profitability of momentum and reversal patterns of pan ...

PROFITABILITY OF MOMENTUM AND REVERSAL PATTERNS OF PAN-EUROPEAN INDUSTRIES PORTFOLIO Mario Toscano*, Giuseppe Torluccio** Department of Management Credif (Centro di Studi e Ricerche sul Credito e la Finanza),

University of Bologna Via Capo di Lucca 34, 40126 Bologna [email protected], [email protected]

August 2009 Abstract This paper seeks to find the existence of momentum and reversal patterns in the 18 industry indexes of DJ Euro Stoxx. We made an analysis of five portfolios over eight retention periods, using monthly data from 31 December 1991 to 30 April 2007. These five portfolios were created on the basis of the returns of past industry indicators 5. Upon definition of the past returns, the sample is divided into quartiles. An additional portfolio represent the difference between worst and best performance portfolios. Once the existence of return patterns is identified, we check whether they should be related to the various industry betas of model portfolios or, on the contrary, should be attributed to bull or bear market trends. Momentum and reversal patterns are present over different building and retention periods. Finally, in order to test a possible structural breakpoints, using bootstrapping methodologies, we compute a comparison between performances before and after the year 2000. Both analysis are performed considering the mean and median as well. Results confirm a presence of patterns (momentum or contrarian) in industry index portfolio strategies.

Keywords: Financial Markets, Momentum, Contrarian, Market efficiency, Bootstrap, Monte Carlo simulations JEL Classifications: G12

*

PhD Student at Dipartimento di Scienze Aziendali – University of Bologna CEFIN (Centro Studi Banca e Finanza – Center for Research in Banking and Finance – Università di Modena e Reggio Emilia).

**

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1. INTRODUCTION Identifying a robust methodology to enable a periodical screening of the financial marked consists of one of the most covered areas of research over the last decades. Since the first works that defined financial markets as efficient systems (efficient market hypothesis, EMH), efforts were made to prove that such rules were a mere utopia. Specifically, more than to destroy the EMH foundations, different researches tried to capture and anticipate the future fluctuations of stock prices and, therefore, to model a portfolio that would allow investors to gain extra-profits and returns. Different studies try to find “abnormal” behaviours in the historic profitability of stock movements. In other words, the greatest part of empirical studies has focussed mainly on the stocks and only recently has it dedicated a small portion to stock index, proving the concrete existence of the profitability anomalies. Part of the research has proved the presence of historical abnormal returns, further known as momentum and reversal. More specifically, defined in detail a portfolio of stocks is a strategy called momentum, if it is built a sub-portfolio with shares, which in a determined referral period (building period), have resulted in a better performance (winner). Regardless of the marketplace where the quoted company or the market represented by indicators operate, it is statistically testes more than once that stocks or indicators which in 3, 6 or 12 previous months have resulted in a better performance than other shares (winner), usually, continue to perform better in the 12 successive months as compared to those with worse performances (loser). At the opposite, to what is described above, a reversal strategy shall be the one where a sub-portfolio is build with stocks that have showed a bad performance (loser) during the building period. Even in this case, it has been statistically tested that by investing in loser stocks (in the 3, 6 or 12 last months) in a long term period, (in any case for no less than 6-12 months) they resulted to perform better than remain stocks previous defined as winner.

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2. REVIEW OF THE LITERATURE Jegadeesh and Titman (JT) are universally known as the authors of momentum pattern. Specifically, an empirical study of the quoted stocks on NYSE over the period from July 1962 to December 1989, the authors show that the strategies which foresee the purchase of winners through selffunding by selling the losers produce significant returns in the successive 312 months. However, the overwhelming bulk is extended in a two-year term. According to the authors, such results should not be blamed on systemic risks or delayed responses towards common factors, but they should be seen as more related to the delayed reaction of the stock prices on specific information related to the company or enterprise. In other words, investors are inclined to under-revaluate the positive information (under-reaction), which is available from to time on determined stocks (winner). As a result, the current price of the winner stock does not duly represent the fair value of the stock itself and it is, thus, bound to adapt so that information is distributed even to the most marginal investors, continuing to over-perform as compared with losers. Similar outcomes are the case, if we took into account the profitability of winner and loser stocks one month upon announcement of quarterly earnings. The under-reaction hypothesis is also the conclusion reached by Hong, Lim and Stein (HLS)1. While making a performance analysis of the quoted NYSE, AMEX and Nasdaq stocks from 1976 to 1996, the authors come to the conclusion that the phenomenon may more concretely be attributed to the under-reaction hypothesis, in particular for the stocks with losing performance in the past. At the same time they reach the conclusion that momentum strategies work quite well for those stocks which, according to analysts, have a negligible coverage. Quite contradictory are the findings of De Bondt and Thaler (DBT)2 after the analysis of common stocks’ profitability quoted in the New York

1

Cfr. Hong H., Lim T., Stein J.C., “Bad news travels slowly: size, analyst coverage and the profitability of momentum strategies”, The Journal of Finance, February 1990. 2 Cfr. De Bondt, W.F.M, Thaler R., “Does the stock market overreact?”, The Journal of Finance, July 1985. De Bondt W.F.M, Thaler R., “Further evidence on Investor overreaction and stock market seasonality”, The Journal of Finance, July 1987.

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Stock Exchange (NYSE) in the period between January 1926 and December 1982. While studying the performance of losers, the authors reach the conclusion that “the earnings of winning and losing firms show reversal pattern that are consistent with over-reaction”. In other words, in a shortterm period, the market is inclined to overrate (overreaction) negative information and, therefore, winners continue to over-perform losers. Whereas in a long-term period where the overreaction is bound to go back to the previous figures, the quotations of the losers are bound to reach more normal values and, thus, to over perform in relation to winners. Authors further investigate the fact whether such anomalies are related to the size of the enterprise (relation between the market and book value) or to the immediate alterations of beta portfolio, which, however, lead to not satisfactory results. Finally, in the analysis of the profitability of the S&P500 stocks from 1970 to 2004, Figelman3 supports the idea that reversal pattern is a phenomenon which, besides the fact that it is confirmed in a long-term period, it also becomes present in short terms (quarterly), whereas the momentum emerges only in mid-term periods. With regard to the short term, the results are related to specific factors that are relevant to a specific stock and in the middle and long term they appear to be a phenomenon related to the ways in which information is disseminated among investors (over/under-reaction effect). The author comes across a sort of seasonbased feature in the January4 effect type of earnings. Similar in many ways with the DBT type are the studies carried out by Chan, Narasimhan and Lakonishok (CNL)5. The latter analyse the stocks quoted in NYSE, AMEX and Nasdaq from January 1977 to January 1993, elaborating on the momentum phenomenon as a under-reaction that no longer is related to past performances by to the data related to past earnings. In particular, stocks which used to result in high (low) profitability are associated to major positive (negative) changes of the earnings in the future. In addition, they

3

Cfr. Figelman I., “Stock Return Momentum and reversal”, The Journal of Portfolio Management, Fall 2007. 4 On January effect topic Cfr. Toscano M., “Verification of Calendar Anomalies on Liffe, Mif e Matif”. AF – Financial Analysis, n° 31. 5 Cfr. Chan L.K.C., Jegadeesh, Lakonishok J., “Momentum strategies”, The Journal of Finance, December 1996.

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are associated with the low book to market ratio portfolios (BV/MV), knowing that winners’ portfolios usually include glamour stocks (growth stocks), as opposed to losers stock value (high book to market ratio). From the analysis of stocks profitability quoted in NYSE, AMEX and Nasdaq, from July 1963 to December 1994, Asness6, confirms that the momentum strategy performs generally well when the investor invests on growth stocks (low BV/MV and/or high P/E ratio). They find, on the other hand, that when stocks choice is value oriented (purchase those with high book-to-market ratio and/or dividend/price), it is losers that generally perform well. The same findings were made previously by Lakonishok, Shleifer and Vishny (LSV)7, who, in an empirical research on NYSE and AMEX stocks from April 1963 to April 1990, grouping stocks into value and growth type , concluding that in the five-years

retain period, the annual and/or

cumulated

profitability of value portfolios results higher as compared to growth stocks. Chopra, Lakonishok, Ritter (CLR)8, Aarts Lehnert (AL)9 identify the existence of an important economic over-reaction effect, even after the portfolios were on basis of some balance ratios and/or betas of specific stocks. The first who performed the analysis of stocks quoted in NYSE from 1926 to 1986, indicate that the reversal effect is quite distinct in the small enterprises affecting the medium/big enterprises and that a significant part of the extra dividends is concentrated in January. They pose likewise the question whether such earnings are a result of anomalies of a compensation which is higher for a different beta in different groups. Results are surprising, as, in fact, only an extra earning of 2.5% over a total of 14% should be attributed to anomalies. The second group, who divides FTSE 350 stocks into different portfolios, depending on whether they are small or large cap and considering the book to market ratio come up with different

6

Cfr. Asness C.S., “The interaction of value and momentum strategies”, Financial Analysts Journal, March-April 1997.

7

Cfr. Lakonishok J., Shleifer A., Vishny R.W., “Contrarian investment, extrapolation, and risk”, The Journal of Finance, December 1994. 8 Cfr. Chopra N., Lakonishok J., Ritter J.R., “Measuring abnormal performance: do stock overreact?”, The Journal of financial Economics, n° 31, 1992. 9 Cfr. Aarts F., Lehnert T., “On style momentum strategies”, Applied Economic Letters, December 2005.

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results, according to the weight attributed to any stock or an equal weight to all stocks (regular) or a differentiated weight according to market capitalization (weighted). Finally, besides proving the existence of the momentum pattern, they find that the profitability of the regular portfolios exceeds that of the weighted ones. Mengoli10, in his analysis of the stocks quoted in the Italian stock market covering the period between 1955 up to 1995 confirms the existence of both momentum and reversal patterns. However, while correlating the effects with various risk measures, he reaches contradictory conclusions. Given that the beta of winner and loser portfolios are equivalent, it would be impossible to examine any effects, considering that in such a case, the construction of a beta zero portfolio (sell loser and buy winner, or vice-versa, with the same beta) we would have to expect a zero performance. Griffin, X. Ji and Martin (GJM)

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studied the indicators of 40

countries and found out that the macro factors cannot explain the earnings of momentum portfolios and that the performances of various markets have a weak correlation among them. This implies that the momentum is riskoriented, thus it is obviously specific to any country. In a further verification12 performed on the same sample, authors correlate the performance of winner and loser portfolios with the past returns. Results are not so different from what has been already proven by the momentum strategy on prices. In other words, if we achieve to obtain indicators which show simultaneously better performance and profitability, the marked is under systematic observation. Finally, they establish a relation with the momentum strategy and market trends (up & down) and with the economic cycle,

coming

to

the

conclusion

that

the

momentum

strategy

is

distinguished from the market trends and economic cycles. Involved in the

10

Mengoli S., “On the source of contrarian and momentum strategies in the Italian equity market”, International Review of Financial Analysis, 13-2004. 11 Cfr. Griffin J.M., Ji S., e Martin J.S., “Momentum investing and business cycle risk: evidence from pole to pole”, Working Paper, march 2002. 12 Cfr. Griffin J.M., Ji S., e Martin J.S., “Global momentum strategies: a portfolio perspective”, Working Paper, july 2004.

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research in the same field, Cooper, Gutierrez, Hameed (CGH)13, after verification of stocks quoted in NYSE and AMEX during the period between January 1926 and December 1995, pose the question whether the momentum and reversal patterns should be seen as results of the bull and bear market trends. Generally, the average extra-profits of momentum strategies during the bull market trend amount to 0.93%, whereas those of bear market trend are negative with 0.37%. The same as (GJM), they also argue that the macro-economic factors cannot explain the outcomes of momentum strategies. When the testing includes assets which are different from stocks, results are not significantly diverse from those of the previous studies. In particular, Chan, Hamed, Tong (CHT)14, who investigate into the weekly-based profit-making opportunity of the momentum pattern on various national indicators, confirm the presence of significant profitability, especially, in quite short retention periods (4 weeks). The strategy results into higher profitability, if it is built on indicators that, in the given building period, have displayed high volumes or when such indicators are further strengthened following an increase of trading volumes. Further, Bianchi, Drew and Polichronics (BDP)15 have observed the existence of the momentum pattern, while analyzing the forex market performances on G7 currencies from 1980 to 2004. They come up with the conclusion that if transaction costs are taken into account, the effect is definitely there. Due to this motive, the application of a momentum-type strategy may be advantageous only to the one who acts on these costs and, as opposed to this, it can in no case be advantageous to the small and medium enterprises, especially the retail ones. Finally, the existence of momentum and reversal patterns has been proven in more than one case and on various assets.

13

Cfr. Cooper J.M., Gutierrez Jr. R.C., Hammed A., “Market states and momentum”, Working Paper, 2003. 14 Cfr. Chan K., Hameed A., Tong W., “Profitability of momentum strategies in the international equity markets”, Journal of Financial and Quantitative Analysis, June 2005. The sample includes 23 countries, nine of which from Asia, 11 from Europe, two from North America and one from Africa. Except for Austria, Indonesia and South Africa, the sample covers a period between January 1980 and June 1995. All indicators are converted in US dollars. 15 Cfr. Bianchi R., Drew M.E., Polichronics J., “A test of momentum trading strategies in foreign exchange markets: evidence from G7”, Queensland University of Technology, Discussion Paper n° 182, July 2004.

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In this study, we pose the question whether these researches have been useful to investors in years to ensure extra-profits or whether they are exclusively academic case studies. An answer to this question was provided by Grinblatt, Titman e Wermers (TW)16, Griffin, Harris e Topaloglu (GHT)17 e Hwang and Rubesam (HR)18. The first, who studied the mutual fund strategies concluded that 77% of the invested funds come “momentum investors” or those who buy previously stocks previously rated as winners. However, they do not sell losers on a regular basis. At the same time, funds which have applied the following strategies have resulted with a better performance as compared to others. In the analysis of Nasdaq 100 market stocks, GHT indirectly confirm what is mentioned above, concluding that winners are bought by institutional investors and they are sold by the retail in 65.2% of the cases against 41.3% of the worst performing stocks. Finally, HR, in the analysis of the of the stocks quoted in NYSE, AMEX and Nasdaq from June 1926 to December 2005, post the question whether the momentum pattern results in extra-profits, regardless of the fact that more than 15 years have passed from the first studies. In this relation, the authors notice two break periods. Following the second period, by the end of 2000, the momentum effect disappeared, according to the authors. This shows that, after being put into practice by the investors, the anomalies under study ceased to produce extra profits. Nevertheless, the question remains of why this anomaly has last for so long after its detection. Perhaps, according to the authors, the answer is in the stock exchange boom in the ‘90s, in particular the technological boom and of the intermediaries, which used to persist during those year and are not likely to repeat themselves in the future.

16

Cfr. Grinblatt M., Titman S., Wermers R, “Momentum investment strategies, portfolio performance, and herding: a study of mutual fund behavior”, The American Economic Review, December 1995. 17 Cfr. Griffin J.M., Harris J., Topaloglu S., “The dynamics of institutional and individual trading”, Working Paper, May 2002. 18 Cfr. Hwang S., Rubesam A., “The disappearance of momentum”, Cass Business School, 2007.

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2. RESEARCH QUESTION AND DATA The novelty in such research, which should lead us towards the macrozone of behavioural finance, stands in the study of the profitability of the industry indicators or in proving the existence of their anomalies. Specifically, such research is aimed at testing the existence of momentum and reversal patterns in the profitability of the industry indicators or at showing, as

it has already

happened in the previously mentioned

researches, whether, an investor who, based on the industry historical performance, would buy what in the past were winners, by means of selffunding through selling the losers or by building a portfolio at zero cost, manages within a mid-term period to achieve positive performances (momentum patterns) from the statistical perspective. At the same time, we try to evaluate the reversal hypothesis, that is, in other words an effort to establish whether an investor who, based on the assets’ historical performances, would buy the losers or, in a better scenario, would invest the latter by self-funding it with the selling of winners (zero-cost portfolio), succeeds

in

having

positive

performances

(reversal

patterns),

from

statistical perspective. Upon testing the existence of both momentum and reversal patterns, we shall perform an analysis on whether they are restricted into the short and medium term the first and into the long-term period the second. Thus, we shall try to indirectly evaluate the findings by Figelman (2007) or the presence of the reversal pattern in a short-term period. Further, we will elaborate on the fact, widely proven by Werner, De Bondt and Thaler (1987), Chan, Hameed and Tong (1999), whether the momentum and reversal patterns in different markets are related to stationary beta character in various market trends (bull and bear) or whether they may have been caused by immediate and unexpected beta changes of the portfolios. The last analysis tends to examine if the Hwang and Rubesam (2007) thesis is applicable also to the industries in question, or prove whether the

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momentum and reversal patterns have been grown weaker by the end of19. In this regard, an analysis shall be performed on the mean and median of returns. At the same time, taking into account the reduced number of observation cases of the 2000-2007 sample and in order to avoid that specific disproportionate data may affect the final outcome, we shall continue via bootstrapping with and without replacement, with the latter to be further subjected to the permutation test. Selection of the industries to be included in the winner and loser portfolios

takes

into

account

only

the

past

performances,

without

determining any other indicators. The valid periods (building periods) to define the past performance and, therefore, the industries to include in the winner and loser portfolios, are 4 (1, 3, 6 and 12 months). On the other hand, there are eight of them (1, 3, 6, 12, 18, 24, 30 and 36 months), which serve for the performance evaluation (retention period). The set of data refers to the historical series of monthly official closing quotations, as reported by Reuters news agency, on DJ Euro Stoxx, or the quotation of the 18 industries it is divided into20. Quotations of 31 December 1991 and 30 April 2007 are the first and last observation, respectively.

19

In this case, focus of the analysis shall be exclusively the zero-cost portfolio. www.stoxx.com/indices/types/sector.html. DJ Euro Stoxx indicator is composed of quotations of the companies with legal headquarters in the countries where the common currency is in use: Automobiles & Bank Basic resources Chemical Construction Parts materials Financial Food & Beverage Industrial Goods & Insurance Media services Services Health Care Personal & Oil & Gas Retail Technology Household Goods Telecom Travel & Leisure Utility Suppliers Each sector is composed of a number of companies ranging from a minimum of 8 (Travel & Leisure) to a maximum of 44 (Bank)

20

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&

4. MOMENTUM AND REVERSAL PATTERNS: MODELS AND EMPIRICAL TESTING 4.1 Models In order to determine the winner and loser stocks, the following process is followed: 1. For each “i” industry (where i=1, 2, ..., 18) is determined the cumulative return (R) during the building period (t-n; t where n=1, 3, 6, 12 months), as a logarithm of the ratio between the closure and opening prices of the indicators in the end (t) and in the beginning of the building period, respectively  Pit     i t n  

(t-n): Ri t  n;t   Ln P

After establishing the past performances for various industries, the series of returns are distributer into quartiles: the one comprising the bestperformance industries (winners) shall be identified as II1, whereas the one with the worst performance (losers) shall be identified as II4. Intermediate quartiles shall be identified as II2 and II3. A fifth portfolio, II5, or the zerocost portfolio shall be result of the difference between II1 and II4. In other words, to test the existence of the momentum patterns, we suppose the purchase of II1 through the sale of II4 and vice-versa for the reversal pattern. Under these circumstances, so that the figure to be invested in the winners (losers) industries is equivalent to the one obtained by the sale of losers (winners), the sum of weights to be attributed to each portfolio, II1 and II4, should be equal to 1. Weights shall be determined according to a following methodology: a. Based on the official closure price at the end of the building period   21;

21

it 

Pit



Pit

con  = (1, 2, 3 e 4) e

   1 it

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b. Based on the official closure price at the end of the building period and on the respective return   22. 2. For each “i” of the industry and, subsequently, for each II shall be calculated the average return applicable for the retention period (t; t+y with y=1, 3, 6,12, 18, 24, 30, 36), as a logarithm of the ratio between the indicator closure and opening price at the end of (t+y) and

beginning

of

the

retention

period

(t),

respectively:

 Pi t  y    Ri t ;t  y   Ln   Pit 

The above steps shall be applicable to any available observation, so as per each n and y, a historical series of returns is taken as a sample, one per each II. The distribution or average, median and statistical meaning of the t-test shall be determined for each industry individually. 4. The above steps shall be applicable to any available observation, so as per each n and y, a historical series of returns is taken as a sample, one per each II. The distribution or average, median and statistical meaning of the t-test shall be determined for each industry individually. 5. In order to prove which one of the two pondering factors is more efficient, for each n and y a new vector we calculate as a difference between average returns differently weighted:    R  t  n;t ;t  y  

 R

t  n; t ;t  y  



Rt n;t ;t  y 



For each separate vector we determine the average of differences and statistical meaning of the t-test to prove whether the pre-selected pondering factor affects significantly the returns of various portfolios (alternative hypothesis) or whether the use of one method or the other is not relevant to performance determination (null hypothesis).

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it 



Pit 1  R it  n ;t 

  





Pit 1  R i t  n;t 



with



= (1, 4) and

   1 it

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4.2 Empirical Analysis Results of the research confirm also other industry indicators, tested by other researchers on other assets. Actually, the analysis of the five portfolios (table 1) may lead to the conclusion that for industrys, as well, the momentum pattern may be extended in a short and middle term period and vice-versa, the reversal pattern, covers the long-term period. These outcomes do not confirm the existence of a short term reversal for the industry indicators, as it applies for the quoted stocks over S&P500 by Figelman (2007). More specifically, regardless of the building and retention period, the average returns of П1, П2, П3 and П4 are always positive. However, with the prolongation of building and retention periods, average returns of II1 shall be subject to decline whereas, on the contrary, those of II4 tend to grow. Focussing on the zero-cost portfolio (П5 = П1 – П4), the existence of the momentum and reversal patterns, respectively restricted into a short and long term period is quite more evident. Nevertheless, as shown in Chart 1 below, the intensity of such phenomena depends on the building and/or retention period. Chart 1: Monthly average cumulative returns for II5 portfolio, in different building and retention periods, for the sample covering January 1993 to April 2007. The weight (λ) that is attributed to the returns, is correlated with the price of “i” of the industry in “t” or the beginning of each retention period.

0,04 0,02 0 -0,02

6

t-1 t-6 2 t-3 t-1

t+ 3

t+ 1 t+ 3 t+ 6 t+ 1 t+ 2 1 t+ 8 2 t+ 4 30

-0,04 -0,06 -0,08

13

For short building periods (1, 3 months), regardless of the retention period, the momentum pattern is always prevalent on the reversal pattern. In more specific terms, the increasingly growing momentum effect for retention periods up to 12 months (except for t-12), shall be cancelled leaving the place to the reversal pattern with the extension of the retention period. In fact, significant average returns may be obtained, particularly in retention periods of 3/6 months. In such case, II1, which over performs





substantially II4 since after 6 months  R t 65;t;t  6  1.21% * * increases reaching the peak that corresponds to t+12

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, the intensity

 R 

5 t  6 ;t ; t 12 



 2,982% * * .

When the building period exceeds 6 months, the momentum loses the intensity of very short retention periods. As a matter of fact, when the building period is more or less long, reversal patterns is inclined to prevail since during the mid-term period (12 months). Specifically, II4, which never overperforms II1 for short-term building periods (t-1), does so mainly during the12-month building periods. In this context, the reversal pattern is truly inclined to provide important statistical results since after 18 months

 R 

 5 t 12; t ; t 18



 3,427% * * , by increasing the intensity and the duration of the

retention period, reaching the climax of 36 months. In the last case, II4 over performs П1  R t125;t ;t  36   7,256% * * * 24. Finally, we can affirm that both the momentum and reversal pattern depend by the building and retention period, which is short-term in the first case and long term in the second. In order to see that such results are not only an outcome of abnormal observations, we performed the test of the median returns (table 2)25. Even in such a case, as the chart 2 below shows, we can affirm that the momentum and reversal patterns are, however, real phenomena and they are related to short/middle and long term retention periods.

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*, **, *** indicate the scale of the significance of median returns, with 90, 95 and 99%, respectively. 24 Given that the test covers a maximum retention period of 36 months, we are not able to determine whether 7.25% is the climax. In fact, it may occur that by extending the retention period, more significant results are obtained. 25 In this case, the median is calculated only on the cumulative returns determined by the pondering factor λ

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Chart 2: Monthly median cumulative return for II5 portfolio for different building and retention periods, for the sample covering January 1993 to April 2007. The weight attributed to the returns depends on the price industry “i” has in “t” or in the beginning of each retention period.

t-1 t-3 t-6 t-12

t+36

t+30

t+24

t+18

t+12

t+6

t+3

t+1

0,04 0,03 0,02 0,01 0 -0,01 -0,02 -0,03 -0,04 -0,05

We can see that when the analysis is focussed on median returns, the reversal pattern loses intensity, whereas the momentum increases it. If the analysis is restricted only in the nodes where the two patterns display significant results, we notice that the median result is higher than the mean one with around 1% in the case of the momentum pattern, while it is lower with about 2.75% in the reversal pattern:

MEAN 

R  5 

 2,982%



R  5 

 7,256



t  6 ; t ;t 12 



MEDIAN

t 12; t ; t  36 

Δ (Mean and Median)

R  5 

 3,82%

0.838%

R  5 

 4,50

-0.2756%

t  6 ; t ;t 12 

t 12; t ; t  36 

Finally, as opposed to what occurs with median returns, where reversal patterns never apply for building periods of one month, when the analysis focuses on the mean, the reversal patterns are never applied when 15

the building period is three months. Subsequently, without changing the fact that the reversal pattern is a long-term related phenomenon, it loses, however, intensity. When the testing takes place by pondering (or calculating) the industries not only based on the price, but also on the cumulative return in the given building period   , the results are not different in essence (tables 3 and 4). Even in this case, it can be confirmed that momentum and reversal patterns are respectively short-medium and long term patterns. However, observing the nodes in which both patterns, pondered according to factor λ, reach a climax which corresponds respectively with (t-6;t;t+12) and (t-12;t;t+36), inclination towards



  shall significantly reduce the



reversal performance    R t125;t ;t  36  1.165% . Summing up the above, we may reach the conclusion that momentum and reversal patterns are phenomena of short/medium term and long term, respectively. Finally, if we wanted to build an investment portfolio that would maximally increase returns, which we recall to be varying according to the investment horizon, the following approaches should be adopted in a strategic perspective:

MOMENTUM REVERSAL 1. Portfolios created in shortPortfolios created in long-term term building periods building periods 1. Portfolios built on the basis of Portfolios built on the basis of median returns median returns 2. Portfolio-building with a view 2. Portfolio building with a longof short/medium term term view 3. Industry pondering without 3. Industry pondering only based distinction, based on the prices on prices and/or past returns

5. BETA OF WINNER AND LOSER PORTFOLIO AND UP-WARD & DOWNWARD MARKET TRENDS26 5.1 Methodology

26

Analysis of portfolio’s betas consist only of the cumulative median returns calculated by means of the pondering factor  

16

The analysis of the beta in various market up and down trends shall be performed only on II1, II4 and II5. However, before starting action in such a meaning and for information purposes, the beta of each portfolio is determined in relation to various building and retention periods. Betas are calculated by reducing the cumulative extra-median returns (net risk free rates) of the portfolios in different retention periods against the cumulative market extra-return for the same period of time:











Rtn ;t ;t  y   r f t ;t  y    t;t  y    Rmt ;t  y   r f t ;t  y    t;t  y 

In order to calculate the betas of portfolios in different market trends (up & down), according to the patterns applied by Werner, De Bondt and Thaler (1987), further reproposed by Chan, Hameed and Tong (1999), the calculation of the following regression is made:















Rtn;t ;t  y   r f t ;t  y    t ;t  y     Rmt ;t  y   r f t ;t  y  Dt    R mt ;t  y   r f t ;t  y  1  Dt    t;t y 

The regression includes a dummy variable (D), which takes value 1, if



in the given retention period (t;t+y) the market return Rmt ;t  y 



is positive

and, vice-versa, it is equivalent to zero: 1 Dt   0

per

R mt ;t  y   0

per

Rmt ;t  y   0

The testing shall reduce the returns II1, II4 and II5 in different retention periods versus DJ Euro Stoxx market indicators during the same time interval.

17

5.2 Empirical testing As shown by table 5, the beta of portfolio II1 is higher as compared to that of II4 for long building periods, 6 and 12 months, and the difference between the two is further distinguished with the extension of the retention period. On the contrary, the II4 beta is comparably higher than that of II1 when the building period is shorter, 1 and 3 months and the difference is more significant for short/medium retention periods. This implies that if during

the

short-term

periods

the

market

would

show

a

positive

performance (bull market), given that II4 beta is higher than that of II1, then, the first should perform more than the second (reversal). Given that the previous tests show that during the short-term period is the latter to over-perform the former (momentum) and, consequently, either the beta is not a valid information to select the industries or the momentum pattern is associated with the bearish market. In fact, under such circumstances, the higher the beta of a industry is, the higher are the losses, or II4 incurs higher average losses than II1. At the contrary, into the case of market negative performance (bear market). If we referred to the nodes in which II1 and II4 have relatively higher cumulative returns, or in (t-6;t;t+12) the first and in (t-12;t;t+36) the second, we see that in both cases, the beta of the first is higher than that of the second. We know, however, that in the first node II1 has a median cumulative return which is higher than II4 and, vice-versa for the second node. The former are supposed to be present in the up-trend market and the latter in down-trend. Given that these results are contradictory to one another, it is clear that beta is not a valid unit to explain returns II1 and II4.

Meantime, in the analysis of II5, we notice that the beta of the latter does not have a constant progress. More specifically, as shown in the chart 3 below, beta marks a positive average for long-term building periods (6 and 12 months) and, vice-versa, it is negative.

18

Chart 3: Beta of II5, pondered by means of factor λ, for different building and retention periods, for the sample of January 1993 to April 2007. П5

t-1 t-3 t-6 t-12

MEDIA

t+36

t+30

t+24

t+12

t+18

t+6

t+1

t+3

0,400 0,350 0,300 0,250 0,200 0,150 Beta 0,100 0,050 0,000 -0,050 -0,100 -0,150

Based on these results, in long-term building and retention periods II5 shall be inclined to win and lose, with a respective bull and bear market. On the contrary, in short-term building and retention periods, it shall be inclined to win and lose in another bull and bear market, respectively. The analysis of betas in different market trends (up/down) (table 6) shows that in the up-trends of the market (+), the beta of II1 is generally inclined to be higher than that of II4. However, given that the former is reduced with the extension of the retention period and the second decreases in the beginning and then increases, therefore, the difference between the two betas is inclined to disappear. On the basis of such results and, mainly, for short-term retention period, it would be relevant to suppose that a bull market II1 should over perform a II4, which would actually confirm the momentum pattern, by over performing the market itself when β>1. During the reverse down-trends of the market (-), the II4 beta is inclined to be generally higher that that of II1. Considering that with the extension of the retention period, the II4 beta increases in the beginning to be reduced afterwards, the difference between the two betas is consequently inclined to be reduced to zero, and, in the most extreme case, to be exchanged for long-term retention periods. These results confirm the 19

fact that these two patterns exist. In fact, when the building and retention period are short-term ones and II4 beta is higher than II1, the former has greater losses compared to the latter in a down market (momentum pattern). On the other hand, for long-term building and retention periods or when the II4 beta is lower than that of II1, the former loses less compared to the latter in a down market (reversal pattern). Making a summary of the above, given that the momentum and reversal patterns are phenomena related with the short/medium term and long term respectively and, given that we just saw that the beta of II1 and II4 have a varying dynamics based on the building and/or retention periods, as well as due to the up/down market trends, thus it is not to be excluded that the above phenomena occur due to market trends and the portfolios’ betas.

Chart 4: II5 portfolio beta, pondered by factor λ, for different building and retention periods and for different market trends, up(+) and down(-), for the sample beginning in January 1993 and ending in April 2007. П5

П5

t-1 t-3 t-6 t-12

MEDIA

t+30

t+36

t+18

t+24

t+6

t+12

t+1 t-1 t-3 t-6 t-12

MEDIA

t+30

t+36

t+18

t+24

t+6

t+12

t+1

t+3

t+3

0,12 0,1 0,08 0,06 0,04 0,02 0 βeta (-) -0,02 -0,04 -0,06 -0,08 -0,1 -0,12

0,18 0,16 0,14 0,12 0,1 0,08 βeta (+) 0,06 0,04 0,02 0 -0,02 -0,04

To confirm what we said above, the analysis of II5 beta in various up and down market trends (table 6), except for the long-term building and retention, when the market is up(+), II5 has an average positive beta. On the other hand, it is negative on average when the market is down(-) (chart 4). These results are aimed at confirming what was found by Werner, De Bondt and Thaler (1987) and by Chan, Hameed and Tong (1999) or that П5 is inclined to behave well both in the up and down market trends.

6. TESTING OF STRUCTURAL BREAKS 20

6.1 Methodology In order to test whether there is a breakpoint for industry indicators by the end of 2000, as identified by Hwang and Rubesam (2007), studies that were aimed at testing the existence of momentum & reversal patterns on a subsample of what was previously available27. If there is a breakpoint in nodes, the marks of average returns and/or of portfolio II5, the returns of both samples shall be different (table 7).

6.2 Empirical testing When analysing the average returns for long-term building and retention periods or when the reversal pattern is applicable, if we exclude few sample exceptions from 2000, the sample of 1993 was recollected more than once. On the other hand, for short-term building and retention periods or when momentum patters have emerged from 1993, they over-perform almost always with those of 2000. However, there are three exceptions where the average of the two samples is the same. These three above-mentioned exceptions obviously support the theory, according to which, a breakpoint was marked in 2000. More specifically, the nodes where breakpoints emerge are as follows (chart 5): 

t-6;t;t+6



t-6;t;t+18



t-12;t;t+6

Chart 5: Median cumulative returns for II5, pondered by factor λ, for different building and retention periods, for the samples beginning in January 1993 and January 2000 to the end of April 2007, respectively.

27

The analysis refers only to II5 portfolio. The first valid data for these tests dates back to 31 December 2000.

21

In the first breakpoint, the median return of the portfolio since 1993 is positive and amounts to 1,476%, while, on the other hand, from 2000, it is negative with -0.593% (Δ=2,069%). Likewise, the second median return of the portfolio since 1993 is positive and equal to 0.299%, while, on the other hand, that of 2000 is negative, with -1.203% (Δ 1,502%). Finally, in the third, the difference is less relevant, but not less significant due to this; from a median positive return of 0.97% in 1993 it switches to a negative median return, with 0.8%, for 2000 (-Δ 1.77%).

In

conclusion,

the

breakpoints exist only and

exclusively for

momentum patterns (with the return becoming negative from positive, with the change of sample). The analysis includes data by the unique sample, starting from 2000, but the size of the sample is, however, quite reduced, thus, it may occur that potential abnormal returns may affect the calculation of the median and, therefore, in determining the breakpoints. In order to avoid these problems, the test was carried out by no longer using the mean of returns, but the median (table 8). Differently from the analysis of the median returns, we may say that for retention periods of less than 18 months, the sample of 1993 always prevails on that of 2000. Unlikely, for retention 22

periods of 18 and 24 months, the latter prevails on the former. There is not always an available answer when the retention period exceeds 30 months. Among others, we have five breakpoints, not three (Chart 6).

Chart 6: Comparison among cumulative median returns, calculated according to the pondering parameter λ, for the II5 portfolio, for different building and retention periods, on samples starting in January 1993 and January 2000 and ending in April 2007.

More specifically: Median

t-1;t+30

t-3;t+30

t-3;t+36

t-6;t+1

t-12;t+24

2000

0.0288

-0.014

-0.0042

-0.0012

0.0223

1993

-0.00725

0.0112

0.0019

0.00335

-0.0063

Δ

3.605

2.52

0.61

0.455

2.86

Thus, when the test focuses on the medians, the breakpoints exist both in momentum (returns changes from positive to negative with the change of sample), as well as in reversal patterns (from negative to positive, with the change of sample).

23

6.3 Bootstrapping and Permutation Test28 In order to validate the test performed up to this moment, we shall try to prove that breakpoints are not simply random or a result of the returns’ time structure. In this regard, we proceeded with a double test, based on bootstrapping techniques. The first technique known as “bootstrapping with replacement” suggests that per each available sample, both that of 2000 and 1993, a sample with the same number of observations compared to the previous is built. The specificity is in the fact that every identified finding in the observed sample may be used more than once in establishing and extracting new samples. More specifically, and only in those previous cases where the presence of the breakpoint in the median returns was tested, each sample is taken with 500 replacements (500 for the sample of 1993 and 500 for that of 2000). For each of these 1000 sub-samples, we calculate the median ( x 93 e x 00 ). Each median x 93 is subtracted x 00 to obtain a vector of median differences of the 500 observations. In order to establish the meaningful statistics of the latter, we calculate its average and compare it to the one observed concretely. The difference between two data is known as bias.

Observed

Mean

Bias

Standard

95% Confidence Interval

Deviation

Lower

Upper

-0,00996

0,03263

-0,02151

-0,01577

0,00066

0,00634

0,00333

0,00445

of median 12x24

-0,02860

-0,01864 (-12,76141)

6x1

0,00455

0,00389

28

See on various bootstrapping techniques: Compare Karolyi G.A., Kho B.C., “Momentum strategies: some bootstrap tests”, Journal of Empirical Finance, 11-2004; Cfr. Efron B., “Bootstrap methods: another look at the jackknife”, Annals of Statistics, 7-1979; Cfr. Efron B., Tibshirami R.J., “An introduction to the bootstrap”, Chapman & Hall 1993, New York; Cfr. Efron B., “The jackknife, the bootstrap and other resampling plan”, Capital city Press 1982, Montpelier; Cfr. Fox J., “Bootstrapping regression model”, Appendix to An R and S-Plus companion to applied regression, January 2002; Cfr. Mills J.A., Zandvakili S., “Statistical inference via bootstrapping for measures of inequality”, Working Paper – University of Cincinnati, 136-1995; Cfr. Fair C.R., “Bootstrapping macroeconometric models”, Discussion Paper – Yale University, 1345-2001; Cfr. Bhlmann P., “Boostraps for time series”, Statistica Science, 1-2002, vol. 17; Cfr. Jegadeesh N., titman S., “Cross sectional and time-series: determinants of momentum returns”, The Review of financial Studies, Sprinter, 1-2002 vol. 15; Cfr. Bianchi R., Drew E.M., Polichronics J., op. cit.

24

(13,70945) 3x36

0,00610

0,00692

-0,00082

0,02692

0,00455

0,00929

0,00569

0,03517

0,01642

0,02261

-0,00459

0,03330

-0,03438

-0,02853

(5,74511) 3x30

0,02520

0,01951 (12,39458)

1x30

-0,03605

-0,03146 (-21,10076)

() value of T-test

A small bias shows that the bootstrap distribution is focused on the first-sample statistics, the sample focused on the population. In other words, breakpoints are not a random result. Such procedure yields good results, if the bootstrap distribution is regular. If the bootstrap distribution is not regular and, in particular, if there is a significant bias in the observations of the two samples, or one of the samples is quite small, the permutation test may be applied (second method)29. Such method does not differ significantly from the previously proposed bootstrapping procedure. The key distinction is into the fact that samples are collected “without replacement”. The two samples of different size will be mixed into a unique sample. From the latter are made as many observations “without replacement” as those of the 1993 sample. The other observations shall compose the sample from 2000. The sample mean is calculated out of the two samples and, then, its bias is identified. The procedure is repeated 500 times, in order to extract 1,000 samples of various coverage, in order to determine a vector of 500 biases. The permutation test is applied on the last vector. This test attempts to determine whether the effect required for study is different from that of the population. In fact, we start from the pre-requisite that the null hypothesis is true and the probability to observe a value that is higher (lower) compared to the one observed in reality, the alternative hypothesis, known as p-value. The low p-value constitutes evidence against the null hypothesis and, therefore, in favour of the alternative hypothesis or a real effect among the population. As a matter of fact, the breakpoint is not just a case

29

Cfr. Hesterberg T., Moore D.S., Monaghan S., Clipson A., Epstein R., “Bootstrap Methods and permutation tests”, in Moore D.S. e McCabe G.P. (eds.), “Introduction to the practice of statistics”, Cap. XIV, Freeman, New York, 2005.

25

or the bias between the two samples is not pondered to zero. In other words, we calculate the percentage of all the times where all the mean of differences are present in favour of the alternative hypothesis, or higher or lower as compared to the observed differences in the previous samples, viewed from the positive or negative angle. Therefore, if the bias observed initially stems from the distribution, the alternative hypothesis is proven. As evident, the distribution of differences tends towards zero. This shows that the samples are more or less similar, and, consequently, there are no breakpoints. Observed

12x24

-0,02860

Mean of differences

Std.

Alternative

p-value

Deviation

Hypothesis

0,00122

0,02942

smaller

0,18838

0,00633

greater

0,22445

0,02241

greater

0,31062

0,03219

greater

0,30862

0,02875

smaller

0,12826

(0,92377) 6x1

0,00455

-0,00073 (-2,56027)

3x36

0,00610

-0,00072 (-0,71469)

3x30

0,02520

0,00257 (1,78597)

1x30

-0,03605

-0,00340 (-2,63988)

()T-test value

Finally, the permutation test p-value is the proportion of 500 samples, displayed in percentages that have respectively: 

a lower value than -0.03605 for t-1 and t+30;



a higher value than 0.02520 and 0.0061 for t-3 and t+30 and t+36, respectively;



a higher value than 0.00455 for t-6 and t+1;



a lower value than -0.02860 for t-12 and t+24. The p-value shows two critical breakpoints from the statistical

perspective, for building periods equal to t-1 and t-12. In fact, under these circumstances, the p-value is 12.8% and 18,8%, respectively. In other words, taking into account the biases, in the first case, there are sample biases in 12.8 cases with one hundred smaller than -3.605% and 18.8 26

cases with one hundred smaller than -2.86% in the second. The low p-value figures prove that the difference between the first two samples is close to the end of the distribution or that such a high value is less likely to be needed, if the null hypothesis is proven true (chart 7).

Chart 7: Distribution of the permutation sample of mean of differences. The black line is the real difference of the observed means.

Making a summary of the above, when the breakpoints are determined

based

on the median return,

the sample shows

three

breakpoints and all require application of the momentum pattern. On the other hand, when defined on basis of the mean return, five breakpoints are present, three of which pertaining to the momentum pattern and two to the reversal pattern. When the bootstrapping is applied on the last nodes, the results differ depending on the fact whether sample collection is with or without replacement. In the first case, all breakpoints are considered meaningful from the statistical perspective, while in the second case, the pvalue test shows that only breakpoints subject to reversal patterns result to be significantly meaningful. 7. CONCLUSIONS Several studies have found more than once and in various markets the existence of the momentum and reversal patterns. With few exceptions, 27

most of these studies look at the first pattern in a short/medium term and the second in a long term. Even the results of this study on the performance of DJ Euro Stoxx industries confirm the absolute prevalence of the shortterm momentum pattern and long-term reversal pattern. When the industries that are part of two winner and loser portfolios are weighted on the basis of the industry quotation from the beginning of the retention period, the momentum pattern shows a quite meaningful result that correspond to six-month building periods and 12-month retention periods. In this case, the winner portfolio results on an average to be 2.98% higher than the loser one. Regarding reversal patterns, the most important results are present during the 12-month building period and 36 months retention period. In this case, the loser portfolio results in 7.25% more as compared to winners. Quite similar results are achieved when the industries are weighted, both for the prices and the profitability in the building and period. However, in this case, results are quite different (better) in the reversal pattern. When the test performed on median returns (instead of mean), there is no change in essence, but the momentum gain intensity compared to losers. If we want to make a strategic investment by following the instructions of a momentum pattern, it would be advisable to build the winner portfolio on the basis of median returns. Otherwise, for a reversal pattern, losers are preferred on the basis of mean returns. By correlating the momentum and reversal patterns with the market up and down trends, we are faced with quite surprising results. In fact, as from the test it results that the zero cost portfolio beta is positive in the upmarket trends and negative in down trends, it is quite clear that the zero cost strategy has an excellent performance. Such results were confirmed previously by the studies of Cooper, Guitierrez and Hammed (2003) and of Griffin, Ji and Martini (2002). The duty to prove it is more difficult, given that from a certain point and onwards both models cease to be efficient in realizing extra-profits. Referring to the work of Hwang and Rubesam (2007), who identify two breakpoints, one of which corresponds to 2000, median returns of the 28

sample studied previously with those of a sub-sample, which started to be observed from 2000. The three breakpoints were identified also in all the three cases where it is unclear whether it is a momentum pattern (return switch from positive to negative with the changing of the sample). While, when the test is performed on mean returns, there are five breakpoints, of which, three correspond with the momentum pattern and the other two with reversal pattern. In order to add to the validity of the results on the median, a double test, based on the bootstrapping methodology, is performed. In the first test, the first sample and sub-sample were collected with

replacement,

500

times

each.

The

mean

of

differences

by

bootstrapping is similar to the one observed originally, adding validity to the breakpoints identified by the means. In the second test (permutation test), both samples are recollected without replacement, 500 times each. The mean of differences by bootstrapping confirms as valid only in the reversal breakpoints (from negative to positive). The existence of momentum and reversal patterns, although they have been subject to studies since a long time, remains a controversial issue. The rationale we tried to bring here is not far from the over or underreaction hypothesis. Above all, the loser and winner portfolio perform positively, these results support the arguments of Hwang and Rubesam (2007), according to whom, if the momentum still exists, this is result of the boom of stock exchanges by the end of 1980s to date. Second, referring also to the tests performed by Chan, Hamed and Tong (2005), Grinblatt, Titman and Wermers (1995) and Griffin, Harris and Topaloglu (2002)

29

REFERENCES

AARTS F., LEHNERT T., “On style momentum strategies”, Letters, December 2005.

Applied Economic

ASNESS C.S., “The interaction of value and momentum strategies”, Financial Analysts Journal, March-April 1997. BIANCHI R., DREW M.E., POLICHRONICS J., “A test of momentum trading strategies in foreign exchange markets: evidence from G7”, Queensland University of Technology, Discussion Paper n° 182, july 2004. BHLMANN P., “Boostraps for time series”, Statistica Science, 1-2002, vol. 17; CHAN K., HAMEED A., TONG W., “Profitability of momentum strategies in the international equity markets”, Journal of Financial and Quantitative Analysis, June 2005. CHAN L.K.C., JEGADEESH, LAKONISHOK J., “Momentum strategies”, The Journal of Finance, December 1996. CHOPRA N., LAKONISHOK J., RITTER J.R., “Measuring abnormal performance: do stock overreact?”, The Journal of financial Economics, n° 31, 1992. COOPER J.M., GUTIERREZ JR. R.C., HAMMED A., “Market states and momentum”, Working Paper, 2003. DE BONDT W.F.M, THALER R., “Further evidence on Investor overreaction and stock market seasonality”, The Journal of Finance, July 1987. DE BONDT, W.F.M, THALER R., “Does the stock market overreact?”, The Journal of Finance, July 1985. EFRON B., “Bootstrap methods: another look at the jackknife”, Annals of Statistics, 7-1979; EFRON B., TIBSHIRAMI R.J., “An introduction to the bootstrap”, Chapman & Hall 1993, New York; EFRON B., “The jackknife, the bootstrap and other resampling plan”, Capital city Press 1982, Montpelier; FAIR C.R., “Bootstrapping University, 1345-2001;

macroeconometric models”, Discussion Paper – Yale

FAMA E., “Efficient capital market”, The Journal of Finance, 25, 1970. FIGELMAN I., “Stock return Momentum and reversal”, The Journal of Portfolio Management, Fall 2007. FOX J., “Bootstrapping regression model”, Appendix to An R and S-Plus companion to applied regression, January 2002; GRIFFIN J.M., HARRIS J., TOPALOGLU S., “The dynamics of institutional and individual trading”, Working Paper, May 2002. GRIFFIN J.M., JI S., E MARTIN J.S., “Global momentum strategies: a portfolio perspective”, Working Paper, july 2004. GRIFFIN J.M., JI S., E MARTIN J.S., “Momentum investing and business cycle risk: evidence from pole to pole”, Working Paper, march 2002.

30

GRINBLATT M., TITMAN S., WERMERS R, “Momentum investment strategies, portfolio performance, and herding: a study of mutual fund behavior”, The American Economic Review, Dicembre 1995. HESTERBERG T., MOORE D.S., MONAGHAN S., CLIPSON A., EPSTEIN R., “Bootstrap Methods and permutation tests”, in Moore D.S. e McCabe G.P. (eds.), “Introduction to the practice of statistics”, Cap. XIV, Freeman, New York, 2005. HONG H., LIM T., STEIN J.C., “Bad news travels slowly: size, analyst coverage and the profitability of momentum strategies”, The Journal of Finance, Febbraio 1990. HWANG S., RUBESAM A., “The disappearance of momentum”, Cass Business School, 2007. JEGADEESH N., TITMAN S., “Cross sectional and time-series: determinants of momentum returns”, The Review of financial Studies, Sprinter, 1-2002 vol. 15; JEGADEESH N., TITMAN S., “Return to buying winner and selling losers: implication for stock market efficency”, The Journal of Finance, March 1993. KAROLYI G.A., KHO B.C., “Momentum strategies: some bootstrap tests”, Journal of Empirical Finance, 11-2004; LAKONISHOK J., SHLEIFER A., VISHNY R.W., “Contrarian extrapolation, and risk”, The Journal of Finance, December 1994.

investment,

MENGOLI S., “On the source of contrarian and momentum strategies in the Italian equity market”, International Review of Financial Analysis, 13-2004. MOSKOWITZ T. J., GRINBLATT M.” Do industries explain momentum” Journal of Finance, 4-1999 MILLS J.A., ZANDVAKILI S., “Statistical inference via bootstrapping for measures of inequality”, Working Paper – University of Cincinnati, 136-1995; RICHARDS, A.J., “Comovements in national stock market returns: Evidence of predictability, but not cointegration”, Journal of Monetary Economics, 36-1995 RICHARDS, A.J “Winner-loser reversals in national stock market indices: Can they be explained?”, Journal of Finance, 52-1997 ROUWENHORST, K.G. “International momentum strategies”, Journal of Finance 53(1)-1998 SHARPE, W.F. “Integrated September/October”, 1987

Asset

Allocation,

Financial

Analyst

Journal,

31

TABLE 1: Mean cumulated returns for portfolios П1, П2, П3, П4, П5=П1-П4, for different periods of detention, sample from January 1993 to April 2007. The weight (λ ) assigned to every industry depends on its current price in the same

(t-1)

(t-3)

(t-6)

(t-12)

t.

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

П1

0,011328***

0,029127***

0,051093***

0,089672***

0,1156***

0,146292***

0,17352***

0,214962***

П2

0,00785**

0,02158***

0,04469***

0,08749***

0,12148***

0,15724***

0,18875***

0,21392***

П3

0,003224

0,019614

0,040912

0,075164

0,109585

0,149930

0,188149

0,210513

П4

0,00748**

0,01857***

0,03698***

0,06669***

0,104687***

0,13287***

0,17213***

0,20310***

П5

0,003849

0,01056**

0,019966

0,02299**

0,010910

0,013407

0,001399

0,011860

П1

0,009173***

0,026097***

0,047746***

0,091437***

0,116087***

0,146249***

0,172167***

0,207854***

П2

0,01032***

0,02499***

0,04989***

0,08997***

0,12182***

0,15434***

0,18163***

0,20472***

П3

0,00862**

0,02605***

0,04705***

0,07799***

0,11733***

0,15397***

0,19427***

0,22477***

П4

0,00402

0,01484**

0,03233***

0,06163***

0,09986***

0,13333***

0,17699***

0,20862***

П5

0,00515*

0,01125**

0,01541**

0,02981**

0,01622

0,01291

-0,00483

-0,00077

П1

0,01068***

0,028629***

0,049928***

0,093414***

0,108548***

0,131611***

0,153535***

0,180984***

П2

0,002138**

0,02222***

0,04531***

0,08101***

0,12437***

0,16531***

0,20164***

0,22712***

П3

0,00576*

0,02266***

0,04592***

0,08539***

0,12343***

0,16133***

0,20391***

0,23633***

П4

0,00588*

0,01652***

0,03516***

0,0635***

0,10556***

0,13986***

0,18032***

0,21630***

П5

0,004799

0,01210**

0,01476**

0,02982**

0,00299

-0,00825

-0,02679

-0,03532***

П1

0,01085***

0,02801***

0,04798***

0,07214***

0,08862***

0,11110***

0,13423***

0,16002***

П2

0,00705**

0,02282***

0,04614***

0,08872***

0,12804***

0,15923***

0,20429***

0,23355***

П3

0,00837**

0,02324***

0,04509***

0,08490***

0,12554***

0,17696***

0,21566***

0,24831***

П4

0,005129

0,01682***

0,03827***

0,08011***

0,12290***

0,15941***

0,19362***

0,23259***

П5

0,00572*

0,01120**

0,009704

-0,007968

-0,03427**

-0,04831***

-0,05939***

-0,07256***

***, **, * identify the significance level of the T-test respectively at 99%, at 95% and at 90%

32

TABLE 2: Mean cumulated returns for portfolios П1, П2, П3, П4, П5=П1-П4, for different periods of detention, sample from January 1993 to April 2007. The weight (ω) assigned to every industry depends on its current price in the same t that corresponds to the beginning of detention period and from the return in detention period.

(t-1)

(t-3)

(t-6)

(t-12)

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

П1

0,01135***

0,02915***

0,051072***

0,0896***

0,115376***

0,145993***

0,173039***

0,214515***

П2

0,00783**

0,021599***

0,04469***

0,087803***

0,12139***

0,15714***

0,18859***

0,21376***

П3

0,003179

0,01959***

0,04092***

0,075167***

0,10960***

0,14989***

0,18810***

0,21048***

П4

0,00749***

0,01858***

0,03693***

0,06667***

0,10478***

0,13312***

0,17247***

0,20339***

П5

0,003862

0,01056**

0,014136

0,02292**

0,010589

0,012871

0,000569

0,011124

П1

0,009223***

0,026187***

0,047754***

0,091618***

0,115939***

0,145773***

0,171402***

0,207266***

П2

0,01034***

0,02496***

0,04982***

0,08996***

0,12184***

0,15442***

0,18161***

0,20467***

П3

0,008639**

0,026052***

0,04703***

0,077929***

0,117391***

0,154032***

0,194231***

0,224837***

П4

0,00415

0,01507**

0,03212***

0,06139***

0,09975***

0,13304***

0,17712***

0,20857***

П5

0,00507*

0,01112**

0,01562**

0,03023**

0,01619

0,01273

-0,00572

-0,00130

П1

0,010679***

0,028631***

0,049992***

0,093397***

0,107793***

0,130791***

0,15213***

0,180029***

П2

0,00855**

0,02220***

0,04528***

0,08098***

0,12429***

0,16511***

0,20138***

0,22664***

П3

0,05731*

0,022585***

0,045813***

0,08529***

0,12343***

0,16125***

0,20389***

0,23628***

П4

0,00608***

0,01656***

0,0343***

0,06289***

0,10472***

0,13864***

0,17946***

0,21530***

П5

0,004597

0,01207**

0,01562**

0,03050**

0,003070

-0,007850

-0,027333

-0,03528**

П1

0,01079***

0,02809***

0,04817***

0,07145***

0,08740***

0,10890***

0,13183***

0,15803***

П2

0,00705**

0,02289***

0,04619***

0,08888***

0,12802***

0,15925***

0,20452***

0,23384***

П3

0,00840**

0,02333***

0,04529***

0,08509***

0,12598***

0,17739***

0,21580***

0,24834***

П4

0,002986

0,01816***

0,03899***

0,07726***

0,11696***

0,15343***

0,18805***

0,21894***

П5

0,00781*

0,009932

0,009189

-0,005802

-0,02955*

-0,04453**

-0,05622**

-0,06091***

33

TABLE 3: Mean and median cumulated returns for portfolio П5, determined as weight parameters λ, for different periods of detention, sample from January1993 to April 2007.

П5

t-1

Mea Median

t-3

Mean

Mean

t+6

0,0038

0,0106

0,0200

0,0028

Mean

0,0043

0,0148

0,0112

0,0081

0,0218

0,0097

0,0298

0,0105 0,0030

0,0382 -0,0080

0,0087

0,0068 0,0162

0,0298

0,0163

0,0123

0,0109

0,0298

0,0170

0,0111

t+18

0,0230

0,0152

0,0046

0,0034

t+12

0,0154

0,0121

0,0057

Median

0,0048 0,0113

0,0048

Median t-12

t+3

0,0052

Median t-6

t+1

0,0014 -0,0343

0,0090

0,0032

t+24

0,0134

t+30

0,0014

0,0087 0,0129

-0,0048

0,0217 -0,0008

0,0112 -0,0268

0,0087 -0,0483

0,0119

-0,0073

0,0034 -0,0083

t+36

0,0019 -0,0353

-0,0079 -0,0594

-0,0063

-0,0101 -0,0726

-0,0367

TABLE 4: Mean and median cumulated returns for portfolio П5, determined as weight parameters λ e ω, for different periods of detention, sample from January 1993 to April 2007.

П5

t-1

t-3

t-6

t-12

ω

0,00386

0,00507

0,00460

0,00781

λ

0,00385

0,00515

0,00480

0,00573

ω

0,01056

0,01112

0,01207

0,00993

λ

0,01056

0,01125

0,01210

0,01120

ω

0,01414

0,01563

0,01562

0,00919

λ

0,01997

0,01541

0,01476

0,00970

ω

0,02292

0,03023

0,03050

-0,00580

λ

0,02299

0,02981

0,02982

-0,00797

ω

0,01059

0,01618

0,00307

-0,02955

λ

0,01091

0,01622

0,00299

-0,03427

ω

0,01287

0,01273

-0,00785

-0,04453

λ

0,01341

0,01291

-0,00825

-0,04831

ω

0,00057

-0,00572

-0,02733

-0,05623

λ

0,00140

-0,00483

-0,02679

-0,05939

ω

0,01112

-0,00130

-0,03528

-0,06091

λ

0,01186

-0,00077

-0,03532

-0,07256

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

34

-0,0450

Table 5: Monthly mean betas for portfolios П1 E П4, weighted as weight parameter λ, for different periods of detention, sample from January 1993 to April 2007.

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

MEAN

П1

1,022

0,990

0,966

0,946

0,932

0,960

0,998

1,002

0,977

П4

0,942

1,002

1,000

1,026

1,028

1,054

1,018

0,993

1,008

BETA

t-1

П5

t-3

-0,034

-0,080

-0,096

-0,094

-0,020

0,009

-0,030

0,921

0,951

0,966

0,980

0,952

0,952

1,016

1,002

0,968

П4

1,028

1,044

1,000

1,030

1,034

1,053

0,977

0,993

1,020

-0,105

-0,088

-0,025

-0,036

-0,064

-0,085

0,053

0,020

-0,041

П1

0,944

1,003

1,012

1,014

0,964

0,979

1,049

1,057

1,003

П4

1,001

0,985

0,953

0,979

0,969

0,963

0,882

0,907

0,955

П5

t-12

-0,007

П1

П5

t-6

0,082

-0,055

0,023

0,065

0,050

0,013

0,032

0,181

0,160

0,059

П1

0,957

1,063

1,007

1,072

1,074

1,145

1,185

1,133

1,080

П4

1,008

0,953

0,950

0,921

0,925

0,861

0,843

0,856

0,915

П5

-0,049

0,114

0,066

0,165

0,167

0,301

0,356

0,288

35

0,176

Table 6: Betas for portfolios П1, П4 e П5,in different market phases up & down, weighted as weight parameter λ, for different periods of detention, sample from January 1993 to April 2007.

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

MEAN

β (+)

1,055

1,0758

0,9272

0,9404

0,8379

0,9131

0,9525

0,9093

0,9514

β (-)

0,997

0,9164

1,0032

0,9509

1,0112

1,0016

1,0408

1,1031

1,003

β (+)

1,0056

0,847

0,9582

0,7562

0,7489

0,8174

0,7403

0,9105

0,848

β (-)

0,8915

1,1361

1,0403

1,2466

1,2624

1,2632

1,2785

1,1072

1,1532

β (+)

0,0524

0,2375**

-0,031

0,1843*

0,089

0,0957

0,2123**

0,1088

0,063

β (-)

0,1053

-0,2187**

-0,0371

-0,2957***

-0,2512**

-0,2616***

-0,2377***

-0,0988

-0,0102

β (+)

1,0267

0,9859

0,9272

0,9977

0,9048

0,9268

0,9455

0,9098

0,953

β (-)

0,8374

0,9207

1,004

0,9653

0,9917

0,9733

1,0824

1,1025

0,9847

β (+)

0,9401

0,8613

0,9582

0,6903

0,6783

0,7625

0,7042

0,8028

0,7997

β (-)

1,0972

1,2019

1,0394

1,3074

1,3324

1,3099

1,2336

1,1993

1,2151

β (+)

0,0899

0,1334

-0,0198

0,3368***

0,2749**

0,2130**

0,2814***

0,1342*

0,0678

β (-)

-0,2597***

-0,2808***

-0,0295

-0,3399***

-0,3480***

-0,3481***

-0,1612'

-0,1045

-0,067

β (+)

1,083784

1,113336

1,000538

1,095567

0,914518

1,010215

0,993836

0,954258

1,0208

β (-)

0,832964

0,907431

1,018328

0,947172

1,006135

0,950681

1,100521

1,168364

0,9914

β (+)

0,925921

0,786065

0,795164

0,619897

0,631644

0,656219

0,669724

0,824627

0,7387

β (-)

1,060288

1,157665

1,104929

1,270895

1,25206

1,233851

1,08239

0,9968

1,1449

β (+)

0,161094

0,3360***

0,2166**

0,5051***

0,3312***

0,4027***

0,3642***

0,1569*

0,1611

β (-)

-0,2272*

-0,2493**

-0,08074

-0,3214***

-0,2532**

-0,2947***

0,008116

0,1638*

-0,0363

β (+)

1,117669

1,272327

1,183589

1,111323

1,002754

1,046464

0,985446

0,891831

1,0764

β (-)

0,829516

0,880756

0,837371

1,039469

1,133761

1,232653

1,373282

1,394898

1,0902

β (+)

0,883105

0,654702

0,619649

0,63614

0,691907

0,757831

0,813935

0,854276

0,7389

β (-)

1,106953

1,212066

1,267452

1,153247

1,120546

0,952653

0,870553

0,856883

1,0675

β (+)

0,2378*

0,6264***

0,5751***

0,5046***

0,3592***

0,3374***

0,2116**

0,06479

0,0648

β (-)

-0,2773**

-0,3304***

-0,4242***

-0,1115

0,005899

0,2685***

0,4927***

0,5303***

-0,0528

П1

t-1

П4

П5

П1

t-3

П4

П5

П1

t-6

П4

П5

П1

t-12

П4

П5

36

TABLE 7: Mean cumulated returns for portfolio П5, determined as two weight parameters  for different periods of detention , for samples beginning respectively in January 1993 and January 2000 and finish in April 2007.

Sample

t1

t3

t6

t12

1993

0,00385

0,00515

0,00480

0,00573

2000

0,00305

0,00653

0,00182

0,00263

1993

0,01056

0,01125

0,01210

0,01120

2000

0,01457

0,01490

0,00381

0,00011

1993

0,01997

0,01541

0,01476

0,00970

2000

0,00747

0,01286

-0,00539

-0,00800

1993

0,02299

0,02981

0,02982

-0,00797

2000

0,02757

0,02783

0,00674

-0,03423

1993

0,01091

0,01622

0,00299

-0,03427

2000

0,02764

0,02537

-0,01203

-0,06224

1993

0,01341

0,01291

-0,00825

-0,04831

2000

0,04336

0,02711

-0,01525

-0,09236

1993

0,00140

-0,00483

-0,02679

-0,05939

2000

0,02227

-0,00138

-0,05181

-0,11988

1993

0,01186

-0,00077

-0,03532

-0,07256

2000

0,03312

-0,00832

-0,05147

-0,10826

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

37

TABLE 8: Median cumulated returns for portfolio П5, as two weight parameters  for different periods of detention, for samples beginning respectively in January 1993 and January 2000 and finish in April 2007.

Sample

t-1

t-3

t-6

t-12

2000

0,00280

0,00295

-0,00120

0,00580

1993

0,00280

0,00425

0,00335

0,00805

2000

0,00560

0,00925

0,00715

0,01200

1993

0,00480

0,00455

0,01105

0,01230

2000

0,01520

0,01070

0,00510

0,00660

1993

0,01520

0,01700

0,01630

0,00870

2000

0,01540

0,02440

0,03650

0,02610

1993

0,02175

0,02980

0,03820

0,00900

2000

0,01490

0,02040

0,00660

0,01760

1993

0,00680

0,01050

0,00140

0,00320

2000

0,02825

0,01720

0,02130

0,02230

1993

0,00870

0,00340

0,00870

-0,00630

2000

0,02880

-0,01400

-0,01830

-0,02060

1993

-0,00725

0,01120

-0,00790

-0,03670

2000

0,04460

-0,00420

-0,03050

-0,02980

1993

0,02165

0,00190

-0,01010

-0,04500

t+1

t+3

t+6

t+12

t+18

t+24

t+30

t+36

38