Dynamic Dependence and Diversification in Corporate Credit

Dynamic Dependence and Diversification in Corporate Credit Abstract We characterize dependence in corporate credit and equity returns for 215 firms using a new class of large-scale dynamic copula models. Copula dependence and especially tail dependence are highly variable and persistent, increase significantly in the financial crisis, and have remained high since. The most drastic increases in credit dependence occur in July/August of 2007 and in August of 2011 and the decrease in diversification potential caused by the increases in dependence and tail dependence is large. Credit default swap correlation dynamics are important determinants of credit spreads. 1. Introduction Characterizing the dependence between credit-risky securities is of great interest for portfolio management and risk management, but not necessarily straightforward because multivariate modeling is notoriously difficult for large cross-sections of securities. In existing work, computationally straightforward techniques such as factor models or constant copulas are often used to model correlations for large portfolios of credit-risky securities; alternatively, simple rolling correlations or exponential smoothers are used. We instead use multivariate econometric models for the purpose of modeling credit correlation and dependence. We use genuinely dynamic copula techniques that can capture univariate and multivariate deviations from normality, including multivariate asymmetries. We demonstrate that by using recently proposed econometric innovations, it is possible to apply copula models on a large scale that is essential for effective credit risk management. We perform our empirical analysis using data on a large cross-section of credit-risky securities, namely 5-year credit default swap (CDS) contracts for 215 constituents of the first 18 series of the CDX North American investment grade index. We use a long time series of weekly data for the period January 3, 2001 to August 22, 2012. The 215 firms enter and leave the sample at different time points, but this can easily be accommodated by the estimation methodology we employ. We compute returns on selling credit protection through CDS and investigate the dependence of these returns across firms as well as their tail dependence. We formally define returns on a short CDS position in the next section. We also analyze dependence in the underlying equity for comparison. Interestingly, the credit and equity return dynamics differ in important aspects. We document several important stylized facts, and substantial differences between credit and equity dependence. We show that the time-varying correlations in CDS returns implied by the copula models we use, which we will refer to as “copula correlations” for short, vary substantially over the sample period, with a substantial increase following the financial crisis in 2007. Equity correlations also increase in the financial crisis, but somewhat later, and the increase is less pronounced and not as persistent. Our estimates indicate fat tails in the univariate credit distributions, but also multivariate non-normalities for CDS returns. Multivariate asymmetries seem to be less important for credit than for equity returns, confirming results from threshold correlations. Credit copula correlations are more persistent than equity copula correlations. This greatly affects how major events such as the subprime funds collapse, the Lehman bankruptcy, and the US sovereign debt downgrade affect subsequent dependence in credit and equity markets. Tail dependence for credit and equity increases substantially during our sample, more so than copula correlations. Surprisingly, the Lehman bankruptcy affects equity (tail) dependence more strongly than credit (tail) dependence. The US sovereign downgrade in mid-2011 is an important credit event, but this is more apparent when analyzing tail dependence, somewhat less so when analyzing copula correlations. The increase in cross-sectional dependence is clearly important for the management of portfolio credit risk. We use our estimates to compute time-varying diversification benefits from selling credit protection. We find that the increase in cross-sectional dependence following the financial crisis has substantially reduced diversification benefits, similar to what happened in equity markets. We use panel regressions to investigate if correlation and tail dependence help explain the variation in firm credit spreads. We find that higher copula correlation and tail dependence are related to higher CDS spreads, even after controlling for well-established determinants of credit spreads at the firm level, such as equity volatility, interest rates, and leverage. Identifying financial and macroeconomic variables that can capture the clustering in default and cross-firm default dependence is of great interest for the purpose of modeling portfolio credit risk, but there is no guidance from theory regarding the economic determinants of credit dependence and tail dependence. We use a regression analysis to identify financial and macroeconomic determinants of the time-series variation in credit dependence. Copula correlations increase with the VIX, the overall level of credit spreads, and inflation, and decrease with the level of interest rates and S&P 500 returns. The effect from VIX is robust when including lagged correlations in the regressions. The empirical analysis proceeds in three steps. The first two steps are univariate. In the first, we remove the short-run dynamics from the raw data by estimating firm-by-firm autoregressive (AR) models on weekly returns. In the second step, we estimate firm-by-firm variance dynamics on the residuals from the first step. We use an asymmetric nonlinear generalized autoregressive conditional heteroskedasticity (NGARCH) model with an asymmetric standardized t-distribution following Hansen (1994).1 Finally, in the third step we provide a multivariate analysis using the copula implied by the skewed t-distribution in Demarta and McNeil (2005). Dynamic copula correlations are modeled based on the linear correlation techniques developed by Engle (2002) and Tse and Tsui (2002).2 Dynamic tail dependence depends on time-varying correlations and degrees of freedom, which we capture using a smooth exponential spline function (see Engle and Rangel, 2008). To alleviate the computational burden, we rely on the composite likelihood technique of Engle, Shephard, and Sheppard (2008) and the moment matching from Engle and Mezrich (1996). Patton (2013) provides an excellent survey of copula models. The remainder of the article is structured as follows. In Section 2, we document some stylized facts in our CDS sample and briefly review existing models of default and default dependence. Section 3 reports the estimation results from the dynamic models for expected credit and equity returns and volatility. It also introduces the dynamic copula models and presents the estimation results for copula and threshold correlations. In Section 4, we present various economic applications of our dynamic copula model. We investigate credit diversification dynamics, relate credit spreads to copula correlations, and relate copula correlations to economic covariates. Section 5 concludes. 2. CDS Markets: Stylized Facts and Default Models We discuss the CDS data we use in our empirical work. We also briefly discuss existing techniques for modeling default dependence. 2.1 CDS Data A CDS is essentially an insurance contract on corporate or sovereign debt, under which the buyer makes regular premium payments and is compensated by the seller in case of default by the underlying entity. One element of the success and resilience of CDS markets has been the creation of market indexes consisting of CDSs, such as the CDX index in North America and the iTraxx index in Europe. Using data from Markit, we consider 5-year CDS contracts on all firms included in the first 18 series of the North American investment grade CDX.NA.IG index. We use the longest possible sample available from Markit for all these firms, starting on January 3, 2001, and ending on August 22, 2012. Many firms do not have CDS quotes available for every day of this sample period. Fortunately, as shown by Patton (2006a), the dynamic multivariate modeling approach we employ in our empirical work allows for individual series to begin (and end) at different time points. We make full use of this and include a firm if it has at least 1 year of consecutive weekly data points. The resulting list of 215 firms is provided in Table I. Table I Company names Using data from Markit, we consider all firms included in the first eighteen series of the CDX North American investment grade index dating from January 1, 2001 to August 22, 2012. Our sample consists of 215 firms. Firms ordered alphabetically within each GIC sector. Consumer discretionary  Consumer discretionary (Cont.)  Industrials  Financials   American Axle & Manuf. Holdings   Tribune Co   Boeing Capital Corp.   ACE Limited   Autozone Inc.   Viacom Inc.   Bombardier Capital Inc.   Allstate Corp.   Belo Corp.   Visteon Corp.   Bombardier Inc.   American Express Co   Black & Decker Corp.   Walt Disney Co   Burlington Northern Santa Fe Corp.   American International Group Inc.   Brunswick Corp.   Wendy's International Inc.   CSX Corp.   Berkshire Hathaway Inc.   CBS Corp.   Whirlpool Corp.   Caterpillar Inc.   Boston Properties LP   CENTEX Corp.   YUM! Brands Inc.   Cendant Corp.   CIT Group Inc.   Carnival Corp.     Deere & Co   Capital One Bank USA Nat. Assoc.   Clear Channel Comms Inc.  Consumer Staples   GATX Corp.   Capital One Financial Corp.   Comcast Cable Communication LLC   Albertsons Inc.   General Electric Capital Corp.   Chubb Corp.   Comcast Corp.   Altria Group Inc.   Goodrich Corp.   Countrywide Home Loans Inc.   Cox Communications Inc.   Beam Inc.   Honeywell International Inc.   EOP Operating LP   DIRECTV Holdings LLC   CVS Caremark Corp.   Ingersoll Rand Co   ERP Operating LP   Darden Restaurants Inc.   Campbell Soup Co   Lockheed Martin Corp.   Federal Home Loan Mortgage Corp.   Delphi Corp.   ConAgra Foods Inc.   Masco Corp.   Federal National Mortgage Assoc.   Eastman Kodak Co   General Mills Inc.   Norfolk Southern Corp.   General Motors Acceptance Corp.   Expedia Inc.   H. J. Heinz Co   Northrop Grumman Corp.   Hartford Financial Services Group   Ford Motor Credit Co   Kraft Foods Inc.   Pitney Bowes Inc.   International Lease Finance Corp.   GAP Inc.   Kroger Co   R. R. Donnelley & Sons Co   Loews Corp.   Gannett Co Inc.   Reynolds American Inc.   Raytheon Co   MBIA Insurance Corp.   Harrah's Operating Co Inc.   Safeway Inc.   Ryder System Inc.   MBNA Corp.   Hilton Hotels Corp.   Sara Lee Corp.   Southwest Airlines Co   Marsh & Mclennan Co Inc.   Home Depot Inc.   Supervalu Inc.   Textron Financial Corp.   MetLife Inc.   J. C. Penney Co Inc.   Tyson Foods Inc.   Union Pacific Corp.   National Rural Util. Coop Fin. Corp.   Johnson Controls Inc.   Wal Mart Stores Inc.   United Parcel Services Inc.   Radian Group Inc.   Jones Apparel Group Inc.       Residential Capital Corp.   Knight-Ridder Inc.  Health Care  Information Technology   SLM Corp.   Kohls Corp.   Aetna Inc.   1st Data Corp.   Simon Property Group Inc.   Lear Corp.   Amgen Inc.   Arrow Electronics Inc.   Vornado Realty LP   Lennar Corp.   Baxter International Inc.   Avnet Inc.   Washington Mutual Inc.   Liberty Media Corp.   Boston Scientific Corp.   CA Inc.   Wells Fargo & Co   Limited Brands Inc.   Bristol Myers Squibb Co   Cisco Systems Inc.   Weyerhaeuser Co   Liz Claiborne Inc.   Cardinal Health Inc.   Computer Sciences Corp.   XLIT Limited   Lowe's Companies Inc.   Cigna Corp.   Dell Inc.   iStar Financial Inc.   M.D.C. Holdings Inc.   McKesson Corp.   Electronic Data System Corp.     Macy's Inc.   Pfizer Inc.   Hewlett Packard Co  Energy   Marriott International Inc.   Quest Diagnostics Inc.   IAC InterActive Corp.   Amerada Hess Corp.   May Department Stores Co   UnitedHealth Group Inc.   IBM Corp.   Anadarko Petroleum Corp.   Maytag Corp.   Universal Health Services Inc.   Motorola Inc.   Canadian Natural Resources Ltd   McDonald's Corp.   Wyeth   Sabre Holdings Corp.   ConocoPhillips   Mohawk Industries Inc.     Sun Microsystems Inc.   Devon Energy Corp.   NY Times Co  Telecommunication Services   Xerox Corp.   Halliburton Co   Newell Rubbermaid Inc.   ALLTEL Corp.     KerrMcGee Corp.   News America Inc.   AT&T Corp.  Materials   Kinder Morgan Energy LP   Nordstrom Inc.   AT&T Inc.   Alcan Inc.   Nabors Industries Inc.   Omnicom Group Inc.   AT&T Mobility LLC   Alcoa Inc.   Transocean Inc.   Pulte Homes Inc.   AT&T Wireless Services Inc.   Barrick Gold Corp.   Valero Energy Corp.   RadioShack Corp.   BellSouth Corp.   Dow Chemical Co   XTO Energy Inc.   Sears Roebuck Acceptance Corp.   CenturyLink Inc.   E. I. du Pont de Nemours & Co     Staples Inc.   Cingular Wireless LLC   Eastman Chemical Co  Utilities   Starwood Hotels & Resorts Inc.   Citizens Communication Co   Freeport McMoran Copper & Gold   American Electric Power Co Inc.   TJX Companies Inc.   Embarq Corp.   International Paper Co   Constellation Energy Group Inc.   Target Corp.   Intelsat Limited   MeadWestvaco Corp.   Dominion Resources Inc.   Time Warner Cable Inc.   Sprint Corp.   Olin Corp.   Duke Energy Carolinas LLC   Time Warner Inc.   Verizon Communications Inc.   Rio Tinto Alcan Inc.   Exelon Corp.   Toll Brothers Inc.   Verizon Global Funding Corp.   Rohm & Haas Co   FirstEnergy Corp.   Toys “R” Us Inc.     Sherwin Williams Co   Progress Energy Inc.       Temple-Inland Inc.   Sempra Energy  Consumer discretionary  Consumer discretionary (Cont.)  Industrials  Financials   American Axle & Manuf. Holdings   Tribune Co   Boeing Capital Corp.   ACE Limited   Autozone Inc.   Viacom Inc.   Bombardier Capital Inc.   Allstate Corp.   Belo Corp.   Visteon Corp.   Bombardier Inc.   American Express Co   Black & Decker Corp.   Walt Disney Co   Burlington Northern Santa Fe Corp.   American International Group Inc.   Brunswick Corp.   Wendy's International Inc.   CSX Corp.   Berkshire Hathaway Inc.   CBS Corp.   Whirlpool Corp.   Caterpillar Inc.   Boston Properties LP   CENTEX Corp.   YUM! Brands Inc.   Cendant Corp.   CIT Group Inc.   Carnival Corp.     Deere & Co   Capital One Bank USA Nat. Assoc.   Clear Channel Comms Inc.  Consumer Staples   GATX Corp.   Capital One Financial Corp.   Comcast Cable Communication LLC   Albertsons Inc.   General Electric Capital Corp.   Chubb Corp.   Comcast Corp.   Altria Group Inc.   Goodrich Corp.   Countrywide Home Loans Inc.   Cox Communications Inc.   Beam Inc.   Honeywell International Inc.   EOP Operating LP   DIRECTV Holdings LLC   CVS Caremark Corp.   Ingersoll Rand Co   ERP Operating LP   Darden Restaurants Inc.   Campbell Soup Co   Lockheed Martin Corp.   Federal Home Loan Mortgage Corp.   Delphi Corp.   ConAgra Foods Inc.   Masco Corp.   Federal National Mortgage Assoc.   Eastman Kodak Co   General Mills Inc.   Norfolk Southern Corp.   General Motors Acceptance Corp.   Expedia Inc.   H. J. Heinz Co   Northrop Grumman Corp.   Hartford Financial Services Group   Ford Motor Credit Co   Kraft Foods Inc.   Pitney Bowes Inc.   International Lease Finance Corp.   GAP Inc.   Kroger Co   R. R. Donnelley & Sons Co   Loews Corp.   Gannett Co Inc.   Reynolds American Inc.   Raytheon Co   MBIA Insurance Corp.   Harrah's Operating Co Inc.   Safeway Inc.   Ryder System Inc.   MBNA Corp.   Hilton Hotels Corp.   Sara Lee Corp.   Southwest Airlines Co   Marsh & Mclennan Co Inc.   Home Depot Inc.   Supervalu Inc.   Textron Financial Corp.   MetLife Inc.   J. C. Penney Co Inc.   Tyson Foods Inc.   Union Pacific Corp.   National Rural Util. Coop Fin. Corp.   Johnson Controls Inc.   Wal Mart Stores Inc.   United Parcel Services Inc.   Radian Group Inc.   Jones Apparel Group Inc.       Residential Capital Corp.   Knight-Ridder Inc.  Health Care  Information Technology   SLM Corp.   Kohls Corp.   Aetna Inc.   1st Data Corp.   Simon Property Group Inc.   Lear Corp.   Amgen Inc.   Arrow Electronics Inc.   Vornado Realty LP   Lennar Corp.   Baxter International Inc.   Avnet Inc.   Washington Mutual Inc.   Liberty Media Corp.   Boston Scientific Corp.   CA Inc.   Wells Fargo & Co   Limited Brands Inc.   Bristol Myers Squibb Co   Cisco Systems Inc.   Weyerhaeuser Co   Liz Claiborne Inc.   Cardinal Health Inc.   Computer Sciences Corp.   XLIT Limited   Lowe's Companies Inc.   Cigna Corp.   Dell Inc.   iStar Financial Inc.   M.D.C. Holdings Inc.   McKesson Corp.   Electronic Data System Corp.     Macy's Inc.   Pfizer Inc.   Hewlett Packard Co  Energy   Marriott International Inc.   Quest Diagnostics Inc.   IAC InterActive Corp.   Amerada Hess Corp.   May Department Stores Co   UnitedHealth Group Inc.   IBM Corp.   Anadarko Petroleum Corp.   Maytag Corp.   Universal Health Services Inc.   Motorola Inc.   Canadian Natural Resources Ltd   McDonald's Corp.   Wyeth   Sabre Holdings Corp.   ConocoPhillips   Mohawk Industries Inc.     Sun Microsystems Inc.   Devon Energy Corp.   NY Times Co  Telecommunication Services   Xerox Corp.   Halliburton Co   Newell Rubbermaid Inc.   ALLTEL Corp.     KerrMcGee Corp.   News America Inc.   AT&T Corp.  Materials   Kinder Morgan Energy LP   Nordstrom Inc.   AT&T Inc.   Alcan Inc.   Nabors Industries Inc.   Omnicom Group Inc.   AT&T Mobility LLC   Alcoa Inc.   Transocean Inc.   Pulte Homes Inc.   AT&T Wireless Services Inc.   Barrick Gold Corp.   Valero Energy Corp.   RadioShack Corp.   BellSouth Corp.   Dow Chemical Co   XTO Energy Inc.   Sears Roebuck Acceptance Corp.   CenturyLink Inc.   E. I. du Pont de Nemours & Co     Staples Inc.   Cingular Wireless LLC   Eastman Chemical Co  Utilities   Starwood Hotels & Resorts Inc.   Citizens Communication Co   Freeport McMoran Copper & Gold   American Electric Power Co Inc.   TJX Companies Inc.   Embarq Corp.   International Paper Co   Constellation Energy Group Inc.   Target Corp.   Intelsat Limited   MeadWestvaco Corp.   Dominion Resources Inc.   Time Warner Cable Inc.   Sprint Corp.   Olin Corp.   Duke Energy Carolinas LLC   Time Warner Inc.   Verizon Communications Inc.   Rio Tinto Alcan Inc.   Exelon Corp.   Toll Brothers Inc.   Verizon Global Funding Corp.   Rohm & Haas Co   FirstEnergy Corp.   Toys “R” Us Inc.     Sherwin Williams Co   Progress Energy Inc.       Temple-Inland Inc.   Sempra Energy  We construct nonoverlapping weekly returns using a fixed day each week. We use Wednesdays, which is the weekday that is least likely to be a holiday. We obtain equity data on the sample firms from the Center for Research in Security Prices (CRSP). Out of the 215 firms, 12 firms do not have at least a consecutive 52-week history of equity prices, and those are dropped from the sample.3 Note that reflecting the growth in market activity, CDS markets underwent a number of changes in April 2009, some of which affect the reporting of spreads (see Markit (2009), our data source, for details). To facilitate the comparison of the estimated correlations with equity market correlations, we use CDS returns, denoted by Rt, for our dependence analysis.4 To further facilitate the exposition of our results, we consider realized returns from selling credit protection throughout the article. Adverse events for a firm, therefore, result in both negative equity returns and negative CDS returns. We compute realized CDS excess returns as in Berndt and Obreja (2010) and Junge and Trolle (2015). In case of a credit event, the weekly return to the protection seller is computed as follows:   Rt=−(1−REC)+CDSt−1(τ−(t‐1))360, (2.1) where CDSt is the CDS spread at time t, τ is the credit event date and REC is the recovery rate in case of default. We obtain these ex post recovery rates from creditfixings.com. In the absence of a credit event, the weekly return to the protection seller is:   Rt=−(CDSt−CDSt−1)(PVBPt(CDSt,REC)−7360)+CDSt−17360, (2.2) where PVBPt(CDSt,REC) is the present value of a risky annuity of one basis point defined as:   PVBPt(CDSt,REC)=∑j=1J((tj−tj−1)360D(t,tj)S(t,tj)−∫max⁡{t,tj−1}tj(u−tj−1)360D(t,u)dS(t,u)), (2.3) with payment at dates t<t1<…<tJ, and where D(t,tj) is the discount factor used from time tj to t, and S(t,tj) is the risk-neutral survival probability up to time tj.5 In our empirical implementation, we obtain the survival probability for each firm by calibrating a constant default intensity λ for the life of the contract. This implies S(t,tj)= exp(−λtj−t360) and dS(t,u)=−λexp(−λu−t360)du in Equation (2.3). We use Bloomberg’s zero curve to compute the risk-free discount rates. While CDS levels display a high level of persistence, the presence of the last term in Equation (2.2) related to the CDS level, CDSt−1, adds only a very small persistent component to CDS returns. The first term in Equation (2.2), which depends on the change in CDS spreads and the present value of the risky annuity, is orders of magnitude larger than the last term. We show in Section 2.2 that the persistence in CDS returns is only slightly higher than the persistence in equity returns. Figure 1 plots the time series of weekly CDS returns for four representative firms with long time series of spreads available: IBM, McDonald’s, Toys “R” Us, and Walt Disney. The return series clearly exhibit episodes of heightened volatility which we model in the following section. The solid black line in Panel A of Figure 2 plots the time series of the median CDS spread across firms, and the gray area represents the interquartile range (IQR). Panel B presents the median and IQR for the CDS return volatilities obtained from GARCH models as discussed in Section 3.2 below. Figure 1. View largeDownload slide Weekly returns on short CDS positions for selected firms. We plot weekly returns on short CDS positions for four selected firms with long time series. We compute returns using Equations (2.2) and (2.3). Figure 1. View largeDownload slide Weekly returns on short CDS positions for selected firms. We plot weekly returns on short CDS positions for four selected firms with long time series. We compute returns using Equations (2.2) and (2.3). Figure 2. View largeDownload slide Quantiles of CDS spreads, equity prices, and return volatilities. We plot weekly quantiles across the 215 firms listed in Table I for CDS spreads, equity prices, and their return volatilities from GARCH models. We use a level-NGARCH model for CDS returns, and a NGARCH model for stock returns. In each panel, the black line reports the median across firms for each week, and the gray area shows the IQR across firms. The vertical lines indicate the major events during the sample period listed in Section 2.1. Figure 2. View largeDownload slide Quantiles of CDS spreads, equity prices, and return volatilities. We plot weekly quantiles across the 215 firms listed in Table I for CDS spreads, equity prices, and their return volatilities from GARCH models. We use a level-NGARCH model for CDS returns, and a NGARCH model for stock returns. In each panel, the black line reports the median across firms for each week, and the gray area shows the IQR across firms. The vertical lines indicate the major events during the sample period listed in Section 2.1. Panels C and D of Figure 2 replicate Panels A and B using the equity data. The equity price in Panel C is normalized to one for each firm at the start of the sample. The vertical lines in Figure 2 denote eight major events during our sample period: the WorldCom bankruptcy in July 2002, the Ford and GM downgrades to junk in May 2005, the Delphi bankruptcy on October 8, 2005, the Bear Stearns subprime funds collapse in July and August 2007, the Bear Stearns bankruptcy in March 2008, the Lehman Brothers bankruptcy in September 2008, the stock market bottom and CDS Big Bang in March and April 2009 (henceforth referred to simply as the stock market bottom), and the US sovereign debt downgrade in August 2011. 2.2 Stylized Facts Figure 2 illustrates some important stylized facts regarding the trends in credit risk during the sample period. Panel A of Figure 2 indicates that the time series of the median CDS spread and the IQR reach their maximums during the peak of the financial crisis in 2008–09. Less dramatic turbulence is also evident during the dot-com bust in 2002 and the US sovereign debt downgrade in 2011. Panel D of Figure 2 indicates that the time series pattern for equity volatility is similar to that for credit spreads in Panel A. The relationship between credit spreads and equity volatility is of course suggested by structural credit risk models such as Merton (1974). Panels A and D also suggest that CDS spreads and equity volatility are highly persistent over time. The IQRs in the four panels of Figure 2 also contain valuable insights into credit and equity dependence. The cross-sectional range of spreads in Panel A is much wider during the financial crisis compared with the pre-crisis years. This effect lingers on to some extent in the post-crisis period. The high post-crisis range in spreads suggests that investors may be able to at least partly diversify credit risk which is a key topic of interest for us. Figure 3 plots the median CDS spread in each industry. The 215 firms in our sample are distributed along the following 10 GIC sectors: Energy (12 firms), Materials (14), Industrials (25), Consumer Discretionary (64), Consumer Staples (16), Health Care (13), Financials (34), Information Technology (15), Telecommunications Services (14), and Utilities (8). For ease of exposition, Figure 3 combines the energy and utility sectors which separately have few firms. Figure 3. View largeDownload slide Median CDS spread by sector. We report the weekly median CDS spread by sector using the GIC sectors in Table I. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Note that the scale differs across sectors. Figure 3. View largeDownload slide Median CDS spread by sector. We report the weekly median CDS spread by sector using the GIC sectors in Table I. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Note that the scale differs across sectors. The impact of the financial crisis is obvious in Figure 3, but interestingly the crisis affected different industries quite differently. Some industries, Information Technology and Telecommunication Services in particular, were affected as much or even more by the 2001–03 dot-com bubble versus the 2007–09 crisis. In Table II, we report sample moments averaged across firms for CDS and equity returns. Panel A of Table II shows the median across firms of the first four sample moments of weekly CDS and equity returns along with the IQR for each moment. Average annualized equity returns are 10.48%. The average annualized returns to selling credit protection are much lower at 0.91%, which is not surprising given our sample period. We also report the Jarque–Bera tests for normality as well as the first two autocorrelation coefficients. Note the strong evidence of non-normality as well as some evidence of dynamics in the weekly CDS returns. We will model both of these features below. Table II Descriptive statistics on CDS and equity returns We report sample moments on weekly CDS and equity returns across firms. Panel A reports sample moments computed using weekly returns for CDS and equity. The CDS return is the return from selling credit protection. Panel B reports average sample correlations across firms using weekly returns. On the diagonal we report the median and IQR of the correlations between each firm and all other firms, using either credit or equity. On the off-diagonal we report the median and IQR of the correlation between CDS and equity returns for the same firm. Panel A: Sample moments on weekly returns    Annualized average (%)  Annualized standard deviation (%)  Skewness  Kurtosis  Jarque–Bera p-value  AR(1) coefficient  AR(2) coefficient    CDS                 Median  0.91  4.42  –0.83  19.79  0  0.10  0.04   IQR   [0.46, 1.70]  [2.56, 7.12]  [–1.79, 0.18]  [14.24, 33.74]  [0.00, 0.00]  [0.03, 0.19]  [–0.02, 0.09]  Equity                 Median  10.48  34.73  0.02  7.46  0  –0.04  –0.02   IQR  [6.19, 16.63]  [27.94, 43.23]  [–0.31, 0.38]  [5.91, 11.06]  [0.00, 0.00]  [–0.07, 0.00]  [–0.06, 0.02]    Panel B: Correlations of weekly returns      CDS  Equity              CDS                 Median  0.343  0.349             IQR  [0.219, 0.457]  [0.243, 0.451]            Equity                 Median    0.322             IQR    [0.248, 0.408]            Panel A: Sample moments on weekly returns    Annualized average (%)  Annualized standard deviation (%)  Skewness  Kurtosis  Jarque–Bera p-value  AR(1) coefficient  AR(2) coefficient    CDS                 Median  0.91  4.42  –0.83  19.79  0  0.10  0.04   IQR   [0.46, 1.70]  [2.56, 7.12]  [–1.79, 0.18]  [14.24, 33.74]  [0.00, 0.00]  [0.03, 0.19]  [–0.02, 0.09]  Equity                 Median  10.48  34.73  0.02  7.46  0  –0.04  –0.02   IQR  [6.19, 16.63]  [27.94, 43.23]  [–0.31, 0.38]  [5.91, 11.06]  [0.00, 0.00]  [–0.07, 0.00]  [–0.06, 0.02]    Panel B: Correlations of weekly returns      CDS  Equity              CDS                 Median  0.343  0.349             IQR  [0.219, 0.457]  [0.243, 0.451]            Equity                 Median    0.322             IQR    [0.248, 0.408]            In Panel B, we report statistics for the distribution of sample correlations between CDS and equity returns. On the diagonal we report the median and IQR of the correlations between each firm and all other firms, using either credit or equity. On the off-diagonal we report the same statistics for the distribution of the correlation between the CDS and equity returns on the same firm. The relatively high and robust positive correlation between weekly equity returns and weekly CDS returns is expected because we consider the returns to selling credit protection. In order to further explore the dependence across firms we compute threshold correlations, following Ang and Chen (2002) and Patton (2004), for example. For a pair of firms i and j, we define the threshold correlation ρ¯ij(x) using standardized returns R¯i and R¯j  ρ¯ij(x)={Corr(R¯i,R¯j|R¯i<x,R¯j<x) when x<0Corr(R¯i,R¯j|R¯i≥x,R¯j≥x) when x≥0,  (2.4) where we have standardized returns by their sample mean and standard deviation, and thus measure x as the number of standard deviations from the mean. The threshold correlation reports the linear correlation between two assets for the subset of observations lying in the bottom-left or top-right quadrant. In the case of the bivariate normal distribution the threshold correlation approaches zero when the threshold x goes to plus or minus infinity. Panels A and C of Figure 4 report the median and IQR of the bivariate threshold correlations computed across all possible pairs of firms. Both the CDS and the equity threshold correlations in Panels A and C are high and show some evidence of asymmetry: large downward moves are more highly correlated than large upward moves. Also, both Panels A and C in Figure 4 show strong evidence of multivariate non-normality. This is evidenced by the large deviations of the solid line (empirics) from the dashed lines (normal distribution). Adequately capturing these non-normalities motivates the non-normal copula approach below. Figure 4. View largeDownload slide Threshold correlations for CDS and equity returns and their AR-GARCH residuals. For each pair of firms we compute threshold correlations on a grid of thresholds defined using the standard deviation from the mean for each firm (horizontal axis). The solid lines show the median threshold correlations across firm pairs, the gray areas mark the IQRs and the dashed lines show the threshold correlations from a bivariate Gaussian distribution with correlation equal to the average for all pairs of firms. Figure 4. View largeDownload slide Threshold correlations for CDS and equity returns and their AR-GARCH residuals. For each pair of firms we compute threshold correlations on a grid of thresholds defined using the standard deviation from the mean for each firm (horizontal axis). The solid lines show the median threshold correlations across firm pairs, the gray areas mark the IQRs and the dashed lines show the threshold correlations from a bivariate Gaussian distribution with correlation equal to the average for all pairs of firms. 2.3 Credit Default Models Measuring default dependence has long been a problem of interest in the credit risk literature. For instance, a bank that manages a portfolio of loans is interested in how the borrowers’ creditworthiness fluctuates with the business cycle. While the change in the probability of default for an individual borrower is of interest, the most important question is how the business cycle affects the value of the overall portfolio, and this depends on default dependence. An investment company or hedge fund that invests in a portfolio of corporate bonds faces a similar problem. Over the last decade, the measurement of default dependence has taken on added significance because of the emergence of new portfolio and structured credit products, and as a result new methods to measure correlation and dependence have been developed. Different techniques are used to estimate default dependence. The oldest and most obvious way to estimate default correlation is the use of historical default data. In order to reliably estimate default probabilities and correlations, typically a large number of historical observations are needed which are not often available. See, for instance, deServigny and Renault (2002). The alternative to historical default data is the combination of a factor model with a model that extracts default intensities or default probabilities. For each of these two tasks, different models have proven especially useful. For publicly traded corporates, a Merton (1974) type structural model is often used to link equity returns or the prices of credit-risky securities to the underlying asset returns and extract default probabilities.6 This approach is usually combined with a one-factor CAPM-style model for the underlying equity return to model the default dependence in credit portfolios. Clearly, the reliability of the default dependence estimate is determined by the quality of the factor model. Alternatively, intensity-based models have become very popular in the academic credit risk literature over the last decade.7 This approach typically models the default intensity using a jump diffusion, and is also sometimes referred to as the reduced-form approach. Within this class of models, there are different approaches to modeling default dependence. One class of models, referred to as conditionally independent models or doubly stochastic models, assumes that cross-firm default dependence associated with observable factors determining conditional default probabilities is sufficient for characterizing the clustering in defaults. See Duffee (1999) for an example of this approach. Das et al. (2007) provide a test of this approach and find that this assumption is violated. Other intensity-based models consider joint credit events that can cause multiple issuers to default simultaneously, or they model contagion or learning effects, whereby default of one entity affects the defaults of others. See, for example, Davis and Lo (2001) and Jarrow and Yu (2001). Jorion and Zhang (2007) investigate contagion using CDS data. Our article instead uses copula methods to model CDS returns. See Joe (1997) and Patton (2009a, 2009b, 2013) for excellent overviews of copula modeling. Copulas have been used extensively for modeling default dependence, especially among practitioners, and for the purpose of CDO modeling. The advantage of the copula approach is its flexibility, because the parameters characterizing the multivariate default distribution, and hence the correlation between the default probabilities, can be modeled in a second stage, after the univariate distributions have been estimated. In some cases the copulas are also parsimoniously parameterized and computationally straightforward, which facilitates calibration. Calibration of the correlation structure is mostly performed using CDO data. The simple one-factor Gaussian copula is often used in the literature, but extensions to multiple factors (Hull & White, 2010), stochastic recovery rates (Hull & White, 2006), and non-Gaussian copulas provide a better fit. In contrast to existing static approaches, in our analysis of CDS returns the emphasis is on modeling dynamic dependence. Our approach also allows for multivariate asymmetries.8 Several existing papers use copulas from the Archimedean family to capture dependence asymmetries (see Patton, 2004, 2006b and Xu and Li, 2009), but this approach is difficult to generalize to higher dimensions, and our focus is on the analysis of a large portfolio of underlying credits. To capture time variation in dependence, some existing papers use regime switching models. See Chollete, Heinen, and Valdesogo (2009); Garcia and Tsafack (2011); Hong, Tu, and Zhou (2007); and Okimoto (2008) for examples. We instead follow the autoregressive approach of Christoffersen and Langlois (2013), Christoffersen et al. (2012), and De Lira Salvatierra and Patton (2015). Oh and Patton (2016b) develop a factor copula approach to analyze dynamic dependence for a large portfolio of underlying credits. Their model assumes a factor structure with dynamic loadings, which provides very flexible dynamics, but requires numerical integration to compute the likelihood. We do not impose a factor structure up front and our closed-form likelihood function enables us to implement the model simultaneously for many underlying credits. Oh and Patton (2016b) find empirical evidence of a structural break in the dynamics of CDS spreads around the Big Bang in April 2009. We do not formally test for a structural break in our model. We instead use our dynamic model to assess the evolution of dependence in the credit market relative to the equity market and to measure changes in diversification benefits over our sample period that includes various important events including the Big Bang. 3. Dynamic Models of Credit and Equity Returns Our dynamic model implementation proceeds in three steps. In the first step, we model the mean dynamics on the univariate time series of each CDS and stock return. In the second step, we model the variance dynamics and the distribution of the time-series residual for each firm. In the third step, we develop dynamic copula models for CDS and equity returns using all the firms in our sample. 3.1 Mean Dynamics The weekly return data contain some short-run dynamics. In order to obtain white-noise innovations required for consistent modeling of correlation dynamics, we fit univariate AR-NGARCH models to the time series. In a first step, we estimate an AR(12) and sequentially remove the least significant lag until all remaining lags are significant and as long as the p-value from a Ljung–Box test remains greater than 5%. This allows for the most parsimonious specification possible. If we select an AR(p) as the preferred specification   Rt=μ+φ1Rt−1+…+φpRt−p+εt, (3.1) where εt is assumed to be uncorrelated with Rs for s < t, the conditional mean for Rt constructed at the end of week t – 1 is simply   μt=μ+φ1Rt−1+…+φpRt−p. 3.2 Variance Dynamics In a second step we use volatility models specific to each instrument. For equity returns we fit an Engle and Ng (1993) NGARCH model to the AR filtered residuals εt. For the majority of firms we rely on the NGARCH(1, 1), but we estimate a NGARCH(2, 2) when the p-value of an ARCH test is lower than 5%. The NGARCH(2, 2) is given by   εt=σtztσt2=w+β1σt−12+β2σt−22+α1σt−12(zt−1−γ1)2+α2σt−22(zt−2−γ2)2w=σ¯2[1−β1−β2−α1(1+γ12)−α2(1+γ22)]zt∼i.i.d.ast(λ,ν) (3.2) where we constrain α1,α2, β1,β2>0, and the variance persistence to be <1, and set the unconditional variance, σ¯2, equal to the sample variance of εt. The i.i.d. return residuals zt are assumed to follow the asymmetric standardized t-distribution from Hansen (1994) which we denote ast(λ,ν). The skewness and kurtosis of the residual distribution are nonlinear functions of the parameters λ and ν. When λ  = 0 the symmetric standardized t-distribution is obtained. When λ = 0 and 1/ν=0, we get the normal distribution. The corresponding cumulative return probabilities are given by   ηt≡Pr⁡t−1(R<Rt)=σt−1∫−∞σt−1(Rt−μt)ast(z;λ,ν)dz. (3.3) Note that the individual residual distributions are constant through time, but the individual return distributions do vary through time because the return mean and variance are dynamic. Using time series observations on εt, the parameters β1, β2, α1, α2, γ1, γ2, λ and ν are estimated using a likelihood function based on (3.2) and ast(z;λ,ν). When estimating the standard NGARCH models on CDS returns we found that in many cases they did not pass standard specification tests. Motivated by Brenner, Harjes, and Kroner (1996) we therefore instead estimate a level NGARCH model defined by   εt=σtztσt2=ΨtCDSt2δΨt=ω+β1Ψt−1+α1Ψt−1(zt−1−γ1)2 (3.4) where the NGARCH mean-reverting dynamic Ψt is multiplied by a power of the current CDS spread level, CDSt2δ. In our empirical application this specification adequately captures the variance dynamics in CDS returns. We estimate the parameters δ, ω, β1, α1, γ1, λ, and ν using a likelihood function based on (3.4) and ast(z;λ,ν). 3.3 Estimates of Mean and Variance Dynamics Panel A of Table III reports for each of the twelve autoregressive models the percentage of firms chosen by our model selection procedure. On average fewer than three lags are used to remove the autocorrelation in CDS returns, but the autocorrelation structure is much richer for some firms. The autocorrelation structure for equity returns is more parsimonious than for CDS returns. Panel A also shows the median and IQR across firms of each AR coefficient estimate. The parameter values vary considerably across firms. Table III Summary of AR-NGARCH estimation on weekly returns For each firm, we estimate in a first step an AR(12) model and remove the least significant lag until all lags are significant as long as a Ljung–Box test on residuals is not rejected at the 5% level. Panel A contains summary statistics on chosen AR specifications. For stock returns, we then estimate a NGARCH(1,1) model on residuals. If the Ljung–Box test on squared standardized equity residuals is rejected at the 5%, we estimate an NGARCH(2,2). For CDS returns, we estimate a Level-NGARCH(1,1). In both cases, the residual distribution is asymmetric t with parameters ν and λ. Panel B contains median and IQRs for NGARCH parameters as well as summary statistics on chosen model specifications. L-B(4) denotes a Ljung–Box test with four lags that the squared residuals (Panel B) are serially uncorrelated. Panel A: Conditional mean dynamics    CDS returns  Equity returns  Lag  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters    1  51%  0.18  [0.124, 0.240]  17%  –0.11  [–0.132, –0.063]  2  21%  0.12  [–0.105, 0.166]  12%  –0.10  [–0.112, –0.070]  3  21%  0.12  [–0.103, 0.159]  10%  0.09  [–0.035, 0.120]  4  23%  –0.13  [–0.163, –0.088]  7%  –0.08  [–0.101, –0.040]  5  14%  –0.09  [–0.151, 0.144]  5%  0.07  [–0.059, 0.101]  6  17%  0.12  [–0.128, 0.171]  7%  –0.06  [–0.121, 0.085]  7  19%  –0.12  [–0.142, –0.067]  26%  –0.11  [–0.133, –0.087]  8  12%  0.11  [0.062, 0.121]  5%  –0.03  [–0.084, 0.088]  9  7%  –0.09  [–0.149, 0.046]  6%  –0.08  [–0.107, –0.048]  10  41%  –0.14  [–0.182, –0.106]  10%  –0.08  [–0.114, –0.067]  11  12%  –0.10  [–0.123, 0.117]  7%  0.07  [–0.016, 0.105]  12  12%  –0.08  [–0.100, 0.092]  6%  –0.09  [–0.113, –0.055]  Average no. of lags used  2.49      1.19      Panel B: Conditional volatility dynamics and return distribution    Parameter estimates    CDS returns  Equity returns          β1  Median  0.861  0.808          IQR  [0.823, 0.902]  [0.674, 0.864]        β2  Median  –  0.276          IQR    [0.002, 0.528]        α1  Median  0.107  0.055          IQR  [0.078, 0.140]  [0.035, 0.094]        α2  Median  –  0.005          IQR    [0.000, 0.014]        γ1  Median  0.407  1.098          IQR  [0.157, 0.622]  [0.711, 1.654]        γ2  Median  –  8.897          IQR    [2.646, 12.518]        ν  Median  3.739  6.839          IQR  [3.210, 4.610]  [5.262, 9.006]        λ  Median  –0.157  –0.040          IQR  [–0.201, –0.102]  [–0.096, 0.008]        δ  Median  0.337  –          IQR  [0.256,0.568]          L-B(4) p-value z2 > 5%  Proportion of firms  93%  92%        NGARCH(1,1)  Proportion of firms  100%  86%        NGARCH(2,2)  Proportion of firms  0%  8%        Panel A: Conditional mean dynamics    CDS returns  Equity returns  Lag  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters    1  51%  0.18  [0.124, 0.240]  17%  –0.11  [–0.132, –0.063]  2  21%  0.12  [–0.105, 0.166]  12%  –0.10  [–0.112, –0.070]  3  21%  0.12  [–0.103, 0.159]  10%  0.09  [–0.035, 0.120]  4  23%  –0.13  [–0.163, –0.088]  7%  –0.08  [–0.101, –0.040]  5  14%  –0.09  [–0.151, 0.144]  5%  0.07  [–0.059, 0.101]  6  17%  0.12  [–0.128, 0.171]  7%  –0.06  [–0.121, 0.085]  7  19%  –0.12  [–0.142, –0.067]  26%  –0.11  [–0.133, –0.087]  8  12%  0.11  [0.062, 0.121]  5%  –0.03  [–0.084, 0.088]  9  7%  –0.09  [–0.149, 0.046]  6%  –0.08  [–0.107, –0.048]  10  41%  –0.14  [–0.182, –0.106]  10%  –0.08  [–0.114, –0.067]  11  12%  –0.10  [–0.123, 0.117]  7%  0.07  [–0.016, 0.105]  12  12%  –0.08  [–0.100, 0.092]  6%  –0.09  [–0.113, –0.055]  Average no. of lags used  2.49      1.19      Panel B: Conditional volatility dynamics and return distribution    Parameter estimates    CDS returns  Equity returns          β1  Median  0.861  0.808          IQR  [0.823, 0.902]  [0.674, 0.864]        β2  Median  –  0.276          IQR    [0.002, 0.528]        α1  Median  0.107  0.055          IQR  [0.078, 0.140]  [0.035, 0.094]        α2  Median  –  0.005          IQR    [0.000, 0.014]        γ1  Median  0.407  1.098          IQR  [0.157, 0.622]  [0.711, 1.654]        γ2  Median  –  8.897          IQR    [2.646, 12.518]        ν  Median  3.739  6.839          IQR  [3.210, 4.610]  [5.262, 9.006]        λ  Median  –0.157  –0.040          IQR  [–0.201, –0.102]  [–0.096, 0.008]        δ  Median  0.337  –          IQR  [0.256,0.568]          L-B(4) p-value z2 > 5%  Proportion of firms  93%  92%        NGARCH(1,1)  Proportion of firms  100%  86%        NGARCH(2,2)  Proportion of firms  0%  8%        Panel B in Table III shows the median and IQR across firms for each of the NGARCH parameters as well as the two parameters of the asymmetric t-distribution. For CDS returns we estimate a Level-NGARCH model for each of the 215 firms. For the equity returns we estimate an NGARCH(2,2) model for eighteen firms and an NGARCH(1,1) model for the remaining firms. The γ1 and γ2 parameters capture the asymmetric volatility response to positive and negative return residuals. For equities, the median γ1 value is 1.1 and the IQR is entirely positive. For CDS returns the median and IQR of γ1 are smaller in magnitude, but also positive. The Ljung–Box test of serial correlation in the zt2 shows that the NGARCH model is able to adequately capture variance dynamics. Equity returns have 8% of NGARCH models rejected by Ljung–Box at the 5% level, which is clearly not drastically above the size of the test. For CDS returns the level NGARCH model is rejected for 7% of the firms at the 5% significance level. The ν parameter has a median of 3.7 for CDS returns and 6.8 for equity returns, respectively, indicating fat tails in the conditional distribution. The asymmetry parameter, λ, is generally negative for both equity and CDS returns and larger in absolute value for CDS returns. Recall again that the CDS returns capture the returns to selling credit protection. As discussed above, Panel B of Figure 2 shows the time paths for the median and IQR of CDS return volatilities. Comparing the path of CDS volatility in Panel B and the path of equity volatility in Panel D is interesting. The time paths of the medians and the IQR are clearly moving together. Note that the relationship between equity volatility and CDS spreads has been extensively studied motivated by the Merton (1974) model. The relation between equity returns and equity volatility has also been extensively analyzed in the empirical literature. Panel C of Figure 2 confirms that equity returns are negatively related to equity volatilities in Panel D. The sample correlation of the medians is –0.65. This stylized fact is usually referred to as the leverage effect, and it leads to negative skewness in the return distribution. However, little is known about the relation between CDS return volatility and CDS returns. Note that our analysis focuses on the returns to selling credit protection. Visual inspection of Panels A and B suggests a positive relation between CDS spread levels and CDS return volatility which from Equation (2.2) implies a negative relation between the return to credit protection selling and CDS volatility. The sample correlation between the median CDS spread levels and the median CDS return volatility is 0.92. Figure 5 plots the median of the weekly level NGARCH dynamic in CDS returns for the nine industries from Figure 3. The variation of CDS return volatility across industries is quite dramatic. The high level of volatility in the financial crisis is apparent, but for the Information Technology and Telecommunication industries credit volatility is even more elevated in the aftermath of the bursting of the technology bubble. Figure 5. View largeDownload slide Median conditional volatility of CDS returns by sector. We report the weekly median conditional volatility of CDS returns by sector using the GIC sectors in Table I. Conditional volatility is estimated using a level-NGARCH model for each firm. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Figure 5. View largeDownload slide Median conditional volatility of CDS returns by sector. We report the weekly median conditional volatility of CDS returns by sector using the GIC sectors in Table I. Conditional volatility is estimated using a level-NGARCH model for each firm. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Panel A of Table IV contains descriptive statistics of the AR-NGARCH model residuals. Skewness and kurtosis are still present after standardizing by the NGARCH model, which motivates our use of the asymmetric standardized t innovations. As expected, the correlations between CDS and equity returns residuals are not materially different from the raw return correlations in Panel C of Table II. Table IV AR-NGARCH residual statistics and dynamic copula parameter estimation We report sample statistics on AR-NGARCH residuals and estimation results for different copula models. Using the AR-NGARCH residuals, z, we compute in Panel A the median and IQR of the skewness, kurtosis, correlations for each pair of firms, and correlations between CDS and equity for each firm. We estimate the DAC and the DSC with and without a time trend for the degree of freedom, and the DNC models on the 215 firms in our sample. Each of the models is estimated on AR-NGARCH residuals from weekly returns from selling the CDS contract and buying the equity. In the last two rows of Panel B and C, we report the Rivers and Vuong (2002) test statistics for model comparison. Panel A: Residual sample moments      Skewness  Kurtosis  Cross-firm correlation  Cross-security correlation      CDS returns  Median  –1.798  14.344  0.333  0.301      IQR  [–2.971, –1.002]  [8.191, 31.458]  [0.245, 0.416]  [0.215, 0.376]    Equity returns  Median  –0.173  4.909  0.295        IQR  [–0.403, –0.004]  [4.058, 6.322]  [0.232, 0.369]        Panel B: DAC estimation        Model I: νc(t) = 4 + δC,0  Model II: νc(t) = 4 + δC,0 exp( δC,1t )  Model III: νc(t) = 4 + δC,0 exp(δC,1t + δC,2t2 )    CDS returns  Equity returns  CDS returns  Equity returns  CDS returns  Equity returns    βC  0.951  0.918  0.957  0.926  0.957  0.926  αC  0.019  0.019  0.016  0.018  0.016  0.018  Correlation persistence  0.971  0.937  0.973  0.944  0.973  0.944  δC,0  12.859  8.944  59.773  13.731  59.773  13.731  δC,1      –3.80E-03  –1.90E-03  –3.80E-03  –1.90E-03  δC,2          1.78E-10  –1.71E-09  λC  –0.082  –0.242  –0.049  –0.218  –0.049  –0.218  Composite log-likelihood  978,813  682,796  980,018  684,464  980,032  684,473  Model comparison t-statistic with DSC Model I  0.10  2.72  0.48  2.47  0.48  2.46  Model comparison t-statistic with DAC Model I  –  –  0.42  0.83  1.19  0.21  Panel C: DSC estimation        Model I: νc(t) = 2 + δC,0  Model II: νc(t) = 2 + δC,0 exp( δC,1t )        CDS returns  Equity returns  CDS returns  Equity returns        βC  0.956  0.913  0.957  0.914      αC  0.016  0.019  0.016  0.019      Correlation persistence  0.972  0.932  0.973  0.933      δC,0  13.455  11.095  31.893  18.338      δC,1      –2.42E-03  –1.69E-03      Composite log-likelihood  978,685  672,099  981,489  673,747      Model comparison t-statistic with DNC  5.73  4.69  5.15  4.41      Model comparison t-statistic with DSC Model I  –  –  1.48  1.12        Panel D: DNC estimation        CDS Returns  Equity Returns            βC  0.956  0.908          αC  0.016  0.018          Correlation persistence  0.971  0.927          Composite log-likelihood  936,795  638,289          Panel A: Residual sample moments      Skewness  Kurtosis  Cross-firm correlation  Cross-security correlation      CDS returns  Median  –1.798  14.344  0.333  0.301      IQR  [–2.971, –1.002]  [8.191, 31.458]  [0.245, 0.416]  [0.215, 0.376]    Equity returns  Median  –0.173  4.909  0.295        IQR  [–0.403, –0.004]  [4.058, 6.322]  [0.232, 0.369]        Panel B: DAC estimation        Model I: νc(t) = 4 + δC,0  Model II: νc(t) = 4 + δC,0 exp( δC,1t )  Model III: νc(t) = 4 + δC,0 exp(δC,1t + δC,2t2 )    CDS returns  Equity returns  CDS returns  Equity returns  CDS returns  Equity returns    βC  0.951  0.918  0.957  0.926  0.957  0.926  αC  0.019  0.019  0.016  0.018  0.016  0.018  Correlation persistence  0.971  0.937  0.973  0.944  0.973  0.944  δC,0  12.859  8.944  59.773  13.731  59.773  13.731  δC,1      –3.80E-03  –1.90E-03  –3.80E-03  –1.90E-03  δC,2          1.78E-10  –1.71E-09  λC  –0.082  –0.242  –0.049  –0.218  –0.049  –0.218  Composite log-likelihood  978,813  682,796  980,018  684,464  980,032  684,473  Model comparison t-statistic with DSC Model I  0.10  2.72  0.48  2.47  0.48  2.46  Model comparison t-statistic with DAC Model I  –  –  0.42  0.83  1.19  0.21  Panel C: DSC estimation        Model I: νc(t) = 2 + δC,0  Model II: νc(t) = 2 + δC,0 exp( δC,1t )        CDS returns  Equity returns  CDS returns  Equity returns        βC  0.956  0.913  0.957  0.914      αC  0.016  0.019  0.016  0.019      Correlation persistence  0.972  0.932  0.973  0.933      δC,0  13.455  11.095  31.893  18.338      δC,1      –2.42E-03  –1.69E-03      Composite log-likelihood  978,685  672,099  981,489  673,747      Model comparison t-statistic with DNC  5.73  4.69  5.15  4.41      Model comparison t-statistic with DSC Model I  –  –  1.48  1.12        Panel D: DNC estimation        CDS Returns  Equity Returns            βC  0.956  0.908          αC  0.016  0.018          Correlation persistence  0.971  0.927          Composite log-likelihood  936,795  638,289          Finally, Panels B and D of Figure 4 plot the median and IQR threshold correlations on the weekly AR-NGARCH residuals. Compared to the threshold correlations from raw returns in Panels A and C, the median threshold correlations in residuals are typically lower, but still higher than the bivariate Gaussian distribution (dashed lines) would suggest. Overall, Figure 4 indicates that by removing univariate non-normality from the data, the AR-NGARCH models are also able to remove some of the multivariate non-normality. Modeling the remaining multivariate non-normality is the task to which we now turn. 3.4 Dynamic Copula Functions From Patton (2006b), who builds on Sklar (1959), we can decompose the conditional multivariate density function of a vector of returns for N firms, ft(Rt), into a conditional copula density function, ct, and the product of the conditional marginal distributions fi,t(Ri,t) as follows   ft(Rt)=ct(F1,t(R1,t),F2,t(R2,t),…,FN,t(RN,t))∏i=1Nfi,t(Ri,t)=ct(η1,t,η2,t,…,ηN,t)∏i=1Nfi,t(Ri,t), (3.5) where Rt is now a vector of N returns at time t, fi,t is the density, and Fi,t is the cumulative distribution function of Ri,t. Following Christoffersen et al. (2012) and Christoffersen and Langlois (2013) we allow for dependence across the return residuals using the copula implied by the skewed t-distribution discussed in Demarta and McNeil (2005). The skewed t copula cumulative distribution function Ct for N firms can be written as   Ct(η1,t,η2,t,…,ηN,t;Ψ,λC,νC,t)=tΨ,λC,νC,t(tλC,νC,t−1(η1,t),tλC,νC,t−1(η2,t),…,tλC,νC,t−1(ηN,t)), (3.6) where λC is a scalar parameter capturing copula asymmetry, νC,t is a time-varying scalar parameter denoting the copula degree of freedom, tΨ,λC,νC,t is the multivariate skewed t density with correlation matrix Ψ, and tλC,νC,t−1 is the inverse cumulative distribution function of the corresponding univariate skewed t-distribution. Note that the copula correlation matrix Ψ is defined using the correlation of the “copula residuals” zi,t∗≡tλC,νC,t−1(ηi,t) and not of the return residuals zi,t. If the marginal distribution in (3.3) is close to the copula tλC,νC,t distribution, then zt∗ will be close to zt. We now build on the linear correlation techniques developed by Engle (2002) and Tse and Tsui (2002) to model dynamic copula correlations. We use the “copula residuals” zi,t∗≡tλ,ν−1(ηi,t) as the model’s building block instead of the return residuals zi,t. In the case of non-normal copulas, the fractiles do not have zero mean and unit variance, and we therefore standardize zi,t∗ before proceeding. The copula correlation dynamic is driven by   Γt=(1−βC−αC)Ω+βCΓt−1+αCz¯t−1∗z¯t−1∗⊤ (3.7) where βC and αC are positive scalars, and z¯t∗ is an N-dimensional vector with typical element z¯i,t∗=zi,t∗Γii,t. The matrix Γt is a weighted average of three components: a constant matrix Ω which captures the average level of correlation, an auto-regressive term Γt−1 reflecting how conditional correlations in the previous period affect current correlations, and the cross-product of z¯t−1∗ which depicts how lagged (transformed) return shocks impact current correlations. The weighting parameters, βC and αC, describe how current correlations are affected by these components; a higher αC indicates that a positive (negative) cross-product of shocks, z¯i,t−1∗z¯j,t−1∗⊤, translates into a larger increase (decrease) in Γij,t compared to a smaller αC value, whereas a higher βC smooths over time the correlation deviations from their average level Ω. The conditional copula correlations are defined via the normalization   Ψij,t=Γij,t/Γii,tΓjj,t. To allow for general patterns in tail dependence, we allow for slowly moving trends in the degrees of freedom. Following Engle and Rangel (2008), who model the trend in volatility, we define the degree of freedom at time t, νC,t, using an exponential quadratic spline   νC,t=νC̲+δC,0 exp ⁡(δC,1t+∑j=1kδC,j+1max⁡(t−tj−1,0)2) (3.8) where νC̲ is the lower bound for the degrees of freedom, which is equal to four for the skewed t copula, δC,0,…,δC,k+1 are scalar parameters to be estimated, and {t0=0,t1,…,tk=T} denotes a partition of the sample in k segments of equal length. The specification of k different segments allows us to capture periods of positive and negative trends in the degrees of freedom. The exponential form ensures that the degrees of freedom are positive and above their lower bound at all times. Note that we model degree of freedom dynamics using splines and not lagged returns, because—unlike for variance and correlation—it is not obvious what the functional form of the lagged return should be when updating the degree of freedom process. In the next section, we investigate the time variation in both correlations and tail dependence. Whereas correlation at time t is driven by the dynamic in Equation (3.7), tail dependence is determined by both the time-varying correlation and the degrees of freedom. Hence, our model allows for changes in tail dependence that are separate from those in correlation. Below we refer to the model using (3.6)-(3.8) as the Dynamic Asymmetric Copula (DAC) model. The special case where λC=0, we denote by the Dynamic Symmetric Copula (DSC). In this case the lower bound for the degree of freedom is νC̲=2. When we additionally impose 1/νC=0, we obtain the Dynamic Normal Copula (DNC). Following Engle, Shephard, and Sheppard (2008), we estimate the copula parameters αC, βC, λC, and δC using the composite likelihood (CL) function defined by   CL(αC,βC,λC,δC)=∑i=1N∑j>1∑t=1Ti,jln⁡ct(ηi,t,ηj,t;αC,βC,λC,δC), (3.9) where ct is the copula density from (3.5) and Ti,j is the sample size available for pair i and j. Note that the CL function is built from the bivariate likelihoods so that the inversion of large-scale correlation matrices is avoided. In a sample as large as ours, relying on the CL approach is imperative. The unconditional correlations are estimated by unconditional moment matching (See Engle and Mezrich, 1996)  Ω^i,j=1Ti,j∑t=1Ti,jz¯i,t∗z¯j,t∗, (3.10) which is another crucial element in the feasible estimation of large-scale dynamic models. As discussed above, the estimation of dynamic dependence models using long time series and large cross-sections is computationally intensive. In our case, estimating the dynamic copula models for 215 firms is possible only because we implement unconditional moment matching and the CL approach. An additional advantage of the CL approach is that we can use the longest time span available for each firm pair when estimating the model parameters, thus making the best possible use of a cross-section of CDS time series of unequal length. 3.5 Copula Correlation and Tail Dependence Estimates Panel B of Table IV contains the Dynamic Asymmetric Copula (DAC) parameter estimates and CLs from fitting the model to the 215 firms in our sample. We again present separate results for models estimated on the weekly residuals of CDS and equity returns. The different GARCH specifications required for CDS and stock returns and the differences in correlation persistence demonstrate the importance of modeling separate dynamics for volatility and correlation. Panel C of Table IV reports the parameter estimates for the Dynamic Symmetric Copula (DSC) model where λC=0 and Panel D reports on the Dynamic Normal Copula (DNC) where we also impose 1/νC,t=0. To formally test for the difference in composite log-likelihood across models, we follow Oh and Patton (2016a) who apply the Rivers and Vuong (2002) tests to copula models estimated by composite log-likelihood maximization. Under the null hypothesis that two competing models fit the data equally well, the standardized difference in composite log-likelihood has a standard normal distribution. We use a Newey–West estimator with T14≈5 lags to estimate the standard error of the difference in composite log-likelihoods. The second-to-last line in Panel C reports t-statistics comparing the DSC models to the DNC models in Panel D. A positive value implies that the model in the top row of the panel is preferred to its DNC version. Each DSC model is significantly preferred to the Normal model. The second-to-last line in Panel B reports t-statistics comparing the DAC models to the DSC models in Panel C. The improvement in fit is largest for equities. This result matches the patterns in the threshold correlations in Figure 4 which show the strongest degree of bivariate asymmetry for equities. We also estimate a model with a simple time trend for the copula degree of freedom. The estimates in columns 3 and 4 of Panel B indicate that degrees of freedom have been trending down for both CDS and equity thus increasing the degree of multivariate non-normality. In columns 5 and 6, we allow for more complex shapes in degrees of freedom by increasing the number of splines in Equation (3.8), and find that the decreasing time trend is robust. Remarkably, the multivariate asymmetry parameters λC are also largely unaffected by the inclusion of a quadratic trend for the degree of freedom dynamics. The last row in Panel B reports the t-statistics obtained when comparing the models with time-varying degrees of freedom (columns 3 to 6) to the models with constant degrees of freedom (columns 1 and 2). Given the modest improvement in likelihood obtained by adding a quadratic segment, in the rest of the article we focus on the DAC specification with a simple log-linear time trend in the degree of freedom (in columns 3 and 4). Figure 6 plots the median and IQR of the DAC copula correlations for CDS and stock returns. The level of the CDS return correlation is higher than that of equity correlation throughout the sample. Credit correlations in Panel A show a pronounced and persistent uptick in 2007 around the time of the subprime funds collapse, and another pronounced uptick in mid-2011 following the US sovereign downgrade. The equity correlations in Panel B show less persistent upticks in late 2008 following the Lehman bankruptcy, and again in mid-2011 following the US sovereign downgrade. The differences in persistence following these major events are of course related to the differences in copula correlation persistence between credit and equity in Panel D of Table IV. Figure 6. View largeDownload slide Quantiles of copula correlations: CDS and equity returns. Using all available pairs of firms we report the weekly median (black line), and IQR (gray area) of the dynamic correlations from the DAC model. Figure 6. View largeDownload slide Quantiles of copula correlations: CDS and equity returns. Using all available pairs of firms we report the weekly median (black line), and IQR (gray area) of the dynamic correlations from the DAC model. Figure 7 plots the median and IQR of the DAC copula tail dependence for CDS and stock returns. Lower tail dependence is formally defined via the probability limit   τi,j,tL=lim⁡ζ→0Pr⁡[ηi,t≤ζ|ηj,t≤ζ]=lim⁡ζ→0Ct(ζ,ζ)ζ (3.11) where ζ is the tail probability.9 Upper tail dependence is defined analogously. We obtain an approximation by numerically integrating the copula density function, and using the 0.001 quantile for lower and the 0.999 quantile for upper tail dependence. We plot the lower tail dependence, because it is the most interesting of the two tails from a risk perspective. Importantly, Figure 7 shows that equity and credit tail dependence increase more over the sample than the copula correlations in Figure 6. Figure 7. View largeDownload slide Quantiles of tail dependence: CDS and equity returns. Using all available pairs of firms we report the median (black line) and IQR (gray area) of the dynamic tail dependence from the DAC model. Figure 7. View largeDownload slide Quantiles of tail dependence: CDS and equity returns. Using all available pairs of firms we report the median (black line) and IQR (gray area) of the dynamic tail dependence from the DAC model. In Figure 8, we plot the median DAC correlation of CDS and equity returns for the nine industries in Figure 3. Figure 9 does the same for the tail dependence of credit and equity. While the uptick in credit correlation during 2007 is evident for most industries, the variation across industries is large. The time paths of credit and equity tail dependence are clearly different from each other and are fairly similar across industries, because all industries share the same trend. Figure 8. View largeDownload slide Median copula correlation within sectors: CDS and equity returns. We report the median dynamic copula correlation by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity correlations. We combine the energy and utility sectors. Figure 8. View largeDownload slide Median copula correlation within sectors: CDS and equity returns. We report the median dynamic copula correlation by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity correlations. We combine the energy and utility sectors. Figure 9. View largeDownload slide Median lower tail dependence within sectors: CDS and equity returns. We report the median lower tail dependence by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity tail dependence. We combine the energy and utility sectors. Figure 9. View largeDownload slide Median lower tail dependence within sectors: CDS and equity returns. We report the median lower tail dependence by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity tail dependence. We combine the energy and utility sectors. 4. Economic Applications In this section, we apply our estimated credit volatilities and correlations to the measurement of portfolio diversification benefits and to credit spread prediction. Finally, we investigate potential economic drivers of the credit return correlations. 4.1 Conditional Diversification Benefits Consider a financial institution that has sold credit protection on a broad array of corporate names, thus collecting the CDS premia while hoping to diversify default risk. While the premia collected are readily observed, the institution needs a model such as that developed above to provide real-time quantification of the expected benefits from diversification. To be specific, we consider an equal-weighted portfolio of the constituents of the on-the-run CDX investment grade index in any given week. We want to assess the diversification benefits of the portfolio using the dynamic, non-normal copula model developed above. As in Christoffersen et al. (2012), we define the conditional diversification benefit by   CDBt(p)≡ES¯t(p)−ESt(p)ES¯t(p)−ES̲t(p), (4.1) where ESt(p) denotes the expected shortfall with probability threshold p of the portfolio at hand, ES¯t(p) denotes the average of the ES across firms, which is an upper bound on the portfolio ES, and ES̲t(p) is the portfolio VaR, which is a lower bound on the portfolio ES. The CDBt(p) measure takes values on the [0,1] interval, and is increasing in the level of diversification benefit. Note that by construction the conditional diversification benefit (CDB) does not depend on the level of expected returns. Expected shortfall is additive in the conditional mean which thus cancels out in the numerator and denominator in (4.1). Note also that CDB takes into account diversification benefits arising from all higher-order moments and not just variance. The CDB measure depends on the threshold probability p. Below we consider p=5%. The CDB measure is not available in closed form for our dynamic copula model and so we compute it using Monte Carlo simulation. Each week t we form an equally weighted portfolio of the 125 companies in the CDX.NA.IG index at time t. We use the longest available history of returns up to week t – 1 to estimate the unconditional correlation matrix for the 125 firms, and then compute the conditional correlations from our DAC model. In order to have sufficient historical data available, we keep only firms with at least 2 years of data, and start on September 22, 2004, which is the first day of Series 3 of the index. The solid black line in Figure 10 shows the CDB(5%) measure for an equal-weighted portfolio selling credit protection as well as for an equal-weighted portfolio of equity returns for the same firms. First, consider Panel A: diversification benefits for CDS have declined from about 80% at the end of 2004 to around 50% at the end of our sample. The majority of the decline took place during the mid-2007 to mid-2008 period and was relatively gradual. Panel B shows that the decline in diversification benefits in equity markets has been smaller in magnitude, from just over 70% in 2007 to around 60% at the end of our sample. The majority of the decline in equity market diversification benefits took place at the end of 2008. The decline in diversification benefits from equity occurred later than the decline in diversification benefits in credit. This is consistent with the correlation and tail dependence patterns in Figures 6 and 7. Figure 10. View largeDownload slide Conditional diversification benefits. Credit and Equity Portfolios. 5% Tail. Using equally weighted portfolios of available on-the-run CDX firms in a given week, we compute the 5% CDB using the DAC model for CDS returns (Panel A) and equity returns (Panel B). The credit portfolio sells credit protection by shorting CDS contracts. The dashed line shows the average correlation (on the left-hand axis) and the gray line shows the average volatility (on the right-hand axis). We use the first 2 years of the sample to estimate the unconditional correlation matrix, and thus plot CDB starting in 2004. Figure 10. View largeDownload slide Conditional diversification benefits. Credit and Equity Portfolios. 5% Tail. Using equally weighted portfolios of available on-the-run CDX firms in a given week, we compute the 5% CDB using the DAC model for CDS returns (Panel A) and equity returns (Panel B). The credit portfolio sells credit protection by shorting CDS contracts. The dashed line shows the average correlation (on the left-hand axis) and the gray line shows the average volatility (on the right-hand axis). We use the first 2 years of the sample to estimate the unconditional correlation matrix, and thus plot CDB starting in 2004. Figure 10 also depicts the average volatilities (in gray, on the right-hand axis) and the average correlations (the dashed line, on the left-hand axis). Intuitively, changes in the diversification measure should be closely (inversely) related to changes in correlation, which captures risk that is more systematic in nature. The relationship with average volatility is less obvious ex ante due to the normalization of the CDB measures. Figure 10 indeed shows that overall diversification benefits are much more closely related to average correlations than to average volatilities. Average correlation and average volatility sometimes co-move strongly—perhaps most obviously during the US sovereign debt downgrade in August 2011 which triggered a large drop in CDB. It is also interesting to relate credit diversification benefits in Panel A to equity volatilities in Panel B. The average volatilities in Panel B are highly correlated with the VIX and with other indicators of turmoil in equity markets. Clearly, the majority of the decline in diversification benefits in credit markets took place well before the peak in equity market volatility. The credit market CDB actually increased a bit during late 2008 and early 2009 when equity market turmoil was most intense. We conclude that while the data confirm the relationship between the level of credit spreads and equity volatility predicted by Merton-type structural models (see Figure 2), credit diversification benefits are—partly by design—more tightly linked with correlations. Above we have discussed various differences and similarities between the returns to holding equity on the one hand and selling credit protection on the other. The differences we find could to some extent be driven by counterparty credit risk priced into the CDS spreads. Using a proprietary data set on multiple dealers, Arora, Gandhi, and Longstaff (2012) indeed find that counterparty credit risk is priced in the CDS market. However, they also find that the effect is “vanishingly” small. We, therefore, do not attempt to identify counterparty credit risk in our empirical analysis. 4.2 Economic Determinants of Credit Spreads We now investigate if our new dynamic credit risk measures predict credit spreads when controlling for the usual economic drivers of credit spreads. The determinants of credit spreads have been extensively studied both theoretically and empirically. Most notably, following the analysis of Merton (1974), structural models of credit risk have established volatility, interest rates, and leverage as prime candidates to explain credit spreads. Partly based on theory, there is an extensive empirical literature regarding the determinants of credit spreads, both using bond data and CDS data. This literature provides some support for structural models of credit risk, and has also documented other macroeconomic and firm-specific determinants of credit risk.10 The definitions of the variables we consider are detailed in the appendix. We investigate if our CDS dependence measures, credit correlation and tail dependence, help predict changes in credit spreads. We also analyze the impact of CDS volatility on credit spread changes. We present results for univariate regressions of credit spread changes on the average copula correlation and tail dependence for each firm with all other firms, but we also present results for multivariate regressions where these dependence measures are added to equity volatility, term structure variables, and leverage, which are the determinants of credit risk according to the Merton (1974) model. Several studies have specifically questioned the ability of regressors suggested by theory to explain time-series variation in spreads (see Collin-Dufresne, Goldstein, and Martin, 2001), so we focus on pooled time-series regressions. We include lagged changes in spreads as regressors because of the persistence in the spreads. The signs of the estimated coefficients do not change when lagged spreads are not included (not reported). Table V presents the results. We run predictive panel regressions with firm fixed effects and double-clustering in which all predictors are lagged by 1 week. A first important conclusion is that CDS correlation and tail dependence in regressions (x) and (xi) are positively and significantly related to changes in credit spreads. We obtain these results even when including additional control variables in the regressions. In univariate regressions (v) and (vi), we see that the first lag of the spread changes captures a larger part of its future variation. CDS correlation and tail dependence are still positively related to future spread changes, but are not as significant as in the multivariate regressions. Finally, we find some support for the theory underlying structural credit risk models. Credit spreads increase with equity volatility, but leverage is not significant. Table V Panel regressions for changes in log CDS spreads We report regression coefficients and adjusted R2 from panel regressions. The left-hand side variable is the weekly change in log CDS spread for each firm. Right-hand side variables include the firm’s leverage ratio, the weekly and 1-year trailing return on the S&P 500 index, the 3-month constant maturity US Treasury rate, the difference between the 10-year and the 3-month constant maturity US Treasury rates, the TED spread, the firm-level equity NGARCH volatility and illiquidity, the firm-level CDS NGARCH volatility and illiquidity, the average equity and CDS illiquidity, the average CDS correlation with all other firms, and the average CDS tail dependence with all other firms. Illiquidity for equity is the Amihud price impact measure. Illiquidity for CDS the modified Amihud (MA) measure. All regressors are lagged, and we also include the first lag of the regressand. We run panel regressions with firm fixed effects. Standard errors are clustered by time and firm. Significance for regression coefficients at 5% and 1% are denoted by * and **, respectively. Estimates for the TED spread are multiplied by 100 for ease of exposition.   (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  Lagged log CDS change  0.051*  0.086**  0.086**  0.088**  0.088**  0.088**  0.051*  0.051*  0.052*  0.054*  0.053*  Leverage  –0.006            –0.006  –0.008  0.004  –0.007  –0.008  S&P 500 return  –0.328*            –0.332*  –0.347**  –0.320*  –0.330*  –0.331*  S&P 500 1-year return  0.029            0.029  0.035*  0.028  0.027  0.019  Interest rate level  0.003            0.003  0.002  0.002  0.007  0.012*  Yield curve slope  0.002            0.002  0.001  0.001  0.006  0.011*  TED spread  0.010            0.009  0.008  0.011  0.005  0.005  Equity volatility  0.010            0.010  0.009  0.016**  0.009  0.010  Equity illiquidity  –0.025            –0.019  –0.026  –0.025  –0.020  –0.023  Equity market illiquidity  –1.983            –1.896  –4.558  –2.130  –2.286  –2.476  CDS illiquidity MA    –0.696          –0.314          CDS market illiquidity MA      –6.755          32.451        CDS volatility        –0.004          –0.006*      CDS correlation          0.026          0.098*    CDS tail dependence            0.031          0.264**  Average Adjusted R2 (%)  1.826  0.647  0.655  0.813  0.709  0.693  1.818  1.938  1.965  2.071  2.207    (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  Lagged log CDS change  0.051*  0.086**  0.086**  0.088**  0.088**  0.088**  0.051*  0.051*  0.052*  0.054*  0.053*  Leverage  –0.006            –0.006  –0.008  0.004  –0.007  –0.008  S&P 500 return  –0.328*            –0.332*  –0.347**  –0.320*  –0.330*  –0.331*  S&P 500 1-year return  0.029            0.029  0.035*  0.028  0.027  0.019  Interest rate level  0.003            0.003  0.002  0.002  0.007  0.012*  Yield curve slope  0.002            0.002  0.001  0.001  0.006  0.011*  TED spread  0.010            0.009  0.008  0.011  0.005  0.005  Equity volatility  0.010            0.010  0.009  0.016**  0.009  0.010  Equity illiquidity  –0.025            –0.019  –0.026  –0.025  –0.020  –0.023  Equity market illiquidity  –1.983            –1.896  –4.558  –2.130  –2.286  –2.476  CDS illiquidity MA    –0.696          –0.314          CDS market illiquidity MA      –6.755          32.451        CDS volatility        –0.004          –0.006*      CDS correlation          0.026          0.098*    CDS tail dependence            0.031          0.264**  Average Adjusted R2 (%)  1.826  0.647  0.655  0.813  0.709  0.693  1.818  1.938  1.965  2.071  2.207  CDS illiquidity and CDS market illiquidity are positively correlated with CDS levels, but the estimates are never significant.11 We conclude that the CDS correlation and tail dependence measures that we estimate using the dynamic copula model are important determinants of the time-series variation in credit spreads. 4.3 Economic Determinants of Credit Dependence Are our new dynamic measures of credit dependence related to traditional economic determinants of credit risk? To answer this question, we now consider predictive regressions of copula correlations on various economic and financial determinants. An important class of credit default models uses observable macroeconomic factors to characterize the clustering in defaults and cross-firm default dependence. We have obtained estimates of default dependence without relying on such observable factors, and it is useful to investigate how closely our estimates are related to economic variables that are commonly used as factors. We focus on explaining median dependence because our first concern is to verify if the economic variables can explain the time variation in the dependence measures. The cross-sectional variation in dependence and the loadings of different firms on the dependence measures are also of interest, but we leave this topic for future work. In the regressions, we limit ourselves to economic variables that are available at the weekly frequency because we have modeled dependence at the weekly frequency, and we want to capture as much of the time-series variation as possible. The variables are again defined in the appendix. Table VI presents the regression results. For the CDX, VIX, S&P 500 returns, illiquidity, and term structure variables, univariate regressions are provided in columns (i)–(viii). Column (ix) presents the results of a multivariate regression including all variables, and column (x) further includes the lagged dependent variable. All regressors are lagged one week and all results are obtained using OLS with Newey–West standard errors using T1/4≈5 lags, where T is sample size. Table VI Time series regressions for median CDS correlations We regress the weekly DAC median CDS correlation on the CDX North American investment grade index, the credit market modified Amihud illiquidity measure, the CBOE implied volatility index, the stock market aggregated Amihud illiquidity measure, the weekly return and the 1-year trailing return on the S&P 500, the 3-month constant maturity US Treasury rate, the difference between the 10-year and the 3-month constant maturity US Treasury rates, the TED spread, the West Texas Intermediate Cushing crude oil spot price, the Aruoba–Diebold–Scotti business conditions index, and the US breakeven inflation rate. All regressors are lagged, and the first lag of the regressand is included in specification (x). We compute Newey–West standard errors, and significance for regression coefficients at 5% and 1% are denoted by * and **, respectively. All regression estimates are multiplied by 10 for ease of exposition, except for the first lag of the regressand.   (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)   Intercept  –0.43  4.02**  0.64*  3.99**  4.17**  4.25**  4.57**  3.79**  –1.87**  –0.26*  First lag                    0.91**  CDX  1.06**                0.60**  0.05  CDS market illiquidity    1361.56**              258.82*  87.98*  VIX      1.20**            0.41**  0.10**  Equity market illiquidity        175.49**          –13.16  –12.42**  S&P 500 return          –1.11        1.46**  –0.42  S&P 500 1-year return            –1.43**      0.22  0.06  Interest rate level              –0.22**    –0.17**  –0.02  Yield curve slope                0.20**  –0.25**  –0.03  TED spread                  –0.00  –0.00  Crude oil price                  0.62**  0.04  Business conditions index                  0.04  0.01  Breakeven inflation                  0.14  0.02  Adjusted R2  0.73  0.19  0.57  0.20  –0.00  0.17  0.41  0.16  0.90  0.99    (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)   Intercept  –0.43  4.02**  0.64*  3.99**  4.17**  4.25**  4.57**  3.79**  –1.87**  –0.26*  First lag                    0.91**  CDX  1.06**                0.60**  0.05  CDS market illiquidity    1361.56**              258.82*  87.98*  VIX      1.20**            0.41**  0.10**  Equity market illiquidity        175.49**          –13.16  –12.42**  S&P 500 return          –1.11        1.46**  –0.42  S&P 500 1-year return            –1.43**      0.22  0.06  Interest rate level              –0.22**    –0.17**  –0.02  Yield curve slope                0.20**  –0.25**  –0.03  TED spread                  –0.00  –0.00  Crude oil price                  0.62**  0.04  Business conditions index                  0.04  0.01  Breakeven inflation                  0.14  0.02  Adjusted R2  0.73  0.19  0.57  0.20  –0.00  0.17  0.41  0.16  0.90  0.99  Table VI shows that the univariate regressions all provide intuitively plausible results: when times are bad and the economy experiences negative shocks, the CDX, the VIX, and market illiquidity are high, stock returns and interest rates are low, and this poor economic environment is associated with higher dependence and fewer diversification opportunities. However, the R-squares indicate that stock market returns explain much less of the time-series variation in copula correlations than the interest rate level and the VIX. Finally, the R-square in regression (i) indicates that the level of credit risk, represented by the CDX index, is a prime candidate for explaining dependence in credit markets.12 So far we have separately analyzed each individual coefficient which may thus be influenced by omitted variables bias. In specification (ix) we, therefore, include all the variables simultaneously and in specification (x) we further include the lagged dependent variable. Surprisingly, in the multivariate regression (ix), the stock market returns have a positive sign but this is not the case in (x). The interest rate level remains statistically significant in (ix) but not in (x). Our most robust finding is that the VIX turns out to be important for explaining credit dependence, in spite of the fact that volatilities do not always co-move with credit correlations, as documented in Figure 10 and Section 4.1. In fact, somewhat surprisingly, in (x) the estimated coefficient on VIX is significant but the coefficient on CDX is not. The insignificance of CDX is partly due to the smaller point estimate of the coefficient—which is still positive—and partly due to the larger standard deviation on the coefficient when all variables are included. Credit market illiquidity remains positive and significant in both cases. In summary, we have found some evidence of economic drivers of the median credit copula correlation, mainly CDS market liquidity and the VIX. However, the strong autocorrelation in the CDS correlations renders steadfast inference difficult. 5. Conclusion This article documents cross-sectional dependence in CDS returns, and compares it with dependence in equity returns. Our results are complementary to existing correlation and dependence estimates, which are typically based on historical default rates or factor models of equity returns, and to existing intensity-based studies, which characterize observable macro variables that induce realistic correlation patterns in default probabilities (see Duffee, 1999; Duffie, Saita, and Wang, 2007). We use econometric techniques that allow us to estimate a model with multivariate asymmetries and time-varying dependence using a long time series and a large cross-section of CDS spreads. We document six important stylized facts. First, copula correlations in CDS returns vary substantially over our sample and increase substantially following the financial crisis in 2007. Equity correlations also increase in the financial crisis, but somewhat later, and the increase is less pronounced and not as persistent. Second, our estimates indicate fat tails in the univariate distributions, but also multivariate non-normalities. Multivariate asymmetries seem to be less important for credit than they are for equities. Third, credit dependence is more persistent than equity dependence, and this greatly affects how major events such as the subprime funds collapse, the Lehman bankruptcy, and the US sovereign debt downgrade affect subsequent dependence in credit and equity markets. Fourth, tail dependence increases more dramatically in our sample period than do copula correlations. Fifth, the CDS dependence and tail dependence measures are related to the time-series variation in credit spreads, even after accounting for other well-known firm-level determinants of spreads. Sixth, VIX is an important driver of credit correlations over time. These stylized facts, and the increase in cross-sectional dependence in particular, have important implications for the management of portfolio credit risk. We illustrate these implications by computing the diversification benefits when selling credit protection. The increase in cross-sectional dependence following the financial crisis has reduced diversification benefits, not unlike what has happened in equity markets. Footnotes 1 Engle (1982) and Bollerslev (1986) developed the first ARCH and GARCH models. Bollerslev (1990) first combined the GARCH model with a t-distribution. 2 See Engle and Kroner (1995) for an early multivariate GARCH model and Engle and Kelly (2012) for a simplified dynamic correlation model. 3 The twelve firms are: AT&T Mobility LLC, Bombardier Capital Inc., Bombardier Inc., Cingular Wireless LLC, Capital One Bank USA National Association, Comcast Cable Communication LLC, General Motors Acceptance Corp., Intelsat Limited, International Lease Finance Corp., National Rural Utilities Coop Financial Corp., Residential Capital Corp., and Verizon Global Funding Corp. 4 In a robustness analysis, we use log-differences in CDS spreads. The dependence results are very similar. 5 To compute this present value, we need recovery rates for the entire cross-section of firms at all times. We obtain these recovery rates from Markit. When Markit does not provide a recovery rate, we use 0.4, which is the industry standard. 6 The structural approach goes back to Merton (1974). See Black and Cox (1976), Leland (1994) and Leland and Toft (1996) for extensions. See Zhou (2001) for a discussion of default correlation in the context of the Merton model. 7 See Jarrow and Turnbull (1995), Jarrow, Lando, and Turnbull (1997), Duffee (1999), and Duffie and Singleton (1999) for early examples of the reduced form approach. See Lando (2004) and Duffie and Singleton (2003) for surveys. 8 We build on Jondeau and Rockinger (2006), who analyze dynamic dependence using symmetric copulas. 9 See Patton (2006b) for an application of the tail dependence measure to exchange rates. 10 See Collin-Dufresne, Goldstein, and Martin (2001); Campbell and Taksler (2003); Cremers et al. (2008); and Ericsson, Jacobs, and Oviedo (2009). 11 The results in Table 5 are obtained using the modified Amihud (MA) illiqudity measure. We obtain similar results (unreported) when using the illiquidity measure in Junge and Trolle (2015). We are grateful to the authors for making their illiquidity measure available online. 12 The impact of the CDX index on copula correlations is to some extent mechanical because (transformations of) the individual credit spreads that constitute the index are used to update the copula correlations in Equation (3.7). 13 The TED spread is an indicator of liquidity in fixed-income markets. The funding liquidity variable in Fontaine and Garcia (2012) provides an alternative liquidity indicator, but it is not available at the weekly frequency. 14 See Blume and Keim (1991); Fons (1991); Helwege and Kleiman (1997); Hillegeist et al. (2004); Jonsson and Fridson (1996); Keenan, Sobehart, and Hamilton (1999); McDonald and Van de Gucht (1999); and Pesaran et al. (2006) for examples of other macroeconomic variables that are useful for explaining and forecasting credit spreads and default. 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( 2009): Estimation of dynamic asymmetric tail dependences: an empirical study on Asian developed futures markets, Applied Financial Economics  19, 273– 290. Google Scholar CrossRef Search ADS   Zhou C. ( 2001): An analysis of default correlation and multiple defaults, Review of Financial Studies  14, 555– 576. Google Scholar CrossRef Search ADS   Appendix In this appendix, we describe the regressors used in the analyses in Sections 4.2 and 4.3. In Table V, we use the following regressors: Leverage is defined as the ratio of long-term liabilities to the sum of long-term liabilities and market value of equity. CDS and equity volatility are the log of the NGARCH volatilities each week. Equity illiquidity is the daily Amihud price impact measure, defined as the absolute daily return divided by daily dollar volume, averaged over each week. Equity market illiquidity is the median equity illiquidity across all firms for a given week. A CDS illiquidity measure modeled after the Amihud equity illiquidity measure. For the CDS market, dollar volumes are not available and we follow Junge and Trolle (2015) by dividing absolute daily return by the number of dealers that provided a quote for that day and average over each week. The CDS market illiquidity measure is the median across all firms. We denote this measure by “MA” for Modified Amihud. The CDS market illiquidity measure of Junge and Trolle (2015) based on the price discrepancy between credit indexes and their CDS constituents (used in unreported results). The return on the S&P 500 captures the changes in stock market capitalization. We use the 1-week return as well as a trailing 1-year return. The term structure is captured by a level variable, the 3-month US Constant Maturity Treasury (CMT) index, and a slope variable, the 10-year CMT index minus the 3-month CMT. The difference between the interest rate on interbank loans and on short-term government debt, that is the TED spread.13 While theory suggests several determinants of credit spreads, there is no acknowledged theory on the selection of economic and financial factors that can capture cross-firm default dependence; perhaps, as a result the existing empirical literature is very extensive, and many different economic variables have been used as factors. Duffie, Saita, and Wang (2007) provide an excellent discussion of the existing literature, and choose 1-year trailing S&P 500 returns and interest rates as macro variables to capture default dependence in their own empirical implementation. Duan and Van Laere (2012) also use stock index returns and interest rates, and Collin-Dufresne, Goldstein, and Martin (2001) use S&P 500 returns, interest rates, and the VIX. Campbell, Hilscher, and Szilagyi, (2008) use S&P 500 returns to normalize firm returns in their analysis of default and credit risk. Doshi et al. (2013) use term structure variables and the VIX, and the latent variable model in Duffee (1999) uses interest rate factors to capture the dependence in credit spreads.14 When selecting economic variables in Table VI, we limit ourselves to variables that are available at the weekly frequency because we model dependence at the weekly frequency, and we want to capture as much of the time-series variation as possible. An analysis of lower frequency macroeconomic variables would be interesting but we leave that for future work. In the absence of explanatory variables suggested by theory, we consider economy-wide measures of risk in equity and default insurance markets, risk-free (government) term structures, and other macro variables that are reasonable additional metrics of the state of the economy, and that have explanatory power for credit spreads documented by the papers cited above. Specifically, we use the following regressors: The S&P 500 return variables, term structure variables, TED spread, and CDS and equity market illiquidity variables used in Table V. The log of the CDX North American investment grade index level is used to proxy for the overall level of risk in credit markets. The log of the VIX index represents equity market risk. The log of crude oil price as measured by the West Texas Intermediate Cushing Crude Oil Spot Price. The Aruoba-Diebold-Scotti (ADS) business condition index from the Federal Reserve Bank of Philadelphia. The breakeven inflation level implied by Treasury Inflation Protected Securities. Unlike standard inflation measures, this series is available at the weekly frequency. © The Authors 2017. Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Review of Finance Oxford University Press

Dynamic Dependence and Diversification in Corporate Credit

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Oxford University Press
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© The Authors 2017. Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com
ISSN
1572-3097
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1573-692X
D.O.I.
10.1093/rof/rfx034
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Abstract

Abstract We characterize dependence in corporate credit and equity returns for 215 firms using a new class of large-scale dynamic copula models. Copula dependence and especially tail dependence are highly variable and persistent, increase significantly in the financial crisis, and have remained high since. The most drastic increases in credit dependence occur in July/August of 2007 and in August of 2011 and the decrease in diversification potential caused by the increases in dependence and tail dependence is large. Credit default swap correlation dynamics are important determinants of credit spreads. 1. Introduction Characterizing the dependence between credit-risky securities is of great interest for portfolio management and risk management, but not necessarily straightforward because multivariate modeling is notoriously difficult for large cross-sections of securities. In existing work, computationally straightforward techniques such as factor models or constant copulas are often used to model correlations for large portfolios of credit-risky securities; alternatively, simple rolling correlations or exponential smoothers are used. We instead use multivariate econometric models for the purpose of modeling credit correlation and dependence. We use genuinely dynamic copula techniques that can capture univariate and multivariate deviations from normality, including multivariate asymmetries. We demonstrate that by using recently proposed econometric innovations, it is possible to apply copula models on a large scale that is essential for effective credit risk management. We perform our empirical analysis using data on a large cross-section of credit-risky securities, namely 5-year credit default swap (CDS) contracts for 215 constituents of the first 18 series of the CDX North American investment grade index. We use a long time series of weekly data for the period January 3, 2001 to August 22, 2012. The 215 firms enter and leave the sample at different time points, but this can easily be accommodated by the estimation methodology we employ. We compute returns on selling credit protection through CDS and investigate the dependence of these returns across firms as well as their tail dependence. We formally define returns on a short CDS position in the next section. We also analyze dependence in the underlying equity for comparison. Interestingly, the credit and equity return dynamics differ in important aspects. We document several important stylized facts, and substantial differences between credit and equity dependence. We show that the time-varying correlations in CDS returns implied by the copula models we use, which we will refer to as “copula correlations” for short, vary substantially over the sample period, with a substantial increase following the financial crisis in 2007. Equity correlations also increase in the financial crisis, but somewhat later, and the increase is less pronounced and not as persistent. Our estimates indicate fat tails in the univariate credit distributions, but also multivariate non-normalities for CDS returns. Multivariate asymmetries seem to be less important for credit than for equity returns, confirming results from threshold correlations. Credit copula correlations are more persistent than equity copula correlations. This greatly affects how major events such as the subprime funds collapse, the Lehman bankruptcy, and the US sovereign debt downgrade affect subsequent dependence in credit and equity markets. Tail dependence for credit and equity increases substantially during our sample, more so than copula correlations. Surprisingly, the Lehman bankruptcy affects equity (tail) dependence more strongly than credit (tail) dependence. The US sovereign downgrade in mid-2011 is an important credit event, but this is more apparent when analyzing tail dependence, somewhat less so when analyzing copula correlations. The increase in cross-sectional dependence is clearly important for the management of portfolio credit risk. We use our estimates to compute time-varying diversification benefits from selling credit protection. We find that the increase in cross-sectional dependence following the financial crisis has substantially reduced diversification benefits, similar to what happened in equity markets. We use panel regressions to investigate if correlation and tail dependence help explain the variation in firm credit spreads. We find that higher copula correlation and tail dependence are related to higher CDS spreads, even after controlling for well-established determinants of credit spreads at the firm level, such as equity volatility, interest rates, and leverage. Identifying financial and macroeconomic variables that can capture the clustering in default and cross-firm default dependence is of great interest for the purpose of modeling portfolio credit risk, but there is no guidance from theory regarding the economic determinants of credit dependence and tail dependence. We use a regression analysis to identify financial and macroeconomic determinants of the time-series variation in credit dependence. Copula correlations increase with the VIX, the overall level of credit spreads, and inflation, and decrease with the level of interest rates and S&P 500 returns. The effect from VIX is robust when including lagged correlations in the regressions. The empirical analysis proceeds in three steps. The first two steps are univariate. In the first, we remove the short-run dynamics from the raw data by estimating firm-by-firm autoregressive (AR) models on weekly returns. In the second step, we estimate firm-by-firm variance dynamics on the residuals from the first step. We use an asymmetric nonlinear generalized autoregressive conditional heteroskedasticity (NGARCH) model with an asymmetric standardized t-distribution following Hansen (1994).1 Finally, in the third step we provide a multivariate analysis using the copula implied by the skewed t-distribution in Demarta and McNeil (2005). Dynamic copula correlations are modeled based on the linear correlation techniques developed by Engle (2002) and Tse and Tsui (2002).2 Dynamic tail dependence depends on time-varying correlations and degrees of freedom, which we capture using a smooth exponential spline function (see Engle and Rangel, 2008). To alleviate the computational burden, we rely on the composite likelihood technique of Engle, Shephard, and Sheppard (2008) and the moment matching from Engle and Mezrich (1996). Patton (2013) provides an excellent survey of copula models. The remainder of the article is structured as follows. In Section 2, we document some stylized facts in our CDS sample and briefly review existing models of default and default dependence. Section 3 reports the estimation results from the dynamic models for expected credit and equity returns and volatility. It also introduces the dynamic copula models and presents the estimation results for copula and threshold correlations. In Section 4, we present various economic applications of our dynamic copula model. We investigate credit diversification dynamics, relate credit spreads to copula correlations, and relate copula correlations to economic covariates. Section 5 concludes. 2. CDS Markets: Stylized Facts and Default Models We discuss the CDS data we use in our empirical work. We also briefly discuss existing techniques for modeling default dependence. 2.1 CDS Data A CDS is essentially an insurance contract on corporate or sovereign debt, under which the buyer makes regular premium payments and is compensated by the seller in case of default by the underlying entity. One element of the success and resilience of CDS markets has been the creation of market indexes consisting of CDSs, such as the CDX index in North America and the iTraxx index in Europe. Using data from Markit, we consider 5-year CDS contracts on all firms included in the first 18 series of the North American investment grade CDX.NA.IG index. We use the longest possible sample available from Markit for all these firms, starting on January 3, 2001, and ending on August 22, 2012. Many firms do not have CDS quotes available for every day of this sample period. Fortunately, as shown by Patton (2006a), the dynamic multivariate modeling approach we employ in our empirical work allows for individual series to begin (and end) at different time points. We make full use of this and include a firm if it has at least 1 year of consecutive weekly data points. The resulting list of 215 firms is provided in Table I. Table I Company names Using data from Markit, we consider all firms included in the first eighteen series of the CDX North American investment grade index dating from January 1, 2001 to August 22, 2012. Our sample consists of 215 firms. Firms ordered alphabetically within each GIC sector. Consumer discretionary  Consumer discretionary (Cont.)  Industrials  Financials   American Axle & Manuf. Holdings   Tribune Co   Boeing Capital Corp.   ACE Limited   Autozone Inc.   Viacom Inc.   Bombardier Capital Inc.   Allstate Corp.   Belo Corp.   Visteon Corp.   Bombardier Inc.   American Express Co   Black & Decker Corp.   Walt Disney Co   Burlington Northern Santa Fe Corp.   American International Group Inc.   Brunswick Corp.   Wendy's International Inc.   CSX Corp.   Berkshire Hathaway Inc.   CBS Corp.   Whirlpool Corp.   Caterpillar Inc.   Boston Properties LP   CENTEX Corp.   YUM! Brands Inc.   Cendant Corp.   CIT Group Inc.   Carnival Corp.     Deere & Co   Capital One Bank USA Nat. Assoc.   Clear Channel Comms Inc.  Consumer Staples   GATX Corp.   Capital One Financial Corp.   Comcast Cable Communication LLC   Albertsons Inc.   General Electric Capital Corp.   Chubb Corp.   Comcast Corp.   Altria Group Inc.   Goodrich Corp.   Countrywide Home Loans Inc.   Cox Communications Inc.   Beam Inc.   Honeywell International Inc.   EOP Operating LP   DIRECTV Holdings LLC   CVS Caremark Corp.   Ingersoll Rand Co   ERP Operating LP   Darden Restaurants Inc.   Campbell Soup Co   Lockheed Martin Corp.   Federal Home Loan Mortgage Corp.   Delphi Corp.   ConAgra Foods Inc.   Masco Corp.   Federal National Mortgage Assoc.   Eastman Kodak Co   General Mills Inc.   Norfolk Southern Corp.   General Motors Acceptance Corp.   Expedia Inc.   H. J. Heinz Co   Northrop Grumman Corp.   Hartford Financial Services Group   Ford Motor Credit Co   Kraft Foods Inc.   Pitney Bowes Inc.   International Lease Finance Corp.   GAP Inc.   Kroger Co   R. R. Donnelley & Sons Co   Loews Corp.   Gannett Co Inc.   Reynolds American Inc.   Raytheon Co   MBIA Insurance Corp.   Harrah's Operating Co Inc.   Safeway Inc.   Ryder System Inc.   MBNA Corp.   Hilton Hotels Corp.   Sara Lee Corp.   Southwest Airlines Co   Marsh & Mclennan Co Inc.   Home Depot Inc.   Supervalu Inc.   Textron Financial Corp.   MetLife Inc.   J. C. Penney Co Inc.   Tyson Foods Inc.   Union Pacific Corp.   National Rural Util. Coop Fin. Corp.   Johnson Controls Inc.   Wal Mart Stores Inc.   United Parcel Services Inc.   Radian Group Inc.   Jones Apparel Group Inc.       Residential Capital Corp.   Knight-Ridder Inc.  Health Care  Information Technology   SLM Corp.   Kohls Corp.   Aetna Inc.   1st Data Corp.   Simon Property Group Inc.   Lear Corp.   Amgen Inc.   Arrow Electronics Inc.   Vornado Realty LP   Lennar Corp.   Baxter International Inc.   Avnet Inc.   Washington Mutual Inc.   Liberty Media Corp.   Boston Scientific Corp.   CA Inc.   Wells Fargo & Co   Limited Brands Inc.   Bristol Myers Squibb Co   Cisco Systems Inc.   Weyerhaeuser Co   Liz Claiborne Inc.   Cardinal Health Inc.   Computer Sciences Corp.   XLIT Limited   Lowe's Companies Inc.   Cigna Corp.   Dell Inc.   iStar Financial Inc.   M.D.C. Holdings Inc.   McKesson Corp.   Electronic Data System Corp.     Macy's Inc.   Pfizer Inc.   Hewlett Packard Co  Energy   Marriott International Inc.   Quest Diagnostics Inc.   IAC InterActive Corp.   Amerada Hess Corp.   May Department Stores Co   UnitedHealth Group Inc.   IBM Corp.   Anadarko Petroleum Corp.   Maytag Corp.   Universal Health Services Inc.   Motorola Inc.   Canadian Natural Resources Ltd   McDonald's Corp.   Wyeth   Sabre Holdings Corp.   ConocoPhillips   Mohawk Industries Inc.     Sun Microsystems Inc.   Devon Energy Corp.   NY Times Co  Telecommunication Services   Xerox Corp.   Halliburton Co   Newell Rubbermaid Inc.   ALLTEL Corp.     KerrMcGee Corp.   News America Inc.   AT&T Corp.  Materials   Kinder Morgan Energy LP   Nordstrom Inc.   AT&T Inc.   Alcan Inc.   Nabors Industries Inc.   Omnicom Group Inc.   AT&T Mobility LLC   Alcoa Inc.   Transocean Inc.   Pulte Homes Inc.   AT&T Wireless Services Inc.   Barrick Gold Corp.   Valero Energy Corp.   RadioShack Corp.   BellSouth Corp.   Dow Chemical Co   XTO Energy Inc.   Sears Roebuck Acceptance Corp.   CenturyLink Inc.   E. I. du Pont de Nemours & Co     Staples Inc.   Cingular Wireless LLC   Eastman Chemical Co  Utilities   Starwood Hotels & Resorts Inc.   Citizens Communication Co   Freeport McMoran Copper & Gold   American Electric Power Co Inc.   TJX Companies Inc.   Embarq Corp.   International Paper Co   Constellation Energy Group Inc.   Target Corp.   Intelsat Limited   MeadWestvaco Corp.   Dominion Resources Inc.   Time Warner Cable Inc.   Sprint Corp.   Olin Corp.   Duke Energy Carolinas LLC   Time Warner Inc.   Verizon Communications Inc.   Rio Tinto Alcan Inc.   Exelon Corp.   Toll Brothers Inc.   Verizon Global Funding Corp.   Rohm & Haas Co   FirstEnergy Corp.   Toys “R” Us Inc.     Sherwin Williams Co   Progress Energy Inc.       Temple-Inland Inc.   Sempra Energy  Consumer discretionary  Consumer discretionary (Cont.)  Industrials  Financials   American Axle & Manuf. Holdings   Tribune Co   Boeing Capital Corp.   ACE Limited   Autozone Inc.   Viacom Inc.   Bombardier Capital Inc.   Allstate Corp.   Belo Corp.   Visteon Corp.   Bombardier Inc.   American Express Co   Black & Decker Corp.   Walt Disney Co   Burlington Northern Santa Fe Corp.   American International Group Inc.   Brunswick Corp.   Wendy's International Inc.   CSX Corp.   Berkshire Hathaway Inc.   CBS Corp.   Whirlpool Corp.   Caterpillar Inc.   Boston Properties LP   CENTEX Corp.   YUM! Brands Inc.   Cendant Corp.   CIT Group Inc.   Carnival Corp.     Deere & Co   Capital One Bank USA Nat. Assoc.   Clear Channel Comms Inc.  Consumer Staples   GATX Corp.   Capital One Financial Corp.   Comcast Cable Communication LLC   Albertsons Inc.   General Electric Capital Corp.   Chubb Corp.   Comcast Corp.   Altria Group Inc.   Goodrich Corp.   Countrywide Home Loans Inc.   Cox Communications Inc.   Beam Inc.   Honeywell International Inc.   EOP Operating LP   DIRECTV Holdings LLC   CVS Caremark Corp.   Ingersoll Rand Co   ERP Operating LP   Darden Restaurants Inc.   Campbell Soup Co   Lockheed Martin Corp.   Federal Home Loan Mortgage Corp.   Delphi Corp.   ConAgra Foods Inc.   Masco Corp.   Federal National Mortgage Assoc.   Eastman Kodak Co   General Mills Inc.   Norfolk Southern Corp.   General Motors Acceptance Corp.   Expedia Inc.   H. J. Heinz Co   Northrop Grumman Corp.   Hartford Financial Services Group   Ford Motor Credit Co   Kraft Foods Inc.   Pitney Bowes Inc.   International Lease Finance Corp.   GAP Inc.   Kroger Co   R. R. Donnelley & Sons Co   Loews Corp.   Gannett Co Inc.   Reynolds American Inc.   Raytheon Co   MBIA Insurance Corp.   Harrah's Operating Co Inc.   Safeway Inc.   Ryder System Inc.   MBNA Corp.   Hilton Hotels Corp.   Sara Lee Corp.   Southwest Airlines Co   Marsh & Mclennan Co Inc.   Home Depot Inc.   Supervalu Inc.   Textron Financial Corp.   MetLife Inc.   J. C. Penney Co Inc.   Tyson Foods Inc.   Union Pacific Corp.   National Rural Util. Coop Fin. Corp.   Johnson Controls Inc.   Wal Mart Stores Inc.   United Parcel Services Inc.   Radian Group Inc.   Jones Apparel Group Inc.       Residential Capital Corp.   Knight-Ridder Inc.  Health Care  Information Technology   SLM Corp.   Kohls Corp.   Aetna Inc.   1st Data Corp.   Simon Property Group Inc.   Lear Corp.   Amgen Inc.   Arrow Electronics Inc.   Vornado Realty LP   Lennar Corp.   Baxter International Inc.   Avnet Inc.   Washington Mutual Inc.   Liberty Media Corp.   Boston Scientific Corp.   CA Inc.   Wells Fargo & Co   Limited Brands Inc.   Bristol Myers Squibb Co   Cisco Systems Inc.   Weyerhaeuser Co   Liz Claiborne Inc.   Cardinal Health Inc.   Computer Sciences Corp.   XLIT Limited   Lowe's Companies Inc.   Cigna Corp.   Dell Inc.   iStar Financial Inc.   M.D.C. Holdings Inc.   McKesson Corp.   Electronic Data System Corp.     Macy's Inc.   Pfizer Inc.   Hewlett Packard Co  Energy   Marriott International Inc.   Quest Diagnostics Inc.   IAC InterActive Corp.   Amerada Hess Corp.   May Department Stores Co   UnitedHealth Group Inc.   IBM Corp.   Anadarko Petroleum Corp.   Maytag Corp.   Universal Health Services Inc.   Motorola Inc.   Canadian Natural Resources Ltd   McDonald's Corp.   Wyeth   Sabre Holdings Corp.   ConocoPhillips   Mohawk Industries Inc.     Sun Microsystems Inc.   Devon Energy Corp.   NY Times Co  Telecommunication Services   Xerox Corp.   Halliburton Co   Newell Rubbermaid Inc.   ALLTEL Corp.     KerrMcGee Corp.   News America Inc.   AT&T Corp.  Materials   Kinder Morgan Energy LP   Nordstrom Inc.   AT&T Inc.   Alcan Inc.   Nabors Industries Inc.   Omnicom Group Inc.   AT&T Mobility LLC   Alcoa Inc.   Transocean Inc.   Pulte Homes Inc.   AT&T Wireless Services Inc.   Barrick Gold Corp.   Valero Energy Corp.   RadioShack Corp.   BellSouth Corp.   Dow Chemical Co   XTO Energy Inc.   Sears Roebuck Acceptance Corp.   CenturyLink Inc.   E. I. du Pont de Nemours & Co     Staples Inc.   Cingular Wireless LLC   Eastman Chemical Co  Utilities   Starwood Hotels & Resorts Inc.   Citizens Communication Co   Freeport McMoran Copper & Gold   American Electric Power Co Inc.   TJX Companies Inc.   Embarq Corp.   International Paper Co   Constellation Energy Group Inc.   Target Corp.   Intelsat Limited   MeadWestvaco Corp.   Dominion Resources Inc.   Time Warner Cable Inc.   Sprint Corp.   Olin Corp.   Duke Energy Carolinas LLC   Time Warner Inc.   Verizon Communications Inc.   Rio Tinto Alcan Inc.   Exelon Corp.   Toll Brothers Inc.   Verizon Global Funding Corp.   Rohm & Haas Co   FirstEnergy Corp.   Toys “R” Us Inc.     Sherwin Williams Co   Progress Energy Inc.       Temple-Inland Inc.   Sempra Energy  We construct nonoverlapping weekly returns using a fixed day each week. We use Wednesdays, which is the weekday that is least likely to be a holiday. We obtain equity data on the sample firms from the Center for Research in Security Prices (CRSP). Out of the 215 firms, 12 firms do not have at least a consecutive 52-week history of equity prices, and those are dropped from the sample.3 Note that reflecting the growth in market activity, CDS markets underwent a number of changes in April 2009, some of which affect the reporting of spreads (see Markit (2009), our data source, for details). To facilitate the comparison of the estimated correlations with equity market correlations, we use CDS returns, denoted by Rt, for our dependence analysis.4 To further facilitate the exposition of our results, we consider realized returns from selling credit protection throughout the article. Adverse events for a firm, therefore, result in both negative equity returns and negative CDS returns. We compute realized CDS excess returns as in Berndt and Obreja (2010) and Junge and Trolle (2015). In case of a credit event, the weekly return to the protection seller is computed as follows:   Rt=−(1−REC)+CDSt−1(τ−(t‐1))360, (2.1) where CDSt is the CDS spread at time t, τ is the credit event date and REC is the recovery rate in case of default. We obtain these ex post recovery rates from creditfixings.com. In the absence of a credit event, the weekly return to the protection seller is:   Rt=−(CDSt−CDSt−1)(PVBPt(CDSt,REC)−7360)+CDSt−17360, (2.2) where PVBPt(CDSt,REC) is the present value of a risky annuity of one basis point defined as:   PVBPt(CDSt,REC)=∑j=1J((tj−tj−1)360D(t,tj)S(t,tj)−∫max⁡{t,tj−1}tj(u−tj−1)360D(t,u)dS(t,u)), (2.3) with payment at dates t<t1<…<tJ, and where D(t,tj) is the discount factor used from time tj to t, and S(t,tj) is the risk-neutral survival probability up to time tj.5 In our empirical implementation, we obtain the survival probability for each firm by calibrating a constant default intensity λ for the life of the contract. This implies S(t,tj)= exp(−λtj−t360) and dS(t,u)=−λexp(−λu−t360)du in Equation (2.3). We use Bloomberg’s zero curve to compute the risk-free discount rates. While CDS levels display a high level of persistence, the presence of the last term in Equation (2.2) related to the CDS level, CDSt−1, adds only a very small persistent component to CDS returns. The first term in Equation (2.2), which depends on the change in CDS spreads and the present value of the risky annuity, is orders of magnitude larger than the last term. We show in Section 2.2 that the persistence in CDS returns is only slightly higher than the persistence in equity returns. Figure 1 plots the time series of weekly CDS returns for four representative firms with long time series of spreads available: IBM, McDonald’s, Toys “R” Us, and Walt Disney. The return series clearly exhibit episodes of heightened volatility which we model in the following section. The solid black line in Panel A of Figure 2 plots the time series of the median CDS spread across firms, and the gray area represents the interquartile range (IQR). Panel B presents the median and IQR for the CDS return volatilities obtained from GARCH models as discussed in Section 3.2 below. Figure 1. View largeDownload slide Weekly returns on short CDS positions for selected firms. We plot weekly returns on short CDS positions for four selected firms with long time series. We compute returns using Equations (2.2) and (2.3). Figure 1. View largeDownload slide Weekly returns on short CDS positions for selected firms. We plot weekly returns on short CDS positions for four selected firms with long time series. We compute returns using Equations (2.2) and (2.3). Figure 2. View largeDownload slide Quantiles of CDS spreads, equity prices, and return volatilities. We plot weekly quantiles across the 215 firms listed in Table I for CDS spreads, equity prices, and their return volatilities from GARCH models. We use a level-NGARCH model for CDS returns, and a NGARCH model for stock returns. In each panel, the black line reports the median across firms for each week, and the gray area shows the IQR across firms. The vertical lines indicate the major events during the sample period listed in Section 2.1. Figure 2. View largeDownload slide Quantiles of CDS spreads, equity prices, and return volatilities. We plot weekly quantiles across the 215 firms listed in Table I for CDS spreads, equity prices, and their return volatilities from GARCH models. We use a level-NGARCH model for CDS returns, and a NGARCH model for stock returns. In each panel, the black line reports the median across firms for each week, and the gray area shows the IQR across firms. The vertical lines indicate the major events during the sample period listed in Section 2.1. Panels C and D of Figure 2 replicate Panels A and B using the equity data. The equity price in Panel C is normalized to one for each firm at the start of the sample. The vertical lines in Figure 2 denote eight major events during our sample period: the WorldCom bankruptcy in July 2002, the Ford and GM downgrades to junk in May 2005, the Delphi bankruptcy on October 8, 2005, the Bear Stearns subprime funds collapse in July and August 2007, the Bear Stearns bankruptcy in March 2008, the Lehman Brothers bankruptcy in September 2008, the stock market bottom and CDS Big Bang in March and April 2009 (henceforth referred to simply as the stock market bottom), and the US sovereign debt downgrade in August 2011. 2.2 Stylized Facts Figure 2 illustrates some important stylized facts regarding the trends in credit risk during the sample period. Panel A of Figure 2 indicates that the time series of the median CDS spread and the IQR reach their maximums during the peak of the financial crisis in 2008–09. Less dramatic turbulence is also evident during the dot-com bust in 2002 and the US sovereign debt downgrade in 2011. Panel D of Figure 2 indicates that the time series pattern for equity volatility is similar to that for credit spreads in Panel A. The relationship between credit spreads and equity volatility is of course suggested by structural credit risk models such as Merton (1974). Panels A and D also suggest that CDS spreads and equity volatility are highly persistent over time. The IQRs in the four panels of Figure 2 also contain valuable insights into credit and equity dependence. The cross-sectional range of spreads in Panel A is much wider during the financial crisis compared with the pre-crisis years. This effect lingers on to some extent in the post-crisis period. The high post-crisis range in spreads suggests that investors may be able to at least partly diversify credit risk which is a key topic of interest for us. Figure 3 plots the median CDS spread in each industry. The 215 firms in our sample are distributed along the following 10 GIC sectors: Energy (12 firms), Materials (14), Industrials (25), Consumer Discretionary (64), Consumer Staples (16), Health Care (13), Financials (34), Information Technology (15), Telecommunications Services (14), and Utilities (8). For ease of exposition, Figure 3 combines the energy and utility sectors which separately have few firms. Figure 3. View largeDownload slide Median CDS spread by sector. We report the weekly median CDS spread by sector using the GIC sectors in Table I. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Note that the scale differs across sectors. Figure 3. View largeDownload slide Median CDS spread by sector. We report the weekly median CDS spread by sector using the GIC sectors in Table I. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Note that the scale differs across sectors. The impact of the financial crisis is obvious in Figure 3, but interestingly the crisis affected different industries quite differently. Some industries, Information Technology and Telecommunication Services in particular, were affected as much or even more by the 2001–03 dot-com bubble versus the 2007–09 crisis. In Table II, we report sample moments averaged across firms for CDS and equity returns. Panel A of Table II shows the median across firms of the first four sample moments of weekly CDS and equity returns along with the IQR for each moment. Average annualized equity returns are 10.48%. The average annualized returns to selling credit protection are much lower at 0.91%, which is not surprising given our sample period. We also report the Jarque–Bera tests for normality as well as the first two autocorrelation coefficients. Note the strong evidence of non-normality as well as some evidence of dynamics in the weekly CDS returns. We will model both of these features below. Table II Descriptive statistics on CDS and equity returns We report sample moments on weekly CDS and equity returns across firms. Panel A reports sample moments computed using weekly returns for CDS and equity. The CDS return is the return from selling credit protection. Panel B reports average sample correlations across firms using weekly returns. On the diagonal we report the median and IQR of the correlations between each firm and all other firms, using either credit or equity. On the off-diagonal we report the median and IQR of the correlation between CDS and equity returns for the same firm. Panel A: Sample moments on weekly returns    Annualized average (%)  Annualized standard deviation (%)  Skewness  Kurtosis  Jarque–Bera p-value  AR(1) coefficient  AR(2) coefficient    CDS                 Median  0.91  4.42  –0.83  19.79  0  0.10  0.04   IQR   [0.46, 1.70]  [2.56, 7.12]  [–1.79, 0.18]  [14.24, 33.74]  [0.00, 0.00]  [0.03, 0.19]  [–0.02, 0.09]  Equity                 Median  10.48  34.73  0.02  7.46  0  –0.04  –0.02   IQR  [6.19, 16.63]  [27.94, 43.23]  [–0.31, 0.38]  [5.91, 11.06]  [0.00, 0.00]  [–0.07, 0.00]  [–0.06, 0.02]    Panel B: Correlations of weekly returns      CDS  Equity              CDS                 Median  0.343  0.349             IQR  [0.219, 0.457]  [0.243, 0.451]            Equity                 Median    0.322             IQR    [0.248, 0.408]            Panel A: Sample moments on weekly returns    Annualized average (%)  Annualized standard deviation (%)  Skewness  Kurtosis  Jarque–Bera p-value  AR(1) coefficient  AR(2) coefficient    CDS                 Median  0.91  4.42  –0.83  19.79  0  0.10  0.04   IQR   [0.46, 1.70]  [2.56, 7.12]  [–1.79, 0.18]  [14.24, 33.74]  [0.00, 0.00]  [0.03, 0.19]  [–0.02, 0.09]  Equity                 Median  10.48  34.73  0.02  7.46  0  –0.04  –0.02   IQR  [6.19, 16.63]  [27.94, 43.23]  [–0.31, 0.38]  [5.91, 11.06]  [0.00, 0.00]  [–0.07, 0.00]  [–0.06, 0.02]    Panel B: Correlations of weekly returns      CDS  Equity              CDS                 Median  0.343  0.349             IQR  [0.219, 0.457]  [0.243, 0.451]            Equity                 Median    0.322             IQR    [0.248, 0.408]            In Panel B, we report statistics for the distribution of sample correlations between CDS and equity returns. On the diagonal we report the median and IQR of the correlations between each firm and all other firms, using either credit or equity. On the off-diagonal we report the same statistics for the distribution of the correlation between the CDS and equity returns on the same firm. The relatively high and robust positive correlation between weekly equity returns and weekly CDS returns is expected because we consider the returns to selling credit protection. In order to further explore the dependence across firms we compute threshold correlations, following Ang and Chen (2002) and Patton (2004), for example. For a pair of firms i and j, we define the threshold correlation ρ¯ij(x) using standardized returns R¯i and R¯j  ρ¯ij(x)={Corr(R¯i,R¯j|R¯i<x,R¯j<x) when x<0Corr(R¯i,R¯j|R¯i≥x,R¯j≥x) when x≥0,  (2.4) where we have standardized returns by their sample mean and standard deviation, and thus measure x as the number of standard deviations from the mean. The threshold correlation reports the linear correlation between two assets for the subset of observations lying in the bottom-left or top-right quadrant. In the case of the bivariate normal distribution the threshold correlation approaches zero when the threshold x goes to plus or minus infinity. Panels A and C of Figure 4 report the median and IQR of the bivariate threshold correlations computed across all possible pairs of firms. Both the CDS and the equity threshold correlations in Panels A and C are high and show some evidence of asymmetry: large downward moves are more highly correlated than large upward moves. Also, both Panels A and C in Figure 4 show strong evidence of multivariate non-normality. This is evidenced by the large deviations of the solid line (empirics) from the dashed lines (normal distribution). Adequately capturing these non-normalities motivates the non-normal copula approach below. Figure 4. View largeDownload slide Threshold correlations for CDS and equity returns and their AR-GARCH residuals. For each pair of firms we compute threshold correlations on a grid of thresholds defined using the standard deviation from the mean for each firm (horizontal axis). The solid lines show the median threshold correlations across firm pairs, the gray areas mark the IQRs and the dashed lines show the threshold correlations from a bivariate Gaussian distribution with correlation equal to the average for all pairs of firms. Figure 4. View largeDownload slide Threshold correlations for CDS and equity returns and their AR-GARCH residuals. For each pair of firms we compute threshold correlations on a grid of thresholds defined using the standard deviation from the mean for each firm (horizontal axis). The solid lines show the median threshold correlations across firm pairs, the gray areas mark the IQRs and the dashed lines show the threshold correlations from a bivariate Gaussian distribution with correlation equal to the average for all pairs of firms. 2.3 Credit Default Models Measuring default dependence has long been a problem of interest in the credit risk literature. For instance, a bank that manages a portfolio of loans is interested in how the borrowers’ creditworthiness fluctuates with the business cycle. While the change in the probability of default for an individual borrower is of interest, the most important question is how the business cycle affects the value of the overall portfolio, and this depends on default dependence. An investment company or hedge fund that invests in a portfolio of corporate bonds faces a similar problem. Over the last decade, the measurement of default dependence has taken on added significance because of the emergence of new portfolio and structured credit products, and as a result new methods to measure correlation and dependence have been developed. Different techniques are used to estimate default dependence. The oldest and most obvious way to estimate default correlation is the use of historical default data. In order to reliably estimate default probabilities and correlations, typically a large number of historical observations are needed which are not often available. See, for instance, deServigny and Renault (2002). The alternative to historical default data is the combination of a factor model with a model that extracts default intensities or default probabilities. For each of these two tasks, different models have proven especially useful. For publicly traded corporates, a Merton (1974) type structural model is often used to link equity returns or the prices of credit-risky securities to the underlying asset returns and extract default probabilities.6 This approach is usually combined with a one-factor CAPM-style model for the underlying equity return to model the default dependence in credit portfolios. Clearly, the reliability of the default dependence estimate is determined by the quality of the factor model. Alternatively, intensity-based models have become very popular in the academic credit risk literature over the last decade.7 This approach typically models the default intensity using a jump diffusion, and is also sometimes referred to as the reduced-form approach. Within this class of models, there are different approaches to modeling default dependence. One class of models, referred to as conditionally independent models or doubly stochastic models, assumes that cross-firm default dependence associated with observable factors determining conditional default probabilities is sufficient for characterizing the clustering in defaults. See Duffee (1999) for an example of this approach. Das et al. (2007) provide a test of this approach and find that this assumption is violated. Other intensity-based models consider joint credit events that can cause multiple issuers to default simultaneously, or they model contagion or learning effects, whereby default of one entity affects the defaults of others. See, for example, Davis and Lo (2001) and Jarrow and Yu (2001). Jorion and Zhang (2007) investigate contagion using CDS data. Our article instead uses copula methods to model CDS returns. See Joe (1997) and Patton (2009a, 2009b, 2013) for excellent overviews of copula modeling. Copulas have been used extensively for modeling default dependence, especially among practitioners, and for the purpose of CDO modeling. The advantage of the copula approach is its flexibility, because the parameters characterizing the multivariate default distribution, and hence the correlation between the default probabilities, can be modeled in a second stage, after the univariate distributions have been estimated. In some cases the copulas are also parsimoniously parameterized and computationally straightforward, which facilitates calibration. Calibration of the correlation structure is mostly performed using CDO data. The simple one-factor Gaussian copula is often used in the literature, but extensions to multiple factors (Hull & White, 2010), stochastic recovery rates (Hull & White, 2006), and non-Gaussian copulas provide a better fit. In contrast to existing static approaches, in our analysis of CDS returns the emphasis is on modeling dynamic dependence. Our approach also allows for multivariate asymmetries.8 Several existing papers use copulas from the Archimedean family to capture dependence asymmetries (see Patton, 2004, 2006b and Xu and Li, 2009), but this approach is difficult to generalize to higher dimensions, and our focus is on the analysis of a large portfolio of underlying credits. To capture time variation in dependence, some existing papers use regime switching models. See Chollete, Heinen, and Valdesogo (2009); Garcia and Tsafack (2011); Hong, Tu, and Zhou (2007); and Okimoto (2008) for examples. We instead follow the autoregressive approach of Christoffersen and Langlois (2013), Christoffersen et al. (2012), and De Lira Salvatierra and Patton (2015). Oh and Patton (2016b) develop a factor copula approach to analyze dynamic dependence for a large portfolio of underlying credits. Their model assumes a factor structure with dynamic loadings, which provides very flexible dynamics, but requires numerical integration to compute the likelihood. We do not impose a factor structure up front and our closed-form likelihood function enables us to implement the model simultaneously for many underlying credits. Oh and Patton (2016b) find empirical evidence of a structural break in the dynamics of CDS spreads around the Big Bang in April 2009. We do not formally test for a structural break in our model. We instead use our dynamic model to assess the evolution of dependence in the credit market relative to the equity market and to measure changes in diversification benefits over our sample period that includes various important events including the Big Bang. 3. Dynamic Models of Credit and Equity Returns Our dynamic model implementation proceeds in three steps. In the first step, we model the mean dynamics on the univariate time series of each CDS and stock return. In the second step, we model the variance dynamics and the distribution of the time-series residual for each firm. In the third step, we develop dynamic copula models for CDS and equity returns using all the firms in our sample. 3.1 Mean Dynamics The weekly return data contain some short-run dynamics. In order to obtain white-noise innovations required for consistent modeling of correlation dynamics, we fit univariate AR-NGARCH models to the time series. In a first step, we estimate an AR(12) and sequentially remove the least significant lag until all remaining lags are significant and as long as the p-value from a Ljung–Box test remains greater than 5%. This allows for the most parsimonious specification possible. If we select an AR(p) as the preferred specification   Rt=μ+φ1Rt−1+…+φpRt−p+εt, (3.1) where εt is assumed to be uncorrelated with Rs for s < t, the conditional mean for Rt constructed at the end of week t – 1 is simply   μt=μ+φ1Rt−1+…+φpRt−p. 3.2 Variance Dynamics In a second step we use volatility models specific to each instrument. For equity returns we fit an Engle and Ng (1993) NGARCH model to the AR filtered residuals εt. For the majority of firms we rely on the NGARCH(1, 1), but we estimate a NGARCH(2, 2) when the p-value of an ARCH test is lower than 5%. The NGARCH(2, 2) is given by   εt=σtztσt2=w+β1σt−12+β2σt−22+α1σt−12(zt−1−γ1)2+α2σt−22(zt−2−γ2)2w=σ¯2[1−β1−β2−α1(1+γ12)−α2(1+γ22)]zt∼i.i.d.ast(λ,ν) (3.2) where we constrain α1,α2, β1,β2>0, and the variance persistence to be <1, and set the unconditional variance, σ¯2, equal to the sample variance of εt. The i.i.d. return residuals zt are assumed to follow the asymmetric standardized t-distribution from Hansen (1994) which we denote ast(λ,ν). The skewness and kurtosis of the residual distribution are nonlinear functions of the parameters λ and ν. When λ  = 0 the symmetric standardized t-distribution is obtained. When λ = 0 and 1/ν=0, we get the normal distribution. The corresponding cumulative return probabilities are given by   ηt≡Pr⁡t−1(R<Rt)=σt−1∫−∞σt−1(Rt−μt)ast(z;λ,ν)dz. (3.3) Note that the individual residual distributions are constant through time, but the individual return distributions do vary through time because the return mean and variance are dynamic. Using time series observations on εt, the parameters β1, β2, α1, α2, γ1, γ2, λ and ν are estimated using a likelihood function based on (3.2) and ast(z;λ,ν). When estimating the standard NGARCH models on CDS returns we found that in many cases they did not pass standard specification tests. Motivated by Brenner, Harjes, and Kroner (1996) we therefore instead estimate a level NGARCH model defined by   εt=σtztσt2=ΨtCDSt2δΨt=ω+β1Ψt−1+α1Ψt−1(zt−1−γ1)2 (3.4) where the NGARCH mean-reverting dynamic Ψt is multiplied by a power of the current CDS spread level, CDSt2δ. In our empirical application this specification adequately captures the variance dynamics in CDS returns. We estimate the parameters δ, ω, β1, α1, γ1, λ, and ν using a likelihood function based on (3.4) and ast(z;λ,ν). 3.3 Estimates of Mean and Variance Dynamics Panel A of Table III reports for each of the twelve autoregressive models the percentage of firms chosen by our model selection procedure. On average fewer than three lags are used to remove the autocorrelation in CDS returns, but the autocorrelation structure is much richer for some firms. The autocorrelation structure for equity returns is more parsimonious than for CDS returns. Panel A also shows the median and IQR across firms of each AR coefficient estimate. The parameter values vary considerably across firms. Table III Summary of AR-NGARCH estimation on weekly returns For each firm, we estimate in a first step an AR(12) model and remove the least significant lag until all lags are significant as long as a Ljung–Box test on residuals is not rejected at the 5% level. Panel A contains summary statistics on chosen AR specifications. For stock returns, we then estimate a NGARCH(1,1) model on residuals. If the Ljung–Box test on squared standardized equity residuals is rejected at the 5%, we estimate an NGARCH(2,2). For CDS returns, we estimate a Level-NGARCH(1,1). In both cases, the residual distribution is asymmetric t with parameters ν and λ. Panel B contains median and IQRs for NGARCH parameters as well as summary statistics on chosen model specifications. L-B(4) denotes a Ljung–Box test with four lags that the squared residuals (Panel B) are serially uncorrelated. Panel A: Conditional mean dynamics    CDS returns  Equity returns  Lag  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters    1  51%  0.18  [0.124, 0.240]  17%  –0.11  [–0.132, –0.063]  2  21%  0.12  [–0.105, 0.166]  12%  –0.10  [–0.112, –0.070]  3  21%  0.12  [–0.103, 0.159]  10%  0.09  [–0.035, 0.120]  4  23%  –0.13  [–0.163, –0.088]  7%  –0.08  [–0.101, –0.040]  5  14%  –0.09  [–0.151, 0.144]  5%  0.07  [–0.059, 0.101]  6  17%  0.12  [–0.128, 0.171]  7%  –0.06  [–0.121, 0.085]  7  19%  –0.12  [–0.142, –0.067]  26%  –0.11  [–0.133, –0.087]  8  12%  0.11  [0.062, 0.121]  5%  –0.03  [–0.084, 0.088]  9  7%  –0.09  [–0.149, 0.046]  6%  –0.08  [–0.107, –0.048]  10  41%  –0.14  [–0.182, –0.106]  10%  –0.08  [–0.114, –0.067]  11  12%  –0.10  [–0.123, 0.117]  7%  0.07  [–0.016, 0.105]  12  12%  –0.08  [–0.100, 0.092]  6%  –0.09  [–0.113, –0.055]  Average no. of lags used  2.49      1.19      Panel B: Conditional volatility dynamics and return distribution    Parameter estimates    CDS returns  Equity returns          β1  Median  0.861  0.808          IQR  [0.823, 0.902]  [0.674, 0.864]        β2  Median  –  0.276          IQR    [0.002, 0.528]        α1  Median  0.107  0.055          IQR  [0.078, 0.140]  [0.035, 0.094]        α2  Median  –  0.005          IQR    [0.000, 0.014]        γ1  Median  0.407  1.098          IQR  [0.157, 0.622]  [0.711, 1.654]        γ2  Median  –  8.897          IQR    [2.646, 12.518]        ν  Median  3.739  6.839          IQR  [3.210, 4.610]  [5.262, 9.006]        λ  Median  –0.157  –0.040          IQR  [–0.201, –0.102]  [–0.096, 0.008]        δ  Median  0.337  –          IQR  [0.256,0.568]          L-B(4) p-value z2 > 5%  Proportion of firms  93%  92%        NGARCH(1,1)  Proportion of firms  100%  86%        NGARCH(2,2)  Proportion of firms  0%  8%        Panel A: Conditional mean dynamics    CDS returns  Equity returns  Lag  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters  Proportion of firms for which lag is chosen  Median lag parameter  IQR of lag parameters    1  51%  0.18  [0.124, 0.240]  17%  –0.11  [–0.132, –0.063]  2  21%  0.12  [–0.105, 0.166]  12%  –0.10  [–0.112, –0.070]  3  21%  0.12  [–0.103, 0.159]  10%  0.09  [–0.035, 0.120]  4  23%  –0.13  [–0.163, –0.088]  7%  –0.08  [–0.101, –0.040]  5  14%  –0.09  [–0.151, 0.144]  5%  0.07  [–0.059, 0.101]  6  17%  0.12  [–0.128, 0.171]  7%  –0.06  [–0.121, 0.085]  7  19%  –0.12  [–0.142, –0.067]  26%  –0.11  [–0.133, –0.087]  8  12%  0.11  [0.062, 0.121]  5%  –0.03  [–0.084, 0.088]  9  7%  –0.09  [–0.149, 0.046]  6%  –0.08  [–0.107, –0.048]  10  41%  –0.14  [–0.182, –0.106]  10%  –0.08  [–0.114, –0.067]  11  12%  –0.10  [–0.123, 0.117]  7%  0.07  [–0.016, 0.105]  12  12%  –0.08  [–0.100, 0.092]  6%  –0.09  [–0.113, –0.055]  Average no. of lags used  2.49      1.19      Panel B: Conditional volatility dynamics and return distribution    Parameter estimates    CDS returns  Equity returns          β1  Median  0.861  0.808          IQR  [0.823, 0.902]  [0.674, 0.864]        β2  Median  –  0.276          IQR    [0.002, 0.528]        α1  Median  0.107  0.055          IQR  [0.078, 0.140]  [0.035, 0.094]        α2  Median  –  0.005          IQR    [0.000, 0.014]        γ1  Median  0.407  1.098          IQR  [0.157, 0.622]  [0.711, 1.654]        γ2  Median  –  8.897          IQR    [2.646, 12.518]        ν  Median  3.739  6.839          IQR  [3.210, 4.610]  [5.262, 9.006]        λ  Median  –0.157  –0.040          IQR  [–0.201, –0.102]  [–0.096, 0.008]        δ  Median  0.337  –          IQR  [0.256,0.568]          L-B(4) p-value z2 > 5%  Proportion of firms  93%  92%        NGARCH(1,1)  Proportion of firms  100%  86%        NGARCH(2,2)  Proportion of firms  0%  8%        Panel B in Table III shows the median and IQR across firms for each of the NGARCH parameters as well as the two parameters of the asymmetric t-distribution. For CDS returns we estimate a Level-NGARCH model for each of the 215 firms. For the equity returns we estimate an NGARCH(2,2) model for eighteen firms and an NGARCH(1,1) model for the remaining firms. The γ1 and γ2 parameters capture the asymmetric volatility response to positive and negative return residuals. For equities, the median γ1 value is 1.1 and the IQR is entirely positive. For CDS returns the median and IQR of γ1 are smaller in magnitude, but also positive. The Ljung–Box test of serial correlation in the zt2 shows that the NGARCH model is able to adequately capture variance dynamics. Equity returns have 8% of NGARCH models rejected by Ljung–Box at the 5% level, which is clearly not drastically above the size of the test. For CDS returns the level NGARCH model is rejected for 7% of the firms at the 5% significance level. The ν parameter has a median of 3.7 for CDS returns and 6.8 for equity returns, respectively, indicating fat tails in the conditional distribution. The asymmetry parameter, λ, is generally negative for both equity and CDS returns and larger in absolute value for CDS returns. Recall again that the CDS returns capture the returns to selling credit protection. As discussed above, Panel B of Figure 2 shows the time paths for the median and IQR of CDS return volatilities. Comparing the path of CDS volatility in Panel B and the path of equity volatility in Panel D is interesting. The time paths of the medians and the IQR are clearly moving together. Note that the relationship between equity volatility and CDS spreads has been extensively studied motivated by the Merton (1974) model. The relation between equity returns and equity volatility has also been extensively analyzed in the empirical literature. Panel C of Figure 2 confirms that equity returns are negatively related to equity volatilities in Panel D. The sample correlation of the medians is –0.65. This stylized fact is usually referred to as the leverage effect, and it leads to negative skewness in the return distribution. However, little is known about the relation between CDS return volatility and CDS returns. Note that our analysis focuses on the returns to selling credit protection. Visual inspection of Panels A and B suggests a positive relation between CDS spread levels and CDS return volatility which from Equation (2.2) implies a negative relation between the return to credit protection selling and CDS volatility. The sample correlation between the median CDS spread levels and the median CDS return volatility is 0.92. Figure 5 plots the median of the weekly level NGARCH dynamic in CDS returns for the nine industries from Figure 3. The variation of CDS return volatility across industries is quite dramatic. The high level of volatility in the financial crisis is apparent, but for the Information Technology and Telecommunication industries credit volatility is even more elevated in the aftermath of the bursting of the technology bubble. Figure 5. View largeDownload slide Median conditional volatility of CDS returns by sector. We report the weekly median conditional volatility of CDS returns by sector using the GIC sectors in Table I. Conditional volatility is estimated using a level-NGARCH model for each firm. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Figure 5. View largeDownload slide Median conditional volatility of CDS returns by sector. We report the weekly median conditional volatility of CDS returns by sector using the GIC sectors in Table I. Conditional volatility is estimated using a level-NGARCH model for each firm. We combine the energy and utility sectors. Each panel title indicates the total number of firms available for each sector throughout the sample. Panel A of Table IV contains descriptive statistics of the AR-NGARCH model residuals. Skewness and kurtosis are still present after standardizing by the NGARCH model, which motivates our use of the asymmetric standardized t innovations. As expected, the correlations between CDS and equity returns residuals are not materially different from the raw return correlations in Panel C of Table II. Table IV AR-NGARCH residual statistics and dynamic copula parameter estimation We report sample statistics on AR-NGARCH residuals and estimation results for different copula models. Using the AR-NGARCH residuals, z, we compute in Panel A the median and IQR of the skewness, kurtosis, correlations for each pair of firms, and correlations between CDS and equity for each firm. We estimate the DAC and the DSC with and without a time trend for the degree of freedom, and the DNC models on the 215 firms in our sample. Each of the models is estimated on AR-NGARCH residuals from weekly returns from selling the CDS contract and buying the equity. In the last two rows of Panel B and C, we report the Rivers and Vuong (2002) test statistics for model comparison. Panel A: Residual sample moments      Skewness  Kurtosis  Cross-firm correlation  Cross-security correlation      CDS returns  Median  –1.798  14.344  0.333  0.301      IQR  [–2.971, –1.002]  [8.191, 31.458]  [0.245, 0.416]  [0.215, 0.376]    Equity returns  Median  –0.173  4.909  0.295        IQR  [–0.403, –0.004]  [4.058, 6.322]  [0.232, 0.369]        Panel B: DAC estimation        Model I: νc(t) = 4 + δC,0  Model II: νc(t) = 4 + δC,0 exp( δC,1t )  Model III: νc(t) = 4 + δC,0 exp(δC,1t + δC,2t2 )    CDS returns  Equity returns  CDS returns  Equity returns  CDS returns  Equity returns    βC  0.951  0.918  0.957  0.926  0.957  0.926  αC  0.019  0.019  0.016  0.018  0.016  0.018  Correlation persistence  0.971  0.937  0.973  0.944  0.973  0.944  δC,0  12.859  8.944  59.773  13.731  59.773  13.731  δC,1      –3.80E-03  –1.90E-03  –3.80E-03  –1.90E-03  δC,2          1.78E-10  –1.71E-09  λC  –0.082  –0.242  –0.049  –0.218  –0.049  –0.218  Composite log-likelihood  978,813  682,796  980,018  684,464  980,032  684,473  Model comparison t-statistic with DSC Model I  0.10  2.72  0.48  2.47  0.48  2.46  Model comparison t-statistic with DAC Model I  –  –  0.42  0.83  1.19  0.21  Panel C: DSC estimation        Model I: νc(t) = 2 + δC,0  Model II: νc(t) = 2 + δC,0 exp( δC,1t )        CDS returns  Equity returns  CDS returns  Equity returns        βC  0.956  0.913  0.957  0.914      αC  0.016  0.019  0.016  0.019      Correlation persistence  0.972  0.932  0.973  0.933      δC,0  13.455  11.095  31.893  18.338      δC,1      –2.42E-03  –1.69E-03      Composite log-likelihood  978,685  672,099  981,489  673,747      Model comparison t-statistic with DNC  5.73  4.69  5.15  4.41      Model comparison t-statistic with DSC Model I  –  –  1.48  1.12        Panel D: DNC estimation        CDS Returns  Equity Returns            βC  0.956  0.908          αC  0.016  0.018          Correlation persistence  0.971  0.927          Composite log-likelihood  936,795  638,289          Panel A: Residual sample moments      Skewness  Kurtosis  Cross-firm correlation  Cross-security correlation      CDS returns  Median  –1.798  14.344  0.333  0.301      IQR  [–2.971, –1.002]  [8.191, 31.458]  [0.245, 0.416]  [0.215, 0.376]    Equity returns  Median  –0.173  4.909  0.295        IQR  [–0.403, –0.004]  [4.058, 6.322]  [0.232, 0.369]        Panel B: DAC estimation        Model I: νc(t) = 4 + δC,0  Model II: νc(t) = 4 + δC,0 exp( δC,1t )  Model III: νc(t) = 4 + δC,0 exp(δC,1t + δC,2t2 )    CDS returns  Equity returns  CDS returns  Equity returns  CDS returns  Equity returns    βC  0.951  0.918  0.957  0.926  0.957  0.926  αC  0.019  0.019  0.016  0.018  0.016  0.018  Correlation persistence  0.971  0.937  0.973  0.944  0.973  0.944  δC,0  12.859  8.944  59.773  13.731  59.773  13.731  δC,1      –3.80E-03  –1.90E-03  –3.80E-03  –1.90E-03  δC,2          1.78E-10  –1.71E-09  λC  –0.082  –0.242  –0.049  –0.218  –0.049  –0.218  Composite log-likelihood  978,813  682,796  980,018  684,464  980,032  684,473  Model comparison t-statistic with DSC Model I  0.10  2.72  0.48  2.47  0.48  2.46  Model comparison t-statistic with DAC Model I  –  –  0.42  0.83  1.19  0.21  Panel C: DSC estimation        Model I: νc(t) = 2 + δC,0  Model II: νc(t) = 2 + δC,0 exp( δC,1t )        CDS returns  Equity returns  CDS returns  Equity returns        βC  0.956  0.913  0.957  0.914      αC  0.016  0.019  0.016  0.019      Correlation persistence  0.972  0.932  0.973  0.933      δC,0  13.455  11.095  31.893  18.338      δC,1      –2.42E-03  –1.69E-03      Composite log-likelihood  978,685  672,099  981,489  673,747      Model comparison t-statistic with DNC  5.73  4.69  5.15  4.41      Model comparison t-statistic with DSC Model I  –  –  1.48  1.12        Panel D: DNC estimation        CDS Returns  Equity Returns            βC  0.956  0.908          αC  0.016  0.018          Correlation persistence  0.971  0.927          Composite log-likelihood  936,795  638,289          Finally, Panels B and D of Figure 4 plot the median and IQR threshold correlations on the weekly AR-NGARCH residuals. Compared to the threshold correlations from raw returns in Panels A and C, the median threshold correlations in residuals are typically lower, but still higher than the bivariate Gaussian distribution (dashed lines) would suggest. Overall, Figure 4 indicates that by removing univariate non-normality from the data, the AR-NGARCH models are also able to remove some of the multivariate non-normality. Modeling the remaining multivariate non-normality is the task to which we now turn. 3.4 Dynamic Copula Functions From Patton (2006b), who builds on Sklar (1959), we can decompose the conditional multivariate density function of a vector of returns for N firms, ft(Rt), into a conditional copula density function, ct, and the product of the conditional marginal distributions fi,t(Ri,t) as follows   ft(Rt)=ct(F1,t(R1,t),F2,t(R2,t),…,FN,t(RN,t))∏i=1Nfi,t(Ri,t)=ct(η1,t,η2,t,…,ηN,t)∏i=1Nfi,t(Ri,t), (3.5) where Rt is now a vector of N returns at time t, fi,t is the density, and Fi,t is the cumulative distribution function of Ri,t. Following Christoffersen et al. (2012) and Christoffersen and Langlois (2013) we allow for dependence across the return residuals using the copula implied by the skewed t-distribution discussed in Demarta and McNeil (2005). The skewed t copula cumulative distribution function Ct for N firms can be written as   Ct(η1,t,η2,t,…,ηN,t;Ψ,λC,νC,t)=tΨ,λC,νC,t(tλC,νC,t−1(η1,t),tλC,νC,t−1(η2,t),…,tλC,νC,t−1(ηN,t)), (3.6) where λC is a scalar parameter capturing copula asymmetry, νC,t is a time-varying scalar parameter denoting the copula degree of freedom, tΨ,λC,νC,t is the multivariate skewed t density with correlation matrix Ψ, and tλC,νC,t−1 is the inverse cumulative distribution function of the corresponding univariate skewed t-distribution. Note that the copula correlation matrix Ψ is defined using the correlation of the “copula residuals” zi,t∗≡tλC,νC,t−1(ηi,t) and not of the return residuals zi,t. If the marginal distribution in (3.3) is close to the copula tλC,νC,t distribution, then zt∗ will be close to zt. We now build on the linear correlation techniques developed by Engle (2002) and Tse and Tsui (2002) to model dynamic copula correlations. We use the “copula residuals” zi,t∗≡tλ,ν−1(ηi,t) as the model’s building block instead of the return residuals zi,t. In the case of non-normal copulas, the fractiles do not have zero mean and unit variance, and we therefore standardize zi,t∗ before proceeding. The copula correlation dynamic is driven by   Γt=(1−βC−αC)Ω+βCΓt−1+αCz¯t−1∗z¯t−1∗⊤ (3.7) where βC and αC are positive scalars, and z¯t∗ is an N-dimensional vector with typical element z¯i,t∗=zi,t∗Γii,t. The matrix Γt is a weighted average of three components: a constant matrix Ω which captures the average level of correlation, an auto-regressive term Γt−1 reflecting how conditional correlations in the previous period affect current correlations, and the cross-product of z¯t−1∗ which depicts how lagged (transformed) return shocks impact current correlations. The weighting parameters, βC and αC, describe how current correlations are affected by these components; a higher αC indicates that a positive (negative) cross-product of shocks, z¯i,t−1∗z¯j,t−1∗⊤, translates into a larger increase (decrease) in Γij,t compared to a smaller αC value, whereas a higher βC smooths over time the correlation deviations from their average level Ω. The conditional copula correlations are defined via the normalization   Ψij,t=Γij,t/Γii,tΓjj,t. To allow for general patterns in tail dependence, we allow for slowly moving trends in the degrees of freedom. Following Engle and Rangel (2008), who model the trend in volatility, we define the degree of freedom at time t, νC,t, using an exponential quadratic spline   νC,t=νC̲+δC,0 exp ⁡(δC,1t+∑j=1kδC,j+1max⁡(t−tj−1,0)2) (3.8) where νC̲ is the lower bound for the degrees of freedom, which is equal to four for the skewed t copula, δC,0,…,δC,k+1 are scalar parameters to be estimated, and {t0=0,t1,…,tk=T} denotes a partition of the sample in k segments of equal length. The specification of k different segments allows us to capture periods of positive and negative trends in the degrees of freedom. The exponential form ensures that the degrees of freedom are positive and above their lower bound at all times. Note that we model degree of freedom dynamics using splines and not lagged returns, because—unlike for variance and correlation—it is not obvious what the functional form of the lagged return should be when updating the degree of freedom process. In the next section, we investigate the time variation in both correlations and tail dependence. Whereas correlation at time t is driven by the dynamic in Equation (3.7), tail dependence is determined by both the time-varying correlation and the degrees of freedom. Hence, our model allows for changes in tail dependence that are separate from those in correlation. Below we refer to the model using (3.6)-(3.8) as the Dynamic Asymmetric Copula (DAC) model. The special case where λC=0, we denote by the Dynamic Symmetric Copula (DSC). In this case the lower bound for the degree of freedom is νC̲=2. When we additionally impose 1/νC=0, we obtain the Dynamic Normal Copula (DNC). Following Engle, Shephard, and Sheppard (2008), we estimate the copula parameters αC, βC, λC, and δC using the composite likelihood (CL) function defined by   CL(αC,βC,λC,δC)=∑i=1N∑j>1∑t=1Ti,jln⁡ct(ηi,t,ηj,t;αC,βC,λC,δC), (3.9) where ct is the copula density from (3.5) and Ti,j is the sample size available for pair i and j. Note that the CL function is built from the bivariate likelihoods so that the inversion of large-scale correlation matrices is avoided. In a sample as large as ours, relying on the CL approach is imperative. The unconditional correlations are estimated by unconditional moment matching (See Engle and Mezrich, 1996)  Ω^i,j=1Ti,j∑t=1Ti,jz¯i,t∗z¯j,t∗, (3.10) which is another crucial element in the feasible estimation of large-scale dynamic models. As discussed above, the estimation of dynamic dependence models using long time series and large cross-sections is computationally intensive. In our case, estimating the dynamic copula models for 215 firms is possible only because we implement unconditional moment matching and the CL approach. An additional advantage of the CL approach is that we can use the longest time span available for each firm pair when estimating the model parameters, thus making the best possible use of a cross-section of CDS time series of unequal length. 3.5 Copula Correlation and Tail Dependence Estimates Panel B of Table IV contains the Dynamic Asymmetric Copula (DAC) parameter estimates and CLs from fitting the model to the 215 firms in our sample. We again present separate results for models estimated on the weekly residuals of CDS and equity returns. The different GARCH specifications required for CDS and stock returns and the differences in correlation persistence demonstrate the importance of modeling separate dynamics for volatility and correlation. Panel C of Table IV reports the parameter estimates for the Dynamic Symmetric Copula (DSC) model where λC=0 and Panel D reports on the Dynamic Normal Copula (DNC) where we also impose 1/νC,t=0. To formally test for the difference in composite log-likelihood across models, we follow Oh and Patton (2016a) who apply the Rivers and Vuong (2002) tests to copula models estimated by composite log-likelihood maximization. Under the null hypothesis that two competing models fit the data equally well, the standardized difference in composite log-likelihood has a standard normal distribution. We use a Newey–West estimator with T14≈5 lags to estimate the standard error of the difference in composite log-likelihoods. The second-to-last line in Panel C reports t-statistics comparing the DSC models to the DNC models in Panel D. A positive value implies that the model in the top row of the panel is preferred to its DNC version. Each DSC model is significantly preferred to the Normal model. The second-to-last line in Panel B reports t-statistics comparing the DAC models to the DSC models in Panel C. The improvement in fit is largest for equities. This result matches the patterns in the threshold correlations in Figure 4 which show the strongest degree of bivariate asymmetry for equities. We also estimate a model with a simple time trend for the copula degree of freedom. The estimates in columns 3 and 4 of Panel B indicate that degrees of freedom have been trending down for both CDS and equity thus increasing the degree of multivariate non-normality. In columns 5 and 6, we allow for more complex shapes in degrees of freedom by increasing the number of splines in Equation (3.8), and find that the decreasing time trend is robust. Remarkably, the multivariate asymmetry parameters λC are also largely unaffected by the inclusion of a quadratic trend for the degree of freedom dynamics. The last row in Panel B reports the t-statistics obtained when comparing the models with time-varying degrees of freedom (columns 3 to 6) to the models with constant degrees of freedom (columns 1 and 2). Given the modest improvement in likelihood obtained by adding a quadratic segment, in the rest of the article we focus on the DAC specification with a simple log-linear time trend in the degree of freedom (in columns 3 and 4). Figure 6 plots the median and IQR of the DAC copula correlations for CDS and stock returns. The level of the CDS return correlation is higher than that of equity correlation throughout the sample. Credit correlations in Panel A show a pronounced and persistent uptick in 2007 around the time of the subprime funds collapse, and another pronounced uptick in mid-2011 following the US sovereign downgrade. The equity correlations in Panel B show less persistent upticks in late 2008 following the Lehman bankruptcy, and again in mid-2011 following the US sovereign downgrade. The differences in persistence following these major events are of course related to the differences in copula correlation persistence between credit and equity in Panel D of Table IV. Figure 6. View largeDownload slide Quantiles of copula correlations: CDS and equity returns. Using all available pairs of firms we report the weekly median (black line), and IQR (gray area) of the dynamic correlations from the DAC model. Figure 6. View largeDownload slide Quantiles of copula correlations: CDS and equity returns. Using all available pairs of firms we report the weekly median (black line), and IQR (gray area) of the dynamic correlations from the DAC model. Figure 7 plots the median and IQR of the DAC copula tail dependence for CDS and stock returns. Lower tail dependence is formally defined via the probability limit   τi,j,tL=lim⁡ζ→0Pr⁡[ηi,t≤ζ|ηj,t≤ζ]=lim⁡ζ→0Ct(ζ,ζ)ζ (3.11) where ζ is the tail probability.9 Upper tail dependence is defined analogously. We obtain an approximation by numerically integrating the copula density function, and using the 0.001 quantile for lower and the 0.999 quantile for upper tail dependence. We plot the lower tail dependence, because it is the most interesting of the two tails from a risk perspective. Importantly, Figure 7 shows that equity and credit tail dependence increase more over the sample than the copula correlations in Figure 6. Figure 7. View largeDownload slide Quantiles of tail dependence: CDS and equity returns. Using all available pairs of firms we report the median (black line) and IQR (gray area) of the dynamic tail dependence from the DAC model. Figure 7. View largeDownload slide Quantiles of tail dependence: CDS and equity returns. Using all available pairs of firms we report the median (black line) and IQR (gray area) of the dynamic tail dependence from the DAC model. In Figure 8, we plot the median DAC correlation of CDS and equity returns for the nine industries in Figure 3. Figure 9 does the same for the tail dependence of credit and equity. While the uptick in credit correlation during 2007 is evident for most industries, the variation across industries is large. The time paths of credit and equity tail dependence are clearly different from each other and are fairly similar across industries, because all industries share the same trend. Figure 8. View largeDownload slide Median copula correlation within sectors: CDS and equity returns. We report the median dynamic copula correlation by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity correlations. We combine the energy and utility sectors. Figure 8. View largeDownload slide Median copula correlation within sectors: CDS and equity returns. We report the median dynamic copula correlation by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity correlations. We combine the energy and utility sectors. Figure 9. View largeDownload slide Median lower tail dependence within sectors: CDS and equity returns. We report the median lower tail dependence by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity tail dependence. We combine the energy and utility sectors. Figure 9. View largeDownload slide Median lower tail dependence within sectors: CDS and equity returns. We report the median lower tail dependence by sector using the GIC sectors in Table I. The black line shows CDS and the gray line equity tail dependence. We combine the energy and utility sectors. 4. Economic Applications In this section, we apply our estimated credit volatilities and correlations to the measurement of portfolio diversification benefits and to credit spread prediction. Finally, we investigate potential economic drivers of the credit return correlations. 4.1 Conditional Diversification Benefits Consider a financial institution that has sold credit protection on a broad array of corporate names, thus collecting the CDS premia while hoping to diversify default risk. While the premia collected are readily observed, the institution needs a model such as that developed above to provide real-time quantification of the expected benefits from diversification. To be specific, we consider an equal-weighted portfolio of the constituents of the on-the-run CDX investment grade index in any given week. We want to assess the diversification benefits of the portfolio using the dynamic, non-normal copula model developed above. As in Christoffersen et al. (2012), we define the conditional diversification benefit by   CDBt(p)≡ES¯t(p)−ESt(p)ES¯t(p)−ES̲t(p), (4.1) where ESt(p) denotes the expected shortfall with probability threshold p of the portfolio at hand, ES¯t(p) denotes the average of the ES across firms, which is an upper bound on the portfolio ES, and ES̲t(p) is the portfolio VaR, which is a lower bound on the portfolio ES. The CDBt(p) measure takes values on the [0,1] interval, and is increasing in the level of diversification benefit. Note that by construction the conditional diversification benefit (CDB) does not depend on the level of expected returns. Expected shortfall is additive in the conditional mean which thus cancels out in the numerator and denominator in (4.1). Note also that CDB takes into account diversification benefits arising from all higher-order moments and not just variance. The CDB measure depends on the threshold probability p. Below we consider p=5%. The CDB measure is not available in closed form for our dynamic copula model and so we compute it using Monte Carlo simulation. Each week t we form an equally weighted portfolio of the 125 companies in the CDX.NA.IG index at time t. We use the longest available history of returns up to week t – 1 to estimate the unconditional correlation matrix for the 125 firms, and then compute the conditional correlations from our DAC model. In order to have sufficient historical data available, we keep only firms with at least 2 years of data, and start on September 22, 2004, which is the first day of Series 3 of the index. The solid black line in Figure 10 shows the CDB(5%) measure for an equal-weighted portfolio selling credit protection as well as for an equal-weighted portfolio of equity returns for the same firms. First, consider Panel A: diversification benefits for CDS have declined from about 80% at the end of 2004 to around 50% at the end of our sample. The majority of the decline took place during the mid-2007 to mid-2008 period and was relatively gradual. Panel B shows that the decline in diversification benefits in equity markets has been smaller in magnitude, from just over 70% in 2007 to around 60% at the end of our sample. The majority of the decline in equity market diversification benefits took place at the end of 2008. The decline in diversification benefits from equity occurred later than the decline in diversification benefits in credit. This is consistent with the correlation and tail dependence patterns in Figures 6 and 7. Figure 10. View largeDownload slide Conditional diversification benefits. Credit and Equity Portfolios. 5% Tail. Using equally weighted portfolios of available on-the-run CDX firms in a given week, we compute the 5% CDB using the DAC model for CDS returns (Panel A) and equity returns (Panel B). The credit portfolio sells credit protection by shorting CDS contracts. The dashed line shows the average correlation (on the left-hand axis) and the gray line shows the average volatility (on the right-hand axis). We use the first 2 years of the sample to estimate the unconditional correlation matrix, and thus plot CDB starting in 2004. Figure 10. View largeDownload slide Conditional diversification benefits. Credit and Equity Portfolios. 5% Tail. Using equally weighted portfolios of available on-the-run CDX firms in a given week, we compute the 5% CDB using the DAC model for CDS returns (Panel A) and equity returns (Panel B). The credit portfolio sells credit protection by shorting CDS contracts. The dashed line shows the average correlation (on the left-hand axis) and the gray line shows the average volatility (on the right-hand axis). We use the first 2 years of the sample to estimate the unconditional correlation matrix, and thus plot CDB starting in 2004. Figure 10 also depicts the average volatilities (in gray, on the right-hand axis) and the average correlations (the dashed line, on the left-hand axis). Intuitively, changes in the diversification measure should be closely (inversely) related to changes in correlation, which captures risk that is more systematic in nature. The relationship with average volatility is less obvious ex ante due to the normalization of the CDB measures. Figure 10 indeed shows that overall diversification benefits are much more closely related to average correlations than to average volatilities. Average correlation and average volatility sometimes co-move strongly—perhaps most obviously during the US sovereign debt downgrade in August 2011 which triggered a large drop in CDB. It is also interesting to relate credit diversification benefits in Panel A to equity volatilities in Panel B. The average volatilities in Panel B are highly correlated with the VIX and with other indicators of turmoil in equity markets. Clearly, the majority of the decline in diversification benefits in credit markets took place well before the peak in equity market volatility. The credit market CDB actually increased a bit during late 2008 and early 2009 when equity market turmoil was most intense. We conclude that while the data confirm the relationship between the level of credit spreads and equity volatility predicted by Merton-type structural models (see Figure 2), credit diversification benefits are—partly by design—more tightly linked with correlations. Above we have discussed various differences and similarities between the returns to holding equity on the one hand and selling credit protection on the other. The differences we find could to some extent be driven by counterparty credit risk priced into the CDS spreads. Using a proprietary data set on multiple dealers, Arora, Gandhi, and Longstaff (2012) indeed find that counterparty credit risk is priced in the CDS market. However, they also find that the effect is “vanishingly” small. We, therefore, do not attempt to identify counterparty credit risk in our empirical analysis. 4.2 Economic Determinants of Credit Spreads We now investigate if our new dynamic credit risk measures predict credit spreads when controlling for the usual economic drivers of credit spreads. The determinants of credit spreads have been extensively studied both theoretically and empirically. Most notably, following the analysis of Merton (1974), structural models of credit risk have established volatility, interest rates, and leverage as prime candidates to explain credit spreads. Partly based on theory, there is an extensive empirical literature regarding the determinants of credit spreads, both using bond data and CDS data. This literature provides some support for structural models of credit risk, and has also documented other macroeconomic and firm-specific determinants of credit risk.10 The definitions of the variables we consider are detailed in the appendix. We investigate if our CDS dependence measures, credit correlation and tail dependence, help predict changes in credit spreads. We also analyze the impact of CDS volatility on credit spread changes. We present results for univariate regressions of credit spread changes on the average copula correlation and tail dependence for each firm with all other firms, but we also present results for multivariate regressions where these dependence measures are added to equity volatility, term structure variables, and leverage, which are the determinants of credit risk according to the Merton (1974) model. Several studies have specifically questioned the ability of regressors suggested by theory to explain time-series variation in spreads (see Collin-Dufresne, Goldstein, and Martin, 2001), so we focus on pooled time-series regressions. We include lagged changes in spreads as regressors because of the persistence in the spreads. The signs of the estimated coefficients do not change when lagged spreads are not included (not reported). Table V presents the results. We run predictive panel regressions with firm fixed effects and double-clustering in which all predictors are lagged by 1 week. A first important conclusion is that CDS correlation and tail dependence in regressions (x) and (xi) are positively and significantly related to changes in credit spreads. We obtain these results even when including additional control variables in the regressions. In univariate regressions (v) and (vi), we see that the first lag of the spread changes captures a larger part of its future variation. CDS correlation and tail dependence are still positively related to future spread changes, but are not as significant as in the multivariate regressions. Finally, we find some support for the theory underlying structural credit risk models. Credit spreads increase with equity volatility, but leverage is not significant. Table V Panel regressions for changes in log CDS spreads We report regression coefficients and adjusted R2 from panel regressions. The left-hand side variable is the weekly change in log CDS spread for each firm. Right-hand side variables include the firm’s leverage ratio, the weekly and 1-year trailing return on the S&P 500 index, the 3-month constant maturity US Treasury rate, the difference between the 10-year and the 3-month constant maturity US Treasury rates, the TED spread, the firm-level equity NGARCH volatility and illiquidity, the firm-level CDS NGARCH volatility and illiquidity, the average equity and CDS illiquidity, the average CDS correlation with all other firms, and the average CDS tail dependence with all other firms. Illiquidity for equity is the Amihud price impact measure. Illiquidity for CDS the modified Amihud (MA) measure. All regressors are lagged, and we also include the first lag of the regressand. We run panel regressions with firm fixed effects. Standard errors are clustered by time and firm. Significance for regression coefficients at 5% and 1% are denoted by * and **, respectively. Estimates for the TED spread are multiplied by 100 for ease of exposition.   (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  Lagged log CDS change  0.051*  0.086**  0.086**  0.088**  0.088**  0.088**  0.051*  0.051*  0.052*  0.054*  0.053*  Leverage  –0.006            –0.006  –0.008  0.004  –0.007  –0.008  S&P 500 return  –0.328*            –0.332*  –0.347**  –0.320*  –0.330*  –0.331*  S&P 500 1-year return  0.029            0.029  0.035*  0.028  0.027  0.019  Interest rate level  0.003            0.003  0.002  0.002  0.007  0.012*  Yield curve slope  0.002            0.002  0.001  0.001  0.006  0.011*  TED spread  0.010            0.009  0.008  0.011  0.005  0.005  Equity volatility  0.010            0.010  0.009  0.016**  0.009  0.010  Equity illiquidity  –0.025            –0.019  –0.026  –0.025  –0.020  –0.023  Equity market illiquidity  –1.983            –1.896  –4.558  –2.130  –2.286  –2.476  CDS illiquidity MA    –0.696          –0.314          CDS market illiquidity MA      –6.755          32.451        CDS volatility        –0.004          –0.006*      CDS correlation          0.026          0.098*    CDS tail dependence            0.031          0.264**  Average Adjusted R2 (%)  1.826  0.647  0.655  0.813  0.709  0.693  1.818  1.938  1.965  2.071  2.207    (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  Lagged log CDS change  0.051*  0.086**  0.086**  0.088**  0.088**  0.088**  0.051*  0.051*  0.052*  0.054*  0.053*  Leverage  –0.006            –0.006  –0.008  0.004  –0.007  –0.008  S&P 500 return  –0.328*            –0.332*  –0.347**  –0.320*  –0.330*  –0.331*  S&P 500 1-year return  0.029            0.029  0.035*  0.028  0.027  0.019  Interest rate level  0.003            0.003  0.002  0.002  0.007  0.012*  Yield curve slope  0.002            0.002  0.001  0.001  0.006  0.011*  TED spread  0.010            0.009  0.008  0.011  0.005  0.005  Equity volatility  0.010            0.010  0.009  0.016**  0.009  0.010  Equity illiquidity  –0.025            –0.019  –0.026  –0.025  –0.020  –0.023  Equity market illiquidity  –1.983            –1.896  –4.558  –2.130  –2.286  –2.476  CDS illiquidity MA    –0.696          –0.314          CDS market illiquidity MA      –6.755          32.451        CDS volatility        –0.004          –0.006*      CDS correlation          0.026          0.098*    CDS tail dependence            0.031          0.264**  Average Adjusted R2 (%)  1.826  0.647  0.655  0.813  0.709  0.693  1.818  1.938  1.965  2.071  2.207  CDS illiquidity and CDS market illiquidity are positively correlated with CDS levels, but the estimates are never significant.11 We conclude that the CDS correlation and tail dependence measures that we estimate using the dynamic copula model are important determinants of the time-series variation in credit spreads. 4.3 Economic Determinants of Credit Dependence Are our new dynamic measures of credit dependence related to traditional economic determinants of credit risk? To answer this question, we now consider predictive regressions of copula correlations on various economic and financial determinants. An important class of credit default models uses observable macroeconomic factors to characterize the clustering in defaults and cross-firm default dependence. We have obtained estimates of default dependence without relying on such observable factors, and it is useful to investigate how closely our estimates are related to economic variables that are commonly used as factors. We focus on explaining median dependence because our first concern is to verify if the economic variables can explain the time variation in the dependence measures. The cross-sectional variation in dependence and the loadings of different firms on the dependence measures are also of interest, but we leave this topic for future work. In the regressions, we limit ourselves to economic variables that are available at the weekly frequency because we have modeled dependence at the weekly frequency, and we want to capture as much of the time-series variation as possible. The variables are again defined in the appendix. Table VI presents the regression results. For the CDX, VIX, S&P 500 returns, illiquidity, and term structure variables, univariate regressions are provided in columns (i)–(viii). Column (ix) presents the results of a multivariate regression including all variables, and column (x) further includes the lagged dependent variable. All regressors are lagged one week and all results are obtained using OLS with Newey–West standard errors using T1/4≈5 lags, where T is sample size. Table VI Time series regressions for median CDS correlations We regress the weekly DAC median CDS correlation on the CDX North American investment grade index, the credit market modified Amihud illiquidity measure, the CBOE implied volatility index, the stock market aggregated Amihud illiquidity measure, the weekly return and the 1-year trailing return on the S&P 500, the 3-month constant maturity US Treasury rate, the difference between the 10-year and the 3-month constant maturity US Treasury rates, the TED spread, the West Texas Intermediate Cushing crude oil spot price, the Aruoba–Diebold–Scotti business conditions index, and the US breakeven inflation rate. All regressors are lagged, and the first lag of the regressand is included in specification (x). We compute Newey–West standard errors, and significance for regression coefficients at 5% and 1% are denoted by * and **, respectively. All regression estimates are multiplied by 10 for ease of exposition, except for the first lag of the regressand.   (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)   Intercept  –0.43  4.02**  0.64*  3.99**  4.17**  4.25**  4.57**  3.79**  –1.87**  –0.26*  First lag                    0.91**  CDX  1.06**                0.60**  0.05  CDS market illiquidity    1361.56**              258.82*  87.98*  VIX      1.20**            0.41**  0.10**  Equity market illiquidity        175.49**          –13.16  –12.42**  S&P 500 return          –1.11        1.46**  –0.42  S&P 500 1-year return            –1.43**      0.22  0.06  Interest rate level              –0.22**    –0.17**  –0.02  Yield curve slope                0.20**  –0.25**  –0.03  TED spread                  –0.00  –0.00  Crude oil price                  0.62**  0.04  Business conditions index                  0.04  0.01  Breakeven inflation                  0.14  0.02  Adjusted R2  0.73  0.19  0.57  0.20  –0.00  0.17  0.41  0.16  0.90  0.99    (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)   Intercept  –0.43  4.02**  0.64*  3.99**  4.17**  4.25**  4.57**  3.79**  –1.87**  –0.26*  First lag                    0.91**  CDX  1.06**                0.60**  0.05  CDS market illiquidity    1361.56**              258.82*  87.98*  VIX      1.20**            0.41**  0.10**  Equity market illiquidity        175.49**          –13.16  –12.42**  S&P 500 return          –1.11        1.46**  –0.42  S&P 500 1-year return            –1.43**      0.22  0.06  Interest rate level              –0.22**    –0.17**  –0.02  Yield curve slope                0.20**  –0.25**  –0.03  TED spread                  –0.00  –0.00  Crude oil price                  0.62**  0.04  Business conditions index                  0.04  0.01  Breakeven inflation                  0.14  0.02  Adjusted R2  0.73  0.19  0.57  0.20  –0.00  0.17  0.41  0.16  0.90  0.99  Table VI shows that the univariate regressions all provide intuitively plausible results: when times are bad and the economy experiences negative shocks, the CDX, the VIX, and market illiquidity are high, stock returns and interest rates are low, and this poor economic environment is associated with higher dependence and fewer diversification opportunities. However, the R-squares indicate that stock market returns explain much less of the time-series variation in copula correlations than the interest rate level and the VIX. Finally, the R-square in regression (i) indicates that the level of credit risk, represented by the CDX index, is a prime candidate for explaining dependence in credit markets.12 So far we have separately analyzed each individual coefficient which may thus be influenced by omitted variables bias. In specification (ix) we, therefore, include all the variables simultaneously and in specification (x) we further include the lagged dependent variable. Surprisingly, in the multivariate regression (ix), the stock market returns have a positive sign but this is not the case in (x). The interest rate level remains statistically significant in (ix) but not in (x). Our most robust finding is that the VIX turns out to be important for explaining credit dependence, in spite of the fact that volatilities do not always co-move with credit correlations, as documented in Figure 10 and Section 4.1. In fact, somewhat surprisingly, in (x) the estimated coefficient on VIX is significant but the coefficient on CDX is not. The insignificance of CDX is partly due to the smaller point estimate of the coefficient—which is still positive—and partly due to the larger standard deviation on the coefficient when all variables are included. Credit market illiquidity remains positive and significant in both cases. In summary, we have found some evidence of economic drivers of the median credit copula correlation, mainly CDS market liquidity and the VIX. However, the strong autocorrelation in the CDS correlations renders steadfast inference difficult. 5. Conclusion This article documents cross-sectional dependence in CDS returns, and compares it with dependence in equity returns. Our results are complementary to existing correlation and dependence estimates, which are typically based on historical default rates or factor models of equity returns, and to existing intensity-based studies, which characterize observable macro variables that induce realistic correlation patterns in default probabilities (see Duffee, 1999; Duffie, Saita, and Wang, 2007). We use econometric techniques that allow us to estimate a model with multivariate asymmetries and time-varying dependence using a long time series and a large cross-section of CDS spreads. We document six important stylized facts. First, copula correlations in CDS returns vary substantially over our sample and increase substantially following the financial crisis in 2007. Equity correlations also increase in the financial crisis, but somewhat later, and the increase is less pronounced and not as persistent. Second, our estimates indicate fat tails in the univariate distributions, but also multivariate non-normalities. Multivariate asymmetries seem to be less important for credit than they are for equities. Third, credit dependence is more persistent than equity dependence, and this greatly affects how major events such as the subprime funds collapse, the Lehman bankruptcy, and the US sovereign debt downgrade affect subsequent dependence in credit and equity markets. Fourth, tail dependence increases more dramatically in our sample period than do copula correlations. Fifth, the CDS dependence and tail dependence measures are related to the time-series variation in credit spreads, even after accounting for other well-known firm-level determinants of spreads. Sixth, VIX is an important driver of credit correlations over time. These stylized facts, and the increase in cross-sectional dependence in particular, have important implications for the management of portfolio credit risk. We illustrate these implications by computing the diversification benefits when selling credit protection. The increase in cross-sectional dependence following the financial crisis has reduced diversification benefits, not unlike what has happened in equity markets. Footnotes 1 Engle (1982) and Bollerslev (1986) developed the first ARCH and GARCH models. Bollerslev (1990) first combined the GARCH model with a t-distribution. 2 See Engle and Kroner (1995) for an early multivariate GARCH model and Engle and Kelly (2012) for a simplified dynamic correlation model. 3 The twelve firms are: AT&T Mobility LLC, Bombardier Capital Inc., Bombardier Inc., Cingular Wireless LLC, Capital One Bank USA National Association, Comcast Cable Communication LLC, General Motors Acceptance Corp., Intelsat Limited, International Lease Finance Corp., National Rural Utilities Coop Financial Corp., Residential Capital Corp., and Verizon Global Funding Corp. 4 In a robustness analysis, we use log-differences in CDS spreads. The dependence results are very similar. 5 To compute this present value, we need recovery rates for the entire cross-section of firms at all times. We obtain these recovery rates from Markit. When Markit does not provide a recovery rate, we use 0.4, which is the industry standard. 6 The structural approach goes back to Merton (1974). See Black and Cox (1976), Leland (1994) and Leland and Toft (1996) for extensions. See Zhou (2001) for a discussion of default correlation in the context of the Merton model. 7 See Jarrow and Turnbull (1995), Jarrow, Lando, and Turnbull (1997), Duffee (1999), and Duffie and Singleton (1999) for early examples of the reduced form approach. See Lando (2004) and Duffie and Singleton (2003) for surveys. 8 We build on Jondeau and Rockinger (2006), who analyze dynamic dependence using symmetric copulas. 9 See Patton (2006b) for an application of the tail dependence measure to exchange rates. 10 See Collin-Dufresne, Goldstein, and Martin (2001); Campbell and Taksler (2003); Cremers et al. (2008); and Ericsson, Jacobs, and Oviedo (2009). 11 The results in Table 5 are obtained using the modified Amihud (MA) illiqudity measure. We obtain similar results (unreported) when using the illiquidity measure in Junge and Trolle (2015). We are grateful to the authors for making their illiquidity measure available online. 12 The impact of the CDX index on copula correlations is to some extent mechanical because (transformations of) the individual credit spreads that constitute the index are used to update the copula correlations in Equation (3.7). 13 The TED spread is an indicator of liquidity in fixed-income markets. The funding liquidity variable in Fontaine and Garcia (2012) provides an alternative liquidity indicator, but it is not available at the weekly frequency. 14 See Blume and Keim (1991); Fons (1991); Helwege and Kleiman (1997); Hillegeist et al. (2004); Jonsson and Fridson (1996); Keenan, Sobehart, and Hamilton (1999); McDonald and Van de Gucht (1999); and Pesaran et al. (2006) for examples of other macroeconomic variables that are useful for explaining and forecasting credit spreads and default. 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A CDS illiquidity measure modeled after the Amihud equity illiquidity measure. For the CDS market, dollar volumes are not available and we follow Junge and Trolle (2015) by dividing absolute daily return by the number of dealers that provided a quote for that day and average over each week. The CDS market illiquidity measure is the median across all firms. We denote this measure by “MA” for Modified Amihud. The CDS market illiquidity measure of Junge and Trolle (2015) based on the price discrepancy between credit indexes and their CDS constituents (used in unreported results). The return on the S&P 500 captures the changes in stock market capitalization. We use the 1-week return as well as a trailing 1-year return. The term structure is captured by a level variable, the 3-month US Constant Maturity Treasury (CMT) index, and a slope variable, the 10-year CMT index minus the 3-month CMT. The difference between the interest rate on interbank loans and on short-term government debt, that is the TED spread.13 While theory suggests several determinants of credit spreads, there is no acknowledged theory on the selection of economic and financial factors that can capture cross-firm default dependence; perhaps, as a result the existing empirical literature is very extensive, and many different economic variables have been used as factors. Duffie, Saita, and Wang (2007) provide an excellent discussion of the existing literature, and choose 1-year trailing S&P 500 returns and interest rates as macro variables to capture default dependence in their own empirical implementation. Duan and Van Laere (2012) also use stock index returns and interest rates, and Collin-Dufresne, Goldstein, and Martin (2001) use S&P 500 returns, interest rates, and the VIX. Campbell, Hilscher, and Szilagyi, (2008) use S&P 500 returns to normalize firm returns in their analysis of default and credit risk. Doshi et al. (2013) use term structure variables and the VIX, and the latent variable model in Duffee (1999) uses interest rate factors to capture the dependence in credit spreads.14 When selecting economic variables in Table VI, we limit ourselves to variables that are available at the weekly frequency because we model dependence at the weekly frequency, and we want to capture as much of the time-series variation as possible. An analysis of lower frequency macroeconomic variables would be interesting but we leave that for future work. In the absence of explanatory variables suggested by theory, we consider economy-wide measures of risk in equity and default insurance markets, risk-free (government) term structures, and other macro variables that are reasonable additional metrics of the state of the economy, and that have explanatory power for credit spreads documented by the papers cited above. Specifically, we use the following regressors: The S&P 500 return variables, term structure variables, TED spread, and CDS and equity market illiquidity variables used in Table V. The log of the CDX North American investment grade index level is used to proxy for the overall level of risk in credit markets. The log of the VIX index represents equity market risk. The log of crude oil price as measured by the West Texas Intermediate Cushing Crude Oil Spot Price. The Aruoba-Diebold-Scotti (ADS) business condition index from the Federal Reserve Bank of Philadelphia. The breakeven inflation level implied by Treasury Inflation Protected Securities. Unlike standard inflation measures, this series is available at the weekly frequency. © The Authors 2017. Published by Oxford University Press on behalf of the European Finance Association. All rights reserved. For Permissions, please email: journals.permissions@oup.com

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Review of FinanceOxford University Press

Published: Mar 1, 2018

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