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## Abstract

Extracting information from daily CDS spreads, we propose a measure of correlated default risk, which we show is a meaningful predictor of bankruptcy clusters. Focusing on U.S. corporate bonds, we also find that our measure of correlated default risk is more pronounced and commands a higher premium during periods of financial distress and for speculative issues. For instance, we find that after controlling for other known determinants of bond pricing, a 0.5 increase in aggregate correlated default risk is associated with a 13-bps increase in credit spreads, and elevates to a 22-bps premium for speculative issues and to a 17-bps premium during periods of financial distress. Overall, our paper provides compelling evidence as to the efficacy of our measure in capturing correlations in the likelihood of default over time, and has important implications for future work in asset allocation and fixed-income pricing.

Determining and predicting the time-varying correlation in default risk is of fundamental interest and importance to academics and practitioners alike. A clearer grasp of correlated default risk has direct implications for fixed-income pricing and portfolio risk management, particularly since correlated default risk undermines diversification efforts and results in a greater likelihood of extreme losses. Overall, a cleaner and more powerful measure of correlated default risk is crucial for properly calibrating fixed-income portfolios and ascertaining the Value-at-Risk (VaR), as well as for determining capital adequacy needs or setting leverage thresholds on investment portfolios.

However, empirical challenges lie in how to measure this unobservable risk that is inherently difficult, if not impossible, to capture directly through observed default intensities. For one, default events are uncommon, particularly for highly rated investment grade issues. Thus, a sample correlation of default intensities provides an unreliable estimate of the actual time-varying correlation in the risk of default. Moreover, the time clustering in corporate defaults has empirically been significantly higher than what common firm-level and macroeconomic covariates imply (Das et al. [2007]; Duffie et al. [2009]),^{1} suggesting the need for a more direct and powerful measure of correlated default risk across firms.

Our purpose is to introduce a well-specified yet broadly observable measure that better captures correlated default risk, and to organize evidence suggesting its efficacy in capturing correlated default risk. To this end, we leverage the important insight that the correlation in the realization of defaults mainly emanates from correlation in default probabilities (Das et al. [2006, 2007]). That is, if we can calculate a more accurate measure of correlations in the likelihood of default, this measure should also capture correlated default risk.

To do so, we propose a method to extract information contained in daily, single-name CDS spreads^{2} to determine the correlation in the likelihood of default. In general, the CDS market is known to be far more liquid and efficient than the corporate bond market, with CDS spreads reflecting changes in credit quality of the reference entity in a more timely manner than the spreads of the corresponding bond issues (Blanco, Brennan, and Marsh [2005]; Ericsson, Jacobs, and Oviedo [2009]). Thus, to the extent that CDS spreads proxy for default probabilities (Friewald, Wagner, and Zechner [2014]), movements in CDS spreads provide us reliable and timely updates as the respective movements in the default probabilities of the corresponding reference entities.

As a result, rolling one-month correlations of daily, single-name CDS spreads provides us a measure of time-varying correlated default risk. That is, in capturing the correlation in the likelihood of default, our method ultimately captures the correlated default risk inherent in each issue, since the correlation in the realization of defaults predominately emanates from the correlation in default probabilities (Das et al. [2006, 2007]). Using CDS spreads in this way has the advantage of enabling us to avoid imposing any structural assumptions under the risk-neutral framework to infer default probabilities, and hence, allows us to avoid the joint hypotheses conundrum: that is, the validity of the structural assumptions vis-à-vis the informativeness of the data. This is mainly due to the fact that we are interested in the *correlation* in default probabilities and not the actual *levels* of default probabilities.

In our sample, which spans 63,377 bond-year-months from July 2002 to August 2013, the average correlation in default probabilities is 0.29, with a standard deviation of 0.24, suggesting substantial variation in correlated default risk across issues and across time. Notably, our comprehensive panel allows us to explore the cross-sectional and cross-temporal aspects of correlated default risk for the U.S. corporate bond market as a whole, as well as various important segments and sub-periods throughout time.

First, in examining how our *Aggregate Default Correlation* metric correlates with actual realizations of bankruptcies in the Unite States, we find that after controlling for various other factors that predict the incidence of bankruptcies, a 0.50 increase in *Aggregate Default Correlation* approximates to an 8% increase in bankruptcy filings, as reported by United States Courts.^{3} That is, our measure of correlated default risk is a meaningful predictor of bankruptcy clusters. Moreover, consistent with theory, *Aggregate Default Correlation* is positively associated with overall bond-market illiquidity,^{4} and is more pronounced in the sub-period spanning the recent financial crisis, defined by December 2007 to June 2009.^{5} We also find that *Aggregate Default Correlation* increases substantially following a market downturn, as measured by the S&P 500 Composite Index. That is, there is an asymmetry in joint asset distributions, whereby correlations tighten considerably in downturns relative to normal conditions (a la Ang and Bekaert [2002]; Das and Uppal [2004]).

Thus, our measure of correlated default risk moves as expected with the clustering in bankruptcy filings, as well as with observable indicators of the correlation in default probabilities. In addition, the literature shows that observable covariates alone are inadequate in their ability to account for the empirical clustering/intensity of defaults (Das et al. [2007]; Duffie et al. [2009]). As such, to the extent that our default-correlation measure is an effective estimator of the correlation in the realization of defaults, we expect that our measure should contain additional information beyond that explained by the same observable covariates, which, among other factors, include structural determinants such as leverage or volatility. Consistent with this idea, we find that the observable covariates account for less than 15% of the variation in our measure.

Furthermore, the pricing implications of our default-correlation metric further suggest its viability as an apt measure of correlated default risk. That is, controlling for pricing factors that are known to determine credit spreads, such as firm leverage, volatility, and bond-issue liquidity, we find that changes in credit spreads are substantially associated with the correlation in default probabilities. Specifically, a 0.50 increase in an individual issue’s average pairwise default correlation (with other issues in the cross-section) is associated with a modest 6 basis-point (i.e., 3.0%) increase in the attendant credit spread, and is elevated to an 11 basis-point (i.e., 5.3%) increase for speculative issues and to a 13 basis-point (i.e., 6.2%) increase during the period spanning the subprime financial crisis.^{6}

This association is markedly intensified when we examine the effects of aggregate economy-wide default correlation. That is, 0.50 increase in *Aggregate Default Correlation* is associated with a 13 basis-point (i.e., 6.4%) increase in credit spreads, and is elevated to a 22 basis-point (i.e., 10.5%) increase for speculative issues and to a 17 basis-point (i.e., 8.2%) increase in credit spreads during the period spanning the recent subprime financial crisis. Intuitively, the economy-wide tightening of correlations in default probabilities undermines diversification attempts and, thus, commands an even higher premium during times of financial distress. We find similar associations between our default correlation metric and changes in CDS spreads.

Overall, a timely and more powerful measure of the correlation in default probabilities is crucial for determining appropriate amounts of leverage for banks, shadow entities, and investment portfolios in general, and should also prove valuable to future work on asset allocation as well as fixed-income pricing with regime shifts, whereby correlations tighten considerably during bad times (a la Ang and Bekaert [2002], Gourieroux et al. [2014], and Elkamhi and Stefanova [2015]). Furthermore, the broadly observable metric of default correlation that we introduce should prove useful to future work concerning cross-sectional determinants of fixed-income and credit-derivatives pricing, and thus extends the work of Collin-Dufresne, Goldstein, and Martin [2001] and Friewald, Jankowitsch, and Subrahmanyam [2012], among others, who examine the determinants of credit spread and fixed-income pricing.

Our study is also related to Anderson [2016], who decomposes the sources of increased co-movement in the CDS spreads on a group of liquid, investment grade reference entities during the recent 2007–2009 financial crisis. Whereas Anderson [2016] seeks to understand the fundamental and non-fundamental sources of the marked spike in correlations during the crisis, we seek to understand the uses of and information conveyed by cross-sectional and time-varying correlations in CDS spreads across reference entities spanning 1,901 distinct issues over an (unbalanced) panel of 11 years. That is, we seek to explore the important default-associated and pricing implications during both crisis and non-crisis years, as well as for a broader array of reference entities (both investment and non-investment grade).

This paper is organized as follows. In the next section we describe our method to measuring correlations in the likelihood of default using CDS data. In the following section we describe our data sources, define our main variables of interest, and provide key summary statistics. Then we explore the determinants of correlated default risk, and then we examine how our measure of *Aggregate Default Correlation* relates to the incidence of bankruptcy filings. In the next two sections we examine how our measure of correlated default risk relates to corporate-bond spreads, and provide additional analyses to test the robustness and validity of our measure. In the final section we discuss and conclude.

## MEASURING CORRELATION IN DEFAULT PROBABILITIES

Given the empirical challenges in directly measuring correlated default risk, we instead propose that the information contained in credit default swap (CDS) spreads be employed to extract the correlation in default risk between reference entities. Specifically, we propose that the time-varying correlation in default probabilities can be captured via correlations in daily single-name CDS spreads, which provides us a reliable metric of correlated default risk, since default correlation predominately emanates from the correlation in default probabilities (Das et al. [2007]).

In general, CDS securities are known to be extremely liquid and efficient, with CDS spreads reflecting changes in the credit risk of the reference entity in a far more timely manner than the corresponding fixed-income issues (Blanco, Brennan, and Marsh [2005]; Ericsson, Jacobs, and Oviedo [2009]). Overall, the notional amount of CDS contracts grew from $0.6 trillion in June 2001 to a peak of $62.2 trillion by the second half of 2007,^{7} and has rapidly become the most prominent and liquid credit derivative, encompassing approximately half of the entire credit-derivatives market (Blanco, Brennan, and Marsh [2005]).

Thus, correlated movements in CDS premiums provide us reliable and timely updates as to the correlated movements in the respective reference entities’ default probabilities. That is, the CDS spread is determined by: the *expected loss*, which captures the reference entity’s likelihood of default times the loss given default, and the issuer’s *counterparty risk*, which accounts for the joint likelihood that the contract writer is unable to pay the promised amount upon default of the reference entity. However, only the likelihood of default is of first-order importance, with the recovery rates and counterparty risk either being time-invariant over our one-month estimation horizon or exhibiting relatively little impact on the CDS spread itself.

That is, recovery rates are known to be extremely persistent, and the majority of work in this arena assumes a constant recovery rate.^{8} Furthermore, the impact of counterparty risk on CDS spreads is exceedingly small and economically negligible (Arora, Gandhi, and Longstaff [2012]).^{9},^{10} Specifically, “the credit spread of a CDS dealer would have to increase by nearly 645 basis points to result in a one-basis-point decline in the price of credit protection.” Equivalently, counterparty risk has negligible impact on the covariance between CDS spreads.

Thus, the covariance between CDS spreads ultimately captures the covariance in default probabilities of the underlying reference entities:

1We emphasize that our method does not require us to impose structural assumptions under the risk-neutral framework to infer default probabilities, which allows us a more direct path to measuring correlated default risk. Our ability to do so is driven by the fact that we need only the *correlation* in default probabilities, and not the actual *levels* of the default probabilities themselves. To the extent that CDS spreads proxy for the risk-neutral default probabilities,^{11} our measure captures the correlation in risk-neutral default probabilities, though we do not rule out that our measure may also be capturing the correlation in physical default probabilities (insofar as the general movements are similar under both the physical and risk-neutral measures).

To lend further support to the validity of our measure as a proxy for the correlation in default probabilities, in later analyses (which we present in a later section), we explore the impact of recovery rates and CDS liquidity on the validity of our default correlation metric. Furthermore, since the correlation in CDS spreads could also reflect systematic movements in the price of default risk, we also explore the impact of time-varying credit-risk premiums on our default correlation metric. Finally, in additional robustness tests, we orthogonalize the correlation in CDS spreads to correlations in CDS liquidity and recovery rates as well as to the excess bond premium (*ebp*). We now proceed to describe our data sources and key variables.

## DATA SOURCES AND DESCRIPTION OF VARIABLES

In this section, we describe our data sources and present key summary statistics.

### Sources

Our sample period spans July 2002 to August 2013 and consists of the U.S. corporate bond issues that lie at the intersection of the Trade Reporting and Compliance Engine (TRACE), Markit, Mergent Fixed Income Securities Database (FISD), OptionMetrics, Federal Reserve Economic Data (FRED), CRSP, and COMPUSTAT datasets. In the following section, we describe our filtering process and provide a discussion of variables alongside dataset references. Ultimately, our final sample consists of a panel dataset of 63,337 monthly observations of 1,901 bond issues.

### Key Definitions and Discussion of Variables

We calculate the pairwise correlation in default probabilities between issues *i* and *j* as the Pearson’s rho correlation in the corresponding daily CDS spreads over the 30 days prior, which we obtain from Markit. *Default Correlation*_{i,t} is then defined as the average pairwise correlation in daily CDS spreads of issue *i* with all other issues in year-month *t*. We also define a macro-level measure, *Aggregate Default Correlation*_{t}, calculated as the average *Default Correlation*_{i,t} across all issues at time *t*.^{12}

In our calculations, we focus on the five-year, single-name CDS contracts, as these are the most common and most liquid format (Hull, Presdescu, and White [2004]). Moreover, to be included in our sample, the number of reported CDS premiums for a given reference entity during any given month must be at least equal to the average number of reported CDS premiums for that month, which is calculated by scaling the total number of reported CDS premiums for that month by the total number of observed reference entities in that month. This additional filter is applied to ensure that we include only those reference entities that are traded frequently enough to calculate meaningful correlations in CDS premiums with other sample reference entities.

In Appendix A, we provide a brief summary sheet of all variables, and in Appendix B, we provide a more detailed outline of variable descriptions and calculations, along with the corresponding data source(s).

### Summary Statistics

In Exhibit 1, we present summary statistics on basic bond, CDS, and firm characteristics. The average default correlation in our sample is 0.29, with a standard deviation of 0.24. Furthermore, the average credit spread is 2.08%, with an average time to maturity of 9.20 years, and 66% of issues are investment grade (i.e., rated at least BBB or higher). We now proceed to explore our default correlation metric in greater detail.

## THE DETERMINANTS OF CORRELATED DEFAULT RISK

To explore the observable covariates explaining the correlation in default probabilities across bond issues, we estimate the following OLS regression:^{13}

*Aggregate Default Correlation*_{t}, the dependent variable, represents the average *Default Correlation*_{i,}_{t} across all bond issues in year-month *t*. *X _{t}*

_{−1}is a vector of the following macro-level variables:

*r*10

_{t−1},

*Yield Slope*

_{t}

_{−1},

*S&P Return*

_{t−1},

*VIX*,

_{t−1}*Aggregate Amihud Illiquidity*

_{t−1},

*Aggregate Leverage*

_{t−1},

*ebp*, and

_{t−1}*Crisis*which are as defined in Appendix A. We employ a lagged regression specification, because

_{t−1}*Default Correlation*

_{i,t}is calculated based on information from the month prior.

*Aggregate Amihud Illiquidity*

_{t−1}and

*Aggregate Leverage*

_{t}

_{−1}represent the average

*Amihud Liquidity*and average

*Leverage*, respectively, across all issues in our sample at time

*t*− 1. T-statistics are calculated using Newey-West standard errors with four lags,

^{14}to account for heteroscedasticity and serial correlation in

*Aggregate Default Correlation*

_{t}.

The results, presented in Exhibit 2, show that *Aggregate Default Correlation* moves as expected with observable structural covariates that we expect to drive the correlation in default probabilities. First, we observe that *Aggregate Default Correlation* increases substantially with overall bond-market illiquidity, consistent with the theory that bond-market illiquidity adversely impacts debt renegotiation (Ericsson and Renault [2006]) as well as rollover losses (He and Xiong [2012]) and thereby raises the default boundary. Specifically, a 0.40 increase in *Aggregate Amihud Illiquity* translates to an increase in *Aggregate Default Correlation* of 0.40 (*t*-statistic = 2.89). For reference, 0.40 represents a one standard-deviation change in the *Amihud Illiquidity* measure. We note that this illiquidity coefficient estimate is substantially abated with the inclusion of the *Crisis*_{t} indicator variable, due to the fact that liquidity effects and flight-to-quality risk are particularly pronounced in times of financial crisis (Dick-Nielsen, Feldhütter, and Lando [2012]; Friewald, Jankowitsch, and Subrahmanyam [2012]).

Furthermore, our aggregate correlation measure increases considerably following negative realizations of the S&P500 Composite Index (coefficient estimate = −0.432, *t*-statistic = −1.72), which comports with the widely recognized principle that asset correlations tighten considerably following market downturns (Ang and Bekaert [2002]; Das and Uppal [2004]). Similarly, we observe that aggregate default correlation is elevated during the period spanning the recent financial crisis (coefficient estimate = 0.121, *t*-statistic = 3.02).

One possible concern is that, by dint of our method of calculation, our default-correlation metric may also be capturing the variation in the risk premium of default over time. That is, if CDS spread correlations increase during times when investors demand a greater price for bearing credit risk, our measure of aggregate default correlation would also pick up these time-varying excess credit-risk premiums in addition to the time-varying correlated default risk. However, in contrast to the *Crisis* indicator and the *S&P Composite Index*, the excess bond premium (i.e., *ebp,* as calculated by Gilchrist and Zakrajsek [2012]), has quite a modest association with *Aggregate Default Correlation*, whereby a one standard-deviation (i.e., 0.79%) increase in *ebp* is associated with a 0.006 decrease in aggregate default correlation (coefficient estimate = −0.077, *t*-statistic = −2.80). That is, our default correlation metric is not simply capturing the time variation in excess credit-risk premiums.

In addition, consistent with earlier findings that empirical covariates have very limited power to explain the empirical intensity or clustering of defaults (Das et al. [2007]; Duffie et al. [2009]), we observe that these standard structural determinants of default probabilities and credit spreads explain less than 15% of the variation in *Aggregate Default Correlation*, further suggesting its viability and validity as a proxy for the correlated risk that causes firms to fail together and drives up credit spreads. We now proceed to examine this idea in greater detail.

## AGGREGATE DEFAULT CORRELATION AND BANKRUPTCY CLUSTERS

In this section, we explore whether our measure of correlated default risk behaves in the way that we expect as economy-wide correlations tighten and relax over time. Specifically, to bolster confidence in our measure, we begin by examining whether our measure of aggregate default correlation is meaningfully associated with the intensity of bankruptcy filings. To start, we estimate the following OLS regression:

3log(*Bankruptcy Filings*_{t}), the dependent variable, represents the log of the sum of new Chapter 7, 11, 12, and 13 business filings at time *t*, as reported by United States Courts.^{15} *Aggregate Default Correlation*_{t} represents the average *Default Correlation*_{i,t} across all bond issues in year-month *t*. *X*_{t} is a vector of the following macro-level variables: *r*10_{t}, which is the yield on the 10-year Treasury note; *Yield Slope*_{t}, which is the difference between the 10-year and the 2-year Treasury yields; *S&P Return*_{t}, which is the monthly return on the S&P500 composite index; *LogGDP*_{t}, which we obtain from the FRED database; *Industrial Production*_{t}, which measures the real output of all relevant establishments located in the United States (also obtained from FRED); *Inflation*_{t}, which we also obtain from FRED; and *Unemployment _{t}*, as reported by the Bureau of Labor Statistics.

In our regressions, we employ a contemporaneous specification as well as a lagged-variables specification, whereby we regress *log(Bankruptcy Filings*_{t}*)* against lagged values of our explanatory variables. Thus, in the contemporaneous specification, we are regressing time *t* bankruptcy filings on aggregate default correlation measured over the prior 30-day lookback period. Similarly, in the lagged specification, we are regressing time *t* bankruptcy filings on aggregate default correlation measure over the 30-day look-back period beginning 60 days prior. *T*-statistics are calculated using Newey-West standard errors with four lags,^{16} to account for potential heteroscedasticity and serial correlation in *Bankruptcy Filings _{t}*.

The results, presented in Exhibit 3, show a substantial clustering of bankruptcy filings when *Aggregate Default Correlation* is high. Specifically, in the contemporaneous specification (Column 1), we see a coefficient estimate of 0.166 (*t*-statistic = 2.41) on *Aggregate Default Correlation*, which translates to an approximate 0.166 × 0.5 = 8.3% increase in the incidence of bankruptcy filings for a 0.50 increase in aggregate default correlation.^{17} Similarly, the coefficient estimate on *Aggregate Default Correlation* in the lagged specification (Column 2) is 0.163 (*t*-statistic = 2.14).^{18} Exhibit 11 demonstrates this relation graphically, whereby we interlay bankruptcy filings with aggregate default correlation throughout our sample period. Consistent with our findings in Exhibit 3, we observe that our measure of aggregate default correlation moves jointly with the incidence of bankruptcy filings, with both experiencing a marked spike during the crisis period.

These results suggest that a spike in our measure of aggregate default correlation subsequently translates to a clustering of realized bankruptcies, as we would expect to see in situations where economy-wide correlations tighten. We now proceed to explore additional implications and validity tests of our measure, namely as it relates to fixed-income and credit-derivatives pricing.

## IMPLICATIONS AND USES OF OUR MEASURE

In this section, we examine the association between credit spreads and the correlation in default probabilities, after accounting for other factors known to determine credit spreads.

We begin, in Exhibit 4, by presenting a comparison of basic summary statistics between crisis and non-crisis periods, as well as a comparison between investment and non-investment grade issues. As expected, we observe that average *Default Correlation* is higher in crisis periods (0.38) than in non-crisis periods (0.27). We also observe that average credit spreads are much higher in crisis years (3.98%) than in non-crisis years (1.64%). We do not observe a material difference in average *Default* *Correlation* within investment grade issues (0.29) versus non-investment grade issues (0.29); we do note, however, that average credit spreads are substantially higher among non-investment grade issues (3.61%) than among investment grade issues (1.31%).

### Credit Spreads and Correlated Default Risk

To further explore the efficacy and validity of our measure, we examine its changes in association with changes in the level of credit spreads via the following pooled OLS regressions:

4 5*Credit Spread*_{i,t}, the dependent variable, is the difference between the yield on bond *i* and the yield on the associated Treasury at the same maturity in year-month *t*. *Default Correlation*_{i,t} represents the average pairwise correlation in default probability of bond *i* with all other bonds at time *t*, as described earlier as well as in Appendix A, and *Aggregate* *Default Correlation*_{t} represents the average *Default Correlation* across all issues at time *t*. *X*_{i,t} represents a vector of control variables guided by prior literature studying this specification, and involves changes in firm-level characteristics (Δ*Amihud Illiquidity*_{i,t}, Δ*Investment Grade*_{i,t}, Δ*Leverage*_{i,t}, Δ*Volatility*_{i,t}) as well as changes in macro-level characteristics (Δ*r*10_{t}, Δ*Yield Slope*_{t}, Δ*ebp*_{t}, and *S**&P Retu**rn*_{t}), which are also described earlier in Appendix A. *T*-statistics are calculated using White-robust standard errors adjusted for firm-level clustering, which account for heteroscedasticity and serial correlation (Peterson [2009]).

The results, presented in Exhibit 5, suggest that a 0.50 increase in *Default Correlation* (Panel A, Column 1) is associated with a 6 basis-point increase (*t*-statistic = 6.97) in the attendant credit spread, and the same increase in *Aggregate Default Correlation* (Panel B, Column 1) is associated with a 13 basis-point increase (*t*-statistic = 8.64) in the attendant credit spread. That is, accounting for other known structural determinants of credit spread changes, such as leverage and volatility, the correlation in default probabilities remains a substantial explanatory factor in fixed-income pricing.

The results also suggest that the correlation in default probabilities commands a far higher premium in crisis years (Column 3) and among non-investment grade issues (Column 5). Specifically, a 0.50 increase in an individual issue’s average pairwise default correlation (with other issues in the cross-section) is associated with a modest 2 basis-point increase in credit spreads in non-crisis periods (*t*-statistic = 4.41), but is elevated to a 13 basis-point increase in credit spreads during the crisis period (*t*-statistic = 4.63).

Similarly, a 0.50 increase in *Aggregate Default Correlation* is associated with a 7 basis-point increase in credit spreads during non-crisis times (*t*-statistic = 7.50), and is elevated to a 17 basis-point increase during the crisis period (*t*-statistic = 5.03). Moreover, we observe that the aggregate correlation in default probabilities commands a substantially higher premium among non-investment grades issues. For instance, a 0.50 increase in *Aggregate Default Correlation* is associated with an 8 basis-point increase in credit spreads of investment grade issues (*t*-statistic = 10.02), but is associated with a 22 basis-point increase in credit spreads of non-investment grade issues (*t*-statistic = 5.66).

The results suggest that the correlation in default probabilities are meaningfully associated with credit spreads, even after controlling for other observable firm and macro characteristics, including the time-varying excess bond premium (as in Gilchrist and Zajrajsek [2012]). Furthermore, we note that in measuring correlated default risk via correlations in CDS spreads, our analyses encompass those firms that are actively CDS traded, which tend to be larger, safer, and more profitable, and have more working capital (Subrahmanyam, Tang, and Wang [2014]). Since correlated default risk likely commands a lower premium among such firms, we posit that any relation we observe between default-risk correlation and credit spreads within our sample likely understates the overall population impact.

### CDS Spreads and Correlated Default Risk

Furthermore, we make similar observations within the context of credit-derivatives pricing, wherein we estimate a pooled OLS regressions of changes in CDS spreads against changes in default correlations:

6 7where *CDS Spread*_{i,t} is the average five-year, single-name CDS spread on reference entity *i* for month *t*, and all other regression variables are as specified above in equations (4) and (5).

The results, which we present in Exhibit 6, show a similar pattern as in the previous set of analyses that studied changes in credit spreads. That is, the change in *Default Correlation* is also a significant determinant of the change in *CDS Spreads* (Panel A, Column 1: coefficient estimate = 0.077; *t*-statistic = 3.44), as is the change in *Aggregate Default Correlation* (Panel B, Column 1: coefficient estimate = 0.182, *t*-statistic = 3.32). Moreover, similar to the credit-spread specification, we continue to observe that the change in correlation in default probabilities is associated with a substantially higher change in CDS spreads during the crisis period (Column 3) and among non-investment grade reference entities (Column 5).

Finally, to be clear, we do not purport to solve the credit-spread puzzle or to identify the latent variable driving credit spreads (a la Collin-Dufresne, Goldstein, and Martin [2001]). In fact, when we conduct a principal component analysis (PCA) of the residuals from our credit-spread regression, the first principal component explains approximately 75% of the residual variation, which is similar in magnitude to that found by Collin-Dufresne et al. [2001]. Instead, our purpose is to introduce a valid and easily quantifiable metric of correlated default risk, and to organize evidence along this regard. To this end, we now proceed to additional analyses to further test the validity of our proposed metric.

## ADDITIONAL ANALYSES

The results thus far point to the validity of our measure in capturing the correlated default risk, and suggest the importance of our measure in fixed-income and credit-derivatives pricing as well as in portfolio risk management. However, a question remains as to whether our measure, *Default Correlation*, is truly reflecting correlated default risk itself, as opposed to time-varying excess bond premiums or correlations in CDS liquidity or recovery rates. Thus, we now proceed to empirically examine the extent to which CDS spreads—and more importantly, *correlations* in CDS spreads—may be affected by these factors. As an additional robustness check, we then proceed to examine the predictive power of our measure in bankruptcy clusters as well as the incremental pricing information it contains when we orthogonalize our measure of default correlation to the correlation in CDS liquidity and the correlation in recovery rates as well as to the excess bond premium (*ebp*).

### A Further Look at the Role of Recovery Rates and CDS Liquidity

**CDS Liquidity.** A growing body of evidence suggests that liquidity risk in the bond market is a significant priced factor in fixed-income credit spreads (Bao, Pan, and Wang [2011]; Dick-Nielsen, Feldhütter, and Lando [2012]; Friewald, Jankowitsch, and Subrahmanyam [2012]). However, the substantial differences between credit default swaps and their underlying fixed income securities make it unclear whether liquidity risk in the CDS market is a priced factor in CDS spreads. For one, as mentioned previously, the single-name CDS market is far more liquid than the bond market, and represents the most liquid credit derivative, accounting for approximately half of the overall credit-derivatives market (Blanco, Brennan, and Marsh [2005]). Furthermore, the notional amount of CDS contracts is technically unlimited, making it unlikely that supply/demand imbalances affect the CDS market as they do the bond market, and since new CDS contracts can be written at any time, the CDS market is less susceptible to “cornering” or “squeezing” (Longstaff, Mithal, and Neis [2005]).^{19}

More recent evidence suggests a strong liquidity component in intermediary asset pricing in general (Brunnermeir and Pedersen [2009]; He and Krishnamurthy [2013]; and Kondor and Vayanos [2014]), and in CDS spreads in particular (Tang and Yan [2007]; Bongaerts, de Jong, and Driessen [2011]; Junge and Trolle [2015]). Specifically, the capital and funding restrictions along with the risk-bearing capacity of the contributors and market-making intermediaries create frictions and illiquidity that would impact the pricing of CDS contracts. However, aside from the issue of whether liquidity is a significant, priced factor in CDS spreads, Junge and Trolle [2015] find that CDS market liquidity is highly persistent, with a first-order autocorrelation of 0.93.

Ultimately whether CDS liquidity is a concern in the implementation and interpretation of our default correlation measure, as operationalized by the correlation in CDS spreads, is an empirical issue. To explore this possibility, we begin by examining the impact of CDS liquidity, which we measure using Markit’s number of contributors. In doing so, we follow a long line of studies that have used this variable to examine or to control for CDS market liquidity. For instance, Ashcraft and Santos [2009] provide evidence that CDS trading has increased the cost of debt financing, controlling for CDS liquidity via the number of contributors. Similarly, Friewald, Wagner, and Zechner [2014] study the cross-sectional relation between equity returns and credit risk, also controlling for Markit’s number of contributors as a proxy for CDS liquidity. Furthermore, Qui and Yu [2012] study CDS liquidity itself, examining the determinants of liquidity provision in the CDS market, as measured by the number of contributors.^{20}

The results, which we present in Exhibit 7, show a significant association between the number of contributors and CDS spreads, with each additional contributor decreasing the corresponding CDS spread by 10.6 basis points (see Exhibit 7, Panel A). For reference, the average CDS spread is 2.0% (untabulated),^{21} and the average number of contributors per contract is 6.5, with a minimum of 2 contributors and a maximum of 31 contributors per contract (untabulated).

However, when we explore the relation between the number of contributors and *correlations* between CDS spreads, we observe a statistically significant but economically negligible association. Specifically, each additional contributor to CDS *i* is associated with a 0.002 decline (*t*-statistic = −2.65) in the average pairwise correlation, measured over the preceding 30-day period, of CDS spread *i* with all other CDS spreads (see Exhibit 7, Panel B.1).

Similarly, we also observe an economically negligible association in the relation between the *correlations* in the number of contributors and *correlations* between CDS spreads. Specifically, a one-standard deviation increase in the correlation in contributors (i.e., 0.06) is associated with an increase of 0.0207 (*t*-statistic = 8.50) in the correlation in CDS spreads (see Exhibit 7, Panel B.2), which represents a move of less than 9% of one standard deviation in CDS spread correlation. For reference, the average correlation in contributors is 0.03, with a standard deviation of 0.06 (untabulated), and the average correlation in CDS spreads is 0.29, with a standard deviation of 0.24, which we presented earlier in Exhibit 1.

**Recovery rate.** We also examine the impact of reported recovery rates on the levels of CDS spreads, and ultimately, on the correlation between CDS spreads. We obtain recovery rates from the Markit database, which provides the recovery rate corresponding to each credit curve as reported to Markit by institutions (i.e., market makers) contributing daily CDS pricing data. As expected, we observe that the recovery rate plays a significant role in the pricing of CDS contracts (see Exhibit 7, Panel A).

Upon further examination, the recovery rate itself is not a statistically significant factor in the *correlation* between CDS spreads (Exhibit 7, Panel B.1). The *correlation* in recovery rates, though, does meaningfully associate with CDS spread correlation (Exhibit 7, Panel B.2). That is, a one standard-deviation (i.e., 0.36) increase in the correlation in recovery rates is associated with a 0.025 increase in correlation in CDS spreads. Nonetheless, the total adjusted R-squared, when accounting for both the *Correlation in Contributors* and the *Correlation in Recovery Rate*, amounts to only 2.4% of the total variation in the correlation in CDS spreads. That is, these factors, though statistically significant, have very limited explanatory power due to the persistence in the number of contributors and the recovery rate over our 30-day estimation horizon.

**Excess bond premium (ebp).** An earlier possible concern that we broached is that our default-correlation metric may be capturing the time-varying excess credit-risk premiums. To further explore this possibility, we also include the excess bond premium, *ebp*, as an additional regressor (Exhibit 7, Panel B.3), and we observe that the excess bond premium is not a significant contributor to the monthly spread correlation once the *Crisis* indicator is accounted for. That is, *ebp* explains only an additional 0.2% of the variation in the monthly spread correlation (coefficient estimate = −0.001, *t*-statistic = −0.47), once accounting for: (1) the *Correlation in Contributors*, (2) the *Correlation in Recovery Rate*, and (3) the *Crisis* indicator.

### Residual Aggregate Default Correlation and Bankruptcy Filings

We now re-estimate Equation 3
(i.e., Exhibit 3), this time taking *Residual Aggregate Default Correlation**t* as our main variable of interest to ensure that it is the correlation in default probabilities itself that is associated with bankruptcy clusters, rather than the correlation in expected recovery rates or correlation in CDS liquidity.

Specifically, we calculate *Residual Default Correlation*_{i,t} as the residual from regressing *Default Correlation*_{i,t} on (1) the average pairwise correlation of the recovery rate on firm *i* with the recovery rate of all other firms at time *t*, and (2) the average pairwise correlation of the number of contributors/market-makers for firm *i* with the number of contributors/market-makers for all other firms at time *t*. We then define *Residual Aggregate Default Correlation*_{t} as the average *Residual Default Correlation*_{i,t} across all firms at time *t*. The results, which we present in Exhibit 8, show that *Aggregate Default* *Correlation* remains a significant predictor of bankruptcy clusters even after removing the portion, if any, that is attributable to correlations in recovery rates or CDS liquidity.

### Residual Default Correlation and Credit Spreads

As an additional robustness check of our measure, we also re-estimate our credit-spread regressions (Equations 4 and 5) and our CDS-spread regressions (Equations 6 and 7), now using *Residual Default Correlation*_{i,t} and *Residual Aggregate Default Correlation*_{t} as our main variables of interest. The results, which we present in Exhibits 9 and 10,^{22} show that both individual default correlation and aggregate default correlation remain significant factors in credit spreads (Exhibit 9) as well as in CDS spreads (Exhibit 10), even after removing the portion, if any, that is attributable to correlations in recovery rates or CDS liquidity. Furthermore, we note that in including the excess bond premium (*ebp*_{t}) as an explanatory variable, these results also indicate that both individual default correlation and aggregate default correlation remain significant factors in credit spreads and CDS spreads, even after removing the portion, if any, that is attributable to time-varying premiums demanded by investors.

## CONCLUSION

This paper proposes a novel method for measuring time-varying correlations in default probabilities, and organizes useful and compelling evidence as to its efficacy and validity in capturing the correlation in the realization of default. In particular, we find that our proposed measure is substantially associated with the incidence of bankruptcy filings. We also provide evidence that our measure is a meaningful determinant of changes in credit spreads and CDS premiums, and we provide evidence that default correlation is more pronounced and commands an even higher premium during periods of financial distress and for non-investment grade issues.

Overall, the broadly observable metric of correlated default risk that we introduce should prove useful to future work concerning cross-sectional determinants of fixed-income pricing and returns, as well as to future work concerning optimal asset allocation in the face of time-varying correlations. Furthermore, as a timely and more powerful measure of the correlation in default probabilities, our measure should prove useful in active risk management and determination of capital adequacy needs in banks, shadow entities, and investment portfolios in general.

## APPENDIX A

### LIST OF VARIABLES

Presented below is an alphabetized list of variables with corresponding definitions. Firm-level variables are identified by subscript *{i,t}* and macro-level variables are identified by subscript *{t}*. A more detailed outline of descriptions and calculations is presented in Appendix B.

## APPENDIX B

### DESCRIPTIONS, DATA SOURCES, AND DISCUSSION OF VARIABLES

In this section, we describe our filtering process and our regression variables, along with the corresponding data source(s).

#### Firm Leverage

For each firm, we calculate leverage each month as the ratio of the firm’s book value of debt to the sum of its market value of equity and book value of debt:

B-1Following Collin-Dufresne, Goldstein, and Martin [2001], we obtain equity valuation from CRSP, which we match to the most recent quarterly book value of debt obtained from COMPUSTAT.

#### Volatility

We use 30-day implied volatilities, obtained from OptionMetrics, to capture a forward-looking measure of the firm’s volatility. We collect this information on the first day of each month in our sample.

#### Market Uncertainty

*VIX*, the CBOE Volatility Index that we obtain from OptionMetrics, is designed to capture overall market volatility, as measured by the implied volatility of options on the S&P500 composite index.

#### Overall Market Condition

Following Collin-Dufresne, Goldstein, and Martin [2001], we use monthly returns on the S&P500 composite index, which we obtain from CRSP, to capture the overall state of the economy.

#### Excess Bond Premium (ebp)

Following Gilchrist and Zakrajsek [2012], *ebp*, the excess bond premium, represents the average price of bearing exposure to credit risk in excess of the compensation for expected defaults over time. We obtain *ebp* data from http://people.bu.edu/sgilchri/Data/data.htm.

#### Riskless Rate (r10)

Following Collin-Dufresne, Goldstein, and Martin [2001] and Ericsson, Jacobs, and Oviedo [2009], among others, we use the yield on the 10-year Treasury note, which we obtain from FRED.

#### Yield Curve Slope

Following Collin-Dufresne, Goldstein, and Martin [2001] and Ericsson, Jacobs, and Oviedo [2009], among others, we calculate the slope of the yield curve as the difference between the 10-year and 2-year Treasury yields, which are also obtained from FRED. Based on the expectations hypothesis of the term structure of interest rates, we interpret this measure as the market perception regarding the expectation of future short-term rates.

#### Bond Illiquidity

Following recent literature studying the impact of bond-market illiquidity on fixed-income pricing (Dick-Nielson, Feldhütter, and Lando [2012]; Friewald, Jankowitsch, and Subrahmanyam [2012]), we construct the Amihud [2002]-based measure of illiquidity as the bond’s average absolute return scaled by trading volume for trades executed during time *t*:

Intuitively, greater price impact indicates less liquidity in the market. We obtain trading activity and returns data from TRACE. Although we employ the *Amihud Illiquidity* measure throughout our tabulated analyses, we check the robustness of our results using alternative liquidity proxies, such as trading volume and trading interval, which we also obtain from TRACE.

#### Credit Spread

Following Bessembinder et al. [2009] and Friewald, Jankowitsch, and Subrahmanyam [2012], among others, we employ the newly available bond-pricing data from TRACE, which provides intra-day price and yield information on individual bond transactions along with associated execution times. Specifically, to calculate daily bond yields, we begin by eliminating trades below $100,000. We then calculate a trade-weighted average of the remaining transaction yields, thereby minimizing the effect of the relatively large bid-ask bounce associated with smaller trades (Bessembinder et al. [2009]). Using these daily-yield calculations, we derive monthly bond yields by taking the trade-weighted average of daily yields.

Next, we merge our monthly-yield data with Mergent FISD to obtain various bond characteristics, such as credit rating and time to maturity. We then proceed to remove bond issues from our sample that (1) have embedded options, (2) have defaulted, (3) belong to issuers in the financial sector or utilities, (4) have a convertibility feature, (5) are denominated in a foreign currency, or (6) are an asset-backed issue. That is, we keep only the plain-vanilla, single-name corporate bond issues.

Finally, we calculate the credit spread for each issue *i* at year-month *t* as the difference between the bond’s yield and the yield on the associated Treasury at the same maturity, which we obtain from FRED:

Specifically, we use the rates obtained from FRED to construct the term structure for each year-month *t*, interpolating between provided rates in the term structure to match the time to maturity of bond issue *i* at time *t*.

## APPENDIX C

## APPENDIX D

## ENDNOTES

We thank Sanjiv Das, Travis Davidson, Assaf Eisdorfer, Yalin Gündüz, Jean Helwege, H.J. Abraham Lin, John McConnell, George Pennacchi, Victor Shen, Roger Stein, William Waller, Junbo Wang, and seminar participants at the Eastern Finance Association (EFA) 2015 Annual Meeting, Financial Management Association (FMA) 2015 Annual Meeting, FMA Applied Finance 2016 Conference, FMA Asia-Pacific 2016 Annual Meeting, 9th Financial Risks International Forum, Indian School of Business, Ohio University, Oklahoma State University, Southern Finance Association (SFA) 2016 Annual Meeting, and Southwestern Finance Association (SWFA) 2016 Annual Meeting for their comments and suggestions.

↵

^{1}See also Lando and Nielsen [2010] for additional discussion and potential pitfalls regarding tests of the doubly stochastic assumption under times to default (i.e., the assumption that default times are conditionally independent, given some structural determinants or underlying state process of default).↵

^{2}A credit default swap (CDS) is a financial swap agreement, whereby the contract writer agrees to compensate the CDS holder in the event that the reference entity defaults. In exchange for this guarantee, the CDS holder makes periodic payments, known as the CDS*spread*or*fee*, to the contract writer.↵

^{3}See http://www.uscourts.gov/statistics-reports/caseload-statistics-data-tables.↵

^{4}For instance, He and Xiong [2012] derive a model where debt-market illiquidity leads to greater credit risk by increasing the default boundary, and Ericsson and Renault [2006] derive a model where bond-market illiquidity affects the likelihood of debt renegotiation.↵

^{5}We follow the period as defined by the Federal Reserve. See “The Great Recession of 2007–2009” (Federal Reserve Bank of New York [2013]). See also peaks and troughs defined by the Federal Reserve Economic Data (FRED) site: https://research.stlouisfed.org/fred2/help-faq/#graph_recessions.↵

^{6}The percentages reported here and in the following paragraph are based on a sample average credit spread of 2.08%.↵

^{7}See ISDA Market Survey Summaries, 2010-1995 (http://www2.isda.org/functional-areas/research/surveys/market-surveys/).↵

^{8}See, for example, page 2436 of Friewald, Wagner, and Zechner [2014]. See also Berndt et al. [2005] and Bharath and Shumway [2008], who also assume a constant recovery rate, and Hull, Predescu, and White [2004], who argue that the estimate of the CDS spread is fairly insensitive to the choice of recovery rate given such low probabilities of default. Other studies assuming a constant recovery rate include: Altman and Kishore [1996]; Keenan, Shtogrin, and Soberhart [1999]; and Eom, Helwege, and Huang [2004].↵

^{9}Similarly, Junge and Trolle [2015] infer from the annual “Margin Surveys” by the International Swaps and Derivatives Associations (ISDA) that “the fraction of credit derivatives trades covered by collateral agreements averaged more than 80% over the sample period.” Accordingly, they do not account for counterparty risk in their CDS pricing model.↵

^{10}This empirical finding is distinct from prior theoretical work focusing on the impact of counterparty risk in the pricing of credit derivatives contracts in a model economy in which contracts liabilities are not collateralized (Jarrow and Yu [2001]; Hull and White [2001]; Kraft and Steffensen [2007]).↵

^{11}See, for instance, page 2436 (footnote 19) of Friewald, Wagner, and Zechner [2014].↵

^{12}Our results are robust to the use of medians versus means in defining our correlation metrics.↵

^{13}We choose OLS estimation for ease of exposition. A double-censored Tobit model produces almost identical results, and our results are robust to alternative specifications.↵

^{14}We employ the formulas 0*.*75 ×*N*^{(1/3)}= 3*.*84 and 4 × (*N/*100)^{(2/9)}= 4*.*26 to arrive at our four-lag specification (e.g., see Newey and West [1994]; Adkins and Hill [2011]). Our results are robust to calculating Newey-West standard errors with five lags.↵

^{15}See http://www.uscourts.gov/statistics-reports/caseload-statistics-data-tables.↵

^{16}We employ the formulas 0*.*75 ×*N*^{(1/3)}= 3*.*84 and 4 × (*N/*100)^{(2/9)}= 4*.*26 to arrive at our four-lag specification (e.g., see Newey and West [1994]; Adkins and Hill [2011]). Our results are robust to calculating Newey-West standard errors with five lags.↵

^{17}Recall:*ln(y)*=*x*implies*dy/y*=*dx*. Similarly, when we employ a levels specification (untabulated), we observe a coefficient estimate on*Aggregate Default Correlation*of 1,489 (*t*-statistic = 2.59), which suggests an increase in bankruptcy filings of 744 and represents 7.5% of the mean. For reference, the average number of bankruptcy filings is 9,966 (untabulated).↵

^{18}In untabulated analyses, we find that greater lags of*Aggregate Default Correlation*continue to have strong explanatory power in predicting subsequent bankruptcy clusters.↵

^{19}In particular, see pages 2219–2220 of Longstaff, Mithal, and Neis [2005] for a discussion regarding the distinguishing features of credit default swaps making them less susceptible to liquidity risk premiums.↵

^{20}See Qui and Yu [2012] for an extensive discussion as to the validity of using Markit’s number of contributors as a proxy for CDS liquidity.↵

^{21}As distinct from the 1.54% reported average CDS spread in our final sample (see Panel B of Exhibit 1).↵

^{22}For ease of exposition, Exhibits 9 and 10 report only the coefficient estimates on our main variables of interest,*Residual Default Correlation*and*Residual Aggregate Default Correlation*. The full array of coefficient estimates is reported in Appendix C (for Exhibit 9) and Appendix D (for Exhibit 10).^{23}We define the period as specified by the Federal Reserve. See “The Great Recession of 2007–09” (Federal Reserve Bank of New York [2013]).^{24}We use the ebp data provided at: http://people.bu.edu/sgilchri/Data/data.htm.

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