Communication and Risk Taking in the Banking Industry

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UNIVERSITY OF GRONINGEN

Communication and Risk Taking in the

Banking Industry

by

J. J. Bosma

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science (MSc).

in the

Research Master Program: Economics & Econometrics Faculty of Economics & Business

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“As long as the music is playing, you have to get up and dance. We’re still dancing.”

Charles Prince, CEO Citigroup, summer 2007.

“I guess I should warn you, if I turn out to be particularly clear, you’ve probably misun-derstood what I’ve said.”

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UNIVERSITY OF GRONINGEN

Abstract

Research Master Program: Economics & Econometrics Faculty of Economics & Business

by J. J. Bosma

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Acknowledgements

I would like to thank Michael Koetter and Lammertjan Dam for reviewing this thesis. Additionally, I am grateful to Michael Koetter for his helpful comments and insights on the topic, and enjoyed his interest in the topic as well as the fruitful discussions. Furthermore, I would like to express my gratitude towards De Nederlandsche Bank (the Dutch central bank) for allowing me to work on this thesis during an internship at their Economic Research and Policy division in the months June through August, 2010. In particular I would like to thank Itai Agur and Wilko Bolt for their comments on the thesis and helpful insights.

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List of Figures

3.1 Game Tree . . . 12

4.1 Equilibrium correspondence of αx and θ∗(z), for given αz . . . 18

4.2 Higher value of the public signal . . . 20

4.3 Imposing higher capital adequacy requirements . . . 22

4.4 Lowering of expected return of banks’ portfolio . . . 25

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List of Tables

A.1 Parameter values for figures.. . . 33

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Abbreviations

cdf cumulative distribution f unction

iff if and only if

LLR Lender of Last Resort

pdf probability density f unction

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Symbols

ai Strategic action of bank i.

A Group of banks investing in risk bearing loans.

B Set of banks.

c1 Costs associated with bailing out non-risky banks.

c0 Costs associated with bailing out risky banks.

Di Deposits held by bank i.

f pdf of ˜r (return on loans).

h Auxiliary function.

i Indexes bank i.

k Capital adequacy ratio.

L Risk free loans.

Li Loans held by bank i.

Qi Equity held by bank i.

rD Return on deposits.

¯

r Maximum value of ˜r.

˜

r Stochastic return on loans for bank i. Ui Profit of bank i.

xi Idiosyncratic signal for bank i.

x∗ Equilibrium value of xi.

z Public signal.

α Sum of αx and αz.

αx Precision parameter in distributing xis.

αz Precision parameter in distributing z.

ε Noise in z.

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Key Symbols x

θ Costs associated with not initiating a bail out. θ∗ Equilibrium value of θ.

λ Risky loans.

ξi Noise in xi.

π Probability of a bail-out program being initiated. ˆ

πi Predicted probability of receiving bail-out support.

ρ Premium for Qi on top of rD

φ Standard normal pdf.

Φ Standard normal cdf.

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Chapter 1

Introduction

It has become apparent during the recent financial turmoil that regulatory authorities, such as central banks, maintain bailout schemes to assist financially distressed banks with capital injections or other forms of support to prevent them from going bankrupt. In this light the main goal of bail-out support programs is to ensure stability in the financial sector and to prevent potential negative spillover effects of the failure of banks to the banking sector or to other sectors of the economy. As argued by Diamond and Dybvig [10] deposit insurance schemes may prevent bank runs that would otherwise lead to forced liquidation of banks. This in turn may prevent other banks from experiencing financial distress as in the framework of Allen and Gale [2]. However, a potential side effect is that bail-out programs may give rise to moral hazard conceptualized by excessive risk taking by banks. Since part of the risk of going bankrupt will be insured by the government. This thesis presents a theoretical model in which the regulator imperfectly communicates its optimal strategy to initiate a bail-out program. The noise term that the regulator introduces in its communication is then evaluated with respect to its effects on overall risk taking by banks, as well as its effects on policy measures other than communication.

The framework is based on a game played between the regulator and the individual banks in the banking industry. In this game, banks choose their preferred risk level; and the regulator weighs the costs associated with bank failure versus the costs arising from initiating bail-out support. The regulator communicates its probability of initiating a bail-out scheme with imperfect information to the banking sector by sending noisy

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Chapter 1. Introduction 2

signals about the true value of the costs of bank failure that contain noise terms. This ensures that banks are not able to perfectly infer whether the regulator will initiate a bail-out program, but rather they partly rely on what they think all other banks predict with regard to the regulator’s action. The modeling technique adopted to create this framework is based on the global games methodology (Carlsson and Van Damme [5], Frankel et al. [12], and Morris and Shin [23]).

The model solves for a unique equilibrium that identifies a proportion of banks that engage in risk taking as a result of banks having imperfect information with regards to the strategy of the regulator. Comparative static exercises reveal that as the regulator communicates its actions with more noise, it may curtail risk taking in the banking sec-tor. Additionally, results indicate that a tradeoff exists between noise in communication and alternative policy measures, such as capital requirements.

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Chapter 2

Literature Review

In this chapter the literature on the link between last resort lending and potential moral hazard are reviewed. An additional section is reserved for evaluating the global games methodology and to motivate why this modelling technique has been adopted in this thesis. Furthermore, throughout this thesis the concepts of ‘central bank’, ‘lender of last resort’ (‘LLR’) and ‘government’ are used interchangeably and are commonly denoted as regulator.

2.1 Moral hazard versus social cost of last resort lending

Within the strand of literature that deals with the implications of last resort lending on moral hazard in banking Goodhart and Huang [17] explicitly model the standard social cost-moral hazard tradeoff associated with bank bailout programs. The intuition behind this tradeoff is that as banks deem it more likely to receive bailout support they ceteris paribus expect a higher probability of their own survival. In turn, this may enhance their incentives to engage in ‘excessive’ risk taking. ‘Excessive’, in the sense that their resulting risk position may be detrimental for the overall stability of the sector1_{, since} the additional risk may increase the bank’s probability of bankruptcy. On the other hand, in the event that a supervisory authority can commit to not initiate a bail-out policy consequences may be that a bank will go bankrupt whereas it would otherwise be saved. Within the framework of Allen and Gale [2] the failure of one bank could lead to

1

See for instance Allen and Gale [2] and Freixas et al. [15] for more on the negative spillover effects of a bank’s failure on the stability of the banking sector.

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Chapter 2. Literature Review 4

the failure of other banks, thereby increasing the costs of letting one bank fail. Although the model presented in this thesis solves for a partial equilibrium that only concerns the banking sector, it is well taken that a bank’s failure may have adverse implications for industries other than the banking industry.

Goodhart and Huang [17] show that last resort lending may be a result of a regulator’s concern about the consequences of adverse contagious effects of individual with bank failures on the financial and non-financial sectors of the economy. Last resort support would then be initiated once these outweigh the moral hazard considerations. On a related investigation, Freixas [14] also argues that a tougher stand of the regulator may help to alleviate the moral hazard.

Cordella and Yeyati [8] show that when a central bank credibly commits ex ante to a bailout policy for insolvent financial institutions in adverse macroeconomic conditions, it can create a value enhancing effect of the bank that may outweigh the moral hazard cost component of the bailout policy. Ultimately this may lower bank risk taking in the presence of a bailout policy. By extending the model of Diamond and Dybvig [10] to allow for moral hazard, Cooper and Ross [7] show that the moral hazard component2can be mitigated by imposing capital adequacy requirements on banks. A more thorough discussion on the moral hazard implications of deposit insurance programs can be found in the work of Kareken and Wallace [19], Freeman [13], and Hazlett [18].

Farhi and Tirole [11] model a mechanism by which banks choice in setting their overall level of risk of their balance sheet constitutes a strategic complement bears close resem-blance to the model presented in this thesis. The complementarity in their approach arises from monetary policy tools of the regulator. In the event that the banking indus-try becomes financially distressed, the regulator sets a lower interest rate to reduce the costs banks incur on their deposits. Thereby granting banks the resources to recover from an adverse shock. Through this implicit subsidy channeled from consumers to banks, distortions arise in banks’ risk taking preferences. When only a marginal frac-tion of banks take on a risky balance sheet, the regulator is not inclined to conduct a monetary bailout by setting a low interest rate, since it would be to the detriment of consumer welfare. However, when a significantly large fraction of banks takes on a risky balance sheet, the regulator would not have an option. In contrast to Farhi and Tirole

2_{Moral hazard enters the model through the existence of a deposit insurance scheme, that reduces}

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[11], this thesis is concerned with the relation between communicating the regulator’s strategy in initiating a bail-out program and risk taking in banking. Whereas Farhi and Tirole [11] do not consider communication and present possible optimal bail-out programs to curtail financial crises.

2.2 Global games methodology

Banks are modeled as risk takers in this thesis and the level of risk they choose constitutes a strategic complement. As one bank takes on more risk, it becomes more attractive for others to follow suit. The reason for the existence of such complementarity in strategies is due to the presence of a regulator that is concerned with the stability of the banking sector as a whole. As banks take on more risk their probability of bankruptcy increases. This implies that in the aggregate more risk taking would translate into a more fragile banking sector. In turn, this induces the regulator to be more inclined to bail out financially distressed banks, in order to safeguard the stability of the sector. Through this channel of bail out policies, a higher probability of receiving bailout support induces other banks to take on more risk. This notion illustrates the too-many-to-fail problem that Acharya and Yorulmazer [1] deal with.

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In their seminal work on global games Carlsson and Van Damme [5] analyze two-player games where each player can choose between two actions. In the resulting two-by-two payoff matrix a parameter is introduced that each agent imperfectly observes through a noisy signal. They show that in the event that each action is strictly dominant for both players to choose, iterative strict dominance would result in one prevailing equilibrium, even as noise becomes small. Frankel et al. [12] extended the results of Carlsson and Van Damme [5] by introducing conditions that allow for strategic complementarities where actions of agents are linearly ordered. The associated theoretical result allowed for establishing the generality of global games and has been applied to solve the issue of multiplicity of equilibria in various contexts. See for instance Morris and Shin [21] on speculative attacks on pegged exchange rates, and Goldstein and Pauzner [16] on finding a unique equilibrium in the model on bank runs by Diamond and Dybvig [10]. Another reason for adopting the global games technique lies in the implementation of communication of imperfect information through the noise parameters. In evaluating the effects of public information on welfare, Morris and Shin [22] consider the beauty contest3 of Keynes [20] in which a social planner cares to ensure that actions of players are close to some state that agents perceive with noise. Their modeling approach follows that of the global games technique described above, in which agents perceive two types of signals of the state, a public and an idiosyncratic signal. In the event that only a public signal is perceived, increases in the noise term with which the signal is dispersed is detrimental for welfare. However, this result changes when the setup is augmented with private signals. In this context, a range of values with which public information is dispersed can actually be welfare enhancing. This point is also stressed by Svensson [25]. Despite that these theoretical results have not been applied in the literature on the lender of last resort and moral hazard implications in the presence of bailout programs, it has received considerable attention in evaluating the policy effectiveness of central banks in inflation targeting. Demertzis and Viegi [9] analyze the implications of changes in a noise term about the inflation target set by a central bank on the central bank’s credibility and success in achieving realizing the set target. An overview of this topic can be found in the work of Blinder et al. [3].

3

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Chapter 3

The Model

The first section of this chapter is concerned with the actions and payoffs of banks and presents the bank’s payoff function, used to set a preferred risk position. Second, the incentives and actions of the regulator are discussed and an objective function is presented to determine whether to initiate a bail-out program. Additionally, the case is considered where banks have perfect knowledge concerning the regulator’s objective function, but have imperfect information with regard to the regulator’s perception of the costs associated with not initiating a bail-out program. The third section models this imperfect information explicitly in the form of signals that banks perceive about their social value. The final sections concludes by presenting the equilibrium analysis, key results from comparative static exercises.

3.1 Actions and payoffs of banks

Consider a continuous set of banks normalized on the unit interval, i.e. B = [0, 1], where the ith element indexes bank i. I follow Chu [6] and assume that the bank’s managers are its shareholders. Thereby abstracting from potential agency problem that may arise between these two stakeholders1. In this game each bank faces the decision to augment its overall loan portfolio with additional earning assets that are assumed to be relatively more risky than the initial loan portfolio. The intuition of Bolt and Tieman [4] is followed by interpreting the bank’s action as a lowering of the acceptance criteria

1

The terms ‘bank’s management’ and the ‘bank’ as an entity are used interchangeably. Both refer to the decision makers within the bank.

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Chapter 3: The Model. 8

for granting loans. This is parameterized by ai, where ai = 1 indicates that a bank has

lowered its criteria and allows for additional loans being granted2 that would otherwise not be, ai = 0. A binary action space, however, comes with a caveat since banks are only

able to choose the extreme values zero and one and no values between these extremes. However, Oury [24] shows that if there are M binary action variables adopted in the framework of Frankel et al. [12] results remain unchanged. This proposition is used here to argue that the final results would be the same as in a model where banks can decide on M different levels of risk position. Large values of M would then reflect the case of continuity in ai between the values zero and one.

In functional form, bank i’s loans can be expressed as

Li(ai) ≡ L + aiλ; L > 0, λ > 0, ai = {0, 1}

where L denotes the market value of the risk-free portfolio, and λ the additional risky loans granted when the acceptance criteria standards have been lowered.

Turning to the payoff of bank i: Let Qi denote its equity; Di its deposit holdings; rD

the fixed return on deposits; ρ the premium received by equity holders, in addition to rD; ˜r denotes the general random return on the loan portfolio. ˆπi : R2 → (0, 1) is the

expected probability about the initiation of a bail-out scheme, defined by ˆπi= ˆπ(xi, z).

Where xi, and z denote respectively a private and public signal received by banks. The

expected probability of receiving bail-out support is strictly increasing in both types of signals3_{. The randomness in ˜}_{r may result in either positive or negative profits for bank} i. However, due to limited liability negative profits can not exceed the amount of equity raised by the bank. Additionally, in the event that a bank enters a state of financial distress, the regulator may decide to bail out the bank and would set its utility equal to zero but allowing it to operate in the future4. On the other hand, if the regulator decides not to bail out the bank the bank will go bankrupt and exit. Furthermore, it is

2

One may also think of the action variable as a decision variable on whether to augment a low-risk portfolio with additional risk bearing assets. This bears no consequences for the final results.

3_{More structure on how these signals influence the expected probability of a bailout is discussed in}

3.4.

4_{The model considered in this thesis is static and does not allow for inferring dynamic implications}

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assumed that Di = (1 − k)Li(αi), and Qi = kLi(αi), where k ∈ (0, 1) denotes the capital

adequacy requirement.

The profit function of bank i can now be expressed as:

Ui(ai|xi, z) ≡ max{˜rLi(ai) − rDDi− (rD+ ρ)Qi, −(1 − ˆπi)(rD+ ρ)Qi}

By noting the capital adequacy requirement, this expression can be simplified into:

Ui(ai, xi, z) ≡ max{(˜r − (rD+ ρk))Li(ai), −(1 − ˆπi)(rD+ ρ)kLi(ai)} (3.1)

Subsequently a probability of bankruptcy can be derived

Prob[Bankruptcy|xi, z] ≡ Prob[˜r < rD(1 − k) + ˆπi(rD+ ρ)k|xi, z] (3.2)

Since ρ > 0 if follows from (3.2) that the probability of bankruptcy is increasing in the capital requirement parameter k when π > _rD+ρrD . This results stems from the assumption that the capital requirements are binding. As the capital requirements increase banks require a higher return on their portfolio to recover the additional costs on equity relative to debt. These costs increase proportional to the capital adequacy requirement, thereby increasing the probability of bankruptcy.

Suppose that ˜r = ¯r ˜X, where ˜X ∼ Bernoulli(p), in which p ≡ Prob[˜r < rD(1−k)+(rD+

ρ)k]. The assumption of independence between the two states of ˜r and the idiosyncratic signals banks perceive ensures tractability. Note that ˜r = ¯r implies positive profits and ˜

r = 0 results in financial distress. Furthermore, these distributional properties ensure that the expected utility of bank i is strictly increasing in xi. This is a necessary condition

for identifying a unique equilibrium. AppendixA.1outlines the necessary conditions for a general functional form of probability density function (pdf) of ˜r, f , to ensure that the expected utility is increasing in xi. This constitutes a necessary condition for identifying

a unique equilibrium in the equilibrium analysis.

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A ≡ Z

i∈B

aidi. (3.3)

3.2 Regulator’s incentives and bailout policy

In the event of financial instability, the regulator is able to weigh the costs associated with initiating a bailout program for the banking sector and the costs arising from not initiating such a program. Respectively these costs are denoted by ψ and Ψ. ψ can be regarded as the necessary effort conducted by the regulator to set up a bailout program as well as the required resources for funding it. One may think of the costs defined by Ψ to reflect the social costs incurred by society. This includes for instance negative spillover effects of the failure of a bank to the banking sector5, see for instance Allen and Gale [2] and Freixas et al. [15]. Additionally, loss of depositors trust in the system may also accrue to Ψ. The probability that the regulator attaches to this event is the probability of distress of the overall sector, which depends on the risk positions of the banks. According to (3.1), in deciding on their level of risk banks take into account the perceived probability of being bailed out when financially distressed. Hence, this implies that their risk taking action would be dependent on the probability of receiving bailout support: ai ≡ a(π), where π denotes the true probability of a bailout program to be

initiated. The assumption that a is increasing in π can be justified by the notion that as it becomes more likely to be bailed out, incentives for taking on more risk also increase. Based on definition (3.3), this implies that A is dependent on π. Therefore A ≡ A(π), with dA(π)_dπ > 0. Consequently, the expected probability of a bank going bankrupt6 can now be denoted by P ≡ P (A(π)).

In combining these two events and their respective payoffs an objective function can be constructed where the regulator maximizes over the probability of initiating a bailout. This function is denoted Ω : [0, 1] → R and defined by

Ω(π) = −P (A(π))(Ψ(1 − π) + ψπ), (3.4)

5

Since this thesis evaluates the banking sector in a partial equilibrium modeling setting, potential negative spillovers to other sectors of the economy are not considered.

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where Ω reflects the regulator’s utility. First the situation is considered where banks have perfect information with regards to Ψ and ψ, and in the subsequent subsection they have imperfect information with regards to their social value. Note that in optimizing (3.4) the regulator is not subjected to a budget constraint.

Maximizing with respect to π yields the first order derivative

dΩ(π)

dπ = −

dP (A(π))

dπ (Ψ(1 − π) + ψπ) − P (A(π))(ψ − Ψ). (3.5)

Note that dP (A(π))_dπ = f (A(π))dA(π)_dπ > 0, where f denotes the density of the return on loans for banks. Given that P (A(π)) > 0, (3.5) is negative if and only if (iff) ψ > Ψ. This implies that when the costs of bailing out a bank outweigh the costs of letting the bank fail the regulator would set π = 0 in order to maximize its objective function for the possible values of π.

In case ψ < Ψ, (3.5) is negative iff dP (A(π))_dπ _{P (A(π))}1 > _{Ψ(1−π)+ψπ}ψ−Ψ . Again, the regulator would set π = 0 to maximize Ω. However, (3.5) is positive whenever dP (a(π))_dπ _{P (A(π))}1 <

ψ−Ψ

Ψ(1−π)+ψπ, which induces the regulator to set π = 1. Hence, in the event where ψ < Ψ

two possible solutions prevail in maximizing (3.4), namely π = 1 or π = 0. This results in the multiplicity of equilibria that are characterized by either all banks investing in risk bearing assets, when π = 1, or none, π = 0.

3.3 Considering imperfect information about Ψ

In the previous section where banks have imperfect information with regards to Ψ and ψ, the corner solutions π = 1 and π = 0 will ensure that either all banks will engage in risk taking behavior or none, respectively. Following up on this, banks are now considered to have imperfect information regarding the two types of costs. First, the social costs, Ψ, associated with not initiating a bailout program are replaced by θ, and the costs of initiating such a program, ψ, are replaced by c1A + c0(1 − A). I assume that the costs

associated with bailing out banks that have taken on additional risk are higher7 than the banks that did not, i.e. c1 > c0. This implies that the condition for the regulator

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to initiate the bailout policy is altered into A < _c1−c0θ−c0 . Since banks do not observe θ with complete knowledge, they will not be able to infer with certainty whether π = 1. Hence, they form a perceived probability about π, that has been introduced above as ˆ

πi and depends on the signals bank i receives. Consequently, A can not depend on π in

the case where banks have incomplete information regarding Ψ and ψ.

3.4 Timing and information.

The game consists out of three stages. In the first stage, θ is determined by nature. I stage two, each bank simultaneously decides whether to invest in additional risk bearing loans and estimated the probability of being saved when it would face financial distress. In determining the expected probability of being bailed out, the bank considers the likelihood that a sufficient portion of banks in the banking sector engages in investing in risky assets. Hence, risk taking constitutes an action that can be regarded as a strategic complement through the probability of being bailed out. In the third stage, the government determines whether to initiate a bailout program. A graphical depiction of the staging of the game considered is displayed in figure3.1.

Figure 3.1: Game Tree

Bank i decides on ai∈ {0, 1}. Nature decides on ˜r. (¯r− (rD+ ρk))Li(ai) p Regulator decides on π ∈ {0, 1}

survival, bailout support. π = 1 Prob[A_{< θ]} i.e. liquidation; π =0 Prob[A ≥ θ] 1 −p ai= {0, 1}

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imperfect and asymmetric. The idiosyncratic signal of θ received by bank i is denoted by xi : R2× R+→ R and defined as xi(θ, ξi, αx) = θ + ξi √ αx , (3.6)

where ξi ∼ N (0, 1), and is independent of θ as well as across banks. The public signal

of θ is denoted by z : R2× R+→ R and follows the definition

z(θ, ε, αz) = θ +

ε √

αz

, (3.7)

where ε ∼ N (0, 1), and is also independent of θ and ξi across all banks. The signal z is

commonly observed by all banks and may act as their prior distribution for inferring a posterior distribution for θ conditional on their idiosyncratic signal xi. In addition, since

the public signal is equal for all banks, it ensures that the role of a bank’s beliefs about other bank’s beliefs about the regulator’s action are included in the bank’s decision on its preferred risk level. Since the distribution of ξi and ε follows the standard normal,

conventional notation for indicating the associated pdf as well as the cumulative density function (cdf) are adhered to: the pdf is indicated by φ(.) and the cdf by Φ(.). Further-more, it is assumed that the precision by which information is dispersed is known to all players, i.e. αxand αz are known. Additionally, the distributional properties associated

with z may act as a conjugate prior for inferring a posterior distribution about θ. By letting α ≡ αx+ αz it can be shown that that this posterior is characterized by8

θ|xi, z ∼ N (

αxxi+ αzz

α , α

−1

). (3.8)

The cdf associated with this distribution is denoted by µ(θ|xi, z), where θ ∈ R.

3.5 Equilibrium analysis

The defining element of the model is the coordination game in which banks coordinate around θ to infer whether risk taking constitutes a profitable action. The resulting

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equilibrium can be summarized by the following definition, and consists of the necessary conditions defined by Morris and Shin [23]:

Definition 3.1. An equilibrium consists of a (symmetric) strategy for bank i, ai : R2 →

{0, 1}; a cdf µ : R3 _{→ [0, 1]; and solves for equilibrium values x}∗_{(z) and θ}∗_{(z) such that:}

• a∗_i(xi, z) = arg maxai∈{0,1}[E[U (ai|xi, z)]];

• µ(θ|xi, z) is obtained from Bayes’ rule;

• and the resulting banks engaging in additional risk taking A : R2 _{→ [0, 1] is defined}

by A ≡ A(θ, z).

The investment decision of bank i can be derived from the expected incremental gain derived from investing in risk bearing assets. Based on (3.1), the distributional properties of ˜X and the definition of loans, it can be stated that

a∗_i = arg max ai∈{0,1} [E[U (1|xi, z)] − E[U (0|xi, z)]] = arg max ai∈{0,1} ai[p(¯r − (rD+ ρk) − (1 − p)(1 − ˆπi)(rD + ρ)k] × λ (3.9)

For a given realization of θ and received z it is supposed that there exists a pivotal x∗(z)

for which banks who receive a signal xi > x∗(z) will find it optimal to invest in a risk

bearing project9. Based on (3.3) and (3.6) the entire mass of banks investing in a risk bearing project, A, can be redefined by A : R2→ B and takes the functional form of

A(θ, z) = Z i∈B ∞ Z x∗_(z) √ αxφ( √ αx(xi− θ))dxidi = 1 − Φ(√αx(x∗(z) − θ)).

The inner integral states the probability that bank i will invest in a risk bearing project and the outer integral sums these probabilities across the banking sector. Note that

9

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A is strictly increasing in θ. Hence, the condition10 _{for initiating a bailout program,} A(.) < θ, can be restated as θ > θ∗_{(z), where θ}∗_{(z) is obtained through}11

A(θ∗(z), z) = θ∗(z)

1 − Φ(√αx(x∗(z) − θ∗(z))) = θ∗(z).

Solving for x∗(z) then yields:

x∗(z) = θ∗(z) − √1 αx

Φ−1(θ∗(z). (3.10)

Given the new condition for a bailout program to be initiated, θ > θ∗(z), it can be stated

that ˆπi= Prob[θ ≥ θ∗(z)|xi, z]. Since each bank forms a different posterior distribution

about θ, as indicated by (3.8), the posterior probability of the event that a bailout program is initiated can be denoted by:

Prob[θ ≥ θ∗(z)|xi, z] = 1 − Φ( √ α(θ∗(z) − αx α xi− αz α z)). (3.11)

Since the term ˆπi is the only term in (3.9) that is dependent on xi and Prob[θ ≥

θ∗(z)|xi, z] is increasing in xi, it can indeed be stated that

ai=      1 if xi ≥ x∗(z) 0 if xi < x∗(z).

Substituting (3.11) for ˆπi in (3.9) and equating to zero allows for identifying the bank

that has received the private signal that renders the bank indifferent whether to lower its monitoring standards. Consequently, the solution yields the threshold value x∗(z).

Φ(√α(θ∗(z) − αx α x ∗_{(z) −} αz α z)) = p(¯r − (rD+ ρk)) (1 − p)(rD+ ρ)k . 10

Without loss of generality, the condition A < θ−c0

c1−c0 is normalized by setting c0= 0 and c1= 1. 11_{Since x}∗

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Following, substituting (3.10) and simplifying the new expression then yields

Γ(θ, z) = γ, (3.12) where Γ(θ, z) ≡ √αz αx(θ − z) + Φ −1_{(θ), and γ ≡}q α αx p(¯r−(rD+ρk))

(1−p)(rD+ρ)k. Given the equilibrium

relationship between x∗(z) and θ∗(z) in (3.10), existence of a unique equilibrium can be established by considering the properties of the function Γ. Note that for any value of z ∈ R, Γ is continuous in θ; with Γ(0, z) = −∞, and Γ(1, z) = ∞. Therefore, to ensure that a unique solution to (3.12) exists, the following condition must hold:

∂Γ(θ, z) ∂θ = αz √ αx + 1 φ(Φ−1_{(1 − θ))} > 0.

Since αz, αx and φ(.) are strictly positive, Γ is indeed strictly increasing in θ. This

implies that a unique solution, θ∗(z) can be found that solves (3.12), provided that αx > 0. This is summarized in the following proposition.

Proposition 3.2. (Unique and monotone equilibrium.)

Let αx and αz denote respectively the precision with which private and public signals are

distributed. There always exists a monotone equilibrium and it is unique iff αx > 0 and

αz ≥ 0. This equilibrium is characterized by identifying a value for θ∗ and x∗. It can be

shown that θ∗ equals the proportion of risk taking banks and x∗ identifies the threshold value that renders a bank indifferent in deciding whether to invest in a risk bearing asset.

Under the conditions imposed on αx and αz to identify a unique equilibrium Morris

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Chapter 4

Results

4.1 Changes in the information structure

It is shown in the previous sections that when the signals (3.6) and (3.7) are dispersed with no noise, the model solves for multiple equilibria. However, when a noise term is included in the distribution of both signals, a unique equilibrium prevails with critical value θ∗. Apart from solving the issue of obtaining multiple equilibria, information plays

an additional role. It is not only the precision with which the signals reflect the true value of θ, but it also matters whether common knowledge among banks exists about the value of θ. That is to say, whether a bank knows that other banks know that θ exceeds its critical threshold value, θ∗. The public signal plays a vital role in the formation of

common knowledge among banks, in the sense that all banks receive the same public signal and know that other banks have received this signal as well. In absence of a private signal, all banks would therefore infer an equal posterior probability for the event that a bail-out program is initiated. This section considers several comparative static exercises on the precision with which the regulator may communicate the two types of signals in the banking sector. Specific attention is paid to the role of communication in the formation of the group of banks that take on risk bearing assets, A.

The condition for which θ∗ and x∗ can be sustained in equilibrium can be restated as:

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Chapter 4. Results 18 Γ(θ∗(z), z) − γ = 0 αz √ αx (θ∗(z) − z) + Φ−1(θ∗(z)) − r 1 + αz αx Φ−1(v) = 0, (4.1)

where v ≡ p(¯_{(1−p)(rD+ρ)k}r−(rD+ρk)) is introduced for ease of exposition. A preliminary evaluation of (4.1) can be conducted by regarding the limiting results associated with private signals being dispersed with relatively high precision; i.e. αx → ∞ for given αz, or αz → 0 for

given αx. In either of these two cases, √αz_αx → 0 and

q

1 +_αxαz → 1. This implies that

(4.1) solves for θ∗(z) → v. Lemma4.1 summarizes this result.

Lemma 4.1. (Limiting results of relatively precise private information.)

By taking the limiting cases: αx → ∞ for given αz, or αz→ 0 for given αx, there exists

a unique monotone equilibrium in which the regulator would initiate a bail out program for the banking sector iff θ > θ∗(z), where θ∗(z) = v.

Figure 4.1: Equilibrium correspondence of αxand θ∗(z), for given αz

0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 −10 −5 0 5 10 θ α_x Γ ( θ ,z)− γ

Figure4.1summarizes the equilibrium values of θ for varying αx along the black line of

which the implicit functional form can be inferred from condition (4.1). The remaining parameter values are summarized in the appended tableA.1. For intermediate values of αx, and given αz, one may observe a nonlinear relation between the equilibrium value

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Chapter 4. Results 19

result that as public information is distributed perfectly, i.e. αz → ∞, then θ∗(z) → z,

which follows from (3.7).

Lemma 4.2. (Limiting result of relatively precise public information.)

By taking the limiting cases: αz → ∞ for given αx, there exists a unique monotone

equilibrium in which the regulator would initiate a bail out program for the banking sector iff θ > θ∗(z), where θ∗(z) = z.

By evaluating the limit results summarized by lemmas 4.1and 4.2the implicit relation between θ∗ and αx, displayed in 4.1, is not monotone. Rather the slope of this implicit

function is dependent on the realized value of the public signal. It can be noted from (3.8) that based on the relative precision with which the public and private signals are dispersed through the sector, banks assign relative weights to their received signals to form the first moment of their posterior distribution about θ. This moment will tend to the value of the public signal as the public signal is distributed with increasingly more precision relative to the precision with which the private signals are dispersed. Based on lemma 4.1 and 4.2, the more the value of z deviates from v, the more pronounced will the curvature of the implicit relation between θ∗(z) and αx be for low values of αx.

In order to derive the slope of the implicit equilibrium relation, let θ∗ : R × R+→ R be defined by θ∗ ≡ θ∗_{(z, α}

x). Totally differentiating (4.1) with respect to θ∗ and αx yields

∂θ∗(z, αx) ∂αx = αz αx√αx(θ ∗_{(z, α} x) − z −√1_αΦ−1(v)) αz √ αx + 1 φ(Φ−1_(θ∗_(z,αx))) (4.2)

For (4.2) to be strictly positive it is required that z < θ∗(z, αx) −√1_αΦ−1(v). Similarly,

z > θ∗(z, αx) −√1_αΦ−1(v) would ensure that (4.2) is strictly negative. Figure 4.2

illus-trates this finding by presenting a downward sloping curve in the αx,θ-plane for a higher

value of z than depicted in figure 4.1. Additionally, lemma 4.3 summarizes this result in a formal manner.

Lemma 4.3. (Slope of equilibrium relation between αx and θ.)

For z < θ∗(z, αx) − √1_αΦ−1(v) it follows that ∂θ ∗_(z,αx) ∂αx > 0; and for z > θ ∗_{(z, α} x) − 1 √ αΦ

−1_{(v) it is the case that} ∂θ∗(z,αx) ∂αx < 0.

Combining the assumption that the regulator is not able to set z, along with lemma4.3

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Figure 4.2: Higher value of the public signal

0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 −15 −10 −5 0 5 θ α_x Γ ( θ ,z)− γ

in strictly lower or higher values of θ∗, depending on the realization of z. Additionally, in equilibrium it is the case that A(θ∗(z), z) = θ∗(z). This would imply that for small αx

and z deviating highly from θ∗(z, αx) −√1_αΦ−1(v) the proportion of banks that invest

in risk bearing assets will be considerably higher or lower compared to the equilibrium characterized by αx→ ∞, in which case A → v. The result that the equilibrium values

of θ∗, and implicitly A, deviate from the value v more as public information is dispersed with more precision for given αx is dubbed by Morris and Shin [23] ‘overreaction to

public information’. By noting that private signals are distributed independently across banks, only the public signal and its distributional properties provide a framework for banks to infer the beliefs of other banks. Since the public signal is equal for all banks. Intuition for the result of ‘overreaction’ is then gained by considering the public signal being distributed relatively more precise, banks posterior about θ will then become more similar. This amplifies banks’ common knowledge of each other beliefs whether θ exceeds it critical value for a bail out program to be initiated. Additionally, all banks know that they have nearly identical information such that one bank knows that other banks will find it profitable to set ai either equal to one or zero. Banks do have perfect knowledge

of each others optimal strategy once αx = 0 and this will consequently result in the

multiplicity of equilibria1 where A = 0 or A = 1.

1_{Note that α}

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Following this discussion and based on lemmas4.1,4.2and4.3the following proposition can be formulated that summarizes the main implications of increases in precision of public information for a given value of αx.

Proposition 4.4. (Implications of precision in the public signal on the proportion of risk taking banks.)

For a given realization of θ, consider the values of precision with which the public signal is distributed, α_z0 and α00_z, where α0_z < α00_z. Corresponding to the two values of precision

with which public signals are distributed are the respective equilibrium values of θ: θ∗0 and θ∗00. For α0_z < α00_z it then holds for all values of αx that |θ∗

00

− θ∗0

| > 0. This illustrates the overreaction to public information in risk taking by banks as public information is distributed more precise, since the equilibrium value of the proportion of risk taking banks equals the equilibrium value of θ.

In light of the research question regarding the effects of precision with which the regulator communicates the signals, two conclusions. First, conditional on the realization of the public signal, overreaction to public information arises by communicating the public signal with more precision for a given precision in communicating private information. This follows from the result that banks place more emphasis on the public signal in deriving the posterior distribution of θ. However, if private signals are not communicated that precise, this may result in a considerable lower or higher proportion of banks that engage in risk taking behavior. Secondly, lemma 4.5 indicates that in the event of ‘overreaction’ the equilibrium value of θ can not exceed the value of z. Since the regulator is not able to set this value, policy parameters will have decreasing influence on the equilibrium value of θ as the relative precision of distributing the public signal increases. In contrast to this, in the event that private information is distributed relatively more precise, v enters the equilibrium value of θ. This implies that the regulator can alter the equilibrium value of A by setting the policy parameters, such as the capital adequacy requirement. This is considered next.

4.2 Changes in the capital adequacy requirements

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changes in k can be inferred from reviewing (3.2). Increases in k imply that the proba-bility of bankruptcy of a bank is enhanced. This reduces the incentive for the bank to engage in more risk taking, since in the event of bankruptcy the potential loss is larger due to increase in obligation to hold relatively more equity.

In order to derive a function for the implicit effect of the parameter k on the equilibrium value of θ, let θ∗ : R × (0, 1) → R be defined by θ∗ ≡ θ∗_{(z, k), and v : (0, 1)}2 _{→ R be}

defined by v ≡ v(p, k). Totally differentiating (4.1) with respect to θ∗ and k then yields2

∂θ∗(z, k) ∂k = q α αx 1 φ(Φ−1_(v(p,k))) ∂v(p,k) ∂k αz √ αx + 1 φ(Φ−1_(θ∗_(z,αx))) . (4.3)

Note that this expression is strictly negative, implying that as the capital adequacy requirements is increased the equilibrium value of θ decreases. Consequently, the pro-portion of banks taking additional risk declines. Based on lemma 4.1 and taking the limit αx → ∞, it can be shown that ∂θ

∗_(z,k) ∂k →

∂v(p,k)

∂k . Figure 4.3 illustrates the

im-plicit equilibrium relation between αx and θ∗ for a higher level of k relative to the value

adopted in figure4.1.

Figure 4.3: Imposing higher capital adequacy requirements

0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 −10 −5 0 5 10 θ α_x Γ ( θ ,z)− γ

An intuition for the result that (4.3) can be obtained by considering (3.2). As k increases, the costs of bankruptcy of the bank increase as well, since banks have to hold relatively

2

Based on the definition of v, it can be shown that ∂v(p,k)_∂k =−p(¯r−rD)

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more equity for a given amount of liabilities. This increases the costs associated with taking on more risk, which confirms the initial observation outlined at the beginning of this section.

In order to evaluate the existence of a tradeoff between the precision with which the public signal is dispersed and the capital adequacy requirements by keeping θ∗ constant, let k : R+→ (0, 1) be defined by k ≡ k(α_z). Totally differentiating (4.1) with respect to αz and k then yields

∂k(αz) ∂αz = 1 √ α(θ ∗_{(z) − z −}_√1 αΦ −1_(v)) q α αx 1 φ(Φ−1_(θ∗_(z))) ∂v(p,k) ∂k . (4.4)

Note that (4.4) is strictly negative when z < θ∗(z) − √1 αΦ

−1_{(v), and strictly positive}

when z > θ∗(z) −√1 αΦ

−1_{(v). Implications of this result can be evaluated by considering}

that for given αx, any increase in αz, when z < θ∗(z) −√1_αΦ−1(v), would imply that k

has to decrease in value to keep the equilibrium value of θ constant. In contrast, when z > θ∗(z) − √1

αΦ

−1_{(v) the opposite is true: an increase in α}

z would result in a higher

value of θ∗, which implies that k has to increase to keep θ∗constant. By using the result presented in4.4 the following proposition can be formulated.

Proposition 4.5. (Tradeoff between precision in distributing the public signal and the capital adequacy requirement.)

For a given realization of θ, consider the values of precision with which the public signal is distributed, α_z0 and α00_z, where α0_z < α00_z. Corresponding to the two values of precision with which public signals are distributed are the respective equilibrium values of θ: θ∗0

and θ∗00. For α0_z < α00_z it then holds for all values of αx that |θ∗ 00

− θ∗0| > 0. Hence,

by lowering αz, the difference between θ∗ 00

− θ∗0

, in the event that θ∗00− θ∗0

> 0, can be offset by lowering k.

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in αz is nullified. A similar result applies when αz → ∞, since the equilibrium value of

θ would then equal z.

Intuition for this result is gained by considering a lowering in the capital requirements. This induces banks to take on more risk, since the potential losses in the event of bankruptcy are reduced due to a lowering of the equity to dept ratio. However, by low-ering the precision with which the public signal is distributed the regulator creates more uncertainty on whether a bail-out program will be initiated. In turn, this increased un-certainty lowers, across all banks, the expected probability of receiving bail-out support, which offsets the benefit of lower capital requirements. Effectively the expected costs of bankruptcy remain unchanged.

4.3 Changes in the expected return on risk bearing assets

Following, the effect of changes in the expected return of banks’ loan-portfolio on the equilibrium value of θ is considered. Based on the distributional properties of the return on loans, an increase in p would imply that the expected return on loans increases. At first sight it can be noted from (3.1) that for higher values of p the gains from investing in risk bearing assets are higher and the costs of bankruptcy are reduced. Whereas lower values p would have opposite effects. This implies that higher expected returns banks would be relatively more inclined to invest in risk bearing assets then when conditions are less favorable. To evaluate this link more formally, let θ∗ : R × (0, 1) → R be defined by θ∗ ≡ θ∗(z, p). Totally differentiating (4.1) with respect to p and θ∗ yields3

∂θ∗(z, k) ∂k = q α αx 1 φ(Φ−1_(v(p,k))) ∂v(p,k) ∂p αz √ αx + 1 φ(Φ−1_(θ∗_(z,αx))) . (4.5)

This expression is strictly positive. Implying that as the expected return increases it would lead to an overall increase in the proportion of banks that engage in risk taking, for all values of αx. Figure4.4illustrates this result by depicting (4.1) in the αx, θ-plane

for a lower value of p than in figure 4.1. Proposition4.6summarizes this result.

3_{Based on the definition of v, it can be shown that} ∂v(p,k) ∂p =

¯ r−rD−ρk

(1−p)2(rD+ρ)k is strictly larger than

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Figure 4.4: Lowering of expected return of banks’ portfolio

0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 −10 −5 0 5 10 θ α_x Γ ( θ ,z)− γ

Proposition 4.6. (Implications of changes in p for A.)

For a given realization of θ, consider the values for the probability of success of the risk bearing investment p0 and p00, where p0 < p00. Corresponding to these two parameter

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Chapter 5

Concluding Remarks

5.1 Conclusion

In this thesis the implications of noisy communication by a regulator are evaluated for its effects on moral hazard in bank behavior, conceptualized by excessive risk taking. We use global games techniques to solve for a unique equilibrium that identifies a proportion of banks that engage in excessive risk taking. The use of this noise assists in identifying this unique equilibrium, but the results also suggest that noise can be introduced for strategic use by the regulator to curtail risk taking by banks. Additionally, the proportion of banks engaging in risk taking is lowered when the expected return on their portfolio diminishes.

More specifically, the main findings suggest that as the regulator communicates its strat-egy with less noise in the signal that is common to all banks, the banking sector tends to overreact to the received information relative to the case in which banks have perfect private information. This result is due to the concept that banks’ beliefs about the reg-ulator’s strategy will depend more on what they belief other banks’ predictions of the regulator’s action are. Additionally banks will put more weight on the received public signal. In the case where there is no private information at all about the regulator’s be-havior multiple equilibria prevail, since all banks have only public information at their disposal and will therefore only rely on their belief about other banks’ predictions to decide on whether to take additional risk or not. This would imply that either all banks engage in excessive risk taking behavior or none, which constitutes the multiplicity and

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Chapter 5. Concluding Remarks 27

reflects the concept of ‘overreaction to public information’. Hence, the precision with which public information is dispersed through the banking industry is effective in influ-encing risk taking behavior. It is also the case that it may be too effective in case it leads to overreaction

Furthermore, it is found that increasing the reserve requirement ratio lowers overall risk taking behavior by banks, regardless of the noise composition in which the regulator communicates. Since a lowering of capital requirements results in lower expected costs of bankruptcy, combined with limited liability this would induce banks to take on more risk. For a given level of precision with which private information is distributed, decreasing the precision with which public information is dispersed would create more uncertainty for banks on whether the regulator will initiate a bail-out program. This increase in uncertainty may offset the lowering of capital adequacy requirements. Thus keeping constant the aggregate level of risk taking in the banking industry. This results suggests a tradeoff between being more strict on risk taking by imposing higher capital requirements or being less precise on whether a bail-out will be initiated.

5.2 Suggestions for further research

Although the model considered in this thesis includes a limited amount of policy mea-sures at the disposal of the regulator1, the result that there may exists a tradeoff between communicating the regulator’s strategy with less precision and increasing the capital ad-equacy requirement is interesting to invite future research to be conducted on other types of policy measures. For instance, one may think of evaluating the implications of penal-ties, or obligatory insurance schemes to curtail moral hazard behavior, where the policy measure being taken is contingent on the risk taking behavior of banks.

Additionally, this model evaluates a continuum of homogeneous banks. Heterogeneity could be introduced by allowing for differences in size, e.g. total assets, between banks, or more general importance for the stability of the sector. One may also consider first mover implications, where one type of bank takes risk before others do.

1_{Namely its precision parameters used for communicating its signals about its strategy to the banking}

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Chapter 5. Concluding Remarks 28

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Appendix A

Derivations and proofs.

A.1 Implications of the return distribution on expected

utility.

To identify a unique equilibrium it is required that the expected utility of bank i is strictly increasing in xi. In order to generalize the set up presented in the outline of

the model general distribution functions for the return on loans are considered. In what follows it is assumed that ˜r ∼ f (˜r|ai), and that the cdf is characterized by F (˜r|ai).

Additionally, it is again assumed that ∂πi_∂xi > 0. First, the probability of bankruptcy is

restated to identify the pivotal value of ˜r

Prob[Bankruptcy|xi, z] ≡ Prob[˜r < rD(1 − k) + πi(rD+ ρ)k|xi, z]

= Prob[˜r < r_i∗|x_i, z],

where r∗_i ≡ rD(1 − k) + πi(rD+ ρ)k. The expected profits of bank i can now be stated

as:

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To evaluate the effect of a change in xi on the expected profits of bank i, the first order

derivative of (A.1) with respect to xi is evaluated. The term _∂xi∂

R∞

r∗_i ˜rf (˜r|ai)d˜r will be

evaluated with the help of Leibniz’s rule1_{. Note that} ∂ri∗

∂xi = (rD+ ρ)k ∂πi

∂xi This allows for

stating the first order derivative of (A.1) as:

Rearranging terms then yields

∂E[U (ai|xi, z)] ∂xi = L(ai)(rD+ ρ)k ∂πi ∂xi [F (r_i∗|ai) − (1 − πi)(rD + ρ)kf (ri∗) − r∗_if (r∗_i|ai) + (rD+ ρk)f (r∗i|ai)]. (A.2)

For _∂xi∂ E[U (ai|xi, z)] > 0 to hold, two sufficient conditions are F (r∗_i|ai) > r∗_if (r∗_i|ai)

and (rD+ ρ)k < rD + ρk. The last of these two conditions holds for k < 1. Since it is

assumed in our model that k ∈ (0, 1), this conditions holds. The former condition can be written as

1_{According to Leibniz’s rule, the first order derivative of F (x) =} Rb(x)

a(x)f (x, t)dt with respect to

x can be stated as: dF (x)_dx = f (x, b(x))b0(x) − f (x, a(x))a0(x) +R_a(x)b(x)_dxdf (x, t)dt. This implies that

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Appendix A: Derivations and proofs 31

F (r_i∗|a_i) f (r∗_i|ai)

− r∗_i > 0.

Suppose that ˜r ∼ U [r

¯(ai), ¯r], where r¯: {0, 1} → R, defined as r¯= r¯(ai) with

d

dair_¯(ai) < 0.

This distributional form ensures that as bank i decides to take on more risk it allows for a higher variance in its asset’s return and for a lower expected return, reflecting the risk enhancing effect associated with lowering monitoring standards. The conditions stated above can now be stated as

F (r_i∗|a_i) f (r∗_i|ai) − ri = r∗_i − r ¯(ai) ¯ r − r ¯(ai) (¯r − r ¯(ai)) − r ∗ i = r∗_i − r ¯(ai) − r ∗ i > 0

Hence, regardless of the action, ai, of bank i, its expected utility will be strictly increasing

in xi when r

¯(ai) < 0.

Generally, let h : R → R be defined by the left hand side of the condition, i.e. h(y) =

F (x)

f (x) − x. Where f is a particular density function and F the associated cumulative

distribution function. It is explicitly assumed that f represents a distribution that has a single central mass of positive probability weights and has a support that covers the real line. First, the first order derivative is evaluated, yielding dh(x)_dx = F0(x)f (x)−F (x)f_{(f (x))}2 0(x) −

1 = −F (x)f_{(f (x))}0(x)2 . Hence, maximizer or minimizer candidates can be identified by solving

f0(x∗) = 0. To identify either a maximum or minimum, the second-order derivative is evaluated, yielding:

d2h(x)

dx2 = −

(f (x)f0(x) + F (x)f00(x))(f (x))2− 2F (x)(f0_(x))2_{f (x)}

(f (x))4 ,

substituting the candidate solution x∗, for which f0(x∗) = 0, then yields:

d2h(x∗)

dx2 = −

F (x∗)f00(x∗) (f (x∗₎₎2 .

Since F and f are both strictly positive, x∗denotes a minimizer when f00(x∗) < 0. This is

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Appendix A: Derivations and proofs 32

e.g. normal density, or densities that have a central mass with positive probability weight.

A.2 Derivation of the posterior distribution of θ.

Bank i observes a single signal that follows the process described by (3.6), where xi ∼ N (θ, α−1x ). Additionally, bank i observes a public signal that, based on (3.7),

is characterized by z ∼ N (θ, α−1_z ). The distribution of z may serve as a prior

distribu-tion for inferring a posterior about θ. Implying that θ|z ∼ N (z, α−1_z ). Furthermore, it is assumed that both αx and αz are common knowledge. According to Bayes’ rule the

posterior distribution, h : R2 → R+_{, of θ would be}

Inserting the functional form of the respective densities then yields

φ(√αx(xi− θ)|θ)φ( √ αz(θ − z)|z) = r αx 2πexp[− αx 2 (xi− θ) 2_] ×r αz 2πexp[− αz 2 (θ − z) 2_] ∝ exp[−αx 2 (xi− θ) 2₋αz 2 (θ − z) 2_] = exp[−1 2(θ 2_(α x+ αz) − 2θ(αxxi+ αzz) + αxx2i + αzz2)].

For ease of exposition, let δ1 ≡ αx+ αz, δ2 ≡ αxxi + αzz and δ3 ≡ αxx2_i + αzz2. The

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Appendix A: Derivations and proofs 33 exp[−δ1 2(θ 2₋2δ2 δ1 θ +δ3 δ1 )] = exp[−δ1 2(θ − δ2 δ1 )2]exp[−δ 2 1 2 ( δ3 δ1 −δ 2 2 δ2₁)] ∝ exp[−δ1 2(θ − δ2 δ1 )2].

This final expression implies that the posterior density of θ is characterized by a normal distribution with mean δ2_δ1 and variance _δ11. Hence,

θ|xi, z ∼ N ( αxxi+ αzz αx+ αz , (αx+ αz)−1), as was to be shown.

A.3 Tables

Table A.1: Parameter values for figures.

Parameter Figure 4.1 Figure4.2 Figure 4.3 Figure4.4

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Communication and Risk Taking in the Banking Industry