UNIVERSITY OF AMSTERDAM
MASTER SCHOOL
Master of Science Finance
Major in Asset Management
Master Thesis
The idiosyncratic volatility puzzle
after the 2008 financial crisis
Supervisor
prof. Liang Zou
Candidate
Nicolò Podestà
ID: 11400528
This document is written by Nicolò Podestà who declares to take full responsibil-ity for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Contents
Summary vii 1 Introduction 1 2 Literature Review 3 2.1 Lottery Preferences . . . 4 2.2 Market Frictions . . . 63 Data and methodology 9 3.1 Analysis of the idiosyncratic volatility puzzle . . . 9
3.2 Analysis of the economic explanations. . . 10
3.3 Economic explanations . . . 11
3.3.1 Idiosyncratic skewness . . . 12
3.3.2 Bid-ask spread . . . 13
4 Empirical results 15 4.1 Sample descriptive statistics . . . 15
4.2 Investigating the existence of the idiosyncratic volatility puzzle . . . . 16
4.3 Fama-MacBeth regression . . . 18
4.4 Robustness tests . . . 21
4.4.1 Idiosyncratic volatility puzzle with CAPM . . . 21
5 Conclusion 25
6 Additional Tables 27
Summary
The objective of this study can be summarised in two parts. Since all the studies related to the idiosyncratic volatility are referring to a time period before the 2008 financial crisis, the first aim is to replicate the model commonly used in the previous literature to investigate if the idiosyncratic volatility puzzle is still present nowadays. My empirical results confirm that the puzzle is still observed in the markets and it even increased slightly. The second aim of this research is to test the two economic variables that in previous studies were observed to be the main driving forces of the puzzle. These two explanations are the expected idiosyncratic skewness of returns and the bid-ask spread. I was able to run a Fama-MacBeth regression to test their relative importance in the most recent dataset, and my results confirms that the puzzle can not be explained entirely by these individual economic explanations. In fact, the expected idiosyncratic skewness can only explain 12.8% of the puzzle, while the bid-ask spread explains 30.2% of it.
Chapter 1
Introduction
Ang, Hodrick, Xing, and Zhang (2006), in their paper “The Cross-Section of Volatil-ity and Expected Returns”, discovered that there exists a negative correlation be-tween idiosyncratic volatility of stocks and stock returns. This empirical result is surprising because asset pricing theories state that either idiosyncratic volatil-ity does not have effect on returns because rational investors have well diversified portfolios or investors demand to be rewarded for the higher risk, thus stock with high idiosyncratic volatility should earn a higher risk premium. This apparent con-tradiction between the theory and empirical results has been named idiosyncratic volatility puzzle.
Several papers have tried to explain the puzzle, but no one seem to have nailed a full explanation. Similar to the analysis of Hou, and Loh (2015), in this research the possible explanations will be considered either as lottery preferences of investors or as market frictions. The main example of a lottery preference that will be analized is stock returns expected idiosyncratic skewness, while the main market friction taken into consideration will be the bid-ask spread.
After the 2008 financial crisis, we might expect some market dynamics to have changed. For instance, investor will probably be more willing to diversify their portfolio; in addition, market frictions became more important, therefore it is not
1 – Introduction
clear whether the economic explanations driving the idiosyncratic volatility puzzle are still present nowadays.
In this research I will investigate this issue. In particular, I will test the hypoth-esis of whether the cross-sectional stock returns do not have a negative relationship with the idiosyncratic volatility anymore. For this purpose, I will first replicate the model proposed by Ang et. al (2006) on a dataset of stocks from January 1963 to December 2008. If the idiosyncratic volatility puzzle is confirmed, I will apply it to a more recent sample of data, from January 2009 to December 2016. I will be able to assess if the idiosyncratic volatility puzzle is still present after the financial crisis by observing the empirical results.
If this negative relationship between idiosyncratic volatility and stock returns is still present, my aim is to investigate which economic explanations can explain this puzzling relationship in the most complete way. Following the empirical results of Hou, and Loh (2015), I chose to analyse one economic relationship related to lottery preferences of investors and one related to market frictions. The former is the expected idiosyncratic skewness of returns, while the latter is the end of day bid-ask spread. I will use a Fama-MacBeth two stage regression to compare the relative contribution of these two variables. The procedure will be explained more in depth in Chapter3. At the end of this analysis, I expect to observe a percentage that represents the proportion of the puzzle which is explained by the individual economic variables.
Chapter 2
Literature Review
The most important paper related to the idiosyncratic volatility puzzle was pub-lished by Ang, Hodrick, Xing, and Zhang (2006); in it, they empirically found a negative correlation between idiosyncratic volatility and stock returns by doing a cross-section analysis. The paper is also important because of the methodology used to compute the idiosyncratic volatility; starting from a linear time series regression of stock returns on the Fama-French (1993) factors, they measured the idiosyncratic volatility as the standard deviation of the residuals. This is the same framework that all the following papers used in trying to explain the economic reasons underlying the puzzle.
The most comprehensive comparison of the different plausible explanations was published by Hou, and Loh (2015). This paper is important because it uses the a single methodology to compare different papers and investigates which explanations (or group of) explain the puzzle better. To compare the robustness of the various economic explanations, they use a Fama-MacBeth procedure; in my research I will use the same framework, but on different time samples.
The potential explanations of the idiosyncratic volatility puzzle are several, thus in order to analyse them it is reasonable to separate them in two main groups: those related to the lottery preferences of investors and those related to market frictions.
2 – Literature Review
2.1
Lottery Preferences
An important paper written by Barberis and Huang (2008) argues that, by looking at investors’ preferences according to cumulative prospect theory, people overes-timate small probabilities of future gains. Cumulative prospect theory, originally proposed by Kahneman and Tversky (1979), states that investors estimate risk fol-lowing a function of gains/losses that is concave in the gain region and convex in the loss region. A particular characteristic of this behavioral theory is related to the overweighting of probabilities related to rare events; in fact, investors tend to overestimate the tails of the returns distribution. This behavior implies a preference for positively skewed securities, which causes securities with this characteristic to be overpriced, eventually earning low returns for investors.
Skewness is usually infuenced by small probability events, therefore it is not a stable measure over time. To circumvent this issue, Boyer, Mitton, and Vorkink (2010) proposed a model for estimating expected skewness, that will be later tested in relationship with the idiosyncratic volatility. In order to compute the expected skewness, they first try to establish how firm-specific characteristics are related to skewness using a cross sectional regression. After having found the skewness value related to each firm at time t, they build another regression to find the value of expected skewness at time t + T , where T is the total investment horizon in terms of months. By following this approach, the relation between skewness and firm characteristics varies through time (as it should) and the future skewness can be estimated for each month. The expected skewness used in this research will be computed following the same procedure. The paper by Boyer, Mitton, and Vorkink (2010) uses an investment horizon of 60 months. With their computations, they prove that idiosyncratic volatility is a better predictor of expected skewness compared to skewness at time t. In addition, they test the relation between the idiosyncratic volatility puzzle and the expected skewness and they propose a possible
2.1 – Lottery Preferences
explanation for the puzzle, linking it to expected skewness. In fact, they conclude that investors are willing to pay a premium for high idiosyncratic volatility stocks because this measure is a very good predictore for future skewness. Some investors may be willing to accept a higher volatility stock (with lower expected returns) with the expectation of a greater return in the future if the long tail of the distribution actually materializes.
Another plausible explanation related to investors’ preferences comes from a pa-per by Bali et al. (2011); following the aforementioned cumulative prospect theory, they examine the role of extreme positive returns in the pricing of stocks. In order to assess this effect, they created portfolios of stock sorted by their relative maxi-mum daily return during the previous month and compared the difference between returns of the portfolio with the highest maximum daily return and the portfo-lio with the lowest. The difference they found is significant and equal to -1.03%, meaning that the stocks with highest extreme positive returns are sought after by investors and their prices increase consequently. The authors of the paper link this results to the cumulative prospect theory, stating that when probability weighting is estimated uncorrectly, investors tend to over-value stocks with a small probability of huge positive returns. They also discover that stocks with extreme results are usually small and illiquid. They subsequently research the relationship between id-iosyncratic volatility and maximum daily returns, with the aim of explaining at least in part the idiosyncratic volatility puzzle. Thus, they run the previous analysis by controlling for idiosyncratic volatility because, by construction, the maximum daily return has a positive correlation with it and might be considered a proxy for that measure. Even after controlling for idiosyncratic volatility, the results of the analysis are similar to what they previously found. Therefore, they conclude by positing that the typical investor does not like stocks with high idiosyncratic volatility but prefers stocks that show a lottery-like payoff, so their prices and returns are influenced abnormally by this kind of preference.
2 – Literature Review
Han and Kumar (2013) examine the role of speculative trading activities of re-tail investors on prices. They proposed an indicator that can be used to describe stocks which are usually preferred by lottery-seeking investors, called “retail trad-ing proportion” (RTP). This measure is computed by dividtrad-ing the monthly dollar value of trades below $5000 by the dollar value of its total trading volume in the same month; the authors proved the efficiency of this measure as a good proxy for identifying stocks that are subject to high levels of speculative trading. In this context, stocks with a high RTP measure are those which show lottery features, thus are preferred by the investors who are risk-seekers. Not surprisingly, in their analysis they found the quintile comprising of high RTP stocks to have the lower risk-adjusted premium return; they also noticed that this result is entirely due to the undeperformance of high RTP stocks and not due to the overperformance of the low RTP stocks.
2.2
Market Frictions
Moving to the explanations related to frictions present in the stock market, the main factor that is thought to influence the correlation between returns and idiosyncratic volatility is the illiquidity. Several papers have proposed different ways to measure it or, more correctly, to create proxies for it.
One of the most used proxy for illiquidity was proposed in an influential paper by Amihud (2002) and it has been called “Amihud Illiqudity Measure” since then. In the paper, Amihud suggested a simple way to compute this proxy, which is basically the ratio of daily stock return to its dollar volume. In addition, Amihud computed a cross-sectional analysis to investigate the role of illiquidity on stock excess returns. The results show that it exists a positive correlation between returns and illiquidity: investors wants to be rewarded to hold a stock that is very illiquid in their portfolios, because illiquidity might be considered as a risk. Not only the paper shows that
2.2 – Market Frictions
expected returns are different across stocks depending on illiquidity, but also that market expected illiquidity affects the stock excess returns. These effects are more pronounced on small stock portfolios.
Han, Hu, and Lesmond (2014) argue that the entire idiosyncratic volatility puzzle is driven by the underlying bid-ask spread, which they consider the best proxy for stocks illiquidity. In fact, they found that the bid-ask spread is negatively correlated with stock returns, which is a puzzling result for the same reasons as the idiosyncratic volatility puzzle.
Chapter 3
Data and methodology
3.1
Analysis of the idiosyncratic volatility puzzle
For the purpose of this research, I used a sample of CRSP common stocks (with share codes 10 or 11), considering a time range from January 1963 to December 2016. Data on stock prices, returns, shares outstanding, bid and ask prices are obtained from the CRSP database. Data on book value and size are obtained from the Compustat database. The Fama-French factors that will be used in the regressions to compute the idiosyncratic volatility of each stock are obtained from Ken French website.
In the analysis I only considered stocks with a non-negative book value and a non-negative size. In addition, I removed the so-called penny stocks by applying a price threshold of 1$, exluding from the analysis all the stocks with a lower price.
Following Ang et al. (2006), the idiosyncratic volatility for each stock is a firm-specific shock that influence stock returns. There is not a single way to estimate the idiosyncratic volatility, but following the previous literature I will use a proxy. The proxy that will be used to estimate it is the standard deviation of the error term
3 – Data and methodology
Fama-French three factor model:
rt= α + β1HMLt+ β2SMBt+ β3Rm+ t [3.1]
where rt is the stock return (not the excess return), HMLt and SMBt are the
factors High Minus Low and Small Minus Big introdued by Fama and French (1993), and Rm is the market risk premium.
After having identified the idiosyncratic volatility for each stock in the sample, I sorted them into 5 different portfolios based on this value, where quintile 1 is the one comprising stocks with the lowest idiosyncratic volatility and quintile 5 is made of stocks with the highest. The portfolios are computed and rearranged monthly.
By creating these 5 portfolios, the relationship between them and the average returns can be easily observed. If the idiosyncratic volatility puzzle is present, Portfolio 1 (the lowest volatility one) will have higher returns than Portfolio 5 (the highest volatility one), or in other words the difference between returns of Portfolio 5 and those of Portfolio 1 will be negative.
3.2
Analysis of the economic explanations
In order to compare the relative importance of the economic explanations, I used a cross sectional Fama-MacBeth regression, which is the most common method used in the previous literature to investigate the relationship between idiosyncratic volatility and stock returns.
The first step of the Fama-MacBeth regression is a cross-sectional regression of individual stock returns on idiosyncratic volatility (IV OL) at time t − 1 as explana-tory variable:
3.3 – Economic explanations
In the second step, IV OL at t − 1 becomes the dependent variable and it is regressed on the possible economic explanation (ECO):
IV OLit−1 = αt−1+ δt−1ECOit−1+ µit−1 [3.3]
Since the economic explanations that I analyse in this research and the idiosyn-cratic volatility are correlated, by running this regression I can assess their relative impact.
By looking at the last regression, two components can be separated: δt−1ECOit−1
is related to the relationship between the economic explanation and the idiosyncratic volatility, while the component αt−1+ µit−1 is the residual component of the
regres-sion and it is not related to the explanation.
In the last step, the estimated coefficient γt from equation [3.2] will be
decom-posed using the linearity of covariances:
γt =
Cov[Rit, IV OLit−1]
Var[IV OLit−1]
= Cov[Rit, αt−1+ δt−1ECOit−1+ µit−1] Var[IV OLit−1]
= Cov[Rit, δt−1ECOit−1] Var[IV OLit−1]
+Cov[Rit, αt−1+ µit−1] Var[IV OLit−1]
= γEt + γtR [3.4]
By decomposing the covariances, γE
t /γt measures the relative impact of the
eco-nomic explanation on the idiosyncratic volatility, while γR
t /γt measures the fraction
of the puzzle that is still unexplained.
3.3
Economic explanations
For the purpose of this analysis, I chose to investigate the role of two economic explanations of the idiosyncratic volatility, one related to the lottery preferences of
3 – Data and methodology
investors and the other related to market frictions; the former explanation is the expected skewness of stock returns and the latter is the stock bid-ask spread. I will use these two explanations in my analysis because Hou, and Loh (2015) discovered that they are the variables with the most promising results when trying to explain the idiosyncratic volatility puzzle.
3.3.1
Idiosyncratic skewness
In order to compute the idiosyncratic skewness, I used the method proposed by Boyer, Mitton, and Vorkink (2010). The first step of this model relates to choosing the time horizon over which investors are expecting to observe an extreme positive outcome. In my computations I chose an investment horizon of 60 months. Then, the model aims to estimate a relationship between stock-specific characteristics and the idiosyncratic skewness of returns:
isi,t = β0+ β1isi,t−T + β2ivi,t−T + γSIZEi,t−T + it [3.5]
where isi,t and ivi,t are the historical estimates of idiosyncratic skewness and
idiosyncratic volatility respectively, considering daily returns from the first day of month t − (T + 1) to the end of month t, and SIZEi,t−T is a control variable
representing the market capitalizion of the stock considered.
Using the parameters estimated in Equation3.5, the expected idiosyncratic skew-ness at time t + T can then be estimated with the following regression:
Et[isi,t+T] = ˆβ0+ ˆβ1isit+ ˆβ2ivit+ ˆγSIZEit [3.6]
This model aims to deal with the issue of using the historical skewness to ex-plain the idiosyncratic volatility puzzle. In fact, historical skewness is a variable that does not vary through time, while in reality the skewness of returns changes greatly.
3.3 – Economic explanations
Hence, the expected idiosyncratic skewness computed with this model allows the re-lationship between stock-specific characteristic and the skewness of return to change through time.
3.3.2
Bid-ask spread
Following the findings of Han, Hu, and Lesmond (2014), I will use the stock bid-ask spread as the variable related to market frictions. They discovered that analysing the pricing of idiosyncratic volatility is closely related to analysing the bid-ask spread. In their analysis, they used the daily percentage spreads which is the difference between the ask and the bid divided by the average of the bid-ask in month t − 1. The bid and ask values are the end of day values found on CSRP.
Chapter 4
Empirical results
4.1
Sample descriptive statistics
The first step of my analysis is to replicate the methodology used by Ang et al. (2006) on the dataset with the characteristics listed in Chapter 3. I divided the entire dataset in two parts: the first one ranging from January 1963 to December 2008, and the second one ranging from January 2009 to December 2016.
This separation into two smaller dataset was made to test if the computations applied to the first yield the same result as in the paper by Ang et al. (2006), since the time ranges considered are similar (in the original paper they used data from July 1963 to December 2000). Results considered comparable would confirm that my methodology correctly follows the original one, thus I can replicate it on the data sample from January 2009 to December 2016 to investigate if the idiosyncratic volatility puzzle is still relevant after the 2008 financial crisis.
Table 6.1 shows some descriptive statistics in the first data sample, while Ta-ble 6.2 shows statistics for the second sample. The tables report the mean, the standard deviation, the sample size, and the percentiles related to stock returns, idiosyncratic volatility, size of the firms, skewness and spread, which are the firm
4 – Empirical results
specific characteristics that will be used in this analysis.
By looking at Tables 6.1 and 6.2 it can be noticed that the mean returns are slightly higher in the most recent sample (2% compared to 1.5%), the idiosyncratic volatility is slightly lower (13.6% against 15.9%) and the average spread decreased heavily (from 2.6% to 0.7%). Another feature that can be observed is that the average market capitalization increased to a great degree, moving from $1862M to $5132M.
Tables 6.3 and 6.4 show the correlations between idiosyncratic volatility and the aforementioned firm-specific characteristics. The main takeaway from these two tables are the following. Bid-ask spread and market capitalizaton (i.e. size) are inversely correlated (-0.098 in Table 6.3 and -0.082 in Table 6.4), because bid-ask spread is a proxy for illiquidity, thus a higher bid-ask spread means that the stock is less liquid, a characteristic that is prominent in small firms. The idiosyncratic volatility is positively correlated with both the bid-ask spread and expected skewness (0.134 and 0.161 respectively, in Table 6.4), and this is a promising correlation be-cause the aim of this analysis is to assess the level of idiosyncratic volatility which is explained by these economic characteristics. Finally, not surprisingly, idiosyncratic volatility is negatively correlated with market capitalization (-0.107 in Table6.3and -0.151 in Table6.4), which is expected because smaller firms are more illiquid thus more volatile.
4.2
Investigating the existence of the idiosyncratic
volatility puzzle
As mentioned in Chapter 3, for each month I formed 5 quintile portfolios based on the stocks’ idiosyncratic volatilities and computed the difference between the portfolio with the highest average idiosyncratic volatility and the portfolio with the
4.2 – Investigating the existence of the idiosyncratic volatility puzzle
lowest. As shown in Table4.1, the results resemble the ones originally found by Ang et al. (2006). The values are not exactly the same because the data samples differ a little, but the general patterns still stand. In fact, the portfolio with the highest volatility (Portfolio 5) yields a lower return for investors and its difference with the lowest volatility portfolio is negative and statistically significant.
Table 4.1. Portfolios Sorted by Idiosyncratic Volatility Relative to FF-3 (the sample period is from January 1963 to December 2008). The portfolios were formed every month by sorting stocks based on their idiosyncratic volatility relative to the Fama-French 3 factors model. Portfolios were formed monthly, based on their daily returns of their previous month. Portfolio 1 (5) is the portfolio of stocks with the lowest (highest) idiosyncratic volatilities. The statistics related to returns, i.e. Mean and Std. Dev., are measured in monthly percentage terms and apply to total, not excess, returns. The column Size is the log market capitalization. The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1 (t-statistic in parentheses). The Alpha column reports the constant from the regressions made with the Fama–French (1993) three-factor model.
Rank Mean Return Std. Dev. Size FF-3 Alpha 1 1.12 3.85 5.21 0.03 2 1.24 4.88 5.93 0.09 3 1.30 5.96 4.98 0.07 4 0.83 7.56 3.12 0.02 5 0.07 8.54 2.69 0.01 5-1 -1.05 (-2.97)
The next step is to make the same analysis on the most recent data sample, to check if the idiosyncratic volatility is still present after the 2008 financial crisis. The results are listed in Table 4.2. The difference between the portfolio with the highest volatility and the one with the lowest is statistically significant and even more pronounced. In fact, in this sample the portfolio of stocks with the highest idiosyncratic volatility offers a returns that is 1.52% lower compared to the portfo-lio of stocks with the lowest idiosyncratic volatility. This result confirms that the
4 – Empirical results
idiosyncratic volatility puzzle is still present in the most recent dataset. Not surpris-ingly, we can also detect other patterns from the other statistics listed in the table. Moving from Portfolio 1 to Portfolio 5 we can observe a decrease in average size and an increase in standard deviation. These results are not puzzling, because in general it is expected for smaller stocks to have a higher idiosyncratic volatility (since they are less anchored to market movements) and a higher standard deviations of returns.
Table 4.2. Portfolios Sorted by Idiosyncratic Volatility Relative to FF-3 (the sample period is from January 2009 to December 2016). The portfolios were formed every month by sorting stocks based on their idiosyncratic volatility relative to the Fama-French 3 factors model. Portfolios were formed monthly, based on their daily returns of their previous month. Portfolio 1 (5) is the portfolio of stocks with the lowest (highest) idiosyncratic volatilities. The statistics related to returns, i.e. Mean and Std. Dev., are measured in monthly percentage terms and apply to total, not excess, returns. The column Size is the log market capitalization. The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1 (t-statistic in parentheses). The Alpha column reports the constant from the regressions made with the Fama–French (1993) three-factor model.
Rank Mean Return Std. Dev. Size FF-3 Alpha 1 1.64 3.91 9.37 0.02 2 1.68 4.64 8.71 0.05 3 1.71 5.78 8.08 0.06 4 0.94 7.32 7.37 0.05 5 0.12 8.40 6.49 0.01 5-1 -1.52 (-3.23)
4.3
Fama-MacBeth regression
Since the puzzle is still present after the 2008 financial crisis, I focus on testing whether the economic explanations that I chose to analyse can explain it. In order to make this analysis, I procedeed in 3 stages that are summarised in Table 4.3.
4.3 – Fama-MacBeth regression
Table 4.3. Fama-MacBeth cross-sectional regressions. In Stage 1, returns at month t are regressed on month t − 1 idiosyncratic volatility (IVOL), as explained in Equation 3.2: Rit = αt + γtIV OLit−1 + it. In Stage 2,
IVOL is regressed on the economic explanations as explained in Equation 3.3:
IV OLit−1 = αt−1 + δt−1ECOit−1 + µit−1. In Stage 3, the two components of
the regressions made in Stage 2 can be decomposed to investigate the relative im-portance of the economic explanations with respect to the idiosyncratic volatility. The time-series average of γtE divided by the time-series average of γt measures the fraction of the negative relationship between idiosyncratic volatility and re-turns explained by the economic explanation, while γtR divided γt measures the
fraction that is still not explained by the explanation. The time-series averages of estimated coefficients are reported with t-statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively.
Stage Variable Exp Skewness Spread 1 Intercept 0.301*** (5.48) 0.391*** (4.37) IVOL -12.826*** (-7.94) -19.152*** (-6.78) 2 Intercept 1.140*** (23.54) 1.734*** (34.36) Explanation 1.232*** (15.45) 22.288*** (19.18) Adj R-squared 14.13% 23.5% 3 Explanation (γE t ) -1.642 12.8%*** (4.80) -5.792 30.2%*** (5.92) Residual (γR t ) -11.184 87.2%*** (22.55) -13.360 69.8%*** (13.26) Total -12.826*** (-7.94) 100% -19.152*** (-6.78) 100%
idiosyncratic volatility in month t − 1. When considering the Expected Skewness explanation, the coefficient for the idiosyncratic volatility is significant and equal to -12.826. In the case of the Spread explanation, the idiosyncratic volatility coefficient is -19.152 and significant as well.
4 – Empirical results
coefficient found in the first stage is regressed on the economic explanation, as ex-plained in Equation 3.3. The coefficient of the Expected Skewness is 1.232 and very significant; the coefficient of Spread is 22.288 and also significant. These re-sults suggest that both these two economic explanations are indeed related to the idiosyncratic volatility, because the interpretation of this regression is that for an increase of one unit of Expected Skewness the idiosyncratic volatility increases by 1.232%, while a unit change of Spread makes idiosyncratic volatility to increase by 22.288% (the increase in this case is very big because an increase of one unit in daily percentage spread would be an abnormal increase). The adjusted R2 of the
two regressions is fairly high, considering the scope of this analysis. In the first regression, the 14.13% of the variation in idiosyncratic volatility can be explained by the Expected Skewness, while in the regression with Spread as explanation the adjusted R2 is higher and equal to 23.5%.
Moving to the third stage of the analysis, I used Equation 3.4 to compute the covariances between the returns and the economic explanations, with the aim of separating the part of idiosyncratic volatility explained by Expected Skewness and
Spread (which is defined as γE
t ) from the part explained by residual components not
related to the economic explanations (defined as γR
t ). Regarding Expected Skewness,
the average γE
t is -1.642, while the average γtR is -11.184 (for a total of -12.826).
By computing the proportion γE
t /γt, the fraction of the coefficient of idiosyncratic
volatility found in Stage 1 explained by Expected Skewness is 12.8%, while the resid-ual component explains 87.2% of the coefficient. When looking at Spread, the value of γE
t is -5.792 and γtRis -13.360 (the total is -19.152). Hence, the proportion γtE/γt
is 30.2%.
At the end of this analysis, the main takeaway is that Expected Skewness explains only 12.8% of the idiosyncratic volatility puzzle, while the Spread shows a more interesting result because it explains 30.2% of the puzzle. A possible interpretation for these very different results might be related to how the possible explanations
4.4 – Robustness tests
variable are formed. By construction, the Spread variable reflects market frictions that strongly drive short-term returns for the investors, and bigger bid-ask spreads are usually a characteristic of stocks which are not very liquid and more volatile; by not being very liquid, prices adjust more slowly, therefore the volatility premium that an investor would expect are not incorporated in prices correctly. On the other hand, Expected Skewness might be a worse explanation for two main reasons. First, it is not an observable variable but it should be estimated with a model. The model has to make assumptions, therefore it might be improved to estimate the expected skewness more efficiently. Second, stocks with a high expected skewness might be priced more quickly by the market because they are not necessarily illiquid, unlike stocks with a high bid-ask spread.
4.4
Robustness tests
4.4.1
Idiosyncratic volatility puzzle with CAPM
In Chapter 4.2 I followed the procedure by Ang et al. (2006) to compute the id-iosyncratic volatility for every stock considerd in the analysis, created 5 monthly portofolios based on their idiosyncratic volatility value and observed the relation-ship between them and returns. This procedure is explained in Chapter 3.1 and uses the Fama-French 3 Factors model to compute the residual errors, which are used to estimate the idiosyncratic volatility. The Fama-French model is just one of the many asset pricing models, therefore it is not certain that this is the best way for estimating the idiosyncratic volatility. As a robustness test, I replicated the en-tire analysis by using another asset pricing model, the Capital Asset Pricing Model (CAPM), to estimate the idiosyncratic volatilities that will influence the formation of the monthly portfolio. The purpose of this test is to check if the idiosyncratic volatility is still present even with a different estimation of the idiosyncratic volatility
4 – Empirical results
characteristic itself.
The regression equation for the CAPM is the following:
rt= rf + β1Rm+ t
where rf is the risk free rate at time t, Rm is the market risk premium, and the
idiosyncratic volatility is computed as the standard deviation of the residual term. The analysis of the idiosyncratic volatility puzzle in the dataset going from Jan-uary 1963 to December 2008 is shown in Table4.4. The difference between Portfolio 5 and Portfolio 1 is negative (and significant), therefore even using the CAPM model to estimate idiosyncratic volatility the puzzle was present in the first part of the dataset.
Table 4.4. Portfolios Sorted by Idiosyncratic Volatility Relative to CAPM (the sample period is from January 1963 to December 2008). The portfolios were formed every month by sorting stocks based on their idiosyncratic volatility relative to the Fama-French 3 factors model. Portfolios were formed monthly, based on their daily returns of their previous month. Portfolio 1 (5) is the portfolio of stocks with the lowest (highest) idiosyncratic volatilities. The statistics related to returns, i.e. Mean and Std. Dev., are measured in monthly percentage terms and apply to total, not excess, returns. The column Size is the log market capitalization. The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1 (t-statistic in parentheses). The Alpha column reports the constant from the regressions made with the Fama–French (1993) three-factor model.
Rank Mean Return Std. Dev. Size CAPM Alpha 1 1.35 3.22 5.15 0.14 2 1.48 3.91 6.04 0.19 3 1.44 4.71 5.79 0.22 4 1.12 6.65 2.65 0.00 5 0.57 8.01 2.21 0.00 5-1 -0.78 (-3.55)
I replicate the same analysis with the second part of the dataset, which has a sample period from January 2009 to December 2016. The results are summarised in
4.4 – Robustness tests
Table4.5. The difference between portfolio 5 and portfolio 1 is equal to -1.06% (with a t-statistic of -5.12). Even in the most recent dataset, the idiosyncratic volatility puzzle is confirmed when using the CAPM instead of the Fama-French model. As expected, also the other characteristics such as standard deviation and size follow the same pattern found with the Fama-French model.
Table 4.5. Portfolios Sorted by Idiosyncratic Volatility Relative to CAPM (the sample period is from January 2009 to December 2016). The portfolios were formed every month by sorting stocks based on their idiosyncratic volatility relative to the Fama-French 3 factors model. Portfolios were formed monthly, based on their daily returns of their previous month. Portfolio 1 (5) is the portfolio of stocks with the lowest (highest) idiosyncratic volatilities. The statistics related to returns, i.e. Mean and Std. Dev., are measured in monthly percentage terms and apply to total, not excess, returns. The column Size is the log market capitalization. The row “5-1” refers to the difference in monthly returns between portfolio 5 and portfolio 1 (t-statistic in parentheses). The Alpha column reports the constant from the regressions made with the Fama–French (1993) three-factor model.
Rank Mean Return Std. Dev. Size CAPM Alpha 1 1.45 2.93 9.23 0.11 2 1.41 3.25 8.49 0.14 3 1.37 4.88 7.63 0.09 4 0.76 6.43 2.22 0.02 5 0.39 7.80 2.03 0.00 5-1 -1.06 (-5.12)
4.4.2
Conventional approach
The method used in this research for investigating the relative importance of the individual economic explanations is different from the conventional approach used by several papers before. In fact, many papers used the following equation:
4 – Empirical results
where the idiosyncratic volatility is simply used as control variable in a regression of returns on economic explanations. The procedure states that if γR
t is different
from zero, its difference with the coefficient of the idiosyncratic volatility γt found
in the first equation of the MacBeth procedure (Equation3.2) would be the fraction of the idiosyncratic volatility puzzle explained by the economic variable. The issue with this procedure lies with the fact that the coefficients are not comparable: γt
is the result of a variation of idiosyncratic volatility itself, while γtR measures the variation that is independent of the economic explanation (because it was introduced as control variable).
Table4.6confirms the issue with the old approach. In fact, if the aforementioned regression is ran, the economic explanations do not have any predictive power if we control for IVOL (the t-statistics are -0.41 and 1.32 for Expected Skewness and Spread respectively). This means that the parts of the explanations that are not related to idiosyncratic volatility are not able to influence returns negatively, therefore it would mean that these variables are not good candidates when investigating the idiosyncratic volatility puzzle, while I proved in Chapter 4.3 that they actually explain a good percentage of the puzzle.
Table 4.6. Conventional approach for measuring the relative importance of the candidate explanations. The table shows the results of the regression: Rit =
α + γtRIV OLit−1+ γtEECOit−1+ it, where the idiosyncratic volatility is used as
control variable. The cross-section coefficients are reported with t-statistics in parentheses. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively.
Variable Exp Skewness Spread Intercept 0.325*** (5.55) 0.368*** (4.65) IVOL -12.132*** (-7.01) -20.654*** (-7.39) Explanation -0.173 (-0.41) 1.994 (1.32)
Chapter 5
Conclusion
In this research, my aim was first to replicate the empirical results found by the previous literature. I used the conventional method proposed by Ang, et al. (2006) to check if I could observe the same idiosyncratic volatility puzzle in my dataset built with stocks before the 2008 financial crisis.
After the confirmation that the model that I set up obtained the same empirical results as the papers who previously analysed the same issue, I replicate the same analysis on a more recent dataset, going from January 2009 to December 2016. The aim of this part of the study was to check if the idiosyncratic volatility was still present after the financial crisis or if some of the underlying factors driving the puzzle changed. The results of this part of the research signaled that, if possible, the idiosyncratic volatility puzzle is still present and slightly more pronounced after the 2008 financial crisis.
In the final section of the research, I moved to analyse the relative importance of two economic explanations that in the previous literature were signaled as the one driving the puzzle the most. These two economic explanations are the expected idiosyncratic skewness of returns and the bid-ask spread. By computing a Fama-MacBeth regression, I was able to assess the relative impact of these two variables; in my empirical results the expected idiosyncratic skewness explains just 12.8% of
5 – Conclusion
the idiosyncratic volatility puzzle, while the bid-ask spread explains the 30.2%. Judging by these results, it is clear that the major part of this puzzle is still left unexplained, therefore it remains difficult to incorporate it during an asset pricing procedure. On the other hand, the economic explanation related to market frictions shows promising results, thus the model can be improved in future researches by keeping this starting point in mind.
Chapter 6
6 – Additional Tables T able 6.1. Sample statistics from Jan uary 1963 -Decem b er 2008. This table sho ws the distribution of some firm sp ecific characteristics. The sample cons ists of all CRSP common sto cks with share prices of at least $1 at the end of the p revious mon th. N is the total n um b er of firm-mon th observ ations. R eturn is the CRSP mon thly return. Idiosyncr atic volatility (IV OL) is the standard deviation of residuals from a regression of mon thly sto ck returns on the F ama-F renc h (1993) factors. Size is measured as in F ama and F renc h (2006). Skew is the sk ewness of ra w mon thly returns. Spr ead is the bid-ask spread, normaliz ed b y the a v erage bid-ask spread. V ariable Mean Std Dev N 1st Pctl 25th Pctl 50th Pctl 75th Pctl 99th Pctl Return 0.015 0.162 1074 193 -0.352 -0.061 0.003 0.074 0.52 0 IV OL 0.159 0.094 1074193 0.000 0.100 0.141 0.193 0.477 Size ($M) 1862.243 11226.430 1074193 3.832 43.3 01 159.620 682.294 30390.995 Sk ew 0.810 1.1 38 1074193 -1.193 0.163 0.610 1.213 4.825 Spread 0.026 0.038 1074193 0.000 0.004 0.014 0.035 0.171
6 – Additional Tables T able 6.2. Sample statistics from Jan uary 2009 -Decem b er 2016. This table sho ws the distribution of some firm sp ecific characteristics. The sample cons ists of all CRSP common sto cks with share prices of at least $1 at the end of the p revious mon th. N is the total n um b er of firm-mon th observ ations. R eturn is the CRSP mon thly return. Idiosyncr atic volatility (IV OL) is the standard deviation of residuals from a regression of mon thly sto ck returns on the F ama-F renc h (1993) factors. Size is measured as in F ama and F renc h (2006). Skew is the sk ewness of ra w mon thly returns. Spr ead is the bid-ask spread, normaliz ed b y the a v erage bid-ask spread. V ariable Mean Std Dev N 1st Pctl 25th Pctl 50th Pctl 75th Pctl 99th Pctl Return 0.020 0.155 297277 -0.304 -0.049 0.010 0.07 2 0.489 IV OL 0 .136 0.095 297277 0.000 0.083 0.118 0.164 0.465 Size ($M) 5132.702 21277.060 297277 9.861 145.464 590.285 2453.380 90111.0 73 Sk ew 0.755 1.075 297277 -0.947 0.090 0.525 1.110 4.973 Spread 0 .007 0.018 297277 0.000 0.000 0.001 0.005 0.084
6 – Additional Tables
Table 6.3. Sample statistics from January 1963 - December 2008. Time-series averages of cross-sectional correlations between firm characteristics
Spread Size IVOL Skewness Spread 1.000
Size -0.098 1.000
IVOL 0.131 -0.107 1.000
Skewness 0.128 -0.066 0.552 1.000
Table 6.4. Sample statistics from January 2009 - December 2016. Time-series averages of cross-sectional correlations between firm characteristics
Spread Size IVOL Skewness Spread 1.000
Size -0.082 1.000
IVOL 0.134 -0.151 1.000
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