The structural model implies that the relationship between equity volatility and the credit spread depends on the level of credit risk. The relationship should be stronger for riskier firms as an increase in the equity volatility of those firms may significantly increase the default probability and hence the potential loss which bondholders face. On the other hand, the potential of equity volatility to meaningfully raise the default probability of high credit quality firms is limited. This implies that it is important to control for the level of credit risk in an analysis of the relationship between the credit spread and equity risk measures. Fixed effects in panel models control for the cross- sectional difference in the level of credit risk to some extent, but since fixed effects are time invariant they cannot capture changes in credit risk over time. A common approach in the literature is to use credit ratings to control for credit risk, which amounts to creating a set of dummy variables taking the value of one if a firm is assigned
a particular rating by a major rating agency or zero otherwise. Although appealing on the basis of its simplicity, this approach limits empirical analysis because of the credit ratings are coarse-grained. To overcome this limitation of credit ratings, the level of credit risk is controlled for with an interaction variable (equity volatility times the distance to default), and alternatively with dummy variables for ranges of the distance to default values.
Table 5.25 presents an estimate of the panel model with the equity volatility and the distance to default interaction variable. The coefficient of the interaction variable is negative as expected. It implies that the impact of changes in equity volatility on the credit spread increases with the level of credit risk (i.e. as the distance to default decreases). Equity volatility is virtually economically insignificant in explaining the credit spread on bonds issued by firms five distances away from the default point, and the economic significance of a one percentage point change in equity volatility increases at the rate of 1.68 basis points per unit change in the distance to default. When compared to the univariate model presented in Table 5.2, the coefficient of equity volatility drops from 1,049 to 835 while the R-squared increases from 39 to 42 per cent. This finding is generally consistent with Campbell and Taksler (2003) and Cremers et al. (2008) who use accounting leverage ratios and credit ratings to proxy for the level of credit risk. It is interesting to note that these studies do not evidence a monotonic increase in the importance of equity volatility as credit ratings and the leverage ratio deteriorate. Campbell and Taksler estimate a larger coefficient for firms with long-term debt to assets ratios between 10 and 25 per cent than more leveraged firms with a ratio of between 25 and 66 per cent. Cremers et al. obtain the same inconsistent result for BBB+/BBB- and BB+ and lower rated firms.
Table 5.25
Interaction between equity volatility and the distance to default in explaining variations in the credit spread
Variable Coefficient Std. Error t-Statistic Prob. Equity Volatility 834.80 59.31 14.07 0.00 Equity Volatility x
Distance-to-Default -168.45 21.96 -7.67 0.00
C 233.48 29.63 7.88 0.00
R-squared 0.42 Mean dependent var 276.78 Adjusted R-squared 0.42 S.D. dependent var 360.72 S.E. of regression 273.97 Akaike info criterion 14.06 Sum squared resid 5.47E+10 Schwarz criterion 14.06 Log likelihood -5.13E+06 Hannan-Quinn criter. 14.06 F-statistic 267,486.20 Durbin-Watson stat 0.03 Prob(F-statistic) 0.00
Dependent Variable: Credit Spread; Method: Panel Least Squares (constant coefficient model); Sample: 8/01/1996 2/18/2011; Periods included: 3797; Cross-sections included: 352; Total panel (unbalanced) observations: 729252; White period standard errors & covariance (d.f. corrected).
To further examine this effect, seven dummy variables are created for values of the distance to default variable. The results are presented in Table 5.26. The dummy variable which indicates the lowest credit risk (i.e. distance to default ≥ six) is dropped to avoid the multicollinearity problem (the dummy variable trap). The impact of equity volatility monotonically increases with an increase in credit risk (i.e. a decrease in the distance to default). All control variables are highly statistically significant. This clearly confirms the prediction of the structural model that the credit spread becomes more sensitive to changes in volatility as the default probability increases. Interestingly, the results suggest that equity volatility has a negative impact upon the credit spread on bonds issued by the highest-quality firms (i.e. firms with a distance to default above five). A one percentage point increase in equity volatility widens the credit spread on the highest-risk bonds (i.e. the group with DD < 1 ) by 9.44 basis points, while it narrows the credit spread on the lowest quality bonds by 3.12 basis points (i.e. the group with DD > 6 ).
Table 5.26
The relationship between the credit spread and equity volatility across distance to default groups
Variable Coefficient Std. Error t-Statistic Prob. Equity Volatility -312.42 94.75 -3.30 0.00 Equity Volatility x I (DD < 1) 1,256.89 106.15 11.84 0.00 Equity Volatility x I (1≤ DD > 2) 950.84 79.86 11.91 0.00 Equity Volatility x I (2≤ DD > 3) 731.83 61.94 11.82 0.00 Equity Volatility x I (3≤ DD > 4) 532.28 50.15 10.61 0.00 Equity Volatility x I (4≤ DD > 5) 356.15 37.00 9.63 0.00 Equity Volatility x I (5≤ DD > 6) 208.51 26.02 8.01 0.00 C 213.87 21.45 9.97 0.00
R-squared 0.43 Mean dependent var 276.78 Adjusted R-squared 0.43 S.D. dependent var 360.72 S.E. of regression 272.03 Akaike info criterion 14.05 Sum squared resid 5.40E+10 Schwarz criterion 14.05 Log likelihood -5.12E+06 Hannan-Quinn criter. 14.05 F-statistic 78,997.99 Durbin-Watson stat 0.04 Prob(F-statistic) 0.00
Dependent Variable: Credit Spread; Method: Panel Least Squares; Sample: 8/01/1996 2/18/2011; Periods included: 3,797; Cross-sections included: 352; Total panel (unbalanced) observations: 729,252; White period standard errors and covariance (d.f. corrected).
Although the distance to default appears to be successful in grouping observations according to the level of credit risk, it implies zero default probability for the majority of estimates, which is unrealistic. Therefore, the distance to default appears to perform well as a credit risk indicator, but its conversion into probability of default, as envisaged by the structural model, is clearly problematic. This explains the finding of Campbell and Taksler that the impact that equity volatility has on the credit spread is larger than predicted by the structural model, and a weak correlation between the credit spread and the default probability implied by the structural model reported by Bharath and Shumway (2008).
As previously noted, Vassalou and Xing (2004) consider the distance to default as a credit risk indicator, and Crosbie and Bohn (2003) note that Moody’s Analytics maps
distance to default values to default probabilities according to an empirical default probability distribution.