Bank capital, liquid reserves, and insolvency risk∗
Julien Hugonnier†
Erwan Morellec‡
November 20, 2015
Abstract
We develop a dynamic model of banking to assess the effects of liquidity and leverage
requirements on banks’ insolvency risk. In this model, banks face taxation, flotation
costs of securities, and default costs and maximize shareholder value by making their
financing, liquid asset holdings, and default decisions in response to these frictions
as well as regulatory requirements. Our analytic characterization of the bank policy
choices shows that imposing solely liquidity requirements leads to lower bank losses in
default at the cost of an increased likelihood of default. Combining liquidity requirements with leverage requirements reduces drastically both the likelihood of default and
the magnitude of bank losses in default.
Keywords: banks; liquidity buffers; capital structure; insolvency risk; regulation
JEL Classification Numbers: G21, G28, G32, G33.
∗
We thank Lucy White for encouraging us to develop this model and for helpful comments, and Darrell
Duffie and Jean-Charles Rochet for stimulating discussions and detailled feedback. We also thank Arnaud
Boot, Pierre Collin-Dufresne, Peter DeMarzo, Ruediger Fahlenbrach, Damir Filipovic, Thierry Foucault,
Sebastian Gryglewicz, Dirk Hackbarth (AFA discussant), Ron Kaniel, Nataliya Klimenko (EFA discussant),
Semyon Malamud, Konstantin Milbradt, Artem Neklyudov, Norman Schuerhoff, Anders Trolle, Francesca
Zucchi, and seminar participants at the 2015 American Finance Association meeting in Boston, Banque de
France, Bocconi University, EPFL, Erasmus University Rotterdam, the 2014 European Finance Association
meeting in Lugano, the Finance Theory Group meeting at the University of Chicago, Imperial College,
INSEAD, the Swiss Finance Institute research days in Gerzensee, the Swiss National Bank, the Swissquote
bank, Tilburg University, the University of Amsterdam, the University of Lausanne, the University of Piraeus,
and the University of Zurich for comments. Financial support from the Swiss Finance Institute and the
Fondation Banque de France is gratefully acknowledged. The usual disclaimer applies.
†
Swiss Finance Institute, Ecole Polytechnique Fédérale de Lausanne and CEPR. Postal: Extranef,
Quartier UNIL Dorigny, 1015 Lausanne, Switzerland. Email: [email protected]
‡
Swiss Finance Institute, Ecole Polytechnique Fédérale de Lausanne and CEPR. Postal: Extranef,
Quartier UNIL Dorigny, 1015 Lausanne, Switzerland. Email: [email protected]
1
Introduction
Banks can impose major risks on the economy. Avoidance of these risks and the associated
costs is the overwhelming concern of prudential regulation. Given the experience in the
recent financial crisis, in which high debt levels and insufficient liquidity buffers led to the
collapse of major players of the financial industry such as Bear Sterns or Lehman Brothers,
the debate on banking regulation has evolved around two main ideas that have been reflected
in proposals for regulatory reform. First, because most bank assets are illiquid and raising
fresh equity is costly, banks should hold a buffer of liquid reserves to be able to cope with
short-term losses. Second, equity capital should be significantly increased, so that if the
value of the banks’ assets were to decline, this would not automatically lead to distress and
the resulting losses would be borne by the bank owners.
While many insightful discussions of liquidity and leverage requirements are available in
the literature, financial theory has made little headway in developing models that can provide
quantitative guidance for bank capital structure decisions or for the effects of regulatory
requirements on these decisions, and the induced insolvency risk. Our objective in this
paper is therefore twofold. First, we seek to develop a dynamic model of banks’ choices
of liquid asset holdings, financing, and default policies in the presence of realistic market
frictions. Second, we want to use this model to characterize the endogenous response of
banks to the imposition of liquidity and leverage requirements, and to measure the effects
of such regulatory requirements on banks’ policy choices and insolvency risk.
We begin our analysis by formulating a dynamic structural model in which banks face
taxation, issuance costs of securities, and default costs and may be constrained by a regulator
to hold a minimum amount of liquid reserves and/or equity capital. In the model, banks
are financed with equity, insured deposits, and risky, subordinated debt. They hold risky,
illiquid assets (e.g. risky loans or derivatives) whose cash flows are subject to small and
frequent shocks as well as large and infrequent negative jumps capturing tail risk. They
can also invest in risk-free, liquid assets (e.g. cash reserves) that can be used to absorb
losses and save on recapitalization costs. Banks earn revenues from their investments as well
as by providing liquidity services to their depositors. They maximize shareholder value by
choosing their buffers of liquid assets as well as their financing and default policies.
1
We first consider a benchmark environment in which banks are unregulated and solve for
a bank’s optimal capital structure, which involves determining how much debt to issue and
how much liquid reserves to hold. In this environment, we show that the costs of external
finance create a wedge between inside and outside equity and generate a precautionary
demand for liquidity as well as an optimal capital structure for banks. In our model, banks
take high debt ratios to exploit the tax deductibility of interest payments and the potential
mispricing of deposits induced by deposit insurance and/or liquidity premia. Our analytic
characterization of the optimal liquidity management policy also reveals that banks absorb
small and intermediate losses using their liquid reserves or by raising outside equity at a
cost, so that such losses do not trigger default. We show, however, that large losses due to
a realization of tail risk may lead to default. Notably, we find that insolvency risk increases
with tail risk, debt levels, and financing costs and decreases with liquid reserves.
After solving for the policy choices of unregulated banks, we examine the effects of
prudential regulation on these policy choices and the resulting insolvency risk. We first
analyze liquidity requirements that mandate banks to hold a minimum amount of liquid
reserves. We show that, when facing such a requirement, banks voluntarily choose to hold
liquid reserves in excess of the required minimum to reduce the costs associated with breaches
of the requirement. More importantly, we demonstrate that for any given capital structure,
liquidity requirements lead to lower bank losses in default and to an increase in the likelihood
of default. Our analysis therefore suggests that while liquidity requirements can reduce
potential capital injections by the regulator by limiting bank losses in default, they are
ineffective at reducing default risk. The combination of these two effects implies that liquidity
requirements may decrease the cost of market debt to banks, leading bank shareholders to
choose higher debt levels that ultimately result in even higher default risk. As a result, the
regulator may need to combine liquidity requirements with leverage constraints to harness
the benefits of the decrease in default losses induced by the liquidity requirement.
In addition to liquidity requirements, banks may indeed be subject to leverage requirements. Such requirements do not impose any restriction on the assets that banks should
hold, but impose constraints on the way they fund their operations. As argued by Admati
and Hellwig (2013), a tightening of leverage requirements transfers a large fraction of the
2
bank’s risks to its shareholders, which otherwise might be passed on to creditors or taxpayers. Leverage requirements generally prescribe how much equity capital banks should
have relative to their total assets.1 Using a calibrated version of the model, we show that
such requirements have significant effects on insolvency risk but little effect on bank losses in
default. For example, imposing a maximum ratio of debt to total assets of 90% decreases the
one-year default probability of an average U.S. bank by 46.61% while decreasing its losses
in default by 0.359%. Importantly, we also show that combining liquidity requirements with
leverage requirements reduces drastically both the likelihood of default and bank losses in
default. For example, our calibration shows that combining a maximum debt ratio of 90%
with a liquidity requirement of 10% of deposits, reduces the one-year default probability of
an average U.S. bank by 46.84% while reducing losses in default by 14.053%.
The model also allows us to investigate the effects of mispriced deposits and depositor
preference in default on banks’ capital structure decisions and insolvency risk. We show that
mispriced deposit insurance and/or the existence of a liquidity premium on deposits makes
it optimal for banks to take on more subordinated (market) debt, leading to a large increase
in insolvency risk and bank losses in default. On the contrary, depositor preference in default
increases the cost of market debt and motivates banks to adopt more conservative financing
policies, thereby reducing insolvency risk and bank losses in default.
The literature on default risk in banks has started with the early contributions of Merton
(1977, 1978), in which the objective is to determine the cost of deposit insurance and loan
guarantees. Although important milestones, these papers suffer from four limitations for our
purpose. First, they assume that banks default whenever their assets-to-deposits ratio falls
below some exogenous barrier that depends neither on the characteristics of the bank nor on
the frictions that it faces. Second, the capital structure of banks is set exogenously and does
not reflect the frictions that they may face such as taxes, default costs, or issuance costs of
securities. Third, the dynamics of the banks’ assets are governed by an exogenous process,
1
For example, banks are expected to maintain a Tier 1 capital to asset ratio of 3% under Basel III. In July
2013, U.S. banking regulators have also proposed a new restriction on borrowing for U.S. banks, requiring a
ratio of (Tier 1) capital to total assets of 6% for 8 Systemically Important Financial Institutions and 5% for
their bank holding companies. Similarly, the Vickers Commission in the U.K. has recommended a 4% ratio
rather than the 3% global standard set by the Basel III regime.
3
implying that there is no connection between the banks’ asset and capital structures. Fourth,
raising equity is costless, so that liquid reserves are irrelevant.
Most of the recent quantitative banking models, including Mella-Barral, and Perraudin
(1997), Bhattacharya, Planck, Strobl, and Zechner (2002), or Décamps, Rochet, and Roger
(2004) among others, examine variants to the first of these assumptions. In these contributions, insolvency is endogenous and triggered by shareholders’ decision to cease injecting
funds in the bank. While identifying some prime determinants of insolvency risk, these
theories assume that asset and liability structures are exogenously given. As a result, they
leave open the question of how financing structure and asset structure interact and jointly
affect insolvency risk. In addition, these models also maintain the assumption that banks
can raise outside funds at no cost, thereby leaving no role for liquid reserves.
Our analysis inherits some of the assumptions of this literature. For example, bank
shareholders are protected by limited liability and the bank’s objective is to maximize
shareholder value. However, it differs from these contributions in three important respects.
First, we consider that at least part of a bank’s assets are illiquid and that it is costly to raise
outside equity, thereby providing a role for liquid reserves. Second, we incorporate some of
the key market imperfections and regulatory requirements that banks face in practice and
relate banks’ payout, financing, and default policies to these frictions. Third, in our model,
financing structure and asset structure interact and jointly affect insolvency risk. We show
that these unique features have important implications. For example, while in most prior
models shareholder value is always increased by making dividend payments, this is not the
case in our model with frictions, in which shareholders have incentives to protect their
franchise value by maintaining adequate liquid reserves.
There exists a large literature analyzing the role of capital in banking regulation (see e.g.
Décamps, Rochet, and Roger (2004), Morrison and White (2005), or Repullo and Suarez
(2013)). But it is only recently that the question of bank optimal capital structure has
begun to be addressed. In our model, banks hold liquid reserves to reduce financing frictions
and expected default costs. They take high debt ratios to exploit the mispricing of deposits
and the tax deductibility of debt payments.2 The notion that bank deposits commands a
2
In a recent study, Schepens (2014) use a difference-in-difference approach to show that the introduction
of an equity tax shield in Belgium in 2006 led banks to significantly increase their capital ratios by increasing
4
liquidity premium is well accepted (see Gorton and Pennacchi (1990) or DeAngelo and Stulz
(2015)). By contrast, tax benefits of debt are often overlooked in the banking literature.
In this literature, our work is most closely related to the papers by Froot and Stein (1988),
Allen, Carletti, and Marquez (2014), Sundaresan and Wang (2015), and Subramanian and
Yang (2014). Subramanian and Yang develop a dynamic model to examine banks’ insolvency
risk, focusing on regulators’ incentives to step in as banks become distressed. Sundaresan
and Wang use a dynamic model to analyze banks’ financing decisions and the effects of
deposit insurance and regulatory closure on bank liability structures. Allen, Carletti, and
Marquez use a static model to show that banks hold capital to reduce expected default costs
and the cost of deposit finance. None of these papers has a role for liquid reserves. Froot and
Stein (1998) build a two-period model in which capital is initially costless but may become
costly in the future. In their model, bank shareholders may hedge future losses and financing
costs by increasing the initial level of equity capital. In our model, banks always face the
same frictions and choose their payout, financing, and default decisions in response to these
frictions as well as regulatory requirements. Another difference is that we consider a dynamic
model. In a static model, a regulatory requirement can only affect optimal behavior if it is
binding. In practice however, regulatory requirements are binding for a minority of banks
and yet seem to influence the behavior of all banks.
From a modeling perspective, our paper relates to the inventory models of Milne and
Whalley (2001), Peura and Keppo (2006), Décamps, Mariotti, Rochet, and Villeneuve (2011),
Bolton, Chen, and Wang (2011), and Hugonnier, Malamud, and Morellec (2015) in which
financing costs lead banks or corporations to maintain liquidity buffers.3 In these models,
uncertainty is solely driven by Brownian shocks so that there is no risk of default if firms can
raise outside equity. Our paper advances this literature in two important dimensions. First,
we incorporate jumps in our analysis to account for tail risk and show that banks may find
their equity capital. See Ashcraft (2008), De Mooij and Keen (2012), De Mooij, Keen, and Orihara (2014),
or Horvath (2013) for additional evidence on the effects of tax rates on bank capital ratios.
3
Our model also relates to the dynamic, discrete-time models of De Nicolo, Gamba, and Lucchetta (2014)
and Vuillemey (2015). These papers numerically solve models in which banks are financed with equity,
insured deposits, and risk-free debt and can invest in risky loans and risk-free bonds (i.e. cash is negative
debt). By contrast, our paper provides an analytical characterization of banks’ policy choices in the presence
of regulatory constraints and allows banks to issue risky debt. That is, banks can have both cash holdings
and debt on their balance sheet and debt can be risky. Our model therefore allows us to examine the joint
effects of risky debt financing, liquidity choices, and regulation on default risk.
5
it optimal to default following large shocks, even if they can raise outside funds. Second,
we endogenize not only banks’ payout decisions but also their financing and default policies.
This allows us to examine the effects of regulation on financing decisions and insolvency
risk. As we discuss below, this problem is more difficult to solve because we have two free
boundaries (the default and payout thresholds) instead of one (the payout threshold), and
an important contribution of this paper is to develop a new method based on fixed-point
arguments to solve such optimization problems.
The remainder of the paper is organized as follows. Section 1 presents the model and
provides an informal description of the optimal strategy. Section 2 presents our analytic
characterization of the value-maximizing payout, financing, and default policies for unregulated banks. Section 3 examines the effects of liquidity and leverage requirements on these
policy choices and insolvency risk. Section 4 discusses the model’s implications. Section 5
concludes. All proofs are gathered in a separate Appendix.
1
Model
1.1
Assumptions
Throughout the paper, time is continuous and all agents are risk neutral and discount cash
flows at a constant rate ρ > 0.
The subject of study is a bank held by shareholders that have limited liability. This bank
is subject to taxation at rate θ ∈ (0, 1). It owns a portfolio of risky assets (e.g. risky loans
or derivatives) as well as liquid reserves (i.e. cash reserves or safe government bonds) and is
financed with equity, insured deposits, and market debt. The bank’s risky assets generate
after-tax cumulative cash flows At that evolve according to:
dAt = (1 − θ)µdt + σdBt − YNt dNt .
In this equation, Bt is a standard Brownian motion, Nt is a Poisson process, (µ, σ) are
positive constants, and (Yn )∞
n=1 is a sequence of independently and identically distributed
random variables that are drawn from (0, ∞) according to an exponential distribution with
6
mean 1/β > 0. The increments of the Brownian motion represent small and frequent shocks
to the bank cash flows. The jumps of the Poisson process represent large losses that may
be due, for example, to defaults across the loan portfolio of the bank (see Acharya, Cooley,
Richardson, and Walter (2009) for an analysis of the role of tail risk in the financial crisis).4
We denote the intensity of the Poisson process by λ ≥ 0, so that over an infinitesimal time
interval there is a probability λβe−βy dt that the bank makes a loss of size y ≥ 0. In a
non-financial firm, dAt is the total earnings. In a bank, it only represents the earnings from
risky assets, not including the income from serving deposit accounts.
In addition to risky assets, the bank can or may be constrained to hold liquid, risk-free
reserves. We denote by St the liquid reserves of the bank at time t ≥ 0. Holding liquid
reserves generally involves deadweight costs. We capture these costs by assuming that the
rate of return on liquid reserves is zero.5 When optimizing its liquid asset holdings, the bank
trades-off the lower returns of these assets with the benefits of liquidity.
As discussed in Sundaresan and Wang (2015), “banks share some common characteristics
with non-financial firms: Both have access to the cash flows generated by their assets and
both finance their assets with debt and equity. Banks, however, differ from non-financial
firms in that they take deposits and provide account services to their depositors [...]” In
many countries, deposits are insured by the regulator and the deposit-taking activity comes
with heavy regulation. Deposits and the associated account services, deposit insurance,
and regulation distinguish the banking business from other businesses and set the capital
structure decision of banks apart from that of other firms.
To capture these important differences, we consider that the liability structure of the bank
comprises equity, deposits, and market debt.6 Deposits are insured against bank failure, have
4
These jumps may reflect trading losses, such as the $8.9 billion loss of Morgan Stanley on credit default
swaps in 2012 or the $7.2 billion loss of Société Générale on index futures in 2008. They may also reflect
losses on poorly performing acquisitions, such as the $10.4 billion loss of Crédit Agricole on Emporiki. Lastly,
they may reflect the payments made by a number of banks to authorities in the United States for having
facilitated tax evasion or for having traded with countries under embargo. For example, on June 30, 2014,
BNP Paribas agreed to pay a fine of $8.9 billion for having violated U.S. sanctions against Cuba, Iran, Sudan,
and other countries. For more details, see “Capital Punishment” in The Economist July 5, 2014.
5
A necessary condition for a well-defined payout policy is that there exists a cost of holding liquidity in
that the rate of return on liquid reserves is strictly less than the risk-free rate of return ρ.
6
As in many recent contributions (e.g., Allen, Carletti, and Marquez (2014), Subramanian and Yang
(2014), or Hanson, Shleifer, Stein and Vishny (2015)), we downplay the vulnerability of deposits to runs and
7
face value D, and require the bank to make a payment cD ≥ 0 per unit of time. The coupon
rate on deposits cD /D may differ from the fair rate of return, reflecting mispriced deposit
insurance and/or the income earned by the bank for providing liquidity services to depositors
as e.g. DeAngelo and Stulz (2015). Market debt is subordinated, requires the bank to make a
payment cL ≥ 0 per unit of time, and has face value L. In summary, the combined payments
of the bank on its market debt and deposits are given by c = cD + cL < µ per unit of time
and its cumulative earnings Ct evolve according to:7
dCt = dAt − (1 − θ)cdt = (1 − θ)(µ − c)dt + σdBt − YNt dNt .
(1)
Because the distribution of the jump magnitudes has unbounded support, the bank will
be unable to withstand large losses with positive probability, leading to default. In the
following, we use a stock-based definition of default whereby the bank services debt as long
as equity value is positive (as in Leland (1994) or Duffie and Lando (2001)). That is, default
is the result of the optimizing behavior of shareholders. We assume that the liquidation
value of the bank’s risky assets in default is given by
Λ = (1 − ϕ)V +
where
Z
∞
−ρt
e
V =E
0
dAt
1
=
ρ
λ
(1 − θ)µ −
β
(2)
denotes the present value of the infinite stream of cash flows generated by these assets,
the constant ϕ ∈ [0, 1] is a haircut related to bankruptcy or liquidation costs, and x+ =
max{0, x} denotes the positive part of a real number x.
In the model, equity capital and liquid reserves serve as buffers against default risk. The
bank can increase its liquid reserves either by retaining earnings or by issuing new equity. To
capture the substantial costs associated with equity issuance, we consider that the bank has
emphasize the opposite aspect of deposits: Relative to other forms of private-money creation that occur in
the shadow-banking sector, deposits are highly sticky and not prone to run at the first sign of trouble.
7
If the total coupon rate c exceeds µ then expected earnings are negative even in the absence of jumps
and, as a result, it is optimal for shareholders to immediately liquidate the bank.
8
to pay a lump-sum cost φ upon raising outside funds. Because of this fixed cost, the bank
will raise fresh equity through lumpy and infrequent issues. In addition, to reduce issuance
costs, the bank will retain earnings and hold liquid reserves.
A payout and financing strategy for the bank is a pair π = (Ptπ , Rtπ ) where Ptπ is an
adapted, left-continuous, and non-decreasing process with initial value zero that represents
the cumulative dividends payments to shareholders, and Rtπ is an adapted, left-continuous,
and non-decreasing process such that
Rtπ =
∞
X
1{t>ξn } rn ,
n=1
for some increasing sequence of stopping times (ξn )∞
n=1 that represent the dates at which
the bank raises funds, and some sequence of nonnegative random variables (rn )∞
n=1 that
represent the net financing amounts raised from outside investors. The liquid reserves process
associated with a strategy π is defined by:
Stπ = s + Ct − Ptπ + Rtπ ,
where Ct is defined in equation (1) and s is the initial level of liquid reserves. This equation
is a general accounting identity which shows that liquid reserves increase with earnings and
with the funds raised from outside investors and decrease with payouts to shareholders and
potential operating losses. The liquidation time associated with the payout and financing
strategy π is then defined by:
π
π
τπ = inf t ≥ 0 : St+ = lim Su ≤ 0 .
u↓t
In the model, bank shareholders make their financing, payout, and default decisions after
observing the increment of the cash flow process. As a result, we use left-continuous processes
in the definition of strategies and right-hand limits in the definition of the liquidation time.
Notably, the occurrence of a cash flow jump
π
∆Ct = Ct − Ct− = Ct − lim Cs ≤ −St−
s↑t
9
that depletes the liquid reserves of the bank only results in default if shareholders do not
provide sufficient funds for the right hand limit
π
π
π
π
St+
= St−
+ ∆Ct + Rt+
− Rtπ − Pt+
− Ptπ
π
π
to be strictly positive. We consider strategies such that Pt+
− Ptπ ≤ (Stπ + Rt+
− Rtπ )+
implying that the bank cannot distribute cash it does not hold.
Management chooses the payout, financing, and default policies of the bank to maximize
the present value of future dividends to existing shareholders, net of the total cost of capital
injections. That is, management solves:
Z
−ρt
e
v(s) = sup Es
π∈Π(s)
τπ
(dPtπ
π
−ρτπ
− dΦt (R )) + e
`
Sτππ
(3)
0
where Π(s) denotes the set of admissible strategies, the operator Es [·] denotes an expectation
conditional on an initial level s of liquid reserves, the function
`(s) = (s + Λ − L − D)+
represents the liquidation payment to shareholders if the bank holds s ≤ 0 in liquid reserves,
and the non-decreasing process
Φt (Rπ ) = Rtπ +
∞
X
n=1
1{t>ξn } φ =
∞
X
1{t>ξn } (rn + φ)
n=1
captures the total contribution of shareholders, including the cumulative cost of financing.
The first term on the right-hand side of (3) captures the present value of dividend payments.
The second term gives the present value of the cash flow to shareholders in default.
1.2
Informal description of the optimal strategy
To gain some intuition on the solution to problem (3) note that, because of the fixed costs
of financing, the bank only considers raising equity when cash flow shocks deplete its liquid
reserves, i.e. when s ≤ 0. In such instances, the bank has to either raise new funds to
10
finance the shortfall and continue operating, or default. The bank chooses to default when
the shortfall, the cost of refinancing, or the liquidation value of risky assets are large. When
defaulting is not optimal, the bank pays the cost φ and raises funds to maximize equity value.
Taking into account these two possibilities shows that the equity value function satisfies:
v(s) = max {f v(s), `(s)} = (max {f v(0), `(0)} + s)+ ,
for s ≤ 0,
(4)
where the financing operator
f v(s) = sup (v(b) − (b − s + φ)) = s + f v(0),
(5)
b≥0
gives the maximal value that shareholders can obtain by refinancing the bank at a point
where its cash reserves are equal to s ≤ 0.
Because the likelihood of costly refinancing or inefficient liquidation decreases as liquid
reserves increase, we expect the marginal value of liquid reserves to be decreasing and,
therefore, the equity value function to be concave on the positive real line. If this conjecture
is verified, there should exist some level b∗0 such that v 0 (s) ≥ 1 if and only if s ≤ b∗0 and the
optimal payout policy should consist in distributing dividends to maintain liquid reserves at
or below the target level b∗0 . Given the target level of liquid reserves, equation (4) shows that
the optimal refinancing policy depends on the comparison between
f v(0) = sup (v (b) − (b + φ)) = v (b∗0 ) − (b∗0 + φ)
b≥0
and the payoff `(0) that shareholders receive if they liquidate the bank at a point where
it has no liquid reserves. If `(0) > f v(0), as illustrated by the left panel of Figure 1,
it is never optimal for shareholders to refinance and the bank is liquidated the first time
that liquid reserves fall below zero. If on the other hand f v(0) ≥ `(0), as illustrated by
the right panel of the figure, shareholders may prefer refinancing over liquidation. In that
case, they refinance the bank back to b∗0 whenever liquid reserves become negative with a
shortfall smaller than f v(0), and otherwise liquidate. Figure 2 shows two trajectories for
liquid reserves corresponding to these two strategies. In the top panel, `(0) > f v(0) and it is
11
Figure 1: Equity value for an unregulated bank
B. Refinancing: f v(0) ≥ `(0)
A. No refinancing: `(0) > f v(0)
=
v0
•◦
v(b∗0 )
•◦
v(b∗0 )
Default
Default
Dividends
◦
•
•◦
f v(0)
0
=
v0
v0
=
1
`(0)
Dividends
1
Equity value v(s)
v0
=
1
v(s)
Retain
Refinance
1
v(s)
Retain
◦
•
−`(0)
•◦
0
0
b∗0
−f v(0)
Liquid reserves s
0
b∗0
Liquid reserves s
Notes. This figure illustrates the shape of the equity value function and the optimal strategy
on each side of the graph correspond respectively
for an unregulated bank. The regions
to liquidation (left) and dividend payments (right) while the intermediate regions
and
correspond respectively to earnings retention and refinancing.
never optimal to refinance. In the bottom panel, f v(0) ≥ `(0) and it is optimal to refinance
shortfalls smaller than f v(0).
The main difficulty in verifying these conjectures is that one needs to simultaneously
determine the values of the constants b∗0 and α0∗ = max{`(0), f v(0)}. To circumvent this
difficulty, we proceed in two steps. In the first step, we fix the value of the constant α and
solve for the optimal payout policy in an auxiliary problem in which the bank cannot raise
funds but produces the payoff (α + s)+ to shareholders when it runs out of liquid reserves
and defaults. In the second step, we show that the constant α can be chosen in such a way
that the value of this auxiliary problem coincides with that of problem (3) and derive the
equity value-maximizing payout, financing, and default policies.
12
Figure 2: Optimal policy for an unregulated bank
A. No refinancing: `(0) > f v(0)
Liquid reserves
−b∗0
0
Default
0
•
q1
q2 = τπ∗
B. Refinancing: f v(0) ≥ `(0)
Liquid reserves
b∗0
0
Refinancing
•◦
−f v(0)
Default
0
q1
q2 = ξ1
•
q3 = τπ∗
Calendar time
Notes. This figure illustrates the dynamics of the optimal policy for an unregulated bank. In
both panels the stopping times (qn )∞
n=1 indicate the occurrence of negative cash flow jumps, the
vertical regions
indicate the times at which dividends are paid and • indicates where the
represents the shortfalls that
bank is liquidated. In the bottom panel, the horizontal region
shareholders are willing to refinance and ◦ indicates where the bank raises outside funds.
Before proceeding with to these two steps, we first determine the equity value and the
bank’s policy choices in a benchmark economy in which there are no refinancing costs. This
allows us to derive a frictionless value of equity that will be of repeated use when solving
shareholders’ optimization problem.
13
2
2.1
Value of an unregulated bank
Frictionless benchmark
When there are no costs of raising funds, in that φ = 0, any loss can be covered by issuing
equity at no cost and there is no need to hold liquid reserves (as in Leland (1994), Duffie
and Lando (2001), or Sundaresan and Wang (2015)). As a result, it is optimal to payout all
positive earnings and the optimization problem of the bank reduces to choosing the default
policy that maximizes equity value. That is, the value of the bank’s equity is given by
v ∗ (s) = (v ∗ + s)+
with the constant v ∗ defined by
Z
∗
e
v = sup E
τ ∈S
τ−
−ρt
−ρτ
dCt + e
`(∆Cτ ) ,
0
where S denotes the set of all stopping times and ∆Cτ ≤ 0 represents the loss leading to
default. Solving this optimal stopping problem leads to the following result.
Proposition 1 (Frictionless benchmark). When issuing equity is costless, the value of equity
and the optimal default time are given by
v ∗ (s) = (max {v0∗ , `(0)} + s)+
and
τ ∗ = 1{v∗ ≥`(0)} inf {t ≥ 0 : |∆Ct | ≥ v0∗ } ,
0
where the strictly positive constant v0∗ is the unique solution to
ρv0∗ = (1 − θ) (µ − c) − λE [min {v0∗ , Y1 }] .
(6)
Equity value in the frictionless benchmark is increasing in the cash flow rate µ, and decreasing
in the coupon rate c, the jump intensity λ, and the mean jump size 1/β.
14
Proposition 1 (which is true for any distribution of losses) shows that absent financing
frictions two cases may occur: Either it is optimal to immediately liquidate, in which case
shareholders get `(0) + s, or it is optimal to continuously refinance the bank until the first
time that the absolute value of a jump of the cash flow process exceeds equity value. In the
later case, the amount |∆Ct | = YNt plays the role of a cost of investment that shareholders
have to pay to keep the bank alive following the occurrence of a large loss.
To better understand this feature, consider the case where v ∗ = v0∗ ≥ `(0) so that it is
not optimal to immediately liquidate the bank. In this case, we can rewrite the Bellman
equation (6) for equity value as
∗
v = p(1 − θ)
µ−c
ρ
with the constant p =
ρ
ρ+λ
+ (1 − p)E (v ∗ − Y1 )+ .
∈ (0, 1]. This equation shows that, when raising equity is costless,
the bank’s problem can be interpreted as a discrete-time, infinite horizon problem in which
shareholders earn (1 − θ) µ−c
each period with probability p and otherwise face a random
ρ
liquidity shock that they can decide to pay, in which case the bank continues, or not, in
which case the bank is liquidated with a zero payoff to shareholders.
To determine the effect of limited liability on the policy choices of the unconstrained
bank, observe that the value of the option to default is v ∗ − v0 where
Z
∞
−ρt
e
v0 = E
0
dCt
1
=
ρ
λ
(1 − θ)(µ − c) −
β
,
gives the equity value in a bank without liquid reserves under the assumption that shareholders never default. Using equation (6), it can be shown that when v0∗ ≥ `(0) this option
value satisfies:
v ∗ − v0 = (λ/ρ) E (Y1 − v ∗ )+ .
This equation shows that, in a viable bank, limited liability only has value if cash flows are
subject to jumps, i.e. only if λ > 0, for otherwise the bank would not be subject to default
risk. Importantly, the fact that the unconstrained equity value of such a bank is strictly
15
positive for all c < µ, irrespective of the frequency and magnitude of jumps, implies that it
may be optimal for the bank to operate even if the present value v0 of its future cash flows
is negative. That is, the option to default associated with limited liability may lead the
shareholders of the unconstrained bank to invest in risky assets that would otherwise have
negative present value.
2.2
Equity value and bank policies with no refinancing
Having determined the value of equity when there are no costs of raising funds, we now turn
to the solution of the auxiliary problem in which the bank has no access to outside funds.
In this case, shareholders cannot refinance, have to default if the bank runs out of liquid
reserves, and can only optimize equity value over the bank’s payout policy.
Let α ≥ 0 be a constant and consider a financially constrained bank whose assets produce
after-tax net cash flows given by dCt as long as it is in operation, and a lump sum (α + s)+
to shareholders if liquidation occurs at a point where St = s. The optimization problem that
determines the equity value of such a bank is given by
Z
−ρt
e
w(s; α) = sup Es
π∈Π0 (s)
τπ
dPtπ
+e
−ρτπ
α+
+
Sτππ
,
(7)
0
where Π0 (s) denotes the subset of strategies such that Rπ = 0 (i.e. no refinancing) and the
stopping time τπ denotes the first time that liquid reserves become negative.
Following the literature on optimal dividend policies (see Albrecher and Thonhauser
(2009) for a survey), it is natural to expect that the optimal payout strategy for shareholders
in the auxiliary problem (7) should be of barrier type. Specifically, we conjecture that for
any given α, there exists a constant barrier b∗ (α) ≥ 0 such that the optimal policy of the
auxiliary bank consists in paying dividends to maintain liquid reserves at or below b∗ (α). In
addition, it is natural to conjecture that if the value of α is sufficiently large, it should be
optimal for shareholders to immediately liquidate and distribute all the available cash. The
following result confirms this intuition.
Lemma 2. If α ≥ v0∗ , then b∗ (α) = 0 and it is optimal for shareholders to immediately
liquidate the auxiliary bank.
16
Given the result in Lemma 2, we consider below the optimization problem (7) for a bank
with 0 ≤ α < v0∗ . To verify our conjecture on the optimal payout policy for such a bank,
we start by calculating the equity value associated with a barrier strategy. Let b > 0 and
consider the strategy πb that consists in paying dividends to maintain liquid reserves at or
below b. The cumulative payout process associated with this strategy is
Ptb = 1{t>0} max (Xu − b)+
0≤u<t
where the process Xt = s + Ct denotes the uncontrolled liquid reserves of the bank (i.e.
assuming that there are no dividend payments), and the corresponding value is defined by
Z
τπ b
e
w(s; α, b) = Es
−ρt
dPtb
+e
−ρτπb
α+
0
Sτππb
b
+ .
Letting ζ0 denote the first time that Xt becomes negative and using the dividend–penalty
identity (see Gerber, Lin, and Yang (2006)) shows that:
w(s; α, b) =

+


 (α + s) ,
ψ(s; α) +



for s ≤ 0,
W (s)
W 0 (b)
0
(1 − ψ (b; α)) , for 0 < s ≤ b,
s − b + w(b; α, b),
(8)
for s > b,
where the function
ψ(s; α) = Es e−ρζ0 (α + Xζ0 )+
gives the present value of the cash flow that the shareholders of the auxiliary bank receive in
liquidation if no dividends are distributed prior to default, and W (s) is the ρ-scale function
of the uncontrolled liquid reserves process. Closed-form expressions for these functions are
provided in the Appendix.
In the payout region where s > b, the bank pays dividends to bring its liquid reserves
down to b and, as a result, equity value grows linearly with liquid reserves. In the earnings
17
retention region where s ≤ b, the term
W (s)
= Es
W 0 (b)
Z
τπb
−ρt
e
dPtb
0
gives the present value of the dividends that the bank pays out to maintain its liquid reserves
at or below b, and the term
+ W (s)
πb
−ρτπb
α + S τπ
= Es e
ψ(s; α) − ψ (b; α) 0
b
W (b)
0
gives the present value of the cash flow that shareholders receive in liquidation, with the
second term on the left hand side reflecting the effects of payouts.
Equation (8) shows that in the earnings retention region, the equity value associated with
a barrier strategy depends on the barrier level only through
H(b; α) =
1 − ψ 0 (b; α)
.
W 0 (b)
In the Appendix, we show that for any 0 ≤ α < v0∗ there exists a unique b∗ (α) > 0 that
maximizes this function over the positive real line and, relying on a verification theorem
for the Hamilton-Jacobi-Bellman equation associated with problem (7), we prove that the
corresponding barrier strategy is optimal not only in the class of barrier strategies but among
all strategies. This leads to the following result.
Proposition 3 (Auxiliary value function). Consider a bank with no access to outside funds
that produces a cash flow (α + s)+ to shareholders in default. The equity value of such a bank
is concave and twice continuously differentiable over (0, ∞) and given by
w(s; α) = w(s; α, b∗ (α)),
where b∗ (α) is the unique solution to H 0 (b∗ (α); α) = 0 when α < v0∗ and b∗ (α) = 0 otherwise.
The optimal policy for shareholders in such a bank is to distribute dividends to maintain
liquid reserves at or below b∗ (α).
18
2.3
Equity value and optimal bank policies
Having solved the problem of the auxiliary bank, we now show how to obtain the optimal
policies for problem (3) by endogenizing the constant α = w(0; α) that gives the equity value
of the auxiliary bank at the point where it runs out of liquid reserves.
Fix b > 0 and consider the strategy π̂b that consists in paying dividends to maintain
liquid reserves at or below b and either liquidating or raising funds back to b whenever liquid
reserves become negative, depending on which is more profitable. Denote by
Z
τπ̂b
vb (s) = Es
−ρt
e
dPtπ̂b
0
π̂b
−ρτπ̂b
` Sτπ̂
− dΦt (R ) + e
π̂b
b
the corresponding equity value. By definition, this function satisfies
vb (s) = max {vb (b) − b + s − φ, `(s)} = (max {vb (b) − b − φ, `(0)} + s)+
for all s ≤ 0. Since the bank does not raise funds before the first time τπb that liquid reserves
are negative, we have that Rtπ̂b = Ptπ̂b − Ptb = 0 for all 0 ≤ t ≤ τπb . Therefore, using the
above equations together with the law of iterated expectations, we get that
Z
τπb
−ρt
dPtb
−ρt
dPtb
e
vb (s) = Es
Z0 τπ
= Es
b
e
−ρτπb
vb
−ρτπb
+e
+e
Sτππb
b
max {vb (b) − b − φ, `(0)} +
0
Sτππb
b
+ = w(s; vb (0), b),
(9)
where the auxiliary equity value function w(s; α, b) is defined as in equation (8). Evaluating
this equation at s = b and at s = 0 gives
vb (b) = w (b; vb (0); b)
vb (0) = max{vb (b) − b − φ, `(0)} = max {w (b; vb (0); b) − b − φ, `(0)}
19
and it follows that the equity value α̂(b) = vb (0) of the bank at the point where it runs out
of liquid reserves solves
α̂(b) = max {w(b; α̂(b); b) − b − φ, `(0)} .
The results of section 2.2 now suggest that to obtain a candidate optimal strategy, we need
to look for a dividend barrier b∗0 such that b∗0 = b∗ (α̂(b∗0 )) or, equivalently, for a fixed point
of the function
g(α; `(0)) = max {w(b∗ (α); α; b∗ (α)) − b∗ (α) − φ, `(0)} .
We show in the Appendix that this fixed point is uniquely given by α0∗ = max{a, `(0)} where
the constant a ∈ [0, v0∗ ) is the unique fixed point of g(α; 0).
After these preparations, let b∗0 = b∗ (α0∗ ) and consider the candidate optimal strategy π̂b∗0 .
If the fixed cost of equity financing exceeds the threshold
φ∗ = (w (b∗ (`(0)); `(0); b∗ (`(0))) − b∗ (`(0)) − `(0))+ ,
then α0∗ = `(0) ≥ a and the candidate optimal strategy is to distribute dividends to maintain
the liquid reserves of the bank at or below the target level b∗0 = b∗ (α0∗ ) = b∗ (`(0)) and to
liquidate the bank the first time that liquid reserves become negative. Otherwise, we have
α0∗ = a > `(0) and the candidate optimal strategy is to maintain the liquid reserves of the
bank at or below the level b∗0 = b∗ (a), to refinance to this level whenever liquid reserves
become negative with a shortfall smaller than a, and to liquidate for larger shortfalls. In this
case, a gives both the bank’s equity value at the point where it runs out of liquid reserves
and the maximum loss that shareholders are willing to refinance.
The identity (9) and the definition of the constants α0∗ and b∗0 imply that the equity value
associated with the candidate optimal strategy is given by
vb∗0 (s) = w(s; α0∗ , b∗0 ) = w(s; α0∗ ).
20
Using the properties of the auxiliary value function derived in the previous section, we show
in the Appendix that for all s ≤ 0 this function satisfies
+
vb∗0 (s) = (α0∗ + s)+ = max f vb∗0 (0), `(0) + s ,
where the financing operator f vb∗0 (s) is defined as in equation (5) and, relying on a verification
theorem for the Hamilton-Jacobi-Bellman equation associated with the bank’s optimization
problem, we prove that the strategy π̂b∗0 is optimal, not only in the class of barrier strategies,
but among all strategies. This leads to the following result.
Proposition 4 (Equity value function). The equity value function is concave and twice
continuously differentiable over (0, ∞) and given by
v(s) = vb∗0 (s) = w (s; α0∗ ) ,
with the constant α0∗ = max{a, `(0)}. The optimal payout strategy is to distribute dividends
to maintain liquid reserves at or below the target level b∗0 = b∗ (α0∗ ). When φ < φ∗ , the bank
raises funds to move to b∗0 whenever liquid reserves become negative with a shortfall smaller
than α0∗ = a, and liquidates otherwise. When φ ≥ φ∗ , the bank never raises equity and
defaults the first time that liquid reserves become negative.
Lemma 2 and Proposition 4 show that if `(0) > v0∗ then b∗0 = 0 so that it is optimal to
immediately distribute all available liquid reserves and liquidate. Otherwise, the target level
of liquid reserves b∗0 is strictly positive and the bank optimally pays dividends to maintain
liquid reserves at or below this level. This is illustrated by the smooth pasting and high
contact conditions (see Dumas (1991)):
0 = v 0 (b∗0 ) − 1 = v 00 (b∗0 ),
which show that liquid reserves are optimally reflected down at b∗0 . When liquid reserves
exceed b∗0 , the bank is fully capitalized and places no premium on internal funds so that it
is optimal distribute the lump sum s − b∗0 . Another feature that emerges from Proposition 4
21
is that, even though shareholders are risk-neutral, financing costs generate an implicit risk
aversion by making equity value concave over the positive real line.
The target level of liquid reserves b∗0 and the value α0∗ of the bank’s equity at the point
where it runs out of liquid reserves both result from the trade-off between the cost of accessing
outside liquidity and the cost of holding liquidity inside the bank. In particular, our analysis
of the bank’s optimization problem allows to show that:
Corollary 5. The target level of liquid reserves b∗0 increases with refinancing costs φ and
decreases with the liquidation value Λ of risky assets. The value α0∗ of equity in a bank without
liquid reserves decreases with refinancing costs φ and increases with the liquidation value Λ
of risky assets.
To conclude this section, note that in our model banks are subject to default risk even
though they can raise outside funds. Indeed, any realization of tail risk that takes liquid
reserves to a point below the critical level
s∗0 = −1{a≥`(0)} a
(10)
results in default because the present value v(b∗0 ) − (b∗0 − s) − φ of the refinancing operation is
negative for all s ≤ s∗0 . Our results are therefore in sharp contrast with those of Browniandriven inventory models, such as Peura and Keppo (2006), Bolton, Chen, and Wang (2011),
or Décamps, Mariotti, Rochet, and Villeneuve (2011) among others, in which there is no
default if banks are allowed to raise outside equity by paying a moderate fixed cost.
3
Prudential bank regulation
In this section, we examine the effects of prudential regulation on bank policy choices and
insolvency risk. To do so, we consider that the regulator can constrain the bank to maintain
its liquid reserves in excess of a given regulatory level and/or can set a lower bound on
the amount of equity capital that the bank must hold at all times. That is, we focus on
the instruments of regulation introduced in the Third Basel Accord, namely liquidity and
leverage requirements.
22
3.1
Liquidity requirements
To study the effects of liquidity requirements, we consider a regulatory constraint that
mandates the bank to hold a minimum amount T ≥ 0 of liquid reserves at all times. The
conventional view behind the imposition of such a constraint is that it should increase the
resilience of banks to adverse shocks and, thereby, reduce both the frequency of default
events and the losses that they impose on society as a whole. As we show below, this view is
not entirely justified because it fails to properly take into account that banks endogenously
adjust their policies in response to the imposition of the constraint.
The value of equity in a bank subject to a minimum regulatory level of liquid reserves is
defined by the optimization problem
Z
v(s; T ) =
e
sup Es
π∈Π(s,T )
τπ,T
−ρt
(dPtπ
π
−ρτπ,T
− dΦt (R )) + e
0
`
Sτππ,T
,
(11)
π
≤ T } denotes the default time associated with the strategy π in
where τπ,T = inf{t ≥ 0 : St+
the presence of a liquidity requirement at the level T , and Π(s, T ) denotes the set of payout
and financing strategies such that
π
π
Pt+
− Ptπ ≤ Stπ − T + Rt+
− Rtπ ,
t ≥ 0.
This constraint reflects the fact that the regulatory amount of liquid reserves T cannot be
distributed to shareholders. It implies that the liquidity requirement effectively increases
the collateral available to creditors by shifting the refinancing point up from 0 to T , and
suggests that the solution to the regulated problem (11) can be obtained by a translation of
the unregulated policy derived in Proposition 4. The following Proposition derives the value
of equity for a bank subject to liquidity requirements and confirms this intuition.
Proposition 6 (Liquidity requirements). Equity value in a bank subject to a minimum
regulatory level T ≥ 0 of liquid reserves is given by:

 v(s − T ; 0),
if `(T ) ≤ a,
v(s; T ) =
 w(s − T ; `(T )), otherwise,
23
and the optimal payout strategy for shareholders is to distribute dividends to maintain liquid
reserves at or below the target level b∗T defined by:

 T + b∗ ,
if `(T ) ≤ a,
0
∗
bT =
 T + b∗ (`(T )), otherwise.
When `(T ) ≤ a, the bank raises outside equity to move to the target level b∗T whenever
liquid reserves fall below the regulatory level with a shortfall smaller than a, and liquidates
otherwise. When `(T ) > a, the bank never raises outside equity, and defaults as soon as
liquid reserves fall below the regulatory level.
Proposition 6 shows in the presence of a liquidity requirement the nature of the optimal
policy depends on the size of the regulatory amount of liquid reserves. When this amount
is such that `(T ) > a, it is never optimal for shareholders to refinance since their cash flow
in liquidation exceeds the continuation value of equity. In this case, the bank distributes
dividends to maintain its liquid reserves at or below the target level and liquidates as soon
as they fall short of the regulatory level. By contrast, when `(T ) ≤ a it is optimal for
shareholders to refinance small and intermediate losses, and default only occurs following a
large loss that takes liquid reserves below T − a.
Combining these results with those of Corollary 5 and the fact that the liquidation cash
flow `(T ) is non-decreasing in T shows that the presence of a liquidity requirement pushes
the default threshold of the bank up from the unregulated level s∗0 defined in (10) to the
regulated level
s∗T = T − 1{a≥`(T )} a ≥ T − 1{a≥`(0)} a ≥ −1{a≥`(0)} a = s∗0 ,
and simultaneously pushes the optimal level of excess liquid reserves down from the unregulated level b∗0 to the regulated level
b∗T − T = b∗ (min{a, `(T )}) ≤ b∗ (min{a, `(0)}) = b∗0 .
24
Figure 3 illustrates these two effects by plotting the value of bank equity as a function of
liquid reserves for different liquidity requirements.
Let now τT∗ denote the default time of a bank subject to a liquidity requirement at T .
The increase in the default threshold and the fact liquidity requirement increases the amount
of collateral available to creditors suggests that the default loss, which is defined by
L∗T = L + D − SτT∗ − Λ
+
,
should decrease as a result of the liquidity requirement. On the other hand, the decrease
in the optimal level of excess liquid reserves and the increase in the default threshold imply
that banks subject to liquidity requirements are generally closer to their default threshold,
which suggests that the imposition of a liquidity requirement will make default events more
frequent. The following Proposition confirms these conjectures by deriving the precise effects
of liquidity requirements on default risk and bank losses in default.
Proposition 7. For any given capital structure (c, L, D), liquidity requirements increase
the frequency of default events and decrease bank losses in default in that the probabilities
PT +b [τT∗ ≤ h] and PT +b [L∗T ≤ k] are both non decreasing in T for all (b, h, k) ∈ R3+ .
Proposition 7 shows that for any given liquidity cushion b above the minimum regulatory
level T , an increase in T leads to an increase in the likelihood of default and to a decrease
in bank losses in default. Thus, while liquidity requirements are effective at reducing the
losses that banks can impose on their creditors, they tend to make default events more
frequent. This result takes the capital structure of the bank as given but also has important
consequences for the choice of an optimal capital structure. Notably, it suggests that by
increasing the collateral available to creditors, liquidity requirements may decrease the cost
of market debt and provide incentives for bank shareholders to choose higher debt levels,
leading to a further increase in insolvency risk (see section 4). To offset this unintended
effect and harness the benefits of the decrease in default losses afforded by the liquidity
requirement, the regulator may thus need to also impose leverage constraints that limit the
debt levels that banks are allowed to take.
25
Figure 3: Equity value for a regulated bank
Equity value
Unregulated
Regulated with `(T0 ) ≤ a
Regulated with `(T1 ) > a
•◦
•◦
•◦
•
`(T1 )
a
0
•
•
s∗0
s∗T1
T0 b∗0
0
b∗T0 T1 = s∗T1
b∗T1
Liquid reserves
Notes. This figure illustrates the change in the equity value that occurs when going from an
unregulated bank with α0∗ = a to a regulated bank with either T0 such that `(T0 ) ≤ a or T1 such
that `(T1 ) > a. On each curve the empty dot indicates the target level of liquid reserves while the
filled dot indicates the point below which the bank is optimally liquidated.
3.2
Endogenous leverage and leverage requirements
We now turn to the analysis of the privately optimal mix between equity and debt financing
and the effects of regulatory constraints on this mix. To do so, we follow the literature on
optimal financing decisions (e.g. Leland (1994) or Duffie and Lando (2001)) and consider
a bank that initially raises an amount D of deposits with coupon cD and issues at par a
perpetual debt contract with face value L and coupon cL .
In our benchmark environment, deposits are correctly priced (i.e. deposit insurance is
fairly priced and there is no liquidity premium) and depositors are senior to other creditors.8
8
In the U.S., Congress has adopted in 1993 a national depositor preference law that elevates the claims of
domestic depositors on the assets of a failed bank over the claims of foreign depositors and general creditors.
26
This seniority rule implies that the payment to depositors in default is min{D, (s + Λ)+ }
while the payment to market debtholders is min{L, (s + Λ − D)+ }. This in turn implies that
the total payment to the creditors of the bank is
min D, (s + Λ)+ + min L, (s + Λ − D)+ = min D + L, (s + Λ)+
while the payment to shareholders is ` (s). These formulas show that the priority rule in
default does no affect the total payment to creditors and, therefore, the total cost of debt
and the total amount of debt that bank shareholders want to issue when deposits are fairly
priced. Accordingly, in our benchmark environment, we optimize the bank value over the
total coupon c = cL + cD and the total amount of debt F = D + L.
When choosing the amount of debt to issue, the objective of shareholders is to maximize
the value of equity after debt has been issued plus the total proceeds from the debt issue (i.e.
the value of the bank), net of the cost of providing the required capital. Assuming that the
bank is subject to a setup cost Ψ ≥ φ that includes the cost of buying assets, we therefore
consider the static optimization problem given by
sup (v (b∗T (c, F ); T |c, F ) + F − b∗T (c, F ) − Ψ) ,
(12)
c≥0
subject to the constraint
"Z
F = Eb∗T (c,F )
τT∗ (c,F )
−ρτT∗ (c,F )
e−ρt cdt + e
0
n
min F, SτT∗ (c,F ) + Λ
#
+ o
.
(13)
In these equations, b∗T (c, F ), τT∗ (c, F ), and v(s; T |c, F ) denote respectively the target level of
liquid reserves, the default time, and the equity value of a bank with total coupon rate c and
total debt F that is subject to a liquidity requirement at level T . Equation (13) requires
that debt is issued at par given that the bank starts from the target level of liquid reserves
and that creditors correctly anticipate the strategy of shareholders.
The main objective of this law is to produce cost savings for the Federal Deposit Insurance Company (FDIC)
and, thereby, limit injections of public funds in the financial sector.
27
An important difference between problem (12) and leverage choices in dynamic corporate
finance models following Leland (1994) is that the initial contribution b∗T (c, F ) + Ψ − F
of shareholders is endogenous and reflects both the liquidity needs of the bank and the
constraints that it faces. Another key difference is that, in our model, the bank determines
its optimal capital structure by balancing tax benefits against both refinancing and default
costs, whereas external financing is assumed to be costless in models, such as Sundaresan
and Wang (2015), that adapt the Leland (1994) framework to the study of banks.
Denote by c∗T the optimal coupon rate and by FT∗ the corresponding total face value of
debt. When all debt is fairly priced, deposits and subordinated market debt come with the
same costs and benefits. As a result, there is no optimal debt structure for bank shareholders
in that any pair (L, D) such that L + D = FT∗ and
"Z
D = Eb∗T (c,F )
τT∗ (c,F )
−ρτT∗ (c,F )
e−ρt cD dt + e
0
#
+ o min D, SτT∗ (c,F ) + Λ
n
(14)
(c,F )=(c∗T ,FT∗ )
for some cD ≤ c∗T yield the same value to shareholders. To determine the mix between
deposits and subordinated debt, we therefore pick the amount D of deposits such that
the ratio D/FT∗ of deposits to total debt matches its empirical counterpart and solve (14)
to obtain the coupon paid cD to depositors. The coupon and face value of the banks’
subordinated market debt are then uniquely determined by setting cL = c∗T − cD and L =
FT∗ − D so that debt is issued at par.
When making their capital structure decisions, banks may be constrained by leverage
requirements. That is, in an attempt to reduce insolvency risk, the regulator may require
the bank to choose a debt ratio, defined as
L+D
,
L + D + v(s; T |c, L + D)
that lies below some fixed upper bound Ω ∈ [0, 1] for any level s ≥ T of liquid reserves. Since
the value of the bank’s equity is increasing in liquid reserves, its debt ratio is maximized
at T and it follows that constraining the bank’s debt ratio to be lower than a fixed upper
28
bound Ω is equivalent to requiring that
v(T, T |c, L + D) ≥
1
− 1 (L + D).
Ω
(15)
To take into account leverage requirements, we simply append this constraint to the static
optimization problem of shareholders. Unfortunately, neither the problem defined by (12)
and (13) nor the augmented problem defined by (12), (13) and (15) can be solved analytically.
In the next section, we circumvent this difficulty by first calibrating the parameters of the
model to match the basic observed characteristics of U.S. banks and then solving numerically
for the privately optimal capital structure under various regulatory scenarios.
4
Model analysis
4.1
Parameter values and implied variables
This section illustrates the effects of regulatory constraints on bank’s policy choices and the
induced insolvency risk. To do so we start by calibrating the parameters of the model to
match the observed characteristics of average U.S. banks.
In our benchmark model, capital structure decisions reflect a trade-off between tax
benefits and refinancing and liquidation costs. The statutory tax rate for U.S. corporations
ranges up to 35%. We thus set θ to this level, which is also consistent with the recent
estimates of De Mooij, Keen, and Orihara (2014) or Langedijk, Nicodème, Pagano, and
Rossi (2014) for the marginal tax rates of banks. The parameter that controls liquidation
costs is set to ϕ = 20% based on the estimates of James (1991) and Flannery (2011) for risky
assets in banks. Boyson, Fahlenbrach, and Stuz (2015) estimate that the average underwriter
spread for bank seasoned equity offerings between 1996 and 2007 is 5.02% (see their Table 1,
Panel C). To match this estimate, we set the refinancing cost to φ = 1.55%, implying that
the cost of refinancing represents on average 5.04% of the amount raised.
The values of the other parameters of the model are set follows. The risk-free rate of
return is set to ρ = 0.04. The mean after-tax cash flow rate is set to (1 − θ)µ = 0.19 and the
after-tax cash flow volatility to σ = 0.10, based on the estimates of Sundaresan and Wang
29
Table 1: Base case parameters and implied variables
Symbol
Value
A. Parameter values
Tax rate
Liquidation costs
Financing cost
Mean cash flow rate
Cash flow volatility
Jump intensity
Mean jump size
Discount rate
Liquidity requirement
Maximal debt ratio
θ
ϕ
φ
(1 − θ)µ
σ
λ
1/β
ρ
T
Ω
0.35
0.20
0.0155
0.19
0.10
0.50
0.09
0.04
0
1
B. Implied variables
Present value of asset cash flows
Liquidation value of assets
Optimal coupon
Face value of subordinated debt
Face value of deposits
Target level of liquid reserves
Default threshold
Equity value at s = 0
Average financing cost (%)
Debt ratio band (%)
Deposits over assets (%)
1Y Default probability at the target (bps)
V
Λ
c∗T
L
D
b∗0
s∗0
α0∗
3.625
2.9
0.1809
1.1191
3.3573
0.2027
−0.4334
0.4334
5.04
[87.29, 91.17]
[65.47, 68.38]
7.475
(2015). The arrival intensity and the mean size of after-tax cash flow jumps are respectively
set to λ = 0.5 and 1/β = 0.09, implying that large losses represent on average 2.5% of the
bank’s risky asset value and occur every other year on average. Lastly, in our base case
environment the bank is unregulated so that the regulatory amount of liquid reserves and
the upper bound on the bank’s debt ratio are initially set to T = 0 and Ω = 1.
Table 1 summarizes the baseline values of the parameters and reports the corresponding
values of the endogenous quantities implied by the model. In our base case environment, the
privately optimal amount of debt is FT∗ = 4.476 and the total coupon rate is c∗T = 0.1809.
We determine the level of deposits by requiring that D = 0.75FT∗ . This implies that deposits
30
represent between 65.46% to 68.38% of total bank value depending on the level of liquid
reserves, consistent with the estimates of Gropp and Heider (2010). Given this optimal
capital structure the model implies that the target level of liquid reserves is b∗0 = 0.2027 and
the value of equity of a bank without liquid reserves is α0∗ = a = 0.4334. This in turn implies
that it is optimal for shareholders to refinance the bank up to the point s∗0 = −0.4334 and it
follows that the maximal amount of capital that they are willing to contribute to keep the
bank alive is b∗0 − s∗0 = 0.6361, which represents 12.4% of the total value of the bank at the
target level of liquid reserves.
Because liquid reserves fluctuate between the minimum regulatory level T = 0 and the
target level of liquid reserves b∗0 , the debt ratio of the bank effectively remains within a
band as in the dynamic capital structure models of Fischer, Heinkel, and Zechner (1989),
Strebulaev (2007), and Morellec, Nikolov, and Schürhoff (2012) among others. In our base
case environment, the debt ratio of the bank fluctuates between 87.29% and 91.17%. We show
in section 4.4 that debt ratios are even higher with mispriced deposits and broadly consistent
with the aggregate liability structure of FDIC-insured banks reported by Sundaresan and
Wang (2015, Table 3). Lastly, the parameters of the model and the induced optimal capital
structure imply that the 1-year default probability at the target level of liquid reserves is 7.47
basis points, consistent with the values reported by Hamilton, Munves, and Smith (2010)
and Crossen and Zhang (2012) for financial firms.
4.2
Target liquid reserves and optimal default decisions
We start our analysis by examining the determinants of cash holdings and default decisions.
To do so, Figure 4 plots the target level of liquid reserves b∗T and the default threshold s∗T
as functions of the intensity of large losses λ, the volatility of after-tax cash flows σ, the
mean cash flow rate µ, and the liquidity requirement T , holding the financing structure of
the bank (c∗T , FT∗ ) fixed at the benchmark values reported in Table 1.
In the model, the target level of liquid reserves results from the trade-off between the
carry cost of liquidity ρ and the cost of raising capital φ. Because an increase in volatility
raises the likelihood of a costly equity issuance, the target level of liquid reserves increases
with σ. An increase in the frequency of large losses λ, or in their expected size 1/β, has two
31
opposite effects. First, it raises the likelihood of a costly refinancing and therefore increases
shareholders’ incentives to build up liquid reserves. Second, it reduces the expected cash flow
from operating the bank’s assets and thus decreases shareholders’ incentives to contribute
capital. As shown by the top-left panel, the first (second) effect dominates for low (large)
values of the jump intensity λ and the expected jump size 1/β.
The effect of the mean cash flow rate µ on the target level of liquid reserves also results
from two opposite effects. On the one hand, a higher mean cash flow rate increases the
bank’s expected cash flows and thus shareholders’ incentives to contribute capital. On the
other hand, it increases the bank revenues and reduces the role of liquid reserves as a buffer
to absorb losses. As shown by the bottom-left panel, the first (second) effect dominates for
low (high) values of µ. Lastly, consistent with Proposition 6, the bottom right panel of the
figure shows that both the target level of reserves and the default threshold grow linearly
with the regulatory amount of liquid reserves as long as refinancing is optimal. When the
regulatory amount of liquid reserves reaches the point where `(T ) = a, refinancing becomes
suboptimal. As a result, the default threshold jumps up to the regulatory amount of liquid
reserves, and the optimal excess level of liquid reserves b∗T − T starts to decrease until its
reaches zero, at which point shareholders should immediately liquidate the bank.
Consider next the decision to default. When there is no liquidity requirement and
refinancing is optimal in that α0∗ = a, default occurs when the bank is hit with a negative
shock that takes its liquid reserves below the default threshold s∗0 = −a. To understand
how this threshold is affected by the parameters of the model observe that, in this case, the
constant a can be written as the difference between the value of equity at the target, the
new provision of capital, and the refinancing cost, i.e.
a = v(b∗0 ; 0) − b∗0 − φ.
By decreasing the value of the claim of incumbent shareholders at refinancing, an increase
in the intensity of large losses λ, in their expected size 1/β, or in the after-tax cash flow
volatility σ leads to an increase in the default threshold. By contrast, an increase in the
mean cash flow rate µ leads to an increase in the value of equity and, therefore, to a decrease
32
Figure 4: Effect of the cash-flow parameters
0.4
0.2
0.1
Liquid reserves
Liquid reserves
0.2
0.0
−0.5
0.0
−0.3
−1.0
−1.5
−0.6
0
0.5
1
1.5
2
0
0.05
0.1
0.15
0.2
σ
λ
0.2
2.5
Liquid reserves
Liquid reserves
0.1
0.0
−1.0
Retain
Refinance
Dividends
Liquidation
−2.5
0
0.1
0.2
1.5
0.5
−0.5
0.3
0.4
0
µ
0.2
0.4
0.6
T /F0∗
Notes. This figure plots the target level of liquid reserves b∗T (solid line) and the liquidation
threshold s∗T (dashed line) as functions of the jump arrival intensity, the mean jump size, the cash
flow volatility and the cash flow drift. In each panel the upper and lower regions
correspond
respectively to dividend payments and liquidation while the intermediate regions
and
correspond respectively to earnings retention and refinancing.
33
in the default threshold. As shown by the bottom left panel of the figure, when µ falls below
c equity becomes worthless and it is optimal to default immediately.
4.3
Regulation, insolvency risk, and value-at-risk
An important question for regulators is whether liquidity and leverage requirements affect
insolvency risk and bank losses in default. Proposition 7 shows that for any given capital
structure, liquidity requirements increase the likelihood of default and reduce bank losses
in default by increasing the collateral available to creditors. If the second effect dominates,
liquidity requirements may effectively reduce the cost of debt and provide incentives for
shareholders to issue additional debt, leading to a further increase in default risk.
To quantify the effects of regulation on the probability of default and the extent of the
default losses loss implied by shareholders’ financing and liquidity management decisions,
we need to compute the probability
fT (s, y, t) = Ps [{τT∗ ≤ t} ∩ {L∗T ≥ y}]
= Ps {τT∗ ≤ t} ∩ SτT∗ ≤ T − `(T + y)
that the bank defaults prior to some horizon t ≥ 0 in a state where the default losses of its
creditors exceed some amount y ≥ 0. Unfortunately, this probability cannot be computed
analytically. To circumvent this difficulty, we derive in the Appendix a closed-form solution
for its Laplace transform
Z
fbT (s, y, k) =
∞
e
0
−kt
h
i
−kτT∗
fT (s, y, t)dt = (1/k) Es e
1{L∗ ≥y} ,
T
and use the Gaver-Stehfest formula (see Stehfest (1970)) to obtain a numerical approximation
of the probability.
We then quantify the effects of regulation by computing both the
probability of default prior to a given horizon and the conditional value-at-risk
VaRT (s, q, t) = inf {Φ ≥ 0 : Ps [{L∗T ≥ Φ}| τT∗ ≤ t] ≤ q}
= inf Φ ≥ 0 : Ps SτT∗ ≤ T − `(T + Φ) τT∗ ≤ t ≤ q .
34
To facilitate the interpretation of our results, we express this conditional value-at-risk as a
fraction of the present value V of the infinite stream of cash flows generated by the risky
assets of the bank, as defined in equation (2).
Table 2 reports the optimal coupon and face value of debt, the optimal debt ratio band,
the one year default probability, the one year conditional value-at-risk at 1%, the target
level of liquid reserves expressed as the value of the bank assets, and the average financing
costs incurred by shareholders in different economic environments. Consistent with the
above discussion, Table 2 shows that liquidity requirements decrease the cost of market
debt, leading bank shareholders to issue more debt and to an increase in default risk. With
our baseline parameters, debt ratios increase by 0.31%, leading to a relative increase of 20%
in the one year default probability. In line with Proposition 7, Table 2 also shows that an
offsetting effect for the regulator is that the loss in default, as measured by the value-at-risk
conditional on default, decreases with liquidity requirements. In our base case environment,
this loss decreases by 13.51% when imposing a minimum liquidity requirement of 10% of
deposits. Another important change triggered by the imposition of a liquidity requirement
is that liquid reserves increase to 9.84% of total assets. This in turn implies that net debt,
defined as FT∗ − b∗T , decreases with liquidity requirements.
We also investigate in Table 2 the effects of asset risk on financing decisions and insolvency
risk. The table shows that a decrease in the frequency or magnitude of tail risk generally
leads to a large increase in privately optimal debt ratios. For example, the debt ratio of the
bank at the refinancing point increases from 91.27% to 93.15% when the arrival intensity of
negative cash flow jumps decreases to λ = 0.25, and from 91.17% to 96.53% when their mean
size decreases to 1/β = 0.045. Despite these more aggressive financing strategies, a decrease
in the frequency or magnitude of tail risk leads to a decrease in the default probability.
Table 2 also shows that volatility risk works differently. Notably, an increase in σ leads to
an increase in the expected cost of refinancing, which in turn leads to an increase in the
target level of liquid reserves and to wider variations in debt ratios with little effect on the
average debt ratio. In all of these alternative environments, liquidity requirements have the
same effects as in our baseline environment. That is, they lead to more aggressive financing
35
36
Coupon
c∗T
0.1809
0.1818
4.476
4.499
Face value
FT∗
[87.29, 91.17]
[87.60, 91.49]
Debt ratio
band (%)
0.2186
0.2205
5.423
5.461
[89.79, 93.15]
[90.29, 93.68]
0.0448
0.0467
1.065
1.103
[64.75, 76.08]
[66.71, 78.11]
0.2303
0.2305
5.734
5.742
[92.91, 96.53]
[92.98, 96.60]
0.1349
0.1364
3.294
3.330
0.1814
0.1823
4.488
4.511
0.1781
0.1790
4.403
4.426
[85.99, 92.51]
[86.30, 92.83]
[88.03, 90.23]
[88.33, 90.57]
[80.28, 84.36]
[80.92, 85.02]
8.026
9.456
8.213
9.667
18.69
22.29
2.183
2.386
52.29
71.74
5.634
7.849
7.475
8.805
1Y Default
probability (bps.)
62.728
53.864
68.612
59.553
67.790
60.139
66.799
56.625
51.928
47.036
68.927
59.409
66.875
57.841
1Y1% VaR
(%)
3.35
3.38
8.31
8.34
5.20
5.24
5.43
5.44
4.75
4.88
5.09
5.17
5.04
5.07
Financing
cost (%)
9.517
18.62
3.051
12.34
5.982
14.05
5.108
15.427
16.69
22.21
4.838
14.56
5.591
14.85
Target level
(%)
—
0.172
—
0.163
—
0.308
—
0.066
—
0.528
—
0.143
—
0.163
Change in bank
value (%)
Notes. This table reports the optimal coupon, the optimal debt ratio band, the one year default probability, the one year value
at risk at 1% as a fraction of the present value of asset cash flows, the average financing costs as a fraction of the amount raised,
the target level of liquid reserves as a fraction of the present value of asset cash flows, and the change in total bank value due
to liquidity requirements under different parametric assumptions. In the first column, the regulatory level of liquid reserves is
expressed as a fraction of amount of deposits that is optimal absent liquidity requirements.
0
10%
F. Higher cash-flow volatility (σ = 0.20)
0
10%
E. Lower cash-flow volatility (σ = 0.05)
0
10%
D. Larger average cash-flow jumps (1/β = 0.135)
0
10%
D. Smaller average cash-flow jumps (1/β = 0.045)
0
10%
C. More frequent cash-flow jumps (λ = 1.50)
0
10%
B. Less frequent cash-flow jumps (λ = 0.25)
0
10%
A. Base case environment
Liquidity
Requirement T
Table 2: Effect of liquidity requirements in different environments
strategies and to an increase in default risk but simultaneously reduce the magnitude of the
losses that the bank imposed on its creditors in default.
Consider next the effects of leverage requirements. The first four columns of Table 3
report the optimal coupon, the optimal debt ratio band, the probability of default over
a one-year horizon, and the conditional 1% value-at-risk at a one year horizon for a bank
subject to different combinations of liquidity and leverage requirements.9 As shown by Panel
A leverage requirements have a strong quantitative effect on the likelihood of default but a
more modest impact on default losses. In our base case environment, reducing the coupon of
the bank from its privately optimal level to a level corresponding to a maximal debt ratio of
90% reduces the one year default probability from 7.475 basis points to 3.991 basis points, but
leaves the conditional value-at-risk essentially unchanged. In effect, leverage requirements
lead to an increase in the cash flow rate to shareholders and, hence, to an increase in their
willingness to absorb losses, but they do not allow for an effective control of default losses
as they have little impact on the collateral available to creditors.
Comparing Panels A and B of Table 3 shows that when combining liquidity requirements
with leverage requirements, the regulator can drastically reduce both the likelihood of default
and the magnitude of default losses. For example, imposing a maximum debt ratio of
90% together with a liquidity requirement of 10% of deposits, reduces the one year default
probability of the bank from 7.475 basis points to 3.974 basis points and simultaneously
lowers its conditional value-at-risk at a one.year horizon from 66.875% to 57.477%. This
confirms that in order to effectively control both the likelihood of bank defaults and the
extent of the losses that banks can impose on their creditors, it is necessary to impose both
liquidity and leverage requirements, as mandated in the Basel III accord.
Another important aspect of regulation relates to its effects on valuations. Table 2 shows
that one surprising implication of our analysis is that liquidity requirements increase bank
value. For example, in our base case environment, requiring that liquid reserves represent
9
The leverage requirement that we consider can be easily mapped into a risk-weighted capital ratio
constraint. For example, assuming that risk-weighted assets represent 57% of total assets, which is the
average number for the financial industry in North America (see e.g. Le Leslé and Avramova (2012)), our
constraint of 90% is equivalent to a risk-weighted capital ratio of 0.10/0.57 = 17.54%. When banks have
access to multiple risky assets, risk-based capital charges constrain asset choices to reflect the trade-off
between risk and return. See Rochet (1992) for an analysis of the benefits and costs of risk-weights.
37
Table 3: Effect of leverage constraints
Maximal debt
ratio Ω
Total coupon
c∗T
Debt ratio
band (%)
1Y Default
probability (bps.)
1Y1% VaR
(%)
Change in bank
value (%)
[87.29, 91.17]
[87.29, 91.17]
[86.18, 90.00]
[81.39, 85.00]
[76.60, 80.00]
7.475
7.475
3.991
0.316
0.029
66.875
66.875
66.635
63.969
60.930
−0.000
−0.000
−0.185
−2.159
−4.400
8.805
8.805
3.974
0.315
0.029
57.841
57.841
57.477
54.715
51.669
−0.163
−0.163
−0.111
−2.154
−4.399
A. Without liquidity requirement
100%
95%
90%
85%
80%
0.1809
0.1809
0.1774
0.1634
0.1502
B. With a liquidity requirement at 10% of deposits
100%
95%
90%
85%
80%
0.1817
0.1817
0.1773
0.1634
0.1502
[87.59, 91.48]
[87.59, 91.48]
[86.61, 90.00]
[81.39, 85.00]
[76.60, 80.00]
Notes. This table reports the optimal coupon, the optimal debt ratio band, the one year default
probability, the one year value at risk at 1% as a fraction of the present value of asset cash flows,
and the change in the total value of the bank at s = 0 implied by different leverage constraints
absent liquidity requirement (Panel A) and in the presence of a liquidity requirement set at 10% of
the amount of deposits that is optimal absent for an unregulated bank (Panel B).
at least 10% of the value of deposits increases the value of the bank by 0.16%. This increase
in value is due to the fact that liquidity requirements allow shareholders to commit that the
bank will hold a minimum amount of liquid reserves at all times, making risky debt cheaper
and allowing the bank to pursue more aggressive financing strategies to reduce its cost of
capital. Of course, the net effect of liquidity requirements on initial shareholder wealth is
negative as shareholders need to contribute additional capital to acquire the bank assets in
that net debt FT∗ − b∗T decreases with the level of liquidity requirement.
By contrast, The last column of Table 3 shows that leverage requirements always reduce
bank value. For example, total bank value is reduced by 4.427% when moving from the
privately optimal capital structure to one with 80% of debt financing. As illustrated by Table
4 this drop in value is magnified when bank deposits are mispriced because privately optimal
debt ratios are higher in such environments, and can reach up to 6.52% when deposits are
38
junior and the bank is subject to a leverage requirement at 80%. In our model, this decrease
in value is due to an increase in the bank’s cost of capital.
4.4
Deposit insurance, depositor preference, and insolvency risk
In our base case environment, all debt is fairly priced and depositors have priority in default.
In this section we allow deposits to be mispriced, either because deposit insurance is mispriced
or because of a liquidity premium on deposits, and investigate the effect of alternative priority
rules in default on financing decisions and insolvency risk. Bank shareholders benefit from
mispriced deposits because they lower debt payments and reduce the liquidation costs priced
in market debt, thereby allowing the bank to follow more aggressive financing strategies
(see DeAngelo and Stulz (2015)). Making market debt senior also matters if deposits are
mispriced, as it leads to a decrease in the cost of market debt without increasing the cost of
deposits and, thus, to an increase in privately optimal debt ratios.
We consider two environments with mispriced deposits: One in which deposits are senior
to market debt in default and another in which they are junior. To solve for the privately
optimal capital structure in such environments, we fix the amount of deposits to its base
case level and let shareholders maximize the value of the bank over market debt given some
exogenous coupon rate on deposits and under the constraints that the debt ratio does not
exceed the upper bound prescribed by the regulator and that market debt is issued at par
when liquid reserves are at the target level. Specifically, we fix cD /D = 0.02 < ρ and
numerically solve the static optimization problem
sup (v (b∗T (c, F ); T | c, F ) − b∗T (c, F ) + F − Ψ) ,
cL ≥0
subject to equation (15) and
"Z
L = Eb∗T (c,F )
τT∗ (c,F )
−ρτT∗ (c,F )
e−ρt cL dt + e
0
n
min L, Λ + SτT∗ (c,F ) − qD
#
+ o
.
where the constant q is set to 1 in the environment where deposits are senior to market debt
in default, and to zero otherwise.
39
40
Ω
100%
100%
[87.29, 91.17]
[87.59, 91.48]
Base
100%
95%
90%
85%
80%
[87.29, 91.17]
[87.29, 91.17]
[86.17, 90.00]
[81.39, 85.00]
[76.60, 80.00]
[90.12, 93.11]
[90.12, 93.11]
[87.13, 90.00]
[82.31, 85.00]
[77.48, 80.00]
[90.12, 93.11]
[90.15, 93.14]
MSD
[92.29, 93.31]
[91.95, 95.00]
[87.13, 90.00]
[82.31, 85.31]
[77.48, 80.00]
[92.29, 93.31]
[93.38, 96.36]
MJD
7.475
7.475
3.991
0.316
0.029
7.475
8.804
100%
95%
90%
85%
80%
[87.59, 91.48]
[87.59, 91.48]
[86.17, 90.00]
[81.39, 85.00]
[76.60, 80.00]
[90.15, 93.14]
[90.15, 93.14]
[87.13, 90.00]
[82.30, 85.00]
[77.48, 80.00]
[93.38, 96.36]
[91.99, 95.00]
[87.13, 90.00]
[82.31, 85.00]
[77.48, 80.00]
8.804
8.804
3.974
0.316
0.028
6.108
6.108
0.669
0.024
0.001
65.25
22.79
0.665
0.024
0.001
29.39
23.19
0.672
0.024
0.001
29.39
65.25
MJD
57.84
57.84
57.48
54.72
51.67
66.87
66.87
66.63
63.97
60.93
66.87
57.84
Base
103.97
103.97
102.14
97.97
93.88
112.82
112.82
111.17
107.01
102.92
112.82
103.97
MSD
106.91
106.39
102.26
97.98
93.88
114.85
114.85
111.27
107.02
102.92
114.85
106.91
MJD
1Y1% VaR (%)
0.163
0.163
−0.111
−2.154
−4.399
—
—
−0.185
−2.159
−4.400
—
0.163
Base
0.129
0.129
−0.812
−3.131
−5.403
—
—
−0.933
−3.248
−5.517
—
0.129
MSD
∆Value (%)
0.641
0.428
−1.802
−4.160
−6.410
—
−0.009
−1.933
−4.277
−6.523
—
0.641
MJD
Notes. This table reports the optimal debt ratio band, the one year default probability, the one year value at risk at 1% as a
fraction of the present value of asset cash flows, and the change in total bank value for different liquidity and leverage requirements.
In column Base we assume that all debt is fairly priced. In columns MSD and MJD we assume that deposits are Mispriced in
that cD /D is fixed at 2% and either Senior or Junior to market debt. When present the liquidity requirement is set at 10% of the
amount of deposits that is optimal for an unregulated bank.
10%
10%
10%
10%
10%
6.015
6.015
0.675
0.024
0.001
6.015
6.108
MSD
1Y DP (bps.)
Base
C. With both a leverage constraint and a liquidity requirement
0
0
0
0
0
B. With a leverage constraint
0
10%
A. Benchmarks
T
Debt ratio band (%)
Table 4: Effect of mispriced deposits and debt seniority
Table 4 reports the optimal debt ratio band, the one year default probability, the one
year conditional value at risk at 1%, and the change in the bank value due to regulation,
for the base case and the two mispriced deposits environments. The table shows that
when deposits are mispriced but remain senior the bank’s privately optimal debt ratio
is higher and fluctuates between 90.12% and 93.11% at the optimal coupon level. Even
though debt ratios increase, the total payment to creditors actually decrease, leading to
an increase in shareholders’ willingness to absorb losses and to a decrease in insolvency
risk. Notably, the one year default probability at the target level of liquid reserves is now
6.015 basis points. However, and as shown by the table, bank losses in default increase
significantly. Importantly, because privately optimal debt ratios are higher with mispriced
deposits, leverage requirements lead to a larger drop in total bank value. With the baseline
parameters for example, constraining the bank debt ratio to be less than 80% leads to a 5.52%
decrease in bank value. By contrast, as in the base case environment, liquidity requirements
reduce the cost of debt and allow bank shareholders to issue more debt, leading to a moderate
increase in the total value of the bank.
Consider next the effects of securing market debt. Securing market debt effectively
implies that market debt has priority over deposits in default. Table 4 shows that in this
case optimal debt ratios and insolvency risk are even higher. This is due to the fact that
security provisions increase the payoff to market debtholders in default and thus the value
of market debt, leading the bank to take on additional debt. That is, combining mispriced
deposits with secured financing encourages high debt ratios and fuels default risk. With our
baseline parameters for example, the optimal debt ratio ratio of the bank fluctuates between
92.29% and 93.31% and the one year default probability increases from 7.475 basis points
to 29.39 basis points, a 300% increase. Since unconstrained debt ratios are higher with
secured market debt, binding leverage requirements lead to an even larger drop in value that
reaches up to 6.52% when the bank is limited to 80% of debt financing. Lastly, Panel C of
Table 4 shows that a robust feature of the model is that combining leverage requirements
with liquidity requirements leads to a sharp drop in both the likelihood of default and the
magnitude of bank losses in default.
41
5
Conclusion
We develop and analytically solve a dynamic model of banking in which banks face taxation,
flotation costs of securities, and default costs and are financed with equity, insured deposits,
and market debt. In this model, shareholders have limited liability and the bank’s liquidity
management, financing and default policies are jointly and endogenously determined to
maximize shareholder value. When raising outside funds is costly, inside and outside equity
are not perfect substitutes and banks find it optimal to hold buffers of liquid assets even in
the absence of regulation. Banks facing higher costs of outside financing have lower debt
ratios, pay less dividends, hold more liquid reserves, and default earlier.
Using this model, we show that, holding capital structure fixed, liquidity requirements
constraining banks to hold a minimum amount of liquid reserves increase default risk. We
also show that liquidity requirements generally increase the collateral value of bank assets,
leading to a decrease in bank losses in default and to an increase in the collateral available
to creditors. This increase in collateral in turn reduces the cost of market debt to the bank
and feeds back in the policy choices of the bank, providing additional incentives for bank
shareholders to choose high debt levels. To offset this effect and harness the benefits from
the potential decrease in bank losses in default, the regulator may thus need to combine
liquidity requirements with leverage constraints. Our analysis indeed shows that leverage
requirements generally increase shareholders’ willingness to absorb losses and reduce default
risk. Ex-ante, however, leverage requirements may significantly reduce total bank value by
increasing its cost of capital. Lastly, we show that combining liquidity requirements with
leverage requirements drastically reduces both the likelihood of default and bank losses in
default. For example, we find that imposing a maximum ratio of debt to total assets of
90% together with a liquidity requirement of 10% of deposits reduces the one-year default
probability by 47% while reducing bank losses in default by 14%.
42
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