**A.2 Leland & Toft (1996)**

**3.5 Empirical Results**

**3.5.2 Modeling CDS Spreads**

A simple comparison of the two average estimated risk premia in Figure 3.11 with the average market CDS spread from Figure 3.1 suggest a link between the CDS spreads and these model implied risk premia. To see if a time varying risk premium can help structural models to explain credit spreads I follow Elkamhi

& Ericsson (2007) and consider the following panel regression for the level of the
CDS spreads^{27}

CDSit = + _{1}Levi;t+ _{2}Evoli;t+ _{3}Ereti;t+ _{4}Slopet+ _{5}rt+ _{6}RP I_{i;t}^{equity}+"i;t;
(3.12)
whereLev denotes the …rm’s leverage, Evol is either the …rm’s 250 day historical
volatility or it’s 30-day option implied volatility,Eret is the daily equity return of
the …rm,Slopeis the di¤erence between the 10- and 2-year constant maturity rate
and r is the 5-year constant maturity rate corresponding to the maturity of the
CDS spreads. RPI^{equity} is the equity implied measure of the credit risk premium
calculated in equation (3.4), and it is thus purely based on the equity market and
the structural model. The regression in (3:12) is run both with and without the
model implied risk premium in order to gauge the gain in explanatory power by
including this variable. The results are shown in Table 3.4, when the regressions
are run on the full sample^{28}. Panel A of Table 3.4 tabulates the results with
the historical volatility included in the regression, while the results with implied
equity volatility are reported in panel B.

We see that including the equity implied risk premium increases the
explana-tory power of the regressions and the coe¢ cients on the risk premium are all
strongly signi…cant^{29}. When the risk premium is included in the regression with
the historical volatility the R-square increases by3% from49:4% to52:4%;while
the R-square increases by 5:5% from 57:4% to 62:9% when the regressions are
run with the implied volatility.

27The regresion in Elkamhi & Ericsson (2007) is performed on corporate bond spreads.

28The same variables are included in all of the regressions, although some of the variables may be insigni…cant at times.

29This is consistent both when the standard errors are clustered by time and by …rm. The OLS standard errors on the risk premium coe¢ cients are very similar to the standard errors when clustering by time, while standard errors are substantially larger when clustering by …rm.

This indicates a …rm e¤ect in the data (see Petersen (2007)). The results are also robust if a weekly time dummy is included, while clustering by …rm. In this case the slope and the interest rate are left out of the regression since they capture a time e¤ect.

Table3.4:ImportanceofRiskPremiumforCDSSpreads ThistablereportstheresultsofthepanelregressionCDSit=+1Levit+2Evolit+3Eretit+4Slopet+5rt+6RPIit+"it. T-statisticsarereportedinparentheses.Evoliseitherthehistoricalequityvolatility,calculatedusing250daysofhistoricalequity returns,ortheimpliedvolatilityon30-dayat-the-moneyputoptions.Leverage(lev)istotalliabilitiesdividedbythesumoftotal liabilitiesandequitymarketcapitalization.Eretisthedailyequityreturn,slopeistheslopeoftheyieldcurveandristhelevel oftheinterestrate.RPIistheequityimpliedmodelriskpremium.*,**and***denotesigni…canceat10,5and1percent, respectively. PanelAHistoricalvolatilityPanelBImpliedvolatility Intercept-191.83-127.96-191.83-127.96-252.2-142.97-252.2-142.97 (-9.51)(-6.53)(-7.38)(-5.17)(-13.3)(-7.94)(-8.79)(-3.88) Lev2.811.902.811.902.671.462.671.46 (39.40)(32.04)(6.95)(5.08)(43.24)(20.97)(7.29)(4.08) Evol.5.844.615.844.617.824.557.824.55 (24.97)(20.39)(7.96)(5.33)(35.81)(15.99)(7.93)(3.25) Eret0.800.990.800.991.952.121.952.12 (0.66)(0.85)(1.48)(1.80)(1.64)(1.84)(3.89)(3.61) Slope-14.18-12.85-14.18-12.85-8.52-5.03-8.52-5.03 (-4.49)(-4.19)(-3.10)(-2.65)(-3.08)(-2.00)(-2.07)(-1.26) r-0.43-1.68-0.43-1.680.853.530.853.53 (-0.09)(-0.34)(-0.1)(-0.41)(0.25)(1.15)(0.21)(1.08) RPIequity -0.60-0.60-1.32-1.32 -(22.19)-(2.71)-(21.37)-(3.45) R2 0.4940.5240.4940.5240.5740.6290.5740.629 N3340133401334013340133401334013340133401 ClusterDateDateFirmFirmDateDateFirmFirm

The results are in line with Elton et al. (2001), who show that there is a nontrivial component of credit spreads, interpreted as a risk premium, which is correlated with factors explaining equity risk premia. Elkamhi & Ericsson (2007) also …nd that risk premia in credit and equity market are closely related, and emphasizes that the nonlinear relationship implied by the structural model plays an important role in establishing the link between the equity premium, the model implied credit risk premium and the credit spread.

On the other hand Berndt, Lookman & Obreja (2006) extract a factor rep-resenting the part of default swap returns, implied by a reduced form credit risk model, that does not compensate for interest rate risk or expected default losses.

They …nd that this factor is priced in the corporate bond market but that they cannot establish with the same con…dence that it is a factor for equity returns.

Their estimate of credit risk premia is based on EDF’s though, which we have seen might give rise to mismeasured credit risk premia.

In Table 3.5 the regressions are run for the investment grade segment. Again the coe¢ cients are highly signi…cant on the risk premium and now the R-square increases by 5:3% from 44:4% to 59:7% with the historical volatility included, while the R-square increases by8:2% from 52:6% to60:8% when the regressions are run with the implied volatility.

In Table 3.6 the regressions are run for the speculative grade segment. Now there is only a marginal increase in the R-square, which increases by 2:5% from 53:8% to 56:3% with the historical volatility included, while there is no increase in the R-square, which stays at 74:7%;when the regressions are run with the im-plied volatility. Furthermore the coe¢ cient on the risk premium is insigni…cant when implied volatility is included. Combined with the regression results for the investment grade segment, this suggest that the risk premium is more important for investment grade …rms than for speculative grade …rms, and also that invest-ment grade …rms have proportionally higher risk premia. This supports results found in e.g. Elkamhi & Ericsson (2007), Berndt et al. (2005) and Huang &

Huang (2003).

Table3.5:ImportanceofRiskPremiumforInvestmentGradeCDSSpreads ThistablereportstheresultsofthepanelregressionCDSit=+1Levit+2Evolit+3Eretit+4Slopet+5rt+6RPIit+"it. Theregressionisrunfortheinvestmentgradequotesinthesample.T-statisticsarereportedinparentheses.Evoliseitherthe historicalequityvolatility,calculatedusing250daysofhistoricalequityreturns,ortheimpliedvolatilityon30-dayat-the-money putoptions.Leverage(lev)istotalliabilitiesdividedbythesumoftotalliabilitiesandequitymarketcapitalization.Eretisthe dailyequityreturn,slopeistheslopeoftheyieldcurveandristheleveloftheinterestrate.RPIistheequityimpliedmodelrisk premium.*,**and***denotesigni…canceat10,5and1percent,respectively. PanelAHistoricalvolatilityPanelBImpliedvolatility Intercept-167.99-110.49-167.99-110.49-215.28-114.21-215.28-114.21 (-11.48)(-7.57)(-7.99)(-6.19)(-16.78)(-9.71)(-8.98)(-4.27) Lev2.041.202.041.201.960.911.960.91 (32.74)(25.15)(7.91)(4.80)(36.04)(20.55)(7.94)(3.37) Evol.4.543.374.543.375.872.865.872.86 (18.72)(15.37)(6.59)(5.18)(27.14)(12.95)(7.88)(3.43) Eret0.520.700.520.701.401.441.401.44 (0.53)(0.76)(0.76)(1.08)(1.73)(1.90)(2.18)(2.22) Slope-8.23-6.58-8.23-6.58-2.870.41-2.870.41 (-2.96)(-2.48)(-1.74)(-1.45)(-1.47)(0.24)(-0.79)(0.14) r7.767.067.767.069.9511.259.9511.25 (1.94)(1.79)(1.86)(1.88)(4.48)(5.38)(2.77)(3.92) RPIequity -0.57-0.57-1.28-1.28 -(23.99)-(3.85)-(20.54)-(3.48) R2 0.4440.4970.4440.4970.5260.6080.5260.608 N3071230712307123071230712307123071230712 ClusterDateDateFirmFirmDateDateFirmFirm

Table3.6:ImportanceofRiskPremiumforSpeculativeGradeCDSSpreads ThistablereportstheresultsofthepanelregressionCDSit=+1Levit+2Evolit+3Eretit+4Slopet+5rt+6RPIit+"it. Theregressionisrunforthespeculativegradequotesinthesample.T-statisticsarereportedinparentheses.Evoliseitherthe historicalequityvolatility,calculatedusing250daysofhistoricalequityreturns,ortheimpliedvolatilityon30-dayat-the-money putoptions.Leverage(lev)istotalliabilitiesdividedbythesumoftotalliabilitiesandequitymarketcapitalization.Eretisthe dailyequityreturn,slopeistheslopeoftheyieldcurveandristheleveloftheinterestrate.RPIistheequityimpliedmodelrisk premium.*,**and***denotesigni…canceat10,5and1percent,respectively. PanelAHistoricalvolatilityPanelBImpliedvolatility Intercept-617.11-387.66-617.11-387.66-699.56-674.35-699.56-674.35 (-9.31)(-5.99)(-3.32)(-1.82)(-15.21)(-13.97)(-5.87)(-3.56) Lev6.614.206.614.205.334.995.334.99 (26.24)(12.15)(4.44)(2.40)(31.30)(20.79)(5.91)(3.61) Evol.6.444.026.444.0212.6412.1312.6412.13 (12.66)(6.09)(3.65)(1.67)(30.06)(19.63)(6.00)(3.27) Eret3.053.103.053.104.354.474.354.47 (0.71)(0.72)(1.33)(1.34)(1.79)(1.85)(1.99)(2.36) Slope63.3162.5063.3162.5037.2937.2637.2937.26 (4.91)(5.18)(2.74)(2.91)(4.38)(4.39)(2.64)(2.62) r49.3333.2949.3333.2937.6138.7037.6138.70 (3.23)(2.32)(1.55)(1.31)(4.01)(4.20)(2.00)(2.42) RPIequity -1.14-1.14-0.19-0.19 -(9.20)-(2.32)-(1.84)-(0.30) R2 0.5380.5630.5380.5630.7470.7470.7470.747 N26892689268926892689268926892689 ClusterDateDateFirmFirmDateDateFirmFirm

In all of the regressions performed in Tables 3.4, 3.5 and 3.6 the R-squares are higher when implied volatility is included instead of the historical volatility and this is especially striking for the speculative grade segment, where the R-squares without the risk premium included are 53:8% and 74:7% respectively.

This con…rms that option implied volatility has a higher explanatory power for credit spreads, and suggest that when measuring the time variation in the risk premia one should use the information contained in option implied volatilities to back out default probabilities. Furthermore the results are also in line with Cao et al. (2006), who …nd the strongest link between option-implied volatilities and CDS spreads among …rms with the lowest rating.