Limited Arbitrage between Equity and Credit Markets

49 Pages·2010·0.56 MB·English

Limited Arbitrage between Equity and Credit Markets
Nikunj Kapadia and Xiaoling Pu1
First Version: March 2009
Current Version: April 2010
1University of Massachusetts and Kent State, respectively. We thank Viral Acharya, Kobi Boudoukh, Sanjiv
Das, Michael Gordy, Jason Kremer, Sanjay Nawalkha, N. R. Prabhala, Akhtar Siddique, Dimitri Vayanos,
and discussants and participants at IDC-Herzliya, Journal of Investment Management Conference, Second Paul
Woolley Conference at the London School of Economics, China International Conference in Finance, Fordham
University and the Indian School of Business. Xuan Che provided excellent research assistance. We thank
Markit for providing us the CDS data. The contents are the sole responsibility of the authors. Please address
correspondence to Nikunj Kapadia Isenberg School of Management, University of Massachusetts, Amherst, MA
01003. Email: [email protected].
Abstract
Proposingastatisticalmeasureofmarketintegration,weinvestigatewhetherlimitstoarbitrageexplain
the level of integration between firms’ equity and credit markets. We find cross-sectional and time-
series variation in equity-credit market pricing discrepancies are explained by variables associated with
market liquidity, funding liquidity, and idiosyncratic risk. Illiquidity of the CDS market is particularly
important in determining cross-sectional variation in integration. Our evidence provides a potential
resolution to the puzzle of why Merton model hedge ratios match empirically observed stock-bond
elasticities(SchaeferandStrebulaev(2008)), andyetthemodelcannotexplaintheintegrationbetween
equity-credit markets (Collin-Dufresne, Goldstein and Martin (2001)).
Keywords: limited arbitrage, market integration, Merton model, capital structure arbitrage
JEL classification: G14, G12, C31
1 Introduction
Merton’s (1974) structural model of credit risk implies that stock price and credit spread changes must
be precisely related to prevent arbitrage. Consistent with academic theory, hedge funds and private
equityfirmsareactiveinavarietyoftradingstrategies-popularlyknownascapitalstructurearbitrage
- that attempt to “arbitrage” across equity and credit markets. Given the theoretical link between
the equity and bond of a firm, active arbitrageurs, and integrated equity and credit markets, stock
returns and changes in credit spreads should be closely related. Instead, recent research finds to the
contrary. In a regression of monthly changes in credit spreads on stock returns and other variables
consistent with a structural framework, Collin-Dufresne, Goldstein and Martin (2001) find adjusted
R2s of the order of 17% to 34%. They conclude: “Given that structural framework models risky debt
as a derivative security which in theory can be perfectly hedged, this adjusted R2 seems extremely low.”
Blanco, Brennan, and Marsh (2005) conduct a similar exercise using weekly changes in spreads of
credit default swaps (CDS), and find three-quarters of the variation remains unexplained.
This low correlation poses a puzzle. First, as opposed to changes, Merton-model factors do a
much better job explaining the cross-sectional variation in the level of credit spreads (Ericsson, Jacobs
and Oviedo (2009), Zhang, Zhou and Zhu (2009)). In our dataset, a cross-sectional regression of the
average five-year credit default swap spread over 2001-05 on the firm’s average debt ratio and stock
return volatility gives an adjusted R2 of 61%. Second, Schaefer and Strebulaev (2008) document that
theempiricalsensitivityofbondreturnstostockreturnsisroughlyoftheorderofmagnitudepredicted
by the Merton model. How can the Merton model provide reasonable hedge ratios and yet not explain
integration of equity-credit markets?
Existing literature provides two hypotheses regarding why equity-credit market correlations may
be low. First, a low correlation may result from changes in firms’ asset volatility or debt.1 Unexpected
changesinafirm’svolatilityanddebtcancausewealthtransfersbetweenshareholdersandbondholders
andreducetheobservedcorrelationbetweenequityreturnsandcreditspreadchanges. Indeed, changes
in equity volatility are an explanation for pricing discrepancies in the equity option market (Bakshi,
Cao and Chen (2000)). Second, the low correlation may be explained by a non-structural pricing
model where the credit market prices a factor that is not priced in the equity market. For example,
the credit market may price a credit market specific systematic liquidity factor. Longstaff, Mithal and
Neis (2005) and Chen, Lesmond and Wei (2007) document liquidity components in corporate bond
1Merton’s (1974) model assumes constant firm volatility anddebt. Attempts to make the Merton (1974)model more
realistic have focused on specifications for the default boundary, recovery, and the stochastic process determining the
underlying firm value or leverage. See Black and Cox (1976), Leland (1994), Leland and Toft (1996), Longstaff and
Schwartz (1995), Anderson and Sundaresan (1996), Collin-Dufresne and Goldstein (2001).
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spreads. By the first hypothesis, wealth transfers between shareholders and bondholders give a false
signal of lack of integration. By the second hypothesis, the low correlation between equity and credit
markets is to be expected because when markets price different factors, by definition, the two markets
are not integrated. The common implication of both these hypotheses is that Merton-model pricing
discrepancies are not anomalies when evaluated against the true credit pricing model.
In this paper, we propose and investigate a third hypothesis that Merton-model pricing discrepan-
cies, at least in part, are anomalies, and the lack of integration of the two markets is a result of limited
arbitrage (Shleifer and Summers (1990) and Shleifer and Vishny (1997)). Limits to arbitrage can
prevent arbitrageurs like hedge funds from investing the capital required to eliminate pricing anoma-
lies, therefore, potentially resolving the puzzle of why structural models cannot explain equity-credit
market integration. Our study is the first to investigate the possibility that limited arbitrage explains
the integration between equity and credit markets.2 The CDS-equity market is a singularly useful
laboratory to test the implications of the recent theoretical literature on limits to arbitrage not only
because of its size but because of the participation of active and sophisticated arbitrageurs.
Limited arbitrage is a natural hypothesis to explore because convergence trades across equity
and credit markets are, in practice, not a zero-capital riskless arbitrage as modeled in structural
models. When faced with mispricing, arbitrageurs will limit the capital deployed because of existing
or potential funding constraints (Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009)),
and liquidity and other costs associated with undertaking the arbitrage (e.g., Pontiff (1996, 2006)). In
addition, mispricingswillpersistovertime, becausearbitragecapitalmayaccumulateslowly(Mitchell,
Pedersen and Pulvino (2007)) in part because of institutional constraints on the availability of risk
capital (Merton (1987)) and in part because arbitrageurs can reduce risk if they synchronize their
activities (Abreu and Brunnermeier (2002)). Arbitrage activity may be expected to reduce pricing
discrepancies, on average, but not necessarily completely nor always (Xiong (2001), Kondor (2009)).
We begin by proposing a statistical measure for market integration based on the presence of pricing
discrepancies. When pricing in two related markets are determined by a single factor, as stocks and
bonds in the Merton model, pricing discrepancies can be identified by concordance of price changes in
thetwomarkets. Therefore,wepropose,ameasureoftheconcordancecanserveasameasureofmarket
integration. Because the concordance is directly linked to pricing discrepancies, there is no ambiguity
in its interpretation unlike measures such as the coefficient of determination used in Collin-Dufresne,
2Limits to arbitrage have been invoked for a wide range of anomalies including the closed end fund discount (Pontiff
(1996)),violationsofput-callparity(Ofek,Richardson,andWhitelaw,(2004))andnegativestubvalues(Mitchell,Pulvino
andStafford(2002),LamontandThaler(2003)). SeealsorelatedworkbyAli,HwangandTrombley(2003)andMitchell,
Pedersen and Pulvino (2007).
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Goldstein and Martin (2001). We provide simulation evidence to show the sensitivity of the measure
in detecting mispricing in datasets of limited time series length as well as its robustness.
We document the following evidence. First, we show that Merton-model pricing discrepancies are
frequent, persistent, and economically large. Over a 5 business day interval, 46.3% of all relative
stock and CDS spread movements are classified as discrepancies. These discrepancies are coincident
with economically large changes in the underlying spread and stock return. For example, pricing
discrepancies over a 50-business day interval are associated with 8.4% mean absolute stock return and
themeanabsolutechangeinCDSspreadis30basispointsinthewrongdirection. Pricingdiscrepancies
decline with horizon. Over 5-business day intervals, on average, 45% of all price movements are
classified as anomalous, almost twice those over the 50-business day interval.
Pricing discrepancies cannot be explained away by changes in equity option implied volatility or
firms’ debt levels. Surprisingly, the majority of mispricings occur when increases (decreases) in implied
volatility are coincident with decreases (increases) in stock prices. Over short horizons of a quarter,
book value of debt changes by less than 2%, on average. Not surprisingly, changes in book value of
debt do not explain pricing discrepancies.
Pricing discrepancies are mispricings. Capital structure arbitrage trades betting on convergence
of equity and credit markets after observation of short-term pricing discrepancies earn average excess
returns of about 1% over a month with Sharpe ratios as high as 4.42. Convergence of the two markets
is evidence against any hypotheses that argues pricing discrepancies are not real anomalies. In the
absence of arbitrage risks or costs, mispricings would constitute unusually profitable and therefore
unlikely, opportunities.
The frequency of mispricings and, therefore, the level of integration varies across the cross-section
of firms. Cross-sectional differences in the integration of equity-credit markets are correlated with the
proxies for costs and risks associated with convergence trades. Specifically, liquidity of both equity and
credit markets - but especially credit market illiquidity - has significant impact on market integration.
Idiosyncraticrisk, whichconstitutesacosttoarbitrageurs(Pontiff(2006)), isalsosignificantforhigher
rated firms. Market-wide changes in pricing discrepancies are correlated in the time series with the
Eurodollar interest rate, the cost of funding. In summary, limits to arbitrage are both statistically and
economically significant in explaining variation in the level of equity-credit market integration.
Our finding that it is possible to identify mispricings using the simplest implication of the Merton
model complements the finding of Schaefer and Strebulaev (2008) that the Merton model is also good
matching the sensitivity of bond returns to stock returns, and helps reconcile their results with those
of Collin-Dufresne, Goldstein and Martin (2001). Even though it is possible to identify economically
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significantmispricingsbasedontheMertonmodel,integrationofequityandcreditmarketswilldepend
on costs associated with convergence trades.
In related work, Duarte, Longstaff and Yu (2007) and Yu (2006) study the profitability of capital
structure arbitrage, and show, as we do, that the returns are positive, positively skewed, and have
high Sharpe ratios. Duarte, Longstaff and Yu (2007) suggest that positive returns may be a result of
deploymentofintellectualcapital. However, wedemonstratethatiftherewerenocostsassociatedwith
arbitragethenevensimpletradingrulesthatrequirethetradertodonothingmorethanobserverelative
price movements in equity-credit markets leads to significant positive returns. Acharya, Schaefer and
Zhang (2008) focus on an interesting episode where downgrades of Ford and GM debt in May 2005
results in a market-wide increase in correlation between firms’ CDS spreads. They interpret these
results as arising from additional funding constraints on market participants in times of crises. Our
analysis complements their work in that it indicates that impediments to arbitrage impact markets
even in more normal times.
We proceed as follows. Section 2 reviews the relevant literature. In Section 3, we propose a
statistical measure of market integration. The data is described in Section 4. Section 5 documents
Merton-model pricing discrepancies across the market. Section 6 empirically tests whether variation
in integration between a firm’s equity and credit markets may be explained by variables impacting
arbitrage activity. The last section concludes.
2 Overview of the Literature and Empirical Implications
We review the existing literature and derive cross-sectional implications for market integration based
on differences in arbitrage costs.
2.1 Funding Liquidity
Arbitrageactivityisimpactedbyanarbitrageur’sabilitytofundthepositions,or,hisfundingliquidity.
Gromb and Vayanos (2002) observe that the inability to cross-margin is a significant constraint on
funding liquidity. Margin has to be posted separately on each leg of the convergence trade - the long
position on one leg of the trade cannot be posted as collateral for the short position on the other
leg. Besides preventing the arbitrageur from taking every potentially profitable opportunity, the risk
of a tightening of a financial constraint and forced liquidation will also reduce the arbitrage capital
allocated to any convergence trade. Acharya, Schaefer and Zhang (2008) provide a case study of an
episode where arbitrageurs face such a liquidity shock. Such risks associated with funding illiquidity
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are also discussed in Attari, Mello and Ruckes (2005). Brunnermeier and Pedersen (2009) provide a
comprehensive discussion of the importance of funding constraints. The literature does not indicate
how funding liquidity is expected to vary in the cross-section. We discuss this below.
Let the arbitrageur’s available equity be W. Let the equity and debt of the firm of asset value A
be P(A) and B(A), respectively. Let |n | and |n | designate the arbitrageur position in stock and
P B
bond as a fraction of firm’s equity and debt, P and B, respectively. The size of the bond position,
|n |B, for a given stock position, |n |P, in the convergence trade will be determined by the elasticity,
B P
(B/P)(∂P/∂A)/ (∂B/∂A). If a change in firm value results in stock and bond returns of 80 bps and
20 bps, respectively, then for every $1 stock position, the arbitrageur will take a $4 bond position. Let
m and m be the margin requirements (“haircut”) on the stock and bond, respectively, defined as a
P B
fraction of the dollar value of the position. Then, the total margin, W, is,
W = m |n |P +m |n |B
P P B B
(cid:18) (cid:19)
∂P/∂AB
= |n |P m +m . (1)
P P B
∂B/∂AP
Foragivenstockposition,|n |P,thetotalmarginrequiredtofundtheconvergencetradeisdetermined
P
by (i) the margin requirement for the stock, m , (ii) the margin requirement for the bond or CDS,
P
m , and (iii) the elasticity, (B/P)(∂P/∂A)/ (∂B/∂A).
B
The margin for the stock position set by Regulation T of the Federal Reserve Board does not
require m to vary across the cross-section of stocks. It appears reasonable to presume arbitrageurs
P
do not face cross-sectionally varying margins on stock. Margin requirements for selling protection with
5-year credit default swaps recently proposed by FINRA depend on credit riskiness as follows,3
Spread (bps) Margin
0-100 4%
100-300 7%
300-500 15%
500-700 20%
> 700 25%
ThemarginrequirementforbelowinvestmentgradeCDSisaboutthreetosixtimesthatforinvestment
grade CDS.
3See FINRA Rule 4240, “Margin Requirements for Credit Default Swaps.” The margin requirements for buying
protection are half for those selling protection, and therefore have similar cross-sectional implications.
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Finally, to understand how the elasticity varies in the cross-section, consider Table 5 of Schaefer
and Strebulaev (2008). They report the sensitivity of the bond return for a given stock return, i.e.,
the reciprocal of (B/P)(∂P/∂A)/ (∂B/∂A). The sensitivity of the bond return decreases rapidly as
the bond gets safer. For an investment grade firm with quasi-leverage of 0.20 and asset volatility of
20%,4 Schaefer and Strebulaev (2008) report a sensitivity of 0.70 compared with 24.17 for a firm with
quasi-leverage of 0.50 and asset volatility of 40%. That is, the bond position required to hedge a given
stock position, (B/P)(∂P/∂A)/ (∂B/∂A), increasing very rapidly as the bond gets safer - in the above
example, the position for the investment grade bond is about 35 times that for below investment grade
bond.
Putting it together, cross-sectional differences in margin requirements are likely to be driven by
cross-sectional differences in elasticity rather than cross-sectional differences in m or m . Because
P B
the elasticity of the bond is related to the firm’s credit risk, the equity required to fund trades cross-
sectionally with the credit-risk of the firm. That is, the credit riskiness of the firm is an indirect proxy
for funding cost associated with the equity required to find trades.
2.2 Market Liquidity
Convergence traders may not be able to enter or unwind the position in a timely manner without
impacting the price. Liquidity of underlying markets are especially important because convergence
trading requires coordinated trades across equity and credit markets. Thus, illiquidity of either market
can constrain an arbitrageur. In addition, liquidity risk may be related to funding illiquidity as it is
likely that arbitrageurs get funding from the same investment banks that also make markets. The
natural hypothesis that follows is that lower liquidity in either the equity or credit markets will reduce
arbitrage activity and the level of integration between the two markets.
Cross-sectional differences in liquidity have been extensively documented in the literature. In
particular, a number of measures using stock daily return data have been proposed for measuring the
different aspects of equity market liquidity. These include Roll (1984), Lesmond, Ogden and Trzinka
(1999) and Amihud (2002). Compared with the equity liquidity literature, the literature documenting
cross-sectional differences in the credit markets is more recent (e.g. Longstaff, Mithal and Neis (2005)
and Chen, Lesmond and Wei (2007)). With the notable exception of Das and Hanouna (2009), there
is no literature that has studied cross-market liquidity across a firm’s equity and credit markets.
4The quasi-leverage is defined as the ratio of the face value of debt discounted at the riskfree rate to asset value.
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2.3 Idiosyncratic Risk
Despite having a long and short position, the arbitrageur is unlikely to be perfectly hedged. For
example, the arbitrageur may not have the correct hedge ratio, or the hedge ratio may be stale if it
is not possible to continuously adjust the hedge. If so, the arbitrageur will have a net long or short
position in the underlying firm, resulting in the arbitrageur being exposed to the total risk of the
firm. Exposure to the systematic component of the firm’s risk is not a cost to the arbitrageur as
the arbitrageur is compensated for this exposure. However, the arbitrageur is not compensated for
non-priced idiosyncratic risk. Pontiff (2006) argues that the exposure to idiosyncratic risk imposes
significant constraint on arbitrageurs.
There is also empirical evidence that relates pricing discrepancies in the equity markets to idiosyn-
cratic risk. Wurgler and Zhuravskaya (2002) demonstrate that stocks with higher idiosyncratic risk
have higher abnormal stock returns when they are included into the S&P 500 index. Ali, Hwang
and Trombley (2003) and Mashruwala, Rajgopal and Shevlin (2006) relate idiosyncratic risk to the
book-to-market anomaly and accrual anomaly, respectively.
If unhedged idiosyncratic risk is costly, arbitrageurs will be less active in markets where the firm
has greater idiosyncratic risk. The level of integration between a firm’s equity and credit markets will
then be negatively correlated to the magnitude of idiosyncratic risk of the firm.
2.4 Slow-Moving Capital and Horizon
Abreu and Brunnermeier (2002) observe that a capital-constrained arbitrageur may not have sufficient
capitaltosingle-handedlyeliminateapricingdiscrepancy. Ifso,theeliminationofpricingdiscrepancies
will require coordinated action by a number of arbitrageurs. Because arbitrageurs are likely to become
aware of a pricing discrepancy slowly over time, arbitrage capital will take time to accumulate and
pricing discrepancies will be resistent to being eliminated in the short-run. Mitchell, Pulvino and
Stafford (2002) provide evidence that is consistent with slow-moving arbitrage capital.
The primary empirical implication of slow-moving capital is that pricing discrepancies will decline
over longer horizons.
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3 Methodological Framework
3.1 Statistical Measure of Market Integration
When the underlying asset return follows a one-dimensional diffusion, the “delta” of a call option is
bounded between 0 and 1. Thus, in the Merton (1974) model, an instantaneous increase (decrease)
in firm value results in an increase (decrease) in both stock prices and credit spreads. We propose a
statistical measure of market integration based on the theory.
Let P and CDS be the stock price and CDS spread, respectively, for firm i. Assume we have
i i
k = 1,...,M instantaneous observations of CDS spreads and stock prices, where the spread is the
cost of insurance on the firm’s zero coupon debt of face value F and time to maturity T. Define
∆CDSτ = CDS (k +τ)−CDS (k) and ∆Pτ = P (k +τ)−P (k), k ≤ M −τ, 1 ≤ τ ≤ M −1.
i,k i i i,k i i
Then, a pair of stock prices and CDS spreads present an arbitrage if ∆CDSτ ∆Pτ > 0. We define
i,k i,k
market integration for a stock i based on the frequency of observations of such arbitrage opportunities.
Specifically, define κˆ as,
i
M−1M−τ
(cid:88) (cid:88)
κˆ = 1 . (2)
i [∆CDSτ ∆Pτ >0]
i,k i,k
τ=1 k=1
For a given M, firm i’s stock and CDS markets are more integrated than those of firm j if κˆ < κˆ .
i j
The measure takes into account all pairs of CDS and stock prices.
Asanillustration, supposestockpricesareP(1) = 100, P(2) = 80andP(3) = 90. Themarketsare
more integrated if corresponding CDS spreads are CDS(1) = 50 bps, CDS(2) = 30 bps, CDS(3) = 60
bps than if they are CDS(1) = 50 bps, CDS(2) = 30 bps, CDS(3) = 40 bps. In the latter case, there
is a (Merton-model) arbitrage across every pair of observations [1,2], [2,3] and [1,3].
This definition of market integration is appealing for three reasons. First, the measure is robust
because (i) it is non-parametric and (ii) not impacted by non-linearities. Second, the measure accounts
for all M pairs of observations and is, therefore, independent of the horizon over which prices are
observed. This is useful as there may be variation in the length of the periods over which pricing
discrepancies develop and disappear. In addition, as we document in simulations below, the measure is
sensitive to mispricing in short datasets.5 Finally, because the measure relies only on the concordance
of stock and CDS prices, there is a one-to-one correspondence between κˆ and the Kendall correlation
5AlthoughtakingintoaccountallpairsofCDSandstockpricesallowsmispricingstobedetectedoversmallsamples,
this benefit comes with the tradeoff. If the observations span a long time period with large changes in stock and CDS
spreads,thenthemeasurewillbelesssensitivetomispricing. Asweshallseeinthenextsection,themeasureissensitive
tolevelsofmispricingwhenthetimeperiodisayear,andthussuitableforourstudywhichismotivatedbyshort-horizon
mispricings.
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