3.6. Equity Volatility
3.6.2. Volatility Components
There are two distinctive components of variations in equity returns. The first is the market or systematic volatility, driven by market-wide events and risks. The second is the idiosyncratic volatility or variations in returns due to firm-specific risks. Campbell et al. (2001) document that these two components of total volatility behave differently over time. The idiosyncratic volatility trended upward, while there was no trend in the systematic volatility over the same period. The market volatility leads the idiosyncratic volatility and increases in recessions. Brandt et al. (2008) point to a sudden drop in the idiosyncratic volatility during the last few years and argue that the time-series behavior of idiosyncratic volatility documented by Campbell et al. (2001) is due to sporadic highly
GARCH
TGARCH
t
1
towned by retail investors are found more likely to have high and rising volatility. Bennett, Sias and Starks (2003) also link the rise of firm-specific risks to ownership but they argue that the growth in institutional ownership is responsible for the increase in the idiosyncratic volatility. Guo and Savickas (2008) provide another explanation unrelated to ownership. They show that idiosyncratic volatility is significant in predicting the market return and argue that the idiosyncratic volatility is a proxy for changes in the investment opportunity set.
The market volatility can be measured as described in the previous section. On the other hand, since the idiosyncratic volatility is not observable, its measurement is not so straightforward. The simplest and most parsimonious measure would be the variation in the excess returns (i.e. returns in excess of the market return).
2 , , , 1 , ( ) n i j t i j i i t n h h s = - =
å
(3.26) whereη
i,j,tis the equity return in excess of the market return (η
i,j,t= EPi,j,t - EPM).i,j
η
is the average excess returnThe major weakness of this simple measure is that it ignores the difference in risk factor loadings among securities and assumes that expected returns for all equities are the same. This issue can be addressed by estimating the expected returns with one of the previously considered models and then measuring the idiosyncratic volatility as a sum of squared errors of the model.
Taking into account that persistence of shocks may vary from very short-term to permanent, another potentially useful decomposition of the equity volatility is to consider it as a sum of the short-term and long-term components. Engle and Lee (1999) propose the following specification:
2 2 2 1 1 1 2 2 1 1 1 ( ) ( ) ( ) t t t t t t t t t t t l s s s l l s a b a e s w r f e s - - - - - - = + = + + - = + + - (3.27) where 2 t
s
is the conditional variance at time ts
t is the short-term component of the conditional variance lt is the long-term component of the conditional varianceε is the unexpected return
Assuming that 0 < (α+β) < 1, the short-term component mean-reverts to zero at a rate of (α+β), while the long-term component follows an auto-regressive process and if 0 <
ρ < 1, it converges to ω/(1- ρ). Since the long-term component should mean-revert at a slower rate, it is assumed that 0 < (α+β) < ρ < 1. Engle and Lee (1999) find that this model produces results consistent with the highly volatile period surrounding the October 1987 market crash. Using the same methodology, Adrian and Rosenberg (2008) find a positive trade-off between returns and both volatility components. The risk premium for the long-term component is estimated to be about 35 per cent higher than the compensation for the short-term component. The authors argue that the short- term component is related to the market skewness risk, whereas the long-term component is closely related to the business cycle risk. Zhu (2009) finds that only the short-term volatility component is positively and significantly related to returns in ten Asia-Pacific equity markets. The long-term component is found to account for about 75 per cent of the total volatility while the short-term component is responsible for the remaining 25 per cent. The empirical results are consistent with the Asian market crisis, using the 1997 data.
3.7. Summary
Markowitz (1952) shows that only the systematic risk, inherent in the entire market or entire market segment, should be priced in the equity markets. Specific risks associated with individual securities are diversifiable and therefore the exposure to these risks
develop the Capital Assets Pricing Model (CAPM) which implies that the expected return of a security only depends on its correlation with the market return. Despite the strong theoretical underpinning of the CAPM, a large body of literature shows that is does not fully explain variations in equity returns. Furthermore, contrary to the CAPM main implication, variables such as the firm size (Banz, 1981), leverage (Bhandari, 1988) and earnings yield (Basu, 1077, 1983) are found to be significant in explaining the variations in returns unaccounted for by the CAPM. These findings spearheaded the development of multi-factor models for equity pricing. Fama and French (1992) propose an empirically inspired model which shows a notable success in explaining the equity returns with three risk factors: the market rate of return, the difference in returns on big and small firms and the difference in returns on high and low book-to-market equity firms.
The models discussed above estimate the expected excess return of individual securities relative to the excess return of the entire market or the equity premium. Fama and French (1989) provide the evidence that the equity premium is countercyclical, i.e. low when economic prospects are good and high during the challenging economic times. In line with this, a number of authors (e.g. Mehra and Prescott, 2008) shows that the equity premium varies over time. Therefore, the static models such as the unconditional CAPM should not be able to satisfactorily explain equity returns. Donaldson, Kamastra and Kramer (2008) argue that the time variation is the most important feature of the equity premium process.
The equity volatility is a major measure of risk. The simplest approach to estimate the equity volatility is to assume that it is constant. In this case, the equity volatility is estimated as a standard deviation of historic equity returns. This approach is widely used although it is an empirical fact that the volatility is not constant but varies over time. Given the importance of equity volatility as a risk measure, the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models should be used for the estimation of the equity volatility to ensure that time series properties are taken into account.
Finally, decomposing the equity volatility provides an opportunity to analyze how each component of the equity volatility impacts equity returns. Campbell et al. (2001)
decompose the volatility into systematic and idiosyncratic components, and document their different time behaviours. Engle and Lee (1999) propose decomposition into short-term and long-term components.
The next chapter reviews the existing literature, develops hypotheses and presents the research methodology for the empirical study of the relationship between the corporate credit spread and changes in the systematic and idiosyncratic risks of corresponding equities.
CHAPTER 4
SYSTEMATIC AND IDIOSYNCRATIC EQUITY RISKS
AS DETERMINANTS OF THE CREDIT SPREAD
4.1. Introduction
The structural model of Merton (1974) provides a theoretical foundation for the analysis of the relationship between the values of equity and debt securities. The model treats the firm’s equity and debt as derivatives written on the firm’s assets, and implies that the default probability is defined by the difference between the value of assets and the value of debt relative to the volatility of the firm’s asset value. As a result, the most important determinants of the difference in the yields on corporate and government bonds, known as the credit spread, should include leverage and asset volatility. Since the latter is unobservable, it is usually derived from equity volatility.
Despite a strong theoretical foundation, the existing empirical evidence on the determinants of credit spread is far from conclusive. Collin-Dufresne, Goldstein and Martin (2001) note that changes in the yields of governmental bonds and equity returns explain about 60 per cent of the variation in the corporate bond yield, and only five per cent of changes in the credit spread. They report that the theoretically relevant variables such as the leverage and equity volatility fail to explain the majority of changes in the credit spread. Furthermore, they find that the residuals from regressing changes in the credit spread on the theoretically derived variables are highly correlated, which leads them to conclude that the credit spread is driven by a common factor. In a widely cited paper, Elton et al. (2001) show that this factor can be proxied by the Fama-French factors (Fama and French, 1993) which are known to be the common factors priced in equity returns. After showing that the Fama-French factors explain as much as 85 per cent of the credit spread not accounted for by the expected default loss and higher taxes paid on corporate bonds relative to governmental bonds, the authors conclude
that the risks inherent in corporate bonds are systematic and, by extension, rewarded with a risk premium.
As noted above, the structural model links the values of the firm’s equity and debt by considering them as options written on the firm’s assets. As in option pricing, the total volatility of assets, which represents systematic as well as idiosyncratic risks, is used to measure the probability that the firm’s asset will fall to the level of debt, thereby triggering bankruptcy. If the credit risk cannot be diversified away as indicated by the above studies, then the systematic risks should be the major drivers of the credit spread. This is important in light of Campbell et al. (2001) who find that idiosyncratic volatility has been trending upwards in recent decades while the market-wide volatility has been stationary.
The purpose of this chapter is to review existing studies on the relationship between the corporate credit spread and changes in the systematic and idiosyncratic risks of the corresponding equities. This literature review guides the formation of this study’s hypotheses, empirically tested on firm-level data covering almost 15 years, including the period of the recent 2007 financial crisis. This chapter goes on to present the research methodology and the dataset. The empirical results are presented in Chapter 5.
The existing literature focuses on the relationship between the credit spread and the volatility of equity returns in excess of the market return. This study aims to extend the literature findings in several ways. First, conditional equity volatility models, such as the Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model, are employed to account for the time series behaviour and the asymmetric nature of equity volatility. Second, systematic and idiosyncratic expected equity returns are estimated using the CAPM and the Fama and French (1993) three factor model. This removes the limitation imposed by the commonly used assumption that all firms’ loadings on systematic risks are equal, i.e. all betas are equal to one. Finally, in addition to using individual theoretically derived variables in the regression analysis, the structural model is explicitly estimated to control for the level of the credit risk, as well as to provide an insight into its performance as compared to individual variables.