CONDITIONAL MEAN DOMINANCE: TESTING FOR SUFFICIENCY OF ANOMALIES

Transcription

1 CONDITIONAL MEAN DOMINANCE: TESTING FOR SUFFICIENCY OF ANOMALIES K. Victor Chow and Ou Hu* ABSTRACT Extensive epirical literature of anoalies suggests that an asset reallocation by buying a subset of the arket portfolio and selling siultaneously another subset of the arket portfolio, according to predictable past history and/or fundaental inforation of firs, generates excess returns. This article deonstrates that evidence of (unconditional) excess returns is necessary but insufficient to prove the existence of anoalies. Instead, the existence of anoalies is confired under the condition that portfolio reallocation generates additional or excess utility for all investors. We show that for all risk-averse investors, conditional ean doinance (CMD) is necessary and sufficient for increasing expected utility fro an asset reallocation strategy. The CMD rule is separated fro an individual preference function and is free fro any odeling specification of the return generating process. Also, the statistical test of CMD is siple and standard. Epirically, the CMD test sufficiently confirs the value and oentu effects but strongly rejects the size effect in the U.S. equity arket. Specifically, there are significantly negative conditional size preius corresponding to a downside arket. In addition, the contrarian effect is weak. Keywords: Anoalies, Value, Size, Moentu, Contrarian, Conditional Mean, Stochastic Doinance. JEL Classification: G11, G14 * College of Business and Econoics, West Virginia University

2 In the past decade, one of the ost extensively studied areas of financial research concentrates on the predictability of cross-section stock returns based on either a fir s fundaental inforation or past price history. For exaple, Faa and French (1992, 1996, 1998) docuent that stocks with high ratios of book-to-arket (B/M), earnings to price (E/P) or cash flow to price (C/P) have higher average returns than low B/M, E/P or C/P stocks. This is called the value effect. Jegadeesh and Titan (1993, 2001), Rouwenhorst (1998), Chan, Jegadeesh and Lakonishok (1996) reveal that short-ter past returns or past earnings predict future returns. Average returns of the best prior perforing stocks (winners) exceed those of the worst prior perforing stocks (losers), so that there is oentu in stock prices. On the other hand, DeBondt and Thaler (1985) find a contrarian effect such that stocks with low long-ter past returns outperfor stocks with high long-ter past returns. Because these patterns in average returns are not explained by the conventional capital asset pricing odel (CAPM) of Sharpe (1964) and Linter (1965), they are typically called anoalies. If anoalies exist, investors are able to create profitable investent strategies by purchasing doinating assets (e.g., the value stocks or the past winners) at the cost of selling doinated assets (e.g., the growth stocks or the losers). Many epirical studies show that this asset reallocation process generates positive (unconditional) average returns. 1 However, despite pervasive evidence of excess returns or anoalies, there reains debate on this issue. Many argue that the epirical evidence ay be siply a data snooping bias in that the anoalies are saple specific results that are unlikely to be observed out of saple. 2 Excess (unconditional) returns ay not be evidence of anoalies in that the preiu between doinating and 1 Again, for the oentu effect, see Jegadeesh and Titan (1993, 2001), Rouwenhorst (1998), Chan, Jegadeesh and Lakonishok (1996). On the contrary, DeBondt and Thaler (1985, 1987), Chopra, Lakonishok, and Ritter (1992), Richards (1997) suggest a profitable contraian strategy of buying the losers and selling the winners. Faa and 1

3 doinated assets is copensation for risk. 3 It can be thus viewed as a risk factor, in equilibriu, priced in addition to the traditional CAPM type systeatic risk. 4 In addition, there could be a isspecification of risk factor in the efficient arket odel. 5 The purpose of this article is twofold: redefining the ter "anoaly" in the fraework of investors' preference choice, and providing a siple ethod to exaine the existence of an anoaly. Without relying on any equilibriu asset-pricing odel, we argue that an anoaly exists if the predictability of stock returns is able to generate excess preferences for all riskaverse investors. Consider that under arket equilibriu, the existing asset allocation in the arket portfolio is optial for expected utility axiizing investors. Any reallocation process of assets inside the arket portfolio thus should provide no additional utility. Analogically, an iproveent of expected utility fro an asset reallocation strategy (e.g. purchasing the pastwinners and selling the past-losers) indicates a non-optial arket and the existence of an anoaly. Based on this siple rationale, we exaine anoalies by testing the condition such that portfolio switching strategies according to financial and/or past-price inforation criterion generate excess expected utility for all investors. Note that the positive expected return of portfolio switching strategies is necessary but insufficient to ensure excess expected utility, and thus is insufficient to conclude the existence of anoalies. That is, if an anoaly exists, all investors prefer the (zero-cost) portfolio switching French (1992, 1996, 1998) argue that positive preiu between value and growth stocks indicates that the investent strategy of longing the value and shorting the growth produces positive returns. 2 See MacKinlay (1995), Knez and Ready (1997) and Loughran (1997). 3 See Brennan, Chordia, and Subrahanya (1998), Chan, Chen and Hsieh (1985), Chan, Karceski and Lakonishok (1998), Chen and Zhang (1998), and Dichev (1998). 4 Faa and French (1993, 1995, 1996, 1998) argue that the value preiu is copensation for systeatic risk. There is no evidence that average returns vary with fir size and B/M in a way that cannot be explained by risk loading, and there is no evidence that variation in risk loadings is uncopensated when it is unrelated to size and B/M. 5 For exaple, in the studies of oentu effect, see Harvey and Siddique (2000), Grundy and Martin (2001), Dittar (2002) and Johnson (2002). 2

4 strategy in that it provides higher utility of returns, and the expected return is positive. However, the positive expected return does not guarantee an increase in investors' utility unless investors are risk-neutral, or the strategies are risk-free. Unfortunately, the assuption of investors' riskneutrality is naïve, and no portfolio strategy can be classified as risk-free without knowing the for of the underlying return distribution of assets and/or the true return generating process. This highlights a potential error in concluding the existence of an anoaly in any epirical studies. To resolve this proble, this paper provides a distribution and preference free testing ethod for exaining the existence of anoalies. Under expected utility axiization, we show that the Conditional Mean Doinance (CMD) criterion is not only necessary but also sufficient for iproving expected utility for all risk-averse investors. Specifically, for instance, the expected utility of risk-averse investors increases by purchasing the winners at the cost of selling losers if and only if the conditional ean returns of winners are always greater than or at least equal to those of the losers, corresponding to the arket (core-portfolio) return distribution. The CMD is siple, intuitive and consistent with the theory of expected utility axiization. Iportantly, the CMD approach requires no odeling specification and assuption about the stock return generating process, and it is separated fro individual utility functions. Furtherore, the statistical inference of the CMD rule is standard and straightforward. Applying the CMD test to investent switching strategies including value vs. growth (value effect), sall vs. large (size effect), short-ter past-winners vs. short-ter past-losers (oentu effect), and long-ter past-losers vs. long-ter past-winners (contraian effect), we find the conditional eans of value preius are significantly non-negative. However, the size effect is strongly rejected. Although the unconditional ean of the size preiu is positive, the conditional size preiu corresponding to the down side arket distribution is, in fact, 3

5 significantly negative. This suggests that iicking portfolios for a factor related to size in the three-factor odel of Faa and French (1993, 1996) ust be used cautiously. The oentu effect is very strong, but the contraian effect is weak. The paper is organized as follows. Section I forally deonstrate that for all risk-averse utility functions, conditional ean doinance (CMD) is a necessary and sufficient condition for iproving expected utility fro an asset reallocation process. We present a siple asyptotically distribution-free inference ethod for testing the CMD. Next, in Section II, we illustrate the procedure by testing the CMD for the following investent strategies: SMB (Sall inus Big) for size effect, HML (Value inus Growth) for value effect, WML ( Short-Ter Past Winner inus Short-Ter Past Loser) for oentu effect, and LMW ( Long-Ter Past Loser inus Long-Ter Past Winner) for contrarian effect. The final section contains concluding rearks. n i= 1 i i r j I. CONDITIONAL MEAN DOMINANCE Suppose investors hold a diversified core-portfolio, and the core-portfolio can be decoposed into a set of n utually exclusive sub-portfolios. For exaple, the core-portfolio can be classified as a cobination of different value and size as suggested by Faa and French (1992) or a coposition of past-winners and past-losers as specified by Jegadeesh and Titan (1993). Without loss of generality, we let the value weighted arket portfolio be the coreportfolio and its return can be written as r = w r, where is the return of the j-th subportfolio, and n w i i= 1 = 1. Assue that investors are axiizing their expected utility of returns. If the existing asset allocation of the core-portfolio is not optial, investors will be able to iprove their expected utility by increasing the holding of one sub-portfolio k fro a decrease of 4

6 the position of another sub-portfolio j. For instance, if anoalies such as the value/size or oentu/contrarian effects do exist, a portfolio reallocation process ipleented by active investent strategies, i.e., buying past winners (or value stocks) at the cost of selling past losers (or growth stocks), will increase investor utility. That is, investors increase wk and decrease w j keeping the su constant, so that (1) dw j dw = 0 + k Fro conventional portfolio theory, we forally identify a criterion to ensure that an asset reallocation increases expected utility for risk-averse investors. Definition 1. For all risk-averse and expected utility axiizing investors, given the existing arket portfolio, the following condition ensures that investors prefer to increase holdings of portfolio k and decrease holdings of portfolio j: d (2) E( U ( W )) = EU '( W )( rk rj ) 0, 6 where dw k W n = 1 + w r i i. i= 1 Follow Shalit and Yitzhaki (1994) and Chow (2001), we deonstrate that conditional ean doinance (CMD) is the necessary and sufficient condition for the arginal increase of expected utility as shown in (2). 6 This is the standard Arrow (1970, p. 101) condition, but for rando wealth. 5

7 Theore 1. Conditional Mean Doinance (CMD). For all risk-averse investors, the inequality (2) holds if and only if τ p (3) r f ( r, r ) dr dr r f ( r, r ) dr dr, or k k k τ p j j j (4) E ( r r ) 0 r τ, k j p for all p, where 0 p 1, E is the expectation operator, and = F 1 ( p). cuulative density function of. r 8 τ p 7 F = p is the It is iportant to note that the inequality (4) is consistent with expected utility axiization without prior knowledge about individual utility functions and the underlying for of the return generating process of assets. Therefore, the CMD rule is separated fro individual investor's utility and is also distribution-free. Lea 1. Necessary Condition of CMD. Let and µ be the ean returns of portfolios k and j respectively. The condition of positive preiu, (5) µ k µ 0, j µ k j is necessary but insufficient to have a positive change in expected utility as referred by the inequality (2). 9 We note that positive unconditional ean-difference is only a necessary condition for increasing expected utility. This highlights the potential probles in any epirical studies in finding the phenoenon of arket anoalies. For exaple, it is obvious that the positive preiu τ p k k k k τ p 7 Since r f ( r, r ) dr dr = pe( r r ), the inequality (4) also holds. 8 Shalit and Yitzhaki (1994) apply the concept of Absolute Concentration Curve (ACC) often used in incoe inequality study to prove the necessary and sufficient conditions of (2). Chow (2001) explicitly shows that the expression of ACC inequality is equivalent to the conditional ean inequality. 9 Since inequality (5) is a necessary condition of inequality (4), the proof of Lea 1 is straightforward. 6

8 calculated by the ean-difference between value (or winner) portfolio and growth (or loser) portfolio is insufficient to show the existence of Value (or Moentu) effects. The purpose of this paper is to provide a testing ethod for the necessity and sufficiency of arket anoalies. Since it is ipossible to exaine an infinite nuber of conditional eans with respect to the entire return distribution of the core portfolio, to siplify the CMD rule, it naturally begins by selecting a set of finite target returns, { t = 1, 2,...,T} corresponding to a set of quantiles of t the core portfolio return distribution. Let I τ t be an indicator variable such that I τ = 1, t if r τ, and, I τ = 0 otherwise. Then, the CMD rule of inequality (3) can be written as t (6) µ ( ) = E[ ( rk rj ) I ] 0, for all τ t. µ ( t τ ) can be characterized as the conditional expected preiu between k and j for all corresponding arket returns below the target but ( ) = 0 for the rest of ); and (3) non-coparability ( τ ) > 0 for at least one, µ. This allows us to develop a test for CMD using standard statistical theory. However, we note that there are three possible outcoes fro the CMD test: (1) equality ( τ ) = 0 for all ) ; (2) doinance ( τ ) > 0 for soe, µ ( t µ ( t µ ( t and ) < 0 for at least one ). Since conventional goodness of fit testing ethods (e.g. µ ( Chi-square and F-test) are unable to distinguish between doinance and non-coparability when the null hypothesis of equality is rejected, a ultiple coparison test becoes necessary. Given a set of N rando saple returns, { r, r, r ),, r, r, r ) }, the saple estiates of CMD ordinates can be expressed as: ( k, 1 j,1, 1 ( k, N j, N, N 7

9 1 1 (7) ˆ µ ( τ t ) = N rk, i I N N N i= 1 i= 1 r j, i I = r I - r I, t = 1,2,, T. k j Let µˆ = ( ( τ ), ( τ,, ˆ µ ( τ )'. We wish to test the null hypothesis that H0: µ = 0. ˆ µ 1 ˆ µ 2 ) k j T ) Theore 2. Sapling Distribution of CMD estiates. Suppose that { ( rk, 1, rj,1, r, 1),, r, r, r ) } are rando saple return with a size of N. Given a set of T targets returns ( k, N j, N, N { = 1, 2,..., T}, N ( µˆ - ) converges in distribution to a T-variate noral distribution t µ with ean zero and a variance-covariance atrix V = JΩJ' = ( σ ij ), where 1 1 J = O O, ( Tx2T ) 1 1 and Ω is a variance-covariance atrix of a set of 2T variables, { r τ 1,, τ T, τ1,r T }. I k k I j I I τ j Proof: Fro direct calculations, (6) and (7) can be used to show that E( µˆ ) =, where µ E( r I k ) = τ ), and E( r µ k ( t j I τ1 ) = µ j ( ). Let Ψ 1x2T = ( r I,, T k I τ τ1 r k, r I j,, r I τt j )' = { ψ i i = 1,...,2T }. For large saples, the Kologorov strong law of large nuber iplies that ψ converges in probability to i ψ i. Fro the Lindeberg-Levy Central Liit Theore, we obtain the result that N ψ ψ ) converges in distribution to N 0, σ ). Further, fro the Craer- ( i i ( ii Wald Theore, it can be shown that N ( Ψ Ψ) converges to a 2T-variate noral distribution, N ( 0, Ω). Finally, using Rao s (1965) theore on the liiting distribution of differentiable functions of rando variables, the liiting distribution of N ( µˆ noral with zero ean and covariance atrix V = JΩJ', where J Tx2T =[ δµ ( τ t ) / δ ψ i ] ]. Ψ= Ψ - ) is also ultivariate µ ˆ 8

10 We note that the full variance-covariance structure of the asyptotic noral distribution of a vector of CMD ordinates depends on only the first and second oents, which can be consistently estiated without any prior specification of the population density underlying the saple data. This fact allows distribution-free inference for hypothesis tests with CMD. To test the null hypothesis, H 0 : ( τ 1 ) = ( τ 2 ) = τ ) = 0, one joint test statistic such as µ µ the chi-square test statistic ay be appropriate. However, when H0 is rejected, further inforation concerning whether the individual CMD ordinates or all ordinates are different fro zero is desirable. This can be tested siultaneously fro a set of sub-hypotheses, { H 01 : k ( τ 1 ) =0, H02: µ ( τ 2 ) =0,, H0T: τ ) =0 }, by exaining ultiple Z test statistics. µ j µ ( T µ ( T (8) Z ( ˆ = N µ / S ), where S ( µ ) N τ / ( ˆ, t N rk i I rj, i I ) i= 1 = 1/ 2, Further, letting the largest absolute value of the test statistic be Z = Max Z * t 1 t T τ, the confidence interval for the extree statistic can be defined as Z * ± SMM ( α; T; ), where SMM ( α; ; ) is the asyptotic critical value of the α point of the Studentized Maxiu Modulus (SMM) distribution with paraeter and degrees of freedo. 10 Thus, the asyptotic joint confidence interval of at least 100(1-α) percent for a set of CMD estiates is: τ (9) Z t ± SMM ( α; T; ) for t=1,2,,τ. 10 See Chow and Denny (1993) 9

11 One can control the size of a ultiple test of CMD estiates by siply coparing the Z- statistics with SMM critical values. 11 The epirical CMD rules using the above inference procedure are suarized as follows: 12 Epirical CMD Inference Rules: (a) Portfolio k doinates (is doinated by) portfolio j, if Z ( ) SMM ( α; T; ) for all, t = 1,2, T, and with at least one strong inequality. (b) No doinance exists otherwise. II. EMPIRICAL ILLUSTRATION To illustrate the CMD test, we exaine the anoalies of size, value, oentu, and contrarian effects in U.S. equity arkets. The data source is twofold: First, onthly returns fro June 1926 to Deceber 2002 for all stocks traded on the New York Stock Exchange and Aerican Stock Exchange were obtained fro the Center for Research on Securities Prices (CRSP). Second, the size preiu (SMB) and the value preiu (HML) were downloaded directly fro Kenneth French's electronic data library. 13 SMB represents the difference between the returns of sall size portfolios and the returns of the large size portfolios. HML is denoted as the return of the high book-to-arket (B/M) ratio stocks inuses that of the low B/M ratio stocks. To test the oentu effect, we construct overlapping oentu portfolios following Jagadeesh and Titan (1993). Since the 6-onth/6-onth portfolio foring strategy is the coon ethod applied by ost studies of the oentu effect, we focus on the case where the sorting period and holding period are both six onths. 14 Specifically, for each onth t starting fro June 1927, all NYSE-AMEX stock returns for onths fro t 11 to t 6 are 11 The SMM table can be found in Hahn and Hendrickson (1971) and Stoline and Ury (1979) 12 Chow (2001) shows that the test is conservative in nature. The test has power to detect doinance for saples with ore than 300 observations. 13 See 10

12 ranked into ten deciles based on the cuulative returns of the foration period between t 11 and t 6. For each decile, an equally weighted portfolio is fored. We then repeat the sae foration of portfolios for the onth t 10 through t 5 and up to t 6 through t 1. The top ranked decile portfolio is designed as the past-winner, and the past-loser is the botto decile portfolio. Therefore, the oentu preiu is the return difference between past-winners and past losers, WML (Winner Minus Loser). For exaining the contrarian effect, we extend the portfolio-holding period fro 6 onths to 36 onths. The portfolio construction process is siilar to that of the oentu strategy. However, the preiu is calculated, contrary to the WML, by taking the return difference between past-losers and past-winners (LMW, Loser Minus Winner). By setting a set of ten equally spaced percentiles such that p = 0.1, p = ,, p = , a set of target returns is deterined by the corresponding saple quantiles of onthly arket portfolio's returns, ( p). For exaple, in Table I, the epirical arket return distribution fro June 1926 to Deceber 2002 can be characterized by a set of τ such that = ˆ 1(0.1) = , τ = ˆ 1(0.2) = ,., = ˆ 1(1.0) = τ 1 Corresponding to the set of arket targets, we then calculate a set of CMD saple estiates according to equation (7) and the associated test statistics according to equation (8). CMD is tested siultaneously fro a set of sub-hypotheses, for instance, { H01: ( τ 1 ) =0, H 02 : µ HML ( τ 2 ) F 2 τ = F ˆ F 1 F µ HML =0,, H 010 : µ ( HML τ 10 ) =0 }. To control the size of these ultiple hypothesis tests jointly, we copare the calculated Z-statistics with the SMM critical value of 2.81 for the 5- percent level of significance. τ See Jagadeesh and Titan (1993, 1996, 2001), Chordia and Shivakuar (2002) 11

13 Table I presents the CMD tests for the onthly value preiu in US equity arkets. Fro the overall saple analysis ( ), the null hypothesis is clearly rejected in that the CMD ordinates are significantly positive corresponding to target arket returns of, τ and τ 10. Since there is no significantly negative ordinate, this indicates that value stocks arginally and conditionally doinate growth stocks. Consequently, one ay conclude that an asset reallocation of switching between value and growth stocks generates excess utility for riskaverse investors. Interestingly, not only the unconditional ean of the value preiu (the CMD ordinate under τ 10 τ 5 6 ) for the saple period is statistically insignificant fro zero, but also seven out of ten CMD ordinates are significantly negative. Obviously, the value effect was negative for the pre-1950, and growth stocks doinated value stocks. By exaining the CMD ordinates for the sub-saple period of post-1950 ( and ), we find that the value effect ainly occurs after This phenoenon of value doinating growth stocks persists strongly for the past two decades. Table II reports the CMD analysis for the US equity size preiu. For all saples and sub-saples, none of the statistics are significantly positive. This deonstrates that there is no size effect in the U.S. equity arket. In fact, the size preiu (SMB) is significantly negative particularly for downside arkets. For instance, fro June 1926 to Deceber 2002, the conditional eans (the CMD ordinates) of SMB according to the down side distribution of the arket portfolio (i.e. for p 0.5) are all negative and significant at the 5 percent level. This iplies that large stocks perfor better than sall firs during down arket conditions. We find that this downside doinance of Big over Sall also exists in the sub-saple periods of pre th and post-1950 th. However, for the current period of , neither Big nor Sall doinates each other, and we fail to reject the null hypothesis of zero CMD ordinates. Our 12

14 evidence of the non-existence of a size effect raises a caution in using Faa and French's (1993, 1996) three-factor risk-return odel. We provide CMD estiates and their associated Z-statistics for the oentu preiu (WML) in Table III. The test strongly rejects the null hypothesis of zero CMD ordinates except for the sub-saple period of pre For the entire saple ( ), alost all statistics are significant at the 5 percent level. The non-negative CMD ordinates for all levels of arket conditions (fro the downside to the upside) indicate that the portfolio switching process of buying past-winners and selling past-losers iproves risk-averse investors' expected utility. 15 We note that the oentu effect is stronger in upside arket conditions (e.g. p 0.5) than in the downside arket. It is iportant to note that none of the statistics calculated fro the pre saple are statistically different fro zero. Therefore, for risk-averse investors, there was indifference between an active oentu investent strategy and the passive rando selection ethod before However, after 1950, oentu portfolios clearly outperfored the arket. For instance, even when onthly arket returns are down to -1.03%, 2.29% and -4.42% during 1950 and 2002, oentu stocks generate average excess returns of 0.35%, 0.21% and 0.14% of excess returns, respectively. Again, a positive oentu preiu persists in the post-1980 period. As shown in Table IV, the contrarian effect appears to be weak in U.S. arkets. Fro the overall saple analysis, although the unconditional ean of the contrarian preiu (LMW) is about 0.69% and is statistically positive, all of the conditional eans for p 0.9 are insignificantly different fro zero. The CMD test fails to reject the null hypothesis for the pre period saples. This indicates there was no contrarian effect. The significantly and 13

15 positively conditional LMW appears in the saple period between 1950 and Interestingly, the overall ean return of LMW during this period is statistically insignificant. We note that conditional eans for the post-1950 LMW are positive, and those of the pre-1950 LMW are negative. This ay explain the insignificant LMWs for the entire saple. In addition, there is a large increase in conditional eans fro p = 0.9 to p = 1.0 (fro to ) in the pre saple. This ay be the cause for the significant ean of the LMW ( at p = 1.0) in the overall saple period. Finally, unlike the value and oentu effects, the contrarian strategy does not generally outperfor the arket in the recent period of III. CONCLUSIONS The positive unconditional ean returns fro switching portfolios according to criteria using firs' financial inforation and/or past-return inforation is necessary but insufficient to prove the existence of anoalies. Without relying on a pre-specified equilibriu asset pricing odel, we deonstrate that an anoaly can be deterined if an asset reallocation process utilizing predictable inforation about stock returns is able to generate excess expected utility for all risk-averse investors. The purpose of this anuscript is to provide a siple ethod to exaine excess expected utility. We show that the Conditional Mean Doinance (CMD) is a necessary and sufficient condition for iproving preference (or for having excess expected utility) for risk-averse investors. The CMD rule is separated fro individual utility in that inforation about the for of the utility function is irrelevant to the preference ranking. In addition, the CMD rule requires no assuption about the for of the return distribution and no specification of the underlying 15 Although the un-conditional ean of (at p = 1.0) is not significantly different fro zero in the joint or ultivariate fraework, the stand-along Z test statistic of 2.29 is statistically significant in the univariate 14

16 return generating process. The statistical inference of the CMD rule is standard and asyptotically distribution-free. Specifically, the statistical CMD rule is as siple as a ultiple Z tests. Epirically, we test the data directly provided by Kenneth French. The value effect is strong in all saple periods. This confirs that the findings in Faa and French (1992, 1996) are robust. However, the size effect is strongly rejected by the CMD tests. The conditional eans of the size preiu (SMB) are generally negative. Since the significantly negative size preiu appears in the left tail of the arket distribution, this indicates that large firs perfor better than sall firs in downside arket conditions. Further, we use the onthly returns fro June 1926 to Deceber 2002 for all stocks traded on the New York Stock Exchange and Aerican Stock Exchange to reconstruct the contraian and oentu portfolio strategies of DeBondt and Thaler (1985) and Jegadeesh and Titan (1993,2001), respectively. The null hypothesis of zero conditional eans (or zero excess expected utility) is statistically rejected in that the CMD ordinates of oentu preiu portfolios (WML) are significantly positive, except for the sub-saple of pre The evidence supports the existence of a oentu effect, where the short-ter (6-onth) return history is predictable for the next 6-onth future returns. Finally, by applying the CMD test to the contraian strategy, we find that vidence of long-ter stock return predictability is weak. The CMD test fails to reject the null hypothesis of zero conditional contraian preiu (LMW) for the current period of post In suary, the CMD approach for testing anoalies is siple, intuitive and straightforward. It is consistent with the fundaental theory of expected utility axiization and iportantly is free fro any further financial odeling specification. fraework. 15

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18 Faa, Eugene F. and Kenneth R. French, 1993, Coon risk factors in the returns on stocks and bonds, Journal of Financial Econoics 33, Faa, Eugene F. and Kenneth R. French, 1995, Size and book-to-arket factors in earnings and returns, Journal of Finance 50, Faa, Eugene F. and Kenneth R. French, 1996, Multifactor explanations of asset pricing anoalies, Journal of Finance 51, Faa, Eugene F. and Kenneth R. French, 1998, Value versus growth: the international evidence, Journal of Finance 53, Grundy, Bruce and J. Spencer Martin, 2001, Understanding the nature and the risks and the sources of the rewards to oentu investing, Review of Financial Studies 14, Hahn, G.J., and R.W. Hendrickson, 1971, A table of percentage points of the distribution of the largest absolute value of k Student t variates and its applications, Bioetrika 58, Harvey, Capbell and Akhtar Siddique, 2000, Tie-varying Conditional Skewness and the Market Risk Preiu, Research in Banking and Finance 1, Jegadeesh, Narasihan, and Sheridan Titan, 1993, Returns to buying winners and selling losers: Iplications for stock arket efficiency, Journal of Finance 48, Jegadeesh, Narasihan, and Sheridan Titan, 2001, Profitability of oentu strategies: an evaluation of alternative explanations, Journal of Finance 56, Johnson, Tiothy C., 2002, Rational oentu effects, Journal of Finance 57, Knez, Peter J. and Mark J. Ready, 1997, On the robustness of size and book-to-arket in crosssectional regressions, Journal of Finance 52, Lintner, John, 1965, The Valuation of Risky Assets and the selection of Risky portfolios in stock portfolios and capital budgets, Review of Econoics and Statistics 47, Loughran, Ti, 1997, Book-to-Market Across Fir Size, Exchange, and Seasonality: Is There an Effect?, Journal of Financial and Quantitative Analysis 32, MacKinlay, A. Craig, 1995, Multifactor odels do not explain deviations fro the CAPM, Journal of Financial Econoics 38, Richards, Anthony J., 1997, Winner-loser reversals in national stock arket indices: Can they be explained?, Journal of Finance 52, Rouwenhorst, K. Geert, 1998, International oentu strategies, Journal of Finance 53,

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20 Table I Conditional Mean Doinance Test for Value Effect in US Equity arket To test the size effect, we calculate CMD ordinates of the value preiu (HML) corresponding to epirical quantiles of arket return distribution, ˆ 1 τ = F ( p ). Monthly value preius and arket returns are obtained directly fro Kenneth French s data library at The corresponding CMD ordinates are statistically different fro zero at the 5 percent level when copared the Z-score with the SMM critical value of The CMD ordinate at p = 1.0 is equivalent to the unconditional ean return. p ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (-0.21) (0.46) (0.53) (1.81) (2.81)* (3.24)* (2.74) (2.53) (2.66) (3.36)* ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (-3.16)* (-4.23)* (-5.34)* (-4.46)* (-3.62)* (-3.58)* (-3.11)* (-2.15) (-1.05) (1.32) ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (5.29)* (6.04)* (6.13)* (7.12)* (7.74)* (7.85)* (6.55)* (5.56)* (5.23)* (3.65)* ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (3.91)* (5.13)* (5.80)* (6.70)* (7.60)* (7.73)* (5.96)* (4.75)* (4.00)* (3.01)*

21 Table II Conditional Mean Doinance Test for Size Effect in US Equity arket To test the size effect, we calculate CMD ordinates of the size preiu (SMB) corresponding to epirical quantiles of arket return distribution, ˆ 1 τ = F ( p ). The onthly size preius and arket returns are obtained directly fro Kenneth French s data library at The corresponding CMD ordinates are statistically different fro zero at the 5 percent level when copared the Z-score with the SMM critical value of The CMD ordinate at p = 1.0 is equivalent to the unconditional ean return ˆ 1 τ = F ( p ) p CMD Ordinates (Z-Score) (-4.93)* (-5.33)* (-5.08)* (-4.93)* (-3.93)* (-3.01)* (-1.37) (-0.82) (0.41) (1.87) ˆ 1 τ = F ( p ) ˆ 1 τ = F ( p ) ˆ 1 τ = F ( p ) (-2.30) (-3.71)* (-3.93)* (-3.89)* (-3.07)* (-2.19) (-1.67) (-1.39) (0.37) (1.55) (-3.91) (-3.78)* (-3.42)* (-3.52)* (-2.69) (-2.10) (-0.38) (-0.03) (0.99) (1.12) (-2.35) (-2.19) (-1.87) (-1.93) (-1.88) (-1.60) (-0.28) (0.14) (0.50) (0.19)

22 Table III Conditional Mean Doinance Test for Moentu Effect in US Equity arket To test the oentu effect, we construct overlapping oentu portfolios following Jagadeesh and Titan (1993). For each onth t starting fro June 1927, all NYSE-AMEX stocks with returns for onths t 11 through t 6 are ranked by 10 deciles based on their foration period (t 11 through t 6) cuulative returns. Then, for each decile, an equal-weighted portfolio is fored. By repeating the sae foration of portfolios for onths t 10 through t 5 up to t 6 through t 1, a oentu preiu (WML, Winner Minus Loser) is calculated by taking the difference between the onthly return of the past-winner (the top ranked portfolio) return and that of the past-loser (the botto ranked portfolio). We then copute CMD ordinates of WML and their associated statistics corresponding to epirical quantiles of arket return distribution, ˆ 1 τ = F ( p ). The corresponding CMD ordinates are statistically different fro zero at the 5 percent level when copared the Z-score with the SMM critical value of The CMD ordinate at p= 1.0 is equivalent to the unconditional ean return of WML. p ** ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (2.40) (3.34)* (4.70)* (4.96)* (5.92)* (6.32)* (6.28)* (6.27)* (4.99)* (2.29) ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (1.66) (2.36) (2.57) (2.40) (2.32) (2.67) (2.53) (2.06) (1.42) (-1.29) ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (2.31) (2.96)* (4.14)* (4.56)* (4.64)* (5.58)* (6.12)* (5.78)* (5.00)* (3.53)* ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (2.64) (2.40) (3.71)* (3.07)* (3.13)* (4.14)* (4.43)* (3.26)* (2.83)* (2.34)

23 Table IV Conditional Mean Doinance Test for Contrarian Effect in US Equity arket For each onth t starting fro Deceber 1930, all NYSE-AMEX stocks with returns for onths t 71 through t 36 are ranked by 10 deciles based on their foration period (t 71 through t 36) cuulative returns. Then, for each decile, an equally weighted portfolio is fored. By repeating the sae foration of portfolios for onths t 70 through t 35 up to t 36 through t 1, a contrarian preiu (LMW, Loser Minus Winner) is calculated by taking the difference between the onthly return of the past-loser (the botto ranked portfolio) and that of the past-winner (the top ranked portfolio). We then copute CMD ordinates of LMW and their associated statistics corresponding to epirical quantiles of arket return distribution, ˆ 1 τ = F ( p ). The corresponding CMD ordinates are statistically different fro zero at the 5 percent level when copared the Z-score with the SMM critical value of The CMD ordinate at p = 1.0 is equivalent to the unconditional ean return of LMW. p ** ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (1.86) (1.76) (1.42) (1.72) (2.35) (2.18) (2.33) (1.91) (2.32) (3.32)* ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (-0.42) (-0.29) (-1.15) (-1.55) (-0.18) (-0.28) (-0.02) (-0.02) (1.09) (2.66) ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (3.63)* (3.40)* (3.27)* (3.25)* (3.50)* (3.26)* (3.36)* (2.97)* (2.68) (2.03) ˆ 1 τ = F ( p ) CMD Ordinates (Z-Score) (1.78) (2.33) (2.26) (2.27) (2.80) (2.07) (1.96) (1.67) (1.40) (0.43)