The structural model implies that the strength of correlation between the returns on firm’s equity and debt depend on the level of credit risk. High quality firms are very unlikely to default hence new information from equity markets has limited importance for the holders of their debt. Therefore, the returns on high quality bonds behave like the returns on government bonds. However, a change in the value of equity becomes increasingly relevant for bond pricing as the level of credit risk increases. At the brink of bankruptcy, bond returns are expected to be highly and positively correlated to equity returns.
The level of credit risk is proxied by the distance to default of Merton (1974). A higher distance to default implies a lower credit risk. Therefore, it is expected that the distance to default has a negative impact upon the correlation between the equity and bond returns.
7.4.1. The Constant Coefficient model
The starting point of the analysis is a constant coefficient panel model. The conditional correlation between the equity and bond returns is regressed on the distance to default, and unique coefficients are estimated for all firms in the sample. The results are presented in Table 7.7.
The coefficient of the distance to default is negative and significant as expected. The effect is significant in an economic sense as well. An improvement in credit quality as measured by one distance to default lowers the correlation by two percentage points. This implies that the equity and bond returns of risky firms (i.e. firms with a lower distance to default) are more positively correlated than for less risky firms. The distance to default explains about six per cent of the variation in the correlation, which is similar to the explanatory power of equity volatility.
Table 7.7
The impact of the distance to default upon the correlation between the equity and bond returns: the constant coefficient model
Variable Coefficient Std. Error t-Statistic Prob. Distance to Default -0.02 0.00 -7.01 0.00
C 0.22 0.02 9.35 0.00
R-squared 0.06 Mean dependent var 0.09 Adjusted R-squared 0.06 S.D. dependent var 0.28 S.E. of regression 0.27 Akaike info criterion 0.20 Sum squared resid 2,426.22 Schwarz criterion 0.20 Log likelihood -3,421.46 Hannan-Quinn criter. 0.20 F-statistic 2,141.28 Durbin-Watson stat 0.35 Prob(F-statistic) 0.00
Dependent Variable: Equity-Bond Returns Correlation; Method: Panel Least Squares (constant coefficient model); Sample: 1996M08 2011M02; Periods included: 175; Cross-sections included: 351; Total panel (unbalanced) observations: 33,855; White period standard errors & covariance (d.f. corrected).
7.4.2. The Cross-sectional Fixed Effects Model
Table 7.8 presents an estimate of the fixed effects panel data model in which each firm in the sample has its own intercept.
Table 7.8
The impact of the distance to default and the correlation between the equity and bond returns: the cross-sectional fixed effects model
Variable Coefficient Std. Error t-Statistic Prob. Distance to Default 0.00 0.00 -1.59 0.11
C 0.10 0.01 11.17 0.00
R-squared 0.61 Mean dependent var 0.09 Adjusted R-squared 0.61 S.D. dependent var 0.28 S.E. of regression 0.17 Akaike info criterion -0.66 Sum squared resid 998.77 Schwarz criterion -0.58 Log likelihood 11,602.71 Hannan-Quinn criter. -0.64 F-statistic 151.08 Durbin-Watson stat 0.83 Prob(F-statistic) 0.00
Dependent Variable: Equity-Bond Returns Correlation; Method: Panel Least Squares (cross-section fixed - dummy variables); Sample: 1996M08 2011M02; Periods included: 175; Cross-sections included: 351; Total panel (unbalanced) observations: 33,855; White period standard errors & covariance (d.f. corrected).
After allowing for fixed effects, the distance to default becomes statistically insignificant. The t-statistic is reduced from 7.01 in the constant coefficient model presented in Table 7.7 to 1.59 which makes it statistically insignificant at the ten per cent level. Further, the coefficient size approaches zero which indicates a low economic significance. This unexpected result is also obtained by Scheicher (2009) who reports that a simple leverage ratio (total debt / total assets) is not a significant determinant of the correlation between the equity returns and the credit default swap premia. As in this study, Scheicher uses a panel model with fixed effects and a White period covariance matrix. The author notes that this finding may be due to a limited data sample which includes only three years of quarterly observations. The above results are based on a data sample with 33,855 observations hence they cannot be explained by a limited sample. Since the distance to default becomes insignificant after the fixed effects are added to the model, an alternative explanation is that the cross-sectional differences in credit risk are captured by the fixed effects. To examine this hypothesis, a constant coefficient model is augmented with dummy variables taking the value of one is the distance to default is within a certain range and zero otherwise, instead of the fixed effects.
Seven dummy variables are created for values of the distance to default variable. The dummy variable which indicates the lowest credit risk (i.e. the distance to default ≥ 6) is dropped to avoid the multicollinearity problem (the dummy variable trap). As shown in Table 7.9, after controlling for the level of the distance to default, changes in that distance are not significant in explaining the correlation between the equity and bond returns. As suspected, it seems that the fixed effects in the panel model presented in Table 7.8 capture this effect. The dummy variables are statistically significant for values of the distance to default up to a value of four. The magnitude of coefficients emphasizes the economic significance of credit risk in determining the correlation. The correlation between equity and bond returns of the lowest-risk firms is virtually zero, while the correlation between the returns of the highest-risk firms is 31 per cent.
Table 7.9
The impact of the distance to default upon the correlation between the equity and bond returns, after controlling for the level of distance to default
Variable Coefficient Std. Error t-Statistic Prob. Distance to Default 0.00 0.00 -0.43 0.67 Distance to Default < 1 0.31 0.04 7.74 0.00 1≤Distance to Default < 2 0.25 0.04 6.64 0.00 2≤ Distance to Default < 3 0.18 0.03 6.10 0.00 3≤ Distance to Default < 4 0.10 0.02 4.49 0.00 4≤ Distance to Default < 5 0.03 0.02 1.65 0.10 5≤ Distance to Default < 6 0.00 0.01 0.26 0.80 C 0.04 0.03 1.29 0.20
R-squared 0.10 Mean dependent var 0.09 Adjusted R-squared 0.10 S.D. dependent var 0.28 S.E. of regression 0.26 Akaike info criterion 0.16 Sum squared resid 2,321.86 Schwarz criterion 0.16 Log likelihood -2,677.26 Hannan-Quinn criter. 0.16 F-statistic 536.91 Durbin-Watson stat 0.37 Prob(F-statistic) 0.00
Dependent Variable: Equity-Bond Returns Correlation; Method: Panel Least Squares (constant coefficient model); Sample: 1996M08 2011M02; Periods included: 175; Cross-sections included: 351; Total panel (unbalanced) observations: 33,855 - Distance-to-Default < 1 (800), 1 ≤ Distance-to-Default > 2 (1811), 2 ≤ Distance-to-Default > 3 (3,430), 3 ≤ Distance-to-Default > 4 (4,615), 4 ≤ Distance-to-Default > 5 (5,611), 5 ≤ Distance-to-Default > 6 (5,387), Distance-to-Default ≥ 6 (12,201); White period standard errors & covariance (d.f. corrected).
The results presented in Table 7.9 indicates that the correlation between the equity and bond returns monotonically decreases with the credit risk up to a certain point after which further changes in the credit risk do not impact upon the correlation.
To further examine this issue, Table 7.10 presents a panel model with fixed effects for observations with a value of distance to default of less than four. The coefficient of the distance to default is negative and significant at the five per cent level. An improvement in credit quality as measured by one distance to default lowers the correlation by one percentage point. This confirms that the statistical insignificance of the distance to default in the model presented in Table 7.8 is explained by the fixed effects capturing cross-sectional differences in credit risk.
Table 7.10
The impact of the distance to default (for observations with the value less than 4) upon the correlation between the equity and bond returns: the cross-sectional fixed model
Variable Coefficient Std. Error t-Statistic Prob. Distance to Default -0.01 0.01 -2.30 0.02
C 0.24 0.02 14.64 0.00
R-squared 0.59 Mean dependent var 0.20 Adjusted R-squared 0.57 S.D. dependent var 0.31 S.E. of regression 0.20 Akaike info criterion -0.33 Sum squared resid 421.84 Schwarz criterion -0.10 Log likelihood 2,085.28 Hannan-Quinn criter. -0.25 F-statistic 43.23 Durbin-Watson stat 0.97 Prob(F-statistic) 0.00
Dependent Variable: Equity-Bond Returns Correlation; Method: Panel Least Squares (cross- section fixed - dummy variables); Sample: 1996M08 2011M02 IF Distance to Default<4; Periods included: 164; Cross-sections included: 339; Total panel (unbalanced) observations: 10,656; White period standard errors & covariance (d.f. corrected).
7.4.3. The Period Fixed Effects Model
To control for the common time variations in the correlation between the equity and bond returns, the model is augmented with a dummy variable for each period (i.e. month) in the sample. The results are presented in Table 7.11.
After controlling for the time variations, the relationship between the distance to default and the correlation between the equity and bond returns remains negative and significant. Relative to the constant coefficient model presented in Table 7.7, the size of the coefficient increased in the absolute value from -0.02 to -0.03 with no substantial change in statistical significance. This implies that an improvement in credit quality as measured by one distance to default lowers the correlation by three percentage points. The results are similar to the results obtained in the model with equity volatility as an independent variable, and are expected because equity volatility and the distance to default are highly correlated. The significance of period fixed effects indicates the influence of common factors on the relationship between the equity and bond returns.
Table 7.11
The impact of the distance to default upon the correlation between the equity and bond returns: the period fixed effects model
Variable Coefficient Std. Error t-Statistic Prob. Distance to Default -0.03 0.00 -6.63 0.00
C 0.26 0.03 8.74 0.00
R-squared 0.10 Mean dependent var 0.09 Adjusted R-squared 0.09 S.D. dependent var 0.28 S.E. of regression 0.26 Akaike info criterion 0.17 Sum squared resid 2,334.35 Schwarz criterion 0.22 Log likelihood -2,768.06 Hannan-Quinn criter. 0.19 F-statistic 20.23 Durbin-Watson stat 0.36 Prob(F-statistic) 0.00
Dependent Variable: Equity-Bond Returns Correlation; Method: Panel Least Squares (period fixed - dummy variables); Sample: 1996M08 2011M02; Periods included: 175; Cross-sections included: 351; Total panel (unbalanced) observations: 33,855; White period standard errors & covariance (d.f. corrected).