Limited Arbitrage between Equity and Credit Markets

Transcription

1 Limited Arbitrage between Equity and Credit Markets Nikunj Kapadia and Xiaoling Pu 1 First Version: March 2009 Current Version: April University of Massachusetts and Kent State, respectively. We thank Viral Acharya, Kobi Boudoukh, Sanjiv Das, 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 Paul Woolley Conference at the London School of Economics, China International Conference in Finance, Fordham University and the Indian School of Business. Xuan Che provided excellent research assistance. We thank Markit for providing us the CDS data. The contents are the sole responsibility of the authors. Please address correspondence to Nikunj Kapadia Isenberg School of Management, University of Massachusetts, Amherst, MA nkapadia@som.umass.edu.

2 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 timeseries 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

3 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 R 2 s 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 R 2 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 on the firm s average debt ratio and stock return volatility gives an adjusted R 2 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 bond 1 Merton 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 the underlying firm value or leverage. See Black and Cox (1976), Leland (1994), Leland and Toft (1996), Longstaff and Schwartz (1995), Anderson and Sundaresan (1996), Collin-Dufresne and Goldstein (2001). 1

4 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 discrepancies, 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 anomalies, 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, 2 Limits 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, Pulvino and Stafford (2002), Lamont and Thaler (2003)). See also related work by Ali, Hwang and Trombley (2003) and Mitchell, Pedersen and Pulvino (2007). 2

5 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 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

6 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

7 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 n P and n B 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, n B B, for a given stock position, n P 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 m P and m B 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 = m P n P P + m B n B B ( ) P/ A B = n P P m P + m B. (1) B/ A P For a given stock position, n P P, the total margin required to fund the convergence trade is determined by (i) the margin requirement for the stock, m P, (ii) the margin requirement for the bond or CDS, m B, 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 m P 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 % % % % > % The margin requirement for below investment grade CDS is about three to six times that for investment grade CDS. 3 See 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

8 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 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 m P or m B. Because the elasticity of the bond is related to the firm s credit risk, the equity required to fund trades crosssectionally 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. 4 The quasi-leverage is defined as the ratio of the face value of debt discounted at the riskfree rate to asset value. 6

9 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 idiosyncratic 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

10 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 P i and CDS i 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 CDS τ i,k = CDS i(k + τ) CDS i (k) and P τ i,k = P i(k + τ) P i (k), k M τ, 1 τ M 1. Then, a pair of stock prices and CDS spreads present an arbitrage if CDSi,k τ P i,k τ > 0. We define market integration for a stock i based on the frequency of observations of such arbitrage opportunities. Specifically, define ˆκ i as, ˆκ i = M 1 τ=1 M τ k=1 1 [ CDS τ i,k Pi,k τ >0]. (2) For a given M, firm i s stock and CDS markets are more integrated than those of firm j if ˆκ i < ˆκ j. 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 ˆκ and the Kendall correlation 5 Although 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 CDS spreads, then the measure will be less sensitive to mispricing. As we shall see in the next section, the measure is sensitive to levels of mispricing when the time period is a year, and thus suitable for our study which is motivated by short-horizon mispricings. 8

11 (Kendall (1938)). Defining κ as the Kendall correlation, κ = 4ˆκ 1. (3) M(M 1) 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 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 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, 6 In 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 equity prices. Even in the absence of mispricings, the coefficient of determination will be less than 1. Moreover, as the magnitude of the bias varies with leverage, it is difficult to use the R 2 to make cross-sectional comparisons across firms. 9

12 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 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, V t, 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 volatility and stochastic interest rates. However, this bias is very small. The maximum bias that is observed is less than For moderate levels of mispricing, the bias is smaller than 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 = MISPRICING + e Merton, κ = MISPRICING + e SV, κ = 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

13 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, 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 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 information 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 of 7 The 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 are defined 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 rating in the BB/Ba category by one or both S&P and Moody s. 8 Markit provides information on both the average agency rating and an implied rating. We use the agency rating averaged over our sample period when available. When the agency rating is unavailable, we use the implied rating. 11

14 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 R 2 of 61%, CDS i = EQVOL i LEV i + e, R 2 =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 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

15 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 Ri 2 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 σ 2 i,ɛ σ 2 i = σ2 i σ2 im σm 2 σi 2 = 1 R 2 i. ( ) 1 R 2 The idiosyncratic measure idiosyn is then defined as the logistic transformation ln i. The Ri 2 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 crosssectional 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 CDS τ i,k P τ i,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 CDSi,k τ P i,k τ = 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

16 Pricing discrepancies are common. Over the entire sample, at a 5-business day frequency, stock prices and spreads co-move as expected ( CDSi,k τ P i,k τ < 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. represent arbitrage opportunities by the Merton model. Even at a 50-business day horizon, 31% of co-movements 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 associated 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

17 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 our 9 Cremers, Driessen, Maenhout and Weinbaum (2008) document that increases in equity implied volatility increases credit spreads. 15

18 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 CDSi,t τ P i,t τ > 0 and if P τ i,t /P i,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). W = m P n P P + m CDS n CDS S, where n P Each position is initiated with a total equity of W = $1, 000, 000 where and n CDS are the number of shares and CDS contracts that form the two legs of the (hedged) convergence trade. The margins for equity and CDS, m P and m CDS, 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

19 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 Observe that the hedge ratio used in the convergence trade is also determined by relative price movements in equity and CDS markets by running a regression of CDS spread changes on stock returns. Thus, the implementation of the strategy requires the trader to do nothing more than be an astute observer of the markets. 11 According to the Journal, the group undertook trades across equity and CDS markets, as well as across corporate bonds and CDS markets. Arbitrage trades across corporate bonds and CDS markets are typically undertaken when there is a negative basis ; when the cost of protection is less than the income produced by the bond. A significant proportion of the 2008 loss at Deutsche Bank appear to be related to the widening of the basis coinciding with lower funding liquidity. 17

20 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 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 κ = 1 2 (1+κ) ln (1 κ). The panel specification is, κ i,t = α + β 1 eqvol i,t + β 2 lnmcap i,t + β 3 lev i,t + β 4 variable i,t + ɛ i,t, (4) where variable i,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 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 κ 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