Unravelling the Halloween puzzle

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Unravelling the Halloween puzzle

Name: Ivan Novakovic

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out of 31 countries studied. There appeared to be a difference in the season returns between developed and emerging markets where the developed markets exhibited a more significant difference in returns between the seasons. Halloween effect has also been studied at the industry level where it also become visible. In the end a comparison has been made between the profitability of the “Sell in May, and go away” investment strategy versus the buy-and-hold strategy.

Introduction

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the Halloween effect means that investors sell their stocks in May, because of the supposedly lower returns in the summer period, and invest the proceeds in risk-free assets such as short-term Treasury bonds. Investors will hold the risk-free assets until the Halloween (31 of October) and will then sell the risk-free assets and invest again in a market portfolio. This strategy protects investors from the lower returns on their market portfolio during the summer period. We must notice however that the very existence of any kind of irregularity which offers seasonal trading as a way to gain higher returns is against the principles of the efficient market theory. According to the efficient market theory investors cannot benefit from market timing activities such as the “Sell in May, and go away” effect because financial markets are supposed to reflect all known information. This way it should not be possible to outperform the market year in year out by constantly using the same “trick” or market trading mechanism because the market should account for this, incorporate it and reflect the known information by adjusting the returns. Nevertheless, stock returns during November-April period are indeed significantly higher than the stock returns for period May-October (Bouman, Jakobsen, 2002). Bouman and Jakobsen even claim that they have found evidence of Sell in May effect for the U.K. stock market as far back as 1694. The existence of Halloween effect has also been noted in the news, magazines and literature over the years, however this has not led to disappearance of the Halloween effect. This contradicts the efficient market theory and brings us to the so-called “Sell in May, and go away” puzzle. We can regard this phenomenon as a puzzle because it should not exist. There are no reasons that one should assume any (significant) difference in stock market returns between the seasons. Looking at this matter statistically we can state that the probability of finding something as the “Sell in May, and go away” should be 50%. Chance that winter returns are higher than summer returns should be random if we are to believe that markets are efficient, hence the 50% probability. In reality this randomness does not hold, the results are biased. For this reason we regard the “Sell in May, and go away” as a puzzle (Bouman, Jakobsen, 2002).

Not everyone however agrees with Bouman and Jakobsen. In their paper Maberly and Pierce (2004) re-examine the Bouman and Jakobsen (2002) paper and find that existence of a Halloween effect in U.S. disappears after certain adjustments are made. Consequently this makes the Halloween effect a not exploitable anomaly and reclaims market efficiency.

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Second reason why the Halloween Effect is interesting to investigate is because it still exists despite the fact that it is known. So far markets have not adequately adjusted for this

phenomenon.

The Halloween Effect and the associated investing strategy become futile if the return/risk trade-off during summer is worse than the return/risk trade-off of risk-free assets. If the

return/risk trade-off during summer is not better than the return/risk trade-off of risk-free assets then there is no reason for investors to sell their stocks in May and leave the market. However, even though Halloween Effect may not be present at the country level it still may be present at industry level. This is the reason why it is important to investigate the possible existence of an Halloween Effect at the industry level. By doing so, the “Sell in May, and go away” investment strategy is likely to become more profitable because investors will know not only when to invest, but also in which type of industry to invest. This way the payoff is likely to increase. Jakobsen and Visaltanachoti (2006) have investigated all U.S. stock market industries and sectors. The reason for this investigation was primarily that although the winter returns in U.S. were higher than the summer returns, summer returns were higher than the short term interest rates. Because of this reason the “Sell in May, and go away” investing is not that interesting for an average U.S. investor. However little was known about the possible presence of an

Halloween effect at the industry level. Perhaps there is a strong and significant Halloween effect in some sectors, while in others not. Jakobsen and Visaltanachoti concluded that all U.S. sectors perform better during the winter period than during the summer period, but that in two thirds of the sectors this difference in performance is statistically significant. Half of the sectors examined showed negative returns during summer. The found differences between the sectors can help investors to choose between industries and reap higher profits by improving the trade off of their returns versus the risk (Jakobsen and Visaltanachoti, 2006).

According to Claessens (1995) emerging markets have higher degree of segmentation

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order to gain the best results. This research topic also deals with the question of integration. Do emerging and mature markets move in line or do they move independently? We would expect to see more and more co-movement in seasonal returns between the both types of markets because more efforts are being put into unification of the world markets.

Problem Statement

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authors believe that Halloween effect is present (Bouman, Jakobsen 2002) while the others state that there is no such thing as Halloween effect (Maberly, Pierce 2004). Maberly and Pierce have conducted the same type of research as Bouman and Jakobsen, however they have focused solely on U.S.. The results obtained by Maberly and Pierce differ from Bouman and Jakobsen. They claim that the found Halloween Effect by Bouman and Jakobsen is exists merely due to outliers in data. According to Maberly and Pierce there would be no evidence of the Halloween Effect in U.S. if the outliers in data would not have been taken into account. Two main outliers are discussed, namely the October 1987 stock market crash and August 1998 in which the Russian government announced moratorium on debt repayment. The first outlier, also known as the Black Monday, happened on the 19th_{of October 1987. It was a stock market}

crash which led to a plummet of 22.6% of the Dow Jones Industrial Average (Browning, 2007). The second outlier was the Russian financial crisis which is also known as the Ruble crisis. This crisis was caused primarily by refusal to pay the taxes by major Russian companies. Russia was, and still is, very dependent on the export of mainly raw materials such as oil (Henry and Nixon, 1998). In 1998 the world price of oil had decreased severely which meant less income for Russian companies and less income for the government. A downward pressure on the Ruble increased because people started selling Rubles. Since Russia was applying a floating peg between the US dollar and the Ruble this meant that they had to spend their foreign reserves in order to purchase Rubles. This way the Ruble could sill float in the allowed range. These factors contributed to the final collapse of the market in August 1998. Because of the fact that both of the outliers fall in the summer period they contribute to the existence of the Halloween effect. If however, the two outliers are controlled for, the Halloween effect

disappears for the U.S. (Maberly, Pierce 2004).

In my research I would like to investigate the existence of the Halloween effect and I would like to see whether the outlier effect can be found in other countries as well. The effect of the outliers on the performance of the “Sell in May, and go away” investment strategy will also become apparent.

Secondly, I would like to research whether there is a difference in the Halloween effect

between emerging and mature markets. As already noted, emerging markets appear to be more volatile than developed markets (De Santis and Imrohoroglu, 1997). It would be interesting to find out whether the possible seasonal returns have the same direction in movement in

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knowledge about the Halloween effect making it easier to adapt the investing strategy and improve the return/risk trade off.

Thirdly, I would like to research the presence of a Halloween effect on industry level. This way it is likely to become more apparent from which industries the Halloween effect is coming from and where the seasonal deviation in returns is the most significant. The investing strategy can be adapted to this information in order to reap higher profits.

Finally, I shall make a comparison between the “Sell in May, and go away” investing strategy and the simple buy-and-hold strategy in order to see which performs better. The second and third objective fulfil the role of improvers of the investing strategy. In the end it matters how, where and when investors can earn the biggest profits. Hopefully I shall be able to at least partly answer this question and maybe even come up with an improved investing strategy.

Literature Review

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1. Economic Significance

The first explanation of existence of any kind of irregularity relates to economic significance. All the irregularities require economic significance in order to continue existing. If an

irregularity would not be economically significant, it would simply not exist. Economic significance depends basically on two factors, economic benefits and economic costs.

Economic costs are dependent on many different variables, transaction costs are one of these variables. Reason for existence of many anomalies is the fact that transaction costs have not been introduced. Without the introduction of transaction costs, anomalies may appear and may continue to exist theoretically because of the lower threshold. However if we were to add transaction costs to the equation, the threshold to become economically significant would automatically increase making it more difficult for an irregularity to become economically significant. This in turn will lead to disappearance of a portion of anomalies and irregularities. An example which suits this story particularly well is the so-called Weekend Effect. Weekend effect is a theory which states that stock returns are often lower on Monday than on the directly previous Friday, and that this difference is often significant. One of the reasons for existence of this anomaly is the fact that companies have the tendency to bring bad news on Friday after the market closure. Even though this effect might hold and might exist, it is not an exploitable anomaly due to high transaction costs involved. In order to pursue this strategy you should sell your stock on Friday, when it reaches its week peak, and buy it back on Monday when its value is the lowest. The price difference between Monday and Friday can be regarded as the benefit of this strategy. These potential benefits however disappear due to transaction costs which fulfil the function of raising the threshold of economic significance.

The important lesson is that transaction costs must be included, only than can one truly test the economic significance of an anomaly. This needs be done also in the case of the Halloween Effect. When investing according to the Halloween Effect transaction costs will occur twice per year, at the beginning of November and May. This means that the Halloween Effect is not nearly subjected to transaction costs as Weekend Effect. Nonetheless the occurred transaction costs do affect the profitability of the investment strategy and need to be included in order to correctly test the Halloween Effect anomaly.

The following function shows the relationship:

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RHalloween Strategy are the returns achieved by the Halloween investment strategy.

RBuy-and-Hold are returns achieved by a simple buy-and-hold strategy

TC are the transaction costs associated with the Halloween strategy.

According to this function the Halloween Effect appears to be an exploitable anomaly if it possesses higher returns than a simple buy-and-hold strategy plus the occurred transaction costs. Only in this case will investors be able to benefit from the Halloween investment strategy and only in this case will the Halloween Effect continue existing.

2. Data Mining

Second explanation of why Halloween effect or any other anomaly could exist is data mining. The term data mining describes the process of discovering knowledge from data-bases stored in data marts or data warehouses (Cooper and Shindler, 2001). Thanks to data-mining we are able to construct forecasting models. However data mining can turn out to be a problem especially when researchers that are going through the data do not mention the number of unsuccessful mining attempts before they proclaim a given pattern (McQueen, Grant and Thorley, Steven, 1999). This way researchers are not showing the full picture of their research. An important question that must be raised deals with the detection of data mining. How can we detect whether a forecasting model is not a victim of data-mining?

In their paper McQueen and Thorley discuss this issue. According to them first warning sign is “Too much digging”. By this they mean that the investigator took too many variables into his research and simply searched long enough to find certain relations. A good example of this is given by Leinweber’s 1998 paper in which he was mining the database of United Nations. Leinweber has found during his research that 75% of the S&P 500 variation can be predicted by the butter production in Bangladesh. This finding may seem silly and useless but it does provide an excellent example of how one can manipulate data in order to draw erroneous conclusions.

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serves as a verification that the Halloween effect is not made up after the empirical findings and hence it does not fall under the data mining fallacy.

The best way to test whether a model falls under the data-mining fallacy is to perform an out-of-sample test. An out-out-of-sample test tests means that the model is tested on out-out-of-sample data. If the model truly works then it should provide the same answer no matter which sample of data is used. If the Halloween Effect would be driven by data-mining than one would expect to find evidence for the Halloween Effect in only a few countries. Also the Halloween Effect should then hold for a short period of time. However, according to Bauman and Jakobsen, the results obtained hold for the majority of countries and over long periods of time (Bauman, Jakobsen, 2002). This serves as evidence that the Halloween Effect does not fall under the data-mining fallacy.

3. Risk

Third explanation of the Halloween Effect would be risk. It seems perfectly natural to question whether the level of risk throughout the year is in line with the expected returns according to the Halloween Effect. Capital Asset Pricing Model (CAPM) states that the expected rates of return which are demanded by investors depend essentially on two different things. First of all, demanded rate of return is depended on the time value of money, and second of all, it is

dependent on the risk premium (Brealey, Myers, Marcus, 2004).

Whatever the investment may be an investor is involved in, he/she must be compensated for the time value of money. Time value of money is approximated by the so-called risk-free rate of return (rf), which is rate of return on an investment with zero risk. Of course not all

investments are risk-free. In order to attract investors to take on riskier investments the associated returns must increase accordingly. The so-called risk premium is defined as the difference between the expected market return and the risk-free rate of return times the beta of the associated investment. The beta (β) measures the volatility of the investment compared to the market as a whole. If the beta is equal to one, it means that the investment’s return will move exactly in line with the market. If the beta is higher than one it means that the investment is more volatile than the market, meaning that it is riskier since the range of

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demanded rate of return be. In their paper Ghysels, Santa-Clara and Valkanov, focused on the trade-off between the variance of stock market return and its mean. They found a positive and significant relationship between risk and return (Ghysels, Santa-Clara, Valkanov, 2004). Their paper serves as evidence that CAPM model holds.

Risk-free rate= rf

Market risk premium= rm – rf

Risk premium= β(rm – rf)

Expected return= r= rf + β(rm – rf)

Halloween Effect states that the returns during the November-April period are higher than the returns during the May-October period. Based on the CAPM model this could mean that the reason for the difference in returns is existence of differences in risk between the two periods. In order to find out whether this truly is the case a comparison of variation in stock returns between the winter and the summer period should be performed. According to the CAPM model, higher returns in the winter period should mean that in this period the beta is higher as well. In their paper, Bauman and Jakobsen(2002) show that standard deviation which is used as a measure of risk, remains rather constant between the winter and the summer period. This means that risk does not serve as a possible explanation of the Halloween Effect.

4. January Effect

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used to offset the potential capital gains which you realized either in the short run (at the end of the same year) or in the long run (gains realized in more than one year time). By offsetting the capital gains you lower your personal tax liability by the amount of your portfolio losses. Investors sell their stocks for this reason at the end of the year. However, once the new year starts (January 1) the investors reinvest their money and buy new stocks. This injection of money in the market by investors in the very beginning of the year causes stock prices to raise. This phenomenon is called the January Effect. In their 2006 article, Haug and Hirschey show evidence of the January effect. They compared returns of value-weighted portfolios in January versus the other 11 months of the year for period 1802-2004. It appeared that the returns in January are higher and are also combined with a lower standard deviation (Haug M., Hirschey M., 2006). The Halloween Effect states that the returns during November-April period are higher than during May-October period. Once we know that January Effect takes place, we might wonder what the effect of the January Effect is on the Halloween Effect. Since January falls in the November-April period, and since the returns are higher in January than the average returns of other 11 months (Haug M., Hirschey M., 2006), we could even state that the

Halloween Effect is actually January Effect in disguise. In order to test whether this is true or not Bauman and Jakobsen performed an additional regression in which they have included a January dummy (Bauman, Jakobsen, 2002). By adding the January dummy the extra returns realized in January compared to May-October period are regarded as the January Effect. This way the extra January returns are controlled for. Nonetheless the Halloween Effect remained present in most markets meaning that the January Effect does not explain the Halloween Effect entirely. It does however explain a portion of the Halloween Effect.

5. Vacations

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automatically because this way the market offers higher expected returns for the remaining investors who are still willing to take on the risk. The second explanation regarding the

vacations takes on a different approach. Investors spend in general more during their vacations than what they usually spend while not being on vacation. The amount of money spend during the summer period is thus higher, which leads to a decrease in the level of liquidity. Once the investors come back from their vacation and realize that their liquidity level has declined they will demand a premium on liquidity. This happens because they are now “stuck” with assets that are less liquid, and in order to keep those assets the investors feel the need to be

compensated for the risk that illiquid assets bring along. The so-called liquidity risk is the risk which is generated by illiquid assets because of the fact that illiquid assets cannot be bought or sold quickly enough in order to prevent a potential loss. Investors demand a liquidity premium during winter for this very reason.

6. Seasonal Affective Disorder

Seasonal affective disorder (SAD) is the sixth factor which can help us to explain the

Halloween Effect. SAD, also known as the winter depression, is a mood disorder manifested every winter. People suffering from SAD experience during the winter period serious mood changes compared to the rest of the year. The psychological explanation behind why SAD would exist lies in the amount of daylight. The reason for this belief is the effectiveness of light therapy against SAD. In their study, Avery et. al., have tested the effectiveness of light therapy against the level of depression as measured by the Hamilton Depression Rating. The outcome was that the number of hours of sunshine during the week received a positive response from the patients (Avery et. al., 2001).

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as portfolios become more risky, the expected return increases due to higher risk premium, leading to an increase in stock returns. SAD has a significant effect on stock market returns, especially in countries positioned on a higher latitude (Kamstra, Kramer, Levi, 2003). It still remains very questionable how much SAD explains the existence of the Halloween Effect, especially because (luckily) not everybody suffers from SAD. In order to determine true impact of SAD on stock market returns it should firstly be determined how many people suffer from SAD in a given country. Only then could we determine how much more risk averse a country as a whole becomes during the winter period. It is this increase in risk aversion of the country as a whole which should be compared to stock market returns, and not the number of hours of daylight. Another important issue is that Kramer et. al. make no difference between the total hours of daylight and sunshine. What happens if the autumn is very sunny and people do receive more hours of sunlight than expected? Does this affect the expected outcome? These issues have not been discussed making SAD a rather weak argument offered to explain the Halloween Effect. However, I should add that even though SAD may not be statistically significant explanation, it certainly does take a portion of the explanation.

7. Optimism Cycle

The optimism cycle is the seventh explanation of the Halloween Effect. The optimism cycle rests on the idea that people, financial forecasters and investors in particular, are in general excessively optimistic (Doeswijk, 2005). According to this theory there is a seasonal cycle in how investors perceive, and the way they feel about the future. As the end of the year

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period of the Halloween Effect. The pessimistic period of the optimism cycle is very similar to the summer period of the Halloween Effect. According to the optimism cycle investors will prefer to invest in cyclical stock during the “positive winter”. This happens because stocks of cyclical companies are characterized by a beta higher than one. This means that their returns are more volatile than the market returns. During winter market is supposed to perform well, for this reason it is wise to invest in cyclical stocks because they will generate an above average return. However, optimism cycle also tells us the opposite story. Once the investors become pessimistic and the market starts performing worse it is wise to invest in non-cyclical stocks, because these stock have a beta that is lower than one. For this reason their return will not be as low as the market return.

8. Weather Effects

Weather could offer yet another explanation of seasonality in stock market returns. Saunders (1993) researched the effect of weather in New York City and the index changes in stock listed companies in NYC. Something we can ask ourselves is why would weather have an effect on index changes in stock listed companies, and how this effect would take place. First of all, stocks that are traded in NYC exchange all represent companies that operate on a global scale. People and agents working for those companies cannot be affected by weather in the same fashion simply because weather is not everywhere the same. This means that the weather effect must take place through the local agents trading stocks in NYC. Brokers who are physically present have the potential to affect the stock prices, and it is exactly these brokers that can be affected by the weather. Weather creates and shapes the environment which affects the mood of the brokers. Finally, mood changes can influence the willingness of investors to take risk, this way weather affects the investing behaviour of investors. According to Saunders (1993) sunny days lead to increases in optimism which eventually leads to higher market returns. Cloudy days on the other hand make investors more pessimistic and less willing to take risk, lower market returns follow. Saunders has also shown that the difference in stock market returns between the most sunny days and most cloudy days is statistically significant. Cao and Wei (2005) examine the relationship between temperature and stock market returns. Temperature is together with length of day and number of hours of sunshine seen as the most influential weather variable. In their paper Schneider et al. (1980) concluded that high temperature is mostly associated with predominant feeling of indifference and lethargy, while cold

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hand, high levels of temperature are linked with lower level of risk taking. This inverse relationship between temperature and stock market returns can explain a possible seasonal cycle in stock market returns.

Research Question

As described in the problem statement I would like to research whether the Halloween effect truly exists and if so, to what extent. This would mean testing the existence of significantly higher winter stock returns compared to the summer stock returns. Answering this question is essential. The second part of the research will mainly concentrate on finding possible

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In the third part of the research possible existence of the Halloween effect on an industrial level will be investigated. The second and the third part of the research serve as possible improvers of the “Sell in May, and go away” investment strategy due to their fine-tuning characteristics. In the fourth and final part the “Sell in May, and go away” strategy will be compared with the simple buy-and-hold strategy. This final part will provide us with the answer whether investors can benefit by investing seasonally.

The main research question is:

What is the evidence of Halloween effect and profitability of associated “Sell in May, and go away” investment strategy?

The sub-questions are:

1. Are the winter returns significantly larger than the summer returns once the outliers are taken into account?

This sub-question will provide the main evidence whether or not Halloween effect is actually present and if so, how significantly. The previously two mentioned outliers, namely October 1987 and August 1998, both fall in the summer period. This leads to lower summer returns compared to the winter returns. The question I am interested in is whether the Halloween effect would be present if these two events would be controlled for. Also January effect needs to be controlled for in order to get unbiased results because due to the January effect winter returns become higher than expected. So far we are dealing with three different variables affecting the outcome and presence of the Halloween effect. All three variables are in favour of the

Halloween effect because they lead to a higher discrepancy between the winter and summer returns. For this reason it is necessary to control for these three variables in order to obtain unbiased results.

2. Is there a difference in seasonal returns between emerging and mature markets?

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in seasonal returns. Furthermore, this question deals with the level of integration of world markets. If the results show evidence of very little co-movement between the mature and emerging markets we could conclude that the level of integration of world markets is low. It is expected however that over time the level of integration should increase which should be illustrated by higher levels of co-linearity of stock market returns between mature and emerging markets.

3. Is there evidence of an industry-level Halloween effect?

This question is interesting because it zooms into the whole issue. This allows us to gather even more specific information and evidence on the Halloween effect and its origins within a given country. Even if Halloween effect would not exist in most countries, it could still be very present in some specific industries or sectors. As mentioned earlier, the “Sell in May, and go away” investing strategy is futile if the stock market summer returns/risk trade-off is more gainful for investors than the return/risk trade-off of risk-free assets during summer. However, if the Halloween effect would be present at an industrial level investors could still benefit from the “Sell in May, and go away” investing strategy due to lower summer return/risk trade-off of some industries compared to the return/risk trade-off of the risk-free assets.

4. Is the “Sell in May, and go away” investment strategy more profitable than simple buy-and-hold strategy?

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Data and Methodology

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than winter returns. In order to answer the main research question I shall answer the first sub-question which deals with the actual evidence of an Halloween effect. In order to find out if the Halloween effect is truly present data is required. Data that is required is the monthly data on stock returns. A regression technique which strongly resembles a mean test will be employed with the purpose of verifying whether there is a difference between the winter and the summer stock returns and whether it is also significant. The significance level taken is 10% since it provides enough evidence in favour of the existence of differences between groups. We will distinguish furthermore between 5% and 1% significance level. Countries where the results are significant at those levels are obviously even better examples of strong differences between the groups examined. The 10% significance level may seem a bit too wide, it does however offer great indication of inter-group differences and should therefore be regarded as useful.

The regression is represented by:

(1) Rt = μ + α1St + εt

- Rt is the dependent variable and it stands for monthly compounded stock returns.

- St is a dummy variable taking on value 1 if the month falls in the winter period

(November-April) and 0 otherwise.

- μ stands for the mean returns over the summer period (May-October)

- μ + α1 stands for the mean returns achieved during the winter period

(November-April)

- α1 indicates the strength of the dummy variable, if α1 is positive and significant it

means that the winter stock returns are significantly higher than the summer stock returns.

- εt is the error term.

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Rt = μ + (α1 x 0)+ ε t= μ + ε t

Because we assume that the error term ε t averages to zero the equation for the

summer group is equal to: Rt = μ. This means that μ stands for the mean returns

over the summer period. When looking at the winter group the regression equation takes on the following form:

Rt = μ + (α1x 1)+ ε t= μ + α1 +ε t= μ + α1

Again we assume that the error term averages to zero. In order to find out what the difference between both groups is we need to subtract the outcomes of the separate groups from each other. The final outcome is that the difference between the winter and the summer group is equal to α1(μ + α1- μ= α1). So if α1 is positive

and significant it means that the winter stock returns are significantly higher than the summer returns and that seasons do matter.

So far formula 1 enables us to see possible existence of the Halloween effect and its significance. However the possible effect of the outliers in the data are still not taken into account. The 1987 market crash and the 1998 Russian Ruble crisis can be regarded as irregular events leading to unusually low returns. As already explained it is needed to control for these events in order to create a clearer and unbiased picture of the real stock market returns and possible Halloween effect because these two events may be exaggerating the size of the Halloween effect. It is possible to control these two events by introducing a second dummy variable Ot. The dummy variable Ot is equal to 1 for the months when the outliers happen,

October 1987 and August 1998. The new regression which takes the outliers into consideration is represented by:

(2) Rt = μ + α1St + α2Ot + εt

- μ + α1 stands for the mean returns achieved during the winter period (November-April)

- α1 indicates the strength of the dummy variable St, if α1 is positive and significant it means

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- Ot is a dummy variable taking on value 1 if the month falls in the period October 1987 and

August 1998, and 0 otherwise.

- α2 indicates the strength of the dummy variable Ot, if α2 is negative and significant it means

that the outliers have a negative and significant influence on the monthly compounded stock returns.

- εt is the error term.

At this stage the regression is extended and it allows us to see whether the Halloween effect is present after the outliers are controlled for. However, there is still another effect which causes the Halloween effect to become larger than it should be. This effect works the opposite way compared to the October 1987 and August 1998 outliers which made the summer returns lower. I am talking about the January effect which leads to higher than expected winter returns. Due to the January effect the discrepancy between the winter and the summer stock returns increases and consequently enhances the size and significance of the Halloween effect. It is therefore needed to control for the January effect. The regression which allows us to control for the January effect adds a dummy variable Jt to the regression equation. The dummy

variable Jt is equal to 1 whenever a month falls in January and is equal to 0 otherwise.

(3) Rt = μ + α1St + α3Jt + εt

Where,

that the winter stock returns are significantly higher than the summer stock returns. - Jt is a dummy variable taking on value 1 if the month falls in January, and 0 otherwise.

- α3 indicates the strength of the dummy variable Jt, if α3 is postive and significant it means that

the January effect has a positive and significant influence on the monthly compounded stock returns.

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This regression tells us whether there is a Halloween after we have controlled for the January effect. The above average January returns which lead to above average winter returns are not taken into consideration in this regression. The outcome is a more precise and unbiased picture of the Halloween effect and the seasonal differences.

The last question I am concerned with is what will happen to the Halloween effect if we take both the outlier dummy and the January dummy and put it into one equation.

This is represented by the following equation:

(4) Rt = μ + α1St + α2Ot + α3Jt + εt

This regression combines both the formula 2 and formula 3 into one regression. Now we can see how the Halloween effect reacts when it is confronted with both dummy variables simultaneously. I expect to see the lowest Halloween effect when using formula 4 since both effects which used to contribute to a higher Halloween effect are now controlled for and cannot exert any influence on the final outcome.

The second sub-question deals with finding out whether there a difference in seasonal returns between emerging and mature markets. Once again regression analysis will be employed to detect possible existence of differences in Halloween effect between the two groups of countries.

(5) Rt = μ + α1St + α5Mt + εt

Where,

- μ stands for the mean returns over the summer period (May-October) for emerging markets.

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- Mt is a dummy variable taking on value 1 if the stock market is mature and developed and 0 if

the it is an emerging market.

- α5 indicates the strength of the dummy variable Mt, if α5 is positive and significant it means

that mature markets have higher monthly compounded returns than emerging markets. - εt is the error term.

According to formula 5 we can see whether there is a (significant) difference between the summer and winter returns between the emerging and mature markets. It is needed to look at how significant the discrepancy between the summer and the winter returns is for firstly the mature markets, and secondly, for emerging markets. The results obtained will then be compared in order to find out if there truly are seasonal differences in stock market returns between the both types of markets.

The third sub-question is concerned with investigating the existence of an industry-level Halloween effect. In order to find out whether this exists data on an industry-level is required. Stock returns from different industries will be compared in a regression analysis in order to find out if the stock returns during different seasons differ between different industries. The following regression is going to be used:

(6) Rt = μ + α1St + α6It + εt

Where,

- μ stands for the mean returns over the summer period (May-October) for different industries . - μ + α1 stands for the mean returns achieved during the winter period (November-April) in

different industries.

that the winter stock returns are significantly higher than the summer stock returns. - It is a dummy variable taking on different values for different industries

- α6 indicates the strength of the dummy variable It

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Results

In this section results associated with the above mentioned regressions and the related research sub-questions will be provided and discussed.

In order to find out the possible existence of a “Sell in May and go away effect” we need to compare the returns obtained during the winter season and the summer season.

(insert figure 1)

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However, the fact that all the 17 countries demonstrate higher winter than summer returns raises much doubt in the theory of equal returns during different seasons. The probability of finding higher winter returns in all the 17 countries is 0.000763%. 1

Another interesting finding is that in seven countries the summer returns are in fact negative. This fact gives power to the idea that the portfolio should be sold during summer and invested in risk-free assets. Winter returns are above eight percent in all except two countries, and exceed even ten percent in eight countries, making winter a great period to hold on to your portfolio and in doing so great profits are to be made. In the group of developed countries the smallest difference between the winter and the summer returns is observed in Hong Kong (2.03% difference) while the biggest difference between the winter and the summer returns is observed in Italy (17.26%, see figure 1). In eight countries the difference in returns between the seasons is higher than 10% giving lots of opportunities for investor to make profits. Another interesting observation worth mentioning is the fact that the biggest difference between the winter and the summer returns is observed in the countries of Western Europe, secondly in North American countries, and finally in developed countries from Asia. In figure 2 the average returns in the summer period (May-October) and the winter period (November-April) are reported for all the emerging markets in the data sample.

(insert figure 2)

In total 14 countries are included in the sample. As we can see in figure 2, winter returns are in general higher than the summer returns also for the group of emerging markets. However there are two countries in which the summer returns are in fact higher than the winter returns. These countries are Bangladesh, and Slovenia. Nonetheless the probability of finding two out of fourteen countries with higher summer than winter returns is still quite small, approximately 0.56%.2

It seems that the group of emerging markets, just as the group of developed markets, shows signs of the Halloween Effect. However, only one country actually displays negative summer returns, this country is Malaysia. Summer returns are actually rather high, in eight countries the summer returns are higher than 5%. On the other hand, in two out of those eight countries the winter returns are above 10% and in four out of those eight countries winter returns are actually above the staggering 20%, making discrepancy between summer and winter nevertheless big. Bangladesh is the country with the biggest difference in returns between seasons in favour of the summer (18.28%, see figure 2). While the biggest difference in returns

1_{The probability is calculated in the following fashion: 0.5^17}

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in favour of the winter is observed in Turkey with astonishing 27.35% (see figure 2). It seems that the intra-group discrepancies are larger in the emerging markets (see figure 2). There is no clear geographical pattern detectable in the group of emerging markets. It seems that the markets are randomly dispersed when it comes to season returns despite the fact where they are located. Overall when looking at both groups of markets we can state that in 29 out of 31 countries winter returns are higher than the summer returns. This fact provides great optimism in the possible benefits of the “Sell in May, and go away” investment strategy. The differences in returns between seasons are in general large as well.

The results obtained so far are economically significant, however one should examine the results more thoroughly before placing too much emphasis on them. The question I am interested in deals in the first place with whether the obtained results are statistically

significant. As explained above, the method used to find this out is a regression which strongly resembles a mean test. Table 1 provides us with the found results.

(Insert table 1)

In the first column the number of observations is shown. Each observation stands for a month. As can be seen, different number of observations is used per country due to unavailability of data. For each country number of observations stands for the total data available, meaning that no data has been left out. Data availability ranges from 144 months (12 years) for Republic of South Africa, to 684 months (57 years) for United States and Japan. The second and third column provide the monthly mean returns and monthly standard deviation in returns expressed as a percentage.

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Results Regression 2

It became apparent from table 1 that in 29 out of 31 countries the winter returns are higher than the summer returns. In 16 of those countries this difference was statistically significant.

However as explained earlier one of possible causes for lower summer returns could be the outliers. Two main outliers are taken into consideration in the second regression. These outliers are the October 1987 stock market crash and August 1998 in which the Russian government announced moratorium on debt repayment. Both of the events had great impact on the stock markets worldwide. It is interesting to note that both the 1987 stock market crash and the 1998 Ruble Crisis, happened in the summer period. For this reason the summer returns worldwide are negatively influenced and consequently this could lead to the existence of a “Sell in May, and go away” effect. The second regression will help us too see whether or not the outliers have impacted the significance of the seasons on monthly returns.

(insert table 2)

When we take a look at the third and fourth column of table 2 we can see the t-values and the correlation coefficients of the outlier dummy with respect to the dependent variable represented by the monthly returns as in regression 2. Evidently in 26 out of 31 countries the t-values of the outlier dummy are significant at a 10% significance level, while in 20 out of those 26 countries the outlier dummy is even significant at a 1% significance level. In all countries, except for Slovakia, the t-values and the value of α2 are negative. The amount of information so far is

plenty in order to conclude that the October 1987 stock market crash and August 1998 Ruble Crisis are negatively related with the performance of the stock exchanges worldwide. This finding gives hope for all the critics of the “Sell in May, and go away” effect since the highly significant t-values of the outlier dummy might explain why the summer returns are so much lower than the winter returns. Even though the outlier dummy is significant in 26 out of 31 countries, still we need to take a look at the effect of the outlier dummy on the monthly mean returns during summer. Since both outliers happened during summer season the mean value of the summer returns should move up in most of the 31 countries after it has been controlled for. Before discussing the rest of results from table 2, let’s take a look at the adjusted summer returns.

(insert table 3)

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summer returns. Slovakia is the only country in the sample where the outliers actually had a positive effect on the summer returns. Before outlier adjustment there were ten countries which had negative summer returns (see table 1). After the adjustment this number has decreased to only four countries, these are Italy, France, Belgium and Austria. Furthermore, outliers had a bigger effect on summer returns in emerging markets than in developed markets (1.81% vs. 1.12%). It seems clear that controlling for the outliers has decreased the gap between the winter returns and the summer returns. The important question is however in how many countries the gap between the winter and the summer returns is still significant. Before the outlier

adjustment there were 16 countries where the winter returns were significantly higher than the summer returns on a 10% significance level (see table 1). In eight out of those 16 countries the winter-summer difference was significant even at a 1% significance level. However this picture has changed after the outliers have been controlled for. As can be seen in table 2, in 9 countries the winter returns are still significantly higher than the summer returns at 10% significance level, while in only three countries they are significant at 1% significance level.

Considering the second regression we can state that the outliers do have a negative effect on the existence of the “Sell in May, and go away” effect since it is no longer statistically

significant in four countries (see table 2). Although the outliers negatively affected the “Sell in May, and go away” effect, we still have to note that the winter returns remain higher than the summer returns in 29 out of 31 countries. Bangladesh and Slovenia are still the countries with higher summer returns than winter returns. In conclusion we can state that even though the outliers have a negative impact on the strength of the “Sell in May, and go away” effect, this impact remains on a statistical basis.

In regression 3 we have tried to find out how the January effect affects the “Sell in May, and go away” effect. Firstly, let’s take a look at what has happened to the gap between the winter returns and the summer returns after adjusting for the January effect.

(insert table 5)

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negative in 23 out of 31 countries. The adjusted winter returns appear to be lower than the non-adjusted winter returns in 14 out of 17 developed markets, and in 9 out of 14 emerging

markets. In 8 out of 31 countries the adjusted winter returns are actually higher than the non-adjusted winter returns. In those countries the January returns were lower than the average summer returns, which has resulted in an increase of the adjusted winter returns. The next question we need to answer is how the controlling for the January effect has influenced the significance of the “Sell in May, and go away” effect.

(insert table 4)

Firstly when looking at the fourth column we find that the January effect is positively related to monthly returns in 24 out of 31 countries. The January effect is significant in eight out of 24 countries, in five countries at a 10% significance level, and in three countries at a 1% significance level. Positive relationship between the monthly mean returns and the January dummy means that once we control for the January effect this positive relationship will lead to lower winter returns. The lower winter returns will in turn lead to a smaller winter-summer gap, which will decrease the strength of the “Sell in May, and go away” effect. In seven countries the January dummy is negatively related to the monthly mean returns. Surprisingly, in Italy this negative relationship is actually significant. When looking at the season dummy, it has remained significant in 13 out of 31 countries, which translates into a decrease of three countries (see table 1 and 4). In four out of those 13 countries the season dummy was

significant at a 1% significance level. 12 out of 13 are developed countries. The only emerging market where the season dummy was significant after controlling for the January effect was Malaysia ( see table 4). In conclusion, we can state that controlling for the January effect has led to a smaller gap between the winter and the summer returns in 23 out of 31 countries (table 5). Controlling for the January effect has also caused the difference in season returns to become insignificant in 3 countries (table 4). The theory seems to be confirmed, January effect does affect the strength of the “Sell in May, and go away” effect positively, once we control for the January effect, the strength of the “Sell in May, and go away” weakens.

Regression 4 shows us the relationship between the monthly mean returns on the one hand and the season differences, the outliers and the January effect on the other. It also shows us what effect the controlling of the outliers and the January effect have on the significance of the “Sell in May, and go away” effect.

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Table 7 recaps the previous results showing the newly adjusted summer and winter mean returns. The last column shows the combined effect of the outliers and the January effect. It seems that in 17 out of 31 countries both effects combined affected the yearly mean returns negatively. 11 of those countries were developed, while 6 were emerging countries.

Interestingly, the overall combined effect is negative for the developed markets overall

(-0.78%) while positive for the emerging markets (2.67%). The real question deals once again with the effect on the significance of the “Sell in May, and go away” effect. How did the two dummy variables, the outlier dummy and the January dummy, affect the significance of the “Sell in May, and go away” effect?

(insert table 7)

Firstly, let’s start by looking at column 3 where we can find the t-values of the outlier dummy. Prior to adding the January dummy to the equation the same test was run (see table 2). It appeared that in 26 out of 31 countries the t-values of the outlier dummy were significant at a 10% significance level. In 20 out of those 26 countries the t-values were even significant at a 1% significance level. After adding the January dummy this has not changed (see table 6). Again the t-values of the outliers are significant in 26 countries at 10% significance level, and in 20 out of those 26 countries at a 1% significance level. The same story holds for the t-values of the January dummy. Just as in table 4, the same countries have significant values at 10% and at 1% significance level. There are once again 8 out of 31 countries with significant t-values for the January effect at a 10% significance level. 3 out of those 8 countries have the t-values significant at a 1% significance level, just as they did in regression 3 (see table 4).

The most important question however deals with the possible changes of the first variable in regression 4, namely the season dummy and its significance. If we take a look at table 6 it becomes clear that the season dummy is significant in 10 out of 31 countries at a 10% significance level. In two out of those ten countries the season dummy is significant at a 1% level. Compared to results from regression 1, regression 2 and regression 4 we can detect, as expected, a declining pattern. When we did not control for anything, the “Sell in May, and go away” effect was significant in 16 out of 31 countries at 10% significance level. In eight out of those 16 at a 1% significance level (see table 1). After controlling for the outliers, this has dropped to 12 countries being significant at 10% and 3 out of those 12 countries at 1%

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significant in 10 countries at a 10%, in two out of those 10 countries at a 1% significance level. It becomes clear that the “Sell in May, and go away” effect looses its statistical significance once we control for certain variables. However, it is still present in almost one third of the countries with statistical significance. Another important thing to note is that all of the 10 countries with significant results for the season dummy in regression 4 are developed

countries. The “Sell in May, and go away” effect has vanished from the three emerging markets where it was significant before controlling for other variables (see table 1 and table 6).

Results regression 5

In regression 5 we attempted to find out the effect of the type of market on the “Sell in May, and go away” effect. Is there a significant difference between the emerging and the developed markets when it comes to the differences in the winter and the summer returns?

(insert table 9)

The first column of table 8 shows us the t-value of the market dummy. This value does not seem to be statistically significant at 10% significance level. In fact the significance level obtained was 12.2%, meaning that the variable was almost significant. Subsequently first thing we can conclude is that the relationship between the obtained monthly mean returns and the type of market (developed vs. emerging) is not significant. The relationship is negative, meaning that if a market is developed, then the relationship with respect to the monthly mean returns is more negative than if a market is emerging. This information basically tells us that the overall mean returns are lower in developed markets than in emerging markets. The third column shows the relationship between the season dummy and the dependent variable. It seems that this relationship is significant and positive. The monthly mean returns are positively related to the winter season for both types of markets.

The true question deals however with answering whether the season differences between market types are significant.

(insert figure 3)

In figure 3 we can see the average winter and summer returns in both types of markets. It becomes clear that the average returns taken over the both seasons are higher in the emerging markets. However, the gap between the season is higher in developed markets. The question is whether the difference in this gap is significantly different between the emerging and the developed markets.

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Table 9 offers the answer to our question. It seems that the discrepancy in season returns is significantly different between the different types op market. This difference is rather strong, being significant at a 5% significance level (see table 10).

In conclusion to the fifth regression and the second sub-question, we can state that the

difference in season returns between developed and emerging markets is significantly different. Emerging markets are characterized by higher average returns over the year. The higher

average returns are mainly caused by the higher summer returns compared to developed markets which are characterized by a much larger discrepancy in returns between the seasons. As seen from previous results, developed markets are the ones with a stronger and more present “Sell in May, and go away” effect. For this reason it seems obvious to expect larger discrepancy between season returns in developed markets. Our expectations have been met.

In order to find out more about the “Sell in May, and go away” effect one does not need to look at country level solely. Regression 6 looks at the “Sell in May” effect further than just the general index level, it takes a look at a sectoral level instead. Three countries are taken in the sample, six industries within each country have been observed. These countries are the

Netherlands, Sweden and the United Kingdom. The six sectors observed are: consumer goods, consumer services, energy, financials, industrials and beverages/food.

(insert table 11)

As can be seen from table 10, different number of observations was available per country. This makes the comparison between the countries hard, on the other hand, it does offer the clearest possible picture per country by taking advantage of the complete time-frame. Table 10 shows the mean winter returns and the mean summer returns per sector per country. It becomes clear that in 15 out of 18 sectors the winter returns are higher than the summer returns (the

probability of finding this is approximately 0.3%).3

The Netherlands has the least number of sectors where the winter returns are higher than the summer returns, namely four. Second place is for Sweden with five sectors out of six where the winter returns are higher than the summer returns. First place is for United Kingdom where all the sectors have higher winter returns than summer returns. This pattern could be linked with the observed time frame of the countries. It seems the longer the timeframe, the more industries exhibit higher winter than summer returns.

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As with previous question our interest lies in finding out whether the season effect is significant, and if so at what level.

(insert table 12)

Table 12 shows the obtained results for regression 6. As we can see in 6 out of 18 industries the winter returns are significantly higher than the summer returns. Two out of those six industries show difference between seasons significant at 10% significance level. These industries are UK industrials sector, and UK beverages/food sector. Three sectors show difference between seasons to be significant at a 5% significance level. These sectors are UK consumer services, Sweden financials and beverages/food sector. Finally, one sector shows difference between the winter and the summer returns to be significant at a 1% significance level. This sector is Sweden industrials. When we look at the obtained results it is striking that in the Netherlands none of the six sectors exhibits significant difference in season returns. In Sweden and UK half of the sectors shows significant difference between the seasons. When we look at the winter and the summer returns it seems that the reason for lack of significant difference in the

Netherlands are not high summer returns (see table 11). The low returns obtained during winter seem to be the main cause. An explanation for this could simply be lower number of

observations which does not capture the true situation in the Netherlands. We should not forget that the general index for the Netherlands did show significant differences between the season (see table 2, 4 and 6) even after controlling for the outlier dummy and the January dummy. Another interesting point is that summer returns were higher than the winter returns in three sectors. Two times it was the energy sector that exhibited higher summer than winter returns ( in the Netherlands and in Sweden). Another example of overlap in findings between the countries is that both industrials sector and the beverages/food sector show the most significant differences between seasons, being significant in both Sweden and United Kingdom. Followed by consumer services and financials sector.

In conclusion we can state that there is some overlap in season differences between different sectors. Sweden and the United Kingdom can be better compared due to almost same

timeframe. It is here that we have found same results for the beverages/food and industrials sectors. Energy sector seems to exhibit no possibilities for “Sell in May, and go away” investment strategy.

The “Sell in May, and go away” versus the buy-and hold

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Afterwards we continued with answering the sub-questions proposed. We controlled for the outliers and for the January effect and concluded that both affect the strength of the “Sell in May, and go away” negatively. However “Sell in May, and go away” effect still remained even statistically present (see tables 1,3,5 and 7). After looking at the country level the research zoomed in on the industry level to investigate possible presence of the Halloween effect. So far we have given the answer to the first part of the main research question. We have shown the evidence of the Halloween effect. Now we shall answer the second part of the main

research question, namely what is the profitability of the “Sell in May, and go away”

investment strategy? In doing so the annual returns will be compared between the two investing strategies. The first strategy is the “Sell in May, and go away”. As explained earlier when an investor chooses this strategy he/she will invest in a market portfolio during the winter period (November-April) and sell the market portfolio at the beginning of the summer period (May-October). The investor will reinvest his money in riskless assets during the summer period. One might argue that there are no riskless assets, in our case the investor will invest his money in six-month treasury bills. The second strategy is the buy-and hold strategy. When an investor chooses this strategy he/she will hold the market portfolio during the whole time. Buy-and-hold strategy has the advantage of not being subjected to the transaction costs. The winter returns are the same for both strategies. This means that the question of profitability comes from the difference during the summer period. The buy-and hold strategy will prove more profitable and win the contest if the returns achieved during the summer period are higher than the interest rate offered on treasury bills minus the transaction costs.

Buy-and-hold “wins” if: rsummer>iT-bill – TC

Where,

- rsummer stands for returns during summer

- iT-bill stands for interest rate on t-bills

- TC stands for transaction costs

The interest rates were taken from the datastream per country investigated. The transaction costs on the other hand were equalized across all countries. The transaction costs were fixed at a rate of 0.1% per single transaction. This number is based on the information provided by the website of ABN AMRO. In addition, transaction costs were estimated to be 0.1% concerning the futures market (Solnik, 1993).

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Table 13 shows the results obtained for the 23 markets investigated. It seems that the “Sell in May” strategy outperforms the buy-and-hold strategy in 19 out of 23 markets. Interestingly, the four markets where the buy-and-hold strategy outperforms the “Sell in May” strategy are all located in Central-Eastern Europe. These countries are Czech Republic, Slovakia, Poland and Slovenia. Explanation for this is first of all short observation period in all four countries. For Slovenia we observed only four years, and from the other countries six years were observed. Short timeframe gives outliers a greater chance to unbalance the entire picture.

(insert table 14)

Furthermore, as table 14 suggests, three countries were characterized by higher summer returns in this period than their average summer returns. Especially in Czech Republic and Slovakia the difference was remarkable. This was not the case only in Slovenia where the average summer returns were higher than the summer returns during the four year period. However, we should remember that it was Slovenia that had higher overall summer returns than winter returns (see figure 2) and hence it did not exhibit evidence of the Halloween effect in the first place. Another point worth mentioning is that Czech Republic and Poland showed no

significant difference between the winter returns and the summer returns (see table 1), in these countries however the winter returns were higher than the summer returns. These overall high summer returns in Slovenia and the high summer returns in the other three countries during the observation period, made the buy-and-hold strategy more profitable for these countries than the “Sell in May, and go away” strategy.

When looking at the average score for the 23 markets we find that the Halloween strategy is definitely the ultimate winner having an average annual mean of 16.57% versus the 11.39% of the buy-and-hold strategy. And while the Halloween strategy has higher returns, it also has standard deviation which is lower more than 4.5% than the standard deviation of the buy-and-hold strategy (see table 14). It appears that the “Sell in May, and go away” strategy

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Conclusions

Our research has made an attempt to find out if a phenomenon such as the Halloween effect truly exists, and if so, if it is possible to make profits from it. The results have shown that indeed, as expected, the winter returns tend to be higher than the summer returns. In 29 out of 31 countries this was the case. The results also confirmed the negative expected relationship between the difference of the seasonal returns on one hand, and the 1987 stock market crash, 1998 Ruble crisis and the January effect on the other hand. The existence of the Halloween effect was even significant in more than half of the countries investigated (see table 1).

Actually, the “world average” winter returns remained higher than the “world average” summer average returns after adjusting the summer mean returns for the outliers and the winter mean returns for the January effect (see table 8).

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An industry-level analysis was also conducted where six industries from three different countries were investigated. It appeared that in 15 out of 18 industries the winter returns were higher than the summer returns (see table 11). The energy sector appeared in two out of three countries as the sector with higher summer returns than winter returns.

In the end we have compared the profitability of the “Sell in May, and go away” investment strategy versus the buy-and-hold strategy. It appeared that in 19 out of 23 countries the “Sell in May” proved to be more profitable. The four countries where this was not the case were all located in Central-Eastern Europe and shared rather short observation period. They were also characterized by higher than average summer returns during their observation period which made the buy-and-hold strategy more profitable. It must be noted however that in doing our calculations the outliers have not been taken into account, neither has the January effect been taken into account. We have not controlled for these variable in order to make the picture real. Since investors do not know when exactly a crisis will take place, they also could not control for the crises which were about to take place. This way we have mimicked the real life, obviously the profitability of the Halloween strategy would decrease if we were to control for the outliers and the January effect.

Several different explanations have been mentioned which could help us explain the existence of the Halloween effect. We have already concluded that data mining does not do a good job in explaining the Halloween effect since the Halloween effect is known long before it is

empirically tested, hence it is not constructed solely on data. Furthermore, as we have found the Halloween effect to be present in almost all the countries over a long period of time, it seems that the out-of-sample test, which would suggest existence of the Halloween effect only in a few countries over a shorter period of time, has been passed as well.

Risk was another explanation which was offered to explain the Halloween effect. When we take a look at table 2 it become clear that the winter returns are on average higher than the summer returns. This finding must therefore imply that the volatility, or the risk, associated with the winter period should also be higher. This is however not the case. It seems that the winter returns offer higher returns accompanied with lower rates of risk.

We offered the January effect also as one of the explanations of the Halloween effect. The January effect does have in most countries a positive effect on the winter returns, and therefore it does enlarge the discrepancy between the seasonal returns. However, once we have

Unravelling the Halloween puzzle