Herding Behavior Evidence from Portuguese Mutual Funds

Herding Behavior - Evidence from Portuguese Mutual Funds - Júlio Lobão* Ana Paula Serra** Keywords : Herding Behavior; Capital Markets; Mutual Funds. This Draft: March 11, 2002 *Júlio Lobão (corresponding author), Instituto de Estudos Financeiros e Fiscais, Edifício Heliântia, Avenida dos Sanatórios, 4405 Valadares; Telephone: 351 227538800; Fax: 351 227624590; Email: [email protected]. ** Ana Paula Serra, CEMPRE1 , Faculdade de Economia do Porto, Universidade do Porto, Rua Dr. Roberto Frias, 4200-456 Porto; Telephone: 351 225571100; Fax: 351 225505050; Email: [email protected]. 1 CEMPRE - Centro de Estudos Macroeconómicos e Previsão - is supported by the Fundação para a Ciência e a Tecnologia, Portugal, through the Programa Operacional Ciência, Tecnologia e Inovação (POCTI) of the Quadro Comunitário de Apoio III, which is financed by FEDER and Portuguese funds. 1 Herding Behavior - Evidence from Portuguese Mutual Funds - Abstract We test for herding by Portuguese mutual funds over the period of 1998 to 2000. We employ the (herding) measure of trading suggested by Lakonishok et al. (1992). We find strong evidence of herding behavior for Portuguese mutual funds. Furthermore, our results suggest that the level of herding is 4 to 5 times stronger than the herding found for institutional investors in mature markets. The herding effect seems to affect, as likely, purchases and sales of stocks. There seems to be a stronger tendency to herd among medium-cap funds rather than very large or very small funds, and among funds with less stocks. Lastly, herding seems to decrease when the stock market is doing well or is more volatile. Keywords : Herding Behavior; Capital Markets; Mutual Funds. JEL: D7; G14; G23 2 1. INTRODUCTION For the last two decades, the importance of mutual funds all over the world has increased enormously. In 1950, institutional investors in the US held 6% of the stock market. Today that share represents over 50% of the stock market capitalization (around US dollars 30 trillion) and mutual funds are the more popular way to invest in the stock market. In Europe, the role of institutional investors is far from what it represents in the US but it is growing at a very fast pace. By the end of the nineties, total assets managed by mutual funds amounted to 70%, 60%, 90%, 60% and 200% of GNP, respectively, in Italy, Germany, France, Spain, the Netherlands and the UK. The growth of Portuguese mutual funds, over the last decade, has also been impressive: by the end of 2001, total assets in management by mutual funds amounted to 20 billion euros, 20% of GNP (12% of the Portuguese stock market capitalization) against less than 5% in 1990. Mutual funds are thus important players in the market and their trading accounts for an increasing share of total trading. More and more, institutional investors trading strategies impact prices. Institutional investors may have incentives to buy and sell the same stocks at the same time. This convergence in trading strategies is commonly known as herding. Herding may affect market prices by driving prices away or close from fundamental values. There are several ways to define herding. Broadly, herding could be defined as correlation in the behavior of investors. Yet the fact that a group of investors trade the same stock, in the same direction, over a period of time does not necessarily mean that investors are influenced by others. Trading together could result because investors are independently influenced by a common information or factor ("spurious herding"). Therefore, a more restrict definition of herding would rule out clustering, caused by some omitted factor, and would take in consideration only correlated trades originated by copying or imitation. In other words, a more restrict definition of herding focuses only on correlation in trades that results from interactions between investors. Herding behavior - rational or irrational - may lead to errors and misvaluation of assets. Investigating and measuring the level of herding on institutional investors trading could shed light on different phenomena such as excess volatility, price momentum, systematic errors in expectations causing systematic market misvaluations, crashes, speculative bubbles, fades, etc. 2 Evidence regarding the impact of herding on prices is therefore valuable to current debates on market efficiency and on the validity of traditional asset pricing models. The fact that, across different investor groups, there are different problem sets and different incentives, implies that, within a certain group, we should observe a more homogeneous trading behavior across investors. Evidence on herding by a particular type of investors, e.g., institutional investors, could thus shed light on whether there are different types or styles of investors, each type with a particular trading pattern. 2 Lakonishok et al. (1992) note that empirical evidence that shows that herding by institutional investors impacts prices, does not imply necessarily that herding causes volatility. If, for example, institutional investors are better informed than individual investors, their trading could drive market prices close to the assets intrinsic values (see, for example, Froot et al. (1992), Bikhchandani et al. (1992), Hirshleifer et al. (1994) and Wermers (1999)). 3 Investigating herding on securities markets could thus be precious for researchers and could have important implications and practical value for investors, traders and regulators. It is commonly accepted among academic researchers that investors trade together or follow similar trading strategies. 3 Why do investors herd? There are two ways to approach the question: the first approach, suggests that herding is irrational and is caused by "herd instinct" or by "investor psychology". The second approach claims that it may be entirely rational to trade together. Several theories may help explain why rational investors trade together: informational cascades, agency reputation based models and information inefficiencies. Informational cascades are the more common explanation for herding. The observation of prior investors' trades can be so informative that investors are better-off disregarding their private information and trading in the same direction (Bikhchandani, Hirschleifer and Welch (1992) and Banerjee (1992)). An alternative explanation for herding is reputational risk: under certain circumstances, asset managers have incentives not to act differently from other competing managers, regardless of their own signals (Scharfstein e Stein (1990)). Finally, herding could result from the way investors deal with information: investors may find attractive to use only private information shared by other investors, and disregard any other private unique information they have. Resource allocation regarding information acquisition is inefficient in this setting (see, for example, Froot, Schaferstein and Stein (1992) and Hirshleifer, Subrahmanyan and Titman (1994)). To what extent do investors herd? Evidence on the extent of herding in securities markets is recent and most of the studies examined herding behavior by institutional investors in the UK and in the US. Lakonishok, Shleifer and Vishny (1992) study 769 US pension funds, managed by 341 different portfolio managers, between 1985 and 1989. They find that the level of herding is not significant. Grinblatt, Titman and Wermers (1995) analyze trading data for 274 mutual funds in the US over the period of 1974-1984 and found evidence of trade convergence for the majority of mutual funds. Wermers (1999) investigates herding over a 20-year period using quarterly portfolio holdings for all mutual funds based in the US from 1975 to 1994. He finds a low level of herding among mutual funds. Yet he finds stronger herding effects among growth-oriented mutual funds and in small and winner stocks. Wylie (2000) examines data on 268 mutual funds operating in the UK from 1986 to 1993 and finds that the level of herding is similar to the herding found for US institutional investors. In this paper we test for herding by Portuguese mutual funds. The paper focuses on mutual funds for several reasons. First, mutual funds are prone to exhibit herding because the different theoretical arguments referred above apply to them and, as such, may allow to disentangle the causes of herding. In 3 The notion that managers and investors are influenced by others has interested economists for some time. Keynes (1936) suggested that investors behave as judges in beauty contests: instead of truly judging the beauty of each contestant, they decide upon their expectation of the other members of the jury vote. 4 particular, herding driven by agency problems makes sense only for institutional investors. Second, it is important to study mutual funds given their increasing importance in the stock market. Third, the availability of data for Portuguese mutual funds provides a good opportunity for an out-of-sample test. Comparing the results for Portugal with the results observed for mature stock markets like the US and the UK may generate insights on the validity of the different theoretical arguments that have been put forward to explain herding. 4 We employ the measure of herding developed by Lakonishok et al. Several studies (Lakonishok et al. (1992), Grinblatt et al. (1995), Wermers (1999) and Wylie (2000)) have used this same measure. This measure compares the effective proportion of funds that bought stocks during a particular period, with the proportion that should be observed if there was no herding. We use quarterly data on the portfolios of 32 Portuguese mutual funds from 1998 to 2000. We find strong evidence of herding behavior for the average fund in our sample. Furthermore, the overall level of herding is much stronger (4 to 5 times) than that found in previous studies for the UK and the US. Portuguese mutual funds exhibit herding either when buying or when selling stocks, but the herding effect is stronger for purchases. Looking at subgroups of funds, we find higher levels of herding among medium cap funds and for funds that hold more stocks in their portfolios. Finally, the positive feedback trading seems to be much weaker when the stock market is doing well and (less significantly) when the market is more volatile. Altogether our results are consistent with an information-based explanation. The remainder of the paper is organized as follows. In section 2 we discuss the hypotheses to test and describe the methodology of our study. Section 3 provides a description of Portuguese mutual funds and of the data. Empirical findings are presented in section 4. Section 5 concludes. 2. HYPOTHESES AND METHODOLOGY 2.1. Hypotheses The main objective of this study is to assess if Portuguese mutual funds exhibit herding and to what extent. In addition, we analyze the behavior of several subgroups of funds. We argue that the existence of different levels of herding behavior by subgroups of funds can enlighten the debate on the causes of herding. In addition to fund specific characteristics (market capitalization, portfolio holdings and rebalancing frequency), we investigate the inference by market conditions (stock market returns and volatility). The different theoretical arguments reviewed above may yield different predictions for the relation between the level of herding and the variables we analyze. 4 For example, one implication of informational cascades models is that, in stock markets that exhibit poorer aggregation of information and where the precision of the public pool of information is lower, the likelihood of departing from a pool should be higher (see, for example, Bikhchandani et al. (1992) or Cao and Hirshleifer, 1997). 5 The implications of the different rational explanations for the variation in herding are discussed below. Market Capitalization Larger funds are more likely to have superior resources to collect and process information. If that is the case, large funds should be imitated by (smaller) funds with more difficult access to information. For large and small funds, within their group, we should expect to see lower levels of herding. Herding should occur mainly between groups with different resources or capabilities and not within groups of funds with the same size. Therefore we should expect to find higher levels of herding for the universe of funds than for subgroups of funds. Medium cap funds should exhibit more herding than very large or very small funds. Reputation risk may predict a different relation. On one hand, managers of funds with a similar size may have incentives to imitate their peers. On the other hand, compensation schemes may be indexed to assets in management so there may be incentives for large funds to herd, and keep their relative positions, and for small funds to deviate from others, implementing distinctive investment strategies to grow. In this setting, we should expect to find higher levels of herding for subgroups of funds than for the universe of funds, and small and medium cap funds should exhibit more herding than large funds. Portfolio Holdings Collecting information and analyzing securities is costly. If funds invest in several asset classes (bonds, stocks, derivatives), they have potentially either higher costs in processing information or less precise signals, leading to greater herding. If this explanation is true, we should observe lower levels of herding for subgroups of funds that are concentrated in one asset class than for more diversified funds. Frequency in Portfolio Rebalancing Information inefficiency models predict higher levels of herding in funds with shorter investment horizons that may find profitable (less costly) using only private information shared with other funds; because they choose to analyze the same information, they end up trading in the same direction. On the contrary, funds that implement buy and hold strategies, because they have long horizons, have fewer incentives to implement "tacit manipulation strategies" given that in the long run prices will be close to fundamentals. They have thus incentive to use all the information that they have (shared or unique) and therefore herd less in trading. We should then observe higher (lower) levels of herding for subgroups of funds that have shorter (longer) horizons and these funds usually show more (less) frequency in portfolio rebalancing5 . Market Stock Returns Informational cascades predict that when markets are doing well, investors are more confident and that may increase the likelihood of using their private signals and 5 Grinblatt et al. (1989, 1995) show that there is a strong relation between the frequency of portfolio rebalancing and momentum strategies. Funds with higher frequency in rebalancing may thus exhibit correlation in trades ("spurious herding"). 6 deviating from others. We should expect thus to find lower levels of herding when markets are doing well. 6 Differently, agency models suggest that when the market is doing very well or very badly, the signals are more precise, it is easier to detect a good from a bad manager and therefore bad managers will try to mimic good managers more often to fool their clients. Thus we should expect higher levels of herding when markets are doing very well or very badly. Market Volatility Most theories predict higher levels of herding when markets are more volatile. For example, higher uncertainty may result in that public information becomes less precise and reliable and therefore (not necessarily) cascades are more likely to occur. Higher uncertainty in private information may result in that cascades start sooner. Therefore we should observe higher levels of herding for periods when market volatility is high. One implication of this argument for the variation in herding across markets is the following: we should find higher levels of herding for Portuguese mutual funds than that found in the US and the UK given that the volatility in the Portuguese stock market is higher7 . Informational cascades models may also predict a negative relation between volatility and the level of herding. The argument is that if investors are not ex-ante identical, the arrival of an individual with deviant information or of very unexpected public information, can dislodge the cascade. Therefore we could also observe lower levels of herding for periods when market volatility is high. 2.2. Measuring herding We use the measure of herding developed by Lakonishok et al. (1992). This measure defines herding as the tendency of funds to trade a given stock together and in the same direction, for whatever reason, more often than would be expected if funds were trading randomly and independently. A group of funds exhibits herding behavior when there is unbalance between funds that buy and funds that sell a given stock (assuming that trades would be balanced if there was no herding). In other words, there is herding behavior when the proportion of funds that trade in a stock in the same direction (buying or selling) is above the expected proportion of funds trading in that direction under the null hypothesis of independent trading decisions by the funds. The measure is defined as : H (i , t ) = p(i , t ) − p (t ) − AF ( i, t ) (1) 6 Additionally, if mutual funds share higher aversion to stocks with high risk than other investors, then when the market is down, they should trade together reflecting their willingness to lower the risk in their portfolio. 7 During the period from 1997 to 2001, the standard deviation of the Portuguese stock market was 24% against 18% in the US and the UK (based on monthly returns of the MSCI country indices). 7 where p (i , t ) = B(i , t ) B(i , t ) + S ( i, t ) and n ∑ p(i , t ) p (t ) = i = 1 . n B(i , t ) [S (i , t )] is the number of funds that buy (sell) the stock i during quarter t, p (i , t ) is the proportion of funds trading stock i that were buyers and p (t ) is a proxy for the expected proportion of buyers under the null of independently trading by funds, E ( p (i , t )) , and is given by the proportion of all stock trades by funds that were purchases during that quarter t. p (t ) is constant for all stocks during a quarter but varies over time. The adjustment factor AF (i , t ) is given by: AF (i , t ) = E [ p( i, t ) − E ( p(i , t ) ]. This factor allows to capture the random variation of p (i , t ) around its expected proportion of buyers, under the null hypothesis of independent trading and assuming B(i , t ) has a binomial distribution with parameter p = p (t ) .8 The null hypothesis states that if herding does not exist, the proportion of buyers (and sellers) has the same expected value for all stocks in a given period and is constant equal to p (t ) [1 − p(t )] . Under the null, H (i , t ) = 0 . Deviations from p (t ) , above the expected AF (i , t ) , signal herding. As N (i , t ) = B( i, t ) + S (i , t ) becomes larger then, under the null, AF (i , t ) will be close to zero. The main reason for including the adjustment factor is to account for bias that would occur if stocks were illiquid and traded only by a few investors. Positive (or negative) significant values of H (i, t ) can be interpreted as the percentage of funds that were buyers (sellers) in a certain stock above the expected proportion. To evaluate herding for a given subgroup of funds, we compute the average of the measure of herding H (i, t ) across all stocks and quarters in that particular group. For example, we can measure the herding effect for a subgroup of funds - large cap funds or for a subset of quarters - when the market is positive. The herding measure can also be computed separately for stocks that observed higher proportion of buyers (sellers) than the overall average. The averaging is done as above. These conditional measures, the "buy herding" measure and the "sell herding" measure, respectively, are defined as: 8 The probability density function of the binomial distribution is given by  N (i , t )   p(t ) p (i , t )[1 − p(t )][ N (i ,t )− p ( i ,t )] where N (i ,t ) is the number of funds trading the b[N (i, t ), p(i, t ), p(t )] =   p(i ,t )  stock i during quarter t . p(i , t ) and p(t ) are defined above. 8 BH ( i, t ) = H ( i, t ) p( i, t ) > p (t ) , SH ( i, t ) = H (i , t ) p (i , t ) < p (t ) . By analyzing these conditional measures BH (i , t ) and SH (i , t ) we can assess whether herding effects are more common when funds are buying or selling. 9 Measuring the herding behavior on the basis of Lakonishok et al. (1992) has important limitations. First, this measure captures correlation in trades but does not, by itself, disentangle the determinants of herding. Second, this measure does not take in consideration whether the correlation trades results from imitation or merely reflects that traders use the same information. Finally, this measure is biased when there are limitations to short selling strategies. If short selling is prohibited, as it was the case for Portugal until mid 1999, or costly, the measure overestimates true herding: the measure may yield a positive value, indicating herding, when herding effectively does not occur. Another concern regards the measurement unit to use to compute the herding measure. If we use funds as the measurement unit, the measure may be biased upward. Research "buying lists" are most of the times used across funds managed by the same company. As a result, a positive value of the herding measure could occur merely reflecting "spurious" herding and not positive feedback trading. 3. DATA 3.1 The Mutual Fund Industry in Portugal By June 2001, there were 260 investment funds in Portugal managed by 18 different companies. Total net assets under management amounted to around 21 thousand million euros. The breakdown of the funds' assets by country was the following: 33% of total assets were invested in Portugal, 69% were invested in European Union countries (excluding Portugal) and the remaining 9% were invested in countries outside the EU. On aggregate, 43% of total assets were invested in Corporate Bonds, 6% in Stocks, 13% in Treasury Bonds, 11% in International Mutual Funds and the remaining were invested in cash. 90% of total assets were denominated in euros and 6% in US dollars. The largest asset management company held 37% of total assets. The three largest asset management companies held 69% of total assets. 3.2 Sample The data used in our study consists of quarterly portfolio holdings for 32 equity mutual funds based in Portugal, between 1998 and 2000. The data is from Bolsa de Valores de Lisboa e Porto. The 32 funds were selected from a total of 53 equity mutual 9 The adjustment factor FA (i, t ) and p(t ) are computed accordingly, for that subset of stock-quarters. 9 funds operating in Portugal at the end of 1997 and for which we could trace data in any quarter starting in 1988. As in previous studies, we selected those funds that had at least 75% of their total assets invested in stocks (domestic or foreign) by the end of 1997. We impose no minimum survival period requirement for a fund to be included in the sample. 10 By the end of 1997, equity mutual funds in Portugal managed around 2000 million euros of which around 1 700 million euros were invested in domestic stocks, accounting for 5% of the Portuguese stock market capitalization. Foreign stocks accounted for less than 5% of total assets11 . The 32 funds in our sample held 1 310 million euros in domestic equities, 78% of the total equities held by equity mutual funds in Portugal. The total net assets for the average fund in our sample amounted to 52 million 12 euros. The largest fund in sample managed 230 million euros and 13 funds managed total net assets above 50 million euros. Table 1 shows that, for the average fund, domestic equity holdings account for 79% of total net assets; foreign equity holdings represent only 5% of total net assets. Domestic securities (mainly stocks and bonds) account for 81% of total assets. 19 funds in our sample did not invest in foreign securities. The 32 funds in sample were managed by 12 different asset management companies, of which 5 accounted for more than 80% of total assets managed by the funds in our sample. 13 We consider 84 distinct stocks out of the holdings of the 32 funds in our sample. We excluded those securities that were traded by only one or two funds over the sample period. Our data set is composed of 32 different panels containing the quarterly shareholdings for each individual fund, from the last quarter of 1997 to the last quarter of 2000. The average quarter has information for 27 funds that, on average, held 27 stocks. Table 1 presents statistics regarding the portfolio holdings of the funds in our sample broken down by size quintiles, using the fund market capitalization at the end of the last quarter of 1997. 14 If there is a relation between fund size and portfolio holdings, we have to be careful when interpreting the results partitioned by portfolio holdings because we may be capturing a size effect instead. 10 See, for example, Grinblatt et al. (1995) and Wermers (1999). Previous research has shown that the impact of survival bias on performance (Grinblatt e Titman, 1989; Brown e Goetzmann, 1995) and herding (Wermers, 1999; Wylie, 2000) is trivial. 11 By the end of 2001, total assets in management are similar. Equity holdings account, as in 1997, for over 90% of total assets but foreign stocks represent now over 75% of total stocks. 12 Around that date, the average fund in the UK and in the US managed, respectively, 175 and 475 million euros. 13 Given that the Portuguese mutual fund industry is concentrated in the hands of a few companies, the upward bias referred above may be severe. To avoid that bias, we could compute the herding measures looking at the aggregate holdings of funds within the same asset management company and measure herding only across different management companies. We do not compute those statistics here because of the limited number of asset managed companies in our sample. 14 Each quintile has the same number of observations, R, except for the third quintile that has R+S observations. 10 The examination of table 1 shows that there seems to be no monotonic relation between fund size and portfolio holdings. The fact that portfolio holdings are similar for large and small funds could result from the fact that the mutual fund industry in Portugal is still incipient. The first equity fund was launched in 1986 but the important growth in the industry started only a few years ago. 15 On the other hand, the fact that most asset companies are held by banks, and given that the banking sector in Portugal has been through a process of reorganization over the last decade, may also explain why we observe so little variation in the funds' portfolio holdings. Table 2 presents the aggregate statistics on the trades of the funds, which we infer from changes in the quarterly portfolio holdings of each fund. Because we focus on changes in holdings that result from trades (purchases and sales), we excluded all stock/quarter observations whose changes were related with shares adjustments (for example, stock issues or splits). We have a total of 31456 changes. 50.5% of the changes that are positive (buys) and 49.5% are negative (sales). When we analyze the changes, quarter by quarter, the balance between purchases and sales is also even. 4. RESULTS 4.1 Overall Levels of Herding by Mutual Funds In table 3, we present the overall levels of herding exhibited by our sample, for the all period and for each of the three years, 1998, 1999 and 2000. The herding measure of 11.38% shown in table 3 is the Lakonishok et al. (1992) measure of herding computed over all stock-quarters during the 3-year period. We can interpret this average herding measure as meaning that, if 100 funds trade a given stock, then approximately eleven more funds trade on one side of the market than would be expected if there was no positive feedback trading between funds. In other words, if the number of changes in holdings was, a priori, equally balanced between positive and negative changes, 61.38% (50%+11.38%) of the funds traded in one direction and the remaining 38.62% (50%-11.38%) traded in the opposite direction. The average herding measure does not vary much across the three years in sample, the lowest level being observed in 1999 (H=9.08%). The overall level of herding we find is much higher than that reported in previous studies using UK and US mutual and pension fund data (please refer to table 4). The overall level of herding in our study is close to what has been reported by Choe, Kho and Stulz (1999) for their study on the herding behavior of foreign individual investors in the Korean stock market (they find no herding measure below 20%). This higher average level of herding for less mature stock markets is consistent with arguments that rely on informational cascades or information inefficiencies arguments. Agency models would not predict different levels of herding across stock markets in different stages of maturity. Information models could predict those 15 In 1990, 1995 and 1998, total net assets in management accounted for, respectively, 1 897, 10 639 and 24 087 million euros. 11 differences: our evidence suggests that, in stock markets that exhibit poorer aggregation of information and where the precision of the public pool of information is lower, herds seem to form more often. One observable implication of these results should be that as stock markets become more mature, the level of herding decreases. We would expect to find a stronger decrease in the herding measure when we require a larger number of funds to trade a given stock-quarter, given that stocks that are traded by many funds usually have more public information and, as such, the herding effect should be lower. When we introduce restrictions on the minimum number of funds trading in a given stock-quarter, there is very little difference in the results. Table 3 shows that if we impose hurdles of up to ten funds (herds of 2, 5 or 10 funds) the measure of herding increases slightly (H=12.44, 13.54, 13.96). For a minimum of 15 funds trading a given stock-quarter the measure slightly decreases to 13.60. 4.2 Buy-Herding and Sell-Herding Table 5 reports average buy-herding and sell-herding measures for the all period and for each individual year. As stated above, the comparison of the two measures of herding reveals whether herds tend to form more often on one side of the market. We find higher levels of herding on the buy side. 16 Although the level of herding is significant in either side of the market, over time, the value of the statistic changes and so does the side of the market where herding is predominant. In particular, herding in sales seems to have been predominant in 1999. The evidence in our study is similar to that reported by previous studies (Wylie, 2000, reports a stronger herding effect on the buy side while Grinblatt et al., 1995, reports a stronger effect on the sell side). 4.3 Herding and Fund Specific Characteristics Size /Market Capitalization Table 6 presents the herding measures averaged over stock-quarters segregated by fund size. Size quintiles are formed each year so that each quintile has the same number of funds except for the mid-quintile. We find that the levels of herding computed for these subgroups of funds are much smaller than that observed for the overall sample. As discussed in section 2, we might think that funds within the same size-class would show less herding because the imitation would occur across different size groups. In particular, small funds would follow large, presumably more informed funds. Yet if the behavior of funds were driven by reputation concerns, funds would herd within their group to preserve their status quo. Another potential reason for low herding for size subgroups of funds, might be that size is not the central characteristic when choosing the fund to imitate. Within the same size group we may have funds with completely different styles and the herds may be formed based on style, not size. 16 The difference between herding on purchase and herding on sales is significant at a 1% significance level except when the hurdle of the number of funds is set above ten funds. 12 Our results suggest that size is not indeed an important factor: herds seem to be formed with funds of different sizes. Medium cap funds exhibit the highest level of herding. The average level of herding is particularly low for the extreme quintiles (very large or very small funds). A lower level of herding among small funds is consistent with the hypothesis that these funds prefer to imitate large funds, that are expected to have superior information resources and therefore more precise signals. A lower level of herding among large funds could be driven by the fact that they use their private information to trade. Lakonishok et al. (1992) report similar results: size subgroups of funds exhibit lower levels of herding than that observed for the overall sample. They also find that the level of herding is lower within the subgroup of small funds. Our results seem to be consistent with cascade or information inefficiency based explanations. The implication of the reputational explanation for the levels of herding across different subgroups is not borne out by the data. Portfolio Holdings We next examine the levels of herding for subgroups formed by portfolio holdings. Quintiles are formed each year so that each quintile has the same number of funds except for the mid-quintile. In table 7, we report these results. We find that the levels of herding computed for these subgroups of funds are also much smaller than that observed for the overall sample. As for size, portfolio holdings do not seem to be the elected factor to choose which fund to imitate. If we exclude the first quintile, the results suggest a negative relation between the proportion of stocks held by a fund and the level of herding: funds that hold more stocks do not seem to trade together so often. This evidence is consistent with the hypothesis that higher costs in processing information on several asset classes result in that funds that are more diversified should exhibit higher levels of herding. Table 7 shows that funds that have less stocks - and therefore hold other assets - herd more often. As above, results are consistent with an explanation that relies on information. The interpretation of this last result should not be taken too far given that the quintiles formed here do not yield very different groups in terms of portfolio holdings. Please recall that the funds in our sample have a minimum of 75% of their portfolio invested in stocks. Frequency in Portfolio Rebalancing To explore whether herding is different for funds with longer or shorter investment horizons (assuming that these correspond to funds that trade less or more frequently), we divided the sample in quintiles formed on the basis of the degree of portfolio rebalancing. To proxy the degree of rebalancing, we computed, for each fundquarter, the average across stocks of a measure of “permanence” of a given stock. As before, quintiles are formed each year so that each quintile has the same number of funds except for the mid-quintile. We find that the levels of herding computed for subgroups of funds are again much smaller than that observed for the overall sample and for the three individual 13 years, suggesting that the level of herding is lower among funds with similar trading patterns. The results in table 8 do not suggest a relation between the level of herding and the degree of rebalancing: herds seem to be formed with funds with different trading strategies. The evidence presented here does not support an informational inefficiency based explanation, that would predict that funds with higher turnover rates and, presumably lower investment horizons, exhibit higher levels of herding. 4.4 Herding and Market Conditions Market Stock Returns In table 9, we segregate stock-quarters by market aggregate returns. For a particular quarter, market returns are calculated using the average daily market returns of the PSI 20. 17 We sorted the quarters and we construct three different subgroups from our original sample. For example, the first subgroup includes the herding measures for the quarters were market returns were low. Each subgroup has the same number of funds except for the mid-group. There seems to be a clear pattern here: the level of trading seems to decrease when market returns are higher. This negative relation is consistent with some of theoretical arguments exposed above. When markets are doing well, institutional investors feel more confident and are thus more likely to use their own signals and therefore trade independently. Our evidence is also consistent with Cai, Gautman and Zeng (2000): they argue that mutual fund trading is highly correlated with market contemporaneous returns. Returns seem to drive trading and not the other way around. Market Volatility We further investigate if feedback trading strategies occur more often in periods of high or low market volatility. Table 10 shows the results for subgroups formed on the basis of the level of aggregate market volatility. For example, the third subgroup includes the herding measures for the more volatile quarters. The results suggest that the level of herding is lower when the market is more volatile. Higher volatility can be caused by more uncertainty about future values. If that is the case, then higher volatility, reflecting less precise information, should result in higher levels of herding. If, alternatively, higher volatility proxies new, unexpected information, then higher volatility, reflecting more information, should result in lower levels of herding. Our evidence supports this latter argument. This interpretation of the results is, one more time, based on information models. 17 PSI20 stands for the Portuguese Stock Market Index that includes the 20 largest (and more liquid) stocks listed on the Portuguese stock exchange. 14 5. CONCLUSION This paper provides additional evidence on the level of herding in the trades of institutional investors. We investigate the existence and magnitude of herding for a sample of 32 Portuguese mutual funds for the period of 1998 to 2000. In addition, we examine herding by subgroups of funds formed on the basis of fund specific characteristics and market conditions. As in previous studies, we use the measure of herding developed by Lakonishok et al. (1992). This measure evaluates if the proportion of funds that trade in a stock in the same direction (buying or selling) is above the expected proportion of funds trading in that direction if there was no herding. The overall level of herding is very significant. For every 100 funds that trade a given stock, approximately eleven trade on one side of the market, above what would have been expected if they were trading independently. The level of herding does not vary much over time or when we impose a minimum number of funds to trade a given stock. The level of herding is significant in either side of the market, purchases or sales. The average level of herding for Portuguese mutual funds is 4 to 5 times higher than that found in previous studies for the US and the UK. This result seems to suggest that herding is higher on more volatile markets. We find that the overall level of herding is much higher than that observed within subgroups of funds. Herds seem thus to be formed with funds of different size, different portfolio holdings and different trading strategies. The low and high cap subgroups of funds exhibit lower levels of herding and funds with less stocks seem to herd more often. Finally, we find lower levels of herding when the market is doing well and when the market is more volatile. Altogether our results are consistently with the implications of information-based models. 15 REFERENCES Banerjee, A. (1992), “A Simple Model of Herd Behavior.” Quarterly Journal of Economics 107, 797-818. Bikhchandani, S., D. Hirshleifer and I. Welch (1992), “A Theory of Fads, Fashion, Custom, and Cultural Change as Informational Cascades.” Journal of Political Economy 100, 992-1026. Brown, S.J. and W.N. Goetzmann (1995), “Performance Persistence.” Journal of Finance 50, 679-98. Cai, F., K. Gautam and L. Zheng (2000), “Institutional Trading and Stock Returns.” Working Paper. Cao, H. and D. Hirshleifer, 1997, "Limited Observability, Reporting Biases and Informational Cascades, Working Paper, Ohio Sate University. Choe, H., B. Kho and R.M. Stultz (1999), “Do Foreign Investors Destabilize Stock Markets? The Korean Experience in 1997.” Journal of Financial Economics 54, 227-64. Froot, K., D. Scharfstein and J. Stein (1992), “Herd on the Street: Informational Inefficiencies in a Market with Short-Term Speculation.” Journal of Finance 47, 1461-84. Grinblatt, M. and S. Titman (1989), “Mutual Fund Performance: An Analysis of Quarterly Portfolio Holdings.” Journal of Business 62, 394-415. Grinblatt, M., S. Titman and R. Wermers (1995), “Momentum Investment Strategies, Portfolio Performance, and Herding: A Study of Mutual Fund Behavior.” American Economic Review 85, 1088-105. Hirshleifer, D., A. Subrahmanyam and S. Titman (1994), “Security Analysis and Trading Patterns When Some Investors Receive Information Before Other.” Journal of Finance 49, 1665-98. Keynes, J.M. (1936), The General Theory of Employment, Interest and Money, MacMillan, London. Lakonishok, J., A. Shleifer, R.H. Thaler and R.W. Vishny (1992), “The Impact of Institutional Trading on Stock Prices.” Journal of Financial Economics 20, 23-43. Scharfstein, D. and J. Stein (1990), “Herd Behavior and Investment.” American Economic Review 80, 465-79. Wermers, R. (1999), “Mutual Fund Herding and the Impact on Stock Prices.” Journal of Finance 54, 581-622. Wylie, S. (2000), “Fund Manager Herding: Tests of the Accuracy of Measures of Herding Using UK Data.” Working Paper, Dartmouth University . 16 Table 1 – Portfolio Holdings for Funds in Sample by Fund Size Quintiles The table shows the percentage of stocks (domestic and foreign) held by the 32 Portuguese mutual funds in our sample, by size quintiles at the end of the last quarter of 1997, and the proportion of domestic/foreign stocks (securities) in total holdings. The portfolio holdings are obtained from Bolsa de Valores de Lisboa e Porto. Total funds in management by the funds in sample accounted to 1310 million euros. Fund Size (4 th Quarter 1997) Quintile 1 (small) Quintile 3 Quintile 4 Quintile 5 (large) Total Quintile 2 75,69% 83,52% 84,44% 69,09% 81,39% 79,17% % Foreign Stocks 7,91% 0,83% 1,54% 14,52% 2,07% 5,14% % Total Stocks 83,60% 84,35% 85,98% 83,61% 83,46% 84,31% % Domestic Securities 79,52% 84,36% 86,48% 72,20% 82,72% 81,40% 7,91% 0,83% 1,54% 14,55% 2,09% 5,15% 95,18% 99,00% 97,64% 95,69% 98,39% 97,26% 99,99% 99,99% 99,99% 99,79% 99,04% 99,81% % Domestic Stocks % Foreign Securities Domestic Stocks Domestic Securities ForeignStocks ForeignSecurities 17 Table 2 –Buys and Sells The table shows the trading data for 32 Portuguese mutual funds. Trades are inferred from changes in quarterly portfolio holdings, for the all sample period, 1998 to 2000, and by quarter. For each average quarter and for the all period, this table documents the number of purchases, sales and aggregate trades and the proportion of buys and sells. The portfolio holdings are obtained from Bolsa de Valores de Lisboa e Porto. Quarter 1998 - 2000 1 st 2 nd 3 rd 4 th Buys 4160 (50,52%) 4037 (50,57%) 3949 (50,55%) 3741 (50,38%) 15887 (50,50%) Sells 4075 (49,48%) 3946 (49,43%) 3863 (49,45%) 3685 (49,62%) 15569 (49,50%) 8235 (100%) 7983 (100%) 7812 (100%) 7426 (100%) 31456 (100%) TOTAL 18 Table 3 – Herding Levels in Portuguese Mutual Funds Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. n is the number of funds required to trade a stock in each quarter used to compute the Lakonishok et al. (1992) herding measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period 1998 1999 2000 1998 – 2000 n ≥1 11,25a (3018) 9,08a (2691) 14,25a (2281) 11,38a (7990) n≥2 12,45a (3000) 9,88a (2676) 15,47a (2263) 12,44a (7939) n≥5 12,42a (2902) 12,07a (2578) 16,83a (2154) 13,54a (7634) n ≥ 10 13,08a (2603) 13,13a (2249) 16,16a (1884) 13,96a (6736) n ≥ 15 12,11a (2132) 12,68a (1865) 16,63a (1610) 13,60a (5607) 5>n≥2 10,57b (98) -3,52 (98) 9,81b (109) 5,77b (305) 10 > n ≥ 5 8,95b (299) 9,62b (329) 11,47a (270) 9,95b (898) 15 > n ≥ 10 14,32a (471) 13,87a (384) 9,90b (274) 13,09a (1129) 19 Table 4 – Comparative Results The table compares the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds with the results reported using US and UK fund data. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. n is the number of funds required to trade a stock in each quarter used to compute the Lakonishok et al . (1992) herding measure.The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. US pension funds data, US mutual fund data and UK mutual fund data are, respectively, from Lakonishok et al. (1992), Wermers (1999) and Wylie (2000). Results from US and UK Studies Number of Funds Trading in the Period 1998 – 2000 US Mutual Funds (1975-1994) US Pension Funds (1985 – 1989) UK Mutual Funds (1986 – 1993) n ≥1 11,38 (7990) - 2,7 (N/A.) - n≥2 12,44 (7939) - - 2,6 (27014) n≥5 13,54 (7634) 3,4 (109486) - 2,5 (10522) n ≥ 10 13,96 (6736) 3,6 (67252) 2,0 (N/A.) 3,3 (3342) n ≥ 15 13,60 (5607) - - 4,3 (1007) 5>n≥2 5,77 (305) - - 2,6 (16492) 10 > n ≥ 5 9,95 (898) - - 2,1 (7180) 15 > n ≥ 10 13,09 (1129) - - 2,8 (2335) 20 Table 5 – Buy and Sell Herding Levels Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds segregated by purchases and sales. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. Buy herding stock-quarters are those where p(i,t)>p(t), that is, the proportion of buyers was greater than the expected proportion of buyers; sell stock-quarters are those where p(i,t)<p(t) meaning the proportion of sellers was greater than the expected proportion of sellers. n is the number of funds required to trade a stock in each quarter used to compute the Lakonishok et al . (1992) herding measure.The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period 1998 1999 2000 1998 – 2000 Buy Sell Buy Sell Buy Sell Buy Sell n ≥1 12,40a (1634) 10,37a (1384) 8,41a (1387) 9,76a (1304) 16,59a (1223) 12,05a (1058) 12,30a (4244) 10,63a (3746) n≥2 12,55a (1611) 12,45a (1389) 7,65a (1392) 12,27a (1284) 15,92a (1250) 15,08a (1013) 11,93a (4253) 13,11a (3686) n≥5 13,28a (1491) 11,74a (1411) 13,74a (1217) 10,66a (1361) 19,09a (1142) 14,14a (1012) 15,15a (3850) 11,99a (3784) n ≥ 10 13,24a (1314) 12,95a (1289) 15,18a (1038) 11,50a (1211) 14,25a (1034) 18,34a (850) 14,14a (3386) 13,79a (3350) n ≥ 15 14,07a (1015) 10,46a (1117) 16,67a (829) 9,62a (1036) 18,10a (783) 15,22a (827) 16,09a (2627) 11,49a (2980) 5>n≥2 10,47a (41) 10,81a (57) -10,60a (51) 11,20a (43) 12,99 (45) 10,30a (62) 3,46 (137) 10,72a (162) 10 > n ≥ 5 8,00a (139) 9,75a (160) 10,13a (150) 9,12a (179) 11,37a (131) 12,42a (139) 9,81a (420) 10,29a (478) 15 > n ≥ 10 14,33a (230) 14,27a (241) 10,42a (190) 16,82a (194) 11,51 (128) 8,84a (146) 12,32a (548) 13,76a (581) 21 Table 6 – Herding Levels Segregated by Fund Size Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds segregated by fund size. Size is measured by total assets under management. Each quintile is formed on the basis of the size of the fund during the quarter prior to the herding measure quarter. Quintiles are recalculated every year. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. The herding measures p( i, t) − p (t) − E p( i, t) − p( t) are averaged separately over stock-quarters belonging to different fund size quintiles. In each stock-period the funds of the sample were divided R times into 5 with a remainder of S. Then the qth quintile contains R observations, except the third quintile which contains R+S observations. We impose no minimum requirement on the number of funds trading a stock in each period. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. 1998 1999 2000 1998 – 2000 Quintile 1 (small) -1,18 (399) 0,88 (358) 3,00b (372) 0,85 (1089) Quintile 2 1,44 (656) 1,04 (355) 3,65b (486) 2,06 (1497) Quintile 3 1,27 (818) 3,14b (727) 1,24 (599) 1,90 (2144) Quintile 4 2,65 (454) 0,15 (423) 5,91b (388) 2,81b (1265) Quintile 5 (large) 0,57 (691) -1,45 (395) -0,15 (436) -0,16 (1522) Fund Size 22 Table 7 – Herding Levels Segregated by Fund Portfolio Holdings Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds segregated by portfolio holdings. Portfolio holdings are measured by the percentage of stocks in the fund portfolio. Each quintile is formed on the basis of the holdings of the fund during the quarter prior to the herding measure quarter. Quintiles are recalculated every year. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. The herding measures p( i, t) − p (t) − E p( i, t) − p( t) are averaged separately over stock-quarters belonging to different fund size quintiles. In each stock-period the funds of the sample were divided R times into 5 with a remainder of S. Then the qth quintile contains R observations, except the third quintile which contains R+S observations. We impose no minimum requirement on the number of funds trading a stock in each period. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Fund Portfolio Holdings 1998 1999 2000 1998 – 2000 Quintile 1 (less stocks) 3,58 (727) 2,12 (442) 0,38 (462) 2,28 (1631) Quintile 2 5,22b (500) 11,84a (473) 10,00a (391) 8,89b (1364) Quintile 3 3,65 (795) 2,81 (562) 9,90a (593) 5,31b (1950) Quintile 4 1,81 (421) -1,42 (359) 3,25b (468) 1,42 (1248) Quintile 5 (more stocks) -2,25 (575) -1,70 (422) 8,13a (367) 0,71 (1364) 23 Table 8 – Herding Levels Segregated by Frequency in Portfolio Rebalancing Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds segregated by portfolio rebalancing frequency. Portfolio rebalancing frequency is defined as ( ( X t1, A − X 1t −1, A ) 2 )  X t1, A + X t1−1, A    2   2 where X 1t ,A represents the number of company A stocks that fund 1 holds on quarter t. The rebalancing frequency is the average of the statistic calculated over the stocks held by the fund in a particular quarter. The value of the statistic increases when portfolio rebalancing is less frequent. Each quintile is formed on the basis of the frequency statistic of the fund during the quarter prior to the herding measure quarter except for the year 1998 that we used the first quarter of 1998. Quintiles are recalculated every year. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. The herding measures p( i, t) − p (t) − E p( i, t) − p( t) are averaged separately over stock-quarters belonging to different fund size quintiles. In each stock-period the funds of the sample were divided R times into 5 with a remainder of S. Then the qth quintile contains R observations, except the third quintile which contains R+S observations. We impose no minimum requirement on the number of funds trading a stock in each period. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. 1998 1999 2000 1998 – 2000 2,70b (462) -2,41b (438) 6,32a (915) 3,29b (1815) Quintile 2 4,97a (452) -0,31 (325) 6,29a (494) 4,13a (1271) Quintile 3 1,28 (783) 3,34a (561) 7,26a (620) 3,76b (1964) Quintile 4 4,27a (578) 0,83 (534) 9,52a (484) 4,71a (1596) -0,64 (433) 3,56a (526) 3,16b (262) 1,98 (1221) Portfolio Rebalancing Quintile 1 (more frequent portfolio rebalancing) Quintile 5 (less frequent portfolio rebalancing) 24 Table 9 – Herding Levels Segregated by Market Stock Returns Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds segregated by market stock returns. Subgroups are formed on the basis of the Portuguese Stock Index (PSI20) averaged daily returns. Sub-group 1 contains the observations for the four quarters with the lowest returns. Subgroup 3 contains observations for the four quarters with the highest returns. Sub-group 2 contains the observations for the remaining quarters. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. n is the number of funds required to trade a stock in each quarter used to compute the Lakonishok et al. (1992) herding measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period Sub-group 1 (lowest returns) Sub-group 2 Sub-group 3 (highest returns) n ≥1 14,27a (2389) 10,75a (2707) 9,60a (2894) n≥2 15,26a (2375) 12,13a (2687) 10,47a (2877) n≥5 16,26a (2249) 14,15a (2592) 10,75a (2793) n ≥ 10 15,29a (1966) 15,41a (2272) 10,90a (2498) n ≥ 15 16,95a (1591) 14,15a (1931) 10,07a (2085) 5>n≥2 10,64 a (126) 0,08 (95) 4,42 (84) 10 > n ≥ 5 15,68a (283) 7,84b (320) 8,06a (295) 15 > n ≥ 10 12,94a (375) 19,98 a (341) 11,58a (413) 25 Table 10 – Herding Levels Segregated by Market Volatility Lakonishok et al. (1992) Measure of Herding The table reports the Lakonishok et al. (1992) herding measure for a sample of 32 Portuguese mutual funds segregated by market volatility. Subgroups are formed on the basis of the Portuguese Stock Index (PSI20) daily returns standard deviation in each quarter. Sub-group 1 contains the observations for the four quarters with the lowest volatility. Sub-group 3 contains the observations for the four quarters with the highest volatility. Sub-group 2 contains the observations for the remaining quarters. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. n is the number of funds required to trade a stock in each quarter used to compute the Lakonishok et al . (1992) herding measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period Sub-group 1 (lowest volatility) Sub-group 2 Sub-group (highest volatility) n ≥1 12,22a (2285) 10,54a (2757) 11,86a (2948) n≥2 13,62a (2267) 11,14a (2744) 13,10a (2928) n≥5 14,95a (2155) 13,59a (2640) 12,61a (2839) n ≥ 10 13,29a (1876) 14,97a (2332) 13,33a (2528) n ≥ 15 15,21a (1530) 13,53a (1955) 12,42a (2122) 5>n≥2 6,26b (112) -2,52 (104) 11,39a (89) 10 > n ≥ 5 17,85a (279) 6,28b (308) 7,46b (311) 15 > n ≥ 10 11,59a (346) 18,82a (377) 14,10a (406) 26 APPENDIX I – Funds in Sample 27 Table I.1 – Funds in Sample Fund Atlântico Acções Barclays FPA Barclays Premier Acções Portugal BCI Acções Portugal / Santander Acções Portugal BCI Iberfundo Acções / Santander Iberfundo Acções BCP Acções BNU Acções BNU PPA BPI Acções BPI Poupança Acções Caixagest Acções Portugal Caixagest Internacional / Caixagest Acções Europa Caixagest Valorização Capital Portugal DB – Investimento ES Portugal Acções Eurocapital – FA FAIMIABV Lisboa Fipor Poupança Investimento Luso – Acções Luso – Capital M Acções Portugal M Capital MG Acções Novo - Fundo Capital Portugal Acções PPA Atlântico PPA BCP-FPA PPA Grupo BFE / PPA Grupo BPI Sotto PPA Totta Acções Unicapital Asset Management Company AF – INVESTIMENTOS, Fundos Mobiliários, S.A. BARCLAYS FUNDOS, S.A. BARCLAYS FUNDOS, S.A. BCI – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. / SANTANDER – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. BCI – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. / SANTANDER – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. AF – INVESTIMENTOS, Fundos Mobiliários, S.A. INVESTIL – Sociedade Gestora dos Fundos de Investimento Mobiliário, S.A. INVESTIL – Sociedade Gestora dos Fundos de Investimento Mobiliário, S.A. BPI FUNDOS – Gestão de Fundos de Investimento Mobiliário, S.A. BPI FUNDOS – Gestão de Fundos de Investimento Mobiliário, S.A. CAIXAGEST – Técnicas de Gestão de Fundos, S.A. CAIXAGEST – Técnicas de Gestão de Fundos, S.A. CAIXAGEST – Técnicas de Gestão de Fundos, S.A. TOTTA Fundos, S.A. / MC – FUNDOS - Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. DB FUNDOS – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. ESAF – Espírito Santo Fundos de Investimento Mobiliário, S.A. BPI FUNDOS – Gestão de Fundos de Investimento Mobiliário, S.A. AF – INVESTIMENTOS, Fundos Mobiliários, S.A. AF – INVESTIMENTOS, Fundos Mobiliários, S.A. BCI – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. / SANTANDER – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. BCI – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. / SANTANDER – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. M FUNDOS – Gestora de Fundos de Investimento Mobiliário, S.A. / MELLO ACTIVOS FINANCEIROS - Gestora de Fundos de Investimento Mobiliário, S.A. M FUNDOS – Gestora de Fundos de Investimento Mobiliário, S.A. . / MELLO ACTIVOS FINANCEIROS - Gestora de Fundos de Investimento Mobiliário, S.A. MG FUNDOS – Sociedade Gestora de Investimento Mobiliário, S.A. AF – INVESTIMENTOS, Fundos Mobiliários, S.A. DB FUNDOS – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. AF – INVESTIMENTOS, Fundos Mobiliários, S.A. AF – INVESTIMENTOS, Fundos Mobiliários, S.A. BPI FUNDOS – Gestão de Fundos de Investimento Mobiliário, S.A. PLURIFUNDOS – Sociedade Gestora de Fundos de Investimento Mobiliário, S.A. TOTTA Fundos, S.A. TOTTA Fundos, S.A. 28 APPENDIX II – Quarter by Quarter Herding 29 Table II.1 – Quarter by Quarter Herding Levels Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of 32 Portuguese mutual funds. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. n is the number of funds required to trade a stock in each quarter we use to compute Lakonishok et al. (1992) measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period 1998 1 st 2 nd 1999 3 rd 4 th 1 st 2 nd 2000 3 rd 4 th 1 st n ≥1 11,14a 7,48b 15,91a 10,51a 8,67a 7,33b 13,25a 6,88 9,88a Number of stock-quarters 826 749 738 705 670 641 696 684 679 a b 2 nd 3 rd 4 th 19,13a 13,58a 14,71a 654 592 356 n≥2 12,44 8,41 17,60 11,35 9,72 8,13 14,40 7,08 11,00 19,34 15,99 15,96a Number of stock-quarters 820 745 733 702 664 636 694 682 673 653 584 353 n≥5 10,05a 10,78a 17,26a 11,92b 11,60a 11,51a 15,19a 9,82a 11,19a 22,19a 19,03a 14,06a Number of stock-quarters 800 722 707 673 633 605 679 661 659 624 558 313 n ≥ 10 12,30a 11,28a 20,81a 8,21b 14,98a 9,07b 17,34a 11,06b 12,02a 22,55a 18,03a 8,73 Number of stock-quarters 753 652 603 595 564 550 562 573 577 543 494 270 a b a a a a a a n ≥ 15 9,60 12,59 19,26 6,76 11,46 13,02 16,37 9,84 14,06 20,23 16,18 15,28b Number of stock-quarters 637 516 527 452 469 409 497 490 506 480 449 175 5>n≥2 26,15a -7,81 19,29b 6,60 -3,33 -5,00 6,25 -8,59 -6,48 9,66 5,21 18,60a Number of stock-quarters 20 23 26 29 31 31 15 21 14 29 26 40 10 > n ≥ 5 -4,01 9,00 4,61 22,51b 1,41 19,14 11,94 7,01 6,72 7,70 9,03 31,28b Number of stock-quarters 47 70 104 78 69 55 117 88 82 81 a a a a a a b a a a 64 43 15 > n ≥ 10 18,64 8,82 24,79 10,48 23,94 3,61 20,32 14,71 2,49 27,79 26,85 -4,44 Number of stock-quarters 116 136 76 143 95 141 65 83 71 63 45 95 b a b a a 30 Table II.2 – Quarter by Quarter Buy and Sell Herding Levels in 1998 Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of Portuguese mutual funds in 1998 segregated by purchases and sales. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. Buy herding stock-quarters are those where p(i,t)>p(t), that is, the proportion of buyers was greater than the expected proportion of buyers and likewise sell stock-quarters are those where p(i,t)<p(t) meaning the proportion of sellers was greater than the expected proportion of sellers. n is the number of funds required to trade a stock in each quarter we use to compute Lakonishok et al. (1992) measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period Quarter 1 st Buy 2 nd Sell Buy 3 rd Sell 4 th n ≥1 Buy Sell 14,15a 8,34b 5,55 9,86 22,10a 11,84a 9,18 11,92a Number of stock-quarters 423 403 421 328 363 375 427 278 n≥2 15,85a 9,52b 6,32 11,27b 13,89a 7,03 16,50a Number of stock-quarters 379 441 444 301 372 427 275 n≥5 13,35a 7,17 10,32b 11,32 Number of stock-quarters 386 414 391 331 23,25a 361 20,22a 353 15,20 a 354 Sell 9,63 Sell 14,31a 361 312 n ≥ 10 15,52 9,40 11,81 10,68 26,78 16,60 2,14 16,44a Number of stock-quarters 355 398 347 305 263 340 349 246 n ≥ 15 12,48b 7,10 10,78 14,93b 28,19a 13,51b 6,45 7,04 Number of stock-quarters 302 335 291 225 214 313 208 244 5>n≥2 18,97b 35,71 -15,63 0,00 41,44b 8,22 6,67 6,55 Number of stock-quarters 10 10 > n ≥ 5 -6,11 Number of stock-quarters 20 15 > n ≥ 10 15,41 Number of stock-quarters a 70 b 10 -2,43 27 23,49 46 b 11 5,80 44 15,11 60 12 13,78 26 3,58 76 a 8 -0,12 34 39,25 27 a 18 6,97 70 17,55 49 12 23,96 41 3,43 73 17 21,06 37 17,52 70 31 Table II.3 - Quarter by Quarter Buy and Sell Herding Levels in 1999 Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of Portuguese mutual funds in 1999 segregated by purchases and sales. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. Buy herding stock-quarters are those where p(i,t)>p(t), that is, the proportion of buyers was greater than the expected proportion of buyers and likewise sell stock-quarters are those where p(i,t)<p(t) meaning the proportion of sellers was greater than the expected proportion of sellers. n is the number of funds required to trade a stock in each quarter we use to compute Lakonishok et al. (1992) measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period Quarter 1 st 2 nd 3 rd 4 th Buy Sell Buy Sell Buy Sell Buy Sell 8,78 8,56b 4,46 10,42b 12,61b 13,78a 7,56 6,31 Number of stock-quarters. 395 275 336 305 344 352 312 372 n≥2 9,80 9,63b 3,89 12,92b 12,14 16,25a 4,57 9,50b Number of stock-quarters 392 272 334 302 326 368 340 342 n≥5 19,52a 5,74 10,39 12,58b 14,41b 15,79a 11,64a 8,23 Number of stock-quarters 268 365 302 303 295 384 352 309 n ≥1 n ≥ 10 21,90 9,69 4,68 14,09 21,55 14,62 15,14 8,00 Number of stock-quarters 242 322 297 253 236 326 263 310 n ≥ 15 16,88b 6,94 16,00 10,64 18,87 14,63a 14,96 6,00 Number of stock-quarters 207 262 181 228 213 284 228 262 5>n≥2 -13,33 6,67 -12,50 6,25 0,00 10,42 -9,38 34,38 Number of stock-quarters 15 16 19 6 9 a 10 > n ≥ 5 11,72 -5,47 15,35 Number of stock-quarters 26 43 29 15 > n ≥ 10 37,68b 15,70 -6,59 Number of stock-quarters 35 60 97 b 12 23,88 26 24,03b 44 b a 11 a 5,11 18,78 59 58 32,70 23 14,12 42 b 12,99 36 15,68 35 6 3,02 52 13,98 48 32 Table II.4 - Quarter by Quarter Buy and Sell Herding Levels in 2000 Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of Portuguese mutual funds in 2000 segregated by purchases and sales. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. Buy herding stock-quarters are those where p(i,t)>p(t), that is, the proportion of buyers was greater than the expected proportion of buyers and likewise sell stock-quarters are those where p(i,t)<p(t) meaning the proportion of sellers was greater than the expected proportion of sellers. n is the number of funds required to trade a stock in each quarter we use to compute Lakonishok et al. (1992) measure. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Number of Funds Trading in the Period Quarter 1 st 2 nd 3 rd 4 th n ≥1 Buy Sell Buy Sell Buy Sell Buy Sell 12,11b 8,17b 20,89b 17,56a 17,27b 10,29 14,23b 15,24a Number of stock-quarters 298 381 379 275 326 266 220 136 n≥2 11,62 10,51a 22,69a 16,47a 14,41b 17,79a 12,44b 20,06a Number of stock-quarters 300 373 378 275 365 219 207 146 n≥5 16,19b 7,99 20,93a 23,51a 22,90a 15,38b 12,34b 16,26 Number of stock-quarters 235 424 407 217 313 245 187 126 n ≥ 10 9,72 13,89 19,81 25,97 16,51 20,32 6,28 12,09 Number of stock-quarters 252 325 317 226 301 193 164 106 n ≥ 15 11,96 15,68b 23,12a 17,35 22,02a 10,87 11,73a 21,50 b 236 210 239 111 64 5,21 -1,67 12,08 11,90 25,30a 20 20 a a Number of stock-quarters 218 288 244 5>n≥2 18,06 -11,39a 31,25 Number of stock-quarters 3 11 10 > n ≥ 5 13,16 0,27 Number of stock-quarters 46 36 15 > n ≥ 10 15,23 -3,88 Number of stock-quarters 27 44 10 a 17 8,38 7,36 25 56 31,45b 27 25,36b 36 a 12 b 14 a -3,46 27,77 34,86 37 27 23 26,43 23 27,28 22 -7,73 51 27,70b 20 -1,16 44 33 Table II.5 – Quarter by Quarter Herding Levels Segregated by Fund Size Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of Portuguese mutual funds segregated by fund size. Size is measured by total assets under management. Each quintile is formed on the basis of the size of the fund during the quarter prior to the herding measure quarter. Quintiles are recalculated every year. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t ) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. The herding measures p( i, t) − p (t) − E p( i, t) − p( t) are averaged separately over stock-quarters belonging to different fund size quintiles. In each stock-period the funds of the sample were divided R times into 5 with a remainder of S. Then the qth quintile contains R observations, except the third quintile which contains R+S observations. We impose no minimum requirement on the number of funds trading a stock in each period. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Fund Size 1998 1 st 2 nd 1999 3 rd 4 th 1 st 2 nd 2000 3 rd 4 th 1 st 2 nd 3 rd 4 th Quintile 1 (small) 1,00 3,48 -1,43 -11,36 2,82 1,68 -0,93 0,04 -0,61 2,82 9,34a 0,91 Number of stock-quarters 142 102 80 75 95 77 92 94 120 97 92 63 -3,91 0,14 5,80b 4,77 -1,47 1,17 5,81 -1,06 -2,11 9,84a 1,95 5,41 171 184 147 154 94 80 88 93 136 134 123 93 0,43 -1,36 2,20 3,09 1,58 4,97 2,17 4,32 -4,27 5,18 3,85 1,01 177 183 226 232 216 172 175 164 188 174 142 95 Quintile 2 Number of stock-quarters Quintile 3 Number of stock-quarters Quintile 4 7,63 b b b 3,07 0,98 -2,92 -0,11 -2,76 4,26 -0,69 1,32 14,62 4,44 -1,19 Number of stock-quarters 140 97 121 96 113 99 100 111 116 113 120 39 Quintile 5 (large) 0,96 -5,01 3,80 3,37 -1,08 0,00 -1,57 -3,20 -2,82 4,81 -2,59 -1,31 Number of stock-quarters 196 183 164 148 100 96 105 94 119 136 115 66 34 Table II.6 – Quarter by Quarter Herding Levels Segregated by Fund Portfolio Holdings Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of Portuguese mutual funds segregated by fund portfolio holdings. Portfolio holdings are measured by the percentage of stocks in the fund portfolio. Each quintile is formed on the basis of the holdings of the fund during the quarter prior to the herding measure quarter. Quintiles are recalculated every year. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. The herding measures p( i, t) − p (t) − E p( i, t) − p( t) are averaged separately over stock-quarters belonging to different fund size quintiles. In each stock-period the funds of the sample were divided R times into 5 with a remainder of S. Then the qth quintile contains R observations, except the third quintile which contains R+S observations. We impose no minimum requirement on the number of funds trading a stock in each period. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Fund Portfolio Holdings 1998 1 st 2 nd 1999 3 rd 4 th 1 st 2 nd 2000 3 rd 4 th 1 st 2 nd 3 rd 4 th Quintile 1 (less stocks) 5,28b 6,81b 2,83 -0,66 -3,05 5,11 2,54 4,48 -2,61 6,12b -4,31 3,49 Number of stock-quarters 196 177 172 182 120 107 106 109 118 110 129 105 2,62 7,12b 9,12a 0,70 15,94a 9,07a 10,78a 11,15a 11,89a 10,61a 10,63a -0,77 144 145 125 86 127 116 116 114 133 124 98 36 -0,77 b 6,85 b 7,53 198 190 196 0,74 b 5,21 3,33 -0,46 -1,38 -1,19 137 77 94 113 92 81 Quintile 2 Number of stock-quarters Quintile 3 Number of stock-quarters Quintile 4 Number of stock-quarters 1,31 b 7,05 2,73 1,21 -0,52 7,14 15,54 9,81 1,41 211 167 113 138 144 199 184 154 56 -5,38 2,22 1,02 5,05 4,46 1,61 92 94 122 139 127 a b b b a a 80 Quintile 5 (more stocks) -0,37 -7,00 1,09 -2,51 -0,97 -4,31 0,46 -2,07 7,40 6,86 12,04 6,52a Number of stock-quarters 151 160 151 113 112 107 108 95 107 97 84 79 a 35 Table II.7 – Quarter by Quarter Herding Levels Segregated by Frequency in Portfolio Rebalancing Lakonishok et al. (1992) Measure of Herding The table reports quarter by quarter Lakonishok et al . (1992) herding measures for a sample of 32 Portuguese mutual funds segregated by portfolio rebalancing frequency. Portfolio rebalancing ( X t1, A − X 1t −1, A ) 2 where X 1t ,A represents the number of company A stocks that fund 1 holds on quarter t. The rebalancing frequency is the average of the statistic 2  X t1, A + X t1−1, A   2    calculated over the stocks held by the fund in a particular quarter. The value of the statistic increases when portfolio rebalancing is less frequent. Each quintile is formed on the basis of the frequency statistic of the fund during the quarter prior to the herding measure quarter except for the year 1998 that we used the first quarter of 1998. Quintiles are recalculated every year. The herding statistic for a given stock-quarter is defined as p( i, t) − p (t) − E p( i, t) − p( t) , where p(i,t) is the proportion of funds trading stock i during quarter t that are buyers and p(t) is the average of frequency is defined as ( ) p(i,t) over all stocks i in quarter t. E p (i, t) − p( t) is the adjustment factor calculated using a binomial distribution under the hypothesis of no herding. The herding measures p( i, t) − p (t) − E p( i, t) − p( t) are averaged separately over stock-quarters belonging to different fund size quintiles. In each stock-period the funds of the sample were divided R times into 5 with a remainder of S. Then the qth quintile contains R observations, except the third quintile which contains R+S observations. We impose no minimum requirement on the number of funds trading a stock in each period. The herding measures are computed in each stock-quarter and then averaged over the constituents of each group. The number of stock-quarters in each subgroup is in parentheses. a indicates statistically significance at the 1 percent level; b indicates statistically significant at the significance of 5 percent level. Portfolio Rebalancing Quintile 1 (more frequent portfolio rebalancing) Number of stock-quarters 1998 1999 2000 1 st 2 nd 3 rd 4 th 1 st 2 nd 3 rd 4 th 1 st 2 nd 3 rd 4 th 2,84 5,71b 0,65 0,88 0,18 -5,80b -4,00 -0,41 2,97 7,58b 9,09a 5,88 131 127 118 86 114 99 114 111 245 239 220 a 211 Quintile 2 1,63 8,04 6,98 3,17 3,21 -4,12 -5,14 2,47 1,18 8,16 10,60 5,22 Number of stock-quarters 142 141 90 79 111 63 75 76 121 115 a b b a 126 132 b a Quintile 3 1,11 1,49 2,66 -0,01 4,74 -0,94 2,12 7,31 1,80 8,98 Number of stock-quarters 245 181 174 183 144 134 145 138 199 172 b a a 10,91 9,56a 163 86 Quintile 4 8,24 2,17 7,27 -0,12 1,09 -0,34 3,41 -1,02 4,72 20,80 6,85 -5,19b Number of stock-quarters 150 158 125 145 137 133 138 126 153 148 146 37 1,34 -3,50 -2,41 1,61 2,47 7,23b 2,14 2,65 3,13 5,13 1,14 0,60 125 108 96 104 143 124 125 134 82 95 63 22 a Quintile 5 (less frequent portfolio rebalancing) Number of stock-quarters a b 36