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Limited Arbitrage between Equity and Credit Markets Nikunj Kapadia and Xiaoling Pu1 1 Associate Professor and Doctoral Student, respectively, University of Massachusetts, Amherst. This document is very preliminary, and all comments are welcome. Please address correspondence to Nikunj Kapadia Isenberg School of Management, University of Massachusetts, Amherst, MA 01003. Email: nkapadia@som.umass.edu. Abstract Why do equity and credit markets not behave as if they are integrated? We examine whether limits to arbitrage help explain why equity and credit markets are not highly correlated. We ﬁnd that the cross-sectional variation in the level of integration between the equity and the credit default swap market is related to a range of proxies for informational sensitivity, liquidity, and idiosyncratic risk. Equity and credit markets are more integrated when a ﬁrm’s securities are more informationally sensitive, are more liquid and have lower idiosyncratic risk. 1 Introduction The primary insight of Merton’s (1974) structural model of credit risk is that stocks and bonds are contingent claims on the underlying ﬁrm, and, therefore, stock returns and changes in credit spreads must be precisely related to ensure the absence of arbitrage. It is thus not surprising that hedge funds and private equity ﬁrms are active in a variety of trading strategies - popularly known as capital structure arbitrage - that attempt to “arbi- trage” across equity and credit markets. For instance, an article in the Wall Street Journal1 comments on private equity deals: Compare junk-bond yields to the earnings yields on stocks, and it seems like stocks are incredibly cheap. “Look at the valuations in the two markets and they’re about as far apart as they’ve ever been,” says M.S. Howells strategist Brian Reynolds. That creates a great arbitrage situation for deal makers, who get to issue expensive-looking bonds to buy cheap-looking stock. As long as that dynamic persists, the deals will continue and stocks will have at least one reason to rally. Given the theoretical link between equity and credit risk and active arbitrage activ- ity, one would expect the equity and credit markets to be closely linked. Instead, recent empirical research ﬁnds stock returns and changes in credit spreads to be weakly corre- lated. In a regression of monthly changes in credit spreads on the stock returns and other variables consistent with the structural framework, Collin-Dufresne, Goldstein and Martin (2001) ﬁnd adjusted R2 s of the order of 17% to 34%, leading them to 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, and ﬁnd that three-quarters of the variation remains unexplained. This low correlation is especially surprising because, on average, the Merton (1974) model does an excellent job of ﬁtting the cross-sectional dispersion of medium horizon credit spreads. In our dataset, a cross-sectional regression of the average ﬁve-year credit default swap spread on the ﬁrm’s average debt ratio and stock return volatility gives an adjusted R2 of 61%. How then does one explain the low correlations between changes in credit spreads and stock returns? Why 1 Justin Lahart, Wall Street Journal, November 21, 2006. 1 does arbitrage activity not create an integrated stock and bond market? In this paper, we examine whether limits to arbitrage can explain the extent to which the equity and credit markets are integrated. Our focus is on investigating the factors that might impact the amount of capital allocated by arbitrageurs to a relative value or convergence trades across the equity and credit markets. Our motivation to test for limits to arbitrage is two-fold. First, limits to arbitrage have emerged as an important paradigm for explaining market anomalies involving violations of the law of one price, and, therefore, it is a natural hypothesis to investigate. Limits to arbitrage have been invoked for a wide range of anomalies such as 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)).2 The existing literature has not, however, used this paradigm to examine the degree of integration of the corporate equity and credit markets, despite the size and importance of these two markets. Second, evidence on the type of limits impacting the integration of equity and credit markets would provide a direction for the development of next-generation structural models of credit risk. Existing attempts at making the Merton (1974) model more realistic have largely focused on speciﬁcations for the default boundary, recovery, and the stochastic process determining the underlying ﬁrm value or leverage.3 Limits to arbitrage, as these relate to more fundamental assumptions of frictionless markets and full information, potentially pose a more serious challenge that makes it necessary to understand the speciﬁc nature of the impediments. As in the literature (e.g. Shleifer and Summers (1990)), we view convergence trades across the equity and credit markets as risk arbitrage trades rather than the zero-capital, riskless arbitrage modeled in structural models of credit risk. The degree of integration of the two markets will then depend on the arbitrage capital that is allocated to such trades. The magnitude of perceived possible proﬁts - the “alpha” of the trade” - will increase the arbitrage capital, and the impediments to arbitrage such as costs or risks associated with implementing the covergence trade will reduce the amount of capital. In short, the degree of integration will depend on perceived potential proﬁts as well as the magnitude of the impediments to arbitrage. The fundamental hypothesis we test is as follows: If limited arbitrage activity impacts the integration of the equity and credit markets, then the co-movement between stock prices 2 See also related work by Ali, Hwong and Trombley (2003) and Mitchell, Pederson and Pulvino (2006). 3 See Black and Cox (1976), Leland (1994), Leland and Toft (1996), Longstaﬀ and Schwartz (1995), Anderson and Sunderesan (1996), Collin-Dufresne and Goldstein (2001). 2 and credit spreads will vary in the cross-section of ﬁrms with variation in factors determining arbitrage activity. It is reasonable to expect cross-sectional variation as both perceived potential proﬁt opportunities as well as impediments to arbitrage should have ﬁrm-speciﬁc components. We examine our hypothesis by considering co-movements between ﬁrms’ stock prices and spreads on the ﬁrms’ credit default swap. (It is much easier to arbitrage using credit default swaps, and we expect that arbitrageurs like hedge funds use these as opposed to the underlying bonds that are diﬃcult to short.) In a market that is perfectly integrated, we expect a positive (negative) stock return to be associated with a decrease (increase) in the spread. We relate cross-sectional variations in this expected co-movement to the determinants of arbitrage activity, i.e., the potential for proﬁts and the impediments to arbitrage. Shleifer and Vishny (1997) note that arbitrageurs rely on their specialized, presumably costly, knowledge when deciding to undertake convergence trades. Although it is diﬃcult to directly observe the arbitrageur’s private information, we may indirectly proxy for it by the informational sensitivity of the security. The more informationally sensitive the security, the more likely it is that the arbitrageur will be able to trade on his information. To illustrate, consider a capital structure arbitrageur who specializes in analyzing credit risk, and enters into a convergence trade whenever relative bond and stock prices diverge because of, say, noise trader activity in the equity markets. He will be more actively involved in convergence trade for riskier debt than for less risky debt, as the latter, being less informationally sensitive, requires a larger amount of noise trading and stock price movement to have the same impact on the bond price. At an extreme, when the debt is riskless, there is no private information that can make an arbitrageur enter into a convergence trade. In summary, we expect that the more informationally sensitive the ﬁrm’s debt or equity, the more integrated will be the two markets. There is now an extensive literature on the costs and risks that impede a convergence trade. Costs include commissions and bid-ask spreads. More generally, we expect liquidity of the underlying securities to impact arbitrage activity. As the convergence trade requires trading both the CDS and the stock, both the liquidity of the equity and credit markets might be relevant. In addition to liquidity risks, an arbitrageur’s trading position is subject to fundamental idiosyncratic risk when he cannot form a perfect hedge. For example, an arbitrageur who is betting on a decline in stock prices or credit spreads cannot hedge against the ﬁrm-speciﬁc risk that the ﬁrm might undertake a corporate action that, in fact, does 3 the opposite as, for example, if the ﬁrm enters into a leveraged buyout transaction. In addition to liquidity and fundamental idiosyncratic risk, arbitrageurs may, in the short- run, limit the amount of capital that can be allocated to a convergence trade because of institutional constraints on the availability of risk capital (Merton (1987)), noise trader risk in conjunction with mis-match in the arbitrageur’s investment horizon and the horizon over which convergence is expected to occur (DeLong, Shleifer, Summers and Waldmann (1990), Shleifer and Vishny (1997)), as well as uncertainty regarding the timing of trades by other arbitrageurs (Abreu and Brunnermeir (2002)). Although this last set of risks has limited cross-sectional implications, they do indicate that, for all stocks, the markets will have slow- moving arbitrage capital (Mitchell, Pederson and Pulvino (2006)), and therefore be more integrated over longer horizons. Shleifer and Summers (1990) and Pontiﬀ (2006) provide an overview and discussion of the literature. In summary, given the above impediments to arbitrage, we expect the market of a ﬁrms’ equity and credit securities to be less integrated with higher liquidity or idiosyncratic risk. We test for these implications across a cross-section of 200 ﬁrms over 2001-05. We begin by verifying Collin-Dufresne, Goldstein and Martin’s (2001) conclusion - stocks and credit default swaps do not behave as if the two markets are integrated. At a weekly frequency, on average in our sample, stock prices and CDS spreads co-move as predicted 53.8% of the times, i.e., co-movements in the two markets are economically not much diﬀerent from being random. Over longer horizons, stocks and spreads co-move more in line with theory, consistent with the notion that arbitrage capital is slow-moving (Mitchell, Pederson and Pulvino (2006)). But even at a bi-monthly frequency, stocks and CDS spreads co-move as expected only about 70% of the times. The averages mask considerable variation in the cross-section. For example, General Motors and Ford co-move as expected about two- thirds of the times on a weekly frequency. Although the integration between the two markets increases with horizon, cross-sectional diﬀerences persist. In our main set of tests, we relate cross-sectional variation in the integration of the two markets to our three sets of implications. We ﬁnd extensive support for the hypothesis that the integration between the two markets is impacted by factors impacting arbitrage activity. First, we ﬁnd that the informational sensitivity of the underlying debt or equity is a signiﬁcant determinant of the level of integration. We measure the informational sensitivity of the credit default swap by its riskiness as proxied by equity volatility, debt level, and 4 the rating (whether or not it is above or below investment grade). We also control for the size of the ﬁrm. Both equity volatility and rating are consistently signiﬁcant with signs that indicate that the ﬁrm with higher credit risk has more integrated equity and credit markets. We proxy the informational sensitivity of equity by the dispersion of analysts’ earnings forecast. We ﬁnd that the greater the dispersion, the greater is the integration of the ﬁrms’ equity and credit market. Thus, both the informational sensitivity of the stock and the credit default swap impacts the correlation between the two markets. Second, we test whether lower liquidity in either credit or equity markets makes it more diﬃcult to arbitrage. We ﬁnd signiﬁcant support for this implication, ﬁnding that the liquidity of the credit market is signiﬁcant in linking the two markets. Equity market liquidity has almost no eﬀect, suggesting that the liquidity of the credit market imposes the binding constraint. Third, we test whether the the two markets are less integrated with greater idiosyncratic risk. Controlling for the total risk, we ﬁnd that a decrease in the idiosyncratic risk of the ﬁrm makes the two markets more integrated. In summary, limits to arbitrage have an impact on the integration of the two markets. We proceed as follows. Section 2 reports the descriptive statistics of our sample and the construction of impediment measures. Section 3 examines the relative movement of CDS spreads and stock prices over diﬀerent intervals. Section 4 empirically tests whether impediments to arbitrage have signiﬁcant impact on the correlation between credit spread changes and stock returns. Conclusions are in Section 5. 2 Data and Measures 2.1 Descriptive Statistics Our dataset consists of credit default swap spreads, equity prices, and relevant accounting information for U.S. non-ﬁnancial ﬁrms over the period January 2, 2001 and December 31, 2005. We obtain daily price data for the ﬁve-year credit default swap (CDS) 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 5 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.4 We speciﬁcally choose ﬁrms that form part of the index to ensure continuity in price quotes. We match the data from Markit to CRSP and Compustat manually to construct an initial sample of 224 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 200 ﬁrms. Of the 200 obligors in our dataset, 95 obligors have an average rating of investment grade (AAA, AA, A, and BBB), and 105 obligors are below investment grade (BB, B, and CCC).5 We obtain daily equity prices, returns, outstanding number of shares, and other equity information from the Center for Research in Security Prices (CRSP). We use the cumulative factor to adjust prices and outstanding number of shares for split events.6 The 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 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 1 reports the summary statistics of the 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 ﬁrst panel presents the descriptive statistics for the 5-year CDS spreads. The mean spread across the entire sample is 215 basis points (bps). The mean across investment grade ﬁrms is 86 bps while that of the high yield is much larger at 331 bps. The second panel presents the statistics for the ﬁrm size, 4 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. 5 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. 6 Split events usually include stock splits, stock dividends, and other distributions with price factors. Outstanding number of shares is only adjusted for stock splits and stock dividends. ‘cfacpr’ and ‘cfacshr’ are adjustment factors for prices and outstanding number of shares in the CRSP. 6 measured in billions of dollars. The average size of investment grade ﬁrms in our sample is $22.3 billion versus $4.9 billion for high yield ﬁrms. The third panel reports the statistics for equity volatility. Across the entire sample, the average equity volatility is 42%. The average for the equity volatility for investment grade ﬁrms is 33%, while the corresponding statistic for high yield ﬁrms is 50%. The last panel reports descriptive statistics for the leverage. As expected, high yield ﬁrms have much higher leverage than investment grade ﬁrms. The overall mean leverage across all ﬁrms in our sample is 0.29. In Figure 1, for each ﬁrm we plot the mean CDS spread over the sample period against the ﬁrm’s average leverage and equity volatility. Consistent with the basic Merton (1974) model, the spread is signiﬁcantly correlated with the volatility and the leverage. In fact, a linear regression of the mean CDS spread on these variables gives an adjusted R2 of 61%, CDSi = -0.0263 + 0.0758 VOLi + 0.0399 LEVi + e, R2 =61%. [-9.1] [12.2] [7.3] 2.2 Analysts’ Forecast Dispersion Analysts’ earnings forecasts are obtained from the Institutional Brokers Estimate System (I/B/E/S) detail ﬁle. As noted by Diether, Malloy, and Scherbina (2002), I/B/E/S uses a split adjustment factor to adjust historical analysts’ forecasts and then rounds the estimate to the nearest cent. We unadjust the forecasts using the adjustment factor provided by I/B/E/S. The unadjusted analysts forecasts are used to construct the forecast dispersion. For each analyst, the most recent 1-year forecast closest to the end of the ﬁrst quarter (March 31st) is used. The dispersion is then deﬁned as the standard deviation across the earnings forecasts scaled by the year-end stock price. If the stock has a price less than ﬁve dollars, then the observation is excluded from the sample. For each ﬁrm, the average of the yearly forecast dispersion in the sample period is used in the regressions. 2.3 Liquidity Measures We construct liquidity measures for both the equity and credit markets. We use daily stock price data from CRSP to construct stock market liquidity measures. Our primary measures are (i) the square root of the Amivest measure (S.Amivest), and (ii) the proportion of zero stock returns, Zprop. To construct the S.Amivest measure, we ﬁrst compute 0.001 ∗ 7 1800 1600 1400 1200 5−Year CDS Spreads 1000 800 600 400 200 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Equity Volatility 1800 1600 1400 1200 5−Year CDS Spreads 1000 800 600 400 200 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Leverage Figure 1: CDS spread vs. volatility and leverage 8 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 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) ﬁnds that the zero return proportion is strongly correlated with transaction costs. The larger the zero proportion measure, the lower the liquidity. We introduce two new credit maket 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 liquidity of the credit default swap. Thus, we deﬁne “market depth” 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, (Zspread), 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 proportion of zero credit spread changes indicates lower liquidity. 2.4 Idiosyncratic Risk We construct our measure of idiosyncratic risk from the the standard market model. We ﬁrst regress the stock’s excess returns, ri,t = αi + βi rm,t + i,t (1) using daily data over our sample period of 2001 to 2005. Next, following Ferreira and Laux (2007), we compute the ratio of the idiosyncratic volatility to the total volatility for each stock i as 2 σim 2 2 σi − σi, 2 σm 2 = 2 σi σi 2 = 1 − Ri . (2) 9 The idiosyncratic measure Idiosync is then deﬁned as the logistic transformation, 2 2 σi, 1 − Ri ln 2 = ln 2 2 . (3) Ri σi − σi, We also deﬁned a second measure based on the Fama-French three-factor model, but do not report the results as they were identical to those from the market model. Table 2 presents the descriptive statistics of the liquidity and idiosyncratic risk mea- sures. On average, the composite quote of the 5-year CDS spread is constructed from 9 contributors. 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), which reﬂects the fact that the equity market is much more liquid than the CDS market. 3 Co-movements of CDS Spreads and Stock Prices Collin-Dufresne, Goldstein and Martin (2001) and Blanco, Brennan, and Marsh (2005) document that the correlation between stock returns and changes in spreads are low, in- consistent with structural models. We provide corroborative evidence using our data set. Although structural models provide a precise relation between changes in underlying ﬁrm value and corresponding changes in bond and stock prices, their application requires mak- ing assumptions about the underlying model and its parameters. Instead, we make use of a common implication of structural models (e.g. Merton (1974)) that when the stock and bond are viewed as contingent claims on the underlying ﬁrm, the “delta” of both are posi- tive. Unless there are wealth transfers between the stock and bondholders (as, for example, when dividend or investment policies change) an increase (decrease) in ﬁrm value increases (decreases) both the stock and bond prices. That is, over any given interval, stock prices and CDS spreads should move in opposite directions. Table 3 reports the number of times stock prices and CDS spreads move in the same and opposite directions over diﬀerent time horizons, ranging from 5 business days to 50 business days. In addition, we also report the average absolute change in CDS spread and stock return for each direction of movement. From the table, it is evident that stock prices and CDS spreads do not always move in opposite directions. At a weekly frequency, stock prices and CDS spreads co-move as predicted only 53.8% of the times. As the time-horizon increases, the co-movement of stocks and CDS spreads is closer to theory, but even at a frequency of 50 business days, the co- 10 movement is as predicted only two-thirds of the times. On average, co-movements for ﬁrms with below investment grade rating is closer to theory than for investment grade ﬁrms, but not by much. At a weekly (monthly) frequency of 5 (25) business day, investment grade ﬁrms co-move as predicted 52.3% (60.7%) of the times in comparison with 55.6% (67.2%) for below investment grade ﬁrms. Overall, the evidence indicates that bond and stock prices do not co-move as expected over all intervals and sub-samples. Do these observed co-movements support a limits to arbitrage hypothesis? Even when arbitrage activity is constrained by risk or costs, the constraints are less binding as the horizon increases. This is because a number of factors such as information costs (Merton (1987)), liquidity (Mitchell, Pedersen, and Pulvino (2006)), and institutional features that constrain a rapid increase in the supply of arbitrage capital are less binding with horizon. All else equal, we should expect to see co-movements more in line with theory at long horizons as arbitrageurs become more eﬀective in correcting mis-pricings. Second, for any given level of risk or transaction cost, we should observe more arbitrage activities associated with larger amounts of mis-pricing. If the arbitrageurs are eﬀective in correcting the mis- pricing, then we should observe that arbitrage activity is positively related to the magnitude of the changes in CDS spreads or stock prices. Both these implications are jointly supported by the results of Table 3. The propor- tion of co-movements that is in line with theory increases monotonically with the horizon. Although over a weekly frequency across our entire sample set, the co-movement appears roughly random with about half the observations showing co-movements opposite to the im- plication of structural models, the proportion of anomalous observations reduce to one-third as the horizon increases to 50 business days. Moreover, a large proportion of anomalous observations for the weekly frequency, 5.7% of all observations, are related to cases where the CDS spreads or the stock prices do not change. Such zero changes reduce as the hori- zon increases. At a frequency of 50 days, zeros constitute only 0.2% of the observations. As zero changes are a measure of liquidity and transaction costs (Lesmond, Ogden and Trzcinka (1999), Chen, Lesmond and Wei (2007)), the relation of zeros to intervals provide additional support to the hypothesis that anomalous observations are related to limited arbitrage activity. The proportion of co-movements that are in line with theory should be related to larger changes in spreads and stock prices than the co-movements opposite to theory. Table 3 reports the average absolute change in CDS spreads and stock prices for each direction of 11 co-movement. At every horizon, when the co-movement is in line with theory, the average change in stock prices or CDS spreads is higher than the corresponding statistics when the co-movement is opposite. For instance, at a horizon of 25 business days for opposite movements, the average change in CDS spreads across all ﬁrms in our sample is 40.7 basis points and the average change in stock prices is 9.9%. In contrast, the average change in CDS spreads and stock prices for movements in the same direction is respectively 20.9 basis points and 6.4%. We observe a similar pattern across every horizon and for each of our sub-samples. Overall, the results are consistent with the hypothesis that co-movements in the two markets are impacted by costs of arbitrage activity. An alterative hypothesis that could potentially explain why stock and CDS spreads move in the same direction is that there are wealth transfers between stock and bond-holders. Wealth transfers can occur when a ﬁrm changes its corporate policies. For example, an increase in dividends would result in transfer of wealth from the bond-holders to the share- holders, resulting in an increase in the stock price and a concomitant decline in the bond price. An increase in the volatility of the investment would likewise result in such wealth transfer. However, we still observe that co-movements where stock prices and CDS spreads move in the same direction are about 40% of the total observations at a weekly or monthly frequency. Given that ﬁrms do not change dividend or investment policies frequently, it appears unlikely that all such co-movements are related to news of such changes in corporate policies. Instead of pursuing the hypothesis of potential wealth transfers, we focus on providing direct evidence relating the integration of the two markets to factors determining arbitrage activity. 4 Integration of Equity and Credit Markets and Limits to Arbitrage In this section, we relate the correlation between stock returns and CDS spread changes to factors that determine the level of arbitrage activity. Our measure of correlation is the Kendall tau,7 which has a natural interpretation of measuring the concordance of stock re- turns and changes in CDS spread. Table 4 provides a summary of the correlation coeﬃcients 7 We compute the Kendall’s tau-b, which is a nonparametric measure of association based on the number of concordances and discordances in paired observations. The number of tied values in any group is excluded from the total number of pair observations. 12 measured over intervals of 5 to 50 business days. the correlation coeﬃcient increases with interval, indicating increasing concordance of stock returns with changes in CDS spreads in longer horizon, consistent with Table 3. In addition, there is a substantial range at every interval, indicating large variations in the cross-section. In our regressions, we use Fisher’s z transformation (David (1949)) of the correlation coeﬃcient, 1 2 ln (1+c) , where c represents the correlation coeﬃcient. However, there is little (1−c) diﬀerence in magnitude between non-transformed and transformed variable. The regression tests are performed for the correlations between stock returns and credit spread changes over 1-day to 50-day intervals. We report the results for 5-, 10-, 25-, and 50-day intervals, corresponding to weekly, bi-weekly, monthly and bi-monthly intervals. 4.1 Informational Sensitivity of Debt and Equity In the absence of any costs or other frictions, every mis-pricing is arbitraged. However, in the presence of information and transaction costs, it is not proﬁtable to detect and trade every relative mis-pricing between the equity and credit markets. The level of arbitrage activity in the market will depend on the expected magnitude of potential proﬁt opportunities, the “alpha” of the convergence trade. In the cross-section, the likelihood that it is proﬁtable for the arbitrageur to implement a trade based on his private information will be impacted by the informational sensitivity of the stock and credit default swap. We proxy the informational sensitivity of the credit default swap by the riskiness of the ﬁrm’s debt. As discussed earlier, the magnitude of the spread depends strongly on the volatility and debt level of the ﬁrm. We use the equity volatility and debt levels are proxies for the riskiness of the debt. In addition, we also include a dummy variable for whether the rating of the ﬁrm is above or below investment grade. We also include the size of the ﬁrm as a control. In univariate regressions that we do not report here, we ﬁnd the equity volatility and the debt level are consistently signiﬁcant with a negative coeﬃcient. That is, the higher the volatility and higher the debt, the closer is the correlation between the two markets to -1. Analysts’ forecasts dispersion is also consistently signifcant with the expected negative sign. Thus, all the variables proxying for the informational sensitivity of the equity and debt are signiﬁcant with the correct sign. The size of the ﬁrm is not consistently signiﬁcant. On average, the variables that have the highest R2 in the univariate regressions are equity 13 volatility and analysts’ forecast dispersion. Table 5 reports the results of the multivariate regression. We include a dummy variable for investment grade, to control for the possibility that the equity volatility and debt level may not completely account for the change of riskiness from investment grade to below investment grade. We report four regressions, corresponding to each of our intervals, weekly, bi-weekl, monthly, and bi-monthly. Overall, we ﬁnd that the informational sensitivity of both the stock and debt have an impact on the correlation of the equity and credit markets. In three of the four intervals, equity volatility, rating, and analysts’ forecast dispersion are signifcant at 99% level. After controling for equity volatility and rating, the debt level is insigniﬁcant. All of these variables have the expected negative sign, indicating that the greater the informational sensitivity, the closer is the correlation to -1. The size also enters with a negative sign in three of the four intervals. The R2 of the regressions range from 17% to 35% suggesting that the informational sensitivity of the underlying securities plays a very important role in determining the integration of the equity and credit markets. In much of the existing empirical literature, the focus has been on understanding costs and risks that inhibit arbitrage activity. We believe we are the ﬁrst to observe that it is just as important, if not more, to consider the sensitivity of the securities to an arbitrageur’s private information. Below, where we consider the role of liquidity and idiosyncratic risk, we continue to control for the informational sensitivity by including the most signiﬁcant variables in the regression.8 4.2 Liquidity Illiquidity of a market increases transaction costs as well as risks of a convergence trade. With increasing illiquidity, we expect lower integration and, therefore, a less negative cor- relation coeﬃcient. We consider both the liquidity of the equity and credit markets. Table 6 presents the impact of the credit market liquidity on the correlation between stock returns and spread changes. Panel A reports the results for the credit market depth, where the market depth is the average number of data providers to the Markit database for the ﬁrm over the sample period. We ﬁnd this proxy for liquidity is signiﬁcant for three of our four regressions. For all four of the regressions, the sign of the coeﬃcient is negative, 8 An additional reason for including volatility and size in the regressions is that the liquidity and idiosyn- cratic risk may be correlated with these variables. 14 indicating that the greater the number of contributors, the more integrated the market. Thus, the results are consistent with the hypothesis that greater liquidity implies more integration and correlations closer to -1. Panel B reports the results for our second proxy, the proportion of zero spread changes. The zero spread is a measure of illiquidity, i.e., higher the proportion, the more illiquid the market market. We also ﬁnd this measure to be signiﬁcant for each of our regressions with the correct sign. Both the two measures are positively correlated as would be expected - the more contributors, the less likely we are to see a zero change in spread. When we include both the market depth and the zero proportion in one regression, we ﬁnd the signiﬁcance of depth is absorbed by the zero proportion. It appears that the zero proportion is a more direct measure of illiquidity. Overall, we ﬁnd strong evidence that the liquidity of the CDS market impacts the integration between the two markets. Table 7 presents the results for the measures of equity market liquidity on the correla- tions. We provide results for two measures, the square root of the Amivest measure and the proportion of zero stock returns. Overall, we ﬁnd less evidence of the impact of equity market liquidity. The coeﬃcients are either insigniﬁcant or very weakly signiﬁcant. We also test for other proxies (including the Amihud measure) and not ﬁnd them signiﬁcant. Overall, the evidence indicates that, in our sample, equity liquidity has little impact on the integration of the two markets, and that it is the risks and costs imposed by trading in the credit markets that determines the integration of the two markets. 4.3 Idiosyncratic Risk In considering measures of idiosyncratic risk, we have to be careful to control for the total risk of the ﬁrm. As discussed earlier, an increase in the equity volatility increases the informational sensitivity of the credit risk, and thus makes arbitrage activity more likely. However, for a given riskiness of the ﬁrm, a higher level of idiosyncratic risk would deter arbitrage activity (Pontiﬀ (2006)). This consideration is also important in how we construct our measure of idiosyncratic risk. We do by estimating the R2 from a linear regression using the market model, and then taking the logistic transformation as noted in equation (3). Our hypothesis is that the greater the amount of idiosyncratic risk, the lower the integration, and less negative the correlation. Table 8 presents the impact of idiosyncratic risk on the correlations. For each of the 15 four intervals, the coeﬃcient on the idiosyncratic risk is signiﬁcant at the 1% level. The coeﬃcient is positive, consistent with the hypothesis that a higher level of idiosyncratic risk will make make the correlation closer closer to 0 than -1. The coeﬃcient to volatility remains signiﬁcant and negative. Thus, although the total volatility of the stock return makes the equity and credit markets more integrated, the idiosyncratic component makes it less integrated. For robustness, we also considered the idiosyncratic risk component estimated from the Fama-French three-factor model and found identical results. Our results add to the literature that observes that idiosyncratic risk has a signiﬁcant impact in capital markets. Pontiﬀ (2006) provides an overview of this literature. 5 Conclusion We examine whether limits to arbitrage impacts the integration between equity and credit markets. If the costs and risks of undertaking cross-market arbitrages are important, then we expect that the level of integration will vary in the cross-section of ﬁrms. We test and ﬁnd extensive support for this implication. First, we ﬁnd that ﬁrms whose securities are more informationally sensitive have more integrated markets. This is consistent with the notion that, with limited capital, arbitrageurs will prefer focusing on markets where their private information is more likely to be valuable. Second, we ﬁnd that liquidity of the underlying credit market is signiﬁcant in determining the level of integration. Markets are less integrated as the credit default swap becomes less liquid. In contrast, equity market liquidity does not appear to important. Finally, idiosyncratic risk matters. Firms with greater idiosyncratic risk, after controlling for total volatility, are less integrated. Should the low correlation between credit spreads and stock returns be construed as damaging evidence against structural models of credit risk? Our results provide both good and bad news. On one hand, it indicates that there may be little to be gained in devel- oping structural models with more realistic default boundaries or stochastic processes for modeling the underlying ﬁrm value. On the other hand, it also indicates that that research should increasingly focus on the development of models that explicitly allow for the role of arbitrageurs. The latter appears to be a far more diﬃcult task as it relates to fundamental assumptions regarding our markets. 16 References [1] Acharya, Viral and Lasse Pedersen, 2005, “Asset Pricing with Liquidity Risk,” Journal of Financial Economics 77 (2). 375-410. [2] 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. [3] Anderson, Ronald W. and Suresh Sunderesan, 1996, “Design and valuation of debt contracts,” Review of Financial Studies 9(1), 37-68. [4] Amihud, Yacov, 2002, “Illiquidity and Stock Returns: Cross-Section and Time-Series Eﬀects,” Journal of Futures Markets 5, 31-56. [5] Ben-David, Itzhak, and Roulstone, Darren, 2007, “Idiosyncratic risk and coporate transactions”, working paper, University of Chicago. [6] Black, Fischer and John C. Cox, 1976, “Valuing corporate securities: some eﬀects of bond indenture provisions,” Journal of Finance 31, 351:367. [7] 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 Vol. 60 (5), pp. 2255-2281. [8] Barberis, N., R. Thaler, 2003, A survey of behavioral ﬁnance, Handbook of the Eco- nomics of Finance, Chapter 18, edited by G. M. Constantinides, M. Harris, and R. Stulz. [9] Brennan, M., and A. Subrahmanyam, 1995, Investment Analysis and Price Formation in Securities Markets, Journal of Financial Economics, 38, 361-381. [10] Chen, Long, David A. Lesmond, and Jason Wei, 2007, “Corporate Yield Spreads and Bond Liquidity,” Journal of Finance 62(1), 119-149. [11] Collin-Dufresne, Pierre and Robert S. Goldstein, 2001, “Do credit spreads reﬂect sta- tionary leverage ratios?” Journal of Finance 56, 1929-1957. [12] Collin-Dufresne, Pierre, Robert S. Goldstein, and J. Spencer Martin, 2001, “The De- terminants of Credit Spread Changes,” Journal of Finance 56, 2177-2207. 17 [13] F. N. David, 1949, The moments of the z and F distributions, Biometrika 36, 394-403. [14] DeLong, J. Bradford, Andrei Shleifer, Lawrence Summers, and Robert Waldmann, 1990, “Noise trader risk in ﬁnancial markets,” Journal of Political Economy 98, 703- 738. [15] Ferreira, Miguel A., and Laux, Paul A., 2007, “Corporate Governance, Idiosyncratic Risk, and Information Flow”, Journal of Finance, 62(2), 951-989. [16] Hasbrouck, J., 2006, Trading Costs and Returns for US Equities: Estimating Eﬀective Costs from Daily Data, working paper, New York University. [17] 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. [18] Larrain, Borja and Motohiro Yogo, 2007, “Does Firm Value Move Too Much to be Justiﬁed by Subsequent Cash Flow?” Journal of Financial Economics, forthcoming. [19] Leland, Hayne E., 1994, “Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads,” Journal of Finance 49, 1213-1252. [20] Leland, Hayne E. and Klaus Toft, 1996, “Corporate debt value, bond convenants, and optimal capital structure,” Journal of Finance 51, 987-1019. [21] Lesmond, David A., Joseph P. Ogden and Charles A. Trzcinka, 1999, “A New Estimate of Transaction Costs,” Review of Financial Studies 12(5), 1113-1141. [22] Longstaﬀ, F., and Schwartz, E., 1995, A Simple Approach to Valuing Risky Fixed and Floating Rate Debt, Journal of Finance 50, 789-821. [23] Mashruwala, C., S. Rajgopal, and T. Shevlin, 2006, “Why is the Accrual Anomaly not Arbitraged Away?”, Journal of Accounting and Economics, 42, 3C33. [24] Merton, R., 1974, On the pricing of corporate debt: the risk structure of interest rates, Journal of Finance 29, 449-470. [25] Merton, R., 1987, “A Simple Model of Capital Market Equilibrium with Incomplete Information”, Journal of Finance 42, 483C510. [26] Mitchell, M., Pulvino, Todd, and Staﬀord, Erik, 2002, “Limited Arbitrage in Equity Markets”, Journal of Finance 57 (2), 551-584. 18 [27] Mitchell, M., Lasse Heje Pedersen, and Todd Pulvino, 2006, “Slow Moving Capital”, The American Economic Review, forthcoming. [28] 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, 305C342. [29] Pastor, L., and Robert F. Stambaugh, 2003, “Liquidity risk and expected stock re- turns”, Journal of Political Economy 111(3), 642-685. [30] Pontiﬀ, Jeﬀrey, 1996, “Costly Arbitrage: Evidence from Closed-End Funds”, The Quarterly Journal of Economics 111 (4), 1135-1151. [31] Pontiﬀ, Jeﬀrey, 2006, “Costly arbitrage and the myth of idiosyncratic risk”, Journal of Accounting and Economics, 42(1-2),35-52. [32] Roll, Richard, 1984, A simple implicit measure of the eﬀective bid-ask spread in an eﬃcient market, Journal of Finance 39(4): 1127-1139. [33] Wurgler, J., and E. V. Zhuravskaya, 2002, “Does Arbitrage Flatten Demand Curves for Stocks?”, Journal of Business, 75, 583C608. [34] Yu, Fan, 2006, “How Proﬁtable Is Capital Structure Arbitrage,” Financial Analysts Journal, 62(5), 47-62. 19 Table 1: Descriptive statistics The sample consists of 200 non-ﬁnancial N. American ﬁrms over the period January 2, 2001 to December 31, 2005, of which 95 ﬁrms have an average rating above investment grade, and 105 ﬁrms have average ratings below investment 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 debt value to the sum of book debt value and market capitalization. For each obligor, we ﬁrst compute the time-series mean of its (daily) 5-year CDS spreads, (daily) market capitalization, and (quarterly) leverage, and then compute the statistics in the cross-section. The equity volatility is computed as the annualized standard deviation of daily returns across the ﬁve-year sample period. 5-year CDS spread (bps) Mean Median Min Max All 215.17 145.53 19.21 1670.58 Investment Grade 86.57 69.01 19.21 430.98 High Yield 331.51 266.84 44.77 1670.58 Size (’000,000,000) Mean Median Min Max All 13.14 6.24 0.40 227.11 Investment Grade 22.30 13.10 1.09 227.11 High Yield 4.87 2.84 0.40 24.66 Volatility Mean Median Min Max All 0.42 0.36 0.19 1.22 Investment Grade 0.33 0.32 0.19 0.72 High Yield 0.50 0.45 0.23 1.22 Leverage Mean Median Min Max All 0.40 0.39 0.04 0.90 Investment Grade 0.29 0.27 0.04 0.79 High Yield 0.50 0.51 0.07 0.90 20 Table 2: Summary Statistics of Factors Determining Limits to Arbitrage The table reports the statistics for the limits to arbitrage factors for our sample of 200 ﬁrms over the period January 2001 to December 2005. Foredisp is the standard deviation of analysts’ fore- casts scaled by the period-end stock price. Depth is the number of contributors for the compos- ite quotes of 5-year CDS spreads on a daily frequency. The average of daily values is computed as the measure for each ﬁrm. Zspread is deﬁned as the proportion of zero daily CDS spread changes among all the non-missing daily changes in the sample period. S.Amivest 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 ratio of daily zero returns among 2 all the non-missing returns in the sample period. Idiosyn is the logistic transformation ln 1−R , R2 where R2 is the coeﬃcient of determination of the regression of daily excess returns on the market. Measure Mean Median Min. Max. Std Foredisp ( x 103 ) 8.53 5.12 0.40 56.2 9.06 Depth 9 9 3 18 3 Zspread (%) 21.90 21.27 0.00 58.66 9.37 S.Amivest 85.33 74.95 7.38 334.99 54.66 Zprop (%) 1.60 1.32 0.24 6.29 1.08 Idiosyn 1.46 1.49 0.12 4.42 0.78 21 Table 3: Co-movement of CDS Spreads and Stock Prices The table reports the direction of movement between CDS spreads and stock prices reported as a percentage of total observations at a given sampling interval. |∆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. ∆CDSi *∆Pi < 0 ∆CDSi *∆Pi > 0 ∆CDSi *∆Pi = 0 Sample Interval Obs. Fraction |∆CDS| |∆P/P | Fraction |∆CDS| |∆P/P | Fraction (Days) (%) (bps) (%) (%) (bps) (%) (%) 5 37,883 53.8 14.2 4.0 40.5 10.2 3.1 5.7 All 10 18,761 58.1 22.1 5.9 39.5 14.2 4.3 2.4 25 7,441 63.7 40.2 9.8 35.6 21.2 6.3 0.7 50 3,685 68.5 65.5 14.9 31.3 30.3 8.4 0.2 5 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 2.7 Grade 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 0.1 5 16,785 55.6 23.8 4.8 39.8 17.6 3.6 4.6 High 10 8,290 60.7 36.6 7.1 37.3 24.9 4.9 2.0 Yield 25 3,290 67.3 65.6 11.9 31.8 36.9 7.5 0.9 50 1,626 71.5 107.6 18.3 28.2 52.1 10.0 0.3 Table 4: Correlation Between Stock Returns and Change in CDS Spread The table reports the descriptive statistics for the Kendall correlation between changes in CDS spreads and stock returns over the period 2001-05 for the sample of 200 ﬁrms. Firms that had less than 15 available observations for computing correlation were excluded. Interval No. of Firms Mean Median Max. Min. Std. 5 200 -0.12 -0.12 0.08 -0.42 0.08 10 200 -0.17 -0.16 0.16 -0.45 0.10 25 188 -0.25 -0.25 0.24 -0.64 0.16 50 139 -0.34 -0.33 0.13 -0.81 0.18 22 Table 5: Informational Sensitivity The table reports the results of the regression of the measure of correlation, tkcorr = α + β1 foredisp + β2 eqvol + β3 lnmcap + β4 lev + β5 rating + . 1 1+kcorr tkcorr is the transformed Kendall correlation, 2 ln( 1−kcorr ), where korr is the Kendall correlation. F oredisp is the standard deviation of analysts’ forecasts scaled by the period-end stock price. Eqvol is the annualized equity volatility in the sample period. Lnmcap is the log of the market capital- ization. The leverage Lev is calculated as the ratio of book debt value to the sum of book debt value and market capitalization. Rating is a dummy variable for investment grade. The t-values are reported in parenthesis. ** and * indicates signiﬁcance at 5% and 10%, respectively. Interval Firm# Intercept Foredisp Eqvol Lnmcap Lev Rating Adj. R2 5 187 0.0874 -2.5470 -0.0794 -0.0176 -0.0502 0.0263 17% (1.44) (-3.65)** (-1.94)* (-2.92)** (-1.38) (1.73)* 10 187 0.1103 -3.8526 -0.1369 -0.0225 -0.0394 0.0456 27% (1.54) (-4.66)** (-2.82)** (-3.16)** (-0.92) (2.53)** 25 178 0.1277 -6.6410 -0.2473 -0.0286 -0.0737 0.0734 29% (1.05) (-4.56)** (-2.93)** (-2.37)** (-1.00) (2.38)** 50 138 -0.1265 -2.9702 -0.5602 -0.0051 -0.1018 0.1092 35% (-0.71) (-1.53) (-4.46)** (-0.30) (-0.98) (2.64)** 23 Table 6: Credit Market Liquidity Panel A reports the results of the regression, tkcorr = α + β1 depth + β2 foredisp + β3 eqvol + β4 lnmcap + β5 rating + . Panel B reports the results of the regression, tkcorr = α + β1 Zspread + β2 foredisp + β3 eqvol + β4 lnmcap + β5 rating + . 1 1+kcorr tkcorr is the transformed Kendall correlation, 2 ln( 1−kcorr ), where korr is the Kendall correlation. Depth is average number of contributors to the composite 5-year CDS spread quotes. Zspread is the proportion of zero daily spread changes among all non-missing observations. F oredisp is the standard deviation of analysts’ forecasts scaled by the period-end stock price. Eqvol is the annualized equity volatility in the sample period. Lnmcap is log of market capitalization. Rating is a dummy variable for investment grade. t-values are reported in parenthesis; ** and * indicates signiﬁcance at 5% and 10%, respectively. Panel A: Depth Interval # of Firms Intercept Depth Foredisp Eqvol Lnmcap Rating Adj. R2 5 187 0.0667 -0.0099 -2.5371 -0.1145 -0.0066 0.0434 23% (1.30) (-3.93)** (-3.87)** (-2.83)** (-1.11) (2.95)** 10 187 0.0998 -0.0107 -3.7720 -0.1748 -0.0114 0.0629 31% (1.63) (-3.57)** (-4.83)** (-3.63)** (-1.61) (3.58)** 25 178 0.0988 -0.0141 -6.5595 -0.3040 -0.0128 0.0960 31% (0.93) (-2.71)** (-4.70)** (-3.57)** (-1.05) (3.16)** 50 138 -0.1433 -0.0104 -3.4514 -0.5989 0.0050 0.1235 36% (-0.88) (-1.30) (-1.84)* (-4.65)** (0.30) (3.08)** Panel B: Zero Spread Interval # of Firms Intercept Zspread Foredisp Eqvol Lnmcap Rating Adj. R2 5 187 -0.0316 0.1858 -2.3111 -0.0766 -0.0113 0.0280 21% (-0.55) (3.10)** (-3.40)** (–1.91)* (-1.96)* (1.92)* 10 187 -0.0252 0.2450 -3.4157 -0.1331 -0.0156 0.0456 31% (-0.37) (3.48)** (-4.28)** (-2.83)** (-2.29)** (2.65)** 25 178 -0.0905 0.3619 -6.0825 -0.2459 -0.0168 0.07402 31% (-0.76) (2.84)** (-4.28)** (-2.98)** (-1.44) (2.49)** 50 138 -0.4853 0.6401 -1.7429 -0.5371 0.01229 0.1133 40% (-2.83)** (3.21)** (-0.92) (-4.41)** (0.77) (2.92)** 24 Table 7: Equity Market Liquidity Panel A reports the results of the regression, tkcorr = α + β1 S.Amivest + β2 foredisp + β3 eqvol + β4 lnmcap + β5 rating + . Panel B reports the results of the regression, tkcorr = α + β1 Zprop + β2 foredisp + β3 eqvol + β4 lnmcap + β5 rating + . 1 1+kcorr tkcorr is the transformed Kendall correlation, 2 ln( 1−kcorr ), where korr is the Kendall correlation. S.Amivest is sqrt(abs(return)/(abs(price) ∗ sharevolume)). The average of the daily values is computed as the measure for the ﬁrm in the sample period. Zprop is the proportion of zero daily stock returns among all the non-missing observations in the sample period. F oredisp is the standard deviation of analysts’ forecasts scaled by the period-end stock price. Eqvol is the annualized equity volatility in the sample period. Lnmcap is the log of the market capitalization. Rating is a dummy variable for investment grade. t-values are reported in parenthesis; ** and * indicates signiﬁcance at 5% and 10%, respectively. Panel A: Square Root of Amivest Interval # of Firms Intercept S.Amivest (10−3 ) Foredisp Eqvol Lnmcap Rating Adj. R2 5 187 0.0471 -0.0008 -2.7836 -0.0798 -0.0152 0.0305 16% (0.63 (0.00) (-4.09)** (-1.91)* (-1.49) (2.03)** 10 187 0.0673 -0.0461 -4.0344 -0.1388 -0.0188 0.04872 26% (0.77) (-0.19) (-5.02)** (-2.81)** (-1.57) (2.74)** 25 178 -0.1196 -0.7634 -6.9237 -0.2741 0.0052 0.0772 29% (-0.79) (-1.77)* (-4.94)** (-3.23)** (0.25) (2.56)** 50 138 -0.3255 -0.3878 -3.3557 -0.5725 0.0171 0.1184 35% (-1.43) (-0.67) (-1.77)* (-4.50)** (0.57) (2.95)** Panel B: Zero Proportion Interval # of Firms Intercept Zprop Foredisp Eqvol Lnmcap Rating Adj. R2 5 187 0.0505 -0.1773 -2.7835 -0.0777 -0.0153 0.0293 16% (0.92) (-0.27) (-4.10)** (-1.86)* (-2.64)** (1.88)* 10 187 0.0866 -0.4312 -4.0383 -0.1323 -0.0209 0.0462 26% (1.34) (-0.55) (-5.03)** (-2.68)** (-3.05)** (2.50)** 25 178 0.0443 1.6207 -6.9557 -0.2776 -0.0241 0.0871 29% (0.40) (1.08) (-4.93)** (-3.15)** (-2.08)** (2.80)** 50 138 -0.2709 4.0900 -3.3018 -0.6181 0.0011 0.1436 37% (-1.73)* (1.82)* (-1.77)* (-4.81)** (0.07) (3.43)** 25 Table 8: Idiosyncratic Risk The table reports the results of the regression, tkcorr = α + β1 Idiosyn + β2 foredisp + β3 eqvol + β4 lnmcap + β5 rating + . 1 1+kcorr tkcorr is the transformed Kendall correlation, 2 ln( 1−kcorr ), where korr is the Kendall correlation. Idiosyn is computed as the logistic transformation of the coeﬃcient of determination from a market model regression, ln (1 − R2 )/R2 . F oredisp is the standard deviation of analysts’ forecasts scaled by the period-end stock price. Eqvol is the annualized equity volatility in the sample period. Lnmcap is the log of the market capitalization. The leverage Lev is calculated as the ratio of book debt value to the sum of book debt value and market capitalization. Rating is a dummy variable for investment grade. t-values are reported in parenthesis. ** and * indicates signiﬁcance at 5% and 10%, respectively. Idiosyncratic Risk Interval Firm# Intercept Idiosyn Foredisp Eqvol Lnmcap Rating Adj. R2 5 187 0.00 0.0256 -2.8957 -0.0656 -0.0153 0.0453 22% (0.00) (3.47)∗∗ (−4.40)∗∗ (-1.64) (−2.74)∗∗ (3.00)** 10 187 0.0317 0.0255 -4.1507 -0.1231 -0.0208 0.0637 29% (0.50) (2.90)∗∗ (−2.76)∗∗ (−5.27)∗∗ (−2.57)∗∗ (3.53)∗∗ 25 178 -0.0468 0.0693 -6.9807 -0.2388 -0.0252 0.1121 36% (-0.45) (4.56)** (-5.22)** (-2.99)** (-2.30)** (3.79)** 50 138 -0.2910 0.0614 -3.6282 -0.5378 -0.0026 0.1435 39% (-1.91)* (2.96)** (-1.98)** (-4.40) (-0.16) (3.61)** 26