Limited Arbitrage between Equity and Credit Markets by jolinmilioncherie

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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
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                                            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)**




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