Usually, a theoretical model does not reflect all facets of reality, but is invariably an approximation of this reality. Nonetheless a theoretical model that is too abstract, regardless of the extent of its mathematical elegance, may be useless in real life. Therefore, there is a trade-off between the amount of abstraction and the degree of realism or practicality, and by implication the relevancy of the model's policy recommendations. However, a theory should have enough built-in rigor and be coupled with as few complexities as possible. Often a very abstract construct can be still fruitful if it is appropriately calibrated, i.e. if the model parameters are properly chosen.
The purpose of this paper is to calibrate a model of the interbank market due to Rochet and Vives (2004). See also Rochet (2008). By calibration is not only meant the selection of realistic figures for the parameters of the model but also optimizing the relations in order to solve for all the underlying unknown variables, which will be identified later, and this under different scenarios. In general, the methodology relies on using the 'solver' command in Excel in solving for these unknown variables, which number two, and are defined by two different equations, one of them being non-linear.
One of the key outputs of the simulations is the unconditional probability of failure for a solvent but illiquid bank. If an insolvent bank fails this is natural in the modern capitalistic economic system. If a solvent bank fails because of illiquidity this seems unfair. Therefore a system must be constructed to prevent, or at least, mitigate the failure of solvent banks. Does the interbank market foster such a system without interference? This is one of the key questions posed in this paper. Other questions relate to whether regulators, by imposing targets for capital and liquidity ratios, or by providing liquidity assistance through the discount window at preferential rates, can minimize the negative externality of the failure of illiquid but solvent banks.
In the theoretical literature on banking institutions there is a controversy about the proper approach to modeling bank crises. One strand believes that bank runs are explained by sunspots and self-fulfilling panics (Bryant, 1980, Diamond and Dybvig, 1983, 2000, and Diamond, 2007). In addition, these authors contend that there is a constant probability of bank failure, the latter being utterly inevitable. Azar (2012) finds, by Monte Carlo simulation, that the implied mean probability of a bank run in such models is quite reasonable and ranges between 4.15% and 4.23%, supporting the contention of these models. However, the other trend in the literature argues that bank crises are related to bad returns and bad fundamentals about bank prospects (Allen and Gale, 1998, Rochet and Vives, 2004, and Rochet, 2008). There is some evidence that crises are predictable by business cycle patterns (Gorton, 1988). Nonetheless, the issue of the cause of bank crises is not yet settled.
This paper considers the model in Rochet and Vives (2004), and in Rochet (2008). This model is constructed on the same time frame as in Diamond and Dybvig (1983, 2000) and in Diamond (2007), which consists of three periods t=0, t=1, and t=2. However there are at least four differences. One of them is that the time frame is long run, taking years, in the DiamondDybvig model, while the time frame is short run, taking days, in the Rochet-Vives model. Second, the Diamond-Dybvig model is applicable to depositor bank runs while the Rochet-Vives model is applicable to the interbank market, where liquidity can be provided by other banks in the system, and where banks monitor each other. Third, in the Rochet-Vives model a bank may never fail even if everybody withdraws, as long as the fundamentals are good. Lastly, the Rochet-Vives model allows for a role to the central bank in providing liquidity through the discount window, or in imposing liquidity and capital ratios.
The paper is organized as follows. In the following section, section 2, the theoretical model of Rochet and Vives (2004) and Rochet (2008) is presented with its implications on the relations between the underlying variables. Section 3 dwells on the justification of the elements of calibration. Section 4 presents the empirical facts and compares them to the theoretical inferences. In addition, in section 4, the most suitable and powerful regulatory mechanisms are discussed. The last part summarizes and concludes.
THE THEORETICAL MODEL
The model assumes three periods: t=0, t=1, and t=2. At time t=0, the bank raises uninsured deposits of D, normalized to unity. To this liability is added a fraction E of equity. The bank invests the proceeds into cash reserves m, and into a portfolio of assets I, or loans I. The gross random return on these loans is R. Hence loans produce RI in dollar returns that help normally to pay back the deposits at t=2, and keep the rest to stockholders. However depositors have the right of withdrawing early at time t=1, if they see fit. Early withdrawal depends on the observation by depositors of private signals on the future realization of the random return R. If early withdrawals exceed m, then the bank is forced to sell some of its assets. It is assumed that such sales are done at a premium X, which is called, by the authors of the model, the cost of 'fire sales' in the market for interbank funds. In other terms banks that are subject to a liquidity shock are penalized. The reason for this penalty is that counterparties are unsure of the purpose of the asset sales. This purpose could be illiquidity but it could also be insolvency (selling bad loans). Hence there is asymmetric information. However, if a bank has recourse regularly and for an extended period of time to the interbank market as a borrower, this will raise the suspicion of other banks about this bank's solvency situation. That is why, in the interbank market, banks are usually successful in monitoring each other, and, in normal times the premium [lambda] reflects accurately the average bank's riskiness. In times of crisis this premium may reach intolerable levels, which may necessitate the intervention of a lender of last resort, like the central bank.
It can be demonstrated that there is a solvency threshold [R.sub.s]. If the actual realization of the random return R is lower than [R.sub.s] then the bank is insolvent. This threshold is equal to: