Source: https://de.scribd.com/document/306728328/Bauer-and-Ryser-2004
Timestamp: 2019-11-18 06:50:35
Document Index: 54669898

Matched Legal Cases: ['art 1', 'arts 2', 'art 3', 'art 3', 'art 1', 'arts 2', 'art 2']

Bauer and Ryser, 2004 | Bank Run | Capital Structure
speichernBauer and Ryser, 2004 für später speichern
union bank fdic order
Project Report Swati
Determinants of Bank Profitability in Ukraine
Journal of Banking & Finance 28 (2004) 331352
, Marc Ryser
Swiss Banking Institute, University of Zurich, Plattenstr. 14, Zurich 8032, Switzerland
RiskLab, ETH Zurich, Zurich, Switzerland
ECOFIN Research and Consulting, Zurich, Switzerland
Accepted 18 November 2002
We analyze optimal risk management strategies of a bank nanced with deposits and equity
in a one period model. The banks motivation for risk management comes from deposits which
can lead to bank runs. In the event of such a run, liquidation costs arise. The hedging strategy
that maximizes the value of equity is derived. We identify conditions under which well known
results such as complete hedging, maximal speculation or irrelevance of the hedging decision
are obtained. The initial debt ratio, the size of the liquidation costs, regulatory restrictions, the
volatility of the risky asset and the spread between the riskless interest rate and the deposit rate
are shown to be the important parameters that drive the banks hedging decision. We further
extend this basic model to include counterparty risk constraints on the forward contract used
for hedging.
 2003 Elsevier B.V. All rights reserved.
JEL classication: G1; G21; G28
Keywords: Bank; Bank risk management; Corporate hedging
The focus of this paper is to study the rationale for banks risk management strategies where risk management is dened as set of hedging strategies to alter the probability distribution of the future value of the banks assets.
Corresponding author. Tel.: +41-1-634-27-22; fax: +41-1-634-49-03.
E-mail addresses: wobauer@isb.unizh.ch (W. Bauer), marc.ryser@econ.ch (M. Ryser).
URL: http://www.math.ethz.ch/~bauer.
0378-4266/$ - see front matter  2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.jbankn.2002.11.001
W. Bauer, M. Ryser / Journal of Banking & Finance 28 (2004) 331352
There is a broad literature on these decisions for rms in general, beginning with
Modigliani and Miller (1959): Their famous theorem states that in a world of perfect
and complete markets, nancial decisions are irrelevant as they do not alter the value
of the shareholders stake in the rm. The only way to increase shareholders wealth
is to increase value of the rms assets. Neither the capital structure nor the risk management decisions have an impact on shareholders wealth.
Some important deviations from the perfect capital markets in the Modigliani
Miller setting have been identied, giving motivations for rms to care about risk
management, such as taxes, bankruptcy costs, agency costs and others (Froot
et al., 1993; Froot and Stein, 1998; Smith and Stulz, 1985; DeMarzo and Due,
1995; Stulz, 1996; Shapiro and Titman, 1986). When these reasons for risk management are incorporated into the rms objective function, one nds the following
basic result: When all risks are perfectly tradeable the rm maximizes shareholder
value by hedging completely (Froot and Stein, 1998; Broll and Jaenicke, 2000; Mozumdar, 2001). 1
However, the ModiglianiMiller-theorem as well as the aforementioned hedging
motives are ex ante propositions: Once debt is in place, ex post nancial decisions
can alter the equity value by expropriating debt holders. This strategy is known as
asset substitution (Jensen and Meckling, 1976). Because of limited liability, the position of equity holders can be considered as a call option on the rm value (Black and
Scholes, 1973). This implies that taking on as much risk as possible is the optimal ex
post risk management strategy. In summary, theory is inconclusive regarding the
question of the optimal hedging strategy of rms.
Turning to the question of optimal hedging and capital structure decisions of
banks, a rst nding is that the analysis within the neoclassical context of the ModiglianiMiller-theorem would be logically inconsistent. Banks are redundant institutions in this case and would simply not exist (Freixas and Rochet, 1998, p. 8). The
keys to the understanding of the role of banks and their nancial decisions are transaction costs and asymmetric information. These features have been dealt with extensively in the banking literature, departing from the neoclassical framework
(Baltensperger and Milde, 1987; Freixas and Rochet, 1998; Merton, 1995; Schrand
and Unal, 1998; Bhattacharya and Thakor, 1993; Diamond, 1984, 1996; Kashyap
et al., 2002; Allen and Santomero, 1998, 2001):
Banks have illiquid or even nontradeable long term assets because of the transformation services they provide.
Part of the illiquidity of banks assets can be explained by their information sensitivity; banks can have comparative informational advantages due to their role as
delegated monitors. Examples include information about bankruptcy probabilities and recovery rates in their credit portfolio. This proprietary information
This result is a consequence of the payo-functions concavity induced by the risk management
motives and the application of Jensens inequality.
can be further improved through long term relationships with creditors (Boot,
2000; Diamond and Rajan, 2000).
In contrast to other rms, banks liabilities are not only a source of nancing but
rather an essential part of their business: Depositors pay implicit or explicit fees
for deposit-related services (i.e. liquidity insurance, payment services, storage).
The leverage in banks balance sheets is thus many times higher.
Bank deposits can be withdrawn at any time. The sequential service constraint on
these contracts and uncertainty about the banks ability to repay can lead to a
bank run situation: All depositors rush to the bank at the same time to withdraw their money, trying to avoid being the last one in the waiting queue. This
threat of bank runs creates an inherent instability for the banks business (Diamond and Dybvig, 1983; Jacklin and Bhattacharya, 1988).
These characteristics highlight the major dierences between banks and other
rms: Banks, in contrast to other corporations, are nanced by deposits. Their ongoing operating value would be lost to a large extent in case of bankruptcy; depositors
can immediately call their claims and run whereas illiquid and information sensitive
assets have to be liquidated by re sales at signicant costs (Diamond and Rajan
(2000, 2001); Shrieves and Dahl (1992); the size of bankruptcy costs of banks was
estimated in James (1991)). However, these features of a bank are ignored by most
of the literature on capital structure and hedging decisions, which usually deals with
nonnancial rms.
In a recent contribution, Froot and Stein (1998) developed a framework to analyze a banks optimal capital allocation, capital budgeting and risk management
decisions. Their motivation for the bank to care about risk management stems from
convex costs of external nancing for a follow-up investment opportunity. This induces the banks objective function to be concave (the authors call this internal risk
aversion): The more dicult it is for the bank to raise external funds, the more risk
averse it behaves. A publicly traded bank in an ecient and complete market does
not reduce shareholder value by sacricing return for a reduction in risk. Thus, risk
reduction is always desirable for the risk-averse bank in the Froot and Stein (1998)setting. Hence, the resulting optimal strategy is to hedge completely. However, the
authors omit the equitys feature of limited liability and the corresponding agency
problems between shareholders and debtholders. Furthermore, since in their model,
there is no depository debt and thus no bank run possibility, potential eects of defaults on capital structure and risk management decisions are ignored.
In this paper, we model the hedging decision of a bank with the aforementioned
characteristics. We assume the capital budgeting decision to be xed. In a one-period-two-states-model, the bank has a given amount of depository debt. The deposit
rate contains a discount due to deposit-related services. The present value of this discount constitutes the banks franchise value. On the other hand, bank runs can force
the bank to sell all of its assets at once, incurring signicant liquidation costs. This
creates an incentive for not having extraordinary high levels of depository debt. Further, we assume that the bank is restricted in its risk taking behavior by a regulator. We also incorporate limited liability for equity. We assume that the banks
management acts in the shareholders interest and maximizes the present value of the
equity. It faces thus conicting incentives for risk management: Regulatory restrictions and liquidation costs in case of bank runs limit the risk taking on one hand.
On the other hand, limited liability creates incentives for risk taking. This setting allows us to identify situations in which well known results from the corporate nance
literature are found: We show that for some banks, it is optimal to hedge completely
as in Froot and Stein (1998). Other banks will take on as much risk as possible to augment shareholder value by expropriating wealth from depositors, a strategy known as
asset substitution (Jensen and Meckling, 1976). For still other banks, the risk management decision is shown to be irrelevant as in Modigliani and Miller (1959).
The remainder of this paper is organized as follows. In Section 2, we present the
model, discuss the banks objective function and derive the optimal hedging strategy.
In Section 3, we discuss the impact of forward counterparty restrictions on the hedging positions of the bank: Since depositors have absolute priority because of their
possibility to withdraw at any time, the forward counterparty can face additional default risk. It may therefore limit its contract size with the bank. Section 4 concludes
the analysis and gives an outlook on further research possibilities.
2. The general model
Let a probability space (X; F; P) be given, where we dene X : fU; Dg,
F : f;; fUg; fDg; Xg and PU p. The model has one period, between time
t 1 and t 2 and T  f1; 2g denotes the set of time indices.
The market consists of two assets: A riskless asset has at time t 1 a value normalized to 1, B1 1, and B2 B1 R at time t 2 where R > 1 is xed and given; further, a risky asset with value P1 > 0 at time t 1 and a value P2 x at time t 2
Pu  P1 u; x U;
Pd  P1 d; x D;
u > R > d:
For hedging purposes, we further introduce a redundant forward contract on the
risky asset: It is entered at time t 1 at no cost and the buyer of the contract has to
buy one unit of the risky asset at time t 2 at the forward price RP1 . Hence, the
value ft of the forward contract is
f1 0;
fu  Pu  RP1 ;
fd  Pd  RP1 ;
Since we have two assets with linearly independent payos and two states of the
world, the market is complete. We dene the unique risk neutral probability Q by
QU : q such that EQ BP22
P1 , where q Rd
ud
2.2. The bank
To derive the banks objective function, we make the following two assumptions
that deal with agency problems: We explicitly exclude agency problems between
shareholders and bank managers as their decision-taking agents. However, since
banks empirically have very high debt levels, we take asset substitution as an agency
problem between shareholders and depositors into account. Therefore, the problem
of choosing risk after the choice of the initial capital structure is especially pronounced (Leland, 1998):
Assumption 1. Managements compensation is structured to align the managers
interests with those of the shareholders. Therefore the rms objective is to maximize
the value of equity.
This objective is based on the completeness of the nancial market. It is therefore
possible to achieve any distribution of wealth across states. The Fisher separation
theorem then states the following: All utility maximizing shareholders agree on the
maximization of rm value as the appropriate objective function for the rm, notwithstanding the diering preferences and endowments (Eichberger and Harper,
1997, p. 150). However, as Jensen and Meckling (1976) pointed out, shareholders
in levered rms can do better behaving strategically. They will prefer investment
or hedging policies that maximize the value of only their claim, if they are not forced
to a precommitment on the investment and hedging strategy.
Assumption 2. When setting its capital structure, the bank cannot precontract or
precommit its hedging strategy. It will choose the hedging strategy ex post, after
deposits have been raised.
At time t 1, the bank has a loan portfolio, which has the same dynamics as the
risky asset. Its value at t 1 equals aP1 , we will thus say it has a prior position of
a > 0 units of the risky asset. 2 The bank has two sources of capital: Depository debt
and equity where the latter has limited liability. The initial amount of depository
debt D1 is given. While it would also be interesting to analyze the banks capital
structure decision, we limit our analysis to the hedging policy, assuming that the
bank has already set its target capital structure.
Through their monitoring activity, banks may be able to generate additional rents on the asset side as
well. These proprietary assets are however often not tradeable. In this situation, the market is incomplete.
This incompleteness creates problems for the determination of a unique objective function for the bank
and we leave the analysis of the case with nontradeable proprietary assets for further research.
2.3. The deposits and the run-threat to equity
In most papers dealing with the capital structure of rms in general, the taxadvantage of debt is a main incentive for rms to carry debt. For banks, however,
there is a more important motivation for carrying depository debt. Depository debt
in banks can be regarded as a real production element (Bhattacharya and Thakor,
1993). Due to deposit-related services (liquidity provision, payment services), the deposit rate will be lower than the rate that fully reects the risk. We assume that the
bank gets a discount of s > 0 on the deposit rate, resulting in
D 2 D 1 RD ;
where RD > 1 is the deposit rate net of the discount that the bank receives. We call
the net present value of these discounts from future periods the franchise value of the
deposits, denoted by FVt , t 2 T :
FV2 0;
where sD
EQ Bs12
Because of its signicant inuence on the banks equity payo, we should highlight another important dierence between bank deposits and traded debt: Asset-return shocks aect market prices of traded debt equally over all debt holders, whereas
the nominal amount of deposits can be withdrawn at any time. However, as
Diamond and Dybvig (1983) pointed out, the sequential service constraint on these
xed-commitment contracts along with sudden shocks in the liquidity needs of
depositors can lead to a situation in which all depositors withdraw their money at
the same time. This is because the amount received by a individual depositor solely
depends on his relative position in the waiting queue. Such a bank run can happen as
a sunspot phenomenon, whenever there is a liquidity shock and even in the absence of risky bank assets. 3 When uncertain asset returns are introduced into the
analysis, there is another reason why bank runs can occur: Whenever the value of
the banks assets is not sucient to repay every depositors full claim, all fully informed rational depositors would run to the bank at the same time and cause a
so-called information based bank run (Jacklin and Bhattacharya, 1988).
Let us assume that there are n depositors with equal amounts of D2 =n of deposits.
We denote by VL the critical asset value below which there will be a bank run. Without liquidation costs, we nd that VL D2 . Indeed, whenever the value V2 of the
banks assets at time t 2 exceeds the nominal deposits D2 , all depositors will receive
their nominal claim. But as soon as the value V2 of the banks assets falls below the
The right to withdraw at any time is an essential prerequisite for the eciency of the deposit contract.
In accordance with Diamond and Rajan (2000), we therefore exclude the possibility of suspension of
convertibility for the bank in our model; the bank cannot deny redemption of deposits as long as there are
any assets left.
Fig. 1. Payos to a depositor in absence of liquidation costs.
nominal value D2 of the deposits, not all depositors can withdraw their full nominal
amount anymore. In the latter case, each depositor faces the problem of choosing
between two compound lotteries: By running, he chooses the lottery LR with payos
depicted in Fig. 1. By not running, he chooses the lottery LNR with payos also depicted in Fig. 1:
When there is a bank run, the rst V2 =D2 percent of the depositors in the waiting
queue receive their full nominal deposit D2 =n. Thus, if the individual depositor
runs, the likelihood of arriving early at the queue (denoted early) is V2 =D2 .
The payo in this case is Dn2 . If he joins the queue in a later position (denoted
late), his payo is 0. When there is no bank run, the individual depositor is
the only to run and he receives his nominal deposit D2 =n or all of the assets
When the individual does not run he either receives 0 if there is a bank run or Vn2 if
there is no bank run because in this case the value of the remaining assets is distributed equally among the depositors.
For V2 < D2 , the payos of the run-strategy LR are higher or equal to those of the
no-run-strategy in all states of the world. Equivalently, the distribution of the runstrategy LR rst-order stochastically dominates the distribution of the no-run-strategy LNR . Hence, every expected utility maximizing depositor with positive marginal
utility will prefer the run-strategy LR (see e.g. Mas-Colell et al., 1995). This leads to
an equilibrium situation which is called information-based bank run.
In run situations, re sales of assets necessary to pay out the depositors may create
signicant liquidation costs (indirect bankruptcy costs) on the other hand (Diamond
and Rajan, 2001): Asset market prices can drastically decline if big blocks of assets
have to be sold immediately. If the bank has to sell all of its assets at once during a
run, we assume that there are liquidation costs of cV2 , 0 < c < 1. The fraction c of
rm value lost in case of bank runs creates a major incentive for the bank to hedge
its risk: Averaging 30% of the banks assets, these losses are substantial in bank failures as James (1991) found in his empirical work.
Since there is always a possibility of sunspot-bank runs due to unexpected
liquidity shocks, the individual depositor is uncertain whether there will be a bank
Fig. 2. Payos to a depositor in the presence of liquidation costs.
run at time t 2 (Diamond and Dybvig, 1983). 4 Because of this uncertainty and the
liquidation costs cV2 , VL , the value of the assets below which an information based
bank run will be triggered, shifts to
1c
for D2 < V2 < 1c
we now have the payos given in Fig. 2. They resemble the payos
shown in Fig. 1 without liquidation costs, but now total value of the assets is reduced
to 1  cV2 instead of V2 in situations of bank runs:
When there is a bank run, the rst V2 1  c=D2 percent of the depositors in the
waiting queue receive their full nominal deposit D2 =n. If the individual depositor
runs, the likelihood of arriving early is V2 1  c=D2 and the payo in that case is
. If he joins the queue in a later position (denoted late), his payo is 0. When
there is no bank run, the individual depositor is the only one to run and he receives his nominal deposit D2 =n or all of the assets remaining.
When the individual does not run, he receives 0 if there is a bank run. Otherwise
he either receives his full nominal amount Dn2 (if D2 6 V2 6 VL ), or a fraction Vn2 of the
remaining assets, which are distributed equally among the depositors.
Again, for V2 < 1c
, the distribution of the run-strategy LR rst-order stochastically dominates the distribution of the no-run-strategy LNR , causing an information-based bank run equilibrium.
Without this bank run-threat, the payo function for the banks equity at time
t 2 would be
V2  D2 ; V2 P D2 ;
SV2 ; D2 
0 6 V2 < D 2 :
We assume that the bank can only raise deposits of the size D1 6 1  caP1 such that bank runs at
time t 1 are excluded. This condition can equivalently be written as aP
6 1  c and interpreted in the
following way: Banks can only raise deposits up to the point where the debt ratio equals the recovery rate
in case of a run.
Fig. 3. Payo function of equity.
This is the payo of an ordinary call option on the rm value with strike D2 . However, in the presence of liquidation costs, a bank run will always take place if V2 < VL .
Thus the residual payo to shareholders drops to zero below VL . Since VL > D2 , the
equity payo changes to
V2  D2 ; V2 P VL ;
0 6 V2 < VL ;
2.4. The optimization problem
At time t 1, the bank chooses a hedging position consisting of h units of the forward contract on the risky asset. As a function of the chosen hedging position h, the
value of the banks assets at time t 1 hence is
V1 h aP1 hf1 FV1 ;
h 2 R;
and the value of the banks assets in state U and D respectively at time t 2 for a
given hedging position h is denoted by
Vu h  aPu hfu ;
Vd h  aPd hfd ;
h 2 R:
To study the impact of regulatory or other restrictions on the risk management,
we introduce lower and upper bounds 5 on the hedging position h,
Liquidation costs cV2 are expressed as a fraction of the nal rm value V2 . Thus, by introducing the
following (merely technical) restriction on the banks hedging decision, admitting only hedging strategies
for which the rm value is always positive, we guarantee nonnegative liquidation costs. This amounts to
the restriction h 2 Z u ; Z d
where Z u  a uR
and Z d  a dR
. Indeed, these constants follow from
solving the inequalities Vu h P 0 and Vd h P 0 using the denitions (5) and (6) of Vu h and Vd h.
V u h P 0 holds for h 2 Z u ; 1 and V d h P 0 holds for h 2 1; Z d
, thus nonnegative rm value is the
outcome for hedging strategies in 1; Z d
\ Z u ; 1. It follows from the denition of Z u that Z u < a
thus a as the lower bound of the set of feasible hedging positions guarantees nonnegative rm value in
state U. Throughout the following we assume that the upper bound is more restrictive than Z d , a1 < Z d .
a 6 h 6 a1 :
The lower bound a is equivalent to a no net-shortsales-constraint.
The payo S to shareholders at liquidation at time t 2 is a function of rm value
and deposits, SV2 ; D2 . Since the nancial market is arbitrage-free and complete, the
present value of equity at time t 1 for a given future value V2 of the assets equals
Q SV2 ; D2
The banks managements goal is to maximize the present value of equity at time
t 1, by choosing a hedging position h. Let
Q SV2 h; D2
Ih  E
denote the objective function. Ih is the value of equity at time t 1 as a function of
the hedging portfolio h. Then, the banks optimization problem at time t 1 is
max Ih max E
a 6 h 6 a1
Q SaP2 hf2 ; D2
2.5. Optimal hedging strategy
The present value of equity as a function of the hedging position h, Ih, has the
following form, depending on whether for a given hedging position h there will be a
positive payo to shareholders in both states U and D or only in state U:
1. For hedging positions h such that the assets total value exceeds the bank run trigger VL in both states U and D, Vu h > VL and Vd h > VL , amounting to a positive
payo to shareholders in both states (we call these hedging portfolios portfolios of
type 1), the objective function is
1q
aPu hPu  RP1  D2
aPd hPd  RP1  D2
2. For hedging positions h such that the assets total value exceeds only in state U
the bank run trigger VL , Vu h > VL , but is smaller than VL in state D, Vd h < VL
(that is, there is a bank run in state D and shareholders receive only in state U
a positive payo; we call these hedging positions portfolios of type 2), the objective function is
Ih aPu hPu  RP1  D2
The net position in the risky asset is restricted to be nonnegative and thus, the cases where
Vu h < Vd h can be omitted.
Thus, in order to solve the optimization problem, we need to nd conditions
which guarantee the existence of hedging positions h of either type 1 or type 2 for
a given market and bank structure.
Lemma 1. The payoff to shareholders is positive in state U for hedging positions h
VL  aPu
: K u :
Pu  RP1
The payoff to shareholders is positive in state D for hedging positions h
VL  aPd
: K d :
Pd  RP1
K u is the minimal hedging position for which shareholders receive a positive payo in state U. K d is the maximal hedging position for which shareholders receive a
positive payo in state D. Hence, the portfolios of type 1 are in the set K u ; K d
Therefore, the relationship among the terms K u and K d will determine whether this
set is empty and whether there are hedging positions of type 1 or only of type 2.
Corollary 1. Hedging positions h of type 1 exist if
VL 6 aP1 R : V :
V is the value of the assets of the fully hedged bank at time t 2, i.e. the value
attained if the bank sells forward its whole position a in the risky asset and the future
value becomes certain, Vu a Vd a aP1 R V . Thus, if the bank run trigger
VL is smaller or equal to the forward price V of the banks prior position, there exist
hedging positions for which shareholders receive a positive payo in both states U
and D. It is obvious that the fully hedged position a would be such a position.
And if the payo to shareholders with this forward position is strictly positive, there
will be other forward positions close to a which also yield a positive payo to
shareholders. Otherwise, if the forward price V of the banks prior position is smaller
than the bank run trigger VL then there are only hedging positions of type 2. Shareholders then receive a positive payo only in state U and a zero payo in state D,
Vu h P VL and Vd h < VL .
The shape of the objective function is further claried by the following
Lemma 2. If VL P V , then the inequality
K d 6 a 6 K u
holds. Otherwise, if VL 6 V , then the inequality
K u 6 a 6 K d
The intuition for (13) and (14) is as follows:
If VL > V , then the fully hedged position a, leading to a value of V in both states
of the world, results in a payo of zero to shareholders, since the rm value is in
this case below the bank run trigger VL . Thus, the minimal (maximal) hedging position at which shareholders receive a positive payo in state U (D), i.e. K u K d ,
would be higher (smaller) than a. This corresponds to (13).
The converse holds if V P VL . Then the fully hedged position a yields a positive
payo to shareholders. Then the minimal (maximal) hedging position at which
shareholders receive a positive payo in state U (D), i.e. K u (K d ), would be smaller
(higher) than a. This corresponds to (14).
Fig. 4 displays the objective function Ih with the no net short sales restriction in
the three cases where VL > V (Fig. 4(a)), VL < V (Fig. 4(b)) and VL V (Fig. 4(c)).
The bold line is the feasible part of the objective function.
Fig. 4. Three types of objective functions with no net short sales restrictions.
1. If the forward price of the banks prior position is less than the bank run trigger,
aP1 R < VL , and if there is a positive payoff to shareholders in state U at the maximal
admissible hedging position, a1 P K u , then the optimal hedging position is
h a1 :
2. Otherwise, if aP1 R > VL , we find the following optimal hedging strategies:
(a) If a1 P K d and a1 > Ju , then h a1 , where Ju is defined by (15) below.
(b) If Ju > a1 > K d , then a 6 h 6 K d .
(c) If a1 < K d , then a 6 h 6 a1 .
3. If aP1 R VL , we find the following optimal hedging strategies:
(a) If a1 < Ju , then h a.
(b) If a1 > Ju , then h a1 .
Ju can be characterized as follows (see Fig. 4(b)): The position h Ju belongs to
the set of hedging positions for which the objective function is increasing; further, Ju
is the position for which the value function equals the value that is attained on the set
where the objective function is constant,
aP1 
Ju 
aPu  D2
P1 u  R
The three optimal hedging decisions in Proposition 1 have the following economic
h a1 is the strategy of maximal speculation.
h a is the case of complete hedging where the bank sells forward its whole initial position.
In the case where K u < K d , the bank is indierent between the hedging strategies
in the range K u 6 h 6 K d .
Part 1 of Proposition 1 covers the case in which the payo to shareholders would
be zero if the bank hedged completely. It is the case in which the forward price of the
prior position is less than the bank run trigger, V < VL . If a positive payo to shareholders in state U is attainable by taking on more risk, a1 P K u , we have a gamble
for resurrection-situation: It is always optimal to take as much risk as possible,
h a1 . The condition VL > V can equivalently be written as
D1 =aP1 > R=RD 1  c;
which says that the initial debt ratio is higher than the recovery rate (in case of a run)
multiplied by the spread between the deposit rate and the riskless interest rate.
Hence, for banks with high initial debt ratio and/or high liquidation costs, it is always optimal to gamble for resurrection.
Parts 2a to 2c of Proposition 1 cover the cases in which shareholders would still
receive a positive payo if the bank hedged completely, VL < V , resp. D1 =aP1 <
R=RD 1  c.
In 2a, the maximal admissible hedging position a1 yields a higher expected payo
than the fully hedged position a. Since shareholders could lock in a sure positive payo by hedging completely, this is not a gamble for resurrection,
although the optimal hedging strategy is the same. Due to equitys nonlinear payo, they can expropriate wealth from depositors by taking on more risk: The increase of the payo in state U overcompensates the liquidation costs in state D.
This strategy is known as asset substitution (Jensen and Meckling, 1976). Hence,
in the current model, banks in this situation have loose regulatory or other restrictions (a large a1 ). They can take that much risk that the bank run threat does not
have a disciplinary eect anymore.
In 2b, the risk management restriction is so constraining, that at the maximal
admissible position a1 , the gain in expected return does not outweigh the expected
liquidation costs of this portfolio. The expected payo to shareholders for this
hedging position is smaller than for the fully hedged position a. However, there
is no unique optimal hedging strategy: Shareholders are indierent with respect to
the hedging strategies in the whole range between a and K d . If the initial debt
ratio aP
is higher than RdD 1  c, then K d < 0 and the optimal hedging strategy
is risk reducing, h < 0. Risk reducing banks in this case are those with a high initial debt ratio, high asset volatility and/or high liquidation costs.
In 2c, the maximal admissible hedging position a1 belongs to the portfolios for
which shareholders receive a positive payment in both states U and D. The expected payo is the same as the one of the fully hedged position a. In this case,
the ModiglianiMiller-result of hedging-irrelevance also holds ex post, after the
determination of the capital structure: Shareholders are indierent with respect
to all admissible hedging strategies. Banks in this case are, however, forced towards a safe behavior: The risk management restrictions prevent asset substitution since they guarantee that the value of banks assets can never fall below
the bank run trigger.
In part 3a, the fully hedged position h a is optimal. Any risk taken by the
bank induces liquidation costs. But the expected return cannot be increased suciently such that the shareholders would receive a higher expected payment at least
in one state since Ju > a1 > K d . Banks in this situation do not have any risk tolerance. They cannot improve the shareholders position by asset substitution. In our
model, only for this special situation, the Froot and Stein (1998)-result of complete
hedging is derived as the unique optimal hedging strategy.
In part 3b, the regulatory constraint a1 is loose enough to allow the bank to take
on enough risk such that the expected return again outweighs the expected liquidation costs.
Overall, for a regulatory restricted bank nanced with deposits that is subject to
liquidation costs in the event of bank runs, the common interpretation of equity as a
call option does not necessarily apply: Equity value is not always increased by an increase in asset-risk. Further, higher liquidation costs lead to an increase of the bank
run trigger. This creates larger downside risk for shareholders that cannot always be
outweighed by a higher expected return, because regulatory restrictions place an
upper limit on risk taking.
On the other hand, depending on how much risk taking regulatory or other
restrictions allow, hedging completely as in Froot and Stein (1998) is almost never
the unique optimal hedging strategy: Over a wide range, all hedging positions can
be equally optimal. Risk shifting to depositors is optimal as long as the higher expected return outweighs the possible downside loss. If risk management restrictions
are set to prevent asset substitution, the value of the banks assets cannot fall below
the bankrun trigger. The result then coincides with the ModiglianiMiller-result of
hedging irrelevance.
3. Impact of counter party risk constraints
We extend the analysis of the previous section by introducing counter party
restrictions on the attainable forward contract size used for hedging. The forward
price RP1 is set such that expected prot from the forward contract is zero under
the risk neutral probability measure Q. Yet, if the bank can default on the forward
contract, the counter party will demand a higher forward price to get compensated
for the additional risk. If we leave the forward price xed, the bank will not be able
to enter every desired forward position any more. The counter party restricts the
hedging decision by oering only forward contracts for which the probability of default does not exceed some threshold. In the current binomial setting, statements on
probabilities correspond to conditions on states U and D:
Zero probability of default is equivalent to no default in both states U and D.
If the probability of default can be positive, then the bank is not allowed to
default either in state U or in state D.
Proposition 2. The bank will not default on the forward contract in state U for contracts of size h P K u . Further, the bank will not default on the forward contract in state
D for contracts of size h 6 K d . Under the requirement that the bank should not default
in any state of the world on its obligations from the forward contract, it will not be able
to enter a forward contract unless K u 6 K d . It will only be offered contracts h such that
Ku 6 h 6 Kd .
The question when the bank is oered both long and short or only long or only
short positions is answered by the following
Lemma 3. If the bank is not allowed to default in state U (state D), it will be offered
short (long) positions if and only if VL < aP1 u VL < aP1 d.
Hence, the restriction not to default in state D may prevent the bank to enter long
positions, namely if D1 =aP1 > 1  cd=RD . These banks either have a high debt
ratio, high liquidation costs and/or a high asset-volatility. They would face a bank
run in state D without hedging and the costs would be borne by the counter party.
On the other hand, the restriction not to default in state U may prevent the bank to
enter short positions if D1 =aP1 > 1  cu=RD . The debt ratio, the liquidation
costs and/or the asset-volatility of these banks is that high that they would face a
bank run already in the good state of the world U and the counter party enforces
the asset substitution in this case. The most important type of restriction 7 is the one
which does not allow default in any state. In the case where VL > V , 8 the bank cannot enter a forward contract. With its combination of deposits, initial position and
liquidation costs, it will not be oered forward contracts due to default risk. Thus,
the gamble for resurrection is not possible any more. When VL 6 V , the bank is prevented from taking on any risk which would trigger a bank-run. It can only enter positions in the forward in the range a 6 h 6 K d . Thus, the bank will always have the
possibility to reduce risk by entering short positions. In the subcase where
VL > aP1 d, it will not be able to obtain long positions (Lemma 3). That is, when the
banks prior position is sucient to prevent a bank run only in state U, but not in
state D, the bank will only be oered contracts that reduce the risk suciently to ensure that there will be no bank run in state D. Without hedging, the bank would face a
bank run in state D. But with the positive cash ow Pd  RP1 from the short position in the forward contract in state D, the banks assets are sucient to prevent a run
in state D. The following Lemma tells when the bank will choose to hedge.
Lemma 4. In the case where VL 6 V and aP1 d < VL , the bank will choose to hedge if
D2 6 aP1 d < VL ; if aP1 d < D2 , then it is optimal for the bank not to hedge.
The reason for this hedging-strategy is the following: By hedging when D2 6
aP1 d < VL , the bank can preserve asset value in the down state D, that otherwise
would be completely lost for the shareholders as liquidation costs. On the other hand,
if aP1 d < D2 , all the remaining asset value up to D2 goes to the depositors anyway. If
the bank hedges, it thus sacrices some payo to shareholders in state U in exchange
The constraint that the bank should not default in state U but is allowed to default in state D is only
meaningful if the risk neutral probability of state D is very low. The forward contract price is then
approximately not aected by the additional default risk. The following results then apply: If VL > V , the
bank can obtain only positions h P K u , since it would default in state U on all other positions. Therefore,
the bank can still follow a strategy of asset substitution by holding long positions in the forward contract.
If even VL P aP1 u, that is, if the bank faced a bank run without hedging in state U, it would only be oered
long positions to hedge and thus be forced to gamble for resurrection: For suciently large hedging
positions, the value of the banks asset is above the bank run trigger in state U (whereas in state D, the
bank will default on its obligation from the forward contract). In the case where VL 6 V , the constraint that
the bank is not allowed to default in state U is not binding: It will be oered any contracts of size h P K u
but the lower bound on its hedging position h already is a where a 6 K u ((14) in Lemma 2).
VL > V implies K d < K u , and if follows from Proposition 2 that the bank will not be oered forward
for securing payos for depositors in state D. The bank can do better for the shareholders by not hedging at all, that is, by keeping the higher expected return of the unhedged position while letting the depositors bear the downside loss in state D.
Overall, the introduction of counterparty-restrictions mitigates risk taking incentives for a bank, since it is not possible to gamble for resurrection anymore.
We have presented a one-period model in which we analyze the banks risk management decision. The bank is regulatory restricted, nanced by deposits and is subject to liquidation costs in the event of a bank run.
We nd that the common interpretation of equity as an ordinary call option does
not apply: Equity value is not always increased by increasing the assets volatility,
since this also raises the likelihood of a bank run. Whenever the expected costs of
such a run for shareholders cannot be outweighed by an increase of the expected return (because regulatory restrictions limit the maximum achievable risk), it is not
optimal to take as much risk as possible. In these cases, safe banks with low debt
ratios and asset volatility can still augment their risk exposure to the point where
downside loss comes into play. However, for banks with a high debt ratio and a high
asset volatility, risk reduction is the optimal strategy.
This deterrence of asset substitution however vanishes in the absence of regulatory
constraints or with a complete deposit insurance (Calomiris and Kahn, 1991): Without the possible downside loss, the equity payo would be that of an ordinary call
option and it would always be optimal for the bank to take as much risk as possible.
Also, without regulatory restrictions, the possible downside loss could always be outweighed by higher expected return through higher risk-exposures.
restrictions allow, it may not be optimal for the bank to hedge completely as in Froot
and Stein (1998): Because equity features limited liability, risk shifting to depositors
is still preferred as long as the higher expected return outweighs the possible downside loss. The less restrictive regulatory restrictions are, the more relevant becomes
this strategy of asset substitution. Without any restrictions of regulators or counter
parties, asset substitution would always be the optimal strategy.
Further, there is one constellation for which the hedging decision is shown to be
irrelevant, which coincides with the result of the ModiglianiMiller-theorem. This,
however, is only a special situation, where the risk management restrictions, the size
of the liquidation costs in case of a bank run and the initial debt ratio are all set such
that risk shifting to depositors is impossible and no bank run takes place.
Among the open questions remains the analysis of the hedging decision in a multiperiod setting. Bauer and Ryser (2002) have looked at the eect that the banks
franchise value of deposits then has. It gives an incentive to reduce risk taking since
the whole stream of future income from deposit services would be lost in a run situation. Furthermore, it would be interesting to analyze the hedging decision in the
presence of a nontradeable proprietary bank asset that generates an extra rent as
in Diamond and Rajan (2000). The market completeness breaks down in this case
and the determination of a unique objective function for the bank is not trivial anymore.
Proof of Lemma 1. Using the denition (5) of Vu h we nd that for a given h
Vu h > VL () h >
K u;
similarly for a given h
Vd h > VL () h <
Proof of Corollary 1. From Lemma 1 we know that Vu h > VL for h 2 K u ; 1 and
Vd h > VL for h 2 1; K d
. Hence, hedging positions h of type 1 (that is, h for
which both Vu h > VL and Vd h > VL ) are h 2 K u ; K d
; this interval is not empty if
Ku 6 Kd :
Using the denitions (10) and (11) of K u and K d this can be written equivalently as
aPd
VL aPu
6 PVdLRP
. Solving for VL yields VL 6 aP1 R V . 
Pu RP1
Proof of Lemma 2. Consider rst the case VL > V . From the denition (12) follows
VL P V aP1 R. Subtracting aPu yields VL  aPu P  aPu  P1 R, dividing by
Pu  P1 R yields K u P  a. The inequalities for K d and for the case where VL 6 V
follow in the same way. h
The following lemma will be useful to prove Proposition 1.
Lemma 5. The sets of candidates for the optimal hedging strategy h are fa1 g, fag
and 1; K d
\ K u ; 1. The values of the objective function evaluated at these candidate points are
< B2 aPu a1 Pu  RP1  D2
; K 6 K 6 a1 or K 6 K < a1 ;
Ia1 aP1  B ;
K 6 a1 6 K ;
K d < a1 < K u ;
Ia
aP1  DB22 ;
Ku < Kd ;
K d < K u;
Ku 6 h 6 Kd ;
K d < h < K u:
Proof of Lemma 5. For convenient notation, we write the constraints (7) a 6 h 6 a1
Ah 6 a;
where A  A1 A2 0 1 1 0 and a  a1 a 0 .
We consider rst the case VL 6 V aP1 R. From VL < V aP1 R and (17) follows
that K u 6 K d . Hence, for any h 2 K u ; K d
holds that Vu h P VL and Vd h P VL . On
this interval, the Lagrangian is as follows:
Lh; l1 ; l2
lk Ak h  ak
a hP1 
hP1 D2 X
l Ak h  ak ;
B2 k1 k
yielding the rst-order conditions
P1 
l Ak 0;
B1 k1 k
Ak h  ak 0;
k 1; 2:
We have the following candidate points for an optimum:
1. l1 l2 0: (22) reduces to the condition P1  BP11 0 where the last equality is
due to B1 1. Thus, in this case any h such that Vu h P VL and Vd h P VL is optimal, that is, the set of candidates is K u ; K d
. From B1 1 follows Ih aP1  DB22 ,
2. If l1 6 0, l2 0 then h a1 . Then, if K u < K d < a1 Ia1 Bq2 aPu
a1 Pu  RP1  D2
since the payo to shareholders is zero in state D for h a1
due to the fact that K d < a1 . If K u 6 a1 6 K d
aPu a1 Pu  RP1  D2
aPd a1 Pd  RP1  D2
aP1  :
3. If l1 0, l2 6 0 then h a. Then, since K u 6 a 6 K d ,
aPu  aPu  RP1  D2
aPd  aPd  RP1  D2
Consider now the case VL > aP1 R when there are only h such that Vu h > VL and
Vd h < VL holds for xed h. The Lagrangian is then
lk Ak h  ak ;
lk Ak 0:
1. l1 l2 0 could hold only if PPu1 R which was excluded in (1).
6 0, l2 0 and h a1 . Then, if K d < K u 6 a1 , Ia1 Bq2 aPu a1 Pu  RP1 
since the payo to shareholders is zero in state D due to the fact that K d < a1 .
If K d < a1 < K u we have Ia1 0 since the payo to equity holders is both zero
in state D (from K d < a1 ) and in state U (from a1 < K u ).
3. l1 0, l2 6 0 and h a. Then, it follows that Ia 0 since it follows from
(13) that K d < a and hence at the position a shareholders receive a zero payo
in state D.
As for the last equality, it is obvious that for h > K d , the payo to shareholders is
zero in state D and for h < K u , the payo to shareholders is zero in state U, hence for
K d < h < K u follows Ih 0. h
Proof of Proposition 1. We consider rst part 1 of the proposition, that is, the case
where VL > aP1 R and a1 P K u . As this is the case when K d < K u , it follows from (20)
that the objective function equals zero for all h 2 K d ; K u including a (since, from
aPu
Lemma 2, K d < a < K u ). Denition (10) of K u yields for a1 P K u a1 P PVuLRP
aPu a1 Pu  RP1 P VL . From (18), Ia1 B2 aPu a1 Pu  RP1  D2
VL > D2 , hence, a1 is the optimum.
We now turn to parts 2 and 3 of the proposition; both cases are covered by the
inequality VL 6 aP1 R or equivalently K u 6 K d , 2 being the case of strict inequality
and 3 the case of equality. Consider rst 2a and 3b respectively. From (20) follows
that for h 2 K u ; K d
(and hence also for h a) Ih aP1  DB22 . From Ju < a1 follows aP1  DB22 < Bq2 aPu a1 Pu  RP1  D2
Ia1 , hence h a1 .
Similarly follows for 2b and 3a when Ju > a1 > K d that aP1  DB22 > Bq2 aPu a1 Pu 
RP1  D2
Ia1 , hence h 2 a; K d
In part 2c, feasible portfolios h are h 2 a; a1
 K u ; K d . Hence, from (20), all
feasible portfolios have the same value of the objective function, Ih aP1  DB22 ,
K u < h < K d , h 2 K u ; K d . h
Proof of Proposition 2. If the bank should not default in state U on the forward
contract, then the banks assets net of the value of debt in state U must be positive,
aPu hfu  D2 P 0. Solving for h yields the required inequality h P K u . In the case
where the bank should not default in state D on the forward contract, the banks
assets net of the value of debt in state D must be positive, aPd hfd  D2 P 0.
Solving for h yields again the required inequality h 6 K d . If the counter party of the
forward contract requires that the bank does not default in any state, then h needs to
be in the intersection of the intervals K u ; 1 and 1; K d
. If K d < K u , then this
intersection is empty, hence the bank will not be able to enter a forward contract.
If K d > K u , then the intersection is exactly K u ; K d
Proof of Lemma 3. By (1) Pu P1 u > RP1 holds and thus P1 u  RP1 > 0. Therefore
aP1 u
K u PV1LuRP
< 0 () VL  aP1 u < 0 () VL < aP1 u. If the bank is not allowed to
default in state U, it will be constrained, by Proposition 2, to hedging strategies
h P K u . Thus, short positions (i.e. h < 0) will only be available if K u < 0, hence the
statement follows. The inequality for K d follows by the same arguments. h
Proof of Lemma 4. In the case where VL 6 V and aP1 d < VL , it follows from Lemmas
2 and 3 that K u 6 K d < 0. Since VL 6 V , there is a positive payo to shareholders
without hedging and the value of the objective function for the decision not to hedge
is I0 Bq2 aP1 u  D2 . The bank is oered only hedging positions in the interval
K u ; K d
for which (by (20) of Lemma 5) the value of the objective function is aP1  DB22
I0 ()
aP1 u 1q
aP1 d  qD
 1qD
. For h 2 K u ; K d
, Ih >
> Bq2 aP1 u  D2 which is, due to the fact
that both 1  q > 0 and B2 > 0, equivalent to aP1 d P D2 . The inequality aP1 d < VL
follows from the fact that we look at the case where K d < 0 and Lemma 3. h
We wish to thank the RiskLab (Z
urich) for the nancial support of the rst
author and ECOFIN Research and Consulting for the nancial support of the second author. This work is part of the RiskLab project titled The Impact of Asymmetric Information on Banks Capital Allocation and Hedging Decisions. We gratefully
acknowledge helpful comments from Rajna Gibson, Rene Stulz, Giorgio P. Szeg
well as an anonymous referee.
Allen, F., Santomero, A.M., 1998. The theory of nancial intermediation. Journal of Banking & Finance
21, 14611485.
Allen, F., Santomero, A.M., 2001. What do nancial intermediaries do? Journal of Banking & Finance 25,
271294.
Baltensperger, E., Milde, H., 1987. Theorie des Bankverhaltens. Springer Verlag, Berlin.
Bauer, W., Ryser, M., 2002. Bank risk management and the franchise value of deposits. Working paper.
Bhattacharya, S., Thakor, A.V., 1993. Contemporary banking theory. Journal of Financial Intermediation
3, 250.
81, 637659.
Boot, A.W.A., 2000. Relationship banking: What do we know? Journal of Financial Intermediation 9, 7
Broll, U., Jaenicke, J., 2000. Bankrisiko, Zinsmargen und exibles Futures-Hedging. Schweizerische
ur Volkswirtschaft und Statistik 136 (2), 147160.
Calomiris, W.C., Kahn, C.M., 1991. The role of demandable debt in structuring optimal banking
arrangements. American Economic Review 81 (3), 497513.
DeMarzo, P., Due, D., 1995. Corporate incentives for hedging and hedge accounting. Review of
Financial Studies 8, 743772.
Diamond, D.W., 1984. Financial intermediation and delegated monitoring. Review of Economic Studies
51, 393414.
Diamond, D.W., 1996. Financial intermediation as delegated monitoring: A simple example. Federal
Reserve Bank of Richmond Economic Quarterly 82 (3), 5166.
Diamond, D.W., Dybvig, P.H., 1983. Bank runs, deposit insurance and liquidity. Journal of Political
Economy 91, 401419.
Diamond, D.W., Rajan, R.G., 2000. A theory of bank capital. Journal of Finance LV (6), 24312465.
Diamond, D.W., Rajan, R.G., 2001. Liquidity risk, liquidity creation and nancial fragility: A theory of
banking. Journal of Political Economy 109 (2), 287327.
Eichberger, J., Harper, I.R., 1997. Financial Economics. Oxford University Press, New York.
Freixas, X., Rochet, J.C., 1998. Microeconomics of Banking. MIT Press, Cambridge, MA.
Froot, K., Stein, J., 1998. Risk management, capital budgeting and capital structure policy for nancial
institutions: An integrated approach. Journal of Financial Economics 47, 5582.
Froot, K.A., Scharfstein, D.S., Stein, J.C., 1993. Risk management: Coordinating corporate investment
and nancing policies. Journal of Finance XLVIII (5), 16291658.
Jacklin, C.J., Bhattacharya, S., 1988. Distinguishing panics and information-based bank runs: Welfare
and policy implications. Journal of Political Economy 96 (3), 568592.
James, C., 1991. The losses realized in bank failures. Journal of Finance 46 (4), 12231242.
Jensen, M.C., Meckling, W., 1976. Theory of the rm: Managerial behavior, agency costs and ownership
structure. Journal of Financial Economics 3, 305360.
Kashyap, A.K., Rajan, R., Stein, J.C., 2002. Banks as liquidity providers: An explanation for the
coexistence of lending and deposit-taking. Journal of Finance 57 (1), 3373.
Leland, H.E., 1998. Agency costs, risk management and capital structure. Journal of Finance 53, 1213
Mas-Colell, A., Whinston, M.D., Green, J.R., 1995. Microeconomic Theory. Oxford University Press,
Merton, R.C., 1995. Financial innovation and the management and regulation of nancial institutions.
Journal of Banking and Finance 19, 461481.
Modigliani, F., Miller, M.H., 1959. The cost of capital, corporation nance, and the theory of investment.
American Economic Review 48, 261297.
Mozumdar, A., 2001. Corporate hedging and speculative incentives: Implications for swap market default
risk. Journal of Financial and Quantitative Analysis 36 (2), 221250.
Schrand, C., Unal, H., 1998. Hedging and coordinated risk management: Evidence from thrift
conversions. Journal of Finance LIII (3), 9791013.
Shapiro, A., Titman, S., 1986. An integrated approach to corporate risk management. In: Stern, J., Chew,
D. (Eds.), The Revolution in Corporate Finance. Basil Blackwell Inc., Cambridge, MA.
Shrieves, R.E., Dahl, D., 1992. The relationship between risk and capital in commercial banks. Journal of
Banking & Finance 16 (2), 439457.
Smith, C.W., Stulz, R.M., 1985. The determinants of rms hedging policies. Journal of Financial and
Quantitative Analysis 20 (4), 391405.
Stulz, R.M., 1996. Rethinking risk management. Journal of Applied Corporate Finance 9 (3), 824.
Dokumente ähnlich wie Bauer and Ryser, 2004
SwatiRay
fbhacka
Lovenish Atal
Handoo - All
2-RatioAnalysis
Structural and Market-related Factors Impacting Profitability. a Cross Sectional Study of Listed Companiesa
What is Corporate Finance4.9.docx
Bloomberg_Businessweek_USA_-_April_22_2019.pdf
FIN 400 Class Notes Set 1
Grow Your Small Savings to One Crore
FAT MCQs