The effect of capital requirements on incentive
compatibility constraints of banks.
Stephan Boxem, s1530089 August 29, 2012
Abstract
This thesis explores the effect of capital requirements on the moral hazard problem in the context of deposit insurance. Four capital re-quirement regimes are analyzed for their effect on the incentive compat-ibility constraints of banks which determine whether the bank invests in a ‘safe’ or ‘risky’ project.
Supervisor: Prof.dr. L.J.R. Scholtens University of Groningen Assessor: Dr. H. Gonenc University of Groningen JEL Codes: G21, G32
Introduction
An appealing service of financial intermediaries is liquidity transformation. This refers to the possibility of banks to transfer money from period to period and individual to individual. If an individual has more cash in his wallet than he requires for consumption in this period, he can put his excess money in a deposit account. This earns the individual a deposit interest rate. The bank uses the money in deposit accounts, to provide loans to individuals or firms who have an excess demand for money (liquidity). The bank earns the margin between the required interest on the loans and the deposit rate. However, the maturity of loans, which determines when the loan must be repaid, does not always match the liquidity demand of de-positors who brought up the liquidity available for investment. To prevent illiquidity, a situation where banks do not have sufficient money to provide individuals with withdrawals from their deposit account, the banks are re-quired to hold a share of all deposit holdings in a reserve account. This is called the fractional reserve system. A large proportion of deposits is available for providing loans but the bank must hold some fraction of the deposits uninvested and hence available for withdrawal. In this fractional reserve system, depositors have two basic uncertainties. The first is that too large a proportion of deposit holdings is ‘invested’ in loans, such that the available liquidity (the reserve account) is insufficient to provide depositors with the desired withdrawals. This is called illiquidity. In this scenario, the demand for withdrawals is too large and the required liquidity is not available in a timely fashion. The value of the assets (the loans) exceeds the value of the liabilities (the deposit accounts). Therefore, if the maturity of the loans is reached, the liquidity is sufficient to provide for the withdrawal of all deposit accounts. Illiquidity therefore refers to the unavailability of money caused by the mismatch between the maturity of loans and the de-mand of withdrawal. The second concern is that the liabilities of the bank (the deposit accounts) in fact exceed the assets of the bank (the loans). This problem of insolvency is observed when debtors cannot repay the loan. As the bank uses the deposit accounts to provide the loans, the money of the depositors is lost. When facing illiquidity, banks can liquidate their investments in order to provide more withdrawals. However, premature liq-uidation often involves a cost, such that illiquidity can eventually lead to insolvency, because the value of assets decreases when the asset is liquidated.
insur-ance, the government guarantees a certain amount of deposit holdings in case of insolvency. This not only reduces the loss in the case of insolvency, but also reduces the fear of insolvency, such that illiquidity is less likely to lead to insolvency. Banks use the deposit accounts of depositors to provide loans. When the loans are repaid, the interest payments accrue to the fi-nancial intermediary. When the debtors default, the assets of the bank are decreased. This increases the likelihood of insolvency, of which the losses are borne by the depositors who lose their deposit account. Or, if deposit insurance is in place, the losses are borne by tax payers who pay for the gov-ernment guarantee of the deposit accounts in case of a bank failure. This illustrates the limited-liability facing financial intermediaries, when making investment decisions. This refers to the notion that the financial liabilities in case of default do not extend to banks and their employees. If the bank faces insolvency, the bank employees are not hurt as they are not personally liable for the losses. In terms of economic theory, depositors face a moral hazard problem. Moral hazard can arise when the investment decision is made by someone else than the party who bears the costs. In the context of bank deposits, the bank chooses which loan is provided, by investing the deposit accounts of individuals. When the investments provide a return, this is yielded by the bank. In case of default on a loan, the deposit holders bear the costs. Moral hazard occurs when for instance, two investment op-portunities exist, and the depositors prefer one investment while the bank prefers the other investment. Although the depositors bear the costs in case of default, the bank determines for which investment a loan will be provided.
risky investment which has a negative expected payoff level but a higher payoff level given that the investment succeeds. The analysis of this the-sis shows whether capital requirements can induce banks to invest in the safe investment rather than the risky invest. Capital requirements refer to the obligation to hold a specific portion of capital uninvested. This capital, which is refered to as regulatory capital, must induce banks to reduce their risk-taking behaviour. The analysis considers the incentive compatibility constraints banks face while deciding which project to invest in. The hy-pothesis is that capital requirements can steer the incentives of banks such that they are inclined to provide a loan to the investment that is preferred by the depositors. First, I provide a theoretical framework in two parts. The first part establishes the ambiguous ability of capital requirements to reduce the risk-taking behaviour of banks. Blum (1998) and Hellman, Mur-dock and Stiglitz (2000) argue that capital requirements can induce banks to invest at a lower risk level by putting assets of the bank at stake. How-ever, as this raises the costs to acquire capital (costs of capital) available for investment, there is also a perverse effect of capital requirement such that risk-taking is increased. The second part introduces a framework on which the model of this thesis is based. The starting point is the seminal paper by Diamond & Dybvig (1983), which states that the existence of financial in-termediaries is desireable for depositors which are uncertain when they need their money for consumption. The article by Hazlett (1995) extends on the Diamond-Dybvig model by introducing a risk component. Additionally, I discuss two papers by Cooper & Ross (1997, 2002), which extend on the investment decision by banks and introduce capital requirements. After the theoretical framework, I will set out a model which will be used to analyze the effects of capital requirements regarding moral hazard. This analysis will be presented in a separate section. Afterwards, I will conclude.
1
Literature
This section will review literature concerning capital requirements and de-posit insurance. First, attention will be devoted to the influence of capital requirements. The effect of capital requirements on risk-behavior of banks, in a setting of moral hazard, is argued to be ambiguous. Secondly, this section provides a theoretical framework of deposit insurance and capital requirements. Starting point of the technical analysis is Diamond and Dy-bvig (1983) which was the first article to establish a bank run to be an equilibrium outcome. The literature review continues by considering the investment return to be uncertain (Hazlett, 1997), which provides further insights in how a bank run can come about. Further improvements on the general model of Diamond & Dybvig are made by Cooper and Ross (1997, 2002). The first article extends upon the investment decisions. The second article introduces a capital requirement for the financial intermediary in the insurance scheme.
1.1 Moral hazard and the effect of capital requirements
Deposit holders can face a moral hazard problem. Krugman (2009) describes moral hazard as any situation in which one person makes the decision about how much risk to take, while someone else bears the costs if things go bad. In the context of deposits, banks can choose the investment in which to invest the deposit holdings of depositors. The profit function of banks, can lead to banks having a tendency to take undue risks as the costs involved are not borne by the bank, but instead by another party effectively facing the risk. As Freixas and Rochet (2008) argue, banks are induced to invest in the project with the lowest probability of succes as the earnings of the bank are not dependent upon the risk the bank takes. Due to limited-liability bank managers are not liable in case of default. If the investment succeeds, the bank benefits whereas the deposit holders receive a fixed deposit rate independent of the payoff of the investment. In the case of failure, the bank might default but employees are not financially liable. The depositors on the other hand, have lost their entire deposit holdings. By introducing deposit insurance, the deposit holdings are guaranteed up to a certain limit. If the investment of the bank does not provide sufficient payoffs, the depositors will be partly protected against severe losses of their deposit holdings.
some extent) guaranteed, the investment decision of banks is not affected. Therefore, the risk of a bank failure has not diminished, such that the prob-ability that the deposit insurance is called upon, has not decreased. In order to affect the investment decision, the deposit insurance should include cap-ital requirements. The effects of capcap-ital requirements are however unclear. As Hellman, Murdock and Stiglitz (2000) argue, capital requirements put bank equity at risk, such that the incentive to take risk, is reduced. The authors however also recognize a perverse effect of capital requirements, as putting bank equity at risk might also encourage risk-taking. The ultimate effect of capital requirements is therefore ambiguous. Blum (1998) provides argumentation which points both in the direction of reducing risk-taking as well as actually inducing risk-taking. Blum argues that capital requirements can increase the cost of capital. Higher costs of capital may increase risk-taking as to compensate the higher cost with higher payoffs. The author also connects excessive risk-taking to limited-liability. Limited-liability is a protection for the employees of a bank. In case of default, the assets of the bank are liquidated but the bank employees cannot be forced to pay for unfulfilled claims. Limited-liability is quite hazardous as the bank man-ager is (at least partially) responsible in the case of default, as he is in charge of the bank’s operations. Bank employees benefit from risk-taking when investments succeed (as risk-taking yields higher payoffs), but do not suffer from investment failure. Formally, the bank manager does not face downside-risk. This creates a moral hazard problem. Whereas the bank is induced to invest the deposit holdings in risky projects as he benefits from the success, depositors are worse off as these projects have a higher failure rate. Contrarily to the bank manager, depositors face the full downside risk and do not benefit from risk-taking as they receive a fixed deposit rate. In the model section I will introduce four capital requirement ‘regimes’ which will be analysed for their ability to reduce the incentive to invest in risky projects and thereby to reduce the moral hazard problem.
1.2 The general framework of deposit insurance
Diamond and Dybvig (1983) have developed a framework for deposit con-tracts which points to the possibility of a bank run. Depositors are endowed with a unit of production which earns them an income. This income is used for consumption. However, depositors do not know when they will consume and hence face an uncertainty regarding their demand for liquidity. The Diamond & Dybvig model encompasses three periods. In the first period (T = 0), the depositors are endowed with one unit of production. Con-sumption occurs only in periods T = 1 and T = 2. Deposit holders are subdivided in a patient and an impatient group, which is modeled as con-suming in either period T = 1 or T = 2. Information regarding the type of consumer the individual is, is revealed at T = 1. At T = 0, depositors have two options concerning their unit of production. Either the unit is stored until T = 1, at which point the individual can consume one unit of con-sumption, or the unit is invested in an illiquid productive technology which yields R > 1 units of consumptions in period T = 2. The illiquid character of the productive technology lies in the fact that in order to consume at period T = 1, the investment needs to be liquidated prematurely, at T = 1, at which point the unit of consumption is only λ < 1. Implicitly, Diamond & Dybvig therefore account for liquidation costs to amount to (1 − λ). The depositor faces uncertainty regarding its consumption type. For impatient consumers it is optimal to choose the storage option. A patient individual on the contrary, is better off by choosing the illiquid investment. He then ends up consuming R > 1 in period T = 2. Although knowledge regarding the consumption type is revealed at T = 1, the proportion of impatient consumers, defined as π, is public information.
1.2.1 Autarky
First, Diamond & Dybvig consider the autarky economy, in which no trade between agents occurs and there are no financial intermediaries. At period T = 0, each individual determines what proportion of its unit of production it will invest (I) in the productive technology, leaving an amount of (1 − I) available for storage. If at T = 1, the consumer turns out to be a type 1 consumer, its consumption will be (recall that λ denotes the amount available for consumption after liquidation of the illiquid investment):
Whereas a type 2 consumer would obtain a consumption level of:
C2= RI + (1 − I). (2)
In period T = 0, without knowledge of their patience level, all individuals have the same utility function:
U (C1, C2) = πU (C1) + (1 − π)U (C2). (3)
Which is an increasing and concave utility function. Since λ < 1 < R, it is found that C1 ≤ 1 and C2 ≤ R with at least one strict inequality. For
impatient consumers it is optimal to set I equal to zero, whereas for pa-tient consumers, the optimal level for I is 1. Unfortunately, the investment decision must be made in period T = 0, before information concerning the consumption type is revealed. Therefore, the investment decision is ineffi-cient. As it turns out, this inefficiency can be mitigated when we allow for trade between agents.
1.2.2 Market
To see how trade would affect the consumption outcome, Diamond & Dybvig turn to the market economy. Although financial intermediaries are still absent, individuals can trade their investments for consumption. At T = 1, individuals can trade T = 1 units against T = 2 units at a price p. This yields the following consumption levels:
C1 = pRI + (1 − I). (4)
C2= RI +
(1 − I)
p . (5)
Type 1 individuals, can trade their investment in the productive technology with a type 2 consumer at price p. The depositors can therefore increase their consumption by an amount of pRI. Consequently, type 2 individuals bought these investments with their amounts of non-investment (1−I) (their investments are illiquid). From equations 4 and 5 it follows that C1 = pC2.
1.2.3 Optimal allocation
There is however another solution which in fact results in optimal risk-sharing. Optimal consumption levels must satisfy the following conditions:
(I) C12∗= C21∗= 0.
Where, Cit∗ is the consumption level of consumer type i in period t. Con-dition (I) states that impatient (patient) individuals consume nothing at T = 2 (T = 1) in the optimal consumption allocation, as consumers do not derive utility from the respective period consumption.
(II) U0(C11∗) = RU0(C22∗).
Which states that marginal utilities are equal. And
(III) πC1∗
1 +
(1−π)C2∗ 2
R = 1.
Which states the resource constraint.
Since R > 1 and utility functions are increasing and concave, it follows from condition (II) that C22∗ > C11∗ (period 2 consumption levels exceeds period 1 consumption levels). This outcome ensures that patient consumers prefer to withdraw their deposits in period two. Additionally, Diamond & Dybvig find that C11∗ > 1 and C22∗ < R. This finding crucially depends on the assumption of Diamond & Dybvig that the relative risk-aversion al-ways exceeds unity. The reasoning is that individuals are willing to insure against the risk of being type 1. The insurance increases consumption at T = 1, however at the expense of some consumption at T = 2. Assuming that individuals are relatively risk-averse yields an optimum of C1∗
1 > 1 and
C22∗< R. This optimal allocation of (C1, C2) can only be achieved by
risk-sharing is based upon public knowledge regarding the proportion of pa-tient versus impapa-tient individuals depositing their money holdings. That enables the bank to choose investment levels appropriately:
Store : πC1∗ Invest : (1 − π)C1∗ (6)
If for some reason, individuals panic, in which case patient depositors sud-denly decide to withdraw their deposits at T = 1, a bank run occurs. The bank is required to liquidate its investments in the productive technology. Although liquidation costs are not specified in the Diamond-Dybvig model, we know that liquidated assets yield a value of λ, such that a proportion of (1 − λ) is lost, and can be considered as liquidation costs. The funds recovered through liquidation are not sufficient to redistribute deposit hold-ings, hence the bank becomes insolvent. The bottom line is that banks provide liquidity through asset-transformation at the expense of vulnerabil-ity to runs.
1.2.4 Deposit insurance
The central bank has a solution at its disposal to prevent bank runs. By providing a deposit insurance, a central bank takes away the risk of losing deposit holdings. If a bank failure occurs, the deposit holdings are guaran-teed by the central bank. In the framework of Diamond & Dybvig, deposit insurance anticipates upon patient consumers who are concerned about re-ceiving their withdrawals in period T = 2. This concern can cause patient consumers to withdraw in period T = 1, but the funds of the bank in that period are only sufficient as to provide the amount of withdrawals of impa-tient consumers. Initially, this creates a illiquidity as the bank does not have sufficient liquid funds. By increasing their liquidity, banks must liquidate their investments. Because of the costs involved in liquidation, the total amount of funds might turn out to be insufficient, at which point the bank faces insolvency. Diamond & Dybvig however do not account for uncertainty regarding the return of the illiquid investment. Therefore, the uncertainty of depositors is, in their model, unjustified.
banks. The model however does not consider a risky investment. The au-thors recognize that including risk in the model additionally allows for moral hazard which affects the investment decision of banks. I will now turn at-tention to an article which does account for uncertainty regarding the return of investments.
1.3 A risky investment
In order to account for risk regarding the return of the illiquid asset in the framework of Diamond & Dybvig, I will follow the article by Hazlett (1995). This article shows how constraints change by taking into account an investment with an uncertain return. At T = 2, the economy will be in either a good state (denoted as g) or a bad state (denoted as b) which will affect the return of the risky investment. For the states of the economy I assume :
s ∈ S = {g, b}, P (s) > 0, P (g) + P (b) = 1. (7)
Where P (s) describes the probability that the economy will be in state s. There is no aggregate uncertainty regarding the types of consumers. In other words, the proportion of impatient people, π, is known. The utility function of the individuals takes the following form:
U (c) = 2 X h=1 P (h)X s∈S P (s)Uh(C1h, C2h(s)). (8)
Investment Return
T = 0 T = 1 T = 2
-1 1 R > 1
-1 r < 1 R(s)
Hazlett assumes that the expected return of the risky investment exceeds or equals the return of the safe investment. However, in the bad state, the safe investment offers a higher return (whereas in the good state, the risky investment provides the highest return). In mathematical terms:
P (g)R(g) + P (b)R(b) ≥ R, R(b) ≤ R ≤ R(g). (9) As before, withdrawals at T = 1 depend on the type of consumers the individuals experience to be. In addition to the type of consumer, T = 2 consumption also depends upon the state of the economy. The share of assets invested by the bank in the safe asset is B, such that the following two resource constraints can be defined:
2
X
h=1
P (h)C1h ≤ B. (10)
Which is the resource constraint for period T = 1 consumption.
2 X h=1 P (h)C2h(s) ≤ B − 2 X h=1 P (h)C1h ! R + (1 − B) R(s). (11)
Which is the resource constraint for period T = 2 consumption, which must hold for all s ∈ S. Additionally, Hazlett formulates an incentive compat-ibility constraint to ensure that, when individuals in T = 1 learn their consumption type, the consumption profile intended for patient consumers is the preferred allocation, compared to the consumption profile intended for the impatient consumers. For a principle-agent problem, incentive com-patibility constraints are requirements which ensure that agents act in ac-cordance with their solution. The incentive compatibility constraint below ensures that patient consumers prefer the consumption allocation intended for patient consumers over the allocation for impatient consumers.
X
s∈S
P (s)Uh(C1h, C2h(s)) ≥X
s∈S
P (s)Uh(C1i, C2i(s)), ∀ h, i ∈ H. (12)
the proportion of impatient consumers is larger than expected, the amount of withdrawals require the bank to liquidate some of its risky asset, as to provide impatient deposit holders their money. As the bank is balancing its budget carefully, liquidating the risky assets prematurely, might cause problems in period T = 2, such that the conditional equation 11 might no longer hold. Although the proportion of patient consumers is lower than expected, the liquidation of assets causes the available funds at T = 2 to be diminished such that even when illiquidity is averted in period T = 1 it might not be avoidable in period T = 2. As is established in equations 10 and 11, not only does liquidity depend upon the uncertain return of the risky asset but also on the proportion of patient consumers and the invest-ment in the safe asset accordingly. Whereas Diamond & Dybvig had a fixed proportion of patient consumers (1 − π), in the Hazlett framework with a risky asset, no such proportion is postulated. In fact, consumer types are modeled following a declaration of type by the consumers. This allows for unexpected changes in consumer types which undermine the stability of the banking system.
By including a risky investment, Hazlett accounts for banks choosing a cer-tain risk level. The paper shows how a bank run can come about when the amount of investment in the risky investment is too large. There exists a moral hazard problem as the existence of deposit insurance can distort the bank behavior, which causes a deposit insurance bailout to be more likely. The author argues that by changing the design of the deposit insurance, the moral hazard can be reduced. This reduction comes however at a price as it involves a higher probability that a bank run occurs. The paper does not address how the bank chooses what project to invest in. This creates a dif-ferent moral hazard which this thesis focuses on. This moral hazard problem does require a risky investment to be available, which was introduced in the Diamond-Dybvig model by Hazlett.
1.4 Extension on investment decisions
a cost. Cooper & Ross define the liquidation costs per unit of investment as (1 − λ). Additionally, there exists a liquid technology which yields a one unit return in the next period per unit of investment. Explicitly defining liquidation costs enables the authors to investigate how these costs affect investment decisions. Another progression is that Cooper & Ross account for public uncertainty concerning the proportion of impatient versus patient consumers. First considering the case without intermediaries and without public uncertainty, consumers choose a level of illiquid investment I as to maximize:
πU (I(1 − λ) + (1 − I)) + (1 − π)U (IR + (1 − I)) . (13) As before, π denotes the proportion of impatient consumers. Obviously, there exists a tradeoff between a high return of the illiquid investment, R, and the liquidation costs λ. If the return is large enough, consumers will choose to invest in the illiquid investment. The specific critical value of liquidation costs can be expressed as:
λc= (1 − π)(R − 1)
π . (14)
For levels of liquidation costs above this critical value, receiving the return when being patient outweighs incurring the liquidation costs in the case of being impatient, hence I > 0. Liquidation occurs with probability π. The authors proceed by forming a planning problem as to frame an insur-ance scheme, which prevents liquidation completely. Financial intermedi-aries are introduced as providers of the insurance contract. Consumer types are considered observable ex-post such that consumption levels are contin-gent upon types. The bank (financial intermediary) will propose a deposit contract which determines a bundle of consumption levels in periods 1 and 2. Recall that consumption optimally occurs either in period 1 for impa-tient types or in period 2 for paimpa-tient types. The consumption levels for the deposit contract are denoted by δ = (C1, C2) and are subject to resource
constraints. The planner’s maximization problem is defined as: max
I πU (C1) + (1 − π)U (C2). (15)
Such that:
πC1 = (1 − I), (1 − π)C2 = IR. (16)
Note that I in this maximization problem is the level of investments in the illiquid investment per deposit holder. The optimal allocation δ∗ satisfies:
This optimal equilibrium does not account for aggregate uncertainty con-cerning the proportion of impatient versus patient types and hence no liq-uidation will occur. As R exceeds unity and the utility function is strictly concave, period 1 consumption levels are lower than those in period 2 in order for equation 17 to hold. As was established earlier, the expected util-ity obtained from this optimal deposit contract will in general exceed the autarky solution utility. This hinges upon the concavity of the utility func-tion which implies that consumers are risk-averse and obtain utility from pooling risk. In that case, the optimality condition in equation 17 is fulfilled for levels of C1 lower than C2.
When Cooper & Ross allow consumption types to be private information, the problem evolves in a three-stage sequence. First, the planner will deter-mine a consumption allocation which specifies both a period 1 consumption level as well as a period 2 level. Second, deposit holders will learn their consumption preference, which will also become clear to the planner. In the third stage, consumers will receive the consumption allocation corresponding to their type as determined by the deposit contract. Cooper & Ross address here the issue of truth-telling. For impatient consumers, truth-telling is the dominant strategy as their utility for period two consumption is zero. For patient consumers, truth-telling being an optimal strategy is contingent on all other late consumers also revealing their true preference. Given a pa-tient consumer which misrepresents itself as being impapa-tient, it will receive C1 which can be stored in the liquid project until period 2. As mentioned
before, C1 < C2 such that this strategy is dominated by the truth-telling
strategy, which offers consumption level C2. Hence, the optimal allocation,
δ∗, which satisfies equation 17, is a Nash equilibrium which is achieved by all consumer honestly representing their consumption type. There exists however a Nash equilibrium in which late consumers collectively misrepre-sent themselves. The misrepremisrepre-sentation leads to a ’ bank run equilibrium’ if the bank has insufficient funds as to repay C1 to all agents. Then, patient
depositors are better off by withdrawing in period T = 1. In the event of a bank run, the authors consider an amount of NR depositors which receive payments as part of the total number of deposit holders N. The resource constraint facing the bank is defined as:
NRC1= (1 − I)N + N I(1 − λ) = N (1 − Iλ). (18)
be written as:
P (λ) = (1 − Iλ) C1
. (19)
Recall that under the optimal contract, δ∗, equation 16 holds, such that the probability of receiving your deposits is defined as:
P (λ)∗ = (1 − Iλ)π
(1 − I) . (20)
Where I is determined by equation 17. It is important to note that if P (λ) is smaller than 1, a bank run is an equilibrium for the optimal allocation, hence the authors regard P (λ) = 1 as a no-run condition. This is because for P (λ) = 1, the bank has sufficient funds as to provide all impatient con-sumer with their contractual consumption allocations. As the value of P (λ) becomes lower, the event of a bank run becomes more likely. What becomes clear from expression 20 is that the probability of receiving your deposits, is dependent upon the liquidation costs (λ) and the investment level in the illiquid project (I). Specifically, P (λ) is decreasing in λ (I is independent of λ). Therefore, if liquidation costs are sufficiently large, bank runs are very likely. The authors investigate how the level of investment in the illiquid technology changes compared to the optimal allocation, δ∗. Additionally, it will be illustrated when it is optimal to choose a runs-preventing con-tract. The authors make the additional assumption that there exists a run-equilibrium under optimal contract δ∗. Moreover, by adding the restriction that P (λ) ≥ 1, truth-telling is the only Nash-equilibrium. The bank can also invest in the liquid project for two periods, which will be denoted by I2. The bank was able to do so previously but this was not desirable when
consumption types were observable. The level of illiquid investments is as before denoted by I. In order for a deposit contract to be runs preventing, it should satisfy the following three conditions:
πC1 = 1 − I − I2, (21)
(1 − π)C2= IR + I2, (22)
C1≤ 1 − Iλ. (23)
types cannot exceed 1. By additionally considering an investment in the liquid project for two periods, the resource constraints are slightly altered. The amount of I2 denotes how much is invested in the safe project for two
consecutive periods. The authors argue that with I2 = 0, the allocations
that satisfy the resource constraints are not necessarily runs-proof. For allocations to be runs-proof there must be a positive level of investments in the liquid investment. As the liquid investment does not involve liquidation costs, these investments can be either used in period 1 or period 2. Moreover, I2 does not come up in the no-runs condition. Cooper & Ross define I2> 0
as banks holding excess liquidity. Although it does not earn a return in period two, it does support a high level of C1 while avoiding runs. Then
attention is turned to the case when a run equilibrium is allowed. The probability of liquidation is denoted by q. The probability of liquidation could be further specified but is regarded by Cooper & Ross as being a representation of economy-wide pessimism which could enforce depositor beliefs into a bank run. Accordingly, economy-wide optimism is observed with probability (1 − q). Whilst taking into account this probability of liquidation, q, the contract must solve:
max (1 − q)[πU (C1) + (1 − π)U (C2)] + q[
U (C1)((1 − Iλ)
C1
]. (24)
Such that:
πC1 = 1 − I − I2, (1 − π)C2 = IR + I2, I ≥ 0, I2 ≥ 0. (25)
The authors wonder whether the bank will hold excess liquidity and find that I2 is positive if:
qλ > (1 − q)(R − 1). (26)
Intuitively, expression 26 can be understood as a cost-benefit requirement. In the event of a run (occurring with probability q), the benefit of investing in the liquid instead of the illiquid project is λ per unit of investment as this is the liquidation costs. The benefit in a bank run scenario translates in a loss without a bank run. If a bank run does not occur (with probability 1 − q) the additional benefit of investing in the liquid project instead of the illiquid is (R − 1). The gain in expected utility with a bank run is:
qλ > U (C1) C1
. (27)
whereas the loss in expected utility for the scenario without a bank run is:
Concavity of the utility function implies that U (C1)/C1 > U0(C2), such
that given the requirement in 26, the gain in expected utility exceeds the loss. When bank runs are likely and liquidations costs are high, depositors benefit from banks holding excess liquidity (I2> 0). When liquidation costs
are low, there are no gains from holding excess liquidity such that I2 = 0.
Rewriting condition 26 as an equality yields the threshold value for q, q∗, which is defined as:
q∗= R − 1
R − (1 − λ). (29)
for values of q exceeding this threshold level, the solution to the maximiza-tion problem is a run-proof contract. For these levels of q, the probability of a bank run is too high such that the depositors benefit from having a runs proof contract.
1.5 Capital requirements : Cooper & Ross (2002)
In their 2002 article, Cooper & Ross start off by recognizing that whilst deposit insurance was implemented to build depositor confidence, it has encouraged excessive risk-taking by financial intermediaries. To overcome the moral hazard problem, Cooper & Ross (2002) suggest to include cap-ital requirements in the deposit insurance system. The authors adopt the Diamond-Dybvig framework. The bank is expected to operate in a compet-itive environment such that it must offer a deposit contract that maximizes the expected utility for consumers ex-ante, subject to a break-even con-straint. Hence the bank solves:
max πU (C1) + (1 − π)U (C2). (30)
Subject to the resource constraint:
1 = πC1+ (1 − π)
C2
R. (31)
Given that confidence in the bank prevails, truth-telling concerning its con-sumption type is the dominant strategy for individuals. Hence, impatient consumers will go to the bank for withdrawal in period T = 1, whereas the patient consumers withdraw in period T = 2. As consumption in period T = 2 does not yield utility for impatient consumers, the only requirement as to ensure truth-telling is that the consumption level in period two exceeds the level of consumption in period one. A higher period two consumption level ensures that patient consumers delay withdrawal until period T = 2. Only if patient consumers lose confidence in the bank, the strategy to with-draw early might be optimal, if resources are insufficient to provide for all withdrawals. Given that liquidation of the illiquid technology in period T = 1 yields a consumption level of 1, only a deposit contract (C1, C2) which
has C1 > 1 is vulnerable to bank runs. Given the risk-aversion assumption
from Diamond & Dybvig, the optimal contract indeed offers a period T = 1 consumption exceeding unity, such that the optimal contract is prone to bank panics. To avoid the vulnerability to panics, banks could choose con-tract (C1, C2) = (1, R), which is nor vulnerable to runs nor optimal from
the consumers perspective.
T = 2 of λR with probability v and 0 otherwise. The value of the in-vestment in case of liquidation in period T = 1 equals that of the safe investment, which is one. For parameter λ is assumed that its value ex-ceeds unity, whereas vλ < 1. Therefore, the return of the alternative (risky) project is higher than the original risky investment but its expected return is lower. Hence, the safe investment is preferred to the risky investment by all risk-averse consumers, which follows from:
vU (λR) + (1 − v)U (0) ≤ U (vλR) < U (R). (32) for any concave U (.).
After the payment to patient consumers in period T = 2, all remaining funds are retained by the bank. Cooper & Ross show that in fact, the bank may choose the risky investment over the safe investment. Without accounting for capital requirements specifically, the risky investment yields a higher return for the bank in case of success. Given a sufficiently generous deposit insurance system, deposit holders will not discipline the bank for choosing the risky investment. In case of a full deposit guarantee, the bank will choose to invest a proportion of i in the risky investment as to maximize the following expression which describes expected profits:
v{iλR + (1 − i − πC1∗)R − (1 − π)C2∗}
+(1 − v){max((1 − i − πC1∗)R − (1 − π)C2∗, 0)}. (33) The optimal contract, which satisfies equation 17 and requires that i = 0, earns the bank zero profit. Also, when the risky investment fails, the return to the bank is zero. Given that the risky project succeeds, the profit of the intermediary is as expressed by the following equation:
Π = v (iλR + (1 − i − πC1∗)R − (1 − π)C2∗) . (34) Differentiating this profit expression for the level of investments in the risky investment, i, I find:
∂Π
∂i = vR(λ − 1). (35)
the maximum operator and assume the profit of the safe investment to be non-negative) I find:
∂E[Π]
∂i = v(λR − R) − (1 − v)R = R(vλ − 1). (36) Which by assumption is negative (as vλ ≤ 1). Hence, although higher values for i raise the profit of the banker, it worsens social welfare.
Additionally, the authors introduce capital requirements. A requirement on the ratio of debt to equity financing will be implemented. For each unit of deposits, the bank is obliged to contribute σ units of (its own) equity to the capital account of the bank. Returning to the maximization problem of the bank, equation 33 is now written as:
v{iλR + ((1 + σ) − i − πC1)R − (1 − π)C2}
+(1 − v){max(((1 + σ) − i − πC1)R − (1 − π)C2, 0)}. (37)
Again, i denotes the optimal level of investment in the risky asset as to maximize the expected profit of the bank. An important development is that the banker can now lose its own assets. As the probability of failure is higher for the risky asset, this provides an incentive for the bank to invest in the safe project. Cooper & Ross define a level of investments i∗ for which:
((1 + σ) − i∗− πC1) R = (1 − π)C2. (38)
bears the risk as the deposit insurance will be called upon. It is then optimal to fully invest in the risky investment. Hence, the bank will choose to either invest fully in the riskless investment project or to fully invest in the risky investment project.
Cooper & Ross offer a proposition as to prove that for adequate levels of capital requirements, banks no longer prefer to invest in the risky asset, thus solving the moral hazard problem. With full deposit insurance, bank runs are completely eliminated and depositors have no incentive to monitor the banks investment decision. If the amount of own-assets required to be involved in the investments exceed the critical value σ∗, banks are induced to invest in the safe project. The critical value of capital requirement σ is found to be:
σ∗= v(λ − 1)
(1 − λv). (39)
For σ ≥ δ∗, the optimal allocation σ∗= (C1∗, C2∗) is achievable without bank runs.
Given the maximization problem of the bank, equation 37, filling in for the optimal contract (C1∗, C2∗) yields:
v (iλR + ((1 + σ) − i − πC1∗)R − (1 − π)C2∗)
+(1 − v)max (((1 + σ) − i − πC1∗)R − (1 − π)C2∗, 0). (40) Using the resource constraint, equation 31, reduces the maximization prob-lem to:
max v (iλR + (σ − i)R) + (1 − v) max{(σ − i)R, 0}. (41) Differentiating this expression for i yields:
R(vλ − 1). (42)
Which is negative such that I find that for higher values of i, the profit for the bank reduces. The extremes of the profit equation can be found by plugging in both i = 0 (full investment in safe project) and i = 1 + σ (full investment in the risky project). The critical value for capital requirement σ, which ensures that bankers fully invest in the safe assets is defined as:
δ ≥ σ∗= v(λ − 1)
Consequently, the bank will choose to fully invest in the safe project, and given that confidence prevails, bank runs are prevented. As the authors point out, this outcome crucially depends on risk-neutrality of the bank as the uncertain returns would concern additional costs for risk-averse banks. In-terestingly, if capital requirements are not at the appropriate level (δ < δ∗), the bank is not restricted in his actions regarding his own capital funds. In other words, given insufficient capital requirements, the bank could wind up fully investing in the risky project. Additionally, adjustments to the pa-rameters that form the critical capital requirements are of interest. As the authors point out, a mean-preserving spread on the risky project’s return, which translate in a higher λ and lower v whilst keeping λv unchanged, will cause the critical level of capital requirements to increase. This result could be anticipated as the probability of success is lower which makes the risky project more of a gamble while the expected return is not higher. Less intu-itive however is the result that increasing either λ or v, which both increase the efficiency of the risky asset, also requires higher capital requirements to prevent the bank from morally hazardous behavior.
2
Model
This chapter presents the general setup of the model. First, I consider the investment option for the bank. As will become clear, there exists a moral hazard problem as the bank does not bear the downside risk, since the projects are funded by deposit holdings. I introduce several capital require-ment regimes facing banks, which are analyzed for their ability to reduce the moral hazard problem. The model is the framework on which the analysis (in the next section) will be based.
2.1 General Setup
B > G. (44) The expected payoff assumption:
φSG > 1 > φRB. (45)
The success probability assumption:
φS> φR. (46)
These assumptions are quite reasonable as investments which inhibit a higher risk level often provide a higher payoff given success. This is also established in this simple investment decision. The success rate of the safe project is higher than that of the risky project such that the probability of a positive payoff is higher if the bank chooses to invest in the safe investment.
Another assumption is that the depositors are all risk-averse. Given their risk-aversion, depositors favour investment in the safe project, as this in-vestment has a positive NPV. In reality, individuals can also exhibit risk-neutrality of risk-loving behavior. However, for such behavior, individuals can resert to other investments than deposit accounts. Individuals can allign their portfolio risk to their own desired risk-level by choosing a combination of higher risk and lower risk investments. A deposit account is chosen as a safe haven and therefore attracts risk-averse individuals.
The Bad project on average generates losses and should not be undertaken, given risk-aversion. Additionally it is assumed that banks have no finan-cial means of their own available for investment. This assumption ensures that, when capital requirements are absent, banks face no down-side risk. Of course, the bank has financial means but for this analysis we can for example expect that its assets are already invested and therefore illiquid. One capital requirement regime in fact requires banks to hold some assets liquid at all times.
The deposit holders place their deposit holdings at the bank which invests those resources in either the safe or the risky project. The bank has limited-liability and hence does not face down-side risk of the investments. The deposit holders require a return on their money holdings, the deposit rate, which I will define as rd (note that returns such as G, B and rd are to be
of unity for investment i). The deposit rate can be interpreted as the cost facing the bank. In order to be able to invest in projects, the bank is de-pendent on depositors placing their money at the bank. The costs involved are the deposit rate per unit of deposit. Given this deposit rate, the profit of the bank per unit of investment are defined as either:
(G − rd) for the good investment, or
(B − rd) for the bad investment.
Where the bank earns the payoff of the investment, which is considered as the loan interest. The profit of the bank is thus the margin between the loan interest rate (G for the safe investment and B for the risky investment) and the deposit rate.
2.2 Moral Hazard
The investment decision is made by the party which does not face the risk of losing money.
2.3 Incentive compatibility constraints and the critical de-posit rate
As in Hellman, Murdock and Stiglitz, I analyse the investment decision of the bank by means of incentive compatibility constraints. The incentive compatibility constraint shows the condition which must be satisfied for the bank to prefer the safe investment. The safe investment is the desireable project to invest in from the depositor’s point of view. If the incentive com-patibility constraint is satisfied, the investment opportunity preferred by depositors, the safe investment, yields the highest expected payoff for the bank. The incentives of both depositors and bank managers are alligned, such that the moral hazard problem is solved. I investigate whether certain capital requirement regimes are able to provide incentive compatibility con-straints which incentivize the bank to invest in the safe rather than the risky project. As the moral hazard problem depends upon the required deposit rate, the problem will not be completely resolved but can be reduced.
In order to compare the capital requirements, I rewrite the incentive com-patibility constraints to find a cap on the deposit rate rd, as do Freixas &
the case without capital requirements. Then I will introduce several capital requirement ’regimes’ with the corresponding incentive compatibility con-straints and critical deposit rates. In the analysis section I will compare the capital requirement regimes as to investigate which requirement reduces the moral hazard problem facing depositors.
Without Capital Requirement: As a benchmark for comparing capital requirement, I first present the case without capital requirements. As as-sumed above, the bank has illiquid financial means which are not available for the investment decision which is considered in this analysis. In other words, the assets of the bank are tied up into investments such that, in order to invest in new projects, the bank must collect capital from deposit holders. The assumption that banks only invest depositor’s money ensures that banks face no downside risk for the investment under consideration. As Hellman, Murdock and Stiglitz argue, banks typically hold too low a level of capital. In the setting without capital requirements, the bank does not hold any capital in liquid form. First, I will provide a benchmark by presenting the incentive compatibility constraint for the case without capi-tal requirements. As I explain above, the bank will choose to invest in the safe investment if the incentive compatibility constraint is satisfied. The constraint is written as:
φS(G − rd) > φR(B − rd). (47)
The left-hand side corresponds to the expected profit per unit of investment in the Good project. In case of success, the per unit payoff is G which is reduced with the cost of capital, the required deposit rate rd. Similarly,
the right-hand side corresponds to the per unit of investment profit given investment in the risky project. As I mention above, φS and φR, describe the probability of success for the safe and the risky investment respectively. Also, G and B are the payoff levels given success of the safe and the risky project. rd is the deposit rate which depositors earn. In order to raise
rdC = φ
SG − φRB
(φS− φR) . (48)
If the required deposit rate exceeds the critical level, the bank will choose to invest in the risky investment. Hazlett, Murdock and Stiglitz refer to the critical deposit rate as ”the threshold interest rate at which gambling occurs”. Moreover, the authors state a ’No Gambling Condition’:
rd< rCd. (49)
For deposit rates lower than the critical level, the bank will choose to invest in the safe investment. The reasoning is that when deposit rates are high, the payoff for the bank is lower. It might then not be worthwhile to invest in the safe project as this has a lower payoff level. So for higher deposit rates, the bank manager might be inclined to invest in the risky project as to compensate the higher deposit rate with a higher payoff, given that the investment provides a payoff. The analysis of the capital requirements, I introduce below, will be based upon condition 49. More specifically, when capital requirements result in a higher critical deposit rate, condition 49 is satisfied for a higher range of deposit rates and hence reduces the moral hazard problem facing deposit holders. Conversely, if a capital requirement yields a lower critical deposit rate, for some levels of deposit rates, condition 49 is not satisfied whereas before it was satisfied for these deposit rates. For these levels of deposit rates, the bank manager is now inclined to in-vest in the risky project, such that the moral hazard problem is increased. As shown in 48, the critical deposit rate depends upon the expected payoff level differential and the success rate differential of the two projects. As the expected payoff of the safe project increases, the bank manager is more inclined to invest in the safe project. The same holds for the risky project. Additionally, when the success rate differential is small, the critical deposit rate becomes very large, such that condition 49 is very likely to be satisfied. Because of the design of the investment opportunities with respect to the expected pay off levels, a small difference in success rates translates in a small difference in payoff levels. The additional payoff clearly is not com-pensated by the higher risk of the investment being a failure.
Assume the following values for the parameters: φS = 0.9, φR = 0.65,
G = 1.2, B = 1.5 and rd = 0.05. The left-hand side of 47 equals 1.035,
net present value. The opposite holds for the expected payoff of the risky investment, the right-hand side. As the condition is satisfied, the bank manager has an incentive to invest in the safe investment. Although having a higher payoff in the case of success, this is mitigated by a higher failure rate.
Capital Requirement 1: Own-asset involvement The first capital re-quirement I consider is the own-asset involvement rere-quirement, introduced by Cooper & Ross. In addition to the deposit holdings, the bank is required to invest a certain proportion of its own assets in the projects. Recall that the bank has own assets tied up into investments. Under this capital re-quirement, banks must have financial assets available in order to invest the money from deposit holders. For each unit of deposit invested in a certain investment, the bank must also invest a proportion of its own assets in the same investment. The capital restraint causes the bank to face the risk of losing its own assets. Without this constraint, the bank can invest the de-posit holdings as it pleases whilst not facing losses if the projects fail. If the incentive compatibility constraint below holds, the bank prefers to invest in the safe project. The proportion of own assets which is required to invest is defined as β. As Hellman, Murdock and Stiglitz argue, if a large enough amount of own capital is at stake for the bank, it will be induced to invest in the safe investment. The capital requirement is desireable if it helps to fulfill the following constraint:
φS((1 + β)G − rd) + (1 − φS)(−β) >
φR((1 + β)B − rd) + (1 − φR)(−β). (50)
As before, the left-hand side corresponds to the expected profit per unit of investment in the Good project. The bank now also has to consider the expected loss of own-assets (β per unit of investment) if the projects fail. The right-hand side represents the expected profit per unit of investment in the risky project. From equation 50 the critical deposit rate can be found to be:
rd< rβd =
φS(1 + β)G − φR(1 + β)B
(φS− φR) + β. (51)
affects the investment decision of the bank manager. When the required amount of own-assets to be invested is increased, the critical deposit rate increases by an equal amount, which induces the bank manager to invest in the safe project.
Capital Requirement 2: Risk-independent ex-ante fund contribu-tions The second capital requirement under consideration is risk-independent ex-ante fund contributions. It requires banks to deposit a proportion of its deposit holdings at the central bank. Hence, part of the deposits are not available for investment and are required to be held liquid at the central bank. The central bank will manage the reserve deposit account, which will be used in case of a bank failure to repay the lost deposit holdings (hence, it functions as an insurance fund). I denote the proportion of deposits which are to be reserved as α. Note that the amount of deposit holdings available for investments are therefore reduced by α percent. For the bank to invest in the safe project, the following incentive compatibility constraint must hold:
φS(1 − α)G − φSrd> φR(1 − α)B − φRrd. (52)
As follows from expression 52, the bank can only invest a proportion of (1−α) per unit of deposit. However, depositors still have a required deposit rate which is unchanged, such that the costs to deposit holders are unchanged. As the amount available for investment as a proportion of deposits has decreased, the expected payoff per unit of deposit has decreased as has the total earnings of the bank. This causes the critical required deposit rate to change to:
rd< rdα= (1 − α)
(φSG − φRB)
(φS− φR) . (53)
Capital Requirement 3: Risk-dependent ex-ante fund contribu-tions Capital requirement three considers risk-dependent ex-ante fund contributions. Similar to the fixed proportion α, banks are now required to hold a proportion of its deposits in a reserve account, which depends on the risk the bank faces. The risk-dependent fund contributions are higher if the bank invests in a project with a higher failure rate. As before, the share of deposit holdings available for investments are reduced by holding deposits in a reserve account. For the bank to prefer investment in the safe project the incentive compatibility constraint below must be satisfied. The proportion of risk-dependent fund contributions is denoted by γ. The effec-tive proportion depends upon the failure rate. The inceneffec-tive compatibility constraint can be written as:
φS (1 − (1 − φS)γ)G − rd > φR (1 − (1 − φR)γ)B − rd . (54)
Similar to capital requirement two, the funds available for investment per unit of deposits is lower than unity. The proportion available, depends upon the risk-level of the bank. For a measure of the risk-level, the failure probability of the investment is taken into consideration. For investments with a higher probability of failure, the risk-level is higher and hence, the bank will be required to hold a larger proportion of deposits liquid. From equation 54, the critical deposit rate is found:
rd< rdγ=
φS 1 − (1 − φS)γ G − φR 1 − (1 − φR)γ B
(φS− φR) . (55)
Expression 55 shows how the critical deposit rate is affected by the risk-dependent fund contributions. The fund contributions depend upon the failure rate of the two investments. If the bank invests in the safe project, the required fund contributions are therefore lower than when the bank invests in the risky project. Instead of requiring banks to contribute a certain proportion of deposits to the insurance fund, this capital requirement in fact can steer the investment decision of banks by making the contributions depend on the risk of the chosen investment. It reduces the margin which the bank earns but in an asymmetric fashion. The margin is more reduced for the investment with a higher failure rate such that this investment becomes less interesting for the bank to invest in.
contributions separately. I now consider both fund contributions in one capital requirement framework. Therefore banks not only must place a proportion α of deposit holdings in a reserve account but also a proportion depending on its risk profile. The proportions are denoted, as before, by α and γ. The incentive compatibility constraint which must be satisfied in order for banks to prefer investing in the safe project is defined as:
φS (1 − (1 − φS)γ − α)G − rd
>
φR (1 − (1 − φR− α)γ)B − rd . (56) As before, α denotes the risk-independent capital requirement, whereas γ denotes the proportion of deposits which are required to be held liquid, which depends on the risk-level of the bank. The critical deposit rate is written as:
rd < rα,γd =
φS(1 − (1 − φS)γ − α)G − φR(1 − (1 − φR)γ − α)B
(φS− φR) .
3
Analysis
In this section I analyse the different capital requirement regimes, based on how the critical deposit rate changes from the benchmark case. If the critical deposit rate increases, the requirement reduces the moral hazard problem. The bank prefers the safe investment over the bad investment for a larger range of deposit rates. The analysis is per capital requirement to test whether the critical deposit rate has increased.
Capital Requirement 1: Own-asset involvement Regarding the first capital requirement, the constraint reduces the moral hazard problem if the critical deposit rate is higher than the critical deposit rate under the benchmark case without capital requirements. This is the case if:
rdC < rβd. (58) which equals: φSG − φRB (φS− φR) < φS(1 + β)G − φR(1 + β)B (φS− φR) + β. (59)
which can be rewritten as:
β (φSG − φRB) | {z } a + (φS− φR) | {z } b > 0. (60)
Capital Requirement 2: Risk-independent ex-ante fund contribu-tions Next, I consider the second capital requirement which requires (risk independent) contributions to the deposit insurance fund. Again, the con-straint reduces the moral hazard problem if the critical deposit rate is higher than the critical deposit rate under the benchmark case without capital re-quirements. This is the case if:
rdC < rαd. (61) which equals: φSG − φRB (φS− φR) < (1 − α) φSG − φRB φS− φR . (62)
As α > 0, it becomes immediately clear from the expression above, that the critical deposit rate with risk-independent deposit rate cannot exceed the critical deposit rate without capital requirements. Therefore, this capital requirement regime is not able to reduce the moral hazard problem.
Capital Requirement 3: Risk-dependent ex-ante fund contribu-tions I now consider the risk-independent fund contributions. To reduce the moral hazard problem, the critical deposit rate must be higher than the critical deposit rate under the benchmark case without capital requirements. Hence: rCd < rγd. (63) Which equals: φSG − φRB (φS− φR) < φS 1 − (1 − φS)γ G − φR 1 − (1 − φR)γ B (φS− φR) . (64)
Which can be rewritten as:
(1 − φR)φRB > (1 − φS)φSG. (65)
I define the success rate differential as .
= φS− φR. (66)
By assumption B > G. More explicitly, I define B = (1 + δ)G. The requirement for the capital requirement to reduce moral hazard, condition 65, can be rewritten as:
(1 − φR)φR(1 + δ) > (1 − φR− )(φR+ ).
(1 − φR)φR(1 + δ) > (1 − φR)(φR+ ) − (φR+ ). (1 − φR)φR(1 + δ) > (1 − φR)φR+ (1 − φR) − (φR+ ).
δ(1 − φR)φR> (1 − 2φR− ).
By differentiating this expression to δ, and φR, the effect of these param-eters on the critical deposit rate can be found. Differentiation with respect to these parameters yields:
∂ ∂δ = (1 − φ R)φR. (68) ∂ ∂ = −1 + 2φ R+ 2. (69) ∂ ∂φR = δ − 2δφ R+ 2. (70)
As 68 is strictly positive, for higher values of δ, condition 67 is more easily satisfied as the left-hand side of 67 increases. The δ parameter defines how much larger the payoff of the risky project is proportional to the payoff of the safe investment. Because the risky project has by assumption a negative net present value, a project with a higher payoff also has a higher failure rate. Therefore, the risk-dependent contributions are higher. As the results above show, for a higher δ, the capital requirement of risk-dependent fund contributions further reduce the moral hazard problem.
The sign of expressions 69 and 70 depend upon the values of δ, φR and .
The effect of these variables on condition 67 is therefore ambiguous. Ex-pression 69 is positive for higher values of φR and . In fact, 69 is positive for values of φS higher than 0.5 as = φS− φR, which is quite reasonable
for a safe project. As 69 is positive, the capital requirement is more likely to reduce the moral hazard problem as increases. Hence, if the success rate differential is larger, which in this setting constitutes the risk levels of the projects, the capital requirement helps to induce the bank to invest in the safe rather than the risky project. The sign of expression 70 also depends on δ, φRand . The sign is likely to be positive for higher values of δ and
Capital Requirement 4: Risk-dependent and indepentent ex-ante fund contributions Above I analyzed two capital requirements seper-ately which will now be analysed in combination. Above I establish that the risk-independent fund contribution is unable to reduce the moral hazard problem. The risk-dependent fund contributions are effective given that the success rate differential is small enough. Regarding the capital requirement which combines the two components, the regime is moral hazard reducing if the critical deposit rate is higher than the critical deposit rate under the benchmark case without capital requirements. This is the case if:
rCd < rdγ,α. Which equals: φSG − φRB (φS− φR) < φS(1 − (1 − φS)γ − α)G − φR(1 − (1 − φR)γ − α)B (φS− φR) .
Which can be rewritten as:
φR (1 − φR)γ + α B > φS (1 − φS)γ + α G. (71)
Since by definition, B = (1 + δ)G and = φS− φR, this requirement can be
rewritten as: φR(1 − φR)γ + αφR (1 + δ) > (φR+ )(1 − φR− )γ + β(φR+ ). φR(1 − φR)γ + αφR (1 + δ) > φR(1 − φR− )γ + (1 − φR− )γ + β(φR+ ). φR(1 − φR)γ + αφR (1+δ) > φR(1−φR)γ−φRγ+(1−φR−)γ+βφR+β. φR(1 − φR)γ + αφR δ > −2φRγ + (1 − )γ + α. φR(1 − φR)γ + αφR δ + 2φRγ − (1 − )γ − α > 0. (72)
If condition 72 is satisfied, capital requirement four reduces the moral hazard problem. Whether the condition is satisfied depends upon the sign of the left-hand side, which depends on the specific values of δ, φR and . By rewriting 72 it can be compared to condition 67. Condition 72 equals:
γ δ(1 − φR)φR− (1 − 2φR− ) + α(φRδ − ) > 0. (73)
δ cannot exceed φR. Therefore, the last term in 73 is zero or negative.
Con-dition 73 is therefore less easily satisfied than 67 which considers the third capital requirement. Differentiation of 72 with respect to the parameters δ, φR and yields: ∂ ∂δ = (1 − φ R)φRγ + αφR. (74) ∂ ∂ = 2φ Rγ − γ + 2γ − α. (75) ∂ ∂φR = δγ − 2φ Rγδ + αδ + 2γ. (76)
Expression 74 is strictly positive, such that for higher values of δ, condition 72 is more easily satisfied. The capital requirement reduces the moral haz-ard problem for higher values of δ. For φR = 0.5, γ = 0.05 and α = 0.05, 74 has a value of 0.0375. The sign of expressions 75 and 76 depends upon the values of δ, φR and . The effect of these variables on condition 72 is unclear. With regard to 75, higher values of φR and and lower values of α increase 75. The effect of γ on 75 depends upon the value of φS as = φS− φR. For values of φS higher than 0.5, higher values of γ lead to
a higher value of expression 75 and vice versa. By plugging in φR = 0.5,
γ = 0.05, = 0.3 and α = 0.05, expression 75 has a value of 0.07. For a positive sign, higher values of increase the left-hand side of 72, such that the capital requirement reduces the moral hazard problem. If the capital requirement requires higher risk-dependent contributions (higher γ) or lower risk-independent contributions (lower α), expression 75 increases. By plug-ging in φR = 0.5, γ = 0.05, δ = 0.2 = 0.3 and α = 0.05, expression 76 is
found to equal 0.04. Again the size of the differential is small and the size is positive. Therefore, for higher values of φR, the capital requirement helps to reduce moral hazard. As the required risk-independent fund contributions are higher (higher α), expression 76 becomes higher such that for higher success rates of the risky investment, the capital requirement regime further reduces the moral hazard problem.
4
Conclusion
This thesis analyzes different capital requirement regimes and their effect on the investment decision of banks. There was a risk component introduced in the framework of deposit insurance by Diamond and Dybvig (1983). This al-lows for a moral hazard problem facing depositors. Literature regarding the effect of capital requirements in order to reduce the moral hazard problem points to both reducing and encouraging risk-taking behavior. While putting the bank itself at risk might induce banks to reduce their risk-taking behav-ior, it also reduces the interest rate margin, which might lead to increased risk-taking. This study has introduced four capital requirement regimes in a simple framework and shows that the design and the investment character-istics are important factors which influence the moral hazard problem facing depositors. The investment decision of banks does not always coincide with the preference of depositors, who cannot invest their own deposit holdins. This thesis investigates which capital requirements can influence the invest-ment decision of banks such that the preferences of depositors and banks are more alligned.
This thesis has focused upon four capital requirement regimes. The first is labeled the ’own-assets involvement requirement’ which requires banks to invest a certain proportion of its own assets in the investments. The second and third requirement require the bank to contribute a proportion of deposit holdings in an insurance fund. Whereas the former is a risk-independent requirement, the latter is dependent upon the risk-profile of the bank, mea-sured by the failure rate of its investments. The fourth capital requirement regime is a combination of both fund contribution requirements.
This thesis finds that the ’own-assets involvement’ requirement helps re-ducing the moral hazard problem. Regardless of the specific characteristics of the investment options, this capital requirement leads to a higher criti-cal deposit rate. The risk-independent fund contribution requirement does not reduce the moral hazard problem. In fact, by decreasing the margin of interest rates which accrue to banks, this requirement induces risk-taking behavior. The risk-dependent fund contribution requirement can reduce the moral hazard problem. This in fact is dependent upon the specific character-istics of the investment options. If the risky project has a success rate above 50%, this capital requirement leads to a higher critical deposit rate, regard-less of the specifics of the safe project (although by assumption the success rate of the safe project exceeds the success rate of the risky project). Addi-tionally, for risky projects with a success rate below 50%, the effect on moral hazard of the capital requirement depends upon the success rate differential (), the success rate of the risky project (φR) and to what extent the risky project yields a higher payoff in case of success (δ). In general, the capital requirement reduces moral hazard for higher values of δ. Hence, if the payoff in case of success for the bad project is relatively high compared to the safe project, the capital requirement helps reducing the moral hazard problem. Also, when the success rate differential increases (higher ) or when the suc-cess rate of the risky project decreases (lower φR), the risk-dependent fund contributions are more likely to reduce moral hazard. Capital requirement four which combines risk-dependent and risk-independent contributions to a deposit insurance fund reduces the moral hazard problem less than the risk-dependent capital requirement. Again the effectiveness of the require-ment depends upon the variables , δ and φR. As before, for higher values of δ, the capital requirement influences the incentive compatibility constraint such that banks are more inclined to invest in the safe project. Also, for higher values of and lower values of φR, the combination capital require-ment is more likely to reduce moral hazard.
Additionally, this thesis shows that the requirement to banks to invest own-assets in their investments does help to reduce the moral hazard problem facing depositors and the risk-taking behavior of banks. By requiring the bank to bear some downside risk, the incentive compatibility constraint which determines the investment decision is affected. The bank manager is induced to invest in the safe investment rather than the risky one. The ability of the two capital requirements which consider risk-dependent fund contributions to reduce moral hazard, is ambiguous. The effect on the in-centive compatibility constraint of the bank is shown to be dependent upong the characteristics of the investment opportunities. Specifically, the payoff differential, the success rate differential and the success rate of the risky project influence whether the bank manager prefers the safe or the risky project. The capital requirement regime which only requires risk-dependent fund contributions is found to further reduce moral hazard than the regime which combines it with risk-independent contributions. This thesis shows that own-asset involvement and risk-dependent contributions reduce moral hazard. Which requirement most reduces moral hazard depends upon the specific characteristics of the investments.
