Patterns and Pricing of Idiosyncratic Volatility in the French Stock Market

We investigate the time series behavior of idiosyncratic volatility and its role in asset pricing in France. We find that both aggregate idiosyncratic and market volatility exhibit regime switching behavior similar to that in the U.S. and other developed countries. Furthermore, we find a marginally significant negative IVOL effect in the French stock market. We add new evidence to the mounting results questioning the ubiquity of the IVOL effect which highlights the importance of country verification of so called anomalies in the US, even in developed markets.


Introduction
In a recent study, Ang et al. [1] confirm the ubiquity of a puzzling negative idiosyncratic volatility (IVOL) effect [1] in 23 developed countries, including the seven largest developed economies (G7) where high volatility stocks earn low risk-adjusted returns. This is puzzling because traditional finance theory suggests that idiosyncratic volatility should not be priced as it could be eliminated at no cost through diversification. In case investors cannot fully diversify, finance theory suggests a positive (not a negative) relationship between idiosyncratic risk and return [2] [3]. Ang et al. [4] report a statistically significant difference in risk-adjusted returns between high and low IVOL portfolios of 1.31% per month across 23 developed markets. 1 However, in their study, they also report that 1 However, some studies suggest that Ang et al.'s findings are not robust to portfolio weighting schemes [5] [6] and controls for short-term reversals [7]. Others argue that a positive relationship exists between idiosyncratic volatility and returns using alternative measures of expected idiosyncratic volatility [8] [9] [10] [11] [12]. Theoretical Economics Letters among G7 countries not only did France show a decrease in the magnitude of the idiosyncratic volatility coefficient when idiosyncratic volatility was computed using a local Fama-French model instead of a world Fama-French model, but also the idiosyncratic volatility coefficient turned insignificant, indicating the absence of an IVOL effect. We investigate this further in this study. To the best of our knowledge, this is the first paper to examine the role of IVOL in pricing French stocks. If there is no significant relationship between returns and idiosyncratic volatility in the French stock market, this would add to mounting evidence questioning the ubiquity of the so called idiosyncratic volatility effect [5] [6] [11] [13]- [19].
Campbell et al. [20] report evidence of an increasing trend in idiosyncratic volatility in the U.S. relative to market volatility in the period 1962 to 1997. This is important since it implies increasing benefits from diversification. Theoretically, if there is an increasing trend in IVOL with a flat market volatility, investors would benefit from portfolio diversification. If IVOL has a decreasing trend or remains flat, then it is hard for investors to capture diversification benefits. In this sense, IVOL may not be priced in the stock market. However, Brandt, Brav, Graham, and Kumar [21] dispute Campbell et al.'s findings of an increasing trend, suggesting instead an episodic pattern in idiosyncratic volatility in the U.S. that is largely driven by the behavior of retail investors. In a related study Bekaert, et al. [22] show for the U.S. and in 22 other developed markets, that instead of a long-term trend, IVOL follows a stationary autoregressive process that occasionally switches to a higher variance regime. The trend in IVOL is an important issue relative to the benefit of portfolio diversification, hence our interest in the historical trend of IVOL in the French stock market, which is also the first evidence on this issue.
In this study, we investigate the behavior of aggregate idiosyncratic and market volatility from 1991 to 2012 in the French stock market. Then we examine the relationship between idiosyncratic volatility and cross-sectional stock returns. There are two reasons why we are interested in the IVOL effect in French stock market. First, most of the previous literature investigates trends and pricing behavior of IVOL in a group of European stock markets, but not in the French stock market on its own. The French stock market is one of the oldest stock markets in the world, and the 2 nd largest stock market ranked by capitalization in Europe following the U.K. stock market. Moreover, the French stock exchange was ranked the 4 th largest exchange in the world, with a total market capitalization of USD $3.5 trillion in November 2014 [23]. Surprisingly, the French stock market as an individual sample has so far been ignored in the literature, especially in the fields of asset pricing and financial anomalies. Therefore, this study is going to fill the gap in the literature. Second, the French stock market, known as the Paris Bourse, has been restructured in September 2000, and it plays the role of a regional stock exchange rather than a stock exchange for an individual country. The early stage of the French stock exchange was comprised by three sections: the official list (the Premier Marché), the lists for me- We contribute to the literature on the idiosyncratic volatility effect in a number of ways. First, we present the first empirical evidence examining the trend in IVOL and its role in asset pricing in the French stock market. Prior studies mostly focused on a group of developed stock markets rather than on individual stock markets. Hence we draw attention for both academia and practitioners on an individual developed stock market. Second, we add new evidence to the mounting results questioning the ubiquity of the IVOL effect. This highlights the importance of country verification of so called anomalies in the US, even in developed markets. Next, we provide empirical evidence on the pricing of IVOL both at the portfolio and firm levels. Previous research only focuses on the role of pricing IVOL on either portfolio level or firm level. Therefore, our results are comprehensive and more convincing. Finally, we confirm earlier evidence that idiosyncratic volatility in the French stock market exhibits a regime switching behavior rather than showing a long-term time trend.
The rest of the paper is organized as follows: Section 2 describes our data and methods; Section 3 presents the empirical results in three parts. First we report volatility patterns over time, then we examine the relation between volatility and market returns, and finally we examine the relation between idiosyncratic volatility and cross-sectional stock returns. Section 4 concludes the paper. Market returns are the value-weighted returns of all firms used in the study.

Data and Methods
We exclude investment trusts, closed-end funds, exchange traded funds, and preferred shares. At the beginning of each month, we exclude stocks that do not have continuous return records in the past 22 trading days. In order to reduce noise in computing IVOL for each stock, we also exclude stocks with daily returns less than −100% and/or monthly return greater than 200% as well as stocks with negative book-to-market (BM) ratio. Stocks with missing accounting data in a particular month were also excluded from the sample in that month.

Estimating Idiosyncratic Volatility
We follow Ang et al. [ where day t refers to the 22 trading days ending on the last trading day of month m-1. R i,t and MKT are excess returns of firm i and the market, respectively, over the risk-free rate. SMB is the excess return of small firms over big firms, and HML is the excess return of high book-to-market (BM) firms over low BM firms. SMB is the return of the upper half less the return of the lower half of all firms ranked in ascending order according to market capitalization while HML is the return of the bottom third less the return of the top third of all firms ranked in ascending order according to BM.

Portfolio Analysis and Fama-MacBeth Regressions
We use both portfolio-level analysis as well as firm-level Fama-MacBeth cross-sectional regressions to examine the relation between IVOL and expected returns. In portfolio-level analysis, firms are first sorted into tertiles at the start of each month based on IVOL and allocated to groups. We then compute each tertile portfolio's equal-and value-weighted raw returns for the current month. We also estimate each tertile portfolio's alpha (α coefficient) from the FF3-factor model (Equation (1)) using each tertile portfolio's full sample of monthly valueor equal-weighted returns.
As a robustness test, we also conduct firm-level Fama-MacBeth regressions to control for various variables. We estimate the following model and its nested R t , is realized stock return in month t. IVOL is realized idiosyncratic volatility as defined previously. SIZE at the end of month t is defined as the log of the firm's market capitalization at the end of month t. BM is the firm's book-to-market ratio six months prior, i.e. at the end of t-6. Following Jegadeesh and Titman [28], MOM at time t is the stock's 11-month past return lagged one month, i.e. return from month t-12 to month t-2. REV in month t is short-term reversal defined as the return on the stock in month t-1, following Jegadeesh [29] and Lehmann [30].

Descriptive Statistics
We report the descriptive statistics for three volatility series in Table 1 Table 1 shows that both IVOL EW and IVOL VW are 0.0188 and 0.0127 respectively, which is only about half compared to the IVOL in China, the biggest emerging stock market in the world [31]. The results support the view that stock markets in developed countries might be more stable than those in emerging stock markets. Moreover, the results also indicate that small firms seem to have higher idiosyncratic volatility than big firms as suggested by the higher mean of IVOL EW compared with IVOL VW . This is consistent with results in other markets particularly the U.S. However, IVOL EW is less variable than IVOL VW as indicated by its lower coefficient of variation (CV). MVOL on the other hand is more variable than IVOL EW having a higher CV, but it has a similar CV as IVOL VW .
As expected, our volatility measures are highly correlated as shown in Panel B, with correlation coefficients ranging from 0.849 to 0.913.
Panel C displays the autocorrelation structure of the three volatility series. As serial correlation is fairly high in all three series, we test for the presence of unit roots using the augmented Dickey and Fuller [32] test. Panel D shows that we can reject the presence of unit roots for all three series, whether or not a trend is included. Hence our analysis of the volatility series will be in levels instead of first differences. Figure 1 plots IVOL EW , IVOL VW , and MVOL. As indicated in Figure 1, there does not appear to be any long-term trend in any of these volatility series.

Does a Time Trend Exist?
where VOL represents IVOL EW , IVOL VW , and MVOL, and t is time. The estimated time trend b 1 parameter and its t-statistic are reported in Table 2. Over the full sample period from 1991: 08 to 2012: 06, the standard t-test shows a statistically significant positive trend in IVOL EW but no trend for both IVOL VW and MVOL. However, Vogelsang [34] points out that when errors in the trend regression are persistent, the student t-value often rejects the null hypothesis of no trend. As a consequence Bunzel and Vogelsang [25] developed the t-dan test which we employ in this study. 3 This test is valid whether or not a unit root exists in the error terms. The t-dan test also has better power than the standard t-test while retaining its good size properties. The t-dan test statistics reported in

Regime Switching Behavior in Idiosyncratic Volatility
In this section, we test for regime-switching behavior in idiosyncratic volatility.
We follow Bekaert et al.'s [22] method to further test whether or not our volatility series in the French stock market is characterized by a stationary process that occasionally switches between high-and low-volatility regimes. Bekaert et al. [22] argue that the upward trends of idiosyncratic volatility in the U.S. and 22 other developed markets were driven by the chosen starting-and ending time points. For example, if the starting point is in a low volatility period, while the end point is in a high volatility period, then the trend test would easily show a positive trend. Bekaert et al. [22] thus suggest that idiosyncratic volatility in the U.S. and 22 other developed markets are best characterized by a stationary process that occasionally switches between high-and low-volatility regimes. A regime-switching behavior in idiosyncratic volatility also appears evident in emerging markets with Nartea, et al. [35] documenting evidence of such behavior in the Chinese stock market, the world's largest emerging market.
To test for regime switching behavior in idiosyncratic volatility in the French stock market, we let volatility, y t , follow an AR(1) model where all parameters can take on one of two values depending on the realization of a discrete regime variable, s t . The regime variable follows a Markov chain with constant transition probabilities. Indexing the current regime by i the model is with e t ~ N (0,1). In the model, we force regime 0 (regime 1) to be the low (high) volatility regime and the mean levels (μ i ) of the volatility series of both regimes to be nonnegative (i.e. μ 1 > μ 0 > 0). The transition probability matrix, Φ, is 2 × 2, where each probability The transition probability matrix is: The transition probability parameters, p11 and p22, are constrained to be in (0,1) over the study period. We also reparameterize to ensure μ 2 > μ 1 > 0. to compute the standard errors. Table 3 shows that levels for both EW and VW IVOL in the low volatility regime (μ 0 ) at 0.0258 (IVOL EW ) and 0.0105 (IVOL VW ) respectively, which are both statistically significant. The corresponding levels in the high volatility regime (μ 1 ) at 0.0163 and 0.0181 are also both statistically significant. The differences between the levels of the two regimes for both volatility series are also statistically significant as indicated by the results of the Wald test.
Our results also show higher volatility in regime 1(σ 1 ) compared with regime 0 (σ 0 ) for both EW and VW IVOL. The EW (VW) volatility in regime 1 is 0.0040 (0.0035) compared with 0.0013 (0.0013) for regime 0. Thus we find that idiosyncratic volatility in the French stock market conforms with a stationary autoregressive process that occasionally switches between high and low-variance regimes. This is consistent with the behavior of idiosyncratic volatility in developed stock markets [22] and in the world's largest emerging market [35]. Figure 2 shows the smoothed probabilities of being in regime 0 for our three volatility series. Unlike the evidence reported by Bekaert et al. [22] in the U.S. stock market, we find that both high-and low-idiosyncratic volatility regimes in the French stock market have the propensity to stay for a period before switching to another. We observe this phenomenon several times over the study period. For example, Panel a of Figure 2 shows that IVOL EW was in a high    Table 2. For example, the absence of a trend in both IVOL VW and MVOL reported in Table 2 is consistent with both panels b and c in Figure 2, where both series start and end in the low volatility regimes over our study period. Panel a of Figure 2 shows that our testing period starts from a high-level of IVOL and ends in a low-level of IVOL, which indicates that the significant positive trend in IVOL EW reported in Table 2 is not due to the choice of sample period. However, results reported in Table 3 still suggest a significant regime-switching behavior for IVOL EW .
We also find it interesting that IVOL VW and IVOL EW exhibit a divergence in the period from 1992 to 1999, with IVOL EW being on a high-volatility regime while IVOL VW was on a low-volatility regime. We suggest that this could be due to the boom in high-tech stocks over this period. As high-tech stocks are normally smaller in size and more volatile than traditional listed firms, we expect IVOL EW to be more volatile than IVOL VW before the high-tech bubble burst around year 2000. We also find that both IVOL VW and IVOL EW were on a high volatility regime during the recent 2008 financial crisis. This is consistent with previous findings in the literature wherein stock markets are more volatile during the financial crisis period than other periods [35]. Finally, we find that both IVOL VW and IVOL EW show a convergent behavior after 2002 in the French stock market.
In sum, the results from Table 3 and Figure 2 indicate evidence of episodic behavior in all three volatility series, consistent with occasional regime shifts throughout the study period.

Portfolio-Level Analysis
In this section we examine the presence of an IVOL effect in the French stock market. Table 4 shows the average monthly returns and FF-3 alpha of EW and VW portfolios sorted according to idiosyncratic volatility. Though both the EW and VW return spreads between high and low IVOL portfolios are consistently negative at −0.91% and −0.71% per month respectively, they are not statistically significant. We only report the FF-3 alpha of each portfolio in the third column.
Stotz et al. [36] report that there are not big differences between Fama and French's alpha and Jensen's alpha. The EW alpha spread is likewise negative at −0.34% per month but also statistically insignificant. The exception is the statistically significant VW alpha spread at −0.58% per month. This appears to be consistent with the anomalous and puzzling evidence documented by Ang et al. [1] [4] and Brockman and Yan [37] for the U.S. market. However it is not as high as the −1.31% per month reported by Ang et al. [1] for the U.S.
Before we test the robustness of this apparent negative IVOL effect in the French stock market, we report the average of the monthly averages of various  characteristics of the IVOL-sorted portfolios in Table 5. We report values for IVOL, size (SIZE), BM (Value), momentum (MOM), and short-term reversal (REV). These variables are as defined previously. The high IVOL portfolio has three times as much IVOL as the low IVOL portfolio and the difference is highly statistically significant as expected. High (low) IVOL stocks also tend to be small (big) stocks. These results are consistent with Drew et al.'s [38] findings in the German and UK stock markets, where they find that small firms have higher IVOLs than big firms. High (low) IVOL stocks also tend to be previous losers (winners) in the past 11 months. However, there is no significant difference in the value and short-term reversal variables between high and low IVOL portfolios. We formally control these variables using firm-level cross-sectional regressions in the next section.

Firm-Level Cross-Sectional Regressions
We begin with univariate regressions on IVOL and our control variables.  trading daily returns data. SIZE is the firms' capitalization at the end of month t; Value is the firm's book-to-market ratio six months prior, i.e. at the end of t-6. Momentum represents the stock's 11-month past return lagged one month by following Jegadeesh and Titman [28], i.e. return from month t-12 to month t-2. REV in month t is short-term reversal defined as the return on the stock in month t-1, following Jegadeesh [29] and Lehmann [30]. T-statistics are reported in parenthesis.  Table 6. Univariate Fama-Macbeth regression results. We perform firm-level Fama-MacBeth cross-sectional regressions on the return on that month with values of the control variables in the previous month for the full sample period. The time-series averages of the slope coefficients and their associated t-statistics are reported in the table. IVOL is the standard deviation of the residuals of the FF3-factor model computed using the past 22 trading daily returns data. SIZE is the firms' capitalization at the end of month t; Value is the firm's book-to-market ratio six months prior, i.e. at the end of t-6. Momentum represents the stock's 11-month past return lagged one month by following Jegadeesh and Titman [28], i.e. return from month t-12 to month t-2. REV in month t is short-term reversal defined as the return on the stock in month t-1, following Jegadeesh [29] and Lehmann [30]. Newey-West T-statistics are reported in parenthesis.  the end of month t; Value is the firm's book-to-market ratio six months prior, i.e. at the end of t-6. Momentum represents the stock's 11-month past return lagged one month by following Jegadeesh and Titman [28], i.e. return from month t-12 to month t-2. REV in month t is short-term reversal defined as the return on the stock in month t-1, following Jegadeesh [29] and Lehmann [30]. Newey-West T-statistics are reported in parenthesis.

Concluding Remarks
In a recent study, Ang et al. [4] confirm the ubiquity of a puzzling negative idiosyncratic volatility (IVOL) effect [1] in 23 developed countries, including the seven largest developed economies (G7). However, in their study, they also report that among G7 countries not only did France show a decrease in the magnitude of the idiosyncratic volatility coefficient when idiosyncratic volatility was computed using a local Fama-French model instead of a world Fama-French model, but also the idiosyncratic volatility coefficient turned insignificant, indicating the absence of an IVOL effect. We investigate this further in this study.
We also investigate the behavior of aggregate idiosyncratic and market volatility in the French stock market in as much as Campbell, et al. [20] report evidence of an increasing trend in idiosyncratic volatility in the U.S. relative to market volatility which is disputed by both Brandt, et al. [21] and Bekaert, et al. [41]. We find that both idiosyncratic and market volatility do not exhibit long-term trends. Instead their patterns are consistent with regime switching behavior similar to that in the U.S. and other developed countries. Though we initially find a negative IVOL effect in the French stock market which is robust in bi-variate Fama-MacBeth regressions, the negative IVOL effect promptly disappears when we control for these well-known effects simultaneously.
We add new evidence to the mounting results questioning the ubiquity of the IVOL effect which highlights the importance of country verification of so-called anomalies in the US, even in developed markets.