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Longstaff and Schwartz (1995) provide a model for the valuation of corporate bonds, which account for two stochastic factors, which accommodate the effects of interest-rates and the firm's asset value. One of the most notable predictions of this and other structural models with an asset-based default process is that credit spreads are negatively related to the riskless rate. Longstaff and Schwartz test this prediction along with their predicted negative impact of the asset factor on credit spreads using Moody’s indices and consider the negative relationship they find as evidence supporting their model’s prediction.
Duffee (1998) argues that most of the bonds in Moody’s indices are callable bonds and thus the negative yield spread – riskless rate relation could be largely due to the negative
relationship between the yield premium attributed to callability and the riskless rate. Instead, Duffee uses noncallable bond portfolios and finds that the negative yield spread – riskless rate relation is much weaker once callability is controlled. Elton et al. (2001), point out that Duffee’s (1998) analysis ignores the effects of state taxes on U.S. bonds, and they argue that differential taxes on corporate and government bonds have an important impact on corporate bond yield spreads.
Canadian corporate bond indices are devoid of tax effects since Canadian corporate and government bonds, unlike U.S. bonds, are subject to the same tax treatment. These indices also contain a call provision that allows for identifying callable and economically noncallable bonds. Using this Canadian bond index data, we find an insignificant yield spread – government yield relation for economically noncallable bonds. For bonds with an economically viable call option, the negative relation we find increases with the moneyness of the call option. Our tests of different models with both real interest rates and nominal interest rates, with both yield spread and real (expected) default rates, consistently show the robustness of our results. We conclude that call risk dominates the negative yield spread – government yield relation. We further conclude that, for investment-grade bonds, the role of the asset factor (manifesting default risk) in influencing the sensitivity of corporate bond yield spreads to government yields is not significant.
These results indicate that a gap remains in our understanding of the default process. Theoretically, structural models suggest that an increase in the riskless rate implies a higher expected future value for the firm’s asset relative to the default threshold, and a lower risk- neutral probability of default and risk neutral credit spread. Empirically, we show that bondholders do not adjust their required default premium for an increase in the riskless rate. Our results provide support for reduced-form models that explicitly define a default hazard process and untie the relation between the firm’s asset value and default probability.
Appendix
***Insert Table A.1 here***
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No. of Mean of Std. Dev. Of Mean of Std. Dev. Of
Observations Credit Spread Credit Spread Relative Spread Relative Spread
AAA 200 0.5761 0.3050 1.0559 0.0331
AA 300 0.6350 0.3184 1.0716 0.0434
A 300 0.8612 0.3533 1.0980 0.0548
BBB 300 1.5359 0.7916 1.1842 0.1266
Table 1
Summary Statistics for Yield Spreads in SCM Long-Term Corporate Bond Indices for the August 1976 to July 2001 Period
The yield spread is the difference between the yield on a long-term index and the yield on the constant maturity, long-term Government of Canada index. The relative spread is the ratio of the yield on a long-term index to the yield on the constant maturity, long-term Government of Canada index. Data for the AAA indices are available only until March 1993.
Index a b c R2 AAA 0.0057 -0.1447 -0.6796 0.10 (0.39) (-4.66) (-2.21) AA 0.0026 -0.1538 -0.6808 0.11 (0.24) (-5.86) (-3.02) A 0.0043 -0.1513 -0.7903 0.12 (0.41) (-5.99) (-3.65) BBB 0.0086 -0.1892 -1.0957 0.08 (0.53 ) (-4.77) (-3.22) Index a b c R2 AAA N/A AA -0.0041 -0.2106 -0.6623 0.14 (-0.27) (-3.12) (-2.26) A 0.0015 -0.1927 -0.6160 0.13 (0.10) (-3.02) (-2.22) BBB 0.0022 0.0442 -1.2557 0.11 (0.10) (0.44) (-2.87) Panel A: 09:1976-07:2001 Panel B: 01:1995-07:2001
Table 2 reports the results of the OLS estimation of regression model (1). This regression model is of the following form:
∆S = a + b∆Y + cI + ε,
where ∆S is the monthly change in the yield spread:the difference between the yield on the relevant SCM index and that of the constant maturity, long-term Government of Canada index, ∆Y is the monthly change in the yield of the constant maturity, long-term Government of Canada index, and I is the monthly return on the Toronto Stock Exchange 300 index. t-values are in parentheses. Panel A reports the estimates for the entire sample, covering the 08:1976-07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds carrying the doomsday call are expected to dominate all indices.
Table 2
Regressions of Changes in the Yield Spread of SCM Long-Term Corporate Bond Indices on Changes in the Yield of the Constant Maturity Long-Term Government of
Canada Index and the Return on the Toronto Stock Exchange 300 Index - OLS Estimation
Table 2 reports the results of the OLS estimation of regression model (1) in which the dependent variable is the monthly change in the absolute yield spread. This regression model is of the following form:
∆S = a + b∆Y + cI + ε,
where ∆S is the monthly change in the absolute yield spread, ∆Y is the monthly change in the yield of the constant maturity, long-term Government of Canada index, and I is the monthly return on the Toronto Stock Exchange 300 index. t-values are in parentheses. Panel A reports the estimates for the entire sample, covering the 08:1976-07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds carrying the doomsday call are expected to dominate all indices.
Table 2
Regressions of Changes in Absolute Yield Spreads of SCM Long-Term Corporate Bond Indices on Changes in the Yield of the Constant Maturity Long-Term Government of Canada Index and the Return on the Toronto Stock Exchange 300
Index a b c R2 AAA 0.0007 -0.1927 -0.0553 0.16 (0.59) (-6.16) (-2.26) AA 0.0005 -0.2304 -0.0801 0.17 (0.44) (-7.76) (-3.56) A 0.0007 -0.2573 -0.0889 0.22 (0.68) (-9.09) (-4.15) BBB 0.0013 -0.3778 -0.1247 0.14 (0.66) (-6.84) (-2.98) Index a b c R2 AAA N/A AA -0.0002 -0.3294 -0.1266 0.23 (-0.08) (-4.42) (-2.69) A 0.0006 -0.3502 -0.1214 0.25 (0.25) (-4.83) (-2.65) BBB 0.0009 -0.2879 -0.2223 0.14 (0.23) (-2.48) (-3.04) Panel A: 09:1976-07:2001 Panel B: 01:1995-07:2001 Table 3
Regressions of Changes in Relative Yield Spreads of SCM Long-Term Corporate Bond Indices on Percentage Changes in the Yield of the Constant Maturity Long- Term Government of Canada Index and the Return on the Toronto Stock Exchange
300 Index - OLS Estimation
Table 3 reports the results of the OLS estimation of regression model (2) in which the dependent variable is the monthly change in the relative yield spread. This regression model is of the following form:
∆R = a + b∆PY + cI + ε,
where ∆R is the monthly change in the relative yield spread, ∆PY is the monthly
percentage change in the yield of the constant maturity, long-term Government of Canada index, and I is the monthly return on the Toronto Stock Exchange 300 index. t-values are in parentheses. Panel A reports the estimates for the entire sample, covering the 08:1976- 07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds carrying the doomsday call are expected to dominate all indices.
Regression Coefficients
Index β0 β1 β2 m p q Norm. Test R2 LM
AAA -0.0027 -0.0442 -0.0806 1, 2, 4 1 1 <0.0001 0.19 <0.0001 (-0.51) (-2.51) (-4.25) AA -0.0074 -0.1231 -0.1893 1, 2, 4 1 1 0.0005 0.18 <0.0001 (-1.82) (-8.72) (-12.56) A -0.0063 -0.1011 -0.1514 1, 2 1 1 0.0001 0.16 <0.0001 (-1.13) (-6.22) (-8.16) BBB 0.0025 -0.1244 -0.1996 - 1 1 <0.0001 0.09 - (0.18) (-3.26) (-4.72) Regression Coefficients
Index β0 β1 β2 m p q Norm. Test R2 LM
AAA N/A AA -0.0076 -0.1732 -0.1751 - - - - 0.08 - (-0.48) (-2.34) (-2.44) A -0.0016 -0.1519 -0.1645 - - - - 0.08 - (-0.11) (-2.17) (-2.42) BBB -0.0037 0.1361 0.0947 - - - - 0.02 - (-0.15) (1.21) (0.87)
AR and GARCH Parameters Goodness of Fit
Panel A: 09:1976-07:2001
AR and GARCH Parameters
Panel B: 01:1995-07:2001 Table 4
Regressions of Changes in Absolute Yield Spreads of SCM Long-Term Corporate Bond Indices on Changes in Government of Canada Yields -
AR-GARCH Estimation
Table 4 reports the results of the AR-GARCH estimation of regression model (3) in which the dependent variable is the monthly change in the absolute yield spread. This regression model is of the following form:
∆S = β0 + β1∆YT-bill + β2∆Slope + ε,
where ∆S is the monthly change in the absolute yield spread, ∆YT-bill is the monthly
change in the three-month Treasury Bill yield, and ∆Slope is the monthly change in the spread between the constant maturity long-term Government of Canada index and the three-month Treasury bill yield. t-values are in parentheses, m gives the degree of the autoregressive process as determined by the stepwise autoregression method, p and q are the GARCH(p,q) parameters, Norm. Test gives the p-value for the normality test for detecting misspecification of the GARCH model, and finally LM gives the p-value for the Lagrange multiplier test. Panel A reports the estimates for the entire sample, covering the 08:1976-07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds carrying the doomsday call are expected to dominate all indices.
Regression Coefficients
Index β0 β1 β2 β3 β4 m p q Norm. Test R2 LM AAA -0.0101 -0.1547 0.0614 -0.0887 -0.0055 1, 2, 4 1 1 <0.0001 0.22 0.0004 (-1.51) (-11.06) (3.80) (-7.40) (-3.61) AA -0.0063 -0.1196 0.0429 -0.0679 -0.0050 1, 2, 4 1 1 0.0018 0.23 0.0067 (-1.47) (-9.67) (4.40) (-6.90) (-4.68) A -0.0067 -0.1208 0.0600 -0.0541 -0.0058 1, 2 1 1 0.0021 0.20 <0.0001 (-1.07) (-8.86) (5.59) (-5.06) (-4.66) BBB 0.0029 -0.1467 0.0447 -0.0701 -0.0109 - 1 1 <0.0001 0.15 - (0.21) (-3.82) (1.58) (-2.67) (-3.30) Regression Coefficients
Index β0 β1 β2 β3 β4 m p q Norm. Test R 2 LM AAA N/A AA -0.0154 -0.1676 0.2605 -0.0041 -0.0070 - - - - 0.16 - (-0.85) (-2.02) (1.31) (-0.08) (-2.31) A -0.0001 -0.1817 0.0541 -0.0041 -0.0063 - - - - 0.13 - (-0.00) (-2.29) (0.29) (-0.08) (-2.17) BBB 0.0047 0.0510 -0.0081 0.0141 -0.0132 - - - - 0.16 - (0.14) (0.43) (-0.03) (0.18) (-3.03)
AR and GARCH Parameters Goodness of Fit Panel A: 09:1976-07:2001
AR and GARCH Parameters
Panel B: 01:1995-07:2001
Table 5
Regressions of Changes in Nominal Yield Spreads of SCM Long-Term Corporate Bond Indices on Changes in Selected Nominal Determinants -
AR-GARCH Estimation
Table 5 reports the results of the AR-GARCH estimation of regression model (4) in which the dependent variable is the monthly change in the nominal yield spread. This regression model is of the following form:
∆S = β0 + β1∆YLT + β2 (∆YLT) 2 + β3∆Slope + β4I + ε,
where ∆S is the monthly change in the nominal yield spread, ∆YLT is the monthly change in the nominal yield on the constant maturity long-term Government of Canada index, (∆YLT)2 is a convexity term, ∆Slope is the monthly change in the nominal spread between the constant maturity long-term Government of Canada index and the three-month Treasury bill yield, and I is the monthly nominal return on the Toronto Stock Exchange 300 index. t-values are in parentheses, m gives the degree of the autoregressive process as determined by the stepwise autoregression method, p and q are the GARCH(p,q) parameters, Norm. Test gives the p-value for the normality test for detecting misspecification of the GARCH model, and finally LM gives the p-value for the Lagrange multiplier test. Panel A reports the estimates for the entire sample, covering the 08:1976-07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds carrying the doomsday call are expected to dominate all indices.
Regression Coefficient
Index β0 β1 β2 m p q Norm. Test R
2 LM AAA -0.0012 -0.2232 -0.1801 1,4 1 1 <0.0001 0.14 <.0001 (-0.15) (-8.02) (-8.47) AA -0.0036 -0.1542 -0.1350 1, 2, 4 1 1 <0.0001 0.16 <.0001 (-0.8) (-6.74) (-9.03) A -0.0049 -0.1527 -0.1204 1, 2 1 1 0.0003 0.14 <0.0001 (-0.82) (-5.99) (-7.11) BBB 0.0018 -0.1822 -0.1454 - 1 1 <0.0001 0.07 - (-0.13) (-3.22) (-3.83) Regression Coefficients
Index β0 β1 β2 m p q Norm. Test R
2 LM AAA N/A AA -0.0079 -0.2082 -0.1778 - - - - 0.08 - (-0.5) (-2.33) (-2.62) A -0.0021 -0.1916 -0.1629 - - - - 0.08 - (-0.14) (-2.27) (-2.53) BBB -0.0045 0.1266 0.1143 - - - - 0.16 - (-0.19) (-0.93) (-1.11) Panel A: 09:1976-07:2001 AR and GARCH Parameters
Panel B: 01:1995-07:2001
AR and GARCH Parameters Goodness of Fit
Table 6
Regressions of Changes in Absolute Nominal Yield Spreads of SCM Long-Term Corporate Bond Indices on Changes in the Real Yield of the Constant Maturity Long-Term Government
of Canada Index and on Changes in the Monthly Inflation Rate
Table 6 reports the results of the AR-GARCH estimation of regression model (5). This regression model is of the following form:
∆S = β0 + β1∆Inf + β2∆YLT + ε,
where ∆S is the monthly change in the nominal yield spread, ∆YLT is the monthly change in the real
yield on the constant maturity long-term Government of Canada index, ∆Inf is the monthly change in the inflation rate. t-values are in parentheses, m gives the degree of the autoregressive process as determined by the stepwise autoregression method, p and q are the GARCH(p,q) parameters, Norm. Test gives the p-value for the normality test for detecting misspecification of the GARCH model, and finally LM gives the p-value for the Lagrange multiplier test. Panel A reports the estimates for the entire sample, covering the 08:1976-07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds carrying the doomsday call are expected to dominate all indices.
Regression Coefficients
Index β0 β1 β2 β3 β4 m p q Norm. Test R2 LM AAA -0.0041 0.0019 0.0002 -0.0418 -0.0005 1, 2 1 1 <0.0001 0.15 <.0001 (-0.55) (0.13) (0.02) (-3.06) (-0.31) AA -0.0040 -0.0178 0.0052 -0.0706 -0.0040 1, 2, 4 1 1 <.0001 0.19 0.0005 (-0.71) (-1.36) (0.66) (-5.91) (-2.90) A -0.0032 0.0005 0.0044 -0.0571 -0.0034 1, 2 1 1 0.0012 0.15 <.0001 (-0.52) (0.04) (0.58) (-4.28) (-2.81) BBB -0.0022 -0.0029 0.0208 -0.0818 -0.0078 - 1 1 <0.0001 0.11 - (-0.15) (-0.21) (1.38) (-3.10) (-2.38) Regression Coefficients
Index β0 β1 β2 β3 β4 m p q Norm. Test R2 LM
AAA N/A AA -0.0180 -0.0104 0.1060 -0.0453 -0.0044 - - - - 0.16 - (-1.15) (-0.29) (2.19) (-0.98) (-1.49) A -0.0115 -0.0130 0.0927 -0.0503 -0.0042 - - - - 0.10 - (-0.64) (-0.37) (1.89) (-1.05) (-1.45) BBB -0.0185 0.0465 0.0997 -0.0131 -0.0122 - - - - 0.13 - (-0.68) (0.86) (1.33) (-0.18) (-2.77)
AR and GARCH Parameters Goodness of Fit Panel A: 09:1976-07:2001
AR and GARCH Parameters
Panel B: 01:1995-07:2001
Table 7
Regressions of Changes in Real Yield Spreads of SCM Long-Term Corporate Bond Indices on Changes in Selected Real Determinants -
AR-GARCH Estimation
Table 7 reports the results of the AR-GARCH estimation of regression model (4) in which the dependent variable is the monthly change in the real yield spread. This regression model is of the following form:
∆S = β0 + β1∆YLT + β2(∆YLT,R)2 + β3∆Slope + β4I + ε,
where ∆S is the monthly change in the real yield spread, ∆YLT,R is the monthly change in the real yield on the constant maturity long-term Government of Canada index, (∆YLT,R)2 is a convexity term, ∆Slope is the monthly change in the real spread between the constant maturity long-term Government of Canada index and the three-month Treasury bill yield, and I is the monthly real return on the Toronto Stock Exchange 300 index. t-values are in parentheses, m gives the degree of the autoregressive process as determined by the stepwise autoregression method, p and q are the GARCH(p,q) parameters, Norm. Test gives the p-value for the normality test for detecting misspecification of the GARCH model, and finally LM gives the p-value for the Lagrange multiplier test. Panel A reports the estimates for the entire sample, covering the 08:1976- 07:2001 25-year period. Data during this sample period are dominated by corporate bonds carrying a standard call provision. Panel B outlines the results for the 01:1995-07:2001 sub-period, in which bonds