Internet Appendix to “Bond Illiquidity and Excess Volatility” Jack Bao and Jun Pan∗ May 3, 2013
1
Introduction
This Internet Appendix provides further results and discussions. In Section 2, we consider alternative volatility calculations for both empirical and model volatilities. In Section 3.1, we consider structural models of default other than the Merton (1974) model. In Section 4, we consider other issues including standard errors and additional empirical results.
2
Alternative Volatility Calculations
2.1
Empirical Bond Volatility without Weighting Prices
In constructing monthly bond returns in the paper, we adopt the convention of using transaction size-weighted prices prior to calculating bond returns. As shown in Edwards, Harris, and Piwowar (2007) and Bao, Pan, and Wang (2011), corporate bonds can have very significant effective bid-ask spreads and this is particularly true for smaller volume trades. Large effective bid-ask spreads imply that there will be a large negative autocovariance when transaction prices are used and this will in turn generate large volatilities. By using transaction size-weighted prices in the main analysis, we weight transactions with lower effective bid-ask Bao is at the Fisher College of Business, Ohio State University, bao
[email protected]. Pan is at the MIT Sloan School of Management, CAFR, and NBER,
[email protected]. ∗
1
spreads more and average across a number of trades (some of which are at bid and others at ask), mitigating these effects. Here, we present results that use the last trade of a month rather than transaction size-weighted prices and show that using last prices will lead to a greater excess volatility. Table IA.1: Data Estimated vs. Model Implied Bond Volatility Straight Callable By Year 2003 2004 2005 2006 2007 2008 2009 2010 By Rating Aaa & Aa A Baa Junk By Time to Maturity 0-2 2-4 4-6 6-8 8 - 10
Merton σ ˆD − σD t-stat 25th median 4.11 0.68 2.03 3.38 0.52 2.19
75th 4.82 5.37
σ ˆD mean 8.58 9.92
Merton σD mean 4.66 5.71
#obs 2,883 9,322
mean 3.91 4.21
94 362 525 473 358 279 347 445
3.53 1.73 3.06 3.17 2.05 8.48 8.41 2.69
2.97 10.60 10.50 15.00 8.63 11.46 2.67 7.14
0.37 0.52 0.70 0.99 0.48 3.01 -0.13 0.42
1.65 1.15 1.92 2.16 1.33 6.34 3.94 1.85
5.10 2.37 4.24 4.38 2.97 12.54 11.60 3.73
9.13 5.64 6.26 5.43 5.52 14.69 19.08 7.36
5.60 3.91 3.19 2.26 3.47 6.21 10.67 4.67
723 1,038 621 451
2.03 2.56 4.53 9.01
4.33 4.80 3.83 2.74
0.25 0.55 1.00 2.49
1.31 1.55 2.63 4.82
3.30 3.49 5.68 9.64
6.53 6.34 9.05 16.26
4.51 3.78 4.51 7.25
882 627 311 264 799
1.98 2.89 2.75 3.33 7.49
6.99 3.47 3.18 4.67 4.12
0.54 0.22 0.38 0.80 2.52
1.37 1.27 1.47 1.98 5.13
2.80 3.14 3.60 4.72 9.39
4.06 7.26 7.92 9.09 14.67
2.08 4.37 5.17 5.76 7.18
The difference between empirically estimated bond volatility and model-implied bond volatility. Volatilities are expressed in annualized % and are calculated each year using monthly returns. The returns used to calculate empirical bond volatilities are based on the last transaction of the month for a bond. t-stats are calculated using standard errors clustered by time and by firm, with the exception of the by-year results which use standard errors clustered by firm.
As compared to the average excess volatility of 2.19% using volume-weighted prices, the average excess volatility here is higher at 3.91%.
2.2
Empirical Volatility using Rolling Windows
In the paper, our primary goal is to compare contemporaneous realized empirical bond volatilities with those implied by the Merton model. In calculating Merton model-implied 2
volatilities, one of the primary inputs is the equity volatility of the issuing firm. For the baseline analysis in our paper, we calculate both bond and equity volatilities using monthly returns each year. Volatilities are calculated using 10-12 monthly returns per year and there is no return overlap between different firm-years and bond-years. Though using non-overlapping returns has some conceptual advantages (volatilities are measured for distinct periods that are different from other observations), it raises the concern that volatilities are measured imprecisely. Thus, we also consider measuring volatilities using rolling windows. While volatilities for 2004 were measured using only returns in 2004 in the main analysis of the paper, we now consider measuring this volatility using returns from both 2003 and 2004 and requiring at least 20 monthly returns. For 2005, we use returns from 2004 and 2005 and so forth. Our empirical results using rolling windows indicate that the results in our paper are robust to this alternative measure of volatility. The mean excess volatility using rolling window volatility calculations is 3.69%, larger than the 2.19% in the base case.1 The correlation between the two versions of excess volatility is 0.78. Higher empirical bond volatility in the rolling window case (8.56% vs. 6.86%) is the primary driver of the higher excess bond volatility.
2.3
Model Bond Volatility with Correlations
In the paper, we assume that the firm value and interest rate processes are dVt = (rt − δ) dt + σv dWtQ Vt
(IA.1)
drt = κ (θ − rt ) dt + σr dZtQ
(IA.2)
where dWtQ and dZtQ are uncorrelated Brownian motions. If we instead assume that the Brownian motions have a correlation, ρ, the relations between the asset volatility and bond 1
The sample for the rolling window volatility is somewhat smaller as volatility is not calculated for 2003.
3
and equity volatilities include and additional term to account for the correlation. In particular, the relation between bond and asset volatility becomes
Merton 2 σD
=
∂ ln Bt ∂ ln Vt
2
σv2
+
∂ ln Bt ∂rt
2
σr2 + 2
∂ ln Bt ∂ ln Bt ρσv σr ∂ ln Vt ∂rt
(IA.3)
The pricing of bonds and equities with a non-zero ρ also has some minor modifications. In both the bond and equity pricing equations, Σ in equation (8) of the paper becomes 2 σr2 2ρσv σr 2σr 2ρσr σv σr2 −2κτ 2 −κτ Σ = τ σv + 2 + + + e − 1 − e − 1 κ κ κ3 κ2 2κ3
(IA.4)
In bond pricing equations (13), d3 becomes ln d3 =
V K
− a(T1 ) − b(T1 )r0 − δT1 − 21 Σ + σr2 b(T2κ−T1 ) −b(T1 ) + √ Σ
exp(−2κT1 )−1 2κ
− ρσv σr b(T2 − T1 )b(T1 ) (IA.5)
The quantitative impact of allowing a non-zero ρ depends on the magnitude of the correlation as small value of ρ will lead to the correlation term being dominated by either the variance from the asset value process or the variance from the interest rate process. Following Eom, Helwege, and Huang (2004), we proxy for ρ by calculating the correlation between changes in the short-term interest rate and equity returns over the previous five years. The mean ρ in our sample is 0.07, in contrast to the -0.15 reported by Schaefer and Strebulaev (2008). This difference is attributable to the sample period. As shown by Baele, Bekaert, and Inghelbrecht (2010), the correlation between equity and Treasury bond returns was positive for almost all periods prior to the late 1990s, but has since become negative for most periods. To the extent that Treasury bond returns are negatively correlated with changes in the short rate, this suggests a positive correlation between equity returns and changes in the short rate after the late 1990s.
4
We apply the mean ρ of 0.07 to the firms in our sample to gauge the effect of including a non-zero ρ in the model. Overall, we find that the model-implied volatility decreases slightly on average from 4.66% with ρ = 0 to 4.59% when ρ = 0.07. If we instead apply a firm-byfirm ρ based on the correlation between a firm’s equity returns and the change in the short rate, we find an average model-implied volatility of 4.53%.
2.4
Model Bond Volatility using a Hybrid Approach
The main equation used for estimating the firm-level asset volatility is:
σE2
=
∂ ln Et ∂ ln Vt
2
σv2
+
∂ ln Et ∂rt
2
σr2
(IA.6)
The most straightforward calculation of σv is to use the estimated values of all of the other parameters of the Merton model and to calculate the value of σv such that equation (IA.6) holds. The realized volatility, σE , is a particularly important input into this calculation. Importantly, σv appears both directly in the equation and also through the equity-asset hedge ratio,
∂ ln Et . ∂ ln Vt
The implicit assumption of backing out a single σv is that the unconditional
asset volatility, which is relevant for the equity-asset hedge ratio,
∂ ln Et , ∂ ln Vt
is the same as the
realized asset volatility (which appears directly in equation (IA.6)), for the period. To ensure that our results are not solely driven by our estimation procedure, we consider a “hybrid” method which we use to de-link the unconditional and conditional asset volatilities. First, we obtain unconditional estimates of equity and Treasury bond volatilities using monthly equity and Treasury bond returns going as far back in history as possible.2 We can then use these unconditional equity volatilities in (IA.6) along with other firm-level parameters to obtain unconditional asset volatilities for each firm. The mean and median estimates of the unconditional volatility in our sample are 20.14% and 19.35%, respectively. This is 2
As robustness, we also consider estimating equity volatility using a GARCH(1,1) model and find similar results.
5
higher than the estimates in our base estimation, which are 16.68% and 13.50%, respectively. Note that a higher asset volatility does not necessarily imply a higher model bond volatility as this simultaneously decreases the sensitivity to interest rates while increasing the sensitivity to asset value. With the unconditional asset volatility, we can then use equation (IA.6) to estimate conditional asset volatilites, plugging in the unconditional asset volatility into the sensitivities (∂ ln E/∂ ln V and ∂ ln E/∂r) and the other appropriate firm-level parameters into equation (IA.6). Specifically, we use the equation 2 σE,t
=
∂ ln E ∂ ln V
2 σv,t
+
∂ ln E ∂r
2 σr,t
where we use t subscripts to emphasize that we are relating the conditional equity volatility to conditional asset and interest rate volatilities. The sensitivities, however, are function of the unconditional asset and interest rate volatilities. Similarly, model-implied bond volatilities can be calculated by applying the unconditional asset volatility to the sensitivities (∂ ln B/∂ ln V and ∂ ln B/∂r), but the conditional volatilities otherwise in equation (3) from the paper. For this estimation, methodology, we find results similar to our previous results. The mean difference between empirical and model bond volatilities is 2.36 percentage points with a t-stat of 2.96.
6
3
Alternative Models
3.1
Stochastic Volatility
To examine the impact of time-varying volatility, in this section, we consider the use of a stochastic volatility model.3 We model the firm value and asset variance processes as: p p dVt (1)Q (2)Q = (r − δ)dt + Ht ρdZt + 1 − ρ2 dZt Vt p (1)Q dHt = κH (θH − Ht )dt + σH Ht dZt
(IA.7) (IA.8)
where Ht is the instantaneous asset variance. Thus, σE2
=
∂ ln E ∂ ln V
2
Ht +
∂ ln E ∂H
2
2 σH Ht
+2
∂ ln E ∂ ln V
∂ ln E ∂H
ρσH Ht
(IA.9)
The value of equity and its partials with respect to V and H can be calculated by noting that equity is a call option on firm value and using the methodology in Duffie, Pan, and Singleton (2000). To infer the set of parameters [κH , θH , σH , ρ], we start with an empirical time-series of equity variances, estimated each month using daily returns. We then follow the following procedure: 1. Use the time-series of equity variance to estimate initial values for [κH , θH , σH ]. As an initial starting point for ρ, use the correlation between the change in equity variance and log equity returns. 2. Calculate a time-series of Ht using the [κH , θH , σH , ρ] and the time-series of equity variances using equation (IA.9). 3. Using the time-series of Ht from (2), solve for [κH , θH , σH ] 3
See Zhang, Zhou, and Zhu (2009) for another application of a structural model of default with stochastic asset volatility.
7
4. Approximate ρ using Ht , Vt , the parameters [κH , θH , σH ], and discretized versions of the firm value and variance processes in (IA.8). If [κH , θH , σH , ρ] are different from the previous iteration, return to step (2). After calibrating [κH , θH , σH , ρ] and a time-series of Ht above, we can then calculate model bond volatilities using: 2 σD
=
∂ ln D ∂ ln V
2
Ht +
∂ ln D ∂H
2
2 σH Ht
+2
∂ ln D ∂ ln V
∂ ln D ∂H
ρσH Ht
(IA.10)
The model value of the CDS part of the synthetic floating rate bond is:
CDS =
20 X t t t−1 t −r 4t r r t−1 − (1 − R) e − e 4 G0,−1 G0,−1 e 4 G0,−1 4 4 4 4 t=1
20 X s t=1
(IA.11) where
G0,−1
ψ(0, H0 , t) 1 − = 2 π
Z
0
∞
Im ψ(−iu, H0 , t)eiu(ln k) du u
(IA.12)
as in Duffie, Pan, and Singleton (2000). In Table IA.2, Panel B, we report the results of the stochastic volatility model.4 Across the full sample, the mean difference between empirically estimated CDS volatility and stochastic volatility model-implied CDS volatility is 1.28%, with a t-stat of 5.13. Thus, for the overall sample, there remains evidence of excess volatility. However, the result is insignificant for both 2009 and 2010. The results for the stochastic volatility model are extremely similar to the results in the paper that use de-linked realized and long-run volatilities. The reason is that θH , the long run mean variance in the stochastic volatility model, plays a similar role to σv2 , the long-run variance in the model in Appendix D of the paper. The two are very 2 highly correlated. The current asset variance, Ht plays a similar role to σv,t , the realized 4
In Panel A, we report a similar subsample, but with stochastic volatility turned off.
8
asset variance in Appendix D and the two are very highly correlated. Thus, the two models generate very similar model-implied CDS volatilities and excess volatility estimates. Table IA.2: Data Estimated vs. Model Implied Volatility for CDS, Stochastic Volatility Model 2004 2005 2006 2007 2008 2009 2010 Full
#obs 2,399 3,572 3,614 3,476 3,268 3,060 2,267 21,656
2004 2005 2006 2007 2008 2009 2010 Full
#obs 2,485 3,638 3,637 3,496 3,317 3,176 2,331 22,080
Panel A: Base Case mean t-stat 25th median 75th 1.79 5.86 0.52 0.85 1.48 2.15 7.56 0.66 1.17 2.33 1.49 9.32 0.52 0.89 1.74 1.83 7.85 0.47 0.97 2.10 2.37 2.64 -2.34 -0.24 2.02 1.77 3.28 -2.09 0.41 1.95 1.40 4.96 0.53 1.27 2.19 1.85 7.86 -0.28 0.74 1.96 Panel B: Stochastic Volatility Case mean t-stat 25th median 75th 2.05 2.43 -0.16 0.45 0.93 2.08 3.87 0.35 0.87 1.72 1.00 6.61 0.30 0.62 1.18 1.48 5.81 0.43 0.85 1.71 1.53 2.33 -2.01 0.97 2.60 0.06 0.15 -2.59 0.03 1.56 0.65 1.58 -0.47 0.78 1.52 1.28 5.13 -0.60 0.65 1.61
σ ˆD 2.49 3.01 2.29 3.02 8.98 8.31 3.86 4.57
Merton σD 0.70 0.85 0.80 1.19 6.61 6.54 2.46 2.72
σ ˆD 3.90 3.78 2.37 3.08 9.29 8.60 4.08 5.00
Merton σD 1.85 1.69 1.38 1.60 7.76 8.54 3.43 3.72
The difference between empirically estimated and model-implied CDS volatility. Volatilities are calculated using daily returns each month, annualized and in %. Panel A reports the difference between empirically estimated and model implied volatilities for the base case. Panel B uses a stochastic volatility model to account for time-varying volatility. t-stats are calculated using standard errors clustered by time and by firm.
3.2
Mean-Reverting Leverage
In this subsection, we consider the sensitivity of our volatility estimates to using a meanreverting leverage model similar to Collin-Dufresne and Goldstein (2001). Define the log asset value and log leverage processes as: 1 2 Kt ¯ λ vt − k − kt + σv dZtQ dvt = r − δ − σv + 2 Vt dkt = λ vt − k¯ − kt dt 9
(IA.13) (IA.14)
Equations (IA.13) and (IA.14) correspond to equations (11) and (12) in Collin-Dufresne and Goldstein (2001) with one main difference. Collin-Dufresne and Goldstein (2001) assume that the proceeds of new debt issuance are used to repurchase equity, leaving firm value unchanged. Instead, we assume that proceeds from new debt issuance increase firm value by exactly the value of the proceeds and do not change the amount of equity outstanding. This seems more in line with what is typically seen. The important parameters to note above are k¯ which is the long-run target value of ln(V /K) and λ, which governs the speed of mean reversion in leverage. Defining xt ≡ vt − kt , we have: 1 2 ¯ dxt = r − δ − σv + λ (exp(−xt ) − 1) (xt − k) dt + σv dZt 2 Pricing a zero-coupon bond in this set-up amounts to solving: f ≡ e−r(T −t) EtQ [1xT >0 ] This can be solved by numerically solving a PDE through finite difference methods. We then calculate the value of a coupon bond by using a term structure of f. On the equity side, we calculate the sensitivity of equity to asset value, V, by simulating dZtQ . To check the sensitivity of our calibrations to a mean-reverting leverage ratio model, we first assume that a mean-reverting leverage ratio model is the true data generating process. With a set of assumed parameters in the model, we can then calculate the true bond and equity volatility under the mean-reverting leverage ratio model. Equipped with the true equity volatility, we can then calculate Merton-model implied asset and bond volatilities if we use the observed equity volatility from the mean-reverting leverage ratio model. Effectively, what we examine is a case where the true data generating process is the mean-reverting leverage ratio model, but an Econometrician estimates a Merton model on this data instead. In our analysis, we examine a 5-year bond with a coupon rate of 5%, and a recovery rate
10
of 50%. We also assume that r = 5% and δ = 3%. The long-run target leverage, K/V is assumed to be 0.4, which corresponds to k¯ = 0.9163. In Table IA.3, we display the results of analysis for different values of
K , V
σv , and λ.
Generally, estimating a Merton model when the true data generating process is a meanreverting leverage ratio model produces a higher model bond volatility than the true bond volatility. Table IA.3: Comparison of Mean-Reverting Leverage Ratio and Merton Model Parameters
K V
0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6
σv 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1
λ 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000
MeanReverting Leverage σD σE 0.1251 0.6112 0.1418 0.7132 0.1562 0.7965 0.1777 0.9249 0.0985 0.5054 0.1072 0.5773 0.1143 0.6346 0.1224 0.7217 0.0372 0.3329 0.0333 0.3627 0.0288 0.3863 0.0195 0.4184 0.0872 0.5313 0.0888 0.5743 0.0889 0.6057 0.0854 0.6487 0.0465 0.4016 0.0423 0.4230 0.0378 0.4379 0.0285 0.4583 0.0025 0.2183 0.0013 0.2236 0.0006 0.2274 0.0001 0.2331
Merton
Parameters
K V
σD 0.1249 0.1387 0.1430 0.1411 0.0982 0.1177 0.1290 0.1394 0.0368 0.0476 0.0563 0.0683 0.0870 0.0987 0.1064 0.1155 0.0463 0.0533 0.0581 0.0647 0.0025 0.0030 0.0034 0.0040
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
σv 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1
λ 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000 0.0000 0.1000 0.2000 0.4000
MeanReverting Leverage σD σE 0.0432 0.4473 0.0393 0.4579 0.0355 0.4657 0.0283 0.4763 0.0097 0.3109 0.0075 0.3146 0.0057 0.3173 0.0031 0.3213 0.0000 0.1567 0.0000 0.1580 0.0000 0.1591 0.0000 0.1607 0.0067 0.3652 0.0066 0.3660 0.0063 0.3667 0.0053 0.3679 0.0001 0.2442 0.0001 0.2444 0.0001 0.2448 0.0001 0.2456 0.0000 0.1221 0.0000 0.1222 0.0000 0.1224 0.0000 0.1227
Merton
σD 0.0431 0.0462 0.0484 0.0515 0.0096 0.0103 0.0108 0.0115 0.0000 0.0000 0.0000 0.0000 0.0067 0.0067 0.0068 0.0070 0.0001 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000
The columns under “Mean-Reverting Leverage” represent the volatilities under the true data generating process, the Mean-Reverting Leverage Ratio Model. Assumed parameters are T = 5, coupon rate = 5%, recovery rate = 50%, δ = 3%, and k¯ = 0.9163. Additional assumed parameters for each row are in the columns labeled “Parameters”. The columns under “Merton” represent the Merton model implied bond volatilities when the equity volatility from the true data generating process is used.
11
4
Additional Issues & Results
4.1
Standard Errors
In determining the statistical significance of excess volatility, our moment condition for M erton estimating excess volatility is µ ˆ − E(ˆ σD − σD ) = 0 where bond and time subscripts are M erton suppressed. Estimates of σD that are noisy, but have errors independent of the errors M erton of the other σD estimates will increase the standard error of µ ˆ. However, given the M erton dependence of σD on firm-level parameters, there is likely correlation within firm. To
the extent that equity volatility is correlated across firms within a period time, there may also be within period error correlation. Thus, we allow for two-way clustering as in Petersen (2008) and Cameron, Gelbach, and Miller (2011). In the notation of Cameron, Gelbach, and Miller (2011), our variance matrix is: ˆ = Vˆ G [β] ˆ + Vˆ H [β] ˆ − Vˆ G∩H [β] ˆ Vˆ [β]
(IA.15)
ˆ is the variance matrix clustered by firm, Vˆ H [β] ˆ is the variance matrix clustered where Vˆ G [β] ˆ is the variance matrix clustered by firm-time. Cameron, Gelbach, and by time, and Vˆ G∩H [β] Miller (2011) show that this variance estimate is consistent under the very mild assumption that errors that do not share a group are uncorrelated. Since cluster-robust standard errors do not require a function form for the cluster error variance matrices, they allow for quite general error correlation, including autocorrelated errors. The main concern when using two-way clustering to account for cross-sectional and timeseries error correlation (other than specifying the correct clusters) is whether there are a sufficient number of clusters. The theory is asymptotic and requires min(G, H) → ∞. This is particularly a concern when we use monthly returns to calculate volatilities each year as there are a small number of cross-sections. To address this, we adopt a wild cluster bootstrap-t procedure, following Cameron, Gelbach, and Miller (2008) that allows for oneway clustering. Empirically, it is the clustering by time that has the greatest effect on the 12
standard errors when monthly returns are used. For CDS volatility, clustering only by time produces a 3.87 t-statistic as compared to a 3.77 t-statistic when clustering by both time and firm. As Cameron, Gelbach, and Miller (2008) show, the wild clustering procedure works well when there are as few as five clusters. We cluster on time and find that our results hold when critical values are calculated using the wild clustering procedure. Table IA.4: Bootstrapped t Statistics Sample Straight bonds (month-end prices) Straight bonds (weighted prices) CDS (monthly returns) CDS (daily returns)
obs 2,883 2,883 1,819 24,900
Merton σ ˆD − σD , s.e. cluster by year mean t-stat 2.5% Boot t 97.5% Boot t 3.91 4.40 -2.34 2.34 2.19 2.92 -2.27 2.27 2.84 3.87 -2.22 2.22 1.92 11.75 -1.98 2.02
Mean excess volatility reported in %. The samples correspond to Tables 5 and 6 in the paper. t-stats clustered by time are reported Bootstrapped t-stat critical values are calculated following the wild cluster bootstrap t-procedure in Cameron, Gelbach, and Miller (2008).
When daily CDS returns are used to calculate volatilities each month, serial correlation is much more of a concern. However, in this data, we are less concerned about the number of clusters as we have 81 months of data and over 400 firms. Though two-way clustering allows for consistent standard errors in very general settings including autocorrelation5 , we also M erton consider taking the average of σ ˆD − σD period-by-period. We then calculate the time-
series average of the 81 cross-sectional averages and use Newey-West t-statistics to account for autocorrelation. We find an estimate of 1.89% with a t-statistic of 8.22. One further concern is that the finite sample properties of cluster-robust standard errors in the presence of skewed data is unclear. While Petersen (2008) and Cameron, Gelbach, and Miller (2011) both run a series of simulations to examine the extent to which different covariance matrix estimates over-reject the null, both papers use fixed effects and errors that are normally distributed. To get a better sense of the effect of skewed data, we run 5 Cameron, Gelbach, and Miller (2011) run Monte Carlo simulations on a case with cross-sectional correlation and also autocorrelation. The sample in their simulation has 21 time periods and 50 observations per cross-section. They find the performance of two-way clustered standard errors to be reasonable as rejection at the 5% level occurs 6.9% of the time. Our sample when daily CDS returns are used is larger in both the time-series and cross-section.
13
simulations similar to those in Petersen (2008) and Cameron, Gelbach, and Miller (2011), but use our data to impose skewed errors. In particular, we define yjt as the de-meaned excess volatility of bond j in year t and specify
yjt = γi + αt + ejt
where γi is a firm fixed-effect, αt is a time fixed-effect, and ejt is an error term. We specify the distribution of values of γi , αt , and ejt by regressing yjt on firm and time dummies and obtaining empirical distributions. In our simulations, we draw (with replacement) γi for each firm, αt for each time, and ejt for each observation. For each simulation, we can then calculate a t-stat using two-way clustering. This allows us to generate a distribution of t-statistics for the null of yjt = 0 (which is true under our data generating process). In 5,000 simulations using our corporate bond sample, we find that using conventional 5% t-statistics of -1.96 and 1.96, the null hypothesis is rejected 15.64% of the time. However, this over-rejection is asymmetric as the t-stat is less than -1.96 in 14.58% of the simulations and greater than 1.96 in only 1.06% of simulations. That is, the likelihood of making a Type I error with positive excess volatility is low. The asymmetry in our simulations is largely due to the fact that the time fixed-effects, αt are also positively skewed. Thus, we also consider simulations where we assume that αt is normally distributed, but continue to sample γi and ejt from the empirical distribution. In 5,000 simulations, we find that conventional 5% t-statistics reject the null hypothesis 10.24% of the time. The t-stat is less than -1.96 in 6.46% of cases and greater than 1.96 in 3.78% of cases. Thus, there is some over-rejection, though it is not as severe for positive excess volatility.
4.2
Sorting
In the paper, we provide regression-based evidence that excess volatility is related to credit ratings, time-to-maturity, and measures of illiquidity. Specifically, we use panel regressions with time fixed-effects, which take out the mean excess volatility and the mean of each 14
regressor, period-by-period, allowing for the variation to be relative to period means. Here, we provide similar evidence that excess volatility is larger for poorer credit ratings, longer maturity bonds, and less liquid bonds using a sequential sorting methodology. Generally the results are consistent with the regressions in that the three variables do not drive each other out. In Tables IA.5 and IA.6, bond are first sorted by rating or time-to-maturity and then liquidity, period-by-period. In Tables IA.7 and IA.8, bond are sorted first by liquidity and then by ratings or time-to-maturity.
4.3
CDS Data
The CDS data used in the paper is from Datastream, which in turn uses CMA Datavision as its datasource through September 2010. In the analysis in the paper, we use the data through the end of September 2010 when calculating CDS volatility each month using daily data. However, we do not use 2010 data when calculating CDS volatility each year using monthly data as we do not have a full year of data. As 2010 represents a period of recovery from the Financial Crisis, it may be interesting to see if volatility levels return to pre-crisis levels when using monthly returns to calculate volatility. In order to examine monthly CDS data in 2010, we supplement the data from Datastream by collecting CDS data from Bloomberg for the period from September 2010 to December 2010. In Bloomberg, we use the CMA New York datasource for the period from September 2010 to December 2010. Pooling the CMA Datavision and CMA New York datasets for 2010 together, we calculate CDS volatility using monthly data.6 For 2010, the mean empirical CDS volatility is 4.50%, much more in-line with the pre-crisis period than the 2008-2009 crisis period. The mean excess volatility for CDS in 2010 is 2.03%, also more in line with the pre-crisis period. 6
We have September 2010 data for both sources. For the change in CDS spreads from August 2010 to September 2010, we use CMA Datavision. For the change in CDS spreads from September 2010 to October 2010, we use CMA New York. That is, the change is never calculated as the difference between CDS spreads in CMA Datavision and CDS spreads in CMA New York. We are are careful to do this as the two sources are not exactly the same in all cases, even though they are very highly correlated and have very similar magnitudes.
15
Table IA.5: Double Sorts on Rating, then Liquidity Rating Aaa/Aa A Baa Junk
Most Liquid 0.20 0.91 1.90 3.90
Rating Aaa/Aa A Baa Junk
Most Liquid -0.14 0.76 1.45 2.69
Rating Aaa/Aa A Baa Junk
Most Liquid -0.21 0.98 1.74 4.48
Rating Aaa/Aa A Baa Junk
Most Liquid -0.13 0.64 1.53 3.11
Rating Aaa/Aa A Baa Junk
Most Liquid -0.43 0.77 1.48 3.54
Rating Aaa/Aa A Baa Junk
Most Liquid -0.02 0.97 1.82 4.22
Panel A: Amihud Measure 3 Least Liquid 0.39 1.40 0.84 1.85 2.61 4.24 4.62 7.27 Panel B: IRC Measure 2 3 Least Liquid -0.51 0.27 2.09 0.45 1.24 2.23 1.25 2.82 5.27 4.35 6.26 7.68 Panel C: SD(Amihud Measure) 2 3 Least Liquid 0.02 -0.12 1.82 0.43 1.11 1.89 2.08 2.31 4.19 3.82 5.15 7.13 Panel D: SD(IRC Measure) 2 3 Least Liquid -0.45 0.49 1.73 0.52 1.17 2.29 1.78 2.69 4.72 3.39 6.37 7.88 Panel E: Quoted Bid-Ask Spread 2 3 Least Liquid 0.35 0.90 1.08 0.94 1.07 2.02 1.68 2.83 4.88 4.08 5.77 7.84 Panel F: SD(Quoted Bid-Ask Spread) 2 3 Least Liquid -0.00 0.65 1.83 1.03 0.86 1.46 1.65 2.29 3.46 3.58 4.61 8.22 2 -0.45 1.02 1.84 5.23
Least - Most 1.20 0.94 2.34 3.37
t-stat 2.00 2.13 3.14 2.12
Least - Most 2.24 1.47 3.82 4.99
t-stat 3.89 2.80 3.22 2.11
Least - Most 2.03 0.91 2.44 2.65
t-stat 2.37 1.34 3.81 1.25
Least - Most 1.86 1.66 3.18 4.77
t-stat 2.66 2.46 2.67 2.39
Least - Most 1.51 1.25 3.41 4.30
t-stat 3.38 2.45 2.85 1.84
Least - Most 1.85 0.50 1.64 4.00
t-stat 3.34 1.54 1.67 1.50
Bonds are first sorted into one of four ratings groups before being sorted by a liquidity measure (labeled at the top of each panel), period-by-period. The average contemporaneous excess volatility within each group is reported as is the difference in average excess volatility between the least liquid and most liquid groups. The t-stat for this difference is also reported, using standard errors clustered by both firm and time.
16
Table IA.6: Double Sorts on Maturity, then Liquidity Maturity 0-2 2-4 4-6 6-8 8+
Most Liquid 0.40 1.21 0.75 1.67 4.06
Maturity 0-2 2-4 4-6 6-8 8+
Most Liquid 0.50 0.34 0.39 1.37 3.06
Maturity 0-2 2-4 4-6 6-8 8+
Most Liquid 0.71 0.75 0.46 1.63 4.13
Maturity 0-2 2-4 4-6 6-8 8+
Most Liquid 0.59 0.25 0.61 1.40 3.53
Maturity 0-2 2-4 4-6 6-8 8+
Most Liquid 0.98 0.31 0.60 1.03 3.27
Maturity 0-2 2-4 4-6 6-8 8+
Most Liquid 0.68 1.51 1.08 2.05 4.19
Panel A: Amihud Measure 3 Least Liquid 0.33 1.51 0.39 1.76 0.68 1.08 1.21 2.19 4.49 6.11 Panel B: IRC Measure 2 3 Least Liquid 0.37 0.27 2.03 0.10 0.55 2.98 0.76 0.71 2.05 0.70 1.27 3.01 4.21 4.66 6.71 Panel C: SD(Amihud measure) 2 3 Least Liquid 0.47 0.32 1.50 0.37 0.34 1.81 0.58 1.57 1.06 0.87 0.86 2.72 3.66 4.03 6.76 Panel D: SD(IRC Measure) 2 3 Least Liquid 0.22 0.34 1.94 0.13 0.63 2.67 0.51 0.94 1.85 0.80 0.72 3.10 3.81 4.35 6.98 Panel E: Quoted Bid-Ask Spread 2 3 Least Liquid 0.21 0.49 1.54 0.12 0.35 3.25 0.64 0.78 2.45 1.02 1.14 3.30 3.16 4.76 7.57 Panel F: SD(Quoted Bid-Ask Spread) 2 3 Least Liquid 0.67 0.50 0.90 0.29 0.48 1.94 0.43 1.72 0.97 0.56 1.12 2.56 3.13 3.87 7.11 2 0.92 0.44 1.28 1.06 4.01
Least - Most 1.12 0.56 0.33 0.52 2.05
t-stat 2.76 2.04 0.63 0.52 1.64
Least - Most 1.53 2.64 1.66 1.64 3.65
t-stat 1.92 2.96 2.11 2.30 2.09
Least - Most 0.79 1.07 0.59 1.09 2.63
t-stat 1.34 1.09 1.57 2.33 1.77
Least - Most 1.36 2.41 1.24 1.69 3.45
t-stat 1.75 1.96 1.22 2.39 1.81
Least - Most 0.56 2.93 1.86 2.27 4.30
t-stat 2.12 2.25 2.99 1.97 2.40
Least - Most 0.21 0.42 -0.11 0.51 2.92
t-stat 1.30 0.92 -0.23 0.42 1.66
Bonds are first sorted into one of five maturity groups before being sorted by a liquidity measure (labeled at the top of each panel), period-by-period. The average contemporaenous excess volatility within each group is reported as is the difference in average excess volatility between the least liquid and most liquid groups. The t-stat for this difference is also reported, using standard errors clustered by both firm and time.
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Table IA.7: Double Sorts on Liquidity, then Rating Most liquid 2 3 Least liquid
Aaa/Aa 0.33 -0.36 0.04 1.48
Most liquid 2 3 Least liquid
Aaa/Aa -0.20 -0.92 0.78 2.11
Most liquid 2 3 Least liquid
Aaa/Aa -0.36 0.08 0.40 1.45
Most liquid 2 3 Least liquid
Aaa/Aa -0.54 -0.29 0.72 1.88
Most liquid 2 3 Least liquid
Aaa/Aa -0.27 0.27 0.79 1.89
Most liquid 2 3 Least liquid
Aaa/Aa 0.06 0.35 0.54 1.94
Panel A: Amihud Measure Baa Junk 1.85 4.98 1.88 3.35 2.68 4.84 4.76 6.82 Panel B: IRC Measure A Baa Junk 0.72 1.09 3.79 0.67 1.83 2.30 1.14 3.03 4.88 2.53 5.62 7.01 Panel C: SD(Amihud Measure) A Baa Junk 0.81 1.49 3.93 0.61 1.82 4.10 1.27 2.22 4.41 1.95 4.83 6.34 Panel D: SD(IRC Measure) A Baa Junk 0.74 1.23 2.66 0.71 1.56 3.00 1.12 3.14 4.13 2.39 5.14 7.35 Panel E: Quoted Bid-Ask Spread A Baa Junk 0.89 1.38 4.34 0.74 1.84 2.54 1.30 2.77 3.68 2.36 6.15 5.88 Panel F: SD(Quoted Bid-Ask Spread) A Baa Junk 1.22 1.90 4.19 0.86 1.36 3.54 0.93 2.16 3.54 1.42 3.53 7.43 A 0.98 1.11 0.55 2.05
Junk-Aaa/Aa 4.65 3.72 4.81 5.34
t-stat 2.14 2.68 2.43 1.63
Junk-Aaa/Aa 3.99 3.21 4.10 4.91
t-stat 2.67 2.70 2.14 1.72
Junk-Aaa/Aa 4.29 4.02 4.00 4.89
t-stat 3.22 2.52 2.51 1.55
Junk-Aaa/Aa 3.20 3.30 3.42 5.47
t-stat 2.06 2.04 3.45 1.82
Junk-Aaa/Aa 4.61 2.27 2.88 3.99
t-stat 2.50 1.70 1.41 1.66
Junk-Aaa/Aa 4.12 3.19 3.01 5.50
t-stat 2.75 1.24 2.41 1.95
Bonds are first sorted into a liquidity measure (labeled at the top of each panel) before being classified by rating, period-by-period. The average contemporaenous excess volatility within each group is reported as is the difference in average excess volatility between the junk and Aaa/Aa groups within each liquidity quartile. The t-stat for this difference is also reported, using standard errors clustered by both firm and time.
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Table IA.8: Double Sorts on Liquidity, then Maturity Most liquid 2 3 Least liquid
0-2 0.72 0.39 0.64 2.09
Most liquid 2 3 Least liquid
0-2 0.43 0.42 1.58 3.08
Most liquid 2 3 Least liquid
0-2 0.65 0.34 0.82 3.46
Most liquid 2 3 Least liquid
0-2 0.48 0.34 1.63 3.29
Most liquid 2 3 Least liquid
0-2 0.59 0.65 1.66 2.56
Most liquid 2 3 Least liquid
0-2 0.80 0.49 0.59 1.38
Panel A: Amihud Measure 2-4 4-6 6-8 8+ 1.38 0.81 2.04 4.10 0.39 0.76 0.78 4.02 0.49 1.09 1.04 3.58 1.97 0.99 2.13 6.06 Panel B: IRC Measure 2-4 4-6 6-8 8+ 0.32 0.18 1.28 3.56 -0.04 0.92 0.87 2.55 1.71 0.74 1.03 3.96 2.80 1.74 2.77 5.79 Panel C: SD(Amihud Measure) 2-4 4-6 6-8 8+ 0.52 0.73 1.13 3.90 0.53 0.66 1.57 4.03 0.70 0.63 0.93 4.28 1.82 1.64 2.10 5.22 Panel D: SD(IRC Measure) 2-4 4-6 6-8 8+ -0.06 0.69 1.38 3.13 0.31 0.67 0.95 4.01 0.90 0.90 0.62 4.13 3.89 1.62 2.59 5.35 Panel E: Quoted Bid-Ask Spread 2-4 4-6 6-8 8+ 0.36 0.53 1.85 3.45 0.06 0.46 1.24 3.13 0.37 0.77 0.77 3.34 3.63 2.19 2.37 6.63 Panel F: SD(Quoted Bid-Ask Spread) 2-4 4-6 6-8 8+ 1.05 1.11 1.74 4.12 0.76 0.98 1.39 3.88 0.55 1.11 0.77 3.34 1.83 1.03 2.20 5.79
Long-Short 3.38 3.62 2.93 3.98
t-stat 3.04 3.38 3.67 2.67
Long-Short 3.14 2.13 2.37 2.71
t-stat 3.20 2.26 3.15 2.58
Long-Short 3.25 3.69 3.46 1.77
t-stat 3.03 3.79 2.66 1.63
Long-Short 2.66 3.67 2.50 2.06
t-stat 4.27 2.65 3.19 1.64
Long-Short 2.86 2.48 1.68 4.06
t-stat 2.42 2.82 2.05 2.70
Long-Short 3.32 3.39 2.75 4.40
t-stat 4.26 2.85 2.91 3.10
Bonds are first sorted into a liquidity measure (labeled at the top of each panel) before being classified by time-to-maturity, period-by-period. The average contemporaenous excess volatility within each group is reported as is the difference in average excess volatility between the longest and shortest maturity groups within each liquidity quartile. The t-stat for this difference is also reported, using standard errors clustered by both firm and time.
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4.4
Investment Grade vs. Speculative Grade
In the paper, we consider the relation between excess bond volatility and firm- and bond-level characteristics for our full sample of bonds. Due to the fact that investment grade and junk bonds may be very different, here we consider the possibility that the relation between excess volatility and a number of regressors may be different for investment grade and speculative grade bonds. Thus, we include interaction terms between most of our regressors and a dummy for ratings of Ba1 and below. The results are presented in Table IA.9. For the vast majority of the regressors, there is no statistically significant difference between the coefficients for investment and speculative grade bonds. There is some evidence that the relation between excess volatility and illiquidity for speculative grade bonds is stronger as both the standard deviation of round-trip costs times the junk dummy and the standard deviation of bid-ask spreads times the junk dummy are statistically significant. Table IA.9: Excess Bond Volatility and Firm- and Bond-Level Characteristics
Rating Maturity Maturity × Junk Age Age × Junk log(Amt) log(Amt) × Junk B/A Spd B/A Spd × Junk SD(B/A Spd) SD(B/A Spd) × Junk
(1) 0.147 (2.45) 0.159 (5.38) -0.0751 (-1.42) 0.00573 (0.20) -0.0323 (-0.43) -0.0414 (-0.51) -0.229 (-1.09) 1.331 (1.62) 4.872 (2.13) -1.956 (-1.08) 20.00 (4.07)
(2) 0.242 (5.79) 0.146 (5.10) -0.0509 (-1.03)
(3) 0.171 (3.95) 0.130 (4.36) -0.0531 (-0.93) 0.00775 (0.28) 0.00469 (0.063) 0.239 (1.40) -0.0544 (-0.27) 0.0429 (0.05) 3.187 (1.25) -1.908 (-1.24) 15.14 (3.77)
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(4) 0.336 (3.07) 0.178 (6.63) -0.0190 (-0.53)
(5) 0.217 (2.86) 0.155 (5.00) -0.0777 (-1.35) 0.00288 (0.12) -0.0852 (-1.04) -0.0115 (-0.16) -0.362 (-0.92) 1.206 (1.32) 4.317 (2.00) -1.576 (-0.83) 18.27 (4.21)
(6) 0.289 (3.29) 0.144 (4.83) -0.0443 (-0.87)
(7) 0.204 (2.79) 0.129 (4.20) -0.0534 (-0.87) 0.00517 (0.21) -0.0280 (-0.35) 0.223 (1.32) -0.00343 (-0.01) 0.0192 (0.02) 3.220 (1.27) -1.722 (-1.07) 14.91 (3.68)
Bond Zero Bond Zero × Junk
0.0133 (3.13) -0.0330 (-1.82)
Amihud
-0.119 (-1.19) -0.0587 (-0.26) 3.727 (3.78) -3.978 (-1.54) 0.0912 (1.09) 0.353 (1.78) -0.874 (-1.42) 4.650 (2.56)
Amihud × Junk IRC IRC × Junk SD(Amihud) SD(Amihud) × Junk SD(IRC) SD(IRC) × Junk
0.0145 (2.66) -0.0388 (-2.59) 0.0425 (0.58) -0.145 (-0.82) 2.695 (2.41) -3.754 (-1.53) 0.115 (1.38) 0.191 (1.24) -0.459 (-0.77) 3.600 (2.60)
EBIT/Assets EBIT/Assets × Junk Cov Ratio Cov Ratio × Junk Sales/Assets Sales/Assets × Junk RE/Assets RE/Assets × Junk NI/Assets NI/Assets × Junk log(Assets) log(Assets) × Junk Observations R-squared Within-group R2
2,552 0.447 0.348
2,609 0.419 0.330
2,432 0.473 0.388
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0.0192 (5.28) -0.0242 (-1.39)
1.052 (0.17) -30.76 (-1.53) 0.00296 (0.20) 0.0582 (0.21) 0.01 (0.04) 0.653 (0.91) 0.0400 (0.10) -1.856 (-1.77) 5.510 (0.64) -5.067 (-0.37) 0.339 (1.48) 0.329 (1.79) 2,818 0.379 0.275
-4.218 (-0.70) -16.49 (-1.00) 0.00956 (0.72) -0.0366 (-0.12) 0.05 (0.32) -0.178 (-0.25) 0.0734 (0.18) -1.497 (-1.38) 5.002 (0.54) 3.824 (0.32) 0.337 (2.07) 0.272 (1.07) 2,537 0.461 0.362
-0.127 (-1.26) -0.0827 (-0.45) 3.874 (4.05) -3.847 (-1.65) 0.0964 (1.15) 0.351 (1.79) -1.048 (-1.87) 4.399 (3.22) -2.627 (-0.42) -7.096 (-0.45) 0.00970 (0.71) -0.0256 (-0.09) 0.10 (0.49) -0.179 (-0.30) -0.133 (-0.34) -1.863 (-1.85) 6.235 (0.69) -6.054 (-0.46) 0.172 (0.92) 0.0888 (0.64) 2,594 0.426 0.339
0.0174 (3.31) -0.0292 (-1.75) 0.0180 (0.25) -0.153 (-0.89) 2.651 (2.38) -3.436 (-1.28) 0.117 (1.40) 0.188 (1.19) -0.506 (-0.87) 3.156 (2.12) -5.400 (-0.91) -10.54 (-0.65) 0.00968 (0.75) 0.0176 (0.06) 0.04 (0.27) -0.326 (-0.49) -0.0505 (-0.12) -1.554 (-1.39) 6.037 (0.63) 2.535 (0.21) 0.165 (0.99) 0.0481 (0.19) 2,417 0.479 0.396
Merton All regressions include time fixed-effects. The dependent variable is σ ˆD − σD , where σ ˆD is the realized Merton volatility of a corporate bond using monthly returns in a calendar year and σD is the volatility implied by the Merton model and realized equity volatility. Both are expressed in annualized % and the difference is winsorized at the 1st and 99th percentiles. EBIT/Assets is defined using Compustat data as OIADP/AT. Coverage Ratio is defined as (OIADP + XINT)/XINT. Sales/Assets is defined as SALE/AT. RE/Assets is retained earnings divided by assets and is defined as RE/AT. NI/Assets is net income divided by assets and is defined as NI/AT. Assets is total book assets in $mm. Rating is a bond’s Moody rating where Aaa = 1 and C = 21. Maturity is a bond’s time to maturity in years. Age is a bond’s time since issuance in years. Amt is a bond’s amount outstanding in $mm face value. B/A Spd is a bond’s bid-ask spread divided by its mid price, scaled by 100. SD(B/A Spd) is the standard deviation of a bond’s bid-ask spread divided by its mid price, scaled by 100. Bond Zero is the percentage of days that a bond does not have at least one trade of $100k, following Dick-Nielsen, Feldhutter, and Lando (2012). Amihud is the Amihud measure and IRC is an implied round-trip cost measure. SD(Amihud) and SD(IRC) are the standard deviations of these measures. Amihud, IRC, SD(Amihud), and SD(IRC) are defined as in Dick-Nielsen, Feldhutter, and Lando (2012), but scaled by 100 here. Junk is a dummy that equals 1 if the Moody’s rating of a bond is Ba1 or worse. t-stats are in parentheses and use standard errors clustered by firm.
4.5
Volatility of Macroeconomic Factors
In Table 9 of the paper, we examine the relation between excess CDS volatility and changing macroeconomic conditions in the time-series. One potential issue is that if CDS returns follow a linear factor model where the factors are changes in macroeconomic variables, CDS return volatilities may then be related to the volatility of macroeconomic factors. The volatility of these macroeconomic factors may then explain excess volatility, though the relation between excess volatility and the volatility of macroeconomic factors is not ex-ante obvious as the volatility of macroeconomic factors may also be related to model volatilities (through a relation to equity returns). In Table IA.10, we regress excess CDS volatility on the volatilities of daily changes of a number of macroeconomic factors. We find that excess CDS volatility is positively correlated with repo rate volatility and negatively correlated with the volatility of credit spread changes. The former result is suggestive of an interest rate effect that exists in CDS, but is not captured by the Merton model. The latter result is consistent with the Merton model being able to better capture volatility during periods of uncertain credit conditions. As these periods tend to coincide with worsening credit conditions, this result is consistent with the results in the 22
paper where we show that excess volatility is worse in periods of improving credit conditions, as measured by the Conference Board’s indicators. However, we note that the results in this section are economically less significant than those found in Table 9 of the paper. The interquartile range of repo volatility is 0.4586 and that of the credit spread is 0.3363. Under the base case, this implies a difference in excess volatility of 0.16 and -0.06, respectively. Compared to the 0.5 to 1.2 differences in excess volatility implied by the interquartile ranges of the Conference board variables and also of volatility of bid-ask spreads reported in the paper, the effect of the variables here is small. Table IA.10: Excess CDS Volatility and the Volatility of Macroeconomic Variables
(1) Base (2) Cond
SD(∆ VIX)
SD(∆ Repo)
SD(∆ LIBOR)
-0.0010 (-0.09) 0.0027 (0.19)
0.3541 (2.63) 0.5938 (4.37)
-0.8317 (-1.60) -0.4308 (-0.78)
SD(∆ Term Spread) -0.5995 (-0.95) -0.3963 (-0.69)
SD(∆ Credit Spread) -0.1705 (-3.25) -0.2153 (-4.68)
SD(∆ CDS Index) 0.0707 (0.55) -0.0529 (-0.42)
Obs
R2
24,900
0.208
Withingroup R2 0.011
22,080
0.180
0.019
Both regressions include firm fixed-effects. The dependent variable is the annualized excess volatility, σˆD − Merton σD for daily CDS returns, expressed in %. All of the dependent variables are annualized volatilities of daily changes in the variables, with all variables expressed in %. t-stats using standard errors clustered by time are reported.
4.6
Excess CDS Volatility and the Volatility of Fundamentals
In Table IA.11, we examine the relation between excess CDS volatility and CDS- and firmlevel characteristics in panel regressions with time fixed-effects. In particular, we add the volatility of cash flows, earnings, leverage, and sales to Table 9 from the main paper, finding that only the volatility of leverage is statistically significant.
23
Table IA.11: Excess CDS Volatility and Firm- and Bond-Level Characteristics VARIABLES CDS Spread EBIT/Assets Coverage Ratio Sales/Assets Retained Earnings/Assets Net Income/Assets log(Assets) Cash flow vol Earnings vol Leverage vol Sales vol
(1) Base 0.0236 (13.32) 5.734 (1.457) -0.00809 (-0.996) 0.350 (1.547) 0.900 (1.384) 1.110 (0.239) 0.534 (3.748) 1.736 (0.651) -6.623 (-1.156) -11.67 (-2.242) -1.386 (-0.299)
CDS B/A SD(CDS B/A) Observations R-squared Within-group R2
1,482 0.587 0.571
(2) Base 0.0189 (5.637)
-0.187 (-2.730) 0.394 (7.041) 1,819 0.634 0.621
(3) Base 0.0223 (6.257) 3.637 (0.998) -0.00688 (-0.890) 0.180 (0.906) 0.451 (0.750) 4.054 (0.966) 0.302 (1.796) -0.227 (-0.0885) -8.296 (-1.368) -10.28 (-2.202) 1.176 (0.291) -0.182 (-2.377) 0.354 (5.773) 1,482 0.632 0.618
Merton All regressions include time fixed-effects. The dependent variable is σ ˆD − σD , where Merton σ ˆD is the realized CDS volatility and σD is the volatility implied by the Merton model and equity volatility. Both are expressed in annualized % and based on monthly returns. CDS Spread is the CDS spread in basis points. EBIT/Assets is defined using Compustat data as OIADP/AT. Coverage Ratio is defined as (OIADP + XINT)/XINT. Sales/Assets is defined as SALE/AT. Retained Earnings/Assets is defined as RE/AT. Net Income/Assets is defined as NI/AT. Assets is the book value of assets in $mm. Cash flow vol is the volatility of the ratio of cash flows to assets. Earnings vol is the volatility of the ratio of earnings to assets. Leverage vol is the volatility of firm leverage. Sales vol is the volatility fo the ratio of sales to assets. All four vol variables are calculated using the last five years of Compustat quarterly data. CDS B/A is the bid-ask spread of CDS in basis points. SD(CDS B/A) is the standard deviation of the CDS bid-ask spread in basis points. t-stats are in parentheses and use standard errors clustered by firm.
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References Baele, L., G. Bekaert, and K. Inghelbrecht (2010). The Determinants of Stock and Bond Return Comovements. Review of Financial Studies 23, 2374–2428. Bao, J., J. Pan, and J. Wang (2011). The Illiquidity of Corporate Bonds. Journal of Finance 66, 911–946. Cameron, A. C., J. B. Gelbach, and D. L. Miller (2008). Bootstrap-Based Improvements for Inference with Clustered Errors. Review of Economics and Statistics 90, 414–427. Cameron, A. C., J. B. Gelbach, and D. L. Miller (2011). Robust Inference with Multiway Clustering. Journal of Business & Economics Statistics 29, 238–249. Collin-Dufresne, P. and R. S. Goldstein (2001). Do Credit Spreads Reflect Stationary Leverage Ratios? Journal of Finance 56, 1929–1957. Dick-Nielsen, J., P. Feldhutter, and D. Lando (2012). Corporate bond liquidity before and after the onset of the subprime crisis. Journal of Financial Economics 103, 471–492. Duffie, D., J. Pan, and K. Singleton (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica 68, 1343–1376. Edwards, A. K., L. E. Harris, and M. S. Piwowar (2007). Corporate Bond Market Transaction Costs and Transparency. Journal of Finance 62, 1421–1451. Eom, Y. H., J. Helwege, and J. Z. Huang (2004). Structural Models of Corporate Bond Pricing: An Empirical Analysis. Review of Financial Studies 17, 499–544. Merton, R. C. (1974). On the Pricing of Corporate Debt: The Risk Structure of Interest Rates. Journal of Finance 29, 449–470. Petersen, M. A. (2008). Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches. Review of Financial Studies 22, 435–480. Schaefer, S. M. and I. A. Strebulaev (2008). Structural Models of Credit Risk are Useful: Evidence from Hedge Ratios on Corporate Bonds. Journal of Financial Economics 90, 25
1–19. Zhang, B. Y., H. Zhou, and H. Zhu (2009). Explaining Credit Default Swap Spreads with Equity Volatility and Jump Risks of Individual Firms. Review of Financial Studies 22, 5099–5131.
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