5.8. Robustness of the Results
6.3.5. Equity Systematic Returns and Volatility
The market-wide returns and volatility are proxied by the S&P 500 index returns and volatility and are calculated as described in sections 6.3.1. and 6.3.3.
To take into account differences in exposure to systematic risk, the systematic equity returns and volatility are also estimated at the firm level. According to the widely used
, 1, ,
(
, ,)
2, , 3, , i t ft i t m t f t i t t i t tr
= +r
b
r
-r
+b
SMB
+b
HML
(6.19) where , i tr
is the equity return of firm i at time t ,f t
r
is the risk-free rate ,mt
r
is the return on the S&P 500 index at time tSMBt is the difference in the returns of big and small firms at time t
HMLt is the difference in the returns of high and low book-to-market equity firms at time t
As described in Section 4.3.4, the conditional betas are estimated with bivariate GARCH- in-mean as proposed by Bollerslev, Engle and Wooldridge (1988). Once the conditional betas are estimated, the systematic equity returns are calculated in a fairly straightforward manner according to Equation 6.19. Finally, it is assumed that the volatility of systematic returns follows the GARCH(1,1) process given in Equation 6.17.
6.3.6. Bond Issue Characteristics
To control for the maturity of bonds, daily duration is calculated according to the following formula: 1
1
(1
)
i i i T i dCF
d
T
B
=Y
=
+
å
(6.20) whereis the dirty bond price (principal + accrued interest) is the cash flow in year i
is the time in years to the ith cash flow
The control variable for the size of the bond issue is the natural logarithm of the bond’s market price multiplied by the number of outstanding bonds.
6.3.7. Panel Data Analysis
The data set consists of the conditional correlation between the equity and bond returns, and a set of independent variables for n firms over T consecutive time periods. The simplest model for analysis of this two-dimensional data set is given by:
2
. . .(0,
)
it it it itCORR
= a+b +e ex
,:i i d
s
(6.21) where itCORR is the conditional correlation between the equity and bond returns of firm i at time t
a is the intercept
b is a k x 1 parameter vector
it
x
is a vector of k explanatory variablesit
e
is a disturbance termThis model is referred to as the constant coefficient model because it imposes the same coefficient for all firms in the sample. This is the most parsimonious panel data model but is severely restricted. Most importantly, by imposing the same intercept for all firms it effectively assumes that other firm-specific determinants of the correlation between the equity and bond returns are the same for all firms. Other firm-specific effects can be taken into account by allowing the intercept to vary in the cross-section. Consider the following model:
2
. . .(0,
)
it i it it itCORR
= a +b +e ex
,:i i d
s
(6.22)The subscript i fora indicates that each firm has its own intercept or fixed effect. This feature of the model controls for time invariant firm characteristics and therefore provides the basis for the analysis of the effect of controlled variables that vary over time.
The correlation between the equity and bond returns may vary across time. As with cross-sectional fixed effects, the constant coefficient panel data model can be extended to control for time effects. Consider the following model:
2
. . .(0,
)
it t it it itCORR
= a + +b +e eg
x
,:i i d
s
(6.23)where
g
t is time-specific, effect. This effect is common in the cross-section so itcaptures all time-varying variables that affect the correlation between the equity and bond returns but are constant in the cross-section.
Following Petersen (2009), and Zhang, Zhou and Zhu (2009) clustered standard errors are used in all models to account for the serial correlation of errors (i.e. cross-sectional clustering).
6.4. Data
This study requires the firm-level equity and bond data. Since bond data points are relatively scarce compared to the equity data, the sample selection starts with all straight corporate bonds issued by non-financial companies in the US market available in the Thomson Reuters Datastream database. When multiple bonds are available from the same issuer, the bond with the maximum number of observations in considered. This is preferred to averaging the data of different bonds with a common issuer because all bonds have different characteristics such as duration and issue size. Bonds with less than 36 monthly observations, asset-backed bonds, bonds with any sort of collateral, or with an average market value of less than USD 10 million are excluded from the sample. Once the bond data is collected, it is matched with the equity data, also obtained from the Thomson Reuters Datastream database. The matched sample consists of 351 firms and 33,870 observations at the monthly level.
The sample covers the period from August 1996 to February 2011. Not all series cover the entire sample period, so the sample is unbalanced. It should be noted that the number of observations available at the beginning of the sample period (1996-2000) is much lower than later in the sample period (2001-2011). However, the beginning sample dataset is still large (1,519 observations for 33 firms) when compared to other studies which deal with bond data.
The accounting data required for the estimation of the distance-to-default is obtained from Compustat. The Fama and French factors are obtained from Kenneth R. French’s web site, and the risk-free interest rate and the S&P 500 index data are downloaded from the Thomson Reuters Datastream database.
6.5. Summary
The structural model of Merton (1974) shows that the equity and debt securities issued by a firm can be considered as contingent claims on firm’s assets with the book value of debt as the strike price. The value of equity resembles a call option, while the risk premium on a bond is modelled as a put option. As the option pricing theory of Black and Scholes (1973) suggests, the values of both equity and debt primarily depend on the value and the underlying assets volatility.
Factors affecting the value of the assets push the values of equity and bonds in the same direction and therefore induce a positive correlation between the returns of these two asset classes. On the other hand, an increase in the volatility of firm’s assets augments the value of equity and depresses the value of bonds, which clearly induces a negative correlation between the returns. The empirical literature generally finds that the correlation between equity and debt returns is positive, which implies that changes in the values of equity and debt are mostly caused by a change in the value of firm's assets. The volatility of assets is found to be the major determinant of the correlation between the equity and bond returns around specific events such as share repurchases and leveraged buyouts when the correlation turns negative.
Another important theoretical prediction arising from the literature is that the strength of correlation between the returns depends on the riskiness of firm’s assets. Information from the equity market has a limited impact upon the value of bonds issued by firms which are stable and with little debt. The sensitivity of the value of debt to changes in the value of equity increases with the riskiness of the assets and therefore the correlation between the returns strengthens. Hotchkiss and Ronen (2002) find that the correlation between the equity and bond returns is not statistically significant
a significant determinant of the correlation between equity returns and the credit default swap premium. Other studies generally confirm that credit risk and equity volatility are important determinants of the relationship between equity and bond returns.
The empirical literature emphasizes that the correlation between returns increases in turbulent times. The structural model does not distinguish between idiosyncratic and systematic risks. Systematic risks, therefore, should be as important as idiosyncratic risks in determining the correlation between equity and bond returns. The only common variable that is explicitly included in the structural model is the risk-free rate. In the risk-neutral framework of the structural model, the value of assets grows at the risk-free rate. As a result, an increase in the risk-free rate increases the value of equity. On the other hand, bonds prices decrease as all future cash flows are discounted at a higher rate. A change in the risk free rate, therefore, negatively affects the correlation between the equity and bond returns.
The overwhelming majority of the existing empirical studies (e.g. Kwan, 1996; Campbell and Taksler, 2003; Cremers et al., 2008) focuses on examining the unconditional correlation between the credit spread or the bond yield and the variables deriving from the structural model of Merton (1974). This study aims to extend the existing literature by examining the time properties of the correlation between the equity and bond returns. This is achieved by estimating the conditional correlation between the equity and bond returns, and then regressing this on a measure of credit risk, on equity volatility and on other variables of interest.
The conditional correlation between the equity and bond returns is estimated by means of a bivariate GARCH model. The data sample consists of the merged equity and bond data sets of 351 firms covering the period from 1/8/1996 to 18/2/2011 (over 33,000 monthly observations).
CHAPTER 7
CORRELATION BETWEEN THE EQUITY AND BOND RETURNS:
AN EMPIRICAL INVESTIGATION
7.1. Introduction
This chapter empirically examines the correlation between equity and bond returns. The statistical validity of hypotheses proposed in Chapter 6 is assessed by regressing the conditional correlation between the equity and bond returns on measures of equity, credit and systematic risks. The empirical analysis commences with the calculation of equity and bond returns. As described in Section 6.3.1, equity returns are calculated as the natural logarithm of the share price at the time t plus the dividends paid out during the period t over the share price at the time t-1, while bond returns are calculated as the exponential bond price returns plus the interest accrued during the observation period. The conditional correlation between the equity and bond returns, estimated by a bivariate Diagonal VECH, is comprehensively examined in a set of panel data models. Section 7.2 evaluates the conditional correlation between the equity and bond returns estimated by the bivariate Diagonal VECH(1,1) model, as well as the asymmetric Diagonal VECH(1,1,1) model. Section 7.3 presents an analysis of the effect of equity volatility on the correlation between the equity and bond returns. The analysis proceeds by regressing the correlation on equity volatility in the constant coefficient panel model. In the second stage, panel data models with cross-sectional and time fixed effects are estimated.
Section 7.4 examines the impact of credit risk upon the correlation between the equity and bond returns. The distance to default measure of Merton (1974) is utilized as an indicator of credit risk. As in the previous section, a set of panel data models is estimated with the correlation as the dependent variable and the distance to default and fixed effects as regressors. Section 7.5 considers the interaction between equity volatility and the distance to default. The analysis is conducted by regressing the
correlation on equity volatility and the interaction variable (equity volatility x the distance to default). The discrete version of this model is also estimated, whereby the interaction variable is replaced with a set of dummy variables taking the value of one if the distance to default is within a predefined range, and zero otherwise.
Section 7.6 examines the relationship between common factors and the correlation between equity and bond returns. The common factors considered are: the S&P 500 index returns, the S&P 500 index volatility, the risk-free rate, and the slope of the risk- free term structure. To take into account differences in firm exposure to systematic risks, firm-level systematic returns are considered instead of the S&P 500 index returns. The Fama and French (1993) model is employed to estimate firm-level systematic returns or expected returns. The final section examines the robustness of the results to changing the conditional correlation estimation model. The base case bivariate Diagonal VECH(1,1) model is expanded to allow for an asymmetric response of the conditional correlation to positive and negative shocks to equity and bond returns. Furthermore, the robustness of the results to controlling for firm size, bond duration and bond issue size is examined.