# Momentum in the Peruvian stock exchange

**Info:** 8483 words (34 pages) Dissertation

**Published:** 9th Dec 2019

**Tagged:**
International Business

__Contents:____Contents__Abstract...................................................................2

- Introduction................................................................3
- The Capital Asset Pricing Model and factor models

- Literature Review............................................................9
- Behavioural finance

- Data 16
- Methodology..............................................................18
- Results...................................................................25
- Empirical findings after controlling for risk ……………………………………………………………………….
- Transaction costs

- Potential Problems and Limitations..............................................31
- Conclusion................................................................33

__Abstract:__**The efficient market hypothesis states that all available information is fully reflected in security prices at any time, negating any predictability of future security prices or the ability to develop profitable trading strategies to obtain abnormal returns. Momentum is This paper investigates the presence of momentum signals in the Peruvian stock exchange, Bolsa de Valores de Lima (BVL) over a period of 10 years from January 2007 to January 2017. Sixteen trading strategies are employed over differing time periods on stocks taken from the BVL General Index to test for momentum profits. An overview of the Peruvian market leads to a discussion on possible behavioural biases which contributes IDV, herd behaviour. Test for robustness are**

__Introduction:__

__The Capital Asset Pricing Model (CAPM) and Conditional CAPM__

*i*, Rfis the risk-free rate, ERmis the return of the market portfolio and βiis the Beta coefficient for stock

*i*. The Beta coefficient measures the stock’s volatility compared to the market as a whole, thus capturing the stock’s sensitivity to market risk. Under the assumptions of the CAPM all investors hold the market portfolio which eliminates firm specific risk through diversification. This entails that the CAPM relationship between expected return on a stock and its risk is linear. The Beta for stock

*i*is defined as: βi=cov(Ri, RM)σ2(RM) where cov(Ri, RM)is the covariance between the returns of stock

*i,*and returns of the market portfolio. σ2(RM)is the variance of the market. The CAPM is a static, single period model which assumes that betas are constant over time. However, assuming that beta remains constant is questionable, for beta and the expected return will most likely be dependent upon information available at any particular point in time and thus be time-varying. For instance, during recessions investors are very risk-averse and require a high return for bearing risk. During economic booms, they are willing to take bigger risks for smaller return. It is therefore natural that the investors would wish to limit their risk exposure during recessions and increase the risk exposure during booms. However, the CAPM is a one-period model, in which such preferences cannot exist. We analyse the failings of the CAPM from a statistical standpoint. The definition of covariance is: Covx,y=E[x-Ey*y-Ey] That is, the covariance of two variables xand yis the expected value of the product between two deviations from the average. If we assume the expectation is linear then the covariance implies that: Covx,y=Ex*y-Ex*E(y) Ex*y=Covx,y+Ex*E(y) If both the beta and the market return are random variables then: Eβi*RM=Eβi*ERM+Cov(βi, RM) Which indicates that the risk premium of stock

*i,*is equal stock to the product of the expected values plus the covariance. The static CAPM assumes that this covariance is 0 (any variation is due to noise). In contrast, the conditional CAPM allows for time varying betas to capture inconsistency in market return variance over time. Irregular variance over time is called, heteroscedasticity and can be seen in return data when the data contains periods of abnormally high volatility (“volatility clustering”) amid periods of relative low volatility. While the static model ignores this heteroscedasticity the conditional CAPM prices in these fluctuations into the pricing equation. If covariance is positive, the static CAPM will underestimate the expected risk premium of the stock. If the covariance is negative, the static CAPM will overestimate it. If the correlation between the two variables increases, the effectiveness of the conditional CAPM increases. This paper uses the conditional CAPM as a test for robustness for momentum effects. The conditional CAPM equation is given by: Rit=αi+βimRmt+εit Where Ritis the realised return of portfolio

*i*at time

*t,*Rmtis the return of the market at time

*t*, and εitis the zero-mean disturbance term. We use this regression method to obtain the beta of each of the respective decile portfolios. The return of the market portfolio mean equally weighted returns of all the securities listed on the BVL.

**2.**

__Literature Review__**:**

**Inspired by the report of DeBondt and Thaler (1985) that contrarian strategies could achieve abnormal returns, an investment style which goes against prevailing market trends, Jegadeesh and Titman (1993) hypothesized that trading strategies which chose stocks based on their past returns should be profitable if stock prices either overreact or underreact to news. They examined U.S stocks between the years 1965 to 1989 using the following methodology. At the beginning of each month, securities are ranked in ascending order based on their returns in the past J months. From these rankings, ten decile portfolios are formed with equal weight of stocks contained in each decile. In each month, the strategy buys the winner portfolio, the bottom decile, and sells the loser portfolio, the top decile, and holds this position for K months. They apply 16 such strategies (J, K=1, 2, 3, or 4 quarters) using daily data from the CRSP (Centre for Research in Security Prices). Additionally, they examine a second set of 16 strategies that skip one week between the portfolios formation period and the holding period. Their results show that firms with higher returns over the past 3-12 months subsequently outperform firms with lower returns over the same period. Furthermore, all their 32 zero-cost strategies are positive and all profits are statistically significant. These strategies are known as momentum trading strategies and the phenomenon of the continuation in a stock’s performance is called the momentum effect. Since then a great deal of research over the last 30 years has gone into quantifying momentum effects in various stock markets around the globe, such as European stock markets (Rouwenhorse (1998), Liu, Strong and Xu (1999), Hon and Tonks (2001), Morelli (2014) Griffin et al. (2003), Antoniou et al. (2007), Asness et al. (2013)); Asian stock markets (Chui et al. (2000), Griffin et al. (2003)), and Latin American markets (Muga and Santamaria (2007)) amongst others. Following Jegadeesh and Titman (1993), Chan, Jegadeesh y Lakonishok (1996) found additional evidence in the US for investor overreaction in explaining momentum profits. They report around 41% superior performance in the first six months of the price momentum strategy occurring around the announcement dates of earnings. They find that if the “market is surprised by good or bad earnings news, then on average the market continues to be surprised in the same direction at least over the next two subsequent announcements.” Chan (1996) Furthermore, Carhart (1997) and Bollen & Busse (2005) and Grinblatt et al. (1995) all find evidence of momentum in mutual funds. Grinblatt et al. (1995) find that 77% of 155 mutual funds in the US over the period 1975-1984 where ‘momentum investors’, buying stocks that were past winners; however not systematically selling past losers. The funds also had the tendency to exhibit ‘herding’ behaviour, i.e., the buying and selling of the same stocks at the same time by investors. Fama and French (1993, 1996) propose a three-factor model for expected returns with two more risk factors added on top of the market risk of the CAPM. These risk factors were size and book-to market ratio. The three-factor model is: ERi=Rf+βiRm-Rf+βi,SMBRSMB+βi,HMLRHML+ei Where SMB, (small minus big) is the difference between the returns on diversified portfolios of small and big stocks, HML, (high minus low) is the difference between the returns on diversified portfolios of high and low Book value/Market Value stocks. The intuition being that smaller companies tend to exhibit greater expected returns but are much riskier. Additionally, they suggest that book-to-market ratios are a proxy for different facets of “distress risks”. However, Fama & French (1996) admit that their three-factor asset pricing model was unable to explain momentum. They found that, in regard to contrarian strategies, long term past losers have higher β,SMBand β,HMLthan long term past winners, and so were expected to generate higher expected returns in the future. Fama & French also show the existence of covariation in the returns on small stocks that is not captured by the market betas and is compensated in average returns. Carhart (1997) introduces a fourth risk factor (PR1YR) to account for momentum. The Fama-French-Carhart four-factor model for the expected return for stock i is: ERi=Rf+βiRm-Rf+βi,SMBRSMB+βi,HMLRHML+βi,PR1YRRPR1YR+ei where RPR1YRis the expected return of a prior one-year momentum portfolio. Carhart formulates the momentum variable as the return on an equally weighted portfolio of the 30% of stocks with the highest past return, minus the return on an equally weighted portfolio of stocks with the lowest return. Carhart concludes that the PR1YR factor, along with the SMB factor, explained most of the persistence found in mutual funds over the period 1962-1993. Fama & French’s (1996) admission that their three-factor model could not explain momentum returns, triggered a number of theories departing from the traditional asset pricing models and agent rationality framework to find possible explanations for momentum returns within the framework of the behavioural finance theory which will be examined later on. Initially they suggested that the anomaly could be due to data snooping and recommended wider international analysis to establish the robustness of the momentum effect. Rouwenhorst (1998) took this observation into consideration by analysing 12 European markets between 1978 and 1995. He found a monthly 1% margin between the returns of winner loser risk adjusted portfolios. Furthermore, momentum effects were present in each of the 12 countries individually and had an intermediate 1-year horizon, similar to that found by Jegadeesh and Titman (1993). The correlation between momentum effects in both Europe and the US led him to believe that a common factor in both markets could cause the anomaly. Antoniou et al. (2007) confirmed that momentum effects were still present in Europe sampling markets in France, Germany and the UK between 1977 and 2002. In the UK, Liu, Strong and Xu (1999) document significant momentum profits period 1977-1996. Using weekly data from stocks quoted on the London Stock Exchange and a similar methodology to Jegadeesh and Titman (1993), they find the most profitable strategy to be a formation period J of 12 months and 3 month holding period K generating an annual return of 24%. Hon and Tonks (2001) also examine a much larger sample of historical returns in the UK over the period 1955-1996. They found strong evidence for momentum over the short to medium term. However, when splitting their time horizon into two sub periods of 1955-1976 and 1977-1996 they found little evidence of momentum in the sub period 1955-1976. The profitable momentum effects they had found in the total sample were almost entirely explained by the momentum effects in the second sub sample. This suggested that the positive correlation in returns was not a general feature of UK stocks, but a feature of certain time periods. They conclude that momentum profits could not be explained by either an adjustment of Beta risk or the size effect. Most recently Morelli (2014) found evidence of momentum in the UK stock market over the period 1980-2010. He found the most profitable strategy to be the J(12)xK(6), generating an average monthly return of 0.852%. Morelli sought to explain these profits using a conditional time-varying Beta calculation which could adjust the systematic market risk in each period. Using heteroscedastic models, namely ARCH, GARCH and GARCH-M models to calculate the conditional variance of the market portfolio and the conditional covariance between winner/loser portfolios and the market portfolio he found that, in general, winner portfolios have a higher systematic risk than loser portfolios over the period. He concludes that while conditional time-varying systematic risk was playing a role in the momentum profits, it could not explain all of the excess returns found. Among studies which investigate momentum in emerging markets Rouwenhorst (1999), Van der Hart et al. (2003) as well as Griffin et al. (2003) are all worth special mention. Rouwenhorst (1999) examines the cross section returns in 20 emerging markets and finds that return factors are qualitatively similar to those in developed markets. That is, small stocks outperform large stocks, value stocks outperform growth stocks and emerging market stocks exhibit momentum. He points out that global beta factors cannot explain the average returns in emerging markets and there is low correlation between country average returns suggesting that the premiums have strong local character. Similarly, Van der Hart et al. (2003) find significant excess returns within a 2851 firm sample spanning 32 emerging markets when using multivariate strategies to rank stocks on multiple indicators including past 6 to 12 month returns (momentum). Griffin, J.M. et al. (2003) find that momentum portfolio profits are large and positive within countries in Africa, America and Europe with slight exceptions in Asia. They find that momentum profits have only a weak co-movement among countries indicating that it is driven by risk and the risk is largely country specific. Furthermore, they find higher returns in more developed markets invalidating the theory that momentum could reflect inefficiencies in the diffusion of information. Ultimately, they cannot explain momentum using macroeconomic factors, Chen et al. (1986), and find that momentum profits reverse soon after the investment period and become negative over longer horizons. In contrast to Griffin, J.M. et al. (2003), Chui et al. (2000) examine momentum profits in eight Asian markets and their results indicate that momentum trading strategies are highly profitable when implemented on Asian stock markets outside Japan. Furthermore, they concluded that the momentum effect was stronger among small companies with a low book to market ratio. The overriding conclusion from this research is that momentum exists in emerging markets, but it is less intense than in more developed markets. Despite this Muga and Santamaria (2007), found momentum effects in 4 Latin-American countries, namely Argentina, Brazil, Chile and Mexico which were comparable to those of developed markets. Additionally, by analysing sub periods they demonstrate that repeated economic crisis can possibly ameliorate the momentum effect because of stock characteristics such as size and turnover, and market features, such as short-selling constraints and analyst coverage. They conclude that crises are so frequent, however, that momentum returns cannot persist. The first studies of market efficiency in the Peruvian stock market (BVL) were conducted by Terrones & Nagamine (1995) and Delgado & Humala (1997). Terrones & Nagamine performed tests of serial correlation within a nascent and largely undeveloped BVL, found Leptokurtic (fat tail) distributions in the error term. They also found high levels of correlation with past prices and evidence for day of the week effects in volatility equations. Delgado & Humala (1997) present a broad overview of the BVL and its initial development from the beginning of the 1990s. Like Terrones & Nagamine (1995) they perform tests for autocorrelation and stationarity as well for cointegration between markets. By using a Q Ljung Box statistic with 1, 5 and 30 lags they are able to reject the null hypothesis (the sum of autocorrelations is zero) with 99% confidence in all of their subperiod samples. Their bivariate analysis of various North and South American indexes showed little evidence of cointegration between the markets. One of their main conclusions is that whilst market inefficiency is usually attributed to noise traders which distort prices, inefficiency in the BVL seems to emanate from the slow and costly diffusion of new information. Chang (2012) finds that turnover velocity and trading volumes in the BVL are significantly lower compared to other Latin-American stock markets. Furthermore, due to a lack of cohesive reporting of company financial and fundamentals across the market he hypothesizes that diffusion of information in the BVL is slow and costly. Because of this, arbitrageurs have a greater chance of abnormal returns in comparison to other markets where price divergences are quickly eliminated. He carries out a trading strategy based on buying stocks with low price to earnings ratio and selling stocks with high price to earnings ratios. His strategy obtains abnormal returns compared to the market while also adjusting for risk and transaction costs for the period 2006-2011. More recently tests of market efficiency have focused on the Latin American Integrated Market (MILA). MILA is a product of the 2010 Pacific Alliance initiative conformed by Colombia, Chile, Peru and Mexico towards the free trade of goods, services, capital and people. The four economies together conform the largest trade block in the region. The main issues these markets contend with separately are asymmetric information costs and low trading volumes. The MILA attempts to facilitate capital movement and access to information, by integrating the Stock Exchanges of Colombia (“Bolsa de Valores de Colombia”), The Stock Exchange of Santiago (“Bolsa de Comercio De Santiago”, Chile) and the Stock Exchange of Lima (“Bolsa de Valores De Lima”, Peru) and the Stock Exchange of Mexico (Bolsa Mexicana de Valores"). Since MILA started operations, any investor or trader can buy and sale stocks in the three stock exchanges via the integrated system thus eliminating the need of using a counterpart trader, reducing trading costs and increasing overall transactions. It is within this context that Fernandez et al. (2016) explore multifactor trading strategies which beat the returns of the S&P MILA Andean 40 market index. The loading factors used in their model are: forward and trailing price to earnings ratio, price to book value and level of gearing. They found that portfolios constructed on the basis of price to book value and gearing levels demonstrated an excess return between 21.1% and 31.7% over a 10-year period. Whilst they found a positive alpha when adjusting for risk they could not conclude that these results were statistically significant. Similarly, Peschiera (2014) investigated momentum profits for 666 stocks from the constituent countries of MILA from 1991 to 2003. The methodology he followed consisted of overlapping portfolio J formation periods of 3,6,9 and 12 months and holding periods K of consecutive months between 1 and 24 months. His results indicated momentum profits were primarily concentrated in the winner portfolios whilst short selling the lose portfolios did not generate significant gains. The excess return of winner portfolios was 2.1% and this figure steadily declined with longer holding periods, in line with much of the literature which indicate mid-term reversals. The marginally higher returns in the winner portfolios could also endorse the hypothesis that the momentum effect is stronger in smaller stocks with slowly diffusing news. Hong & Stein (1999). Galarza, F. (Ed.) (2014) En este sentido, en el presente trabajo se analizan las estrategias de inversión de momentum y value investment durante enero de 2000 y julio de 2013 para Perú, Colombia, Chile y México.**

**2.1**

__Explanations for Momentum and Behavioural Finance__**It is therefore clear that the momentum effect has been extensively documented and empirically proven for various decades now within the academic literature. Adhering to the semi-strong form of EMH, the findings of academic studies constitute public information which should be rapidly incorporated into market prices. However, the profitability of momentum trading strategies continues to persist in markets across the globe. Explanations for the continuing presence of momentum are roughly divided into rational and irrational explanations. Rational explanations follow traditional finance theory which holds the concept of efficient markets as its core. Some of these explanations are: transaction costs, data mining, short-selling constraints, model misspecifications, time-varying risk and inadequate measures of risk. One of the earliest rational explanations for momentum is proposed by Conrad and Kaul (1998). They assert that momentum profits are attributable to cross-sectional variation in expected returns, meaning that stocks with high/low unconditional expected rates of return in adjacent time periods are expected to have high/low realized rates of returns in both periods. In response, Jegadeesh and Titman (2001) argue that if Conrad and Kaul’s hypothesis were true momentum profits should be similar in any post-ranking period. This is because Conrad and Kaul (1998) hypothesize that stock prices follow random walks with drifts and that it is this (unconditional) drift that varies across stocks. Jegadeesh and Titman (2001) test Conrad and Kaul’s assertion and find that momentum returns increase monotonically for approximately one year and then decline for the following four years. The momentum strategy generates an average profit of 1.01% per month in the first year but registers losses ranging from 0.23 to 0.31% in years 2-5, thus disproving Conrad and Kaul’s (1998) findings. Chordia and Shivakumar (2002) find that momentum profits can be explained by lagged macroeconomic variables linked to the business cycle, such as inflation. The authors argue that momentum returns may be attributable to time-varying expected returns which are informed by the economic cycles and are therefore compatible with rational models. Karolyi and Kho (2004) find that 75-80% of momentum profits can be explained by market-wide and macroeconomic instrumental variables. However, Griffin et al. (2003) find that macroeconomic risk cannot explain momentum profits and show that such profits are large and significant in good and bad economic states. Furthermore, Siganos and Chelley-Steeley (2006) find that momentum returns are stronger following bear markets in the UK and Muga and Santamaria (2009) find that momentum returns are significantly positive in Spain following both up and down market states. Thus, it is difficult to draw conclusions on the relation of market states and momentum returns. Momentum strategies heavily rely on the ability to short sell underperforming stocks in order to be profitable. Short selling may not always be possible and at the very least incurs transaction costs. Lesmond et al. (2004) assert that previous studies documenting significant momentum profits (such as Jegadeesh and Titman, 1993) under-estimate transaction costs. Lesmond et al. (2004) argue that momentum strategies require frequent trading in rendering most ‘profits’ found in previous studies insignificant. To prove this they, re-assess the returns to the momentum strategy documented by Jegadeesh and Titman (1993 and 2001) for a different time period (1980-1998) and find that the majority of the trading returns (ranging from 53% to 70%) are generated by short selling the loser portfolio. Because short sold stocks in the loser portfolio are characteristically small, low liquidity and high beta they have an elevated trading cost which would eliminate any reported momentum profit. The probability of the momentum strategy generating positive post-cost abnormal returns increases when one uses a relatively long holding period and focuses on low transaction shares. However, Jegadeesh and Titman (2001) conclude that the argument that momentum profits should disappear for larger stocks (but not for smaller ones due to transaction stocks) is not supported by their data. The profits from trading in past winners are not eliminated to a greater degree than those of past losers. Agyei-Ampomah (2007) shows that only momentum strategies with holding periods greater than six months are capable of generating statistically and economically significant post-cost returns, while Li et al. (2009) generate similar returns when concentrating on low transaction-cost shares. Rey and Schmid (2007) show that significant post-cost returns can be generated by focusing solely on large capitalisation companies. The failure of rational models to fully account for momentum profits has led to irrational (behavioural) explanations of the anomaly. Behavioural finance theory states that share prices partially deviate from their true fundamental value and these deviations are due to investors acting not fully rational. This invalidates the weak form EMH. Using the theories from behavioural finance, there are several possible explanations for the observed momentum effect. The models from behavioural finance are based on investor biases which lead them to suboptimal decision-making. Some of these biases are: overconfidence, representativeness, herding, anchoring and conservatism. De Bondt and Thaler (1985) argue that overreaction occurs because investors place too much weight on recent news (especially bad news) and prices are thus based too much on current earning power and too little on long-term dividend-paying power (the ‘recency effect’). Investors extrapolate too far into the future on the basis of the present, consistent with the representativeness heuristic (Tversky and Kahneman, 1982, p.31). In other words, naïve investors become excessively pessimistic (optimistic) about the prospects of firms that experienced some form of bad (good) news. Overreaction is consistent with the so-called ‘bandwagon effect’ and ‘speculative bubbles’. It may also be explained by the ‘hot-hand hypothesis’, which states that traders attempt to unearth trends in stock prices and thereby overestimate the autocorrelation in the series. In an experimental setting, Offerman and Sonnemans (2004) show that overreaction is more consistent with the hot-hand hypothesis than the recency hypothesis. Daniel, Hirshleifer and Subrahmanyam (1998) propose the biases of overconfidence and self-attribution bias to explain the momentum effect and long-term reversals. Overconfidence means that people overestimate their ability to estimate quantities and probabilities. According to the attribution theory, people tend to attribute the success of their actions to their ability and the failure of their actions to bad luck. This is known as self-attribution bias. In the context of financial markets, investors are overconfident about their ability to generate and analyse information related to the value of firm. They overestimate the precision of their own private information and underestimate the forecast’s errors. When a positive private signal arrives, investors tend to overweight this information and push the stock prices too high compared to its fundamental value. Due to self-attribution bias this investor’s overconfidence increases following the arrival of confirming news. Hence, investors do not update the confidence in their own skill rationally. Subsequent to the arrival of confirming news, investor increases the belief in their ability, while they attribute adverse market movements to external factors. This increase in overconfidence promotes the initial overreaction and generates the return momentum. The overreaction in prices will eventually be corrected in the longer run as investors observed future news and realized their mistakes, leading to long run reversals Beside the work of Daniel et al. (1998), DeLong et al. (1990) also presented another model that presents evidence for the overreaction hypothesis. The study proposed a framework in which the positive feedback traders cause momentum and long-term reversals. Positive feedback traders are investors who buy (sell) more of a stock that has recently increased (decreased) in value. During the period of good news, when prices rise, positive feedback traders buy the asset in the subsequent period. This leads to momentum. However, this lead to stock prices increase more than its fundamental value triggering on an average lower return in the following periods and therefore generating long-term reversals.**

**Chan, Jegadeesh and Lakonishok (1996) put forward a behavioral explanation for the profitability of momentum and earnings revisions strategies, based on the idea that financial markets respond only gradually to new information, to earnings-related Chang study feeds of this Most recently Bornholt & Malin (2015) explore the relationship between traded volume and momentum in 51 countries. They attempt to explore e that there is little consensus on how past volume information should be interpreted. More importantly, they argue that even less is known about how trading volume interacts with past price movement in the prediction of cross-sectional returns.**

**3.**

__Data__**:**Stocks were selected from the S&P/BVL Peru General Index and the S&P/BVL Lima 25 Index. Both are modified market cap-weighted indexes designed to represent the most liquid, actively traded companies on the BVL and serve as broad benchmark for the Peruvian stock market. All data was retrieved from Datastream

^{©}[1]. Closing price data was compiled for the universe of stocks for every second Wednesday of the month from July 2007 to July 2017. This was done to reduce the Monday effect, where stock prices are often lower on Mondays and the month effect where stock prices tend to increase at the end of the month relative to the middle. Although the selection enables us to avoid the serious problems that can arise from infrequent trading in the markets, past international evidence shows that the results may be affected, as several studies report stronger momentum in small securities (see Forner and Marhuenda 2003; Hameed and Kusnadi 2002; Hong et al. 2000; Jegadeesh and Titman 2001; Muga and Santamaría 2007).

**3.**

__Methodology__**:**This paper adopts a methodology similar to that used by Jegadeesh and Titman (1993) in their seminal paper on the momentum effect. Jegadeesh and Titman (1993) analyse J x K trading strategies so as to determine whether these strategies result in momentum profits. The trading strategy involves firstly ranking securities based on their performance over the previous J months. In month t, a winner/loser portfolio is formed conformed of the top and bottom decile of ranked stocks. The strategy then requires the investor goes long (purchases) the winner portfolio whilst short selling the stocks in the loser portfolio. This amounts to a zero-sum position as the purchased portfolios are fully funded by the short selling of the loser portfolios. Under the assumption that the momentum effect exists and is persistent, self-financing trading strategies are a form of statistical arbitrage with cumulative discounted value

*v(t).*Statistical arbitrages reject the market as being in any economic equilibrium which is an important prerequisite for an efficient market (Jarrow 1988). By definition, a statistical arbitrage satisfies four conditions: (1) it is a zero-initial cost (v(0)=0) self-financing trading strategy, that in the limit has (2) positive expected discounted profits, (3) a probability of a loss converging to zero, and (4) a time-averaged variance converging to zero if the probability of a loss does not become zero in finite time. The fourth condition implies that a statistical arbitrage opportunity eventually produces riskless incremental profit, with an associated Sharpe ratio increasing monotonically through time (Hogan, Jarrow, Teo and Warachka (2004)):

- vt-1=0,
- limt→∞EP[vt>0,
- limt→∞Pvt<0=0,
- limt→∞VarP[v(t)t=0if Pvt<0>0 ∀t<∞

*i*in month t-1. Each month all the stocks are ranked in ascending order according to their past performance obtained by calculating their cumulative continuous returns (CCR):

**CCRi=∑t=-1-JRit Differing from the methodology of Jegadeesh and Titman (1993) and following that of Muga and Santamaria (2007), the ranked stocks are divided into quintiles with equal weighting, thus establishing the winner portfolio as the highest performing 20 percent of securities and the loser portfolio of the 20 percent worst performing stocks over the same J formation period. The increased number of securities in each portfolio is due to the limited number of securities in the universe and the need for a certain degree of portfolio diversification. Having created winner and loser portfolios, the remaining part of the trading strategy consists in buying the winner portfolios and short selling the loser portfolios and consequently holding onto the winner portfolios for a period of K months. The final step is to determine the profits of a winner minus loser portfolio where the mean monthly returns from past loser (R̅loser) are subtracted from mean monthly returns of past winner portfolios R̅winner: R̅winner-loser=R̅winner-(R̅loser ) The process is repeated for 16 momentum trading strategies (given that there are 4 periods over which the CCRs are calculated (J=3,6,9,12) and 4 holding periods of different length (K=3,6,12,24)) over the ten year sample period. The use of overlapping time periods avoids the loss of information, and furthermore, avoids the possibility of the economic cycle influencing the outcome of the momentum trading strategies. If the difference in the average return of the two portfolios, R̅winner-loser is significantly greater than zero then there will be evidence to support the existence of the momentum effect in the BVL and also reject the null hypothesis of the EMH, EũWtFt-1-EũLtFt-1=0which predicts returns should be equal to zero[2] assuming that transaction costs do not influence the momentum profits. A T-test is to determine the statistical significance of each winner-loser return. Portfolio variances were calculated for each of the winner and loser portfolios each month as well as for the market portfolio. The market portfolio is the equal weighted average returns of every stock in the S&P/BVL Peru General Index and the S&P/BVL Lima 25 Index. Covariances were also calculated. To illustrate the momentum effect using the above described methodology we take the 3x12 strategy for March 2013 as an example. The cumulative continuous returns of all the securities are calculated from January 2012 to March 2013. At time**

*t*(Feb 2013) the stock returns are ranked in ascending order and divided into quintiles. The top quintile goes onto form the winner portfolio and the bottom quintile forms the loser portfolio. A short position is taken in the loser portfolio whilst going long the winner portfolio. The self-financing portfolio’s performance cumulative continuous returns are tracked over the next 12 months. Figure 1 illustrates the momentum effect that we are trying to detect by plotting the prices of Alicorp, a food processing company taken from the winner portfolio in this period, against the market index. The graph starts tracking the monthly price data for Alicorp on January 2013 producing a 3-month formation period to March 2013. Six months later while the market has fallen by 16%, Alicorp’s stock price has increased by 3.25%. However more than a year later both the market return and Alicorp have fallen by similar percentages. This illustrates the momentum effect which is the tendency of stocks which have been previously performed well to continue this performance into the near future whilst reverting over longer time frames.

**3.**

__Results__**:**

**3.1**

__Empirical finds adjusting for risk__**:**

**We expect riskier investments generally to yield higher returns than less risky investments, so that the results from the previous section, which have shown that returns on winner portfolios dominate returns on loser portfolios may be because the securities in the winner portfolio are riskier. We now use the capital asset pricing model (CAPM) to quantify of the trade-off between risk and expected return. With the market portfolio as exogenous and conditional on the realised return of individual assets, the CAPM model offers a testable prediction of betas. Thus, to investigate whether a time varying beta explains the phenomenon observed, the ordinary least squares (OLS) estimator of the slope coefficient in the market model is used to estimate the respective portfolio betas:**

**Momentum:**There isn’t a solid theory behind why momentum exists. So behavioural fallacies.

- Explain data selection, you begin with a Wednesday to avoid the Monday effect.
- January effect?
- Do a pie graph to percentage composition of each industry in the stocks you are selecting
- Say that you are selecting these stocks because they conform the IGBVL, representing the broad market in Peru.
- Peru is skewed towards mining
- Previouse works on Peru, the Chang thing which complained about how the Peruvian market was undeveloped with the poor dissemination of company information and performance coupled with a lack of liquidity.
- However you say that according to the S&P article liquidity has been increasing and therefore that the potential profit

**Si bien Chang obtuvo retornos superiores al mercado peruano en su investigación, es posible que su estrategia no haya dado los mismos resultados en el MILA debido a la mayor liquidez y profundidad del Mercado An undeveloped stock market will exacerbate inefficiencies, because there are less arbitrageurs to eliminate noise.**

**Exacto. Porque si más del 50% del índice es minero, tu ves que la economía peruana sube 7% y la bolsa no sube, porque la mitad depende del precio de los commodities. Si sacáramos a las mineras, el mercado hubiese crecido más de 10% en el 2012.**

**References**Grinblatt, M., Titman, S. and Wermers, R., 1995. Momentum investment strategies, portfolio performance, and herding: A study of mutual fund behavior.

*The American economic review*, pp.1088-1105.

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