The following provides a short description of the DEA methodology.18 Assume that there are k inputs and m outputs for each of the n banks. For the i-th bank these are represented by the vectors xi and yi, respectively. The k x n input matrix, X, and the m x n output matrix, Y, represent the data of all n banks. It is also assumed that banks are operating with constant returns to scale (CRS). For each bank, the purpose is to obtain a measure of the ratio of all outputs over all inputs, such as u’yi/v’xi, where u is an m x 1 vector of output weights and v is k x 1 vector of input weights (superscript’ indicates transpose).
To select the optimal weights, the following mathematical programming problem has to be solved:
To avoid infinite solutions to the above problem, the constraint v’xi = 1 is imposed, which leads to:
where the notation of the weights has changed from u and v to u. and v, respectively, in order to reflect the transformation.
Using the duality in linear programming, an equivalent envelopment form of the above problem can be derived:
Where θ is a scalar and λ is a n x 1 vector of constraints. The value of 9 is the efficiency score for the i-th bank, which ranges between 0 and 1. Therefore the problem has to be solved n times, one for each bank, in order to have the full picture.
However, the CRS assumption is rather restrictive. A number of factors, including imperfect market competition, may cause a bank to be not operating at optimal scale, i.e. along the flat portion of the long-run average cost curve. To allow variable returns to scale (VRS), it is necessary to add to the problem in equation (4) the convexity constraint:
where I is n x 1 vector of ones.
The difference between the efficiency scores calculated under the VRS and the CRS assumptions provides an indicator of scale inefficiency. In other words, the difference between the two efficiency scores indicates the additional gain in efficiency that could be achieved if banks were operating at the long-run equilibrium CRS.
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Prepared by A. Giustiniani. I am grateful to Kevin Ross for his comments and technical help. All the flaws and errors remain mine.
A SIng savings banks pool resources (e.g., capital, liquidity, risk management) with a central entity while maintaining some practical some practical and legal independence, also called “cold-merger,” is a sort of joint-venture in which participating. In the note, especially in the empirical section, the term “merger” will be used indistinctively.
Recently one of the original SIP (Banco Base) broke up. While one of the participating savings banks (Caja del Mediterráneo) is currently seeking a new partnership, the other savings banks decided to form a new SIP (Effibank). Moreover, the three Basque savings banks (Kutxa, BBK, and Vital) are negotiating a possible merger.
The FROB has been authorized to acquire stakes in banks’ share capital for a limited period of time (no longer than 5 years) to strengthen their own funds. The beneficiary institutions have to implement a recapitalization plan, approved by the BdE. In case of a savings bank or an SIP, the lending activity has to be transferred to a bank by the mechanisms stipulated by the law (indirect exercise of financial activity or conversion into a foundation owing a bank).
In spinning-off their banking business, two institutions so far (BFA-Bankia and Caixa) have segregated their impaired real-estate assets in a separate company (either credit institution or other financial entity) together with other profitable assets to compensate for the low income stream of the former group of assets.
Given the context, perhaps the number of branches and employees would have been more direct variables to consider. Unfortunately, those data were not available for all credit institutions in the considered period. Other specifications of the model comprising flow and balance sheet variables have been tested without significant improvements.
The analysis does not take into account the recent breakdown of the SIP at the basis of Banco Base as well as the potential merger between Basque savings banks.
Geometrically, the scale score would be represented by the cotangent of the angle formed by the ray joining the axes origin with the bank-data-point.
T. Kohers, M. Huang, and N. Kohers (2000).
As for the other results, one savings bank, BBK, is now fully efficient under the CRS and the VRS model; while the other two savings banks from the Basque region (Vital and Kutxa) could improve their efficiency by scaling up their activities. Ibercaja and Pollensa are not locally efficient any longer and this would explain the drop in their technical efficiency, since the scale factor is equal to 1.