12.3 DEA in Practice: Insurance Agencies

In order to illustrate a practical application of DEA we consider an example from the empirical study of  Scheel (1999). This concrete data analysis is about the efficiency of 63 agencies of a German insurance company, see Table 12.1. The input $ X\in\mathbb{R}_+^4$ and output $ Y\in\mathbb{R}_+^2$ variables were as follows:

$ X_1$
: Number of clients of Type A,
$ X_2$
: Number of clients of Type B,
$ X_3$
: Number of clients of Type C,
$ X_4$
: Potential new premiums in EURO,
$ Y_1$
: Number of new contracts,
$ Y_2$
: Sum of new premiums in EURO.

Clients of an insurance company are those who are currently served by the agencies of the company. They are classified into several types which reflect, for example, the insurance coverage. Agencies should sell to the clients as many contracts with as many premiums as possible. Hence the number of clients ($ X_1$, $ X_2$, $ X_3$) are included as input variables, and the number of new contracts ($ Y_1$) and the sum of new premiums ($ Y_2$) are included as output variables. The potential new premiums ($ X_4$) is included as input variables, since it depends on the clients' current coverage.

Summary statistics for this data are given in Table 12.2. The DEA efficiency scores and the DEA efficient levels of inputs for the agencies are given in Tables 12.3 and 12.4, respectively. The input efficient score for each agency provides a gauge for evaluating its activity, and the efficient level of inputs can be interpreted as a 'goal' input. For example, agency 1 should have been able to yield its activity outputs ($ Y_1=7$, $ Y_2=1754$) with only 38% of its inputs, i.e., $ X_1=53$, $ X_2=93$, $ X_3=4$, and $ X_4=108960$. By contrast, agency 63, whose efficiency score is equal to 1, turned out to have used its resources 100% efficiently.


Table 12.1: Activities of $ 63$ agencies of a German insurance company
  inputs   outputs
Agency $ X_1$ $ X_2$ $ X_3$ $ X_4$   $ Y_1$ $ Y_2$
1 138 242 10 283816.7   7 1754
2 166 124 5 156727.5   8 2413
3 152 84 3 111128.9   15 2531
. . . . .   . .
. . . . .   . .
. . . . .   . .
62 83 109 2 139831.4   11 4439
63 108 257 0 299905.3   45 30545


Table 12.2: Summary statistics for $ 63$ agencies of a German insurance company
  Minimum Maximum Mean Median Std.Error
X1 42 572 225.54 197 131.73
X2 55 481 184.44 141 110.28
X3 0 140 19.762 10 26.012
X4 73756 693820 258670 206170 160150
Y1 2 70 22.762 16 16.608
Y2 696 33075 7886.7 6038 7208


Table 12.3: DEA efficiency score of the 63 agencies
Agency Efficiency score
1 0.38392
2 0.49063
3 0.86449
. .
. .
. .
62 0.79892
63 1
19456 STFnpa03.xpl


Table 12.4: DEA efficiency level of the 63 agencies
  Efficient level of inputs
Agency $ X_1$ $ X_2$ $ X_3$ $ X_4$
1 52.981 92.909 3.8392 108960
2 81.444 60.838 2.4531 76895
3 131.4 72.617 2.5935 96070
. . . . .
. . . . .
. . . . .
62 66.311 87.083 1.5978 111710
63 108 257 0 299910
19462 STFnpa03.xpl