It is expected that equity volatility has a positive impact upon the conditional correlation between the equity and bond returns. This expectation is inconsistent with the prediction of the structural model that a change in volatility of the value of firm’s assets has an opposite effect on the values of equity and debt. As holders of a call option on firm’s assets, equity holders stand to benefit from the upside potential associated with higher volatility, whereas debt holders face only a higher default probability caused by an increase in volatility. Therefore, a change in asset volatility should give rise to a negative correlation between equity and bond returns. This result is derived under an assumption that the value of the underlying assets remains the same. In this case, a change in asset volatility causes a redistribution of value between equity and debt holders.
In the empirical data it is hard to find an observation when the asset volatility changes while the asset value remains unchanged. In the sample used in this study, the volatility
of underlying assets is negatively correlated with equity and bond returns. This indicates that an increase in volatility is accompanied by a decrease in the values of debt and equity.
7.3.1. The Constant Coefficient Model
In the first step, the relationship between the equity volatility and the conditional correlation between equity and bond returns is examined in a constant coefficient panel data model which estimates unique coefficients for all firms in the sample. The results are presented in Table 7.3.
The relationship between equity volatility and the correlation between the equity and bond returns is positive as expected. The coefficient is statistically significant and its size implies that a percentage point increase in annual equity volatility raises the correlation by 0.3 per cent. Equity volatility explains about six per cent of variations in the correlation.
This result is generally consistent with Scheicher (2009) who reports that an increase in equity volatility negatively impacts upon the correlation between the equity returns and credit default swap premia. A negative coefficient of equity volatility in the Scheicher study is equivalent to a positive coefficient in this study because bond returns and credit default swap premia are negatively correlated.
Table 7.3
The impact of equity volatility upon the correlation between the equity and bond returns: the constant coefficient model
Variable Coefficient Std. Error t-Statistic Prob. Equity Volatility 0.33 0.03 10.14 0.00
C -0.03 0.01 -1.76 0.08
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 2,428.70 Schwarz criterion 0.20 Log likelihood -3,432.80 Hannan-Quinn criter. 0.20 F-statistic 2,128.31 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: 33,870; White period standard errors & covariance (d.f. corrected); Equity volatility is annualized.
7.3.2. The Cross-sectional Fixed Effects Model
The constant coefficient model by construction does not allow cross-sectional differences in a relationship. In this case it is too restrictive as the correlation between equity and bond returns should be, in addition to equity volatility, influenced by bond maturity, credit risk and other factors. To account for those other factors, each firm in the sample is allowed to have its own intercept or fixed effect.
The results, which are presented in Table 7.4, indicate large cross-sectional differences in the correlation between the equity and bond returns. The relationship between equity volatility and the correlation remains positive and significant as in the constant coefficient model presented in Table 7.3, but the economic and the statistical significance of equity volatility is substantially reduced. The results imply that a one percentage point increase in equity volatility raises the correlation by 0.1 percentage point. Further, the fixed effects increase the R-squared of the model from six per cent to 61 per cent.
Table 7.4
The impact of equity volatility upon the correlation between the equity and bond returns: the cross-sectional fixed effects model
Variable Coefficient Std. Error t-Statistic Prob. Equity Volatility 0.06 0.02 2.77 0.01
C 0.07 0.01 9.36 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 -0.67 Sum squared resid 996.51 Schwarz criterion -0.58 Log likelihood 11,653.73 Hannan-Quinn criter. -0.64 F-statistic 151.87 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: 33,870; White period standard errors & covariance (d.f. corrected).
Formal tests strongly reject the hypothesis that fixed effects are redundant. As shown in Table 7.5, the F-test and the
c
2 test assign zero probability to a hypothesis that the fixed effects are redundant.Table 7.5
The test for the redundancy of the fixed-effects
Test cross-section fixed effects Statistic d.f. Prob. Cross-section F 137.63 -3.50E+07 0.00 Cross-section Chi-square 30173.04 350.00 0.00
The tests evaluate the joint significance of the fixed effects using sums-of-squares (F-test) and the likelihood function (Chi-square test).
It is noted that the fixed effects are substantially more important in the analysis of the correlation between the equity and bond returns than in the analysis of the credit spread, as presented in Chapter 5. In the latter analysis, the fixed effects improve the model’s R-squared by 14 percentage points and do not substantially reduce the size of the coefficient of equity volatility.
7.3.3. The Period Fixed Effects Model
The constant correlation panel model is augmented with period fixed effects to control for time variations in the relationship between equity volatility and the correlation.
Similar to cross-sectional fixed effects, which capture firm specific factors, period fixed effects are dummy variables which take the value of one if an observation is in a particular month, and zero otherwise. The results are presented in Table 7.6.
The relationship between equity volatility and the correlation between the equity and bond returns remains positive and statistically significant after controlling for the period effects. The economic significance of equity volatility (the magnitude of the coefficient) is increased by 43 percentage points relative to the coefficient in the constant coefficient model presented in Table 7.3. The estimated coefficient implies that a one percentage point increase in equity volatility raises the correlation by 0.5 percentage point. The period fixed effects improve the model’s adjusted R-squared from six per cent to nine per cent. Formal tests strongly reject the hypothesis that the period effects are redundant. It should be noted that the improvement in explanatory power is modest in comparison to the improvement associated with the cross-sectional fixed effects. Consistent with the results presented in Chapter 5 and other studies (e.g. Ericsson, Jacobs and Oviedo, 2009), this result highlights the challenges in explaining the correlation between the equity and bond returns in cross-section.
Table 7.6
The impact of equity volatility upon the correlation between the equity and bond returns: the period fixed effects model
Variable Coefficient Std. Error t-Statistic Prob. Equity Volatility 0.47 0.05 9.66 0.00
C -0.08 0.02 -4.16 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 2,330.59 Schwarz criterion 0.22 Log likelihood -2,734.45 Hannan-Quinn criter. 0.19 F-statistic 20.71 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: 33,870; White period standard errors & covariance (d.f. corrected).