Muestra probabilística o aleatoria

2.3.1

Players and action sets

The model extends the results of Repullo (2004) by introducing a regulator who requires a levy on secured or unsecured debt. We consider the following groups of risk-neutral players: shareholders who own n = 2 banks, secured (unsecured) depositors, and a regulator who decides about the level of the levy ex ante. Each group of players only lives for one period, does not have alternative investment possibilities and desires to consume at the end of its life, i.e. at the end of the investment period, when the payo¤s are distributed among shareholders and depositors. Henceforth we sometimes write "banks" (i.e. bank managers) instead of shareholders. As we do not take any principal-agent-problems into account, shareholders and banks have the same incentives. Hence, a di¤erentiation between shareholders and banks is not necessary.

Banks

The two banks are located on the ends of a "street" of length one. At the beginning of the investment period (discrete time), shareholders invest an in…nitesimal amount of equity capital in the bank. In excess of this in…nitesimal amount of equity capital, shareholders have to pay a levy. This levy might be required ex ante (Section 2.4.1) or ex post of the bank’s investment (Section 2.4.2). If this levy has to be paid ex ante, the shareholders have to pay the levy before the bank is allowed to invest. Shareholders have then invested the equity capital and the levy, but only the equity capital appears on the balance sheet, i.e. can be invested in projects by the bank. If this levy has to be paid ex post, shareholders only have to pay the in…nitesimal amount equity capital before the bank is allowed to invest. In the ex post case, the levy has to be paid only if the bank’s project has been successful.

Each bankj and j chooses its deposit rate r and either invests in a prudent asset or in a gambling asset.6 _{All characteristics of the prudent asset (e.g. return, probability of success)}

are indexed with "P". The characteristics of the gambling asset are indexed with "G". The balance sheet total of bankj consists of the in…nitesimal amount of equity capital and deposits denoted by Dj.

6_{The equity capital investment does not have an impact on the banks’ behaviour. The investment} only clari…es that the banks are owned by shareholders who receive the net return of the investment.

The prudent investment yields a rate of return (henceforth we often write return) P 2

(0;1)with probabilitypP _{and a return of zero, otherwise. The gambling investment generates}

a return G ₂ (0;1) with probability pG and a return of zero otherwise. Consequently, the banks can choose between a project with a relatively high return, but low probability of success, and a project with a relatively low return, but a high probability of success. The return function R can be summarized as follows, where RG _{is the return for the gambling investment and R}P

for the prudent investment:

RG = ( G _{with probability p}G 0 with probability 1 pG RP = ( P _{with probability p}P 0 with probability 1 pP .

We assume that the return of the gambling asset is higher than the return of the prudent asset, i.e. G _> P_{, but the expected return of the gambling asset is lower, i.e. p}P P _{> p}G G_.

Limited liability of banks provides incentives to invest in the gambling asset and thus to cause moral hazard.

The residual of the project payo¤ (after depositors are paid) is distributed among the banks’ shareholders. We further assume that shareholders have a time preference for consumption, which is represented by the discount factor >0.

Depositors

The depositors are continuously distributed on the "street".7 The aggregate market volume of deposits is constant to one. Depositing funds in a bank causes transportation costs per unit of distance, >0, which can be interpreted as the heterogeneity among banks. Imagine, for instance, that interest rates on deposits are not the only feature a customer takes into account when depositing his funds in a bank. Advisory services or/and the number of cashpoints might also play an important role for his decision. Following the cashpoint argument, we can assume that potential depositors are located between two cashpoints, owned by di¤erent banks, and customers choose the bank with the nearest one.

The amount of depositors that bankj attracts depends on its o¤ered deposit rate, on the deposit rates o¤ered by its neighbour and on the transportation costs . We denote the deposit rate of bank j by rj 2 (0;1) and of bank j’s neighbour by r j. The banks’ investment is

private information and cannot be observed by depositors, except in Section 2.5.1. A depositor located between bankj and bank j is indi¤erent between these banks if he receives the same net return from both banks, i.e. if

rj z =r j [1 z],

where z is the distance between the indi¤erent depositor and bank j (or its cashpoint) and 1 z the distance between the indi¤erent depositor and bank j.8 _{We do not explicitly}

exclude negative net payo¤s for depositors. This is in line with Repullo (2004). Solving for z reveals the "amount" (mass) of depositors on the line between bank j and j who decide to deposit their funds in bank j:9

z(rj; r j) = Dj(rj; r j) =

1 2+

rj r j

2 ; (2.1)

where Dj(rj; r j) 0 (a negative amount of depositors is impossible). The demand of

deposits, Dj, decreases in and r j and increases in rj. The more heterogenous banks are,

i.e. the higher the distance between the cashpoints (higher ), the less the impact of deposit rates on deposit demand.

Finally, the insurance premium is normalized to zero.10

Regulator

In the initial period, the regulator announces: the bank levy rate ,

the minimum amount of equitykDj each bank has to hold, wherek is greater than the

in…nitesimal amount of equity (Section 2.6.2), the pro…t tax rate (Section 2.6.2).

Dj is supposed to be observable and veri…able. Hence, the announcement to require a

levy is credible as both parties can sign a contract concerning the duty to pay the levy. If the project fails, shareholders receive zero payment and depositors are paid by the regulator unless they are unsecured.

2.3.2

Sequence of events

Figure 2.1 illustrates the sequence of events for the case, in which the levy has to be paid ex ante. At the beginning of the investment period, the regulator announces the bank levy rate (or the tax rate and the minimum amount of capital 2.6). Thereafter, the banks o¤er a deposit rate and decide about the type of investment simultaneously. Then, depositors choose a bank 8_{Note that the comparison between transportation costs and returns is possible because of the nor-} malization of payo¤s and returns.

9_{Note that the demand of deposits does not depend on the banks’ investment risk because deposits} are covered by a deposit insurance.

10_{An insurance premium of zero is the extreme of an unfairly priced insurance. Unfairly priced means} that the insurance premium is not equal to the expected payment of the insurance to the depositors. The introduction of a fairly priced insurance premium is impossible if several banks are insured and the characteristics of the banks are private information (Chan, Greenbaum & Thakor (1992)). For the impact of deposit insurance on risk-taking, see, for instance, Matutes & Vives (2000) and Cordella & Yeyati (2002).

in order to store their funds. Finally, payo¤s are realized and distributed among depositors and shareholders.

Figure 2.1: Sequence of events

t Regulator announces the required bank levy 1 = t 0 = t

Banks offer a deposit rate and decide about their type of investment simultaneously

Returns realized and distributed among shareholders and depositors Depositors choose a bank simultaneously 2 = t

2.3.3

Equilibrium concept

We focus on symmetric equilibria: a prudent equilibrium, where both banks choose the prudent asset P and the optimal deposit rate in equilibrium rP_{, and a gambling equilibrium, where}

both banks choose the gambling assetGand the optimal deposit rate in gambling equilibrium rG_{. A prudent (a gambling) equilibrium, respectively, exists if neither bank j nor bank j}

has an incentive to deviate unilaterally by choosing the gambling (prudent) investment and the respective optimal deposit rate and if depositors do not have an incentive to change their decision about where to deposit their funds.

Competition between the banks only takes place in the deposit market because the in- vestment of bank j does not have an impact on bank j. We solve the game by backward induction, beginning with the second stage where depositors choose between bank j and bank j. Taking the decision of depositors into account, we proceed with the …rst stage of the game where the banks decide simultaneously about the asset type and the deposit rate they o¤er. Since the banks choose their assets and the deposit rate simultaneously, they cannot observe their neighbour’s choice.

2.4

The impact of a bank levy on banks’risk-taking