The negative relationship between the cross-section of expected returns and lagged idiosyncratic volatility. The German stock market 1990-2016

Master's Thesis, 2018

32 Pages, Grade: 1.0


Table of contents

1 Introduction

2 Literature

3 Data

4 Methodology

5 Results
5.1 Estimating Idiosyncratic Volatility
5.2 Patterns in Average Returns January 1990 - June 2016
5.3 Patterns in Average Returns January 2003 - June 2016

6 Conclusion

7 Appendix
7.1 Portfolio Strategies January 1990 - June 2016
7.2 Portfolio Strategies January 2003 - June 2016

List of Tables

2 Autocorrelation 1 Month d-Estimates

3 Correlation 1-Month-Estimates

4 1/0/1 Value-Weighted Portfolios January 1990 - June 2016

5 1/0/1 Equal-Weighted Portfolios January 1990 - June 2016

6 3/0/1 Portfolios January 1990 - June 2016

7 3/0/3 Portfolios January 1990 - June 2016

8 1/0/1 Portfolios January 2003 - June 2016

9 Value-weighted Portfolios: Controlling for Size January 2003 - June 2016

10 Equal-weighted Portfolios: Controlling for Size January 2003 - June 2016

11 1/0/3 Portfolios January 1990 - June 2016

12 1/0/3 Portfolios January 2003 - June 2016

13 3/0/3 Portfolios January 2003 - June 2016

14 3/0/1 Portfolios January 2003 - June 2016

List of Figures

1 Weights of Portfolio 1 — 1/0/1- Strategy

List of Symbols

Abbildung in dieser Leseprobe nicht enthalten


Similar to the findings of Ang et al. (2006) for the US stock market this paper shows that there is a significant difference in returns relative to the Fama-French three-factor model, between portfolios of stocks with high and portfolios of stocks with low past idiosyncratic volatility. Although for the period 1990 - 2016 no relationship between lagged idiosyncratic volatility and the cross-section of stock returns has been found, the Idiosyncratic Volatility Puzzle reveals itself for the sub-period 2003 - 2016, when the respective portfolios of stocks with different levels of idiosyncratic volatility are controlled for size.

1 Introduction

Because empirical evidence indicates that the Capital Asset Pricing Model (CAPM) is not able to explain the variation in stock returns (see, among others: Black et al., 1972; Douglas, 1969; Fama and French, 1992), many researchers investigated potential risk factors other than the market risk. On the basis of empirical research Fama and French (1993, 1996) developed a broadly accepted factor model, which extents the classical CAPM by two ad­ditional factors. In contrast to the empirical approach, Levy (1978) theoretically shows that expected returns of individual securities are not fully determined by the price of systematic risk and the risk free rate, in case investors are not able to hold the market portfolio. The latter seems reasonable in the presence of transaction costs, limited tradeability of securities, governmental regulations and restrictions (eg.liquidity constraints) and other structural fac­tors. In this context idiosyncratic risk can then be rationalised by investors to compensate for the lack of diversification in their portfolios (Malkiel and Xu, 2002). More precisely, firms with high idiosyncratic volatility will need to pay a risk premium because they increase the non-diversified risk in the investors portfolio. Similar to the work of Levy (1978) this im­plication is also formally derived by Merton (1987), who assumes incomplete information among investors and Hirshleifer (1988), who theoretically investigates the impact of trading costs in the commodity future markets.

Since these theoretical findings have been revealed, several empirical studies look at the influence of idiosyncratic volatility on asset pricing. More recent empirical findings by Ang et al. (2006) indicate that there is a negative relationship between idiosyncratic volatility and the cross section of expected stock returns for the US market. Moreover, in a follow up paper Ang et al. (2009) provide further evidence for the relationship and show that the negative relationship is not only country specific, but an international phenomenon, by extending their analysis to 23 markets. Among them are the markets of the G71 countries, for which the negative relationship is statistically and economically significant in each case. It seems that investors do prefer stocks with higher past idiosyncratic volatility. Because of the conflicting theoretical predictions and empirical findings, the common literature refers to this problem set, as the Idiosyncratic Volatility Puzzle.

The main goal of this paper is to examine whether the negative relationship between the cross-section of expected returns and lagged idiosyncratic volatility also can be found for the German stock market for the period of January 1990 through June 2016, by sorting stocks into portfolios on the basis of their idiosyncratic volatility estimates. This procedure follows Ang et al. (2006). The paper is organised as follows. The next section reviews the empirical literature on the Idiosyncratic Volatility Puzzle. Section 3 is designated to the sample data. Section 4 explains the methodology. Section 5 presents the results and shows that the Idiosyncratic Volatility Puzzle is present in the German stock market and Section 6 concludes.

2 Literature

In the early years of the CAPM, researchers mainly focused on empirically testing the im­plication and conclusions of the market model. Fama and MacBeth (1973) tested the two- parameter model using monthly average total return data of all common stocks listed on the NYSE during the period January 1926 trough June 1968. Their basic approach was to use a combination of time-series and cross-sectional regressions to test the three condi­tions of the Sharpe-Lintner CAPM. They used this regression procedure, later commonly applied and referred to as the Fama-McBeth-Regression, to produce results which are con­sistent to cross-sectional correlation. In their regressions they assume s p(êi) to proxy for stock's idiosyncratic volatility. Hereby, s p(êi) denotes the average idiosyncratic volatility of stocks within beforehand formed portfolios relative to the CAPM. These portfolios are formed based on past beta estimates of individual stocks.1 s p (êi) is used as a regressor to test, whether the average monthly stock returns are able to be explained by additional risk exposure other than beta. Beta commonly refers to the factor loading of the market risk in the context of the CAPM. Although they find significant systematic effects of this proxy for idiosyncratic risk on period-by-period returns, they can not reject the hypothesis, that these effects are zero on average.2 3 The results of Fama and MacBeth (1973) are seen to be the reason why the systematic roles of idiosyncratic volatility and idiosyncratic risk on asset pricing have not been further analysed (Malkiel and Xu, 2002). Some studies regarding id­iosyncratic risk where undertaken by Lehmann (1990), Tinic and West (1986) and Malkiel and Xu (1997), but they either show inconsistent results or do not report any significance level (Bali and Cakici, 2008).

Malkiel and Xu (2002) reproduce the study of Fama and MacBeth (1973) by replicat­ing their sample and methodology. Moreover they find significant positive coefficients of S p( f^ i), after lifting the number of preformed portfolios from 20 to 50. Using the same methodology, the authors also analyse an extension ofthe sample used byFama and French (1992), covering a more recent period of stock data from 1963 to 20004. They construct 100 beta / size -portfolios, similar to those of Fama and French (1992). Again in the regres­sions the coefficients of s p( £ i) often are significantly positive. The results also seem to hold after some robustness checks. Apparently the work of Malkiel and Xu (2002) supports the theoretical implications of Merton (1987) and Lintner (1965), since evidence for a positive idiosyncratic risk premium is found. However s p( £ i) is seen to be a problematic proxy for individual stock's idiosyncratic risk, since it is the average idiosyncratic volatility over mul­tiple stocks, that are similar in past beta estimates only.

This could be the reason why Ang et al. (2006) obtain contradictory results indicating that idiosyncratic risk demands a negative risk premium relative to the Fama-French three- factor-model (FF-3 model) or the CAPM. In their work the authors sort stocks based on past idiosyncratic volatility estimates into portfolios. The average monthly portfolios' returns and Jensens alphas are then used to examine differences between the portfolios. In contrast to Malkiel and Xu (2002) and Fama and MacBeth (1973) they use within monthly data of each stock to estimate the respective idiosyncratic volatility relative to the FF-3 model. Portfolios are constructed using different strategies, but each strategy leads to the same results, which indicate that idiosyncratic riskis negatively priced inthe cross-section of expected stockre­turns. In particular, in their main analysis, they compare one portfolio containing 20% ofthe stocks with the highest past idiosyncratic volatility and one portfolio containing 20% of the stocks with the lowest past idiosyncratic volatility, where idiosyncratic volatility is estimated using within monthly data ofthe previous month and where the portfolios are value-weighted and rebalanced every month. For these portfolios they find a highly significant difference in average returns of-1.06%permonth with a t-statistic of-3.10, respectively. Ahighly signif­icant difference in Jensen's alphas between both portfolios also shows, that these differences in returns are persistent even when the portfolio returns are risk-adjusted relative to the FF-3 model. Furthermore, Ang et al. (2006) show that these abnormally low returns of stocks with high idiosyncratic volatility can not be explained by other risk factors or anomalies commonly known in finance literature. For these checks, the authors use double-sorted port­folios. For example they consider momentum effects, liquidity risk, analyst coverage etc.. For each double-sorted portfolio the relationship between lagged idiosyncratic volatility and stock returns stays significantly negative. They further analyse sub-samples covering every decade of the sample period July 1964 through December 2000. In each sub-sample they are able to observe significant differences in the returns between the two portfolios. Finally, they are able to rule out other possible explanations for the findings, including asymmetries in return distributions across business cycles and wrong expectations of agents, regarding the concurrence of recessions in the sample period. If the results are correct, a negative pricing of idiosyncratic volatility is tantamount with investors preferring riskier investments. In a sense this validates the assumption of risk averse investors1, on which modern portfolio the­ory is grounded. However this is only true if there is no other risk factor, which is omitted and which can be made account for this anomaly and if lagged idiosyncratic volatility is a correct proxy of future idiosyncratic risk.

Some further empirical investigations ofthe problem set were undertaken by Bali and Ca- kici (2008). The results of Ang et al. (2006) were confirmed by these independent authors, but Bali and Cakici (2008) further examine the robustness using different portfolio construc­tion settings. They find that different portfolio breakpoints, different weighting schemes and a different measure of idiosyncratic risk influence the overall results. More precisely, dur­ing the analysis, the authors use, in addition to the CRSP5 6 breakpoints, NYSE breakpoints, and breakpoints that lead to the construction of portfolios, which each represent the same proportion of the market. In addition to value-weighting stocks the authors also analyse equal-weighted portfolios.7 Further, for the estimation of idiosyncratic risk Bali and Cakici (2008) use, besides the measure also used by Ang et al. (2006), a measure that is only based on monthly stock data of the previous 24-60 months. It is claimed that this estimate is less exposed to microstructure problems such as asynchronous trading and bid-ask bounce (see Bali and Cakici, 2008, p.49). Finally, the authors show that their results are not driven by other firm characteristics, by either screening for size, price or liquidity. With regards to their results, especially the significance of the difference in average raw returns between the portfolio of stocks with the highest idiosyncratic volatility and the portfolio of stocks with the lowest idiosyncratic volatility changes for these varying portfolio construction settings. Thereby going to an equal-weighting scheme seems to be most influential. However, when looking at the difference in FF-3 alphas between the portfolio of stocks with the highest id­iosyncratic volatility and the portfolio of stocks with the lowest idiosyncratic volatility the significance of the difference mostly continuous throughout the varying portfolio construc­tion settings, which again indicates that idiosyncratic volatility is negatively priced relative to the FF-3 model. For a sub-sample of NYSE stocks only, even equal-weighted portfolios show these significant differences in alphas. Regarding the portfolios, that are independent of the represented market share (which represent 20% of the market share each), Bali and Cakici (2008) can show that the negative relationships between idiosyncratic volatility and future returns found by Ang et al. (2006) mostly vanishes. However this also might be drawn back to a decrease in the intended divergence of idiosyncratic risk between the portfolios' stocks. Still the work of Bali and Cakici (2008) shows that the robustness of the results of Ang et al. (2006) is prone to different portfolio weighting and construction schemes. Upon these findings, Bali and Cakici (2008) conclude that no clear return pattern can be identified and hence there is no negative trade-off between idiosyncratic risk and return.

A second paper of Ang et al. (2009) regarding idiosyncratic volatility deals with the prob­lems that come along with the methodology of portfolio construction and instead they use a Fama-McBeth regression style to compute the risk premium for idiosyncratic volatility. Some advantages of this approach are that it is independent of portfolio breakpoints and that it allows for simultaneous controls for multiple firm characteristics. Whenever simultaneous controls are performed using the portfolio methodology (hence via multi-sorted portfolios), portfolios tend to only contain a few stocks, which produce a lot of noise. The Fama-Mcbeth regression is conducted in two stages. Ang et al. (2009) use the following procedure. Cross­sectional regressions of the monthly excess returns on lagged idiosyncratic volatility, a set of factor-loadings, multiple firm characteristics and a constant for every month produce a time series of coefficient estimates for each regressor in the first stage. The inclusion of multiple firm characteristics in the first stage regression is done to control for variables that could bias the results, if omitted. Ang et al. (2009) especially use log firm size, book-to-market and lagged return, besides country specific dummies as control variables. Idiosyncratic volatil­ity, in the context of the first stage regressions, is again estimated using within monthly data of the previous month/months1. In the second stage each coefficient time series is regressed onto a constant. The resulting average represents the respective risk premia of each coef­ficient. T-statistics are used to test whether these are significantly different from zero. In addition to the different methodology the authors investigate the relationship between past idiosyncratic volatility and returns internationally using global, regional and country specific FF-3 models. The sample therefore is extended to daily returns of 23 developed markets for the period January 1980 through December 2003. They find evidence that the Idiosyncratic Volatility Puzzle is not only present for the US market, but rather for mostly all of the anal­ysed 23 markets and especially for each of the G7 countries. Moreover they can also show that these results hold regardless of which weighting scheme is implemented. Additionally, the authors report that there is a high international comovement in returns of stocks with high idiosyncratic volatility, which indicates, that the low returns of stocks with high idiosyncratic risk are not easily diversifiable (Ang et al., 2009).

More light on the relationship between expected idiosyncratic risk and expected returns is shed by Fu (2009). In his work he shows, that lagged idiosyncratic volatility is not an appropriate proxy for expected idiosyncratic risk, since standard residual errors of stock re­turns regressed on the FF-3 factors and a constant do not seem to follow a Gaussian Random Walk. Rather it seems that idiosyncratic volatility varies substantially over time and the dif­ference in logarithm idiosyncratic volatility follows a first-order moving average process.8 9 Hence a rational investor is not likely to only use lagged idiosyncratic volatility to form ex­pectations of future idiosyncratic volatility. Based upon this finding, Fu (2009) implements an EGARCH model to estimate conditional idiosyncratic volatility. Under the assumption that this estimate is a better proxy for idiosyncratic risk, he shows that the relationship be­tween idiosyncratic risk and expected returns is statistically significant and positive in value. Some others who adopt the EGARCH model to estimate conditional idiosyncratic volatility are Spiegel and Wang (2005) and Eiling (2013). Both papers also findapositive pricing of idiosyncratic risk in the cross-section of expected returns. All three papers constitute em­pirical support for the theoretical predictions of Merton (1987) and Levy (1978). However, although, as shown by Fu (2009), the results of Ang et al. (2006) and Ang et al. (2009) might not be appropriate to draw precise conclusions between expected idiosyncratic risk and expected returns, they are still puzzling, particularly if the true relationship between id­iosyncratic risk and the cross-section of stock returns is positive. As stated by Ang et al. (2009) themselves:

”[...]There are potentially large trading returns possible in going long (short) stocks with low (high) idiosyncratic volatility in international markets.” 10

Besides this statement, Ang et al. (2009) also contradict Fu (2009) by arguing that past idiosyncratic volatility is correlated with future idiosyncratic volatility and hence will be used by investors to form their expectations of future idiosyncratic risk, at least to some extent (Ang et al., 2009, p.3).

With regards to these puzzling and contradictory findings, efforts were made to explain the low returns of stocks with high idiosyncratic volatility. Huangetal.(2009)notethatomitting previous month's stock returns can negatively bias the relationship between lagged idiosyn­cratic volatility and returns. Hereby the omitted variable is a combination of short-term return reversals, which occur on a monthly basis (found by Jegadeesh, 1990) and the posi­tive relationship between monthly absolute stock returns and idiosyncratic volatility (found by Duffee, 1995). High absolute returns of the previous month drive up stock's lagged id­iosyncratic volatility. Hence by sorting on past idiosyncratic volatility, the portfolios of stocks with high idiosyncratic volatility are likely to contain an increased number of stocks with extremely high or low past returns. These get reversed the following month. Herewith Huang et al. (2009) explain the differing results of Ang et al. (2006) and Bali and Cakici (2008). In value-weighted portfolios more weight is given to the stocks with higher market capitalisation and consequently more relative weight is given to stocks with high past re­turns. These tend to reverse the following month and the reversal exceeds in magnitude the upwards return reversal of the past losers, causing significantly low portfolio returns. Incase of equal-weighted portfolios winner and looser stocks receive the same weights and the re­versals cancel out(Huang et al., 2009, p.166). Upon this explanation, Fu (2009)and Huang et al. (2009) show evidence that some results of Ang et al. (2006) can be explained by sim­ply controlling for return reversals. Reported is the setting, where idiosyncratic volatility is estimated relative to the FF-3using within monthly data ofthe previous month. However, if return reversal also can explain the low returns of portfolios with high idiosyncratic volatile stocks formed using longerestimation periods of idiosyncratic volatility seems questionable and is not reported.

Bali et al. (2011) use a different approach to explain the results of Ang et al. (2006), Bali and Cakici (2008), Ang et al. (2009). They report, that there is a significant and positive relationship between past extreme returns and future returns. They argue, that the low returns ofstocks with high idiosyncratic volatility might be driven by a group ofinvestors that have a preference for lottery-like payoffs. Hence these investors prefer to some extend riskier investments. Related to this explanation are among others Barberis and Huang (2008) and Boyer et al. (2009). For example Boyer et al. (2009) argue that investors use idiosyncratic volatility as an indication for high future skewness exposure and therefore are willing to pay a premium in return for the increased chance of extremely high future returns (see Boyer et al., 2009, p.200).

There are several more papers trying to explain the puzzle, but it seems, that none are able to fully explain all findings.1 Clearly, it is possible that the anomaly emerges as a result of different interlocking factors including several firm-characteristics, risk premia or market frictions, that comove with high idiosyncratic volatility. If such simultaneous reasons cause the puzzle to exist, it is challenging for econometricians to disentangle the interdependencies and to finally solve the puzzle. Besides solving the puzzle, it is a special interest, whether the anomaly persists after it is discovered. This is where this paper partly contributes to the literature.

3 Data

The analysis incorporates stocks traded at the Frankfurt Stock Exchange in the segment ”General Standard” and ”Prime Standard”. These stocks are part of the CDAX index, which is operated by the Deutsche Borse AG. The daily total return index and the market value time­series for each stock are obtained from Datastream11 12 for the period January 1990 through June 2016. The total return index data is used to calculate daily returns. Daily and monthly Fama- French factors (FF-3 factors) for the German market are provided by Prof. Richard Stehle from the Humboldt University Berlin and can be found at the university's website13. There is more than one factor set available, differing in the sample of stocks that is used to construct the FF-3 factors. This paper uses the monthly and daily FF-3 factors ”ALL” and ”Without Tax Credit”. Further information regarding the difference between the FF-3 factor sets can be found in Briickner et al., 2015.

To produce a reliable sample, a screening procedure similar to Lee (2011) is used. Because the total return indices are reported to the nearest tenth, values less than 0.01 are set to missing (Lee, 2011), so that the daily returns for the corresponding day and the following day are missing as well. Stocks with an initial public offering after May 2016 (practically identified if they have less than 30 non-missing values) are excluded from the sample because they wont be used in the portfolio construction. Furthermore non-common stocks, detected by name filters1, are also excluded. Dropping any day as a non-trading day if more than 95% of stocks have missing returns is not essential for later analysis, but is done to align the return time-series with the factor time-series and to identify stocks which are likely to be less liquid. The last step in the screening procedure is to set the daily returns for t and t — 1 to missing if Ri) t Ri) t —1 < 0.5 and Ry or Rijt —1 is at least 200%. Ry hereby denotes the gross return of stock i at time t. This step corrects for stock splits, which may have not been adjusted in the total return index.

After the screening, the sample contains 393 stocks and covers 6699 daily return observa­tions of 317 months. One disadvantage of the used sample is that only stocks are incorpo­rated which are listed at the time the data was accessed via Datastream. This might lead to a survivorship bias of later results, since stocks, which where delisted or downlisted14 15 before the date of access are not taken into account. Another consequence is that it also reduces the number of stocks with available return data for the early sample period. For example the average number of firms with available return data is 83 in 1990, 103 in 1995 and 232 in 2000 (after screening). Together this means increased noise and increased tendency towards a survivorship bias for the early years of the sample period. Therefore, to check robustness and for further examinations a sub-period of January 2003 until June 2016 is also analysed.

Other quite influential effects, regarding the sample are the Dot-com bubble, which peaked in 2000, and the financial crisis around 2008. It is noticeable that stocks have increased volatility during both time periods. This also transfers to the portfolios, which are con­structed during the analysis. Although the sample does not include every stock of the Ger­man stock universe it should reflect the average Euro of the overall (common) German stock market quite well.


1 Group of 7: Germany, France, Italy, Japan, Canada, the United Kingdom and the United States.

2 For more details see Fama and MacBeth, 1973, p. 614ff.

3 See Fama and MacBeth (1973) Table 3 and Table 5

4 A similar sample to Fama and French (1992) was later used by Ang et al. (2006),Bali and Cakici (2008) and Ang et al. (2009). All authors obtained the sample from the CRSP (Center for Research in Security Prices)

5 This is one one the Markowitz assumptions, see Markowitz, 1952.

6 Center for Research in Security Prices

7 Also a third weighting scheme is used, for which stocks are weighted by their inverse idiosyncratic volatility. This is done to give lower weights the the stocks with higher idiosyncratic volatility.

8 Similar to the procedure in their previous paper (2006), they examine the robustness for various idiosyncratic volatility estimation periods. For detailed information see Ang et al., 2009, p. 7f. .

9 See Fu, 2009, p.27, Table 1.

10 Ang et al., 2009, p.16

11 Hou and Loh (2016) concluded that every explanation explains less than 10% of the Idiosyncratic Volatility Puzzle.

12 The data can be found using the Datastream code LCDAXGNI.

13, 15/07/2018

14 Following Lee, 2011 the exclusion is performed by filtering stock names for ”REAL.ESTATE”, ”REIT”, ”GDR”, ”PF”, ”PREF”, ”PRF”, ”REAL.EST”, because these could represent real estate investment trusts, global depository receipts or preferred stocks. Afterwards a manually selection is undertaken to re-include for example ”DT.PFANDBRIEFBANK.-.TOT.RETURN.IND.”.

15 A stock gets downlisted, whenever the legal requirements for the exchange segment are not fulfilled anymore.

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The negative relationship between the cross-section of expected returns and lagged idiosyncratic volatility. The German stock market 1990-2016
University of Hannover  (Institute of Financial Markets)
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idiosyncratic volatility, cross-section of stock returns, market frictions
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Lasse Homann (Author), 2018, The negative relationship between the cross-section of expected returns and lagged idiosyncratic volatility. The German stock market 1990-2016, Munich, GRIN Verlag,


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