Essays on Credit Risk and Credit Derivatives
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Bajlum, Claus Doctoral Thesis Essays on Credit Risk and Credit Derivatives PhD Series, No. 12.2008 Provided in Cooperation with: Copenhagen Business School (CBS) Suggested Citation: Bajlum, Claus (2008) : Essays on Credit Risk and Credit Derivatives, PhD Series, No. 12.2008, ISBN 9788759383612, Copenhagen Business School (CBS), Frederiksberg, https://hdl.handle.net/10398/6520 This Version is available at: https://hdl.handle.net/10419/208695 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/
ISSN 0906-6934 ISBN 978-87-593-8361-2 Essays on Credit Risk and Credit Derivatives Essays on Credit Risk and Credit Derivatives Claus Bajlum PhD Series 12.2008 PhD School in Economics and Business Administration CBS / Copenhagen Business School CBS PhD omslag nr. 12 - Claus Bajlum.indd 1 30/04/08 12:46:43
Essays on Credit Risk and Credit Derivatives
Claus Bajlum Essays on Credit Risk and Credit Derivatives CBS / Copenhagen Business School PhD School in Economics and Business Administration PhD Series 12.2008
Claus Bajlum Essays on Credit Risk and Credit Derivatives 1. edition 2008 PhD Series 12.2008 © The Author ISBN: 978-87-593-8361-2 ISSN: 0906-6934 Distributed by: Samfundslitteratur Publishers Rosenørns Allé 9 DK-1970 Frederiksberg C Tlf.: +45 38 15 38 80 Fax: +45 35 35 78 22 [email protected] www.samfundslitteratur.dk All rights reserved. No parts of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without permission in writing from the publisher.
Contents Introduction ix 1 Accounting Transparency and the Term Structure of Credit Default Swap Spreads 1 1.1 Introduction.............................. 3 1.2 Hypotheses .............................. 7 1.3 Measuring Accounting Transparency . . . . . . . . . . . . . . . . 15 1.4 Data.................................. 16 1.5 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6.1 The Term Structure of Transparency Spreads . . . . . . . 24 1.7 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.1 Alternative Econometric Speci…cations . . . . . . . . . . . 30 1.7.2 Individual Maturity Classes . . . . . . . . . . . . . . . . . 37 1.8 Conclusion............................... 48 A Du¢e&Lando(2001) ........................ 49 A.1 Pricing the CDS . . . . . . . . . . . . . . . . . . . . . . . 51 B The Accounting Transparency Measure . . . . . . . . . . . . . . . 54 2 Capital Structure Arbitrage: Model Choice and Volatility Calibration 57 2.1 Introduction.............................. 59 2.2 TradingStrategy ........................... 63 2.2.1 CDSPricing.......................... 64 iii
2.2.2 Implementation of the Strategy . . . . . . . . . . . . . . . 66 2.2.3 Trading returns . . . . . . . . . . . . . . . . . . . . . . . . 67 2.3 Data.................................. 68 2.4 Model Choice and Volatility Calibration . . . . . . . . . . . . . . 71 2.4.1 CreditGrades . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.2 Leland & Toft (1996) . . . . . . . . . . . . . . . . . . . . . 73 2.4.3 Model Calibration and Implied Parameters . . . . . . . . . 76 2.5 CaseStudies.............................. 79 2.5.1 Sears, Roebuck and Company . . . . . . . . . . . . . . . . 79 2.5.2 Time Warner and Motorola . . . . . . . . . . . . . . . . . 82 2.5.3 Mandalay Resort Group . . . . . . . . . . . . . . . . . . . 85 2.6 GeneralResults............................ 87 2.6.1 Capital Structure Arbitrage Index Returns . . . . . . . . . 91 2.7 Conclusion............................... 95 A Appendix ............................... 97 A.1 CreditGrades ......................... 97 A.2 Leland & Toft (1996) . . . . . . . . . . . . . . . . . . . . . 99 3 Credit Risk Premia in the Market for Credit Default Swaps 103 3.1 Introduction.............................. 105 3.2 Measuring Credit Risk Premia (RPI) from Yield Spreads . . . . . 108 3.2.1 Yield Spread Components . . . . . . . . . . . . . . . . . . 109 3.2.2 Measuring the Risk Premium from CDS Spreads . . . . . . 110 3.3 Data.................................. 112 3.4 Empirical Implementation . . . . . . . . . . . . . . . . . . . . . . 114 3.4.1 Calibrating the Leland & Toft (1996)-Model . . . . . . . . 115 3.4.2 Estimating the Asset Value Risk Premia . . . . . . . . . . 118 3.5 EmpiricalResults........................... 120 3.5.1 Decomposing the Credit Spread . . . . . . . . . . . . . . . 123 3.5.2 Modeling CDS Spreads . . . . . . . . . . . . . . . . . . . . 135 3.6 Conclusion............................... 141 A Leland&Toft(1996)......................... 143 A.1 Survival Probabilities . . . . . . . . . . . . . . . . . . . . . 145 A.2 Pricing the Credit Default Swap without a Risk Premium 146 iv
A.3 Pricing the Credit Default Swap with a Risk Premium Included ............................. 147 Summary 149 v
Coauthored with Peter Tind Larsen, School of Economics and Management, University of Aarhus Abstract1 This paper estimates the impact of accounting transparency on the term structure of CDS spreads for a large cross-section of …rms. Using a newly developed measure of accounting transparency in Berger et al. (2006), we …nd a downwardsloping term structure of transparency spreads. Estimating the gap between the high and low transparency credit curves at the 1, 3, 5, 7 and 10-year maturity, the transparency spread is insigni…cant in the long end but highly signi…cant and robust at 20 bps at the 1-year maturity. Furthermore, the e¤ect of accounting transparency on the term structure of CDS spreads is largest for the most risky …rms. These results are strongly supportive of the model by Du¢ e & Lando (2001), and add an explanation to the underprediction of short-term credit spreads by traditional structural credit risk models. 1We thank Lombard Risk for access to the credit default swap data. We thank Christian Riis Flor, Peter Løchte Jørgensen, David Lando, Mads Stenbo Nielsen, Thomas Plenborg and participants at the Danish Doctoral School of Finance Workshop 2007 for valuable comments and insights. Any remaining errors are our own. 2
1.1 Introduction Traditional structural credit risk models originating with Black & Scholes (1973) and Merton (1974) de…ne default as the …rst passage of a perfectly measured asset value to a default barrier. While later extensions that allow for endogenous default and debt renegotiations have increased predicted spread levels, it is well-known in the empirical literature that structural models underpredict corporate bond credit spreads, particularly in the short end.2Reasons for the poor performance may lie in shortcomings in the models as well as factors other than default risk in the corporate bond credit spread. As noted in Du¢ e & Lando (2001), it is typically di¢ cult for investors in the secondary credit markets to observe a …rm’s assets directly, either because of noisy or delayed accounting reports or other barriers to monitoring. Instead, investors must draw inference from the available accounting data and other publicly available information. As a consequence they build a model where credit investors are not kept fully informed on the status of the …rm, but receive noisy unbiased estimates of the asset value at selected times. This intuitively simple framework has a signi…cant implication for the term structure of credit spreads. In particular, for …rms with perfectly measured assets credit spreads are relatively small at short maturities and zero at zero maturity, regardless of the riskiness of the …rm. However, if …rm assets periodically are observed with noise, credit spreads are strictly positive under the same limit because investors are uncertain about the distance of current assets to the default barrier. This paper contributes to the existing literature by estimating the component of the term structure of credit spreads associated with a lack of accounting transparency.3To this end, credit default swap (CDS) spreads at the 1, 3, 5, 7 and 10-year maturity for a large cross-section of …rms are used together with a newly developed measure of accounting transparency by Berger et al. (2006). We relate this transparency measure to CDS spreads in two ways. First, it is used to estimate a gap between the high and low transparency credit curves. This gap interpreted as a transparency spread is estimated at 20 bps at 2See e.g. Jones, Mason & Rosenfeld (1984), Ogden (1987), Huang & Huang (2003) and Eom, Helwege & Huang (2004). 3Consistent with the literature, we use the terms "accounting noise" and "accounting transparency" interchangeably. If the noise in the reported asset value is low, the accounting transparency is high. 3
the 1-year maturity and narrows to 14, 8, 7 and 5 bps at the 3, 5, 7 and 10-year maturity, respectively. The downward-sloping term structure of transparency spreads is highly signi…cant in the short end but most often insigni…cant above the 5-year maturity. Furthermore, the e¤ect of accounting transparency is largest for the most risky …rms. These results are robust across alternative econometric speci…cations controlling for within cluster correlations and a large set of control variables. Second, we analyze each maturity class in isolation using the raw transparency measure and a rank transformation. In this speci…cation, the equal maturities across …rms …xed through time in the CDS data allow the control variables to impact spreads di¤erently across maturity classes. Since insights from above are preserved, the results are supportive of hypotheses derived from Du¢ e & Lando (2001) and add an explanation to the underprediction of short-term credit spreads by traditional structural models. However, the explanatory power of accounting transparency and a typical set of control variables is small for less risky …rms. This observation is supportive of the problems in earlier studies, when explaining the credit spreads of low-yield …rms using structural models. This paper suggests that variables other than accounting transparency are needed, also in the short end. The results contradict an earlier study by Yu (2005), who analyzes corporate bond credit spreads in 1991 to 1996 using the AIMR analyst ranking of corporate disclosure. He attributes a u-shaped transparency spread with the largest a¤ect at longer maturities to a discretionary disclosure hypothesis, where …rms hide information that would adversely a¤ect their long-term outlook. While Du¢ e & Lando (2001) assume an exogenous unbiased accounting noise, the theory of discretionary disclosure starting with Verrecchia (1983) suggests that withheld information may signal hidden bad news about a company. Consistent with the term structure implications in Du¢ e & Lando (2001), our study shows that the transparency spread is downward-sloping in the CDS market. Although a close relation exists between corporate bond and CDS spreads (Du¢ e (1999)), the latter are preferable from several perspectives when analyzing the determinants of the shape of the credit curve. First, the …xed maturities in CDS contracts make term structures directly comparable across …rms and time. There is no maturity shortening as there would be with corporate bonds, and we are not forced to interpolate maturities to compare spreads in the cross-section. 4
Second, quotes at di¤erent maturities should be compared on the same curve, and a study of multiple maturity observations for a given …rm at a given date is in e¤ect only possible in the CDS market. Third, a use of CDS spreads avoids any noise arising from a misspeci…ed risk-free yield curve (Houweling & Vorst (2003)). Fourth, as shown in Lando & Mortensen (2005) and Agrawal & Bohn (2005), the shape of the corporate bond credit curve depends on deviations from par under the realistic recovery of face value assumption. As Yu (2005) focuses on secondary market yields this technical e¤ect may in‡uence his results. The same e¤ect is not present in the CDS market as CDS spreads are closely related to par bond spreads. Fifth, CDS contracts are less likely to be a¤ected by di¤erences in contractual arrangements such as embedded options, guarantees, covenants and coupon e¤ects. Although bonds with e.g. call features may be deliverable in default, this e¤ect is likely to be present across the term structure of CDS spreads. Sixth, several recent studies …nd that CDS spreads are a purer measure of credit risk and represent more timely information than corporate bonds. Nondefault components stemming from asymmetric taxation and illiquidity have been compared across corporate bond and CDS markets.4However, the component due to imprecisely observed assets, let alone the term structure implications, is much less understood. A reason for the lack of evidence on the impact of accounting transparency is the di¢ culty in constructing an empirical measure of a …rm’s overall information quality. The accounting literature explaining e.g. the cost of capital has relied on the AIMR analyst ranking of corporate disclosure. Analyzing the cost of debt, Sengupta (1998) …nds a negative relationship between the AIMR measure and o¤ering yields. This measure is also adopted by Yu (2005), with a resulting sample almost entirely made up of investment grade …rms. As the measure ends in 1996, it cannot be related to CDS curves. However, a newly developed measure of accounting transparency by Berger 4Blanco, Brennan & Marsh (2005) …nd that the CDS market leads the corporate bond market. Longsta¤, Mithal & Neis (2005) …nd a signi…cant non-default related component in the corporate bond credit spread correlated with illiquidity proxies. Ericsson, Reneby & Wang (2006) …nd this not to be present in CDSs. Elton, Gruber, Agrawal & Mann (2001) document a tax premium of 29 to 73 percent of the corporate bond credit spread, depending on the rating. Related studies on corporate bonds include Delianedis & Geske (2001) and Huang & Huang (2003). 5
et al. (2006) can be readily calculated for a large sample of …rms. This allows us to study a large set of credit curves across rating categories. The idea behind the measure is that given the idiosyncratic cash ‡ow volatility, the better a …rm’s information quality the higher its …rm-speci…c equity return volatility. Berger et al. (2006) conduct several tests to assess their measure, and …nd results in accordance with intuition. Our application in the credit derivatives market provides additional evidence to the validity of the measure. This paper is related to Sarga & Warga (1989), Fons (1994), Helwege & Turner (1999), Lando & Mortensen (2005) and Agrawal & Bohn (2005) who analyze the slope of the credit curve as a function of credit quality. Ignoring noisy asset reports, standard theory predicts an upward-sloping credit curve for high quality …rms and a humped shaped or mostly downward-sloping credit curve for low quality …rms. However, these papers are silent on decomposing the curve and the e¤ect of accounting transparency. Early studies mainly analyze the 5-year maturity, which is considered the most liquid point on the curve. This paper contributes to an increasing literature analyzing the entire term structure of CDS spreads. In addition to Lando & Mortensen (2005) and Agrawal & Bohn (2005) this includes Huang & Zhou (2007), who conduct a consistent speci…cation analysis of traditional structural models. Although the 5-year maturity dominates our data, a signi…cant number of observations are found at the 1, 3, 7 and 10-year maturity. Finally, the paper is related to studies on the determinants of credit spreads such as Collin-Dufresne, Goldstein & Martin (2001), Campbell & Taksler (2003), Ericsson, Jacobs & Oviedo (2005), Cremers, Driessen, Maenhout & Weinbaum (2006) and Cao, Yu & Zhong (2006). These papers analyze the explanatory power of traditional structural variables such as leverage, asset volatility and risk-free interest rates, but are silent on di¤erent maturity classes and accounting transparency. Finally, Güntay & Hackbarth (2007) study the relation between corporate bond credit spreads and the dispersion of equity analysts’ earnings forecasts. The outline of the paper is as follows. Section 1.2 reviews the Du¢ e & Lando (2001) model and motivates the hypotheses. This section also shows a formula for the CDS spread that avoids a double integral and is easily comparable with the case of perfect information. Section 1.3 outlines the accounting transparency measure developed in Berger et al. (2006), while section 1.4 presents the data. 6
The descriptive statistics are presented in section 1.5, while section 1.6 and 1.7 contain the empirical results and a robustness analysis. Section 1.8 concludes. Appendix A and B give details behind the Du¢ e & Lando (2001) model and the transparency measure, respectively. 1.2 Hypotheses In traditional structural credit risk models, default is de…ned as the …rst hitting time of a perfectly observed di¤usion process on a default barrier. This default barrier can be exogenously determined as in e.g. Black & Cox (1976) and Longsta¤ & Schwartz (1995) or endogenously derived as in e.g. Leland (1994) and Leland & Toft (1996). As shown in Leland (2004), these models do a reasonable job in predicting longer horizon default rates while the prediction of short-term default rates is far to low. The problem is that conditional on the …rm value being above the barrier, the probability that it will cross the barrier in the next tis o(t)and the conditional default probability converges to zero as time goes to zero. Du¢ e & Lando (2001) argue that it is typically di¢ cult for investors in the secondary credit markets to perfectly observe the …rm’s assets and introduce accounting noise into a Leland (1994)-type model. More speci…cally, the value of the …rm’s assets is assumed to follow a geometric Brownian motion unobservable to the credit investors. Instead, the …rm periodically issues noisy unbiased accounting reports, which makes investors uncertain about the distance of the assets to the default barrier. Conditional on the accounting reports and the fact that the …rm has not defaulted investors are able to compute a distribution of the value of assets. This conditional distribution of assets is reproduced in Figure 1.1 for various degrees of accounting noise aand a set of base case parameters. The crucial parameter ameasures the standard deviation of the normal noise-term added to the true asset value. A lower athus represents a higher degree of accounting transparency and less uncertainty about the true asset value. When aapproaches zero the distribution will eventually collapse around the latest reported asset value. 7
According to Du¢ e & Lando (2001) this simple mechanism of uncertainty surrounding the true asset value is enough to produce a default probability within the next tthat is of the same order as t. In fact, they show that the default stopping time has an intensity. The Du¢ e & Lando (2001) model is further described in appendix A. Figure 1.1: Conditional Asset Density The …gure illustrates the conditional asset density for varying accounting precisions, reproducing the base case in Du¢ e & Lando (2001). The tax rate = 0:35, volatility = 0:05, risk-free rate r= 0:06, drift m= 0:01, payout ratio = 0:05 and default cost = 0:3. The coupon rate C= 8:00 and the default barrier VB(C) = 78. A noise-free asset report V(t1) = ^ V(t1) = 86:3is assumed together with a current noisy asset report ^ V(t) = 86:3. The standard deviation ais assumed at 0:05,0:1and 0:25 and measures the degree of accounting noise. 8
The payments in a CDS …t nicely into a continuous-time framework since the accrued premium must also be paid if a credit event occurs between two payment dates. In appendix A we show that with continuous payments the CDS spread with maturity Tcan be written as c(0; T) = r(1 R)R1 G(x; T)g(x)dx 1erT R1 (1 (T; x )) g(x)dx R1 G(x; T)g(x)dx; (1.1) where ris the risk-free interest rate and Ris the recovery rate.5(T; x ) denotes the probability of …rst passage time of a Brownian motion with constant drift and volatility parameter from an initial condition (x)>0to a level below zero at time T, where xand denote the logarithm of the asset value and default barrier, respectively. The formulas for (T; x )and G(x; T)are given in closed form in the appendix together with the conditional density function of the logarithm of assets g(x)at the time of issuance of the CDS. In the case of perfect information the integral and the density function g(x) simply disappears, leading to a closed form solution for the CDS spread known from traditional structural credit risk models. In Figure 1.2, the term structure of CDS spreads in equation (1.1) is shown for the associated conditional distribution of assets in Figure 1.1 and the various degrees of accounting noise a. Also depicted is the traditional case of perfect information a= 0, where the spread approaches zero as maturity goes to zero. However, this is not the case when noisy reports are introduced. As abecomes larger, the probability that the asset value is, in fact, close to the default barrier and may cross in a short period of time increases, resulting in higher short-term spreads. The di¤erence in spreads due to a lack of accounting transparency is less pronounced at longer maturities. Figure 1.3 and 1.4 depict the case of a lower leverage and a lower asset volatility, respectively. This captures the e¤ect of accounting transparency on CDS spreads for less risky …rms than the base case. The spreads are compressed compared to Figure 1.2, indicating that we should expect a lower absolute e¤ect of accounting transparency for less risky …rms. 5The formula in Du¢ e & Lando (2001) is based on semiannually payments and a double integral over time and the asset density. The assumption of continuous payments implies that it is only necessary to calculate a single integral numerically to evaluate the CDS spread. 9
Figure 1.2: CDS Spreads for Varying Accounting Precisions The …gure illustrates the CDS spreads associated with the conditional asset densities for varying accounting precisions, reproducing the base case in Du¢ e & Lando (2001). The tax rate = 0:35, volatility = 0:05, risk-free rate r= 0:06, drift m= 0:01, payout ratio = 0:05, default cost = 0:3and recovery rate R= 0:5. The coupon rate C= 8:00 and the default barrier VB(C) = 78. A noise-free asset report V(t1) = ^ V(t1) = 86:3is assumed together with a current noisy asset report ^ V(t) = 86:3. The standard deviation ais assumed at 0:05,0:1and 0:25 and measures the degree of accounting noise. 10
Figure 1.3: CDS Spreads For a Low Leverage Firm The …gure illustrates the CDS spreads for varying accounting precisions in Du¢ e & Lando (2001). A higher current and lagged asset report are assumed, capturing a lower leverage ratio. The tax rate = 0:35, volatility = 0:05, risk-free rate r= 0:06, drift m= 0:01, payout ratio = 0:05, default cost = 0:3and recovery rate R= 0:5. The coupon rate C= 8:00 and the default barrier VB(C) = 78. A noise-free asset report V(t1) = ^ V(t1) = 90:0is assumed together with a current noisy asset report ^ V(t) = 90:0. The standard deviation ais assumed at 0:05,0:1and 0:25 and measures the degree of accounting noise. 11
transparency scores distributed across 368 …rms from May 2002 to September 2004.11 1.5 Descriptive Statistics Table 1.1 illustrates the distribution of the annual accounting transparency measure. Panel A represents statistics based on the pooled measure across …rms and years, while statistics in Panel B are calculated after averaging the measure for each …rm in the time-series. The pooled mean and median are 0.50 and 0.29, respectively. A few high transparency scores drive up the average, and about 10 percent of the sample …rm-years have scores larger than the theoretical upper bound of 1. A similar result based on a larger set of …rms is found in Berger et al. (2006), who attribute it to possible time-varying expected returns. Table 1.1: Summary Statistics of Accounting Transparency This table reports summary statistics for the accounting transparency measure developed in Berger, Chen & Li (2006) and calculated in section 1.3. Panel A represents statistics when pooling the measure across …rms and years, while panel B displays statistics after averaging the measure in the time-series for each …rm. In panel A, N denotes the number of …rm-years with su¢ cient data to calculate the accounting transparency measure and with associated CDS data. In panel B, N denotes the number of unique …rms. N Mean Std.dev. Min 25% 50% 75% 99% Max Panel A. Statistics on the pooled transparency measure 890 0.50 0.61 0.00 0.16 0.29 0.60 3.23 5.65 Panel B. Statistics on the time-series average transparency measure 368 0.50 0.57 0.01 0.16 0.30 0.62 2.84 4.44 11One …rm is excluded, Colgate Palmolive, as the transparency measure is calculated at 10.23, 11.56 and 11.89 in year 2002-2004. This persistently large score far above the remaining …rms might indicate a data problem speci…c to the …rm. 18
The standard deviation is 0.61 and the inter-quartile range is 0.44. The same variation is observed in Panel B after averaging the measure in the time-series, indicating a large variation in accounting transparency across the …rms. The data allow for a maximum of 3 consecutive annual transparency scores with associated CDS data for each …rm. An untabulated mean and median annual absolute change of 0.17 and 0.04, respectively, indicate a somewhat persistent transparency measure in the time-series. Table 1.2 presents summary statistics of key variables across the senior unsecured credit rating from Standard & Poor’s. The variables presented are averages across time and across …rms. Consistent with the predictions of structural credit risk models, a lower rating is associated with a higher credit spread level represented by the 5-year CDS spread, a higher equity volatility and a higher leverage. The equity volatility is calculated using 250 days of equity returns, and leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. Table 1.2: Summary Statistics of Major Variables This table reports averages of key variables across …rms and time. The statistics are presented across the senior unsecured credit rating from Standard & Poor’s. The 5-year spread represents the overall spread level and is averaged over …rms and end-of month observations. The volatility is calculated at month-end using 250-days of historical equity returns. The associated leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. The accounting transparency measure is developed in Berger, Chen & Li (2006) and calculated in section 1.3. NR means not rated. 5yr spread Volatility Leverage Transparency AAA 23 0.29 0.28 0.92 AA 26 0.28 0.21 0.88 A 48 0.33 0.34 0.60 BBB 128 0.36 0.49 0.40 BB 392 0.49 0.61 0.39 B 658 0.74 0.76 0.20 NR 137 0.33 0.31 0.66 19
A better credit rating is associated with a higher accounting transparency. This observation and a correlation of 0.16 in Table 1.3 provide additional evidence to the validity of the transparency measure as documented empirically in Berger et al. (2006). As noted in Sengupta (1998) and Yu (2005), credit agencies claim to have incorporated the quality of information disclosure in the credit ratings. Hence, we follow Sengupta (1998) and Yu (2005) and use credit ratings with caution when controlling for the cross-sectional determinants of credit spreads other than accounting transparency. We use an alternative set of control variables from studies on the determinants of credit spreads such as equity volatility, leverage, liquidity and the risk-free yield curve. However, we also analyze whether credit ratings absorb the e¤ect of accounting transparency on the term structure of CDS spreads. As a …nal remark, the correlation between the accounting transparency measure and leverage and volatility, respectively, is estimated at -0.16 and -0.08. This is of similar sign and magnitude as the correlations found in Yu (2005) based on the AIMR measure in 1991 to 1996. Table 1.3: Average Correlations Among Major Variables This table reports the Spearman rank correlation coe¢ cients between the major variables. The correlations are calculated each month, and the resulting average correlations are reported. The volatility is calculated at month-end using 250-days of historical equity returns. The associated leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. The accounting transparency measure is developed in Berger, Chen & Li (2006) and calculated in section 1.3. The senior unsecured credit ratings from Standard & Poor’s are transformed to a numerical scale, where …rms rated AAA are assigned the highest number, AA the next highest and so forth. 5yr spread Volatility Leverage Transp Volatility 0.57 Leverage 0.62 0.25 Transp. -0.11 -0.08 -0.16 Rating -0.76 -0.41 -0.55 0.16 20
The distribution of the CDS spreads across credit ratings and maturities is illustrated in Table 1.4 Panel A. The mean consensus quote across time and …rms is found in the …rst row, while the number of observations and the mean relative quote dispersion are found in the second and third row, respectively. Panel B contains the statistics for full month-end curves with observations at all maturities at month-end for a given …rm. By considering full curves, the mean consensus quotes within a given rating class are comparable across maturities, since all averages are calculated from the same set of dates and …rms. As expected, the mean consensus quotes increase monotonically with maturity for high credit quality …rms and decrease monotonically with maturity for the lowest credit quality …rms.12 The 5-year maturity accounts for the highest number of observations, but even the least observed 1-year maturity accounts for almost 15 percent of the observations. Across ratings the lower end of the investment grade segment has the highest number of observations. However, we are able to study a signi…cant proportion of sample spreads across maturities in the low credit quality segment. For BB-rated …rms the sample consists of 449 to 757 month-end quotes for each maturity and 342 full curves, while the number of quotes for B-rated …rms ranges from 66 to 87 with 50 full curves.13 Lando & Mortensen (2005) interpret the relative quote dispersion as a proxy for liquidity. The more agreement about a quote, the higher the liquidity for that particular credit. Adopting this liquidity proxy, we see a liquidity smile for a …xed rating across maturities. This is consistent with the fact that the 5-year maturity is considered the most liquid point on the curve. However, the di¤erence in the mean relative quote dispersion across maturities is small. 12Theory predicts an upward-sloping credit curve for high quality …rms and a humped shaped or mostly downward-sloping credit curve for low quality …rms. While the …rst is well-established in the empirical literature, the latter is more controversial. See Sarga & Warga (1989), Fons (1994), Helwege & Turner (1999), Lando & Mortensen (2005) and Agrawal & Bohn (2005). 13For comparison, Yu (2005) studies 0 speculative grade bonds in 1991-1994, 4 in 1995 and 15 in 1996. 21
Table 1.4: Summary Statistics by Credit Rating and Maturity This table illustrates the distribution of month-end CDS quotes across credit ratings and maturities. The mean consensus quote across time and …rms is found in the …rst row for each rating category, while the number of observations and the mean relative quote dispersion are found in the second and third row, respectively. The latter is calculated as the standard deviation of collected quotes divided by the consensus quote. Panel A reports the statistics for unrestricted curves, while Panel B reports statistics for full curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. 1yr 3yr 5yr 7yr 10yr Total Panel A. Unrestricted curves AAA 24 25 25 33 38 29 34 59 92 66 45 296 0.13 0.13 0.13 0.13 0.13 0.13 AA 24 24 26 29 35 28 146 264 351 297 226 1,284 0.14 0.14 0.12 0.12 0.13 0.13 A 45 44 48 52 59 50 1,177 1,930 2,136 1,856 1,658 8,757 0.14 0.12 0.09 0.11 0.12 0.11 BBB 131 126 128 127 131 128 1,732 2,568 2,736 2,365 2,234 11,635 0.13 0.11 0.08 0.09 0.11 0.10 BB 419 407 392 390 368 395 449 702 757 559 567 3,034 0.11 0.10 0.09 0.09 0.10 0.10 B 761 712 658 613 615 672 66 82 87 76 70 381 0.12 0.11 0.08 0.09 0.10 0.10 NR 142 137 137 184 183 154 31 53 55 35 38 212 0.10 0.11 0.09 0.09 0.07 0.09 Total 141 136 133 129 139 3,635 5,658 6,214 5,254 4,838 0.13 0.12 0.09 0.10 0.11 22
Table 1.4: Summary Statistics by Credit Rating and Maturity (cont.) This table illustrates the distribution of month-end CDS quotes across credit ratings and maturities. The mean consensus quote across time and …rms is found in the …rst row for each rating category, while the number of observations and the mean relative quote dispersion are found in the second and third row, respectively. The latter is calculated as the standard deviation of collected quotes divided by the consensus quote. Panel A reports the statistics for unrestricted curves, while Panel B reports statistics for full curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. 1yr 3yr 5yr 7yr 10yr Total Panel B. Full curves AAA 33 44 54 56 61 49 18 18 18 18 18 90 0.14 0.12 0.09 0.11 0.12 0.12 AA 28 35 39 41 46 38 94 94 94 94 94 470 0.14 0.13 0.10 0.11 0.12 0.12 A 48 55 60 63 69 59 893 893 893 893 893 4,465 0.14 0.12 0.09 0.11 0.12 0.12 BBB 133 140 143 144 146 142 1,428 1,428 1,428 1,428 1,428 7,140 0.13 0.11 0.07 0.09 0.11 0.10 BB 428 425 413 403 390 412 342 342 342 342 342 1,710 0.11 0.10 0.08 0.08 0.10 0.10 B 690 690 668 642 626 663 50 50 50 50 50 250 0.12 0.10 0.08 0.09 0.10 0.10 NR 210 219 219 231 222 220 12 12 12 12 12 60 0.10 0.10 0.08 0.08 0.08 0.09 Total 148 154 155 155 157 2,837 2,837 2,837 2,837 2,837 0.13 0.11 0.08 0.10 0.11 23
In the end, the measure developed in Berger et al. (2006) allows us to relate accounting transparency to CDS curves for a large cross-section of …rms. Importantly, the distribution of CDS spread observations across credit quality and maturity is desirable in our attempt to understand the impact of accounting transparency on the term structure of CDS spreads. The accounting transparency varies considerably in the large cross-section but less in our relatively short timeseries. Furthermore, some evidence indicates that credit spread changes in the time-series are mostly driven by market factors that tend to overwhelm the effect of …rm-level characteristics.14 Hence, cross-sectional regressions form our benchmark approach. This makes the results comparable to Yu (2005), as crosssectional regressions constitute the only regression framework in his study. Later, various econometric speci…cations are introduced to ensure that the results are not driven by spurious correlations. 1.6 Empirical Results First, we estimate a gap between the high and low transparency credit curves. This allows us to directly estimate the term structure of transparency spreads. We then study a restricted set of full curves and estimate the transparency spread term structure for high and low risk …rms. 1.6.1 The Term Structure of Transparency Spreads Du¢ e & Lando (2001) predict accounting transparency to be an important variable in explaining credit spreads in the short end. At reasonable parameter values, the model does not predict a signi…cant impact of accounting transparency above the 5-year maturity. However, discretionary disclosure may still imply an e¤ect in the long end. The corporate bond data used in Yu (2005) consists of bonds with unequal and shortening maturities and durations. This forces him to construct a piecewise linear function of bond maturity across the …rms at each month-end. He then estimates the level of the credit spread at the constructed and arti…cial knot points. 14The results in Collin-Dufresne et al. (2001) suggest that the time-series variation in corporate bond credit spreads is mainly determined by local supply and demand shocks independent of credit risk factors and liquidity proxies. Huang & Zhou (2007) …nd that …ve popular structural models cannot capture the time-series behavior of CDS spreads. 24
As a starting point, we adopt a comparable speci…cation and estimate the gap between the high and low transparency credit curves. However, we estimate the gap between the two curves at the equal, …xed and therefore directly comparable maturities in the CDS data, and interpret the gap as a transparency spread term structure. In particular, de…ne das a dummy variable that equals 1 if a …rm’s transparency measure calculated in equation (1.4) in a given year ranks above the median score. Furthermore, de…ne mTas a dummy variable that attains a value of 1 if the CDS spread has a maturity of Tand zero otherwise. Hence, in the linear combination 1m1+2m3+3m5+4m7+5m10 the coe¢ cient irepresents the level of the term structure at maturities 1, 3, 5, 7 and 10 years. Now, de…ne dmTas the product of the transparency dummy dand mT. The regression coe¢ cient in front of this term can be directly interpreted as the transparency spread, i.e. the gap between the high and low transparency credit curves at the given maturity. Hence, we run monthly cross-sectional regressions of CDS spreads on the transparency variables, volatility (vol), leverage (lev) and relative quote dispersion (Qdisp)15 SpreaditT =1m1it +2m3it +3m5it +4m7it +5m10it (1.5) +6dm1it +7dm3it +8dm5it +9dm7it +10dm10it +11V olit +12Levit +13QdispitT +"itT : The coe¢ cient estimates are averaged in the time-series and standard errors are calculated following Fama & MacBeth (1973). Table 1.5 displays the results. Focusing on the …rst column, the transparency spread is highly signi…cant and estimated at 23 bps at the 1-year maturity and 20, 13, 13 and 11 bps at the remaining maturities. Particularly the transparency spread in the short end represents a considerable part of the average CDS spread level of 130 to 140 bps across maturities as reported in Table 1.4. 15To facilitate interpretation the regression equation does not include an intercept term. Hence, the R2is not reported under this empirical speci…cation. 25
Table 1.5: Estimation of the Term Structure of Transparency Spreads This table reports the results of monthly cross-sectional regressions when estimating the gap between high and low transparency CDS spread curves. The coe¢ cient estimates are averaged in the time-series. T-statistics are reported in parentheses and are based on the standard error in Fama & MacBeth (1973). dis a dummy variable equal to 1 if the transparency measure developed in Berger, Chen & Li (2006) and calculated in section 1.3 in a given year ranks above the median score. mTis a dummy that attains a value of 1 if the CDS maturity equals T. The regression coe¢ cient in front of the product dmT can be directly interpreted as the transparency spread. The volatility is calculated using 250 days of historical equity returns, and leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. Quote dispersion is the standard deviation of collected quotes divided by the consensus quote. Full curves are a restricted set of curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. The monthly regressions are SpreaditT =1m1it +2m3it +3m5it +4m7it +5m10it +6dm1it + 7dm3it +8dm5it +9dm7it +10dm10it +11V olit +12Levit +13QdispitT +"itT : *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. (1) (2) (3) (4) Unrestr. Unrestr. Full curves Full curves m1-293.64*** -299.10*** -315.01*** -333.24*** (-11.21) (-12.78) (-11.48) (-13.84) m3-292.11*** -297.06*** -312.26*** -328.17*** (-11.17) (-12.66) (-10.78) (-12.54) m5-293.64*** -297.26*** -316.80*** -328.18*** (-10.87) (-11.94) (-10.82) (-12.00) m7-296.34*** -300.50*** -315.20*** -328.85*** (-10.74) (-11.87) (-10.36) (-11.74) m10 -295.43*** -300.12*** -311.45*** -327.26*** (-10.37) (-11.55) (-9.90) (-11.44) dm1-22.66*** -22.35*** -23.56*** -24.31*** (-4.22) (-4.11) (-3.91) (-4.29) dm3-20.04*** -19.98*** -20.52*** -20.94*** (-6.58) (-6.44) (-3.57) (-3.67) dm5-13.15*** -13.24*** -17.61*** -18.18*** (-5.56) (-5.54) (-3.11) (-3.21) dm7-12.88*** -13.05*** -14.67** -15.78*** (-5.98) (-5.75) (-2.71) (-2.82) dm10 -10.94*** -10.82*** -13.08** -14.06** (-5.26) (-5.21) (-2.47) (-2.59) Volatility 805.44*** 805.59*** 873.06*** 874.86*** (16.50) (16.53) (12.92) (12.99) Leverage 317.20*** 318.99*** 315.71*** 321.00*** (12.98) (13.14) (11.80) (12.29) Qdisp -33.37 -122.68** (-1.07) (-2.37) 26
As expected, the volatility and leverage are highly signi…cant in explaining credit spreads. However, the relative quote dispersion varies in signi…cance and has a negative coe¢ cient estimate. If proxying for liquidity, the coe¢ cient is expected to be positive. Hence, although the variable allows for reasonable interpretations on average as liquidity in Table 1.4, it is questionable whether the relative quote dispersion captures di¤erences in liquidity as suggested in Lando & Mortensen (2005). As the control variable only has a minor impact on the remaining coe¢ cient estimates and signi…cance, we keep it in our future regressions16. Firms usually have corporate bonds outstanding with just a few (or one) maturities. Hence, studying multiple maturity observations for a given …rm at a given date is in e¤ect only possible in the CDS market, and therefore not pursued in Yu (2005). Table 1.5 also contains the regression results for a restricted set of full month-end curves with observations at all maturities at month-end for a given …rm. This makes CDS spreads directly comparable across maturities as all observations are from the same set of dates and …rms. As noted in Helwege & Turner (1999) …rms with heterogenous credit quality are known to populate di¤erent ends of the corporate bond credit curve. This maturity bias is avoided when studying full curves in the CDS market. A highly signi…cant downward-sloping term structure of transparency spreads also emerges from a study of full curves. From a transparency spread of 24 bps at the 1-year maturity it decreases to 13 bps at the longest maturity. The results in Table 1.5 to some extend support the …ndings in Yu (2005). While agreeing on the statistically and economically signi…cant transparency spread in the short end, Yu (2005) …nds a widening transparency spread at longer maturities. In fact, he …nds the transparency spread larger in the long end than short end. He attributes this observation to the discretionary disclosure hypothesis where …rms hide information that would adversely a¤ect their long-term outlook.17 In alternative econometric speci…cations building on the 16Unreported results show that the presence or omission of relative quote dispersion has no impact on any results reported in the paper. 17Although Yu (2005) has only few observations in the longest end, he calculates a transparency spread at the 30-year knot point coinciding with the maximum corporate bond maturity. Hence, this estimate is likely to be less reliable. However, while our transparency spread term-structure remains downward-sloping, his exhibits a u-shape already at the 10-year knot point. More precisely, he estimates a transparency spread of 11, 3, 9 and 13 bps at the 0, 5, 10 and 30-year knot points. 27
Table 1.8 repeats the speci…cations in Table 1.7, but includes the senior unsecured credit rating from Standard & Poor’s as an additional control variable in equation (1.5). As noted in Sengupta (1998) and Yu (2005), credit agencies claim to have incorporated the quality of information disclosure in the credit ratings. The results show that credit ratings do not absorb the e¤ect of accounting transparency on the term structure of credit spreads. After accounting for the information content in credit ratings, the transparency spread continues to be highly signi…cant at the 1-year maturity and downward-sloping. However, now the gap between the high and low transparency credit curves is insigni…cant after the 5-year maturity. As expected, the credit rating is highly signi…cant and a one notch increase in rating lowers the CDS spread by approximately 50 bps. Unreported results based on full curves support these …ndings. Consistent with empirical …ndings in Du¤ee (1998), structural models such as Longsta¤ & Schwartz (1995) predict an inverse relationship between the risk-free rate and credit spreads. An increase in the risk-free rate increases the risk-neutral drift of the asset value process and reduces the risk-neutral default probability. If an increase in the slope of the risk-free yield curve increases the expected future short rate, then by the same argument as above it implies a decrease in credit spreads. From a di¤erent perspective, as noted in Collin-Dufresne et al. (2001), a decrease in the slope of the risk-free yield curve may imply a weakening economy with decreasing expected recovery rates and higher default rates. Once again, a negative relationship between the slope of the risk-free yield curve and credit spreads is expected. The risk-free term structure variables are constant across all …rms in a given month. Hence, they cannot be included in the empirical speci…cations from Table 1.7 based on Fama & MacBeth (1973) or when including monthly dummies. Table 1.9 presents the results from including the slope of the yield curve in addition to credit ratings in equation (1.5). The slope is de…ned as the di¤erence between the 10 and 1-year constant maturity treasury yields.23 The slope of the risk-free yield curve is highly signi…cant and estimated with a negative coe¢ cient. However, the transparency spread continues to be highly signi…cant in the short end, downward-sloping and insigni…cant after the 5-year maturity. 23The level of the risk-free yield curve is discussed in section 1.7.2, where individual maturity classes are studied. 34
Table 1.8: The Term Structure of Transparency Spreads and Credit Ratings This table estimates the gap between the high and low transparency CDS curves under various econometric speci…cations. (1) is a pooled OLS regression with White errors, while (2) and (3) control for residual dependence by estimating cluster-robust errors by curves and time, respectively. Regression (4) to (6) extend (1) to (3) by including monthly dummies. The Fama & MacBeth estimates are reported in (7), and Fama & MacBeth standard errors adjusted for a dependence between the cross-sections by Abarbanell & Bernard (2000) are reported in (8). T-statistics are reported in parantheses. The senior unsecured credit ratings from S&P are transformed to a numerical scale, where …rms rated AAA are assigned a score of 10, AA a score of 9 and so forth. The regressions are SpreaditT =1m1it +2m3it +3m5it +4m7it +5m10it +6dm1it +7dm3it +8dm5it +9dm7it +10dm10it + 11V olit +12Levit +13QdispitT +14Ratingit +"itT :*, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. (1) (2) (3) (4) (5) (6) (7) (8) White Cluster Cluster White Cluster Cluster F-M Adj. F-M m1-m10 supp. supp. supp. supp. supp. supp. supp. supp. dm1-15.93*** -15.93*** -15.93*** -14.28*** -14.28*** -14.28** -18.90*** -18.90*** (-2.76) (-2.81) (-2.89) (-2.66) (-2.62) (-2.57) (-3.52) (-2.68) dm3-11.76*** -11.76*** -11.76*** -11.83*** -11.83*** -11.83*** -17.23*** -17.23*** (-3.07) (-3.25) (-3.24) (-3.19) (-3.41) (-3.33) (-6.18) (-3.52) dm5-6.43** -6.43** -6.43** -5.69*-5.69** -5.69** -10.16*** -10.16** (-2.09) (-2.17) (-2.45) (-1.92) (-2.02) (-2.16) (-5.17) (-2.26) dm7-4.77 -4.77 -4.77** -4.48 -4.48 -4.48*-8.87*** -8.87*** (-1.45) (-1.50) (-2.10) (-1.42) (-1.48) (-1.99) (-4.88) (-4.20) dm10 -3.02 -3.02 -3.02 -1.51 -1.51 -1.51 -7.45*** -7.45*** (-0.94) (-0.98) (-1.30) (-0.49) (-0.51) (-0.65) (-3.80) (-2.81) Volatility 639.80*** 639.80*** 639.80*** 694.29*** 694.29*** 694.29*** 702.68*** 702.68*** (32.33) (19.44) (9.38) (26.06) (16.55) (9.14) (13.65) (4.05) Leverage 263.48*** 263.48*** 263.48*** 262.34*** 262.34*** 262.34*** 228.60*** 228.60*** (43.04) (21.85) (10.26) (44.96) (22.70) (11.21) (11.67) (3.47) Qdisp 74.71*** 74.71** 74.71*126.28*** 126.28*** 126.28*** 72.30** 72.30 (3.30) (2.45) (1.83) (5.38) (4.01) (3.04) (2.73) (1.55) Rating -49.70*** -49.70*** -49.70*** -46.62*** -46.62*** -46.62*** -48.73*** -48.73*** (-37.41) (-20.70) (-15.14) (-27.60) (-16.66) (-12.73) (-17.54) (-5.96) Cluster - Curve Month - Curve Month - - Dummy - - - Month Month Month - - 35
Table 1.9: The Term Structure of Transparency Spreads and the Yield Curve This table estimates the gap between the high and low transparency CDS curves under various econometric speci…cations. (1) is a pooled OLS regression with White errors, while (2) and (3) control for residual dependence by estimating cluster-robust errors by curves and time, respectively. T-statistics are reported in parantheses. The senior unsecured credit ratings from Standard & Poor’s are transformed to a numerical scale, where …rms rated AAA are assigned a score of 10, AA a score of 9 and so forth. The slope of the yield curve is the di¤erence between the 10 and 1-year constant maturity treasury rates. Panel A displays the results for unrestricted curves, while Panel B displays results for full curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. The regressions are SpreaditT =1m1it +2m3it +3m5it +4m7it +5m10it + 6dm1it +7dm3it +8dm5it +9dm7it +10dm10it +11V olit +12Levit +13QdispitT + 14Ratingit +15Slopet+"itT :*, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel A. Unrestricted curves Panel B. Full curves (1) (2) (3) (1) (2) (3) White Cluster Cluster White Cluster Cluster m1-m10 supp. supp. supp. supp. supp. supp. dm1-15.48*** -15.48*** -15.48*** -13.93** -13.93** -13.93*** (-2.70) (-2.75) (-2.77) (-2.30) (-2.35) (-3.16) dm3-11.60*** -11.60*** -11.60*** -9.02*-9.02*-9.02** (-3.05) (-3.23) (-3.20) (-1.72) (-1.78) (-2.13) dm5-6.27** -6.27** -6.27** -6.22 -6.22 -6.22* (-2.05) (-2.14) (-2.37) (-1.33) (-1.38) (-1.72) dm7-4.68 -4.68 -4.68** -3.76 -3.76 -3.76 (-1.43) (-1.49) (-2.05) (-0.84) (-0.87) (-1.10) dm10 -2.89 -2.89 -2.89 -1.50 -1.50 -1.50 (-0.91) (-0.95) (-1.23) (-0.36) (-0.37) (-0.42) Volatility 649.90*** 649.90*** 649.90*** 682.60*** 682.60*** 682.60*** (32.52) (19.57) (9.56) (35.31) (17.12) (7.99) Leverage 263.51*** 263.51*** 263.51*** 270.80*** 270.80*** 270.80*** (43.31) (22.04) (10.83) (34.71) (15.91) (10.58) Qdisp 94.90*** 94.90*** 94.90** 75.96** 75.96 75.96 (4.18) (3.09) (2.29) (2.25) (1.54) (1.13) Rating -49.38*** -49.38*** -49.38*** -57.97*** -57.97*** -57.97*** (-37.37) (-20.74) (-13.66) (-35.79) (-16.72) (-13.58) Slope -0.51*** -0.51*** -0.51*** -0.65*** -0.65*** -0.65*** (-22.26) (-11.09) (-3.19) (-18.99) (-8.89) (-2.82) Cluster - Curve Month - Curve Month Dummy - - - - - - 36
1.7.2 Individual Maturity Classes When included in equation (1.5), the control variables are only allowed to induce a parallel shift in the term structure of CDS spreads. As a …nal exercise, we allow the control variables to impact CDS spreads di¤erently across maturities. For that purpose, we analyze each maturity class in isolation using the raw transparency measure calculated in equation (1.4) and a rank transformation. This is possible since the data consists of CDS spreads with equal and …xed maturities. For each maturity class T, Table 1.10 Panel A presents the results of monthly cross-sectional regressions of CDS spreads on the transparency measure, volatility, leverage and relative quote dispersion Spreadit =0+1Transpit +2V olit +3Levit +4Qdispit +"it:(1.6) The coe¢ cient estimates are averaged in the time-series and standard errors are calculated following Fama & MacBeth (1973). The average adjusted R2ranges from 0.58 to 0.60 and accounting transparency is signi…cant or highly signi…cant at all maturities. From a coe¢ cient of -13.45 at the 1-year maturity, the coe¢ cient on accounting transparency decreases to -6.75 and -6.68 at the 3 and 5-year maturity, respectively. After this point a u-shape kicks in with coe¢ cients of -8.49 and -9.56 at the 7 and 10-year maturity, respectively. The variation in accounting transparency in each maturity class is similar to the variation reported in Table 1.1 for the entire sample. Hence, a one standard deviation increase in transparency reduces the spread by approximately 8, 4, 4, 5 and 6 bps across the curve. Table 1.10 Panel B contains the regression results for the restricted set of full curves with observations at all maturities at month-end for a given …rm. The resulting coe¢ cients on accounting transparency are all highly signi…cant and larger at -22.28, -21.32, -19.75, -12.15 and -17.90 at maturities of 1, 3, 5, 7 and 10 years, respectively. A one standard deviation increase in transparency reduces the spread by approximately 14, 13, 12, 7 and 11 bps across the curve, and main insights from the unrestricted curves in Panel A are preserved. Under alternative econometric speci…cations and a broader set of control variables, the impact of accounting transparency is later shown to strictly decrease with maturity. 37
Table 1.10: Fama & MacBeth Regressions on Isolated Maturity Classes This table reports the results of monthly cross-sectional regressions when analyzing each maturity class in isolation. The coe¢ cient estimates are averaged in the time-series. T-statistics are reported in parentheses and are based on the standard error in Fama & MacBeth (1973). The accounting transparency measure is developed in Berger, Chen & Li (2006) and calculated in section 1.3. The volatility is calculated using 250 days of historical equity returns, and leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. Quote dispersion is the standard deviation of collected quotes divided by the consensus quote. Panel A displays the results for unrestricted curves, while Panel B displays results for full curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. The monthly regressions for each maturity class are Spreadit =0+1Transpit +2V olit + 3Levit +4Qdispit +"it. *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel A. Unrestricted curves 1-year maturity 3-year maturity 5-year maturity 7-year maturity 10-year maturity Intercept -317.92*** -368.62*** -329.58*** -333.73*** -328.56*** -299.57*** -263.25*** -271.89*** -232.29*** -254.03*** (-8.17) (-10.24) (-10.31) (-12.47) (-14.79) (-15.71) (-10.07) (-12.80) (-11.48) (-14.72) Transp -13.45** -10.92** -6.75*** -6.92*** -6.68*** -6.57*** -8.49*** -8.62*** -9.56*** -9.51*** (-2.58) (-2.56) (-2.86) (-3.01) (-4.01) (-3.75) (-4.27) (-4.57) (-4.20) (-4.23) Volatility 928.25*** 941.46*** 860.30*** 861.16*** 802.93*** 802.58*** 744.61*** 749.34*** 717.25*** 716.69*** (11.94) (12.07) (15.95) (16.06) (19.82) (19.66) (16.79) (16.95) (17.55) (17.52) Leverage 325.54*** 331.19*** 334.08*** 335.64*** 335.77*** 325.38*** 292.83*** 296.24*** 293.93*** 298.19*** (11.17) (11.45) (11.55) (11.94) (14.92) (14.66) (11.97) (12.56) (14.64) (14.89) Qdisp -350.11*** -24.15 265.73*** -60.51 -172.06*** (-5.37) (-0.43) (5.08) (-0.94) (-3.29) Adj. R20.59 0.58 0.59 0.59 0.60 0.60 0.59 0.59 0.60 0.60 38
Table 1.10: Fama & MacBeth Regressions on Isolated Maturity Classes (cont.) This table reports the results of monthly cross-sectional regressions when analyzing each maturity class in isolation. The coe¢ cient estimates are averaged in the time-series. T-statistics are reported in parentheses and are based on the standard error in Fama & MacBeth (1973). The accounting transparency measure is developed in Berger, Chen & Li (2006) and calculated in section 1.3. The volatility is calculated using 250 days of historical equity returns, and leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. Quote dispersion is the standard deviation of collected quotes divided by the consensus quote. Panel A displays the results for unrestricted curves, while Panel B displays results for full curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. The monthly regressions for each maturity class are Spreadit =0+1Transpit +2V olit + 3Levit +4Qdispit +"it. *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel B. Full curves 1-year maturity 3-year maturity 5-year maturity 7-year maturity 10-year maturity Intercept -311.87*** -382.08*** -347.54*** -358.88*** -373.66*** -331.24*** -293.47*** -308.58*** -257.89*** -285.08*** (-7.83) (-10.23) (-10.20) (-12.84) (-14.03) (-14.16) (-11.71) (-14.18) (-10.63) (-14.56) Transp -22.28*** -23.23*** -21.32*** -20.80*** -19.75*** -18.82*** -12.15** -12.16*** -17.90*** -17.85*** (-3.66) (-4.15) (-3.66) (-3.81) (-4.97) (-3.72) (-2.08) (-3.71) (-3.75) (-3.78) Volatility 956.21*** 968.14*** 907.03*** 930.93*** 872.92*** 874.33*** 812.82*** 830.24*** 790.30*** 792.69*** (11.28) (11.44) (13.14) (13.18) (13.33) (13.63) (12.31) (13.26) (13.32) (13.37) Leverage 310.69*** 321.69*** 337.53*** 331.23*** 340.91*** 328.65*** 320.43*** 321.37*** 302.95*** 309.63*** (9.31) (9.85) (11.19) (11.74) (12.97) (13.00) (13.96) (13.53) (13.85) (14.47) Qdisp -468.32*** -66.81 452.42*** -126.51 -198.98*** (-7.63) (-0.56) (3.75) (-1.10) (-3.33) Adj. R20.62 0.61 0.63 0.63 0.64 0.64 0.63 0.63 0.64 0.64 39
A concern is that the accounting transparency measure is a noisy estimate of "true" accounting transparency, where an interpretation of the distance between two scores in a cardinal manner is unreasonable. Hence, we transform the annual accounting transparency measure to evenly spaced observations on the unit interval [0,1], and only interpret the annual ranking ordinally. A transformed score of 1(0) in a given year is assigned to the …rm with highest(lowest) transparency. Table 1.11 Panel A presents highly signi…cant coe¢ cient estimates of -36.69, -27.73, -20.11, -26.93 and -26.89 across the curve. If a …rm is able to improve its accounting transparency from the lowest to a median ranking, say, the result is a reduction in CDS spreads of 18, 14, 10, 13 and 13 bps at maturities of 1, 3, 5, 7 and 10 years, respectively. A similar conclusion is reached from full curves in Panel B. Table 1.12 analyzes the impact of accounting transparency for high and low risk …rms using the annual transparency ranks. Consistent with the results in the previous section, the e¤ect of accounting transparency is small and most often insigni…cant when based on …rms with a low leverage and a low volatility in Panel B. However, for the most risky …rms with a high leverage and a high volatility in Panel A, the coe¢ cient estimates are -99.02, -83.78, -68.09, -70.84 and -66.29 and highly signi…cant. Hence, if a risky …rm is able to improve its accounting transparency from the lowest to a median ranking, say, the result is a reduction in CDS spreads of 50, 42, 34, 35 and 33 bps at maturities of 1, 3, 5, 7 and 10 years, respectively. Note the large R2of 0.59 to 0.63 for the risky …rms and the much smaller R2of 0.14 to 0.20 for the …rms with low leverage and low volatility. This observation is supportive of the problems in earlier studies when explaining the credit spreads of low-yield …rms using structural models. This paper suggests that variables other than accounting transparency are needed - also in the short end. 40
Table 1.11: Fama & MacBeth Regressions on Isolated Maturity Classes Based on Transparency Rankings This table reports the results of monthly cross-sectional regressions when analyzing each maturity class in isolation. The coe¢ cient estimates are averaged in the time-series. T-statistics are reported in parentheses and are based on the standard error in Fama & MacBeth (1973). The accounting transparency measure is developed in Berger, Chen & Li (2006) and calculated in section 1.3. The transparency measure is subsequently transformed to evenly spaced observations on the unit interval [0,1]. A transformed score of 1(0) in a given year is assigned to the …rm with highest(lowest) transparency. The volatility is calculated using 250 days of historical equity returns, and leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. Quote dispersion is the standard deviation of collected quotes divided by the consensus quote. Panel A displays the results for unrestricted curves, while Panel B displays results for full curves with an observation at a maturity of 1, 3, 5, 7 and 10 years. The monthly regressions for each maturity class are Spreadit =0+1Transpit +2V olit +3Levit +4Qdispit +"it. *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel A. Unrestricted curves Panel B. Full curves 1-year 3-year 5-year 7-year 10-year 1-year 3-year 5-year 7-year 10-year Intercept -302.23*** -316.63*** -320.59*** -252.54*** -221.81*** -295.67*** -333.10*** -360.83*** -283.41*** -246.09*** (-7.91) (-10.02) (-14.41) (-9.85) (-11.04) (-7.55) (-9.65) (-13.41) (-11.37) (-10.23) Transp -36.69*** -27.73*** -20.11*** -26.93*** -26.89*** -39.91*** -37.48*** -33.06*** -25.87*** -31.33*** (-4.33) (-6.60) (-9.14) (-7.15) (-6.74) (-3.85) (-4.89) (-6.43) (-3.23) (-4.13) Volatility 928.39*** 858.59*** 802.09*** 743.96*** 715.29*** 947.32*** 902.76*** 869.96*** 810.88*** 786.03*** (11.87) (15.90) (19.74) (16.80) (17.54) (11.27) (13.11) (13.27) (12.41) (13.26) Leverage 317.56*** 331.05*** 334.49*** 290.78*** 292.83*** 308.24*** 333.69*** 337.15*** 317.24*** 300.85*** (10.70) (11.35) (14.74) (11.79) (14.72) (9.22) (11.04) (12.58) (13.79) (13.78) Qdisp -346.94*** -27.19 263.40*** -62.00 -176.22*** -483.29*** -79.69 425.67*** -132.83 -212.79*** (-5.53) (-0.49) (5.01) (-0.96) (-3.33) (-8.02) (-0.68) (3.50) (-1.17) (-3.56) Adj. R20.59 0.59 0.61 0.59 0.60 0.63 0.64 0.64 0.64 0.64 41
Table 1.12: Isolated Maturity Classes and High and Low Risk Firms This table reports the results of monthly cross-sectional regressions when analyzing each maturity class in isolation. The coe¢ cient estimates are averaged in the time-series. T-statistics are reported in parentheses and are based on the standard error in Fama & MacBeth (1973). The accounting transparency measure is developed in Berger, Chen & Li (2006) and calculated in section 1.3. The transparency measure is subsequently transformed to evenly spaced observations on the unit interval [0,1]. A transformed score of 1(0) in a given year is assigned to the …rm with highest(lowest) transparency. The volatility is calculated using 250 days of historical equity returns, and leverage is total liabilities divided by the sum of total liabilities and equity market capitalization. Quote dispersion is the standard deviation of collected quotes divided by the consensus quote. The monthly regressions for each maturity class are Spreadit =0+1Transpit +2V olit +3Levit +4Qdispit +"it. Each month, the …rms are separated into high and low leverage and volatility groups by the respective medians. Pabel A displays the results when running the regression on …rms with both a high leverage and a high volatility, while Panel B is for …rms with low leverage and low volatility. *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel A. High leverage, high volatility Panel B. Low leverage, low volatility 1-year 3-year 5-year 7-year 10-year 1-year 3-year 5-year 7-year 10-year Intercept -579.49*** -643.09*** -563.56*** -546.239*** -469.89*** -22.81 -12.14 -22.77*** -13.20** -7.49 (-6.70) (-9.98) (-14.30) (-9.78) (-12.93) (-1.64) (-1.12) (-2.94) (-2.36) (-0.7) Transp -99.02*** -83.78*** -68.09*** -70.84*** -66.29*** -2.72 -7.05** -3.60 -6.57*** -11.62* (-3.55) (-5.60) (-8.06) (-5.61) (-5.37) (-0.39) (-2.29) (-1.28) (-3.18) (-1.81) Volatility 1193.6*** 1210.0*** 1123.5*** 1028.4*** 1027.6*** 129.47** 178.10*** 206.55*** 211.15*** 207.01*** (10.47) (10.38) (10.73) (10.13) (9.91) (2.33) (7.97) (12.58) (11.54) (5.72) Leverage 625.08*** 622.28*** 548.87*** 588.45*** 507.47*** 160.80*** 105.85*** 91.88*** 99.34*** 127.57*** (6.76) (6.85) (8.14) (8.23) (7.85) (5.80) (10.36) (10.89) (10.30) (10.33) Qdisp -593.92*** 133.95 298.77** 8.946 -73.94 -186.42*** -170.67*** -135.06*** -187.27*** -189.97*** (-4.00) (0.81) (2.72) (0.06) (-0.54) (-2.91) (-4.10) (-4.83) (-6.65) (-7.58) Adj. R20.62 0.62 0.63 0.60 0.59 0.16 0.14 0.16 0.20 0.18 42
Finally, we allow the broader set of control variables to impact spreads differently across the curve under the alternative econometric speci…cations introduced earlier.24 The conclusion is a downward-sloping impact of accounting transparency across maturities that is highly robust in the short end. Across all speci…cations, a move from the lowest to a median transparency ranking, say, reduces the 1-year spread by approximately 15 bps. In particular, Table 1.13 presents the results from including the credit rating as a control variable. In the cross-sectional regressions in Panel A and B, the coe¢ cients on accounting transparency are insigni…cant or only weakly signi…cant after the 5-year maturity. The remaining speci…cations in Panel C to F support a highly signi…cant e¤ect of accounting transparency at the 1-year maturity and a declining coe¢ cient with varying signi…cance at longer maturities. The credit rating is highly signi…cant in all speci…cations, and R2increases to 0.68 compared to an R2around 0.60 without credit ratings in the Fama & MacBeth (1973) regressions in Table 1.10. Table 1.14 presents the results from including the slope of the yield curve in addition to credit ratings.25 As before, this variable can only be included in a subset of the empirical speci…cations. While estimated with a highly signi…cant negative coe¢ cient, the slope of the yield curve only increases R2marginally. Accounting transparency continues to be highly signi…cant in the short end, and the impact continues to decline as maturity increases. 24As each maturity class is analyzed in isolation, the various econometric speci…cations do not include standard errors robust to within cluster correlation at the curve level. 25Including the maturity-matched constant maturity treasury yield in addition to the slope implies that both are estimated insigni…cantly. However, coe¢ cients and signi…cance of the transparency gap are unchanged. 43
The resulting equity value is S(V; C) = V rVB(C) rV VB(C) + (1) C r"1V VB(C)#; (1.11) while the value of the consol bond is d(V; C) = (1 )VB(C) rV VB(C) +C r"1V VB(C)#:(1.12) Finally, the optimal coupon is chosen such that the initial total value of the …rm S(V; C) + d(V; C)is maximized. After issuance, bond and CDS investors are not kept fully informed on the status of the …rm. They do understand that equity owners will force liquidation when the asset value falls to VB, but they cannot observe the asset process V directly. Instead, they receive an accounting report at selected times t1; t2:::; ti< t in terms of a noisy estimate of the asset value given by b Vt, where log b Vtand log Vt are joint normal. Speci…cally, Y(t) = log b Vt=Z(t) + U(t);(1.13) where U(t)is independent of Z(t)and normally distributed with mean u= a2 2=E(Ut)and variance a2=V ar(Ut):Hence, the standard deviation aof Ut measures the degree of accounting noise. Also observed at each tis whether the …rm has defaulted or not. For simplicity, it is assumed that equity is not traded in the public market and equity owners are precluded from trading in the credit market.27 Based on the information available, it is possible for the investors to calculate the conditional distribution of assets Vt. With the simple case of having observed only a single noisy asset report at time t=t1, the density g( j Yt; z0; t)of Ztcan be computed conditional on the noisy observation Yt, a lagged noise-free report z0and > t. With ey=yu,ex=xand ez=z0, where log(VB) = , 27Hence, the information …ltration in the credit market is de…ned as Ht= Y(t1); ::::; Y (tn);1f(VB)sg: 0 stfor the largest nsuch that tnt: 50
the density is shown to be g(xjy; z0; t) = q0 exp (J(ey; ex; ez0)) 1exp 2exez0 2t exp 2 1 4031 p20exp 2 2 4032 p20; (1.14) where J(ey; ex; ez0) = (eyex)2 2a2+(ez0+mt ex)2 22t;(1.15) 0=a2+2t 2a22t;(1.16) 1=ey a2+ez0+mt 2t;(1.17) 2=1+ 2 ez0 2t;(1.18) 3=1 2 ey a2+(ez0+mt)2 2t!(1.19) and is the standard normal distribution function. Conditional on survival up to time t, this density gives us the conditional distribution of assets as g(V)=V , depicted in Figure 1.1. The conditional survival probability q(t; s) = Q( > s j Ht) to some future time s > t is q(t; s) = Z1 (1 (st; x )) g(xjYt; z0; t)dx: (1.20) (st; x )at time tdenotes the probability of the …rst passage of a Brownian motion with drift mand volatility parameter from an initial condition (x)>0to a level below zero at time s. This probability is known as 1(st; x )(1.21) = (x) + m(st) p(st)!exp 2m(x) 2 (x) + m(st) p(st)!: A.1 Pricing the CDS A CDS is an insurance contract against credit events such as the default on a corporate bond (the reference obligation) by a speci…c issuer (reference entity). 51
In case of a credit event, the seller of insurance is obligated to buy the reference obligation from the protection buyer at par. For this protection, the buyer pays a periodic premium to the protection seller until the maturity of the contract or the credit event, whichever comes …rst. Since the accrued premium must also be paid if a credit event occurs between two payment dates, the payments …t nicely into a continuous-time framework. The present value of the premium payments can be calculated as EQcZT 0 exp Zs 0 rudu1f>sgds, (1.22) where cdenotes the annual premium known as the CDS spread, Tthe maturity of the contract, rthe risk-free interest rate, the default time of the obligor and EQdenotes the expectation under the risk-neutral pricing measure. Assuming independence between the default time and the risk-free interest rate, this can be written as cZT 0 P(0; s)q(0; s)ds, (1.23) where P(0; s)is the price of a default-free zero-coupon bond with maturity s, and q(0; s)is the risk-neutral survival probability until time sat the time of issuance, derived in equation (1.20). The present value of the credit protection is equal to EQ(1 R) exp Z 0 rudu1f<Tg, (1.24) where Ris the recovery of bond market value measured as a percentage of par in the event of default. Maintaining the assumption of independence between the default time and the risk-free interest rate and assuming a constant R, this can be written as (1 R)ZT 0 P(0; s)q0(0; s)ds, (1.25) where q0(0; t) = dq(0; t)=dt is the probability density function of the default time. The CDS spread is determined such that the value of the contract is zero at initiation 0 = cZT 0 P(0; s)q(0; s)ds + (1 R)ZT 0 P(0; s)q0(0; s)ds, (1.26) 52
and hence c(0; T) = (1 R)RT 0P(0; s)q0(0; s)ds RT 0P(0; s)q(0; s)ds . (1.27) As mentioned, the model assumes a constant interest rate r, implying that c(0; T) = (1 R)RT 0ersq0(0; s)ds RT 0ersq(0; s)ds :(1.28) Integrating the denominator by parts yields c(0; T) = r(1 R)RT 0ersq0(0; s)ds 1erT q(0; T) + RT 0ersq0(0; s)ds:(1.29) We …nd q0(0; s)by di¤erentiating equation (1.20) inside the integral. To ease notation, we denote b=x,g(x) = g(xjYt; z0; t)and t= 0, implying that a noise-free report is received one period before. Since g(x)does not depend on s, we only need to di¤erentiate 1(s; b)with respect to syielding @(1 (s; b)) @s =b p2s3exp 1 2(b+ms) ps2!=f(x; s);(1.30) where f(x; s)is the …rst hitting time density of a Brownian motion with drift m and volatility parameter . Therefore, q0(0; s) = Z1 f(x; s)g(x)dx; (1.31) and hence ZT 0 ersq0(0; s)ds =ZT 0 ers Z1 f(x; s)g(x)dxds (1.32) =Z1 g(x)ZT 0 ersf(x; s)dsdx; again since g(x)does not depend on s: The inner integral RT 0ersf(x; s)ds is the integral of a discounted …rst hitting time density known from Reiner & Rubinstein 53
(1991) and Leland & Toft (1996) in closed form as G(x; T) = ZT 0 ersf(x; s)ds (1.33) = exp ((c+z)b) (h1(T)) + exp ((cz)b) (h2(T)) ; where h1(T) = (bz2T) pT;(1.34) h2(T) = (b+z2T) pT;(1.35) c=m 2;(1.36) and z=(m2+ 2r2) 1 2 2:(1.37) In the end, to calculate the CDS spread we only need to evaluate a single integral numerically c(0; T) = r(1 R)R1 G(x; T)g(x)dx 1erT q(0; T)R1 G(x; T)g(x)dx (1.38) =r(1 R)R1 G(x; T)g(x)dx 1erT R1 (1 (T; x )) g(x)dx R1 G(x; T)g(x)dx: B The Accounting Transparency Measure The basic idea in Berger et al. (2006) is that when pricing equity, investors perceive a …rm’s permanent earnings as a geometrically weighted average of reported earnings and industry average earnings. Investors put more weight on the …rm’s reported earnings when the accounting transparency is high. Denote e Ej;t as investors’perception of …rm j0spermanent earnings in year t,Ej;t as the …rm’s reported earnings and EI;t as the industry average earnings. Scaling the earnings by …rm asset Aj;t and industry assets AI;t, the permanent earnings perceived by investors is formally written as e Ej;t Aj;t1 =Ej;t Aj;t1EI;t AI;t11 ;(1.39) 54
where 2[0;1] is the weight put on …rm-speci…c information. Taking logarithms and …rst-order di¤erences yields eej;t =ej;t + (1 )eI;t + (1 )ln Aj;t1 AI;t1ln Aj;t2 AI;t2:(1.40) Lower case letters denote the log-growth rate of the variable eej;t = ln e Ej;t e Ej;t1, ej;t = ln Ej;t Ej;t1,eI;t = ln EI;t EI;t1and Aj;t AI;t represents the …rm’s share of the industry assets. Assuming this share does not change much from year t2to t1, we approximately have eej;t =ej;t + (1 )eI;t:(1.41) The equity price Pj;t is determined by investors’ perception of permanent earnings, and with the assumption of a constant cost of capital jand a constant expected growth rate gj, we have Pj;t =e Ej;t jgj :(1.42) Hence, a …rm’s equity return equals its permanent earnings growth rate rj;t = eej;t, implying that the idiosyncratic variance of the return must equal the idiosyncratic variance of the perceived permanent earnings. Idiosyncratic is de…ned relative to the industry, and the following relations between …rm and industry returns and between …rm and industry earnings, respectively, are assumed rj;t =eej;t =r+rI;t +"r j;t (1.43) ej;t =ae+beI;t +"e j;t:(1.44) Finally, using equations (1.41), (1.43) and (1.44), the idiosyncratic variance of the perceived earnings growth equals 2times the idiosyncratic variance of the reported earnings growth var("r j) = 2var("e j);(1.45) and the measure of accounting transparency is calculated as the idiosyncratic volatility of equity returns divided by the idiosyncratic volatility in earnings 55
growth =vol("r j) vol("e j):(1.46) 56
Chapter 2 Capital Structure Arbitrage: Model Choice and Volatility Calibration 57
Coauthored with Peter Tind Larsen, School of Economics and Management, University of Aarhus Abstract1 When identifying relative value opportunities across credit and equity markets, the arbitrageur faces two major problems, namely positions based on model misspeci…cation and mismeasured inputs. Using credit default swap data, this paper addresses both concerns in a convergence-type trading strategy. In spite of di¤erences in assumptions governing default and calibration, we …nd the exact structural model linking the markets second to timely key inputs. Studying an equally-weighted portfolio of all relative value positions, the excess returns are insigni…cant when based on a historical volatility. However, relying on an implied volatility from equity options results in highly signi…cant excess returns. The gain is largest in the speculative grade segment, and cannot be explained from systematic market risk factors. Although the strategy may seem attractive at an aggregate level, positions on individual obligors can be very risky. 1We thank Lombard Risk for access to the credit default swap data. We are grateful to Peter Løchte Jørgensen, David Lando, Hayne Leland, Svein-Arne Persson, Stephen Schaefer, Ilya Strebulaev, Carsten Sørensen, participants at the C.R.E.D.I.T. 2006 Doctoral Tutorial in Venice, the Danish Doctoral School of Finance Workshop 2007, a Credit Risk Workshop at Aarhus School of Business, the Nordic Finance Network Workshop 2007 in Helsinki, the European Financial Management Association’s 2007 meeting in Vienna and seminar participants at University of Aarhus and Aarhus School of Business for useful discussions and comments. Any remaining errors are our own. 58
2.1 Introduction Capital structure arbitrage refers to trading strategies that take advantage of the relative mispricing across di¤erent security classes traded on the same capital structure. As the exponential growth in the credit default swap (CDS) market has made credit much more tradable and traditional hedge fund strategies have su¤ered declining returns (Skorecki (2004)), important questions arise for hedge funds and proprietary trading desks. In particular, do credit and equity markets ever diverge in opinion on the quality of an obligor? What is the risk and return of exploiting divergent views in relative value strategies? Although trading strategies founded in a lack of synchronicity between equity and credit markets have gained huge popularity in recent years (Currie & Morris (2002) and Zuckerman (2005)), the academic literature addressing capital structure arbitrage is very sparse. This paper conducts a comprehensive analysis of the risk and return of capital structure arbitrage using CDS data on 221 North American obligors in 2002 to 2004. When looking at one security in order to signal the sale or purchase of another, the resulting link and initiation of a trade depends on the chosen model relating the markets. We address two major problems facing the arbitrageur, namely relative value opportunities driven by model misspeci…cation or mismeasured inputs. Duarte, Longsta¤ & Yu (2005) analyze traditional …xed income arbitrage strategies such as the swap spread arbitrage, but also brie‡y address capital structure arbitrage. Yu (2006) cites a complete lack of evidence in favor of or against strategies trading equity instruments against CDSs. Hence, he conducts the …rst analysis of the strategy by implementing the industry benchmark CreditGrades using a historical volatility, which is a popular choice among professionals.2 We show that the more comprehensive model by Leland & Toft (1996) only adds an excess return of secondary order. However, when exploiting a wider array of inputs and securities in model calibration and identi…cation of relative value opportunities, the result is a substantial improvement in strategy execution and returns. 2That CreditGrades is the preferred framework among professionals is argued in Currie & Morris (2002) and Yu (2006), while the CreditGrades Technical Document by Finger (2002) advocates for the 1000-day historical volatility. 59
where Nis the number of shares outstanding.5Hence, tis de…ned as the dollaramount of shares bought per dollar notional in the CDS. The choice of underlying model-framework and calibration is discussed in section 2.4. 2.2.2 Implementation of the Strategy Using the market value of equity, an associated volatility measure and the liability structure of the obligor, the arbitrageur uses a structural model to gauge the richness and cheapness of the CDS spread. Comparing the daily spread observed in the market with the equity-implied spread from the model, the model helps identify credits that either o¤er a discount against equities or trade at a very high level. If e.g. the market spread at a point in time has grown substantially larger than the model spread, the arbitrageur sees an opportunity. It might be that the credit market is gripped by fear and the equity market is more objective. Alternatively, he might think that the equity market is slow to react and the CDS spread is priced fairly. If the …rst view is correct, he should sell protection and if the second view is correct, he should sell equity. Either way, the arbitrageur is counting on the normal relationship between the two markets to return. He therefore takes on both short positions and pro…ts if the spreads converge. In the opposite case with a larger model spread, the arbitrageur buys protection and equity. This relative value strategy is supposed to be less risky than a naked position in either market, but is of course far from a textbook de…nition of arbitrage. Two important caveats to the strategy are positions initiated based on model misspeci…cation or mismeasured inputs. Such potential false signals of relative mispricing are exactly what this paper addresses. We conduct a simulated trading exercise based on this idea across all obligors. Letting be the trading trigger, c0 tthe CDS spread observed in the market at date tand ctshort-hand notation for the equity-implied model spread, we initiate 5This calculation deviates slightly from the one in Yu (2006), since we formulate all models on a total value basis and not per share. Equation (2.8) follows from a simple application of the chain rule. 66
a trade each day if one of the following conditions are satis…ed c0 t>(1 + )ctor ct>(1 + )c0 t:(2.9) In the …rst case, a CDS with a notional of $1 and shares worth $t1are shorted.6In the second case, the arbitrageur buys a CDS with a notional of $1 and buys shares worth $t1as a hedge. Since Yu (2006) …nds his results insensitive to daily rebalancing of the equity position, we follow his base case and adopt a static hedging scheme. The hedge ratio in equation (2.8) is therefore …xed throughout the trade and based on the model CDS spread ctwhen entering the position. Knowing when to enter positions, the arbitrageur must also decide when to liquidate. We assume that exit occurs when the spreads converge de…ned as ct=c0 tor by the end of a pre-speci…ed holding period, which ever comes …rst. In principle, the obligor can also default or be acquired by another company during the holding period. Yu (2006) notes that in most cases the CDS market will re‡ect these events long before the actual occurrences, and the arbitrageur will have ample time to make exit decisions.7Speci…cally, it is reasonable to assume that the arbitrageur will be forced to close his positions once the liquidity dries up in the underlying obligor. Such incidents are bound to impose losses on the arbitrageur. 2.2.3 Trading returns The calculation of trading returns is fundamental to analyze how the risk and return di¤er across model assumptions and calibration methods. Since the CDS position has a zero market value at initiation, trading returns must be calculated by assuming that the arbitrageur has a certain level of initial capital. This assumption allows us to hold …xed the e¤ects of leverage on the analysis. The initial capital is used to …nance the equity hedge, and is credited or deducted as a result of intermediate payments such as dividends or CDS premia. Each trade 6tis, of course, negative. 7This argument seems to be supported in Arora, Bohn & Zhu (2005), who study the surprise e¤ect of distress announcements. Conditional on market information, they …nd only 11 percent of the distressed …rms’equities and 18 percent of the distressed bonds to respond signi…cantly. The vast majority of prices are found to re‡ect the credit deterioration well before the distress announcement. 67
is equipped with this initial capital and a limited liability assumption to ensure well-de…ned returns. Hence, each trade can be thought of as an individual hedge fund subject to a forced liquidation when the total value of the portfolio becomes zero.8 Through the holding period the value of the equity position is straightforward, but the value of the CDS position has to be calculated using equation (2.7) and market CDS spreads c0(t; T)and c0(0; T). Since secondary market trading is very limited in the CDS market and not covered by our dataset, we adopt the same simplifying assumption as Yu (2006), and approximate c0(t; T)with c0(t; t +T). That is, we approximate a CDS contract maturing in four years and ten months, say, with a freshly issued 5-year spread. This should not pose a problem since the di¤erence between to points on the curve is likely to be much smaller than the time-variation in spreads. Yu (2006) …nds his results insensitive to the exact size of transaction costs for trading CDSs. We adopt his base case, and assume a 5 percent proportional bidask spread on the CDS spread. The CDS market is likely to be the largest single source of transaction costs for the arbitrageur. We therefore ignore transaction costs on equities, which is reasonable under the static hedging scheme. 2.3 Data Data on CDS spreads is provided by the ValuSpread database from Lombard Risk Systems, dating back to July 1999. This data is also used by Lando & Mortensen (2005) and Berndt, Jarrow & Kang (2006). The data consists of midmarket CDS quotes on both sovereigns and corporates, with varying maturity, restructuring clause, seniority and currency. For a given date, reference entity and contract speci…cation, the database reports a composite CDS quote together with an intra-daily standard deviation of collected quotes. The composite quote is calculated as a mid-market quote by obtaining quotes from up to 25 leading market makers. This o¤ers a more reliable measure of the market spread than using a single source, and the standard deviation measures how representative the mid-market quote is for the overall market. 8This is reminiscent of potential large losses when marked to market, triggering margin calls and forcing an early liquidation of positions. 68
We con…ne ourselves to 5-year composite CDS quotes on senior unsecured debt for North American corporate obligors with currencies denominated in US dollars. Indeed, the 5-year maturity is the most liquid point on the credit curve (see e.g. Blanco et al. (2005)). Regarding the speci…cation of the credit event, we follow Yu (2006) and large parts of the literature in using contracts with a modi…ed restructuring clause. The frequency of data on CDS quotes increases signi…cantly through time, re‡ecting the growth and improved liquidity in the market. To generate a subsample of the data suitable for capital structure arbitrage, we apply several …lters. First, we merge the CDS data with quarterly balance sheet data from Compustat and daily stock market data from CRSP. The quarterly balance sheet data is lagged one month from the end of the quarter to avoid the look-ahead bias in using data not yet available in the market. We then exclude …rms from the …nancial and utility sector. Second, for each obligor in the sample, daily data on the 30-day at-the-money put-implied volatility is obtained from OptionMetrics. OptionMetrics is a comprehensive database of daily information on exchange-listed equity options in the U.S. since 1996. OptionMetrics generates the 30-day at-the-money put-implied volatility by interpolation. Third, in order to conduct the simulated trading exercise, a reasonably continuous time-series of CDS quotes must be available. In addition, the composite quote must have a certain quality. Therefore, we de…ne the relative quote dispersion as the intra-daily standard deviation of collected quotes divided by the mid-market quote. All daily mid-market quotes with an intra-daily quote dispersion of zero or above 40 percent are then deleted.9For each obligor, we next search for the longest string of more than 100 daily quotes no more than 14 calender days apart, which have all information available on balance sheet variables, equity market and equity options data.10 As noted in Yu (2006), this should also yield the most liquid part of coverage for the obligor, forcing the arbitrageur to 9One could argue for a cut-o¤ point at a lower relative dispersion, but on the other hand a trader is likely to take advantage of high uncertainty in the market. The vast majority of quotes have a relative dispersion below 20 percent. 10As discussed below, this may give rise to a survivorship issue. However, we try to minimize this by requiring a string of only 100 spreads, far less than Yu (2006). In any case, this should not pose a problem, since the focus of the paper is on relative risk and return across models and calibration methods, and not absolute measures. 69
close his positions once the liquidity vanishes. Finally, the 5-year constant maturity treasury rate and the 3-month treasury bill rate are obtained from the Federal Reserve Bank of St. Louis. The 5-year interest rate is used to calculate the equity-implied 5-year CDS spread, while the 3-month interest rate is chosen when calculating daily excess returns from the trading strategy11. Applying this …ltration to the merged dataset results in 221 obligors with 65,476 daily composite quotes, dating back to July 2002 and onwards to the end of September 2004. Table 2.1 presents summary statistics for the obligors across the senior unsecured credit rating from Standard & Poor’s when entering the sample. The variables presented are averages over time and then …rms. The majority of …rms are BBB rated, and 16 …rms are in the speculative grade segment, including one non-rated obligor. A lower spread is associated with a lower leverage and volatility, which is in line with predictions of structural credit risk models. We implement the trading strategy using the implied volatility from equity options (IV), and a 250-day volatility from a historical time-series of equity values (HV). On average these volatilities are similar, but it turns out that the dynamics of option prices provide the arbitrageur with superior information. The average correlation between changes in the spread and the equity value is negative as expected from a structural viewpoint, but fairly low. This is consistent with Yu (2006) and correlations ranging from minus 5 to minus 15 percent quoted by traders in Currie & Morris (2002). This indicates that the two markets may drift apart and hold divergent views on obligors, which fuels the arbitrageur ex ante. Ex post, it suggests that the equity hedge may be ine¤ective. 11This choice of short-term interest rate is consistent with Yu (2006). Changes in shorter maturity rates are to a larger extend driven by idiosyncratic variation (see Dufee (1996)). 70
Table 2.1: Sample Characteristics This table reports sample characteristics for the 221 obligors. First, the average characteristics are calculated for each obligor over time, then averaged across …rms. The statistics are presented across the senior unsecured credit rating from Standard & Poor’s. Nis the number of obligors and spread is the 5-year composite CDS quote. While the historical equity volatility HV is calculated from a 250-day rolling window of equity returns, the implied equity volatility IV is inferred from 30-day at-the-money put options. The leverage ratio lev is total liabilities divided by the sum of total liabilities and equity market capitalization, and size is the sum of total liabilities and equity market capitalization in millions of dollars. Finally, corr is the correlation between changes in the CDS spread and the equity value, averaged across ratings. Rating N Spread HV IV Lev. Size Corr. AAA 4 16 0.284 0.227 0.197 142,619 -0.107 AA 11 23 0.267 0.257 0.216 95,237 -0.050 A 80 40 0.305 0.293 0.354 40,274 -0.089 BBB 109 103 0.346 0.337 0.502 25,431 -0.124 BB 15 270 0.386 0.377 0.524 13,667 -0.056 B 1 355 0.554 0.555 0.564 34,173 -0.261 NR 1 172 0.229 0.219 0.450 11,766 -0.129 2.4 Model Choice and Volatility Calibration Having the trading strategy and data explained, next we introduce the two underlying models and the associated calibration. The formulas for each model including the risk-neutral survival probability qt(s), the CDS spread c(0; T), the contract value (t; T)and the equity delta (t; T)are described in the appendix. Further details on the models can be found in Finger (2002) and Leland & Toft (1996). 2.4.1 CreditGrades The CreditGrades model is jointly developed by RiskMetrics, JP Morgan, Goldman Sachs and Deutsche Bank with the purpose to establish a simple framework linking credit and equity markets. As noted by Currie & Morris (2002) and Yu (2006), this model has become an industry benchmark widely used by traders, preferably calibrated with a rolling 1000-day historical volatility as advocated in 71
Finger (2002). It loosely builds on Black & Cox (1976), with default de…ned as the …rst passage time of …rm assets to an unobserved default barrier. Hence, deviating from traditional structural models, it assumes that the default barrier is an unknown constant drawn from a known distribution. This element of uncertain recovery increases short-term spreads, but cannot do so consistently through time.12 Originally, the model is built on a per-share basis taking into account preferred shares and the di¤erences between short-term versus long-term and …nancial versus non-…nancial obligations, when calculating debt per share. Like Yu (2006), we only work with total liabilities and common shares outstanding. Therefore, we formulate the model based on total liabilities and market value of equity. Under the risk-neutral measure, the …rm assets Vare assumed to follow dVt=VVtdWt, (2.10) where Vis the asset volatility and Wtis a standard Brownian motion. The zero drift is consistent with the observation of stationary leverage ratios in Collin- Dufresne & Goldstein (2001). The default barrier is LD, where Lis a random recovery rate given default, and Ddenotes total liabilities. The recovery rate Lfollows a lognormal distribution with mean L, interpreted as the mean global recovery rate on all liabilities, and standard deviation . Then, Rin equation (2.6) is the recovery rate on the speci…c debt issue underlying the CDS. Instead of working with a full formula for the value of equity S; CreditGrades uses the linear approximation V=S+ LD, (2.11) which also gives a relation between asset volatility Vand equity volatility S V=S S S+ LD:(2.12) The model is easy to implement in practice. In particular, Dis the total liabilities from quarterly balance sheet data, Sis the market value of equity calculated as the number of shares outstanding multiplied by the closing price, 12A theoretically more appealing approach is given by Du¢ e & Lando (2001). 72
and ris the 5-year constant maturity treasury yield. Furthermore, the bondspeci…c recovery rate Ris assumed to be 0:5and the standard deviation of the global recovery rate is 0:3. All parameters are motivated in Finger (2002) and Yu (2006). The volatility measure is a key input to the pricing of credit. Instead of using a rolling 1000-day volatility Sfrom historical equity values as Yu (2006), we implement the strategy using a 250-day historical volatility and the implied volatility from equity options. According to Cremers et al. (2006) and Cao et al. (2006), the implied volatility contains important and timely information about credit risk di¤erent from the historical measure. This may potentially lead the arbitrageur to superior entry and exit decisions and trading returns. We expect the gain to be mostly pronounced for the speculative grade sample, where obligors typically experience large variations in spreads. Here, historical volatilities may lag true market levels and send a false signal of mispricing to the arbitrageur. Finally, we follow Yu (2006) in using the mean global recovery rate Lto align the model with the credit market before conducting the trading exercise. In particular, we infer Lby minimizing the sum of squared pricing errors using the …rst 10 CDS spreads in the sample for each …rm. Now, all parameters are in place to calculate the time-series of CDS spreads underlying the analysis, together with hedge ratios and values of open CDS positions. 2.4.2 Leland & Toft (1996) This model assumes that the decision to default is made by a manager, who acts to maximize the value of equity. At each moment, the manager must address the question if meeting promised debt service payments is optimal for the equityholders, thereby keeping their call option alive. If the asset value exceeds the endogenously derived default barrier VB, the …rm will optimally continue to service the debt - even if the asset value is below the principal value or if cash ‡ow available for payout is insu¢ cient to …nance the net debt service, requiring additional equity contributions. In particular, …rm assets Vare assumed to follow a geometric Brownian motion under the risk-neutral measure dVt= (r)Vtdt +VVtdWt, (2.13) 73
where ris the constant risk-free interest rate, is the fraction of asset value paid out to security holders, Vis the asset volatility and Wtis a standard Brownian motion. Debt of constant maturity is continuously rolled over, implying that at any time sthe total outstanding debt principal Pwill have a uniform distribution over maturities in the interval (s; s + ). Each debt contract in the multi-layered structure is serviced by a continuous coupon. The resulting total coupon payments Care tax deductible at a rate , and the realized costs of …nancial distress amount to a fraction of the value of assets in default VB. Rolling over …nite maturity debt in the way prescribed implies a stationary capital structure, where the total outstanding principal P, total coupon C, average maturity 2and default barrier VBremain constant through time. To determine the total value of the levered …rm v(Vt), the model follows Leland (1994) in valuing bankruptcy costs BC(Vt)and tax bene…ts resulting from debt issuance TB(Vt)as time-independent securities. It follows, that (Vt) = Vt+TB(Vt)BC(Vt)(2.14) =S(Vt) + D(Vt), where S(Vt)is the market value of equity and D(Vt)the market value of total debt. To implement the model, we follow Ericsson et al. (2006) in setting the realized bankruptcy cost fraction = 0:15, the tax rate = 0:20 and the average debt maturity 2= 3:38.13 Furthermore, as above, Pis the total liabilities from quarterly balance sheet data, Sis the market value of equity and ris the 5-year constant maturity treasury yield. We also follow Ericsson et al. (2006) in assuming that the average coupon paid out to all debtholders equals the risk-free interest rate, C=rP.14 The asset payout rate is calculated as a time-series mean of the weighted average historical dividend yield and relative interest expense from 13The choice of 15 percent bankruptcy costs lies well within the range estimated by Andrade & Kaplan (1998). 20 percent as an e¤ective tax rate is below the corporate tax rate to re‡ect the personal tax rate advantage of equity returns. Stohs & Mauer (1996) …nd an average debt maturity of 3.38 years using a panel of 328 industrial …rms with detailed debt information in Moody’s Industrial Manuals in 1980-1989. 14A …rm’s debt consists of more than market bonds, and usually a substantial fraction of total debt is non-interest bearing such as accrued taxes and supplier credits. Furthermore, corporate bonds may be issued below par, which also opens up for this approximation. 74
balance sheet data =Interest expenses Total liabilities L+ (Dividend yield)(1 L)(2.15) L=Total liabilities Total liabilities +Market equity. Contrary to CreditGrades, the default barrier VBis endogenously determined and varies with fundamental characteristics of the …rm such as leverage, asset volatility, debt maturity and asset payout rates. Due to the full-blown relationship between equity and assets, the estimation of the asset value Vand asset volatility Vis a more troublesome exercise in Leland & Toft (1996). Hence, when analyzing the trading strategy with a 250-day historical volatility, we use the iterative algorithm of Moody’s KMV outlined in Crosbie & Bohn (2003) and Vassalou & Xing (2004) to infer the unobserved time-series of asset values and asset volatility. This iterative algorithm is preferable over an instantaneous relationship between asset volatility Vand equity volatility S, governed by Ito’s lemma. The latter underlies the implementation of CreditGrades in equation (2.12), and is used in Jones et al. (1984). As noted in Lando (2004), the iterative algorithm is particularly preferable when changes in leverage are signi…cant over the estimation period. In short, the iterative scheme goes as follows. The value of equity Stis a function of the asset value Vt, asset volatility Vand a set of parameters in equation (2.31), i.e. St=f(Vt; V; ). We use a 250-day window of historical equity values to obtain an estimate of the equity volatility S, by viewing the value of equity as a geometric Brownian motion. Given this initial estimate of the asset volatility Vand quarterly balance sheet data, we calculate the value of the default barrier. Using the daily market values of equity and the equity pricing formula we then back out an implied time-series of asset values Vt(V) = f1(St; V; ). Next, the daily asset values allow us to obtain an improved estimate of the asset volatility V, which is used in the next iteration. This procedure is repeated until the values of Vconverge. When analyzing the trading exercise based on implied volatilities from equity options, we do not face the problem of changing leverage in a historical estimation 75
2.5.2 Time Warner and Motorola Simulating the trading strategy on Time Warner and Motorola supports the former insights. Figure 2.2 depicts the fundamentals behind Time Warner, rated BBB by S&P and Baa1 by Moody’s. In August 2002 just prior to the beginning of the sample, Moody’s changes their outlook to negative as the SEC investigates the accounting practices and internal controls. As markets recover in late 2002, CreditGrades with historical volatility indicates that protection is cheap relative to equity, while spreads in Leland & Toft (1996) are more neutral. Although equity prices increase throughout 2003, many losing trades are initiated as market spreads are more than cut by half within few months and Moody’s changes their outlook back to stable. Again, the historical volatility lags the market following the episode, while the implied volatility is more responsive. In October and November 2002, where market spreads have already tightened substantially, model spreads inferred from implied volatilities suggest that protection is expensive relative to equity and should tighten further. Selling protection at 339 bps and equity at $14.75 on October 31, 2002 result in convergence and 15 percent returns on December 12, where the CDS and equity are trading at 259 bps and $13.56, respectively. However, spreads inferred from implied volatilities are volatile, resulting in rather noisy estimates of credit outlooks and a frequent liquidation of positions as market spreads tighten. Operating with a very low trigger may reverse positions several times during this period, while a trigger of 0.5 results in only few positions. In Figure 2.3, the key variables for Motorola rated BBB by S&P are depicted. Building on historical volatilities the arbitrageur initiates many trades and su¤ers losses, while implied volatilities suggest the two markets to move in tandem and hold similar views on the obligor. In the latter case, only few relative value opportunities are apparent. 82
Figure 2.2: Time Warner This …gure illustrates the fundamentals behind capial structure arbitrage. In panel A, we depict market CDS spreads together with model spreads in Leland & Toft (1996) LT inferred from historical HV and option-implied volatilities IV . In panel B, the corresponding spreads are depicted based on CreditGrades CG. Panel C depicts the historical and option-implied volatility, where the …rst is calculated from a rolling 250- day window of equity returns and the latter is inferred from 30-day at-the-money puts. Finally, panel D illustrates the total market value of equity in millions of dollars. 83
Figure 2.3: Motorola This …gure illustrates the fundamentals behind capial structure arbitrage. In panel A, we depict market CDS spreads together with model spreads in Leland & Toft (1996) LT inferred from historical HV and option-implied volatilities IV . In panel B, the corresponding spreads are depicted based on CreditGrades CG. Panel C depicts the historical and option-implied volatility, where the …rst is calculated from a rolling 250- day window of equity returns and the latter is inferred from 30-day at-the-money puts. Finally, panel D illustrates the total market value of equity in millions of dollars. 84
2.5.3 Mandalay Resort Group Capital structure arbitrage is very risky when based on individual obligors, and the arbitrageur may end up in severe problems irrespective of model choice and calibration. Figure 2.4 presents the fundamental variables behind Mandalay Resort Group, rated BB by S&P. Throughout the coverage, spreads in Leland & Toft (1996) based on historical volatilities diverge from market spreads in a smooth manner, while spreads in CreditGrades diverge more slowly. In both cases the arbitrageur sells protection and equity as hedge, but su¤ers losses as positions are liquidated after the maximum holding period. Based on implied volatilities, May and June 2004 are particularly painful as model spreads plunge and stay tight throughout the coverage. On June 4, 2004 the competitor MGM Mirage announces a bid to acquire Mandalay Resort Group for $68 per share plus assumption of Mandalay’s existing debt. Moody’s places the rating on review for a possible downgrade due to a high level of uncertainty regarding the level of debt employed to …nance the takeover. As a result, the equity price increases from $54 to $69 over a short period, the implied volatility plunges and the CDS spread widens from 188 bps to 227 bps.17 On June 15, 2004 a revised o¤er of $71 per share is approved, and the transaction is completed on April 26, 2005. This opposite reaction in equity and credit gives the arbitrageur short in both markets a painful one-two punch similar to the one experienced by hedge funds in May 2005, where General Motors is downgraded while the equity price soars.18 Luckily, not many trades are open during the takeover bid as model and market spreads recently converged. However, the short positions initiated in May 2004, where credit seems expensive relative to equity, su¤er large losses on both legs. 17Implied volatilities from at-the-money calls plunge as well. 18This case study is discussed in Duarte et al. (2005). 85
Figure 2.4: Mandalay Resort Group This …gure illustrates the fundamentals behind capial structure arbitrage. In panel A, we depict market CDS spreads together with model spreads in Leland & Toft (1996) LT inferred from historical HV and option-implied volatilities IV . In panel B, the corresponding spreads are depicted based on CreditGrades CG. Panel C depicts the historical and option-implied volatility, where the …rst is calculated from a rolling 250- day window of equity returns and the latter is inferred from 30-day at-the-money puts. Finally, panel D illustrates the total market value of equity in millions of dollars. 86
2.6 General Results In this section, we simulate the trading strategy for all 221 obligors. Following Yu (2006), we assume an initial capital of $0.5 for each trade and $1 notional in the CDS. The strategy is implemented for trading triggers of 0.5 and 2, and maximum holding periods of 30 and 180 days. Naturally, absolute trading returns will vary with the above characteristics, as well as the particular period studied and how to account for vanishing liquidity. However, these characteristics are all …xed when studying the relative risk and return across models and calibration methods. Therefore, a scaling of returns with the amount of initial capital is unlikely to in‡uence our conclusions.19 Indeed, although based on a di¤erent dataset, the benchmark results for CreditGrades with a historical volatility are similar to the …ndings in Yu (2006). Table 2.3 and 2.4 present the summary statistics of holding period returns based on CreditGrades and Leland & Toft (1996), respectively. A longer maximum holding period leads to more converging trades, fewer trades with negative returns and higher average returns. This fundamental result underlies both models and volatility measures. Consistent with Yu (2006), although the distribution of returns becomes less dispersed, a higher trading trigger does not necessarily lead to higher mean returns. When identifying relative value opportunities from implied not historical volatilities, the number of initiated trades rises for investment grade obligors and falls for speculative grade obligors. This results from both models, although the absolute number of trades is larger in Leland & Toft (1996). This is consistent with …ndings in Finger & Stamicar (2005a) and Cao et al. (2006), where the advantage of implied volatility in tracking market spreads with CreditGrades is concentrated among speculative grade obligors. We …nd this measure to identify fewer relative value opportunities on obligors with larger variations in spreads. The results clearly show a di¤erence in risk and return across models and volatility input. Identifying relative value opportunities on speculative grade 19Yu (2006) also conducts his analysis with an initial capital of $0.1. The resulting returns are scaled up accordingly. Unreported results with this initial capital and other trading triggers leave our conclusions unchanged. 87
obligors in CreditGrades with a historical volatility, a maximum holding period of 180 days and a trading trigger of 2 yields a mean holding period return of 2.64 percent. However, simulating the trading strategy with option-implied volatilities increases the return to 4.61 percent.20 The corresponding numbers based on Leland & Toft (1996) are 3.14 and 5.47 percent. The gain from implied volatilities across trading triggers and maximum holding periods is also apparent from the number of trades ending in convergence and the fraction of trades with negative returns. However, the incremental return is much smaller for investment grade obligors. On top of this, the mean holding period return and dispersion are both higher on speculative grade obligors compared to the investment grade sample. This supports the similar result in Yu (2006) and happens irrespective of model choice and volatility measure. Although more likely to su¤er from vanishing liquidity and default, this supports his observation that the aggregate success of the strategy depends on the availability of large variations in spreads. For such obligors, the more timely implied volatility results in incremental trading returns from superior entry and exit decisions. The holding period returns are more favorable when Leland & Toft (1996) is used to identify relative value opportunities. However, in practice it is hard to discern exactly where the di¤erence arises, as the models di¤er in many respects and enter in all parts of the strategy. While model choice does matter, it seems second to properly measured key inputs. 20While the average pro…tability increases when identifying relative value opportunities from implied volatilities, so does the volatility of returns. As the mean holding period return consists of many overlapping holding periods, the statistical signi…cance of trading returns is analyzed from a return index below. 88
Table 2.3: Holding Period Returns Based on CreditGrades This table shows the holding period returns resulting from CreditGrades CG with a historical volatility HV and option-implied volatility IV . The maximum holding period HP is either 30 or 180 days. Trigger denotes the minimum threshold between the market and model spread before positions are initiated. Rating denotes whether the strategy is implemented on investment grade or speculative grade obligors. Nis the number of trades, Nconv the number of trades ending in convergence, and Neg is the percentage of trades ending in negative return. The mean and median returns are in percentages. Model HP Trigger Rating N Nconv Neg. Mean Median Std.dev. Min Max Panel A. CreditGrades Holding Period Returns (30 Days) CG HV 30 0.5 Inv 45,190 789 0.45 0.01 0.04 1.16 -24.39 25.42 0.5 Spec 1,860 0 0.47 0.28 0.08 3.38 -11.07 16.66 CG IV 30 0.5 Inv 53,559 2,787 0.32 0.33 0.13 1.75 -23.22 66.22 0.5 Spec 1,598 231 0.27 2.46 0.74 10.20 -28.57 90.18 CG HV 30 2 Inv 27,212 46 0.41 0.05 0.06 0.84 -9.09 25.42 2 Spec 824 0 0.38 0.84 0.47 2.76 -6.63 12.41 CG IV 30 2 Inv 46,179 727 0.32 0.28 0.13 1.38 -17.80 38.64 2 Spec 380 19 0.11 2.06 1.36 4.96 -3.31 89.76 Panel B. CreditGrades Holding Period Returns (180 Days) CG HV 180 0.5 Inv 45,190 7,088 0.40 -0.08 0.15 3.10 -36.25 35.88 0.5 Spec 1,860 3 0.43 0.17 0.38 5.50 -24.02 15.18 CG IV 180 0.5 Inv 53,559 6,231 0.27 1.13 0.39 3.58 -25.18 89.74 0.5 Spec 1,598 665 0.18 6.58 2.08 16.83 -28.46 124.99 CG HV 180 2 Inv 27,212 1,488 0.30 0.26 0.26 2.00 -26.99 14.84 2 Spec 824 0 0.16 2.64 1.48 3.76 -6.42 15.18 CG IV 180 2 Inv 46,179 2,504 0.27 0.95 0.38 3.14 -12.87 58.71 2 Spec 380 103 0.08 4.61 2.61 8.12 -0.63 100.52 89
Table 2.4: Holding Period Returns Based on Leland & Toft This table shows the holding period returns resulting from Leland & Toft (1996) LT with a historical volatility HV and optionimplied volatility IV . The maximum holding period HP is either 30 or 180 days. Trigger denotes the minimum threshold between the market and model spread before positions are initiated. Rating denotes whether the strategy is implemented on investment grade or speculative grade obligors. Nis the number of trades, Nconv the number of trades ending in convergence, and Neg is the percentage of trades ending in negative return. The mean and median returns are in percentages. Model HP Trigger Rating N Nconv Neg. Mean Median Std.dev. Min Max Panel A. Leland & Toft Holding Period Returns (30 Days) LT HV 30 0.5 Inv 50,196 1,857 0.41 0.15 0.07 1.39 -19.05 38.39 0.5 Spec 2,496 81 0.33 0.99 0.61 4.35 -40.44 94.04 LT IV 30 0.5 Inv 54,708 3,305 0.32 0.31 0.13 1.64 -31.46 65.21 0.5 Spec 2,438 337 0.27 2.48 0.88 10.04 -37.16 106.59 LT HV 30 2 Inv 36,673 260 0.38 0.13 0.08 0.99 -16.06 17.05 2 Spec 1,598 4 0.31 0.86 0.59 2.37 -14.00 13.59 LT IV 30 2 Inv 45,139 1,239 0.32 0.24 0.12 1.22 -21.61 21.55 2 Spec 1,370 101 0.21 2.76 0.88 11.17 -16.73 106.59 Panel B. Leland & Toft Holding Period Returns (180 Days) LT HV 180 0.5 Inv 50,196 7,964 0.35 0.49 0.22 3.01 -86.92 55.60 0.5 Spec 2,496 226 0.21 2.90 1.67 5.96 -20.39 94.04 LT IV 180 0.5 Inv 54,708 7,966 0.28 1.07 0.37 3.62 -47.01 89.32 0.5 Spec 2,438 673 0.13 7.02 2.41 17.20 -39.32 147.05 LT HV 180 2 Inv 36,673 2,428 0.31 0.42 0.25 2.02 -86.92 42.43 2 Spec 1,598 11 0.16 3.14 1.67 4.16 -14.00 16.22 LT IV 180 2 Inv 45,139 3,921 0.28 0.82 0.34 2.94 -39.70 49.71 2 Spec 1,370 194 0.09 5.47 2.26 12.66 -2.94 117.00 90
2.6.1 Capital Structure Arbitrage Index Returns As illustrated in the previous sections, capital structure arbitrage is very risky at the level of individual trades. The hedge may be ine¤ective and the markets may continue to diverge, resulting in losses and potential early liquidations. However, when initiated on the cross-section of obligors, the strategy may be pro…table on average depending on the particular implementation. Having established this …nding, the next step is to understand the sources of the pro…ts, i.e. whether the returns are correlated with priced systematic risk factors. Hence, we construct a monthly capital structure arbitrage excess return index from all individual trades, following Duarte et al. (2005) and Yu (2006). Speci…cally, we compute daily excess returns for all individual trades over the entire holding period. On a given day, thousands of trades may be open. By essentially assuming that the arbitrageur is always invested in an equally-weighted portfolio of hedge funds, where each fund consists of one trade, we calculate an equally-weighted average of the excess returns on a daily basis. These average daily excess returns are then compounded into a monthly frequency. Table 2.5 presents the summary statistics of monthly excess returns based on a maximum holding period of 180 days, covering 24 months in 2002-2004. However, some strategies result in months with no trades. In this case, a zero excess return is assumed. Again, although also present in the investment grade segment, the bene…t of option-implied volatilities is concentrated among speculative grade obligors. Additionally, timely inputs are relatively more important than the exact structural model underlying the strategy. In particular, when based on CreditGrades with option-implied volatilities and a trading trigger of 2, the mean excess return is 0.44 percent on investment grade and 1.33 percent on speculative grade obligors. These numbers are highly signi…cant after correcting for serial correlation. The corresponding numbers when Leland & Toft (1996) is used to identify relative value opportunities are 0.27 and 2.39 percent, respectively, both highly signi…- cant. The excess returns resulting from a historical volatility are much smaller and most often insigni…cant. Indeed, the mean excess return from this measure may turn negative and signi…cant at a lower trading trigger of 0.5, while it continues to be positive and signi…cant based on implied volatilities. 91
where H(T) = er (G(T+)G()) , (2.23) G(T) = dz+1=2ln d VpTzVpT +dz+1=2ln d VpT+zVpT, (2.24) =2 2 V , (2.25) z=s1 4+2r 2 V , (2.26) and G(T)is given in Reiner & Rubinstein (1991). When determining the hedge ratio, we follow Yu (2006) and approximate the contract value in equation (2.7) by (0; T)=(c(0; T)c)ZT 0 ersq(s)ds (2.27) =c(0; T)c rq(0) q(T)erT H(T), where c(0; T)is a function of the value of equity Sin equation (2.22), and cis the CDS spread at initiation.23 Using equation (2.8) and the product rule, the hedge ratio is found as 0=Nd (0; T) dS =N r @c (0; T) @S q(0) q(T)erT H(T), (2.28) where Ndenotes the number of shares outstanding. The second term in the product rule is zero, since by de…nition cis numerically equal to c(0; T), evaluated at the equity value S. Finally, @c(0;T) @S is found numerically. 23Yu (2006) interprets this equation in his appendix. Equation (2.27) represents the value of a contract entered into one instant ago at spread c, that now has a quoted spread of c(0; T) due to a change in the value of equity. 98
A.2 Leland & Toft (1996) Equation (2.14) may be written as (Vt) = Vt+C r 1Vt VBx!VBVt VBx , (2.29) with the value of debt D(Vt) D(Vt) = C r+PC r1er rI()+(1 )VBC rJ() , (2.30) and equity S(Vt) S(Vt) = Vt+C r 1Vt VBx!VBVt VBx C rPC r1er rI()(2.31) (1 )VBC rJ() , and default barrier VB VB= C rA rBAP rCx r 1 + x (1 )B. (2.32) The components of the above formulae are A= 2aeraVp2zzVp(2.33) 2 VpzVp+2er VpaVp+ (za), B=2z+2 z2 VzVp(2.34) 2 VpzVp+ (za) + 1 z2 V, I() = 1 rK() erF(), (2.35) K() = V VBa+z (j1()) + V VBaz (j2()) , (2.36) 99
F() = (h1()) + V VB2a (h2()) , (2.37) J() = 1 zVp V VBa+z (j1()) j1() +V VBaz (j2()) j2()!, (2.38) j1() = (bz2 V) Vp;j2() = (b+z2 V) Vp, (2.39) h1() = (ba2 V) Vp;h2() = (b+a2 V) Vp, (2.40) a=(r(2 V=2)) 2 V , (2.41) b= ln Vt VB, (2.42) z=r(a2 V)2+ 2r2 V 2 V , (2.43) x=a+z: (2.44) ()and ()denote the density of the standard normal distribution and the cumulative distribution function, respectively. The CDS Spread and Hedge Ratio Using equation (2.37), the risk-neutral survival probability at horizon tis q(t) = 1 F(t)(2.45) = 1 (h1(t)) + V VB2a (h2(t))!: Assuming constant interest rates, the CDS spread for maturity Tis found by inserting the survival probability (2.45) in equation (2.6), yielding 0 = c(0; T)ZT 0 ersq(s)ds + (1 R)ZT 0 ersq0(s)ds: (2.46) 100
Integrating the …rst term by parts, yields 0 = c(0; T) r1erT q(T) + ZT 0 ersq0(s)ds+ (1 R)ZT 0 ersq0(s)ds, (2.47) where the integral RT 0ersq0(s)ds is given by K(T)in equation (2.36), following Reiner & Rubinstein (1991). Then, 0 = c(0; T) r1erT q(T)c(0; T) r+ (1 R)K(T), (2.48) which allows us to obtain a closed-form solution for the CDS spread c(0; T) = r(1 R)K(T) (1 erT q(T)K(T)):(2.49) When determining the hedge ratio, we again follow Yu (2006) and approximate the contract value in equation (2.7) by (0; T)=(c(0; T)c)ZT 0 ersq(s)ds: (2.50) =c(0; T)c r1erT q(T)K(T), where c(0; T)is a function of the value of equity S, and cis the CDS spread at initiation. Similar to CreditGrades, the hedge ratio is found using equation (2.8) 0=N r @c (0; T) @S 1erT q(T)K(T). (2.51) However, in Leland & Toft (1996) the CDS spread is not an explicit function of the equity value. Therefore, @c(0;T) @S is found numerically using @c (0; T) @S =@c (0; T) @V @V @S =@c (0; T) @V 1 @S @V :(2.52) 101
102
Chapter 3 Credit Risk Premia in the Market for Credit Default Swaps 103
Abstract1 This paper estimates the time-series behavior of credit risk premia in the market for Credit Default Swaps for the period 2001 to 2006. A structural model is used to back out objective default probabilities. The results indicate that risk premia might be incorrectly estimated, when expected losses are based on a historical equity volatility measure as opposed to implied volatility. This e¤ect is largest following the peak in credit spreads and risk premia in the second half of 2002. Secondly, when default probabilities are based on implied volatility, the risk premia tend to be countercyclical in the sense that the risk premium is high when expected losses are high. Finally, using linear regressions, I …nd that augmenting the set of variables predicted by structural models with equity-implied credit risk premia signi…cantly increases the explanatory power. This echoes the results found in Elkamhi & Ericsson (2007) and suggests the need for time varying risk premia in structural models. 1I thank MarkIt for access to credit default swap data. I am grateful to David Lando and Jesper Rangvid for useful comments. All remaining errors are my own. 104
3.1 Introduction This paper estimates the time-series behavior of credit risk premia in the market for Credit Default Swaps (CDS) for the period 2001 until the end of 2006. More speci…cally the structural model by Leland & Toft (1996) is used to back out objective default probabilities from the equity market, and the market CDS spread is then decomposed into an expected loss component and a risk premium component. Not much empirical work has been done on the time variation of risk premia in credit markets. Berndt, Douglas, Du¢ e, Ferguson & Schranz (2005) and Berndt, Lookman & Obreja (2006) use expected default frequencies (EDF) from Moody’s KMV together with CDS spreads to extract historical and risk neutral default intensities respectively. The ratio of these is interpreted as a measure of the default risk premium observed in the marketplace. They document substantial time-series variation in risk premia for the period from 2000-2004 with a peak in the third quarter of 2002 and a subsequent dramatic drop. Elkamhi & Ericsson (2007) develop a methodology to study the linkages between equity and corporate bond risk premia and apply it to a panel of corporate bond transactions data for the period 1995 - 2005. They …nd a time-series behavior and degree of time variation in credit risk premia similar to Berndt et al. (2005), although their study is based on di¤erent data, a di¤erent …nancial instrument and a di¤erent methodology. An obvious problem when estimating credit risk premia is the measurement of objective default probabilities and expected losses. Elkamhi & Ericsson (2007) base the default probabilities on a historical volatility measure, while Berndt et al. (2005) and Berndt, Lookman & Obreja (2006) use the EDF measure, and their estimated default probabilities are thus essentially also based on historical volatility2. I contribute to the existing literature by applying the methodology developed in Elkamhi & Ericsson (2007) to a large panel of CDS quotes, but contrary to them I also back out the default probabilities using option implied volatility. This should give a better view of the uncertainty in the market, especially when the uncertainty changes rapidly. According to Cremers et al. (2006) and Cao et al. (2006), the implied volatility contains important and timely in- 2See Crosbie & Bohn (2003) and Berndt et al. (2005) for a discussion of the EDF measure. 105
formation about credit risk di¤erent from the historical measure, while Finger & Stamicar (2005a) and Finger & Stamicar (2005b) show how model spreads based on historical volatilities lag the market when spreads increase, while overpredicting the market as spreads recover. The computation of objective default probabilities is done as in Elkamhi & Ericsson (2007) by estimating …rm speci…c equity risk premia using the Fama & MacBeth (1973) approach, and then the equity risk premia are "delevered" into asset value risk premia. The measure of the credit risk premium is then the part of the CDS spread in excess of the expected loss component. Although a close relation exists between corporate bonds and CDS spreads (Du¢ e (1999)), the latter are preferable from several perspectives. The use of CDS spreads avoids any noise arising from a misspeci…ed risk-free yield curve (Houweling & Vorst (2003)) and several recent studies …nd that CDS spreads are a purer measure of credit risk compared to corporate bond credit spreads3. Furthermore, while the corporate bonds used in Elkamhi & Ericsson (2007) are of di¤erent maturity and coupon, all of the CDS spreads in this paper have a 5-year maturity and are e¤ectively new par-coupon credit spreads on the underlying …rm. The results are also expected to be more robust compared to Elkamhi & Ericsson (2007) since the data in this paper are larger in the cross section, and the same …rms are followed over time. I …nd that the estimated credit risk premia appear more volatile when default probabilities and expected losses are based on the historical volatility measure compared to implied volatility. Similar to earlier results I …nd that the risk premia peak in the third quarter of 2002, but the subsequent drop in risk premia is not as dramatic, when expected losses are based on implied volatility. Furthermore there is a high degree of uncertainty in the option market in the second half of 2002 as measured by the implied volatility. This result is consistent across industries and ratings (investment grade and speculative grade), and suggests that it may be inappropriate to base expected losses on a historical volatility measure, when estimating credit risk premia. Secondly, when expected losses are based on implied volatility, the credit risk premium is high in times of high default probabilities and low in times of low default probabilities. This suggests that the credit risk premium is countercycli- 3See e.g. Longsta¤ et al. (2005). 106
cal. Furthermore the expected loss ratio and the risk premium ratio behave quite di¤erently from one another over time. Interestingly, when based on implied volatility, the expected loss ratio peaks in late 2002, when credit spreads soared and the credit risk premium peaked. The expected loss ratio is actually higher than 50% at certain points in this period. On the other hand the risk premium ratio tends to be high in times of low credit spreads and low default probabilities. Thirdly I show that there is a close relation between VIX and expected losses, when the asset volatilities are based on implied volatility. Earlier papers such as Collin-Dufresne et al. (2001) and Schaefer & Strebulaev (2004) have showed that VIX is an important explanatory variable for changes in credit spreads, although they do not pin down an explanation for the role of VIX, while Berndt et al. (2005) …nd that VIX is related to the credit risk premium. The results of this paper suggest that VIX is indeed a measure of systematic volatility. Finally I carry out a regression analysis similar to Ericsson et al. (2005), Collin- Dufresne et al. (2001) and Campbell & Taksler (2003). A benchmark regression is performed including standard variables implied from structural models. Augmenting the regressions with an equity implied measure of the credit risk premium improves the explanatory power for the levels of the credit spread, while the co- e¢ cient on this purely model implied risk premium is highly signi…cant. With the historical volatility as part of the variables in the regressions the R-square is 49:4%, and it increases by 3% to 52:4%, when the risk premium is included, while the R-square increases by 5:5% from 57:4% to 62:9%, when the regressions are performed with implied volatility. The increase in the explanatory power is substantially higher, when the risk premium is included for the investment grade segment compared to the speculative grade segment.This suggests that investment grade …rms have proportionally higher risk premia and that risk premia are more important for investment grade …rms than for speculative grade …rms. Similar results are found in Huang & Huang (2003) and Berndt et al. (2005). The regression results echo results in Elkamhi & Ericsson (2007), and combined with the other results of the paper, it suggests that structural models should contain a time varying and countercyclical risk premium. The results also suggest a link between equity risk premia and credit spreads, when the equity risk premium is properly translated to the credit risk premium through a structural model. This is 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 107
Table 3.1: Sample Characteristics This table reports sample characteristics for the 142 obligors. The sample characteristics are averages over the number of quotes. The statistics are presented across the senior unsecured credit rating from Standard & Poor’s. Nis the number of quotes and spread is the avererage 5-year CDS quote. While the historical equity volatility HV is calculated from a 250-day rolling window of equity returns, the implied equity volatility IV is inferred from 30-day at-the-money put options. The leverage ratio Lev is total liabilities divided by the sum of total liabilities and equity market capitalization, and size is the sum of total liabilities and equity market capitalization in billions of dollars. Equity prem: is the estimated equity premium. Rating N Spread HV IV Lev. Size Equity prem. AAA 956 21 0.284 0.261 0.274 375.086 0.067 AA 2580 26 0.289 0.273 0.211 98.894 0.074 A 11671 49 0.315 0.302 0.346 44.080 0.075 BBB 15505 120 0.349 0.331 0.508 29.403 0.076 BB 2069 321 0.454 0.422 0.559 19.094 0.100 B 620 522 0.629 0.507 0.706 25.701 0.106 3.4 Empirical Implementation The structural model by Leland & Toft (1996) is used to obtain risk neutral and objective survival probabilities. In Leland & Toft (1996), …rm assets Vare assumed to follow a geometric Brownian motion under the risk-neutral measure dVt= (r)Vtdt +VVtdWt, (3.5) with dVt=uVVtdt +VVtdWt(3.6) under the objective measure11.ris the risk free interest rate, is the payout ratio and Vis the asset volatility. The …rm defaults when the asset value hits the endogenously derived default barrier VB. To obtain the survival probabilities the unobserved asset value Vt;and asset volatility Vare needed, and to get the objective survival probabilities estimates of the expected asset return uVis needed as well. In the next two sections I describe how these unobserved parameters are 11In this case Vincludes the payout ratio . 114
inferred. A more detailed description of the model by Leland & Toft (1996) is given in appendix A. 3.4.1 Calibrating the Leland & Toft (1996)-Model To infer the unobserved asset value Vt;and asset volatility Vthe model is calibrated to equity market data in two ways. Firstly using a 250-day historical volatility and secondly using an implied volatility from equity options. According to Cremers et al. (2006) and Cao et al. (2006), the implied volatility contains important and timely information about credit risk di¤erent from the historical measure. To implement the model, we follow Ericsson et al. (2006) in setting the realized bankruptcy cost fraction = 0:15, the tax rate = 0:20 and the average debt maturity 2= 3:38.12 Furthermore Pis the total liabilities from quarterly balance sheet data, Sis the market value of equity and ris the 5-year constant maturity treasury yield. We also follow Ericsson et al. (2006) in assuming that the average coupon paid out to all debtholders equals the risk-free interest rate, C=rP .13 The asset payout rate is calculated as a time-series mean of the weighted average historical dividend yield and relative interest expense from balance sheet data =Interest expenses Total liabilities L+ (Dividend yield)(1 L)(3.7) L=Total liabilities Total liabilities +Market equity. The iterative algorithm of Moody’s KMV outlined in Crosbie & Bohn (2003) and Vassalou & Xing (2004) is used to infer the unobserved time-series of asset values and asset volatility. This iterative scheme goes as follows. The value of equity Stis a function of the asset value Vt, asset volatility Vand a set of parameters (see equation (3.17) in appendix A), i.e. St=f(Vt; V; ). A 250- 12The choice of 15 percent bankruptcy costs lies well within the range estimated by Andrade & Kaplan (1998). 20 percent as an e¤ective tax rate is below the corporate tax rate to re‡ect the personal tax rate advantage of equity returns. Stohs & Mauer (1996) …nd an average debt maturity of 3.38 years using a panel of 328 industrial …rms with detailed debt information in Moody’s Industrial Manuals in 1980-1989. 13A …rm’s debt consists of more than market bonds, and usually a substantial fraction of total debt is non-interest bearing such as accrued taxes and supplier credits. Furthermore, corporate bonds may be issued below par, which also opens up for this approximation. 115
day window of historical equity values is used to obtain an estimate of the equity volatility S, by viewing the value of equity as a geometric Brownian motion. Given this initial estimate of the asset volatility Vand quarterly balance sheet data, the value of the default barrier can be calculated. Using the daily market values of equity and the equity pricing formula we then back out an implied timeseries of asset values Vt(V) = f1(St; V; ). Next, since the daily asset values follow a geometric Brownian motion we obtain an improved estimate of the asset volatility V, which is used in the next iteration. This procedure is repeated until the values of Vconverge. When using implied volatilities from equity options, the instantaneous relationship given by St=f(Vt; V; )(3.8) S=@St @Vt V Vt St (3.9) is solved numerically for the unknown asset value Vtand asset volatility V, where (3.9) follows from Ito’s lemma on St. Table 3.2 gives summary statistics for the calibrated parameters. For each rating category the average calibrated parameters look reasonably similar across the two calibration methods, but as we will see later, they will behave di¤erently over time. At …rst sight it may seem surprising that the average calibrated asset volatilities are smallest for the lower ratings, which is consistent for both calibration methods. The reason is that these …rms have very large leverage ratios as seen in Table 3.1. So even though these …rms have higher equity volatilities they end up with lower calibrated asset volatilities due to the high leverage. If we look at the average distance to default measure DD, calculated as VVB VV;we also see that the better rated …rms have a larger distance to default, and thus a smaller risk of defaulting. As described in the introduction the di¤erence in calibration method could give rise to di¤erent implied credit risk premia, and less volatile risk premia are expected, when the survival probabilities and expected losses are based on the model implemented with the option implied volatility. Suppose e.g. that the uncertainty in the market suddenly increases. This implies both a higher option implied volatility and a higher realized equity volatility. The di¤erence is that a change in uncertainty is immediately captured in the implied volatility but only 116
slowly in the historical volatility. This leads to di¤erences in the calibrated asset volatility V;and thus in expected losses. Since a change in uncertainty is also immediately captured in the market CDS spread cmarket t, the use of default probabilities based on historical volatility, when calculating cno risk(0; T)from equation (3.1) might give rise to credit risk premia that are mismeasured. Expected losses in both Berndt et al. (2005), Berndt, Lookman & Obreja (2006) and Elkamhi & Ericsson (2007) are based on historical volatilities. I will therefore check if the use of implied volatility when calculating expected losses leads to better measured and less volatile risk premia. Table 3.2: Descriptive Statistics of Implied Parameters This table reports the central implied parameters from Leland & Toft (1996), calibrated with a historical volatility HV in panel A and option-implied volatility IV in panel B. While the …rst measure is calculated from a 250-day rolling window of equity returns, the latter is implied from 30-day at-the-money put options. The descriptive statistics are averaged over the number of quotes. The asset value and default barrier (barrier) are measured in billions of dollars. Asset vol: is the implied asset volatility and DD is the measure of distance to default calculated as (VVB)=(VV). Variable Asset value Asset vol. Barrier DD Panel A. Leland & Toft HV AAA 357.000 0.209 121.274 4.064 AA 95.754 0.234 15.304 4.021 A 41.933 0.209 12.604 3.924 BBB 27.857 0.171 15.282 3.753 BB 17.981 0.189 8.112 3.203 B 24.002 0.171 15.350 2.562 Panel B. Leland & Toft IV AAA 354.573 0.190 124.643 4.431 AA 95.627 0.218 15.579 4.197 A 41.789 0.200 12.844 3.961 BBB 27.962 0.161 15.929 3.854 BB 18.018 0.179 8.247 3.272 B 24.366 0.146 16.359 2.858 117
Having found the asset value Vand asset volatility Vwe can calculate risk neutral survival probabilities. We now go on to estimate the expected return on assets uVin equation (3.6) to be able to calculate objective probabilities and price the CDS in a market without risk premia cno risk(0; T). 3.4.2 Estimating the Asset Value Risk Premia The expected asset return uVis found as in Elkamhi & Ericsson (2007). More speci…cally we link the risk premium on assets to the risk premium on equity14 uVr= (RV(t)r) = S(RS(t)r);(3.10) where (RS(t)r)is the estimated equity risk premium, RS(t)dt =EP[dSt(Vt) St(Vt)]: RV(t)dt =EP[dVt Vt ] is the expected asset return uV;and S= @St(Vt) @VtVt St(V)!1 (3.11) is found numerically using the structural model by Leland & Toft (1996) and depending on, whether the model parameters (V&v) are calibrated with the historical volatility or option implied volatility. The equity premium (RS(t)r)is estimated for each CDS quote in the dataset. The approach by Fama & MacBeth (1973) is used together with the Fama & French (1993) market factor15. Using a history of 1250 daily stock returns betas are estimated for each CDS quote. A cross sectional regression of the individual stock returns on the betas is then run each day, which yields a daily market risk premium. A moving average over 1250 days is then used as the factor risk premium on the given day16. 14The proof of equation (3:10) can be found in Campello, Chen & Zhang (2008). 15This factor can be found on Kenneth French’s website. 16Running monthly cross sectional regressions instead, followed by averages over 60 month 118
For each stock on each day the equity premium is then found as the beta multiplied by the market risk premium, and the expected asset return uVis then found using equation (3.10). In the last column of Table 3.1 the average equity premia are given across rating and the average equity premium over time is given in Figure 3.2. The average risk premium across all observations is 7:76% and we see from Table 3.1 that better rated …rms have a lower average risk premium. This is similar to the average equity risk premia across ratings in Huang & Huang (2003), although the average estimated risk premia in this paper generally are a bit higher. Figure 3.2: Equity Risk Premium The …gure illustrates the average equity risk premium over time calculated as averages over the cross section of weekly risk premia. The risk premia are measured using the Fama & MacBeth (1973) methodology. does not change the estimated risk premia signi…cantly. Nor did the risk premia change signi…- cantly, when all three Fama & French (1993) factors where used, but the factor risk premia on the SMB and HML where insigni…cant in a large part of the cross sectional regressions. 119
3.5 Empirical Results With all estimated parameters in place we are able to calculate both objective and risk neutral default probabilities. Furthermore CDS spreads can be calculated both with and without the risk premium included, which allows us to decompose the spread. Before we decompose the CDS spreads and calculate the credit risk premium, a comparison of the estimated objective default probabilities with actual default rates is in place. We thus start out by comparing the objective default probabilities across rating categories and horizon with the historical default rates from Moody’s (Hamilton et al. (2007)). Although the sample periods are di¤erent it will give us an indication of whether the estimated objective default probabilities are reasonable. Table 3.3 shows the estimated objective default probabilities together with the average Moody’s default rates for the period 1920-200617. The historical default rates are generally higher than the model implied default probabilities and the model implied default probabilities based on implied volatility are generally lower than the default probabilities based on historical volatility. If we look at the …ve year horizon, which is the maturity of the CDS spreads in the sample, the implied default probabilities match the historical default rates quite well, although the default rates are underestimated for the speculative grade segment. Looking at the Aand BBB ratings, which constitutes the majority of the quotes in the sample, the default rates are also not that dissimilar. Earlier work by Berndt et al. (2005) and Berndt, Lookman & Obreja (2006) have relied on KMV expected default frequencies (EDF) as measures of objective default probabilities. KMV also uses a structural model to estimate a distance to default measure, which is then mapped into the EDF measure using historical default rates18. Since the applied methodology in this paper is the same as in Elkamhi & Ericsson (2007) and given the reasonable size of the estimated objective default probabilities it is expected that the results of this paper can be compared to the three mentioned papers. 17The Moody’s ratings are transferred to the S&P ratings with Aaa =AAA; Aa =AA and so forth. 18See Crosbie & Bohn (2003). 120
Table 3.3: Historical and Model Implied Default Probabilities This table reports the historical and model implied default probabilities by rating category and horizon. HV is based on the historical volatility while IV is based on option-implied volatility. The model implied default probabilities are averaged over the number of quotes, while the historical default probabilities (Actual) represents Moody’s cumulative default rates for the period 1920-2006. Rating/Horizon 1 3 5 7 10 15 20 AAA (Actual) 0.00 0.02 0.16 0.37 0.89 1.44 1.82 AAA (HV) 0.00 0.08 0.33 0.73 1.50 2.96 4.42 AAA (IV) 0.00 0.07 0.29 0.59 1.15 2.14 3.11 AA (Actual) 0.06 0.29 0.72 1.34 2.31 4.29 5.3 AA (HV) 0.00 0.08 0.34 0.75 1.48 2.77 3.96 AA (IV) 0.00 0.05 0.25 0.55 1.13 2.19 3.21 A (Actual) 0.07 0.51 1.13 1.80 2.9 4.91 6.39 A (HV) 0.05 0.59 1.41 2.32 3.63 5.53 7.06 A (IV) 0.01 0.24 0.79 1.49 2.58 4.24 5.61 BBB (Actual) 0.3 1.61 3.26 4.85 7.29 10.87 13.45 BBB (HV) 0.24 1.66 3.26 4.72 6.57 8.88 10.53 BBB (IV) 0.06 0.70 1.68 2.72 4.13 6.00 7.39 BB (Actual) 1.38 5.47 9.83 13.64 18.79 25.81 30.81 BB (HV) 0.58 3.41 5.98 8.03 10.36 13.00 14.75 BB (IV) 0.27 2.22 4.20 5.87 7.82 10.06 11.55 B (Actual) 4.32 14.23 22.45 28.58 34.86 42.11 46.09 B (HV) 2.42 8.51 12.75 15.76 18.91 22.18 24.20 B (IV) 0.56 3.52 6.28 8.45 10.84 13.42 15.05 121
One of the main drivers of the default probabilities and expected losses is the volatility of the …rm’s assets v. To understand what drives the di¤erences in the time-series behavior of the estimated credit risk premia later on, we also take a look at the average time-series behavior of the calibrated asset volatilities from section 3.4. In Figure 3.3 the average calibrated asset volatilities are shown over time, when based on historical and implied volatility respectively. It is clear, that the calibrated volatilities do not move together and that the asset volatility based on historical volatility is much smoother than the one based on implied volatility. In the second half of 2002 the asset volatility based on implied volatility rises substantially, while this rise in uncertainty is only partly captured by the asset volatility based on historical volatility. Subsequently the asset volatility based on implied equity volatility falls faster than the more rigid asset volatility based on historical equity volatility. Figure 3.3: Asset Volatilities The …gure illustrates the average calibrated asset volatilities over time based on historical and option implied equity volatilities respectively. The means are calculated as averages over the cross section of weekly volatilities. 122
As we will see in the next section this di¤erent behavior of the calibrated asset volatilities has implications for the way the expected loss component behaves over time and thus for the time-series behavior of the estimated credit risk premium. From the beginning of 2004 we see that the two calibrated volatilities move together. We now go on to decompose the CDS spread into an expected loss component and a risk premium component. 3.5.1 Decomposing the Credit Spread To decompose the CDS spread and examine the time-series behavior of the credit risk premium an estimate of the expected loss component is needed. In Figure 3.4 the average expected loss component cno risk tcalculated in equation (3.1) is plotted over time together with the average market CDS spread cmarket tfrom Figure 3.1. In panel A of Figure 3.4 the expected loss component is calculated with asset volatilities based on historical equity volatility, while the expected loss component is calculated with asset volatilities based on implied equity volatility in panel B. Similar to the market spreads, the expected loss components vary considerably through the sample period, and when based on implied equity volatility the component peaks around the same time as the market spread, although there is a tendency for the peaks in the expected loss component to appear a little earlier than in the spread. The expected loss component peaks somewhat later, when based on historical equity volatility. When the spreads start to fall the expected loss component based on implied volatility falls as well, while this happens with a lag for the expected loss component based on historical volatility. From a comparison with Figure 3.3 we see that the di¤erent behavior of the asset volatilities in Figure 3.3 to a large degree is re‡ected in the movement of the respective expected losses in Figure 3.4. 123
Figure 3.8: Expected Loss and Risk Premium Ratios The …gure illustrates the expected loss ratio and the risk premium ratio over time. In Panel A the expected loss component is based on historical volatility and in Panel B it is based on implied volatility. 130
Berndt et al. (2005) o¤er possible explanations for the time variation in the risk premia, which I will relate to the …ndings in this paper. One explanation is that the variation in risk premia is partly caused by sluggish movement in risk capital across sectors. Berndt et al. (2005) argue that variations of the supply and demand for risk bearing are exacerbated by limited mobility of capital across di¤erent classes of asset markets, implying that risk premia would tend to adjust so as to match the demand for capital with the supply of capital that is available to the sector. Proxying for market volatility they …nd that VIX20 adds signi…cantly to the explanation of CDS spreads after the EDF measure has been accounted for, and they suggest that credit risk premia strongly depend on market volatility/VIX. If the market volatility goes up, a given level of capital available to bear risk represents less and less capital per unit of risk to be borne. If replacement capital does not move into the corporate debt sector immediately, the supply and demand for risk capital will match at a higher price per unit of risk. In Figure 3.9 the average calibrated asset volatilities from Figure 3.3 are plotted together with VIX. In panel A the average calibrated asset volatilities are based on the historical volatility, while panel B plots the average asset volatilities based on the implied volatility together with VIX. Looking at panel A we see that VIX is much more volatile than the asset volatility based on historical volatility, and VIX also spikes in late 2002 just as the market CDS spreads in Figure 3.1. It is a di¤erent story in panel B. We see that the average calibrated asset volatilities based on implied volatility and the VIX move very closely together throughout the entire period, suggesting that the asset volatilities and expected losses based on implied volatility and VIX are related. In theory the average calibrated asset volatilities should contain both systematic volatility and idiosyncratic volatility, and the systematic volatility should explain part of the expected losses but also the credit risk premia21. What the results of Figure 3.9 suggest is that VIX is indeed a measure of systematic volatility and also an important driver of expected losses, when these are measured with implied volatility2223. This also suggests 20VIX is an index of option implied volatility on the S&P 500. 21Elkamhi & Ericsson (2007) also includes a discussion of this topic, and relate their results to Campbell & Taksler (2003). 22Unreported results also show that VIX adds explanatory power to the credit spreads when both leverage and volatility have been accounted for. 23In panel B the asset volatilities are "delevered", while the VIX volatility is not, and the 131
that the EDF measure and expected losses based on historical volatility do not adequately capture the probability of default implied by the market. Earlier papers such as Collin-Dufresne et al. (2001) and Schaefer & Strebulaev (2004) have shown that VIX is an important explanatory variable for changes in credit spreads, although they did not pin down an explanation for the role of VIX. Figure 3.9: Asset Volatilities and VIX Volatility The …gure illustrates the average asset volatilities and the VIX volatility over time. In panel A the VIX volatility is depicted together with the asset volatility based on historical equity volatility, and in Panel B the VIX volatility is depicted together with the asset volatility based on implied volatility. VIX volatility is thus higher than the average asset volatilities during the main part of the sample period. Interestingly, the average asset volatilities are higher than VIX towards the end of 2005 and the beginning of 2006, suggesting a lot of idiosyncratic volatility in this period. 132
We have not yet discussed the assumption of a constant expected recovery rate Rof 40%. In Figure 3.10, where the average model spread and the average market spread are plotted, we see that the structural model underestimates the market spreads during large parts of the sample period24. This suggests that the assumed recovery rate could be too large. Lowering the recovery rate would raise the spreads, but from equation (3.1) we see that loss given default (LGD) is multiplied onto the part of the calculated spread that is determined by the default probabilities. A lower recovery rate would thus have a small e¤ect on the size of the calculated spreads in times of low default probabilities, and it is in exactly these periods that the model underestimates the spreads. Consequently, as long as the recovery rate is within a reasonable range the results of the paper would not change25. As discussed in Berndt et al. (2005), there could also be correlation between loss given default/recovery rates and the probability of default and in fact Moody’s (Hamilton et al. (2007) ) estimate a negative correlation between annual corporate default rates and recovery rates. The possibility of a negative correlation between default probabilities and the expected recovery rate could lower the time variation in the estimated risk premia, when the default probabilities are based on implied volatility. If we look at Figure 3.4 and 3.5 again, a negative correlation between the recovery rate and the default probabilities based on implied volatility would increase the expected loss component in late 2002, and make it even smaller in 2003 leading to less variation in the resulting average risk premium. But this would also imply a larger underestimation by the structural model of the average market spread in times of low spreads as seen in Figure 3.1026. Based on the above discussion and decomposition of the CDS spreads I conclude that the risk premia estimated in earlier papers such as Berndt et al. (2005) and Elkamhi & Ericsson (2007) might be inappropriate since these risk premia are based on historical volatility, and one should be careful when drawing conclusions on risk premia based on expected losses estimated with a historical volatility. 24Elkamhi & Ericsson (2007) …nd similar results for the same period. 25If anything, a lower recovery rate would enhance the di¤erence in the estimated risk premia based on historical and implied volatility, since a lower recovery rate would raise the expected loss component in times of high default probabilities. 26Assuming that the correlation is of similar size under P and Q. Introducing a time varying recovery rate would also imply a risk premium on the recovery rate. 133
This is especially important in times of high uncertainty. More speci…cally the expected losses based on historical volatility tend to be to smooth and there tends to be an overprediction of expected losses in 2003 following the period of high uncertainty in late 2002. Actually Bohn, Arora & Korablev (2005) report that the EDF’s predicted too many defaults in 2003 consistent with the results in this paper. In the next section this conclusion is supported by a regression analysis showing that option implied equity volatility does a better job in explaining CDS spreads compared to the 250-day historical equity volatility. We will also discuss the possibility of adding a time varying risk premium to structural models. Figure 3.10: Market Spreads and Model Spreads The …gure illustrates the average market spread and model spread over time. In panel A the model is calibrated with the historical volatility, and in Panel B the model is calibrated with implied volatility. The means are calculated as averages over the cross section of weekly spreads. 134
3.5.2 Modeling CDS Spreads The typical structural model predicts, that the level of credit spreads mainly depends on asset volatilities and leverage, while most models are silent on risk premia. Notable exceptions are Chen et al. (2006), Bhamra et al. (2007) and Chen (2007). Chen et al. (2006) e.g. consider whether existing asset pricing models that have proven successful in explaining equity returns can explain the level and volatilities of credit spreads. They have some success with models that exhibit time varying risk premia. Leland (2004) and Huang & Huang (2003) have also studied risk premia in the context of structural models, but they do not consider their dynamics. Figure 3.11 plots the average model implied credit risk premia (RPIequity) over time calculated in equation (3.4). These model implied risk premia stem from the equity market through the translation of the equity risk premium via the structural model. The …gure shows the risk premia when the structural model is calibrated with the historical volatility and the implied volatility respectively. The two risk premia do not move in exactly the same way, which follows from the di¤erences in the calibration of the structural model, but there are large similarities. Figure 3.11: Model Implied Risk Premia The …gure illustrates the average model implied credit risk premia over time, when the structural model is calibrated with the historical and implied equity volatility respectively. The means are calculated as averages over the cross section of weekly spreads. 135
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 spreads27 CDSit =+1Levi;t +2Evoli;t +3Ereti;t +4Slopet+5rt+6RPIequity i;t +"i;t; (3.12) where Lev 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, Slope is the di¤erence between the 10- and 2-year constant maturity rate and ris the 5-year constant maturity rate corresponding to the maturity of the CDS spreads. RPIequity 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 sample28. 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 explanatory power of the regressions and the coe¢ cients on the risk premium are all strongly signi…cant29. When the risk premium is included in the regression with the historical volatility the R-square increases by 3% from 49:4% to 52:4%;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. 136
Table 3.4: Importance of Risk Premium for CDS Spreads This table reports the results of the panel regression CDSit =+1Levit +2Evolit +3Eretit +4Slopet+5rt+6RPIit +"it. T-statistics are reported in parentheses. Evol is either the historical equity volatility, calculated using 250 days of historical equity returns, or the implied volatility on 30-day at-the-money put options. Leverage (lev) is total liabilities divided by the sum of total liabilities and equity market capitalization. Eret is the daily equity return, slope is the slope of the yield curve and ris the level of the interest rate. RPI is the equity implied model risk premium. *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel A Historical volatility Panel B Implied volatility 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) Lev 2.81 1.90 2.81 1.90 2.67 1.46 2.67 1.46 (39.40) (32.04) (6.95) (5.08) (43.24) (20.97) (7.29) (4.08) Evol. 5.84 4.61 5.84 4.61 7.82 4.55 7.82 4.55 (24.97) (20.39) (7.96) (5.33) (35.81) (15.99) (7.93) (3.25) Eret 0.80 0.99 0.80 0.991.95 2.121.95 2.12 (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 -1.68 0.85 3.53 0.85 3.53 (-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) R20.494 0.524 0.494 0.524 0.574 0.629 0.574 0.629 N 33401 33401 33401 33401 33401 33401 33401 33401 Cluster Date Date Firm Firm Date Date Firm Firm 137
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 representing 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 by 8:2% from 52:6% to 60:8% 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 implied 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 investment 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). 138
Table 3.5: Importance of Risk Premium for Investment Grade CDS Spreads This table reports the results of the panel regression CDSit =+1Levit +2Evolit +3Eretit +4Slopet+5rt+6RPIit +"it. The regression is run for the investment grade quotes in the sample. T-statistics are reported in parentheses. Evol is either the historical equity volatility, calculated using 250 days of historical equity returns, or the implied volatility on 30-day at-the-money put options. Leverage (lev) is total liabilities divided by the sum of total liabilities and equity market capitalization.Eret is the daily equity return, slope is the slope of the yield curve and ris the level of the interest rate. RPI is the equity implied model risk premium. *, ** and *** denote signi…cance at 10, 5 and 1 percent, respectively. Panel A Historical volatility Panel B Implied volatility 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) Lev 2.04 1.20 2.04 1.20 1.96 0.91 1.96 0.91 (32.74) (25.15) (7.91) (4.80) (36.04) (20.55) (7.94) (3.37) Evol. 4.54 3.37 4.54 3.37 5.87 2.86 5.87 2.86 (18.72) (15.37) (6.59) (5.18) (27.14) (12.95) (7.88) (3.43) Eret 0.52 0.70 0.52 0.70 1.401.441.40 1.44 (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 -2.87 0.41 -2.87 0.41 (-2.96) (-2.48) (-1.74) (-1.45) (-1.47) (0.24) (-0.79) (0.14) r 7.767.067.767.069.95 11.25 9.95 11.25 (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) R20.444 0.497 0.444 0.497 0.526 0.608 0.526 0.608 N 30712 30712 30712 30712 30712 30712 30712 30712 Cluster Date Date Firm Firm Date Date Firm Firm 139