Abstract
This chapter provides an in-depth analysis of the cross-frontier methodology, an innovative approach based on data envelopment analysis (DEA), for estimating the relative efficiency of alternative organizational forms in an industry, and testing hypotheses primarily founded on the agency theory arguments on the coexistence in the insurance industry of two organizational forms—stock insurers, owned by stockholders and mutual insurers, owned by policyholders. The analysis involves estimating the efficiency of the firms in each group not only with respect to a reference frontier consisting only of firms from its own group but also with reference to the other group’s frontier. This allows calculating cross-to-own efficiency ratios which measure the distance between the stock and mutual frontiers. These ratios are key statistics to test the superiority of one technology over the other. Linear optimization procedures are used to estimate production, cost and revenue frontiers, both for the standard own-frontiers setups as well as the cross-frontiers models.
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Notes
- 1.
Sorting is predicted to occur through the natural operation of the market as firms compete with one another in terms of price, risk management and claims settlement services, product offerings, and another product and service dimensions (Cummins et al. 2004).
- 2.
Empirical evidence has shown that the access to capital is one of the main motivations for the conversion of mutuals to stocks (e.g. Viswanathan and Cummins 2003; Zanjani 2007; Erhemjamts and Leverty 2010). Several studies (e.g. Mayers and Smith 1988; Lamm-Tennant and Starks 1993) have shown that stocks work in riskier lines than mutuals.
- 3.
Regarding the estimation methodology, there are two main approaches in efficient frontier analysis: the econometric approach and the mathematical programming approach. The econometric approaches specify a production, cost, revenue or profit function with a specific shape and usually make assumptions about the distributions of the inefficiency and error terms. There are three principal types of econometric frontier approaches: the stochastic frontier approach (SFA), the distribution free approach (DFA) and the thick frontier approach (TFA). The mathematical programming approaches put significantly less structure on the specification of the efficient frontier and do not decompose the efficiency and error terms. Data envelopment analysis (DEA) is the most used mathematical programming approach which employs linear programming to measure the relationship of produced outputs to assigned inputs and determines the efficiency score as an optimization result. The free disposal hull (FDH) approach is a special configuration of DEA where the convexity assumption on the efficient frontier is relaxed. The ranged-adjusted measure DEA (RAM-DEA) is non-radial in the sense that it does not preserve the mix between inputs in movements toward the frontier (see e.g. Cummins and Weiss 2013).
- 4.
“Best practice” efficient frontier consists of the dominant firms of a reference set. The efficiency values of each firm are measured relative to best practice efficient frontiers. Technical efficiency is defined as the ratio of the input usage of a fully efficient firm producing the same output vector to the input usage by the analyzed firm. Cost efficiency for a specific firm is calculated as the ratio of the costs of a fully efficient firm with the same output quantities and input prices to the specific firm’s actual costs. Cost efficiency is the product of technical and input allocative efficiency. Thus, input allocative efficiency is the ratio of cost efficiency to technical efficiency and gives information on whether the firm uses the optimal mix of inputs. Revenue efficiency is defined as the ratio of the revenues of a specific firm to the revenues of a fully efficient firm with the same input vector and the same output prices. Revenue efficiency is the product of the output technical efficiency to the output allocative efficiency. Therefore, the output allocative efficiency can be calculated by the ratio revenue efficiency to output technical efficiency and gives information on whether the firm uses the optimal combination of outputs.
- 5.
Technology is defined as “including the contractual relationships comprising the firm, organizational, management, and hierarchical structures, and physical technologies” (Cummins et al. 2004, p. 3116).
- 6.
Studies using cross-frontier analysis (e.g. Cummins et al. 1999, 2004; Biener and Eling 2012) provide evidence that the two groups of firms (stocks and mutuals) use different technologies and operate with different frontiers. Thus, comparing efficiencies based on the pooled frontier is not informative.
- 7.
Output allocative efficiency gives information on the success of the firm in choosing the revenue maximization output combination.
- 8.
The constant returns to scale approach (CRS) is used most commonly in literature and measures departures from optimal scale as inefficiency. It represents the optimal outcome from an economic perspective. That is, with CRS, firms are not consuming unnecessary resources because they are too large or too small (see e.g. Aly et al. 1990).
- 9.
Since the cross-frontier analysis assumes constant returns to scale (CRS), the output and input orientations will provide equivalent measures of technical efficiency. So, we use the same notation to express technical efficiency although, as we explained above, technical efficiency is calculated input-oriented DEA and revenue efficiency is calculated output-oriented DEA.
- 10.
The four size quartile dummy variables could be constructed in the following way: quartiles are formed based on the overall sample including all stocks and mutuals. The measure of size could be total assets, total premiums or total output. Quartile 1 could include the smallest firms and quartile 4 the largest. The quartile 1 size dummy variable takes 1 if the firm is classified by its size in quartile 1and 0 otherwise.
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Acknowledgments
The authors wish to express their gratitude to two anonymous referees and Minwir Al-Shammari and Hatem Masri (Editors) for their valuable and helpful comments, which have contributed to improving the quality of this chapter. This research has been partially funded by the Ministerio de Economía y Competitividad (project ECO 2011-26996) and by the Junta de Andalucía (projects SEJ-417 and P10-TIC-6618).
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Rubio-Misas, M., Gómez, T. (2015). Cross-Frontier DEA Methodology to Evaluate the Relative Performance of Stock and Mutual Insurers: Comprehensive Analysis. In: Al-Shammari, M., Masri, H. (eds) Multiple Criteria Decision Making in Finance, Insurance and Investment. Multiple Criteria Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-319-21158-9_4
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