Do Extreme Returns Matter in Emerging Markets? Evidence from the Chinese Stock Market

Do extreme returns matter in emerging markets? Evidence from the

Chinese stock market

Gilbert V. Nartea

Senior Lecturer in Finance

Department of Accounting, Economics and Finance

Faculty of commerce

Lincoln University

Lincoln 7647

Christchurch, NEW ZEALAND

Email: [email protected]

Phone: 64-3-325-3627

Fax: 64-3-325-3847

Ji Wu*

Assistant Professor of Finance

Institute for Financial & Accounting Studies

Xiamen University

Fujian, 361005 P.R.CHINA

Email:[email protected]

Mobile: 86 18650431259

Fax: 86-592-2181787

*Corresponding author



Abstract

Recent evidence in the U.S. indicates that stocks with high maximum daily returns in

the previous month, perform poorly in the current month. We investigate the presence

of a similar effect in the Chinese stock markets with portfolio-level analysis and firm-

level Fama-MacBeth cross-sectional regressions. Unlike Bali et al’s (2011) findings,

we find no evidence of a MAX effect if we hold portfolios for just one month.

However, we find evidence of a negative relationship between extreme positive

returns (MAX) and stock returns in the Chinese stock market if we extend the holding

period to three and six months. Interpreted together with the strong evidence of risk-

seeking behaviour among Chinese investors, our results are consistent with the

suggestion that the negative MAX effect is driven by investor preference for stocks

with lottery-like features. Our results underscore the importance of in-country

verification, especially in emerging markets, of apparent anomalies initially

discovered in developed stock markets.

JEL classification: F39; G12

Keywords: Cross-section of stock returns; Extreme returns; Predictability; China.

Conference category: Economics and Finance



1. Introduction

Motivated by recent evidence from Kumar (2009) that investors in the U.S. stock

markets exhibit a preference for stocks with lottery-like characteristics, Bali, Cakici,

and Whitelaw (2011) investigate the role of extreme positive returns in the cross-

sectional pricing of stocks in the U.S. Considering stocks that exhibit extreme positive

returns to be lottery-like, Bali, et al. (2011) find that stocks with the highest maximum

daily returns in the previous month (MAX), tend to perform poorly in the following

month. For their decile portfolios, they report negative raw and risk-adjusted return

spreads between portfolios with the highest and lowest maximum daily returns

exceeding 1% per month. The negative relationship is robust even as they control for

size, book-to-market, momentum, short-term reversals, liquidity, and skewness. Bali

et al. (2011) explain the apparent negative MAX effect as a result of investor

preference for stocks with lottery-like characteristics, in particular those with the

potential to produce high maximum daily returns albeit with low probability. A

preference for these stocks leads to overpayment which eventually results to

underperformance in the following month. Such behaviour is consistent with two

descriptive models of decision making under uncertainty -- Tversky and Kahneman’s

(1992) cumulative prospect theory (CPT), as recently extended by Barberis and

Huang (2008) and Kothiyal et al. (2011) and the optimal expectations framework of

Brunnermeier and Parker (2005) and Brunnermeier et al. (2007). Cumulative prospect

theory is a non-expected utility model that accommodates overweighting of tails of

distributions as a modelling device that captures the common preference for lottery-

like (positively skewed) wealth distributions (Barberis and Huang, 2008). In the

optimal expectations framework of Brunnermeier and Parker (2005) and

Brunnermeier et al. (2007) decision makers deliberately choose to distort their beliefs

by overestimating the probabilities of events in which their investments pay off well.

Brunnermeier et al. (2005) show that this model leads to a) portfolios that are

underdiversified, b) investors exhibiting a preference for lottery-like assets and c) that

these lottery-like assets tend to have lower returns.

Empirical evidence of the MAX effect in other markets is still very sparse. Apart from

Nartea, Wu, and Liu (2011) who also document a negative MAX effect in the South

Korean stock market, we are not aware of studies done in other emerging stock

markets. By focussing on China we not only examine the world’s largest emerging

market but we are also presented with a unique opportunity to test Bali et al.’s (2011)

“preference for lottery stocks” explanation of the MAX effect, in as much as Chinese

investors have been shown to exhibit risk-seeking behaviour (see for example Ma,

1996; Ng and Wu, 2006; Lee and Wong, 2009; Fong, Wong, and Yong, 2010).

Studies have also shown that social gambling is considered an acceptable form of

entertainment within the Chinese culture (Raylu, and Oei, 2004; Loo, Raylu, and Oei,

2008) which leads to a predisposition for lottery-like stocks. If Bali et al.’s (2011)

explanation is valid, we expect to document a negative MAX effect in the Chinese

stock markets.

In less than twenty years China’s two stock markets, located in Shanghai and

Shenzhen, have grown from a handful of listed stocks to collectively become the

second largest stock market in the world by the end of 2011 behind only the US stock

markets and the largest emerging stock market. Inspite of this remarkable progress

there is still enormous potential for growth since the proportion of their market

capitalisation to China’s GDP is only 49% as of the end of 2009 compared with 86%

for the U.S. stock markets. As China continues to open its markets to foreign investors,

understanding the factors that drive stock price movements in its stock markets has

become an important issue.

For comparability we follow Bali, et al’s (2011) portfolio sorting approach. We sort

stocks according to maximum daily returns in the previous month, form portfolios on

this basis and track the returns of these portfolios in the succeeding month, reforming

portfolios monthly. We also vary the portfolio holding period to three and six months

and confirm the robustness of our results with a double-sort procedure to control for

various cross-sectional effects including size, book-to-market, momentum, short-term

reversal, illiquidity, idiosyncratic volatility, and skewness. In addition to portfolio

analysis, we perform firm-level Fama-MacBeth cross-sectional regressions as further

robustness tests.

Unlike Bali et al. (2011), we find no evidence of a MAX effect if we hold portfolios

for just one month. However, we find evidence of a negative relationship between

extreme positive returns (MAX) and stock returns if we extend the holding period to

three and six months. Our results support the findings of Bali, et al. (2011) and Nartea

et al. (2011) of similar negative MAX effects in the U.S. and Korean stock markets,

respectively. Interpreted together with the strong evidence of risk-seeking behaviour

among Chinese investors documented in the extant literature, our results are also

consistent with Bali, et al.’s (2011) suggestion that the negative MAX effect is driven

by investor preference for stocks with lottery-like features. Insofar as the negative

MAX effect is driven by overpayment of investors for high MAX stocks which then

leads to underperformance, we suggest that this could be due to price trading limits

which cause lags in the adjustment of prices back to fundamental levels, by up to six

months. We further suggest that the negative MAX effect persists due to short-selling

constraints that limit the opportunity to trade this effect away.

The rest of the paper is organized as follows: Section 2 describes our data and

discusses our estimation procedures. It describes the single-sort method of portfolio

analysis and the double-sort procedure that is used to control for various known

effects. Section 3 reports the empirical results with section 3.1 dealing with results of

portfolio-level analysis while section 3.2 reports results of firm-level Fama-MacBeth

cross-sectional regressions. We use an alternative measure of extreme returns in

section 3.3 and extend the portfolio holding period to three and six months in section

3.4. Section 4 concludes.

2. Data and Methods

Daily and monthly stock returns and accounting data for individual firms were

obtained from DataStream. We use A-shares listed in both the Shanghai and Shenzhen

stock exchanges. The data set covered the period from August 1993 with 83 firms, to

November 2007 with 1,201 firms with an average of 741 firms per month resulting in

a total of 128,325 firm-month observations. The risk-free rate which is defined as the

demand deposit rate was also obtained from DataStream. Market returns are the

value-weighted returns of all firms used in the study.

Following common practice in the existing literature we eliminated investment trusts,

closed-end funds, exchange traded funds, and preferred shares from the sample. We

also deleted stocks with daily returns less than -100% and stocks with monthly returns

greater than 200% to avoid the influence of extreme returns and possible data

recording errors.

At the end of each month, we form tertile portfolios according to MAX, defined as the

maximum daily return in the past calendar month-- high MAX (HMAX), medium

MAX (MMAX), and low MAX (LMAX). We apply holding periods of one, three and

six months and determine the raw and risk-adjusted returns (alpha) of each portfolio.

Portfolios are reformed every month. The risk-adjusted return refers to the Fama-

French (1993) three-factor model alpha (FF-3 alpha) estimated using the full sample

of the time-series of value- or equal-weighted returns for each portfolio. We relate

MAX in t with raw and risk-adjusted returns in month t+1, the three-month return

ending in month t+3, and the six-month return ending in month t+6.

We control for several variables including size, book-to-market (BM), intermediate-

term momentum, short-term reversals, illiquidity, skewness, and idiosyncratic

volatility using dependent bi-variate sorts similar to that employed by Bali et al.

(2011). First we sort on the control factor (i.e., size, value, momentum, and so on) into

tertiles. Within each tertile we sort further into tertiles based on MAX. Then we

average within each MAX category resulting in three portfolios with variation in

MAX but similar levels in the control variable. For example, to control for size,

stocks are first sorted into tertiles according to market capitalisation – Big, Medium,

Small. Within each size category, stocks are sorted again into tertiles according to

MAX – high (HMAX), medium (MMAX), and low MAX (LMAX). Therefore, nine

size-MAX portfolios are formed, namely Big-HMAX, Big-MMAX, Big-LMAX,

Middle-HMAX, Middle-MMAX, Middle-LMAX, Small-HMAX, Small-MMAX, and

Small-LMAX. To control for size, we construct a size-neutral portfolio by averaging

the return (or alpha) spreads within each MAX category. To illustrate, a size-neutral

HMAX portfolio is constructed by averaging the alpha spreads of the three HMAX

portfolios within each size tertile, i.e., Big-HMAX, Middle-HMAX and Small-

HMAX so that we have a high MAX portfolio which contains all sizes. We do the

same for the MMAX, and LMAX portfolios. This process results in three portfolios

with variation in MAX but similar levels in the control variable -- size. We replicate

this procedure for the other control variables.

The size variable at the beginning of month t is defined as the log of the firm’s market

capitalization at the end of month t-1, book-to-market is the firm’s book-to-market

ratio six months prior, i.e. at the end of t-6.1 Following Jegadeesh and Titman (1993),

the momentum variable at time t is the stock’s 11-month past return lagged one month,

i.e. return from month t-12 to month t-2. The short-term reversal variable is defined

following Jegadeesh (1990) as the stock’s one month past return, i.e. return in month

t-1. Skewness of stock i as of the beginning of month tis computed using daily returns

in the past 22 trading days. Idiosyncratic volatility (IV) of stock i at the beginning of

month t is defined as the standard deviation of daily residuals from the Fama-French

three factor model (1) estimated using daily returns in month t-1. Ri,t and MKT are

excess returns of firm i and the market, respectively, over the risk-free rate. We

generate daily values of SMB by sorting stocks at the beginning of every month t into

three groups according to size (Small, Medium, Big). SMB is the difference in daily

returns between the small- and large-stock portfolios. Similarly we generate daily

values of HML by sorting stocks into three groups according to their book-to-market

(BM) ratio in month t-6 (High-, Medium-, and Low-BM). HML is the difference in

daily returns between the High- and Low-BM stock portfolios. Portfolios are

reformed every month.

Ri,t = α + βMKT, i, mMKTt + βSMB, i ,mSMBt + βHML, i ,mHMLt + εi,t (1)

3. Empirical Results

1Clubb and Naffi (2007) also show that expected book-to-market ratios also explain UK stock returns.

However, in this study we only deal with past book-to-market ratios. Michou (2009) also reports that

the predictive power of the book-to-market spread depends on portfolio formation strategies and the

relative proportion of large-caps, small-caps, value, and growth stocks in the portfolio.

3.1 Portfolio Analysis

3.1.1 Univariate sorting

Table 1 shows the returns and FF-3 alpha of portfolios sorted on the maximum daily

returns in the past month (MAX). We report results for both value- and equal-

weighted portfolios. Columns 1 and 3 indicate the absence of MAX effect based on

both equal- (EW) and value-weighted (VW) returns since while the difference

between the returns of the high and low MAX portfolios is negative for both EW and

VW returns, these return spreads are statistically insignificant. However, column 2

indicates a negative MAX effect based on the FF-3 alpha of EW portfolios. A

statistically significant alpha spread of -0.74% suggests that on average, low MAX

portfolios outperform high MAX portfolios. However, this alpha spread is

significantly lower compared with the alpha spread documented by Bali, et al. (2011)

for the U.S. markets as well as the alpha spread documented by Nartea, et al. (2011)

for Korea. The VW alpha spread also suggests a negative MAX effect, but at -0.51%,

it is only marginally significant.

(Insert Table 1 about here)

Table 2 presents the characteristics of the MAX-sorted portfolios. Table 2 indicates

that there appears to be no difference between high and low MAX stocks in terms of

average size, BM ratio, momentum, and illiquidity. However, Table 2 also shows that

high MAX stocks tend to be winners in the previous month, have positively skewed

return distributions and have higher IV than low MAX stocks. These characteristics

all point toward lower returns in the succeeding month based on the short-term

reversal effect (Jegadeesh, 1990; Lehman, 1990), negative IV effect (Ang et al., 2006,

2009), and investor preference for positive skewness (Golec and Tamarkin, 1998;

Mitton and Vorkink, 2007) which implies a negative skewness effect. Therefore, these

variables could potentially explain the negative MAX effect. We test this formally

using dependent bivariate sorts and cross-sectional regressions and report the results

in later sections.


3.1.2 Bivariate sorting

Now we control for size, BM, momentum, short-term reversal, illiquidity ratio,

skewness and IV to test the robustness of the apparent negative MAX effect for EW

portfolios using a battery of bivariate sorts and report the results in Table 3. Following

Bali, et al. (2011) we focus our attention on the alphas since they control for the

standard set of systematic factors.


Our results show that the apparent negative MAX effect we document earlier is not

robust. Though the negative MAX effect survives as we control for size, momentum,

illiquidity, and skewness, and IV, the bivariate sorts show that the MAX effect could

potentially be explained short-term return reversals. Panels A, C, E, F and G show

that controlling for size, momentum, illiquidity and skewness, and IV does not

eliminate the significantly negative average alpha spread at -0.73%, -0.77%, -0.51%,

-0.60%, and -0.45% per month respectively. Controlling for BM by averaging across

BM categories, reduces alpha spread to -0.47% and but it is still marginally

significant. However, if we control for return reversal (panel D), the negative MAX

effect disappears, as the average alpha spread becomes insignificant. It is interesting

to note that in panel D the alpha spread in the current month for last month’s winners

(WNR) is positive while the alpha spread for last month’s losers (LSR) is negative

with both being marginally significant. This result suggests a positive MAX effect for

last month’s winners and a negative MAX effect for last month’s losers. This is

consistent with Bali et al. (2011). However, if we control for the past month’s return

(REV) by averaging the alpha spreads within each MAX category, the average alpha

spread turns very close to zero and becomes statistically insignificant, as the alpha

spreads of the WNR and LSR categories cancel each other out. Therefore our results

show that once we control for past returns (REV), the apparent MAX effect also

disappears. The result of this bivariate sort is consistent with our earlier observation

from Table 2 (column 4) that the negative MAX effect could be due to short-term

reversals given that high MAX stocks also tend to be winners (WNRs) while low

MAX stocks tend to be losers (LSRs).2 In sum, the bivariate sorts appear to indicate

that the negative MAX effect could potentially be explained by short-term return

reversals.

3.2 Firm-level cross-sectional regressions

Since dependent bi-variate sorts cannot be used to control for multiple effects

simultaneously we also conduct firm-level Fama-MacBeth regressions. The portfolio

analysis conducted earlier also loses too much information through aggregation. We

estimate the following model and its nested versions:

Ri,t = β0,t-1 + β1,t-1 MAXi, t-1 + β2,t-1 SIZEi, t-1 + β3,t-1 BMi, t-1 + β4,t-1 MOMi, t-1

2 Crucial to this interpretation is evidence of the existence of a short-term reversal effect in the Chinese

stock market which we investigate by sorting stocks into tertiles each month over the sample period

according their returns in the previous month. We then determine the returns of these portfolios in the

current month. The results ?not reported here clearly indicate evidence of a short-term (one-month)

reversal effect. Winner stocks in the previous month have an average EW (VW) monthly return of

10.67% (10.61%), but the return of this portfolio drops to 0.27% (0.05%) in the current month.

Likewise, loser stocks in the previous month have an average EW (VW) monthly return of -8.74% (-

9.01%), but the return of this portfolio rises to 0.93% (0.65%) in the current month.

+ β5,t-1 REVi, t-1 + β6,t-1 ILLIQi, t-1 + β7,t-1 SKEWi, t-1 + β8,t-1 IVi, t-1 + εi,t-1 (2)

Realized stock return in month t, Ri,t , is regressed on one-month lagged values of the

maximum daily return in the previous month (MAX), log of market capitalization

(SIZE), book-to-market ratio (BM), momentum (MOM), short-term reversal (REV),

illiquidity ratio (ILLIQ), and skewness (SKEW), and realized idiosyncratic volatility

(IV). The variables are as defined earlier. Table 4 reports the time series averages of

the slope coefficients over the 116 months from 1993:04-2007:11 for univariate

regressions. The Newey-West t-statistics are given in parenthesis. The univariate

regression shows a statistically significant negative relation between MAX and the

cross-section of one-month ahead stock returns. The results also show a significant

negative size effect, negative IV, negative short-term reversal, a negative skewness

effect, and a marginally significant positive BM effect consistent with expectations.

The rest of the variables -- MOM, ILLIQ have the expected positive coefficients but

they are not statistically significant.


The result of the bivariate regressions with MAX reported in Table 5 shows that the

MAX effect survives when we control for the variables individually, which is

consistent to some extent with the results of the bi-variate portfolio level analysis

reported in Table 3, except when we control for BM and REV. However, if we control

for all nine variables simultaneously, the negative MAX effect disappears. It is

interesting to note that only the factors related to size and illiquidity are statistically

significant in our multi-variate model and their coefficients are consistent with

expectations. This indicates that the apparent negative MAX effect observed in both

the univariate and bivariate cross-sectional regressions could be explained by both the

size and illiquidity effects. SIZE has a negative coefficient indicative of small firms

outperforming big firms while ILLIQ has a positive coefficient which implies that

investors demand compensation for holding on to relatively illiquid stocks. Therefore,

consistent with our results from portfolio analysis, our firm-level cross-sectional

regression results also indicate the absence of a MAX effect if we restrict the holding

period to one month.


3.3 Alternative measure of extreme returns

Bali et al. (2011) report that the negative MAX effect is stronger if they use the

average of the five highest daily returns in the previous month to sort portfolios. They

call this measure MAX(5). Hence we also test for evidence of a MAX effect using

MAX(5) to sort portfolios. The results reported in Table 6 shows that though both EW

and VW return spreads are negative, they are insignificant. However, the EW alpha

spreads is negative and marginally significant.


As an additional test we also perform firm-level Fama-MacBeth cross-sectional

regressions as in equation 2 and report the results in Table 7. Similar to our results

with MAX we also find in the univariate regression, a statistically significant negative

relation between MAX(5) and the cross-section of one-month ahead stock returns.

The result of the bivariate regressions with MAX(5) also reported in Table 7 shows

that if we control for the variables individually, the MAX effect remains significantly

negative. However just as in case with MAX, if we control for all nine variables

simultaneously, though the MAX(5) coefficient remains negative, it is no longer

significant. It is interesting to note that SIZE, REV and ILLIQ are statistically

significant in our multi-variate model with coefficient signs consistent with

expectations. This indicates that apparent negative MAX(5) effect observed in both

the univariate and bivariate cross-sectional regressions could be explained by the size,

short-term reversals and illiquidity. Therefore whether we use MAX or MAX(5) we

find no evidence of a MAX effect in the Chinese stock market when MAX portfolios

are held for one-month.


3.4 Extended holding periods

As the Chinese stock market is subject to trading price limits3, it could be argued that

stock prices may not have fully adjusted within the one holding month period that we

have so far imposed in our analysis. If the negative MAX effect is driven by investors

who overpay for stocks with extreme positive returns which then results in

underperformance when prices settle to fundamental levels, we expect to find

evidence of a stronger MAX effect when we employ a longer holding period. Hence

we also investigate the presence of the MAX effect for extended holding periods of

three and six months. Table 8 shows the three-month returns and FF-3 alpha of

portfolios sorted on the maximum daily returns in the past month (MAX). Columns

one and three of Table 8 show that similar to the results with a one-month holding

period, the equal- and value-weighted raw return spreads are negative but

insignificant even as we increase the holding period to three months. However unlike

in the case of the one-month holding period, both EW and VW alpha spreads are

3 To reduce stock market’s volatility and protect retail investors, the China Securities Regulatory

Commission imposed trading price limits, which only allows stock prices to move up or down on a

single trading day by a maximum of 10% from the last closing price. The policy artificially makes a

barrier for the stock price to efficiently reflect any news impacting stocks, limiting the efficiency of the

Chinese stock market.

significantly negative. This suggests a negative MAX effect in a three-month holding

period and also suggests at least a three-month lag in the adjustment of prices back to

fundamental levels.


Even when we control for the various effects with bivariate portfolio sorts in Table 9,

the alpha spreads remain significantly negative.


The results become more interesting when we employ firm-level Fama-MacBeth

cross-sectional regressions. The results of univariate regressions reported in Table 10

show a significantly negative MAX effect. We also observe significant size, reversal

and skewness effects with expected signs. Interestingly we also observe a highly

significant anomalous negative IV effect consistent with the findings of Nartea and

Wu (2012).


In Table 11 we report the results when we control for various effects individually with

firm-level bivariate regressions. Table 11 shows that the MAX effect remains

significantly negative in bivariate regressions. In fact it is even stronger when

controlled for the size effect. Of particular interest, we find that the negative MAX

effect appears to neutralise the anomalous negative IV effect. Hence our results are

broadly consistent with Bali et al. (2011) who report that MAX reverses the negative

IV effect documented by Ang et al. (2006, 2009). More importantly, the bottom of

Table 11 shows that the negative MAX effect survives even in a multivariate

regression that controls all variables simultaneously. We note that the anomalous

negative IV effect also does not survive in a multivariate setting. We also observe a

size and momentum effect with the expected signs.


Finally, we also examine the MAX effect in portfolios with a holding period of six

months. The results are similar to the three-month holding period. Table 12 reports the

six-month returns and FF-3 alpha of portfolios sorted on MAX and shows that the EW

and VW return spreads are negative but insignificant. However, the alpha spreads are

negative and highly significant, indicative of a negative MAX effect. The results of

bivariate portfolio sorts reported in Table 13 shows that none of the control variables

can eliminate the negative alpha spreads hence none can explain the negative MAX

effect. But more importantly, the Fama-Mac Beth regressions in Tables 14 and 15

indicate a significantly negative MAX effect in univariate, bivariate and multivariate

regressions. Our results suggest that the negative MAX effect could last as long as six

months.





Overall, we find evidence of a negative MAX effect in the Chinese stock market but

only in extended holding periods. This MAX effect is also apparently weaker than in

the U.S. markets. The alpha spreads that we document range from 0.49% to 0.74% per

month compared with around 1% per month in the U.S. We suggest that the

persistence of the negative MAX effect stems from constraints to short-selling in the

Chinese stock markets which limit the opportunity for arbitrage.

Our results are consistent with a market driven by investors with a preference for

lottery-like stocks – those that have exhibited extreme positive returns in the past

month. Indeed apart from anecdotal evidence, there is a body of literature that

suggests risk-seeking behaviour among Chinese investors. Ng and Wu (2006) report

that Chinese investors tend to prefer stocks with large betas and high idiosyncratic

risk based on a comprehensive analysis of 64.22 million trades of 6.8 million

institutional and individual investors in mainland China. Lee and Wong (2009) also

suggest that Chinese investors tend to trade more heavily on riskier stocks based on an

analysis of panel data drawn from the Shanghai stock market. This is consistent with

Fong, Wong, and Yong (2010) who find evidence that mainland Chinese investors are

more speculative and have higher risk appetites than Hong Kong and international

investors. In an earlier study, Ma (1996) also documents evidence of risk-seeking

behavior among mainland Chinese investors by establishing a positive relationship

between share prices and domestic beta risk. In the psychology literature, Raylu and

Oei (2004) report that gambling is an acceptable form of social activity in Chinese

communities while in a related study, Loo, Raylu, and Oei (2008) find widespread

social gambling among Chinese communities as it is a preferred form of

entertainment. The results of both studies suggest a predisposition among Chinese

investors to prefer lottery-like stocks. In sum, our results support the suggestion of

Bali et al. (2011) that investor preference for lottery-like stocks drives the apparent

negative MAX effect.

4. Concluding remarks

Motivated by Bali et al.’s (2011) findings of a significant role of extreme returns in

the U.S. stock markets, we investigate the existence of the same in the Chinese stock

markets. Due to its novelty there is an obvious lack of studies investigating the so-

called MAX effect in other markets. Insofar as the Chinese stock markets have been

characterised as highly speculative, with the extant literature indicating a

predisposition among Chinese investors to prefer riskier stocks, there is reason to

believe that the negative MAX effect would also be evident in this market. However,

unlike Bali et al’s (2011) findings, we find no evidence of a MAX effect if we hold

portfolios for just one month. But we find evidence of a negative relationship between

extreme positive returns (MAX) and stock returns in the Chinese stock market if we

extend the holding period to three and six months. We suggest that this could be due

to price trading limits resulting in a lag in the price adjustment back to fundamental

levels. Our results underscore the importance of country verification, especially in

emerging markets, of apparent anomalies initially discovered in developed stock

markets. However, interpreted along with the strong evidence of risk-seeking

behaviour among Chinese investors, our results support the suggestion of Bali et al.

(2011) that the negative MAX effect is driven by investor preference for stocks with

lottery-like features.

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Table 1. Returns on portfolios sorted by MAX

At the beginning of every month we sort stocks into tertiles according to their maximum daily

return (MAX) in the past calendar month. We compute each portfolio’s equal- and value-

weighted raw returns for the current month. We also estimate each portfolio’s alpha (α

coefficient) from the FF3-factor model estimated using the full sample of monthly value- or

equal-weighted returns for each portfolio. The last row shows the difference in monthly

returns and differences in alpha between the high and low MAX portfolios. T-statistics are

reported in parenthesis. We conduct the analysis for the full sample period 1993:08-2007:11.

EW portfolios VW portfolios

Average return FF-3 alpha Average return FF-3 alpha

High MAX 0.0025

(0.3174)

-0.0076

(-3.6006) 0.0002

(0.0285)

-0.0049

(-2.5760)

Medium MAX 0.0093

(1.1932)

-0.0015

(-0.7593) 0.0089

(1.1712)

0.0041

(2.9050)

Low MAX 0.0110

(1.4645)

-0.0002

(-0.0943) 0.0065

(0.9420)

0.0002

(0.1015)

High- Low -0.0085

(-0.7892)

-0.0074

(-2.4331) -0.0063

(-0.6242)

-0.0051

(-1.8009)

Table 2. Characteristics of portfolios sorted by MAX


return in the past calendar month (MAX). The table reports for each tertile the average of the

monthly averages of various characteristics of the MAX-sorted portfolios over the period

1993:08-2007:11. Size at the end of month t is defined as the log of the firm’s market

capitalization at the end of month t, BM is the firm’s book -to-market ratio six months prior,

i.e. at the end of t-6. Following Jegadeesh and Titman (1993), Momentum at time t is the

stock’s 11-month past return lagged one month, i.e. return from month t-12 to month t-2.

REV in month t is short-term reversal defined as the return on the stock in month t-1,

following Jegadeesh (1990) and Lehmann (1990). Illiquidity is measured following Amihud

(2002) as the ratio of the absolute monthly stock return and its dollar trading volume.

Skewness is the third standardized moment of returns of the past 22 trading days. IV is the

standard deviation of the residuals of the FF3-factor model, using daily data for the previous

22 trading days. The last row is the difference between the high and low MAX portfolio. T-

statistics are reported in parenthesis.

Size BM Momentum REV Illiquidity Skewness IV

High MAX 7.9025

(184.2036)

0.3586

(34.1126)

0.0646

(2.4185)

0.0497

(5.3525)

3.70E-04

(3.3204)

0.5138

(11.909)

0.0218

(44.3045)

Med MAX 7.9153

(192.8663)

0.3591

(35.5905)

0.0570

(2.3469)

0.0006

(0.0733）

9.32E-06

(8.7902)

0.0826

(1.9210) 0.0166

(39.0706)

Low MAX 7.9966

(184586)

0.3630

(33.7229)

0.0455

(1.9802)

-0.0312

(-4.8894)

1.36E-04

(2.2541)

-0.1537

(-3.1905)

0.0141

(38.9329)

High- Low -0.0941

(-1.5596)

-0.0044

(-0.2930)

0.0190

(0.5402)

0.0808

(7.1806)

2.35E-04

(1.856)

0.6676

(10.321)

0.0076

(12.505)

Table 3.Alpha of equal-weighted double-sorted portfolios

At the end of each month over 1993:08-2007:11, stocks are double-sorted 3x3, first by the

control factor (size, BM, momentum, REV, illiquidity, skewness, and IV) into three portfolios

and then within each portfolio we sort stocks again by their maximum daily return in the past

calendar month (MAX). The alpha of each portfolio is presented with t-statistics in

parenthesis. Alpha refers to the FF-3model alpha using the full sample of monthly returns for

each portfolio. To control for a particular factor, we average the alpha within each MAX

category ending up with three portfolios with dispersion in idiosyncratic volatility but

containing all values of the factor being controlled. Size, BM, momentum, REV, illiquidity,

skewness, and IV are as defined in Table 2. LMAX, MMAX , HMAX refer to low, medium,

and high MAX portfolio, respectively; BIG, big size; MED, medium size; SMA, small size;

HBM, MBM , LBM refer to high, medium, low book-to-market, respectively; WNR, winner;

MID, middle; LSR, loser. HIR, MIR, LIR refer to high, medium, low illiquidity ratio,

respectively; HSK, MSK, LSK refer to high, medium, and low skewness. LIV, MIV, HIV refer

to low, medium, and high idiosyncratic volatility, respectively.

Panel A. Double sort on size (market capitalisation) and MAX LMAX MMAX HMAX HMAX-LMAX

BIG -0.0026

(-1.1767)

0.0005

(0.2631)

-0.0060

(-2.6479)

-0.0034

(-1.0683)

MED -0.0019

(-0.8198)

-0.0046

(-1.9500)

-0.0091

(-3.5626)

-0.0072

(-2.074)

SMA 0.0024

(0.9779)

0.0001

(0.0629)

-0.0089

(-3.9272)

-0.0113

(-3.3264)

AVE -0.0007

(-0.5189)

-0.0013

(-2.1050)

-0.0080

(-5.7635)

-0.0073

(-3.7714)

Panel B. Double sort on value (book-to-market) and MAX HBM 0.0011

(0.3112)

-0.0010

(-0.2675)

-0.0103

(-3.0827)

-0.0114

(-2.4060)

MBM -0.0075

(-2.3670)

0.0021

(0.6396)

-0.0028

(-0.8834)

0.0047

(1.0549)

LBM -0.0004

(-0.1107)

-0.0009

(-0.3279)

-0.0077

(-2.1730)

-0.0073

(-1.4748)

AVE -0.0023

(-1.1653)

0.0001

(0.0526)

-0.0069

(-3.6346)

-0.0047

(-1.7100)

Panel C. Double sort on momentum (11/1/1) and MAX WNR 0.0028

(0.9920)

0.0033

(1.1774)

-0.0012

(-0.3795)

-0.0004

(-0.9576)

MID -0.0033

(-1.2111)

-0.0022

(-0.8766)

-0.0103

(-3.5876)

-0.0070

(-1.7700)

LSR -0.0003

(-0.1011)

-0.0080

(-2.7287)

-0.0124

(-3.8524)

-0.0121

(-2.6738)

AVE -0.0003

(-0.1588)

-0.0023

(-1.4547)

-0.0080

(-4.4959)

-0.0077

(-3.1543)

Panel D. Double sort on one-month past return and MAX WNR 0.0908

(26.215)

0.0934

(26.356)

0.1008

(28.170)

0.0100

(1.9920)

MID -0.0089

(-4.4694)

-0.0094

(-4.7834)

-0.0096

(-4.8731)

-0.0007

(-0.2475)

LSR -0.0929

(-27.839)

-0.0923

(-29.007)

-0.1019

(-31.830)

-0.0090

(-1.9600)

AVE -0.0037

(-2.1120)

-0.0028

(-1.6126)

-0.0036

(-2.0520)

0.0001

(0.0407)

Panel E. Double sort on illiquidity and MAX

HIR 0.0009

(0.2790)

0.0050

(1.5746)

-0.0033

(-0.8720)

-0.0042

(-0.8454)

MIR -0.0027

(-1.2024)

-0.0049

(-2.1440)

-0.0102

(-3.8289)

-0.0075

(-2.1150)

LIR -0.0037

(-1.880)

-0.0035

(-1.720)

-0.0074

(-3.9405)

-0.0037

(-1.3413)

AVE -0.0018

(-1.2446)

-0.0011

(-0.7694)

-0.0070

(-4.1519)

-0.0051

(-2.2990)

Panel F. Double sort on skewness and MAX HSK -0.0041

(-1.6234)

-0.0046

(-1.8700)

-0.0097

(-3.9357)

-0.0056

(-1.5839)

MSK 0.0006

(0.2396)

0.0007

(0.3081)

-0.0086

(-3.5049)

-0.0092

(-2.6547)

LSK 2.94E-06

(-0.0012)

-0.0002

(-0.0625)

-0.0033

(-1.2289)

-0.0033

(-0.8976)

AVE -0.0012

(-0.8184)

-0.0014

(-0.9704)

-0.0072

(-4.8555)

-0.0060

(-2.9348)

Panel G. Double sort on IV and MAX HIV -0.0037

(-1.5419)

-0.0084

(-3.3755)

-0.0129

(-4.7170)

-0.0092

(-2.5470)

MIV 0.0007

(0.3147)

-0.0013

(-0.5948)

-0.0021

(-0.8752)

-0.0028

(-0.8600)

LIV 0.0001

(0.0382)

0.0004

(0.1532)

-0.0013

(-0.5245)

-0.0014

(-0.3957)

AVE -0.0010

(-0.6960)

-0.0031

(-3.7200)

-0.0054

(-3.7583)

-0.0045

(-2.2280)

Table 4. Univariate Fama-MacBeth regression results.

Each month from 1993:08 to 2007:11 we run a firm-level univariate Fama-MacBeth cross-

sectional regression of the return on that month with one-month lagged values of the MAX

and other control variables. Each row reports the time-series averages of the slope coefficients

and their associated t-statistics. MAX and the other control variables are defined in Table 2.

Intercept MAX SIZE BM MOM REV ILLIQ SKEW IV

0.016

(1.96)

-0.157

(-3.53)

0.044

(2.48)

-0.005

(-2.61)

-0.003

(-0.36)

0.024

(1.78)

0.009

(1.12)

0.007

(1.40)

0.007

(0.85)

-0.040

(-2.92)

0.005

(0.65)

0.004

(1.66)

0.008

(0.95)

-0.004

(-3.32)

0.014

(1.69)

-0.385

(-2.59)

Table 5. Bivariate and multi-variate Fama-MacBeth regression results with

MAX.

Each month from 1993:08 to 2007:11 we run a firm-level bi-variate and multi-variate Fama-

MacBeth cross-sectional regression of the return on that month with one-month lagged values

of the MAX and other control variables. Each row reports the time-series averages of the

slope coefficients and their associated t-statistics. MAX and the other control variables are

defined in Table 2.


0.060

(3.44)

-0.178

(-4.24)

-0.006

(-3.03)

0.010

(1.13)

-0.207

(-3.60)

0.018

(1.60)

0.020

(2.33)

-0.183

(-4.98)

0.007

(1.43)

0.013

(1.60)

-0.167

(-3.60)

-0.018

(-1.17)

0.016

(2.04)

-0.168

(-4.51)

0.005

(1.94)

0.0144

(1.70)

-0.1191

(-2.37)

-0.0017

(-1.11)

0.017

(2.04)

-0.114

(-2.48)

-0.184

(-1.11)

0.041

(1.66)

-0.143

(-1.32)

-0.005

(-2.20)

0.033

(1.15)

0.009

(1.57)

-0.019

(-0.88)

0.006

(3.13)

-0.003

(-0.72)

-0.520

(-0.76)

Table 6. Returns on portfolios sorted by MAX(5)

At the beginning of every month we sort stocks into tertiles according to the average of the

five highest daily returns (MAX5) in the past calendar month. We compute each portfolio’s

equal- and value-weighted raw returns for the current month. We also estimate each

portfolio’s alpha (α coefficient) from the FF3-factor model estimated using the full sample of

monthly value- or equal-weighted returns for each portfolio. The last row shows the

difference in monthly returns and differences in alpha between the high and low MAX

portfolios. T-statistics are reported in parenthesis. We conduct the analysis for the full sample

period 1993:08-2007:11.



High IV 0.0020

(0.2505)

-0.0063

(-2.7024) -0.0007

(-0.0859)

-0.0033

(-1.5381)

Medium IV 0.0083

(1.0450)

-0.0023

(-1.1915) 0.0069

(0.9018)

0.0016

(0.9375)

Low IV 0.0092

(1.2564)

-0.0001

(-0.0382) 0.0053

(0.7862)

0.0006

(0.2501)

High- Low -0.0072

(-0.6624)

-0.0062

(-1.9095) -0.0060

(-0.5775)

-0.0039

(-1.2535)

Table 7. Fama-MacBeth regression results with MAX (5).


sectional regression of the return on that month with one-month lagged values of the MAX (5)

and bivariate and multivariate cross-sectional regression of the return on that month with one-

month lagged values of the MAX (5) with other control variables. Each row reports the time-

series averages of the slope coefficients and their associated t-statistics. MAX and the other

control variables are defined in Table 2.

Intercept MAX (5) SIZE BM MOM REV ILLIQ SKEW IV

0.0183

(2.18)

-0.2933

(-3.32)

0.0641

(3.55)

-0.3301

(-3.90)

-0.006

(-3.08)

0.0096

(0.96)

-0.2845

(-2.13)

0.018

(1.47)

0.0189

(2.13)

-0.2661

(-2.91)

0.0062

(1.30)

0.0159

(1.85)

-0.3600

(-2.95)

0.0012

(0.06)

0.0161

(2.04)

-0.2721

(-3.14)

0.0004

(1.82)

0.0179

(2.00)

-0.2329

(-2.38)

-0.0032

(-2.45)

0.0189

(2.19)

-0.2579

(-2.44)

-0.0479

(-0.27)

0.236

(1.27)

0.029

(1.09)

-0.013

(-1.70)

-0.329

(-0.96)

0.059

(1.16)

-0.102

(-1.72)

0.005

(1.90)

-0.004

(-1.64)

-0.702

(-1.18)

Table 8. Returns on portfolios sorted by MAX, three-month holding period.


return in the past calendar month (MAX). We compute each portfolio’s equal- and value-

weighted raw returns for a three-month holding period. We also estimate each portfolio’s

alpha (α coefficient) from the FF3-factor model estimated using the full sample of value- or

equal-weighted returns for each portfolio. The last row shows the difference in monthly

returns and differences in alpha between the high and low MAX portfolios. T-statistics are

reported in parenthesis. We conduct the analysis for the full sample period 1993:08-2007:11.



High MAX 0.0127

(0.9163)

-0.0173

(-5.0302) 0.0053

(0.4051)

-0.0132

(-4.1172)

Medium MAX 0.0256

(1.8465)

-0.0049

(-1.6545) 0.0210

(1.5401)

0.0044

(1.5804)

Low MAX 0.0302

(2.2396)

0.0006

(0.1474) 0.0226

(1.7980)

0.0049

(1.4245)

High- Low -0.0175

(-0.9096)

-0.0179

(-3.4596) -0.0173

(-0.9541)

-0.0181

(-3.8167)

Table 9.Alpha of double sorted portfolios, three-month holding period.

At the end of each month over 1993:08-2007:11, stocks are double-sorted 3x3, first by the control

factor (size, BM, momentum, REV, illiquidity, skewness, and IV) into three portfolios and then within

each portfolio we sort stocks again by their maximum daily return in the past calendar month (MAX).

The alpha of each portfolio is presented with t-statistics in parenthesis. Alpha refers to the FF-3model

alpha using the full sample of 3-month returns for each portfolio. To control for a particular factor, we

average the alpha within each MAX category ending up with three portfolios with dispersion in

idiosyncratic volatility but containing all values of the factor being controlled. Size, BM, momentum,

REV, illiquidity, skewness, and IV are as defined in Table 2. LMAX, MMAX , HMAX refer to low,

medium, and high MAX portfolio, respectively; BIG, big size; MED, medium size; SMA, small size;

HBM, MBM , LBM refer to high, medium, low book-to-market, respectively; WNR, winner; MID,

middle; LSR, loser. HIR, MIR , LIR refer to high, medium, low-illiquidity ratio, HSK, MSK, LSK refer

to high, medium, and low skewness. LIV, MIV, HIV refer to high idiosyncratic volatility, respectively.

Panel A. Double sort on size (market capitalisation) and MAX

Equal-weighted Value -weighted

LMAX MMAX HMAX HMAX-

LMAX


LMAX

BIG -0.0027

(-0.6509)

-0.0004

(-0.1179)

-0.0165

(-4.4103)

-0.0138

(-2.4686)

0.0035

(0.9448)

0.0071

(2.1445)

-0.0106

(-2.8920)

-0.0141

(-2.6585)

MED -0.0017

(-0.3918)

-0.0076

(-2.0005)

-0.0170

(-4.0553)

-0.0153

(-2.5759)

-0.0015

(-0.3535)

-0.0079

(-2.0892)

-0.0168

(-3.9492)

-0.0153

(-2.5454)

SMA 0.0070

(1.6062)

-0.0051

(-1.6006)

-0.0193

(-4.9042)

-0.0263

(-4.4731)

0.0050

(1.1531)

-0.0060

(-1.8751)

-0.0200

(-4.9744)

-0.0250

(-4.2042)

AVE 0.0009

(0.3544)

-0.0044

(-2.2371)

-0.0176

(-7.6780)

-0.0185

(-5.5098)

0.0023

(0.9760)

-0.0023

(-1.1402)

-0.0158

(-6.8288)

-0.0181

(-5.4504)

Panel B. Double sort on value (book-to-market) and MAX

HBM -0.0079

(-1.3800)

-0.0103

(-1.2279)

-0.0228

(-3.9541)

-0.0149

(-1.8323)

-0.0011

(-0.1593)

-0.0046

(-0.4729)

-0.0154

(-2.5898)

-0.0143

(-1.6018)

MBM -0.0046

(-0.7520)

-0.0035

(-0.6332)

-0.0105

(-1.7939)

-0.0059

(-0.7009)

0.0037

(-0.5359)

-0.0031

(-0.4754)

-0.0092

(-1.3666)

-0.0129

(-1.3316)

LBM -0.0031

(-0.5658)

-0.0089

(-1.9737)

-0.0285

(-4.4054)

-0.0254

(-3.0058)

-0.0041

(-0.5858)

-0.0027

(-0.4462)

-0.0242

(-3.0662)

-0.0201

(-1.9043)

AVE -0.0052

(-1.5690)

-0.0076

(-2.0537)

-0.0206

(-5.9050)

-0.0154

(-3.2004)

-0.0005

(-0.1261)

-0.0035

(-0.7997)

-0.0163

(-4.0743)

-0.0158

(-2.8020)

Panel C. Double sort on momentum (11/1/1) and MAX

WNR 0.0061

(1.2003)

0.0010

(0.1778)

-0.0105

(-1.9678)

-0.0166

(-2.235*)

0.0134

(2.5923)

0.0133

(2.317*)

-0.0016

(-0.2844)

-0.0150

(-1.9635)

MID -0.0037

(-0.7560)

-0.0098

(-2.023*)

-0.0147

(-2.7338)

-0.0110

(-1.5085)

0.0011

(0.2222)

-0.0122

(-2.261*)

-0.0128

(-2.2647)

-0.0139

(-1.8452)

LSR -0.0044

(-0.8376)

-0.0133

(-2.6921)

-0.0300

(-5.5702)

-0.0256

(-3.3834)

-0.0044

(-0.8319)

-0.0097

(-1.5979)

-0.0330

(-5.5746)

-0.0286

(-3.6745)

AVE -0.0007

(-0.2263)

-0.0074

(-2.465*)

-0.0184

(-5.9018)

-0.0177

(-4.1342)

0.0034

(1.1213)

-0.0029

(-0.8600)

-0.0158

(-4.8578)

-0.0192

(-4.3299)

Panel D. Double sort on one-month past return and MAX

WNR -0.0094

(-1.9872)

-0.0134

(-3.3920)

-0.0299

(-6.1979)

-0.0205

(-3.0516)

0.0038

(0.7247)

-0.0025

(-0.5108)

-0.0220

(-4.0101)

-0.0258

(-3.4086)

MID -0.0010

(-0.2328)

-0.0062

(-1.7015)

-0.0127

(-3.3421)

-0.0117

(-2.0136)

0.0016

(0.3151)

0.0013

(0.3266)

-0.0095

(-2.0614)

-0.0111

(-1.6162)

LSR 0.0048

(0.9839)

0.0081

(1.7432)

-0.0052

(-1.1859)

-0.0100

(-1.5185)

0.0056

(1.0727)

0.0111

(2.262*)

-0.0027

(-0.5564)

-0.0083

(-1.1499)

AVE -0.0019

(-0.6921)

-0.0038

(-1.6221)

-0.0159

(-6.3402)

-0.0141

(-3.8159)

0.0037

(1.2212)

0.0033

(1.2203)

-0.0114

(-3.9381)

-0.0151

(-3.6124)

Panel E. Double sort on Illiquidity ratio and MAX

HIR -0.0036

(-0.6943)

-0.0021

(-0.4720)

-0.0131

(-2.6073)

-0.0095

(-1.3169)

-0.0085

(-1.4494)

0.0017

(0.3026)

-0.0143

(-2.5601)

-0.0058

(-0.7149)

MIR 0.0034

(0.8919)

-0.0071

(-1.9863)

-0.0202

(-5.0105)

-0.0236

(-4.2775)

0.0066

(1.7169)

-0.0003

(-0.0685)

-0.0170

(-3.5308)

-0.0236

(-3.8159)

LIR 0.0016

(0.3149)

-0.0048

(-2.0166)

-0.0166

(-3.4636)

-0.0182

(-2.6258)

0.0101

(2.1226)

0.0040

(0.8095)

-0.0089

(-2.0206)

-0.0190

(-2.9512)

AVE 0.0005

(0.1717)

-0.0048

(-2.0166)

-0.0166

(-6.2356)

-0.0171

(-4.4903)

0.0027

(0.9736)

0.0018

(0.6397)

-0.0134

(-4.6808)

-0.0161

(-4.0236)

Panel F. Double sort on skewness and MAX

HSK -0.0079

(-1.8028)

-0.0116

(-3.0869)

-0.0252

(-6.0974)

-0.0173

(-2.8765)

-0.0044

(-1.0138)

-0.0046

(-1.1116)

-0.0193

(-3.9275)

-0.0149

(-2.2625)

MSK 0.0007

(0.1685)

-0.0038

(-1.0791)

-0.0179

(-4.2482)

-0.0186

(-3.1315)

0.0113

(2.1946)

0.0022

(0.4936)

-0.0139

(-3.0779)

-0.0252

(-3.6645)

LSK 0.0054

(1.2019)

0.0019

(0.4377)

-0.0043

(-1.0222)

-0.0097

(-1.5758)

0.0034

(0.7518)

0.0048

(1.0158)

0.0004

(0.0773)

-0.0030

(-0.4229)

AVE -0.0006

(-0.2379)

-0.0045

(-2.0450)

-0.0158

(-6.5675)

-0.0152

(-4.3609)

0.0034

(1.2531)

0.0008

(0.3121)

-0.0109

(-3.8280)

-0.0144

(-3.6299)

Panel G. Double sort on IV and MAX

HIV -0.0030

(-0.7478)

-0.0133

(-3.2537)

-0.0296

(-6.2075)

-0.0266

(-4.2572)

-0.0043

(-1.0657)

-0.0119

(-2.6674)

-0.0257

(-4.7018)

-0.0214

(-3.1467)

MIV -0.0028

(-0.6757)

-0.0053

(-1.5755)

-0.0104

(-2.6871)

-0.0076

(-1.3431)

0.0006

(0.1400)

0.0004

(0.0926)

-0.0036

(-0.8244)

-0.0042

(-0.6905)

LIV 0.0010

(0.1905)

0.0016

(0.3980)

-0.0047

(-1.0911)

-0.0057

(-0.8447)

0.0064

(1.1826)

0.0084

(1.8262)

0.0079

(1.6649)

0.0015

(0.2095)

AVE -0.0016

(-0.6205)

-0.0057

(-2.5752)

-0.0149

(-5.9342)

-0.0133

(-3.6952)

0.0009

(0.3407)

-0.0010

(-0.4091)

-0.0071

(-2.5273)

-0.0080

(-2.0780)

Table 10. Univariate Fama-MacBeth regression results, three-month holding

period.


sectional regression of the three-month return (t+1 to t+3) with one-month lagged values (t) of

the MAX and other control variables. Each row reports the time-series averages of the slope

coefficients and their associated t-statistics. MAX and the other control variables are defined

in Table 2.


0.043

(2.01)

-0.384

(-4.51)

0.116

(2.56)

-0.013

(-2.58)

-0.008

(-0.40)

0.090

(2.14)

0.029

(1.34)

0.014

(1.16)

0.020

(0.96)

-0.066

(-2.63)

0.022

(1.04)

0.002

(0.71)

0.025

(1.17)

-0.009

(-4.22)

0.041

(1.84)

-0.972

(-2.79)


MAX, three-month holding period.

Each month from 1993:08 to 2007:11 we run a firm-level bivariate and multivariate Fama-

MacBeth cross-sectional regression of the three-month return (t+1 to t+3) with one-month

lagged values (t) of the MAX and other control variables. Each row reports the time-series

averages of the slope coefficients and their associated t-statistics. MAX and the other control

variables are defined in Table 2.


0.144

(3.27)

-0.397

(-4.97)

-0.014

(-2.82)

0.008

(0.31)

-0.327

(-2.60)

0.084

(1.97)

0.045

(2.02)

-0.335

(-4.41)

0.015

(1.29)

0.037

(1.73)

-0.399

(-4.60)

-0.018

(-0.59)

0.043

(1.99)

-0.367

(-4.82)

0.002

(0.65)

0.040

(1.83)

-0.312

(-3.00)

-0.005

(-1.67)

0.048

(2.18)

-0.275

(-3.69)

-0.504

(-1.41)

0.202

(2.92)

-0.423

(-2.37)

-0.019

(-2.48)

0.010

(0.23)

0.029

(2.17)

0.055

(0.84)

0.002

(0.73)

-0.001

(-0.10)

-0.888

(-1.84)

Table 12. Returns on portfolios sorted by MAX, six-month holding period.


return in the past calendar month (MAX). We compute each portfolio’s equal- and value-

weighted raw returns for a six-month holding period. We also estimate each portfolio’s alpha

(α coefficient) from the FF3-factor model estimated using the full sample of value- or equal-

weighted returns for each portfolio. The last row shows the difference in monthly returns and

differences in alpha between the high and low MAX portfolios. T-statistics are reported in

parenthesis. We conduct the analysis for the full sample period 1993:08-2007:11.



High MAX 0.0270

(1.3963)

-0.0286

(-6.0482) 0.0164

(0.8947)

-0.0239

(-5.3212)

Medium MAX 0.0501

(2.4896)

-0.0061

(-1.5079) 0.0392

(1.9980)

0.0108

(2.4499)

Low MAX 0.0578

(2.8446)

0.0010

(0.2029) 0.0433

(2.2453)

0.0082

(1.6078)

High- Low -0.0308

(-1.0996)

-0.0296

(-4.2230) -0.0269

(-1.0129)

-0.0321

(-4.7196)

Table 13.Alpha of double sorted portfolios, six-month holding period.

At the end of each month over 1993:08-2007:11, stocks are double-sorted 3x3, first by the control

factor (size, BM, momentum, REV, illiquidity, skewness, and IV) into three portfolios and then within

each portfolio we sort stocks again by the maximum daily return in the past calendar month (MAX).

The alpha of each portfolio is presented with t-statistics in parenthesis. Alpha refers to the FF-3model

alpha using the full sample of six-month returns for each portfolio. To control for a particular factor,

we average the alpha within each MAX category ending up with three portfolios with dispersion in

idiosyncratic volatility but containing all values of the factor being controlled. Size, BM, momentum,

REV, illiquidity, skewness, and IV are as defined in Table 2. LMAX, MMAX , HMAX refer to low,

medium, and high MAX portfolio, respectively; BIG, big size; MED, medium size; SMA, small size;

HBM, MBM , LBM refer to high, medium, low book-to-market, respectively; WNR, winner; MID,

middle; LSR, loser. HIR, MIR, LIR refer to high, medium, low illiquidity ratio, respectively; HSK,

MSK, LSK refer to high, medium, and low skewness. LIV, MIV, HIV refer to high idiosyncratic

volatility, respectively.

Panel A. Double sort on size (market capitalisation) and MAX

Equal-weighted Value -weighted


LMAX


LMAX

BIG -0.0062

(-1.1639)

0.0015

(0.3109)

-0.0265

(-4.7534)

-0.0203

(-2.6328)

0.0064

(1.0957)

0.0120

(2.2136)

-0.0151

(-2.7718)

-0.0215

(-2.6881)

MED -0.0046

(-0.8198)

-0.0144

(-2.6579)

-0.0265

(-4.7875)

-0.0219

(-2.7649)

-0.0046

(-0.8198)

-0.0141

(-2.5804)

-0.0266

(-4.8137)

-0.0220

(-2.7775)

SMA 0.0131

(2.1643)

-0.0081

(-1.8153)

-0.0276

(-5.3538)

-0.0407

(-5.0776)

0.0131

(2.1643)

-0.0081

(-1.8153)

-0.0276

(-5.3538)

-0.0407

(-5.0776)

AVE 0.0008

(0.2326)

-0.0070

(-2.4825)

-0.0269

(-8.5604)

-0.0276

(-6.0713)

0.0050

(1.4575)

-0.0034

(-1.1493)

-0.0231

(-7.4533)

-0.0281

(-6.0931)

Panel B. Double sort on value (book-to-market) and MAX

HBM -0.0095

(-1.1638)

-0.0208

(-1.9626)

-0.0304

(-3.6850)

-0.0209

(-1.7913)

0.0059

(0.6212)

-0.0063

(-0.5084)

-0.0230

(-2.6559)

-0.0289

(-2.2564)

MBM 0.0169

(1.9036)

0.0029

(0.3317)

-0.0032

(-0.3374)

-0.0201

(-1.5615)

0.0242

(2.4862)

0.0075

(0.7490)

-0.0022

(-0.2033)

-0.0264

(-1.8280)

LBM -0.0061

(-0.8170)

-0.0132

(-2.0666)

-0.0464

(-5.0724)

-0.0403

(-3.3952)

-0.0180

(-1.7262)

-0.0068

(-0.8766)

-0.0392

(-3.5261)

-0.0212

(-1.3937)

AVE 0.0004

(0.0913)

-0.0104

(-2.0551)

-0.0267

(-5.1638)

-0.0271

(-3.8639)

0.0040

(0.7098)

-0.0019

(-0.3170)

-0.0215

(-3.6378)

-0.0255

(-3.1127)

Panel C. Double sort on momentum (11/1/1) and MAX

WNR -0.0084

(-1.2809)

-0.0185

(-3.1936)

-0.0385

(-6.1144)

-0.0301

(-3.2989)

0.0104

(1.2708)

0.0018

(0.2495)

-0.0338

(-4.8751)

-0.0442

(-4.1244)

MID -0.0025

(-0.4358)

-0.0054

(-1.1508)

-0.0171

(-3.1005)

-0.0146

(-1.8392)

0.0065

(0.9692)

0.0072

(1.3423)

-0.0109

(-1.4147)

-0.0174

(-1.6938)

LSR 0.0027

(0.4149)

0.0038

(0.5988)

-0.0165

(-2.7590)

-0.0192

(-2.1716)

0.0038

(0.4651)

0.0087

(1.2279)

-0.0157

(-2.2701)

-0.0195

(-1.8255)

AVE -0.0027

(-0.7503)

-0.0067

(-2.044*)

-0.0240

(-7.0050)

-0.0213

(-4.2563)

0.0069

(1.5398)

0.0059

(1.5610)

-0.0201

(-4.8591)

-0.0270

(-4.4294)

Panel D. Double sort on one-month past return and MAX

WNR -0.0084

(-1.2809)

-0.0186

(-3.1936)

-0.0385

(-6.1144)

-0.0301

(-3.2989)

0.0104

(1.2708)

0.0018

(0.2495)

-0.0338

(-4.8751)

-0.0442

(-4.1244)

MID -0.0025

(-0.4358)

-0.0054

(-1.1508)

-0.0171

(-3.1005)

-0.0146

(-1.8354)

0.0065

(0.9692)

0.0072

(1.3423)

-0.0109

(-1.4147)

-0.0174

(-1.6938)

LSR 0.0027

(0.4149)

0.0038

(0.5988)

-0.0165

(-2.7590)

-0.0192

(-2.1715)

0.0038

(0.4651)

0.0087

(1.2279)

-0.0157

(-2.2706)

-0.0195

(-1.8289)

AVE -0.0027

(-0.7503)

-0.0067

(-2.0442)

-0.0240

(-7.0050)

-0.0213

(-4.2563)

0.0069

(1.5398)

0.0059

(1.5610)

-0.0201

(-4.8591)

-0.0270

(-4.4294)

Panel E. Double sort on Illiquidity ratio and MAX

HIR -0.0155

(-2.2983)

-0.0081

(-1.2582)

-0.0331

(-4.6521)

-0.0176

(-1.8029)

-0.0267

(-3.7152)

-0.0098

(-1.3395)

-0.0414

(-5.3603)

-0.0147

(-1.3944)

MIR 0.0027

(0.5314)

-0.0105

(-2.2518)

-0.0342

(-6.8792)

-0.0369

(-5.1151)

0.0036

(0.6158)

-0.0085

(-1.6207)

-0.0388

(-6.3623)

-0.0424

(-4.9962)

LIR 0.0143

(2.0230)

0.0042

(0.6806)

-0.0204

(-3.1650)

-0.0347

(-3.6048)

0.0251

(3.0089)

0.0225

(3.1920)

-0.0103

(-1.5677)

-0.0354

(-3.3383)

AVE 0.0005

(0.1356)

-0.0048

(-1.4360)

-0.0292

(-8.8051)

-0.0297

(-5.7578)

0.0007

(0.1604)

0.0014

(0.3659)

-0.0302

(-7.6470)

-0.0308

(-5.3801)

Panel F. Double sort on Skewness and MAX

HSK -0.0087

(-1.3732)

-0.0130

(-2.3826)

-0.0341

(-5.7811)

-0.0254

(-2.9428)

0.0023

(0.3533)

-0.0038

(-0.5400)

-0.0317

(-4.4631)

-0.0340

(-3.5074)

MSK 0.0018

(0.3126)

-0.0014

(-0.2934)

-0.0279

(-4.9577)

-0.0297

(-3.7169)

0.0124

(1.5824)

0.0118

(1.7828)

-0.0220

(-3.5720)

-0.0344

(-3.4525)

LSK 0.0058

(1.0094)

-0.0052

(-0.8953)

-0.0180

(-3.0195)

-0.0238

(-2.8520)

0.0055

(0.7853)

-0.0066

(-0.9446)

-0.0054

(-0.6895)

-0.0109

(-1.0262)

AVE -0.0004

(-0.1069)

-0.0065

(-2.0882)

-0.0267

(-7.9146)

-0.0263

(-5.4709)

0.0067

(1.6235)

0.0005

(0.1182)

-0.0197

(-4.8054)

-0.0264

(-4.5328)

Panel G. Double sort on IV and MAX

HIV -0.0111

(-2.0333)

-0.0252

(-4.7049)

-0.0448

(-7.5052)

-0.0337

(-4.1403)

-0.0129

(-2.3860)

-0.0265

(-4.7736)

-0.0436

(-6.3163)

-0.0307

(-3.5038)

MIV -0.0016

(-0.2853)

-0.0083

(-1.8176)

-0.0190

(-3.4453)

-0.0174

(-2.1967)

0.0013

(0.2025)

0.0045

(0.7400)

-0.0076

(-1.2215)

-0.0089

(-0.9750)

LIV 0.0038

(0.5439)

0.0073

(1.3002)

-0.0017

(-0.2778)

-0.0055

(-0.5972)

0.0137

(1.7035)

0.0232

(3.2014)

0.0192

(2.8784)

0.0055

(0.5271)

AVE -0.0030

(-0.8472)

-0.0087

(-2.8990)

-0.0218

(-6.4395)

-0.0189

(-3.8708)

0.0007

(1.7035)

0.0004

(0.1099)

-0.0107

(-2.7965)

-0.0114

(-2.0792)

HIV -0.0111

(-2.0333)

-0.0252

(-4.7049)

-0.0448

(-7.5052)

-0.0337

(-4.1403)

-0.0129

(-2.3860)

-0.0265

(-4.7736)

-0.0436

(-6.3163)

-0.0307

(-3.5038)

Table 14. Univariate Fama-MacBeth regression results, six-month holding

period.


sectional regression of the six-month return (t+1 to t+6) with one-month lagged values (t) of

the MAX and other control variables. Each row reports the time-series averages of the slope

coefficients and their associated t-statistics. MAX and the other control variables are defined

in Table 2.


0.082

(2.19)

-0.669

(-6.08)

0.213

(2.94)

-0.023

(-2.98)

-0.013

(-0.36)

0.140

(1.90)

0.055

(1.55)

0.016

(0.82)

0.047

(1.31)

-0.046

(-1.35)

0.045

(1.25)

0.002

(0.50)

0.049

(1.36)

-0.012

(-4.67)

0.077

(2.07)

-1.878

(-4.07)


MAX, six-month holding period.

Each month from 1993:08 to 2007:11 we run a firm-level bivariate and multi-variate Fama-

MacBeth cross-sectional regression of the six-month return (t+1 to t+6) with one-month

lagged values (t) of the MAX and other control variables. Each row reports the time-series

averages of the slope coefficients and their associated t-statistics. MAX and the other control

variables are defined in Table 2.


0.264

(3.66)

-0.687

(-6.56)

-0.025

(-3.21)

0.009

(0.19)

-0.493

(-2.78)

0.142

(1.85)

0.083

(2.16)

-0.535

(-4.70)

0.018

(0.92)

0.083

(2.23)

-0.742

(-5.62)

0.044

(1.00)

0.081

(2.12)

-0.616

(-6.44)

0.002

(0.45)

0.079

(2.09)

-0.627

(-4.12)

-0.002

(-0.53)

0.092

(2.43)

-0.440

(-4.05)

-1.142

(-2.22)

0.313

(3.35)

-0.748

(-2.54)

-0.030

(-2.68)

0.059

(1.12)

0.043

(2.06)

0.143

(1.49)

0.005

(0.65)

0.005

(0.63)

-0.030

(-1.51)

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