Stochastic dominance in stock market 3

Chapter International Momentum Strategy: A Stochastic Dominance Approach∗ 3.1 Introduction Jegadeesh and Titman (1993) discover that past winners outperform past losers by 1% per month in US markets. Since then, this original result has subsequently been extended in numerous studies. The term “momentum strategy” is widely used and debate still continues about its abnormal returns and the sources of profits in finance literature. Momentum strategy means to purchase (sell) stocks that have recently increased (decreased) significantly in price on the belief that they will continue to increase (decrease). Momentum strategy continues to be profitable in US markets. For example, Chan, Jegadeesh and Lakonishok (1996) find momentum profits based on earnings announcements in US markets. Moskowitz and Grinblatt (1999) show momentum profits emerging across industry-sorted portfolios. Hong, Lim and Stein (2000) detect that the momentum profits vary according to firm size and analyst coverage. Jegadeesh and Titman (2001) disclose that similar momentum profits have continued to appear in the 1990’s. Lewellen (2002) reports that size and book-to-market portfolios exhibit momentum profits that are as strong as that found in individual ∗ This chapter has been accepted by Journal of Financial Markets, co-author with Fong W.M. and Wong W.K. 47 stocks and industries. Chen and Hong (2002) document the existence of momentum profits according to size and book-to-market portfolios. Momentum profits exist globally. Besides the US markets, some studies have evaluated the profitability of momentum strategy for international equities. Richards (1997) documents similar results of winner-loser reversals in indices from 16 countries. Rouwenhorst (1998) finds medium-term return continuation in European markets. Liu, Strong and Xu (1999) show existence of significant momentum profits in the UK stock market. Chan, Hameed and Tong (2000) report the profitability of momentum strategy in 23 national stock markets indices for short holding periods. Griffin, Ji and Martin (2003) observe that the occurrences of momentum profits are economically significant around the world. The momentum effect seems to suggest that returns of winners are too high and returns of losers are too low. Thus, the hypothesis that the portfolio of winners (W) dominates the portfolio of losers (L) will be tested in this study. Asset pricing models play an important role in financial economics. Asset pricing models assume that investors behave rationally. If stocks are priced rationally, then differences in expected returns must reflect differences in risk. A high expected return serves as compensation for high risk. However, empirical studies document that stocks sorted on momentum have returns which cannot be explained by their covariance with the market portfolio. It has been found that past winners can have higher returns and lower variances. The failure of the asset pricing models may be caused by omitted moments of the returns distribution. 48 There are different interpretations and explanations on the momentum effect. This will be discussed in the next section. Despite many extensive studies, the actual causes for momentum profits are still controversial and inconclusive. All the standard asset pricing models such as CAPM and Fama-French three-factor models fail to explain the existence of momentum profits. Nonetheless, the search for more general asset pricing models continues on the premise that existing models may be inadequate because of omitted risk factors. This study intends to re-examine the profitability of momentum strategies in international stock markets using the stochastic dominance (SD) approach. The SD approach differs from the traditional approach of studying momentum profits. It endorses the minimum assumptions of investor’s utility function and study the distribution of returns directly. SD approach is applied to distinguish between the hypothesis that there exists some (more general) asset pricing model that can explain the momentum effect versus the alternative hypothesis that there is no asset pricing model consistent with risk-averse investors that can rationalize the momentum effect. If W stochastically dominates L, then it is unlikely that the problem is due to omitted risk factors but more likely that momentum reflects market inefficiency. By applying the SD approach, this study seeks to provide an alternative interpretation to the momentum puzzle in international equity markets. Moreover, it may shed some light on the relationship between short-term momentum and market efficiency. The market efficiency groups believe that new information is anticipated, thus, opportunity to use the new information to earn abnormal return is limited. If W stochastically dominates L at first order, for example, investors can increase their wealth by short selling L and go long with W. Such a simple investment strategy is inconsistent with efficient market conditions. The market is considered to be efficient 49 by SD rules if it is impossible for investors, independent of their utility functions, to increase their individual expected utility respectively. SD has important advantages over the traditional market rationality test. Traditional tests of market rationality compare observed returns to expected returns generated by some equilibrium models such as CAPM. Thus, abnormal returns that indicate market inefficiency may be attributed to omitted risk factors or model misspecification. In contrast, SD approach examines the entire distribution of returns and eliminates the need to identify a particular mathematical form of utility function. Falk and Levy (1989), Bernard and Seyhun (1997) and Larsen and Resnick (1999) suggest that SD is an alternative method to be considered in testing market efficiency. SD approach is used to by-pass the strong assumptions in the conventional asset pricing models. For instance, the CAPM model which is derived within a meanvariance framework need a strong assumption on the investors’ preferences or the distribution of asset returns. These assumptions are too restrictive because they imply that higher order moments of asset returns are irrelevant with respect to the investor’s welfare. By contrast, the SD approach employs the whole distribution of returns; hence, risk and returns are incorporated into the distribution itself. In addition, the momentum profits are highly non-normal in this study. The substantial deviation from normality and the relevance of higher order moments of portfolio returns to investor’s welfare motivates the adoption of the SD approach in this study. From the results of this study, it is known that the profitability of momentum strategy continues to exist globally. W is preferred to L in the lower range of returns 50 while the reverse is true in the upper range. It appears that stocks with higher downside risk have higher expected returns than stocks with lower downside risk. Most importantly, the fact that W dominates L at second- and third-order SD, implies that all risk-averse and decreasing absolute risk-averse investors prefer W to L. Overall, the evidence is discouraging for the advocates of the efficient market hypothesis, for it means that asset pricing models based on the assumption that investors who are risk averse, cannot explain the momentum effect. In other words, the failure of existing asset pricing models to explain momentum may have less to with the possibility of omitted risk factors than their ability to capture the irrational aspects of momentum investors (e.g., Daniel, Hirshleifer and Subrahmanyam 1998; Hong and Stein 1999). The results of this study are robust to two different sub-periods with different risk and returns characteristics and survive reasonable transaction costs for international index funds. This chapter is organized as follows. The next section reviews the literature. Section explains SD theory and tests. Section covers data and methodology. Section shows the empirical results for SD tests; section constructs on the robustness tests. Section presents the mean-variance analysis and section concludes this chapter. 3.2 Literature Review Much research has focused on whether momentum profits can be explained by risk. Jegadeesh and Titman (1993) find that momentum cannot be explained by exposure to market risk alone. Fama and French (1996) propose two other risk measures. The first is a high minus low book-to-value (HML) risk factor, which is usually interpreted as a 51 firm distress proxy. The second is a small minus large firm size (SML) risk factor to proxy for the higher risk and lower liquidity of small firms. Fama and French (1996) find that their unconditional three-factor model cannot account for momentum profits. Grundy and Martin (2001) show that a conditional three-factor model is unable to explain momentum as well. Conrad and Kaul (1998) conjecture that the three-factor model may be misspecified and that momentum returns may reflect an omitted component of returns that differs across stocks. In other words, the existence of momentum profits may simply be due to cross-sectional dispersions in unconditional expected returns. They show that the momentum strategy’s average profits reflect the result of buying highmean-return securities and selling low-mean-return securities. If the differences in the unconditional mean returns can be attributed to variations in expected returns, the momentum profits are due to cross-sectional dispersions in risk. Grundy and Martin (2001) test the Conrad-Kaul conjecture by using each stock as its own risk control. They find that the momentum strategy still yields excess returns of 9.24% per annum in the period of 1966-1995. Jegadeesh and Titman (2002) also show that differences in unconditional expected returns explain very little of the momentum profits. Moskowitz and Grinblatt (1999) argue that momentum profits may also be due to industry risk exposures that are not captured by standard factor models. Grundy and Martin (2001) test for industry risk but find that it cannot fully explain the momentum effect. In particular, a random industry strategy still earns statistically significant returns in months other than January. Johnson (2002) proposes a single-firm model which can produce such effects when expected dividend growth rates vary over time. 52 Business cycle risk is another potential explanation of the momentum effect. If the expected return to the momentum strategy is related to business cycle risk, then one would expect momentum profits to be lower in poor economic states than in good economic states. Griffin, Ji and Martin (2003) provide international evidence against this hypothesis: momentum profits appear to be large and statistically significant across good and bad economic states. They also show that the multifactor macroeconomic model by Chordia and Shivakumar (2002) cannot explain momentum across markets. In short, current risk-based explanations fail to fully account for the momentum effect. As Grundy and Martin (2001) summarize succinctly, “A full understanding of the source of the risk-adjusted profitability of the momentum strategy remains an open question”. The failure of risk-based models has led to research which attempts to explain momentum in terms of behavioral inconsistencies of investors. The behavioral group like Chan, Jegadeesh and Lakonishok (1996), Barberis, Shleifer and Vishny (1998), Hong and Stein (1999), Jegadeesh and Titman (2001) etc. conclude that the existence of momentum profits is inconsistent with the efficient market hypothesis. They try to explain these abnormal profits by the investors’ over-reaction and under-reaction towards information. For example, Chan, Jegadeesh and Lakonishok (1996) show that due to market responding gradually to new information, future returns are predictable from their prior returns and prior news about earnings. Daniel, Hirshleifer and Subrahmanyam (1998) attribute momentum to the fact that investors are over 53 confident and tend to over-react to confirming news while attributing wrong forecasts to external noise. Over confidence following the arrival of confirming news leads to further over-reaction, generating momentum in stock prices. Hong and Stein (1999) also attribute momentum to over-reaction, and add that over-reaction is a delayed response to the slow diffusion of private information. They also offer conjecture that the less risk-adverse investors are, the greater is the degree of over-reaction. This implies that momentum should be stronger in good economic states than in bad economic states. There is increasing evidence supporting some aspects of behavioral explanations of momentum. For example, behavioral models imply that momentum profits should be better explained in terms of stock-specific risks than systematic risks. Consistent with the hypothesis, Grundy and Martin (2001) show that a momentum strategy that defines W and L based on stock-specific returns (residuals from the three-factor model) is significantly more profitable than a strategy that ranks stocks on total returns. Other studies focus on more specific aspects of idiosyncratic risk. For example, Chan, Jegadeesh and Lakonishok (1996) show that momentum profits coexists with earnings momentum, while Lee and Swaminathan (2000) document stronger momentum effects in stocks with high turnover. Coopers, Gutierrez and Hameed (2004) test the Hong-Stein hypothesis by looking at momentum profits during the different states of the market. They find that momentum profits arise only following up-market periods. They interpret this finding as consistent with the Hong-Stein hypothesis that there is more delayed over-reaction 54 and, hence, stronger momentum following periods when the market is up and investors are less risk averse. Although these evidences point to investor irrationality as primary cause of momentum, many aspects of behavioral theories are intrinsically hard to test. If momentum is due to over-reaction, momentum returns must eventually reverse as investors correct the mispricing. Unfortunately, this hypothesis is difficult to refute since behavioral theories not specify a precise time frame for such reversals to occur. Thus, to behavioral skeptics, the issue of whether momentum profits are compensation for risk or reflect investors’ irrationality remains an open question. The goal of this study is to shed light on this debate by asking whether the momentum effect is consistent with any rational asset pricing model. To address this issue, a general testing framework that depends only on primitive assumptions is needed. SD theory is useful in this regard as it provides rules for unambiguous ranking of risky choices based on utility theory rather than narrowly specified or adhoc asset pricing models. 3.3 Stochastic Dominance Theory and Tests SD provides a general framework for studying economic behavior under uncertainty. It is a nonparametric analysis which can let the data “speak for themselves” (Post 2003). SD rules examine the whole distribution of returns and no need to identify a specific risk measure. SD criteria not depend on the typical risk-return trade-off imposed by conventional models. The SD rules are usually stated so that one is preferred to the other. SD approach has proved to be one of the useful tools in the 55 investment decision-making under uncertainty. It can rank the portfolios with fewer restrictions on investor’s utility function and the distribution of asset returns. SD does not require the assets being compared to be mutually exclusive, uncorrelated nor independent. One may refer to Theorem and in Hadar and Russell (1971), Theorem in Fishburn (1974) and Theorem in Wong and Li (1999). There are also many empirical studies that compare correlated and dependent assets or assets which are not mutually exclusive, for examples, Levy and Sarnat (1970), Joy and Porter (1974), Kroll and Levy (1979), Levy (1985) and Barrett and Donald (2003). SD comparisons allow for interactions between different assets. The advantage of the SD methodology is that we can evaluate investor’s utility preferences towards alternative investment strategies without the need to rely on asset pricing benchmarks. Although SD is not as popular as asset pricing model to empirical applications in finance literature, there are many successful examples. Levy and Hanoch (1970), Levy and Sarnat (1970, 1972), Porter and Gaumnitz (1972), Porter (1973, 1974), Joy and Porter (1974) etc. conduct different studies in various stock markets and mutual funds to compare the efficient set between SD and Mean-Variance (MV). Levy and Kroll (1979) and Kroll and Levy (1979) test SD with riskless asset criterion. Falk and Levy (1989) demonstrate SD technique via the examination of market reaction to quarterly earnings announcements. Seyhun (1993) uses SD approach to test the January effect in NYSE. Larsen and Resnick (1999) compare SD analysis with the traditional event study methodologies and find that SD analysis is better than the latter. Best, Best and Yoder (2000) use SD approach to test the performance of value stocks. 56 Figure 3.4: DD Statistics of Winner-Loser Portfolios for 9-Month Test Period (January 1989 to December 2001) 15 DD Statistics 10 -0.3 -0.2 -0.2 -0.2 -0.1 -0.1 -0 -0 0.02 0.06 0.1 0.13 0.17 0.2 0.24 0.28 0.31 0.35 0.38 0.42 -5 -10 -15 -20 Returns (%) T1 T2 T3 Figure 3.5: DD Statistics of Winner-Loser Portfolios for 1-Year Test Period (January 1989 to December 2001) 15 DD Statistics 10 -0.3 -0.3 -0.2 -0.2 -0.2 -0.1 -0.1 -0 -0 0.02 0.05 0.09 0.12 0.15 0.19 0.22 0.26 0.29 0.32 0.36 -5 -10 -15 Returns (%) T1 T2 T3 Figures 3.1 to 3.5 show the changes of DD statistics over the distribution of returns at each grid point. These figures give a visual representation of the DD test results. The plots show that in general, T1 moves from negative to positive along the distribution of returns for all test periods (except the one-month test period). This implies that W is preferred to L in the lower range of returns while L is preferred to W in the upper range. The above result of T1 moves from negative to positive may be 75 attributed to the higher expected returns for bearing higher downside risk. It is consistent with the result of Ang, Chen and Xing (2001) which suggests that some part of the momentum profits can be explained as compensation for bearing greater downside risk. All T2 and T3 are negative along the distribution of returns (except the onemonth test period) and most are found to be significant at 5% level. In the one-month test period, T2 and T3 are positive for the beginning 19% and 25% of the distribution of returns respectively. Hence, W stochastically dominates L in the higher orders over a wide range of returns. To summarize, W is indifferent (or not comparable) to L for all the non-satiation investors in international equity markets. However, the riskaverse and DARA investors prefer W to L. 3.5.2 Sub-Periods It is also interesting to test whether there is persistence in international momentum profits. Thus, the stationarity and persistency of the W and L are examined in different sub-periods. To this, the whole sample period is separated into two sub-periods. The first sub-period is from January 1989 to December 1996 and the second subperiod is from January 1997 to December 2001. The first sub-period is considered a bullish period for most stock markets. The second sub-period is relatively more bearish and volatile as compared to the first sub-period. There are some major events like the Asian financial crisis (July 1997), Internet stocks bubble (March 2000) and so on during this period. 76 Table 3.5: Descriptive Statistics of Daily Buy-Hold Portfolio Returns for Various Test Periods in Two Sub-Periods 1st Sub-Period (January 1989 – December 1996) L W WL 2nd Sub-Period (January 1997 - December 2001) L W WL 1-Month Mean t-statistic Std. Dev. Skewness ** * 0.0272 2.07 0.2344 -0.4677 0.0511 4.09 0.2219 -0.2263 Mean t-statistic Std. Dev. Skewness * 0.0387 4.17 0.1551 -0.3893 * 0.0274 3.17 0.1441 -0.6558 Mean t-statistic Std. Dev. Skewness * 0.0326 4.89 0.1100 0.1930 * 0.0453 6.63 0.1121 -0.7400 Mean t-statistic Std. Dev. Skewness * 0.0214 4.08 0.0860 0.3270 * 0.0406 7.65 0.0871 -0.4018 Mean t-statistic Std. Dev. Skewness * * 0.0239** 2.08 0.2210 0.2544 -0.0050 -0.19 0.3543 -0.1714 0.0496** 2.17 0.3176 0.2604 0.0545* 2.79 0.2902 0.0599 0.0139 0.76 0.2405 0.0891 0.0425* 2.85 0.1974 0.0109 0.0286*** 1.91 0.2039 -0.4929 -0.0072 -0.59 0.1599 0.2383 0.0631* 7.16 0.1150 0.3473 0.0703* 5.78 0.1594 -0.4889 -0.0019 -0.17 0.1434 0.8399 0.0299* 4.17 0.0932 -0.0184 0.0319** 2.45 0.1681 0.1786 3-Month -0.0113 -1.36 0.1436 -0.0043 6-Month 0.0127*** 1.69 0.1239 -0.1690 9-Month 0.0193* 3.01 0.1055 0.1169 1-Year 0.0261 0.0341 0.0080 0.0067 0.0076 0.0009 5.81 7.27 1.37 0.66 1.25 0.08 0.0738 0.0769 0.0964 0.1312 0.0788 0.1577 0.1847 -0.1060 0.0960 0.3453 -0.3999 0.4906 Note: L is the portfolio of losers, W is the portfolio of winners and WL is the momentum portfolio. t-statistics are adjusted by Newey-West method. * significant at 1% level, ** significant at 5% level, *** significant at 10% level. Table 3.5 shows that momentum portfolios in the first sub-period have positive mean returns except for the 3-month test period. Both W and L have positive returns indicate that momentum for W but reversals for L. In the second sub-period, all momentum portfolios have positive mean returns. When compared between the two sub-periods, the momentum profits decrease for the 1-year test period only, but 77 increase for the others. L has positive mean returns in the first sub-period but the returns decrease for all test periods in the second sub-period. Some test periods (1month, 6-month, and 9-month) even have negative returns in the second sub-period. W have positive mean returns in the second sub-period confirms that W persists over the two periods. Although the mean returns for W decrease for certain test periods (1month, 9-month, and 1-year) in the second sub-period, the reductions are much lower than the quantum reductions of L (except for the 1-year test period). Consequently, the momentum strategy is more profitable in the second sub-period than the first. The standard deviations of momentum portfolios in the second sub-period are bigger than those in the first sub-period. Higher standard deviations may indicate that it is riskier to implement momentum strategies in the bearish period due to uncertainty in the future. On the other hand, it may be argued that higher returns are expected to compensate for bearing higher risks. It appears that momentum profits continue to exist in the second sub-period. Although it is riskier to implement momentum strategies, the momentum profits are higher in the second sub-period than the first sub-period. The summary of DD test results for both sub-periods is reported in Table 3.3. There is no evidence of W dominance in the first sub-period. In fact, W dominates L for the “1-Month and 1-Month” strategy only. One interpretation of this would be that the first sub-period is a bullish period. In a bull market, even losers have positive returns. Thus, this reduces the profitability of the momentum strategy and hence the dominance of W. 78 In the second sub-period, the distributions of DD statistics are smoother than those in the first sub-period. There is no first-order SD between W and L for all test periods except the one-month test period. In addition, the second- and third-order SD results are almost similar to those in the whole sample period. The dominance of W in this period can be adequately explained by the continued rise in the returns of W in some test periods and the fall in the returns of L in all test periods. 3.6 Robustness Tests To make sure the results are robust, five robustness tests are conducted in this study: non-synchronous trading, exchange rate, market capitalization, January effect, and transaction costs. 3.6.1 Non-synchronous Trading Returns are likely to be autocorrelated when there is non-synchronous trading. In addition, given that stock markets operate in different time zones with different trading hours, their rate of returns on a particular calendar day may represent the returns over different time periods. Some component stocks of an index may trade infrequently, index returns will tend to serially correlated. It may cause the momentum strategy to be more profitable. To the extent that stocks trade at least once a week, returns computed with weekly prices may reduce the effect of nonsynchronous trading. Therefore, momentum strategy on weekly returns is conducted to reduce the potential estimation biases arising from non-synchronous data. 79 The results reported in section are based on momentum portfolios formed by skipping one day following the end of the formation period to avoid bid-ask bounce biases. It may be argued that one day is too short for bid-ask bounce effects to dissipate. Furthermore, a one-day skip may not be long enough to eliminate spurious momentum profits due to non-synchronous trading of index component stocks. To address these concerns, all tests are repeated using a one week skip period. W is bought and L is sold five trading days after the formation periods. Delaying the portfolio formation indeed increase the payoff to buy W and sell L because of the higher returns to W and the lower returns to L. Table 3.6: Results of Non-synchronous Trading Tests for Various Test Periods Test Period Mean First-order SD Second-order SD Third-order SD % DD0 % DD0 % DD0 5-Day Lag 1-month 3-month 6-month 9-month 1-year ** 0.0232 0.0018 0.0352* 0.0237* 0.0039 29 19 59 50 37 25 13 32 31 74 41 89 97 69 0 0 68 64 85 96 93 0 0 Weekly Returns 1-month 3-month 6-month 9-month 1-year 0.0473 12 21 23 -0.0184 0 0 0.1830* 31 62 55 0.1192*** 29 18 76 73 0.0061 15 26 18 22 Note: * significant at 1% level, ** significant at 5% level, *** significant at 10% level. % DD < (>) is the percentage of DD statistics which are significantly negative (positive) at the 5% significant level based on the asymptotic critical value of the studentized maximum modulus (SMM) distribution. Table 3.6 reports the mean returns and DD test results for momentum strategies with 5-day lag and momentum strategies on weekly returns. Except for the 1-month and 3-month test periods, the momentum returns with 5-day lag are quite 80 robust to the momentum returns without 5-day lag. For example, the average daily return for the popular 6-month momentum strategy is 0.0352% with 5-day lag, which is very close to the 0.0348% return using 1-day skip (Table 3.1). The same holds for the 9-month momentum strategy (0.0237% versus 0.0241%). Moreover, the momentum profits on weekly returns are larger than those on daily returns (except 3month test period). There is no first-order SD between W and L for both momentum strategies with 5-day lag and momentum strategies on weekly returns. W still dominates L at higher orders for all test periods. Similar to evidence in Chan, Hameed and Tong (2000), it is unlikely that all the momentum profits can be explained by microstructure biases or non-synchronous trading. Table 3.7: DD Test Results of Exchange Rate Effect for Various Test Periods Test Period 1-month 3-month 6-month 9-month 1-year First-order SD % DD < 28 45 67 55 35 % DD > 0 20 30 41 Second-order SD % DD < 84 77 88 98 60 % DD > 0 0 0 Third-order SD % DD < 79 69 83 97 89 % DD > 0 0 0 Note: % DD < (>) is the percentage of DD statistics which are significantly negative (positive) at the 5% significant level based on the asymptotic critical value of the studentized maximum modulus (SMM) distribution. 3.6.2 Exchange Rate Table 3.7 shows the DD test results for W and L in US dollar. These results are similar to those with local currencies. For first-order SD, some DD statistics are found to be significantly positive and negative at 5% significant level for all test periods except the one-month test period. Thus, the results invalidate the hypotheses that W stochastically dominates L or vice versa at first order. For the higher orders, namely second and third orders, no DD statistics are significantly positive while 60% - 98% 81 of DD statistics are significantly negative at 5% significant level for all test periods. Therefore, the results confirm that exchange rate movement does not influence the momentum effect in international equity markets. This result is consistent to Chan, Hameed and Tong (2000), and Hameed and Kusnadi (2002) which document that exchange rate fluctuation not add much to momentum profits. 3.6.3 Market Capitalization Momentum profits may come from emerging markets. In emerging markets, stock returns have higher autocorrelation and are more predictable than returns in developed markets (see Harvey 1995 and Bekaert, Erb, Harvey and Viskanta 1997). Low liquidity in emerging markets may cause the momentum profits to be spurious. If the momentum profits are due solely to small, less open markets, it cannot be claimed that they are abnormal. In order to mitigate the effect of emerging markets on the momentum strategy, DD test is conducted for W and L on developed markets only. Based on the average market capitalization for the whole sample period, Indonesia, Korea, Malaysia, South Africa, Singapore, Spain, Taiwan and Thailand are grouped as emerging markets and discharged from this robustness test. Table 3.8: Results of Developed Stock Markets for Various Test Periods Test Period 1-month 3-month 6-month 9-month 1-year Mean First-order SD % DD0 10 0 10 Second-order SD % DD0 23 0 23 Third-order SD % DD0 0.0044 0.0035 24 0.0264* 0.0290* 0.0160* 27 Note: * significant at 1% level, ** significant at 5% level, *** significant at 10% level. % DD < (>) is the percentage of DD statistics which are significantly negative (positive) at the 5% significant level based on the asymptotic critical value of the studentized maximum modulus (SMM) distribution. 82 Table 3.8 shows the mean momentum returns and DD test results for momentum strategies in the developed markets. From Table 3.8, the mean momentum returns in developed markets are smaller than the mean momentum returns in all markets for all test periods except the 9-month and 1-year. Nonetheless, the 6-month and 9-month mean daily returns for developed markets are still quite large (0.0264% and 0.0290% respectively). This result is consistent with the evidence in Chan, Hameed and Tong (2000) and Richards (1997). Consistent with the liquidity story, there is weaker evidence of momentum effect for developed markets than the overall sample. However, W still stochastically dominates L at second and third orders for the 1-month, 6-month and 9-month horizons. Although market capitalization has affected some of the DD test results, they cannot be considered an emerging market phenomenon only. Imperfect integration or some other form of market imperfection in the emerging markets may cause these differences. 3.6.4 January Effect A lot of empirical evidence shows that average returns in January are higher than the average returns in any other months. Jegadeesh and Titman (1993, 2001) document negative January returns but positive abnormal returns in each of the other months. However, Richards (1997) find that the international winner-loser reversals not appear to be a January phenomenon. To test the impact of January effect in momentum strategy, momentum returns are computed separately for January and nonJanuary. DD tests are conducted for W and L for non-January. 83 Table 3.9: Results of January Effect for Various Test Periods Test Period 1-month 3-month 6-month 9-month 1-year Mean January Non-Jan First-order SD % DD0 Second-order SD % DD0 Third-order SD % DD0 -0.0391 0.0426* 46 72 65 -0.0051 0.0049 17 22 47 78 0.0344** 0.0349* 56 11 85 81 0.0225 0.0243* 49 31 95 94 -0.0104 0.0067 40 33 86 95 Note: * significant at 1% level, ** significant at 5% level, *** significant at 10% level. % DD < (>) is the percentage of DD statistics which are significantly negative (positive) at the 5% significant level based on the asymptotic critical value of the studentized maximum modulus (SMM) distribution. Table 3.9 reports negative mean returns for momentum strategies in January for most of the test periods. Non-January has positive mean returns for all test periods and its mean returns are higher than January’s mean returns. The momentum returns excluding January’s returns are higher than the momentum returns for all months in all test periods. Interestingly, this finding is consistent with Jegadeesh and Titman (1993) in the study of US markets. Table 3.9 shows that the DD statistics between W and L for non-January have similar results with those for all months. Therefore, it is likely that the international momentum profits are not due to January effect. This finding is consistent with Wong, Thompson, Wei and Chow (2004) in the study of US markets. 3.6.5 Transaction Costs Lesmond, Schill and Zhou (2004) have pointed out the economic significance of transaction costs on momentum strategy. Momentum strategies implemented on equally-weighted portfolio of stocks can incur non-trivial transaction costs in terms of direct brokerage costs, bid-ask spreads and price impact and these may negate gross 84 momentum profits. The effect of transaction costs depends on the design of the momentum strategy (e.g. frequency of portfolio rebalancing, number and market size of securities in W and L etc.). Transaction cost impact has been mitigated by focusing on the value-weighted stock indices rather than individual stocks. More importantly, in contrast to previous momentum studies which rebalances portfolios monthly, this study minimizes the frequency of rebalancing to just twice for any holding period (i.e., at the start and the end of each period). As such, transaction costs are unlikely to reverse the profitability of all momentum strategies. Nevertheless, to have a feel of the impact of transaction costs on momentum profits, sensitivity tests using round trip trading costs of 1% and 2% are performed here. The benchmark study of Jegadeesh and Titman (1993) assumes a round-trip trading cost of 1%. For international stock index funds, Engle and Sarkar (2002) estimate an average quoted bid-ask spread of 1.2% for seven MSCI Exchange Traded Funds, which includes one emerging market (Brazil) and 1.04% for developed markets. These spreads appear large, but they are actually quite close to the quoted spreads for the largest quintile of NYSE stocks (see Lesmond, Schill and Zhou 2004). The 2% cost assumes a spread of 1.2% plus roundtrip commissions of 0.8% for transaction amount of $50,000 taken from Lesmond, Schill and Zhou (2004). The per-dollar transaction costs are assumed the same in all sample markets. 85 Table 3.10: Results of Transaction Costs for Various Test Periods Test Period First-order SD Mean % DD0 Second-order SD % DD0 Third-order SD % DD0 66 78 81 94 95 0 0 Round Trip Transaction Cost = 1% * 1-month 3-month 6-month 9-month 1-year -0.0553 -0.0267* 0.0194* 0.0139** -0.0024 1-month 3-month 6-month 9-month 1-year * 32 18 56 49 36 20 11 31 32 73 45 85 95 76 0 0 Round Trip Transaction Cost = 2% -0.1463 32 73 66 -0.0574* 18 20 45 78 0.004 56 11 85 81 0.0037 49 31 95 94 -0.0101 36 32 76 95 Note: * significant at 1% level, ** significant at 5% level, *** significant at 10% level. % DD < (>) is the percentage of DD statistics which are significantly negative (positive) at the 5% significant level based on the asymptotic critical value of the studentized maximum modulus (SMM) distribution. Table 3.10 reports that the mean momentum returns with transaction costs decrease as compared to the mean momentum returns without transaction costs. The profitability of the 6-month and 9-month momentum strategies is left intact. Higher transaction cost will further reduce the momentum profits. However, the DD test results remain unchanged with those examples without transaction costs. Thus, it appears that transaction costs cannot completely explain away the momentum effect in international equity markets. Grundy and Martin (2001) point out, transaction costs are hardly relevant for a choice between two portfolios that differ only in their prior returns. The zero marginal costs of such a choice naturally induce a preference for past winners over past losers. This may well explain the persistence of momentum profits over the years. 86 3.7 Mean-Variance Analysis The SD approach is always comparing to the Mean-Variance (MV) analysis which is developed by Markowitz (1959). Suppose two assets, X and Y with CDFs, F1 and G1 each with mean and variance, ( µ F , σ F2 ) and ( µG , σ G2 ) . The MV rule states that asset X is preferred to asset Y if and only if µ F ≥ µG and σ F2 ≤ σ G2 with at least one strictly inequality. When X dominates Y by MV rule, X has become a member of efficient set while Y is relegated to the inefficient set. A rational investor would like to select investments from the efficient set than the inefficient set. A portfolio is efficient in terms of MV if no other portfolio with the same mean has a smaller variance; and no other portfolio with the same variance has a larger mean. Although the MV analysis is much simpler and inexpensive in practice, it is valid only when asset returns are normally distributed or when the investor’s utility function is quadratic. This would restrict its popularity and general usefulness in empirical financial analysis. However, some works have been done to study the relationship between MV and SD efficient sets. See for example, Meyer (1987), Levy (1989), and Wong (2002). Empirical studies like Levy and Sarnat (1970), Kroll and Levy (1979), Kjetsaa and Kieff (2003) apply the MV rule as the complement of applying the SD criteria. In this study, the MV analysis is applied to the whole sample period and two sub-periods. From Table 3.1, W has higher mean returns and lower standard deviations than L for all test periods. Thus, W dominates L by MV rule in the whole sample period. Table 3.5 shows the results of MV are mixed in the first sub-period. W has higher mean returns for all test periods except the 3-month. However, the standard deviations of W are also greater than L for 6-month, 9-month and 1-year test periods. This is consistent with the findings in Richards (1997) that there is no evidence that L 87 is riskier than W, and may indeed be less risky. Hence, W only dominates L by MV rule for 1-month test periods in the first sub-period. The MV results for the second sub-period are similar to the results in whole sample period; W dominates L by MV rule for all test periods. If we compare with the DD test results, although W dominates L by MV rule, it does not imply that W must stochastically dominate L especially for the first sub-period. The result of this study is consistent with the literature that the MV efficient set is different from the SD efficient set. 3.8 Conclusion Momentum strategy means to purchase (sell) stocks that have recently increased (decreased) significantly in price on the belief that they will continue to increase (decrease). This study re-examines the profitability of momentum strategies in the international stock markets using the SD approach. Specifically, this study would like to distinguish the hypothesis that there exist some general asset pricing models that can explain the momentum effect versus the alternative hypothesis that there is no asset pricing model consistent with risk-averse investors that can rationalize the momentum effect. From the results of this study, it is known that the profitability of momentum strategy continues to exist globally and remains so in recent years. Some literature views this anomaly as market inefficiency. The results of this study show that W is preferred to L in the lower range of returns while the reverse is true in the upper range. It appears that stocks with higher downside risk have higher expected returns than stocks with lower downside risk. Most important is W dominates L at the second- and third-order SD, implying that all risk-averse and DARA investors prefer W to L. 88 Overall, these results indicate that the momentum effect remains an anomaly for the efficient market hypothesis and standard equilibrium asset pricing models, for it implies that asset pricing models which are based on the assumption that investors who are risk averse cannot explain the momentum effect. Although this study shows that the profitability of momentum strategy continue to exist globally, it is difficult to believe that the momentum profits still exist when they are not associated with risk factors. The results of this study have implications that investors with (strictly) concave increasing utility function are confident that W will deliver higher returns than L. As pointed out by Griffin, Ji and Martin (2003), the momentum cannot be explained by macroeconomic environments but can be explained by an unknown form of risk. Hence, some “yet to be discovered models” that can explain the underlying economic mechanism are recommended for future research. It would definitely be beneficial if more refined aspects of investors’ behavior towards risk can be incorporated into such models. However, capturing all salient aspects of investors’ risk preferences in a single model that is suitable for risk adjustments will be a challenging task. Next, the study also addresses the question of how robust the empirical findings in this study are. There are no non-synchronous trading effect, no exchange rate effect, no January effect, and no transaction cost effect on the international momentum strategies. Although there is weaker evidence of the momentum effect for developed markets than the overall sample, the momentum effect is not unique to emerging markets. Overall, the empirical findings in this study are robust to two 89 different sub-periods with different risk and return characteristics and survive reasonable transaction costs for international index funds. The principal disadvantage of SD approach is that it may be sensitive to outliers. A single large negative return in the portfolio can undermine the dominance results. Normally, outliers affect results only towards the end points and/or in a very short interval. However, it is noted that W and L dominates each other for long intervals in the distribution. Moreover, the dominances of W and L appeared in different portions of the distribution (Figures 3.1 to 3.5). Hence, this shows that the DD test results are not affected by the outliers. Therefore, the disadvantage of the SD approach should not pose a significant problem in this study. 90 [...]... 0 .34 30 -0.2859 0.0412* 3. 20 0.27 63 0 .35 81 0.0480* 3. 45 0 .31 33 0. 231 0 0.0 037 0 .31 0.2404 -0.9 135 0.0180** 2.15 0.1700 -0.0542 0.01 43 1 .39 0.2 133 0. 938 4 -0.0069 -0. 83 0.1 638 -0.8722 0.0404* 7.20 0.11 13 0.1675 0.0472* 5.47 0.1716 0.6801 -0.0041 -0.58 0. 139 8 -0 .30 53 0.0269* 5.76 0.09 23 -0.22 13 0. 031 0* 3. 82 0.15 93 0.5558 0.0119*** 1.80 0.1296 -0.4029 0.0095** 2.16 0.0866 -0 .30 63 -0.0024 -0 .33 0.1440 0.4006 1-Month... 0.0496** 2.17 0 .31 76 0.2604 0.0545* 2.79 0.2902 0.0599 0.0 139 0.76 0.2405 0.0891 0.0425* 2.85 0.1974 0.0109 0.0286*** 1.91 0.2 039 -0.4929 -0.0072 -0.59 0.1599 0. 238 3 0.0 631 * 7.16 0.1150 0 .34 73 0.07 03* 5.78 0.1594 -0.4889 -0.0019 -0.17 0.1 434 0. 839 9 0.0299* 4.17 0.0 932 -0.0184 0. 031 9** 2.45 0.1681 0.1786 * 3- Month -0.01 13 -1 .36 0.1 436 -0.00 43 6-Month 0.0127*** 1.69 0.1 239 -0.1690 9-Month 0.01 93* 3. 01 0.1055... 1-month 3- month 6-month 9-month 1-year 32 18 56 49 36 0 20 11 31 32 73 45 85 95 76 0 0 0 0 0 66 78 81 94 95 0 0 0 0 0 0 18 33 27 0 46 0 27 46 0 0 0 44 35 0 0 0 0 0 0 77 81 81 94 95 0 0 0 0 0 1st Sub-Period 1-month 3- month 6-month 9-month 1-year 19 0 30 44 23 0 23 28 24 5 55 0 49 60 46 2nd Sub-Period 1-month 3- month 6-month 9-month 1-year 28 35 63 51 38 0 13 11 36 38 82 86 85 95 62 Note: % DD < (>) 0... -0.2112 0.0040 0.49 0.1704 -0.2051 0.01 73* 2.58 0. 132 8 0.0 630 0.0521* 9.09 0.1 135 -0.2 939 0. 034 8* 4.91 0.1414 -0.1896 0.0124** 2.18 0.1122 0.6096 0. 036 5* 8.04 0.0897 -0.2464 0.0241* 3. 57 0. 133 2 0. 238 4 0.0187* 3. 65 0.10 03 0.1809 0.0 239 * 5.99 0.0787 -0.22 93 0.00 53 0.84 0.1 237 0 .37 72 1-Month Mean t-statistic Std Dev Skewness 3- Month Mean t-statistic Std Dev Skewness 6-Month Mean t-statistic Std Dev Skewness... second-order SD exist An investor’s expected utility is always higher for the dominating portfolio than the dominated portfolio Therefore, the dominated portfolio will never be chosen by the investor 3. 3.2 Almost Stochastic Dominance Investment criteria like MV and SD rules may not differentiate between two assets or portfolios that “most” investors would prefer Leshno and Levy (2002) introduce Almost SD... -0.4 -0 .3 -0.2 -0.1 -0 0. 03 0.11 0.18 0.26 0 .34 0.41 0.49 0.56 0.64 0.72 -4 -6 -8 Returns (%) T1 T2 T3 Figure 3. 3: DD Statistics of Winner-Loser Portfolios for 6-Month Test Period (January 1989 to December 2001) 10 DD Statistics 5 0 -0.5 -0.5 -0.4 -0.4 -0 .3 -0 .3 -0.2 -0.2 -0.1 -0.1 -0 0.02 0.07 0.11 0.16 0.21 0.26 0 .3 0 .35 0.4 -5 -10 -15 -20 Returns (%) T1 T2 T3 74 Figure 3. 4: DD Statistics of Winner-Loser... -0 .3 -0.2 -0.2 -0.2 -0.1 -0.1 -0 -0 0.02 0.06 0.1 0. 13 0.17 0.2 0.24 0.28 0 .31 0 .35 0 .38 0.42 -5 -10 -15 -20 Returns (%) T1 T2 T3 Figure 3. 5: DD Statistics of Winner-Loser Portfolios for 1-Year Test Period (January 1989 to December 2001) 15 DD Statistics 10 5 0 -0 .3 -0 .3 -0.2 -0.2 -0.2 -0.1 -0.1 -0 -0 0.02 0.05 0.09 0.12 0.15 0.19 0.22 0.26 0.29 0 .32 0 .36 -5 -10 -15 Returns (%) T1 T2 T3 Figures 3. 1... level In the one-month test period, T2 and T3 are positive for the beginning 19% and 25% of the distribution of returns respectively Hence, W stochastically dominates L in the higher orders over a wide range of returns To summarize, W is indifferent (or not comparable) to L for all the non-satiation investors in international equity markets However, the riskaverse and DARA investors prefer W to L 3. 5.2... 0. 234 4 -0.4677 0.0511 4.09 0.2219 -0.22 63 Mean t-statistic Std Dev Skewness * 0. 038 7 4.17 0.1551 -0 .38 93 * 0.0274 3. 17 0.1441 -0.6558 Mean t-statistic Std Dev Skewness * 0. 032 6 4.89 0.1100 0.1 930 * 0.04 53 6. 63 0.1121 -0.7400 Mean t-statistic Std Dev Skewness * 0.0214 4.08 0.0860 0 .32 70 * 0.0406 7.65 0.0871 -0.4018 Mean t-statistic Std Dev Skewness * 0.0 239 ** 2.08 0.2210 0.2544 -0.0050 -0.19 0 .35 43 -0.1714... reduced by holding the portfolio for longer period 67 Table 3. 1: Descriptive Statistics of Daily Buy-Hold Portfolio Returns for Various Test Periods in Whole Sample Period: January 1989 – December 2001 L W WL 0.0148 1.11 0.2869 -0 .34 92 0.0505* 4.12 0.2628 0.0896 0. 035 7* 3. 30 0.25 03 0.1927 0.0292* 3. 04 0.1928 -0.1411 0. 033 2* 4.01 0.1668 -0.2112 0.0040 0.49 0.1704 -0.2051 0.01 73* 2.58 0. 132 8 0.0 630 0.0521* . 47 Chapter 3 International Momentum Strategy: A Stochastic Dominance Approach ∗ 3. 1 Introduction Jegadeesh and Titman (19 93) discover that past winners outperform past losers by 1% per month in. (2000) report the profitability of momentum strategy in 23 national stock markets indices for short holding periods. Griffin, Ji and Martin (20 03) observe that the occurrences of momentum profits. international stock markets using the stochastic dominance (SD) approach. The SD approach differs from the traditional approach of studying momentum profits. It endorses the minimum assumptions of investor’s