Funding Liquidity and Arbitrage Efficacy

38 Pages·2017·0.8 MB·English

Funding Liquidity and Arbitrage Efficacy(cid:73),(cid:73)(cid:73)
Jingzhi Chen1, Yongcheol Shin2
Abstract
Traders with severe funding problem are forced to reduce its position by selling assets at a distressed
prices, while arbitrageurs, subject to endogenous leverage constraint, capitalize on these mispricing
opportunities. We study the arb’s funding liquidity and arbitrage efficacy by looking at the marginal
leveragepositionandpercentageofmispricingcorrectionwithrespecttoadditionalmispricings. When
leverage constraint is slack, they tend to lever up to bear against larger mispricings (called stabilizing
leverage), and able to achieve higher percentage of mispricing correction (called efficacious arbitrage),
whichpreventaliquidationatdistressedprices. However,extremelylargemispricingcanmakeleverage
constraint binds, and funding liquidity is severely deteriorated. More importantly, arbitrage efficacy
drops below zero, i.e. arbitrage becomes inefficacious, which leads to large persistent mispricings and
high pricing volatility in the future. To test our model predictions, we design an empirical strategy
to estimate the implied arbitrage efficacy of the S&P 500 index-future arbitrage relationship during
1997-2015,andwefindthat1. theimpliedarbitrageefficacyissignificantlyassociatedwithotherbroad
measure of funding liquidity, such as TED spread, VIX index of implied volatility and the dividend
yield of S&P 500 index; 2. innovation in the implied arbitrage capacity predicts market volatility in
thefuture,especiallyvolatilityriskpremia,andthepredictabilityismostprominentduringtheperiod
of inefficacious arbitrage; 3. inefficacious arbitrage ex ante signals binding funding constraint, as it
triggerstheamplificationeffectandcoincideswiththemajormarketcrashesduringthesampleperiod.
Keywords: Funding liquidity, Arbitrage efficacy, Binding funding constraint, Index arbitrage
1. Introduction
The recent financial crisis during 2007-2008 and the subsequent great recession have put spotlight
on the leveraged financial intermediaries, and the importance of their ability to obtain funding for
operation, i.e. funding liquidity, to both asset prices and the whole economy. Among others, hedge
funds are often thought as sophisticated and rational arbitrageurs (called arbs in short hereafter), as
they largely employ quantitative modeling in making investment decision and heavily use leverage to
support their daily management, such as liquidity provision and mispricing correction. The industry
(cid:73)ThisDraft,March2017,JC.
(cid:73)(cid:73)ThispaperisaPhDworkingpaperinDepartmentofEconomicsandRelatedStudies(DERS),UniversityofYork. It
presentspreliminaryfindingstostimulatediscussionandcriticalcomments. Anyerrorsoromissionsaretheresponsibility
oftheauthors.
1Ph.dinEconomics,DepartmentofEconomicsandRelatedStudies,UniversityofYork,YorkYO105dd,UK,E-mail:
[email protected]
2Professor of Economics, Department of Economics and Related Studies, University of York, York YO105dd, UK,
E-mail: [email protected]
Preprint submitted to RES Junior Symposium March 13, 2017
2
of hedge funds has witnessed a rapid growth since 1990, whose asset on a global basis increases from
39billionUSdollarto1.93trillionin20083. Therelativelyhighuseofleveragecanenhancethehedge
funds’ ability to capitalize on mispricings opportunity and reduce pricing anomalies, referred to as
“smart money” by Akbas et al. (2015). However, they are also exposed to higher funding risks, which
may amplify the market uncertainty. As the funding problem spread to the hedge fund sector, they
will find it more difficult to manage their leverage and react to transient discrepancy in market prices
by implementing arbitrage strategy, which leads to significant asset pricing consequence: large and
persistent pricing anomalies. Therefore, funding liquidity of the arbs plays an important role in the
effectiveness of the arbitrage strategies and the efficiency of the financial market.
Alargenumberofstudiesillustratetheimportanceoffundingilliquidity(orfundingconstraint)as
a limit to arbitrage, and how it affects market liquidity and market efficiency. Duffie (2010) describe
the causes of slow-moving capital that leads to unusual price deviation, which vanishes as funding
condition improves. Akbas et al. (2015, 2016) show that when capital are sufficient for hedge funds,
theyareabletoconductarbitragemoreeffectively, whichleadstolowerpotentialpricinganomaliesin
thefuture. Someempiricalstudiesdocumentthesignificantrelationbetweentrader’sfundingcondition
andthepersistentmispricings,suchasthepricedeviationinTreasurysecuritiesthathavesimilarcash
flows but different ages (Fontaine and Garcia, 2011), price deviation from Credit default swap and
corporate bond (Garleanu and Pedersen, 2011), return of liquidity provision (Nagel, 2012) and the
return spread between low- and high- beta stock (Chen and Lu, 2015).
Undercertaincondition,anamplificationeffectmightatwork,suchthatevenamodesttriggercan
result in large spillovers across the financial system. Shleifer and Vishny (1997) suggest a loss spiral
thatoccursunderbindingfundingconstraint. Arbswholosecapitalduetodeepeningmispricingsmay
face funding withdrawal from lenders and are forced to liquidate their position, which leads to further
pricedistortions. BrunnermeierandPedersen(2009)showsthemarginspiralinextremecircumstance,
suchthatlossincapitalwillforcetraderstoliquidatetheirpositionwhichenlargethepricedeviations
and market illiquidity, while market illiquidity will raise margin requirement which further deteriorate
trader’s funding condition. See also Mitchell, Pedersen, and Pulvino (2007) and Mitchell and Pulvino
(2010)forempiricalevidenceofextremelylargepricinganomaliescausedbyslowmovingcapitalduring
the financial crisis.
Understandingandidentifyingwhereamplificationandspilloverswilloccurisvitalforpolicymakers
during the financial crisis. Although empirical studies have document the large pricing impact due to
trader’s funding illiquidity, a key question that fail to address is when funding illiquidity is so severe
that it will lead to amplification and spillovers. Our paper attempts to fill this gap.
We following the work of Cai et al. (2016) and narrow our focus on the arbs who capitalize on
the mispricing opportunity that results from the fire sales of funding constrained traders. Cai et al.
suggests that arb’s reaction towards mispricing error, i.e. mispricing correction, reveals the limits to
arbitrage they face: arbitrage cost and funding constraint. In particular, we augment the model of
Shleifer and Vishny (1997) and Stein (2009) by allowing arbs to exploit the mispricing opportunity
subject to leverage constraint endogenously set by outside financiers. In our model, mispricings are
3DatafromStein(2009).
3
generated by the liquidation of traders with poor funding condition. The worse is funding condition
among liquidity trader, the larger mispricings will occur. Arbs enter the market to smooth price
fluctuationandcorrectthemispricingerrorintheuseofleverage. Leveragedebtcanbefinancedfrom
outside financiers, who set a maximum leverage limit to protect their own capital from adverse price
movement. We derive the competitive equilibrium of the model where arbs capitalize on mispricings
subject to leverage constraint, and explore its implication on the their funding liquidity and arbitrage
efficacy.
We evaluate the arb’s ability to obtain leverage, i.e. funding liquidity, by the marginal leverage
debt raised by the arb’s in order to bear against additional mispricings. Leverage is (de)stabilizing
when marginal leverage is (negative) positive, such that arbs are able to raise (less) more leverage
debt with mispricings. Moreover, we define the effectiveness of arbitrage, i.e. arbitrage efficacy, by
the marginal percentage of mispricing correction achieved by the arbs with additional mispricings.
Arbitrage is called (in)efficacious when marginal correction is (negative) positive.
Ourmodelsuggeststhat(i)Whenleverageconstraintisloose,thearb’sfundingliquidityispositive
, i.e. stabilizing leverage, such that they can raise more funding with higher mispricing errors. Ample
funding enable them to achieve higher mispricing correction, i.e. efficacious arbitrage. As the size
of mispricing increases, the arb’s funding liquidity and arbitrage efficacy slightly drops but remain
positive, which results in a gentle grow in the persistent mispricing and price volatility in the future.
(ii) However, extremely large mispricings make leverage constraint binds, where arbs are forced to
take the maximum leverage limit set by financiers. Arbs retain a positive but far lower funding
liquidity when financiers are more informed, and willing to offer a higher leverage limit as mispricing
increases. However, destabilizing leverage may occur when financiers are less informed, and tend to
misinterpret the initial mispricings as higher future uncertainty. This exaggerates their estimates of
thefuturemispricings,anddrivearbstodeleveragewithhighermispricings. Inthepresenceofbinding
leverage constraint, arbitrage becomes inefficacious, such that arb’s fail to achieve a higher percentage
of correction. (iii) In the presence of stabilizing leverage and efficacious arbitrage, arbs can effectively
absorb varying selling pressure and smooth volatile prices, which prevents a liquidation in distressed
prices. However,destabilizingleverageandinefficaciousarbitragecanresultinsignificantassetpricing
effects: largemispricingpersistenceandhighpricevolatility. Insuchsituation, themispricingscaused
by liquidity traders with poor funding condition is too large that even arbs can becomes liquidity
traders and have to liquidate to lower their leverage. (iv) As arbs are close to binding leverage
constraint, small increment in mispricings may lead to amplification effects due to discontinuity in
funding liquidity and arbitrage efficacy. If arbs are forced to adopt max-leverage strategy after the
smallincrementinmispricings,itleadstolargedeclineintheirfundingliquidityandarbitrageefficacy,
whichresultsinsharpincreasesinthepersistentmispricingandpricevolatility. (v)Moreimportantly,
the model implies that the sign of arbitrage efficacy can be viewed as an ex ante indicator of whether
leverageconstraintisbindingornot,whichoffersagreattoolforpolicymakerstomonitorthefunding
liquidity condition in the financial market.
After establishing the theoretical framework and deriving a number of predictions, we design an
empirical strategy to capture the arbitrage efficacy as the first-difference estimator of regressing daily
mispricing corrections on daily mispricing errors, where mispricing correction can be estimated from
4
1.00 80
0.75
2002 Market downturn Collapse of
Lehman Brother 60
911 Attack Global financial crisis Black Monday
0.50 2010 Flash crash
40
0.25
0.00 20
-0.25
Treasury Flash crash
Capital outflow 0
from EM
-0.50 First sign of Russian crisis
subprime crisis
-20
AArrbbiittrraaggee eeffffiiccaaccyy VVIIXX iinnddeexx QE1 QE2 QE3
-0.75
2002 2004 2006 2008 2010 2012 2014
Figure1: TheplotofarbitrageefficacyandVIXindex,September7,2000toJune30,2015
The figure plots the time series of arbitrage efficacy (left axis) implied by the arbitrage relationship
betweenS&P500indexandE-minifuture,andtheVIXindex(rightaxis)ofimpliedvolatilityinS&P
500 index options. Major financial market events that causes large jump on VIX index are pointed
out in the figure, while the shaded periods mark the three round of Quantitative Easing announced
by Federal Reserve.
the Error Correction Model. The empirical methodology has three merits. First, arbitrage efficacy
mayalsodependonotherarbitragecost,whilethefirst-differenceestimatormitigatethenoiseofthese
omitted variables; Second, the estimator also wipe out the temporary variation in funding liquidity,
andprovideabetterrepresentationofitsfundamental; Third, themethodology, withproperarbitrage
relationship, can apply to assets, financial sectors, and countries, totestbroad implications of funding
liquidity.
Toexamineourmodelpredictions,weapplythestrategytotheindexarbitragerelationshipbetween
S&P500 index and index future market over the period of September 1997 (the earliest possible time
for E-mini future) to June 2015, and obtain the implied arbitrage efficacy, denoted as AE. It offers
the following advantages. First, the index-future arbitrage relationship can be easily identified with
cost of carry model; Second, the E-mini S&P500 future contracts are one of the most traded future
contrasts,itcontainsalargenumberofhedgefundsandfinancialinstitutionswhoaimtocapitalizeon
the mispricing opportunity. The contract also offers a number of advantages that reduce the potential
arbitragecosts,suchashighmarketliquidity,lowshort-sellingconstraintandlowmarginrequirement;
Also Third, the sample period covering a long period of time, including some major market events.
Thus,weexpectthattheoverallresultsareabletorepresentthemarket-widearbitrageefficacyachieved
by the financial intermediaries.
Figure 1 illustrate some key findings in our empirical application, where we plot the implied arbi-
trage efficacy (solid line, left axis) along with the VIX index of implied volatility in S&P 500 index
options (dashed line, right axis). We show that (vi) Through the daily innovation of the implied ar-
5
bitrage efficacy, the periods of negative AE (inefficacious arbitrage) match the major market crashes
during the sample period. For example, AE dropped sharply below zero in June 2007 and stayed
negative during the financial crisis in 2007-2008. It rebounded back to zero only after the first round
ofquantitativeeasingbyFederalReserve. AlsoAE stayspositiveduringtheFlashcrashonMay2010
and October 2014, which later prove to be a temporary freeze of liquidity. (vii) The effectiveness of
Fed’s monetary policy and lending facilities on arbitrage efficacy is varying. While the first and third
round of Quantitative Easing announced by Federal Reserve have prominent effect on the implied
arbitrage efficacy, the second round of QE leads to a drop in AE.
To further verify the validity of AE capturing the information of funding condition among the
financial intermediaries, we also document (viii) the significant association between the implied arbi-
trage efficacy AE and other broad measures of funding condition, such as TED spread, the VIX index
ofimpliedvolatilityandthedividendyieldofS&P500index. Wefindthathighfundingcostsreflected
by TED spread, tend to be the dominate factor to explain innovations in AE, when arbitrage is inef-
ficacious, i.e. funding constraint is binding; VIX index, on the other hand, becomes the dominating
explanatory variable when AE is positive.
The focus of our paper is then to verify the consequence of the innovation of arbitrage efficacy on
thefinancialmarket. (ix)Onaggregate,wefindevidencethatchangesinAE aresignificantpredictors
for daily innovation of the market volatility, measured by the VIX index of implied volatility in S&P
500 index option, and the volatility risk premia, measured by the difference between the implied and
realized volatility of S&P 500. A drop in AE leads to a significant increment in the implied market
volatility, and volatility risk premia. We also document that (x) the magnitude of the negative effect
is nonlinear conditional on the sign of AE. It is most prominent during the period of inefficacious
arbitrage, i.e. binding leverage constraint, while negligible when AE becomes positive, which reflects
the amplification effect under binding leverage constraint.
Ourpaperrelatestoagrowingliteratureontheimpactoffinancialconstraintsandfundingliquidity
on asset pricing and arbitrage efficacy, both theoretically and empirically. Shleifer and Vishny (1997)
suggest that financiers tend to withdraw funds from arbs with poor performance, which enhance the
downward pressure on asset price. Vayanos and Weill (2006), Garleanu and Pedersen (2011) and
FontaineandGarcia(2012)linkthepricedeviationbetweentwosecuritieswithidenticalcashflowsto
arbitrage costs and funding constraints. Brunnermeier and Pedersen (2009) illustrate the interaction
between an asset’s market liquidity, measured as price deviation from fundamentals, to the traders’
fundingliquidity,measuredastheshadowcostofcapital4. Instead,Duffie(2009)andCaietal. (2016)
tend to focus on the association of asset price dynamics and the funding impediment faced by traders,
suchthatsufficientarbitragecapitalcanspeedupthemeanreversionandpriceconvergence. Similarly,
we relates the innovation in arbs activities towards mispricings to the funding constraint they face.
In this case, the funding condition of the intermediaries would affect their implemented strategies to
arbitrage mispricings away.
We also contributes to the empirical studies which aim to provide measurements about funding
4Severalotherpapers,includingHeandKrishnamurthy(2012,2013),Acharyaetal. (2009)andGrombandVayanos
(2010),providetheoreticalevidenceoffundingconstraintaslimitsofarbitrage.
6
liquidity,andfurtheranalysisofitsimpact. TheTreasury-Eurodollar(TED)spreadisoneofthemost
used, which captures the difference between the three-month LIBOR rate and the three-month T-Bill
yield, a proxy for overall funding costs faced by market intermediaries. As the LIBOR rate reflects
the credit risk of lending to commercial banks and T-Bill rate represents the risk-free rate, a rise in
the TED spread is a sign that the risk of default on interbank loans is increasing, and thus lenders
demand a higher rate of interest. However, TED spread is often questioned for its validity to precisely
represent the funding condition in the market5. Another measure of a similar kind is Drehmann and
Nikolaou (2013). They measure the funding liquidity by the bank’s aggressive bidding, which reflects
how much a bank would pay to gain liquidity, at central bank auctions during 2005-20086.
Ourmeasureoftheimpliedarbitrageefficacyisclosertothosestudiesoffundingliquidityasalimit
ofarbitrage,liquidityprovisionandmarketmakingpurpose. Comerton-Fordeetal. (2008)andAdrian
and Shin (2010) identify intermediaries’ funding liquidity by directly looking at the balance sheet of
them. Comerton-Fordeetal. investigatethepositionofNYSEspecialistasthemajorliquidityprovider
in the market, while Adrian and Shin explore the repurchase agreement (repo) in the balance sheet of
financial intermediaries, which they claim is the primary tool for funding borrowing and lending for
investment banks and hedge funds. Akbas et al. (2015, 2016) rely on the available arbitrage capital,
proxying by capital flow to hedge funds that conduct arbitrage7.
Whilethesepaperstendtofocusontheavailabilityofarbitragecapital,others,includingGarleanu
and Pedersen (2011), Fontaine and Garcia (2012) and Nagel (2012), pay attention to the persistent
pricinganomaliesduetofundingilliquidity. GarleanuandPedersenarguethatfundingilliquiditygive
risetothepricedeviationbetweensecuritieswithidenticalcash-flowbutdifferentmarginrequirements.
Therefore,theyshowthatthemispricingsbetweencreditdefaultswapandthecorrespondingcorporate
bondareassociatedwiththefundingilliquidityamongarbs. FontaineandGarciameasurethefunding
liquidity in U.S. treasury market through the price deviation between bonds with different ages but
similarcashflows,whichisbelievedtobecausedbyfundingliquidityacrossdifferentbonds. Nagel,on
theotherhand,constructaproxyforreturnofliquidityprovisionintheequitymarket(NYSE,AMEX
and Nasdaq stocks) from a reversal strategy that buy (sell) stocks with poor (good) past performance
and find supporting evidence of its relationship with funding costs, supply and other broad measures.
The empirical findings of this paper also relate to earlier work showing the pricing consequence
of the innovation in arb’s funding liquidity. First, on the average (aggregate) level, funding liquidity
leads to asset pricing consequences and predicts the risk appetite, proxied by VIX index of finan-
cial intermediaries (Adrian and Shin, 2010); It is also predicted by the VIX index (Nagel, 2012), since
BrunnermeierandPedersen(2009)arguethathighermarketvolatilityenhancesthescarcityofcapital;
Funding liquidity is associated with the return premia in fix-income securities, i.e. increase in funding
liquidity results in lower excess bond returns (Fontaine and Garcia, 2012). Akbas (2015, 2016) docu-
ments that high capital flows to hedge funds leads to stronger mean reversion in stock prices, which
5SeediscussioninDrehmannandNikolaou(2013)formoredetail.
6Schuster and Uhrig-Homburg (2015), in their empirical analysis, use three measure of the kind for intermediaries
capital condition: market volatility, TED spread and dividend yield, which capture the frictions and scarcity of inter-
mediaries’capital.
7Chordia et al. (2005)andFleckensitein et al. (2014) usethe flows into bond, equity funds and hedge fundsas the
measureforfundingconditioninfinancialintermediaries.
7
leads to lower pricing anomalies. Second, studies like Comerton-Forde et al. (2008), Schuster and
Uhrig-Homburg (2015) and Drehmann and Nikolaou (2013) provide evidence for the nonlinear pricing
consequence during different market circumstances. They document that lack of funding liquidity
leads to future market illiquidity, and the effect is most sensitive when funding constraint is binding.
While we identify the period of binding funding constraint by the sign of implied arbitrage efficacy,
and find statistically significant empirical evidence of the nonlinear consequence, Comerton-Forde et
al. select the threshold for binding funding constraint at its 25th percentile exogenously; Schuster and
Uhrig-Homburg determine the nonlinear relationship endogenously within a regime-switching model;
Drehmann and Nikolaou determine the stress regime of binding funding constraint by the ex post
events, such as the 2007-2008 financial crisis.
The paper proceeds as follows. In section 2, we introduce the marginal correction as a measure
for funding liquidity under the limited arbitrage framework of Shleifer and Vishny (1997) and Stein
(2009), while in section 3 we provide the empirical design to best capture the marginal correction
fromthetheoreticalwork,aswellasthedatadescriptionandapplicationofS&P500indexandE-mini
future. Section 4 summarize the empirical results, including the time series plot of arbitrage efficacy
with cases studies in the crisis period, the relationship with other measures of funding liquidity and
the aggregate and conditional impact of innovation in funding liquidity. Finally section 5 concludes.
2. The Model
2.1. Market structure
We consider the market structure similar to Shleifer and Vishny (1997), where an asset with a
fundamental value, V, trades for three periods, t = 1, 2, 3. At period 1, a pessimistic liquidity shock
ofsize,s pushestheassetpriceawayfromfundamental,whichcorrespondstothefiresalesamongthe
1
financial institutions suffering from severe funding constraint. Then arbs without funding problems
yet attempt to correct the mispricing error, and prevent a liquidation of assets at distressed prices.
Denoting the arb’s total (arbitrage) fund as f , we derive the market clearing price by
1
P =V −s +f . (1)
1 1 1
There exist two different market states at period 2. First, under a bad state, selling pressure deepen
such that sb >s with a probability, q >0, in which case the price becomes:
2 1
Pb =V −sb +f , (2)
2 2 2
where sb is the size of the liquidity shock and f is the total investment fund available at period 2.
2 2
Next, under a good state with a probability, 1−q, liquidity shock disappear (i.e. sg = 0), and the
2
asset price thus converges towards fundamental:
Pg =V. (3)
2
2.1 Market structure 8
Finally, at period 3, price is assumed to revert to fundamental:
P =V (4)
3
We follow Shleifer and Vishny (1997) and maintain the assumption that arbs are risk-neutral.
Further,wefollowStein(2009)andallowarbstoemployleveragetoexploitthemispricingopportunity
as most hedge funds seek arbitrage capital from outside financiers in practice. Arbs hold an equity, fe
1
and borrow funds through short-term debt, fd8. Thus, the total arbitrage fund available at period 1
1
becomes the sum of equity and debt:
f =fe+fd. (5)
1 1 1
Hedge funds maintain leverage mainly through short-term borrowing. If arbs would be able to access
external capital without any friction, then they would be able to always eliminate any mispricing,
and guarantee the law of one price. Thus, the leverage position of arbs will become an important
determinant behind arbitrage efficacy.
In practice, however, they are faced with and limited by several financial constraints. To accom-
modate such financial constraints observed in the real world, we introduce the equity constraint and
theleverageconstraint. First,giventhatthe(long-term)equityprovidersdonotwithdrawtheirfunds
at period 2, we assume that the equity is constrained by
fe <sb, (6)
1 2
such that equity supply cannot be guaranteed to cover the potentially deepening liquidity shock in
the future9. For an ample equity supply fe ≥sb, arbs are able to fully correct the mispricing without
1 2
leverage and enforce the law of one price by investing, f = s at period 1, and f = sb at period 2.
1 1 2 2
Therefore, by imposing the equity constraint, we can focus on the situation where mispricing errors
cannot be corrected with equity funding, and leverage is adopted to support.
Next, we define leverage as the ratio of total asset to equity:10
TotalAsset (cid:40) 1+ f1d for fd ≥0 (cid:41)
L= = f1e 1
Equity 1 for fd <0
1
We then allow arbs to strategically determine the level of short term debt, fd. We assume that fd is
1 1
repaid in full at period 2. This implies that arbs can only invest their equity (i.e., f = fe) under a
2 2
bad state with sb > s in period 211. We now introduce the upper and lower bounds of fd, denoted
2 1 1
8Withoutlossofgeneralityweassumezerointerestrate
9OursettingofequitysupplyisdifferentfromthatinStein(2009). WhiletheequitysupplyinStein’smodelcanbe
infinitebutwithacapitalcostperdollar,ourmodelimposeaconstraintonthesizeofequity,butequityisfreetoarbs.
The result is however similar, such that an equity constraint or a positive cost of capital will prevent arbs from fully
mispricing correction. Under the equity constraint imposed in our model, arbs are induced to use leverage to exploit
mispricings.
10Whenlending,assetisalwaysequaltoequity.
11Hedge funds’ capital consists of equity capital supplied by the investors and (possible) long-term debt financed
by?? during a potential funding crisis (e.g. Bunnermeier and Pedersen (2009) and Ang et al. (2011)). Investors can
withdrawtheircapital,soequityisnotalwayslockedintothefirmindefinitely. Thus,inordertomaintainandtoprotect
2.2 Optimization problem 9
D and D such that
U L
D ≤fd ≤D (7)
L 1 U
ThelowerboundisgivenasD =−fd,indicatingthatarbscanlendtheirequityinfulltootherarbs.
L 1
The upper bound imposes the leverage constraint above which arbs are not able to raise the leverage
fund.
2.2. Optimization problem
Hedgefundmanagersmakeanoptimalleveragedecisionasafunctionoftheinvestmentstrategies,
the risk-return trade-offs and the cost of leverage, all subject to the leverage constraint imposed by
external investors. Similarly, arbs in our model ex ante manipulate their leverage in response to the
riskyarbitrageopportunitysubjecttotheequityandleverageconstraint. Theyarefacedwithasimple
trade-off: arbs are induced to raise as much short-term debt as they can, to invest in period 1 and to
exploit the positive return when price converge towards fundamental value. On the other hand, arbs
may take a cautious leverage position, in order to capitalize on a better opportunity in period 2 if bad
state occurs. The equity available at period 2 under good and bad states, denoted fg and fb, can be
2 2
expressed as
(cid:18)V (cid:19) (cid:18)Pb (cid:19)
fg =fe+f −1 and fb =fe+f 2 −1 . (8)
2 1 1 P 2 1 1 P
1 1
Arbs maximize their expected total wealth at the end of the period under perfect competition, which
is given by
V
E(fe)=(1−q)fg+q fb. (9)
3 2 Pb 2
2
Maximizing the expected total wealth at period 3 in (9) subject to the leverage constraint, (7), we
derive the optimal amount of the short-term debt, fd by the first order conditions:
1
 
fd =D for R1 <R2
 1 L 
D ≤fd <D for R1 =R2 (10)
L 1 U
 fd =D for R1 >R2 
1 U
where R1 = V − 1 is the return of investing in period 1 and holding to price convergence, and
(cid:16) P1(cid:17)
R2 =q V −1 representstheexpectedreturnofinvestinginperiod2. Itisclearfrom(10)thatarbs
Pb
2
canselectoneofthreeequilibriumleveragestrategies(seealsoStein, 2009). ForR1 <R2, theoptimal
decision is not to enter the market at period 1 (the waiting strategy with f = 0), since waiting for
1
the future opportunity provide a higher expected return. For R1 >R2, arbs opt to borrow as much as
they can to exploit the return of investing in period 1, but subject to the binding funding constraint
(the max-leverage strategy with fd = D ). Only when the two returns are indifferent, R1 = R2, the
1 U
fundingliquidity,theyimposeinitiallock-upperiodsandredemptionperiodspriortowithdrawal. Otherarrangements
such as side pocket, gate limits and withdrawal suspensions are also employed. Hedge funds can also raise capital on
liabilityside. Themainsourceofleverageforhedgefundsare(1)collateralizedborrowingfinancedthroughrepomarket;
(2) collateralized borrowing financed by the hedge fund’s prime broker; (3) implicit leverage using derivatives, either
exchangetradedoroverthecounter. Leverageplaysacentralroleinhedgefundmanagement. Hedgefundsuseleverage
totakeadvantageofmispricingopportunitiesbybuyingtheunderpricedandshortingtheoverpriced. Hedgefundsalso
manipulateleveragetorespondtochanginginvestmentopportunityset.
2.3 Leverage setting 10
partial or cautious investment strategy becomes optimal with D ≤fd <D . The partial-investment
L 1 U
strategy contains two sub-strategies; the dry powder strategy and the partial-leverage strategy. Arbs
with the dry powder strategy are unlevered and invest only a part of their equity, i.e. D ≤ fd < 0,
L 1
while arbs with the partial-leverage strategy are willing to be partially levered with 0≤fd <D .
1 U
As leverage is not actively used under waiting and dry powder strategy, our further investigation,
withoutlossofgenerality,willfocusonpartial-andmax-leveragestrategy. Whiletheoptimalleverage
strategy for partial-leverage strategy can be easily found under first order condition by R1 =R2, the
strategy for max-leverage strategy is strongly determined by the upper leverage limit, D , as we will
U
discuss next.
2.3. Leverage setting
It is important to understand how the financiers set the upper leverage limit, D , and arbs then
U
raise the short-term leverage up to D to exploit the arbitrage opportunity. In practice, such leverage
U
setting depends critically on the financiers own predictions on future price movements. Brunnermeier
and Pedersen (2009) assume that the estimated future price volatility consists of fundamental and
liquidity volatility; Informed financiers are able to distinguish the two different type of volatility and
obtainthecorrectestimationofthefuturepricevolatility,whileuninformedfinanciersonlyacknowledge
the total price volatility and thus exaggerate the future price volatility.
Similarly in our model, future price movement is associated with the adverse liquidity shock under
bad state at period 2, sb. We first assume that outside financiers acknowledge the arbitrage strategy
2
at period 1, but may be less informed than arbs about the future, such that they have to predict sb
2
conditionalontheavailableinformation12. Denotingtheirestimateass˜b,financiersareabletopredict
2
the future distressed price P˜b and arbitrage equity f˜b in bad state, such that
2 2
P˜b = V −s˜b +f˜b (11)
2 2 2
(cid:32) (cid:33)
f˜b = fe−(cid:0)fe+fd(cid:1) P˜2b −1 (12)
2 1 1 1 P
1
Secondly, in order to determine the size of leverage offering to arbs, we assume that competitive
financiers set the rate of return as the riskless rate (zero in our model). In other words, financiers
must ensure that the potential loss in a bad state at period 2 must be recovered by arb’s equity. This
no-defaultconditionatperiod2canbepresentedas: f˜b ≥0,suchthatfinancierssetlimitsonleverage
2
debt fd to protect themselves against the adverse future price movements13. The upper leverage limit
1
12In Shleifer and Vishny (1997), outside investors are assumed to be blank about the arbitrage strategy due to the
required specialized knowledge and opacity within institutions, and update their belief about arbs from their past
performance. In our model, financiers provide short-term leverage debt to arbs. In practise, short-term leverage debt
aremostlyfinancedthroughrepomarket,wherearbscanbefinanciers,whichissimilartoourmodelwhenarbsadopt
thewaitinganddry-powderstrategy. Thusfinanciershavetheinformationaboutstrategiesadoptedbyarbs.
13Theleveragesettinginourmodelissimilartothemarginsettinginothertheoreticalpaper,suchasBrunnermeier
and Pedersen (2009) and Gromb and Vayanos (2010). Margin or haircut is defined as the difference between the asset
priceandthecollateralvalue,whichmustbefinancedthrougharbitrageur’sowncapital. Leverage,ontheotherhand,
capturestheratioofthetotalassettoequity,thatis,assetpriceovermargin. Thushigherleveragelimitimplieslower
marginrequirement,andthusmorecapitalofferingtoarbs.