# Behaviour of Idiosyncratic Risk and Systematic Volatility

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Published: *Wed, 14 Mar 2018*

This paper investigates the behaviour of idiosyncratic risk and systematic volatility and examines using the panel data model whether the firm level volatility has moved higher than the systematic volatility during the period 1992-2010. I have not found a significant rise in the idiosyncratic volatility compared to the systematic volatility during the period 1992-2008, but found that the idiosyncratic volatility significantly declined in comparison with the systematic risk in the recent past (2008-2010).

## 1. Introduction and Literature review

## 1.1 Introduction

The time varying behaviour of the idiosyncratic volatility and systematic, aggregate market volatility has long been recognised. The CAPM suggests that the idiosyncratic risk should not be priced as it can be diversified away and so have most of the empirical studies focussed on the systematic risk. But usually investors hold incomplete diversified portfolios and even while diversifying their portfolios they hold a limited number of assets to trim down the transaction costs. The overall stock market volatility or systematic volatility is most important to investors with completely diversified portfolios whereas both firm level and systematic volatility are important for investors with incomplete diversified portfolios.

On theoretical grounds it is possible that the idiosyncratic volatility has increased whereas the market volatility as a whole has remained stable. In my study, I analyse long term trends in both idiosyncratic and systematic volatility during the period 1992 to 2010. I show that the firm level volatility for all the ten firms chosen for my dissertation have increased over the years whereas the systematic volatility has either remained constant or has fallen over the years.

To compare firm level volatility with the systematic volatility I select 10 stocks listed on FTSE 100 stock price index. In my methodology I use CAPM to compute the volatility estimates. My study is important for all those investors holding equity portfolios particularly the ones who hold greater proportions of their portfolios in stocks. To study the long term trends in the two components of volatility, I divide the sample period into two time intervals. The first time interval ranges from November, 1992 to October, 2008 and the second from October, 2008 to December, 2009.

To estimate whether the idiosyncratic volatility has moved higher than systematic volatility, I compute the ratio of idiosyncratic volatility to systematic volatility. I adopt the panel data model in my study and perform linear regression between the ratio as the dependent variable and time as the independent variable. The results of the regression vary for the two time periods. For the time period November 1992 to October 2008, the regression gives a positive yet an insignificant slope coefficient implying that I do not find a significant upward trend in the ratio of volatilities. By graphically analysing the trends of the two volatilities, I find that the idiosyncratic risk has increased over the time period whereas the systematic risk has either remained at the same level or has fallen over the time period. This has further implications for the correlations among individual stocks. A decline in the systematic volatility implies that the correlations among individual stocks have declined.

For the time interval from October 2008 to December 2009 the results of the panel linear regression are different from the results obtained for the first time interval. The slope coefficient from the regression is negative and statistically significant. A significant downward trend in the ratio of the volatilities implies that either of the volatilities or both have a significant trend. This contradicts the stationary behaviour of the volatility. As the sample period is small in comparison to the earlier one and as both upward & downward sloping spikes are found in the graph, it can help preserving the stationarity of the volatility. By analysing the graphs of the idiosyncratic volatility and the systematic volatility, I conclude that the idiosyncratic volatility has declined for all the firms whereas the systematic volatility has remained the same for most of the firms, although having peaks and troughs at various times.

## 1.2 Literature Review

There is astonishingly little empirical research on volatility at the firm level. However, there are many reasons to be interested in the firm level volatilities. Firstly, many investors have large holdings of individual stocks and they may fail to diversify them in order to eliminate the firm level risk. These investors are affected by idiosyncratic volatility just as much as market volatility. Secondly, arbitrageurs trying to exploit the mispricing of the individual stocks will be more interested in the idiosyncratic return volatility compared to market volatility. Lastly, firm level volatility is related to the aggregate output. Caballero and Hammour (1994), Eden and Jovanovic (1994) used models of “cleansing recessions” to show that an exogenous increase in the rate of revelation of information about the management quality may temporarily reduce output as it increases the pace of reallocation of resources from low quality firms to high quality firms.

Recently, the interest in idiosyncratic volatility has increased and the researchers are investigating the relationship between stock returns and idiosyncratic volatility. For example, Campbell, Lettau, Malkiel, and Xu (2001) used a disaggregated approach to study the volatility of common stocks at the market, industry and firm levels. They found that the idiosyncratic volatility increased over the period 1962-1997, whereas the stock market as a whole was not that volatile during this period. They found evidence of a positive deterministic trend in idiosyncratic firm-level volatility during the period 1962-1997. There was no similar trend found in industry or market volatility. This implies that the correlations among individual stock returns have declined over the past few decades. The R squared of the market model for a stock has declined and the number of stocks required to obtain a given amount of portfolio diversification have increased.

Goyal and Santa-Clara (2003) found a significant positive relation between the average stock variance which is largely idiosyncratic and the portfolio returns on the NYSE/Amex/Nasdaq stocks for the sample period of 1963:8-1999:12. In contrast, they find that the variance of the market has no forecasting power for the market return. Since traditional asset pricing models suggest that only market risk determine the returns, their findings seem rather contradictory.

In contrast, Bali, Cakici, Yan and Zhang (2005) show that the above results do not hold for the extended sample of 1963:08 to 2001:12 and for the NYSE/AMEX and NYSE stocks. They also show that a significant positive relation between the idiosyncratic risk and portfolio returns is mainly due to the small stocks traded on the NASDAQ and a liquidity premium associated with them.

Malkiel and Xu (1999) found that the most volatile stocks display a pattern of increasing volatility over the time period of late 1960s through the 1990s. Their estimates of the idiosyncratic volatility of the 500 stocks in the market index show a pattern of increasing volatility.

Malkiel, Xu (2002) have made some progress in understanding the role of idiosyncratic risk in asset pricing both theoretically and empirically. By assuming that not all investors are willing or able to hold the market portfolio, it was shown that idiosyncratic risk affects the asset returns. The empirical results show that the idiosyncratic volatility alone is important in explaining cross-sectional expected return difference. Its explanatory power is not taken away by other variables such as the size, the book-to-market ratio and liquidity variables. The results were found to be robust to Japanese stock return data. More importantly, the cross-sectional results explain that idiosyncratic volatility is more powerful in explaining the cross section of returns than either beta or size measures.

Jiang and Lee (2006) claim that by regressing excess returns on one-lagged values of idiosyncratic volatility provides only a limited picture of the dynamic effect of the volatility, which tends to be persistent over time. After correcting for the serial correlation in idiosyncratic volatility, they found that it has a significant positive relation with the stock returns. Their finding seems robust for various firm size portfolios, sample periods and measures of idiosyncratic risk. They also find a delayed response of market returns to innovations in idiosyncratic volatility suggesting that stock markets misprice idiosyncratic risk.

Poti and Kearney (2008) examined the dynamics of idiosyncratic risk, market risk and stock return correlations in the European equity markets using 3515 stocks listed on the Euro area stock markets over the period 1974-2004. They found a rise in idiosyncratic volatility further implying that it takes more stocks now to eliminate the idiosyncratic risk. However, they found that the market risk was trended upwards during the period and the correlations are not trended downwards. Both the volatility and correlation estimates were found to be pro-cyclical as they rise during the time of low market returns. They also find that there is a significant contemporaneous impact of market returns and market volatility on idiosyncratic volatility which suggests that even positions taken to eliminate market risk like long-short relative value trades may prove to be more volatile during bearish market and at times when the market is highly volatile.

Huang, Liu, Rhee and Zhang (2009) found a negative cross-sectional relationship between the returns and idiosyncratic risk when estimated using daily returns. This relationship does not exist once return reversals are controlled for. In contrast, they found a significantly positive relationship between the idiosyncratic volatility and expected returns computed using the monthly data.

## 2. Data & Methodology

## 2.1 Data

The data used in the study comprises of the daily stock prices for ten firms listed on UK stock price index FTSE100. All the stock prices are expressed in GBP and have been obtained from Thomson Reuters DataStream. For the risk free return used to compute the idiosyncratic volatility, daily yields on UK Treasury Bills of 3 month maturity have been used. These have been obtained from Bloomberg. These have been further discounted to compute the daily risk free rate which has been done by dividing the 3 month yield with the number of trading days in 3 months.

The data used is daily and the sample period runs from November 1992 to December 2009. I have divided my sample period into two time periods; November 1992 – September 2008 and October 2008- December 2009. The daily stock returns are computed from the stock prices by taking the log of the ratio of current period’s price to the previous period’s price.

The returns on stock price index FTSE 100 are used as market returns. Data on daily betas for the ten firms are obtained from Bloomberg.

## 2.2 Estimation

To estimate the systematic risk and idiosyncratic volatility for the firms, I used the CAPM equation which implies that one can write the return on any asset as

] + (1)

Where are the equity returns of the firms & the subscript ‘i’ has been used to represent the firms, is the rate on UK Treasury Bill of 3 month maturity, return on stock price index FTSE 100 has been used to represent the return on market, is an error term with mean zero & uncorrelated with.

Taking the variance of LHS & RHS of the above equation gives the following

(2)

The first term on the RHS represents systematic risk that is market related risk & cannot be eliminated by diversification. The second term on the RHS is idiosyncratic risk which is not related to the market and can be eliminated by diversification.

Based on this return decomposition I compute the two volatility components and then calculate the ratio of idiosyncratic risk to the systematic risk for each of the firms.

To determine whether the stocks have become more volatile over the years, I will be using panel data modelling. The advantage of using panel data is that it makes it possible to analyse changes at the individual level. Panel data sets are usually larger than cross-sectional and time series data sets and the variables vary over both time and individuals making the estimators based on panel data sets more accurate than the rest of the models.(Marno Verbeek, 2008) The linear regression model that I have made use of is the following,

(3)

where is the ratio of idiosyncratic risk to systematic risk & I have estimated the model using random effects.

## 2.3 Methodology

Given the daily stock returns, market returns, risk free rates and betas for the firms, I computed the idiosyncratic return εi,t for all the firms using the CAPM equation. By taking the standard deviation of the daily residuals over the month, I measured the monthly idiosyncratic volatility for the firm. Hence, I computed the monthly idiosyncratic volatilities for all the firms in the same manner.

For computing the systematic risk I required data on monthly betas. Due to the unavailability of data on those, I computed monthly betas by taking an average of daily betas over the month. The daily betas are obtained from Bloomberg. To compute the systematic risk I multiplied the monthly beta of the firm with the standard deviation of the excess returns on FTSE 100 over that month. Similarly, I computed the systematic risk for all the firms. The two volatility estimates therefore computed have a monthly frequency.

The ratio of idiosyncratic volatility to systematic volatility is computed for all the firms to estimate the linear regression model in panel data. For panel data analyses I made use of the STATA 9.2 software.

While performing the linear regression in panel data one needs to distinguish between fixed and random effects. The fixed effects model is the linear regression model in which the intercept term is different for individual units where it is assumed that all the regressors are independent of the error terms for the individual units (firms in my example). Another assumption is that the time invariant characteristics of the individual units should not be correlated with other individual characteristics. Hence, the error term and the intercept of the individual firms should not be correlated with others.

In the random effects model the variation across individuals is assumed to be random & it is uncorrelated with the independent variable. If the variation or differences across the individuals have an influence on the dependent variable then one should use random effects model.

The choice between the fixed effects and random effects approach is not straightforward. The fixed effects approach is conditional on the intercept where the intercept is estimated. Intuitively, it is logical if the individuals in the sample are similar. On the contrary, the random effects model is not conditional upon the individual intercept. It allows one to make interpretations with respect to the characteristics of the population as unlike fixed effects approach, one is not interested in a particular individual’s intercept but it focuses on the arbitrary individuals with certain characteristics.

One would use the fixed effects approach when it is important to identify the individual units or when some interest lies in αi, which makes sense if the number of individuals are small and are one of a kind. However, even if the sample population is large though random effects approach seems appropriate, one would prefer fixed effects. The reason for this is that if the intercept and the independent variable are correlated the random effects model would not take that into account and provide inconsistent estimators. Whereas the fixed effects model would remove this problem of correlation.

Hausman (1978) suggested a test to distinguish between fixed effects and random effects. Under the null hypothesis the αs and Xs are not correlated which implies that both random effects and fixed effects are consistent but random effects are more efficient. Under the alternative, the αs and Xs are correlated which implies that fixed effects are consistent whereas random effects are not. If the difference between the fixed effects estimator and random effects estimator is significant then it indicates that you reject the null hypothesis.

## 3. Results

## 3.1 Results (1992-2008)

Table 1

Results of Hausman Test

The Prob>chi2 is greater than 0.05 indicating that the difference between the estimators is not significant which implies that the random effects approach is more appropriate.

Following are the results of panel data estimation using GLS estimator & robust standard errors.

Table 2

GLS regression between the ratio of the volatilities and time

The panel data regression for the period 1992-2008 with the dependent variable as the ratio of idiosyncratic volatility to systematic volatility and time as the independent variable, gives a positive coefficient. The coefficient is not statistically significant as the P value is above 0.05 suggesting that the null hypothesis that the coefficient is zero should be accepted. Hence, there is no significant upward trend in the ratio of the volatility estimates during this period.

A significant upward trend would have implied that either of the volatilities have a significant trend which suggests that the volatility is non stationary. To observe the trend in the idiosyncratic volatility during this period, I have used panel data model for regressing idiosyncratic volatility with time. Following are the results:

Hausman test gives Prob>chi2 greater than 0.05 which implies that estimates from random effects are more efficient. The regression using GLS and robust standard errors give the following results:

Table 3

GLS regression between the idiosyncratic volatility and time

I get a small but significantly positive coefficient for the idiosyncratic volatility. The presence of a significant upward trend contradicts the stationarity of the volatility measure. Although the coefficient is found to be significantly different from zero but it is very small. The graphical analyses of the volatility (Figure.1) will show that it experiences peaks in some periods and troughs in others further maintaining the stationarity of the volatility.

Figure.1. Idiosyncratic volatility of all the ten firms for time period 1992-2008(monthly)

Figure.2 Ratio of the volatilities of ten firms for the period 1992-2008 (monthly)

Figure.3. Systematic volatility of all the ten firms for the time period 1992-2008(monthly)

It can be seen from the graphs (above) that the idiosyncratic risk has increased for most of the firms over the years whereas the systematic risk has decreased or remained the same over the time period. We can further conclude that the idiosyncratic risk has gone up for all the ten firms whereas the systematic risk has gone down for some firms or it has remained the same over time.

## 3.2 Results (2008-2010)

Similarly, I computed the volatility estimates and their ratio for all the firms from October, 2008 to December, 2010. To determine the movement of the ratio during this time I ran the linear regression for panel data model with ratio as the dependent variable and time as the independent variable.

Table 4

Hausman test Results (ratio of the volatilities)

Prob>chi2 obtained from Hausman test is greater than 0.05 indicating that the difference between the estimators is not significant and it is appropriate to use the random effects model.

Following are the results of the regression using GLS and robust standard errors:

Table 5

Results for GLS regression between the ratio and time

The slope coefficient for the regression is negative implying that the ratio has declined over the time period. The coefficient so obtained is statistically significant with P- value as 0 meaning that we reject the null hypothesis that the coefficient is zero. By looking at the figure which plots the ratio over time, I find that it declines for most of the firms except for Tesco and Smith it remains almost the same. For some firms like Royal Dutch Shell and LSG the decline does not seem significant from the graph.

To foreground the trend of the idiosyncratic volatility during this time, I run the panel data regression with idiosyncratic volatility as the dependent variable and time as the independent variable. The result of Hausman test is in favour of the random effects approach as the difference between the estimates is not significant. Following are the results of the panel data regression using GLS and robust standard errors:

Table 6

Panel data linear regression between the idiosyncratic risk and time

The panel data regression of idiosyncratic volatility against time gives a statistically significant negative coefficient as the P value is less than 0.05, although the coefficient is found to be very small. This implies that there is a significant negative trend in the volatility during this period. As the time period over which the trend of the volatility has been evaluated is small compared to the earlier time interval, it seems plausible to find a trend in the volatility.

The figure plotting the idiosyncratic volatility (figure.5) shows a downward trend in the volatility for most of the firms. For some of the firms the systematic risk has slightly increased whereas for the rest it has remained the same over the period.

Figure.4. Ratio of the idiosyncratic risk to systematic risk for the ten firms (monthly, 2008-2010)

Figure.5. Idiosyncratic risk for the ten individual firms(monthly, 2008-2010)

Figure.6. Systematic risk plotted for the firms (monthly, 2008-2010)

## 4. Analysis

## 4.1 Analysis (1992-2008)

## 4.1.1 Theoretical Framework

The aforementioned results obtained for the volatility estimates indicate that the firm level volatility rose steadily during the period November 1992 to October 2008 whereas the systematic volatility did not. This finding further suggests that the decrease in the systematic risk is attributed to a decline in the correlations of individual stocks.

Campbell, Lettau, Malkiel, and Xu (2001) and Wei and Zhang (2006) have suggested reasons for the increase in the idiosyncratic risk. One amongst them is the disposition of the conglomerates for breaking up into smaller and more focussed companies specializing in single industry, leading to a shift from internal capital markets to external capital markets. The firms are now separately listed and their idiosyncratic risk is separately measured whereas before it was a part of a diversified conglomerate.

Another reason is the tendency for firms to access the stock markets at an earlier stage in their development. Firm age has been believed to be a possible reason for explaining the trend in the volatility. It has been argued that the value of equity for firms depend on their cash flows which appear to be distant and uncertain for younger firms compared to the older ones. (Cao, C., et al., (2006))

The companies have started to issue their stocks at an early stage of their life cycle, often at a time when their earnings and profitability are not clearly known and there is uncertainty about their long term prospects.

Mock et al. (1999) provide interesting evidence for the significance of these factors. In their study on variation across countries regarding the explanatory power of their market, they found that the market model has a greater explanatory power in the less developed markets. They argue that these markets rely on internal finance with cross-holdings and internal subsidization such that the individual firm’s stock prices are unable to reflect information regarding the value of their operations.

Also, changes in the executive compensation scheme leads to increasing cash- flow variability attributing to the increasing firm level volatility. The less diversified conglomerates can also be a reason for increasing idiosyncratic volatility.

Increasing institutionalisation of the equity ownership combined with discrepancy between the individual and institutional investor’s sentiment, could describe the excess trading and volatile stock prices. Xu and Malkiel (2003) found proof of a positive relationship between the idiosyncratic risk of US stocks and the institutionalisation of the ownership of the US stocks.

Increase in leverage could be another reason as it would intensify the volatility of stock returns. When leverage increases, the stockholders bear a higher risk related to the firm’s cash flow and the volatility of the stock return increases. Leverage accounts for high volatility during the time of recession. This further gives rise to the relationship between the idiosyncratic volatility and the investment decisions of corporate managers.

Cao, simin and zao establish a theoretical link between growth options available to the managers and idiosyncratic volatility. To relate the idiosyncratic volatility with the decisions of corporate managers, they used the corporate finance model of Galai and Masulis (1976) to indicate that a major portion of the trend in the idiosyncratic volatility can be explained by the level and variance in the growth options. They argue that the moral hazard problem motivates the managers of the leveraged firms to choose the projects which increase the idiosyncratic variance of the firms at the expense of the debt holders. By doing this they can increase the value of the firm’s equity and reduce the market risk of the equity.

There is empirical evidence in support of the hypothesis that the level and variance of growth options explains the trend in idiosyncratic volatility. After controlling for the growth options it was found that the upward trend in the idiosyncratic volatility disappears and becomes negative. After proving that the growth options play a significant role in explaining the trend in the firm specific volatility, one would like to know the reasons behind the change in the level and variance of growth options. Zhingales (2000) has discussed how the nature of firms has changed. Breaking up of large conglomerates into smaller firms, increasing divestitures and decline in diversification of assets and investment possibilities increases the risk of the investment opportunity set.

Globalisation is known to be an obvious reason for increase in the growth options. It leads to more open capital financial and goods markets providing greater opportunities for growth. Zingales (2000) argues that increasing globalisation has led to increased competition which demands improvement in quality and increasing innovation. Globalisation of markets provides greater opportunities of growth forcing the managers to adopt riskier projects which increase their share value. Pontiff (2004) shows that increasing competition attributing to riskier growth options leads to increasing variability of cash flows, thereby increasing the idiosyncratic volatility.

Li, Morck, Yang and Yeung (2004) show that the lower market model R squared is related with greater capital market openness. Fama and French (2004) suggest that the lower cost of capital has induced the smaller firms to raise capital in the market.

The growth options proxies and their time series variances are positively related to the idiosyncratic volatility and explain 63% of the variation in the volatility. (Cao, C., et al., (2006) pp 2600)

Pastor and Veronesi (2003) relate idiosyncratic volatility with the firm profitability given by return on equity. They find that firms with more uncertain future profits and volatile profitability tend to have a greater firm level volatility. Zhang (2006) shows that the declining ROE and the increasing volatility of ROE attribute to an increasing idiosyncratic volatility.

Chan, Lakonishok and Sougiannis (2001) find a positive relationship between the idiosyncratic volatility and R&D intensity of the firm. Since firm R&D and innovations are proxy for firm’s growth options it corroborates the relationship between the idiosyncratic volatility and growth options.

The increasing idiosyncratic volatility can also be explained by the legal and institutional structures which are more favourable to investment as they offer greater growth opportunities. (Cao, C., et al., (2006) )

Financial innovation like introduction of new derivatives markets could have affected the availability of information regarding the future cash flow of the security. They have a significant impact on the price of the security. Usually derivatives markets are known to improve the availability of information of the future cash flows and reduce volatility. But according to Stein (1987) it is plausible for the derivatives markets to change the pattern of trading by the speculators in a way which would reduce the information reflected by the prices and increase the volatility. However, there is little empirical evidence for this effect. On the contrary, studies by Kumar et al (1998) and others have reported that the stock with options written on them have on an average experienced a statistically significant decline in their volatility relative to the market.

Shocks to investor’s discount rates are another factor affecting the idiosyncratic volatility. Within the framework of CAPM, the change in betas can change the discount rates further altering the prices. Therefore, volatile betas can help explain the increasing firm level volatility. However, there is not much empirical evidence on this result.

## 5.1.2 Implications

The nature of the volatility of the equity returns is fundamental to portfolio management, option valuation and asset pricing. According to CLMX (2001), given a random portfolio selection strategy higher the average idiosyncratic risk greater is the number of stocks required for complete diversification.

Poti and Kearney (2008) findings suggest that increasingly more stocks are required to reduce idiosyncratic risk to any level. They found that a greater number of stocks were needed to reduce the idiosyncratic volatility in the second half of their sample period compared to the first half. For Example, to reduce the idiosyncratic volatility to 5%, 261 and 166 stocks were required in the second semester of 2002 and 2003 respectively and 154 stocks in the first semester of 2004. In contrast, 35, 43 and 93 stocks were required in the first semester of 1974 and in the second semester of 1989 and 1997 respectively. Alternatively, lower the correlation amongst stocks, higher the proportion of the average total variance given by the idiosyncratic volatility increasing the potential benefit from diversification.

The level of systematic volatility has remained the same over the period in contrast to the idiosyncratic volatility, though there have been times of increasing systematic volatility as volatility is stochastic and never constant. The volatility is presumed to be pro-cyclical. It has been validated by Wu (2001) and Bekart and Wu (2000) that volatility increases with negative returns and vice versa. It can be seen from the graph for systematic volatility that it peaked during 1992. The events known as Black Wednesday took place during September 1992 in UK when Conservative Government was forced to withdraw the pound sterling from the European Exchange Rate Mechanism (ERM) as they were unable to keep sterling above its agreed lower limit. It led to inc

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