Safe Harbor or Automatic Stay? A Structural Model of

Repo Runs∗

Xuyang Ma†

∗For the most recent version of the paper, please check http://people.stern.nyu.edu/xma/Research.html.I am greatly indebted to my committee: Viral Acharya, Matthew Richardson, Thomas Philippon and Alexi Savov fortheir valuable advice and enormous support on this project. I also thank Jennifer Carpenter, Stephen Figlewski, XavierGabaix, Douglas Gale, Kose John, Anthony Lynch, Bruce Tuckman, Robert Whitelaw, Shengxing Zhang for helpfulcomments and suggestions. All remaining errors are my own.

†New York University, Stern School of Business. Email: xma@stern.nyu.edu

Abstract

The granting of safe harbor (SH), i.e., exemption from automatic stay (AS) of bankruptcy

law, to qualified financial contracts (QFCs) such as sale and repurchase agreements (repos), is

regarded as having significantly contributed to the 2007-2009 financial crisis. Relative to auto-

matic stay, safe harbor promotes liquidity of collateral when the counterparty defaults, which

is valued by short-term creditors; however, the immediate liquidation of large volumes of se-

curities in a short period can result in costly and inefficient “fire sales”. To study the ex-post,

and more importantly, the ex-ante impacts of the two bankruptcy provisions (SH and AS), I

build a structural model that incorporates two layers of strategic interactions: 1) the coordina-

tion problem among infinitesimal creditors, who enjoy a liquidity premium from short-horizon

investments, but when facing deteriorating collateral value and limited available liquidity, can

“rush for the exits”; and, 2) interaction between equity holders and creditors, wherein equity

holders may file for bankruptcy endogenously to pre-empt the creditors run.

The model has rich implications as to when and how the automatic stay should be imposed.

The presence of a stay effectively violates the absolute priority rule and motivates equity hold-

ers of a distressed firm to file for bankruptcy in a more timely manner, thus shortening the

period during which the firm can extract the liquidity premium from short-term creditors. For

highly liquid securities, the fire sale risk with safe harbor (SH) is small and is outweighed by

the cost under automatic stay (AS) of a shorter period in which to exploit the liquidity premium;

ex ante, SH gives higher firm value than AS. For less liquid securities, AS results in higher firm

value, both ex post and ex ante, since the gain from avoiding fire sales under AS dominates the

loss associated with an earlier bankruptcy. Furthermore, higher debt capacity and higher lever-

age ratio arise endogenously with safe harbor, especially when the firm doesn’t endogenize the

fire sale cost. The model also provides implications of firm volatility, firm profitability and

debt rollover frequency for the choice between AS and SH, as well as for the optimal length of

automatic stay.

1 Introduction

In the aftermath of the 2007-2009 financial crisis, how to regulate the financial markets to prevent

a repeat of the crisis has become an important challenge for academic research and policy-making.

Safe harbor (or exemptions from automatic stay of bankruptcy law) provisions, which allow one

party holding “qualified financial contracts” (QFCs, e.g., derivatives and repurchase agreements)

to immediately terminate the contracts and seize the collateral when the counterparty defaults,

are considered to be an important force that drove the wide-spread systemic risk of the 2007-

2009 financial crisis. Safe harbor (SH) promotes liquidity of collateral when the counterparty

defaults, which is valued by short-term investors; however, immediate liquidation of a large volume

of securities in a short period may result in costly and inefficient “fire sales”, which not only hurts

sellers of the securities, but also endangers the stability of other financial firms holding the same or

similar assets.

Among the QFCs, the repo market has become a reliable, inexpensive source of financing for

financial firms. Repos, or repurchase agreements, are the sale of securities together with an agree-

ment for the seller to buy back the securities at a future date. Equivalently, the seller of repos may

be regarded as borrower, and the buyer as creditor, and the securities sold as collateral backing the

transaction. A typical repo contract in shown in Figure 1.1. From the financial perspective, repos

are collateralized secured debts; however, unlike secured debts, repos are “QFCs” and enjoy the

safe harbor provisions in bankruptcy. Normally, bankruptcy procedures impose an automatic stay,

which halts actions by creditors to collect debts from a debtor who has filed for bankruptcy. The

safe harbor exempts creditors from such stay and allows them to seize and liquidate the collat-

eral immediately upon default. Initially, the safe harbor provisions included only U.S. Treasuries,

which are backed by the “full faith and credit of the U.S. government”. However, after the 1998

collapse of Long-Term Capital Management, practitioners advocated to expand the safe harbor

provisions in order to “mitigate counterparty losses and reduce the likelihood of instability in the

financial markets”. This recommendation was finally enacted as “Bankruptcy Abuse Prevention

and Consumer Protection Act of 2005”, with which the safe harbor provisions expanded to include

3

Figure 1.1: A Typical Repo Contract.

mortgage-backed securities and mortgage loans1. Following this expansion, the repo market grew

dramatically before the financial crisis of 2008. As shown in Figure 1.2a, just before the financial

crisis, primary dealers’ total repos outstanding reached a peak of US$4.5 trillion, with two-thirds

of these overnight repos. Most of the U.S. overnight repos are open and roll over automatically

until either party chooses to exit. Over time, primary dealers not only became increasingly reliant

on repo financing, but also more reliant on repos backed by less liquid collateral, such as mortgage-

backed securities and mortgage loans. Figure 1.2b shows that the share of less liquid collateral in

the repo market increased from around 45% in 2005 to a peak of almost 60% at the end of 2008.

Accounting for both growth in the repo market and growth in the share of less liquid collateral,

from 2005 to the end of 2008, the market capitalization of repos backed by less liquid collateral

almost doubled and reached US$2.5 trillion.

The safe harbor was granted and expanded on grounds that it would promote liquidity of collat-

eral, mitigate contagion from a failing financial firm and reduce systemic risk. However, the failure

of Bear Stearns and Lehman Brothers in 2008 and the ensuing market-damaging massive fire sales

1The definition of “repurchase agreement” is amended to include the following additional instruments according tothe Bankruptcy Abuse Prevention and Consumer Protection Act of 2005: mortgage loans; mortgage related securities(as defined in section 3 of the Securities Exchange Act of 1934); interests in mortgage-related securities or mortgageloans; and qualified foreign government securities (defined as a security that is a direct obligation of, or that is fullyguaranteed by, the central government of a member of the Organization for Economic Cooperation and Development).

4

(a) Primary Dealers’ Outstanding Repos (July 6, 1994 - July22, 2009). Source: Federal Reserve Bank of New York(Adrian, Burke, and McAndrews (2009))

(b) Prevalence of Less Liquid Collateral in Primary Deal-ers’ Repo Transactions (January 5, 2005 - July 22, 2009).Source: Federal Reserve Bank of New York (Adrian, Burke,and McAndrews (2009))

Figure 1.2: The Growth of Repo Financing and the Prevalence of Less Liquid Collateral Since 2005

brought into question the effectiveness of safe harbor in attaining the goal of reducing systemic

risk. Copeland, Martin, and Walker (2010), Copeland, Martin, and Walker (2011) and Gorton and

Metrick (2012) document repo runs on financial institutions during the financial crisis. Acharya

and Oncu (2012), Duffie and Skeel (2012), Tuckman (2010) and many others argue that the safe

harbor provisions of QFCs such as repos may exacerbate systemic risk, especially QFCs backed

by illiquid assets. Lawyer (2014) proposes that “the more recent expansion of the exemption to

mortgage-backed securities should be reversed”.

This paper addresses the debate on whether to repeal the safe harbor and re-impose the auto-

matic stay on QFCs. It formalizes the ex-post tradeoff between the two bankruptcy provisions,

namely, safe harbor (SH) and automatic stay (AS), and more importantly, examines the ex-ante

implications of SH and AS, which is important for achieving financial stability. To do so, I build a

structural model that incorporates two layers of strategic interactions: 1) interaction between equity

holders and creditors, wherein equity holders may file for bankruptcy endogenously to pre-empt

creditor runs; and, 2) the coordination problem among atomistic creditors, who enjoy a liquidity

premium from short-horizon investments, but when facing deteriorating collateral value and limited

available liquidity, may “rush for the exits”.

Specifically, the capital structure of financial firms consists of equity, short-term repo debts, and

5

pre-established credit lines (CLs). Following Leland and Toft (1996) and He and Xiong (2012a),

the firm utilizes a stationary staggered debt structure, which means that, at every instant, only a

small portion of debt matures and the maturing creditor decides to roll over or to run; if such

creditor decides to run, the debt is replaced by newly issued debt with the same structure sold

at market price, which consequently causes a rollover loss to the firm. The firm relies on pre-

established, but non-perfectly reliable, credit lines to cover the rollover loss, which may expose

the firm to bankruptcy risk: Once CLs fail, the CLs providers demand an upfront payment from

equity holders, to cover the possible loss CLs face if they continue to provide liquidity to the firm;

however, equity holders may find it not profitable to restore the failed CLs and keep the firm alive,

in which case they simply file for bankruptcy endogenously. Creditors must take into account

equity holders’ bankruptcy decisions when making their rollover choice and vice versa. In addition

to the interaction between creditors and equity holders, creditors do not act collectively as one,

but rather, they compete strategically among themselves: because earlier runs protect the principal

value but forgo coupons associated with their debts, while later runs collect coupons but face the

risk of not recovering their principal in full in bankruptcy, each individual creditor makes his/her

rollover decision based on other creditors’ strategy, to maximize his/her own gains. The collective

choice of running creditors poses an externality to all parties, since it makes bankruptcy risk more

imminent. At the point of bankruptcy, I distinguish two bankruptcy rules: 1) SH, where creditors

immediately liquidate the collateral and take all the liquidated collateral value until full recovery;

and, 2) AS, which allows equity holders to avoid liquidation by making a “take-it-or-leave-it” offer

to creditors who value liquidity higher, and thus the value of collateral is effectively shared between

the creditors and equity holders.

The model has rich implications as to when and how the automatic stay should be imposed. SH

gives creditors the right to immediately liquidate the collateral of a defaulting counterparty, which

protects creditors by effectively granting them absolute priority, but this incurs fire sale risk. On the

contrary, with AS, creditors face the risk of liquidity being tied up in bankruptcy. The presence of

stay allows renegotiation between equity holders and creditors who value liquidity more, and thus

6

avoids costly and inefficient fire sales. Renegotiation under AS essentially violates the absolute

priority rule (APR), in the sense that equity holders could gain some positive value, even before

creditors are paid back in full. Violation of APR motivates equity holders of a distressed firm to

file for bankruptcy in a more timely manner, to obtain their share of value at bankruptcy before it

further deteriorates. This earlier bankruptcy under AS shortens the period during which the firm

can extract short-term creditors’ liquidity premium. Consequently, debt capacity and leverage ratio

with SH are greater than that those under AS, since AS effectively violates the APR ex post and

hampers creditors’ incentive to lend ex ante.

For highly liquid securities, ex post, AS differs from SH only in the distribution of firm value

at bankruptcy: SH strictly follows APR while AS violates APR. Under AS, when facing an APR

violation, creditors have incentive to run at higher fundamental value to preserve their debt value,

and equity holders have incentive to file for bankruptcy “earlier” too, to obtain their share of value

at bankruptcy before it further deteriorates. Thus, on average, under AS, the period during which

the firm can exploit the liquidity premium is shorter relative to the case with SH. Taking these ex-

post incentives into account, the ex-ante firm value under AS is lower than that with SH for highly

liquid securities.

For less liquid securities, ex post, SH suffers severe fire sale risk, while AS can avoid fire

sales via renegotiation between equity holders and creditors, where creditors’ outside option in the

bargaining game is constrained by the liquidity of the securities pledged as repo collateral. For

the same collateral value, repo creditors face higher bankruptcy loss under AS, since not only do

they effectively still face fire sale risk (entering as an outside option in the renegotiation), they

also lose the liquidity premium they normally would enjoy from short-horizon investments, due to

the stay imposed on the collateral. Again, this greater bankruptcy loss would induce creditors to

stop rolling-over at higher fundamental values; violation of APR would then give equity holders

incentive to endogenously default at higher fundamental values as well. However, the damage

caused by this “earlier creditor runs” and “earlier endogenous bankruptcy” under AS, is less than

the damage caused by fire sales with SH. Thus, ex ante, AS yields higher firm value than SH, for

7

less liquid securities.

With SH, the firm essentially faces a liquidity spiral: as liquidation is immediate upon bankruptcy,

it would push down the market price of the securities of the bankrupt firm, which endangers the

stability of other financial firms holding the same or very similar securities and may pushes more

firms to bankruptcy region, and thus magnifying the fire sale cost the failing firm faces. But ex-

ante the firm doesn’t endogenize the fire sale cost, the financial firm has higher debt capacity and

would endogenously choose a higher leverage ratio with SH. When the firm mistakenly takes the

“optimal” debt level ex-ante and faces a much higher fire sale risk ex-post, the firm value with

SH becomes significantly lower than the firm value under AS, since in this way, the firm with SH

maximizes the ex-ante firm value with misspecified fire sale risk.

With respect to financial market conditions, in general, a deteriorating market amplifies the ad-

vantage of AS over SH for less liquid securities. Even for the same security, fire sale risk increases

during systemic risk periods, which renders the security less liquid, and as analyzed previously,

AS outperforms SH more for less liquid securities. The more volatile the collateral, the more com-

petition for liquidity among creditors. When equity holders come into play, under AS, they file

for bankruptcy at much higher fundamental value than with SH, since under AS, equity holders

wish to preserve firm value to obtain a higher equity value at bankruptcy; however, with SH, equity

holders are almost sure to be wiped out in bankruptcy, so they postpone bankruptcy to profit from

the higher option value of their equity when facing higher volatility. Lower firm profitability still

favors AS, as AS not only avoids fire sale risk, but also spurs equity holders to file for bankruptcy

in a much more timely manner. Higher rollover frequency lowers ex-ante firm value, as it causes

more competition among creditors, who rush for the exits, leaving the firm with greater rollover

loss; furthermore, equity holders become more reluctant to contribute funding to restore the failed

credit lines, since they still face greater rollover loss even if the credit lines are restored. As for the

optimal length of the automatic stay, so long as the stay prevents fire sales, it is best to have as short

a stay as possible, to minimize the incentive distortion caused to creditors while avoiding fire sales.

8

Literature Review

The present paper is part of the body of literature on bank runs. In the classic Diamond and Dy-

bvig (1983) model, multiple equilibria exist and a self-fulfilling bank run equilibrium arises due

to the coordination problem among synchronous depositors. He and Xiong (2012a) examine a

dynamic “rat race” among creditors, who essentially compete for liquidity within the firm; the

setting of time-varying fundamentals and staggered debt structure allows a unique threshold equi-

librium. Acharya, Gale, and Yorulmazer (2011) present a model in which bad news can lead to

a “market freeze” due to high rollover risk, even if the fundamental value of the assets is high

in all states of the world. Brunnermeier and Pedersen (2009) link an asset’s market liquidity and

traders’ funding liquidity and show that the two are mutually reinforcing under certain conditions,

which leads to liquidity spirals. Brunnermeier and Oehmke (2013) show that short-term financing

is an equilibrium outcome of a maturity rat race where both borrower and creditors inefficiently

choose short-term financing. Gertler and Kiyotaki (2013) present a macroeconomic model where

bank balance sheets and an endogenously determined liquidation price for bank assets together pin

down whether a bank run equilibrium exists or not. Hellwig and Zhang (2012) study how changes

in expectations about market liquidity and information rents can trigger market runs, even when

fundamentals remain unchanged. Martin, Skeie, and Von Thadden (2013) develop a dynamic equi-

librium model of repo runs and show that the instability of repo financing may be a consequence of

market-wide changes in expectations. Zhang (2014) builds a dynamic model of the bilateral repo

market and emphasizes the illiquidity contagion channel, which interacts with bank valuation of

the repo contracts and triggers massive defaults.

Regarding bankruptcy, evidences show that equity holders time the bankruptcy decisions: The

majority of Chapter 11 petitions are filed on a voluntary basis. Moreover, Franks and Torous

(1989), LoPucki and Whitford (1990) and Betker (1995) document that almost 75% of Chapter 11

reorganizations involve violations of the APR. Longhofer and Carlstrom (1995) build a two-period

model to analyze the impact of APR violations on financial contracts and their policy implications.

Anderson and Sundaresan (1996) show a model of strategic debt service and endogenous reorga-

9

nization boundary in which debt holders and equity holders behave non-cooperatively. Leland and

Toft (1996) build a structural model in which equity holders endogenously default if their equity ap-

preciation is lower than their required contribution to keep the firm solvent. He and Xiong (2012b)

include illiquidity of the secondary debt market to account for rollover risk and credit risk. Cheng

and Milbradt (2012) extend He and Xiong (2012a) to incorporate risk shifting of equity holders,

but there is no endogenous bankruptcy decision faced by equity holders, as they work in a cash-free

model. I build a framework based on Leland and Toft (1996) and He and Xiong (2012a), where

creditors compete for liquidity when making their rollover decisions, and equity holders endoge-

nously default when the value appreciation of keeping the firm alive cannot justify the additional

capital contribution from them. Auh and Sundaresan (2013) build a structural model to show that

short-term debt transfers credit risk to long-term debt and hinders long-term debt capacity, although

overall debt capacity for the firm increases. In their model, the authors essentially make the repos

default-free, and study the interaction between equity holders and subordinated long-term debt.

von Thadden, Berglöf, and Roland (2010) show that the non-cooperative game between creditors

ex post creates a commitment for the firm to repay more in good states to pre-empt individual cred-

itor foreclosure rights, and thus raising the firm’s debt capacity. Broadie, Chernov, and Sundaresan

(2007) provide valuations of debt and equity when borrowers are able to control ex ante reorganiza-

tion timing, and show that lenders can restore the first-best outcome via ex post transfer of control

rights.

Another strand of the literature involves the bankruptcy provisions of QFCs. Roe (2010) points

out that safe harbor provisions to derivatives warp creditors’ incentive to monitor and discipline

derivatives counterparties, thus spreading, instead of preventing, contagion in the 2007-2009 finan-

cial melt-down. Tuckman (2010) proposes that safe harbor only apply where it reduces systemic

risk. Bolton and Oehmke (2011) build a model of derivative hedging, and show that safe harbors

can transfer credit risk to debt holders whereas this risk is borne more efficiently in the deriva-

tive markets. Acharya and Oncu (2012) propose a resolution focusing on systemically important

assets and liabilities (SIALs) to induce market discipline and reduce systemic risk. Duffie and

10

Skeel (2012) discuss the benefits and costs of the exemption from automatic stay for QFCs, and

advocate safe harbor to QFCs backed by liquid securities and automatic stay to QFCs backed by

illiquid assets. Antinolfi, Carapella, Kahn, Martin, Mills, and Nosal (2012) construct a model of

repo transactions and show that policy-makers “face a trade-off between the benefits of investment

activity and the benefits of liquid markets for collateral”. Schwarcz and Sharon (2013) give a path-

dependence analysis and conclude that safe harbor is “a sequence of industry-lobbied legislative

steps, each incremental and in turn serving as apparent justification for the next step, without a

rigorous and systemic vetting of the consequences”; accordingly, safe harbor cannot be taken for

granted to minimize systemic risk.

The remainder of this paper proceeds as follows. Section 2 provides assets and capital struc-

ture of the firm, based on which I analyze the threshold strategies of creditors and equity holders,

respectively, and distinguish the two provisions, SH and AS at the time of bankruptcy. Section 3

gives the solution of the model and shows its optimality. Then I set forth numerical calibrations

and comparative statics to investigate the benefits and costs of SH and AS for different securities

and under different market conditions. Section 4 discusses the assumptions of the model and looks

forward to future extensions. Section 5 concludes.

2 Model

2.1 Firm Assets and Capital Structure

Firm Assets

A financial institution invests in a portfolio of risky securities, called “assets” of the financial insti-

tution or the firm. These securities held by the firm generate cash flow Xt (for t ≥ 0), which follows

11

a geometric Brownian motion process2:

dXt

Xt= µdt +σdZQ

t

where µ represents the growth rate and σ the volatility, and ZQt is a Wiener process under the risk-

neutral measure. All work is done under the risk-neutral measure, and there is a risk-free asset,

yielding a constant rate of return r > µ . At time 0, the firm begins with cash flow X0.

Assume the cash flow generated by the securities is independent of the capital structure of the

firm. Then, the value of the un-levered all-equity financial firm VU (Xt) satisfies:

rVU (Xt) = Xt +µXtVU ′ (Xt)+12

σ2X2

t VU ′′ (Xt)

with the solution given by VU (Xt) =Xt

r−µ.

Repo Financing: stationary staggered debt structure

To finance the purchase of the risky securities, the firm borrows in the repo markets by pledging

its securities as collateral. The repo creditors of the firm consist of a continuum of measure 1 of

infinitesimal creditors.

The repo creditors are short-term investors, who at the same time enjoy the safety of collateral

backing, earn interest, and preserve the liquidity of their cash holdings due to the extreme short-

term nature of repo contracts. Following Krishnamurthy and Vissing-Jorgensen (2012), risk-free

Treasury securities have “money-like” properties, i.e., liquidity and “absolute security of nominal

return”, which results in lower yield requirements from Treasury securities investors. After ac-

counting for maturities, Greenwood, Hanson, and Stein (2010) argue that, due to short horizons,

short-term T-bills are “completely riskless”, since they face virtually no interest rate exposure;

2Here, µ is the risk-neutral drift of the cash payout flow rate; if under the actual physical measure P, the cashflow follows the geometric Brownian motion process dXt

Xt= µPdt +σdZP

t , with µP the actual cash payout rate, thenby assuming the representative agent has a power-function utility, I can transform from the physical measure to therisk-neutral measure by setting µ = µP−Γσ , with Γ as the risk premium.

12

thus, short-term T-bills demonstrate extra “moneyness”, “above and beyond whatever money-like

attributes longer-term Treasuries may already have”. Furthermore, Greenwood, Hanson, and Stein

(2010) argue that not only the government can issue money-like debt claims, the private sector,

such as financial intermediaries, is also actively engaged in money creation and captures the mon-

etary convenience premium, by issuing collateralized short-term debts. Among these short-term

debts, asset-backed commercial paper and over-night repos are salient examples. This “money-

like” property makes short-horizon investments a cheaper way of financing for financial firms,

especially primary dealers. This also brings up the optimal leverage choice for financial firms:

They wish to extract the “money-like” property as much as possible over time. To account for

the “money-like” property of short-term repos, I assume repo creditors enjoy a monetary/liquidity

premium, so that they discount their short-horizon investments at rate r−ρ .

For tractability, I utilize a “staggered stationary debt structure” as in Leland and Toft (1996)

and He and Xiong (2012a). The repo contract with each creditor specifies the principal value P and

coupon rate c (so the total coupon payment is C = cP). The maturities of the continuum of creditors

are spread over time: Each repo creditor is awakened by an exogenous, independent Poisson shock

with arrival rate η , and each current debt contract matures upon arrival of the Poisson shock. Then

the maturing creditor decides to roll over its current repo contract, or to “run”. Thus, once the

maturing creditor decides to run, the firm must raise funds to satisfy the rollover loss, and the new

contract (with same principal P and coupon rate c, different debt price) replaces the old one3. In this

way, the debt structure (P, c, η) remains stationary over time. The average maturity is determined

by the exogenous Poisson shocks, and given by 1η

. From the law of large numbers, a fraction ηdt

of repo creditors are active during interval [t, t +dt] and can choose to roll over or to run.

The staggered stationary debt structure has two advantages: 1) it helps spread the debt maturi-

ties over time, thus reducing the liquidity pressure faced by the firm, if all of a sudden all creditors

demand redemption at the same time; and, 2) the stationary feature makes the solution independent

3It does not matter whether the “specific” old creditor stays with the new contract, or the old creditor leaves with theprincipal value and the firm sells the new contract at the market. All that matters is that the total measure of creditorsremains 1, and the total debt outstanding P, coupon payments C and rollover frequency η remains unchanged overtime, i.e., the debt structure remains stationary.

13

of time, from which it is easier to derive analytical solutions. However, this structure also creates

a coordination problem among creditors. Currently maturing creditor runs create an externality to

future maturing creditors in two ways: 1) creditor runs push the firm into the region that it has to

rely on depletable credit lines to cover the rollover loss, since the credit lines could fail at random

time once drawn down, the “locked-in” creditors are exposed to the risk that credit lines fail before

their debts mature; and, 2) more importantly, more runs means higher aggregate rollover loss, so

once the credit lines fail, equity holders face such high cost to restore the failed credit lines and

save the failing firm that they become less willing to do so, which exposes the remaining creditors

to higher bankruptcy risk that they cannot recover the full value of their debts. Likewise, when

current maturing creditors decide to “rollover”, they must take into account the possibility that dur-

ing their next “locked-in” period, future maturing creditors may begin to “run” and thereby pose a

negative externality to them. The situation where early “running” creditors are able to recover their

full debt value, but leave the remaining “locked-in” creditors bear the possible default risk, makes

every maturing creditor “rush for the exits” when collateral value deteriorates. I call this “strate-

gic interactions among creditors”. It is precisely this “strategic interaction” among creditors that

gives creditors more incentive to rush for the exits, to compete with each other to regain liquidity

before other creditors and before liquidity is exhausted.

It is natural to conjecture that the value of the repo contracts are monotonic and increasing in

the firm’s fundamental value, i.e., the cash flow Xt . Similarly as in He and Xiong (2012a), I use a

“guess and verify” approach to calculate the equilibrium. Based on the cash flow generated by the

firm’s securities and the implied assets value, maturing creditors may choose to roll over their debts

and continue earning interest, or to “run”. Following creditors’ choice, I may divide cash flow Xt

into two regions:

I. Rollover Region: All maturing creditors choose to roll over their current repo contracts; the

cash flow generated by firm’s securities is sufficient to satisfy the coupon payments of repos, and

the excess cash flow accrues to equity holders as dividend payments.

II. Run Region: All maturing creditors stop rolling over and choose to “run”; the rollover loss

14

must be covered to continue the firm. The cash flow of the firm alone is not enough to cover the

coupon payments and rollover cost. The firm stops all dividend payments to equity holders. It also

needs to rely on pre-established credit lines, together with the cash flow generated by the securities,

to payoff repo coupons and rollover loss from maturing creditors.

Credit Lines: liquidity insurance and “aggregate margin call”

Since each creditor is infinitesimally small, an individual creditor’s run action cannot significantly

affect the liquidity of the firm; rather, it is the aggregate run action (across creditors, and over time)

that matters. Due to the equity issuance cost, the firm does not issue equity to meet the rollover

loss every single time a maturing creditor decides to run; rather, the firm relies on pre-established

credit lines to satisfy the rollover loss. However, the credit lines are not perfectly reliable, i.e., they

cannot endlessly provide liquidity to the firm. Once drawn down, the credit lines aggregate and

pay the rollover loss from repo creditor runs, until an exogenous Poisson shock with arrival rate θ

(independent of the Poisson shocks that wake up creditors) arrives. Once shock θ arrives, the credit

lines “fail”: the CLs providers demand that the firm pay off the “aggregate margin call” first4, i.e.,

the CLs providers require additional collateral (or cash) to secure their exposures, otherwise they

simply cease providing liquidity and the firm has to file for bankruptcy5.

The parameter θ may be regarded as the frequency of making the “aggregate margin call” by

CLs providers. The larger θ is, the more frequently CLs demand the “aggregate margin call”

to continue functioning. Since the probability of bankruptcy increases with the frequency of the

aggregate margin call, creditors face higher externality (e.g., default risk) once other creditors start

to run. Therefore, larger θ also means more competition for liquidity among creditors, thus the

firm is more vulnerable to creditor runs.

Credit lines serve as contingent financing for the firm, as they pay the rollover loss when repo

creditors run. In this sense, CLs act as liquidity insurance for the firm. However, once drawn down,

4The credit lines value measures precisely the difference between the market value and principal value of debts, soit could serve as “aggregate margins” required by repo creditors.

5In what follows, I use the terms “credit lines fail” and “aggregate margin call by credit lines” interchangeably.

15

they make the “aggregate margin call” and threaten to force the firm into bankruptcy at rate θ .

From this angle, CLs also serve as a liquidity constraint faced by the firm.

Equity Holders: dividends and endogenous bankruptcy

As seller of the securities, equity holders have the right to the cash flow associated with those

securities: They are entitled to the cash flow Xt generated by the repo securities during the term of

the repo. They have to make coupon payments C of the repo out of the cash flow Xt . When there is

no creditor runs, the residual value Xt−C accrues to equity holders as dividends.

In the “Rollover Region”, since all maturing creditors choose to roll over their current debts, the

firm has sufficient cash flow to make the coupon payments, credit lines of the firm are not drawn

down and thus will not fail, and equity holders benefit from the option value while keeping the firm

alive6.

When creditors start to run, the firm is close to bankruptcy and relies on credit lines to meet

maturing creditors’ rollover losses. Following the spirit of Hart (2000) and Broadie, Chernov,

and Sundaresan (2007), once in the Run Region, I assume that equity holders are penalized by

suspension of dividends. The cash flows Xt now accrues to CLs providers, who are responsible for

making up the rollover losses and making the coupon payments of repos.

Therefore, equity holders receive 0 dividends in the Run Region, until shock θ arrives and

credit lines demand the “aggregate margin call”. If equity holders choose to meet the margin call

by issuing new equity, the failed credit lines are kept alive and the firm functions as before; in this

way, the option value of keeping the firm alive to equity holders is preserved. if equity holders

choose not to meet the margin call, upon shock θ , the firm files for bankruptcy. I assume that

it incurs a fixed equity issuance cost Φ every time the equity holders need raise new financing via

equity issuance, so it is optimal for equity holders to rely on credit lines when they are alive, instead

of continuously issuing equity to cover the rollover loss directly.

6Equity holders effectively face an option with dividend payments, so early exercise of the option cannot be ruledout for granted. I set the parameters such that it is optimal for equity holder to not declare bankruptcy before creditorruns.

16

In the “Run Region”, once the credit lines fail and the “aggregate margin call” is made to equity

holders, equity holders effectively compare the benefits (the extra option value of keeping the firm

alive) and costs (a lump-sum negative dividend to meet the “aggregate margin call”, plus the equity

issuance cost) and make an endogenous optimal choice.

If equity holders choose not to meet the aggregate margin calls and the firm files for bankruptcy,

I distinguish two bankruptcy provisions: a) Automatic Stay (AS), which protects the firm filing for

bankruptcy by stopping all debt-collecting actions, since the collateral is in lien of the firm; and, b)

Safe Harbor (SH), which prevails in repo markets currently: SH exempts the usual AS provisions of

bankruptcy codes, grants the counterparty of a defaulting party the right to immediately terminate

the contract, seize the collateral, and liquidate it in the market. With SH provisions, equity holders

can do nothing and creditors liquidate the firm in the market. However, with AS provisions, equity

holders could take advantage of repo creditors’ extreme aversion to illiquidity and make a “take-

it-or-leave-it” offer to creditors, which, in equilibrium creditors accept, and liquidation is avoided

after the stay.

From the above analysis, equity holders are brought into play in terms of whether to issue

additional equity to meet the “aggregate margin call” when all maturing creditors decide to “run”

and consequently the credit lines fail to continue covering the rollover losses. Conversely, creditors’

rollover decisions depend critically on what creditors can get in case of bankruptcy, which in turn

depends on equity holders’ decision to meet the aggregate margin calls or to ignore the calls and file

for bankruptcy. Therefore, creditors and equity holders make decisions based on the other party’s

decisions, and they affect each other as well. I call it “strategic interactions between creditors

and equity holders”.

Equilibrium Conjecture

I use the indicator ISH to represent that I am working with safe harbor provisions, and indicator IAS

to show that I am dealing with automatic stay provisions.

First I define the stopping times needed in calculating the value functions and deriving the

17

equilibrium:

(τη − t) |t : = inf{τ ≥ t : Poisson shock η arrives at τ} ∼ Exponential(η) : the first moment

after t that one specific repo creditor wakes up and decides to roll over or to run;

(τθ − t) |t : = inf{τ ≥ t : Poisson shock θarrives at τ} ∼ Exponential(θ) : the first moment af-

ter t that all maturing creditors decide to run, credit lines fail to continue covering the rollover loss

and CLs providers demand an “aggregate margin call” for equity holders.

Here, I distinguish the two independent Poisson shocks η and θ : η is the idiosyncratic Poisson

shock that awakes the individual repo creditor under consideration, who then decides whether to

roll over or to run; θ is the Poisson shock at the aggregate creditors level, where all maturing repo

creditors decide to run and collectively pose an externality of making an “aggregate margin call”

and imposing possible bankruptcy risk.

“Guess and verify” equilibrium: It is quite natural to conjecture that the value functions

of the firm are monotonic and increasing in the fundamentals of the firm, i.e., the cash flow Xt

generated by the firm’s securities. I thus conjecture that the firm transits from “Rollover Region”

to “Run Region” as the cash flow falls below a threshold value X∗R7, i.e., the firm shifts to “Run

Region” and relies on credit lines to pay the rollover loss when Xt < X∗R; once the credit lines fail,

equity holders choose to meet the “aggregate margin call” and revive the firm only when the cash

flow is greater than a threshold value X∗B , they will not meet the margin calls and leave the firm

bankrupt if Xt < X∗B8. Here it is possible either X∗R ≥ X∗B or X∗R < X∗B . If X∗R ≥ X∗B , creditor runs are

essentially triggered when the firm’s cash flow falls below X∗R , but as long as they remain above X∗B ,

equity holders would always replenish the failed credit lines; in this case, bankruptcy is possible

only when Xt < X∗B , i.e., when equity holders refuse to meet the “aggregate margin call” made by

CLs providers. If X∗R < X∗B9, it means creditor runs occur when Xt < X∗R , and if the credit lines

fail due to creditor runs (which occurs only when Xt < X∗R , since otherwise the firm stays in the

7Subscript “R” stands for “Runs”.8Subscript “B” stands for “Bankruptcy”.9If this is the case (X∗R < X∗B), together the parameter constraints such that it is never optimal for equity holders to

declare bankruptcy before creditor runs, then for the result X∗R < X∗B , X∗B is only conceptual, and it only reflects the factthat equity holders would never choose to meet the margin calls when credit lines fail due to creditor runs. WLOG, Iset X∗B = X∗R for the case X∗R < X∗B .

18

“Rollover Region” and the CLs will not be drawn down, thus remaining alive), it is never optimal

for equity holders to re-establish the credit lines. For a given pair of (X∗B,X∗R), I can get the value

functions of debt, equity and credit lines (which are functions of (X∗B,X∗R)); then, after deriving the

value functions, I simply apply the decision rules for creditors and equity holders, which separately

determine the threshold values X∗R and X∗B . The equilibrium (X∗B,X∗R) is calculated as a fixed point

of the system.

Once I solve for the equilibrium, similarly as in He and Xiong (2012a) and Cheng and Milbradt

(2012), I am also able to prove that such equilibrium does exist, and is unique, under appropriate

parameter constraints.

The aforementioned equilibrium is calculated for given debt principal P; at time 0, equity hold-

ers choose the optimal amount of debt to maximize the firm value ex ante.

Parameters Constraints and Debt Covenants

First, I need µ < r so that the un-levered all-equity firm value VU (Xt) =Xt

r−µdoes not explode.

The coupon rate c must be such that c > r−ρ , to give creditors the incentive to lend via repo

contracts. I also set the coupon rate c such that: c < r(a1+φ)a1(1−φ) , with φ the proportional fire sale cost at

liquidation and a1 =(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r)

σ2 , to rule out the possibility of the unappealing equi-

librium X∗B > X∗R for the case of SH (for a sufficiently large coupon rate c, creditors choose to stay

with the firm, even when facing equity holders’ endogenous bankruptcy strategy, see Appendix).

In the following, when dealing with SH, I always assume X∗B < X∗R .

With AS, in order to guarantee the existence of equilibrium (especially for the very illiquid se-

curities, i.e., fire sale cost φ very high), I need a debt covenant: If equity holders file for bankruptcy

before creditors start to run, they must give creditors the full principal value; otherwise if equity

holders wait and do not file for bankruptcy until creditors start to run, they may renegotiate and

give creditors the value accounting for illiquidity of the collateral φ . Without the debt covenant,

there might not be equilibrium for illiquid securities. To see why, simply consider the extreme case

where the securities are not liquid at all even after the stay, i.e., φ = 1. Then with the bankruptcy

19

sharing rule under AS, at the point of bankruptcy, creditors get nothing and equity holders get the

entire firm value10 (i.e., in total violation of the APR). Suppose that equity holders decide to file for

bankruptcy at value X1, then the debt value at X1, from the boundary condition, becomes 0. How-

ever, creditors start to run at X2, when the debt value equals the principal value, i.e., D(X2) = P; by

monotonicity I must have X2 > X1. Then, equity holders get the equity value at X2 if they do not file

for bankruptcy until X1; however, they will receive the total firm value (with small discounting) at

X2 if they file for bankruptcy at X2, which is higher than the equity value at X211; so equity holders

would file for bankruptcy at X2 instead of X1, which again makes creditors’ value at X2 equal to 0,

and thus creditors have to run at an even higher value X3, etc. In this way, no equilibrium could

possibly exist without a debt covenant. Since firms cannot borrow without an equilibrium, they

would also have incentive to put in a debt covenant ex ante.

2.2 Decisions of Creditors and Equity Holders

In this section, I assume the capital structure has already been established, i.e., the principal value

of repos P is fixed; and for this given capital structure, I solve for the optimal threshold strategies

of creditors X∗R and of equity holders X∗B . After solving the equilibrium, I can look for optimal P

that maximizes total firm value ex ante.

Since the value functions of repo D(Xt), equity E (Xt) and credit lines L(Xt) all depend implic-

itly on the threshold values of creditor runs X∗R and of equity holders’ endogenous bankruptcy X∗B;

and since once in the “Run Region”, the failure of credit lines arrives randomly as Poisson shocks,

the status that whether the CLs remain alive or fail, would affect the value functions as well; I

express these dependencies by writing the value functions explicitly as D(Xt ;X∗R,X∗B|CL alive),

E (Xt ;X∗R,X∗B|CL alive) and L(Xt ;X∗R,X

∗B|CL alive) when the credit lines are alive; and when the

credit lines fail, D(Xt ;X∗R,X∗B|CL fails), E (Xt ;X∗R,X

∗B|CL fails) and L(Xt ;X∗R,X

∗B|CL fails) sepa-

rately 12.

10With small discounting to account for the time value of assets during the stay period.11This is ensured by parameter constraints.12In the following, when there is no ambiguity, I might depress the explicit dependence of value functions on X∗R , X∗B

20

First I analyze the value functions at t when the credit lines are alive. Due to risk neutrality, the

individual repo creditor’s value is given by:

D(Xt ;X∗R,X∗B|CLs alive) = E {[

∫τ=min{τη ,τθ}

te−(r−ρ)(s−t)Cds

+ e−(r−ρ)(τ−t) maxrollover or run

(D(Xτ ;X∗R,X∗B|CLs alive) , P)I{τ=τη}

+

e−(r−ρ)(τ−t)ISHD(Xτ |firm goes bankrupt with SH)

+ e−r(τ−t)IASD(Xτ |firm goes bankrupt under AS)

I{τ=τθ , Xτ<X∗B}]|Ft}

i.e., the repo creditor continues receiving the coupon C per unit of time, until the Poisson shock η

arrives at τη , when the creditor decides to roll over or to run; or until the Poisson shock θ arrives

at τθ , when the CLs demand an “aggregate margin call” and equity holders refuse to meet the call

(given by the condition Xτ < X∗B), so the firm files for bankruptcy, and then the repo creditor gets

its value in bankruptcy following different bankruptcy provisions. It should be noted that for AS,

due to the stay, creditors lose the monetary/liquidity premium ρ and discount the bankruptcy value

at rate r as long-term investors.

The equity holders’ value function follows:

E (Xt ;X∗R,X∗B|CLs alive) = E {[

∫τ=τθ

te−r(s−t)[(Xs−C)I{Xs>X∗R}

+ e−r(τ−t)E(Xτ ;X∗R,X∗B|CLs fail)I{τ=τθ , Xτ<X∗R}]|Ft}

with the equity value when CLs fail given by:

E(Xτ ;X∗R,X∗B|CLs fail) = max

meet the margin call or not

E (Xτ ;X∗R,X

∗B|CLs alive)− [−L(Xτ ;X∗R,X

∗B|CLs alive)]−Φ;

ISHE (Xτ |CLs fail & firm goes bankrupt with SH)

+ IASE (Xτ |CLs fail & firm goes bankrupt with AS)

that is, equity holders receive the residuals of the cash flow minus coupon payments as dividends,

and CL alive/fails; however, at any instant, this dependence always exists.

21

so long as there is no creditor runs; once creditors start to run and an “aggregate margin call” is

made by the CLs, equity holders choose between keeping the firm alive by satisfying the margin

call and contributing −L(Xτ ;X∗R,X∗B|CLs alive)+Φ, and filing for bankruptcy, in which case they

get the equity value in bankruptcy, following different bankruptcy provisions.

Similarly I obtain the value of credit lines:

L(Xt ;X∗R,X∗B|CLs alive) = E {[

∫τ=τθ

te−r(s−t)IXs≤X∗R [(Xs−C)−η (P−D(Xs;X∗R,X

∗B))]ds]|Ft}

The credit lines are not drawn down until creditors start to run; once creditors run (given by the con-

dition Xs≤X∗R), the firm must rely on the credit lines to pay the rollover loss [P−D(Xs;X∗R,X∗B)]ηds,

with ηds the mass of running creditors. In addition, in line with principal-agent theory, I suppose

the firm suspends any dividend payments, and the residuals accrue to the credit lines, once a run

occurs.

I. Rollover Region: D(Xt)≥ P⇐⇒ Xt ≥ X∗R , creditors rollover, no need to draw down CLs.

In the “Rollover Region”, since every maturing creditor chooses to roll over, the credit lines are not

drawn down, they never fail (i.e., no “aggregate margin call” is made by credit lines) in this region.

So I always work with the condition “CLs alive” for Xt ≥ X∗R .

Due to the staggered stationary debt structure and the Poisson arrival processes, and free of

bankruptcy risk, the value functions are time-homogeneous, and thus time-independent in the

“Rollover Region” (similarly as in He and Xiong (2012a)). Therefore, I derive ODEs for the

Hamilton-Jacobi-Bellman (H-J-B) equations of the value functions, as opposed to the usual PDEs

with the Brownian motion processes.

For a single creditor, when considering the change in the creditor’s continuation value in the

22

time interval [t, t +dt], the value of its repo contract is given by the H-J-B equation:

(r−ρ)D(Xt ;X∗R,X∗B|CLs alive) = C+µXtD′(Xt)+

12

σ2X2

t D′′(Xt)

+η maxrollover or run

{0, P−D(Xt ;X∗R,X∗B|CLs alive)}

with C = cP13 the coupon payment to an individual creditor with principal P. The LHS is the

required return for repo creditors, taking into account the monetary/liquidity premium of extreme

short-term debts enjoyed by repo creditors. The sum of the terms on RHS is the expected incre-

ment in the continuation value from holding the repo contracts; this should be equal to the required

return on LHS under risk-neutral probability. The C on RHS is the instantaneous coupon payment

to creditors; the second and third term on RHS represent the value changes due to the geomet-

ric Brownian motions of the cash flow Xt ; and the last term is the value change when the repo

contract matures, which arrives as Poisson shocks with arrival rate η . Repo creditors choose to

stay with the firm and accrue coupons if their expected value from rolling over the repo contracts,

D(Xt ;X∗R,X∗B|CLs alive), is greater than P, the value they can get by running before the firm de-

faults. Following the threshold assumption, max [0, P−D(Xt ;X∗R,X∗B|CLs alive)] = 0 for Xt ≥ X∗R .

Since there is no creditor runs, the creditor under investigation faces no externality caused by other

creditors.

For equity holders, since in the “Rollover Region”, the dividend payment is always positive,

and the equity value is given by the H-J-B equation:

rE (Xt ;X∗R,X∗B|CLs alive) = (Xt−C)+µXtE ′(Xt)+

12

σ2X2

t E ′′(Xt)

Similarly, the LHS is the required return for equity, where the required return rate is the risk-free

rate r under the risk-neutral measure. The sum of the terms on RHS is the expected return from

13The coupon rate c is chosen s.t. X∗R−C+ηb [D(X∗R)−P] = 0, i.e., the moment that creditors decide to run, is thesame moment that the CLs of the firm are drawn down, thus the firm enters into a liquidity distress state.

This approach is just assumed to simplify the calculations, and is not essential to the analysis. Otherwise, it justcomplicates the calculations, without adding much insights.

23

holding equity; it should be equal to the required return on LHS under risk-neutral probability. For

the RHS, the first term Xt−C represents dividend payments to equity holders, which is the residual

of cash flow deducted by coupon payments of repos. The last two terms of the RHS represent the

value changes due to the geometric Brownian motions of the fundamental Xt . Since all creditors

roll over, the firm has sufficient funding and sufficient cash flow to cover the coupon payments,

there is no need to dilute existing equity.

For the credit lines, in the “Rollover Region”, CLs providers get 0 payment, so the H-J-B

equation for CLs is:

rL(Xt ;X∗R,X∗B|CLs alive) = µXtL′(Xt)+

12

σ2X2

t L′′(Xt)

Again, the LHS represents the required return for CLs providers, who discount future cash flows at

rate r. The sum of the terms on the RHS is the expected return from providing the credit lines to the

firm; it should be equal to the required return on the LHS under risk-neutral probability. Since there

are no other payments or draw-downs to credit lines in the “Rollover Region”, the RHS includes

only the value changes caused by the geometric Brownian motions of the fundamental Xt .

II. Run Region: D(Xt)< P⇐⇒ Xt < X∗R , all maturing creditors decide to run, the firm relies

on CLs to cover the rollover loss

In the “Run Region”, since the firm begins to rely on the credit lines to fulfill the coupon payments

and, more importantly, to cover the rollover loss, the credit lines might fail following a Poisson

shock of arrival rate θ . Thus I consider the two cases following the status of the credit lines: a)

CLs alive; b) CLs fail.

Unlike in the “Rollover Region”, following different bankruptcy provisions14, the values at

bankruptcy might become dependent on time; thus the value functions in the “Run Region” might

depend on the passage of time as well, so I return to the usual case of PDEs for the H-J-B equations

of the value functions.14See analysis in the section “Two Bankruptcy Provisions: SH v.s. AS”.

24

II.a) Run Region Xt < X∗R and CLs Alive:

For equity holders, in the “Run Region”, the dividend payment reduces to 0 so long as the CLs

are alive; if the CLs fail (with Poisson arrival rate θ ) and impose an “aggregate margin call” to

equity holders, equity holders choose to meet the margin call, or to file for bankruptcy; then the

H-J-B equation for equity value is given by:

rE (Xt ;X∗R,X∗B|CLs alive) = µXt

∂E∂Xt

+12

σ2X2

t∂ 2E∂X2

t+

∂E∂ t

+θI{Xt<X∗R} [E(Xt ;X∗R,X∗B|CLs fail)−E(Xt ;X∗R,X

∗B|CLs alive)]

The additional ∂E∂ t term on the RHS represents the change in equity value due to passage of time.

The last term on the RHS represents the changes caused by the failure of credit lines, which would

arrive at rate θ in the “Run Region”. This is the externality to equity holders caused by creditor

runs, which initiate the reliance on credit lines and may deplete the liquidity within the firm (and

margin call to equity holders).

For a single creditor, the value of repo follows the H-J-B equation:

(r−ρ)D(Xt ;X∗R,X∗B|CLs alive) = C+µXt

∂D∂Xt

+12

σ2X2

t∂ 2D∂X2

t+

∂D∂ t

+η maxrollover or run

{0, P−D(Xt ;X∗R,X∗B|CLs alive)}

+θI{Xt<X∗R} [D(Xt ;X∗R,X∗B|CLs fail)−D(Xt ;X∗R,X

∗B|CLs alive)]

Again, the additional ∂D∂ t term on the RHS represents the change of value in repo contracts due to

passage of time. Comparing with the ODEs for repos in the “Rollover Region”, on the RHS, the

maturing creditor’s choice becomes

max{0, P−D(Xt ;X∗R,X∗B|CLs alive)}= P−D(Xt ;X∗R,X

∗B|CLs alive)

instead of 0, i.e., the maturing creditor would choose to run instead of rolling over; moreover, the

25

last term is the change to repo value caused by the failure of credit lines. This last term represents

the externality to remaining “locked-in” creditors who mature in the future, which is caused by the

previously maturing creditor runs. Due to this term, creditors essentially compete for liquidity as

the firm enters into the “Run Region”.

For CLs providers, the H-J-B equation of CLs is as follows:

rL(Xt ;X∗R,X∗B|CLs alive) = {Xt−C−η [P−D(Xt ;X∗R,X

∗B|CLs alive)]}

+µXt∂L∂Xt

+12

σ2X2

t∂ 2L∂X2

t+

∂L∂ t

+θI{Xt<X∗R} [L(Xt ;X∗R,X∗B|CLs fails)−L(Xt ;X∗R,X

∗B|CLs alive)]

The additional ∂L∂ t term on the RHS represents the change in CLs value due to passage of time. In

addition to the changes caused by the geometric Brownian motion (represented by the second line),

there is an extra “deficit” term Xt −C−ηb [P−D(Xt ;X∗R,X∗B|CLs alive)] covered by credit lines,

since the firm’s cash flow Xt alone cannot cover both the coupon payments and the rollover loss.

The last term is the change caused by the failure of credit lines, which arrives at rate θ .

II.b) Run Region Xt < X∗R and CLs Fail:

Now it remains to determine the values of equity, of repos and of CLs when the CLs fail:

E(Xt ;X∗R,X∗B|CLs fail), D(Xt ;X∗R,X

∗B|CLs fail) and L(Xt ;X∗R,X

∗B|CLs fail). These values depend

critically on equity holders’ choice when they face the failure of CLs and are required to meet

the “aggregate margin call”.

As previously analyzed, equity holders face two choices when CLs fail and the “aggregate

margin call” is made: 1) to issue external equity to replenish the failed CLs and meet the “aggregate

margin call”, then the firm is kept alive and the option value of equity is preserved; and, 2) not to

meet the “aggregate margin call”, and the firm files for bankruptcy voluntarily. Since equity holders

have the option to replenish the failed CLs and revive the firm, it should be noted that “CLs fail” is

not the same as “firm goes bankrupt”.

26

Following a standard “benefit v.s. costs” analysis, equity holders (EHs) choose to replenish the

failed CLs if and only if the net benefit of doing so, exceeds what equity holders receive from filing

for bankruptcy:

E(Xt |CLs fail & EHs meet the aggregate margin call)

= E(Xt |CLs alive)− [−L(Xt |CLs alive)]−Φ

> E(Xt |CLs fail & firm goes bankrupt)

i.e., after CLs replenishment, equity holders benefit from the equity value E(Xt |CLs alive) while

keeping the CLs alive, but they must raise sufficient funds (which amounts to a lump-sum negative

dividend; this funding requirement can be satisfied by issuing additional equity, which dilutes the

value of existing equity holders) to pay the negative market value of the CLs −L(Xt |CLs alive),

so that the CLs providers always break even when they restart to extend the credit lines15, and the

equity issuance cost Φ. If equity holders choose not to re-establish the failed CLs, the firm simply

files for bankruptcy. Evidently, equity holders choose to replenish the failed CLs only when the

gain (the option value of keeping the firm alive) exceeds the costs (the payments to CLs plus equity

issuance cost):

E(Xt |CLs alive)−E(Xt |CLs fail & firm goes bankrupt)> Φ−L(Xt |CLs alive).

By conjecture of the threshold equilibrium, equity holders choose to replenish the failed CLs iff

Xt > X∗B; otherwise they simply file for bankruptcy at the point that CLs fail. The threshold value

15It does not matter whether the firm turns to other CLs providers (so that the value of previously failed CLs goesto 0, since there is no future cash flow), or to the same CLs providers as previously (then the previously failed CLsproviders start again from 0, since equity holders pay to CLs providers the amount−LL(Xt |CLs alive) s.t. CLs providersbreak even immediately after replenishment, starting from the point 0 = LL(Xt |CLs alive)+ [−LL(Xt |CLs alive)], withLL(Xt |CLs alive) the value of CLs after replenishment, and −LL(Xt |CLs alive) the lump-sum cash transfer from equityholders to CLs providers.

27

X∗B < X∗R is given by the indifference condition of equity holders:

E(X∗B|CLs alive)− [−L(X∗B|CLs alive)]−Φ≡ E(X∗B|CLs fail & firm goes bankrupt).

As stated in Section 3, to fully solve for X∗B , I need the continuity and smooth-pasting conditions,

i.e., both the value function and the 1-st order derivative of E(X∗B|CLs alive) are continuous at X∗B .

If CLs fail, I compare the two threshold values X∗R and X∗B: 1) X∗R ≥ X∗B , then depending on the

value Xτθat which CLs fail: if X∗B ≤ Xτθ

≤ X∗R , equity holders choose to refill the failed CLs, and

equity value becomes E(Xt |CLs fail & firm meets the aggregate margin calls); if Xτθ< X∗B , equity

holders prefer to leave the CLs failed and the firm files for bankruptcy; 2) X∗R < X∗B , then regardless

of the value Xτθat which CLs fail, equity holders will never choose to replenish the failed CLs, and

the firm files for bankruptcy.

If external equity is raised and the failed CLs are replenished, repo creditors pay nothing (so

long as equity holders choose to refill the CLs, creditors are not affected by the fact that CLs are

drawn down and failed previously; they continue to receive coupon C until their repo contracts

mature, or until CLs fail again), but their value function D(Xt ;X∗R,X∗B|CLs fail) becomes the value

function when the CLs are still alive16; otherwise, CLs are not replenished and the firm files for

bankruptcy:

D(Xt |CLs fail & EHs meet the aggregate margin calls) = D(Xt |CLs alive);

D(Xt |CLs fail & EHs not meet the aggregate margin calls) = D(Xt |CLs fail & firm goes bankrupt).

The equity value changes to

E(Xt |CLs fail & EHs not meet the aggregate margin calls) = E(Xt |CLs fail & firm goes bankrupt)

16Similar concept to debt overhang: Equity holders need to raise external financing to keep the firm alive, butthe value gained from it, is also shared with creditors. Thus equity holders would not make the ’first-best’ choice,instead, they just compare the cost with their option value of keeping the firm alive, not with the total value that firm(equity+debt) gains from it.

28

when equity holders choose not to replenish the failed CLs, and the firm files for bankruptcy.

Regardless of the choice of equity holders, L(Xt ;X∗R,X∗B|CLs fail) = 0, since CLs providers

either stop paying any cash flows (if the firm files for bankruptcy), or start again from the point of

break-even (if equity holders pay to replenish the CLs and meet the “aggregate margin call”).

2.3 Two Bankruptcy Provisions: SH v.s. AS

Again, to calculate the values at bankruptcy, debt value D(Xt |CLs fail & firm goes bankrupt) and

equity value E(Xt |CLs fail & firm goes bankrupt), I need to analyze the two different bankruptcy

provisions: safe harbor or automatic stay. At the point of bankruptcy, if under AS, since collateral

is held by the stay, equity holders could choose to raise funding (by issuing equity) to make a

“take-it-or-leave-it” offer to creditors, which is accepted in equilibrium and liquidation is avoided;

if with SH, collateral is immediately seized by creditors, and equity holders do not have the chance

to renegotiate.

Denote the moment that CLs fail and the firm files for bankruptcy as a stopping time τ .

At bankruptcy τ , if with SH, creditors could immediately seize and liquidate the collateral

securities in the market, yielding the payoff

ISH min(P, (1−φ)VU (Xτ)

)= ISH

[P−max

(0, P− (1−φ)VU (Xτ)

)]where φ is the proportional fire sale cost, and (1−φ) represents the recovery rate from liquidation;

the expected bankruptcy loss to creditors is ISH max(0, P− (1−φ)VU (Xτ)

); and equity holders

get the residual ISHmax((1−φ)VU (Xτ)−P, 0

). The liquidation incurs a proportional loss φ ,

which measures the extent of fire sales in a distressed market.

If under AS, the collateral securities cannot be liquidated until after a stay T . Due to the

stay, sellers can find the best buyers over time, so unlike with SH, the fire sales are avoided un-

der AS. However, the main investors of repos, such as Money Market Funds (MMFs) are more

29

liquidity-sensitive investors17, so they lose the monetary/liquidity premium ρ with the Automatic

Stay imposed at bankruptcy, which means they discount cash flows after the stay T at rate r,

while they continue get the contracted repo rate c; so after the stay period T , for every realiza-

tion of Xτ+T , creditors would demand IAS min(ecT P, (1−φ)VU (Xτ+T )

)18, where c is the coupon

rate specified in the repo contracts, then creditors’ expected discounted payoff at τ is given by

IASE[e−rT min

(ecT P, (1−φ)VU (Xτ+T )

)|Fτ

]. It is obvious that

IASE[e−rT min

(ecT P, (1−φ)VU (Xτ+T )

)|Fτ

]= IASE

[e−rT (ecT P−max

(0, ecT P− (1−φ)VU (Xτ+T )

))|Fτ

]= IAS

{P−

((1− e(c−r)T

)P+E

[e−rT max

(0, ecT P− (1−φ)VU (Xτ+T )

)|Fτ

])}

The expected bankruptcy loss to creditors is(

1− e(c−r)T)

P+E[e−rT max

(0, ecT P− (1−φ)VU (Xτ+T )

)|Fτ

],

with E[e−rT max

(0, ecT P− (1−φ)VU (Xτ+T )

)|Fτ

]evidently a European put option19 with strike

value ecT P, maturity T , and underlying spot price given by Sτ = (1−φ)VU (Xτ), which follows a

geometric Brownian motion with volatility σ , and constant effective dividend payout ratio r−µ20.

Then the value of the European put option is easily calculated via the Black-Scholes formula.

Compare to what creditors get with SH, with the same φ and P, the debt value at bankruptcy

under AS is lower (i.e., the bankruptcy loss to creditors is higher with AS, see 2.1) due to two

reasons: 1. creditors lose the monetary benefits ρ under AS; 2. the creditors’ expected loss with

automatic stay IAS

((1− e(c−r)T

)P+E

[e−rT max

(0, ecT P− (1−φ)VU (Xτ+T )

)|Fτ

])is higher

than the loss with safe harbor ISH max(0, P− (1−φ)VU (Xτ)

), following Jensen’s inequality, the

expectation of maximums is greater than the maximum of expectations. It is well-known that the

value of a put option increases with its maturity T and volatility σ , and increase with the strike

17Either due to lack of expertise, or due to regulation requirements, MMFs hesitate to hold long-term securities.18To simplify the calculations while focus on the essence, the cash flow accumulated during the stay period [τ, τ +T ]

is assumed to be spent out as bankruptcy expenses.19Due to the options values caused by the stay, the value functions close to bankruptcy state become time-dependent,

and I have to work with the usual PDEs.20That is, the cash flow generated during the stay [τ, τ +T ] is dispensed in the bankruptcy procedure and not accrues

to the firm value.

30

1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

Xτ

Cre

dito

rs L

oss

Expected Loss to Creditors in Bankruptcy

SH, φ=0SH, φ=0.2AS, φ=0AS, φ=0.2

Figure 2.1: Expected Loss to Creditors in Bankruptcy, given by ISH max(0, P− (1−φ)VU (Xτ)

)and IAS

((1− e(c−r)T

)P+E

[e−rT max

(0, ecT P− (1−φ)VU (Xτ+T )

)|Fτ

])respectively. The

figure is plotted with P = 50, r = 0.065, ρ = 0.035, c = 0.05, µ = 0.015, σ = 0.15 and T = 1.

value ecT P, so the greater uncertainty during the stay, and higher principal value P, will both

hamper creditors’ value more seriously at bankruptcy. All these ex post effects at bankruptcy will

impact creditors’ ex ante incentive to lend. In addition, for higher fire sale cost φ , it is more likely

that the put option will end up in the money (for the same P), so higher put option value, which

means the creditors’ incentive to lend will be hurt more for less liquid securities.

With AS, if liquidation is carried out (out of the equilibrium path), equity holders would get

the expected discounted value IASE[e−rT max

(0, (1−φ)VU (Xτ+T )− ecT P

)|Fτ

]in expectation,

which is just a European call option with strike value ecT P, maturity T , and spot price given by

Sτ = (1−φ)VU (Xτ), which follows a geometric Brownian motion with volatility σ , and constant

effective dividend payout ratio r−µ .

Instead of letting creditors liquidate the firm after the stay, equity holders could work out a

“reorganization plan” during the stay and make a “take-it-or-leave-it” offer to recapitalize the firm

right after the stay: equity holders offer the amount IAS min(ecT P, (1−φ)VU (Xτ+T )

)to creditors,

31

incurring a fixed equity issuance cost Φ, and take the whole firm value VU (Xτ+T ) at time τ+T . The

surplus from the reorganization is: φVU (Xτ+T )−Φ, so equity holders would choose to reorganize

if and only if φVU (Xτ+T )−Φ ≥ 0. If the offer is accepted by creditors, equity holders’ expected

discounted gain at time τ would become:

IAS

{E[e−rT max

(0, φVU (Xτ+T )−Φ

)|Fτ

]+E

[e−rT max

(0, (1−φ)VU (Xτ+T )− ecT P

)|Fτ

] }

with the last term the equity value at bankruptcy without renegotiation; and the first term the option

value for equity holders to reorganize the firm, similarly it is a European call option. Due to the

existence of the equity issuance cost Φ, equity holders will make the offer only when φVU (Xτ+T )−

Φ ≥ 0, i.e., for larger fire sale cost φ , it is more likely for equity holders to reorganize the firm;

otherwise for small φ , equity holders are more likely to liquidate the firm to avoid costly equity

issuance; and I assume once equity holders propose to reorganize, creditors always take the offer

since they are indifferent21. Reorganization under AS is effectively a violation of the “absolute

priority rules” (see Longhofer and Carlstrom (1995)), in the sense that it gives equity holders some

positive value even before creditors fully recover their debts. This might hurt creditors’ incentive to

lend. But it also avoids inefficient fire sales and gives equity holders incentive to file for bankruptcy

more promptly, which helps preserve firm value at bankruptcy.

To summarize, when the CLs fail, if equity holders do not have the opportunity to raise external

equity to avoid liquidation (with SH), or if they refuse to meet the “aggregate margin call” and

re-establish the failed CLs (under AS and Xt < X∗B), the firm files for bankruptcy, then the repos

21Here, as in Anderson and Sundaresan (1996), all the bargaining power in the renegotiation goes to equity holdersand equity holders exploit all the surplus from reorganization (“first move advantage”). I could also utilize a classicRubinstein bargaining game, where the bargaining power to creditors is p and to equity holders is 1− p.

32

value becomes:

D(Xt |CLs fail & firm goes bankrupt)

= ISH[P−max

(0, P− (1−φ)VU (Xτ)

)]+ IAS

{e(c−r)T P−E

[e−rT max

(0, ecT P− (1−φ)VU (Xτ)

)|Fτ

]};

and equity value becomes:

E(Xt |CLs fail & firm goes bankrupt)

= ISH max(0, (1−φ)VU (Xτ)−P

)+ IAS

{E[e−rT max

(0, φVU (Xτ+T )−Φ

)|Fτ

]+E

[e−rT max

(0, (1−φ)VU (Xτ+T )− ecT P

)|Fτ

] }.

The threshold value X∗R at which creditors choose not to roll over is given by:

D(X∗R;X∗B,X∗R|CLs alive)≡ P;

I choose coupon rate c such that CLs begin to be drawn down at the same time that creditors start

to run: X∗R ≡ cP+ηb (P−D(X∗R|CLs alive)) = cP22; and the threshold value X∗B of endogenous

bankruptcy by equity holders is given by:

E(X∗B|CLs alive)− [−L(X∗B|CLs alive)]−Φ≡ E(X∗B|CLs fail & firm goes bankrupt).

22This setting is somewhat artificial and just set to simplify the calculations. It captures the main feature that: itis precisely the “run” behavior of creditors that places the firm under liquidity pressure, and that causes externalityto remaining “locked-in” creditors and equity holders. Without this condition, I simply need to distinguish the twothresholds, one is the threshold at which creditors decide to run, the other is the threshold at which credit lines begin tobe drawn down. However, this would complicate the calculations without adding further insight.

33

3 Solution and Equilibrium Analysis

3.1 Solution of the Model

I solve the value functions given by the H-J-B equations in Section 2 by first taking the thresh-

old values X∗R and X∗B as given, with appropriate boundary conditions, continuity conditions, and

smooth-pasting conditions; then, once get the value functions, I simply apply the decision rules

for creditors and equity holders which separately pin down the threshold values X∗R and X∗B . The

equilibrium (X∗B,X∗R) is calculated as a fixed point of the system.

Proposition 1. For given frequency θ of credit lines making the “aggregate margin call”, the repo

debt value D(Xt ;X∗R,X∗B), equity value E (Xt ;X∗R,X

∗B), and value of credit lines L(Xt ;X∗R,X

∗B) are

given in Appendix 1. For X∗B < X∗R , the running threshold for repo creditors X∗R is given by setting

D(X∗R;X∗B,X∗R|CLs alive)≡ P. (3.1)

The threshold value X∗B < X∗R for equity holders to meet the “aggregate margin call” from credit

lines when creditors start to run, if it exists, is given by:

E(X∗B;X∗B,X∗R|CLs alive)− [−L(X∗B;X∗B,X

∗R|CLs alive)]−Φ≡ E(X∗B;X∗B,X

∗R|CLs fail & firm goes bankrupt).

(3.2)

If for any value Xt < X∗R , this condition (3.2) for X∗B doesn’t hold, and

E(Xt ;X∗R,X∗R|CLs alive)− [−L(Xt ;X∗R,X

∗R|CLs alive)]−Φ < E(Xt ;X∗R,X

∗R|CLs fail & firm goes bankrupt),

it simply means that it is never optimal for equity holders to meet the margin call, WLOG, I set

X∗B = X∗R , and the value functions become independent of X∗B , X∗R can be calculated directly using

creditors’ decision rule (3.1).

After solving the repo debt value D(Xt ;X∗R,X∗B), equity value E (Xt ;X∗R,X

∗B), and value of credit

34

Parameter Value InterpretationGeneral Environment

r 0.065 annual risk-free rateρ 0.035 monetary/liquidity premium for repo creditorsX0 5 initial cash flowφ 0.20 proportional fire sale costT 120

360 with AS, length of the stayFirm Characteristics

µ 0.015 growth rate of the firm’s cash flowσ 0.15 volatility of the firm’s cash flowΦ 2 fixed equity issuance cost

Repo Debt Characteristicsη 100 rollover frequency (or 1

maturity ) of repo creditorsCredit Lines Characteristics

θ 360 frequency of credit lines making margin calls

Table 1: Benchmark Parameters

lines L(Xt ;X∗R,X∗B), I can prove the existence and uniqueness of the threshold values X∗R and X∗B .

Then, I show that such threshold strategies are indeed optimal for individual creditors and equity

holders, if others all follow the same threshold strategies specified by (X∗B,X∗R).

Proposition 2. (Optimality of the threshold strategies): The threshold strategies X∗R and X∗B specified

in Proposition 1 are indeed optimal for individual creditors and equity holders, if others all follow

the same strategies specified by (X∗B ,X∗R).

(See details and proofs in the online appendix.)

Model Parameters

Table 1 states the benchmark values of parameters I use in the numerical calculations.

According to Federal Reserve data, the 10-year Treasury interest rate has averaged 6.5% since

the 1960s; thus, the risk-free interest rate is set to r = 6.5%. The 1-year Treasury rate has averaged

3.39% since the 1980s. Motivated by this fact, I set the monetary/liquidity premium to ρ = 3.5%, to

obtain a discount rate of 3.0% for extremely short-term investors. I set the growth rate of the firm’s

securities at µ = 1.5%, the same as in He and Xiong (2012a). According to Veronesi and Zingales

(2010) “the cost of bankruptcy is about 22% of enterprise value”, so I choose the benchmark fire

35

sale cost φ = 0.20. The average assets volatility for a range of financial firms is estimated to be

approximately 10.1% in Veronesi and Zingales (2010), so I have σ = 0.15. I choose X0 s.t. the

normalized initial unlevered firm value X0r−µ

equals to 100.

For the length of the stay, I use the typical stay period in Chapter 11 bankruptcy, i.e., 120

days. I choose the rollover frequency to be η = 100, to account for the extremely short nature of

repo debts, for which the majority are overnight repos. Further I set the frequency of credit lines

providers making “aggregate margin calls” θ = 360, i.e., the CLs providers monitor the firm every

day in stressed periods.

Optimal Leverage

Keeping the coupon rate constant, a higher principal P has two off-setting effects: On the one

hand, a higher P means the firm could benefit more from exploiting creditors’ monetary/liquidity

premium per unit of time while the firm remains solvent; on the other hand, it makes creditors

more prone to run (since the run condition is given by D(Xt)< P), which could in turn trigger the

bankruptcy at higher fundamental values, thus shortening the length of time during which the firm

could extract repo creditors’ monetary/liquidity premium. Following the traditional tradeoff theory,

optimal leverage is the leverage ratio which maximally exploits the monetary/liquidity premium

under bankruptcy risk.

Figure 3.1a demonstrates the tradeoff, and gives the optimal P∗ and the corresponding optimal

leverage. For the benchmark case, the optimal leverage level (defined as D(X0)D(X0)+E(X0)+L(X0)

) with

SH is higher than the optimal leverage level with AS.

Maximum Debt Capacity

Now I turn to the maximum level of debt that creditors are willing to extend ex ante. The principal

value P here again has two off-setting effects: For a given coupon rate, a higher P means creditors

enjoy higher coupon payments per unit of time, thus increasing the value of debt; however, a higher

P increases the bankruptcy risk, which reduces debt value. When balancing these two effects, I have

36

30 40 50 60 70 8080

85

90

95

100

105

110

115

120

125

130Total Ex−ante Firm Value for Different Debt Principal Value P

Debt Principal P

Ex−

ante

Firm

Val

ue

SHASAll−Equity

(a) Optimal P∗ that maximizes the ex ante firm value.

30 40 50 60 70 8055

60

65

70

75

80

85

90Ex−ante Debt Value for Different Debt Principal Value P

Debt Principal P

Ex−

ante

Deb

t Val

ue

SHAS

(b) P̂ that maximizes the ex ante debt value.

Figure 3.1: Optimal Leverage and Maximum Debt Capacity

P̂, which maximize the debt capacity ex ante.

Figure 3.1b shows the tradeoff for the maximum debt capacity. For the benchmark case, the

maximum debt capacity D(X0) and the corresponding leverage (defined as D(X0)D(X0)+E(X0)+L(X0)

) with

SH are higher than those under AS. Comparing with Figure 3.1a, to obtain the maximum debt

capacity, the firm needs a higher principal value P̂, which indeed increases ex-ante debt value, but

at the expense of higher bankruptcy risk, thus damaging total firm value.

Equilibrium Outcome for the Benchmark Case

Figure 3.2a and Figure 3.2b plot the equilibrium outcome X∗B and equity values, and X∗R and repo

values, respectively, with SH and under AS. I use the benchmark parameters and principal P = 45.

From Figure 3.2a, we observe that with AS, equity holders face a higher ex-post value at bankruptcy,

which motivates them to file for bankruptcy more promptly at higher X ; consequently, creditors run

at higher fundamental values, as shown in Figure 3.2b.

Figure 3.3 plots total firm value for different cash flow Xt , with the parameters at the benchmark

value, and the principal value P set at the ex-ante optimal level of P∗ for SH and AS, respectively

(e.g., for the benchmark case φ = 0.20, I have for SH, P∗ = 42.9975 and for AS, P∗ = 42.7830).

Since for SH and AS, the optimal P∗ might be different, I compare the optimal cases that generate

37

2 2.5 3 3.5 4 4.5 50

5

10

15

20

25Benefit v.s. Cost for Equity Holders of Keeping the Firm Alive

Cashflow Xt

Equ

ity V

alue

SH: BenefitSH: CostAS: BenefitAS: Cost

(a) Equilibrium X∗B and Equity Value

2 2.5 3 3.5 4 4.5 530

35

40

45

50

55

60

65

70

75

80Repo Values

Cashflow Xt

Deb

t Val

ue

SHASDebt Principal P

(b) Equilibrium X∗R and Debt Value

Figure 3.2: Equilibrium X∗R and X∗B for SH and AS

the optimal ex-ante firm value under SH and AS, respectively, to avoid the effects of a randomly

chosen value of P. Total firm value is given by the sum of values of equity, repo debts, and credit

lines (since to start/sell the firm at time t, equity holders must pay the CLs providers s.t. CLs

providers break even at time t, so the value of CLs must be included in the calculations of firm

value).

From Figure 3.3a, for the benchmark case, with φ = 0.20, the optimal firm value with AS

always dominates that of SH, ex ante and ex post, although the ex-ante (for higher value of cash

flowX) difference is small relative to the ex-post (for lower X) difference. So for moderately liquid

or illiquid securities, fire sale risk dominates the risk of liquidity being tied up in bankruptcy, not

only ex post for the realization of low cash flows, but also ex ante, when creditors take the ex-post

effects into account. On the contrary, for highly liquid securities (φ = 0), Figure 3.3b shows that

the risk of liquidity being tied up is much larger than the fire sale risk, so creditors start to run at

higher fundamental values under AS, shortening the period during which the firm can extract the

liquidity premium, which generates a lower firm value relative to SH, ex post and ex ante.

Another observation justifies the utilization of VU (X) in calculating firm value at bankruptcy:

For sufficiently low cash flow X (low enough such that creditors start to run), the levered firm value

could be equal (AS) or lower (SH) than the non-levered all-equity firm value VU (X). Although

38

2 2.5 3 3.5 4 4.5 520

40

60

80

100

120

140Total Firm Value

Cashflow X

Tot

al F

irm V

alue

SH: Firm Value with Repos E+D+LAS: Firm Value with Repos E+D+LAll−Equity Firm Value X/(r−µ)

(a) Benchmark case, for φ = 0.2.

2 2.5 3 3.5 4 4.5 530

40

50

60

70

80

90

100

110

120

130Total Firm Value, q=1

Cashflow X

Tot

al F

irm V

alue

SH: Firm Value with Repos E+D+LAS: Firm Value with Repos E+D+LAll−Equity Firm Value X/(r−µ)

(b) For φ = 0.

Figure 3.3: Firm value w.r.t. change of cash flow Xt .

there is monetary/liquidity premium enjoyed by short-term creditors, the interactions among credi-

tors and between creditors and equity holders make bankruptcy risk so high at low value of X , that

the value of the firm is even lower than VU (X), which does not enjoy the monetary/liquidity bene-

fits, but avoids the bankruptcy risks. Thus, it is WLOG to use firm value VU (Xt) at bankruptcy23.

Margin Spirals v.s. Repo Rate Spirals

Since in the model I use a stationary debt structure, i.e., the principal outstanding of debt P is

kept constant throughout the model, we do not observe the repo margin spirals as seen during the

financial crisis (see Gorton and Metrick (2012)). Rather, I have an increase in the repo rate once

creditors start to run (new creditors put in the debt value D(Xt), accumulate coupons, and demand

P > D(Xt) at maturity, if the firm has not failed by then). So effectively, creditors demand a higher

repo rate while keeping margin constant. However, since I work in the risk-neutral world, all

agents care only about the expected payoff, a higher repo rate together with constant margin may

23Although this result is self-fulfilling here, i.e., I start with the firm value at bankruptcy VU (Xt), and show that it isindeed the best value the firm can get in case of bankruptcy.

This special treatment could be expected in periods of aggregate risk, i.e., no other financial firms are well-capitalizedenough to take over the failed firm and take advantage of the liquidity premium, so the failed firm has to be liquidatedto the outside non-financial firms, who lose the liquidity premium and only value the assets at non-levered value.

39

be regarded as equivalent to higher margin together with a constant repo rate.

3.2 Comparative Statics

With SH, ex post, when the firm files for bankruptcy, creditors could immediately liquidate the

securities. Since they do so in a short period and in huge volumes, creditors face severe fire sale

risk. Although SH guarantees creditors access to liquidity when facing bankruptcy risk, this might

not be optimal ex ante when taking the ex-post fire sales into account.

With AS, on the contrary, creditors face the risk of liquidity being tied up in bankruptcy. Due

to the stay, equity holders have the chance to propose renegotiation with creditors, thus avoiding

the costly and inefficient liquidation. I assume that equity holders extract all the surplus from the

renegotiation, which is effectively a violation of the APR: Although in equilibrium there is no fire

sales for the firm, creditors effectively still face the fire sale risk, since their outside option in the

bargaining game is constrained by the securities’ illiquidity φ .

First of all, I fix the principal value of debt P = 45, and calculate the optimal bankruptcy thresh-

old X∗B for this fixed level of P, varying φ , σ , µ , or η , the results are plotted in Figure 3.4: 1) it is

quite intuitive that the optimal bankruptcy threshold X∗B increases with fire sale cost φ , since higher

φ would induce more creditor runs when facing deteriorating fundamentals, and consequently eq-

uity holders file for bankruptcy at higher levels; 2) as volatility increases, the optimal bankruptcy

threshold X∗B increases as well. This result is different from the typical Leland model, where the

bankruptcy threshould decreases with σ , as higher volatility makes the option value for equity

higher and quity holders become more willing to wait longer before declaring bankruptcy. How-

ever, here when taking into the coordination problem among creditors into account, with higher

σ , assuming the same creditor running threshold X∗R , it takes shorter period for the fundamental

value to fall below X∗R that creditors start to run24, which essentially means the maturing creditor

faces higher rollover risk from other creditor runs, and it makes every maturing creditor choose

24Although the real option value of the maturing creditor choosing to roll over or to run increases with higherσ , which would induce a lower running threshold X∗R , this effect is dominated by the coordination problem amongcreditors.

40

a higher running threshold X∗R , which in turn makes the bankruptcy risk more imminent, i.e., the

optimal bankruptcy threshold X∗Bincreases with σ . This result also shows that for short term debt,

it is the liquidity risk that drives the insolvency risk. 3) when the productivity µ decreases, the

attractiveness of the securities decrease, and equity holders file for bankruptcy at higher threshold

values; and 4) when creditors incease their rollover frequency, essentially every maturing creditor

is facing higher rollover risk from other creditors, which results in a higher running threshold X∗R

and a higher optimal bankruptcy threshold X∗B .

Another observation from Figure 3.4 is that for the same principal value P, under all circum-

stances, equity holders would file for bankruptcy at higher threshold under AS relative to the case

with SH. This is also quite intuitive: from creditors’ perspective, since the bankruptcy loss is higher

under AS, creditors would stop rolling over at higher threshold X∗R , this early running would pose a

pressure for equity holders to default earlier; from equity holders’ perspective, under AS, due to the

violation of APR, equity holders have incentive to default earlier to take advantage of their higher

value at bankruptcy. These two effects move in the same direction and result in a higher optimal

bankruptcy threshold X∗B .

However, ex-ante, the firm can choose different principal value of debt P when it faces different

market conditions. Therefore, I calculate the optimal level of P∗ as changing the parameters which

gives the highest firm value ex ante. The results are demonstrated as follows.

3.2.1 Fire Sale Cost (Illiquidity of Securities) φ

I plot the equilibrium outcome w.r.t. change of the collateral’s proportional fire sale cost φ in

Figure 3.5, while all other parameters are kept at benchmark values. Here, I compare the ex-ante

optimal equilibrium with SH and under AS, i.e., P is set at the optimal level, and is different for

different fire sale cost φ , or for SH and AS.

Figure 3.5a compares the ex-ante optimal firm value with SH v.s. the ex-ante optimal firm value

under AS. First, optimal firm value declines when firm’s securities become more illiquid, as the firm

faces more severe creditor runs due to higher fire sale risk. When facing sufficiently high fire sale

41

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72

2.5

3

3.5

4

4.5

5

5.5Bankruptcy Threshold for Different φ, with Same P=45

Fire Sale Cost φ

Cas

hflo

w X

t

SHAS

(a) Optimal Bankruptcy Threshold for Different φ

0.1 0.12 0.14 0.16 0.18 0.22.7

2.75

2.8

2.85

2.9

2.95

3

3.05

3.1Bankruptcy Threshold for Different σ, with Same P=45

Volatility σ

Cas

hflo

w

SHAS

(b) Optimal Bankruptcy Threshold for Different σ

−0.01 −0.005 0 0.005 0.01 0.015 0.02 0.0252

2.5

3

3.5

4

4.5Bankruptcy Threshold for Different µ, with Same P=45

Productivity µ

Cas

hflo

w X

t

SHAS

(c) Optimal Bankruptcy Threshold for Different µ

0 50 100 150 200 250 300 350 4002.6

2.65

2.7

2.75

2.8

2.85

2.9

2.95

3Bankruptcy Threshold for Different η, with Same P=45

Rollover Frequency η

Cas

hflo

w X

t

SHAS

(d) Optimal Bankruptcy Threshold for Different η

Figure 3.4: Comparative Statics for Fixed Debt Principal Value P

42

risk, the optimal P∗ under AS gives higher ex-ante firm value than the optimal P∗ with SH. To see

why, I examine Figure 3.5b for the ex-post firm value at bankruptcy: By avoiding costly liquidation,

AS gives higher ex-post firm value than SH. However, the value distributions differ: with SH, all

firm value at bankruptcy goes to creditors; while under AS, creditors only get part of the ex-post

firm value, and the remainder goes to equity holders. Since equity holders enjoy some claims at

bankruptcy, they have incentives to file for bankruptcy earlier, at higher fundamental value, before

their share of bankruptcy value deteriorates further, as shown in Figure 3.5d. Again, since the firm

files for bankruptcy more promptly, the value to creditors at bankruptcy with AS is actually higher

than that with SH, although the share of creditors value declines. Thus, under AS, the average time

during which the firm could exploit the liquidity premium from creditors becomes shorter, but the

cost is outweighed by the benefit of avoiding costly liquidation; therefore, AS gives higher firm

value ex ante for higher φ . Figure 3.5c shows that debt capacity goes down with illiquidity φ as

well, and SH would give higher debt capacity than AS.

3.2.2 Volatility σ

I plot the equilibrium outcome w.r.t. change of the collateral’s volatility σ in Figure 3.6, while all

other parameters are kept at benchmark values. Here I compare the ex-ante optimal equilibrium

with SH and under AS, i.e., P is set at the optimal level, and is different for different volatilities σ ,

or for SH and AS.

In Figure 3.6a, I compare the ex-ante optimal firm value with SH v.s. the ex-ante optimal firm

value under AS. First, optimal firm value declines when firm’s securities become more volatile,

since the firm faces more severe creditor runs due to greater uncertainty. When facing greater un-

certainty, the benefit of AS relative to SH becomes even larger, as shown by greater dispersion in

Figure 3.6a. To see why, I examine Figure 3.6b for ex-post firm value at bankruptcy: First, with

greater uncertainty, ex-post firm value declines, since the firm files for bankruptcy at lower funda-

mental values (see Figure 3.6d), as equity holders’ option value of keeping the firm alive increases

with higher volatility; secondly, by avoiding costly liquidation, AS gives higher ex-post firm value

43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7121

122

123

124

125

126

127

128

129

130Ex−Ante Optimal Firm Value

Fire Sale Risk φ

Tot

al F

irm V

alue

SHAS

(a) ex-ante Optimal Firm Value for Different φ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60Ex−Post Total Firm Value at Bankruptcy

Fire Sale Risk φ

Firm

Val

ue

SH: Equity ValueSH: Debt ValueAS: Equity ValueAS: Debt Value

(b) ex-post Firm Value and Debt Value for Different φ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.770

75

80

85

90

95

100

105Ex−Ante Maximal Debt Value

Fire Sale Risk φ

Max

imal

Deb

t Val

ue

SHAS

(c) Maximum Debt Capacity for Different φ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3Equity Holders Bankruptcy Threshold

Fire Sale Risk φ

Cas

hflo

w X

t

SHAS

(d) Equity Holders’ Optimal Bankruptcy Threshold

Figure 3.5: Equilibrium for different fire sale cost φ .

44

than SH. However, the value distributions differ: with SH, all firm value at bankruptcy goes to

creditors; while under AS, creditors get only part of the ex-post firm value, and the remainder goes

to equity holders. Since equity holders enjoy some claims at bankruptcy, they have incentives to file

for bankruptcy earlier at higher fundamental value, before their portion of bankruptcy value dete-

riorates further, as shown in Figure 3.6d. The greater the uncertainty, the sooner equity holders file

for bankruptcy under AS relative to SH. Again, since the firm files for bankruptcy more promptly,

the value to creditors at bankruptcy under AS is actually higher than that with SH, although the

share of creditors value declines. Thus, with AS, the average time during which the firm could

exploit the liquidity premium from creditors shortens, but the cost is outweighed by the benefit

of avoiding costly liquidation; therefore, AS gives higher firm value ex ante for high values of σ .

Figure 3.6c shows that debt capacity declines with volatility σ as well (due to more severe creditor

runs), and SH would result in higher debt capacity than AS.

3.2.3 Growth Rate µ

When the actual profitability of the securities declines (e.g., the profitability of MBS goes down

when facing more defaults in mortgages during the financial crisis), the growth rate µ declines

as well, which certainly affects firm value (as shown in Figure 3.7) in several ways: 1) Optimal

firm value and debt capacity both increase with µ , when µ becomes negative, the debt capacity

with SH and under AS become almost equal; 2) Ex-post firm value increase with µ as well, the

percentage loss of SH relative to AS ex post increases as µ becomes more negative, which makes

AS more favorable in times of low profitability; and, 3) For lower µ , it is optimal for the firm to

choose a lower principal P ex ante, to prolong the expected time during which the firm can exploit

the liquidity premium from creditors, thus, equity holders declare bankruptcy at lower fundamental

values, for both SH and AS. The gap between the bankruptcy thresholds increases as µ deteriorates,

which means that under AS, the firm tends to file for bankruptcy in a more timely manner, especially

when facing low profitability.

45

0.1 0.12 0.14 0.16 0.18 0.2115

120

125

130

135

140

145Ex−Ante Optimal Firm Value

Volatility σ

Tot

al F

irm V

alue

SHAS

(a) ex-ante Optimal Firm Value for Different σ

0.1 0.12 0.14 0.16 0.18 0.20

10

20

30

40

50

60Ex−Post Total Firm Value at Bankruptcy

Volatility σ

Firm

Val

ue

SH: Equity ValueSH: Debt ValueAS: Equity ValueAS: Debt Value

(b) ex-post Firm Value and Debt Value for Different σ

0.1 0.12 0.14 0.16 0.18 0.275

80

85

90

95

100

105Ex−Ante Maximum Debt Capacity

Volatility σ

Deb

t Cap

acity

SHAS

(c) Maximum Debt Capacity for Different σ

0.1 0.12 0.14 0.16 0.18 0.22.2

2.4

2.6

2.8

3

3.2

3.4

3.6Equity Holders Bankruptcy Threshold

Volatility σ

Cas

hflo

w

SHAS

(d) Equity Holders’ Optimal Bankruptcy Threshold

Figure 3.6: Equilibrium for different volatility σ .

46

−0.005 0 0.005 0.01 0.015 0.02 0.02580

90

100

110

120

130

140

150

160Ex−Ante Optimal Firm Value

Profitability µ

Tot

al F

irm V

alue

SHAS

(a) ex-ante Optimal Firm Value for Different µ

−0.005 0 0.005 0.01 0.015 0.02 0.0250

10

20

30

40

50

60Ex−Post Total Firm Value at Bankruptcy

Profitability µ

Firm

Val

ue

SH: Equity ValueSH: Debt ValueAS: Equity ValueAS: Debt Value

(b) ex-post Firm Value and Debt Value for Different µ

−0.005 0 0.005 0.01 0.015 0.02 0.02560

65

70

75

80

85

90

95

100

105Ex−Ante Maximum Debt Capacity

Profitability µ

Deb

t Cap

acity

SHAS

(c) Maximum Debt Capacity for Different µ

−0.005 0 0.005 0.01 0.015 0.02 0.0252.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3Equity Holders Bankruptcy Threshold

Profitability µ

Cas

hflo

w

SHAS

(d) Equity Holders’ Optimal Bankruptcy Threshold

Figure 3.7: Equilibrium for different growth rate µ .

47

3.2.4 Rollover Frequency η (or Equivalently, Average Debt Maturity 1η

)

From a single creditor’s perspective, if the rollover frequency for other creditors is kept unchanged,

it is optimal for the creditor itself to shorten the debt maturity and increase the rollover frequency,

since higher rollover frequency essentially increases the choice set and the creditor could opt to run

more promptly than others. However, the coordination game among creditors makes it impossible

to keep other creditors’ rollover frequency unchanged. When every creditor opts for higher rollover

frequency, the strategic interaction among creditors is further amplified, which results in higher

threshold value for creditors to “run”; consequently equity holders file for bankruptcy at higher

threshold value as well. Thus higher rollover frequency hurts firm value, as show in Figure 3.8.

3.2.5 Length of the Stay T

Length of the stay T does not affect the equilibrium outcome with SH. For AS, after accounting for

illiquidity of the securities, length of the stay effectively serves 1) to measure the time value loss

of firm’s assets during the stay; 2) as a means to allocate firm value between creditors and equity

holders, since the value of the put option depends critically on length of the stay T . Ex post, a

longer stay allocates lower value at bankruptcy to creditors and distorts creditors’ incentive more,

which results in even lower debt capacity and ex-ante firm value. So long as the stay is long enough

to avoid fire sales, a shorter stay is better, “just sufficient for the failing firm to make arrangements

to avoid inefficient liquidations”, as shown in Figure 3.9.

3.2.6 Endogeneity of Fire Sale Cost φ

As explained in Brunnermeier and Pedersen (2009), there is a liquidity spiral when the market

deteriorates and the bankrupt firm liquidates its assets. In the model, under AS, since the liquidation

is either avoided (for securities with high fire sale cost), or the liquidation is carried out slowly over

time without incurring significant price impacts (for highly liquid securities), the fire sale cost φ is

more hypothetical and serve as an outside option for creditors in the renegotiation game. However,

with SH, since liquidation is immediate upon bankruptcy, it would push down the market price of

48

0 50 100 150 200 250 300 350 400150.5

151

151.5

152

152.5

153

153.5

154

154.5

155Ex−Ante Optimal Firm Value

Rollover Frequency η

Tot

al F

irm V

alue

SHAS

(a) ex-ante Optimal Firm Value for Different η

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

30

35

40

45Ex−Post Total Firm Value at Bankruptcy

Rollover Frequency η

Firm

Val

ue

SH: Equity ValueSH: Debt ValueAS: Equity ValueAS: Debt Value

(b) ex-post Firm Value & Debt Value for Different η

0 50 100 150 200 250 300 350 400125.5

126

126.5

127

127.5

128Ex−Ante Optimal Firm Value

Rollover Frequency η

Tot

al F

irm V

alue

SHAS

(c) Maximum Debt Capacity for Different η

0 50 100 150 200 250 300 350 4002.6

2.65

2.7

2.75

2.8

2.85

2.9

2.95Equity Holders Bankruptcy Threshold

Rollover Frequency η

Cas

hflo

w

SHAS

(d) Equity Holders’ Optimal Bankruptcy Threshold

Figure 3.8: Equilibrium for different rollover frequency η .

49

0 0.1 0.2 0.3 0.4 0.5126

126.2

126.4

126.6

126.8

127

127.2

127.4Ex−Ante Optimal Firm Value

Length of Stay T

Tot

al F

irm V

alue

SHAS

(a) ex-ante Optimal Firm Value for Different T

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

40

45Ex−Post Total Firm Value at Bankruptcy

Length of Stay T

Firm

Val

ue

SH: Equity ValueSH: Debt ValueAS: Equity ValueAS: Debt Value

(b) ex-post Firm Value and Debt Value for Different T

0 0.1 0.2 0.3 0.4 0.584.5

85

85.5

86

86.5

87

87.5

88Ex−Ante Maximum Debt Capacity

Length of Stay T

Deb

t Cap

acity

SHAS

(c) Maximum Debt Capacity for Different T

0 0.1 0.2 0.3 0.4 0.52.62

2.64

2.66

2.68

2.7

2.72

2.74

2.76

2.78

2.8

2.82Equity Holders Bankruptcy Threshold

Length of Stay T

Cas

hflo

w

SHAS

(d) Equity Holders’ Optimal Bankruptcy Threshold

Figure 3.9: Equilibrium for different length of stay T .

50

the securities of the bankrupt firm, which endangers the stability of other financial firms holding

the same or very similar securities and may pushes more firms to bankruptcy region, and thus

magnifying the fire sale cost the failing firm faces. In this way, for a “hypothetical” fire sale cost φ ,

the bankrupt firm may effectively face a fire sale cost λφ , with λ ≥ 1 representing the real fire sale

risk when the firm endogenizes the fire sale risk it poses to the market.

Figure 3.10 plots the optimal leverage and optimal firm value w.r.t. different fire sale risk φ .

As shown in Figure 3.10a, for the same (hypothetical) φ , the firm would optimally choose higher

leverage with SH (green line) than under AS (black line); however, when ther firm endogenizes the

fire sale risk (i.e., λ = 2), it might endogeneously choose a lower leverage with SH (blue line) than

AS for securities with high fire sale cost; moreover, it is likely that the firm chooses the optimal P∗

ex-ante when it doesn’t endogenize the fire sale risk, but when the negative shock materializes, the

firm may find itself facing a much higher fire sale risk (i.e., λ = 2), which would result in a much

higher leverage ratio (red line).

Figure 3.10b shows the optimal firm value. As explained previously, for the same φ , SH (green

line) outperforms AS (black line) for securities with lower fire sale cost φ , and we denote the

crossing point as φ1; when the firm endogenizes the fire sale risk, the domain where SH (blue line)

dominates AS shrinks, as indicated by a lower crossing point φ2 < φ1; moreover, when the firm

mistakenly takes the “optimal” principal debt value P∗ ex-ante and faces a much higher fire sale

risk ex-post, the firm value with SH (red line) becomes significantly lower than the firm value under

AS, since in this way, the firm with SH maximizes the ex-ante firm value with misspecified fire sale

risk.

3.3 Credit Risk

The model developed here has an interesting application: it helps explain the credit risk for ex-

tremely short-term debts, and provides a decomposition which quantifies the importance of insol-

vency risk and illiquidity risk, respectively:

Credit Risk (X∗B (X∗R)) = Pure Insolvency Risk (XB−LT ) + Pure Illiquidity Risk (X∗R (XB−LT )) +

51

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85Ex−Ante Optimal Leverage Ratio

Fire Sale Risk φ

Opt

imal

Lev

erag

e

SH, with φAS, with φSH, with 2φ

SH, with 2φ and P*(φ)

(a) ex-ante Optimal Leverage for Different φ

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3580

85

90

95

100

105

110

115

120

125

130Ex−Ante Optimal Firm Value

Fire Sale Risk φ

Opt

imal

Firm

Val

ue

SH, with φAS, with φSH, with 2φ

SH, with 2φ and P*(φ)

(b) ex-ante Optimal Firm Value for Different φ

Figure 3.10: Equilibrium when firm endogenizes/doesn’t endogenize the fire sale risk φ .

Interaction of Illiquidity and Insolvency Risk, where the Pure Insolvency Risk is simply calculated

from Leland-Toft model: the debt value is determined by XB−LT where there is no liquidity risk

(all rollover cost is coverd by costless equity issuance in Leland-Toft model); the Pure Illiquidity

Risk is represented by the creditors’ rollover threshold X∗R when there is no insolvency risk, i.e.

we pose the firm’s optimal bankruptcy threshold X∗B = VB−LT , since absent liquidity risk, the firm

will not declare bankruptcy until VB−LT - we then calculate the optimal rollover threshold X∗R for

creditors given VB−LT and the resulting credit spread. The residual of credit risk is the interaction

of illiquidity and insolvency risk.

Credit Risk Decomposition for Benchmark Case

Credit Spread Pure Insolvency Risk Pure Illiquidity Risk Interaction ofInsolvency andIlliquidity Risk

Spread (bps) 57.5 0.1 48.9 8.5Percent (%) 100 0 85 15

Table 2: Credit Risk Decomposition for Benchmark Case

Once we get the optimal bankruptcy threshold X∗B , we can predict the 1-year expected default

52

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

years

defa

ult p

roba

bilit

y

Default Probability for Benchmark Case

SHAS

Figure 3.11: Term Structure of Default Probabilitis

probability:

DP(t) = N

[−b−

(µ + e−0.5σ2) t

σ√

t

]+ e−2b(µ+e−0.5σ2)/σ2

N

[−b+

(µ + e−0.5σ2) t

σ√

t

]

where b = ln(

X0X∗B

)and e represents the equity risk premium which translates the risk neutral prob-

ability of default into actual physical probability of default, and for the benchmark case we set

e = 0.04. The default probability predictions for the benchmark case with initial X0 = 5 are plot-

ted in Figure 3.11. For the benchmark case, we anticipate more bankruptcies under AS relative to

SH, since the optimal bankruptcy threshold under AS is higher than that with SH, which makes

bankruptcies more likely. However, since with SH, bankruptcy always results in liquidation and

incurs fire sales, on the contrary, AS could avoid fire sales through renegotiation, the ex-ante firm

value under AS still dominates the case with SH.

53

4 Discussion and Extensions

4.1 Credit Lines

Employment of randomly depletable credit lines can be found in He and Xiong (2012a) and Cheng

and Milbradt (2012), although with a slightly different interpretation of θ 25. This approach is a

compromise of institutional setup, economic intuition, and tractability. There are several rationales

for incorporating credit lines: DeMarzo and Sannikov (2006) show that credit lines are an important

ingredient of optimal capital structure in a principal-agent setting; in Bolton, Chen, and Wang

(2014), credit lines could optimally arise due to costly external financing; moreover, credit lines

provide liquidity issuance among financial institutions, and are commonly used in the real world.

In contrast to Bolton, Chen, and Wang (2014), I do not model the size of credit lines and the change

of drawdowns explicitly: here I am not dealing with perpetuities, but rather, with extremely short-

term debt with frequent rollovers. The rollover cost is effectively borne by credit lines drawdowns

(which should serve as a state variable if explicitly modeled), so the change of drawdowns will

depend on the market value of debts (value function), this dependence of the state variable on value

function makes it a highly non-linear system, which only complicates the analysis and solutions

without providing further economic significance. On the contrary, with the Poisson shocks nature

of credit lines, I am able to derive analytical solutions. Although the Poisson failure rate of credit

lines is only theoretic, it makes the probability of failure (1−e−θ t) increase with time, and captures

the feature that if the firm’s assets deteriorate so much that the firm needs to rely on the credit lines

for a very long period, credit lines capacity may be exhausted and repayment may be demanded

first; on the other hand, if the assets fall in value only for a short while and recover soon enough,

the credit lines drawdowns may be paid back by the firm before exhaustion and rebound to full

capacity.

25State their interpretations of θ here:

54

4.2 Firm’s Fundamental Value

The model can easily accommodate systemic risk and idiosyncratic risk. In periods of aggregate

risk, i.e., periods in which no other financial firms are sufficiently well-capitalized to take over

the failed firm and take advantage of the liquidity premium, the failed firm must be liquidated

to outside, non-financial firms, who lose the liquidity premium and value the assets only at non-

levered value VU (Xt) =Xt

r−µ. If there is only idiosyncratic risk, other well-capitalized financial

firms still enjoy the liquidity premium when the failed firm’s assets are liquidated, then the total

firm value at liquidation is given by the sum26:

F (Xt ;X∗B,X∗R|CLs alive)

= D(Xt ;X∗B,X∗R|CLs alive)+E (Xt ;X∗B,X

∗R|CLs alive)+L(Xt ;X∗B,X

∗R|CLs alive)

In the above comparative statics, I mainly investigate the case with systemic risk, i.e., with

the bankruptcy firm value given by un-levered VU (Xt) =Xt

r−µ. When comparing the benchmark

case with systemic risk and idiosyncratic risk, the implications for the choice between SH and AS

remain largely the same, with the idiosyncratic case bringing higher optimal leverage and maximum

leverage.

4.3 Benefits and Costs of Safe Harbor

As discussed in Duffie and Skeel (2012), there are several benefits and costs of safe harbor provi-

sions to QFCs, where the benefits include: (1) “a reduction of the incentives of repo and derivatives

counterparties to “run” as soon as the debtor’s financial condition is suspect, accelerating a default

or even causing a self-fulfilling expectation of default that need not otherwise occur”; (2) “it in-

creases the ability of a firm to rely on critical hedges”; and, (3) “safe harbors from stays reduce

the risk of costly delivery gridlocks in securities markets that could otherwise occur at the failure

26Note: the expected loss to CLs L(Xt ;X∗B ,X∗R |CLs alive) must be included when calculating the firm value, as the

new owners have to pay CLs s.t. CLs break even ex ante.

55

of one or more systemically important financial institutions”; and the costs are: “(1) lowering the

incentives of counterparties to monitor the firm; (2) increasing the ability of, or incentive for, the

firm to become too big to fail, with the attendant moral hazard of relying on bailouts; (3) inefficient

substitution away from more traditional forms of financing; (4) increasing the market impact of

collateral fire sales; and (5) lowering the incentives of a distressed firm to file for bankruptcy in a

timely manner.”

My model explicitly discusses benefits (1) and costs (4) and (5) of safe harbor: For benefit

(1), with SH, the liquidity of collateral is assured and creditors’ absolute priority at bankruptcy

is protected, which results in lower incentive for creditors to “run”; for cost (4), inefficient fire

sales could be avoided under AS if the distress period does not last too long, or if it is only an

idiosyncratic shock; for cost (5), due to the violation of APR, the distressed firm has more incentive

to file for bankruptcy under AS.

This paper also implicitly addresses benefit (3) and costs (2) and (3): For benefit (3), the gain

from SH is included in the liquidity premium, while under AS, creditors lose the liquidity premium

and must discount at risk-free rate r, which incurs a possibly larger bankruptcy loss for creditors;

for cost (2), with SH, the firm has higher debt capacity, and even the optimal leverage ratio is

higher; for cost (3), since there is only one class of creditors here, there are no substitution effects

of different debts, however, the higher leverage associated with SH does shed some light on using

repos to replace equity financing.

Benefit (2) is not modeled here since I mainly deal with repos, which serve as an important

means of funding, and not directly linked with risk hedging. Bolton and Oehmke (2011) discuss

the hedge issues with safe harbor for derivatives. Moreover, as pointed out by Tuckman (2013),

derivatives also serve as contingent financing for financial firms, so discussion of the impacts of SH

versus AS on firm’s funding also applies to derivatives contracts.

Cost (1) is not directly addressed here. However, since I deal mainly with extremely short-term

repos in the model, the high rollover frequency essentially serves as creditors’ means to monitor

the firm, as pointed out in Cheng and Milbradt (2012).

56

4.4 Extension with Dynamic Capital Structure

In the current model, for tractability, I assume the firm chooses an optimal principal value of debt

P ex ante, and the debt structure is kept staggered stationary until the firm declares bankruptcy.

This is a standard approach with structural models. However, financial firms are actively engaged

in balance sheet adjustments (see Adrian and Shin (2010)), i.e., financial firms could choose to

expand or contract their debt outstanding. So it is useful to extend the model such that the firm can

change the principal outstanding P as the environment changes.

Insert the extended model here:

4.5 Extension with Subordinated Debts

This paper studies how to contain systemic risk by changing the bankruptcy provisions. Another

approach would be to maintain the current safe harbor provisions to QFCs, but increase the liquidity

cushion of the firm during bad times. Furthermore, there is only one class of creditors in the model:

repo creditors. It is of interest to study how the presence of safe harbored repos affects other

subordinated creditors’ incentive to lend; and inversely, how to stabilize repo creditors run by using

both equity and subordinated debts as capital buffers. Flannery (2009), Sundaresan and Wang

(2011) and many others have proposed the use of “contingent capital” by systemically important

financial institutions (SIFIs), which is debt in normal times, but must convert into equity during

times of distress. The provision of “contingent capital” reduces liabilities and enhances the capital

strength of SIFIs, which, in turn, reduces short-term creditors’ (e.g., repos) incentive to run and

preserves the value of SIFIs during distressed periods. The interesting issue is to study the ex-

ante optimal trigger of contingent capital in the presence of short-term debt given the strategic

interactions highlighted in this model.

57

5 Conclusion

The shadow banking system is a collection of non-bank financial intermediaries (e.g., asset-backed

commercial paper conduits, repos markets, money market mutual funds, securitization vehicles,

etc.) that provide services similar to traditional commercial banks (such as maturity, credit and

liquidity transformation). Shadow banking is generally subject to little government regulation and

does not have explicit access to central bank liquidity or public sector credit guarantees. The

shadow banking system is an important part of the financial intermediation sector, but is considered

a significant contributor to the 2007-2009 financial crisis, its high leverage leading to the first runs

of the crisis. Therefore, how to regulate the shadow banking system has become an important

challenge for academic research and policy-making.

This paper studies the special U.S. bankruptcy treatment granted to important liabilities in

shadow banking. As we have seen, creditor runs on repos and derivatives contributed a lot to

the collapse of large financial firms such as Bear Stearns and Lehman Brothers. And this special

bankruptcy treatment granted to qualified financial contracts is regarded as one important factor

leading to the high leverage of the financial sector and the system-wide creditor runs that followed.

This treatment is referred to as “safe harbor”, i.e., the exemption of automatic stay from bankruptcy

law, granted to qualified financial contracts (QFCs) such as sale and repurchase agreements (repos)

and derivatives. The paper studies how SH and AS impact the endogenous decisions of the finan-

cial firm to file for bankruptcy and the decisions of its creditors to run or to rollover their debts, and

how the interaction of these decisions affects the leverage choice of financial firms.

Specifically, the paper formalizes the tradeoff between the two bankruptcy provisions: Auto-

matic Stay is imposed by the U.S. Bankruptcy law, which means when a firm files for bankruptcy,

creditors cannot terminate the contracts with the firm, or liquidate the collateral in the market; in-

stead, they have to wait until after the stay. However, for qualified financial contracts such as repos,

creditors are granted the “safe harbor”, which means they have the right to terminate the contracts,

seize and liquidate the collateral immediately at bankruptcy. In this way, safe harbor promotes

liquidity of collateral when the counterparty defaults; however, the immediate liquidation of large

58

volumes of securities in a short period can result in costly and inefficient "fire sales", which not

only hurt sellers of the securities, but also endanger the stability of other financial firms holding

the same or very similar securities. Automatic stay, on the other hand, exposes creditors to the risk

of liquidity being tied up in bankruptcy, but it may avoid the inefficient fire sales by liquidating

collateral in a more "orderly" manner.

So virtually we are facing the tradeoff of SH v.s. AS, namely, liquidity v.s. fire sale risk. And

ex-ante, it is not obvious to say which one is better. To study this issue, I build a structural model

that incorporates two layers of strategic interactions:

1) the coordination problem among infinitesimal creditors. Because the repos are collateralized

debts and are very short-term, investors in repos enjoy safety and liquidity of their short-horizon

investments, which I model as a liquidity premium. Creditors do not act collectively as one, instead,

when facing falling collateral value, creditors essentially compete with each other for liquidity

within the firm and rush for the exits. The running behavior of maturing creditors may trigger the

bankruptcy of the firm, and cause a negative externality to creditors who remain with the firm.

2) the interaction between equity holders and creditors. Creditor runs result in a rollover loss.

Normally this rollover loss is covered by the liquidity reserve of the firm. But the liquidity reserve

is not perfectly reliable and may fail at random times. Once the liquidity reserve fails, equity

holders face two choices: first to contribute money to replenish the liquidity reserve and save the

firm, or second, to default on their debt obligations. They will file for bankruptcy endogenously if

the benefit of keeping the firm alive is smaller than the cost.

At bankruptcy, I distinguish the two provisions, SH and AS. With SH, it is simple: creditors

just liquidate the collateral immediately, which incurs "fire sale" cost, and equity holders are al-

most surely to be wiped out at bankruptcy; However, under AS: creditors can only liquidate the

collateral after the stay, which is costly to creditors because they lose the liquidity premium and

face the uncertainty risk of collateral value. Equity holders take advantage of this, make a "take it

or leave it" offer to creditors, avoid fire sales, and seize the surplus from the reorganization. This

reorganization under AS effectively violates the Absolute Priority Rule, in the sense that equity

59

holders get some positive value, even before creditors are paid back in full. This violation mo-

tivates equity holders to file for bankruptcy in a more timely manner, at higher threshold values.

This earlier bankruptcy shortens the period during which the firm can extract the liquidity premium

from short-term creditors, and may hurt the ex-ante firm value.

From this structural model, I get three main results:

1) the desirability of SH v.s. AS depends on the liquidity of the collateral, for highly liquid

securities (such as Treasuries), fire sale risk is small, and SH is better than AS; however, for less

liquid securities (such as Mortgage Backed Securities), the benefit of avoiding costly fire sales

is large relative to the cost of earlier bankruptcy, and AS dominates SH. This result provides a

theoretical foundation for the recent regulatory change to the financial sector as embedded in the

Title II of Dodd-Frank Act. The Title II of Dodd-Frank Act is effectively a hybrid of SH and

AS. Normally the bankruptcy procedure follows the current Bankruptcy Code, where QFCs are

exempted from AS and enjoy SH. However, once the failed financial firm is placed into receivership

by the FDIC, creditors are subject to AS, the bridge bank could liquidate Treasuries and other highly

liquid securities very quickly; for illiquid securities, the bridge bank could wait and liquidate slowly

over time, to avoid the costly fire sales.

2) since the firm doesn’t endogenize the fire sale cost, the financial firm has higher debt capacity

and would endogenously choose a higher leverage ratio with SH. As an example, the Bankruptcy

Abuse Prevention and Consumer Protection Act of 2005 extended SH to include mortgage backed

securities and mortgage loans. The repo market grew dramatically since the expansion of safe

harbor, especially the repos backed by illiquid securities. This high leverage might appear desirable

to the firm itself, but it does not take into account the possibility of higher fire sale risk associated

with SH. And the higher leverage may hurt the financial sector more severely when negative shock

comes, just as vividly shown by the recent financial crisis.

3) AS tends to outperform SH more in a deteriorating market, which is characterized by higher

volatility, lower profitability, greater fire sale risk and greater rollover frequency. Moreover, for the

stay to be efficient and cause less distortion to creditors’ incentive to run early, it is important to

60

limit the length of the stay and the uncertainty for creditors.

An important application of the model is to calibrate the short term (i.e., overnight) credit spread

and predict the expected default frequency of a given firm.

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64

Appendix:

Appendix A. Equilibrium Solutions

Define the following global variables, i.e., their values are kept unchanged throughout the model:

−a1 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r)

σ2 and m1 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r)

σ2 ;

−b0 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r−ρ)

σ2 and n0 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r−ρ)

σ2 ;

−b1 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r−ρ+ηb)

σ2 and n1 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r−ρ+ηb)

σ2 ;

−z1 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r)

σ2 and l1 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r)

σ2 ;

−a2 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r+θηb)

σ2 and m2 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r+θηb)

σ2 ;

−b2 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r−ρ+ηb+θηb)

σ2 and n2 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r−ρ+ηb+θηb)

σ2 ;

−z2 =−(µ− 1

2 σ2)−√(µ− 1

2 σ2)2+2σ2(r+θηb)

σ2 and l2 =−(µ− 1

2 σ2)+√(µ− 1

2 σ2)2+2σ2(r+θηb)

σ2 .

Threshold Values Definition:

The threshold value X∗R at which creditors choose not to roll over is given by:

D(X∗R ;X∗B ,X∗R |CLs alive)≡ P;

I choose coupon rate c such that CLs begin to be drawn down at the same time that creditors start to run: X∗R ≡

cP+ηb (P−D(X∗R |CL alive)) = cP; and the threshold value X∗B of endogenous bankruptcy by equity holders is given

by:

E(X∗B |CLs alive)− [−L(X∗B |CLs alive)]−Φ≡ E(X∗B |CLs fail & firm goes bankrupt).

With ISH , the thresholdX̄ is defined as X̄ = ISH(r−µ)P

1−φ, then ISH min

(P, (1−φ)VU (Xτ)

)= ISH (1−φ)VU (Xτ) iff

Xτ < X̄ .

L(Xt ;X∗R ,X∗B |CLs fail) = 0

D(Xt |CLs fail & EHs meet the aggregate margin calls) = D(Xt |CLs alive);

D(Xt |CLs fail & EHs not meet the aggregate margin calls) = D(Xt |CLs fail & firm goes bankrupt).

E(Xt |CLs fail & EHs meet the aggregate margin call) = E(Xt |CLs alive)− [−L(Xt |CLs alive)]−Φ;

E(Xt |CLs fail & EHs not meet the aggregate margin calls) = E(Xt |CLs fail & firm goes bankrupt).

65

D(Xt |CLs fail & firm goes bankrupt)

= ISH[P−max

(0, P− (1−φ)VU (Xτ)

)]+ IAS

{e(c−r)T P−E

[e−rT max

(0, ecT P− (1−φ)VU (Xτ)

)|Fτ

]};

E(Xt |CLs fail & firm goes bankrupt)

= ISH max(0, (1−φ)VU (Xτ)−P

)+ IAS

{E[e−rT max

(0, φVU (Xτ+T )−Φ

)|Fτ

]+E

[e−rT max

(0, (1−φ)VU (Xτ+T )− ecT P

)|Fτ

] }.

I. Value Functions with Safe Harbor (SH):

I. A: X∗B < X∗L and X∗B ≤ X̄

D(Xt ;X∗R ,X∗B |CLs alive) =

B2X−b0

t + D2(r−ρ)−µ

Xt +D1

(r−ρ)

B4X−b1t +B3Xn1

t + D4(r−ρ+ηb)−µ

Xt +D3

(r−ρ+ηb)

B5Xn2t + D6

(r−ρ+ηb+θηb)−µXt +

D5(r−ρ+ηb+θηb)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for Xt < X∗B

with D2Xt +D1 =C, D4Xt +D3 =C+ηbP, and D6Xt +D5 =C+ηbP+θηbD(Xt |CLs fail & firm goes bankrupt).

L(Xt ;X∗R ,X∗B |CLs alive)

=

R2X−z1

t + L2r−µ

Xt +L1r

R4X−z2t +R3X l2

t + L4−γ2D4(r+θηb)−µ

Xt +L3−γ2D3(r+θηb)

+ γ2D(Xt |CLs alive)

R5X l2t + L6−γ3D6

(r+θηb)−µXt +

L5−γ3D5(r+θηb)

+ γ3D(Xt |CLs alive)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for Xt < X∗B

66

with L1 = 0 and L2 = 0, γ1 = 0; L4Xt +L3 = (Xt −C−ηbP), γ2 =ηb

θηb+ρ−ηb; L6Xt +L5 = (Xt −C−ηbP), γ3 =

ηbρ−ηb

.

E (Xt ;X∗R ,X∗B |CLs alive)

=

A2X−a1

t + E2r−µ

Xt +E1r

A4X−a1t +A3Xm1

t + E4−λ2L4−β2D4r−µ

Xt +E3−λ2L3−β2D3

r +λ2L(Xt |CLs alive)+β2D(Xt |CLs alive)

A5Xm2t + E6

(r+θηb)−µXt +

E5(r+θηb)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for Xt < X∗B

with E2Xt + E1 = (Xt −C), λ1 = 0, β1 = 0; E4Xt + E3 = θηb [−(1+ψ) f (θ)], λ2 = θηb(1+ψ)−θηb

= −(1+ψ), β2 =

−λ2ηb

r−(r−ρ+ηb)= (1+ψ) ηb

ρ−ηb; E6Xt + E5 = θηbE(Xt |CLs fail & firm goes bankrupt) (equity holders not to avoid

liquidation since X∗B < X∗AS, and X∗B < X̄), λ3 = 0, β3 = 0.

The coefficients B2, B3, B4, B5; R2, R3, R4, R5; and A2, A3, A4, A5 are determined by the continuity con-

ditions and smooth-pasting conditions of the value functions D(Xt ;X∗R ,X∗B |CLs alive), L(Xt ;X∗R ,X

∗B |CLs alive) and

E (Xt ;X∗R ,X∗B |CLs alive) at the cutoffs X∗R and X∗B , respectively; for details, please refer to the online appendix at

http://people.stern.nyu.edu/xma/Research.html.

I. B: X̄ < X∗B < X∗R

D(Xt ;X∗R ,X∗B |CLs alive)

=

B2X−b0t + D2

(r−ρ)−µXt +

D1(r−ρ)

B4X−b1t +B3Xn1

t + D4(r−ρ+ηb)−µ

Xt +D3

(r−ρ+ηb)

B6X−b2t +B5Xn2

t + D6(r−ρ+ηb+θηb)−µ

Xt +D5

(r−ρ+ηb+θηb)

B7Xn2t + D8

(r−ρ+ηb+θηb)−µXt +

D7(r−ρ+ηb+θηb)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for X̄ ≤ Xt < X∗B

for Xt < X̄

with D2Xt +D1 =C; D4Xt +D3 =C+ηbP; D6Xt +D5 =C+ηbP+θηbP;

andD8Xt +D7 =C+ηbP+θηbD(Xt |CLs fail & firm goes bankrupt).

L(Xt ;X∗R ,X∗B |CLs alive)

=

R2X−z1t + L2

r−µXt +

L1r

R4X−z2t +R3X l2

t + L4−γ2D4(r+θηb)−µ

Xt +L3−γ2D3(r+θηb)

+ γ2DL (Xt |CLs alive)

R6X−z2t +R5X l2

t + L6−γ3D6(r+θηb)−µ

Xt +L5−γ3D5(r+θηb)

+ γ3D(Xt |CLs alive)

R7X l2t + L8−γ4D8

(r+θηb)−µXt +

L7−γ4D7(r+θηb)

+ γ4D(Xt |CLs alive)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for X̄ ≤ Xt < X∗B

for Xt < X̄

67

with L1 = 0 and L2 = 0, γ1 = 0; L4Xt +L3 = (Xt −C−ηbP), γ2 =ηb

θηb+ρ−ηb; L6Xt +L5 = (Xt −C−ηbP), γ3 =

ηbρ−ηb

;

L8Xt +L7 = (Xt −C−ηbP), γ4 =ηb

ρ−ηb.

E (Xt ;X∗R ,X∗B |CLs alive)

=

A2X−a1t + E2

r−µXt +

E1r

A4X−a1t +A3Xm1

t + E4−λ2L4−β2D4r−µ

Xt +E3−λ2L3−β2D3

r +λ2L(Xt |CLs alive)+β2D(Xt |CLs alive)

A6X−a2t +A5Xm2

t + E6(r+θηb)−µ

Xt +E5

(r+θηb)

A7Xm2t + E8

(r+θηb)−µXt +

E7(r+θηb)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for X̄ < Xt ≤ X∗B

for Xt < X̄

with E2Xt + E1 = (Xt −C), λ1 = 0, β1 = 0; E4Xt + E3 = θηb [−(1+ψ) f (θ)],λ2 = θηb(1+ψ)−θηb

= −(1+ψ), β2 =

−λ2ηb

r−(r−ρ+ηb)= (1+ψ) ηb

ρ−ηb;

E6Xt +E5 = θηbE(Xt |CLs fail & firm goes bankrupt), λ3 = 0, β3 = 0;

E8Xt +E7 = θηbE(Xt |CLs fail & firm goes bankrupt), λ4 = 0, β4 = 0.

The coefficients B2, B3, B4, B5, B6, B7; R2, R3, R4, R5, R6, R7; and A2, A3, A4, A5, A6, A7 are deter-

mined by the continuity conditions and smooth-pasting conditions of the value functions D(Xt ;X∗R ,X∗B |CLs alive),

L(Xt ;X∗R ,X∗B |CLs alive) and E (Xt ;X∗R ,X

∗B |CLs alive) at the cutoffs X∗R and X∗B , respectively; for details, please refer to

the online appendix at http://people.stern.nyu.edu/xma/Research.html.

II. Value Functions under Automatic Stay (AS):

II. A. X∗B < X∗R

D(Xt ;X∗R ,X∗B |CLs alive) =

B2X−b0

t + D2(r−ρ)−µ

Xt +D1

(r−ρ)

B4X−b1t +B3Xn1

t + D4(r−ρ+ηb)−µ

Xt +D3

(r−ρ+ηb)

B5Xn2t +FD (Xt)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for Xt < X∗B

with D2Xt +D1 =C, D4Xt +D3 =C+ηbP, and FD (Xt)=C+ηbP+θηbPr−ρ+ηb+θηb

− θηb−ρ+ηb+θηb

E[e−rT max

(0, erT P− (1−φ)VU (Xt+T )

)].

L(Xt ;X∗R ,X∗B |CLs alive)

=

R2X−z1

t + L2r−µ

Xt +L1r

R4X−z2t +R3X l2

t + L4−γ2D4(r+θηb)−µ

Xt +L3−γ2D3(r+θηb)

+ γ2D(Xt |CLs alive)

R5X l2t +FL (Xt)+ γ3D(Xt |CLs alive)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for Xt < X∗B

68

with L1 = 0 and L2 = 0, γ1 = 0; L4Xt +L3 = (Xt −C−ηbP), γ2 =ηb

θηb+ρ−ηb; L6Xt +L5 = (Xt −C−ηbP), γ3 =

ηbρ−ηb

,

and FL (Xt) =L6

(r+θηb)−µXt +

L5−γ3(C+ηbP+θηbP)(r+θηb)

+ γ3E[e−rT max

(0, erT P− (1−φ)VU (Xt+T )

)].

E (Xt ;X∗R ,X∗B |CLs alive)

=

A2X−a1

t + E2r−µ

Xt +E1r

A4X−a1t +A3Xm1

t + E4−λ2L4−β2D4r−µ

Xt +E3−λ2L3−β2D3

r +λ2L(Xt |CLs alive)+β2D(Xt |CLs alive)

A5Xm2t +FE (Xt)

for Xt ≥ X∗R

for X∗B ≤ Xt < X∗R

for Xt < X∗B

with E2Xt + E1 = (Xt −C), λ1 = 0, β1 = 0; E4Xt + E3 = θηb [−(1+ψ) f (θ)], λ2 = θηb(1+ψ)−θηb

= −(1+ψ), β2 =

−λ2ηb

r−(r−ρ+ηb)= (1+ψ) ηb

ρ−ηb; and FE (Xt) =

θηbVU (Xt )r+θηb−µ

+ −θηbPr+θηb

+E[e−rT max

(0, erT P− (1−φ)VU (Xt+T )

)].

The coefficients B2, B3, B4, B5; R2, R3, R4, R5; and A2, A3, A4, A5 are determined by the continuity con-

ditions and smooth-pasting conditions of the value functions D(Xt ;X∗R ,X∗B |CLs alive), L(Xt ;X∗R ,X

∗B |CLs alive) and

E (Xt ;X∗R ,X∗B |CLs alive) at the cutoffs X∗R and X∗B , respectively; for details, please refer to the online appendix at

http://people.stern.nyu.edu/xma/Research.html.

II. B. X∗B ≥ X∗R (conceptually, exists due to the debt covenant)

D(Xt ;X∗R ,X∗B |CLs alive) =

B2X−b0

t + D2(r−ρ)−µ

Xt +D1

(r−ρ)

B5Xn2t +FD (Xt)

for Xt ≥ X∗R

for Xt < X∗R

with D2Xt +D1 =C, and FD (Xt) =C+ηbP+θηbPr−ρ+ηb+θηb

− θηb−ρ+ηb+θηb

E[e−rT max

(0, erT P− (1−φ)VU (Xt+T )

)].

L(Xt ;X∗R ,X∗B |CLs alive)

=

R2X−z1

t + L2r−µ

Xt +L1r

R5X l2t +FL (Xt)+ γ3D(Xt |CLs alive)

for Xt ≥ X∗R

for Xt < X∗R

69

with L1 = 0 and L2 = 0, γ1 = 0; and FL (Xt)=Xt

r+θηb−µ+ −C−ηbP−γ3[C+ηbP+θηbP]

r+θηb+γ3E

[e−rT max

(0, erT P− (1−φ)VU (Xt+T )

)].

E (Xt ;X∗R ,X∗B |CLs alive)

=

A2X−a1

t + E2r−µ

Xt +E1r

A5Xm2t +FE (Xt)

for Xt ≥ X∗R

for Xt < X∗R

with E2Xt +E1 =(Xt −C), λ1 = 0, β1 = 0; and FE (Xt)=θηbVU (Xt )r+θηb−µ

+ −θηbPr+θηb

+E[e−rT max

(0, erT P− (1−φ)VU (Xt+T )

)].

The coefficients B2, B5; R2, R5; and A2, A5 are determined by the continuity conditions and smooth-pasting

conditions of the value functions D(Xt ;X∗R ,X∗B |CLs alive), L(Xt ;X∗R ,X

∗B |CLs alive) and E (Xt ;X∗R ,X

∗B |CLs alive) at the

cutoffs X∗R and X∗B , respectively; for details, please refer to the online appendix at http://people.stern.nyu.edu/

xma/Research.html.

70