1 HELSINKI SCHOOL OF ECONOMICS Department of Accounting and Finance
IDIOSYNCRATIC VOLATILITY, MARKET VOLATILITY, AND EXPECTED STOCK MARKET RETURNS IN EUROPE
Finance Master’s Thesis Pasi Hyttinen Spring 2007
Approved by the Council of the Department ____ / ____ 20____ and awarded the grade_______________________________________________________
2 Helsinki School of Economics Master’s Thesis Pasi Hyttinen
IDIOSYNCRATIC VOLATILITY, MARKET VOLATILITY, AND EXPECTED STOCK MARKET RETURNS IN EUROPE
ABSTRACT In this study I take a look at whether stock market returns can be predicted using measures of market risk and idiosyncratic risk. Idiosyncratic risk is defined as the variance of the firm-specific return that cannot be explained by classical capital asset pricing model (CAPM). I find that in two out of six countries examined market risk and average idiosyncratic risk combined significantly help to explain future stock market returns while results for the rest of the countries point into the same direction but are not statistically significant. Neither market risk nor idiosyncratic risk alone has a significant relation with future stock market returns in any of the countries. The results, while in contrast to traditional financial theory, are in line with some previous empirical studies.
DATA This study covers all major Western European stock markets. Countries included in the study are France, Germany, Italy, Spain, Switzerland, and the UK. I use Morgan Stanley Capital International (MSCI) country indexes within each of the countries as a proxy for the stock market portfolio. The sample period ranges from January 2001 through June 2006.
ACKNOWLEDGEMENTS I am grateful to my thesis instructor Matti Suominen for useful guidance. Especially, I thank Veli-Pekka Heikkinen for many helpful comments and advice along the way.
KEYWORDS Market risk, idiosyncratic risk, risk-return relationship, expected returns
3
TABLE OF CONTENTS:
TABLE OF CHARTS AND TABLES: ......................................................................4 1.
INTRODUCTION................................................................................................5 1.1. MOTIVATION FOR THE STUDY...............................................................................7 1.2. OBJECTIVES OF THE STUDY ..................................................................................9 1.3. STRUCTURE OF THE STUDY .................................................................................10
2.
LITERATURE REVIEW .................................................................................11 2.1. ON THE RELATION BETWEEN MARKET RISK AND RETURN ...................................11 2.2. ON THE RELATION BETWEEN IDIOSYNCRATIC RISK AND RETURN........................12
3.
DATA ..................................................................................................................16
4.
METHODOLOGY ............................................................................................19 4.1. HIGH-FREQUENCY MEASURES OF IDIOSYNCRATIC RISK AND MARKET RISK ........20 4.2. GARCH(1,1) MODEL IN ESTIMATING MARKET RISK...........................................23
5.
HYPOTHESES AND TESTS ...........................................................................26 5.1. HYPOTHESES ......................................................................................................26 5.2. PREDICTIVE REGRESSIONS AND REGRESSION ASSUMPTIONS ...............................27
6.
ANALYSIS AND EMPIRICAL RESULTS ....................................................30 6.1. OVERVIEW OF THE RESULTS ...............................................................................30 6.2. ANALYSIS OF VOLATILITY SERIES ......................................................................30 6.1.1. France ........................................................................................................33 6.1.2. Germany.....................................................................................................35 6.1.3. Italy ............................................................................................................37 6.1.4. Spain ..........................................................................................................39 6.1.5. Switzerland.................................................................................................41 6.1.6. UK ..............................................................................................................43 6.3. ANALYSIS OF REGRESSION RESULTS ...................................................................45 6.2.1. France ........................................................................................................47 6.2.2. Germany.....................................................................................................48 6.2.3. Italy ............................................................................................................48 6.2.4. Spain ..........................................................................................................49 6.2.5. Switzerland.................................................................................................50 6.2.6. UK ..............................................................................................................51
7.
SUMMARY AND CONCLUSIONS ................................................................53
REFERENCES: .........................................................................................................56 APPENDIX A .............................................................................................................60 APPENDIX B .............................................................................................................62 APPENDIX C .............................................................................................................64
4
Table of Charts and Tables:
TABLE 1.
Yearly Number of Constituents………………………………………16
FIGURE 1.
Volatility Series in France……………………………………………33
TABLE 2.
Descriptive Statistics in France………………………………………34
FIGURE 2.
Market Variance in France…………………………………………...35
FIGURE 3.
Volatility Series in Germany…………………………………………35
TABLE 3.
Descriptive Statistics in Germany……………………………………36
FIGURE 4.
Market Variance in Germany………………………………………...37
FIGURE 5.
Volatility Series in Italy……………………………………………...37
TABLE 4.
Descriptive Statistics in Italy……………….………………………...38
FIGURE 6.
Market Variance in Italy……………………………………………...39
FIGURE 7.
Volatility Series in Spain……………………………………………..39
TABLE 5.
Descriptive Statistics in Spain………………………………………..40
FIGURE 8.
Market Variance in Spain…………………………………………….41
FIGURE 9.
Volatility Series in Switzerland………………………………………41
TABLE 6.
Descriptive Statistics in Switzerland…………………………………42
FIGURE 10. Market Variance in Switzerland……………………………………...43 FIGURE 11. Volatility Series in UK……………………………………………….43 TABLE 7.
Descriptive Statistics in UK………………………………………….44
FIGURE 12. Market Variance in UK………………………………………………45 TABLE 8.
Predictive Regression Results in France……………………………..47
TABLE 9.
Predictive Regression Results in Germany…………………………..48
TABLE 10.
Predictive Regression Results in Italy………………………………..49
TABLE 11.
Predictive Regression Results in Spain………………………………50
TABLE 12.
Predictive Regression Results in Switzerland………………………..50
TABLE 13.
Predictive Regression Results in UK………………………………...51
5
1. Introduction Compared to the first three years of this decade, both market risk and average idiosyncratic risk1 are currently at substantially lower levels in all major Western European stock markets2. Further, the trends in the measures of market risk and idiosyncratic risk have been analogous in all of the countries examined in the study. More specifically, after being in a moderate level for 2001-2002, the measures of market risk and idiosyncratic risk peaked around mid-2003, and since then have been steadily declining until recently. After the global stock market plunge and the consequent turbulence from April through May 2006, however, the risk measures show some indications of starting to increase again. Although the main trends in the volatility series have been similar, the absolute levels of market risk and idiosyncratic risk differ somewhat across the countries examined. However, in recent years the differences have become smaller, as the absolute risk levels have declined in Europe. This could also be partly due to the increased integration within Western European stock markets. Further, within the countries examined correlations between the measures of market risk and idiosyncratic risk seem to be quite strong indicating that times of high market risk coincide with times of high average firm-specific risk. At the same time, stock markets throughout Western Europe have experienced two longer trends. From 2001 through the beginning of 2003 there was a long period of bear market, only to be followed by even longer bull market from 2003 until today. From the countries examined, these trends were the most (less) severe in Germany (the UK), where the MSCI Germany (MSCI UK) stock market index fell -60.4% (38.7%) in local currency terms from January 2001 through March 2003, but after 1
In this study market risk simply refers to variance of the stock market return while idiosyncratic risk refers to variance of the firm-specific return that cannot be explained by classical capital asset pricing model (CAPM). 2 The stock markets examined in this study include France, Germany, Italy, Spain, Switzerland, and the UK. Together these countries comprise 87% of the total domestic market capitalization of Western Europe as of the end of 2005. Western Europe is here considered to include the following stock exchanges: Athens Exchange, BME Spanish Exchanges, Borsa Italiana, Deutsche Borse, Euronext, Irish SE, London SE, Luxembourg SE, OMX, Oslo Bors, Swiss Exchange, and Wiener Borse. (Source: World Federation of Exchanges)
6 hitting the bottom has increased +138.5% (+87.5%) until the end of June 2006. These developments have been similar in all of the major Western European stock markets. Characteristic of the long (and steady) bull market, which began in March 2003 (in October 2002 in Spain) and still continues, has been low volatility both at firm and at market level. For active portfolio managers (consider e.g. stock pickers) who are seeking for alpha by over- and underweighting stocks relative to their benchmark index low average idiosyncratic risk of today naturally means more difficult investment environment. For example, in an extreme case (that is purely hypothetical) when average idiosyncratic risk is equal to zero, there would be no sources of active returns when all stocks would yield just the systematic return. In a more reasonable setting, when average idiosyncratic risk goes down, active managers on average have to take bigger bets if they want to be as “active” as before. So it is valuable for active portfolio managers to learn about the developments of average idiosyncratic risk to be able to adjust the magnitude of their active bets to reflect the desired level of active risk in their portfolios3. From an individual investors’ viewpoint, the above discussion could be interpreted so that even when assuming imperfect capital markets it might be more reasonable to invest in index-tracking funds (that have lower management fees) at least in times of low average idiosyncratic volatility when the sources of active returns are scarce. This is especially true in a situation when one believes that his/her portfolio manager is not taking into account the diminished level of return dispersion appropriately. When looking at the predictability of stock market returns by the measures of market risk and idiosyncratic risk, I find no significant time-series relation between either market risk and future stock market return or idiosyncratic risk and future stock market return in any of the countries investigated. Interestingly, however, I find that in two out of six stock markets studied, market risk and idiosyncratic risk combined significantly help to explain future stock market returns. More specifically, I find that in a multiple predictive regression, the measure of market risk is significantly positively related to future stock market return while the measure of average 3
It is, however, another question how this type of “adjustment” in active bets should be made and this study does not go into that.
7 idiosyncratic risk is significantly negatively related to future stock market return both in France and in the UK. Further, also in all other countries examined the results are similar, though not statistically significant, which however might be due to the fact that the sample period in this study is just too short. Above results clearly contradict the traditional financial theory, which suggests a positive relation between market risk and expected stock market returns, while saying that idiosyncratic risk should have no predictive power in explaining future stock market returns. At the same time the obtained results are interesting because they give empirical evidence (at least for some countries) that we can better predict future stock market returns when we instead of concentrating solely on market risk take into account both market risk and idiosyncratic risk.
1.1. Motivation for the study The relation between risk and return in stock markets is interesting and current topic both for academics and practitioners. At the moment, especially the role of idiosyncratic risk in asset pricing is an area of research and debate for academics. At the same time, many practitioners are exposing themselves to higher companyspecific risks when seeking sources of active returns4, which is not surprising as capital markets become more and more efficient. So far, most of the empirical studies concerning the role of idiosyncratic risk, both academic and practitioner-oriented, have used U.S. stock market data. To my best knowledge, the European empirical studies in this area are scarce. Further, the existing studies in Europe have examined either one specific country (see, e.g., Angelidis and Tessaromatis, 2005) or Europe as a whole (see, e.g., Kearney and Poti, 2004), without any comparisons between different countries.
4
An interesting study by Cremers and Petajisto (2006) on active portfolio management indicates that in the past (1990-2003) active managers in the U.S. have been able to create excess returns (over their benchmark index) by bearing firm-specific idiosyncratic risk.
8 Over time, numerous research papers have tried to empirically explore the relation between risk and return in the cross-section of stocks as well as across time. However, the results are still mostly confounding and contradictory. Further, while defining return is quite straightforward, there is some debate on how risk should be measured5. Standard asset pricing models suggest a positive relationship between risk and expected return. For example, according to capital asset pricing model (CAPM) introduced by Sharpe (1964) and Lintner (1965), there is a positive relation between risk and expected return, but only the market risk should matter to investors. This is because investors are assumed to be able to diversify away the idiosyncratic risk of individual companies by holding well-diversified portfolios. Many of the early studies have consequently focused on the relation between market risk and expected returns. In practise, however, some investors might be obliged to hold insufficiently diversified portfolios for various reasons (see, e.g., Malkiel and Xu, 2002). Further, some investors might unintentionally fail to diversify enough, even if they wanted to, or when diversifying their portfolios they might just use simple “rules of thumb” (see, e.g., Bernartzi and Thaler, 2001). Thus, it might be the case that idiosyncratic risk is priced in equilibrium. Therefore investors, whose portfolios are less than perfectly diversified, might demand a risk premium to bear idiosyncratic risk. This would imply a positive relation between idiosyncratic risk and expected return across stocks. Further, there also may be a positive time-series relation between idiosyncratic risk and expected stock market return. If such a trade-off existed, it would be possible to use average idiosyncratic volatility, or some other measure of aggregate idiosyncratic risk, to forecast stock market returns. Goyal and Santa-Clara (2003) propose various explanations why such a trade-off might be in place. One explanation is that nontraded assets, which risk is assumed to be positively correlated with the risk of traded assets, add background risk to investors’ traded-portfolio decisions. In times when the risk of non-traded assets increases, investors would be less willing to hold traded risky assets, and demand higher risk premium to hold market portfolio. Another explanation would be viewing equity in a firm as a call option on its assets, following 5
In this study, risk represents the variance of returns. Throughout the study, I use such concepts as risk, variance, and volatility interchangeably.
9 Black and Scholes (1973) and Merton (1974). When average idiosyncratic risk of stocks goes up, this is benefiting average stockholder at the expense of average debt holder, and we would expect to see a positive relation between average idiosyncratic risk and stock market return (which is the value-weighted average return on all stocks). On the other hand, some authors have even argued that there should be a negative time-series relation between idiosyncratic risk and expected stock market returns. For example, Guo and Savickas (2005) argue that when considering idiosyncratic risk as reflecting the dispersion of opinion among investors in Cao et al. (2005) setting6, we should indeed find a negative relation between idiosyncratic risk and excess stock market return. However, a positive relation between risk and expected returns seems also intuitively appealing both across stocks and across time. If rational investors under perfect information are maximizing their utility, the required risk premium of an asset should increase whenever the risk (volatility) of the asset increases and investors have also other investment opportunities to choose from. Thus, if rational investors for some reason cannot perfectly diversify their portfolios in the stock market, we should find a positive relation between the total risk (which consists of both market risk and idiosyncratic risk) and expected returns across stocks. Similarly, in times when the total risk is higher in the stock markets, we would expect investors to demand a higher equity risk premium.
1.2. Objectives of the study Purpose of this study is to empirically explore the time-series relation between average idiosyncratic risk and future stock market returns within the main Western European stock markets. I investigate whether average idiosyncratic risk (alone or together with market risk) helps explain return on stock market portfolio across time. In other words, I study whether or not investors on average demand higher risk 6
Cao et al. (2005) study limited market participation and argue that when there is a lot of uncertainty (or dispersion of opinion) in the market, this in fact leads to a lower, not higher, equity uncertainty premium.
10 premium to hold stock market portfolio in times when average idiosyncratic risk is higher. My intention, however, is not to study the pricing of idiosyncratic risk in the cross-section of stock returns. At the same time I explore the dynamics of idiosyncratic volatility and stock market volatility in all major Western European stock markets from January 2001 through June 2006. I take a look at the trends, variability, and persistence of the measures of idiosyncratic volatility and market volatility and compare them between the countries examined. I also investigate how strongly the measures of idiosyncratic volatility and stock market volatility are correlated with each other within the countries, i.e. whether the times of high idiosyncratic volatility coincide with the times of high market volatility or not. This study contributes to the existing empirical literature in the area of disaggregated volatility studies by providing results from all major Western European stock markets. To my best knowledge, this is the first study that includes comparisons of the developments and the role of idiosyncratic volatility across different stock markets in Europe. In case differences are found between countries, this type of information would prove to be valuable for investors, many of whom are today considering Western Europe as one unified region. In addition, to my best knowledge, this is the first study in this research area to use generalized autoregressive conditional heteroskedasticity (GARCH) model to estimate market risk.
1.3. Structure of the study The remaining of the study is structured as follows. In Section 2, I take a brief look at the existing literature on the relation between risk and expected returns. Section 3 provides with the description of the data set, while Section 4 introduces the methodologies employed. In Section 5, I present the hypotheses and related tests. All the main empirical results are presented and analysed in Section 6. Finally, Section 7 provides with a brief summary and conclusions.
11
2. Literature review The relation between risk and return has been empirically studied both in the crosssection of stock returns and across time (or intertemporally). Earlier empirical studies tended to take into account only the systematic risk (or market risk), whereas many of the more recent papers study also the role of idiosyncratic risk in explaining returns. While there exists more evidence for a trade-off between risk and return across stocks in a given time period, the question, whether an intertemporal relation between risk and return exists, still remains more or less open. Although the existing literature I discuss in this section includes both cross-sectional and time-series studies, the purpose of this study is to investigate explicitly the time-series relation between idiosyncratic risk and expected stock market returns.
2.1. On the relation between market risk and return Early attempts to empirically investigate the cross-sectional relation between risk and return, or early empirical tests of CAPM, (see, e.g., Blume and Friend, 1973; Fama and MacBeth, 1973) seem to support a positive relation between market risk and expected returns across stocks. However, these early studies consistently reject CAPM because the estimates of risk-free interest rate and market risk premium from empirical regressions tend to be untenable. Moreover, empirical studies from Basu (1977), Banz (1981), and Rosenberg, Reid and Lanstein (1985), among others, present evidence that the market beta cannot capture all the dimensions of risk, as predicted by CAPM. Later, Fama and French (1992) present updated evidence that also size, leverage, B/M-ratio, and P/E-ratio, in addition to market beta, help to explain expected returns in the cross-section of stocks. The early studies seem to support a positive relation between risk and return in the cross-section of stocks, although CAPM might be unable to satisfactorily explain it. On the other hand, as Roll (1977) has pointed out, it is practically impossible to test CAPM, because the market portfolio can never be defined completely.
12 There are also numerous early studies, which investigate the time-series (or intertemporal) relation between risk and return. In his exploratory study, Merton (1980) proposes “preliminary” models to estimate expected market return in a context, where risk is positively related to expected returns. Pindyck (1984) demonstrates that the increases in variance of stock returns can explain much of the decline in stock prices from 1965-1981 in his framework. French, Schwert and Stambaugh (1987) take a look at the time-series relation between risk and expected returns in the U.S. stock market from 1928-1984. They find a positive relation between expected excess stock market return and predictable volatility, employing a GARCH model, under conditional normality, in volatility estimation. Baillie and DeGennaro (1990) use GARCH with a conditional student t density, which they argue is more appropriate, and set aside the results of French et al. (1987) by showing no evidence for significant intertemporal relation between risk and return. More recently, Glosten, Jagannathan and Runkle (1993) present results, which support a negative time-series relation between risk and expected returns. On the other hand, Whitelaw (1994) offers empirical evidence for a positive relation between a lagged volatility measure and future expected returns. Moreover, Scruggs (1998) contradicts the results of Glosten et al. (1993) by refined model specification and suggests a positive partial relation between market variance and market risk premium. In spite of numerous empirical attempts to establish intertemporal relation between market risk and return, there is no clear consensus whether investors on average demand a higher risk premium in times when stock market is more risky (or volatile).
2.2. On the relation between idiosyncratic risk and return In last few years, many authors (see, e.g., Bernartzi and Thaler, 2001; Malkiel and Xu, 2002) have suggested that in real world investors do not always diversify perfectly and thus both systematic and idiosyncratic risk might matter to investors. This clearly contradicts CAPM, which implies that all idiosyncratic risk can and will be diversified away by rational investors. Next, I briefly discuss some recent studies, which take this viewpoint into account. In fact, these studies are more relevant for my purposes than those, which investigate merely the relationship between market risk and expected returns. The amount of studies, which look into the relation between
13 idiosyncratic risk and expected returns, is still quite limited and the results are ambiguous. An important study from Campbell, Lettau, Malkiel and Xu (2001) empirically explores the volatility of the U.S. stocks from 1962-1997 in market, industry, and firm levels. Campbell et al. (2001) find that while market and industry level volatilities have remained quite stable in their sample period, the average firm level volatility has approximately doubled. Correspondingly to the rise in firm level volatility, the average correlation between individual stocks has decreased substantially. Campbell et al. (2001) do not study the pricing of idiosyncratic risk across stocks or its relevance in forecasting future stock market returns, but apparently their study rekindled the interest in the role of idiosyncratic risk. Studies from Malkiel and Xu (2002) and Ang, Hodrick, Xing and Zhang (2006) investigate the pricing of idiosyncratic risk in the cross-section of stock returns. Malkiel and Xu (2002) argue that investors, who are not able to hold the market portfolio, might require a risk premium for bearing idiosyncratic risk. They find empirical evidence to support their view, that there is a positive relation between idiosyncratic risk and stock returns in the cross-section of stocks. On the other hand, Ang et al. (2006) argue to the contrary. They find that stocks with high idiosyncratic risk have abysmally low average returns. The most relevant studies from my viewpoint are those, which investigate the intertemporal relationship between idiosyncratic risk and future stock market returns. Empirical results from these studies are also far from unanimous as can be perceived from the following. In the U.S., Goyal and Santa-Clara (2003) study the time-series relation between idiosyncratic volatility and stock market returns for a sample period from 1962-1999. They find that idiosyncratic volatility helps to forecast future stock market returns, while the relation between market volatility and future stock market return is negative, but insignificant. Both these results are clearly violating CAPM. Bali, Cakici, Yan and Zhang (2005) take a critical look at the Goyal and Santa-Clara (2003) study, and argue that their findings do not hold when extending the sample period till
14 2001. Further, Bali et al. (2005) show that results of Goyal and Santa-Clara (2003) are mainly driven by a “small-firm bias” in the measure of idiosyncratic volatility, and partly due to a liquidity premium. Bali et al. (2005) cannot establish any significant relation between risk and future stock market return for their full sample period from 1962-2001. Guo and Savickas (2005) find that market risk and idiosyncratic risk combined help to explain future expected returns, while individually both have only negligible explaining power. Guo and Savickas (2005) also argue that the reason, why many early studies have not been able to find a positive relation between market risk and expected return (as predicted by CAPM), might be because they have omitted idiosyncratic risk as an explanatory variable. In Europe, using extensive EMU equity market data, Kearney and Poti (2004) investigate the volatility of stock returns similarly to Campbell et al. (2001) in firm, industry, and market levels. Kearney and Poti (2004) find that both market and idiosyncratic volatilities have trended upwards in EMU countries from 1974-2004. Converse to Goyal and Santa-Clara (2003) and similarly to Guo and Savickas (2005), Kearney and Poti (2004) find that lagged stock market volatility is positively related to stock market returns while lagged idiosyncratic volatility is negatively related to stock market returns. Using UK stock market data from 1980-2003, Angelidis and Tessaromatis (2005) study the properties of idiosyncratic volatility and the role of idiosyncratic volatility in explaining future stock market returns. They find that idiosyncratic volatility (either value-weighted or equally weighted) is not related to value-weighted stock market returns. However, the idiosyncratic volatility of small capitalization stocks, which is highly correlated with the average stock variance measure used by Goyal and Santa-Clara (2003), is positively related to the future small-large equity premium. Thus, Angelidis and Tessaromatis (2005) propose that idiosyncratic volatility has predictive power for future returns but only among small capitalization stocks. Following Campbell et al. (2001), Maukonen (2004) studies volatility of publicly listed Finnish firms using a disaggregated approach. He finds that idiosyncratic volatility has increased also in Finland from 1970-2001, while there is no significant trend in market volatility series in this period.
15 In Japan, Hamao, Mei and Xu (2003) explore the dynamics of market volatility and idiosyncratic volatility in extreme conditions, i.e. during the decade-long recession in 1990s. Conversely to Campbell et al. (2001) findings in the U.S., Hamao et al. (2003) find that in Japan the firm-specific volatility has in fact decreased during the recession period. They propose that this might be due to the group protection, which is characteristic of Japanese equity markets, and may have actually deepened the recession while hindering capital allocation between healthy and unhealthy companies.
16
3. Data In this study I explore six Western European stock markets, each of them separately. The sample period ranges from January 2001 through June 2006. I obtain the data from Datastream, Bloomberg, and MSCI-Barra. The countries represented in the study are France, Germany, Italy, Spain, Switzerland, and the UK. Within each country, I use the Morgan Stanley Capital International (MSCI) country index as a proxy for the stock market portfolio. I chose to use MSCI country indexes as they are widely used both among academics and practitioners worldwide and also freely available for academic purposes. From all Western European countries I include only those, in which the proxy for the market portfolio (i.e. country-specific MSCI index) is comprised of at least 25 constituent companies throughout the sample period. This criterion, of course, leads to exclusion of smaller European stock markets from the study (even if the number of listed companies within a country has been above 25 throughout the sample period). Nevertheless, I wanted to include only such countries, where the proxy for the market portfolio is at least quite well diversified and thus the results can be expected to be more meaningful. In addition, I wanted to be consistent when using MSCI country indexes for all countries in the sample (i.e. not to use local indexes as a market proxy, e.g. OMX Helsinki for Finland) and thus facilitate comparability among countries as well as future research in this area.
Country index
2001
2002
2003
2004
2005
2006H1
average
median
MSCI France MSCI Germany MSCI Italy MSCI Spain MSCI Switzerland MSCI UK
53 48 40 33 36 113
54 50 42 27 38 136
54 47 42 30 36 128
59 47 39 28 31 141
57 47 41 32 35 152
62 50 39 33 38 155
57 48 41 31 36 138
56 48 41 31 36 139
Table 1.
Yearly number of constituents. Number of constituent companies in each of the country portfolios as of the end of previous year. For example, market portfolio for France for year 2001 consisted of 53 companies, which is the number of constituents in MSCI France in the end of December 2000.
17 I construct the “market portfolio” for a country j for year y from the constituent companies in the country-specific MSCI index as of the end of year y-1 (i.e. market portfolios are revised yearly). Above table shows the last-year-end number of constituents in each of the country indexes. Not surprisingly, MSCI UK is the most diversified and is comprised of well above 100 companies each year, while MSCI Spain has the least constituent companies every year. However, on average all of the indexes have more than 30 constituent companies and for all indexes the number of constituents is above 25 throughout the sample period. Sample period in the study ranges from January 1, 2001 to June 30, 2006 (i.e. 66 months). The sample period is rather short compared to the earlier studies in this area. A longer sample period would enable the analysis of longer-term trends in the volatility measures, which would be interesting. In addition, the methods used in the study would probably give more reliable results, when applied to a longer time series of data. However, the length of the sample period is due to data availability from MSCI-Barra, from where I obtained the historical firm-level constituent lists for MSCI country indexes, which are not publicly available7. Nevertheless, as I run regressions between monthly stock market returns and monthly risk measures, a 66month sample period should be long enough to discover the possible relation, if there were any. For estimating monthly market risk with GARCH, I use monthly MSCI index-level return data, which dates back to the end of 1987. The index-level return data for MSCI indexes is freely available since the inception. I obtain the firm-level daily total return data (RI) as well as firm-level daily market capitalization data (MV) from Datastream. I obtain both total return data and market capitalization data in each country’s local currency, because I investigate the riskreturn relationship only within the countries. The total return data from Datastream is adjusted for dividends, stock-splits, and other corporate actions. To verify the reliability of Datastream total return data and market capitalization data, I also conducted random cross-checks with data from Bloomberg.
7
I am grateful to MSCI-Barra for providing me with these data.
18 As a risk-free interest rate, I use the 3-month Treasury rate for the UK, and the 3month Euribor rate for the remaining countries (also for Switzerland, although it is not a euro-country). To get daily risk-free interest rate, I simply divide the prevailing yearly risk-free interest rate on each day by the number of trading days (260) in a year. All risk-free interest rate data I obtain from Bloomberg. I calculate daily value-weighted stock market return for each country from daily firmlevel total return data and daily firm-level market capitalization data. I don’t use the index-level return data on MSCI country indexes except from GARCH estimation, where longer history of monthly return data is needed. I obtain the MSCI index-level return data for GARCH estimation from Datastream. However, the correlations between self-calculated market returns and MSCI-provided index-level returns (both daily and monthly) are close to unity.
19
4. Methodology As my intention is to investigate the intertemporal relation between idiosyncratic risk and stock market returns, I have to define the way to estimate idiosyncratic risk. Naturally, both idiosyncratic return and idiosyncratic risk are model-specific, because they are measured relative to “systematic” return predicted by the model in use. Earlier studies have employed different methods to estimate idiosyncratic risk. For example, Campbell et al. (2001) use a simplified market model under the assumption that betas of all securities equal unity. Campbell et al. (2001) simply calculate idiosyncratic return on a firm as the difference between that firm’s return and market return and then use this firm-specific excess return to calculate idiosyncratic volatility. Goyal and Santa-Clara (2003) use equally weighted average stock volatility, which actually gauges total risk, as a proxy for idiosyncratic risk. Bali et al. (2005) use various alternative idiosyncratic and total risk measures to uncover the possible relation between risk and return. These examples give evidence, that there is no clear consensus at the moment, how the idiosyncratic risk should be measured. However, if a significant relation between idiosyncratic risk and future stock market return really exists, it should be captured regardless of the precise estimation process. Below, I introduce the measures that I use in this study to estimate average idiosyncratic volatility and stock market volatility. I define idiosyncratic return on a firm as the return, which cannot be explained by classical CAPM. Then, I construct measures of average idiosyncratic risk on market level based on these firm-specific excess returns. The estimation process is somewhat different than for example in any other study mentioned above, but the idea is very similar to that in Campbell et al. (2001), with the difference that I use CAPM instead of their simplified market model8. 8
In fact, I also estimate idiosyncratic volatility series under the simplified market model (i.e. assuming unit betas) similarly to Campbell et al. (2001). This procedure is exactly the same as the one described below with the only exception that betas are assumed to be equal to unity for all firms. The results under simplified market model are however very similar to the CAPM method and therefore I report these results only in the appendixes (see, Appendix A and Appendix B).
20
I use two different methods to estimate stock market volatility. First, I proceed similarly to many earlier studies (see, e.g., Merton, 1980; Schwert, 1989; Andersen et al., 2003; Campbell et al., 2001) and treat monthly volatilities as observable ex post. This means that I calculate monthly volatility measure based on within-month daily stock market returns and then use this as a proxy for next month’s volatility. This method is fairly straightforward and avoids estimating complex parametric models. However, it yields an estimate of market risk, which is very volatile and might e.g. react sometimes too strongly to current events. Therefore I also estimate stock market volatility with generalized autoregressive conditional heteroskedasticity (GARCH) model. When estimating market volatility with GARCH, I use historical market returns with fairly long dataset to find the parameters that I then use to estimate stock market risk from monthly (not daily) return data for the sample period. Consequently, GARCH estimates of market volatility are somewhat more stable compared to the other method. Below, I use letters i, j, and m to refer to a firm, to a country, and to market portfolio, correspondingly. R and ER refer to return and excess return (over risk-free rate of return), respectively, on individual firm or on market portfolio. Return on market portfolio is calculated as market capitalization weighted average return on all firms within the market portfolio for a given period. Further, w denotes market capitalization based weight, t denotes month, and d denotes day.
4.1. High-frequency measures of idiosyncratic risk and market risk I refer to the monthly measures of idiosyncratic risk and market risk, which are based on daily returns, as high-frequency risk measures9. I first define the way how I estimate average idiosyncratic volatility. I start with a classical CAPM, which implies that the following equation holds for an individual firm:
9
It is probably more conventional to refer to intra-day (i.e. tick-by-tick) data as “high-frequency”. In this study, I however refer to risk measures that are based on daily return data as “high-frequency” measures to distinguish them from GARCH market risk measure, which is based on monthly return data.
21
Ri ,d = r f ,d + β im ( Rm ,d − r f ,d ) + ε i ,d ,
(1)
where Ri,d is return on firm i, rf,d is risk-free rate of return, βim is beta of firm i relative to market, Rm,d is return on market portfolio, and εi,d is idiosyncratic return on firm i and d denotes day. From equation (1) we see that the return on a firm can be decomposed into three different components: risk-free component, market component, and idiosyncratic component. When defining idiosyncratic risk, it is the idiosyncratic component of the return that we are interested in. Now, it is more convenient to consider excess returns over the risk-free rate of return and we can rewrite equation (1) as: ERi ,d = β im ERm,d + ε i ,d .
(2)
This is an expression, where the excess return on a firm is decomposed into market component and idiosyncratic component, but the content is precisely the same as in equation (1). I assume that the expression of equation (2) holds within different countries j. This basically means that I don’t consider different countries together to constitute a unified “Western European stock market”. On the other hand, this enables me later to compare the empirical results across countries. When estimating monthly average idiosyncratic risk (variance), I start by running the regression of equation (2) from daily data for all individual firms i within each market j. From these regressions I get as residuals the daily firm-specific idiosyncratic returns. By running the regression for the full period of data for each firm, I implicitly assume that firm-specific betas relative to market remain constant over time. I then calculate the average squared idiosyncratic return within a market based on the daily firm-specific idiosyncratic returns. I denote equally weighted average squared idiosyncratic return as EASIRj,d and value-weighted average squared idiosyncratic return as VASIRj,d, where d stands for day and j stands for country. Formally, I calculate EASIRj,d and VASIRj,d as follows:
EASIR j ,d =
1 n (ε ij ,d ) 2 , ∑ n i =1
and
(3)
22
n
VASIR j ,d = ∑ wij ,d −1 (ε ij ,d ) 2 ,
(4)
i =1
where n is the number of firms in market portfolio of country j on day d, and wij,d-1 is market capitalization based weight of firm i within country j on day d-1. Then, based on EASIRj,d and VASIRj,d, I construct measures of average monthly idiosyncratic variance, both equally weighted and value-weighted, within each market j. Formally I proceed as follows: Dt
EAIV j ,t = ∑ EASIR j ,d ,
and
(5)
d =1 Dt
VAIV j ,t = ∑ VASIR j ,d ,
(6)
d =1
where Dt is the number of trading days in month t. EAIVj,t and VAIVj,t are then, correspondingly, equally weighted and value-weighted measures of monthly average idiosyncratic variance. Next, I show how I define the high-frequency measure of market risk. Here, I proceed in a similar fashion with Campbell et al. (2001). Simply, based on daily stock market excess returns, I estimate market variance MktVarj,t for a country j in month t as follows: Dt
MktVar j ,t = Var ( ERmj ,t ) = ∑ ( ERmj ,d ) 2 ,
(7)
d =1
where Dt is again the number of trading days in month t. Compared to the more widely used market volatility measures (e.g. 60-month rolling standard deviation) the measure of equation (7) is naturally very volatile, since it uses only the daily returns within a single month. However, the advantage is that a long history of return data is not needed to calculate this measure. Further, the quick
23 responsiveness to current events might also be an advantage of this risk measure when it is used to explain next month’s stock market return. Now, equations (5), (6), and (7) give the high-frequency measures of equally weighted average idiosyncratic variance, value-weighted average idiosyncratic variance, and market variance, correspondingly. Strictly speaking, these are not variance measures, because I do not demean the daily returns. However, as pointed out in Goyal and Santa-Clara (2003), for this type of high-frequency measures, the impact of subtracting the mean return is minimal. Also French et al. (1987) and Scruggs (1998) support this view and therefore I feel free to pass the issue here.
4.2. GARCH(1,1) model in estimating market risk In addition to the high-frequency measure of market risk introduced in equation (7), I use GARCH(1,1) model to estimate monthly market risk. I first estimate GARCH(1,1) parameters for each country from historical monthly stock market return data. Then I use these parameters to calculate monthly GARCH(1,1) estimates of market risk for the sample period. The GARCH(1,1) model was originally introduced by Bollerslev (1986). A good overview of the GARCH(1,1) model is provided in Hull (2006), but refer e.g. to Greene (1993) for a more comprehensive discussion of the more general GARCH(p,q) model. The latter part of the notation GARCH(1,1) simply means that when estimating variance for period t, I use the latest (t-1) observations of return innovation and variance rate. This is the simplest way to estimate the model, but also the most widely used in practise. Following the presentation in Hull (2006), the equation for GARCH(1,1) can be written as follows:
σ t2 = γVL + αu t2−1 + βσ t2−1
(8)
where σ2t is estimate of the variance of returns for period t, γ is the weight assigned to long-term return variance VL, and α and β are the weights assigned to period t-1 squared return u2t-1 and to period t-1 variance rate σ2t-1, respectively. Further, the
24 weights γ, α and β in equation (8) must sum up to unity, and for a stable GARCH(1,1) process α + β < 1, so that the weight γ assigned to the long-term variance rate VL becomes positive. I use maximum likelihood method to estimate the parameters (γ, α, β) in equation (8). While assuming normally distributed monthly returns (ut), I find such parameters that maximize the logarithm of the probability density function of the return distribution. In other words, this maximizes the probability of the historical return data occurring. In practise, I do this by iterating the parameter values until the maximum of the log likelihood function is found. In all empirical GARCH(1,1) estimations in this study, I use such restrictions for the parameters that γVL ≥ 0.001, α ≥ 0.15 and β ≥ 0.510. I first estimate the parameters of equation (8) within each country from historical monthly stock market returns. The period I use to estimate the parameters ranges from the end of 1987 until the end of 2000 (i.e., since the inception of the MSCI country indexes until the beginning of the sample period). Then, I use these parameters to calculate monthly GARCH(1,1) estimates of market variance for the sample period from monthly stock market returns. This method implicitly assumes that the parameters in equation (8) remain stable over time and that investors can use them to estimate market risk. However, I find this to be a reasonable assumption. Both the high-frequency measure of equation (7) and GARCH(1,1) of equation (8) respond to changes in market volatility, but one important difference is that while I use daily returns to calculate monthly high-frequency estimate, monthly GARCH(1,1) estimate is based on monthly returns. This makes it possible that when there is a lot of variation in daily stock market returns within a month and the high-frequency measure reacts by giving a high estimate of market risk for that month, GARCH(1,1) estimate need not react (so noticeably) if these daily observations to some extent offset each other at the end of the month. On the other hand, in a month when stock markets show small but consistent positive (or negative) daily returns day after day and consequently the high-frequency market risk estimate is low, GARCH(1,1) 10
Because the restrictions have to be chosen (more or less) arbitrarily, I estimated the model also without restrictions as well as tried out some different limiting values and found out that while giving very similar results in most of the cases, in some cases the restrictions described above help to obtain more reasonable estimates for the parameters (γ, α, β).
25 measure should go up instead. It is important to keep in mind this difference when interpreting the empirical results later. Having said that, it would be surprising to find out that the contemporaneous correlation between the market risk measures of equations (7) and (8) is very low. Later in the discussion, I refer to the GARCH(1,1) estimate of market variance as GARCH_MktVar.
26
5. Hypotheses and tests As stated above, my intention is to investigate whether idiosyncratic risk alone or together with market risk can help to explain future stock market returns across Western European countries. Next, I present my null hypotheses, which are based on traditional financial theory (possibly except from Hypothesis 3). Then, I introduce the predictive regressions that I exploit to empirically test my hypotheses, i.e. to empirically explore the relationship between the different risk measures and future stock market returns. I also introduce a few tests, which can be utilized to test the assumptions behind the predictive regressions.
5.1. Hypotheses I make the following three hypotheses. Hypothesis 1 and Hypothesis 2 are simply based on traditional financial theory. However, it is not so clear what financial theory suggests about Hypothesis 3, so it is based on the above discussion on the relation between total risk (which consists of both market risk and idiosyncratic risk) of stocks and expected stock market returns.
Hypothesis 1 There is a positive time-series relation between market variance and excess return on the stock market portfolio.
Hypothesis 2 There is no significant time-series relation between average idiosyncratic variance and excess return on the stock market portfolio.
Hypothesis 3 Market variance and average idiosyncratic variance combined can significantly help to explain future excess return on the stock market portfolio.
27
5.2. Predictive regressions and regression assumptions I explore the time-series relation between the different risk measures and future stock market returns by regressing the realized stock market returns on the lagged measures of volatility. Here I proceed similarly to Goyal and Santa-Clara (2003) and Bali et al. (2005). I use the most common method, ordinary least squares (OLS), to estimate the parameters. In the following regressions, I use monthly excess returns on the stock market portfolio and monthly volatility measures. I run the regressions separately for each of the countries j. Formally, the predictive regressions can be written as follows: ERmj ,t +1 = α + βMktVar j ,t + et +1
(9)
ERmj ,t +1 = α + β GARCH _ MktVar j ,t + et +1
(10)
ERmj ,t +1 = α + βEAIV j ,t + et +1
(11)
ERmj ,t +1 = α + βVAIV j ,t + et +1
(12)
ERmj ,t +1 = α + β1 MktVar j ,t + β 2 EAIV j ,t + et +1
(13)
ERmj ,t +1 = α + β1 MktVar j ,t + β 2VAIV j ,t + et +1
(14)
ERmj ,t +1 = α + β1GARCH _ MktVar j ,t + β 2 EAIV j ,t + et +1
(15)
ERmj ,t +1 = α + β1GARCH _ MktVar j ,t + β 2VAIV j ,t + et +1
(16)
The fitted values of the above regressions give the expected stock market excess return conditional on different measures of volatility. Equations (9) and (10) are both related to Hypothesis 1. If Hypothesis 1 were to hold, these regressions should give significantly positive estimates for the parameter β. Equations (11) and (12) in turn relate to Hypothesis 2, which states that in these regressions the estimates for β should not be significantly different from zero. Finally, the multiple regressions in equations (13), (14), (15), and (16) are related to Hypothesis 3. However, it is not clear what kind of estimates (positive or negative, or zero) we should expect in advance for the parameters β1 and β2 in these multiple regressions. This is especially the case when we can expect that there might be multicollinearity present, i.e. when the explanatory variables might be highly correlated with each other.
28
In theory, the assumptions behind the linear regression model are quite stringent, but are occasionally disregarded (or at least not reported) in practise. Following Greene (1993) these assumptions are: 1. Functional form: yi = α + βxi + ei 2. Zero mean of the disturbance: E[ei] = 0, for all i. 3. Homoskedasticity: Var[ei] = σ2, a constant, for all i. 4. Nonautocorrelation: Cov[ei, ej] = 0, i ≠ j. 5. Uncorrelatedness of regressor and disturbance: Cov[xi, ej] = 0, for all i and j. 6. Normality: ei ~ N[0, σ2] In this study, I assume that all these assumptions are fulfilled in the predictive regressions described above. I also explicitly test normality and homoskedasticity by Jarque-Bera test and autoregressive conditional heteroskedasticity (ARCH) test, respectively (see, Greene, 1993). Full results for these tests for each of the predictive regressions and each of the countries are reported in the Appendix C, but I also briefly comment the results later in the text. As stated above, when running linear regressions we assume that the disturbance terms (ei) are normally distributed. Jarque-Bera test is a measure of departure from normality for these disturbances based on sample skewness and kurtosis. The test statistic is defined as follows:
JB =
n⎛ kurt 2 ⎞ ⎟ ⎜⎜ skew 2 + 6⎝ 4 ⎟⎠
(17)
where JB is the test statistic, n is the number of observations, skew is the sample skewness, and kurt is the sample kurtosis. The null hypothesis is that the data (i.e. disturbances) is normally distributed and the statistic follows a chi-squared distribution with two degrees of freedom.
29 Further, when running the predictive regressions, we assume that there is no heteroskedasticity present. This means that in time-series regressions we assume that the variance of the disturbances (et) remains constant through time. ARCH test can be used for testing heteroskedasticity in time-series regressions and it is conducted in its simplest form (i.e. with one lag) by running the following regression:
et2 = α + βet2−1 + ε t
(18)
where et is the disturbance term for period t for the original predictive regression, here equations (9) to (16). Now, the test statistic is calculated based on equation (18) as a product of the number of observations and the R squared from this auxiliary regression, or formally nR2. In equation (18) there is just one lagged disturbance as an explanatory variable in the auxiliary regression, but there could be more. The null hypothesis is that there is no heteroskedasticity present, i.e. the homoskedasticity assumption is fulfilled. In “large” samples the test statistic has a chi-squared distribution with degrees of freedom that correspond to the number of lagged disturbance terms (here one).
30
6. Analysis and empirical results In this section I present the main empirical findings and results of this study. After a brief overview, I first explore the dynamics of idiosyncratic volatility and market volatility series in the sample countries through 2001-2006. Then, I analyse the results from the predictive regressions between stock market return and the different risk measures and compare the results between countries.
6.1. Overview of the results Currently both market risk and idiosyncratic risk are in low levels within the sample countries, and the trends in the volatility series have been similar across the countries during the sample period. Further, correlation between market risk and idiosyncratic risk seems to be high suggesting that periods of high market risk coincide with periods of high idiosyncratic risk. Both market risk and idiosyncratic risk series are quite persistent across time. In France and in the UK, market risk and idiosyncratic risk combined help to explain future stock market returns. Results for the remaining countries are similar, though not statistically significant. I find that neither market risk nor idiosyncratic risk alone can significantly explain stock market returns in any of the countries examined.
6.2. Analysis of volatility series Below, I present the high-frequency volatility series and GARCH(1,1) estimated market volatility series and corresponding statistics and correlations for each country at a time. For all of the countries I present a figure showing the series of equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV) and market variance (MktVar) as well as corresponding 10-month simple moving averages (SMA) for the period from 2001-2006. I also report the
31 descriptive statistics and cross-correlations for the variance series (EAIV, VAIV,
MktVar, GARCH_MktVar) and for corresponding annualized standard deviation series (SDEW, SDVW, SDMkt, SDGARCH_MktVar) as well as for market excess return series (ERMkt) for the full sample period each country at a time. I report the numbers for annualized standard deviation series as well, because in my opinion it is easier to get an understanding of the magnitude of risk when annualized standard deviation is used instead of monthly variance. For each country, I also present the high-frequency measure of market variance (MktVar) versus the GARCH(1,1) estimated market variance series GARCH_MktVar for the full sample period. To facilitate comparisons between countries, I report all the described figures for each of the countries at a time, although I may not always analyze all the figures for every country. The trends in the EAIV, VAIV and MktVar series, which can be observed from the moving averages, are very similar within all of the countries studied. In general, the volatility series were in a moderate level for 2001-2002, then substantially increased peaking around the mid-2003, and have been steadily declining since then. In Germany and in Switzerland, however, the moving average of equally weighted idiosyncratic variance was in a high level throughout the first half of the sample period, which is mostly due to stronger-than-average reaction of EAIV series to 9/11 terrorist attacks in these countries. Further, in Italy the decline in equally weighted idiosyncratic variance SMA is not as steady after mid-2003 as it is in other countries, because there is a sizeable peak in the EAIV series in December 2003, which is due to a country-specific event and I will come back to this later. In the very end of the sample period, from May through June 2006, both market volatility and idiosyncratic volatility have again increased somewhat throughout the countries, which can be seen both from the actual volatility series and from their moving averages. This increase in market risk and idiosyncratic risk is obviously due to the global instability in the stock markets in April and May 2006. The absolute levels of idiosyncratic risk and market risk differ somewhat between countries. On average, market risk and value-weighted idiosyncratic risk have been the lowest in Italy and the highest in France and Germany through the whole sample period. Equally weighted idiosyncratic variance has been the lowest in Spain and the highest in the UK. In general, the series of value-weighted idiosyncratic variance and
32 market variance seem to have been roughly in the same levels and to go pretty much hand-in-hand, especially in Switzerland and in the UK, while the equally weighted measure of idiosyncratic variance has been in a higher level. In all of the countries, except from Spain, both MktVar and VAIV series have been below EAIV series throughout the whole sample period. In every country, including Spain, the EAIV series has been above the VAIV series for the full sample period, which gives support to the notion that idiosyncratic risk on average is higher for small-cap stocks than for large-cap stocks. The cross-correlations between EAIV, VAIV, MktVar and GARCH_MktVar series are quite high within every country. As stated earlier, this is an indication that there is multicollinearity present in the multiple predictive regressions, which makes it more difficult to interpret the regression results later. One would expect that the correlation between equally weighted and value-weighted idiosyncratic variance is higher than either the correlation between equally weighted idiosyncratic variance and market variance or the correlation between value-weighted idiosyncratic variance and market variance, because EAIV and VAIV try to capture the same thing. This is indeed the case in all countries except from Italy, where the correlation between VAIV and
MktVar is higher than the correlation between EAIV and VAIV. So even if MktVar and VAIV series visually seem to go hand-in-hand in most of the countries as stated above, EAIV and VAIV series are actually more correlated with each other. Further, the EAIV, VAIV, MktVar and GARCH_MktVar series seem to be quite persistent over time as measured by the first-order autocorrelation in these series. This applies for every sample country and is in line with the general notion in the financial literature that volatility tends to be persistent over time. The first-order autocorrelation in EAIV, VAIV and MktVar series is the highest in France and Germany and the lowest in Italy. In addition, it seems to be the case in most of the countries that idiosyncratic volatility is more persistent than market volatility as measured by the high-frequency measure MktVar. However, the difference is not immense in every country, if it is even present. Market volatility, when measured by GARCH_MktVar, is generally more persistent than idiosyncratic volatility measures (EAIV or VAIV) or highfrequency market risk measure MktVar.
33 What comes to the high-frequency measure of market variance (MktVar) versus GARCH(1,1) estimate of market variance (GARCH_MktVar), it seems that the highfrequency measure is much more volatile in all of the countries. This is especially the case in Switzerland and in the UK. However, it should not be surprising, because
MktVar is based on daily return data while GARCH_MktVar is based on monthly returns. On the other hand, correlation between MktVar and GARCH_MktVar is fairly high in all of the countries and the trends in both of the measures are similar.
6.2.1. France
As can be observed from Figure 1 below, the developments in the volatility series in France are similar to the other countries examined. The moving averages of all volatility series peak in March 2003, and have been in low levels since the beginning of 2004. For the full sample period, the SMA of EAIV series runs above the SMAs of both VAIV and MktVar series. The effect of 9/11 is easily observable in all volatility series in September 2001. In October 2002 there is a sizeable peak in EAIV series, which must be due to smaller companies, because it is much less observable in VAIV series. Further, the beginning of the war in Iraq clearly shows in the MktVar series, while not so clearly in EAIV and VAIV series, in March 2003.
0.040
0.018
0.035
0.016
0.014
0.030
0.012
0.025 0.010
0.020 0.008
0.015 0.006
0.010 0.004
0.005
0.000 Jan-01
0.002
Jul-01
Jan-02
Jul-02 EAIV
Figure 1.
Jan-03
Jul-03 VAIV
Jan-04
Jul-04
Jan-05 MktVar
Jul-05
Jan-06
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV SMA
Jan-03
Jul-03
Jan-04
VAIV SMA
Jul-04
Jan-05
Jul-05
Jan-06
MktVar SMA
Volatility series in France. Figure on the left plots equally weighted (EAIV) and valueweighted (VAIV) average idiosyncratic variance and market variance (MktVar) for MSCI France from 2001-2006, calculated as in equations (5), (6) and (7). Figure on the right plots the corresponding 10-month simple moving averages (SMA).
Table 2 below shows that in France, EAIV series has been more volatile than the other variance series, as measured by its standard deviation, while the GARCH_MktVar has
34 been the least volatile. EAIV has also been in a higher level on average than VAIV or
MktVar, which could already be seen from Figure 1. All volatility series have been highly persistent in France, as measured by the first-order autocorrelation, as shows from the last column of the Panel A in the table below. Further, the correlations between the volatility series have been high during the sample period.
PANEL A: statistics MSCI France
mean
median
min
max
st.dev.
skewness
kurtosis
AR1
EAIV SDEW
0.0074
0.0053
0.0019
0.0353
0.0063
2.1313
5.6941
0.7040
0.2751
0.2489
0.1516
0.6507
0.1095
1.2148
1.2920
0.8025
VAIV SDVW
0.0051
0.0040
0.0013
0.0179
0.0038
1.4560
1.8684
0.6795
0.2304
0.2107
0.1260
0.4636
0.0854
0.8784
0.0322
0.7987
MktVar SDMkt
0.0045
0.0026
0.0006
0.0251
0.0055
2.3148
4.9129
0.6729
0.2010
0.1741
0.0880
0.5485
0.1131
1.4896
1.7730
0.7623
GARCH_MktVar SDGARCH_MktVar
0.0034
0.0030
0.0025
0.0077
0.0011
2.0305
4.5379
0.7676
0.2006
0.1903
0.1718
0.3032
0.0292
1.6492
2.7401
0.7917
ERMkt
-0.0017
0.0104
-0.1674
0.1309
0.0557
-0.4204
0.8074
0.0342
PANEL B: correlations MSCI France EAIV SDEW
EAIV
SDEW
VAIV
SDVW
MktVar
SDMkt
GARCH_MktVar SDGARCH_MktVar
0.9739
1
VAIV SDVW
0.9321
0.9417
1
0.9080
0.9626
0.9705
1
MktVar SDMkt
0.8557
0.8404
0.8310
0.8080
1
0.8686
0.8872
0.8551
0.8634
0.9729
1
GARCH_MktVar SDGARCH_MktVar
0.8625
0.8481
0.7370
0.7496
0.7484
0.7738
1
0.8551
0.8544
0.7471
0.7675
0.7468
0.7813
0.9961
1
ERMkt
-0.1737
-0.2046
-0.3215
-0.2950
-0.3176
-0.3189
-0.1037
-0.1111
Table 2.
ERMkt
1
1
Descriptive statistics in France. Descriptive statistics and cross-correlations for equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV), high-frequency market variance (MktVar) and GARCH(1,1) estimated market variance (GARCH_MktVar) series and for corresponding annualized standard deviation series as well as for monthly excess market return series in France for the period 2001-2006. AR1 refers to first-order autocorrelation.
The figure below plots the MktVar and GARCH_MktVar series in France for the full sample period. The high-frequency measure seems to have much more variability than the GARCH(1,1) measure, which also shows from the standard deviations that are 0.55% and 0.11%, correspondingly. The trends are generally similar in both of the market risk measures and the contemporaneous correlation is 0.75. Interestingly, in May 2005 there is an increase in GARCH_MktVar series that does not really show in
MktVar series. This, however, is due to small positive consecutive daily returns which together comprise a sizeable positive monthly return. As discussed above, this leads to an increase in GARCH_MktVar, while the high-frequency MktVar does not respond much. The parameter estimates of equation (8) for GARCH_MktVar series are γ = 0.27, α = 0.15, and β = 0.58. Further, the long-term monthly variance rate (VL) is
35 0.37% corresponding to an annualized standard deviation rate of 21.0%, which seems reasonable.
0.030
Figure 2. 0.025
Market variance in France. Highfrequency monthly market variance (MktVar) and GARCH(1,1) estimated monthly market variance (GARCH_MktVar) for MSCI France for the period 2001-2006.
0.020
0.015
0.010
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Jan-04
Jul-04
GARCH_MktVar
Jan-05
Jul-05
Jan-06
MktVar
6.2.2. Germany
Figure 3 below plots the EAIV, VAIV and MktVar series and the corresponding moving averages for the period from 2001-2006 in Germany. The developments in the volatility series, which can best be observed from the moving averages, have been very similar to those in France. Equally weighted idiosyncratic variance has been above both VAIV and MktVar series for the full sample period, when looking at the moving averages. EAIV series reacts more strongly to 9/11 in Germany than in other countries, excluding only Switzerland. Therefore, also the SMA of equally weighted average idiosyncratic variance is in a high level for the first half of the sample period.
0.040
0.020
0.018
0.035
0.016 0.030
0.014 0.025
0.012
0.020
0.010
0.008
0.015
0.006 0.010
0.004 0.005
0.000 Jan-01
0.002
Jul-01
Jan-02
Jul-02 EAIV
Figure 3.
Jan-03
Jul-03 VAIV
Jan-04
Jul-04
Jan-05 MktVar
Jul-05
Jan-06
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV SMA
Jan-03
Jul-03
Jan-04
VAIV SMA
Jul-04
Jan-05
Jul-05
Jan-06
MktVar SMA
Volatility series in Germany. Figure on the left plots equally weighted (EAIV) and value-weighted (VAIV) average idiosyncratic variance and market variance (MktVar) for MSCI Germany from 2001-2006, calculated as in equations (5), (6) and (7). Figure on the right plots the corresponding 10-month simple moving averages (SMA).
36
As shows from Table 3 below, all volatility series have been quite persistent in Germany as measured by their first-order autocorrelation. Also the cross-correlations between the volatility series are high for the full sample period, as in all other countries examined.
PANEL A: statistics MSCI Germany
mean
median
min
max
st.dev.
skewness
kurtosis
AR1
EAIV SDEW
0.0092
0.0075
0.0023
0.0371
0.0077
1.7340
2.8786
0.6988
0.3088
0.3000
0.1644
0.6675
0.1232
1.0302
0.4744
0.7964
VAIV SDVW
0.0049
0.0037
0.0010
0.0178
0.0037
1.5525
2.1613
0.6875
0.2276
0.2121
0.1107
0.4626
0.0822
0.9264
0.1974
0.7850
MktVar SDMkt
0.0049
0.0025
0.0006
0.0260
0.0056
2.0729
4.0351
0.6649
0.2134
0.1729
0.0840
0.5585
0.1163
1.2453
0.9659
0.7515
GARCH_MktVar SDGARCH_MktVar
0.0044
0.0035
0.0026
0.0134
0.0022
2.2785
5.6906
0.7183
0.2234
0.2045
0.1781
0.4015
0.0489
1.7650
3.0185
0.7705
ERMkt
-0.0013
0.0104
-0.2595
0.2054
0.0722
-0.5125
2.5054
0.0482
EAIV
SDEW
VAIV
PANEL B: correlations MSCI Germany EAIV SDEW
SDVW
MktVar
SDMkt
GARCH_MktVar SDGARCH_MktVar
0.9823
1
VAIV SDVW
0.8836
0.9026
1
0.8847
0.9267
0.9852
1
MktVar SDMkt
0.7386
0.7478
0.8520
0.8236
1
0.7469
0.7810
0.8393
0.8429
0.9737
1
GARCH_MktVar SDGARCH_MktVar
0.6091
0.6312
0.6644
0.6694
0.7656
0.7692
1
0.6271
0.6578
0.6820
0.6967
0.7736
0.7901
0.9928
1
ERMkt
-0.1631
-0.1995
-0.2088
-0.2294
-0.2902
-0.3072
-0.0887
-0.0797
Table 3.
ERMkt
1
1
Descriptive statistics in Germany. Descriptive statistics and cross-correlations for equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV), high-frequency market variance (MktVar) and GARCH(1,1) estimated market variance (GARCH_MktVar) series and for corresponding annualized standard deviation series as well as for monthly excess market return series in Germany for the period 2001-2006. AR1 refers to first-order autocorrelation.
Figure 4 below plots the high-frequency market risk measure MktVar and GARCH(1,1) estimated market risk measure GARCH_MktVar in Germany. MktVar series is again more volatile than GARCH_MktVar series. Further, there seems to be more variability in GARCH_MktVar series than in any other country, and correspondingly the standard deviation of GARCH_MktVar (0.22%) is the highest among the countries examined. Contemporaneous correlation between MktVar and
GARCH_MktVar is 0.77, and the two series seem to go pretty nicely hand-in-hand. Similarly to France, there is an increase in GARCH_MktVar measure in May 2005, which is not observable in the high-frequency measure. The parameter estimates for
GARCH_MktVar series are γ = 0.27, α = 0.15, and β = 0.58, and the long-term monthly variance rate (VL) is 0.39%, corresponding to an annualized standard deviation of 21.7%.
37
0.030
Figure 4. 0.025
Market variance in Germany. High-frequency monthly market variance (MktVar) and GARCH(1,1) estimated monthly market variance (GARCH_MktVar) for MSCI Germany for the period 2001-2006.
0.020
0.015
0.010
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Jan-04
Jul-04
GARCH_MktVar
Jan-05
Jul-05
Jan-06
MktVar
6.2.3. Italy
From the Figure 5 below we see that in Italy monthly market volatility peaks in September 2001 being 2.74% for that month. For some reason it seems that in Italy, the market volatility increased more than in other countries due to 9/11, but on the other hand the increase in market risk was short-lived. Another interesting peak in volatility in Italy is the one in the equally weighted average idiosyncratic variance measure in December 2003, when EAIV series reached its peak at 3.45%. This peak was in turn due to Parmalat plunging more than -60% on two consecutive trading days in 19th and 22nd of December 2003, and shows how country-specific events can also drastically impact the observed average idiosyncratic volatility, at least the equally weighted measure.
0.040
0.014
0.035
0.012
0.030 0.010 0.025 0.008 0.020 0.006 0.015 0.004 0.010
0.002
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV
Figure 5.
Jan-03
Jul-03 VAIV
Jan-04
Jul-04
Jan-05 MktVar
Jul-05
Jan-06
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV SMA
Jan-03
Jul-03
Jan-04
VAIV SMA
Jul-04
Jan-05
Jul-05
Jan-06
MktVar SMA
Volatility series in Italy. Figure on the left plots equally weighted (EAIV) and valueweighted (VAIV) average idiosyncratic variance and market variance (MktVar) for MSCI Italy from 2001-2006, calculated as in equations (5), (6) and (7). Figure on the right plots the corresponding 10-month simple moving averages (SMA).
38
As can be observed from the seventh column in Table 4 below, the excess kurtosis of
EAIV and MktVar series has been especially high in Italy, compared to the other countries examined. If the measures were normally distributed the excess kurtosis would be zero, and positive excess kurtosis basically means that their distributions have thicker tails. On the other hand, the excess kurtosis of GARCH_MktVar series is less than two in Italy being around five in all other countries.
PANEL A: statistics MSCI Italy
mean
median
min
max
st.dev.
skewness
kurtosis
AR1
EAIV SDEW
0.0064
0.0048
0.0019
0.0345
0.0053
2.9558
11.8659
0.3499
0.2616
0.2408
0.1515
0.6430
0.0927
1.6470
3.7763
0.5326
VAIV SDVW
0.0035
0.0027
0.0009
0.0138
0.0024
1.9895
5.3194
0.5466
0.1950
0.1792
0.1057
0.4076
0.0609
1.1234
1.3304
0.6605
MktVar SDMkt
0.0031
0.0016
0.0003
0.0274
0.0043
3.3744
14.9630
0.4675
0.1662
0.1374
0.0643
0.5737
0.0991
1.7794
3.6659
0.6500
GARCH_MktVar SDGARCH_MktVar
0.0043
0.0039
0.0033
0.0071
0.0009
1.5937
1.6785
0.7815
0.2246
0.2172
0.2002
0.2924
0.0234
1.4436
1.1746
0.7975
ERMkt
0.0009
0.0100
-0.1526
0.1036
0.0486
-0.9087
1.3414
-0.0406
EAIV
SDEW
VAIV
PANEL B: correlations MSCI Italy EAIV SDEW
SDVW
MktVar
SDMkt
GARCH_MktVar SDGARCH_MktVar
0.9745
1
VAIV SDVW
0.7654
0.8413
1
0.7728
0.8668
0.9834
1
MktVar SDMkt
0.5928
0.6625
0.8749
0.8159
1
0.6117
0.7157
0.8797
0.8619
0.9591
1
GARCH_MktVar SDGARCH_MktVar
0.5328
0.6192
0.7343
0.7369
0.6857
0.7303
1
0.5339
0.6238
0.7375
0.7440
0.6823
0.7308
0.9988
1
ERMkt
-0.1910
-0.2357
-0.3355
-0.3274
-0.4516
-0.4325
-0.1695
-0.1702
Table 4.
ERMkt
1
1
Descriptive statistics in Italy. Descriptive statistics and cross-correlations for equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV), high-frequency market variance (MktVar) and GARCH(1,1) estimated market variance (GARCH_MktVar) series and for corresponding annualized standard deviation series as well as for monthly excess market return series in Italy for the period 2001-2006. AR1 refers to first-order autocorrelation.
Figure 6 below shows the MktVar and GARCH_MktVar series in Italy for the period from 2001-2006. GARCH(1,1) estimated market variance is much less volatile than the high-frequency measure, and for example GARCH_MktVar only slightly increases in September 2001, while MktVar reaches its peak. After mid-2003 MktVar series has been in clearly lower levels than GARCH_MktVar, which at first sight might seem like an estimation error. However, we have to keep in mind that GARCH(1,1) incorporates part of the long-term historical volatility in every monthly variance estimate, and in Italy this term (γVL) alone has been 0.0011. This explains why it is possible that MktVar has been as low as 0.0003, but the minimum in GARCH_MktVar is only 0.0033. Parameter estimates of equation (8) for GARCH_MktVar series are γ =
39 0.20, α = 0.15, and β = 0.65. The estimated long-term monthly stock market variance (VL) is 0.56% corresponding to an annualized standard deviation of 25.8%.
0.030
Figure 6. 0.025
Market variance in Italy. Highfrequency monthly market variance (MktVar) and GARCH(1,1) estimated monthly market variance (GARCH_MktVar) for MSCI Italy for the period 2001-2006.
0.020
0.015
0.010
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Jan-04
Jul-04
GARCH_MktVar
Jan-05
Jul-05
Jan-06
Mkt Var
6.2.4. Spain
Figure 7 below shows that in Spain the high-frequency market volatility measure
MktVar peaks in July 2002 at 2.45%. A similar peak can be observed from the MktVar measure also in other countries, but in Spain the peak is the most dominant. Due to this peak, also the moving average of MktVar runs above the EAIV SMA for couple of months. Otherwise, the volatility series have similar characteristics as in other countries examined. Already from mid-2003 the EAIV, VAIV and MktVar series have been in low levels in Spain, though they have increased somewhat in the last two months of the sample period.
0.030
0.012
0.025
0.010
0.020
0.008
0.015
0.006
0.010
0.004
0.005
0.002
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV
Figure 7.
Jan-03
Jul-03 VAIV
Jan-04
Jul-04
Jan-05 MktVar
Jul-05
Jan-06
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV SMA
Jan-03
Jul-03
Jan-04
VAIV SMA
Jul-04
Jan-05
Jul-05
Jan-06
MktVar SMA
Volatility series in Spain. Figure on the left plots equally weighted (EAIV) and valueweighted (VAIV) average idiosyncratic variance and market variance (MktVar) for MSCI Spain from 2001-2006, calculated as in equations (5), (6) and (7). Figure on the right plots the corresponding 10-month simple moving averages (SMA).
40
In Spain, the correlation between EAIV and VAIV measures is the highest among the countries examined being 0.95 for the full sample period, which shows from Table 5 below. Also from Figure 7 we can see that the movements in these two series are very similar. The correlation between value-weighted idiosyncratic variance and highfrequency market variance (0.89) is also very high. Together with Italy, Spain is the only country where the monthly average excess stock market return has been positive for the period from 2001-2006.
PANEL A: statistics MSCI Spain
mean
median
min
max
st.dev.
skewness
kurtosis
AR1
EAIV SDEW
0.0049
0.0043
0.0015
0.0178
0.0034
1.6174
2.8520
0.6117
0.2314
0.2265
0.1347
0.4615
0.0768
0.9010
0.4277
0.7063
VAIV SDVW
0.0028
0.0022
0.0008
0.0110
0.0021
1.9577
4.3438
0.5544
0.1714
0.1642
0.0989
0.3632
0.0614
1.1191
1.1229
0.6679
MktVar SDMkt
0.0038
0.0023
0.0004
0.0245
0.0043
2.4625
7.7359
0.6899
0.1882
0.1664
0.0702
0.5427
0.1013
1.2282
1.4078
0.7750
GARCH_MktVar SDGARCH_MktVar
0.0041
0.0036
0.0029
0.0097
0.0014
2.2272
5.1400
0.8467
0.2192
0.2076
0.1853
0.3408
0.0342
1.8498
3.3894
0.8471
ERMkt
0.0047
0.0092
-0.1686
0.1289
0.0555
-0.6848
1.2404
-0.0480
EAIV
SDEW
VAIV
PANEL B: correlations MSCI Spain EAIV SDEW
SDVW
MktVar
SDMkt
GARCH_MktVar SDGARCH_MktVar
0.9845
1
VAIV SDVW
0.9487
0.9271
1
0.9498
0.9586
0.9811
1
MktVar SDMkt
0.8202
0.8056
0.8892
0.8589
1
0.8480
0.8601
0.8762
0.8834
0.9656
1
GARCH_MktVar SDGARCH_MktVar
0.6310
0.6426
0.6317
0.6581
0.6352
0.6704
1
0.6589
0.6736
0.6576
0.6871
0.6555
0.6947
0.9959
1
ERMkt
-0.1672
-0.1769
-0.2559
-0.2475
-0.3158
-0.3100
-0.0659
-0.0703
Table 5.
ERMkt
1
1
Descriptive statistics in Spain. Descriptive statistics and cross-correlations for equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV), high-frequency market variance (MktVar) and GARCH(1,1) estimated market variance (GARCH_MktVar) series and for corresponding annualized standard deviation series as well as for monthly excess market return series in Spain for the period 2001-2006. AR1 refers to first-order autocorrelation.
Figure 8 below plots the MktVar and GARCH_MktVar series in Spain for the period 2001-2006. The correlation between the series is only 0.64, which is the lowest among the countries examined. Similarly to Italy, from mid-2004 GARCH(1,1) estimated market risk has been well above the high-frequency market risk measure, but in the end of the sample period MktVar series again rises above GARCH_MktVar series. In Spain, the parameter estimates for GARCH_MktVar are γ = 0.21, α = 0.15, and β = 0.64. The long-term variance rate (VL) is 0.47% and corresponds to an annualized standard deviation of 23.7%.
41
0.030
Figure 8. 0.025
Market variance in Spain. Highfrequency monthly market variance (MktVar) and GARCH(1,1) estimated monthly market variance (GARCH_MktVar) for MSCI Spain for the period 2001-2006.
0.020
0.015
0.010
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Jan-04
Jul-04
GARCH_MktVar
Jan-05
Jul-05
Jan-06
Mkt Var
6.2.5. Switzerland
Figure 9 below plots the EAIV, VAIV and MktVar series and corresponding moving averages in Switzerland for the period from 2001-2006. Similarly to Germany, the
EAIV measure particularly strongly reacts to the 9/11 terrorist attacks in September 2001, while VAIV and MktVar series do not show a similar shock. Further, EAIV series is much more volatile as measured by its standard deviation than either VAIV or
MktVar, which can be seen also from the table below. From the beginning of 2004 until currently all of the volatility series have been in low levels in Switzerland.
0.060
0.025
0.050 0.020
0.040 0.015
0.030
0.010 0.020
0.005 0.010
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV
Figure 9.
Jan-03
Jul-03 VAIV
Jan-04
Jul-04
Jan-05 MktVar
Jul-05
Jan-06
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV SMA
Jan-03
Jul-03
Jan-04
VAIV SMA
Jul-04
Jan-05
Jul-05
Jan-06
MktVar SMA
Volatility series in Switzerland. Figure on the left plots equally weighted (EAIV) and value-weighted (VAIV) average idiosyncratic variance and market variance (MktVar) for MSCI Switzerland from 2001-2006, calculated as in equations (5), (6) and (7). Figure on the right plots the corresponding 10-month simple moving averages (SMA).
42
From the last column of Table 6 below we see that in Switzerland the first-order autocorrelation in stock market excess return has been extraordinarily high (0.24) for the period from 2001-2006. For other sample countries, the first-order autocorrelation in stock market excess return has been very close to zero, either slightly positive or slightly negative, as it theoretically should be. Basically, the result obtained for Switzerland would mean that to some extent there is persistence in the stock market returns. For example, if the stock market goes up this month, it would mean that the probability of stock market continuing to go up also in the next month is higher than one half. However, whether this is economically significant result or not, is another question.
PANEL A: statistics MSCI Switzerland
mean
median
min
max
st.dev.
skewness
kurtosis
AR1
EAIV SDEW
0.0098
0.0070
0.0021
0.0566
0.0101
2.5323
7.6621
0.6522
0.3118
0.2898
0.1587
0.8240
0.1437
1.4273
2.1341
0.7573
VAIV SDVW
0.0036
0.0027
0.0009
0.0146
0.0028
2.0506
5.3288
0.5855
0.1973
0.1803
0.1044
0.4182
0.0702
1.0793
1.2807
0.6855
MktVar SDMkt
0.0031
0.0014
0.0004
0.0198
0.0041
2.5533
6.5964
0.5195
0.1659
0.1309
0.0677
0.4869
0.0974
1.6464
2.3337
0.6364
GARCH_MktVar SDGARCH_MktVar
0.0027
0.0025
0.0022
0.0051
0.0006
2.2364
5.8662
0.6020
0.1789
0.1733
0.1616
0.2463
0.0172
1.9255
4.1778
0.6355
ERMkt
-0.0001
0.0097
-0.1241
0.1137
0.0425
-0.6149
0.7938
0.2420
EAIV
SDEW
VAIV
PANEL B: correlations MSCI Switzerland EAIV SDEW
SDVW
MktVar
SDMkt
GARCH_MktVar SDGARCH_MktVar
0.9720
1
VAIV SDVW
0.8427
0.8816
1
0.8353
0.9059
0.9787
1
MktVar SDMkt
0.6608
0.6930
0.8161
0.7768
1
0.6871
0.7400
0.8195
0.8096
0.9720
1
GARCH_MktVar SDGARCH_MktVar
0.4721
0.5320
0.6484
0.6355
0.7137
0.7348
1
0.4856
0.5480
0.6544
0.6470
0.7158
0.7438
0.9979
1
ERMkt
-0.2116
-0.2641
-0.3380
-0.3530
-0.4663
-0.4752
-0.2613
-0.2582
Table 6.
ERMkt
1
1
Descriptive statistics in Switzerland. Descriptive statistics and cross-correlations for equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV), high-frequency market variance (MktVar) and GARCH(1,1) estimated market variance (GARCH_MktVar) series and for corresponding annualized standard deviation series as well as for monthly excess market return series in Switzerland for the period 2001-2006. AR1 refers to first-order autocorrelation.
Figure 10 below presents the MktVar series versus GARCH_MktVar series in Switzerland for the period 2001-2006. Contemporaneous correlation between MktVar and GARCH_MktVar series is 0.71 for the full sample period, and the standard deviations are 0.41% and 0.06%, correspondingly. The parameter estimates for
GARCH_MktVar are γ = 0.33, α = 0.15, and β = 0.52. The estimated long-term
43 monthly stock market variance (VL) is 0.31%, which corresponds to an annualized standard deviation of 19.2%.
0.025
Figure 10. 0.020
Market variance in Switzerland. High-frequency monthly market variance (MktVar) and GARCH(1,1) estimated monthly market variance (GARCH_MktVar) for MSCI Switzerland for the period 2001-2006.
0.015
0.010
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02
Jan-03
Jul-03
Jan-04
Jul-04
GARCH_MktVar
Jan-05
Jul-05
Jan-06
Mkt Var
6.2.6. UK
Figure 11 below shows that for the first half of the sample period the EAIV series has been clearly higher in the UK relative to both VAIV and MktVar series. Further, until mid-2003 the EAIV series has been higher in the UK relative to other countries, possibly excluding Switzerland. As mentioned above, the fact that EAIV series has been above VAIV series within a country indicates that on average the small-cap stocks have had more idiosyncratic risk than the large-caps. This occurrence seems to be particularly strong in the UK for the first half of the sample period. Otherwise, the volatility series have the same characteristics as in other countries examined.
0.060
0.025
0.050 0.020
0.040 0.015
0.030
0.010 0.020
0.005 0.010
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV
Jan-03
Jul-03 VAIV
Jan-04
Jul-04
Jan-05 MktVar
Jul-05
Jan-06
0.000 Jan-01
Jul-01
Jan-02
Jul-02 EAIV SMA
Jan-03
Jul-03
Jan-04
VAIV SMA
Jul-04
Jan-05
Jul-05
Jan-06
MktVar SMA
Figure 11. Volatility series in UK. Figure on the left plots equally weighted (EAIV) and valueweighted (VAIV) average idiosyncratic variance and market variance (MktVar) for MSCI UK from 2001-2006, calculated as in equations (5), (6) and (7). Figure on the right plots the corresponding 10-month simple moving averages (SMA).
44
Table 7 below shows that in the UK, the variability of the stock market return is quite low relative to other countries. This can be observed both from MktVar (mean 0.0029) and from ERMkt (standard deviation 0.0409). Also the GARCH_MktVar measure, on average, is the lowest among the countries studied. This is not surprising, however, since the stock market portfolio is also the most diversified in the UK. Interestingly, the correlation between EAIV and VAIV (0.85) is much higher than the correlation between VAIV and MktVar (0.69), although in Figure 11 VAIV and MktVar seem to go pretty much hand-in-hand.
PANEL A: statistics MSCI UK
mean
median
min
max
st.dev.
skewness
kurtosis
AR1
EAIV SDEW
0.0104
0.0061
0.0025
0.0557
0.0094
2.2727
7.3206
0.6187
0.3263
0.2713
0.1726
0.8177
0.1363
1.1784
1.3316
0.7564
VAIV SDVW
0.0048
0.0034
0.0015
0.0163
0.0032
1.3966
1.6171
0.6779
0.2290
0.2017
0.1343
0.4424
0.0732
0.8956
-0.0459
0.7660
MktVar SDMkt
0.0029
0.0015
0.0003
0.0225
0.0039
2.8511
9.7666
0.5431
0.1599
0.1322
0.0626
0.5197
0.0962
1.6302
2.6139
0.6773
GARCH_MktVar SDGARCH_MktVar
0.0025
0.0023
0.0021
0.0045
0.0005
2.2493
5.0034
0.7171
0.1712
0.1652
0.1573
0.2331
0.0169
2.0336
3.8553
0.7332
ERMkt
-0.0001
0.0057
-0.1290
0.0897
0.0409
-0.7611
1.1841
0.0349
EAIV
SDEW
VAIV
PANEL B: correlations MSCI UK EAIV SDEW
SDVW
MktVar
SDMkt
GARCH_MktVar SDGARCH_MktVar
0.9755
1
VAIV SDVW
0.8475
0.8901
1
0.8469
0.9097
0.9895
1
MktVar SDMkt
0.7424
0.7434
0.6868
0.6778
1
0.7958
0.8235
0.7582
0.7670
0.9652
1
GARCH_MktVar SDGARCH_MktVar
0.6824
0.6718
0.5326
0.5499
0.7182
0.7485
1
0.6897
0.6845
0.5452
0.5638
0.7227
0.7581
0.9985
1
ERMkt
-0.0281
-0.1053
-0.2123
-0.2286
-0.3561
-0.3513
-0.2274
-0.2213
Table 7.
ERMkt
1
1
Descriptive statistics in UK. Descriptive statistics and cross-correlations for equally weighted average idiosyncratic variance (EAIV), value-weighted average idiosyncratic variance (VAIV), high-frequency market variance (MktVar) and GARCH(1,1) estimated market variance (GARCH_MktVar) series and for corresponding annualized standard deviation series as well as for monthly excess market return series in UK for the period 2001-2006. AR1 refers to first-order autocorrelation.
Figure 12 below plots the MktVar market variance and GARCH_MktVar market variance series in the UK from 2001-2006. The correlation between the series is 0.72 for the full sample period. There is much more variability in the MktVar series compared to the GARCH_MktVar series, and in fact the standard deviation of the latter (0.05%) is the lowest among the countries examined. In the UK, the parameter estimates of equation (8) for the GARCH_MktVar are γ = 0.35, α = 0.15, and β = 0.50, while the long-term monthly variance rate (VL) is only 0.29%, corresponding to an annualized standard deviation of 18.5%.
45
0.025
Figure 12. 0.020
Market variance in UK. Highfrequency monthly market variance (MktVar) and GARCH(1,1) estimated monthly market variance (GARCH_MktVar) for MSCI UK for the period 2001-2006.
0.015
0.010
0.005
0.000 Jan-01
Jul-01
Jan-02
Jul-02
Jan-03 GARCH_MktVar
Jul-03
Jan-04
Jul-04
Jan-05
Jul-05
Jan-06
Mkt Var
6.3. Analysis of regression results Below, I present the results from predictive regressions of equations (9) through (16) for each country at a time. In the text as well as in the tables, I refer to the equations from (9) to (12) as linear regressions, to make a difference to the multiple regressions of equations from (13) to (16). However, all of these predictive regressions are of course linear in this case. In addition to the fitted values of the parameters (α, β, β1, and β2), I report the associated t-statistics and p-values, as well as adjusted R squared for each of the regressions. When discussing the results, I generally use the expression
statistically significant to refer to such results that are significant at five percent confidence level. In the Appendix C, I report the full results for Jarque-Bera normality test as well as for ARCH heteroskedasticity test, but I briefly comment on these results also below. In none of the countries examined, market risk or idiosyncratic risk alone can explain future stock market returns. In other words, there is no statistically significant timeseries relation between future stock market returns and EAIV, VAIV, MktVar or
GARCH_MktVar series in any of the sample countries. Clearly, this result does not support CAPM, since it seems that no significant intertemporal relation can be found between market risk and expected stock market returns from my dataset. On the other hand, the empirical observation that there is no significant time-series relationship
46 between average idiosyncratic risk (either equally weighted or value-weighted) and future stock market returns is in line with the traditional financial theory, which suggests that rational investors diversify away the idiosyncratic risk by holding stock market portfolio. More interestingly, I find that in France and in the UK the high-frequency measure of market risk (MktVar) and the value-weighted measure of average idiosyncratic risk (VAIV) combined help to explain future stock market returns. Within these countries, in a multiple regression with future stock market return, MktVar has a significantly positive and VAIV has a significantly negative coefficient. In all other countries the results are similar, but not statistically significant. Using EAIV (instead of VAIV) with
MktVar in a multiple regression to explain future stock market returns, gives similar results in all of the countries, though these results are not statistically significant in any country at five percent confidence level. Further, using GARCH_MktVar (instead of MktVar) with either EAIV or VAIV to explain future stock market returns yields similar, but not statistically significant, results in all of the countries excluding only Switzerland. The obtained results are in line with those presented in Guo and Savickas (2005) for the U.S. stock market and in Kearney and Poti (2004) for EMU region. Guo and Savickas (2005) propose that it is possible that earlier studies have not been able to find a positive relation between future stock market return and market risk, as stated by CAPM, because they have omitted idiosyncratic risk as an explanatory variable. The same conclusion can also be drawn from my dataset. What comes to the model diagnostics, generally we can say that heteroskedasticity is not a problem in the predictive regressions as measured by the first-order ARCH test. From the countries examined, it seems that only in Spain there is heteroskedasticity present in some of the regressions. On the other hand, generally across the countries studied the disturbances in the all of the predictive regressions are non-normally distributed, as measured by Jarque-Bera test. However, this is not the case in all of the predictive regressions in France and in Switzerland. The full results for Jarque-Bera and ARCH tests for every country are reported in the Appendix C.
47 6.3.1. France
From the first four columns in the Table 8 below we see that linear regressions give no statistically significant results in France. In other words, it seems that either market risk or idiosyncratic risk alone has no predictive power for future stock market returns.
Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
-0.0038
0.0030
-0.0069
-0.0194
-0.0005
0.0122
-0.0385
-0.0365
t-statistic [α]
-0.3510
0.2601
-0.7601
-0.8412
-0.0428
1.0386
-1.2230
-1.4632
p-value [α]
0.7268
0.7956
0.4500
0.4034
0.9660
0.3030
0.2260
0.1485
β
0.2629
-0.9546
1.1161
5.1101
-
-
-
-
t-statistic [β]
0.2365
-0.5239
0.8748
0.7985
-
-
-
-
p-value [β]
0.8138
0.6022
0.3850
0.4276
-
-
-
-
β1
-
-
-
-
3.2711
5.6980
14.8952
16.6893
t-statistic [β1 ]
-
-
-
-
1.3111
2.5154
1.1733
1.7840
p-value [β1 ]
-
-
-
-
0.1947
0.0145
0.2452
0.0793
β2
-
-
-
-
-2.1727
-7.7616
-1.9601
-4.4526
t-statistic [β2 ]
-
-
-
-
-1.0051
-2.4086
-0.8930
MSCI France
p-value [β2 ] Adj. R squared
Table 8.
-1.6766
-
-
-
-
0.3187
0.0190
0.3753
0.0987
-1.497%
-1.147%
-0.368%
-0.570%
-0.352%
6.740%
-0.894%
2.241%
Predictive regression results in France. The fitted values and corresponding t-statistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) in France. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16).
Multiple regressions, on the other hand, give much more interesting results in France. From the sixth column of Table 8, we see that MktVar is positively and VAIV is negatively related to future stock market return. Both of these results are statistically significant with associated t-statistics of 2.52 and -2.41, respectively, and the adjusted R squared of this regression is as high as 6.74%. Regressing stock market return on
MktVar and EAIV (instead of VAIV) gives similar results, though not statistically significant. This, however, supports the finding that there is indeed a positive timeseries relation between market risk and stock market returns while the average idiosyncratic risk is negatively intertemporally related to stock market returns. When I use GARCH_MktVar instead of MktVar to measure market risk I obtain results that point into the same direction, although these results are not statistically significant at five percent confidence level. However, when regressing stock market returns on GARCH_MktVar and VAIV, we see that the coefficient of market risk is again positive and the coefficient of idiosyncratic risk is negative, and both of these
48 coefficients are significant at ten percent confidence level. Further, the adjusted R squared from this regression is quite high, 2.24%.
6.3.2. Germany
None of the linear regressions give statistically significant results in Germany, as can be seen from the Table 9 below. Again, it seems that none of the risk measures alone can help explain future stock market returns.
Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0016
-0.0056
-0.0079
-0.0246
0.0019
-0.0013
-0.0251
-0.0247
t-statistic [α]
0.1148
-0.3717
-0.6551
-1.2198
0.1356
-0.0848
-1.2487
-1.2172
p-value [α]
0.9090
0.7114
0.5148
0.2271
0.8925
0.9327
0.2165
0.2281
β
-0.3838
0.7538
1.2036
5.1889
-
-
-
-
t-statistic [β]
-0.3267
0.3051
0.7520
1.2547
-
-
-
-
p-value [β]
0.7450
0.7613
0.4548
0.2142
-
-
-
-
β1
-
-
-
-
3.5031
2.8691
9.5261
7.7485
t-statistic [β1 ]
-
-
-
-
1.4838
0.9338
1.8440
1.3968
MSCI Germany
p-value [β1 ]
-
-
-
-
0.1429
0.3540
0.0700
0.1674
β2
-
-
-
-
-2.2763
-3.0058
-2.0069
-2.2795 -0.6960
t-statistic [β2 ]
-
-
-
-
-1.3184
-0.6362
-1.3836
p-value [β2 ]
-
-
-
-
0.1922
0.5270
0.1714
0.4891
-1.415%
-1.437%
-0.683%
0.889%
0.483%
-1.644%
2.307%
0.071%
Adj. R squared
Table 9.
Predictive regression results in Germany. The fitted values and corresponding tstatistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) in Germany. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16).
The results from multiple regressions point into the same direction as in France and in the UK, but are not statistically significant in Germany. However, it seems that market risk is positively related to stock market returns while idiosyncratic risk is negatively related to stock market returns, regardless of the measures used. In fact, when explaining stock market returns with GARCH_MktVar and EAIV, the coefficient of market risk is significantly positive at ten percent confidence level and also the adjusted R squared from this regression is quite high, 2.31%.
6.3.3. Italy
In Italy, linear regressions do not give any statistically significant results, as we can see from Table 10 below. None of the risk measures alone can significantly explain
49 stock market returns across time. Further, from the last four columns of Table 10 we can see that multiple regressions give similar results in Italy as in France and in the UK. Regardless of the measures used, it seems that market risk is positively related and idiosyncratic risk is negatively related to stock market returns across time, but the coefficients in multiple regressions are not statistically significant.
Multiple regressions
Linear regressions EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
-0.0040
-0.0044
-0.0048
-0.0342
-0.0040
0.0067
-0.0348
-0.0435
t-statistic [α]
-0.4141
-0.3991
-0.6444
-1.2145
-0.4167
0.5156
-1.1726
-1.3253
p-value [α]
0.6802
0.6912
0.5217
0.2291
0.6783
0.6080
0.2454
0.1899
β
0.6931
1.3909
1.6958
8.1345
-
-
-
-
t-statistic [β]
0.6046
0.5368
1.2165
1.2616
-
-
-
-
p-value [β]
0.5476
0.5933
0.2283
0.2117
-
-
-
-
-
-
-
-
1.8364
4.4155
8.4111
12.0514
MSCI Italy
β1 t-statistic [β1 ]
-
-
-
-
1.0532
1.5355
1.0975
1.2638
p-value [β1 ]
-
-
-
-
0.2963
0.1297
0.2767
0.2110
β2
-
-
-
-
-0.1935
-5.7238
-0.0919
-2.1247
t-statistic [β2 ]
-
-
-
-
-0.1361
-1.0809
-0.0681
p-value [β2 ] Adj. R squared
-0.5601
-
-
-
-
0.8921
0.2839
0.9459
0.5774
-1.001%
-1.125%
0.744%
0.916%
-0.826%
1.009%
-0.675%
-0.175%
Table 10. Predictive regression results in Italy. The fitted values and corresponding t-statistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) in Italy. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16).
6.3.4. Spain
From the first four columns of Table 11 below we can observe that the results from linear regressions are not statistically significant in Spain. It seems that neither market risk nor idiosyncratic risk has any predictive power for future stock market returns. Again, the results from multiple regressions, while not statistically significant, point into the same direction as in France and in the UK. Market risk (either MktVar or
GARCH_MktVar) seems to be positively related to stock market returns, and idiosyncratic risk (either EAIV or VAIV) seems to be negatively related to stock market returns across time. However, the adjusted R squared for all predictive regressions is well below zero in Spain. More accurately, also the results from multiple regressions would probably imply that there is no relationship at all between any of the risk measures and expected stock market returns (i.e. the coefficients in all of the predictive regressions are zero).
50
Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0005
0.0001
-0.0007
-0.0097
0.0031
0.0031
-0.0103
-0.0101
t-statistic [α]
0.0395
0.0096
-0.0773
-0.4665
0.2505
0.2633
-0.4851
-0.4691
p-value [α]
0.9686
0.9924
0.9387
0.6425
0.8030
0.7932
0.6293
0.6406
β
0.5550
1.1271
1.0284
3.1443
-
-
-
-
t-statistic [β]
0.2800
0.3525
0.6557
0.6570
-
-
-
p-value [β]
MSCI Spain
-
0.7804
0.7256
0.5144
0.5136
-
-
-
-
β1
-
-
-
-
2.0409
2.5621
3.8284
3.4629
t-statistic [β1 ]
-
-
-
-
0.7392
0.7424
0.6143
0.5554
p-value [β1 ]
-
-
-
-
0.4626
0.4606
0.5412
0.5806
β2
-
-
-
-
-1.5551
-3.5074
-0.4461
-0.3352
t-statistic [β2 ]
-
-
-
-
-0.4470
-0.4997
-0.1734
-0.0807
p-value [β2 ]
-
-
-
-
0.6564
0.6191
0.8629
0.9360
-1.461%
-1.387%
-0.899%
-0.896%
-2.197%
-2.115%
-2.474%
-2.513%
Adj. R squared
Table 11. Predictive regression results in Spain. The fitted values and corresponding t-statistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) in Spain. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16).
6.3.5. Switzerland
As we see from the first four columns in the Table 12 below, linear regressions give no statistically significant results in Switzerland. Based on these results, we can say that there seems to be no significant time-series relationship between stock market returns and either market risk or idiosyncratic risk alone.
Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0014
0.0037
-0.0005
0.0132
0.0011
0.0068
0.0132
0.0103
t-statistic [α]
0.1922
0.4248
-0.0757
0.5029
0.1458
0.7351
0.4786
0.3502
p-value [α]
0.8482
0.6724
0.9399
0.6168
0.8845
0.4650
0.6339
0.7274
β
-0.1248
-0.9641
0.2303
-4.8228
-
-
-
-
t-statistic [β]
-0.2365
-0.5048
0.1755
-0.5058
-
-
-
-
p-value [β]
0.8138
0.6155
0.8612
0.6148
-
-
-
-
β1
-
-
-
-
0.7782
2.3458
-4.8259
-2.9400
t-statistic [β1 ]
-
-
-
-
1.0277
-0.4434
p-value [β1 ]
-
-
-
-
0.6606
0.3081
0.6590
0.8161
β2
-
-
-
-
-0.3325
-3.7658
0.0004
-0.5834
t-statistic [β2 ]
-
-
-
-
-0.4684
-1.1315
0.0006
-0.2313
MSCI Switzerland
p-value [β2 ] Adj. R squared
0.4412
-0.2335
-
-
-
-
0.6411
0.2622
0.9995
0.8178
-1.497%
-1.178%
-1.538%
-1.176%
-2.812%
-1.088%
-2.808%
-2.720%
Table 12. Predictive regression results in Switzerland. The fitted values and corresponding tstatistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) in Switzerland. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16).
51
When using MktVar to estimate market risk the results from multiple regressions in Switzerland are similar to those in France and in the UK, though they are not statistically significant. Similarly to Spain, it would probably be more accurate to interpret the results so that there seems to be no intertemporal relation whatsoever between the different risk measures and stock market returns both in linear and in multiple regressions.
6.3.6. UK
Linear regressions give no statistically significant results for the UK stock market as we see from Table 13 below. None of the EAIV, VAIV, MktVar or GARCH_MktVar measures can alone predict future stock market returns. These results are in line with those presented in Angelidis and Tessaromatis (2005) for the period from 1980-2003, also for the UK stock market.
Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0022
0.0064
-0.0035
-0.0170
0.0035
0.0108
-0.0344
-0.0253
t-statistic [α]
0.2878
0.6938
-0.5541
-0.6954
0.4562
1.1681
-1.2184
-1.0180
p-value [α]
0.7744
0.4903
0.5815
0.4893
0.6499
0.2472
0.2277
0.3126
β
-0.2197
-1.3471
1.1870
6.8419
-
-
-
-
t-statistic [β]
-0.4015
-0.8427
0.9098
0.7077
-
-
-
-
p-value [β]
0.6894
0.4026
0.3664
0.4817
-
-
-
-
β1
-
-
-
-
3.5279
3.6982
17.7342
15.5547
t-statistic [β1 ]
-
-
-
-
1.8327
2.1092
1.3473
1.3749
p-value [β1 ]
-
-
-
-
0.0716
0.0390
0.1828
0.1741
β2
-
-
-
-
-1.3139
-4.4647
-0.9021
-2.7160
-
-1.6359
-2.0799
MSCI UK
t-statistic [β2 ] p-value [β2 ] Adj. R squared
-
-
-
-1.2140
-1.4495
-
-
-
-
0.1069
0.0417
0.2293
0.1522
-1.328%
-0.455%
-0.270%
-0.786%
2.329%
4.759%
-0.034%
0.945%
Table 13. Predictive regression results in UK. The fitted values and corresponding t-statistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) in UK. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16).
On the other hand, the results from multiple regressions prove to be more interesting. As reported for France above, also in the UK the high-frequency measure of market risk (MktVar) is positively and value-weighted measure of average idiosyncratic risk (VAIV) is negatively related to future stock market returns. The associated t-statistics
52 are 2.11 for MktVar and -2.08 for VAIV, while the adjusted R squared for the regression is as high as 4.76%. Replacing VAIV with EAIV in this regression still yields similar results, though not statistically significant at five percent confidence level anymore. However, in the latter regression the coefficient of market variance is statistically significant at ten percent confidence level and also the adjusted R squared for the regression is high, 2.33%. Angelidis and Tessaromatis (2005) do not test the joint explanatory power of idiosyncratic risk and market risk in their study.
53
7. Summary and conclusions In this study I have explored the properties of stock market volatility and idiosyncratic volatility in all major Western European stock markets for the period from 2001 through 2006. I have also investigated whether idiosyncratic volatility or market volatility or both of them combined can help explain stock market returns across time within the sample countries, which include France, Germany, Italy, Spain, Switzerland, and the UK. The sample period in this study is rather short, only five and a half years (or 66 months), compared to the other studies conducted in this research area. This might have affected the obtained results to some extent. There are several reasons why idiosyncratic risk could be related to expected stock market return, although traditional financial theory doesn’t support it. For example, idiosyncratic risk could be positively related to future stock market returns if holding non-traded assets add background risk to investors’ traded portfolio decisions, or if we consider equity as being a call option on company’s assets (see, Goyal and SantaClara, 2003). On the other hand, if we consider idiosyncratic risk as reflecting the dispersion of opinion among investors, the relation between idiosyncratic risk and future stock market returns could be even negative. Further, it has also been suggested that there should be a relation between the total risk of stocks and expected returns, meaning that only idiosyncratic risk and systematic risk combined can significantly explain the return on the stock market portfolio across time (see, Guo and Savickas, 2005). In all of the countries examined, both the absolute levels and the developments of idiosyncratic volatility and market volatility series have been quite similar through the sample period. Compared to the first years of this decade, both idiosyncratic risk and market risk are currently in low levels in each of the countries. The equally weighted average idiosyncratic variance has been above the value-weighted measure in all of the countries for the full sample period, which gives evidence that on average there is more idiosyncratic risk in small-cap stocks than in large-caps. Further, the
54 contemporaneous correlation between market risk and idiosyncratic risk has been quite high within the countries examined, suggesting that periods of high market risk coincide with periods of high idiosyncratic risk. Also the persistence in both market volatility and idiosyncratic volatility series has been rather high, which is in line with the previous observations in financial literature that volatility is persistent through time. In my opinion the observation that the absolute levels and shorter-term trends in the volatility series seem to be quite analogous across the countries is not surprising considering the high level of integration between Western European stock markets today. However, this has not always been the case and a longer sample period would enable the analysis of longer-term trends in the volatility series, which would be interesting. What comes to the relation between risk and expected return, I have reported that in France and in the UK, idiosyncratic risk and market risk combined help to significantly explain future stock market returns. In more detail, in a multiple regression, the high-frequency measure of market variance is positively related and the value-weighted average idiosyncratic variance is negatively related to expected stock market returns. Also in all other countries the results are similar, but not statistically significant. On the other hand, I report that in none of the countries market risk or idiosyncratic risk alone can explain future stock market returns. The obtained results clearly contradict traditional financial theory, which suggests a positive relation between market risk and expected stock market returns while stating that idiosyncratic risk should have no predictive power for future returns. Earlier, Guo and Savickas (2005) and Kearney and Poti (2004) have found similar results for the U.S. and for the EMU area, respectively. In contrast, Goyal and Santa-Clara (2003) report a positive time-series relation between idiosyncratic risk and market returns, while Bali et al. (2005) cannot establish any significant relationship between various measures of idiosyncratic risk and future market returns, both for the U.S. stock market. It would be interesting to see whether a longer sample period would affect the obtained results. In this study, I have reported statistically significant regression results for two countries out of six studied, while the results for the remaining four
55 countries have consistently pointed into the same direction, but have not been statistically significant. I find it possible that also in the remaining countries market risk and idiosyncratic risk combined can significantly explain stock market returns across time, but the sample period in this study has just been too short to significantly discover this relationship. So far, whether this conjecture is true or not remains an open question, and I leave it for future research.
56
References: Andersen, T., Bollerslev, T., Diebold, F., Labys, P., 2003. Modeling and forecasting realized volatility. Econometrica, 71, 579-625. Ang, A., Hodrick, R., Xing, Y., Zhang, X., 2006. The cross-section of volatility and expected returns. Journal of Finance, 61, 259-299. Angelidis, T., Tessaromatis, N., 2005. Equity returns and idiosyncratic volatility: UK evidence. Unpublished working paper. University of the Aegean, Chios, Greece. Baillie, R., DeGennaro, R., 1990. Stock returns and volatility. Journal of Financial and Quantitative Analysis, 25, 203-214. Bali, T., Cakici, N., Yan, X., Zhang, Z., 2005. Does idiosyncratic risk really matter? Journal of Finance, 60, 905-929. Banz, R., 1981. The relationship between return and market value of common stocks. Journal of Financial Economics, 9, 3-18. Basu, S., 1977. Investment performance of common stocks in relation to their priceearnings ratios: A test of the efficient market hypothesis. Journal of Finance, 32, 663682. Bernartzi, S., Thaler, R., 2001. Naive diversification strategies in defined contribution saving plans. American Economic Review, 91, 79-98. Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637-654.
57 Blume, M., Friend, I., 1973. A new look at the capital asset pricing model. Journal of Finance, 28, 19-33. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307-328. Campbell, J., Lettau, M., Malkiel, B., Xu, Y., 2001. Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk. Journal of Finance, 56, 1-43. Cao, H., Wang, T., Zhang, H., 2005. Model uncertainty, limited market participation and asset prices. Review of Financial Studies, 18, 1219-1251. Cremers, M., Petajisto, A., 2006. How active is your fund manager? A new measure that predicts performance. Unpublished working paper. Yale School of Management, New Haven, Connecticut, US. Fama, E., MacBeth, J., 1973. Risk, return, and equilibrium: Empirical tests. Journal of Political Economy, 81, 607-636. Fama, E., French, K., 1992. The cross-section of expected stock returns. Journal of Finance, 47, 427-465. French, K., Schwert, G. W., Stambaugh, R., 1987. Expected stock returns and volatility. Journal of Financial Economics, 19, 3-29. Glosten, L., Jagannathan, R., Runkle, D., 1993. On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48, 1779-1801. Goyal, A., Santa-Clara, P., 2003. Idiosyncratic risk matters! Journal of Finance, 58, 975-1007.
58 Greene, W., 1993. Econometric Analysis. Macmillan Publishing Company, Don Mills, Ontario, Canada. Guo, H., Savickas, R., 2005. Idiosyncratic volatility, stock market volatility, and expected stock returns. Unpublished working paper. Federal Reserve Bank of St. Louis, Montana, US. Hamao, Y., Mei, J., Xu, Y., 2003. Idiosyncratic risk and the creative destruction in Japan. Unpublished working paper. National Bureau of Economic Research, Cambridge, Massachusetts, US. Hull, J., 2006. Options, Futures, and Other Derivatives. Pearson Prentice Hall, Upper Saddle River, New Jersey, US. Kearney, C., Poti, V., 2004.
Have European stocks become more volatile? An
empirical investigation of volatilities and correlations in EMU equity markets at the firm, industry and market level. Unpublished working paper. Dublin City University Business School, Dublin, Ireland. Lintner, J., 1965. Valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 47, 13-37. Malkiel, B., Xu, Y., 2002. Idiosyncratic risk and security returns. Unpublished working paper. University of Texas at Dallas, Richardson, Texas, US. Maukonen, M., 2004. Three essays on the volatility of Finnish stock returns (Doctoral Thesis). Publications of the Swedish School of Economics and Business Administration, 130. Merton, R., 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449-470. Merton, R., 1980. On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics, 8, 323-361.
59
Pindyck, R., 1984. Risk, inflation, and the stock market. American Economic Review, 74, 335-351. Roll, R., 1977. A critique of the asset pricing theory’s tests. Part 1: On past and potential testability of the theory. Journal of Financial Economics, 4, 129-176. Rosenberg, B., Reid, K., Lanstein, R., 1985. Persuasive evidence of market inefficiency. Journal of Portfolio Management, 11, 9-17. Schwert, G. W., 1989. Why does stock market volatility change over time? Journal of Finance, 44, 1115-1153. Scruggs, J., 1998. Resolving the puzzling intertemporal relation between the market risk premium and conditional market variance: A two-factor approach. Journal of Finance, 52, 575-603. Sharpe, W., 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19, 425-442. Whitelaw, R., 1994. Time variations and covariations in the expectation and volatility of stock market returns. Journal of Finance, 49, 515-541.
60
Appendix A
Idiosyncratic volatility series under simplified market model. Monthly estimates of equally weighted average idiosyncratic variance (EAIV) of equation (5) and value-weighted average idiosyncratic variance (VAIV) of equation (6) under CAPM and simplified market model (SMM) for each of the countries. The last two rows of each table report the correlation between EAIV series under CAPM and SMM as well as the correlation between VAIV series under CAPM and SMM.
France
CAPM:
SMM:
EAIV
VAIV
EAIV
VAIV
Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01 Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04 Nov-04 Dec-04 Jan-05 Feb-05 Mar-05 Apr-05 May-05 Jun-05 Jul-05 Aug-05 Sep-05 Oct-05 Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Correl (EAIV)
1.429% 0.742% 1.536% 1.047% 0.732% 0.903% 0.757% 0.571% 2.126% 1.476% 1.144% 0.535% 0.671% 0.558% 0.539% 0.525% 0.741% 1.095% 2.518% 1.525% 2.157% 3.528% 1.829% 0.922% 1.200% 1.020% 1.258% 1.007% 1.017% 0.887% 0.741% 0.409% 0.555% 0.507% 0.289% 0.307% 0.484% 0.395% 0.335% 0.389% 0.271% 0.197% 0.263% 0.231% 0.310% 0.274% 0.330% 0.248% 0.300% 0.343% 0.223% 0.244% 0.191% 0.216% 0.350% 0.208% 0.245% 0.275% 0.263% 0.195% 0.434% 0.361% 0.382% 0.324% 0.492% 0.532% 0.997
1.295% 0.744% 1.192% 0.650% 0.570% 0.595% 0.581% 0.422% 1.649% 0.917% 0.790% 0.425% 0.532% 0.476% 0.471% 0.386% 0.538% 0.819% 1.791% 1.135% 1.081% 1.451% 0.976% 0.712% 0.904% 0.701% 0.932% 0.749% 0.450% 0.536% 0.424% 0.286% 0.354% 0.282% 0.195% 0.185% 0.414% 0.308% 0.267% 0.302% 0.202% 0.146% 0.172% 0.135% 0.197% 0.176% 0.190% 0.176% 0.200% 0.239% 0.150% 0.181% 0.132% 0.186% 0.263% 0.154% 0.185% 0.228% 0.199% 0.160% 0.342% 0.293% 0.341% 0.200% 0.353% 0.441%
1.575% 0.878% 1.626% 1.254% 0.813% 1.003% 0.913% 0.673% 2.231% 1.821% 1.310% 0.680% 0.741% 0.600% 0.576% 0.587% 0.832% 1.273% 2.858% 1.911% 2.473% 3.998% 2.039% 1.041% 1.259% 1.141% 1.405% 1.082% 1.086% 0.923% 0.800% 0.417% 0.596% 0.549% 0.300% 0.324% 0.492% 0.417% 0.382% 0.405% 0.292% 0.206% 0.286% 0.260% 0.322% 0.297% 0.333% 0.254% 0.300% 0.343% 0.227% 0.237% 0.200% 0.215% 0.354% 0.213% 0.255% 0.265% 0.267% 0.202% 0.459% 0.373% 0.388% 0.326% 0.436% 0.525%
1.409% 0.858% 1.290% 0.743% 0.599% 0.648% 0.647% 0.475% 1.736% 1.096% 0.876% 0.500% 0.566% 0.520% 0.510% 0.433% 0.601% 0.948% 2.004% 1.346% 1.278% 1.762% 1.120% 0.789% 0.924% 0.774% 1.002% 0.786% 0.485% 0.534% 0.450% 0.282% 0.374% 0.297% 0.195% 0.196% 0.419% 0.321% 0.305% 0.312% 0.220% 0.150% 0.185% 0.150% 0.203% 0.187% 0.187% 0.179% 0.200% 0.238% 0.153% 0.178% 0.137% 0.187% 0.262% 0.155% 0.189% 0.215% 0.200% 0.163% 0.356% 0.291% 0.341% 0.202% 0.318% 0.429%
Correl (VAIV)
0.997
Germany
CAPM:
SMM:
EAIV
VAIV
EAIV
VAIV
Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01 Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04 Nov-04 Dec-04 Jan-05 Feb-05 Mar-05 Apr-05 May-05 Jun-05 Jul-05 Aug-05 Sep-05 Oct-05 Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Correl (EAIV)
2.358% 1.307% 1.713% 1.325% 0.827% 0.943% 2.552% 1.128% 2.887% 3.713% 1.435% 0.786% 0.799% 0.945% 0.990% 0.849% 0.982% 1.129% 2.152% 2.163% 2.075% 3.251% 1.624% 1.020% 0.959% 1.001% 1.736% 1.247% 0.828% 0.915% 0.791% 0.827% 0.768% 0.668% 0.732% 0.329% 0.549% 0.399% 0.432% 0.401% 0.415% 0.273% 0.305% 0.409% 0.253% 0.383% 0.307% 0.225% 0.323% 0.389% 0.341% 0.358% 0.329% 0.324% 0.285% 0.316% 0.271% 0.259% 0.311% 0.227% 0.452% 0.443% 0.537% 0.295% 0.632% 0.482% 0.997
1.341% 0.843% 1.027% 0.820% 0.416% 0.454% 0.608% 0.682% 1.307% 0.967% 0.650% 0.316% 0.531% 0.452% 0.345% 0.415% 0.584% 0.540% 1.404% 0.741% 1.077% 1.783% 0.865% 0.536% 0.662% 0.780% 1.402% 0.698% 0.492% 0.538% 0.437% 0.409% 0.403% 0.494% 0.273% 0.220% 0.370% 0.259% 0.284% 0.233% 0.207% 0.155% 0.200% 0.211% 0.157% 0.195% 0.170% 0.102% 0.200% 0.188% 0.200% 0.170% 0.193% 0.181% 0.207% 0.175% 0.188% 0.180% 0.210% 0.145% 0.312% 0.253% 0.379% 0.222% 0.364% 0.319%
2.465% 1.420% 1.778% 1.499% 0.893% 1.004% 2.559% 1.193% 3.163% 3.846% 1.486% 0.862% 0.882% 1.068% 1.050% 0.903% 1.106% 1.268% 2.460% 2.451% 2.241% 3.837% 1.933% 1.216% 1.108% 1.145% 1.986% 1.386% 0.932% 1.004% 0.891% 0.863% 0.843% 0.736% 0.777% 0.350% 0.564% 0.414% 0.475% 0.431% 0.449% 0.293% 0.344% 0.428% 0.268% 0.397% 0.311% 0.232% 0.332% 0.390% 0.347% 0.357% 0.336% 0.325% 0.293% 0.340% 0.275% 0.250% 0.310% 0.232% 0.469% 0.437% 0.524% 0.298% 0.604% 0.464%
1.377% 0.890% 1.024% 0.938% 0.439% 0.487% 0.649% 0.728% 1.483% 1.103% 0.712% 0.376% 0.578% 0.509% 0.368% 0.442% 0.654% 0.633% 1.587% 0.879% 1.147% 2.204% 1.028% 0.622% 0.725% 0.809% 1.522% 0.761% 0.534% 0.582% 0.497% 0.423% 0.430% 0.523% 0.296% 0.226% 0.379% 0.266% 0.294% 0.247% 0.216% 0.165% 0.224% 0.226% 0.168% 0.203% 0.169% 0.103% 0.205% 0.187% 0.199% 0.166% 0.193% 0.179% 0.210% 0.180% 0.185% 0.165% 0.207% 0.145% 0.327% 0.249% 0.373% 0.221% 0.340% 0.302%
Correl (VAIV)
0.995
Italy
CAPM:
SMM:
EAIV
VAIV
EAIV
VAIV
Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01 Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04 Nov-04 Dec-04 Jan-05 Feb-05 Mar-05 Apr-05 May-05 Jun-05 Jul-05 Aug-05 Sep-05 Oct-05 Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Correl (EAIV)
1.032% 0.538% 0.857% 0.462% 0.619% 0.582% 0.892% 0.623% 1.898% 1.372% 1.142% 0.661% 0.682% 0.643% 0.562% 0.525% 0.648% 0.819% 1.423% 1.051% 1.149% 2.166% 1.283% 0.807% 0.758% 0.796% 1.256% 0.600% 0.629% 0.595% 0.466% 0.364% 0.500% 0.396% 0.433% 3.445% 0.609% 0.324% 0.558% 0.284% 0.464% 0.196% 0.224% 0.212% 0.422% 0.245% 0.262% 0.269% 0.282% 0.273% 0.336% 0.326% 0.345% 0.242% 0.270% 0.191% 0.331% 0.345% 0.224% 0.228% 0.338% 0.310% 0.351% 0.340% 0.450% 0.361% 0.996
0.704% 0.447% 0.597% 0.375% 0.366% 0.395% 0.513% 0.303% 1.384% 0.694% 0.591% 0.271% 0.399% 0.378% 0.425% 0.329% 0.398% 0.424% 0.855% 0.446% 0.661% 1.016% 0.683% 0.397% 0.440% 0.492% 0.755% 0.355% 0.316% 0.294% 0.243% 0.161% 0.235% 0.178% 0.208% 0.490% 0.346% 0.188% 0.364% 0.148% 0.182% 0.120% 0.118% 0.093% 0.157% 0.134% 0.164% 0.175% 0.153% 0.183% 0.198% 0.194% 0.204% 0.185% 0.182% 0.140% 0.172% 0.239% 0.184% 0.170% 0.220% 0.217% 0.264% 0.153% 0.223% 0.225%
1.061% 0.564% 0.907% 0.479% 0.629% 0.591% 0.933% 0.647% 1.861% 1.413% 1.096% 0.602% 0.606% 0.630% 0.573% 0.518% 0.632% 0.744% 1.291% 0.894% 0.999% 2.164% 1.414% 0.774% 0.742% 0.798% 1.104% 0.557% 0.603% 0.591% 0.435% 0.342% 0.500% 0.389% 0.444% 3.486% 0.603% 0.311% 0.539% 0.258% 0.445% 0.191% 0.224% 0.212% 0.411% 0.237% 0.253% 0.241% 0.270% 0.254% 0.311% 0.311% 0.322% 0.224% 0.242% 0.164% 0.315% 0.297% 0.207% 0.209% 0.301% 0.272% 0.305% 0.284% 0.341% 0.293%
0.724% 0.464% 0.622% 0.385% 0.368% 0.399% 0.524% 0.310% 1.406% 0.695% 0.582% 0.264% 0.390% 0.399% 0.443% 0.332% 0.407% 0.420% 0.849% 0.447% 0.671% 1.063% 0.778% 0.418% 0.459% 0.529% 0.802% 0.362% 0.325% 0.304% 0.244% 0.156% 0.246% 0.184% 0.216% 0.510% 0.355% 0.190% 0.369% 0.146% 0.196% 0.120% 0.122% 0.096% 0.157% 0.136% 0.168% 0.172% 0.151% 0.176% 0.190% 0.190% 0.202% 0.184% 0.179% 0.129% 0.172% 0.218% 0.181% 0.167% 0.220% 0.220% 0.260% 0.150% 0.210% 0.221%
Correl (VAIV)
0.998
61
Spain
CAPM:
SMM:
EAIV
VAIV
EAIV
VAIV
Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01 Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04 Nov-04 Dec-04 Jan-05 Feb-05 Mar-05 Apr-05 May-05 Jun-05 Jul-05 Aug-05 Sep-05 Oct-05 Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Correl (EAIV)
1.417% 0.619% 0.765% 0.542% 0.549% 0.455% 0.558% 0.506% 1.775% 1.101% 0.699% 0.325% 0.572% 0.465% 0.433% 0.587% 0.641% 0.755% 1.393% 0.930% 0.889% 1.325% 0.654% 0.492% 0.975% 0.637% 0.877% 0.453% 0.673% 0.765% 0.636% 0.214% 0.381% 0.255% 0.235% 0.250% 0.321% 0.344% 0.440% 0.214% 0.236% 0.187% 0.175% 0.205% 0.151% 0.179% 0.155% 0.194% 0.200% 0.308% 0.198% 0.164% 0.168% 0.245% 0.203% 0.215% 0.372% 0.294% 0.266% 0.171% 0.298% 0.349% 0.481% 0.210% 0.422% 0.489% 0.981
0.729% 0.388% 0.364% 0.250% 0.216% 0.245% 0.269% 0.214% 1.013% 0.463% 0.399% 0.207% 0.378% 0.218% 0.205% 0.327% 0.381% 0.463% 1.099% 0.623% 0.536% 0.722% 0.388% 0.350% 0.498% 0.376% 0.721% 0.223% 0.265% 0.313% 0.230% 0.091% 0.234% 0.122% 0.100% 0.145% 0.171% 0.172% 0.227% 0.115% 0.112% 0.093% 0.110% 0.088% 0.082% 0.095% 0.096% 0.094% 0.094% 0.143% 0.103% 0.086% 0.093% 0.139% 0.118% 0.104% 0.246% 0.227% 0.163% 0.101% 0.223% 0.270% 0.238% 0.125% 0.250% 0.264%
1.557% 0.763% 0.959% 0.848% 0.635% 0.586% 0.808% 0.687% 2.115% 1.675% 0.951% 0.556% 0.747% 0.607% 0.454% 0.659% 0.746% 0.913% 1.987% 1.374% 1.222% 1.623% 0.799% 0.609% 1.102% 0.794% 1.115% 0.548% 0.747% 0.846% 0.672% 0.234% 0.399% 0.298% 0.251% 0.261% 0.336% 0.361% 0.456% 0.210% 0.252% 0.192% 0.186% 0.200% 0.146% 0.192% 0.164% 0.197% 0.200% 0.300% 0.199% 0.153% 0.165% 0.237% 0.195% 0.213% 0.371% 0.271% 0.259% 0.169% 0.300% 0.340% 0.474% 0.201% 0.366% 0.463%
0.842% 0.420% 0.458% 0.349% 0.240% 0.297% 0.407% 0.293% 1.286% 0.717% 0.538% 0.302% 0.467% 0.288% 0.224% 0.380% 0.432% 0.566% 1.511% 0.942% 0.831% 0.995% 0.519% 0.424% 0.585% 0.536% 0.932% 0.304% 0.348% 0.368% 0.284% 0.104% 0.231% 0.153% 0.113% 0.150% 0.181% 0.186% 0.225% 0.114% 0.140% 0.096% 0.123% 0.085% 0.079% 0.107% 0.102% 0.096% 0.093% 0.134% 0.099% 0.072% 0.090% 0.131% 0.112% 0.098% 0.243% 0.194% 0.155% 0.097% 0.227% 0.256% 0.234% 0.120% 0.188% 0.234%
Correl (VAIV)
0.987
Switzerland
CAPM:
SMM:
EAIV
VAIV
EAIV
VAIV
Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01 Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04 Nov-04 Dec-04 Jan-05 Feb-05 Mar-05 Apr-05 May-05 Jun-05 Jul-05 Aug-05 Sep-05 Oct-05 Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Correl (EAIV)
1.095% 0.769% 1.447% 1.161% 0.748% 0.874% 1.931% 0.993% 3.501% 5.658% 2.130% 1.326% 1.466% 1.249% 0.813% 0.836% 0.744% 0.920% 3.112% 2.606% 1.960% 4.233% 2.553% 0.798% 0.882% 1.475% 1.729% 0.977% 1.030% 0.936% 1.057% 0.835% 0.657% 0.616% 0.393% 0.348% 1.064% 0.384% 0.410% 0.471% 0.330% 0.236% 0.298% 0.393% 0.234% 0.441% 0.278% 0.218% 0.262% 0.480% 0.253% 0.323% 0.220% 0.232% 0.247% 0.286% 0.210% 0.440% 0.296% 0.240% 0.471% 0.418% 0.359% 0.357% 0.565% 0.397% 0.999
0.647% 0.433% 0.632% 0.441% 0.354% 0.465% 0.584% 0.461% 0.827% 0.789% 0.356% 0.343% 0.392% 0.385% 0.346% 0.314% 0.241% 0.382% 1.457% 0.914% 0.641% 1.442% 0.637% 0.278% 0.519% 0.671% 0.802% 0.412% 0.465% 0.383% 0.365% 0.310% 0.260% 0.218% 0.208% 0.120% 0.542% 0.250% 0.228% 0.314% 0.148% 0.107% 0.187% 0.212% 0.121% 0.209% 0.145% 0.125% 0.113% 0.244% 0.145% 0.181% 0.096% 0.091% 0.109% 0.134% 0.116% 0.246% 0.174% 0.118% 0.230% 0.248% 0.163% 0.132% 0.264% 0.195%
1.076% 0.796% 1.548% 1.256% 0.785% 0.926% 2.016% 1.120% 3.927% 5.941% 2.195% 1.436% 1.497% 1.284% 0.826% 0.856% 0.780% 1.049% 3.325% 2.905% 2.177% 4.643% 2.741% 0.891% 1.047% 1.633% 1.915% 1.043% 1.080% 0.981% 1.105% 0.854% 0.676% 0.658% 0.413% 0.364% 1.063% 0.403% 0.451% 0.475% 0.368% 0.242% 0.299% 0.402% 0.241% 0.439% 0.273% 0.215% 0.262% 0.477% 0.252% 0.321% 0.223% 0.229% 0.249% 0.279% 0.202% 0.422% 0.301% 0.245% 0.476% 0.438% 0.368% 0.367% 0.579% 0.415%
0.606% 0.459% 0.772% 0.496% 0.363% 0.462% 0.625% 0.465% 0.999% 0.844% 0.409% 0.339% 0.406% 0.421% 0.373% 0.327% 0.261% 0.471% 1.688% 1.045% 0.760% 1.751% 0.749% 0.316% 0.599% 0.770% 0.983% 0.478% 0.502% 0.399% 0.384% 0.318% 0.274% 0.242% 0.210% 0.126% 0.538% 0.263% 0.245% 0.312% 0.162% 0.109% 0.185% 0.219% 0.121% 0.203% 0.135% 0.124% 0.113% 0.240% 0.143% 0.179% 0.097% 0.088% 0.110% 0.127% 0.112% 0.230% 0.176% 0.123% 0.249% 0.266% 0.167% 0.138% 0.298% 0.224%
Correl (VAIV)
0.996
UK
CAPM:
SMM:
EAIV
VAIV
EAIV
VAIV
Jan-01 Feb-01 Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01 Jan-02 Feb-02 Mar-02 Apr-02 May-02 Jun-02 Jul-02 Aug-02 Sep-02 Oct-02 Nov-02 Dec-02 Jan-03 Feb-03 Mar-03 Apr-03 May-03 Jun-03 Jul-03 Aug-03 Sep-03 Oct-03 Nov-03 Dec-03 Jan-04 Feb-04 Mar-04 Apr-04 May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04 Nov-04 Dec-04 Jan-05 Feb-05 Mar-05 Apr-05 May-05 Jun-05 Jul-05 Aug-05 Sep-05 Oct-05 Nov-05 Dec-05 Jan-06 Feb-06 Mar-06 Apr-06 May-06 Jun-06 Correl (EAIV)
1.639% 1.108% 1.851% 1.812% 1.209% 1.314% 1.698% 0.822% 2.987% 2.916% 1.873% 1.035% 0.889% 1.059% 1.867% 1.028% 1.166% 1.054% 2.864% 2.287% 1.531% 5.572% 2.764% 2.479% 1.150% 1.394% 2.426% 1.190% 0.934% 1.094% 0.795% 0.544% 0.789% 0.605% 0.622% 0.423% 0.520% 0.504% 0.522% 0.372% 0.409% 0.290% 0.321% 0.371% 0.557% 0.387% 0.353% 0.341% 0.315% 0.319% 0.294% 0.309% 0.309% 0.248% 0.367% 0.257% 0.412% 0.438% 0.453% 0.271% 0.551% 0.401% 0.514% 0.336% 0.562% 0.537% 0.999
1.174% 0.809% 1.110% 0.971% 0.574% 0.768% 0.997% 0.448% 1.631% 0.961% 0.861% 0.422% 0.479% 0.603% 0.496% 0.518% 0.682% 0.436% 1.053% 0.877% 0.669% 1.236% 0.984% 0.539% 0.652% 0.646% 0.599% 0.466% 0.462% 0.416% 0.396% 0.304% 0.337% 0.340% 0.298% 0.206% 0.325% 0.310% 0.282% 0.271% 0.280% 0.189% 0.211% 0.246% 0.285% 0.219% 0.215% 0.213% 0.182% 0.212% 0.187% 0.212% 0.188% 0.150% 0.241% 0.180% 0.215% 0.301% 0.308% 0.182% 0.275% 0.283% 0.344% 0.228% 0.333% 0.254%
1.740% 1.178% 2.082% 1.995% 1.321% 1.384% 1.833% 0.880% 3.173% 3.107% 1.985% 1.115% 0.950% 1.116% 1.920% 1.065% 1.205% 1.088% 3.070% 2.454% 1.691% 5.795% 2.877% 2.553% 1.191% 1.507% 2.601% 1.283% 0.986% 1.123% 0.831% 0.567% 0.812% 0.629% 0.648% 0.432% 0.527% 0.528% 0.550% 0.376% 0.416% 0.298% 0.337% 0.384% 0.569% 0.400% 0.358% 0.344% 0.319% 0.323% 0.296% 0.302% 0.314% 0.248% 0.365% 0.260% 0.414% 0.419% 0.454% 0.270% 0.567% 0.402% 0.512% 0.337% 0.539% 0.554%
1.267% 0.865% 1.259% 1.070% 0.621% 0.808% 1.113% 0.483% 1.732% 1.040% 0.908% 0.458% 0.513% 0.637% 0.514% 0.548% 0.707% 0.457% 1.149% 0.954% 0.787% 1.387% 1.074% 0.593% 0.666% 0.681% 0.638% 0.494% 0.478% 0.416% 0.407% 0.316% 0.342% 0.348% 0.310% 0.212% 0.326% 0.330% 0.301% 0.273% 0.288% 0.190% 0.222% 0.249% 0.288% 0.226% 0.214% 0.216% 0.182% 0.211% 0.187% 0.208% 0.191% 0.148% 0.237% 0.180% 0.215% 0.294% 0.315% 0.179% 0.287% 0.288% 0.345% 0.227% 0.340% 0.272%
Correl (VAIV)
0.999
62
Appendix B
Predictive regression results under simplified market model. The fitted values and corresponding tstatistics and p-values as well as adjusted R squared for the predictive regressions from equations (9) to (16) under simplified market model in each of the countries. Monthly excess stock market return is regressed on the lagged risk measures presented in the first row of the table. First four columns report the results for linear regressions of equations (9) to (12) and last four columns for multiple regressions of equations (13) to (16). Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
-0.0017
0.0043
-0.0069
-0.0194
0.0017
0.0129
-0.0528
-0.0467
t-statistic [α]
-0.1609
0.3849
-0.7601
-0.8412
0.1616
1.1759
-1.6350
-1.8097
p-value [α]
0.8727
0.7016
0.4500
0.4034
0.8722
0.2441
0.1071
0.0752
β
-0.0192
-1.1191
1.1161
5.1101
-
-
-
-
t-statistic [β]
-0.0199
-0.7060
0.8748
0.7985
-
-
-
-
p-value [β]
0.9842
0.4828
0.3850
0.4276
-
-
-
21.3517
MSCI France
β1
-
-
-
-
4.4156
6.6911
21.5104
t-statistic [β1 ]
-
-
-
-
1.7761
2.9384
1.6709
2.1776
p-value [β1 ]
-
-
-
-
0.0806
0.0046
0.0998
0.0333
β2
-
-
-
-
-2.8871
-8.1526
-2.8354
-5.2010
t-statistic [β2 ]
-
-
-
-
-1.5404
-2.8876
-1.4640
-2.1438
p-value [β2 ]
-
-
-
-
0.1285
0.0053
0.1482
0.0360
Adj. R squared
-1.587%
-0.790%
-0.368%
-0.570%
1.772%
10.103%
1.223%
4.861%
MSCI Germany
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0020
-0.0042
-0.0079
-0.0246
0.0031
0.0006
-0.0266
-0.0260
t-statistic [α]
0.1436
-0.2907
-0.6551
-1.2198
0.2241
0.0419
-1.3302
-1.2826
p-value [α]
0.8863
0.7722
0.5148
0.2271
0.8234
0.9667
0.1883
0.2044
β
-0.3922
0.4351
1.2036
5.1889
-
-
-
-
t-statistic [β]
-0.3682
0.2025
0.7520
1.2547
-
-
-
-
p-value [β]
0.7140
0.8402
0.4548
0.2142
-
-
-
-
-
-
-
-
4.2054
3.8250
10.4560
8.6940
t-statistic [β1 ]
-
-
-
-
1.6701
1.1707
1.9641
1.5281
p-value [β1 ]
-
-
-
-
0.0999
0.2462
0.0540
0.1316
β2
-
-
-
-
-2.5609
-4.0222
-2.0962
-2.6250
t-statistic [β2 ]
-
-
-
-
-1.5333
-0.9206
-1.5462
-0.8986
p-value [β2 ]
-
-
-
-
0.1303
0.3608
0.1272
0.3723
-1.369%
-1.521%
-0.683%
0.889%
1.430%
-0.928%
3.030%
0.585%
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
-0.0030
-0.0039
-0.0048
-0.0342
-0.0035
0.0061
-0.0358
-0.0455
t-statistic [α]
-0.3157
-0.3660
-0.6444
-1.2145
-0.3763
0.4896
-1.2090
-1.3662
p-value [α]
0.7533
0.7156
0.5217
0.2291
0.7080
0.6262
0.2312
0.1768
β
0.5545
1.2483
1.6958
8.1345
-
-
-
-
t-statistic [β]
0.4864
0.5008
1.2165
1.2616
-
-
-
-
p-value [β]
0.6284
0.6182
0.2283
0.2117
-
-
-
12.7660
Linear regressions
β1
Adj. R squared
Multiple regressions
Linear regressions MSCI Italy
Multiple regressions
β1
-
-
-
-
1.9115
4.3149
8.8930
t-statistic [β1 ]
-
-
-
-
1.1277
1.5553
1.1688
1.3178
p-value [β1 ]
-
-
-
-
0.2638
0.1250
0.2469
0.1924
β2
-
-
-
-
-0.3122
-5.3616
-0.2551
-2.3827
t-statistic [β2 ]
-
-
-
-
-0.2274
-1.0913
-0.1916
-0.6430
p-value [β2 ]
-
-
-
-
0.8209
0.2794
0.8487
0.5226
-1.207%
-1.184%
0.744%
0.916%
-0.773%
1.044%
-0.622%
-0.015%
Adj. R squared
63
Linear regressions
Multiple regressions
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0018
0.0003
-0.0007
-0.0097
0.0061
0.0032
-0.0115
-0.0103
t-statistic [α]
0.1609
0.0249
-0.0773
-0.4665
0.5153
0.2914
-0.5303
-0.4605
p-value [α]
0.8727
0.9802
0.9387
0.6425
0.6082
0.7717
0.5978
0.6467
β
0.2416
0.8784
1.0284
3.1443
-
-
-
-
t-statistic [β]
0.1630
0.3840
0.6557
0.6570
-
-
-
-
p-value [β]
0.8711
0.7022
0.5144
0.5136
-
-
-
-
-
-
-
-
3.7591
3.7340
4.5078
3.4597
t-statistic [β1 ]
-
-
-
-
p-value [β1 ]
-
-
-
-
0.2735
0.4070
0.4768
0.5958
β2
-
-
-
-
-2.9004
-4.2069
-0.6541
-0.2249
t-statistic [β2 ]
-
-
-
-
-0.9047
-0.6464
-0.3364
-0.0727
MSCI Spain
β1
p-value [β2 ]
1.1048
0.7158
0.8349
0.5332
-
-
-
-
0.3691
0.5204
0.7377
0.9423
-1.544%
-1.350%
-0.899%
-0.896%
-1.190%
-1.840%
-2.337%
-2.515%
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0017
0.0025
-0.0005
0.0132
0.0013
0.0050
0.0127
0.0127
t-statistic [α]
0.2317
0.3028
-0.0757
0.5029
0.1806
0.5717
0.4578
0.4220
p-value [α]
0.8175
0.7630
0.9399
0.6168
0.8573
0.5696
0.6487
0.6745
β
-0.1440
-0.5792
0.2303
-4.8228
-
-
-
-
t-statistic [β]
-0.2954
-0.3622
0.1755
-0.5058
-
-
-
-
p-value [β]
0.7686
0.7184
0.8612
0.6148
-
-
-
-
-
-
-
-
0.9355
2.2722
-4.5135
-4.5104
t-statistic [β1 ]
-
-
-
-
p-value [β1 ]
-
-
-
-
0.6075
0.3643
0.6824
0.7262
β2
-
-
-
-
-0.3823
-2.9325
-0.0327
-0.0791
t-statistic [β2 ]
-
-
-
-
-0.5678
-0.9671
-0.0583
-0.0368
Adj. R squared
Linear regressions MSCI Switzerland
β1
p-value [β2 ]
Multiple regressions
0.5163
0.9139
-0.4112
-0.3518
-
-
-
-
0.5722
0.3372
0.9537
0.9708
-1.447%
-1.376%
-1.538%
-1.176%
-2.642%
-1.642%
-2.803%
-2.806%
EAIV
VAIV
MktVar
GARCH_MktVar
MktVar & EAIV
MktVar & VAIV
GARCH_MktVar & EAIV
GARCH_MktVar & VAIV
α
0.0020
0.0059
-0.0035
-0.0170
0.0033
0.0098
-0.0339
-0.0263
t-statistic [α]
0.2603
0.6534
-0.5541
-0.6954
0.4385
1.0951
-1.1960
-1.0497
p-value [α]
0.7955
0.5159
0.5815
0.4893
0.6626
0.2777
0.2362
0.2979
β
-0.1888
-1.1607
1.1870
6.8419
-
-
-
-
t-statistic [β]
-0.3660
-0.8060
0.9098
0.7077
-
-
-
-
p-value [β]
0.7156
0.4233
0.3664
0.4817
-
-
-
-
-
-
-
-
3.5843
3.7352
17.3431
15.6941
t-statistic [β1 ]
-
-
-
-
p-value [β1 ]
-
-
-
-
0.0722
0.0393
0.1934
0.1742
β2
-
-
-
-
-1.2507
-4.0302
-0.8184
-2.4315
t-statistic [β2 ]
-
-
-
-
-1.6236
-2.0610
-1.1665
-1.4279
Adj. R squared
Linear regressions MSCI UK
β1
p-value [β2 ] Adj. R squared
Multiple regressions
1.8293
2.1054
1.3149
1.3745
-
-
-
-
0.1095
0.0435
0.2479
0.1583
-1.372%
-0.550%
-0.270%
-0.786%
2.268%
4.646%
-0.212%
0.849%
64
Appendix C
Results for Jarque-Bera test and ARCH test. The test statistics for Jarque-Bera normality test and ARCH heteroskedasticity test for each of the predictive regressions and for each of the countries. The critical values for Jarque-Bera test are 9.210 and 5.991 at one percent and five percent confidence levels respectively. The critical values for first-order ARCH test are 6.635 and 3.841 at one percent and five percent confidence levels respectively.
PANEL A Jarque-Bera
ARCH
EAIV
3.927
2.713
VAIV
2.124
3.975
MktVar
7.885
2.270
GARCH_MktVar
4.698
1.332
MktVar & EAIV
12.711
3.181
MktVar & VAIV
18.442
0.980
GARCH_MktVar & EAIV
2.895
0.861
GARCH_MktVar & VAIV
1.751
0.435
MSCI Germany
Predictive regressions
Predictive regressions
MSCI France
PANEL B Tests
ARCH
EAIV
16.942
1.829
VAIV
19.299
1.412
MktVar
25.805
1.027
GARCH_MktVar
25.018
0.097
MktVar & EAIV
23.854
1.060
MktVar & VAIV
35.433
0.844
GARCH_MktVar & EAIV
19.153
0.000
GARCH_MktVar & VAIV
28.534
0.007
PANEL C
PANEL D Tests
Jarque-Bera
ARCH
EAIV
16.047
1.642
VAIV
15.653
1.495
MktVar
22.959
0.374
GARCH_MktVar
17.355
0.252
MktVar & EAIV
22.744
0.364
MktVar & VAIV
34.176
0.135
GARCH_MktVar & EAIV
17.043
0.253
GARCH_MktVar & VAIV
15.264
0.311
MSCI Spain
Predictive regressions
Predictive regressions
MSCI Italy
ARCH
EAIV
12.285
5.292
VAIV
13.035
4.997
MktVar
16.857
3.488
GARCH_MktVar
13.551
3.266
MktVar & EAIV
18.485
3.278
MktVar & VAIV
18.704
2.958
GARCH_MktVar & EAIV
12.878
3.342
GARCH_MktVar & VAIV
13.133
3.312
PANEL F Tests
Jarque-Bera
ARCH
EAIV
5.759
0.174
VAIV
5.428
0.351
MktVar
6.574
0.091
GARCH_MktVar
5.093
0.394
MktVar & EAIV
7.009
0.134
MktVar & VAIV
9.677
0.514
GARCH_MktVar & EAIV
5.093
0.394
GARCH_MktVar & VAIV
5.060
0.451
MSCI UK
Predictive regressions
Predictive regressions
Tests Jarque-Bera
PANEL E MSCI Switzerland
Tests Jarque-Bera
Tests Jarque-Bera
ARCH
EAIV
7.601
3.062
VAIV
6.471
3.477
MktVar
16.419
1.744
GARCH_MktVar
12.973
1.103
MktVar & EAIV
15.850
1.757
MktVar & VAIV
21.282
1.979
GARCH_MktVar & EAIV
9.021
0.192
GARCH_MktVar & VAIV
10.638
0.439