Limited Arbitrage between Equity and Credit Markets

Nikunj Kapadia and Xiaoling Pu1

First Version: March 2009

Current Version: April 2010

1University of Massachusetts and Kent State, respectively. We thank Viral Acharya, Kobi Boudoukh, SanjivDas, Michael Gordy, Jason Kremer, Sanjay Nawalkha, N. R. Prabhala, Akhtar Siddique, Dimitri Vayanos,and discussants and participants at IDC-Herzliya, Journal of Investment Management Conference, Second PaulWoolley Conference at the London School of Economics, China International Conference in Finance, FordhamUniversity and the Indian School of Business. Xuan Che provided excellent research assistance. We thankMarkit for providing us the CDS data. The contents are the sole responsibility of the authors. Please addresscorrespondence to Nikunj Kapadia Isenberg School of Management, University of Massachusetts, Amherst, MA01003. Email: nkapadia@som.umass.edu.

Abstract

Proposing a statistical measure of market integration, we investigate whether limits to arbitrage explain

the level of integration between firms’ equity and credit markets. We find cross-sectional and time-

series variation in equity-credit market pricing discrepancies are explained by variables associated with

market liquidity, funding liquidity, and idiosyncratic risk. Illiquidity of the CDS market is particularly

important in determining cross-sectional variation in integration. Our evidence provides a potential

resolution to the puzzle of why Merton model hedge ratios match empirically observed stock-bond

elasticities (Schaefer and Strebulaev (2008)), and yet the model cannot explain the integration between

equity-credit markets (Collin-Dufresne, Goldstein and Martin (2001)).

Keywords: limited arbitrage, market integration, Merton model, capital structure arbitrage

JEL classification: G14, G12, C31

1 Introduction

Merton’s (1974) structural model of credit risk implies that stock price and credit spread changes must

be precisely related to prevent arbitrage. Consistent with academic theory, hedge funds and private

equity firms are active in a variety of trading strategies - popularly known as capital structure arbitrage

- that attempt to “arbitrage” across equity and credit markets. Given the theoretical link between

the equity and bond of a firm, active arbitrageurs, and integrated equity and credit markets, stock

returns and changes in credit spreads should be closely related. Instead, recent research finds to the

contrary. In a regression of monthly changes in credit spreads on stock returns and other variables

consistent with a structural framework, Collin-Dufresne, Goldstein and Martin (2001) find adjusted

R2s of the order of 17% to 34%. They conclude: “Given that structural framework models risky debt

as a derivative security which in theory can be perfectly hedged, this adjusted R2 seems extremely low.”

Blanco, Brennan, and Marsh (2005) conduct a similar exercise using weekly changes in spreads of

credit default swaps (CDS), and find three-quarters of the variation remains unexplained.

This low correlation poses a puzzle. First, as opposed to changes, Merton-model factors do a

much better job explaining the cross-sectional variation in the level of credit spreads (Ericsson, Jacobs

and Oviedo (2009), Zhang, Zhou and Zhu (2009)). In our dataset, a cross-sectional regression of the

average five-year credit default swap spread over 2001-05 on the firm’s average debt ratio and stock

return volatility gives an adjusted R2 of 61%. Second, Schaefer and Strebulaev (2008) document that

the empirical sensitivity of bond returns to stock returns is roughly of the order of magnitude predicted

by the Merton model. How can the Merton model provide reasonable hedge ratios and yet not explain

integration of equity-credit markets?

Existing literature provides two hypotheses regarding why equity-credit market correlations may

be low. First, a low correlation may result from changes in firms’ asset volatility or debt.1 Unexpected

changes in a firm’s volatility and debt can cause wealth transfers between shareholders and bondholders

and reduce the observed correlation between equity returns and credit spread changes. Indeed, changes

in equity volatility are an explanation for pricing discrepancies in the equity option market (Bakshi,

Cao and Chen (2000)). Second, the low correlation may be explained by a non-structural pricing

model where the credit market prices a factor that is not priced in the equity market. For example,

the credit market may price a credit market specific systematic liquidity factor. Longstaff, Mithal and

Neis (2005) and Chen, Lesmond and Wei (2007) document liquidity components in corporate bond1Merton’s (1974) model assumes constant firm volatility and debt. Attempts to make the Merton (1974) model more

realistic have focused on specifications for the default boundary, recovery, and the stochastic process determining theunderlying firm value or leverage. See Black and Cox (1976), Leland (1994), Leland and Toft (1996), Longstaff andSchwartz (1995), Anderson and Sundaresan (1996), Collin-Dufresne and Goldstein (2001).

1

spreads. By the first hypothesis, wealth transfers between shareholders and bondholders give a false

signal of lack of integration. By the second hypothesis, the low correlation between equity and credit

markets is to be expected because when markets price different factors, by definition, the two markets

are not integrated. The common implication of both these hypotheses is that Merton-model pricing

discrepancies are not anomalies when evaluated against the true credit pricing model.

In this paper, we propose and investigate a third hypothesis that Merton-model pricing discrepan-

cies, at least in part, are anomalies, and the lack of integration of the two markets is a result of limited

arbitrage (Shleifer and Summers (1990) and Shleifer and Vishny (1997)). Limits to arbitrage can

prevent arbitrageurs like hedge funds from investing the capital required to eliminate pricing anoma-

lies, therefore, potentially resolving the puzzle of why structural models cannot explain equity-credit

market integration. Our study is the first to investigate the possibility that limited arbitrage explains

the integration between equity and credit markets.2 The CDS-equity market is a singularly useful

laboratory to test the implications of the recent theoretical literature on limits to arbitrage not only

because of its size but because of the participation of active and sophisticated arbitrageurs.

Limited arbitrage is a natural hypothesis to explore because convergence trades across equity

and credit markets are, in practice, not a zero-capital riskless arbitrage as modeled in structural

models. When faced with mispricing, arbitrageurs will limit the capital deployed because of existing

or potential funding constraints (Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009)),

and liquidity and other costs associated with undertaking the arbitrage (e.g., Pontiff (1996, 2006)). In

addition, mispricings will persist over time, because arbitrage capital may accumulate slowly (Mitchell,

Pedersen and Pulvino (2007)) in part because of institutional constraints on the availability of risk

capital (Merton (1987)) and in part because arbitrageurs can reduce risk if they synchronize their

activities (Abreu and Brunnermeier (2002)). Arbitrage activity may be expected to reduce pricing

discrepancies, on average, but not necessarily completely nor always (Xiong (2001), Kondor (2009)).

We begin by proposing a statistical measure for market integration based on the presence of pricing

discrepancies. When pricing in two related markets are determined by a single factor, as stocks and

bonds in the Merton model, pricing discrepancies can be identified by concordance of price changes in

the two markets. Therefore, we propose, a measure of the concordance can serve as a measure of market

integration. Because the concordance is directly linked to pricing discrepancies, there is no ambiguity

in its interpretation unlike measures such as the coefficient of determination used in Collin-Dufresne,2Limits to arbitrage have been invoked for a wide range of anomalies including the closed end fund discount (Pontiff

(1996)), violations of put-call parity (Ofek, Richardson, and Whitelaw, (2004)) and negative stub values (Mitchell, Pulvinoand Stafford (2002), Lamont and Thaler (2003)). See also related work by Ali, Hwang and Trombley (2003) and Mitchell,Pedersen and Pulvino (2007).

2

Goldstein and Martin (2001). We provide simulation evidence to show the sensitivity of the measure

in detecting mispricing in datasets of limited time series length as well as its robustness.

We document the following evidence. First, we show that Merton-model pricing discrepancies are

frequent, persistent, and economically large. Over a 5 business day interval, 46.3% of all relative

stock and CDS spread movements are classified as discrepancies. These discrepancies are coincident

with economically large changes in the underlying spread and stock return. For example, pricing

discrepancies over a 50-business day interval are associated with 8.4% mean absolute stock return and

the mean absolute change in CDS spread is 30 basis points in the wrong direction. Pricing discrepancies

decline with horizon. Over 5-business day intervals, on average, 45% of all price movements are

classified as anomalous, almost twice those over the 50-business day interval.

Pricing discrepancies cannot be explained away by changes in equity option implied volatility or

firms’ debt levels. Surprisingly, the majority of mispricings occur when increases (decreases) in implied

volatility are coincident with decreases (increases) in stock prices. Over short horizons of a quarter,

book value of debt changes by less than 2%, on average. Not surprisingly, changes in book value of

debt do not explain pricing discrepancies.

Pricing discrepancies are mispricings. Capital structure arbitrage trades betting on convergence

of equity and credit markets after observation of short-term pricing discrepancies earn average excess

returns of about 1% over a month with Sharpe ratios as high as 4.42. Convergence of the two markets

is evidence against any hypotheses that argues pricing discrepancies are not real anomalies. In the

absence of arbitrage risks or costs, mispricings would constitute unusually profitable and therefore

unlikely, opportunities.

The frequency of mispricings and, therefore, the level of integration varies across the cross-section

of firms. Cross-sectional differences in the integration of equity-credit markets are correlated with the

proxies for costs and risks associated with convergence trades. Specifically, liquidity of both equity and

credit markets - but especially credit market illiquidity - has significant impact on market integration.

Idiosyncratic risk, which constitutes a cost to arbitrageurs (Pontiff (2006)), is also significant for higher

rated firms. Market-wide changes in pricing discrepancies are correlated in the time series with the

Eurodollar interest rate, the cost of funding. In summary, limits to arbitrage are both statistically and

economically significant in explaining variation in the level of equity-credit market integration.

Our finding that it is possible to identify mispricings using the simplest implication of the Merton

model complements the finding of Schaefer and Strebulaev (2008) that the Merton model is also good

matching the sensitivity of bond returns to stock returns, and helps reconcile their results with those

of Collin-Dufresne, Goldstein and Martin (2001). Even though it is possible to identify economically

3

significant mispricings based on the Merton model, integration of equity and credit markets will depend

on costs associated with convergence trades.

In related work, Duarte, Longstaff and Yu (2007) and Yu (2006) study the profitability of capital

structure arbitrage, and show, as we do, that the returns are positive, positively skewed, and have

high Sharpe ratios. Duarte, Longstaff and Yu (2007) suggest that positive returns may be a result of

deployment of intellectual capital. However, we demonstrate that if there were no costs associated with

arbitrage then even simple trading rules that require the trader to do nothing more than observe relative

price movements in equity-credit markets leads to significant positive returns. Acharya, Schaefer and

Zhang (2008) focus on an interesting episode where downgrades of Ford and GM debt in May 2005

results in a market-wide increase in correlation between firms’ CDS spreads. They interpret these

results as arising from additional funding constraints on market participants in times of crises. Our

analysis complements their work in that it indicates that impediments to arbitrage impact markets

even in more normal times.

We proceed as follows. Section 2 reviews the relevant literature. In Section 3, we propose a

statistical measure of market integration. The data is described in Section 4. Section 5 documents

Merton-model pricing discrepancies across the market. Section 6 empirically tests whether variation

in integration between a firm’s equity and credit markets may be explained by variables impacting

arbitrage activity. The last section concludes.

2 Overview of the Literature and Empirical Implications

We review the existing literature and derive cross-sectional implications for market integration based

on differences in arbitrage costs.

2.1 Funding Liquidity

Arbitrage activity is impacted by an arbitrageur’s ability to fund the positions, or, his funding liquidity.

Gromb and Vayanos (2002) observe that the inability to cross-margin is a significant constraint on

funding liquidity. Margin has to be posted separately on each leg of the convergence trade - the long

position on one leg of the trade cannot be posted as collateral for the short position on the other

leg. Besides preventing the arbitrageur from taking every potentially profitable opportunity, the risk

of a tightening of a financial constraint and forced liquidation will also reduce the arbitrage capital

allocated to any convergence trade. Acharya, Schaefer and Zhang (2008) provide a case study of an

episode where arbitrageurs face such a liquidity shock. Such risks associated with funding illiquidity

4

are also discussed in Attari, Mello and Ruckes (2005). Brunnermeier and Pedersen (2009) provide a

comprehensive discussion of the importance of funding constraints. The literature does not indicate

how funding liquidity is expected to vary in the cross-section. We discuss this below.

Let the arbitrageur’s available equity be W . Let the equity and debt of the firm of asset value A

be P (A) and B(A), respectively. Let |nP | and |nB| designate the arbitrageur position in stock and

bond as a fraction of firm’s equity and debt, P and B, respectively. The size of the bond position,

|nB|B, for a given stock position, |nP |P , in the convergence trade will be determined by the elasticity,

(B/P )(∂P/∂A)/ (∂B/∂A). If a change in firm value results in stock and bond returns of 80 bps and

20 bps, respectively, then for every $1 stock position, the arbitrageur will take a $4 bond position. Let

mP and mB be the margin requirements (“haircut”) on the stock and bond, respectively, defined as a

fraction of the dollar value of the position. Then, the total margin, W , is,

W = mP |nP |P +mB|nB|B

= |nP |P(mP +mB

∂P/∂A

∂B/∂A

B

P

). (1)

For a given stock position, |nP |P , the total margin required to fund the convergence trade is determined

by (i) the margin requirement for the stock, mP , (ii) the margin requirement for the bond or CDS,

mB, and (iii) the elasticity, (B/P )(∂P/∂A)/ (∂B/∂A).

The margin for the stock position set by Regulation T of the Federal Reserve Board does not

require mP to vary across the cross-section of stocks. It appears reasonable to presume arbitrageurs

do not face cross-sectionally varying margins on stock. Margin requirements for selling protection with

5-year credit default swaps recently proposed by FINRA depend on credit riskiness as follows,3

Spread (bps) Margin

0-100 4%

100-300 7%

300-500 15%

500-700 20%

> 700 25%

The margin requirement for below investment grade CDS is about three to six times that for investment

grade CDS.3See FINRA Rule 4240, “Margin Requirements for Credit Default Swaps.” The margin requirements for buying

protection are half for those selling protection, and therefore have similar cross-sectional implications.

5

Finally, to understand how the elasticity varies in the cross-section, consider Table 5 of Schaefer

and Strebulaev (2008). They report the sensitivity of the bond return for a given stock return, i.e.,

the reciprocal of (B/P )(∂P/∂A)/ (∂B/∂A). The sensitivity of the bond return decreases rapidly as

the bond gets safer. For an investment grade firm with quasi-leverage of 0.20 and asset volatility of

20%,4 Schaefer and Strebulaev (2008) report a sensitivity of 0.70 compared with 24.17 for a firm with

quasi-leverage of 0.50 and asset volatility of 40%. That is, the bond position required to hedge a given

stock position, (B/P )(∂P/∂A)/ (∂B/∂A), increasing very rapidly as the bond gets safer - in the above

example, the position for the investment grade bond is about 35 times that for below investment grade

bond.

Putting it together, cross-sectional differences in margin requirements are likely to be driven by

cross-sectional differences in elasticity rather than cross-sectional differences in mP or mB. Because

the elasticity of the bond is related to the firm’s credit risk, the equity required to fund trades cross-

sectionally with the credit-risk of the firm. That is, the credit riskiness of the firm is an indirect proxy

for funding cost associated with the equity required to find trades.

2.2 Market Liquidity

Convergence traders may not be able to enter or unwind the position in a timely manner without

impacting the price. Liquidity of underlying markets are especially important because convergence

trading requires coordinated trades across equity and credit markets. Thus, illiquidity of either market

can constrain an arbitrageur. In addition, liquidity risk may be related to funding illiquidity as it is

likely that arbitrageurs get funding from the same investment banks that also make markets. The

natural hypothesis that follows is that lower liquidity in either the equity or credit markets will reduce

arbitrage activity and the level of integration between the two markets.

Cross-sectional differences in liquidity have been extensively documented in the literature. In

particular, a number of measures using stock daily return data have been proposed for measuring the

different aspects of equity market liquidity. These include Roll (1984), Lesmond, Ogden and Trzinka

(1999) and Amihud (2002). Compared with the equity liquidity literature, the literature documenting

cross-sectional differences in the credit markets is more recent (e.g. Longstaff, Mithal and Neis (2005)

and Chen, Lesmond and Wei (2007)). With the notable exception of Das and Hanouna (2009), there

is no literature that has studied cross-market liquidity across a firm’s equity and credit markets.4The quasi-leverage is defined as the ratio of the face value of debt discounted at the riskfree rate to asset value.

6

2.3 Idiosyncratic Risk

Despite having a long and short position, the arbitrageur is unlikely to be perfectly hedged. For

example, the arbitrageur may not have the correct hedge ratio, or the hedge ratio may be stale if it

is not possible to continuously adjust the hedge. If so, the arbitrageur will have a net long or short

position in the underlying firm, resulting in the arbitrageur being exposed to the total risk of the

firm. Exposure to the systematic component of the firm’s risk is not a cost to the arbitrageur as

the arbitrageur is compensated for this exposure. However, the arbitrageur is not compensated for

non-priced idiosyncratic risk. Pontiff (2006) argues that the exposure to idiosyncratic risk imposes

significant constraint on arbitrageurs.

There is also empirical evidence that relates pricing discrepancies in the equity markets to idiosyn-

cratic risk. Wurgler and Zhuravskaya (2002) demonstrate that stocks with higher idiosyncratic risk

have higher abnormal stock returns when they are included into the S&P 500 index. Ali, Hwang

and Trombley (2003) and Mashruwala, Rajgopal and Shevlin (2006) relate idiosyncratic risk to the

book-to-market anomaly and accrual anomaly, respectively.

If unhedged idiosyncratic risk is costly, arbitrageurs will be less active in markets where the firm

has greater idiosyncratic risk. The level of integration between a firm’s equity and credit markets will

then be negatively correlated to the magnitude of idiosyncratic risk of the firm.

2.4 Slow-Moving Capital and Horizon

Abreu and Brunnermeier (2002) observe that a capital-constrained arbitrageur may not have sufficient

capital to single-handedly eliminate a pricing discrepancy. If so, the elimination of pricing discrepancies

will require coordinated action by a number of arbitrageurs. Because arbitrageurs are likely to become

aware of a pricing discrepancy slowly over time, arbitrage capital will take time to accumulate and

pricing discrepancies will be resistent to being eliminated in the short-run. Mitchell, Pulvino and

Stafford (2002) provide evidence that is consistent with slow-moving arbitrage capital.

The primary empirical implication of slow-moving capital is that pricing discrepancies will decline

over longer horizons.

7

3 Methodological Framework

3.1 Statistical Measure of Market Integration

When the underlying asset return follows a one-dimensional diffusion, the “delta” of a call option is

bounded between 0 and 1. Thus, in the Merton (1974) model, an instantaneous increase (decrease)

in firm value results in an increase (decrease) in both stock prices and credit spreads. We propose a

statistical measure of market integration based on the theory.

Let Pi and CDSi be the stock price and CDS spread, respectively, for firm i. Assume we have

k = 1, ...,M instantaneous observations of CDS spreads and stock prices, where the spread is the

cost of insurance on the firm’s zero coupon debt of face value F and time to maturity T . Define

∆CDSτi,k = CDSi(k + τ) − CDSi(k) and ∆P τi,k = Pi(k + τ) − Pi(k), k ≤ M − τ , 1 ≤ τ ≤ M − 1.

Then, a pair of stock prices and CDS spreads present an arbitrage if ∆CDSτi,k∆Pτi,k > 0. We define

market integration for a stock i based on the frequency of observations of such arbitrage opportunities.

Specifically, define κ̂i as,

κ̂i =M−1∑τ=1

M−τ∑k=1

1[∆CDSτi,k∆P τi,k>0]. (2)

For a given M , firm i’s stock and CDS markets are more integrated than those of firm j if κ̂i < κ̂j .

The measure takes into account all pairs of CDS and stock prices.

As an illustration, suppose stock prices are P (1) = 100, P (2) = 80 and P (3) = 90. The markets are

more integrated if corresponding CDS spreads are CDS(1) = 50 bps, CDS(2) = 30 bps, CDS(3) = 60

bps than if they are CDS(1) = 50 bps, CDS(2) = 30 bps, CDS(3) = 40 bps. In the latter case, there

is a (Merton-model) arbitrage across every pair of observations [1, 2], [2, 3] and [1, 3].

This definition of market integration is appealing for three reasons. First, the measure is robust

because (i) it is non-parametric and (ii) not impacted by non-linearities. Second, the measure accounts

for all M pairs of observations and is, therefore, independent of the horizon over which prices are

observed. This is useful as there may be variation in the length of the periods over which pricing

discrepancies develop and disappear. In addition, as we document in simulations below, the measure is

sensitive to mispricing in short datasets.5 Finally, because the measure relies only on the concordance

of stock and CDS prices, there is a one-to-one correspondence between κ̂ and the Kendall correlation5Although taking into account all pairs of CDS and stock prices allows mispricings to be detected over small samples,

this benefit comes with the tradeoff. If the observations span a long time period with large changes in stock and CDSspreads, then the measure will be less sensitive to mispricing. As we shall see in the next section, the measure is sensitiveto levels of mispricing when the time period is a year, and thus suitable for our study which is motivated by short-horizonmispricings.

8

(Kendall (1938)). Defining κ as the Kendall correlation,

κ =4κ̂

M(M − 1)− 1. (3)

The advantage of this correspondence is that we can use standard statistical theory to interpret κ.

In particular, using κ as a measure of market integration provides a simple null hypothesis. In the

absence of mispricings, κ = −1. A more positive κ corresponds to a less integrated market.6

3.2 Simulation Evidence

In practice, one does not have M instantaneous observations of CDS spreads and stock prices. Instead

we observe stock prices and 5-year CDS spreads on M different dates. In this case, the estimate of κ

will be biased. How significant is this bias? We provide simulation evidence from the Merton (1974)

model to provide an estimate of the bias. We also use the simulation to investigate the sensitivity of

κ to the probability of mispricing.

For dates t = (k − 1)τ , k = 1, 2, ...,M , let B(t;F (t), T ) be the price for the zero-coupon bond of

face value F (t) and time to maturity T . On each date, the firm re-finances its existing debt of face

value F (t− τ) and remaining time to maturity T − τ with new debt of face value F (t) and maturity T

of identical value, i.e., F (t) is chosen so that B(t;F (t), T ) = B(t;F (t− τ), T − τ). Replacing existing

debt of remaining time to maturity T − τ by debt of maturity T in this manner has no impact on the

firm value or the stock price, and there are no wealth transfers across bond and stock holders. We allow

for mispricings by assuming that at any time t there is a fixed probability of mispricing. Mispricings

last only one period. We simulate one year of data at a weekly and bi-weekly frequency and compute

κ. For comparison, we also compute κ under the assumption that M stock prices and credit spreads

values are observed instantaneously (the “unbiased” κ). Parameters are chosen to be representative of

our sample; more detail on the simulation procedure is in Appendix A.

Figure 2 plots the two estimates of κ against the probability of mispricing for weekly frequency. As

noted, when multiple stock and credit spreads are not simultaneously observed, κ is biased. However,

the bias is small relative to the impact of the probability of mispricing. The maximum bias is about

0.055 when there is no mispricing and is much lower at about 0.01 for moderate levels of mispricing. It

is only slightly higher when stock and CDS spreads are observed at a bi-weekly frequency. For example,6In contrast, the coefficient of determination from a linear regression used in Collin-Dufresne, Goldstein and Martin

(2001) and Blanco, Brennan, and Marsh (2005) is impacted by the non-linear relation between credit spreads and equityprices. Even in the absence of mispricings, the coefficient of determination will be less than 1. Moreover, as the magnitudeof the bias varies with leverage, it is difficult to use the R2 to make cross-sectional comparisons across firms.

9

the bias is 0.02 when the probability of mispricing is 40%. In contrast, for either frequency, κ is very

sensitive to the prevalence of mispricing. As the frequency of mispricing increases from 0% to 40%,

κ increases from -1 to -0.75. That is, the bias is too small to impact inference about cross-sectional

differences in the level of integration.

3.3 Robustness to Stochastic Volatility and Interest Rates

Asset volatility and the risk-free interest rate are assumed to be constant in the Merton (1974) model.

For a given firm value, Vt, a change in asset volatility causes wealth transfers across stock and bond

holders. Changes in the risk-free rate impacts the risk-neutral drift of the firm, with an increase

(decrease) making the firm less (more) likely to default. How is κ impacted by the stochasticity of

interest rates or volatility?

We extend the simulation of the previous section by letting asset volatility and the risk-free interest

rate be stochastic. As in the previous section, we simulate one year of data at a weekly and bi-weekly

frequency to estimate κ. For comparison, we also provide the Merton-model estimate of κ. The

Merton-model is nested in the model with stochastic volatility (SV), which in turn is nested in the

model with both stochastic volatility and stochastic interest rates (SIV). Details of the simulation are

provided in the Appendix. The results of the simulation are reported in Table 1.

The simulation exercise demonstrates that there is a positive bias on κ because of stochastic volatil-

ity and stochastic interest rates. However, this bias is very small. The maximum bias that is observed

is less than 0.02. For moderate levels of mispricing, the bias is smaller than 0.01. Stochastic volatility

does not have much of an impact because the volatility mean-reverts to its long-term mean fast enough

to make little impact on debt of maturity five years. Similarly, for typical parameters, the change in

the risk-free rate also has a small impact relative to the impact of mispricing. More importantly, the

cross-sectional relation between κ and the probability of mispricing is not impacted by the presence of

stochastic volatility or interest rates. A linear regression of κ on the probability of mispricing provides

the following estimates,

κ = −1.01 + 0.823 MISPRICING + e Merton,

κ = −1.00 + 0.826 MISPRICING + e SV,

κ = −1.00 + 0.824 MISPRICING + e SIV.

There is little difference in the estimate of the coefficients across the three models.

We should note, however, that whether in actuality changes in volatility or interest rates do impact

10

market integration is an empirical question. It is possible, for example, that the markets view changes

in volatility as permanent shocks. If so, innovations in volatility would have a much greater impact

both on spreads and κ. The simulation exercise simply demonstrates that κ is robust given the typical

assumptions made in the literature.

4 Data

4.1 Credit Default Swap

Our dataset consists of credit default swap spreads, equity prices, and relevant accounting information

for U.S. non-financial firms over the period from January 2, 2001 to December 31, 2005.

We obtain daily price data for the five-year credit default swap (CDS) on senior, unsecured debt

of non-financial firms from Markit Group, the leading industry source for credit pricing data. Markit

Group collects CDS quotes from a large number of contributing banks, and then cleans it to remove

outliers and stale prices. The obligors that enter our sample are components of the Dow Jones CDX

North America Investment Grade (CDX.NA.IG), the Dow Jones CDX North America High Yield

(CDX.NA.HY) and the Dow Jones North America Crossover (CDX.NA.XO) indices.7 We specifically

choose firms that form part of the index to ensure continuity in price quotes. The time period of our

sample is 2001-2005. We match the data from Markit to CRSP and Compustat manually to construct

an initial sample of 223 North American non-financial firms, from which we eliminate 22 firms that

were delisted over this period and another 2 firms that had less than a year of data of spread and

stock price data. Our final sample set consists of 199 firms of which 95 obligors have an average rating

of investment grade (AAA, AA, A, and BBB), and the remaining 104 obligors are below investment

grade (BB, B, and CCC).8

We obtain daily equity prices, returns, outstanding number of shares, and other equity informa-

tion from the Center for Research in Security Prices (CRSP). Accounting data is obtained from the

COMPUSTAT Quarterly database. We construct three firm level variables: size, leverage, and equity

return volatility. The market capitalization (size) of the firm is calculated as the product of stock

prices and outstanding number of shares. Leverage is computed as the ratio of book debt value to the

sum of book debt value and market capitalization. The book value of debt is defined as the sum of7The IG index consists of 125 equally weighted investment grade entities, the HY of 100 equally weighted entities of

rating below investment grade, and the XO of 35 equally weighted entities with cross-over ratings. Cross-over ratings aredefined as a rating of BBB/Baa by one of S&P and Moody’s, and in the BB/Ba rating category by the other, or a ratingin the BB/Ba category by one or both S&P and Moody’s.

8Markit provides information on both the average agency rating and an implied rating. We use the agency ratingaveraged over our sample period when available. When the agency rating is unavailable, we use the implied rating.

11

long term debt (data51) and debt in current liabilities (data45). Equity volatility is the annualized

standard deviation of daily stock return over the sample period.

Table 2 reports summary statistics of CDS spreads and firm characteristics. In computing these

statistics, we first average over our sample period for each obligor, and then take a second average

across all the firms. The mean CDS spread across the entire sample is 215 basis points (bps). The

mean across investment grade firms is 87 bps while that of the high yield is much larger at 333 bps.

The average size of investment grade firms in our sample is $22.3 billion versus $4.9 billion for high

yield firms. As might be expected, below investment grade firms have higher equity volatility and

leverage than investment grade firms.

Figure 1 plots the mean CDS spread over the sample period for each firm against the firm’s average

leverage and equity volatility. Consistent with the basic Merton (1974) model, the spread is significantly

correlated with volatility and leverage. In fact, a linear regression of the mean CDS spread on these

variables gives an adjusted R2 of 61%,

CDSi = -0.0263 + 0.0760 EQVOLi + 0.0399 LEVi + e, R2=61%.

[-9.1] [12.2] [7.3]

4.2 Liquidity and Idiosyncratic Risk

We construct liquidity measures for each firm separately for equity and credit markets. Equity market

liquidity measures are estimated from daily stock price data from CRSP. Our primary measures are

(i) the square root of the Amivest measure (sqrtamivest), and (ii) the proportion of zero stock returns

(zprop). To construct the Amivest measure, we first compute 0.001 ∗√

price ∗ sharevolume/|return|

from daily data for each firm over our sample period. The time-series mean of the daily estimate is

then used as our measure. The higher the square root of Amivest measure, the higher is the liquidity

of the stock. The zero return proportion is calculated as the ratio of the number of days with zero

returns to the total number of days with non-missing observations (Lesmond, Ogden, and Trzcinka

(1999)). The higher the proportion of zeros, the more illiquid is the stock.

We introduce two new credit market liquidity measures. Our first measure is based on the number

of contributors that provide quotes to Markit on any given date. As contributors are required by

Markit to have firm tradeable quotes, the greater the number of contributors, the greater should be

the depth and liquidity of the CDS. Thus, our first measure (depth) is computed as the mean of the

daily number of contributors for each firm. Second, analogous to the equity liquidity measure, we use

the proportion of zero spread changes, (spreadzero), defined as the ratio of zero daily spread changes

to the total number of non-missing daily CDS changes. As with the equity market measure, a larger

12

proportion of zero spread changes indicates lower liquidity. Markit does not provide bid-ask spreads,

and therefore we cannot use liquidity measures based on bid-ask spreads.

We construct the measure of idiosyncratic risk from the standard market model. We first get the

coefficient of determination R2i from a regression of excess returns for stock i on the market using daily

returns. Next, following Ferreira and Laux (2007), we compute the ratio of the idiosyncratic volatility

to the total volatility for each stock as

σ2i,ε

σ2i

=σ2i −

σ2imσ2m

σ2i

= 1−R2i .

The idiosyncratic measure idiosyn is then defined as the logistic transformation ln(

1−R2i

R2i

). The

corresponding measure computed using the Fama-French three-factor model had a correlation of 0.97

with the measure of idiosyncratic risk from the market model. We only report results from the market

model.

Panel A of Table 3 presents the descriptive statistics. Each variable exhibits considerable cross-

sectional variation. For example, the number of contributors to the CDS quote ranges from 3 to

18. There is also variation across equity and credit market illiquidity. The mean of the proportion

of zero spread changes (0.22) is much larger than the mean of the proportion of zero returns (0.02)

consistent with the perception that the equity market is more liquid than the CDS market. Panel B

reports pair-wise correlations amongst the variables. The signs of the correlations are as expected.

The liquidity and illiquidity measures are negatively correlated with each other in each of the two

markets. Liquidity in the equity market (sqrtamivest) is positively correlated with liquidity in the

credit markets (depth). Firms with higher leverage and equity volatility also have higher idiosyncratic

risk. The highest correlations are observed to be between equity market liquidity and credit market

liquidity, and between leverage and equity liquidity measures.

5 Pricing Discrepancies and Mispricings

5.1 Existence of Pricing Discrepancies

Table 4 reports the frequency of pricing discrepancies defined by ∆CDSτi,k∆Pτi,k > 0, for different

horizons, τ ∈ {5, 10, 25, 50}, corresponding to weekly, bi-weekly, monthly, and bi-monthly horizons,

respectively. We also report frequency of ‘zeros’, corresponding to ∆CDSτi,k∆Pτi,k = 0. Similar exercises

in other markets are conducted in Bakshi, Cao and Chen (2000), Buraschi and Jiltsov (2007), and

Buraschi, Trojani and Vedolin (2009).

13

Pricing discrepancies are common. Over the entire sample, at a 5-business day frequency, stock

prices and spreads co-move as expected (∆CDSτi,k∆Pτi,k < 0) only 53.7% of the times. These are not

explained away by lead-lag relationships across the equity and CDS market as there are significant

pricing discrepancies over all horizons. Even at a 50-business day horizon, 31% of co-movements

represent arbitrage opportunities by the Merton model. The pricing discrepancies are associated

with large and economically significant movements in the underlying markets - discrepancies over

50 business days occur with an average (absolute) stock return of 8.4% and an average (absolute)

change in CDS spread of 30 bps in the wrong direction. These discrepancies are not specific to the

CDS market; Buraschi, Trojani and Vedolin (2009) document that corporate bond spreads also exhibit

pricing discrepancies.

Pricing discrepancies decline over longer horizons. Including zeros, the fraction of discrepancies for

5, 10, 25 and 50 business day intervals are 47%, 42%, 36%, and 31% respectively. This observation is

consistent with the implication of Abreu and Brunnermeier (2002) and Mitchell, Pedersen and Pulvino

(2007) that if arbitrage capital takes time to be deployed, pricing discrepancies should decline with

time. A large proportion of anomalous observations at the weekly frequency, 5.7% of all observations,

are related to cases where CDS spreads or stock prices do not change. Such zero changes reduce as the

horizon increases; at a frequency of 50 days, zeros constitute only 0.2% of the observations. As zero

changes are associated with higher trading costs (Lesmond, Ogden and Trzinka (1999)), it provides

further support for the hypothesis that the level of discrepancies are related to limited arbitrage activity.

There are substantial cross-sectional differences. For every time-interval, investment grade firms

have more frequent pricing discrepancies, but the size of the discrepancy is of smaller magnitude. For

example, 10 (25) business-day pricing discrepancies for investment grade firms is 41.3% vs. 37.4%

(38.6% vs. 31.8%) for below investment grade firms. However, the mean absolute stock return as-

sociated with the 10 (25) day pricing discrepancy is 3.8% vs 4.9% (5.6% vs. 7.5%), respectively. To

illustrate the cross-sectional differences, Figure 3 plots the CDS spread against the stock price over

our sample period for three firms: Alcoa, Hilton and General Motors. GM has a below investment

grade rating of B and one of the highest average spread of 309 bps in our sample, while Alcoa has a

high rating of AA and an average spread of 35 bps. Hilton is rated BB with an average spread of 197

bps. As can be observed, there is a strong relation between GM’s stock price and CDS spread, while

at the other extreme, Alcoa’s stock price and CDS spread appear to be totally unrelated. The relation

between Hilton’s stock price and CDS spread is clearly discernible unlike Alcoa’s, but it is noisier than

that of GM’s. On a 5-business day frequency, pricing discrepancies are 49%, 34%, and 32% for Alcoa,

Hilton, and GM, respectively. The evidence suggests that the puzzle is not just that the equity and

14

credit markets are not integrated on average, but why the level of integration differs across firms.

5.2 Pricing Discrepancies and Changes in Volatility and Leverage

Before we examine whether these pricing discrepancies are mispricings, it is useful to first check whether

they can be explained away by unexpected changes in volatility or leverage. To do so, we construct

a contingency table, dividing pricing discrepancies according to whether they are coincident with

increases or decreases in stock prices and implied volatility/debt. Pricing discrepancies would be

explained by changes in volatility or leverage if the mispricings predominantly occur in periods where

increases (declines) in stock prices are coincident with increases (declines) in volatility or debt.

We proxy unexpected changes in volatility by changes in equity option implied volatility.9 We

defined implied volatility as the Black-Scholes implied volatility of the at-the-money near-month option.

The change in implied volatility is measured as the change in implied volatility from the first date of

the interval to the last date of the interval. We use a horizon of exactly 1-month to avoid overlapping

intervals.

Panel A of Table 5 reports the results. To explain pricing discrepancies, we should observe more

pricing discrepancies in periods when an increase (decrease) in stock price is coincident with an increase

(decrease) in volatility (the right diagonal of the contingency table). In fact, a significantly greater

proportion of discrepancies - over 59% - are in periods when an increase in stock price coincides with

a decrease in volatility and a decrease in stock price coincides with an increase in volatility. In fact,

changes in implied volatility make these pricing discrepancies appear even more anomalous. Even

if all of the remaining 41% of pricing discrepancies are attributed to changes in implied volatility, a

significant proportion of pricing discrepancies remain to be explained.

Do unexpected changes in debt levels cause wealth transfers across equity and bondholders? Panel

B of Table 5 reports pricing discrepancies conditioned on stock price changes and changes in the book

value of debt. The book value of debt is taken from quarterly financial statements, and we also compute

pricing discrepancies over a quarter so as to match the horizon with the financial statements.

We find no evidence to indicate that pricing discrepancies are related to changes in the face value

of debt. The proportion of pricing discrepancies when stock price increases (decreases) coincide with

increases (decreases) in debt is 51% compared with 49% for the other periods. The two proportions

are not significantly different. This is not entirely surprising. Unlike volatility, firm debt levels do not

change much over short horizons - the median quarterly change in the debt ratio for the firms in our9Cremers, Driessen, Maenhout and Weinbaum (2008) document that increases in equity implied volatility increases

credit spreads.

15

sample is less than 2%.

5.3 Are Pricing Discrepancies Mispricings?

Merton-model pricing discrepancies are not mispricings if they are a result of wealth transfers or if

pricing kernels in the equity and bond markets are distinct as, for example, if a systematic liquidity

factor is priced in the bond market but not priced in the equity market. If so, arbitrageurs would not

make excess returns trading on such pricing discrepancies. On the other hand, significant excess returns

would serve as evidence against either of these hypothesis and indicate that pricing discrepancies are

mispricings.

We investigate the magnitude of excess returns using the following trading rules based on Table

4. We initiate a trade across a firm’s stock and CDS markets if there is a pricing discrepancy over

the previous τ business days coincident with a large change in stock prices, and if there is no existing

open trade. Specifically, the trade is initiated at time t if ∆CDSτi,t∆Pτi,t > 0 and if |∆P τi,t/Pi,t| > 3%

(long both stock and CDS if stock returns are negative, and short if stock returns are positive). We

maintain consistency with Table 4 by choosing τ to be either 5- or 10-business days. The 3% threshold

is chosen to match the average absolute stock return for τ = 5 in Table 4. To avoid overlapping

intervals, each firm has only one open trade at any point in time (i.e., a trade is not initiated if a

previous trade is open). Each position is initiated with a total equity of W = $1, 000, 000 where

W = mP |nP |P + mCDS |nCDS |S, where nP and nCDS are the number of shares and CDS contracts

that form the two legs of the (hedged) convergence trade. The margins for equity and CDS, mP and

mCDS , are given in Section 2.1. The hedge ratio is determined empirically by running a regression

of changes in monthly CDS spreads on stock returns as in Schaefer and Strebulaev (2008), and does

not make any model assumption. Details are provided in Appendix B. Once the trade is initiated, the

hedge ratio is not adjusted dynamically. The position is closed at the end of a pre-determined trading

horizon, T , or if the cumulative loss in the position exceeds 10% of the original equity (i.e., $100,000),

whichever occurs first. Trading horizons are 1- or 2-months. Using the trading rule, we implement

convergence trades for each firm in our sample, and compute the average cumulative excess return for

each firm. Excess return is defined as the total gains less interest costs for the trade divided by the

initial equity position.

Summary statistics across the 199 firms are reported in Table 6. Panel A reports results for the first

trading rule with τ = 5 and a horizon of 1-month, and Panel B for the second rule with τ = 10 and a

horizon of 2-months. Given the high frequency of pricing discrepancies, there are ample opportunities

to execute a convergence trade. When the maximum trading horizon is 1-month (2-months), the mean

16

number of trades for each firm are 28.2 (18.6) trades per firm over the five years. There is more trading

for investment grade firms than below investment grade firms because of the higher frequency of pricing

discrepancies in investment grade firms.

Convergence trading, on average, earns statistically significant positive excess returns. In Panel A,

a convergence trade on the average firm earns an excess return of 1.04% over the 1-month horizon.

In Panel B, over the longer horizon of two months, the convergence trade earns only a little more

at 1.23%, suggesting that most of the profits accrue over the first month. The excess profits are also

economically significant. In Figure 4, we plot the cumulative profits over the five years on an investment

of $1 million by compounding the mean daily return over all open positions. Over the five year, the

cumulative return is over 70% and 40% for the one-month and 2-month horizons, respectively. The

excess return distribution is positively skewed. In Panel A, across the 199 firms, the average maximum

return on a trade is over 15% while the minimum return is only -5%.

Significant excess returns suggest convergence of CDS and equity markets soon after the onset

of a pricing anomaly, and is evidence against both the hypotheses that divergence across CDS and

equity markets are caused by wealth transfers or because of differing pricing kernels. In addition, the

magnitude of excess profits also support the notion of limited arbitrage. Hypothetical portfolios that

invest equally in each firm have Sharpe ratios as high as 4.42 (in sub-samples, a higher Sharpe ratio is

associated with 1-month horizon for below-investment grade firms). In the absence of arbitrageur risks

and costs, these profits from trades based on the simplest implication of the Merton model that requires

no information apart from observation of relative price movements in equity and credit markets should

be considered highly anomalous.10

At the very least, convergence trades are risky. If traders had a stop-loss criteria of 10% of initial

capital, a significant proportion of trades would be closed early. For example, in Panel A, about 8 of

the average 28 trades -almost 30% - were closely prematurely because the loss threshold. In a recent

article, The Wall Street Journal (February 6, 2009) reported that the capital structure arbitrage group

at Deutsche Bank made an estimated $900 million in 2006, $600 million in 2007 but lost it all in 2008.11

10Observe that the hedge ratio used in the convergence trade is also determined by relative price movements in equityand CDS markets by running a regression of CDS spread changes on stock returns. Thus, the implementation of thestrategy requires the trader to do nothing more than be an astute observer of the markets.

11According to the Journal, the group undertook trades across equity and CDS markets, as well as across corporatebonds and CDS markets. Arbitrage trades across corporate bonds and CDS markets are typically undertaken when thereis a ‘negative basis’; when the cost of protection is less than the income produced by the bond. A significant proportion ofthe 2008 loss at Deutsche Bank appear to be related to the widening of the basis coinciding with lower funding liquidity.

17

6 Integration of Equity and Credit Markets and Limits to Arbitrage

Existence of significant mispricing supports the interpetation of κ as a measure of integration. Table 7

provides summary statistics for κ across the cross-section of firms. For each firm, κ is estimated from

stock prices and CDS spreads at a weekly (τ = 5) and bi-weekly (τ = 10) intervals for each of the five

years in our dataset. The descriptive statistics are of the average over the annual estimates. The mean

cross-sectional estimates of κ for both weekly and bi-weekly horizons are much below -1 indicating

presence of significant mispricings. Below investment grade firms have a more negative κ consistent

with the lower frequency of mispricing documented in Table 4.

6.1 Cross-sectional Determinants of Market Integration

To investigate the hypothesis whether integration of firms’ equity and credit markets is related to

arbitrage costs, we estimate a panel regression using the annual estimates of κ. In implementing our

regressions, we use Fisher’s z transformation (David (1949)) of κ, defining the transformed variable

κ̄ = 12 ln (1+κ)

(1−κ) . The panel specification is,

κ̄i,t = α+ β1eqvoli,t + β2lnmcapi,t + β3levi,t + β4variablei,t + εi,t, (4)

where variablei,t refers to each of the variables of interest, viz., the CDS liquidity measure (depth), CDS

illiquidity measure (spreadzero), equity liquidity measure (sqrtamivest), equity illiquidity measure

(zprop), and idiosyncratic risk (idiosyn). We include the size of the firm as a control variable. Proxies

for credit risk, equity volatility (eqvol) and leverage (lev), serve as indirect proxies for arbitrage costs.

We include a dummy variable for investment grade rating of the firm. Each variable is measured

annually for each of the five years in our sample over the period 2001-05. To measure significance, we

use robust standard errors clustered by firm. Fixed effects are included. The hypothesis is examined

by the significance and sign of the variables of interest. As discussed in Section 3, the impact of the

impediment to arbitrage should be to make κ̄ more positive. Table 8 reports the results for the entire

sample of 199 firms for twelve specifications.

We begin by considering each of our direct proxies for arbitrage costs individually. Both equity

and credit liquidity has a significant impact on market integration. The credit market illiquidity

measure spreadzero is significant at the 99% for both τ = 5 and τ = 10. The sign of the coefficient is

positive indicating that firms with more illiquid credit markets and more zero spread changes have less

integrated markets. The second measure of credit market liquidity depth does not show significance.

18

With respect to equity liquidity, the Amivest measure is significant at the 95% level with the expected

negative sign for both intervals; firms with higher Amivest measures and higher liquidity have more

integrated markets. For τ = 10, idiosyncratic risk is significant at the 90% level with a positive sign,

indicating that an increase in idiosyncratic risk of the firm decreases the firm’s equity-credit market

integration.

Amongst the control variables, size is insignificant in 6 of the 12 specifications. The rating proxy

for investment grade is significant and positive for four of the twelve specification, and equity volatility

in another four specifications, suggesting that riskier firms have more integrated markets.

Next, we estimate an encompassing regression including all variables that were significant individ-

ually. Credit market liquidity is strongly significant at the 99% level, but equity market liquidity is

no longer significant. Given that credit markets much less liquid than equity markets (see our Table

3), it is not surprising that liquidity cost in the credit market is the bigger constraint on arbitrage

activity. Idiosyncratic risk remains significant for τ = 10 at the 90% level. Although idiosyncratic risk

has a lower statistical significance, the economic impact is of the same magnitude as credit market

illiquidity. A one-standard deviation change in spreadzero and idiosyn changes κ for τ = 10 by 0.035

and 0.037, respectively. This impact is economically significant - given the standard deviation of κ of

0.18, each of these two variables explain roughly 20% of the variability.

To understand differences across investment grade and below investment grade firms, we run the

regressions separately for each set of firms. Table 9 presents the results. In Panel A, we report results

for investment grade firms. Interestingly, credit market illiquidity (spreadzero), equity market liquidity

(sqrtamivest) and idiosyncratic risk (idiosyn) are all significant with the correctly hypothesized signs.

In addition, all three of these variables remain significant when they are considered together. Next,

considering the sample of only below investment grade firms in Panel B, it is evident that the impact

of arbitrage costs varies across the investment and below-investment grade firms. In particular, for the

latter set of firms, equity market liquidity and idiosyncratic risk are not significant. The only significant

arbitrageur cost is credit market illiquidity. There are two possible hypotheses why only credit market

illiquidity is significant - either costs associated with the credit markets are more significant than other

costs in determining arbitrage activity in below investment grade firms or that our proxies for equity

liquidity and idiosyncratic risk measures are not adequate for below-investment grade firms. We tried

other equity liquidity measures, but did not find them significant. In addition, the intercept term when

all variables are included [specification [11] and [12]) is not significant. It appears, therefore, that the

reason why other arbitrageur costs are not significant may be because credit market illiquidity is the

primary cost of trading in below-investment grade markets.

19

For both investment grade firms and below-investment grade firms, variables that proxy for credit

risk of the firm are, on average, statistically significant. Leverage is significant with a negative sign for

investment grade firms in all regressions, and equity volatility is significant with a negative sign in 10 of

the 12 regressions for below investment grade firms. The negative sign of the coefficients indicate that

riskier firms have more integrated markets. The significant impact of credit risk on market integration

provides additional - albeit indirect - support for our hypothesis. Recall from our prior discussions

that (i) the average size of the pricing discrepancy is larger for below-investment grade firms, and (ii)

for a given magnitude of stock return discrepancy, the equity required to implement a convergence

trade is lower for below-investment grade firm than for above-investment grade firms. That is, for a

given amount of equity capital, there is potentially more money to be made implementing convergence

trades in riskier firms. Arbitrage activity will be driven towards more profitable trades and, therefore,

towards markets of riskier firms.

6.2 Time-Series Determinants of Market Integration

Pricing discrepancies may be correlated across firms. In particular, interest rates may impact economy-

wide pricing discrepancies for two reasons. First, it is known that interest rates are correlated with

credit spreads. For example, long-term interest rates may impact corporate buyout activity and thus

impact market-wide levels of credit spreads. Second, interest rates will impact funding costs of the

leveraged position. The latter is consistent with limits to arbitrage, while the former is an alternative

hypothesis for explaining market-wide pricing discrepancies. We investigate each hypothesis by re-

gressing economy-wide pricing discrepancies against the 10-year Treasury and the 3-month Eurodollar

rate, respectively.

We measure economy-wide discrepancies as the fraction of the N firms in our sample that have a

pricing discrepancy at any time t,

pospropt =1Nt

Nt∑i=1

1[∆CDSτi,t∆P τi,t>0]. (5)

Figure 5 plots the fraction of the total number of firms that have pricing discrepancies at a given point

in time over our sample period. For horizons of over a week, pricing discrepancies show a downward

trend in the first two years of our sample period corresponding to the time period over which the CDS

market matured. From about 2003 onwards, market-wide mispricings do not appear related to the

development of the CDS market. Our objective is to understand what might explain the variation in

market-wide mispricings.

20

First, we regress changes in market-wide discrepancies on changes in the 10-year Treasury rate,

∆pospropt = α+ β∆Treasuryt + et, (6)

where ∆Treasuryt = Treasuryt+h − Treasuryt. Panel A of Table 10 reports the results. Changes in

the 10-year Treasury yield do not explain pricing discrepancies; all the regressions show insignificant

coefficients.

Next, we regress changes in market-wide discrepancies on changes in the Eurodollar rate,

∆pospropt = α+ β∆Eurodollart + et, (7)

where ∆Eurodollart = Eurodollart+h−Eurodollart. Panel B of Table 10 reports the results. The sign of

the coefficient is significant and positive in two of the three regressions. The positive coefficient indicates

that an increase in the funding costs increases the proportion of firms with pricing discrepancies.

Overall, the result provides direct evidence of the importance of funding costs in determining market-

wide pricing discrepancies.

How economically significant is funding cost relative to market liquidity and idiosyncratic risk?

An increase in the Eurodollar rate by 1% increases pricing discrepancies by 0.02%. If we make a

linear approximation of the relation between the frequency of mispricings and κ in Figure 2, then a

1% increase in funding costs increases κ by 0.018. This increase is about 10% of the cross-sectional

standard deviation of κ. Recall that a 1-standard deviation change in spreadzero and idiosyn explains

about 20% of the standard deviation of κ. Thus, the impact of direct funding cost of leverage is

economically smaller than the impact of market liquidity or idiosyncratic risk.12

7 Conclusion

Why are firms’ equity and credit markets not highly correlated? We provide evidence to support

the theory that limits to arbitrage result in mispricings that do not vanish instantaneously. As a

result, pricing discrepancies are common, and vary across firms. Cross-sectional variation in market

integration is related to cross-sectional differences in arbitrageur costs. Liquidity in both equity and12Note that the result does not imply that overall funding illiquidity is of lower economic importance than other

costs. Funding cost relates to both the cost of funding leverage as well as the cost of equity required to implement theconvergence trade. As per our previous discussion, the equity required to implement a convergence trade for a givenequity mispricing increases as the firm gets safer. Consistent with this analysis, we find firms that are less risky haveless integrated markets, suggesting that funding liquidity related to the equity needed to fund a trade is of first-ordereconomic importance.

21

credit markets, especially the latter, impacts market integration. Idiosyncratic risk is also significant.

There is less evidence in support of alternative hypotheses. It is unlikely that mispricings are a result

of unexpected changes in volatility, debt, or long-term Treasury interest rates. It is unlikely that

equity and credit markets are not integrated because of differing pricing kernels or wealth transfers as

the frequency of pricing discrepancies decline over longer time intervals and convergence trades make

significant profits.

Our findings indicate that pricing discrepancies across equity and credit markets are anomalies that,

in time, may be corrected. If so, empirical tests of structural models of credit risk need to be imple-

mented over horizons that are sufficiently long to ensure integration. It is not entirely surprising that

extant empirical studies using weekly or monthly frequencies find it difficult to explain co-movements

in the equity and credit markets.

Our findings also provide some of the strongest empirical support for recent theoretical literature on

limits to arbitrage, especially because the credit default swap market is precisely the market dominated

by sophisticated arbitrageurs like hedge funds. In particular, our findings indicate that the variables

posited in the theoretical literature to impact arbitrage activity are indeed economically significant.

Limited arbitrage activity is not just important in times of crises, but also in times of relative tranquility.

Our findings raise additional questions. First, our results indicate that mispricings are common.

This raises the question of how and why mispricings arise. An useful avenue to pursue here is whether

differences of opinion across market participants (Basak and Croitoru (2000, 2006)) impact the preva-

lence of mispricings. Second, we do not consider the impact of limited arbitrage activity on pricing.

In particular, how should the supply and movement of arbitrage capital across markets impact the

pricing of securities? A fuller evaluation and analysis of the role of hedge funds and other arbitrageurs

remains an issue for ongoing research.

22

Appendix A Design of Simulation

A.1 Simulation of Section 3.2

We simulate stock and bond prices using the Merton (1974) model. Let A(t) and P (t) be the time t

value of the (non-dividend paying) firm’s assets and equity, respectively. Assume that on each time t

the firm has debt of face value of debt F (t) and remaining time to maturity T . Let B(t;F (t), T ) be

the price of the bond. The Merton-model value of the equity P (t) is,

P (t) = A(t) Π1 − e−rτF (t) Π2, (8)

where

Π1 =ln(A(t)/F (t)) + (rf + 1

2V )T√V√T

Π2 = Π1 −√V T , (9)

where rf is the riskfree rate and√V is the asset volatility. The value of the bond is,

B(t) = A(t)− P (t), (10)

and the credit spread is defined as CDS = 1T ln (F (t)/B(t))− rf .

The simulation is implemented as follows. For dates t = (k− 1)τ , k = 1, 2, ...,M , we simulate asset

values by,

Akτ = A(k−1)τ exp(

(rf + µ− 12V )τ +

√V τ z̃1

), (11)

where rf +µ is the expected return on the firm and z̃1 is a standard normal variate. On each date, the

firm re-finances its existing debt of face value F (t− τ) and maturity T − τ with new debt of face value

F (t) and maturity T of identical value, i.e., F (t) is chosen so that B(t;F (t), T ) = B(t;F (t− τ), T − τ).

In the absence of mispricing, stock and bond prices are given by (8) and (10), respectively.

We assume there is a fixed probability of observing mispricing in the equity market. When there

is mispricing, the observed stock price is,

P (t) =

P (t− τ) ∗ 1.03 if A(t) < A(t− τ)

P (t− τ) ∗ 0.97 if A(t) > A(t− τ)

instead of the true stock price given by equation (8). The magnitude of mispricing is chosen to match

23

the average mispricing observed in our Table 3. The bond is correctly priced from equations (8) and

(10), and, therefore, the mispricing results in ∆CDSτi,k∆Pτi,k > 0. κ is estimated using these simulated

stock and bond (spread) prices. We vary the probability of mispricing to estimate κ as a function of

the probability.

The parameters for the simulation are as follows: A(0) = 100, F (0) = 50, T = 5, µ = 0.10,

rf = 0.03,√V = 0.40, τ = 1/52, M = 52. For a given probability of mispricing, κ is estimated from

N = 1000 simulations.

A.2 Simulation of Section 3.3

We extend the simulation described above by allowing asset volatility and interest rates to be stochastic,

Vkτ = V(k−1)τ + κv(θv − V(k−1)τ )τ + σv√V (k−1)τ (ρz̃1 +

√1− ρ2z̃2) (12)

rkτ = r(k−1)τ + κr(θr − r(k−1)τ )τ + σr√r(k−1)τ z̃3, (13)

where z̃1, z̃2 and z̃3 independent standard normal variates. Under the assumption that both asset

volatility and interest rates follow a square-root diffusion, the value of equity is,

P (t) = A(t)P1 − F (t)P2, (14)

where P1 and P2 are provided, for example, in Pan (2003, pp. 35). Given the value of the equity, Bt

is computed from equation (10).

The parameters for the volatility process are taken from Pan (2003): κv = 5.3, σv = 0.38 and

ρ = −0.57. The parameters for the interest rate process are those used in Chapman and Pearson

(2000): κr = 0.21459 and σr = 0.07830. The initial values for both volatility and interest rates are

equated to their long-term means, V (0) = θv = 0.16, and r(0) = θr = 0.03.

The rest of the procedure follows the simulation procedure described in A.1. On each date, the

firm refinances its existing debt of face value F (t − τ) with new debt of face value F (t) of identical

value. Mispricings are generated in the same manner as previously described.

24

Appendix B Capital Structure Arbitrage Hedge Ratio

To determine the relation between changes in CDS spread and stock return, we run a regression of

the change in CDS spread on the stock return. Denote ∆CDSji,t = CDSji,t − CDSji,t−1 and ∆P ji,t =

P ji,t−Pji,t−1 as change in CDS spread and stock prices for firm i and rating j. Then, following Schaefer

and Strebulaev (2008), we first run the following regression for each firm i,

∆CDSji,t = αi + βji∆P ji,tP ji,t−1

+ ei,t, (15)

and then average the coefficient over the N j firms within the rating class to get bj = 1Nj

∑Nj

i βji .

To determine the hedge ratio, we need to determine the value of the credit default swap at a

given spread. We use the market standard model recommended and maintained by ISDA and Markit

(http://cdsmodel.com) to determine the cash settlement value, Sji,t, of the spread. For small changes

in the spread, we approximate the relation between the change in the value of the spread, ∆Sji,t =

Sji,t − Sji,t−1 and ∆CDSji,t by,

∆Sji,t = ∆CDSji,tDji,t, (16)

where we define Dji,t as the duration of the spread. The duration of the spread will depend on the level

of the spread and is determined by the pricing model. For example, when the spread is 50 bps, the

duration is about 4,300, i.e., the value of the CDS changes by $4,300 for a 1 bps change in spread.

The hedge ratio, δji,t(Pji,t) defined as the dollar amount of equity required to hedge 1 CDS contract

is then given by,

δji,t

(P ji,t

)= bjDj

i,t. (17)

25

References

[1] Abreu, D., and Markus Brunnermeier, 2002, “Synchronization risk and delayed arbitrage,” Journal

of Financial Economics 66, 341-360.

[2] Acharya, Viral, Stephen Schaefer and Yili Zhang, 2008, “Liqudity and correlation risk: A clinical

study of the General Motors and Ford downgrade of May 2005, Working paper, London Business

School and CEPR.

[3] Ali, Ashiq, Lee-Seok Hwang, and Mark A. Trombley, 2003, “Arbitrage risk and the book-to-market

anomaly,” Journal of Financial Economics 69, 355-373.

[4] Amihud, Yakov, 2002, “Illiquidity and stock returns: cross-section and time-series effects,” Journal

of Financial Markets 5, 31-56.

[5] Anderson, Ronald W. and Suresh Sundaresan, 1996, “Design and valuation of debt contracts,”

Review of Financial Studies 9(1), 37-68.

[6] Attari, Mukarram, Antonio S. Mello, and Martin Ruckes, 2005, “Arbitraging arbitrageurs,” Jour-

nal of Finance 60, 2471-2511.

[7] Bakshi, Gurdip, Charles Cao, and Zhiwu Chen, 2000, “Do Call Prices and the Underlying Stock

Always Move in the Same Direction?” Review of Finanial Studies 13 (3), pp. 549-584.

[8] Basak, Suleyman and Benjamin Croitoru, 2000, “Equilibrium mispricing in a capital market with

portfolio constraints,” Review of Financial Studies 13, 715-748.

[9] Basak, Suleyman and Benjamin Croitoru, 2006, “On the role of arbitrageurs in rational markets,”

Journal of Financial Economics 81, 143-173.

[10] Black, Fischer and John C. Cox, 1976, “Valuing corporate securities: some effects of bond inden-

ture provisions,” Journal of Finance 31, 351-367.

[11] Blanco, Roberto, Simon Brennan, and Ian Marsh, 2005, “An empirical analysis of the dynamic

relation between investment-grade bonds and credit default swaps,” Journal of Finance 60(5),

2255-2281.

[12] Brunnermeier, M. K. and L. H. Pedersen, 2009, “Market liquidity and funding liquidity,” Review

of Financial Studies, 22(6), 2201-2238.

[13] Buraschi, Andrea and Alexi Jiltsov, 2007, “Model Uncertainty and Option Markets with Het-

erogenous Beliefs,” Journal of Finance 61 (6), 2841-2897.

[14] Buraschi, Andrea, Fabio Trojani, and Andrea Vedolin, 2009, “Economic uncertainty, disagreement

and credit markets,” Working paper, Imperial College.

26

[15] Chapman D., and N. D. Pearson, 2000, “Is the short rate drift actually nonlinear?” Journal of

Finance 55, 355-388.

[16] Chen Long, David A. Lesmond, and Jason Wei, 2007, “Corporate yield spreads and bond liquid-

ity,” Journal of Finance 62(1), 119-149.

[17] Collin-Dufresne, Pierre and Robert S. Goldstein, 2001, “Do credit spreads reflect stationary lever-

age ratios?” Journal of Finance 56, 1929-1957.

[18] Collin-Dufresne, Pierre, Robert S. Goldstein, and J. Spencer Martin, 2001, “The determinants of

credit spread changes,” Journal of Finance 56, 2177-2207.

[19] David, F. N., 1949, “The moments of the z and F distributions,” Biometrika 36, 394-403.

[20] Das, Sanjiv, and Paul Hanouna, 2009, “Hedging credit: equity liquidity matters,” Journal of

Financial Intermediation 18(1), 112-123.

[21] Ericsson, Jan, Kris Jacobs and Rodolfo Ovideo, 2009 “The determinants of credit default swap

premia,” Journal of Financial and Quantitative Analysis 44 (1), 109-132.

[22] Diether, K., C. Malloy, and A. Scherbina, 2002, “Differences of opinion and the cross-section of

stock returns,” Journal of Finance 57, 2113-2141.

[23] Duarte, J., Longstaff, F., and Yu, F., 2007, “Risk and return in fixed income arbitrage: nickels in

front of a steamroller?” Review of Financial Studies, 20(3), 769-811.

[24] Ferreira, Miguel A., and Laux, Paul A., 2007, “Corporate governance, idiosyncratic risk, and

information flow,” Journal of Finance 62(2), 951-989.

[25] Gromb, Denis, and Dimitri Vayanos, 2002, “Equilibrium and welfare in markets with financially

constrained arbitrageurs,” Journal of Financial Economics 66, 361-407.

[26] Kendall, M. G., 1938, “ A new measure of rank correlation,” Biometrika 30 (1/2), 81-93.

[27] Kondor, Peter, 2009, “Risks in dynamic arbitrage: The price effects of convergence trading,”

Journal of Finance 64 (2), 631-655.

[28] Lamont, Owen and Richard Thaler, 2003, “Can the market add and subtract? Mis-pricing in tech

stocks carve-outs,” Journal of Political Economy 111, 227-268.

[29] Leland, Hayne E., 1994, “Corporate debt value, bond convenants, and optimal capital structure,”

Journal of Finance 49, 1213-1252.

[30] Leland, Hayne E. and Klaus Toft, 1996, “Optimal capital structure, endogenous bankruptcy, and

the term structure of credit spreads,” Journal of Finance 51, 987-1019.

27

[31] Lesmond, David A., Joseph P. Ogden and Charles A. Trzcinka, 1999, “A new estimate of trans-

action costs,” Review of Financial Studies 12(5), 1113-1141.

[32] Liu, Jun and F. Longstaff, 2004, “Losing money on arbitrage: Optimal dynamic portfolio choice

in markets with arbitrage opportunities,” Review of Financial Studies 17 (3), 611-641.

[33] Longstaff, F., S. Mithal, and E. Neis, 2005, “Corporate yield spreads: default risk or liquidity?

New evidence from the credit default swap market,” Journal of Finance 60, 2213-2253.

[34] Longstaff, F., and Schwartz, E., 1995, “A simple approach to valuing risky fixed and floating rate

debt,” Journal of Finance 50, 789-819.

[35] Mashruwala, C., S. Rajgopal, and T. Shevlin, 2006, “Why is the accrual anomaly not arbitraged

away?” Journal of Accounting and Economics, 42, 3-33.

[36] Merton, R., 1974, “On the pricing of corporate debt: the risk structure of interest rates,” Journal

of Finance 29, 449-470.

[37] Merton, R., 1987, “A simple model of capital market equilibrium with incomplete information,”

Journal of Finance 42, 483-510.

[38] Mitchell, M., Pulvino, Todd, and Stafford, Erik, 2002, “Limited arbitrage in equity markets,”

Journal of Finance 57(2), 551-584.

[39] Mitchell, M., Lasse Heje Pedersen, and Todd Pulvino, 2007, “Slow moving capital,” The American

Economic Review 97(2), 215-220.

[40] Ofek, E., Matthew Richardson, and Robert F. Whitelaw, 2004, “Limited arbitrage and short sales

restrictions: evidence from the options markets,” Journal of Financial Economics 74, 305-342.

[41] Pontiff, Jeffrey, 1996, “Costly arbitrage: evidence from closed-end funds,” The Quarterly Journal

of Economics 111(4), 1135-1151.

[42] Pontiff, Jeffrey, 2006, “Costly arbitrage and the myth of idiosyncratic risk,” Journal of Accounting

and Economics, 42, 35-52.

[43] Roll, Richard, 1984, “A simple implicit measure of the effective bid-ask spread in an efficient

market,” Journal of Finance 39(4), 1127-1139.

[44] Wurgler, J., and E. V. Zhuravskaya, 2002, “Does arbitrage flatten demand curves for stocks?”

Journal of Business 75, 583-608.

[45] Schaefer, Stephen and Ilya Strebulaev, 2008, “Structural Models of Credit Risk are Useful: Evi-

dence from Hedge Ratios on Corporate Bonds”, Journal of Financial Economics 90(1), 1-19.

28

[46] Shleifer, Andrei and Robert W. Vishny, 1997, “The limits of arbitrage,” Journal of Finance 52

(1), 35-55.

[47] Shleifer, Andrei and Summers, Lawrence H., 1990, “The noise trader approach to finance,” The

Journal of Economic Perspectives, 4(2), 19-33.

[48] Yu, Fan, 2006, “How profitable is capital structure arbitrage?” Financial Analysts Journal, 62(5),

47-62.

[49] Xiong, W., 2001, “Convergence trading with wealth effects: An amplification mechanism in finan-

cial markets,” Journal of Financial Economics 62, 247-292.

[50] Zhang, Benjamin Y., Hao Zhou, and Haibin Zhu, 2009, “Explaining credit default swap spreads

with the equity volatility and jump risks of individual firms,” Review of Financial Studies forth-

coming.

29

Table 1: Simulation-Based Estimates of κ

The table reports the simulation-based estimates of κ as a function of the probability of mispricing. Thesecond column reports the Merton model κ when asset volatility and interest rates are constant. The third andfourth columns report κ when asset volatility (SV) and both volatility and interest rates are stochastic (SVI),respectively.

P(mispricing) Merton SV SVI0.00 -0.95 -0.93 -0.930.10 -0.89 -0.89 -0.880.20 -0.84 -0.84 -0.830.30 -0.79 -0.79 -0.780.40 -0.73 -0.73 -0.720.50 -0.67 -0.66 -0.660.60 -0.60 -0.59 -0.580.70 -0.50 -0.48 -0.480.80 -0.35 -0.35 -0.350.90 -0.12 -0.11 -0.10

30

Table 2: Descriptive statistics

The sample consists of 199 non-financial N. American firms over the period January 2, 2001 to December 31,2005, of 95 firms have an average rating above investment grade, and 104 firms have average ratings belowinvestment grade. Volatility is the annualized standard deviation of the stock return over the sample period.Size is the market capitalization measured in billions of dollars. Leverage is the ratio of book value of debt to thesum of book value of debt and market capitalization. For each obligor, we first compute the time-series meanof its (daily) 5-year CDS spreads, (daily) market capitalization, and (quarterly) leverage, and then compute thestatistics in the cross-section. The equity volatility is computed as the annualized standard deviation of dailyreturns across the five-year sample period.

5-year CDS spread (bps)Mean Median Min Max Std

All 215.20 145.17 19.21 1670.58 226.50Investment Grade 86.57 69.01 19.21 430.98 66.24High Yield 332.69 267.57 44.77 1670.58 255.78

Size (’000,000,000)Mean Median Min Max Std

All 13.20 6.25 0.40 227.11 23.87Investment Grade 22.30 13.10 1.09 227.11 31.78High Yield 4.90 2.87 0.40 24.66 5.27

Equity VolatilityMean Median Min Max Std

All 0.42 0.36 0.19 1.22 0.17Investment Grade 0.33 0.32 0.19 0.72 0.08High Yield 0.50 0.45 0.23 1.22 0.20

LeverageMean Median Min Max Std

All 0.40 0.38 0.04 0.90 0.20Investment Grade 0.29 0.27 0.04 0.79 0.14High Yield 0.50 0.51 0.08 0.90 0.19

31

Table 3: Summary Statistics

Panel A reports summary statistics for equity and credit markets liquidity measures and idiosyncratic risk for199 firms over the period January 2001 to December 2005. depth is the number of dealers who contribute tothe composite CDS quote. The average of daily values is computed as the measure for each firm. spreadzerois the proportion of zero daily CDS spread changes among all the non-missing daily changes in the sampleperiod. sqrtamivest is constructed as 0.001

√price ∗ sharevolume/|return| from daily data. The mean of the

daily measures is computed as the measure for the firm in the sample period. zprop is the proportion of zerodaily returns among all non-missing returns in the sample period. idiosyn is the idiosyncratic risk computed asthe logistic transformation of coefficient of determination from a regression of daily excess returns on the market,ln(

1−R2

R2

).

Panel B of the table reports the cross-sectional correlations between the variables.

Panel A: Descriptive Statistics

Mean Median Min Max Std

depth 9 9 3 18 3

spreadzero (%) 21.97 21.35 0.00 58.66 9.36

sqrtamivest 85.49 75.73 7.38 334.99 54.76

zprop (%) 1.60 1.35 0.24 6.29 1.08

idiosyn 1.46 1.49 0.12 4.42 0.78

Panel B: Correlation Matrix

depth spreadzero sqrtamivest zprop idiosyn lev eqvol

depth 1.00

spreadzero –0.45 1.00

sqrtamivest 0.50 -0.08 1.00

zprop -0.35 -0.004 -0.42 1.00

idiosyn -0.33 0.03 -0.22 0.44 1.00

lev -0.25 -0.17 -0.51 0.44 0.28 1.00

eqvol -0.47 -0.03 -0.39 0.48 0.19 0.37 1.00

32

Table 4: Pricing Discrepancies

The table reports the co-movements between CDS spreads and stock prices, ∆CDSτ∆P τ > 0, ∆CDSτ∆P τ < 0,and ∆CDSτ∆P τ = 0 as a proportion of total observations measured over non-overlapping time intervals τ ∈{5, 10, 25, 50}. |∆CDS| is the mean of absolute spread changes. |∆P/P | is the mean of absolute stock returns.“Obs” is the total number of non-missing pairs of spread and price changes in the sample. Co-movements arepricing discrepancies if ∆CDSτ∆P τ > 0.

∆CDS∆P < 0 ∆CDS∆P > 0 ∆CDS∆P = 0

Sample Interval Obs. Fraction |∆CDS| |∆P/P | Fraction |∆CDS| |∆P/P | Fraction(Days) (%) (bps) (%) (%) (bps) (%) (%)

5 37,643 53.7 14.2 4.0 40.6 10.2 3.1 5.7All 10 18,643 58.0 22.1 5.9 39.6 14.2 4.3 2.4

25 7,392 63.7 40.2 9.8 35.6 21.3 6.3 0.750 3,661 68.5 65.6 14.9 31.3 30.2 8.4 0.25 21,098 52.3 6.1 3.4 41.1 4.5 2.7 6.6

Invt. 10 10,471 56.0 9.8 4.9 41.3 6.6 3.8 2.7Grade 25 4,151 60.8 17.9 7.9 38.6 10.9 5.6 0.6

50 2,059 66.2 29.6 12.0 33.7 15.9 7.4 0.15 16,545 55.5 23.9 4.8 39.9 17.7 3.6 4.6

High 10 8,172 60.6 36.8 7.1 37.4 25.0 4.9 2.0Yield 25 3,241 67.3 65.9 12.0 31.8 37.3 7.5 0.9

50 1,602 71.5 108.5 18.4 28.3 52.1 10.1 0.2

33

Table 5: Discrepancies Conditioned on Changes in Volatility and Debt

Panel A reports the fraction of total pricing discrepancies conditioned on direction of changes in stock price andimplied volatility on a monthly frequency. The implied volatility is the average of the nearest to the moneycall and put for a maturity of 30 days. The total number of pricing discrepancies at the monthly frequency are2722. Panel B reports fraction of total pricing discrepancies between CDS spreads and stock prices conditionedon direction of changes in stock price and book value of debt on a quarterly frequency. The book value of debtis defined as sum of long-term and short-term debt. The total number of pricing discrepancies at the quarterlyfrequency are 903.

Panel A: Volatility

Sample ∆Pi > 0 ∆Pi < 0

Increase in Vol 0.162 0.313

Decrease in Vol 0.280 0.244

Panel B: Debt

Sample ∆Pi > 0 ∆Pi < 0

Increase in Debt 0.197 0.219

Decrease in Debt 0.267 0.317

34

Tab

le6:

Cap

ital

Stru

ctur

eA

rbit

rage

Exc

ess

Ret

urns

For

each

firm

,ba

sed

onth

etr

adin

gru

ledo

cum

ente

din

the

text

,w

eco

mpu

teth

eav

erag

eex

cess

retu

rnpe

rtr

ade

over

2001

-05.

The

tabl

ere

port

sde

scri

ptiv

est

atis

tics

for

the

mea

nex

cess

retu

rnov

erth

ecr

oss-

sect

ion

of19

9fir

ms.

Pan

elA

repo

rts

resu

lts

whe

nth

etr

adin

gru

leis

base

don

pric

ech

ange

sov

erth

epr

evio

usfiv

ebu

sine

ssda

ysan

dth

epo

siti

onis

held

for

am

axim

umpe

riod

of1-

mon

th.

Pan

elB

repo

rts

resu

lts

whe

nth

etr

adin

gru

leis

base

don

pric

ech

ange

sov

erth

epr

evio

us10

busi

ness

days

and

the

posi

tion

ishe

ldop

enfo

ra

max

imum

peri

odof

2-m

onth

s.T

heSh

arpe

rati

ois

com

pute

dfo

ra

hypo

thet

ical

port

folio

that

inve

sts

ineq

ually

inal

lth

efir

ms

inth

esa

mpl

e. Pan

elA

Sam

ple

#F

irm

s#

Tra

des

Exc

ess

Ret

.Sh

arpe

Rat

ioM

edia

nSt

d.D

ev.

Skew

Kur

tosi

sM

in.

Max

.#

ofP

rem

atur

eC

losu

res

All

199

28.2

1.04

%4.

420.

66%

3.11

%1.

183.

35-5

.26%

15.4

2%7.

86

Invt

.G

rade

9530

.90.

73%

2.41

0.38

%2.

97%

1.27

3.81

-5.2

0%14

.05%

7.12

Hig

hY

ield

104

25.8

1.32

%3.

591.

48%

3.59

%1.

072.

97-5

.26%

15.4

2%8.

53

Pan

elB

Sam

ple

#F

irm

s#

Tra

des

Exc

ess

Ret

.Sh

arpe

Rat

ioM

edia

nSt

d.D

ev.

Skew

Kur

tosi

sM

in.

Max

.#

ofP

rem

atur

eC

losu

res

All

199

18.6

1.23

%3.

850.

76%

4.53

%0.

963.

18-1

3.21

%22

.83%

7.87

Invt

.G

rade

9520

.71.

45%

2.41

0.79

%4.

28%

2.04

7.69

-5.6

3%22

.83%

7.53

Hig

hY

ield

104

16.5

91.

03%

2.22

0.42

%4.

75%

0.27

0.35

-13.

21%

13.0

7%8.

18

35

Table 7: Descriptive Statistics for κ

The table reports descriptive statistics for κ across the cross-section of 199 firm for τ ∈ {5, 10}. The κ for eachfirm is the time-series average of the measure computed annually over 2001-05.

Panel A: The Whole Sample

τ Mean Median Max. Min. Std.5 -0.35 -0.33 0.23 -0.76 0.17

10 -0.36 -0.35 0.19 -0.78 0.18

Panel B: Investment Gradeτ Mean Median Max. Min. Std.5 -0.31 -0.30 0.18 -0.58 0.15

10 -0.31 -0.31 0.17 -0.60 0.15

Panel C: Below Investment Gradeτ Mean Median Max. Min. Std.5 -0.39 -0.39 0.23 -0.76 0.18

10 -0.40 -0.40 0.19 -0.78 0.19

36

Tab

le8:

Pan

elR

egre

ssio

nfo

rth

eW

hole

Sam

ple

The

tabl

ere

port

sth

ere

sult

sof

the

pane

lre

gres

sion

,

κ̄i,t

=α

+β

1eq

vol i,t

+β

2ln

mca

p i,t

+β

3le

v i,t

+β

4ra

ting

i,t

+β

5va

riab

lei,t

+ε i,t.

κ̄is

the

tran

sfor

med

Ken

dall

corr

elat

ion,

1 2ln

(1+κ

1−κ

),w

hereκ

isth

eK

enda

llco

rrel

atio

n.T

hese

tofvariable

incl

udes

cred

itm

arke

till

iqui

dity

(spreadzero)

,cr

edit

mar

ket

liqui

dity

(depth

),st

ock

mar

ket

liqud

ity

(sqrtamivest)

,st

ock

mar

ket

illiq

uidi

ty(zprop),

and

idio

sync

rati

cri

sk(idiosyn

).eqvol

isth

ean

nual

ized

equi

tyvo

lati

lity

inth

esa

mpl

epe

riod

.lnmcap

isth

elo

gof

the

mar

ket

capi

taliz

atio

n.T

hele

vera

gelev

isca

lcul

ated

asth

era

tio

ofbo

okva

lue

ofde

btto

the

sum

ofbo

okde

btva

lue

and

mar

ket

capi

taliz

atio

n.rating

isa

dum

my

vari

able

for

inve

stm

ent

grad

efir

ms.

Stan

dard

erro

rsar

eW

hite

stan

dard

erro

rsro

bust

tow

ithi

ncl

uste

rco

rrel

atio

n.T

het-

valu

esar

ere

port

edin

pare

nthe

sis.

***,

**an

d*

indi

cate

sign

ifica

nce

at1%

,5%

and

10%

,re

spec

tive

ly.

37

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

κ̄(τ

=5)

κ̄(τ

=10

)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)

eqvo

l-0

.424

∗∗∗

-0.4

78∗∗∗

-0.1

02-0

.138

-0.1

43-0

.150

-0.1

02-0

.097

-0.0

72-0

.068

-0.4

22∗∗∗

-0.4

71∗∗∗

(-3.

66)

(-3.

43)

(-0.

75)

(-0.

85)

(-1.

22)

(-1.

05)

(-0.

78)

(-0.

61)

(-0.

59)

(-0.

46)

(-3.

61)

(-3.

32)

lnm

cap

0.11

7∗0.

143∗∗

0.08

20.

108

0.13

1∗0.

154∗∗

0.07

00.

091

0.09

90.

120

0.15

7∗∗

0.18

6∗∗

(1.7

5)(1

.97)

(1.1

1)(1

.40)

(1.7

9)(2

.01)

(0.9

6)(1

.17)

(1.3

7)(1

.57)

(2.2

3)(2

.43)

lev

-0.4

43-0

.464

-0.4

72-0

.495

-0.3

94-0

.414

-0.4

74-0

.502

-0.4

49-0

.476

-0.3

88-0

.407

(-1.

44)

(-1.

44)

(-1.

43)

(-1.

45)

(-1.

19)

(-1.

20)

(-1.

44)

(-1.

46)

(-1.

36)

(-1.

39)

(-1.

27)

(-1.

26)

rati

ng0.

019

0.00

70.

053∗

0.04

50.

050

0.04

20.

056∗

0.04

90.

075∗∗

0.06

9∗∗

0.03

90.

030

(0.6

3)(0

.23)

(1.7

2)(1

.35)

(1.6

2)(1

.25)

(1.6

8)(1

.35)

(2.4

0)(2

.02)

(1.3

1)(0

.90)

spre

adze

ro0.

332∗∗∗

0.38

5∗∗∗

0.31

7∗∗∗

0.36

9∗∗∗

(4.7

9)(4

.91)

(4.4

7)(4

.60)

dept

h-0

.001

-0.0

03

(-0.

27)

(-0.

71)

sqrt

amiv

est

-0.8

60∗∗

-0.9

49∗∗

-0.3

46-0

.358

(-2.

28)

(-2.

35)

(-0.

75)

(-0.

77)

zpro

p-0

.950

-0.6

48

(-0.

51)

(-0.

33)

idio

syn

0.04

20.

050∗

0.04

00.

048∗

(1.4

6)(1

.67)

(1.4

6)(1

.71)

Inte

rcep

t-1

.204

∗-1

.423

∗-0

.907

-1.1

01-1

.281

∗-1

.476

∗-0

.791

-0.9

78-1

.131

-1.3

21∗

-1.5

82∗∗

-1.8

42∗∗

(-1.

79)

(-1.

95)

(-1.

23)

(-1.

44)

(-1.

75)

(-1.

92)

(-1.

06)

(-1.

22)

(-1.

52)

(-1.

68)

(-2.

28)

(-2.

43)

N83

282

883

282

883

282

883

282

883

282

883

282

8

R2

34.1

%34

.9%

30.5

%30

.8%

31.0

%31

.2%

30.6

%30

.7%

30.9

%31

.1%

34.4

%35

.3%

38

Tab

le9:

Pan

elR

egre

ssio

nfo

rth

eIn

vest

men

tG

rade

and

Bel

owIn

vest

men

tG

rade

Sam

ple

The

tabl

ere

port

sth

ere

sult

sof

the

regr

essi

onfo

rea

chra

ting

clas

s,

κ̄i,t

=α

+β

1eq

vol i,t

+β

2ln

mca

p i,t

+β

3le

v i,t

+β

4va

riab

lei,t

+ε i,t.

κ̄is

the

tran

sfor

med

Ken

dall

corr

elat

ion,

1 2ln

(1+κ

1−κ

).T

hese

tofvariable

incl

udes

cred

itm

arke

till

iqui

dity

(spreadzero)

,cr

edit

mar

ket

liqui

dity

(depth

),st

ock

mar

ket

liqud

ity

(sqrtamivest)

,st

ock

mar

ket

illiq

uidi

ty(zprop),

and

idio

sync

rati

cri

sk(idiosyn

).eqvol

isth

ean

nual

ized

equi

tyvo

lati

lity

inth

esa

mpl

epe

riod

.lnmcap

isth

elo

gof

the

mar

ket

capi

taliz

atio

n.T

hele

vera

gelev

isca

lcul

ated

asth

era

tio

ofbo

okva

lue

ofde

btto

the

sum

ofbo

okde

btva

lue

and

mar

ket

capi

taliz

atio

n.St

anda

rder

rors

are

Whi

test

anda

rder

rors

robu

stto

wit

hin

clus

ter

corr

elat

ion.

The

stan

dard

erro

rsar

ead

just

edfo

rhe

tero

sked

asti

city

.T

het-

valu

esar

ere

port

edin

pare

nthe

sis.

***,

**an

d*

indi

cate

sign

ifica

nce

at1%

,5%

and

10%

,re

spec

tive

ly.

39

Inve

stm

ent

Gra

de (1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

κ̄(τ

=5)

κ̄(τ

=10

)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)

eqvo

l-0

.672

∗∗∗

-0.6

70∗∗∗

0.13

80.

185

-0.1

11-0

.003

0.10

40.

219

0.12

20.

248

-0.6

63∗∗∗

-0.6

28∗∗∗

(-3.

26)

(-3.

11)

(0.5

5)(0

.69)

(-0.

54)

(-0.

01)

(0.5

1)(1

.07

)(0

.58)

(1.1

5)(-

2.95

)(-

2.74

)

lnm

cap

0.06

70.

104

-0.0

180.

018

0.07

60.

102

0.00

20.

032

0.02

70.

062

0.14

50.

184∗

(0.6

8)(0

.98)

(-0.

16)

(0.1

6)(0

.71)

(0.9

3)(0

.02)

(0.3

1)(0

.26)

(0.5

7)(1

.45)

(1.6

9)

lev

-1.1

10∗∗∗

-1.1

81∗∗∗

-1.3

10∗∗∗

-1.4

06∗∗∗

-1.2

10∗∗∗

-1.3

08∗∗∗

-1.3

28∗∗∗

-1.4

29∗∗∗

-1.2

58∗∗∗

-1.3

49∗∗∗

-1.0

26∗∗∗

-1.0

99∗∗∗

(-3.

22)

(-3.

34)

(-3.

28)

(-3.

57)

(-3.

12)

(-3.

40)

(-3.

26)

(-3.

50)

(-3.

26)

(-3.

46)

(-3.

08)

(-3.

23)

spre

adze

ro0.

387∗∗∗

0.44

4∗∗∗

0.32

8∗∗∗

0.37

9∗∗∗

(4.3

7)(4

.39)

(3.5

5)(3

.53)

dept

h0.

002

0.00

02

(0.3

5)(0

.04)

sqrt

amiv

est

-1.2

45∗∗∗

-1.2

23∗∗∗

-0.9

76∗∗

-0.9

13∗∗

(-2.

88)

(-2.

96)

(-2.

32)

(-2.

25)

zpro

p2.

523

3.08

1

(0.7

1)(0

.93)

idio

syn

0.09

2∗∗

0.11

0∗∗∗

0.06

5∗0.

080∗∗

(2.6

1)(3

.16)

(1.8

3)(2

.22)

Inte

rcep

t-0

.551

-0.9

070.

133

-0.1

83-0

.534

-0.7

95-0

.044

-0.3

48-0

.360

-0.7

30-1

.254

-1.6

47

(-0.

56)

(-0.

84)

(0.1

2)(-

0.16

)(-

0.50

)(-

0.72

)(-

0.04

)(-

0.32

)(-

0.35

)(-

0.65

)(-

1.25

)(-

1.49

)

N45

144

945

144

945

144

945

144

945

144

945

144

9

R2

31.0

%32

.3%

26.5

%27

.1%

27.7

%28

.1%

26.6

%27

.3%

28.2

%29

.3%

32.5

%33

.9%

40

Bel

owIn

vest

men

tG

rade

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

κ̄(τ

=5)

κ̄(τ

=10

)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)κ̄

(τ=

5)κ̄

(τ=

10)

eqvo

l-0

.403

∗∗∗

-0.4

73∗∗∗

-0.2

79∗

-0.3

46∗

-0.2

37-0

.295

-0.2

87∗

-0.3

39∗

-0.2

41∗

-0.2

95∗

-0.3

90∗∗

-0.4

62∗∗

(-2.

70)

(-2.

61)

(-1.

79)

(-1.

84)

(-1.

61)

(-1.

63)

(-1.

91)

(-1.

83)

(-1.

67)

(-1.

68)

(-2.

55)

(-2.

49)

lnm

cap

0.18

8∗0.

214∗

0.23

8∗∗

0.26

6∗∗

0.23

4∗0.

271∗∗

0.20

3∗0.

235∗

0.22

2∗0.

251∗∗

0.13

90.

173

(1.7

5)(1

.90)

(2.1

5)(2

.29)

(1.8

8)(2

.00)

(1.6

9)(1

.83)

(1.9

4)(2

.10)

(1.1

2)(1

.33)

lev

0.15

40.

197

0.43

90.

493

0.47

20.

547

0.44

10.

513

0.45

00.

519

0.08

60.

141

(0.3

0)(0

.36)

(0.8

3)(0

.88)

(0.8

7)(0

.94)

(0.8

1)(0

.89)

(0.8

3)(0

.90)

(0.1

6)(0

.26)

spre

adze

ro0.

329∗∗

0.36

2∗∗

0.34

8∗∗∗

0.37

8∗∗∗

(2.5

9)(2

.57)

(2.7

8)(2

.75)

dept

h-0

.005

-0.0

06

(-0.

72)

(-0.

90)

sqrt

amiv

est

0.11

2-0

.103

0.95

00.

801

(0.1

3)(-

0.10

)(1

.04)

(0.9

4)

zpro

p-2

.546

-2.4

48

(-1.

10)

(-0.

97)

idio

syn

-0.0

23-0

.025

-0.0

12-0

.012

(-0.

53)

(-0.

53)

(-0.

28)

(-0.

26)

Inte

rcep

t-1

.978

∗-2

.201

∗-2

.454

∗∗-2

.685

∗∗-2

.496

∗∗-2

.814

∗∗-2

.152

∗-2

.439

∗-2

.351

∗∗-2

.611

∗∗-1

.599

-1.8

80

(-1.

85)

(1.9

6)(-

2.24

)(-

2.32

)(-

2.09

)(-

2.18

)(-

1.78

)(-

1.88

)(-

2.03

)(-

2.15

)(-

1.33

)(-

1.49

)

N38

137

938

137

938

137

938

137

938

137

938

137

9

R2

36.4

%36

.5%

33.8

%34

.0%

33.6

%33

.7%

34.0

%34

.0%

33.7

%33

.8%

36.6

%36

.7%

41

Table 10: Time Series Regression

The table reports the time series regression of changes in market-wide pricing discrepancies, ∆pospropt, defined interms of the fraction of firms that have a pricing discrepancy at any time t, pospropt = 1

N

∑Ni=1 1[∆CDSτi,t∆P τi,t>0].

Panels A and B report the results of the regression of ∆posprop on changes in the 10-year Treasury yield, andchanges in the three-month Eurodollar rate, respectively. ** and * indicates significance at 5% and 10%,respectively.

Panel A: Treasury Yield

Interval # of Obs. Intercept ∆Yield Adj. R2

5 250 0.0004 -0.0017 0%(0.69) (-0.43)

10 124 -0.0011 0.0004 0%(-1.27) (0.08)

25 49 -0.0037 0.0087 2%(-1.85)* (1.35)

Panel B: Eurodollar Rate

Interval # of Obs. Intercept ∆Eurodollar Adj. R2

5 250 0.0003 -0.0054 0%(0.66) (-0.81)

10 124 -0.0009 0.0222 9%(-1.06) (3.58)**

25 49 -0.0034 0.0203 10%(-1.81)* (2.53)**

42

0 0.2 0.4 0.6 0.8 1 1.2 1.40

200

400

600

800

1000

1200

1400

1600

1800

5−Y

ear

CD

S S

prea

ds

Equity Volatility

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

200

400

600

800

1000

1200

1400

1600

1800

5−Y

ear

CD

S S

prea

ds

Leverage

Figure 1: CDS spread vs. volatility and leverage

43

0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.8

0.6

0.4

0.2

0.0

Kendall Correlation

Probability of Mispricing

Kend

all C

orre

latio

n

UnbiasedBiased

Figure 2:κi vs. Probability of Mispricing

44

0 5 10 15 20 25 30 35 40 45 5015

20

25

30

35

40

45

50

55

60

65Alcoa

stock price

CD

S S

prea

d (b

ps)

0 5 10 15 20 250

50

100

150

200

250

300

350

400

450

500Hilton

stock price

CD

S S

prea

d (b

ps)

0 10 20 30 40 50 60 700

200

400

600

800

1000

1200

1400GM

stock price

CD

S S

prea

d (b

ps)

Figure 3: CDS Spread vs. Stock Price

45

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Cum

ulat

ive

prof

it (m

illio

n do

llars

)

5 business days, 1 month10 business days, 2 months

2001 2002 2003 2004 2005 2006

All 199 firms

Figure 4: Cumulative Profits from Capital Structure Arbitrage. Cumulative profits are computed by com-pounding the mean daily return over all open positions.

46

a

40

50

60

% Firms with Discrepancies

0

10

20

30

2001 2002 2003 2004 2005Year

5 days 10 days

40

50

60

70

% Firms with Discrepancies

0

10

20

30

2001 2002 2003 2004 2005Year

25 days 50 days

Figure 5: Percentage of Firms with Discrepancies Over Time

47