The Full Story of Runs∗

Jun Kyung Auh† and Hayong Yun‡

July 2017

ABSTRACT

We examine the dynamics of credit supply during a repo run. Upon the receipt of an initialbad signal, lenders relax their requirements to help the borrower avoid default. As morenegative news arrives, the lenders’ patience is exhausted and they rapidly reduce the creditsupply until the borrower defaults. Lenders with a greater exposure and a closer lendingrelationship have a longer period of patience, and their credit subsequently contracts at amore gradual pace. The initial patience of lenders is consistent with a feedback channelbetween endogenously-chosen collateral requirements and lenders’ decisions about collateralwhen supplying credit. Substantial collateral provides lenders a higher recovery rate uponliquidation, but it simultaneously incentivizes lenders to abandon their borrower prema-turely. The feedback exacerbates a run, as lenders further tighten collateral requirementsupon a weak signal about borrowers’ fundamentals. Lenders factor in this destabilizing roleof collateral when setting ex-ante collateral requirements. Our findings suggest a novel roleof collateral: a medium of lender (mis)coordination. They also highlight the importance oflenders’ incentives in collateralized loans.

Keywords: Bank Run, Collateral Requirement, Lender Coordination, Lending Relationship

JEL classification: D86, G33, G34, K22.

∗We thank Jennie Bai, Sandeep Dahiya, Charles Hadlock, Naveen Khanna, and seminar participants at GeorgetownUniversity, and Michigan State University for helpful comments; and Hojong Shin and Chris Yun for data assistance.All errors are our own. Please address correspondence to the authors via email.†Finance Department, McDonough School of Business, Georgetown University. 37th and O Streets NW, Hariri

583, Washington, DC 20037. Phone: 202-687-2783. Email: junkyung.auh@georgetown.edu.‡Department of Finance, Eli Broad College of Business, Michigan State University, 645 North Shaw Lane, Room

339, East Lansing, MI 48824, Phone: 517-884-0549. Email: yunhayon@msu.edu.

How do lenders behave (e.g., adjust their prices and non-price contractual features) following a

sequence of negative signals about their borrower’s ability to repay the debt? Especially, when the

loan is secured by specific collateral, how do lenders dynamically adjust their credit supply before

a run is triggered? Several episodes in the repurchase agreement (repo) market from the 2008-2009

financial crisis call for a better understanding of these questions. However, due to a lack of available

data, little has been documented on the ex-ante dynamic behavior of lenders with collateralized

loans. In this paper, we fill this gap by examining lenders’ decisions on credit contraction during a

period of borrower stress, using micro-level data from repo contracts. Repo is a type of short-term

collateralized borrowing, and is a dominant source of short-term liquidity for financial and non-

financial firms. In addition to its systematic importance, repo contracts provide an ideal laboratory

in which to study lenders’ behavior because of its short-term nature of repo: lenders make rollover

decisions and reset contract terms at a high frequently.1

Economists pay attention to a potential destabilizing mechanism of the collateral requirement in

a repo contract: the haircut, or the degree of overcollateralization. Lenders use the haircut to

adjust their credit supply and risk exposure. Several studies show that raising haircut during

stress periods can destabilize the credit market empirically (Copeland et al., 2014; Gorton and

Metrick, 2012; Krishnamurthy, Nagel, and Orlov, 2014), as well as theoretically (Martin, Skeie, and

Thadden, 2014a,b). These works document a sharp increase in average aggregate haircuts during

the 2008-2009 crisis, which eventually led to the collapse of the repo market.2 These run studies

mostly focus on aggregate post-run behavior (or the onset of runs). However, these papers do not

consider an important ex-ante aspect of runs: what do lenders do when faced with a possible run in

the near future? Lenders could immediately jump to a run, or they could try to diffuse a potential

run by providing financial slack, such as relaxed covenants or reduced collateral requirements.

Moreover, such decisions on whether to run or to attempt to rescue borrowers could be driven

by lenders’ heterogeneous incentives. Our study focuses on dynamic lender behavior the pre-run

period, especially the impact of lender incentives on their credit contract decisions.1Prior to the 2008-2009 crisis, aggregate outstanding repo activities were estimated to be around $5 trillion to

$10 trillion (Copeland, Martin, and Walker, 2014; Gorton and Metrick, 2012). Currently, they are estimated to bearound $5 trillion (Baklanova, Copeland, and McCaughrin, 2015).

2For example, Figure 4 in Gorton and Metrick (2012) shows a steep increase in the repo-haircut index (equallyweighted average haircut) during 2007-2009.

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Using daily repo contract data for a large hedge fund (we refer to it as Fund X hereafter) that

eventually collapsed, we find that haircut trends exhibited a U-shaped pattern during the three-

month period prior to the end of this fund’s life. Despite the negative news regarding the fund, we

observe an initial decrease in haircuts: at its lowest, the reduction in the average haircut was around

2% (out of an average haircut of 3.7%). However, as negative news continue to accumulate, the

pattern flipped: haircuts began to increase rapidly, which eventually triggered the fund’s failure.

Our results contrast with the monotonic increase in haircuts in the U.S. repo market in the 2007-

2009 period (e.g., Gorton and Metrick (2012)). This increase in aggregate average haircuts reflect

both changes in the composition of collateral (e.g., the liquidation of high quality assets associated

with low haircuts during adverse market conditions) and the tightening of lending conditions (e.g.,

the increase in lenders’ risk aversion for the same collateral). To examine the latter (i.e., pure

haircut changes for rolled-over loans), we focus on micro (lender and loan) level haircut changes in

a matched loan sequence.

The relaxation of credit supply restrictions upon receiving negative news about the borrower is con-

sistent with lender coordination to collectively avoid exacerbating runs. One of the key trade-offs

in the rollover decision is the payoff from forcing the borrower to foreclose the project versus the

payoff from taking a risk by rolling over and hoping for higher returns when the project succeeds.

That is, collateral provides an outside option for lenders when they make rollover decisions. Ex-

cessively high and liquid collateral, therefore, may cause runs even when economic conditions are

strong. Given this trade-off, lenders internalize the knowledge that a larger collateral size increases

the likelihood of a run, and use that knowledge when setting a collateral amount ex-ante.

We then examine whether there is variation in the timing of haircut changes across lenders, and

if this variation is explained by certain lender characteristics. Regarding decision timing, we focus

on two particular points: the moment when lenders start decreasing haircuts after negative news

on the borrower surfaces (Initial Response), and the duration of the decreased haircuts before the

pattern reverses (Lender Patience). We find that each lender significantly drops haircuts at different

times, and there is a substantial variation in the duration of the relaxed credit supply via lowered

haircut. Out of the sixteen lenders that Fund X had lending relationships with during our sample

period, eight had a U-shape haircut pattern during the last three months of this fund’s life. Four

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lenders monotonically decreased haircuts till the end, and four did not significantly change their

haircuts. News that Fund X might be in trouble because of its sub-prime mortgage investments

began to spread approximately three months prior to the eventual shutdown of the fund. Lenders

start decreasing their haircut as early as 70 days prior to the collapse and as late as 32 days prior

to the collapse (excluding those who did not significantly change their haircuts). The duration of

Lender Patience ranged from 0 to 57 days.

To examine how the timing of these actions are associated with lender properties, we consider two

lender-specific characteristics: loan exposure to Fund X and the strength of the lending relationship

(measured by the number of days since the lender’s first trade with Fund X). These variables

measure lenders’ vested interest in a borrower’s financial health and its expected impact on lenders’

haircut decisions. In four different specifications (linear, Cox, Weibull, and accelerated failure time

(AFT) models), we find that loan exposure (measured by the log of the total loan amount) shortens

the Initial Response and lengthens Lender Patience. For example, in the linear specification, a one

standard deviation increase in loan exposure leads to a three week faster Initial Response and

two more weeks of Lender Patience. Similarly, the strength of the lender’s relationship with the

borrower tends to shorten the Initial Response delay and lengthen the duration of the haircut-

reducing period: a one standard deviation increase in the strength of the lending relationship leads

to one week faster Initial Response and one additional week of Lender Patience.

We further investigate variations in Initial Response and Lender Patience at the loan level, tracking

rollover points within each holding position. In addition to the two lender-level incentive variables

(lender-level principal and the strength of the lending relationship), we add loan-level characteris-

tics: (log of) the principal of each loan, and a short-term loan indicator (below median maturity).

Even at the loan level, the total principal aggregated at the lender level is significantly and neg-

atively related to Initial Response and significantly and positively related to Lender Patience.

However, at the loan level, the lender-level strength of relationship (while still the same sign as the

lender-level regressions) loses its statistical significance. Instead, we find that loan-level variables

are significantly related to Initial Response and Lender Patience. We find earlier reduction in hair-

cut and longer Lender Patience period when loan principals are larger and when loan maturities

are shorter. These findings are consistent with the notion that lenders with a strong incentive to

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support the survival of the borrower (larger loans and a long relationship with borrower) are more

willing to give favorable lending terms to the borrower in the face of a possible run, up to a certain

point. They step in earlier and continue the rescue effort longer.

Finally, we show that collateral type also plays an important role in lenders’ actions. While the

dominant asset class in our sample is structured finance securities, government and corporate bonds

take some fraction of the collateral. Setting government bonds as a benchmark, we show that lenders

step in earlier and continue the rescue attempt longer when the collateral is from a less liquid asset

class, such as structured finance. Further, structure finance, the main asset class in the sample, can

be broken into two categories: collateralized debt obligation (CDO) and pass-through mortgage-

backed security (MBS). Due to additional layers of securitization, CDOs are typically more opaque

and illiquid than vanilla MBSs. Using this liquidity-related variation within the structured finance

category, we find a similar pattern: loans against CDOs are associated with longer Lender Patience

than those against MBSs. We repeat this analysis with the contemporaneous credit ratings of

each structured finance asset as of t0, and find that loans against lower ratings are associated with

greater rescuing attempts from lenders. These results consistently suggest that lenders who have

hard-to-sell assets as collateral are more incentivized to save the borrower by relaxing the credit

supply. In other words, if lenders’ loans have relatively liquid collateral, the collateral provides

better liquidation options, and they have less interest in the borrower’s survival.

Our paper relates to several strands of the literature and makes at least two new contributions.

First, this is the first paper to document non-monotonic (U-shaped haircut trends) behavior during

creditor runs. Prior studies on repo market failure document a unidirectional rundown deterioration

during the 2008-2009 financial crisis (Gorton and Metrick, 2012). Our work shows that at the

individual lender level, the “run” phenomenon is more subtle: upon the arrival of bad news about

a borrower, lenders initially decrease their haircut, hoping to revive the borrower. However, after

a period of patience with accumulating negative signals, lenders give up hope, and they rapidly

increase their haircuts until default. That initial period of credit relaxation and patience has never

before been documented in the repo contract literature.

Second, this paper identifies a distinct role of collateral that has not been studied in the extant lit-

erature. The conventional view of collateral is that it complements or substitutes for loan covenants

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and thereby influences banks’ monitoring/screening incentives. Unlike conventional collateral of-

fered by industrial borrowers, collateral for the co-funding of financial assets is not indispensable to

the borrower’s whole production. Therefore, repo collateral provides an outside option for lenders

during rollovers, which can cause coordination failure among lenders. Despite the prominent role

of collateral (haircuts) in repo borrowing and the large impact of the repo market on financial

markets, the way in which lenders choose haircuts has not been extensively studied.3 Our study

provides new insight on lenders’ choice of the amount of collateral (e.g., haircuts in repo loans),

while accounting for its destabilizing impact during rollovers, i.e., the likelihood of repo runs.

The road map of the paper is as follows. Before our empirical analysis, Section I provides institu-

tional details for the repo market, a description of the extant literature on the subject, and state

hypotheses for the empirical tests. Section II describes the sample and variables used for this study.

Section III presents empirical findings on lender behavior via collateral requirements during periods

of disstress for the borrower. Section IV concludes.

I. Background

A. The Repo Market and Haircutss

Repo it the primary financing channel for financial institutions. It is composed of two inter-

connected markets: the tri-party and the bilateral repo markets. Through the tri-party repo

market, money market mutual funds (and security lenders) lend cash to dealer banks against

a particular set of collateral (typically safe assets such as U.S. Treasury securities). Using the

proceeds from the tri-party repo market, dealer banks make loans (in the bilateral repo market) to

hedge funds who use these funds to finance acquisition of risky assets. One of the distinguishing

feature between these two markets is the existence of a centralized clearing house: the tri-party

repo market consists of three parties (a lender, a borrower and a clearing house) whereas bilateral

repo market only involves a lender and a borrower without any clearing system.

Repo is a type of secured lending contract. A repo borrower borrows cash from a repo lender by3Exceptions are Martin et al. (2014a) and Martin et al. (2014b) who, in a dynamic general equilibrium model,

link collateral constraints to conditions on a run.

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pledging collateral. At that time, the borrower promises to purchase back the collateral from the

lender (thus it is called a repurchase agreement or repo) at a predetermined price after a certain

time period. For example, suppose a lender borrows $90 by pledging an asset whose market value

is $100. The borrower makes a promise to buy back the asset by paying $90.45 after 1 month.

As seen in this illustration of a 1-month loan, the loan price can be calculated by the difference

between the lending amount ($90) and the promised amount to repurchase ($90.45), yielding 50

bps per month or a 6% per annum repo rate. Also, the value of the collateral at the time of loan

initiation ($100 in our example) is typically larger than the loan amount ($90). Such a difference

measures the degree of overcollateralization: we say the haircut is 10% ($100/$90-1) in this case.

As a repo contract rolls over, the lending terms can also dynamically change. In particular, the

lender can effectively run on the borrower by increasing the haircut. Deciding not to roll over is a

special case of making the haircut 100%.

In the middle of 2007, pressured by big losses in the subprime mortgage market, Fund X collapsed.

Table 1 summarizes the major news regarding Fund X during this period. About 90 days prior

to the failure, negative news on this fund started to emerge, and by about 45 days prior to the

collapse, the market became widely aware of the potential risk of the fragile subprime mortgage

market. In the last month of the fund’s life, negative news about it accumulated. In the final

week before the fund’s collapse, multiple news items appeared regarding the details of the fund’s

losses, lender negotiations, and restructuring plans. However, the hedge fund could not recover,

and collapsed in the middle of 2007.

[Place Table 1 about here]

B. Hypotheses

What is unique about this episode is that the overcollateralization (the haircut on the repo) ex-

hibited a non-monotonic change over time during the final days of the fund.4 Figure 1 shows the

changes in overall haircut (Panel A), spread (Panel B), and duration (Panel C) during the final4According to Table 2 of Gorton and Metrick (2012), the average haircut of the U.S. repo market was constant

during the first half of 2007, which coincides with our sample period.

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three months of the fund. In the figure, each panel also displays the number of negative news items

about the fund (bar chart) and the outstanding total loan amount (dotted line) over the same

period. Panel A shows that the haircut significantly decreases in response to the initial negative

news. However, after a period with favorable margin requirements (the period of lender patience),

lenders rapidly increase the margin requirement until the borrower’s default, which results in a U-

shaped pattern for the haircut. This haircut raise is contemporaneously accompanied by complete

withdrawals of lending (rollover failures), as seen by the rapid drop in the loan balance. This kind

of loan withdrawal would correspond to a haircut of 100%, but would also disappear from the data.

Therefore, the increasing part of the U-shaped pattern is substantially understated. As opposed

to the clear pattern in the haircut, spread and duration do not show any comparable dynamics.

This stark contrast implies that, in a period of extreme borrower stress, lenders use this particular

non-price term, the haircut, rather than other lending terms, to manage their risk exposure.5

[Place Figure 1 about here]

The relaxation of credit supply restrictions through a reduction in haircut upon receiving negative

news about the borrower is consistent with lenders’ effort to avoid exacerbating runs.6 The amount

of collateral is a (mis)coordination device: when the liquidation of collateral (or declining to roll

over a loan) before the completion of an investment project is inefficient, greater collateral exacer-

bates the coordination failure and helps trigger a run. In other words, excessively strict collateral

requirements raise the risk of a coordination failure among lenders and a premature termination of

the loan.

The key trade-off in the rollover decision is the payoff from forcing the borrower to foreclose the

project versus the potential payoff from taking a risk by rolling the loan over and hoping for higher

returns when the project succeeds.7 In this setting, collateral provides an outside option for lenders

when they make roll over decisions. Excessively high collateral, therefore, may cause runs even5Prior works on credit rationing (e.g., Stiglitz and Weiss (1981)) also emphasize the prominent role of non-price

terms (i.e., credit rationing) when prices (i.e., interest rates) cannot fully resolve market imperfections.6The observed lender patience, as shown in Figure 1 is also consistent with lenders postponing a run (an increase

in haircut) in order to gather more information on the borrower’s bad news. This reason is not mutually exclusiveand can coexist with the collateral-lender coordination channel. However, this information acquisition channel cannotexplain why only the haircut (not prices nor maturity) is used for learning. The collateral-lender coordination channelcan provide a more direct explanation for why mainly haircut is adjusted during the lender patience period.

7A strategic-complementarity-based model to illustrate this tradeoff is provided in the Appendix.

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when economic conditions are strong. Given this trade-off, lenders internalize the knowledge that

a greater collateral size increases the likelihood of a run, and use that knowledge when setting a

collateral amount ex-ante.

Because of the results showns in Figure 1, we focus on haircut dynamics in the following analysis.

Moreover, beyond this overall pattern, we examine how the timing and pace of the haircut changes

differ by lenders and collateral types.

Prior studies on the lending relationship, such as Petersen and Rajan (1994, 1995) show that lenders

with a vested interest in a borrower’s continuation tend to offer more favorable lending terms when

the borrower is in trouble. For this study, we measure lenders’ vested interest in the borrower using

their outstanding loan amount and the length of their lending relationship with the borrower. The

impact of lenders’ incentives on lender patience is summarized in the following hypothesis:

Hypothesis 1. Lenders with larger loans and a longer lending history with the borrower offer

more favorable lending terms when the borrower is in trouble.

The dominant asset types of collateral in repo contracts are financial securities, which are more eas-

ily separable from the borrower’s operation than non-financial assets. This characteristic contrasts

with corporate debt contracts (e.g., corporate bonds), which typically pledge indivisible production

assets as collateral. Seizing or auctioning them off can, therefore, lead to a disruption in the pro-

duction process and have a negative impact on the borrowing firm’s credit quality. Through this

channel, conventional collateral constrains creditors to stay with the firm and mitigates liquidation

bias. However, repo is secured by easily separable assets that could be sold without foreclosing the

whole operation. Together with the short-term nature of repo contracts, the separability impedes

coordination among lenders and makes repo collateral very different from corporate loan collateral.

While greater collateral provides increased recovery upon foreclosure, it makes it harder for lenders

to coordinate with each other to avoid premature foreclosure; in other words, the probability of a

self-fulfilling run increases with the size of the collateral. In this sense, repo collateral plays a role

similar to the one that credit default swap (CDS) plays in corporate debt contracts: it provides

outside options for lenders and thereby weakens incentives for lenders to stay with the borrower in

8

times of financial distress (Bolton and Oehmke, 2011). The impact of asset separability on lender

coordination (and lender patience) is summarized in the following hypothesis:

Hypothesis 2. The asset separability (liquidity) of collateral is inversely proportional to lender

patience.

In Section III, we explain what drives this heterogeneity in haircut dynamics across lenders upon

receiving negative news about the borrower, based on these two hypotheses.

C. Extant Literatures

Bank run, or running on the borrower, is one of the central themes in economics, especially in

the financial intermediation literature. Early theories describe the run as a bistable equilibrium

(Diamond and Dybvig, 1983). Recently, Acharya, Gale, and Yorulmazer (2011), Brunnermeier and

Oehmke (2013), Gorton and Ordonez (2014), He and Xiong (2012), and Bebchuk and Goldstein

(2011) make further theoretical contributions, with models that explain a run in a dynamic context.

Empirically, however, observing such dynamic runs at an individual loan level has been difficult due

to a scarcity of run events and data limitations. Several prior studies have made an attempt to do

this. Schmidt, Timmermann, and Wermers (2016) document a run in the money market. Iyer and

Puri (2014) show a link between geographic ties and runs in local banks. Hertzberg, Liberti, and

Paravisini (2011) consider lender coordination during credit registry expansions. Using repo data,

Copeland et al. (2014), Krishnamurthy et al. (2014), and Gorton and Metrick (2012) document

the repo run during the 2007 to 2008 crisis, showing that repo lenders actively increased the

haircut during the run. Most theories about and empirical evidence on runs describe a monotonic

relationship between borrowers’ creditworthiness and lending terms: deterioration in a borrowers’

financial health leads to rapidly worsening lending conditions (fewer loans and/or higher loan prices)

until the borrower collapses.

On the other hand, theories on collateral have mainly been motivated by the substitution effect

of lender screening (Bester, 1985; Inderst and Mueller, 2007; Rajan and Winton, 1995).8 In these8Prior literature also extensively studies other contractual features, such as maturity (Berglof and Thadden, 1994),

9

models, more collateral reduces the incentive for a lender to screen a borrower because the ex-ante

loss upon default is low when the loan is sufficiently collateralized. In relation to these two strands of

literature, this paper provides empirical evidences that are consistent with the interaction between

collateral and lenders’ run behavior from the perspective of strategic complementarity, which was

pioneered by Carlsson and van Damme (1993) and further developed by Morris and Shin (1998).9

II. Data

A. Sample Construction

We use proprietary data that consists of the complete repo position of multiple fixed-income hedge

funds. From 2004 to 2007, the funds actively traded mostly securitized bonds and other structured

finance securities, typically backed by underlying loans such as mortgages (mortgage backed secu-

rities or MBSes), commercial loans (collateralized loan obligations or CLOs) and other securitized

products themselves (collateralized debt obligation or CDO). The funds took leverage positions on

such assets by borrowing money from multiple dealer banks via secured loans, specifically bilateral

repo contracts. In other words, those funds were borrowers and the dealer banks were secured

lenders. They themselves were the in-house hedge funds of a large dealer bank, and they differ only

in leverage ratios. Therefore, we treat them as a single borrower without explicitly distinguishing

them. The raw data covers 2004 to 2007 and consists of 297,606 daily observations (16,807 repo

agreements, 54 lenders, and 1590 unique CUSIPs). While the full data spans the three-year time

period, we limit our focus to the last three-month period leading up to the collapse of the fund.

Hence, the data that we use in this paper is a subset of the data used by Auh and Landoni (2016),

except that we include loans from the affiliated lender that are excluded from their study because

they focus on interactions among contract terms in a general context. The selection of the three-

month study period is based on news flows. The negative news regarding this borrower started

and covenants (Rajan, 1992). Recent empirical studies on collateral include Calomiris, Larrain, Liberty, and Sturgess(2017), Campello and Larrain (2016), and Vig (2013).

9Other notable works on global games include Angeletos and Werning (2006), Angeletos, Hellwig, and Pavan(2007), Campos (2013), Chen, Goldstein, and Jiang (2010), Ciccone and Costain (2004), Dasgupta (2007), Frankel(2012), Frankel, Morris, and Pauzner (2003), Goldstein and Pauzner (2005), Guimaraes and Morris (2007), He andManela (2016), Izmalkov and Yildiz (2010), Morris and Shin (2006), and Sakovics and Steiner (2012).

10

appearing about 90 days prior to the fund’s collapse. In order to identify rollover contracts, we

follow a procedure proposed by Auh and Landoni (2016): We sequentially match contracts that

finance the same asset position, where the later contract is a roll-over of an earlier one. In addition,

we require the later contract to follow the earlier one in rapid succession (within one day). These

matched rollover contracts have the same lender and the fundamental loan risk at succession is

nearly identical (they are at most one day apart with the same borrower, lender, and underlying

collateral). The final sample leaves 584 loans from 16 lenders.10 We refer to Auh and Landoni

(2016) for detailed information regarding the general data structure, variables and construction

methods.

During the sample period, this fund took out repo loans from sixteen different lenders, including

the affiliated lender. On average, each lender made loans against 89 different items of collateral

to finance the fund’s investment positions. Each position is a set of consecutive individual loans

because, as a current loan matures, the fund rolls over the loan into the subsequent one to keep

its holding position. During the sample period, on average, each position consists of 5 loans, i.e.,

the fund rolled over each loan for 4 times on average, and the initial loan is also included in the

sample. While the loan amount changes over time, as of the start date of our sample, the average

loan size was $28.4 million per each position.

B. Description of Variables

The key variable of our study is the timing of events. The starting point of our sample is when the

first significant negative news item on this firm surfaces. As shown in Table 1, the first negative

news appeared 88 days before the collapse. Hence, we set out a reference point (t0) as the day three

months prior to the failure date. This date is common to all lenders. From this reference date,

the period of main interest includes moments that lenders actively decreased and subsequently

increased their haircuts. Our result shows that the earliest date of the initial haircut drop occurred

about two weeks after t0. We refer to all dates during our sample period as days relative to the

reference point t0.10The composition of collateral types is CDO (55.31%), corporate bond (12.84%), government security (2.05%),

MBS (13.18%), and other structured finance (16.61%).

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There are three key event dates that characterize the run dynamics: the first date that each lender

decreases its haircuts after t0 is defined as t1; the date that each lender increases its haircuts after

t1 is defined as t2; and the date that loans by all lenders are terminated is defined as t3 (hence, t3

is common to all lenders). These four dates are illustrated in Figure 2.

[Place Figure 2 about here]

Table 2 shows event dates for each lender. These dates are computed based on the average daily

haircuts of each lender. To pin down the timing, we rely on the following procedures. For each day,

we compare the haircuts of new repo contracts with those of the maturing contracts that the new

contracts replace. This methodology has the merit of controlling for the potential effect of asset

composition changes on haircut dynamics. Under adverse financing conditions, a borrower may

liquidate high quality assets, which are likely to be associated with lower haircuts. In this case,

without any action by lenders, the haircuts would appear to increase, because the remaining assets

are financed with a higher haircut. Therefore, we need to be able to shut off this channel to observe

a pure haircut move as a consequence of the lender’s reaction. The initial response of a haircut

drop has great variation. t1 ranges from t0 + 19 at the earliest to t0 + 55 at the latest (excluding

no change case). The haircuts start increasing again after the dropping period: t2 occurs mostly

ranging from t0 + 55 to t0 + 68. Loan termination occurred mostly on the last day (t3), but some

lenders terminated as early as t0 + 74.

[Place Table 2 about here]

The lender-driven haircut dynamics during this period of borrower stress can be classified by four

patterns: (i) U-shaped (haircuts initially decrease and then subsequently increase), (ii) monotonic

decrease until the end, (iii) monotonic increase until the end, and (iv) no significant change. As

shown in Table 2, most lenders exhibit the U-shaped haircut pattern, while four lenders have no

change, and three lenders show a monotonic decrease in haircuts until the end of the sample period.

For the U-shaped pattern, starting from t0, all three dates t1, t2, and t3, are well defined. For a

monotonically decreasing pattern, t1 and t3 are well defined, and we set t2 = t3 (or t2 = t3−ε more

rigorously), implying that a lender attempts to increase their haircut in the last minute before the

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fund’s collapse. For the pattern without a significant change in haircuts, we set t1 = t2 = t3 (or,

more rigorously, t1 = t2 − ε; t2 = t3 − ε), indicating that the lender attempts to decrease and then

immediately increase their haircuts, but it is too late, since by the time of the attempt, the loan is

terminated by the fund’s bankruptcy.

We now construct two key variables of interest corresponding to the timing variables: Initial Re-

sponse, which is defined as the time it takes for a lender to make a haircut drop, i.e., t1 − t0, and

Lender Patience, which is defined as the length of the period that a lender gives favorable lending

terms, i.e., t2 − t1. To explain variations in these two variables, we consider several lender-specific

variables. First, we measure lenders’ loan exposure to the borrower by taking the natural logarithm

of their total principal outstanding as of t0. Also, we measure the lending relationship between a

lender and the borrower using the natural logarithm of the number of days since the first loan to

the borrower by each lender (as of t0), assuming that if a lender made loans to the borrower for

a longer period, it has a closer relationship with the borrower (Petersen and Rajan (1994), and

Petersen and Rajan (1995)). Given that our original data starts with this fund’s inception point,

we have full information to calculate this variable.

Column V of Table 2 shows the pattern of Initial Response, sorted by the outstanding loan amount

as of t0. Lenders with a larger exposure to the borrower appear to have a smaller value for Initial

Response, indicating that they respond more quickly to negative news by decreasing their haircuts

at some point after t0. Column VI of Table 2 shows the pattern of Lender Patience, sorted by

outstanding loan size as of t0. Lenders with a greater loan exposure tend to have a larger value

for t2 − t1. This result implies that lenders with a greater loan exposure maintain their reduced

haircuts for longer before they give up and raise haircuts until the borrower fails.

Table 3 shows summary statistics on the variables considered in this paper. On average, lenders

waited for 6 weeks (42.5 days) following t0 before decreasing their haircuts. More than half of the

lenders waited 31 days before decreasing their haircuts. One reason for such a long delay is that we

set the time of the initial bad news outbreak conservatively, to account for potential private leaks

to lenders before the information appeared in news available to the public. Informal discussions

with repo market participants suggest that, a little before the first public news surfaced, market

participants were already speculating that the fund’s heavy exposure to subprime mortgages could

13

cause some trouble. Based on the news summary shown in Table 1, negative news articles started

appearing more frequently one month after t0. Also, note that there is substantial heterogeneity

in Initial Response, with a standard deviation of 23.85 days. This large variation appears to be

driven by lender characteristics related to their vested interest in the borrower: lenders with a

greater interest in the borrower’s survival are incentivized to offer more favorable lending terms

(by reducing their haircuts early). Lender Patience lasts, on average, almost a month (28.81 days),

and there is also a large variation here, with a standard deviation of 18.72 days. Most banks start

reducing their haircuts by t0+31, and negative news starts coming out more frequently after t0+45.

It is remarkable that lenders continue to lower the haircut and keep it at this lower level despite

the negative news for a substantial period of time.

[Place Table 3 about here]

Among other variables, the average loan amount outstanding is above $1 billion (with a mean of

$1.03 billion and a median of $0.772 billion). However, there are lenders with very small exposure

and lenders with very large exposure: a 5th percentile loan amount is $15 million and 95th percentile

loan amount is $2.5 billion. Despite truncation before t0, there is a great variation in relationship

length as well. The mean of relationship length is 930 days, with a standard deviation of 214 days.

The 5th percentile is 140 days, whereas the 95th percentile is 1000 days, which is the same as the

median (suggesting truncation). Panel B shows loan-level summary statistics. The average loan

size is $28 million, with great variation: the bottom 5th percentile is $1.31 million and the top 5th

percentile is $95.04 million. The mean maturity is 25.78 days, with a bottom 5th percentile of 3.22

days and top a 5th percentile of 40.11 days.

Table 4 shows correlations among the key variables used in this paper. Initial Response is negatively

related to the other key variables. The very strong negative correlation (-90%) between t1− t0 and

t2 − t1 suggests that lenders stepping in early to reduce haircuts are also likely to maintain the

decreased haircuts for longer before giving up and running on the fund. The correlation between

t1 − t0 and log(principal) is -0.81, suggesting that lenders with a larger loan exposure tend to

step in earlier to decrease their haircuts despite negative news against the borrower. Similarly,

the correlation between t1 − t0 and log(relationship) is -0.41, demonstrating that lenders with a

14

longer history with the borrower tend to help the borrower during hard times by reducing haircuts.

Lender Patience (t2−t1) is positively related with log(principal) and with log(relationship), showing

a correlation of 0.7 and 0.38, respectively. This suggests that lenders with a greater vested interest

(in terms of a larger loan exposure and longer lending history) tend to wait more patiently for the

borrower’s revival. Finally, log(principal) and log(relationship) have a very low correlation, which

indicates that these two variables measure two distinct aspects of lenders’ vested interest in the

borrower. As shown in Panel B, correlations of loan-level variables are consistent with those at the

lender level, shown in Panel A.

[Place Table 4 about here]

III. Determinants of Lender Behavior Dynamics

In this section, we explore lenders’ decisions about supplying credit when they know there is the

potential for a run. We investigate their dynamics though their haircut changes because the degree

of overcollateralization (or collateral requirements) characterize the lenders’ willingness to extend

credit against a unit of collateral. Since Fund X eventually experienced failure at the end of the

sample period, one may expect, at the aggregate level, lending terms to be tightened at some point.

However, our high frequency and micro-level data allow us to observe the full dynamics of the credit

rationing pattern by each lender, rather than just the aggregate eventual outcome. Furthermore,

such granularity facilitates investigation into the sources of the cross-lender variation in haircut

dynamics.

A. Lender-Specific Analysis

We first hypothesize that a lender with a larger vested interest in a borrower’s continuation is

incentivized to offer more favorable lending terms when the borrower is in trouble, as predicted by

Petersen and Rajan (1994, 1995). For this study, we measure lenders’ vested interest in the borrower

by outstanding loan amount and the length of their lending relationship with the borrower. Figure

3 shows survival rates on Initial Response (Panel A) and Lender Patience (Panel B) for lenders of

15

large (above median) and small (below median) loan amounts. Survival rates for Initial Response

are lower for lenders with a large loan amount (i.e., those lenders have shorter t1 − t0), and those

of Lender Patience are higher for lenders with a large loan amount (i.e., those lenders have longer

t2 − t1). Both figures, therefore, suggest that lenders with a vested interest in the borrower (as

measured by the size of loan) tend to act early to offer relief when the borrower is in trouble and

tend to have more patience before abandoning the borrower and running on their repo loans.

[Place Figure 3 about here]

For more formal analysis, we consider various empirical specifications, and examine the effect of

lenders’ vested interest on their behavior upon receiving negative signals about the borrower. The

dependent variables that describe lender behavior are Initial Response and Lender Patience. The

key explanatory variables we use to measure lenders’ vested interest in the borrower are loan

exposure (i.e., the natural logarithm of the principal amount) and the strength of the lending

relationship (i.e., the natural logarithm of the number of days since the first day of loaning to the

borrower from the inception of the fund).

For each lender behavior (Initial Response and Lender Patience), we examine four different empirical

specifications: linear, Cox proportional hazard, Weibull, and accelerate failure time (AFT) models

(Cleves, Gutierrez, Gould, and Marchenko (2010)). The regression specification for the linear model

is as follows.

∆tj = α+ β · xj + εj , (1)

where ∆tj is Initial Response (t1 − t0) or Lender Patience (t2 − t1) of lender j, xj is a vector of

lender j’s characteristics, and εj is the error term. Although linear models are very intuitive, one

of their shortcomings is that they assume normally distributed residuals, which is unlikely to be

satisfied. In order to account for more realistic distributional assumptions for the residuals, we

consider the Cox proportional hazard model, which fits a nonlinear model of hazard function, as

specified below:

h (t|xj) = h0 (t) exp (xjβx) , (2)

where h0 (t) is the base hazard (not estimated), xj is the explanatory variable (loan exposure and

16

strength of lending relationship) for lender j, and βx is the hazard rate for variable x. The Cox

regression model does not make parametric assumptions on base hazard, h0 (t). As an alternative,

Weibull regressions do make parametric assumptions on the base hazard function, with the following

specification:

h (t|xj) = ptp−1exp (xjβx) , (3)

where xj , and βx are defined the same way as in the Cox model. Parameter p determines the shape

of the base hazard function (h0 (t) = ptp−1). While Weibull regression requires further parametric

assumptions, it permits explicit estimates on failure time. For both Cox and Weibull specifications,

a positive hazard rate (βx) indicates increasing hazard (reduced time to failure) for increasing x,

and a negative hazard rate indicates decreasing hazard (increased time to failure) for increasing

x. Finally, we consider an AFT specification because the Weibull model has an accelerated failure

time interpretation, i.e., there is a one-to-one mapping between these two models (Cleves et al.

(2010)). AFT estimates the following specification:

log (tj) = xjβx + β0 + uj , (4)

where uj follows the extreme value distribution. The advantage of this specification is that param-

eter estimates measure marginal effects on log-failure time, i.e., βx measures the increase in the

expected value of log(time to failure) due to an increase in underlying variable xj (accelerated stop-

ping time is τ j = tje−xjβx , so that when an acceleration parameter is larger than 1, e−xjβx > 1, time

passes more quickly for the lender and failure is expected to occur sooner). For duration models

(Cox, Weibull, and AFT), robust standard errors are used.

Table 5 reports the results of the Initial Response and Lender Patience regressions specified in

Equations 1 to 4. Columns I through IV show the results of Initial Response regressions for each

respective specification. The linear model in Column I shows that both loan exposure and the

strength of the lending relationship have a significant and negative effect on Initial Response: one

standard deviation increase in log(principal) (1.49 from Table 3) causes a nearly three week (19.1

days) decrease in the time of the lender’s Initial Response (to reduce haircuts), and a one standard

deviation increase in log(relationship) (0.49 from Table 3) speeds up Initial Response by more

17

than a week (9.3 days). Columns II and III report results from the hazard rate model in the

Cox (Equation (2)) and Weibull (Equation (3)) specifications. They show that both lender vested

interest variables, log(principal) and log(relationship), have significant and positive effects on the

hazard of failure, i.e., a larger principal amount and a longer relationship leads to an earlier Initial

Response. In order to make a time-dimensional interpretation, we show the results of Weibull-AFT

regressions (Equation (4)) in Column IV. Estimates on both explanatory variables are significantly

negative, indicating that a lender’s large vested interest contributes to an earlier Initial Response:

a one standard deviation increase in log(principal) leads to an earlier haircut reduction by the

estimated acceleration parameter of 0.67 (e−1.49 ×-0.265). This result means the Initial Response

(42.5 days) will be shortened to 28.6 days (0.67×42.5), resulting in a two week (42.5-28.6) reduction

from the mean. Similarly, a one standard deviation increase in log(relationship) leads to an earlier

reduction in haircut by an acceleration parameter of 0.85 (e−0.49 ×-0.33). This indicates that the

mean Initial Response (42.5 days) will be shortened to 36.23 days, resulting in a one week (42.5-

36.23) reduction from the mean. A comparison of the AFT results - two weeks for loan exposure

and one week for relationship - with those from the linear model - three weeks for loan exposure

and one week for relationship - shows that they are in the same direction with similar size.

[Place Table 5 about here]

Columns V to VIII of Table 5 show the results of Lender Patience regressions for each respective

specification. In Column V, the estimation of the linear model (Equation (1)) shows that both

lender interest variables, the log of the principal amount and log of the relationship, are significantly

positive: a one standard deviation increase in loan exposure leads to a two week or 12.9 days

(1.49×8.67) longer period of patience, t2 − t1 and a one standard deviation increase in lending

relationship leads to a one week or 6.9 days (0.49×14.05) longer period of patience, t2−t1. Columns

VI and VII show that hazard ratios estimated by the regression specifications in Equation (2) and

(3), respectively, are significantly negative for both lender interest variables (loan exposure and

the strength of the lending relationship), i.e., a lower hazard of failure or longer Lender Patience

period. For a time-dimensional interpretation, we estimate the AFT regression (Equation (4)) in

Column VIII, and show a significant and positive acceleration parameter of 3.07 (e1.49 ×0.75) for

loan exposure and 2.16 (e0.49 ×1.57) for lending relationship per one standard deviation increase in

18

each lender interest variable. These quantities imply that, from the mean of Lender Patience (28.81

days), a one standard deviation increase in loan exposure leads to 88.44 total days (28.81×3.07) of

Lender Patience, resulting in a 59.63 day (88.44-28.81) longer Lender Patience period. Similarly,

a one standard deviation increase in the strengths of the lending relationship leads to 62.23 days

(28.81×2.16) of Lender Patience, resulting in a 33.42 day (62.23-28.81) longer Lender Patience

period. These estimates are larger than those from the linear model, suggesting that the Weibull

model can be sensitive to parametric assumptions. Nevertheless, all regression specifications (linear,

Cox, Weibull, and AFT) strongly suggest that the lender’s increased vested interest in the borrower

leads to increased effort to rescue the troubled borrower via decreased Initial Response and longer

Lender Patience period.

B. Loan-Level Analysis

To fully exploit the granularity of our data, we further examine the impact of loan-level character-

istics on Initial Response and Lender Patience beyond the explanations provided by lender-specific

variables. The first four columns of Table 6 show the results on Initial Response (t1 − t0), using

combinations of lender-level and loan-level characteristics. As before, we consider four specifica-

tions from Equations (1) to (4): linear (Column I), Cox (Column II), Weibull (Column III), and

AFT (Column IV), respectively. As with the lender-level regressions, we find earlier reduction in

haircut (shorter Initial Response) when lender’s aggregate principal amount is larger (in log-scale).

For example, a one standard deviation increase in the log of the lender level aggregate principal

amount (1.25) leads to an Initial response that is 6 days (-4.848×1.25) earlier. In the AFT regression

(Column IV), a one standard deviation increase in log(Principal) (1.25) leads to an earlier haircut

reduction associated with the acceleration parameter of 0.94 (e1.25 ×-0.051). This result implies

that, from the mean (54.89 days), the Initial Response will be reduced to 51.37 days (0.936×54.89),

lowering the Initial Response by a half of a week (54.89-51.37).

[Place Table 6 about here]

Columns I to IV of the same table show that the parameter estimates of loan-level relationship

(the log of relationship length) variables have the same sign as those in the lender-level regressions

19

(Columns V to VIII in Table 5), but they are statistically insignificant. The impact of the loan-level

variables on Initial Response are weaker than those from the lender-level regressions. This might be

due to the fact that loan-level characteristics explain some of the lender-level effects. In other words,

the Initial Response of each loan may not be entirely driven by lender-specific variables, and the

lender may make different choices for each loan based on individual loan characteristics, ensuring

that aggregate changes across all loans for each lender meets its (lender-level) strategic goal. For

example, among loan-level (therefore collateral-specific) variables, the parameter estimates of loan

principal are significantly positive. In the AFT specification (Column IV), a one standard deviation

increase in the log of loan principal (1.37) leads to an earlier Initial Response by the acceleration

parameter of 0.92 (e−0.064 4×1.37). We can interpret this result as follows: the Initial Response

will be reduced to 50.5 days (54.89×0.92) from its mean (54.89 days). Overall, lenders start

dropping their haircuts (upon initial acknowledgment of bad news on Fund X) earlier for larger

loans (principal amount) and higher quality (investment grade) loans.

Columns V to VIII of Table 6 show the impact of lender-level and loan-level attributes on Lender

Patience (t2− t1). While lender-level loan exposure (the log of the lender-level aggregate principal)

is positively related to Lender Patience, the strength of the lender’s relationship with the borrower

is positive but statistically insignificant. For example, in the AFT specification, a one standard

deviation increase in lender loan exposure (1.25) is associated with a 1.78 (e0.461 ×1.25) times

longer Lender Patience period. This suggests that Lender Patience will be increased 20 days

(25.45×1.78-25.45) from its mean of 25.45 days. On the other hand, all three loan-level attributes

have a significantly positive impact on Lender Patience: larger loans, and shorter loans tend to have

longer Lender Patience periods. This result can be translated as follows: a one standard deviation

increase in the log of the loan principal (1.37) is associated with a Lender Patience period that is

1.18 (e0.123 ×1.37) times longer, and short-term loans (below the median maturity) are associated

with a Lender Patience period that is 1.36 (e0.309) times longer. These correspond to a Lender

Patience period that is 4.6 days (25.45×0.18) and 9 days (25.45×0.36) longer, respectively, for a

one standard deviation increase in the log of the loan principal and shortness of term from the

mean of 25.45 days.

Overall, the results in Tables 5 and 6 show that there is a large variation in lenders’ dynamic

20

patterns of credit supply during a period of borrower distress, and that can be largely explained

by lenders’ vested interests, which make each lender’s incentives different from the others’.

C. Collateral-Level Analysis

In this section, we study the effect of the asset separability (or liquidity) of collateral on Initial

Response and Lender Patience. When collateral can easily be sold in the secondary markets, we

expect lenders to be less incentivized to rescue a borrower from financial difficulties, because they

can liquidate the collateral without incurring much loss.

[Place Table 7 about here]

Table 7 shows results using Weibull accelerated failure time models. Columns I to III show Initial

Response based on collateral characteristics. Column I considers all 584 loans included in our

sample. Relative to loans against Treasury securities, ones pledged using structured finance assets

(CDO, MBS, and other structured finance products) and corporate bonds are associated with a

faster Initial Response. For example, loans against structured finance have an Initial Response that

is 24 days earlier than the sample mean (54.89 days × (1− e−0.578)). Loans against corporate bonds

have an Initial Response that is 20 days earlier (54.89 days × (1− e−0.453)). This result implies

that the time in takes for the lender to begin its attempt to rescue the borrower is associated with

illiquidity of collateral. Specifically, we find that loans against structured finance, followed by those

against corporate bonds, lead to an earlier Initial Response than the much more liquid case of

Treasury securities.

Since the dominant asset class of Fund X is structured finance, we focus only on this class of asset,

which mostly contains CDOs and pass-through MBSs.11 Relative to pass-through vanilla MBSs,

CDOs are much more opaque and illiquid due to additional layers of securitization (i.e., a CDO is a

pool of MBSs). Column II shows no significant variation of Initial Response based on the collateral

type within the structured finance class.

We repeat the previous analysis using the credit ratings of the structured finance assets as of t0 in11There is a small fraction of hard-to-classify sub-asset classes, such as asset-backed securities (ABS).

21

Column III. We expect collateral with higher credit ratings to be more liquid and, hence, lenders to

have fewer reasons to be borrower-friendly (so, a slower Initial Response). As expected, collateral

with a high (AAA or AA) credit rating has a slower Initial Response. For example, AAA-rated

collateral has an eight day slower Initial Response (54.89 days × (e0.140 − 1)).

Columns IV to VI show Lender Patience based on collateral characteristics. Consistent with our

previous results on Initial Response, lenders are more patient and maintain their reduced haircut

for longer for loans against less liquid collateral. Column IV considers Lender Patience by collat-

eral types for all loans. Loans with structured finance (CDO, MBS, and other structured finance)

collateral have longer Lender Patience periods than ones pledged with Treasury securities or cor-

porate bonds. In contrast to our results on Initial Response (in Column II), the results in Column

(V) show that, within structured finance, CDO-pledged loans are associated with greater Lender

Patience: CDO-pledged loans have an 11 day (25.45 × (e0.373 − 1)) longer Lender Patience period

than vanilla MBS pledged loans. Also, Column VI indicates that loans pledged with collateral with

high credit ratings receive less Lender Patience. For example, loans with AAA-rated collateral

have an 8.6 day shorter Lender Patience period than those pledged with A-rated collateral (25.45

× (1− e−0.413)).

Overall, the results in Table 7 show that lenders start decreasing their haircuts earlier and keep

decreasing haircuts for longer for loans with less liquid collateral.

IV. Conclusion

In this paper, we show that the amount of collateral (or haircuts in repo contracts) has a destabi-

lizing feedback effect by increasing the likelihood of a run. Because of this destabilizing feedback,

lenders tend to be patient when deciding on the amount of collateral: when choosing a high level

collateral, lenders must balance the potential for better recovery should the borrower default with

their increased likelihood of prematurely abandoning the project (by running the borrower). Dur-

ing the last months of a large hedge fund, we observe a U-shaped pattern in the haircut trend:

lenders initially decrease haircuts to avoid exacerbating the borrower’s default risk. After a favor-

able period with a relaxed credit supply (a reduced haircut), as more bad news about the borrower

22

accumulates, lenders fail to coordinate and rapidly increase their haircuts until the borrower de-

faults. Closer examination shows that the incentive of lenders to keep the borrowing firm alive is

critical in their haircut choices: lenders with a larger loan exposure and a stronger prior lending

relationship with the borrower extend the relaxed credit supply more quickly, and they are more

patient with credit contraction, raising their haircut later than those lenders that had less loan

exposure to the borrower and a weaker relationship with them.

While this paper exploits empirical evidence in the repo market, our insights have general impli-

cations. The risk that insuring against a future downside can create time-inconsistency problems

(e.g., a premature run) once misfortune (e.g., borrower default) occurs is applicable to a wide vari-

ety of financial contracting circumstances. Also, the new role of collateral as a lender coordination

mechanism requires further study. Casual observation suggests that the impairment of a borrower’s

production by collateral liquidation is tied to the separability of the collateral, which is determined

by the nature of asset type. This, in turn, influences the degree to which collateral requirements

interact with lender coordination regarding rollover decisions. Further investigation is needed to

clarify interrelations among these factors. We leave it for future works.

23

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Vig, Vikrant, 2013, Access to Collateral and Corporate Debt Structure: Evidence from a NaturalExperiment, Journal of Finance 106, 881–2928.

26

Figure 1. Average Change in Haircut by Repo Lenders Until Default: This figure shows the relativechanges in haircut (Panel A), spread (Panel B), and duration (Panel C) by lenders, Fund X’s outstanding loan amount(dotted line), and the number of negative news articles (bar graph) about Fund X. Haircut (spread, duration) as oft0 is normalized to zero for each lender. Specifically, we estimate the following regression:

Terml,t = αl + Σ0i=−88βi · Ii + εl,t,

where Term refers to each lending term (Haircut, Spread and Duration), αl is lender fixed effects and It is a dummyvariable that is 1 only on each corresponding day. The number of news articles is computed by counting the numberof negative news articles on Fund X found in a Factiva search.

05

1015

20Nb

r. of

Arti

cle /

Out

stan

ding

Loa

n Am

t. (in

bn.

)

-6

-4

-2

0

2Ha

ircut

Cha

nges

-88 -80 -72 -64 -56 -48 -40 -32 -24 -16 -8 0Time From Failure

95% CI Mean # Articles Loan Amt.

(a) Haircut

05

1015

20Nb

r. of

Arti

cle /

Out

stan

ding

Loa

n Am

t. (in

bn.

)

-.05

0

.05

.1

.15

Spre

ad C

hang

es

-88 -80 -72 -64 -56 -48 -40 -32 -24 -16 -8 0Time From Failure

95% CI Mean # Articles Loan Amt.

(b) Spread

05

1015

20Nb

r. of

Arti

cle /

Out

stan

ding

Loa

n Am

t. (in

bn.

)

-60

-40

-20

0

20

Dura

tion

Chan

ges

-88 -80 -72 -64 -56 -48 -40 -32 -24 -16 -8 0Time From Failure

95% CI Mean # Articles Loan Amt.

(c) Duration

27

Figure 2. Timeline for a Repo Run: The starting point of our sample is one day before the first significantnegative news on this firm surfaced (88 days before default), which we set as t0. After this date, there are three keyevent dates during the repo run: the first date on which each lender decreased their haircut after t0 is defined as t1,the date on which each lender increased their haircut after t1 is defined as t2, and the date when all loans from eachlender were terminated is defined as t3.

t0 t1 t2 t3

28

Figure 3. Kaplan-Meier Plot Across Fund X’s Lenders: This figure shows the Kaplan-Meier survivalestimates for Initial Response and Lender Patience for two groups based on loan exposure (below and above themedian loan amount against Fund X).

����

����

����

����

����

6XUYLYDO�5DWHV

� �� �� �� ��,QLWLDO�5HVSRQVH��'D\V

6PDOO�/RDQ/DUJH�/RDQ

.DSODQ�0HLHU�6XUYLYDO�(VWLPDWHV

(a) Initial Response (t1 − t0)

0.00

0.25

0.50

0.75

1.00

Surv

ival

Rat

es

0 20 40 60Lender Patience: Days

Small LoanLarge Loan

Kaplan-Meier Survival Estimates

(b) Lender Patience (t2 − t1)

29

Tab

le1.

Neg

ativ

eN

ews

Art

icle

son

Fund

X:

Thi

sta

ble

show

sth

eda

tes

(Col

umn

I),n

umbe

r(C

olum

nII

),an

dco

nten

ts(C

olum

nII

I)of

new

sar

ticle

sre

late

dto

Fund

Xar

ound

the

perio

dw

hen

itco

llaps

ed(8

8to

0ev

entd

ates

).N

ews

artic

les

are

colle

cted

from

Fact

iva

usin

ga

keyw

ord

sear

chfo

rFu

ndX

’spa

rent

com

pany

’sna

me

inm

ajor

new

sou

tlets

(e.g

.,W

SJ,N

YT

,WP,

and

Forb

es).

We

drop

new

sar

ticle

sth

atar

eno

tre

late

dto

Fund

Xits

elf,

such

asFu

ndX

anal

yst’s

com

men

tson

othe

rst

ocks

.

Eve

ntD

ate

No.

ofA

rtic

les

Con

tent

(I)

(II)

(III

)

871

Fund

Xm

akes

loss

from

lend

ing

tom

ortg

age

inve

stm

ent

firm

.85

1C

omm

ent

byFu

ndX

onsu

bprim

em

ortg

age

trou

ble.

671

Fund

Xst

ock

drop

sdu

eto

subp

rime

mor

tgag

etr

oubl

e.52

1Fu

ndX

com

men

tsas

subp

rime

mor

tgag

etr

oubl

esu

rfac

es.

492

Fund

Xin

volv

edin

insi

der

trad

ing.

452

Fund

Xw

illw

rite

dow

ndu

eto

deva

luat

ion

ofon

eof

itsdi

visi

on.

391

Fund

Xm

arke

ttim

ing

scan

dal.

351

Fund

Xea

rnin

gsdo

wng

rade

.24

1Fu

ndX

sued

infix

ing

trad

ing

cost

.22

1Fu

ndX

subp

rime

mor

tgag

etr

oubl

e.17

1Fu

ndX

hurt

bysu

bprim

em

ortg

age

loan

.15

3Fu

ndX

isfa

cing

mor

tgag

elo

sses

.14

4Fu

ndX

drop

inne

tin

com

e.12

1Fu

ndX

incu

rbi

glo

ss.

111

Fund

Xle

nder

sm

eet

toke

epfu

ndafl

oat.

92

Fund

Xfa

cesh

utdo

wn.

87

fund

X’s

pare

ntco

mpa

nyst

aves

offco

llape

ofits

fund

s;Fu

ndX

shar

edr

ops.

71

Fund

Xsu

rviv

esfo

rno

w.

66

Fund

X’s

pare

ntco

mpa

nyw

illpu

t$3

.2B

tosa

veits

hedg

efu

nd;n

egat

ive

impa

cton

stoc

k&

junk

bond

mar

ket.

53

Fund

Xre

scue

plan

for

one

ofits

hedg

efu

nd;l

ende

rsse

llco

llate

ral.

41

Mor

em

eltd

owns

like

Fund

Xex

pect

ed.

35

Fund

Xne

gativ

ely

impa

cts

stoc

km

arke

t;IP

Oca

ncel

latio

nby

Fund

Xre

late

dfir

m.

26

SEC

Fund

XC

DO

prob

e;Fu

ndX

’spa

rent

com

pany

won

’tpr

ovid

ead

ditio

nalf

undi

ngfo

rfa

iling

hedg

efu

nd.

16

Fund

Xco

llaps

ed.

01

Rep

utat

ion

dam

age

topa

rent

com

pany

ofFu

ndX

30

Tab

le2.

Sum

mar

yof

Eve

ntD

ates

Dur

ing

aR

epo

Run

in20

07:

Thi

sta

ble

sum

mar

izes

maj

orev

ent

date

s.C

olum

nI

show

sth

eto

tal

amou

ntof

prin

cipa

lout

stan

ding

atth

ebe

ginn

ing

ofth

esa

mpl

epe

riod

(88

days

prio

rto

defa

ult)

.C

olum

nII

show

sth

eda

teon

whi

chea

chle

nder

star

tsde

crea

sing

thei

rha

ircut

.C

olum

nII

Ish

ows

whe

nea

chle

nder

star

tsin

crea

sing

thei

rha

ircut

.C

olum

nIV

show

sw

hen

each

lend

erte

rmin

ated

thei

rlo

ans.

Col

umn

Vsh

ows

Initi

alR

espo

nse,

whi

chis

the

num

ber

ofda

ysbe

twee

nth

ebe

ginn

ing

ofsa

mpl

epe

riod

(whe

nne

gativ

ene

ws

first

com

esou

t)an

dw

hen

each

lend

erst

arts

decr

easi

ngha

ircut

.C

olum

nV

Ish

ows

Lend

erPa

tienc

e,w

hich

isth

enu

mbe

rof

days

betw

een

whe

nea

chle

nder

star

tsde

crea

sing

thei

rha

ircut

,and

whe

nea

chle

nder

star

tsin

crea

sing

thei

rha

ircut

agai

n.C

olum

nV

IIis

the

hairc

utpa

tter

n,w

hich

isU

-sha

peif

the

hairc

utin

itial

lyde

crea

ses

and

then

incr

ease

sag

ain,

noch

ange

ifth

ere

isno

sign

ifica

ntch

ange

sin

the

hairc

utun

tillo

ante

rmin

atio

n;an

dde

crea

seif

the

hairc

utde

crea

ses

until

loan

term

inat

ion.

Lend

erC

hara

cter

istic

sE

vent

date

sH

airc

utC

hara

cter

istic

s

Lend

erP

rinci

pal

Hai

rcut

Hai

rcut

Loan

Initi

alR

espo

nse

Lend

erPa

tienc

eH

airc

utPa

tter

nID

($M

illio

n)D

ecre

ase

(t1)

Incr

ease

(t2)

Term

inat

ion

(t3)

t 1−t 0

(Day

s)t 2−t 1

(Day

s)

(I)

(II)

(III

)(I

V)

(V)

(VI)

(VII

)

1424

95.3

758

21

3056

Dec

reas

e13

2453

.07

5928

329

31U

-sha

pe5

2183

.70

7029

018

41U

-sha

pe9

2085

.74

7030

018

40U

-sha

pe4

1610

.26

5928

129

31U

-sha

pe8

901.

9565

231

2342

U-s

hape

188

0.07

6529

823

36U

-sha

pe17

803.

1152

21

3650

Dec

reas

e10

741.

1435

1711

5318

U-s

hape

1162

0.18

5222

836

30U

-sha

pe3

590.

8458

1211

3046

Dec

reas

e12

496.

089

87

791

No

chan

ge6

344.

9016

1514

721

No

chan

ge16

300.

4452

1615

3636

Dec

reas

e2

23.4

92

10

861

No

chan

ge15

15.2

56

54

821

No

chan

ge

31

Table 3. Summary Statistics: This table shows number of observations (Column I), mean (Column II), standarddeviation (Columns III), and 5/50/95-percentiles (Columns IV to VI) of the key variables used in this study. PanelA shows lender-level summary statistics for the 16 lenders of this study. Panel B shows loan-level summary statistics.Event time variables are Initial Response and Lender Patience. Initial Response is the first time each lender reducestheir haircut after the bad news began minus the time when the bad news first came out (t1− t0). Lender Patience isthe time when each lender increases their haircut minus the time when each lender decreases their haircut (t2 − t1).Variables on lender characteristics are loan amount, natural logarithm of the principal, relationship (in days), andthe natural logarithm of relationship. Variables on loan characteristics are the principal amount and its logarithmand maturity (in days). Loan amount is the total outstanding principal from each lender as of the beginning of thesample period (88 days prior to default or event date 88). Relationship is the number of days (as of event date 88)since a lender first had dealings with the borrower in the database, which starts in 2004.

Panel (a)

Variables N Mean Std.Dev. p5 p50 p95

(I) (II) (III) (IV) (V) (VI)

Event timesInitial Response, t1− t0 (Days) 16 42.5 23.85 18 33 86Lender Patience, t2− t1 (Days) 16 28.81 18.72 1 33.5 56

Lender characteristicsPrincipal ($Million): Lender 16 1030 849 15 772 2500Log(Principal): Lender 16 20.17 1.49 16.54 20.46 21.64Relationship (Days) 16 930 214 140 1000 1000Log(Relationship) 16 6.77 0.49 4.94 6.91 6.91

Panel (b)

Variables N Mean Std.Dev. p5 p50 p95

(I) (II) (III) (IV) (V) (VI)

Event timesInitial Response, t1− t0 (Days) 584 54.89 25.97 21 52 86Lender Patience, t2− t1 (Days) 584 25.45 23.56 1 28.5 63

Lender characteristicsPrincipal ($Million): Lender 584 1511.48 792.22 23.49 1610.26 2495.37Log(Principal): Lender 584 20.76 1.25 16.97 21.2 21.64Relationship (Days) 584 982.3 102.14 950 1000 1000Log(Relationship) 584 6.87 0.23 6.86 6.91 6.91

Loan characteristicsPrincipal ($Million): Loan 584 28.28 57.22 1.31 11.25 95.04Log(Principal): Loan 584 16.24 1.37 14.09 16.24 18.37Maturity (Days) 584 25.78 13.43 3.22 30.26 40.11

32

Table 4. Correlations: This table shows correlations among major event periods during the repo run and keydeterminants. Panel A shows the correlations for the lender-level variables. Panel B shows correlations among theloan-level variables. Major event periods are Initial Response (t1 − t0) and Lender Patience (t2 − t1). In PanelA, correlations among event times, the natural logarithm of principal and the natural logarithm of relationship areshown. In Panel B, correlations among event times, the log of principal (lender), the log of relationship (lender), logof principal (loan), and the short-term repo (below median) indicators are shown.

Panel (a)

Initial Response Lender Patience Log(Principal)t1− t0 (Days) t2− t1 (Days) Lender

Lender Patience, t2-t1 (Days) -0.90Log(Principal): Lender -0.81 0.70Log(Relationship) -0.41 0.38 0.02

Panel (b)

Initial Lender Log- Log-Response Patience Principal: Log- Principal:

t1 − t0 (Days) t2 − t1 (Days) Lender Relationship Loan Short-term

Lender Patience (t2 − t1) -0.91Log(Principal): Lender -0.24 0.25Log(Relationship) -0.04 0.06 0.08Log(Principal): Loan -0.14 0.04 0.06 -0.11Short-term -0.02 0.06 0.19 0.05 -0.18

33

Tab

le5.

Det

erm

inan

tsof

Init

ialR

esp

onse

and

Len

der

Pat

ienc

eat

the

Len

der

Lev

el:

Thi

sta

ble

show

sth

ede

term

inan

tsof

the

maj

orev

ent

perio

dsdu

ring

the

repo

run

inth

ele

nder

-leve

ldat

a.D

epen

dent

varia

bles

are

maj

orev

ent

perio

ds:

Initi

alR

espo

nse

(t1−t 0

)an

dLe

nder

Patie

nce

(t2−t 1

).Fo

rbo

thde

pend

ent

varia

bles

we

use

four

diffe

rent

spec

ifica

tions

:or

dina

ryle

ast

squa

res

(Col

umn

Ifo

rt 1−t 0

and

Col

umn

Vfo

rt 2−t 1

),th

eC

oxpr

opor

tiona

lhaz

ard

mod

el(C

olum

nII

fort 1−t 0

and

Col

umn

VI

fort 2−t 1

),W

the

eibu

llre

gres

sion

(Col

umn

III

fort 1−t 0

and

Col

umn

VII

fort 2−t 1

),an

dth

eW

eibu

llre

gres

sion

inth

eac

cele

rate

dfa

ilure

time

(AFT

)m

etric

(Col

umn

IVfo

rt 1−t 0

and

Col

umn

VII

Ifo

rt 2−t 1

).In

the

Cox

prop

ortio

nalh

azar

dm

odel

s(C

olum

nsII

and

VI)

,w

efit

h(t|x

j)

=h

0(t

)exp

(xjβ

x),

whe

reh

0(t

)is

base

haza

rd,x

jis

the

expl

anat

ory

varia

ble

for

lend

erj,

andβ

xis

the

haza

rdra

tefo

rva

riabl

ex

.C

olum

nsII

and

VI

repo

rtha

zard

rate

.A

posi

tive

haza

rdra

tein

dica

tes

anin

crea

sing

haza

rd(r

educ

edtim

eto

failu

re)

for

incr

easi

ngx

,and

ane

gativ

eha

zard

rate

indi

cate

sa

decr

easi

ngha

zard

(incr

ease

dtim

eto

failu

re)

for

incr

easi

ngx

.In

the

Wei

bull

regr

essi

ons,

we

estim

ate,h

(t|x

j)

=ptp−

1exp

(xjβ

x).

We

repo

rtha

zard

rate

san

dlogâĄ

ą(p)in

Col

umns

III

and

VII

.The

acce

lera

ted

failu

retim

etr

ansf

orm

sth

eW

eibu

llha

zard

mod

elin

tolo

g(t

j)=

xjβ

x+β

0+u

j,w

hereu

jfo

llow

sth

eex

trem

eva

lue

dist

ribut

ion.

For

dura

tion

mod

els,

robu

stst

anda

rder

rors

are

used

.St

anda

rder

rors

are

show

nin

pare

nthe

ses

and

are

sign

ifica

ntat

the

1%(*

**),

5%(*

*),

and

10%

(*)

leve

ls.

Dep

ende

ntVa

riabl

esIn

itial

Res

pons

e(t

1−t 0

)Le

nder

Patie

nce

(t2−t 1

)

(I)

(II)

(III

)(I

V)

(V)

(VI)

(VII

)(V

III)

Log(

Prin

cipa

l)-1

2.78

988*

**1.

3434

2***

0.95

823*

**-0

.265

14**

*8.

6747

9***

-0.6

0383

***

-1.2

1017

***

0.75

367*

**[1

.944

40]

[0.2

8481

][0

.148

37]

[0.0

4822

][2

.131

33]

[0.1

4232

][0

.344

51]

[0.1

9534

]Lo

g(R

elat

ions

hip)

-19.

0419

3***

1.26

958*

**1.

1765

5***

-0.3

2554

***

14.0

5126

*-0

.976

48**

*-2

.524

03**

*1.

5719

2***

[5.9

3764

][0

.380

26]

[0.2

2609

][0

.043

91]

[6.5

0845

][0

.267

83]

[0.5

6409

][0

.116

31]

Con

stan

t42

9.35

038*

**-4

0.85

056*

**11

.303

14**

*-2

41.2

5901

***

36.5

7800

***

-22.

7801

3***

[55.

5937

8][4

.407

17]

[0.9

6606

][6

0.93

820]

[8.2

6205

][3

.724

85]

Log(

p)1.

2848

4***

1.28

484*

**0.

4735

6**

0.47

356*

*[0

.127

22]

[0.1

2722

][0

.185

29]

[0.1

8529

]

Mod

elLi

near

Cox

Wei

bull

Wei

bull

(AFT

)Li

near

Cox

Wei

bull

Wei

bull

(AFT

)Lo

gps

eudo

likel

ihoo

d-2

2.49

-3.4

5-3

.45

-27.

95-1

7.63

-17.

63W

ald

chi2

22.2

866

.34

99.7

018

.03

24.5

528

5.44

Pro

b>

chi2

0.00

0.00

0.00

0.00

0.00

0.00

R-s

quar

ed0.

8074

60.

6246

0O

bser

vatio

ns16

1616

1616

1616

16

34

Tab

le6.

Det

erm

inan

tsof

Init

ialR

esp

onse

and

Len

der

Pat

ienc

eat

the

Loa

nL

evel

:T

hist

able

show

sde

term

inan

tsof

maj

orev

entp

erio

dsdu

ring

the

repo

run

inth

elo

an-le

veld

ata.

Dep

ende

ntva

riabl

esar

em

ajor

even

tper

iods

:In

itial

Res

pons

e(t

1−t 0

)and

Lend

erPa

tienc

e(t

2−t 1

).Fo

rbot

hde

pend

entv

aria

bles

we

use

four

diffe

rent

spec

ifica

tions

:or

dina

ryle

asts

quar

es(C

olum

nIf

ort 1−t 0

and

Col

umn

Vfo

rt2−t 1

),th

eC

oxpr

opor

tiona

lhaz

ard

mod

el(C

olum

nII

fort

1−t 0

and

Col

umn

VI

fort 2−t 1

),th

eW

eibu

llre

gres

sion

(Col

umn

III

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35

Table 7. The Effect of Collateral on Initial Response and Lender Patience: This tables shows theeffect of the separability of the assets used as collateral on Initial Response and Lender Patience. All regressions usethe Weibull accelerated failure time (AFT) model, where robust standard errors are used. In Weibull regressions,we estimate, h (t|xj) = ptp−1exp (xjβx). The accelerated failure time transforms the Weibull hazard model intolog (tj) = xjβx +β0 +uj , where uj follows the extreme value distribution. Structured Finance is an indicator variablethat is one if the collateral is a CDO, MBS, or other structured finance product (CLO or ABS, etc.). Corporate isone if the collateral is a corporate bond. CDO is one if the collateral type is a collateralized debt obligation. MBS isone if the collateral type is mortgage backed security. Each credit rating indicator (AAA, AA, BBB, BB, and B) isone if the collateral has the corresponding credit rating as of t0. Log(Principal):Loan is the natural logarithm of theloan amount. Short-term is one if the loan maturity is below the median, and zero otherwise. We report log(p) forthe hazard rate estimate of the Weibull model. Standard errors are shown in parentheses and are significant at the1% (***), 5% (**), and 10% (*) levels.

Dependent Variables Initial Response (t1 − t0) Lender Patience (t2 − t1)

(I) (II) (III) (IV) (V) (VI)

Structured Finance -0.57793*** 3.91747***[0.06105] [0.16019]

Corporate -0.45295*** 3.41386***[0.06516] [0.19259]

CDO 0.00199 0.37254***[0.05098] [0.12820]

AAA 0.13995** -0.41298***[0.06071] [0.14542]

AA 0.10884* -0.14962[0.06503] [0.15373]

BBB -0.05597 -0.04055[0.09674] [0.16424]

BB -0.15481 0.34959[0.23186] [0.28790]

B -0.65455*** 0.38069***[0.05623] [0.10601]

Log(Principal): Lender -0.05205*** -0.07058*** -0.04883*** 0.45522*** 0.63742*** 0.46795***[0.00757] [0.00876] [0.00847] [0.08272] [0.04347] [0.08407]

Log(Relationship) -0.08591** -0.11431** -0.12705*** 0.23334 0.97181 0.48052[0.04376] [0.05821] [0.03763] [0.35044] [0.65883] [0.46473]

Log(Principal): Loan -0.06097*** -0.04985*** -0.06516*** 0.13895*** 0.05091 0.14267***[0.01345] [0.01636] [0.01490] [0.03501] [0.03739] [0.04119]

Short-term -0.04921 0.01368 -0.02887 0.28779*** 0.0681 0.15544[0.03844] [0.04551] [0.04045] [0.10374] [0.10885] [0.10801]

Constant 7.35287*** 7.12746*** 6.94956*** -14.26366*** -17.99945*** -12.05268***[0.47526] [0.57300] [0.42170] [2.83307] [4.65337] [3.39690]

Log(p) 0.86750*** 0.84110*** 0.85452*** -0.26945*** -0.16351*** -0.24110***[0.02441] [0.02617] [0.02574] [0.03597] [0.04790] [0.04003]

Log pseudoliklihood -413.75 -287.95 -350.75 -1098.20 -723.20 -909.89Wald chi2 147.90 85.58 1871.64 1013.51 260.15 299.19Prob >chi2 0.00 0.00 0.00 0.00 0.00 0.00Observations 584 400 491 584 400 491

36

Appendix-A. Model

In this section, we develop a theoretical framework to offer an insight into collective lender behav-

ior. Specifically, in a similar spirit to Morris and Shin (2004), we propose a lender coordination

channel through which lenders exhibit borrower-friendly behavior through collateral requirement

determination, i.e., the haircut. While data limits us to providing direct empirical evidence that

the observed haircut dynamics are driven by the strategic interaction of lenders, the model provides

general implications consistent with our empirical findings presented in later sections.

A. Model Setup

At date 0, a firm borrows from a continuum of lenders (total mass of 1) to finance a project. At date

2, the debt contract pays 1 +R if the debt rolls over and the project succeeds, and pays zero if the

project fails. If a lender chooses not to rollover at date 1, he collects the collateral, K0 ∈ [0, 1].12

The notation of K0 emphasizes the fact that the collateral level was predetermined at date 0, before

the intermediate (date 1) signal is realized.13 The success of the project depends on the number of

lenders rolling over and the underlying economic fundamental, θ. Each lender independently and

simultaneously decides whether to rollover or run (foreclose) the loan. We denote the fraction of

lenders deciding to not to rollover as l1 ∈ [0, 1]. The project succeeds if enough lenders choose to

roll over the loan, θ ≥ l1, and fails if too many lenders choose to foreclose the loan, θ < l1. Note

that 0 ≤ l1 ≤ 1, and hence, the project will always succeed if the economic fundamental is very

strong (θ ≥ 1) and will always fail when the economic fundamental is very weak (θ ≤ 0).

We assume that prior beliefs about the current period’s economic fundamental, θ, are normally

distributed with mean y0 and standard deviation σ0 (i.e., precision is τ0 = 1(σ0)2 ). While lenders

cannot directly observe the economic fundamental, they receive a private signal, x1 = θ + σ1 · ε1,

where ε1 follows a standard normal distribution (i.e., the precision of σ1·ε1 is τ1 = 1(σ1)2 ). We assume

12Figure A.1 shows the relationship between haircut and collateral in this model.13Here we assume that, upon deciding not to rollover, the lender seizes the collateral and liquidates it. In reality,

the borrower can liquidate the pledged collateral and repay the lender. However, these two scenarios are economicallyequivalent, as long as termination of the loan requires the borrower to foreclose the pledged asset. In other words,under the assumed restriction that the borrower cannot get any additional capital injection in the interim period,these two are effectively identical.

37

Figure A.1. Haircut and Collateral: At date 0, a borrower pledges asset, A, as collateral. The liquidation ofthe asset at an interim date 1 is inefficient and will only be worth L1, The asset at full maturity (date 2) can generateeither V or L2 depending on the realized state of economy at date 2. Assuming L2 < L1 < A forces lenders to facerisky choices between seizing higher value of collateral (L1) at date 1, or take risk for a gamble at date 2, whose worstcase will yield a lower payoff, L2. Note that majority of collateral in our data are non-government bond (98% suchas CDO, MBS) and incur liquidity cost. Given that the date 1 liquidation value is less than the amount of originalcollateral value (A), lenders may decide to request haircut (H), where 0 ≤ H ≤ A − L1, to reduce risk exposure oftheir loan (L), i.e., H +L = A. Lending L for collateral A(> L) implies overcollateralization at date 0 due to future(dates 1 and 2) possible depreciation of collateral value. Rescaling L − L2 = 1, L2 = 0, and K = L1−L2

L−L2transform

haircut choice into an equivalent collateral choice (K) of a unit loan.

A

H

L

L2

L1

K

1

Original scale(Loan size=L)

Modified scale(Loan size=1)

Le

that ε1 is identical and independently distributed across lenders. After receiving the private signal,

each agent’s updated posterior belief on the economic fundamental follows a normal distribution

with mean y1 = τ0y0+τ1x1τ0+τ1

and precision τ0 + τ1.

B. Endogenous Choice of Collateral (K0)

Once news on the economic fundamental arrives, each lender updates their belief on the economic

fundamental from their prior one, specified by a normal distribution, with a mean of y0 and τ0.

Based on their belief and the information structure at the moment that the rollover decision (rollover

or run) has to be made, each lender sets their collateral amount (K0) to maximize the expected

38

payoff from the rollover decision and subsequent payoff upon completion of the project. When the

collateral level is endogenously determined, similar steps as in Proposition 3 show that the critical

state is proportional to collateral amount, θ∗0 = K01+R (see Corollary 1 and its proof for details). Since

the critical state increases in collateral level (K0), lenders will endogenously determine the collateral

requirement (K0) by taking into account that a higher collateral level makes a run (because of a

higher critical state) more likely and reduces the chance of receiving a reward upon success (1+R).

Proposition 1 characterizes the optimal choice of collateral by lenders using their prior belief as

follows:

PROPOSITION 1: [Optimal Collateral Choice] The endogenous choice of collateral takes extreme

values, i.e., either K0 = 0 or K0 = 1. When R = 0, lenders’ optimal choice of collateral is K0 = 1.

When R > 1, lenders’ optimal choice of collateral is K0 = 0. When 0 < R < 1, there is a switching

state (economic fundamental), θ0, at which lenders are indifferent between either of the (optimal)

extreme collateral choices, K0 = 0 or K0 = 1. Below this switching state, lenders prefer K0 = 1,

and above it lenders prefer K0 = 0.

The first part of Proposition 1 describes the convex nature of this optimization problem: the

expected payoff in case of default increases with collateral, and expected payoff in case of non-

default decreases with collateral. Figure A.2 shows an example of the expected payoff for different

choices of collateral (K0). When the lender’s prior belief is that the economic fundamental (y0) is

low, a payoff from default is more likely, and full collateral maximizes the expected payoff (solid

line in the figure). On the contrary, when the lender’s prior belief is that the economic fundamental

(y0) is high, the payoff from non-default is more likely, and no collateral becomes the optimal

choice. The gradual change in collateralization can be expressed in terms of switching state, θ0,

which is the value of the economic fundamental at which lenders are indifferent between having no

collateral (K0 = 0) and full collateral (K0 = 1). When the true economic fundamental falls below

this switching state, lenders prefer K0 = 1; otherwise lenders prefer K0 = 0.

The second part of Proposition 1 shows that such a switching state exists when 0 < R < 1. 14 Panel14The proof shows that there is a unique switching state if 0 < R < 1, because V (y0, K0 = 1) − V (y0, K0 = 0)

starts above zero and has a negative slope at y0 = 0 and eventually becomes zero at y0 = ∞. The optimalityconditions of this function is the intersection of two scaled normal probability distribution functions, which intersectwhen 0 < R < 1, implying that the curve will have a minimum at a finite y0. The only possible shape that starts

39

Figure A.2. Expected Payoff vs. Collateral Size: This figure shows the lender’s ex-ante expected payoff(with no cost of collateral) as a function of collateral level (K0). A pessimist, with a low prior on the economicfundamentals (y0 = 0), is shown with the solid line, and the optimist, with a high prior (y0 = 1), is shown with thedashed line.

A in Figure A.3 shows the difference in the expected payoff from full collateral, V (y0, K0 = 1), and

zero collateral, V (y0, K0 = 0), with varying prior beliefs on the economic fundamental, y0. When

R = 0 (the solid line), then the payoff upon the successful completion of the project is the one that

is the same amount as the loan. Any positive probability of project failure (with a payoff that is

less than one) makes lenders lose money on average. Hence, full collateralization is optimal. When

R > 1 (the dash-dot line), the lender’s gain from continuing (and receiving the reward, 1+R, upon

success) far outweighs the immediate seizure of collateral. Hence, no collateralization is optimal.

With the intermediate value of the reward upon success, 0 < R < 1 (dashed line), the expected

payoff from full collateral outweighs that of no collateral when the economic fundamental (or the

prior belief about the economic fundamental, y0) is zero. As the economic fundamental improves,

the payoff from no collateral becomes larger and, at some value of y0, the expected payoffs from

full and zero collateral become identical, i.e., V (y0, K0 = 1)− V (y0, K0 = 0) = 0.

Furthermore, Panel B of Figure A.3 compares switching states for a given prior belief with less

above zero, and that has the minimum value at a finite value, and that approaches zero at infinity is the dashed curvein Figure A.3.

40

Figure A.3. Difference in Expected Payoff from Full and Zero Haircuts: This figure shows the differencein expected payoff from choosing full collateral and zero collateral. Panel A shows the effect of R (R = 0, 0 < R <1, R > 1). Panel B shows the effect of uncertainty (σ = 0.25 and σ = 0.5).

𝑉 𝑦#$, 1 −𝑉 𝑦#$, 0

R=0

0<R<1

R>1

𝜃

(a) Effect of R (R = 0, 0 < R < 1, R > 1) (b) Effect of Uncertainty (σ = 0.25 and σ = 0.5)

(σ = 0.25) and more (σ = 0.5) uncertainty. It shows that, with higher uncertainty in the lender’s

prior belief, the switching state is larger. Even moderately negative news, y0, can lead lenders to

switch from zero to full collateral. This is because with great uncertainty, precision (τ t−1) becomes

small, and hence it needs a larger value of y0 to obtain the same value of√τ0(y0 − 1

1+R

)or√τ0y0,

which determines expected payoffs.

Finally, we show that the destabilizing feedback of collateral (i.e., higher collateral level leads to

a higher probability of a run) makes lenders collectively patient. Without the feedback, lenders

would have required equal or higher level of collateral. We summarize this in the next proposition:

PROPOSITION 2: [Patient Lenders] An optimal level of collateral with the feedback effect be-

tween collateral and critical state θ∗0 = K01+R is less than, or equal to, the case without the feedback

effect in which θ∗0 is independent of K0.

The intuition of Proposition 2 is straightforward: if θ∗0 is independent of collateral K0, i.e., there is

no feedback from collateral to the critical state, then, as shown in the proof, full collateralization

is always optimal (K0 = 1). In other words, the feedback effect of collateral (K0) on the critical

state (θ∗0 = K01+R) lowers the optimal collateral level if beliefs about the economic fundamental are

sufficiently strong (i.e., K0 = 0 if y0 > θ0), and lenders become more lenient and more patient upon

receiving of negative news.

41

C. The Model’s Implications for Action Timing

We summarize the rollover game: Proposition 1 shows that lenders endogenously choose collateral

requirements depending on their prior beliefs about the economic fundamental y0: if y0 < θ0, then

lenders set the optimal collateral level to K0 = 1 (full collateral); otherwise, lenders set the optimal

collateral level to K0 = 0 (no collateral). After observing their private signal x1, lenders follow the

triggering strategy (θ∗1 and x∗1), according to Proposition 3.

The key insight of this result is that knowing that larger collateral can exacerbate a self-fulfilling

run (the feedback channel), lenders are incentivized to keep the haircut at a lower level, rather

than immediately require a higher haircut upon the first arrival of negative news. Proposition 2

formalizes this observation, that the feedback between collateral and critical state tend to decrease

the ex-ante collateral choice relative to the case without the feedback channel.

The model has several implications for action timing. For this task, we relate the switching state (θ0)

to the timing of two lender actions as suggested by Figure 1. In the context of the coordination game,

we are interested in a condition under which lender coordination to lower collateral requirements

can begin. To ease the interpretation, we define the following variables:

• Initial Response: The time it takes for a lender to switch to a zero collateral requirement

from a full collateral requirement, starting from an arbitrary point in time.

• Lender Patience: The time it takes for a lender to switch to a full collateral requirement from

a zero collateral requirement, starting from an arbitrary point in time.

Furthermore, we consider sequential repo contracts (each with a duration of ∆t) with an inde-

pendent random draw of the economic fundamental (θ) from a cumulative distribution, F (θ0) =

Prob(θ < θ0) for each repo contract duration.15

C.1. Expected Initial Response and Expected Lender Patience

To estimate the expected Initial Response, we start with a state of full collateral and then compute

the expected time until lenders switch to zero collateral when a new repo contract rollover is15This section heuristically links the switching theta and action timing. Formal analysis involving repeated games

requires further analysis similar to Angeletos et al. (2007), which is beyond the scope of this paper.

42

repeated with a random draw of the economic fundamental. When the economic fundamental is

above the switching state (θ0), lenders will switch from full collateral (K0 = 1) to zero collateral

(K0 = 0), and the Initial Response can be measured by the survival time of the full collateral state.

Note that the collateral decision is repeated every ∆t. Denoting the probability of maintaining full

collateral as

p = F (θ0) = Prob(θ < θ0),

we can derive the expected survival time of full collateral (or Initial Response) as follows:

E [Initial Response] = ∆t×{

1 + p+ p2 + · · ·}

= ∆t1− p. (A.1)

We follow similar steps to estimate the expected Lender Patience. Here, we start with a state of

zero collateral and compute the expected time until lenders switch to full collateral when a new

repo contract rollover is repeated with a random draw of the economic fundamental. When the

economic fundamental falls below the switching state (θ0), lenders will switch from zero collateral

(K0 = 0) to full collateral (K0 = 1), and Lender Patience can be measured by the survival time

of the zero collateral state. The resulting expected survival time of the zero collateral state (or

Lender Patience) is obtained as follows:

E [Lender Patience] = ∆t×{

1 + (1− p) + (1− p)2 + · · ·}

= ∆tp. (A.2)

C.2. Empirical Implications

Interpretation of the theoretical results with respect to the time domain provides implications that

are potentially observable. Without the feedback channel, full collateral is always optimal. This

implies that θ0 is infinitely large, i.e., p = 1. The Initial Response will therefore also be infinity

according to Equation (A.1), implying that a drop in the haircut will never happen. Also, Lender

Patience will be ∆t by Equation (A.2), indicating that lenders would immediately require full

collateral after the current contract with zero collateral requirement matures (after ∆t).

On the other hand, Proposition 2 predicts a stark contrast to the benchmark case when the feedback

43

effect is taken into the consideration. The proposition shows that, in the presence of destabilizing

feedback between collateral and the critical state, the switching state (θ0) is finitely smaller, ob-

taining 0 < p < 1. In this case, there will be a point in time when lenders require zero collateral

(i.e., Initial Response < ∞), and lenders will not immediately require full collateral (i.e., Lender

Patience > ∆t).

44

Appendix-B. Proofs

A. The Effect of Collateral on Lender Coordination

Once collateral is set at date 0 toK0, lenders make rollover decisions based on the private signal (x1).

Following steps similar to those in Morris and Shin (2004), the equilibrium can be characterized by

a triggering strategy, which is summarized in the following proposition:

PROPOSITION 3: Suppose that the lender’s signal satisfies τ0√τ1<√

2π. For a given collateral

level, K0, there exists a unique Bayesian Nash Equilibrium in which all lenders with a signal larger

than x∗1 roll over the loan and all others foreclose. The investment project succeeds if and only if

the economic fundamentals are above the threshold, 0 ≤ θ∗1 ≤ 1, which is given by

θ∗1 = Φ(τ0√τ1

(θ∗1 − y0 −

√τ0 + τ1τ0

Φ−1(

1− K01 +R

))). (B.1)

The corresponding critical private signal, x∗1, is given by

Φ (√τ1 (x∗1 − θ∗1)) = θ∗1. (B.2)

In addition, the trigger state is increasing in collateral amount (K0) and decreasing in payoff upon

success (R), i.e., ∂θ∗1(K0)∂K0

≥ 0 and ∂θ∗1(R)∂R ≤ 0.

The first part of Proposition 3 states that there is a critical state (economic fundamental) above

which all lenders expect the project to succeed. Accounting for the noise in the signal each lender

receives, lenders will rollover the repo loan when they receive a signal above x∗1. The critical state, θ∗1

(given by Equation (B.1), and illustrated in Figure B.1) is central to understanding the equilibrium

of the economy, and it is closely related to collateral level K0 and reward upon success (R), the

dependence of which are described in the second part of the proposition.

The second part of Proposition 3 indicates that the critical state below which a run occurs is

increasing in the level of collateral (K0) and decreasing in the amount of reward upon success

(R). The key insight from this proposition arises from a lender’s tradeoff when making a rollover

decision. The state at which lenders are indifferent between foreclosure and rollover is reached

45

Figure B.1. An Illustration of Determining θ∗1 in Equation (B.1): This figure shows the procedure tosolve Equation (B.1) for the critical state, θ∗1, which is found by the intersection of the 45o line (solid line) and thecumulative normal distribution function (dashed line).

y 𝜃# = Φ��(𝜏#� 𝜃#− 𝑦,( −

��( +𝜏#�

��(Φ.# 1 −

𝐾(1 +𝑅

y 𝜃# = 𝜃#

𝜃#∗

Economic Fundamentals (𝜃#)

y(𝜃#)

when the expected payoff from liquidating the collateral (foreclosing) is the same as the expected

payoff from rolling over and receiving payoff 1 +R when the project succeeds. When the collateral

is set very high, then even with a strong economic fundamental, lenders are tempted to seize the

collateral rather than take the risk to receive the reward upon the project’s success. This tradeoff is

described in the equilibrium outcome: a high collateral level (K0) pushes the critical state upward.

Likewise, when the payoff upon success (1 + R) is large, lenders will tolerate the risk of rolling

over even with a low economic fundamental. That is, a large R pushes the critical state downward.

Lenders choose the collateral to maximize their payoff from lending (or minimize their losses from

the borrower’s failure). Therefore, when the economic condition worsens, lenders may choose to

raise collateral levels. However, the fact that high collateral levels also elevate the critical state (i.e.,

runs are more likely to occur even when economic fundamentals are strong) makes the inefficient

run more likely. To avoid such self-fulfilling runs, lenders may choose to set ex-ante collateral lower

than a case without such destabilizing feedback, which is explored in the next subsection.16

16Endogenous contracts accounting for interim period hold up problems have been considered in other applicationssuch as product market competition (Khanna and Schroder, 2010; Khanna and Tice, 2000, 2005) and venture capitals(Khanna and Mathews, 2015).

46

B. Proof of Proposition 3

The first part: The proof follows arguments similar to those in Morris and Shin (2004); The

equilibrium of this economy can be determined by the following two conditions:

(i) Critical Mass Condition:

Conditional on the (true) economic fundamental, θ, the probability that a lender will receive a

signal below x1 < x∗1 is

Prob(x1 < x∗1|θ) = Φ(x∗1 − θσ1

),

where Φ (·) is a standard normal cumulative distribution function. Since the total mass of lenders

is one, this probability is equal to the fraction of lenders foreclosing the loan (l1). The project

succeeds when this fraction, l1, is less than the economic fundamental, θ. Hence, the critical level

of the economic fundamental that is exactly enough to succeeds is l1 (θ∗) = θ∗1, i.e.,

Φ (√τ1 (x∗1 − θ∗1)) = θ∗1. (B.3)

This condition is often referred to as the critical mass condition in the global games literature

(Corsetti, Dasgupta, Morris, and Shin, 2004).

(ii) Optimal Trigger Condition:

For a given economic fundamental threshold, θ∗1, a lender who receives a signal, x1, has a conditional

probability of successful rollover,

Prob (θ < θ∗1|x1) = 1− Prob (θ ≥ θ∗1|x1) = 1− Φ(θ∗1 − x1σ1

),

and hence will only decide to roll over the loan if the expected payoff from the rollover (i.e., the

probability of a successful rollover, in which the project succeeds in time and the lender receives

the payoff that results from the project’s success, 1 + R) is at least as large as the payoff from

foreclosure (K0). The critical threshold signal (x∗1) is the point above which a lender is indifferent

between rolling over and foreclosing to collect the collateral. For a given economic fundamental

threshold, θ∗1, a lender who receives a signal, x1, has a posterior distribution of θ being a normal

47

distribution with mean y1 = τ0y0+τ1x1τ0+τ1

and precision τ0 + τ1. Also, each lender knows that the

project succeeds if and only if the economic fundamental is above θ∗1. This leads to the following

optimal trigger condition:

{1− Φ

(√τ0 + τ1

(θ∗1 −

τ0y0 + τ1x1τ0 + τ1

))}· (1 +R) = K0.

This condition can be written as

θ∗1 = τ0y0 + τ1x1τ0 + τ1

+Φ−1

(1− K0

1+R

)√τ0 + τ1

which obtains

(τ0 + τ1) θ∗1 = τ0y0 + τ1x1 +√τ0 + τ1Φ−1

(1− K0

1 +R

)

that leads

θ∗1 − x∗1 = −τ0 (θ∗1 − y0)τ1

+√τ0 + τ1Φ−1

(1− K0

1+R

)τ1

.

Substituting this into the critical mass condition gives,

θ∗1 = Φ(τ0√τ1

(θ∗1 − y0 −

√τ0 + τ1τ0

Φ−1(

1− K01 +R

))). (B.4)

As shown in Figure 5, the LHS (θ∗1) is a 45 degree line passing through the origin. The RHS will

cross this 45 degree line once if the slope of the LHS is less than one. The slope of the RHS is

given by τ0√τ1φ (·) where the maximum value of φ (·) is 1√

2π . Hence, the RHS’s slope is less than

one everywhere if τ0√2πτ1

< 1.

The second part: According to Equation (B.4) (and illustrated in Figure B.1), the trigger state

is determined by the intersection of a 45 degree line (θ∗1) and the normal distribution function. The

48

argument inside the cumulative distribution function is

z1 = τ0√τ1

(θ∗1 − z3)

z2 = 1− K01 +R

z3 = y0 +√τ0 + τ1τ0

Φ−1 (z2) .

Note that z2 is decreasing in collateral and increasing in payoff upon success, i.e., ∂z2(K0)∂K0

= − 11+R <

0 and ∂z2(R)∂R = K0

(1+R)2 > 0. Since the inverse cumulative distribution function is smooth, continuous,

and monotonically increasing between -1 and 1, Φ−1 (z2) decreases in collateral (K0) and increases

in payoff upon success (R), i.e., ∂Φ−1(z2)∂K0

= ∂Φ−1(z2)∂z2

∂z2(K0)∂K0

< 0 and ∂Φ−1(z2)∂R = ∂Φ−1(z2)

∂z2∂z2∂R > 0.

Therefore, z3 is decreasing in collateral and increasing in payoff upon success:

∂z3∂K0

=√τ0 + τ1τ0

∂Φ−1 (z2)∂K0

< 0

∂z3∂R

=√τ0 + τ1τ0

∂Φ−1 (z2)∂R

> 0.

Now, Equation (B.4) takes the form

θ∗t = Φ(τ0√τ1

(θ∗1 − z3)).

If there is an increase in collateral decreases z3, which shifts the cumulative distribution function,

Φ(τ0√τ1

(θ∗1 − z3)), to the left as shown in Figure 5. As a result, the intersection of the 45 degree

line and the cumulative distribution function (θ∗1) shifts to the right, i.e., ∂θ∗1(K0)∂K0

≥ 0. Similarly, an

increasing payoff upon success (R) increases z3, which shifts the cumulative distribution function,

Φ(τ0√τ1

(θ∗1 − z3)), to the right as shown in Figure 5. As a result, the intersection of the 45 degree

line and the cumulative distribution function (θ∗1) shifts to the left, i.e., ∂θ∗1(R)∂R ≤ 0. QED �

49

COROLLARY 1: Trigger strategies based on prior beliefs are as follows:

θ∗0 = K01 +R

x∗0 = K01 +R

−Φ−1

(1− K0

1+R

)√τ0

.

Proof of Corollary 1

Denoting the trigger strategy and trigger state using prior beliefs as x∗0 and θ∗0, we can derive the

equilibrium as follows:

(i) The critical mass condition (given the true economic fundamental, θ, the probability of lenders

below x∗0 must be less than or equal to θ. This holds with equality at the critical state)

Φ(√

τ0(x∗0 − θ

∗0

))= θ

∗0

from which we obtain

x∗0 = θ∗0 +

Φ−1(θ∗0

)√τ0

. (B.5)

(ii) The optimal trigger condition (the probability that the economic fundamental is above the

trigger state, which is normal with mean y0 and precision τ0)

{1− Φ

(√τ0(θ∗0 − x∗0

))}· (1 +R) = K0.

This condition can be written as

θ∗0 = x∗0 +

Φ−1(1− K0

1+R

)√τ0

.

Plugging that into critical mass condition in Equation (B.5) gives

θ∗0 = Φ

((−Φ−1

(1− K0

1 +R

)))= 1− Φ

((Φ−1

(1− K0

1 +R

)))= 1−

(1− K0

1 +R

)

50

and

θ∗0 = K0

1 +R.

QED �

C. Proof of Proposition 1

The first part: The optimal collateral choice at date 1 maximizes the expected payoff for the

next period, date 2, based on information held before observing the private signal, i.e., using only

prior beliefs.

The expected payoff can be broken down into two terms: the expected payoff in case of default (if

the creditors run), VD, and in case of non-default (if the project succeeds), VND, which are shown

as:

VD = K0Φ(√

τ t−1(θ∗0 − y0

))VND =

{1− Φ

(√τ0(θ∗0 − y0

))}· (1 +R) ,

where Φ(√

τ0(θ∗0 − y0

))is the probability that the true economic state (based on prior beliefs;

this is normally distributed with mean y0 and precision τ0) is below trigger state (θ∗0), i.e., it is the

probability of failure.

The first term (VD) is increasing in collateral because

∂θ∗0

∂K0= ∂

∂K0

(K0

1 +R

)= 1

1 +R> 0,

and∂Φ

(√τ0(θ∗0 − y0

))∂K0

=∂Φ

(√τ0(θ∗0 − y0

))∂θ∗0

∂θ∗0

∂K0> 0,

and∂VD

∂K0= Φ

(√τ0(θ∗0 − y0

))+ K0

∂Φ(√

τ0(θ∗0 − y0

))∂K0

> 0. (B.6)

51

The second term (VND) is decreasing in collateral because

∂VD

∂K0= − (1 +R)

∂Φ(√

τ0(θ∗0 − y0

))∂K0

< 0. (B.7)

Adding these two terms in Equation (B.6) and (B.7) leads to a convex payoff function whose

maximizing choice is one of the extreme collateral choices. That is, K0 = 0 if the second (non-

default payoff) is larger, or K0 = 1 if the first (default payoff) is larger.

The second part: The expected payoff for a given K0 as a function of the economic fundamental

based on prior belief (y0) is

V(y0, K0

)= K0Φ

(√τ0(θ∗0 − y0

))+{

1− Φ(√

τ0(θ∗0 − y0

))}· (1 +R)

= K0 ·{

1− Φ(√

τ0(y0 − θ

∗0

))}+ Φ

(√τ0(y0 − θ

∗0

))· (1 +R)

or, substituting for the prior-belief-based critical economic fundamental that triggers run, θ∗0 = K01+R ,

is

V(y0, K0

)= K0 ·

{1− Φ

(√τ0

(y0 −

K01 +R

))}+ Φ

(√τ0

(y0 −

K01 +R

))· (1 +R) .

Considering both optimal choices of haircuts, when K0 = 0 (zero haircut), it becomes,

V (y0, 0) = Φ(√

τ0·y0

)· (1 +R) , (B.8)

and when K0 = 1 (full haircut), it becomes

V (y0, 1) ={

1− Φ(√

τ0

(y0 −

11 +R

))}+ Φ

(√τ0

(y0 −

11 +R

))· (1 +R) . (B.9)

Using Equation (B.8) and (B.9), we define the difference in the expected payoff from these haircut

52

choices as

∆V1−0 (y0) = V (y0, 1)− V (y0, 0)

={

1− Φ(√

τ0

(y0 −

11 +R

))}+ Φ

(√τ0

(y0 −

11 +R

))· (1 +R)

−Φ(√

τ0·y0

)· (1 +R)

= 1 +R · Φ(√

τ0

(y0 −

11 +R

))− Φ

(√τ0·y0

)· (1 +R) .

and its derivative with respect to the economic fundamental, y0, is

d (∆V1−0 (y0))dy0

=√τ0R · φ

(√τ0

(y0 −

11 +R

))−√τ0φ

(√τ0·y0

)· (1 +R)

The first order condition (in this case, the value of y that minimizes the expected payoff) becomes

R · φ(√

τ0

(y0 −

11 +R

))− φ

(√τ0·y0

)· (1 +R) = 0.

For R = 0, the first order condition (FOC) is

−φ(√

τ0·y0

)< 0,

i.e., FOC has no solution, and ∆V1−0 (y0) monotonically decreases. For R > 0, FOC has one

solution as shown in Figure A.1 (R = 0.5, τ0 = 2). This is because the first term is a rescaled

(by R < 1) bell-shaped function shifted by 11+R from the origin, and the second term is a rescaled

(by 1 + R > 1) bell-shaped function centered at origin. The latter is a monotonically decreasing

function from 1+R√2π to zero at range yt−1 > 0, and the former has a positive value everywhere with

a peak value of R√2π hence, both terms will cross each other at a point as long as R > 0. Note that

as R approaches zero, the peaks of both bell-shaped curves will approach each other and the first

term will be centered near zero. Hence, in the limit (of R = 0), they will coincide each other.

The slope at zero economic fundamental is negative,

d (∆V1−0 (y0))dy0

=√τ0R · φ

(√τ0

(− 1

1 +R

))−√τ0φ (0) · (1 +R) < 0

53

because φ (·) is at its maximum at zero (i.e., φ (0) > φ(√

τ0(− 1

1+R

))) and R < 1 +R. Hence, the

second term is larger than the first term.

The difference in the expected payoff, ∆V1−0 (y0), at y0 = 0 is positive if 0 < R < 1, which means

that the expected payoff from setting a full haircut (K0 = 1) is larger than that of setting zero

collateral (K0 = 0). To see this, at y0 = 0, we have

∆V1−0 (y0 = 0) = 1 +R · Φ(√

τ0

( 11 +R

))− Φ (0) · (1 +R) .

Since Φ (0) = 12 , we have

∆V1−0 (y0 = 0) = 12 +R ·

{Φ(√

τ0

( 11 +R

))− 1

2

}.

For 0 < R < 1, ∆V1−0 (y0 = 0) is positive because

−12 < Φ

(√τ0

( 11 +R

))− 1

2 < 0.

For R > 1, it is possible for ∆V1−0 (y0 = 0) to be negative which implies that the expected payoff

from zero collateral is higher than for full collateral, even with weak low economic fundamentals

(i.e., y0 = 0).

At the other extreme of y0 → ∞, the difference in the expected payoff approaches zero because

both Φ(√

τ0(y0 − 1

1+R

))and Φ

(√τ0·y0

)approach unity, and we get limyt−1→∞∆V1−0 (y0) =

1 +R− (1 +R) = 0.

For values of 0 < R < 1, ∆V1−0 (y0) starts at a positive value at y0 = 0. Then it monotonically

decreases below zero as y0 increases. When the FOC is zero (which is the case when R > 0),

the trend reverses and ∆V1−0 (y0) monotonically increases as y0 increases. Eventually, when y0 →

∞, ∆V1−0 (y0) approaches zero. For values of R > 1, ∆V1−0 (y0) starts at a negative value at

y0 = 0. Then it monotonically decreases below zero as y0 increases, and reverses the trend whend(∆V1−0(y0))

dy0= 0. Eventually, it approaches zero as y0 → ∞ (Figure 7). As shown in Panel A of

Figure 7, when R = 0, V1−0 (y0) is always positive and K0 = 1 is optimal. When 0 < R < 1,

54

V1−0 (y0) is initially positive where K0 = 1 is optimal. But for strong economic fundamentals

(y0 > θ), V1−0 (y0) is negative and K0 = 0 is optimal. The switching economic fundamental (θ) is

where V1−0 (y0) = 0, and lenders are indifferent between K0 = 0 and K0 = 1. For R > 1, V1−0 (y0)

is always negative and K0 = 0 is optimal. QED�

D. Proof of Proposition 2

If θ∗0 is independent of collateral, K0 , the expected payoff from zero collateral is

V(y0, K0 = 0

)={

1− Φ(√

τ0(θ∗0 − y0

))}· (1 +R) ,

and the expected payoff from full collateral is

V(y0, K0 = 1

)= Φ

(√τ0(θ∗0 − y0

))+{

1− Φ(√

τ0(θ∗0 − y0

))}· (1 +R) .

The difference between these two expected payoffs is always positive, that is,

∆V1−0 (y0) = V (y0, 1)− V (y0, 0) = 1− Φ(√

τ0(y0 − θ

∗0

))> 0.

Proposition 2 shows that full collateralization is always optimal without feedback effects. In other

words, the feedback effect of collateral (K0) on the critical state (θ∗0 = K01+R) lowers the optimal

collateral level if (for a sufficiently large R and low uncertainty (a large τ0) to have θ < 1) the

belief about the economic fundamental are sufficiently strong (i.e., K0 = 0 if y0 > θ), and lenders

become patient. QED�

55