

06

08

10

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

10 20 30 40 50 60 70 80

00

02

04

06

08

10

10 20 30 40 50 60 70 80

Arr

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80

Std

Size Mean












000

005

010

015

020

025

030

10 20 30 40 50 60 70 80001

002

003

004

005

10 20 30 40 50 60 70 80

00

01

02

03

04

05

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

Arr Std

Size Mean


0080

0084

0088

0092

0096

1 2 3 4 50019

0020

0021

0022

0023

0024

0188

0190

0192

0194

0196

0198

0200

00031

00032

00033

00034

00035

00036

00037

1 2 3 4 5

1 2 3 4 51 2 3 4 5

Arr Std

Size Mean







000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13


Pmean RJV size











































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903119894119905+1

= 120572119894119905+ 120574119872120573119894119872

MV119905+ 120574119867120573119894119867

IV119905+ 120577119894119905+1

(13)










119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905 + 120577

119872119898+1

(14)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

(15)


119903119875119898+1

= 120572119875+ 120573119875size119875Size RJV

119898+ 120573119875std119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119875119898+1

(16)



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120574


119898

+ 120574119894stdStd RJV


119898+ 120577119894119905+1

(17)

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898

+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(18)119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Std119875Std RJV

119898+ 120573119875Arr119875Arr RJV119898

+ 120574SMBSMB119898+ 120574HMLHML

119898+ 120577119894119898+1

(19)


119898lowastArr RJV

119898 Std RJV

119898lowastArr RJV

119898 into





00

02

04

06

08

10

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

10 20 30 40 50 60 70 80

00

02

04

06

08

10

10 20 30 40 50 60 70 80

Arr

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80

Std

Size Mean












000

005

010

015

020

025

030

10 20 30 40 50 60 70 80001

002

003

004

005

10 20 30 40 50 60 70 80

00

01

02

03

04

05

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

Arr Std

Size Mean


0080

0084

0088

0092

0096

1 2 3 4 50019

0020

0021

0022

0023

0024

0188

0190

0192

0194

0196

0198

0200

00031

00032

00033

00034

00035

00036

00037

1 2 3 4 5

1 2 3 4 51 2 3 4 5

Arr Std

Size Mean







000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13


Pmean RJV size











































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

02

04

06

08

10

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

10 20 30 40 50 60 70 80

00

02

04

06

08

10

10 20 30 40 50 60 70 80

Arr

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80

Std

Size Mean












000

005

010

015

020

025

030

10 20 30 40 50 60 70 80001

002

003

004

005

10 20 30 40 50 60 70 80

00

01

02

03

04

05

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

Arr Std

Size Mean


0080

0084

0088

0092

0096

1 2 3 4 50019

0020

0021

0022

0023

0024

0188

0190

0192

0194

0196

0198

0200

00031

00032

00033

00034

00035

00036

00037

1 2 3 4 5

1 2 3 4 51 2 3 4 5

Arr Std

Size Mean







000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13


Pmean RJV size











































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










000

005

010

015

020

025

030

10 20 30 40 50 60 70 80001

002

003

004

005

10 20 30 40 50 60 70 80

00

01

02

03

04

05

0000

0002

0004

0006

0008

0010

0012

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

Arr Std

Size Mean


0080

0084

0088

0092

0096

1 2 3 4 50019

0020

0021

0022

0023

0024

0188

0190

0192

0194

0196

0198

0200

00031

00032

00033

00034

00035

00036

00037

1 2 3 4 5

1 2 3 4 51 2 3 4 5

Arr Std

Size Mean







000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13


Pmean RJV size











































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














000100200300400500600700800901

31-Ja

n-07

31-Ju

l-07

31-Ja

n-08

31-Ju

l-08

31-Ja

n-09

31-Ju

l-09

31-Ja

n-10

31-Ju

l-10

31-Ja

n-11

31-Ju

l-11

31-Ja

n-12

31-Ju

l-12

31-Ja

n-13

31-Ju

l-13


Pmean RJV size











































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































5 Robustness Tests

















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















6 Conclusion






Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 15





















Appendix



119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119905+120577119872119898+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898+ 120573119894StdStd RJV

119898

+ 120573119894ArrArr RJV119898 + 120573

119875Size119875Size RJV119898

+ 120573119875Size119875Size RJV

119898+ 120573119875Std119875Std RJV

119898

+ 120577119872119905+1

119903119894119898+1

= 119862 + 120573119894SizeSize RJV

119898

+ 120573119894StdStd RJV

119898+ 120573119894ArrArr RJV119898



119898



119898


00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

01

02

03

04

05

06

10 20 30 40 50 60 70 80000

001

002

003

004

005

006

007

00

02

04

06

08

0000

0004

0008

0012

0016

0020

10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

Arr Std

Size Mean


+ 120573119875Size119875Size RJV

119898+ 119875Std RJV

119898

+ 120573119875Arr119875Arr RJV119898 + 120577

119894119898+1

(A1)



Acknowledgment


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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