Average Efficiency Levels and Returns to Scale

As presented in section 3.3, scale efficiency results are derived from the cost efficiency estimations with CRS and VRS. Table 3.1 shows the average cost efficiency under CRS (CE^CRS) and VRS (CE^VRS), as well as the scale efficiency (SE) and profit efficiency scores (PE^VRS). Mean values are presented by year for all insurers and by total output size quartiles.

We first focus on efficiency results for the entire sample, then discuss the results for the single size quartiles. Since efficiency scores were estimated separately for every year in the

43 The only exceptions are the years 2003 and 2004: In 2003, direct insurers show slightly higher mean value for premium income as compared to independent agent insurers; and in 2004, they show slighlty higher average output and investment income.

observation period, we can draw conclusions about the development of efficiency between groups during the observation period. Because firms exited and entered the market during the observation period, the annual subsamples do not include the same number of observations each year. Thus, the specific efficiency scores for each year cannot be compared individually to one another. However, we can analyze the development of the differences in cost and profit efficiency levels between the different groups over time.

The results show that CE^CRS ranged between 0.27 and 0.46 during the observation period.

CE^VRS ranged between 0.40 and 0.53, and PE^VRS ranged from 0.58 to 0.63. Because of the discrepancy between CE^CRS and CE^VRS, it is important to analyze scale efficiency to determine how firms can improve their efficiency by adjusting their size. Scale efficiency ranges between 0.73 and 0.90 on average, meaning that firms could improve their efficiency by 10-27 percent by moving to the optimal size.

Table 3.1: Average Cost, Scale and Profit Efficiency by Size Quartiles, 1997-2005 Note: Estimated with “DEA Excel Solver”. Source: Own estimations.PE VRS 0.6004 0.5886 0.5932 0.5968 0.5753 0.6130 0.6290 0.5990 0.5966 1997 1998 1999 2000 2001 2002 2003 2004 2005 Quartile 1 CE CRS 0.1548 0.1577 0.1652 0.1965 0.2569 0.2618 0.2980 0.2698 0.2398 CE VRS 0.4183 0.3916 0.3874 0.3910 0.4428 0.3950 0.4569 0.4327 0.4689 SE0.4224 0.4357 0.4821 0.5363 0.5798 0.6613 0.6838 0.6563 0.5592 PE VRS 0.5856 0.3804 0.4148 0.4715 0.4536 0.4599 0.5166 0.4828 0.5112 Quartile 2 CE CRS 0.2822 0.2800 0.2383 0.3137 0.3527 0.4354 0.4547 0.4060 0.4532 CE VRS 0.3329 0.3269 0.2706 0.3481 0.3647 0.4515 0.4907 0.4392 0.4616 SE0.8476 0.8498 0.8818 0.8495 0.9595 0.9632 0.9278 0.9277 0.9838 PE VRS 0.6631 0.4964 0.4727 0.4663 0.4097 0.5172 0.5333 0.5164 0.5273 Quartile 3 CE CRS 0.3491 0.3673 0.3006 0.4296 0.4114 0.4624 0.5385 0.4603 0.4461 CE VRS 0.3583 0.3765 0.3071 0.4366 0.4366 0.4662 0.5483 0.4655 0.4593 SE0.9710 0.9737 0.9763 0.9820 0.9440 0.9915 0.9807 0.9878 0.9726 PE VRS 0.6526 0.6368 0.6476 0.6234 0.6128 0.6912 0.6829 0.6172 0.5925 Quartile 4 CE CRS 0.4858 0.4382 0.3586 0.5254 0.4599 0.4911 0.5631 0.4462 0.4743 CE VRS 0.6674 0.6514 0.6402 0.6638 0.5685 0.5164 0.6125 0.5899 0.6111 SE0.6926 0.7322 0.6098 0.7976 0.8402 0.9692 0.9453 0.7679 0.7871 PE VRS 0.4859 0.8410 0.8352 0.8368 0.8313 0.7836 0.7833 0.7855 0.7553 Total CE CRS 0.2958 0.3108 0.2660 0.3645 0.3688 0.4127 0.4636 0.3940 0.4033 CE VRS 0.4359 0.4366 0.4003 0.4575 0.4530 0.4573 0.5271 0.4812 0.5002 SE0.7288 0.7479 0.7401 0.8027 0.8278 0.8963 0.8844 0.8327 0.8256 PE VRS 0.6004 0.5886 0.5932 0.5968 0.5753 0.6130 0.6290 0.5990 0.5966

Figure 3.4 shows that, for German insurers, scale efficiency increases with firm size: the figure presents a scatter plot of scale efficiency scores against the logarithm of total output, which measures the size of the insurance firms for the year 2000.⁴⁴ Scale efficiency increases up to an output size of approximately 1.18 billion €, where firms show a scale efficiency of 1 (i.e., where they operate at optimal size). For firms larger than 1.18 billion €, scale efficiency decreases again.

Figure 3.4: Scale Efficiency versus Size, Year 2000

0,00 0,20 0,40 0,60 0,80 1,00 1,20

0 2 4 6 8 10 12

size (log (output in €))

Source: Own calculations.

An analysis of the nature of returns to scale for every firm reveals that all firms up to 1.18 billion € show increasing returns to scale, while larger firms show decreasing returns to scale.

These results are confirmed for the whole observation period. Table 3.2 illustrates the number of insurers operating under increasing, constant, and decreasing returns to scale (in percent) for every year by size quartiles, where firm size increases from quartile 1 to quartile 4. The results show that increasing returns to scale apply to the majority of the German life insurance firms, with only a few firms operating under constant or decreasing returns to scale.

44 Because of space limitations, Figure 3.4 presents the scatter plot only for 2000, but the results are consistent and stable over the whole observation period.

1.20

1.00

0.80

0.60

0.40

0.20

0.00

scale efficiency

Table 3.2: Returns to Scale by Size Quartiles, 1997-2005 1997 1998 1999 2000 2001 2002 2003 2004 2005 Quartile 1% IRS 100 100 100 100 100 100 100 100 100 % CRS 0 0 0 0 0 0 0 0 0 % DRS 0 0 0 0 0 0 0 0 0 Quartile 2 % IRS 100 100 100 100 85.0 100 100 100 100 % CRS 0 0 0 0 5.0 0 0 0 0 % DRS 0 0 0 0 10.0 0 0 0 0 Quartile 3 % IRS 100 100 100 100 0 100 100 100 90.0 % CRS 0 0 0 0 0 0 0 0 5.0 % DRS 0 0 0 0 100 0 0 0 5.0 Quartile 4 % IRS 0 21.78.74.8080.080.05.00 % CRS 5.6 4.3 4.3 4.8 0 5.0 5.0 5.0 0 % DRS94.473.987.090.510015.0 15.0 90.0100 Total % IRS 78.380.477.477.9 46.9 95.0 95.0 76.572.5 % CRS 1.21.11.11.1 1.2 1.3 1.310.21.3 % DRS20.518.521.521.851.9 3.83.8 22.226.3 Note: Estimated with “DEA Excel Solver”. Source: Own estimations.

Over the whole observation period, only the largest 20 percent or so show decreasing returns to scale on average.⁴⁵ With regards to the underlying technology, the maintained hypothesis should be VRS.

An analysis of the average efficiency scores and returns to scale determination for the single size quartiles underscores these results. The results in table 3.1 show a positive relationship between cost and profit efficiency levels and size quartile, with larger firms being more cost and profit efficient. This relationship holds true for the whole observation period with only few exceptions.

Scale efficiency also increases from quartile 1 to quartile 3, and then decreases in quartile 4 (with the only exceptions in 2001 and 2005, where it already decreases in the third quartile).

The reason for this decline is that the majority of the firms in quartile 4 exhibit decreasing returns to scale, i.e., they have exceeded their optimal size. These results are also confirmed by the nature of returns to scale by quartiles, wherein all firms in the two smallest quartiles (1 and 2) show increasing returns to scale, and the majority of firms in quartile 3 also represent increasing returns to scale, with the only exception being year 2001. Constant or decreasing returns to scale can be found only in quartiles 3 and 4, while the majority of the firms showing decreasing returns to scale belong to quartile 4.

45 The only exceptions are the years 2001-2003: In 2001, the majority of the firms showed decreasing returns to scale, while in 2002 and 2003 the number of firms showing decreasing returns to scale is very small (3.8 percent). The results are broadly consistent for cost and profit efficiency estimations. Firms reach their optimal size at a similar size in the case of profit efficiency, and the number of firms operating under DRS slightly increases.