Turn-of-the-year effects

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Turn-of-the-year effects

Rijksuniversiteit Groningen, December 2005 Faculty of Economics

Department of Finance, Investment and Accounting William Ford

wford@home.nl (s1256947)

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Abstract - 2 -

Abstract

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Contents - 3 -

Acknowledgements

This is the thesis I have written to complete my study Economics, specialization Finance and Investment, at the Rijksuniversiteit Groningen. Its subject is stock price behavior around the turn of the year. Ever since a few of my friends and I wrote a short paper on the December effect in order to fulfill all the requirements to complete the course “Financiering” in the third year of my study, this subject has held my interest. Accordingly, the selection of a subject for this thesis was not difficult. The scope of the subject of this thesis might appear limited at first sight. However, due to their troublesome relation with the fundamental notion of market efficiency, support for the existence of turn-of-the-year effects in stock returns may have implications that range far beyond evidence of a possibility to earn a return of a few percentage points around the turn of the year. Accordingly, my personal interest lies not in a demonstration of an easy way to get rich fast by trading stocks around the turn of the year, but in the link between apparent stock return anomalies around the turn of the year and fundamental concepts of financial economics, such as risk and the efficient market hypothesis. First of all, I would like to thank my supervisor, Robert Lensink, for all his invaluable comments and suggestions that helped to reshape and improve this thesis. This is also the place to express my immense gratitude to my parents, who have always supported me during my studies and will hopefully continue to do so in the future. Lastly, I would like to thank the two friends mentioned previously, Timo Gerrits and Pieter de Graaf, for their share in sparking my interest in the turn-of-the-year effects and keeping it alive. All remaining errors in this thesis are, of course, my own.

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Section 1 - Introduction - 6 -

1 Introduction

“A brief examination of the Dow-Jones Industrial Average from 1927 to 1942 shows that this index of thirty stocks displayed frequent bullish tendencies from December to January; to be exact, an appreciation in value in eleven of these fifteen years.” (Wachtel [1942], p. 185) “October. This is one of the peculiarly dangerous months to speculate in stocks. The others are July, January, September, April, November, May, March, June, December, August and February.” (Twain [1986], p. 144)

The story of the turn-of-the-year effects starts with a seminal paper written by Wachtel [1942]. He shows a persistent tendency of the Dow Jones Industrial Average (DJIA) to rise at the end of December and in early January. Notwithstanding the plethora of papers devoted to the topic ever since, puzzles and mysteries still surround the turn-of-the-year effects. Several explanations have been proposed for the existence of these effects, but the relative importance of these explanations has yet to be established with certainty.

This thesis attempts to solve a small piece of the puzzle by examining stock return patterns around the turn of the year and the explanations put forward for their existence. In particular, the objective of this thesis is to examine and characterize turn-of-the-year effects on the Dutch, English, French, German and U.S. stock exchanges and to identify factors responsible for the size and presence of the turn-of-the-year effects, if found to exist, in large market-capitalization stocks traded on the Dutch stock exchange. This research objective can be decomposed into three separate questions. The first question is whether abnormal stock returns are present around the turn of the year. Perhaps, this presence differs from stock exchange to stock exchange, or even between different categories of stocks. In order to understand why turn-of-the-year effects are present or absent on a stock exchange, a thorough understanding of the explanations offered for the turn-of-the-year effects is indispensable. The second question asks for the existence and the nature of factors able to explain any of-the-year effect present in stock returns. As is demonstrated in section three, each explanation for the the-year effects predicts the presence or absence of a relation between specific factors and turn-of-the-year effects. The third question is whether it is actually possible to exploit any apparent opportunity to earn abnormal profits around the turn of the year. This is the litmus test of any apparent return anomaly.

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Section 1 - Introduction - 7 -

particular, only a combined examination of both effects allows for a complete test of such an explanation. The window-dressing hypothesis, for example, predicts that winner stocks will show a positive return in December and loser stocks in January. Adding a cross-sectional dimension by including several stock markets makes it possible to determine whether turn-of-the-year effects are restricted to the particular settings and characteristics of one or a small number of stock exchanges or, alternatively, international phenomena. Obviously, these two alternatives have important implications for market efficiency and for the nature of an appropriate explanation of the turn-of-the-year effects. Second, a M-GARCH(1,1) model is used to deal with heteroscedasticity and the relation between risk and return in the context of turn-of-the-year effects. As stock returns generally follow a non-normal distribution, the use of traditional estimation methods such as ordinary least squares (OLS), prevalent in most previous studies, is likely to lead to biased results. Third, this thesis pays attention to small-firm and large-small-firm return patterns around the turn of the year using data that has not previously been available and therefore not thoroughly analyzed as yet. Specifically, it uses indices of Dutch, English, French and German small-firm stocks that are constructed in a similar manner. In addition, recent large-firm stock index returns are included that have not been analyzed in previous studies. Using new data sets makes it possible to come up with a definite rejection of data mining and selection bias as causes of the turn-of-the-year effects, and to determine the degree of persistency of these effects. Fourth, attention is paid to the relationship between the turn-of-the-year effects on the Dutch stock exchange and several factors predicted to be related to these effects. Although the existence of such a relationship has already been examined on the basis of U.S. stock returns and for factors relevant in the U.S. institutional context, the breadth and depth of similar studies for other stock markets is severely restricted.1

The structure of the remainder of this thesis is as follows. Section two presents the empirical evidence on the existence of a January anomaly and a December anomaly in stock returns. In section three, alternative explanations of the two turn-of-the-year effects are discussed. Section four examines the concept of market efficiency and clarifies its relevance in the light of the turn-of-the-year effects. Several hypotheses are formulated in the same section and tested empirically in the two subsequent sections. Section five presents the data sets and the methodology used in the empirical tests. These tests include a pooled time-series cross-sectional regression analysis and several time-series regression analyses with seasonal dummy variables. The empirical results are also discussed and interpreted in section five. Section six turns to the question whether there are any trading strategies that allow an investor to exploit patterns in stock returns around the turn of the year. Section seven concludes and provides some recommendations and directions for future research.

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Section 2 - The effects - 8 -

2 The effects

In this section and the remainder of this thesis, the following definitions of the January effect, the December effect and the turn-of-the-year effects are used. The January effect is defined as the tendency of stock returns to be unusually high in January. The December effect is the tendency of stock returns to be unusually high in December. Statistically, and for the purposes of this thesis, this means that the January (December) effect is the tendency of mean daily stock returns in January (December) to be positive and significantly different from zero. As stock returns tend to be especially high at the end of December, some researchers, like Lakonishok and Smidt [1988] and Van der Sar [2003] call the December effect the end-of-December effect. In this thesis, and unless noted otherwise, the December effect refers to the entire month. Together, the January effect and the December effect constitute the turn-of-the-year effects, a term first coined by Roll [1983].

2.1 The January effect

As discussed in section one, Wachtel [1942] is credited with being the first to notice a surprising pattern in stock returns around the turn of the year. Since the findings reported by Wachtel, academic attention in the January effect waned temporarily. In 1976, a paper by Rozeff and Kinney [1976] sparked the renewal of academic interest in seasonal patterns present in stock returns over all twelve months of the year in general, and during the months surrounding the turn of the year in particular. A few years later, Banz [1981] discovered the size effect. This is the tendency of small-firm stocks to outperform large-firm stocks, even when taking differences in risk into account. Reinganum [1981 and 1982] and many others subsequently confirmed his findings. The link between the January effect and the size effect is established by Keim [1983]. Keim estimates that nearly fifty percent of the average magnitude of the size effect is due to excess returns in January. Even more surprisingly, he concludes that more than fifty percent of the January premium is attributable to large abnormal returns during the first five trading days of the year, and in particular on the first trading day. Roll [1983a] presents similar results.

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used, the full-year size effect is half as large as previously reported, and is confined to the month of January. Roll [1983b] also favors the use of buy-and-hold returns instead of any method implicitly involving periodic rebalancing when measuring compounded returns on small-firm portfolios.

The international dimensions of the January effect are demonstrated by Gultekin and Gultekin [1983]. They show that the January effect is not limited to stock markets in the U.S., but is also present in other stock markets around the world. Similar results and conclusions are presented by Agrawal and Tandon [1994]. Kato and Schallheim [1985] report excess January returns on Japanese small-firm stocks.

According to Bhardwaj and Brooks [1992], the January effect is not so much related to firm size as to stock price. They show that a low stock price is related to a low market capitalization and a high January return, with the relationship between price and January return being stronger than the one between size and return. A second conclusion of Bhardway and Brooks is that the January effect has disappeared in the 1977-1986 period. Gu and Simon [2003] find evidence that the January effect is disappearing in the stock market of the U.K. However, papers such as those by Sias and Starks [1997], Cox and Johnston [1998] and Chen and Singal [2003 and 2004] show that the January effect is alive and well, at least in the U.S. stock market. Haugen and Jorion [1996] even conclude that the magnitude of the effect has not changed significantly since 1926 and that no significant trend predicts its disappearance.

2.2 The December effect

Several papers on the January effect include tables with estimations of average stock returns in each month of the year, including December. Where such tables appear, December returns are generally statistically insignificant. Moreover, the results of Reinganum [1983], Keim [1983], Rogalski and Tinic [1986] and many others examining turn-of-the-year effects in returns of portfolios constructed on the basis of market capitalization indicate that there is no apparent relationship between firm size and stock return in December.

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to small-firm stocks. Ariel [1990] finds very high returns on the last trading day before each holiday, including Christmas and New Year’s Day.

Just as Wachtel [1942], Lakonishok and Smidt [1988] focus on the returns on the Dow Jones. They find an average rate of return in the second half of December of 1.54 percent. This is the highest return of any half-month in the entire year, and seems to be persistent through time. Lakonishok and Smidt denote this phenomenon as the end-of-December effect and conclude that such a consistently high rate of return for very large companies over such a short period of time calls for further attention. Chen and Singal [2003] find high returns for large winners in December, with winners being the stocks with a high return over the eleven months prior to December. Agrawal and Tandon [1994] report that high returns at the end of the year are present on the stock markets of many countries. 2.3 Comparison

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Section 3 - Explanations - 11 -

3 Explanations

In this section, several explanations of the turn-of-the-year effects are presented and discussed. Table 3.1 provides an overview of the various explanations and their relevance for each of the two effects. 3.1 Tax-loss and tax-gain selling

The tax-loss selling hypothesis is the most thoroughly examined explanation of the January effect. Its counterpart with respect to the December effect is the tax-gain selling hypothesis. Both tax-selling hypotheses attribute turn-of-the-year anomalies to the influence of the capital gains tax rules on investor behavior. In the U.S., capital losses can be used to offset capital gains and even ordinary income, but the latter only up to a certain limit. This makes it attractive to realize capital losses before the end of the tax year, as delaying their realization means reducing the present value of the tax benefits associated with the losses. In contrast, realizing gains after the turn of the tax year is more attractive than before the turn of the year, as this defers the payment of capital gains taxes for another year. In the U.S., the tax year and the calendar year coincide.

In addition to the distinction between the current and the next tax year, the U.S. tax code distinguishes between term and short-term capital gains and losses. The distinction between long-term and short-long-term gains and losses is based on the holding period of the investment, with the critical value being twelve months since 1989 and six months between 1984 and 1989. Long-term gains are taxed at a substantially lower rate than term gains. Short-term losses can be used to offset short-term gains and long-short-term gains. Long-short-term losses may be used to offset short-short-term gains, but must be used to offset long-term gains first. This feature of the U.S. tax code makes it even more attractive to realize losses as soon as possible, but at least before the end of the year.

According to the tax-loss selling hypothesis, taxable investors try to realize their short-term losses before the end of the year. This results in selling pressure on the stocks on which many investors have unrealized short-term capital losses. After the turn of the year, this selling pressure abruptly ends, and stock prices rebound up to their equilibrium value. The counterpart of the tax-loss selling hypothesis, the tax-gain selling hypothesis, states that taxable investors try to defer the realization of short-term capital gains at the end of the year to the next year, in order to decrease the present value of the capital gains tax due. Hence, this hypothesis predicts a decrease in tax-gain selling at the end of the year and an increase in tax-gain selling at the beginning of the year. If the decrease and subsequent increase in selling pressure were to affect stock prices, the prices of stocks with large unrealized short-term capital gains should rise prior to the turn of the year, and decrease immediately after the turn of the year. Thus, both hypotheses assume that tax-related selling takes place and has an impact on stock prices.

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tax rate equals the long-term tax rate and transaction costs are absent. In a later paper, Constantinides [1984] describes the optimal trading policy when the two rates differ. He shows that, in the absence of transaction costs, the optimal trading strategy under the U.S. tax code consists of deferring gains and realizing losses as soon as possible. With transaction costs, his model predicts that tax-loss selling increases gradually from January to December. However, Constantinides also argues that investors need to be irrational or at least ignorant of the stock price seasonality if the selling pressure is to affect stock prices at the beginning of January. After all, rational investors would anticipate and exploit the seasonality. This adjustment in investor behavior would cause the January effect to disappear. The same argument holds with respect to the December effect.

The empirical evidence with respect to the tax-loss selling hypothesis is mixed. Five different approaches to testing the tax-selling hypotheses can be distinguished in the literature.

3.1.1 U.S. stock returns

According to the tax-loss selling hypothesis, investors are likely to sell those stocks at the end of the year on which they have high unrealized short-term losses in order to realize these losses for tax purposes. Hence, a number of papers focus on stock returns in the U.S. around the turn of the year and try to relate the January return on a stock to some measure of the potential for tax-related selling of that stock before the turn of the year. Usually, short-term past performance is used to capture the potential for tax-loss selling. Among many others, Branch [1977], Roll [1983a], Reinganum [1983], Givoly and Ovadia [1983], Chan [1986], De Bondt and Thaler [1987], Jegadeesh and Titman [1993] and Poterba and Weisbenner [2001] all find evidence of a negative relationship between January returns and short-term past performance.

Table 5.1

Explanations of turn-of-the-year effects

This table shows which explanation is able to explain the January effect, the December effect, or both. A “yes” means that the theory is able to explain the effect, a “no” indicates it does not provide an explanation for the effect. The last column indicates whether the explanation is consistent or inconsistent with the efficient market hypothesis (EMH).

January effect December effect EMH

Tax-loss selling hypothesis Yes No Inconsistent

Tax-gain selling hypothesis No Yes Inconsistent

Window-dressing hypothesis Yes Yes Inconsistent

Market microstructure effects Yes No Consistent

Risk mismeasurement Yes Yes Consistent

Parking-the-proceeds hypothesis Yes No Inconsistent

Excess cash balances Yes No Inconsistent

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Both Roll [1983a] and Reinganum [1983] come up with a tax-related explanation of why small firms are more likely to experience high turn-of-the-year returns than large firms. Their first argument is that stock returns of small firms are more volatile than those of large firms. Therefore, the former are more likely to incur large unrealized losses than the latter, making them more likely to be subject to tax-related selling pressure at the end of the year and to a greater price increase at the beginning of the year. The second argument of Roll and Reinganum for the link between firm size and January returns is that, due to their lower liquidity and additional risk, bid-ask spreads are relatively high for small-firm stocks, allowing the January effect to survive. That is, high bid-ask spreads for small-firm stocks render transactions designed to exploit the January effect unattractive. Third, Reinganum adds that (tax-exempt) institutional investors have relatively small holdings in small-firm stocks. As most of the owners of these stocks are taxable individual investors, they are relatively more likely to be subject to tax-loss selling. However, as the small-firm stocks without any potential for tax-loss selling also experience abnormal returns in January, Roll and Reinganum are led to conclude that the tax-loss selling hypothesis is unlikely to explain the entire January effect.

3.1.2 Trading volume

The second strand in the literature on the tax-selling hypotheses consists of studies on trading volumes around the turn of the year. The tax-loss selling hypothesis predicts abnormally high trading volume in December for stocks on which many investors have unrealized losses. Compared to December, the trading volume in January should be relatively low. The tax-gain selling hypothesis predicts relatively low trading volume in December for stocks that represent unrealized short-term gains in the portfolios of individual investors. In January, the trading volume of these winner stocks should be relatively high. Dyl [1977] and Lakonishok and Smidt [1984 and 1986] present evidence supporting these predictions.

3.1.3 International stock markets

Third, several papers examine stock returns in markets other than the U.S. market. If tax-related trading around the turn of the year is the sole explanation of the turn-of-the-year effects, these seasonalities should be absent in the stock markets of countries without capital gains taxation. However, as noted by Brown et al. [1983a] and by Reinganum and Shapiro [1987], if international stock markets are perfectly integrated and investors taxable in countries with capital gains taxes have a pervasive influence on stock returns in capital markets of countries without capital gains taxes, tax-induced turn-of-the-year effects may also be present in the latter markets.

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countries without capital gains taxation. These findings are hard to reconcile with the tax-selling hypothesis as the complete explanation of the turn-of-the-year effects.

3.1.4. U.S. returns before capital gains taxation

If the tax-selling hypotheses are correct, turn-of-the-year effects should be absent before the introduction of capital gains taxes. Capital gains taxation was introduced in the U.S. in 1917. Schultz [1985] and Jones et al. [1987 and 1991] examine stock returns before the introduction of capital gains tax in the U.S. Schultz concludes that the January effect was absent before 1917. Jones et al. [1987] contest the validity of findings of Schultz. However, in a subsequent paper, Jones et al. [1991] revise their prior conclusions in a way that is consistent with those of Schultz.

3.1.5 Individual investor trading activity around the turn of the year

One of the predictions of the tax-selling hypotheses is that individual investors will realize short-term losses on the stocks present in their stock portfolios prior to the end of the tax year, and defer the sale of winners to the beginning of the next year. Therefore, if the hypotheses are correct, trading activity of individual investors should change markedly around the turn of the year. It should be noted that the same pattern in trading activity is also predicted by an explanation of the January effect known as the parking-the-proceeds hypothesis (see subsection 3.5). Be that as it may, the findings of Badrinath and Lewellen [1991], Eakins and Sewell [1993] and Griffiths and White [1993] all support the conjecture that tax-induced individual investor trading activity is closely related to the turn-of-the-year effects. 3.2 Window dressing

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small-Section 3 - Explanations - 15 -

firm loser stocks in January. In addition, the window-dressing hypothesis argues that returns on the stocks with a positive performance in the previous year should be positive at the end of December. An empirical test by Lakonishok et al. [1991] lends support to the window-dressing hypothesis.

The window-dressing hypothesis and the tax-selling hypotheses predict largely the same pattern in stock returns around the turn of the year, which makes it hard to distinguish between the hypotheses. Both predict a rise in the stock price of winners at the end of the year, and high returns on small-firm stocks with a negative past performance at the beginning of the year. One respect in which they differ is the hypothesized cause of the patterns in stock returns around the turn of the year. The window-dressing hypothesis places the blame squarely on the trading activity of institutional investors. Contrarily, the tax-selling hypotheses point to the trading activity of investors subject to capital gains taxation. In the U.S., institutional investors are generally exempt from any capital gains taxes, which leaves individual investors as the cause of the turn-of-the-year effects from the point of view of the tax-selling hypotheses. Sias and Starks [1997] and Ng and Wang [2004] show that both individual-investor and institutional-individual-investor trading behavior around the turn of the year contribute to the effects. 3.3 Market microstructure effects

Differences in transaction costs exist between low-price stocks and high-price stocks. The sources of transaction costs are brokerage commissions and the bid-ask spread. The latter is the difference between the bid price and the ask price of a stock. Round-trip transaction costs are the costs involved in the purchase and the subsequent sale of a stock. In an efficient market, there are no opportunities to earn returns that are disproportionate to the risk incurred. Hence, if the efficient market hypothesis holds, abnormal returns will not exceed round-trip transaction costs. Keeping other things equal, relative bid-ask spreads, one component of transaction costs, tend to be higher for low-price stocks than for high-price stocks. To the extent that low-price stocks are small-firm stocks and high-price stocks large-firm stocks, differences in transaction costs can explain the existence of differences in returns, including January returns, between large-firm and small-firm stocks. Besides transaction costs, low trading volume can also explain the existence of non-exploitable differences in returns between small-firm stocks and large-firm stocks. Together, the influences of low stock price and low trading volume on stock return behavior are called market microstructure effects.

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3.4 Risk mismeasurement

Rogalski and Tinic [1986] conclude that seasonal differences in systematic risk underlie seasonal differences in returns. However, Ritter and Chopra [1989] contest the findings presented by Rogalski and Tinic. According to the former, statistical inferences about seasonalities in the relation between risk and return are heavily dependent on whether equally-weighted or value-weighted portfolios of stocks are used to estimate betas. Using value-weighted portfolios, Ritter and Chopra [1989] find that there is a positively sloped January risk-return relation for small-firm stocks in both up and down markets. This underscores the observation of Schultz [1985] that there is little risk involved in buying small-firm stocks in December if they always provide a positive return in January.

Kramer [1994] uses a multifactor model to account for risk and concludes that the high January returns are an appropriate reward for a higher risk. In contrast, Seyhun [1993] uses a stochastic dominance approach and concludes that his results reduce the plausibility of any explanation of the January effect that points at seasonally varying risk factors.

3.5 Parking the proceeds

Ritter [1988] develops the parking-the-proceeds hypothesis to explain the January effect. In his opinion, it is a generalization of the tax-loss selling hypothesis. The hypothesis starts with the assumption that the turn-of-the-year effects are caused by the buying and selling behavior of individual investors. At the end of the year, individuals sell stocks in order to turn unrealized losses on the stocks in their portfolios into tax-deductible losses. Some of the proceeds of these sales are not reinvested immediately, but parked until January. The reinvestment of these parked funds in January increases the prices of small-firm stocks, in which individual investors typically invest. The hypothesis depends on three crucial assumptions. First, when individual investors buy and sell stocks, they are assumed to buy and sell a disproportionate number of small-firm stocks. Second, buying and selling activities are assumed to influence small-firm stock prices. Third, individual investors wait before they reinvest the proceeds of their stock sales in December. Ritter, Dyl and Maberly [1992], Eakins and Sewell [1993], Griffiths and White [1993], Lakonishok and Smidt [1984] all provide evidence supporting the parking-the-proceeds hypothesis.

3.6 Excess cash balances

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principal repayments and interest payments. The subsequent investment of these funds induces a rise in stock prices around the turn of the month. Ogden hypothesizes that the magnitude of the price rise for a given month is positively related to the magnitude of the aggregate liquid profits that are realized in that month, and is able to present supporting evidence. He states that, as the excess cash balances and liquid profits of investors tend to be exceptionally large just before the turn of the year due to increased business activity around the Christmas holidays and year-end salary bonuses, the abnormally large returns in early January are just another manifestation of the monthly liquidity effect, be it of increased magnitude.

3.7 Behavioral explanations

One of the explanations of the turn-of-the-year effects proposed by Wachtel [1942] would be classified as a behavioral explanation nowadays. He states that the advance in stock prices around the turn of the year might be due to “the general feeling of good fellowship and cheer existing throughout the Christmas holidays” (p. 186). In the behavioral finance literature, more recent explanations can be found why, in the presence of taxes, stock selling by individual investors changes markedly around the turn of the year and, in general, why turn-of-the-year effects in stock returns are present. Shefrin and Statman [1985] argue that investors may have a tendency to retain losing investments in their stock portfolios because they are reluctant to admit their mistakes. They call this tendency the disposition effect, and argue that it derives from the aversion of investors to losses. Kahneman and Tversky [1979] provide a theoretical underpinning of loss aversion with their prospect theory.

The reasoning behind the relation between the disposition effect and the turn-of-the-year effects is as follows. During the year, investors are unwilling to realize losses and admit that they made a bad investment decision. At the end of the tax year, and with the help of a deadline, investors overcome their initial reluctance and realize at least part of the losses on their stock portfolio for tax purposes. The closer the end of the year draws near, the greater the tax-induced selling of losers. This explains why selling activity by individual investors is especially prevalent on the last trading day of the year. Thus, tax rules are the reason why the January effect exists, but the disposition effect and loss aversion cause it to be as large as it actually is. As investors are less reluctant to realize gains than losses, they are less likely to have unrealized gains than unrealized losses in their stock portfolios at the end of the year. This would explain the absence of a general price increase for winners prior to the end of the year. Hence, the disposition effect and loss aversion are also able to explain why the December effect may be smaller than the January effect.

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selling activity on the last trading day of the year. Besides the disposition effect, they present another behavioral explanation. According to Dyl and Maberly [1992], the end of the tax year provides individual investors with the deadline they need to overcome their natural tendency to procrastinate. 3.8 Synthesis

Obviously, there is no single explanation for the turn-of-the-year effects. Each of the explanations presented in the preceding subsections is supported, at least to some extent, by empirical evidence. The different explanations are hard to separate, as they often have similar predictions. The possibility that all of the explanations play a role in the complete explanation of the turn-of-the-year effects in a given stock market cannot be excluded. However, the same holds for the possibility that none of them identifies the main cause of the effects. Still, some general inferences can be drawn.

First, a number of papers identify the trading activities of individual investors as the main cause of the rise in stock prices in December and January. Their trading activities may be inspired by tax rules, excess liquidity, behavioral idiosyncrasies or a combination of these and other factors. Whatever the reason, it induces individual investors to invest in low-price, small-firm stocks in January and in large-firm stocks at the end of December. Window dressing by portfolio managers may help to reinforce the size and significance of the turn-of-the-year effects, but is unlikely to be the main culprit, at least in the U.S. stock market.

Second, several characteristics, or factors, serve to identify the stocks likely to experience high turn-of-the-year returns. These factors include stock price, market capitalization and previous performance. Thus, in the U.S., the January effect seems to be limited to low-price stocks, increasing the likelihood that the effect is a consequence of microstructure biases in stock returns. However, although different characteristics are associated with different explanations for the turn-of-the-year effects, they are also closely correlated. Consequently, an examination of the relation between one or more of these characteristics and turn-of-the-year returns often provides little help when trying to single out the main explanation of the December and January effects.

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Section 4 - Hypotheses - 19 -

4 Hypotheses

One of the most fundamental propositions of financial economics is the efficient market hypothesis. According to Fama [1970], an efficient market is a market in which prices fully reflect all available information. Applied to a securities market, the efficient market hypothesis implies that all available information is taken into account in determining security prices. This includes information about historical prices and seasonal return patterns, such as the turn-of-the-year effects. Thus, a logical and conservative starting point is that there are no turn-of-the-year effects. In particular, daily returns on international stock market indices in the two months around the turn of the year do not differ significantly from average daily returns throughout the year.

H 4.1: The January effect is absent in the returns of all international stock indices. H 4.2: The December effect is absent in the returns of all international stock indices.

The January effect seems to be predominantly a small-firm effect in the U.S. stock market. However, there are large institutional differences between the U.S. stock market and other markets. These include the number of stocks that are traded, the average market capitalization of the firms whose stocks are traded, the applicable legislation, the system of corporate governance and the trading rules. These institutional differences, combined with the absence of an explanation why the January effect should be limited to small-firm stocks, imply that there are no convincing reasons to predict a January effect in non-U.S. stock markets. More than that, it is not even clear that the abnormal returns on U.S. small-firm stocks found in earlier data sets should be reproducible in later data sets. Hence the null hypothesis that turn-of-the-year effects, if found to be present in a U.S. or non-U.S. stock market, are unrelated to firm size.

H 4.3: If turn-of-the-year effects are present, they do not differ for indices of large-firm and small-firm stocks.

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do not provide a complete explanation for the turn-of-the-year effects, significant anomalous patterns around the turn of the calendar year should remain.

H 4.4: If seasonalities and turn-of-the-year effects are present in stock returns, they are unrelated to tax rules.

Keim [1983], Roll [1983a], Ritter [1988], Chen and Singal [2003] and many others show that the mean daily returns on small-firm stocks tend to be especially high on the first trading days of January. In addition, there are some indications that stock returns during the second half of December might be higher than during the first half, and that returns on some trading days might be higher than on others. For example, Lakonishok and Smidt [1988] and Ariel [1990] show that returns tend to be exceptionally high on the trading days before Christmas and New Year’s Day. However, as there is no sound and widely accepted economic theory that justifies systematic return differences between trading days in December, the absence of such temporal differences is an appropriate null hypothesis. If the returns are, in fact, highly concentrated on a small number of trading days, this constitutes evidence against the hypothesis that the apparently abnormal returns around the turn of the year can be explained by a proper adjustment for risk. That is, it seems unlikely that the risk of investing in stocks will suddenly rise on a small number of days.

H 4.5: If there is a January (December) effect, the higher January (December) returns are evenly spread over the entire month.

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In sum, many of the explanations of the turn-of-the-year effects predict a specific relation between turn-of-the-year stock returns and some stock characteristic, or factor. Consequently, any examination of the existence and nature of a relation between such a factor and turn-of-the-year returns will be an examination of the validity and relative importance of the explanation involved. However, due to the correlation between the different factors, it can be difficult to make a clear distinction between the different explanations. For example, the stocks of the smallest firms tend to have the lowest return over the previous year, the highest tax-loss selling potential, the highest level of individual-investor ownership and the lowest prices. Still, as the evidence is inconclusive for all the explanations, the appropriate null hypothesis is that none of the factors is related to the turn-of-the-year effects.

H 4.6: Turn-of-the-year effects in stock returns are not related to any factor potentially able to explain their existence and magnitude.

In an efficient market, information cannot be used to earn excess returns. This includes any information relating to the time of the day, week or year. However, finding exceptionally high returns around the turn of the year does not necessarily lead to the conclusion that markets are inefficient. The high returns can be perfectly normal, given the risk of investing in securities in this period of the year. In addition, high transaction costs can make it impossible to profit from any excess returns around the turn of the year that remain after an appropriate adjustment for risk has been made.

In the case that the January effect presents a profit opportunity because it gives rise to abnormal returns, it would disappear rapidly in an efficient market. In December, informed investors would bid up the prices of the stocks expected to rise in January in order to exploit the January effect. The resulting December effect would change into a November effect as soon as information comes to the market of abnormal December returns caused by investors trying to exploit the January effect. In an efficient market, this temporal movement in price increases would continue until each current stock price includes the present value of all future excess January returns of that stock and no January effect would remain. The same holds for any abnormal returns in December. An implication of this line of reasoning is that evidence of trading strategies persistently yielding abnormal profits around the end of the year is evidence against weak-form market efficiency as defined by Fama [1991].

H 4.7: No persistence in high turn-of-the-year returns.

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Section 5 - Empirical analysis - 22 -

5 Empirical analysis

5.1 Data

The daily returns of Dutch, English, French, German and American stock indices constitute the data set used in this and the following subsections. Roll [1983b] demonstrates that the choice of return compounding method can impart a serious bias in the estimated small-firm stock returns. Hence, daily returns are used and reported. Exceptionally, daily returns are compounded over longer time intervals by way of example. For each country, both a stock index dominated by large-firm stocks and a stock index including mainly small-firm stocks are examined. With these two categories of indices, size-based differences can be controlled for that would affect the results if indices spanning the entire market were used. Furthermore, a distinction between large-firm and small-firm indices makes it possible to single out the influence of firm size on turn-of-the-year stock returns. Using indices instead of individual stocks or portfolios of individual stocks reduces some of the problems related to portfolio formation, such as selection bias and survivorship bias.

The set of large-firm stock indices consists of the Dutch AEX, the English FT 30, the French CAC-40, the German DAX, the American Dow Jones Industrial Average. The Dow Jones and the FT 30 are price-weighted indices, while the other three are value-weighted. Unlike the other four indices, the DAX is a total return index by default. This means that its returns are calculated on the basis of the assumption that all dividends are reinvested in the index. The MSCI small-cap indices are selected to represent firm stocks for the four European countries. The S&P 600 is used as the U.S. small-firm stock index. The biases in small-small-firm stock returns observed by Blume and Stambaugh [1983] are unlikely to have a large effect on the results reported here, as all the small-firm indices are value-weighted.

Datastream is the source of all data on index returns. All days on which the stock exchange was closed due to a holiday but would have been open in the absence of such a holiday are deleted by hand from the data set. Table 5.1 presents descriptive statistics of the ten indices. These statistics include the mean and median daily return for each index, standard deviations, the number of observations, the start and end of the return series and whether the return data include dividends. Returns are reported in local currency and in percent. If both returns including dividends and returns excluding dividends are available starting on January 1st_{, 1983, the former are preferred over the latter. Otherwise, the goal of}

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Table 5.1 Descriptive statistics

This table presents descriptive statistics of the ten indices. AEX, CAC-40, DAX, DJIA and FT 30 indicate the Dutch, French, German, U.S. and U.K. large-firm index, respectively. MS NL stands for the MSCI Netherlands Small-Cap Index, MS FR for the MSCI France Small-Cap Index, MS GR for the MSCI Germany Small-Cap Index and MS UK for the MSCI United Kingdom Small-Cap Index. The first column contains the name of the index and the second the time period over which data are collected. The third column shows the number of observations. The fourth column indicates whether the index is a total return index (TRI) or a price index (PI). The fifth, sixth and seventh column show the mean daily return, the median daily return and the standard deviation of each index. Returns are in percent and in local currency. The last two columns contain the Bera-Jarque test statistic and the accompanying probability. The Bera-Jarque test statistic tests whether the index returns are drawn from a normal distribution.

Time period Number of

observations TRI or PI Mean daily return (%) Median daily return (%) deviation (%) Standard Bera-Jarque test statistic Probability

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distributed has to be rejected at the 0.01 significance level. Graphical representations of the return distributions (not reported here) show that they tend to be leptokurtic.2

Table 5.2 shows the correlation coefficients between the ten indices, calculated on the basis of price index return data. The time difference between the U.S. on the one hand and the European countries on the other hand reduces the meaning of the contemporaneous correlation between U.S. stock returns and the other stock indices. Hence, both the contemporaneous correlation and the correlation between U.S. returns and European returns on the next trading day are shown. All correlations are positive. In general, the stock indices are highly correlated. The correlations between pairs of large-firm stock indices tend to be higher than those between large-firm and small-firm stock indices and between pairs of small-stock indices. Still, the correlation coefficients of the small-firm stock indices are remarkably similar to those of the large-firm stock indices. Both the contemporaneous and the lagged U.S. returns are highly correlated with those of the other indices. The high correlation coefficients seem to indicate that there is at least some degree of integration between the stock markets of the five countries.

5.2 Seasonalities and turn-of-the-year effects 5.2.1 Methodology and results

The following equation can be used to test for seasonal patterns in stock returns like the January effect and the December effect:

= + = 12 1 , m im mt it it a D R

ε

i = 1,…,10. (5.1)

In equation 5.1, Rit is the return on stock index i for day t, m is the month of the year, Dmt is a seasonal

dummy taking a value of one in calendar month m and zero in all other months and it is an error term.

The coefficient aim is the average return on index i in month m. If there are no seasonal patterns in

stock returns, all aim’s are equal. If at least one of the monthly return averages differs significantly

from the others, seasonal patterns are present. According to hypotheses 4.1 and 4.2, the coefficients of the January and December dummy variables will be equal to those of the other months. A slightly more restrictive version of the hypothesis that there are no seasonal patterns is that all aim’s are jointly

equal to zero. Similarly, alternative and more restrictive versions of hypotheses 4.1 and 4.2 posit that

ai1 and ai12 are equal to zero, respectively. It should be noted that these adjustments bias the results

slightly in favor of rejecting hypotheses 4.1 and 4.2.

Table 5.3 presents the results of the ten regressions, one for each stock index. As the residuals of

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Table 5.2

Correlations between indices

This table shows the correlations between the ten indices included in the tests. AEX, CAC-40, DAX, DJIA and FT 30 indicate the Dutch, French, German, U.S. and U.K. large-firm index, respectively. MS NL, MS FR, MS GR and MS UK indicate the Dutch, French, German and U.K. MSCI small-cap index, respectively. The correlations are computed with daily local-currency returns over the period starting on January 1st_{, 1993 and ending June 2}nd_{, 2005. Price index return data are used when available, which}

means for all indices except the DAX. Due to non-overlapping trading hours, two different correlations are computed between European U.S. indices. The correlation between the contemporaneous returns appears in the lower-left triangle, the correlation between the European return and the U.S. return lagged one day appears in the upper-right triangle.

AEX MS NL CAC-40 MS FR DAX MS GR DJIA S&P 600 FT 30 MS UK

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Table 5.3

Daily stock index returns (t-statistics) by month and tests of equality of means

This table shows the mean daily return for each month and each index. Returns are in percent and in local currency. The observation period for each index is the period shown in table 5.1. The t-statistics for each mean daily return are in parentheses. The parametric F-test tests whether the daily returns for all months are equal to zero. The non-parametric Kruskal-Wallis [1952 and 1953] (K-W) test tests whether the twelve daily returns are drawn from the same population. For each test, test statistics are shown in parentheses. All t- and F-statistics are based on Newey-West [1987] heteroscedasticity and autocorrelation consistent standard errors.

Daily returns by month F-test K-W test

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initial regressions show signs of autocorrelation and heteroscedasticity, the method proposed by Newey and West [1987] is used to compute the standard errors for regression statistics. Given the violations of the assumptions of the classical linear regression model, a non-linear model would be more appropriate. However, in order to maintain consistency and comparability with previous findings, the use of such a regression model is deferred to later subsections. Table 5.3 shows that, for all indices, the mean daily return is positive in January. The mean January returns of the French and German MSCI small-cap indices are significantly different from zero at the 0.01 significance level. The daily return of the U.S. small-cap index is also statistically significant. In addition, the Dow Jones and the FT 30, two large-cap indices, have a significant mean daily return in January. The CAC-40 is the only large-firm stock index without a statistically significant December return. The only small-firm index with a significant December return is the S&P 600 index. Replacing the returns on the S&P 600 with the returns on the Russell 2000 index shows that the latter has a highly statistically significant and positive December return (not reported here), just as the U.S. large-firm index. Furthermore, the mean January return on the Russell 2000 is not significantly different from zero. The monthly stock price behavior of the MSCI small-cap 1750 index is virtually identical to that of the Russell 2000.

Surprisingly, mean returns tend to be significantly positive in April and significantly negative in September for most of the countries. For all small-cap indices and the FT 30, an F-test for equality of means rejects the hypothesis that the mean daily return for each month is equal to zero. Testing the hypothesis that the mean daily return for each month is equal to the overall mean daily return yields essentially similar results (not reported here). Instead of an F-test, the Kruskal-Wallis [1952 and 1953] test can be used to examine monthly seasonalities in stock returns. Unlike the former, the latter is a non-parametric test. As a consequence, its results are not sensitive to outliers. Kruskal and Wallis [1952] point out that another advantage of their test lies in the absence of restrictive assumptions about the distribution of stock returns. An F-test, on the other hand, assumes that stock returns follow an approximately normal distribution.

Table 5.3 shows that the Kruskal-Wallis test corroborates the conclusions about the presence of seasonal patterns based on the parametric test. All results that are already statistically significant become even more significant with the Kruskal-Wallis test. In contrast, the non-parametric test does not cause any of the F-test probabilities that are above the 0.10 significance level to drop to a value below this level. Hence, both tests lead to similar conclusions about the presence of seasonal patterns in the returns of the ten indices examined here. Similarly, Gultekin and Gultekin [1983] conclude that parametric and non-parametric tests performed on their data sample of international stock market indices lead to the same inferences about the presence of seasonal patterns. In sum, turn-of-the-year effects are present for most of the ten indices examined here and hypotheses 4.1 and 4.2, stating that they are never present, have to be rejected.

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rule out effective international arbitrage. Given that many of the explanations for the turn-of-the-year effects assume that effective arbitrage does not exist or is at least severely limited on individual national stock exchanges, and given the extensive evidence supporting these explanations, the assumption that effective international arbitrage is limited is not that far-fetched.

5.2.2 Comparison with previous studies

Assuming that January has 21 trading days and compounding daily, the French, German and U.S. mean monthly returns for small-firm stocks come down to about 5.0, 4.5 and 3.6 percent, respectively. These returns are quite high, although not adjusted for risk. The January returns are similar in size to the values reported by Gultekin and Gultekin [1983] for their sample of broad-based stock market indices (3.7, 3.1 and 5.1 percent, respectively). Agrawal and Tandon [1994] report returns of 4.4 and 2.2 percent for France and Germany. The comparability of these two studies with the results presented in this subsection is limited, however, as the nature of the indices included and the return calculation methods differ. In addition, they do not take dividends into account.

Assuming a 20-trading-day average for December, the Dutch, German, U.S. and English large-firm index returns are 3.2, 3.1, 2.1 and 2.5 percent, respectively. These returns are considerably higher than the December returns reported by Gultekin and Gultekin and Agrawal and Tandon for the broad stock indices examined in their papers. The difference is at least 0.6 percent, which makes it unlikely that dividends are the main explanation. The FT 30 December return is higher than the December returns on three broad English indices reported by Reinganum and Shapiro [1987]. The December return on the AEX is higher than the equally-weighted average 2.1 percent return for the 61 stocks traded on the Dutch stock exchange between 1966 and 1982 calculated by Van den Bergh and Wessels [1985]. Van der Sar [2003] reports a value-weighted daily return of 0.143 on a total return index of all the stocks listed on this stock exchange between 1980 and 1998. This is equal to a return of 2.9 percent for the entire month of December, and comparable to the 3.2 percent return on the AEX reported here. The Dow Jones December return of 2.1 percent is higher than the statistically significant 1.4 percent return reported by Lakonishok and Smidt [1988] for the 1952-1986 period. Thus, the high December returns on the Dow Jones seem to be persistent.

5.3 Heteroscedasticity and risk

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version of the ARCH(q) model introduced by Engle [1982]. Bollerslev et al. [1992] provide an overview of the literature on the ARCH(q) model and the GARCH(p,q) model, including practical applications of these models. Brooks [2004] and Bollerslev et al. [1992] point out that, in general, the GARCH(1,1) model will be sufficient to capture the clustering of volatility in the data.

Traditional equilibrium asset-pricing models, such as the capital asset pricing model of Sharpe [1964], Lintner [1965] and Black [1972] and the arbitrage pricing theory (APT) developed by Ross [1976], are based on the fundamental notion that risk and return are related. That is, risk-averse investors will require a higher return for bearing more risk. One way to incorporate risk is by including the conditional variance among the variables on the right-hand side of regression equation (5.1). This is the differentiating element of an extended version of the GARCH(1,1) model known as the M-GARCH(1,1) or GARCH-in-mean(1,1) model. The following regression equation results:

= + + = 12 1 2 _, m im mt it it it it a D b h u R i = 1,…,10 (5.2)

where Rit, aim and Dmt are defined as before, hit is the standard deviation of index i on day t and uit is an

error term. The error term is assumed to follow a normal distribution with zero mean and a time- varying conditional variance equal to h_it2. The coefficient of the variance can be interpreted as a risk premium. Adding a proxy for risk changes the interpretation of the aim’s. Now, they are the mean daily

returns per month after controlling for the relation between risk and return. The following regression equation serves to calculate the time-varying conditional variance:

2 1 , 2 1 , 2 − −

+

=

ic ir it iv it it

V

u

V

h

i = 1,…,10 (5.3)

where ui,t-1 is the error term in regression equation (5.2) of index i on trading day t-1 and hi,t-1 is the

variance of index i on trading day t-1. If the sum of the regression coefficients Vir and Viv is greater

than one, the variance is said to be stationary or explosive. As conditional variances are non-negative by definition, each of the regression coefficients Vic, Vir and Viv is required to be

non-negative. The quasi-maximum likelihood estimation method described by Bollerslev and Woolridge [1992] is used to calculate standard errors and t-statistics. Akgiray [1989] concludes that the GARCH(1,1) model fits very satisfactorily to a data sample of daily returns on the CRSP equally-weighted and value-equally-weighted stock indices. Milionis and Moschos [2000] point out that the M-GARCH(1,1) model can be used to test for weak-form market efficiency. They argue that the model is more general than the initial framework used by Fama [1970], as it accounts explicitly for risk. Berument and Kiymaz [2001 and 2003] use a similar model to that presented by equations (5.2) and (5.3) in the context of another seasonality in stock returns, to wit the day-of-the-week effect.

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return. A comparison of table 5.4 with table 5.3 shows that, instead of eliminating any seasonal anomaly, the GARCH-M model has the effect of making the seasonal patterns in stock returns even more explicit. After controlling for temporal variations in risk, all small-firm indices and the Dutch and English large-firm indices show exceptionally high returns in January. December returns that were significantly different from zero in table 5.3 continue to be so after controlling for risk, although the significance level decreases slightly. Furthermore, the results of the Wald test reported in table 5.4 show that seasonal patterns are absent only for the CAC-40 and the Dow Jones.

Table A.1 in the first appendix indicates that the M-GARCH(1,1) model is well-specified. All the variables reported in the table are highly statistically significant. Additionally, the coefficients are never negative and the sum of the coefficients on the first lag of the squared error (Vir) and the first lag

of the conditional variance (Viv) is smaller than one for all indices. Still, for each index, the sum of Vir

and Viv is very close to unity. This indicates that shocks to the conditional variance are highly

persistent. In sum, taking risk into consideration by including a variance term in the regression equation and allowing for temporal variations in variance does not alter any of the conclusions drawn in subsection 5.2. In addition, as long as the conditional variance is an accurate proxy for risk, one can conclude that risk is not responsible for the turn-of-the-year effects. Given the additional insights provided by the M-GARCH(1,1) model, it will be used in the subsequent subsections of this section unless noted otherwise.

5.4 Seasonalities and size

The results presented in table 5.3 and 5.4 seem to warrant the conclusion that large-firm and small-firm stock returns follow distinct seasonal patterns. December returns tend to be bigger for larger small-firms than for smaller firms, while the opposite holds for January. The results of a more formal test are presented in tables 5.5 and A.2. Table 5.5 shows the mean daily difference between the large-cap and the small-cap index for each month and each country. The results are obtained by replacing the index return Rit in regression equation (5.2) by the difference between the large-firm and the small-firm

index for each country. Due to a lack of return data with respect to the MSCI small-cap indices, all the return difference series start on the second trading day of 1993. The index data used exclude dividends, except for the data on the German large-firm index. The German return differences are obtained by subtracting the return on a small-firm price index from the return on a large-firm total return index. Thus, the results for the German stock market should be interpreted with caution.

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large-Table 5.4

Daily stock index returns (z-statistics) by month estimated with a GARCH(1,1) model

This table shows the mean daily return for each month and the coefficient of the conditional variance for each index. Returns are in percent and in local currency. The coefficient of the variance is adjusted in such a way that it is equal to the additional return (in percent) for a unit increase in the conditional variance. The z-statistics for each mean daily return and the conditional variance coefficients are in parentheses. The parametric Wald test tests whether the daily returns for all months are equal to zero. The F-statistics of the Wald test are shown in parentheses below the accompanying probability.

Daily returns by month Wald test

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Table 5.5

Difference in daily returns between large-cap and small-cap index (z-statistics) by month and a test of equality of means

This table shows the mean difference in daily returns by month between the large-firm and the small-firm index for each country. For all indices, return series starting on January, 4th_{1993 and ending on June, 2}nd_{2005 are used. Returns are in percent and in local currency. Except for the return on the DAX, dividends are not included in the}

calculation of index returns. For the DAX, which is a total return index by default, price index data are unavailable. “GARCH” indicates the coefficient of the conditional variance. This coefficient is adjusted in such a way that it is equal to the additional return (in percent) for a unit increase in the conditional variance. The z-statistics for each mean daily return difference and conditional variance are in parentheses. The parametric Wald test tests whether the daily return differences for all months are equal to zero. The F-statistics of the Wald test are reported in parentheses below the accompanying probability.

Daily return differences by month Wald test

Country GARCH Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec Prob. France 0.024 -0.219*** -0.089 -0.006 -0.002 -0.044 0.015 0.104* -0.117* -0.020 0.009 0.013 0.073 0.004 (0.747) (-3.585) (-1.529) (-0.126) (-0.040) (-0.786) (0.272) (1.772) (-1.860) (-0.335) (0.154) (0.202) (1.150) (2.473) Germany 0.017 -0.148*** -0.005 0.018 0.005 -0.015 0.044 0.101** 0.040 0.005 0.093* 0.134** 0.121** 0.008 (0.744) (-2.701) (-0.094) (0.374) (0.082) (-0.307) (0.894) (2.211) (0.763) (0.117) (1.947) (2.408) (2.343) (2.311) Netherlands 0.033 -0.118*** -0.058 -0.059* -0.037 -0.030 -0.011 0.004 0.003 -0.026 0.074** 0.006 0.005 0.022 (1.412) (-4.174) (-1.273) (-1.703) (-0.755) (-0.728) (-0.248) (0.097) (0.060) (-0.517) (1.998) (0.135) (0.108) (2.038) U.K. -0.000 -0.132*** -0.030 -0.043 -0.029 0.006 0.030 0.054 -0.020 -0.032 0.052 0.059 -0.022 0.042 (-0.130) (-3.240) (-0.726) (-1.156) (-0.638) (0.141) (0.703) (1.301) (-0.483) (-0.728) (1.256) (1.291) (-0.486) (1.847) U.S. -0.029 0.052 0.035 0.017 0.052 -0.007 -0.058 0.090* -0.066* -0.053 0.078* 0.057 -0.032 0.019 (-0.674) (1.309) (0.854) (0.420) (1.297) (-0.158) (-1.313) (1.946) (-1.649) (-1.296) (1.855) (1.440) (-0.620) (2.083) * Significant at the 0.10 level.

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firm index and the small-firm index is negative and statistically significant in January. The difference between large-firm December returns and small-firm December returns is positive and significant for Germany. For each market, the hypothesis that the mean daily return difference is equal for each month has to be rejected at conventional significance levels on the basis of a Wald test. Hence, after controlling for risk as captured by the conditional variance, a January effect is present in the return differences for four of the five countries and hypothesis 4.3, stating that the turn-of-the-year effects are unrelated to firm size, has to be rejected.

Surprisingly, there is no sign of a January effect in the U.S. return differences. This deviation from previous results, such as those reported by Banz [1981], Keim [1983], Reinganum [1983] and, more recently, Booth and Keim [2000], may be due to methodological differences. These differences concern, for example, the start and end of the observation period and the composition of the size-based portfolios whose returns are examined. The S&P 600 (Dow Jones) may not be a close proxy for the small-firm (large-firm) decile used to calculate return differences in most of the papers on the January effect in the U.S. stock market. The lower limit of $300 million for inclusion in the S&P 600 is rather high when compared to the average market capitalization $28 million for the decile of the smallest firms traded on the New York Stock Exchange, as reported by Keim [1989]. However, replacing the S&P 600 by the Russell 2000 index or the MSCI small-cap 1750 index yields essentially the same results (not reported here). Hence, it seems that, in the U.S., the January effect is limited to the smallest firms. An alternative explanation for this finding might be the instability of the size premium through time, which is already noted by Banz [1981]. Of course, it is also possible that arbitrage activities are responsible.

5.5 Seasonalities and tax rules

Hypothesis 5.5 predicts that seasonal patterns in stock returns, if present, are unrelated to tax rules. This hypothesis has the following implications. First, seasonalities and turn-of-the-year effects should be present in the stock markets of countries with and without capital gains tax. Second, short-term holding periods and turn-of-the-year effects should be unrelated. Lastly, a difference between the end of the calendar year and the end of the tax year should not be accompanied by a shift of the turn-of-the-year effects to the months surrounding the end of the calendar year. The discussion in this subsection assumes that the Dutch, English, French, German and U.S. stock markets are not perfectly integrated. In particular, in all these markets, the influence of foreign investors, subject to other tax regimes, on domestic stock prices is assumed to be of minor importance.