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Data Envelopment Analysis and its Use in Banking

This article is based on a talk given by Dr Emmanuel Thanassoulis of Warwick Business School to the Mathematical Programming Study Group.

Data Envelopment Analysis

Data Envelopment Analysis (DEA) is a way of assessing the comparative performance of units within an organisation, e.g. branches of a bank, schools within a Local Education Authority, sales outlets of a retailer. These units must perform broadly comparable functions but may vary in size, environment, resources used and results achieved. DEA seeks to measure their efficiency in terms of how well each unit performs when compared with its peers.

Consider an example. A bank has 4 branches within which there are two types of labour: supervisors and trainees, and a single measure of output: thousands of transactions processed.

The data are as shown in the table.

Branch Inputs Outputs
Supervisory Hours Trainee Hours Transactions (000)
1 2 3 1
2 4 1 1
3 2 2 1
4 1 4 1

Clearly branch 1 is doing worse than branch 3 because it uses the same number of supervisory hours but more trainee hours to achieve the same output. But you cannot make direct comparisons with branches 2 and 4. These results are typical: pairwise comparison normally doesn't get very far.

In order to make progress we make some assumptions. Consider two possible sets of inputs and outputs, P1, P2. In our example they could represent branches 1 and 2, i.e. the sets { 2, 3; 1 }, { 4, 1; 1 }. Then we assume that:

  • given any two possible sets of inputs and outputs, P1, P2, any weighted average of these is also possible, i.e. we can define new sets of inputs and outputs of the form [ αP1 + (1 - α)P2 ] for 0 < α < 1;
  • there are constant returns to scale, i.e. if you double the inputs you can double the outputs;
  • you can dispose of excess inputs and outputs at zero cost.

Assumption 1 means that we can make a new possibility by weighting branch 3 by 0.5 and branch 4 by 0.5 to yield:

0.5 * { 2, 2; 1 } + 0.5 * { 1, 4; 1 } = { 1.5, 3; 1 }

This combination can now be compared directly with branch 1. It uses the same trainee hours but fewer supervisory hours to achieve the same result and so is clearly more efficient.

Using other values of α and considering combinations of branches 2 and 3 as well as branches 3 and 4 we can define the efficient frontier, B2 B3 B4 as shown in Figure 1.

Figure 1: Combinations of Trainee and Supervisory Hours to Achieve 1 Unit of Output

These are possible ways of producing 1 unit of output which are efficient in the sense that for each ratio of supervisory to trainee hours it is not possible to achieve the output with fewer hours' labour.

Assumption 3 means that any point in the shaded area above the efficient frontier is also a possible way of achieving the outputs. It is known as the production possibility set. Points strictly within the interior, such as B1, are inefficient. Their efficiency is defined as the proportion by which their inputs could be reduced while retaining the mix of inputs, i.e. how far one could move along the radius from the origin before reaching the efficient frontier. The efficiency of B1 is thus the ratio OM/OB1 = 0.857. The point M is known as the target and B3 and B4 are B1's efficient peers.

Issues with the DEA Approach

In using DEA one is making a fundamental leap of faith with assumption 1, that convex combinations of observed possibilities will work. DEA should only be used where this is credible. On the other hand, assumption 2, about constant returns to scale, can be relaxed (and is in the Warwick DEA software). Assumption 3 is more generally applicable, although there will be some situations where it is not.

DEA tackles problems which might also be tackled using regression. DEA offers the advantage that it identifies an efficient rather than an average level of output against which the performance of individual units is judged. Further, while in regression we must specify in advance the functional form linking inputs to output, that is not necessary in DEA. This makes it possible to consider multiple inputs against multiple outputs in DEA while in regression we must either have a single input with multiple outputs or a single output with multiple inputs. Set against this there is a greater risk of distortion of results by outliers. It is normal to do several runs with different sets of inputs and outputs and check that the results are robust.

Uses in Banking

Two approaches are used in applying DEA in banking:

  • the production view, in which branches are viewed as using labour, capital, space, etc to process transactions, make sales of financial products, etc;
  • the intermediation view, in which branches are viewed as collecting funds and deposits from the neighbourhood and intermediating them into loans and other income-earning activities.

These two views are complementary and can be integrated into an overall assessment, as shown in Figure 2.

Figure 2: Integrating DEA Assessments in Banking

The intermediation view was used in a sales maximization model at a UK bank. Branches were considered as using resources of sales people, opening hours, market size, customer base, transactions processed and facilities. The outputs were mortgage applications, insurance sales and savings accounts sales. This model assumed fixed returns to scale and market size was one of the main inputs. The model served to measure the efficiency of branches in generating business within their markets and comparisons were made primarily among branches with similar types of market.

This model was supplemented by a resource optimization model which took direct staff costs as its input and considered mortgage applications, insurance sales, savings account sales and transactions as outputs. This model used variable returns to scale and was used to set target values to emphasise to branches the potential to improve performance.

The Bank of Finland (i.e. the Finnish central bank) uses DEA in its Financial Markets Department to monitor banks operating in Finland. It uses the production view and seeks to assess the efficiency of banks' payment and account transaction services. The inputs are the number of branches, number of ATMs, the use of labour and the number of computer terminals used. The outputs are the number of transactions handled by clerks, the number of ATM transactions, cash withdrawals and loans processed. Its model uses variable returns to scale and seeks to identify banks' overall efficiency and decompose it into that part which is attributable to the scale at which the unit is operating and the rest, which is attributable to management.

As well as studies across banks, the Bank of Finland also does cross-time studies to compute Malmquist indices. These decompose changes in a unit's productivity over time into that due to the unit's becoming more efficient and that due to shifting of the boundary.

Outcome from DEA Assessments

The main outcome from DEA assessments tends to be the identification of efficient peers as role models for each inefficient unit and the setting of targets. This gets away from the underlying theory of DEA and its applicability, which may well be open to challenge. It therefore acts as a spur to improvement within the normal processes of management. At a higher level, DEA assessments may be used to gauge the level of returns to scale with a view to long-term changes in the structure of an organisation, e.g. whether to reduce the number of branches or concentrate transaction processing.

Related articles include Data Envelopment Analysis and How to Minimize Capital Gains Tax. To find other articles, refer to the MP in Action page.