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What Moves the Correlation between Equity and CDS Markets? *

By

Zilong Liu

Department of Finance

Kent State University

zilong728@gmail.com

Xiaoling Pu

Department of Finance

Kent State University

xpu2@kent.edu

and

Xinlei Zhao

Credit Risk Analysis Division

Office of the Comptroller of the Currency

xinlei.zhao@occ.treas.gov

May 2015

* The views expressed in this paper are those of the authors and do not necessarily represent the views of the Office

of the Comptroller of the Currency or the U.S. Department of Treasury. The authors are responsible for all errors.

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Abstract

We document substantial correlation dynamics between equity returns and CDS spread

changes at the firm level, which is critical for cross-market hedging and arbitrage strategies.

Using the implied cost of capital approach, we decompose the unexpected equity returns into

cash flow and discount rate news, and examine the impact of the shocks on the correlations. We

find discount rate news explains the majority fraction. However, at longer horizons and in

periods when cash flow news is more negatively related with CDS spread changes, the cross-

market integration is stronger. In addition, firms with more cash flow news exhibit stronger

correlations between equity returns and credit spread changes, and the structural model can

explain more variations of credit spread changes in these firms.

JEL code: G11, G12

Key words: correlation, equity return decomposition, cash flow news, discount rate news, credit

default swap spreads, and regime-switching models

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1. Introduction

Dynamic cross-market hedging plays a central role in portfolio management, and

therefore, it is crucial to understand the economics of joint price formation of securities in equity

and credit markets and the time variation in the cross-market asset return correlations. Most

studies investigating the joint stock-bond pricing have focused on the aggregate level (e.g.,

Campbell and Ammer (1993), Connolly, Stivers, and Sun (2005), Baele, Bekaert, and

Inghelbrecht (2010)), and evidence at the firm level has been quite scarce. In the past decade,

cross-market hedging and arbitrage strategies, such as capital structure arbitrage, have become

popular, partly driven by the exponential growth in the credit default swap (CDS) market (Currie

and Morris (2002)). Since such trading strategies exploit mispricing between a firm’s equity and

debt over the short term, understanding the relation between equity returns and credit spread

changes at the firm level becomes critical.

Earlier work (e.g., Keim and Stambaugh (1986), Fama and French (1989), Shiller and

Beltratti (1992), Bekaert and Grenadier (1999)) on joint stock-bond pricing has taken a

fundamental approach to examine common factors determining pricing in two markets. Along

the line, a number of papers investigate the joint distribution of equity and bond returns at an

aggregate level. Campbell and Ammer (1993) employ a vector autoregressive (VAR) model to

decompose monthly stock and Treasury bond returns, and explain the low correlation between

excess stock and bond returns. They find that stock and bond returns are largely driven by news

about future excess returns and inflation while interest rates have little impact. Connolly, Stivers,

and Sun (2005) move the literature forward by showing that stock market uncertainty is related

to the time variation in the comovements of daily stock and Treasury bond returns. Baele,

Bekaert, and Inghelbrecht (2010) use a dynamic factor model with fundamental factors to

explain the time series variations in the stock-bond return correlations at a quarterly frequency.

In this paper, we extend prior work by examining the asset return relation between equity

and credit markets at the firm level. Similar as the empirical phenomenon documented at the

aggregate level (Baele, Bekaert, and Inghelbrecht (2010)), correlations between equity returns

and changes in CDS spreads at firm level also display substantial time variations. Figure 1 plots

the annual average correlations measured using monthly equity returns and CDS spread changes

from 2001 to 2013. The correlation ranges from -9 percent to -37 percent, and the large negative

spike in 2008 is associated with a crash in stock market and credit crunch in the credit market.

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It is noticed that correlation between a firm’s equity and debt is important for the

implementation of the capital structure arbitrage strategy, which exploits mispricings between a

firm’s equity and debt over short term and is one of the most popular fixed income trading

strategies. The industry trend also attracts much academic attention, for example, Yu (2006) and

Duarte, Longstaff, and Yu (2007) examine the risk factors in the equity and credit markets and

their impacts on the return and risk characteristics of the strategy.

The relative importance of cash flow (CF) news and discount rate (DR) news in equity

returns is a central issue in the stock market. Although previous studies (e.g., Campbell (1991))

find that most of the aggregate equity return innovation is driven by DR news, recent work has

shown that CF news can be important with different sample periods or alternative measures (e.g.

Larrain and Yogo (2008)). It will be interesting to examine how credit spread changes correlate

with the two components in stock returns, since equities and corporate bonds are contingent

claims on the same underlying firm value. We use the decomposition method proposed by Chen,

Da, and Zhao (2013) as the traditional VAR approach is shown to be quite sensitive to model

specification.

In the analysis, we first document a substantial time variation in the correlation dynamics

and find most of the fraction in the correlations is explained by DR news, especially over short

horizons. In DR news, risk premium news plays a major role compared with risk free rate news.

Recent literature (Goyal and Welch (2008), Chen and Zhao (2009), Chen, Da, and Zhao (2013))

finds that DR news is dominant in driving equity returns over short horizons while the

importance of CF news increases in long term. Similarly, we find that the importance of CF news

for CDS spread changes increases with horizons. Our results show that when CF news is more

incorporated into stock prices and CDS spreads over longer horizons, correlations between the

two markets become stronger.

Second, we find that correlations between equity returns and CDS spread changes move

with business cycles and exhibit different dynamics between crisis and non-crisis periods. The

presence of regimes should be exploitable in active asset allocations, and a number of studies

(e.g., Ramchand and Susmel (1998), Guidolin and Timmermann (2006), Ang and Bekaert (2004))

consider how the existence of two regimes and return and risk characteristics in the two states

affect the mean-variance asset allocation. In a two-regime-switching model, we present sharp

contrast in the correlations between equity returns and credit spread changes in two states. We

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find that cross-market correlation does become stronger when CF news is more negatively

related with CDS spread changes in the second regime (regime one), which is correspondent to

the crisis period. Our finding suggests that arbitrage strategies heavily employing pricing relation

between equity and debt securities would exhibit different patterns of return and risk

characteristics in two states.

Third, correlations between equity returns and CDS spread changes exhibit variations

across firms and are in much larger magnitudes in non-investment grade firms. In structural

model (Merton (1974)) regressions following the specifications of Collin-Dufresne, Goldstein,

Martin (2001), we find that firms with more CF news in the equity returns have larger

correlations between equity returns and credit spread changes. We run the regressions in five

groups sorted by cash flow beta, discount rate beta, and risk premium beta. Our results are

consistent with the low explanatory power of the regressions using structural model variables on

the credit spread changes (Jones, Mason, and Rosenfield (1984), Eom, Helwege, and Huang

(2004), Huang and Huang (2012), Collin-Dufresne, Goldstein, and Martin (2001), Blanco,

Brennan, and Marsh (2005)). But the more interesting finding is that firms with more cash flow

news or more uncertainties in the cash flow news have larger R-squared in the regressions. Yet

there is no such pattern found for discount rate news or risk premium news.

The rest of the paper is organized as follows. Section 2 describes our data set and

summary statistics. Section 3 provides the return decomposition setup. Section 4 presents the

correlation dynamics in time series. Section 5 reports the correlations between CDS spread

changes and equity returns/components cross-sectionally. Finally, section 6 concludes.

2. Data and summary statistics

We use four main data sources: the CRSP (Center for Research in Security Prices),

Quarterly Compustat, I/B/E/S (Institutional Brokers’ Estimate System), and Markit. Our sample

period is from 2001 to 2013. We get the daily five-year CDS spreads on senior, unsecured debt

of 988 firms from Markit. After removing firms with less than 100 daily observations in Markit,

we merge the firms with CPSP by using the linking table between Redcode (from Markit) and

historical CUSIP. Then we merge the firms with quarterly Compustat. The procedure yields 895

firms with 572 investment grade (AAA, AA, A, and BBB) and 323 speculative grade (BB, B,

and CCC) firms. Markit provides both average rating and implied rating at a daily frequency, and

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we use the average rating for each firm in the sample period. The implied rating is used instead if

the average rating is not available.

Table 1 presents the summary statistics for the sample with 895 firms, and the correlation

trend is presented in figure 1. Book leverage is the ratio of total debt to total assets using data

from Compustat. Market leverage is computed as the ratio of the book debt value (sum of long-

term debt and debt in current liabilities) to the sum of book debt value and market capitalization.

Equity volatility is the annualized standard deviation of daily returns. The average size of the

sample is about 13 billion dollars with average market leverage 36 percent.

It is necessary to obtain analyst forecast data to perform equity decomposition. Thus, we

merge the 895-firm sample with I/B/E/S, and eliminate firms that were delisted during the

sample period. We keep firms with at least 16-month CDS spread observations, and firms with

no missing quarterly forecast dispersion. Our final sample contains 516 firms, in which 389 firms

are investment grade and the remaining 127 firms are speculative grade.

Table 2 presents the summary statistics for the 516-firm sample. There are several

important observations from the comparison between tables 1 and 2. First, firms in the whole

sample are riskier than those in the 516-firm sample with larger CDS spreads, smaller size, and

higher leverage. The average CDS spread is about 58 basis points higher, and the average market

capitalization is five billion dollars smaller in the 895-firm sample. The 516-firm sample has the

average size from 0.58 billion to 228 billion dollars with large standard deviations. Second, the

equity volatility and book-to-market ratios are similar across two samples, and the 895-firm

sample exhibits larger cross-sectional variations. The standard deviation of the book to market

ratio in the 895-firm sample is 0.62, while the correspondent statistics in the 516-firm sample is

only 0.34. It appears that firms with analyst coverage usually are larger and less leveraged. We

will use the 516-firm sample in our following analysis.

3. Equity return decomposition

A seminal work of Campbell and Shiller (1988) suggests that unexpected equity returns

can be decomposed into CF and DR news through a log-linearization approximation of the

present value formula. Along the line, Campbell (1991), Campbell and Vuolteenaho (2004), and

others advocate the VAR approach to directly model the DR news and back out the CF news as

the residual. However, the VAR approach suffers from misspecification errors and sensitivity of

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state variables choices (Chen and Zhao (2009)). Chen, Da, and Zhao (2013) propose a forward-

looking approach with implied cost of capital (ICC), which identifies CF and DR news by using

direct cash flow forecasts. Thus, this study employs the robust ICC approach to differentiate CF

and DR news in equity returns and pins down the component that drives the co-movement across

the equity and credit markets.

Equity returns come from two components: CF and DR news. Positive returns are usually

associated with good news of cash flow or discount rate decrease. When future cash flow

increases, firm wealth increases although the investment opportunities stay the same; when

investors decrease discount rate, firm wealth also increases but future investment opportunities

may diminish.

Campbell and Shiller (1988) provide a log-linear approximate present-value relation, for

analyzing cash flow and discount rate shocks. Following the approach, Campbell (1991) solves

the return decomposition as:

1 1 1 1 1 1 , 1 , 1

0 1

( ) ( )j j

t t t t t t j t t t j CF t DR t

j j

r E r E E d E E r N N

(1)

where 1tr is the stock return at time 1t , tE is the expectation operator at time t , td is the

dividend growth rate, is a constant ( 1 ), and CFN and DRN represent the unexpected shocks

from future cash flows and discount rates. The VAR model is employed to estimate expected

return 1t tE r and DR news 1 1

1

( ) j

t t t j

j

E E r

, and then we can use 1tr and equation (1) to back

out the CF news. However, Chen and Zhao (2013) show that the approach has limitations since

CF news could be contaminated by the large misspecification error from the measurement of DR

news.

We apply the ICC approach from Chen, Da, and Zhao (2013) to estimate CF and DR

news from equity returns. Comparing the traditional return decomposition approach (Campbell

and Shiller (1988), Campbell and Vuolteenaho (2004)), the ICC approach relies on firm-specific

market consensus earnings forecasts and equity prices to back out the discount rates. Then CF

and DR news are identified without resorting to predictability. In addition, the ICC method does

not do linearization approximation. Following Chen, Da, and Zhao (2013), we decompose the

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unexpected return into CF and DR news. Assuming jRetx is the price difference between t j

and t for firm j, then

( , ) ( , )t j t

t j t t j t

j j j

t t

P P f c q f c qRetx CF DR

P P

(2)

( , ) ( , ) ( , ) ( , )/ 2

t j t t j tt j t j t t

j

t t

f c q f c q f c q f c qCF

P P

(3)

( , ) ( , ) ( , ) ( , )/ 2

t t t j t j

t j t t j t

j

t t

f c q f c q f c q f c qDR

P P

(4)

where tc is the cash flow estimates from the earnings forecast and tq is the implied cost of capital.

To examine how the stock-CDS relation reacts to different sources of discount rate news, we

further decompose DR news into risk premium (RP) and risk free rate (RF) news in the same

framework, where td represents risk premium and tr represents the risk free rate in the following

equations.

( , , ) ( , , ) ( , , ) ( , , )

/ 4( , , ) ( , , ) ( , , ) ( , , )

t j t j t t j t t t

t j t j t t j t j t t t

t t t t

t j

t t

j t j t j t j t j

t t

f c d r f c d r f c d r f c d r

P PRP

f c d r f c d r f c d r f c d r

P P

(5)

( , , ) ( , , ) ( , , ) ( , , )

/ 4( , , ) ( , , ) ( , , ) ( , , )

t j t j t j t t t j t t

t j t j t j t t t j t t

t t t t

t t

j t j t j t j t j

t t

f c d r f c d r f c d r f c d r

P PRF

f c d r f c d r f c d r f c d r

P P

(6)

Table 3 presents the return decomposition results following the ICC approach over one-,

three-, six-, and 12-month horizons at the firm level. The decomposition over one-month shows

that DR news is 0.69 percent, which is the major component in the total returns. At the three-

month horizon, in the 2.37 percent unexpected equity return, about 0.86 percent is CF news. Our

decomposition over one quarter shows that DR news explains about 70 percent of the unexpected

returns, which is much higher than CF news. In a longer horizon of 12-month, DR news

decreases to less than 40 percent of the total equity returns. This is consistent with the finding in

Chen, Da, and Zhao (2013), which shows that CF news has less weight over shorter horizons,

and the explanatory ability of CF news increases with horizons.

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In discount rate news, we find risk premium news plays a dominant role in the short

horizons (one-month and three-month). In a 12-month horizon, the risk free rate news has a

larger fraction than risk premium news in the total discount rate news. Overall, the risk premium

news has much higher variations compared with risk free rate news, which suggests that

variability in DR news mainly comes from risk premium news. For example, the variance of the

risk premium news is 18.75 percent while the variance of the risk free rate news is only 1.36

percent at six-month horizon.

Figure 2 depicts the one quarter ahead value weighted stock returns and cash flow news.

We use the ICC approach to compute the numbers at the firm level and then aggregate the data

into market level. In most of the time, the two time series are related, but they do not match as

well as that in figure 2 of Chen, Da, and Zhao (2013), which presents the cash flow news and

equity return at one- and two-year horizons and shows that cash flow news tracks actual returns

closely in longer horizons. Our quarterly graph exhibits larger volatility of equity returns in a

shorter horizon.

4. Correlations dynamics in time series

4.1. Descriptive statistics of the correlations

Panel A of table 4 reports the yearly average correlation between equity returns and the

components measured using monthly data. At one-month horizon, DR news has a significantly

higher correlation with returns than CF news, which is consistent with the observation in table 3

that DR news drives equity returns in short horizon. In contrast, CF news has a much smaller

correlation with returns, and the sign of the correlations flips around in the sample period. The

finding is consistent with the empirical findings documenting that discount rate shocks explain a

large fraction of stock returns since stock market weakly reacts to aggregate earnings news over

short term (Kothari, Lewellen, and Warner (2006)).

The risk premium news is a dominant component in discount rate news. Across all years,

the correlations between equity returns and risk premium news range from 42 percent to over 70

percent. However, the correlations between risk free rate news and equity returns are uncertain.

Equity returns have the largest correlation with discount rate news in 2008, and similar pattern

has been observed for the correlation between equity returns and risk premium news. On the

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other hand, cash flow news and risk free rate news have a weak relation with equity returns and

do not exhibit stronger relation in the crisis.

Panel B presents the correlations between CDS spread changes and equity

returns/components. Overall, the two markets are not closely related as Merton (1974) model has

predicted. The correlations between CDS spread changes and equity returns are from -14 percent

to -38 percent. The largest magnitude of correlation occurs in 2008 and the lowest is in 2006,

which is in line with the business cycle. Consistent with panel A, discount rate shocks are more

related with the credit market and cash flow news does not have a persistently negative relation

with the CDS spread changes. In the 2008 crisis, both DR and CF news become more negatively

related with CDS spread changes. Similar as the equity market, the correlations between CDS

spread changes and discount rate news are mainly driven by risk premium news.

We compute the yearly average of all the monthly correlations between CDS spread

changes and equity returns, and plot the time series from 2001 to 2013 in the first graph of figure

3. The correlation exhibits a dip in the 2002 internet crash. Then the equity and credit markets

are loosely connected between 2004 and 2006, and the smallest magnitude of the correlation is

observed in 2006. The largest negative peak is associated with the 2008 financial crisis. These

observations are consistent with those documented in Kapadia and Pu (2012), which finds the

integration across the two markets is higher in the crisis. Clearly, the correlation patterns in the

first graph of figure 3 and figure 1 are similar.

In the next two graphs of figure 3, we plot the correlation trend between CDS spread

changes and CF/DR news, respectively. Consistent with the statistics in table 4, DR news

exhibits negative correlations with CDS spread changes all the time, and the strongest co-

movement occurs in 2008. The pattern is similar as the correlations between CDS spread

changes and equity returns. CF news has low correlations with CDS spread changes in most of

the years, and it appears that the two markets are more integrated in the periods when CF news

becomes more important.

The monthly correlation dynamics shows that neither equity nor credit market responds

strongly to cash flow news over short horizon. Discount rate news is more important to explain

equity returns or CDS spread changes and the cross-market correlation. Since the correlations

exhibit different patterns across years, we investigate the co-movement in regime-switching

models in next session.

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4.2. Regime-switching models of the correlations between CDS spread changes and

equity returns/components

The previous analysis shows that correlations between equity returns and CDS spread

changes are larger in times when both CF and DR news have a strongly negative relation with

CDS spread changes. Since both equity returns and credit spread changes are more related with

cash flow news in the crisis, we use a regime-switching model to explore the shifts in the time

series of the correlations. Our purpose is to examine whether CF or DR news is the main driver

to affect the probability of switching from one regime to another.

Regime-switching models are well established since Hamilton (1989), and they have

been applied in various areas of financial economics. Previous studies (e.g., Garcia and Perron

(1996), Gray (1996), Bekaert, Hodrick, and Marshall (2001), and Ang and Bekaert (2002a))

employ the empirical models of regime switches in interest rates. The model application has also

been explored in equity returns, bond pricing, and asset allocation (e.g., Kim and Nelson (2001),

Ang and Bekaert (2002b) and Ang and Bekaert (2004)), option pricing (e.g., Duan, Popova, and

Ritchken (2002)), expected equity returns and volatility (Whitelaw (2000)), and stock-bond

correlations (e.g., Connolly, Stivers, and Sun (2005), Baele, Bekaert, and Inghelbrecht (2013)).

We first estimate the constant two-state regime-switching models given by

0 1 1 2CDS CDS Rets s

t t t ta a a (model 1)

0 1 1 2CDS CDS CFs s

t t t ta a a (model 2)

0 1 1 2CDS CDS DRs s

t t t ta a a (model 3).

CDSt , Ret t , CFt , DR t are the CDS spread changes, stock returns, CF news and DR news,

which are the cross-sectional average in each month. t is the residual. The s

ia are estimated

coefficients from the switching models. The superscript s on 0a and 2a indicates regime zero or

regime one, where s can be regarded as an unobserved state variable that follows a two-state,

first-order Markov process. The transition probability matrix is

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1

1

p pX

q q

,

where 1Pr( 0 | 0)t tp s s , and 1Pr( 1| 1)t tq s s . We refer to this model subsequently as

the constant transition probability regime-switching (CTP-RS) model. We document the

statistical association in return co-movements between the two markets instead of investigating

economic causality in this model.

Figure 4 displays the smoothed probability of being in regime one from the basic regime-

switching models. The solid, dashed, and dotted lines are for model 1, 2, and 3, respectively. The

probabilities from the three models follow a similar trend. We find that probability of switching

regimes has two highest peaks between 2007 and 2009, which closely matches the crisis period.

Table 5 presents the results of the model estimates. In panel A, we estimate the relation

between CDS spread changes and stock returns at monthly frequency from 2001 to 2013, and

find strong evidence of regime-switching behavior between two states. The probabilities in two

states are large ( 0.96; 0.91p q ), which suggests that both regimes are persistent. Both of the

estimated coefficients on stock returns in two regimes are negative and significant at one percent.

We find that the estimated coefficient on stock returns ( 0

2a ) is -0.53 in the first regime (regime

zero), while the magnitude of 1

2a coefficient (-2.16) on stock returns is about four times larger in

the second regime (regime one). This shows a substantial contrast between two regimes. The

market integration is weaker in the non-crisis period. The correlation between CDS spread

changes and equity returns is -0.31 in the first regime and decreases to -0.86 in the second

regime.

In panel B, we estimate the regime-switching model for the CDS spread changes and

cash flow news. Interestingly, the estimated coefficient on stock returns is positive in the first

regime (regime zero) while it becomes negative and significant in the second regime (regime

one). This is consistent with the second plot in figure 3, in which cash flow news has positive

correlations with CDS spread changes in tranquil periods and becomes more negatively related

with CDS spread changes in the crisis. In panel C, we estimate the model for the CDS spread

changes and discount rate news. Both of the estimated coefficients on stock returns are negative

and significant. The magnitude of the estimate is much smaller in regime zero, and the signs are

similar as those in panel A. The evidence is in line with the correlation statistics in table 4, which

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shows that discount rate news is a persistent driving force of the cross-market integration. The

results in the three panels also imply that the large correlation between CDS spread changes and

equity returns in the second regime is mainly driven by the combination effect of cash flow and

discount rate news. In the first regime, CF and DR news play opposite roles, which leads to a

low correlation between equity returns and CDS spread changes. In the second regime, the

integration across the two markets gets stronger since both CF and DR news move negatively

with CDS spread changes.

Overall, the results in table 5 show a substantial contrast in the correlations in two

regimes. Although DR news plays a dominant role in the integration across equity and credit

markets, only the combination effect of DR and CF news drives the probability of switching

between two regimes. The result explains the stronger correlation in the crisis when both CF and

DR news have a more negative relation with CDS spread changes.

Since forecast dispersion is found to be positively related with credit spreads (Güntay and

Hackbarth (2010)), we estimate an extended regime-switching model with time-varying

transition probabilities and investigate whether the uncertainty of cash flow news, measured as

the lagged forecast dispersion, can affect the probability of switching regimes.1 We find that

uncertainties in cash flow news do not significantly affect the regime shifting probabilities. This

is consistent with our finding of a weak role of cash flow news played in the cross-market

correlation dynamics. The model estimates are comparable to the constant regime-switching

model, and the results are similar as those in table 5. Both models find that cash flow news has a

weaker role compared with discount rate news and the correlation becomes stronger in regime

two when both equity components contribute to the cross-market integration.

4.3.Cash flow news over various horizons

Previous literature shows that CF news explains more of the equity returns over longer

horizons (Chen, Da, and Zhao (2013)), and we find similar evidence in panel A of table 6. The

correlations between equity returns and CF news increase monotonically over horizons while DR

news exhibit an opposite pattern. Over a 12-month horizon, the correlation between CF news and

equity returns is 32 percent while the correspondent statistics is only two percent over one-month

1 The likelihood ratio test statistics indicates indifferent performance between the constant and time-varying regime-

switching models. Connolly, Stivers, and Sun (2005) and Baele, Bekaert, and Inghelbrecht (2010) find the time-

varying model slightly fits better than the constant one, but the estimates are very similar in their studies.

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horizon. Risk premium news is dominant in driving the correlation between DR news and equity

returns.

Panel B reports the correlations between CDS spread changes and equity

returns/components. Overall, DR news explains the majority of the correlation dynamics, and CF

news explains a small fraction of the correlations. However, the magnitude of the correlations

between CF news and CDS spread changes increases monotonically with horizons. Risk

premium news plays a more important role than risk free rate news in driving the correlations

between CDS spread changes and DR news.

5. Correlation dynamics in cross sections

5.1. Correlations in different rating groups

Since we find that integration between two markets becomes stronger when CF news is

more negatively related with CDS spread changes, we next explore whether correlations between

CDS spread changes and equity returns exhibit various patterns across different firms. For

example, we observe that correlations between equity returns and CDS spread changes are more

negative in non-investment grade firms, in which cash flow news is important to determine the

firm value.

To explore the impact of equity components cross-sectionally, we investigate the

explanation ability of the structural model in both investment and non-investment grade firms at

monthly frequency.2 Following Collin-Dufresen, Goldstein, and Martin (2001), we estimate the

regression for each firm i at date t:

1 2 3 4 5 6 7CDS Ret eqvol lev vix term yield S&Preti i i i i i i i i i i i i

t t t t t t t t t (7)

where CDS is the change in five-year CDS spreads, Ret is the equity return, eqvol is the

change in equity volatility, lev is the change in leverage, and vix is the change in VIX, which

is the CBOE (Chicago Board Options Exchange) implied volatility of Standard and Poor’s 500

Index. term is the change in the term spread, which is defined as the difference between ten-

year Treasury bond yields and two-year Treasury note yields. yield is the change in ten-year

Treasury bond yields. We also include S&P returns in the regression. For N firms in each group,

we then report the averages of the coefficients from the N regressions as the reported coefficient

2 Since leverage (ratio of book debt to the sum of market capitalization and book debt) is quarterly observations, we

use linear interpolation to estimate the monthly leverage.

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values on the variables. We compute the t-values as the ratio of the reported coefficient value to

the standard deviation of the N estimates and scale the ratio by N .

In table 7, we report the regression results for firms sorted by ratings. We find the

adjusted R-squared in the non-investment grade firms is about six percentage points higher than

that of the investment grade firms. The results suggest that integration between the equity and

credit markets is significantly higher in firms in which CF news is more important.

For robustness check, we sort firms into five groups by leverage, equity volatility, and

forecast dispersion.3 Since firms with higher leverage, more volatile equity returns, or more

uncertainties in cash flow news (higher forecast dispersions) are more risky, their equity returns

might be more affected by CF news and more of the CDS spread changes could be explained by

the variables from the structural model. We find monotonically increasing R-squared from low

leverage/equity volatility/forecast dispersion firms to high leverage/equity volatility/forecast

dispersion ones.

5.2.Correlations in groups sorted by equity components

In this section, we examine the explanatory ability of the structural models in five groups

sorted by cash flow beta, discount rate beta, and risk premium beta. Based on the one-quarter

ahead equity return decomposition, we run the regressions of stock returns on the CF/DR/RP

news and the coefficient is identified as the CF/DR/RP beta at the firm level.4 The firms are

sorted into five groups by cash flow or discount rate beta, with the smallest beta in group one and

largest in group five. Then we run the regressions of the credit spread changes on the variables

suggested by the structural model (Merton (1974)) in five groups at monthly frequency.

Following Collin-Dufresne, Goldstein, Martin (2001), we estimate the regression model

(equation (7)) in five groups sorted by cash flow beta in panel A of table 8. The average adjusted

R-squared monotonically increases from 23 percent in group one to 28 percent in group five. The

low R-squared is consistent with the literature (e.g., Collin-Dufresne, Goldstein, Martin (2001)

and Blanco, Brennan, and Marsh (2005)) that structural model variables can only explain less

than thirty percent of the credit spread changes. Our results show that R-squared is lower in firms

with smaller amount of CF news in equity returns, which is in line with our previous finding that

3 The results are available upon request. 4 Our CF/DR/RP beta estimates using monthly decomposition are similar as those reported here.

16

correlations between equity returns and credit spread changes are larger in firms in which CF

news is more important.

The stock returns are negatively significant in all the groups at one percent level.

Changes in volatility and leverage are both significant for the CDS spread changes with positive

coefficients, which is in line with the structural model (Merton (1974)) stating that firm risk is an

important factor of default probability. Leverage is less important in explaining credit spread

changes compared with volatility. S&P returns are negatively related to the credit spread changes,

and the coefficients are significant in three out of five groups. The yield is negatively significant

for the spread changes, which reflects that a higher risk free rate lowers credit spread changes

(Longstaff and Schwartz (1995)). The term spread is positively significant in some of the groups,

but not in all the groups.

The regression coefficients in other panels are in similar magnitude and significance level

as those in panel A. In panel B, we report the regression results in groups sorted by the discount

rate beta, the difference in R-squared among five groups is not substantial. The variation in DR

news mainly comes from the risk premium news, and we observe that the pattern of the R-

squared in panel C is similar as that in panel B. We also examine the regressions in groups sorted

by risk free beta,5 and have not found relation between R-squared and the level of risk free rate

news. The comparison among the three panels implies a strong tie between CF news and cross-

sectional difference in the correlations between stock returns and CDS spread changes.

6. Conclusions

We document substantial variations in the correlation dynamics between equity returns

and CDS spread changes at the firm level. After decomposing the unexpected equity returns into

CF and DR news, we examine how the correlations between equity returns and credit spread

changes can be linked to the shocks in the equity market. Our results have important implications

for cross-market dynamic hedging and capital structure arbitrage activities, which are critical in

asset allocation and risk management research.

Over horizons less than a year, we find DR news is the main driver of the correlations,

and CF news plays a weaker role than DR news. In DR news, risk premium news is the dominant

factor affecting the correlations between equity returns and CDS spread changes. In time series,

5 The results are not tabulated, but available upon request.

17

the integration across the two markets gets stronger when CF news has a more negative relation

with CDS spread changes. Our results from a set of constant probability transition regime-

switching models show that there are substantial contrasts between two states, and the correlation

is stronger in the crisis regime when CF news becomes more important in equity returns.

In cross sections, the correlations between equity returns and CDS spread changes have

larger magnitudes in firms with more CF news or uncertainties in CF news. In addition,

structural model variables can explain more variations for the credit spread changes in these

firms, such as low credit rating firms. However, DR news cannot explain the cross-sectional

difference in the correlations between equity returns and CDS spread changes.

18

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22

Table 1 Descriptive statistics

The table reports the summary statistics of the 895-firm sample from 2001 to 2013. Market leverage is the ratio between book debt over the sum of

book debt (current liabilities in debt plus long term debt) and market capitalization. Book leverage is measured as the ratio of book debt over total

assets. Equity volatility is the annualized standard deviation of the stock returns. The average statistics are first computed for each firm, then the

summary statistics is calculated across all firms.

Variable Firm# Mean Std.dev. Min p25 p50 p75 Max

Market capitalization ($ billions) 895 13.14 24.24 0.19 2.33 5.05 14.12 193.42

Market leverage 895 0.36 0.22 0.01 0.20 0.32 0.50 0.94

Book leverage 895 0.33 0.19 0.01 0.20 0.30 0.43 0.93

Equity volatility 895 0.35 0.15 0.09 0.25 0.32 0.41 1.20

Book-to-market (BM) ratio 895 0.62 0.62 -1.19 0.34 0.52 0.79 4.73

CDS spreads (basis points) 895 215.02 250.97 15.89 67.17 121.89 263.25 1,824.44

23

Table 2 Descriptive statistics

The table reports the summary statistics of the 516-firm sample from 2001 to 2013. Market leverage is the ratio between book debt over the sum of

book debt (current liabilities in debt plus long term debt) and market capitalization. Book leverage is measured as the ratio of book debt over total

assets. Equity volatility is the annualized standard deviation of the stock returns. The average statistics are first computed for each firm, then the

summary statistics is calculated across all firms.

Variable Firm# Mean Std.dev. Min p25 p50 p75 Max

Market capitalization ($ billions) 516 18.70 32.19 0.58 3.38 8.23 18.57 228.14

Market leverage 516 0.30 0.18 0.00 0.17 0.27 0.41 0.87

Book leverage 516 0.29 0.15 0.01 0.18 0.28 0.38 0.68

Equity volatility 516 0.36 0.12 0.17 0.27 0.33 0.42 0.90

Book-to-market (BM) ratio 516 0.62 0.34 0.05 0.36 0.55 0.83 1.96

CDS spreads (basis points) 516 157.81 138.12 17.06 62.53 107.24 197.70 786.80

24

Table 3 Return decomposition using ICC approach

This table reports the mean and variance of cumulative capital gain return (retx), cash flow (CF) news,

discount rate (DR) news, risk free rate (RF) news, and risk premium (RP) news over various horizons at

the firm level. The decomposition details are described in section 3.

1-month 3-month 6-month 12-month

Variable Mean Variance Mean Variance Mean Variance Mean Variance

Equity returns (%) 0.81 0.85 2.37 2.52 4.88 5.51 9.80 11.85

CF news (%) 0.13 1.82 0.86 6.65 2.43 14.74 6.33 32.66

DR news (%) 0.69 2.70 1.62 7.96 2.72 16.00 3.88 29.40

RF news (%) 0.19 0.16 0.62 0.68 1.30 1.36 3.72 2.05

RP news (%) 0.46 2.90 0.98 9.01 1.41 18.75 0.10 34.65

25

Table 4 Correlation statistics

Panel A presents the correlations between monthly equity returns and the components. Panel B presents

the correlations between monthly CDS spread changes and equity returns/components. The equity returns

are decomposed at one-month horizon.

Panel A: Correlations between returns and the components

Year (Ret, CF news) (Ret, DR news) (Ret, RP news) (Ret, RF news)

2001 -0.12 0.72 0.70 -0.21

2002 -0.01 0.68 0.66 -0.23

2003 -0.09 0.61 0.48 0.22

2004 -0.02 0.55 0.49 0.03

2005 0.02 0.54 0.47 0.15

2006 -0.01 0.56 0.52 0.09

2007 0.01 0.63 0.63 -0.28

2008 -0.06 0.76 0.71 0.16

2009 0.04 0.55 0.52 0.09

2010 -0.10 0.55 0.56 -0.33

2011 0.03 0.53 0.59 -0.45

2012 -0.04 0.57 0.58 -0.28

2013 0.05 0.45 0.42 -0.08

Panel B: Correlations between CDS spread changes and equity returns/components

Year (∆CDS, Ret) (∆CDS, CF news) (∆CDS, DR news) (∆CDS, RP news) (∆CDS, RF news)

2001 -0.20 0.01 -0.13 -0.16 0.15

2002 -0.26 0.06 -0.21 -0.24 0.23

2003 -0.19 0.05 -0.12 -0.05 -0.15

2004 -0.20 0.09 -0.18 -0.21 0.13

2005 -0.18 -0.05 -0.04 -0.09 0.15

2006 -0.14 -0.03 -0.06 -0.07 0.02

2007 -0.22 -0.05 -0.10 -0.16 0.25

2008 -0.38 -0.06 -0.24 -0.30 0.18

2009 -0.17 0.02 -0.13 -0.18 0.20

2010 -0.25 0.06 -0.18 -0.24 0.27

2011 -0.34 -0.07 -0.14 -0.21 0.33

2012 -0.36 -0.02 -0.19 -0.25 0.36

2013 -0.33 0.03 -0.19 -0.15 -0.08

26

Table 5 Relation between changes in CDS spreads and equity returns/components in a regime-

switching model

This table presents the estimation results from the two-state regime-switching models as the following

three equations in panel A, B, and C, respectively:

0 1 1 2CDS CDS Rets s

t t t ta a a (model 1)

0 1 1 2CDS CDS CFs s

t t t ta a a (model 2)

0 1 1 2CDS CDS DRs s

t t t ta a a (model 3).

CDSt , Ret t, CFt

, DR t are the CDS spread changes, stock returns, cash flow (CF) news and

discount rate (DR) news, which are the cross-sectional average in each month. t is the residual.

The s

ia are estimated coefficients from the switching models. The superscript s on 0a and 2a

indicates regime zero or regime one, where s can be regarded as an unobserved state variable that

follows a two-state, first-order Markov process. The transition probability matrix is

1

1

p pX

q q

,

where1Pr( 0 | 0)t tp s s , and

1Pr( 1| 1)t tq s s . The t-statistics are in the parentheses.

27

Table 5 Relation between CDS spread changes and equity returns/components in a regime-

switching model (continued)

Panel A: Coefficient estimates 0 1 1 2CDS CDS Rets s

t t t ta a a

Coefficient estimates t-value

𝑎00 -0.003 (-0.41)

𝑎01 0.02 (2.29)

𝑎1 0.16 (2.82)

𝑎20 -0.53 (-2.80)

𝑎21 -2.16 (-9.03)

p 0.96

q 0.91

duration 0 22.25

duration 1 11.55

Sample moments ∆𝐶𝐷𝑆 stock return correlation

Mean Std.dev. Mean Std.dev.

all -0.001 0.092 0.0064 0.044 -0.62

regime zero -0.007 0.011 0.007 0.005 -0.31

regime one 0.010 0.130 0.005 0.007 -0.86

28

Table 5 Relation between CDS spread changes and equity returns/components in a regime-

switching model (continued)

Panel B: Coefficient estimates 0 1 1 2CDS CDS CFs s

t t t ta a a

Coefficient estimates t-value

𝑎00 -0.02 (-2.89)

𝑎01 0.21 (7.66)

𝑎1 0.11 (1.55)

𝑎20 0.67 (3.42)

𝑎21 -1.05 (-1.70)

p 0.97

q 0.50

duration 0 29.18

duration 1 2.00

Sample moments ∆𝐶𝐷𝑆 CF news correlation

Mean Std.dev. Mean Std.dev.

all -0.001 0.092 0.002 0.030 0.11

regime zero -0.007 0.067 0.002 0.030 0.27

regime one 0.240 0.076 -0.002 0.040 -0.66

29

Table 5 Relation between CDS spread changes and equity returns/components in a regime-

switching model (continued)

Panel C: Coefficient estimates 0 1 1 2CDS CDS DRs s

t t t ta a a

Coefficient estimates t-value

𝑎00 -0.002 (-0.43)

𝑎01 0.02 (1.16)

𝑎1 0.23 (4.03)

𝑎20 -0.51 (-4.77)

𝑎21 -2.15 (-9.88)

p 0.96

q 0.82

duration 0 27.61

duration 1 5.46

Sample moments ∆𝐶𝐷𝑆 DR news correlation

Mean Std.dev. Mean Std.dev.

all -0.001 0.092 0.004 0.058 -0.54

regime zero -0.007 0.061 0.005 0.055 -0.40

regime one 0.037 0.190 -0.005 0.070 -0.89

30

Table 6 Correlations over different horizons

Panel A reports the correlations between equity returns and the components. Panel B presents the

correlations between CDS spread changes and equity returns/components. The horizons are from one

month to 12 months.

Panel A: Correlations between equity returns and the components

Horizon (Ret, CF news) (Ret, DR news) (Ret, RP news) (Ret, RF news)

1-month 0.02 0.57 0.56 -0.08

3-month 0.13 0.48 0.50 -0.23

6-month 0.22 0.42 0.46 -0.34

12-month 0.32 0.34 0.39 -0.35

Panel B: Correlations between CDS spread changes and equity returns/components

Horizon (∆CDS, Ret) (∆CDS, CF news) (∆CDS, DR news) (∆CDS, RP news) (∆CDS, RF news)

1-month -0.30 -0.02 -0.16 -0.20 0.17

3-month -0.40 -0.05 -0.19 -0.26 0.27

6-month -0.47 -0.07 -0.23 -0.31 0.34

12-month -0.51 -0.09 -0.26 -0.33 0.37

31

Table 7 Regression results in two rating groups

This table reports the results of the monthly regressions in two rating groups, in which changes in CDS

spreads ( CDS ) are regressed on equity return ( Ret ), changes in equity volatility ( eqvol ), changes in

leverage ( lev ), change in VIX ( vix ), change in term spread ( term ), change in ten-year Treasury

bond yield ( yield ), and S&P returns. VIX is the CBOE (Chicago Board Options Exchange) implied

volatility of S&P 500 index. The term spread is the difference between ten-year Treasury bond yields and

two-year Treasury note yields. The regression takes the form as:

1 2 3 4 5 6 7CDS Ret eqvol lev vix term yield S&Preti i i i i i i i i i i i i

t t t t t t t t t .

The t-statistics are reported in the parenthesis. *, **, and *** represent statistical significance at the 10%,

5%, and 1% levels, respectively.

Groups Investment grade Non-investment grade

Ret -0.459*** -1.146***

(-9.20) (-8.57)

∆eqvol 0.297*** 0.439***

(14.72) (4.85)

∆lev 1.267***

3.093***

(4.96) (3.09)

∆vix 0.001** 0.001

(2.29) (0.79)

∆term 0.086*** 0.269***

(6.69) (3.09)

∆yield -0.104*** -0.198***

(-9.03) (-3.31)

S&P ret -0.373*** -1.110**

(-3.98) (-2.25)

Constant 0.004*** 0.007

(3.25) (1.06)

Adj. R squared 23.15% 29.02%

32

Table 8 Regression results in groups sorted by equity components

This table reports the results of the monthly regressions in two rating groups, in which changes in CDS

spreads ( CDS ) are regressed on equity return ( Ret ), changes in equity volatility ( eqvol ), changes in

leverage ( lev ), change in VIX ( vix ), change in term spread ( term ), change in ten-year Treasury

bond yield ( yield ), and S&P returns. VIX is the CBOE (Chicago Board Options Exchange) implied

volatility of S&P 500 index. The term spread is the difference between ten-year Treasury bond yields and

two-year Treasury note yields. The regression takes the form as:

1 2 3 4 5 6 7CDS Ret eqvol lev vix term yield S&Preti i i i i i i i i i i i i

t t t t t t t t t .

Panel A, B, and C report the regression results sorted by CF (cash flow) beta, DR (discount rate) beta, and

RP (risk premium) beta. The t-statistics are reported in the parenthesis. *, **, and *** represent statistical

significance at the 10%, 5%, and 1% levels, respectively.

Panel A: regression results sorted by CF beta

Groups 1 2 3 4 5

Ret -0.58*** -0.43*** -0.59*** -0.68*** -0.62***

(-5.72) (-4.94) (-7.79) (-6.22) (-5.10)

∆eqvol 0.36*** 0.30*** 0.27*** 0.35*** 0.34***

(6.13) (7.41) (4.73) (6.34) (6.28)

∆lev 1.66*** 0.87** 1.79*** 1.53*** 1.66

(4.09) (1.96) (3.83) (3.96) (1.50)

∆vix 0.001 0.001 0.001 0.001 0.001

(0.85) (0.72) (0.84) (1.39) (0.58)

∆term 0.14* 0.05*** 0.10*** 0.08** 0.07

(1.89) (3.39) (2.89) (2.06) (0.63)

∆yield -0.17*** -0.11*** -0.13*** -0.09*** -0.10

(-3.02) (-6.08) (-5.06) (-3.38) (-1.01)

S&P ret -0.53** -0.49*** -0.64*** -0.26 -0.88**

(-2.50) (-2.83) (-3.28) (-1.09) (-2.23)

Constant 0.01** 0.001* 0.001 0.001 0.01

(2.17) (1.69) (1.11) (0.59) (1.24)

Adj. R squared 23.38% 23.81% 23.95% 25.03% 28.47%

33

Table 8 Regression results in groups sorted by equity components (continued)

Panel B: regression results sorted by DR beta

Groups 1 2 3 4 5

Ret -0.67*** -0.64*** -0.56*** -0.47*** -0.56***

(-5.32) (-6.31) (-6.96) (-5.13) (-5.78)

∆eqvol 0.35*** 0.33*** 0.25*** 0.37*** 0.31***

(6.60) (6.34) (6.62) (5.00) (7.65)

∆lev 1.39 1.99*** 1.32*** 0.92* 1.89***

(1.28) (4.56) (3.39) (1.77) (5.21)

∆vix 0.001 0.001 0.001 0.001 0.001

(0.57) (0.44) (0.39) (0.30) (1.41)

∆term 0.07 0.02 0.12*** 0.09*** 0.15**

(0.61) (0.46) (4.43) (2.80) (2.08)

∆yield -0.11 -0.07** -0.14*** -0.11*** -0.17***

(-1.14) (-2.42) (-7.84) (-4.36) (-3.16)

S&P ret -0.79** -0.58** -0.35*** -0.64*** -0.43**

(-2.03) (-2.18) (-2.61) (-3.19) (-2.05)

Constant 0.01 0.001 0.001 0.001 0.01

(1.08) (0.53) (0.29) (1.41) (2.08)

Adj. R squared 27.51% 25.46% 25.02% 22.73% 23.86%

34

Table 8 Regression results in groups sorted by equity components (continued)

Panel C: regression results sorted by RP beta

Groups 1 2 3 4 5

Ret -0.62*** -0.69*** -0.66*** -0.43*** -0.51***

(-4.75) (-7.08) (-7.18) (-5.31) (-5.39)

∆eqvol 0.37*** 0.28*** 0.35*** 0.31*** 0.29***

(6.79) (5.77) (5.57) (5.57) (7.45)

∆lev 1.69 1.71*** 1.01** 1.25*** 1.87***

(1.53) (4.15) (2.16) (2.98) (5.05)

∆vix 0.001 0.001 0.001 0.001 0.001

(0.66) (0.22) (0.28) (0.65) (1.27)

∆term 0.06 0.05 0.12*** 0.07*** 0.15*

(0.56) (1.32) (3.50) (3.04) (1.95)

∆yield -0.11 -0.09*** -0.12*** -0.11*** -0.17***

(-1.16) (-3.26) (-5.40) (-5.50) (-3.02)

S&P ret -0.79** -0.59** -0.47*** -0.46*** -0.50**

(-2.00) (-2.18) (-2.88) (-2.77) (-2.36)

Constant 0.001 0.001 0.001 0.001* 0.01**

(0.62) (1.03) (0.23) (1.81) (2.21)

Adj. R squared 27.12% 26.34% 24.31% 24.25% 22.53%

35

Figure 1 Correlations between CDS spread changes and equity returns for the 895-firm sample

This figure graphs average yearly correlations measured using monthly equity returns and CDS spread

changes for the 895-firm sample from 2001 to 2013.

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

2000 2002 2004 2006 2008 2010 2012 2014

corr (∆CDS, Ret)

36

Figure 2 Returns and cash flow news

The figure shows the trend of one-quarter ahead return and the corresponding CF news. The numbers are

calculated using the ICC approach at the aggregate level. The data period is from 2001 to 2013. The solid

line is corresponding to the return series and the bar line is corresponding to the cash flow (CF) news.

37

Figure 3 Correlations between CDS spread changes and equity returns/components for the 516-

firm sample

This figure graphs average yearly correlations measured using monthly CDS spread changes and equity

returns/components for the 516-firm sample from 2001 to 2013.

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

2000 2002 2004 2006 2008 2010 2012 2014

corr(∆CDS, Ret)

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

2000 2002 2004 2006 2008 2010 2012 2014

corr (∆CDS, CF news)

38

Figure 3 Correlations between CDS spread changes and equity returns/components for the 516-

firm sample (continued)

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

2000 2002 2004 2006 2008 2010 2012 2014

corr(∆CDS, DR news)

39

Figure 4 Regime probabilities for the constant transition probability model

The plot displays the smoothed probabilities of being in regime one (the second regime) from the three

constant transition regime-switching models, respectively. The solid, dotted, and dashed lines are

corresponding to model 1, 2, and 3, respectively. The models are defined the same as those in table 5.