# Combining ANP and TOPSIS Concepts for Evaluation the Performance of Property-Liability Insurance Companies

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2 Ration of changes for direct premium, SC11 Business Index (C1) Ration of changes for direct paid loss, SC12 Ration of changes for retain premium, SC13 Retain premium / Shareholders, SC21 Gross premium / Shareholders, SC22 Efficiency of property-liability insurance Whole company Operating Index (C2) Net reinsurance comm. / Shareholders, SC23 Total reserve / Shareholders, SC24 Ratio of shareholder changes, SC25 Special claim reserve/ Shareholders, SC26 Profit Ability Index (C3) Return on Shareholders, SC31 Loss ration of retain earn premium, SC32 Fig. 1: Criteria carcass of property-liability insurance efficiency comparisons also need to be created and an eigenvector can be obtained for each element to show the influence of other elements on it. The relative importance values are determined with a scale of 1 to 9, where a score of 1 represents equal importance between the two elements and a score of 9 indicates the extreme importance of one element (row component in the matrix) compared to the other one (column component in the matrix) [10,11]. The supermatrix concept is similar to the Markov chain process [7]. To obtain global priorities in a system with interdependent influences, the local priority vectors are entered in the appropriate columns of a matrix, known as a supermatrix. As a result, a supermatrix is actually a partitioned matrix, where each matrix segment represents a relationship between two nodes (components or clusters) in a system [11]. In essence, this solution algorithm derives weights that account for component interaction, which is a clear benefit of the dynamic ANP model over static models. TOPSIS methodology: The TOPSIS was first proposed by Hwang and Yoon [5]. The underlying logic of TOPSIS is to define the ideal solution and the negative ideal solution. The ideal solution is the solution that maximizes the benefit criteria and minimizes the cost criteria; whereas the negative ideal solution maximizes the cost criteria and minimizes the benefit criteria. The optimal alternative is the one, which is closest to the ideal solution and farthest to the negative ideal solution. The ranking of alternatives in TOPSIS is based on the relative similarity to the ideal solution, which avoids from the situation of having same similarity to both ideal and negative ideal solutions. PROCEDURE OF EVALUATING THE PROPERTY-LIABILITY INSURANCE COMPANIES The modified Delphi method is then adopted to summarize the expert opinions in order to construct an evaluation model to assess the property-liability insurance companies. Based on factors to evaluate property-liability insurance, i.e., whole company operating, business and profit ability, ANP is used to illustrate the problems and combine the three factors to establish the model for performance evaluation in this study. Finally, 14 renowned property-liability insurance 57

4 C 1 C 2 C 3 Fig. 2: Inner dependence among criteria SC 11 SC 12 SC 13 SC 21 SC 22 SC 23 SC 24 SC 25 SC 26 SC 31 SC 32 Fig. 3: Inner dependence among sub-criteria Table 1: The Un-weighted Super-matrix Goal C1 C2 C3 SC11 SC12 SC13 SC21 SC22 SC23 SC24 SC25 SC26 SC31 SC32 C C C SC SC SC SC SC SC SC SC SC SC SC As discussed in the appendix and shown by the dotted bracket in Fig. 3, the supermatrix in this study covers all the elements in the network. The supermatrix, inserted with respective vectors and matrices obtained before is shown in Table 1. A weighted supermatrix is transformed first to be stochastic. After entering the normalized values into the supermatrix and completing the column stochastic, the supermatrix is then raised to sufficient large power until convergence occurs [7,11]. The current supermatrix reached convergence and attained unique eigenvector. Step 5: Estimate the relative weighted of the elements of each level: After entering the normalized 59 values into the supermatrix and completing the column stochastic, the supermatrix is then raised to sufficient large power until convergence occurs [7,11]. The current super-matrix reached convergence and attained unique eigenvector. Table 2 provides the final limit this limit matrix is column stochastic and represents the final eigenvector. The property-liability insurance to manage weight of main-criteria are in order C1 (0.435)>C3 (0.336)>C2 (0.229) and weight of sub-criteria are in order SC32 (0.290)>SC13 (0.131)>SC21 (0.129). Step 6: Determine the property-liability insurance company s performance by TOPSIS Establishing and normalizing the D matrix: According to 2005 disclosure information management method of

5 property-liability insurance get 14 number of propertyliability insurance companies sub-criteria primary data [14] in Table 5. After the decision matrix was determined. Based on Table 3 normalize matrix by Eq. 3. Since the sub-criteria weights (W ANP ) have been obtained from ANP, the weighted normalized performance matrix can be calculated by Eq. 4. Table 2: weights of the criteria, sub-criteria Name Normalized by cluster Name Normalized by cluster C SC C SC C SC SC SC SC SC SC SC SC SC Table 3: The decision-making matrix Company SC11 SC12 SC13 SC21 SC22 SC23 SC24 SC25 SC26 SC31 SC32 Taiwan Fire and Marline Chung Kuo Taiping Fubon Zurich Taian Mingtai Central The First Union Shonkong South-China Cathay Century Tokio Marznrzue Newa Table 4: Resultant of S i and Si Compamy S i Si Taiwan Fire and Marline Chung Kuo Taiping Fubon Zurich Taian Mingtai Central The First Union Shonkong South-China Cathay Century Tokio Marznrzue Newa * i Determining the C : First, determine the ideal solution and negative ideal solution using formulae 7 and 8. Table 4 displays those results. Next, calculate the relative closeness to the ideal solution of each * alternative, C i, using Eq. 9. According to Table 5 the ranking order of the three property-liability insurance companies are Tokio Marznrzue Newa (0.8767), Central ( ) and Fubon (0.7085). Table 5: Summary of the TOPSIS Rank Company 1 Tokio Marznrzue Newa Central Fubon South-China Cathay Century Chung Kuo Shon-kong The First Zurich Mingtai Taiwan Fire and Marline Taian Union Taiping DISCUSSION This proposal further combines the concepts of the ANP and TOPSIS models to evaluate and rank property-liability insurance company performance. The ANP and TOPSIS -based decision-making method for constructing an evaluation method can provide property-liability insurance company decision makers or administrators with a valuable reference for evaluating the company performance. In particularly, investors and administrators frequently lack objective C * i 60

6 decision-making procedures and assessment criteria. Moreover, there are simply too many property-liability insurance companies and thus selecting insurance companies with optimal efficiency is extremely difficult. Finally, by applying ANP in obtaining criteria weight and TOPSIS in ranking on those results, the Tokio Marznrzue Newa (0.8767) insurance company is identified as the optimal insurance company. This study used limited financial data for the performance evaluation and each property-liability insurance company organized its statements differently, creating further limitations. Moreover, regarding of the non- financial factor can accede to in the treat discussion item, for example: service quality of property-liability insurance, customer's satisfaction etc. Second, follow-up research can be the same method but adopt during different study, to judge the stability of the study results; and the present approach can be applied to other industries, for example: Life Insurance Industry, electronics industry etc. REFERENCES 1. Brockett, P.L., W.W. Cooper, L.L. Golden, J.J. Rousseau and W. Yuying, Evaluating solvency versus efficiency performance and different forms of organization and marketing in US property-liability insurance companies. Eur. J. Oper. Res., 154 (2): Chen, T.Y. and T.L. Yeh, A measurement of bank efficiency, ownership and productivity change in Taiwan. Service Ind. J., 20 (1): Chen, Z., H. Li and C.T.C. Wong, Environal planning: Analytic network process model for environmentally conscious construction planning. J. Cons. Eng. Manag., 131 (1): Cummins, D.J., N.A. Doherty and A. Lo, Can insurers pay for the big one? Measuring the capacity of the insurance market to respond to catastrophic losses. J. Banking and Finance, 26: Hwang, C. and K. Yoon, Multiple Attribute Decision Making: Methods and Application, New York: Springer Publications. 6. Gatzert, N. and H. Schmeiser, The influence of corporate taxes on pricing and capital. 7. Saaty, T.L., Decision Making with Dependence and Feedback: The Analytic Network Process, Pittsburgh: RWS Publications. 8. Saaty, T.L. and M. Takizawa, Dependence and Independence: From Linear Hierarchies to Nonlinear Networks. Eur. J. Oper. Res., 26: Saaty, T.L., The Analytic Hierarchy Process, McGraw Hill Publications. 10. Saaty, T.L., Rank from Comparisons and from Ratings in the Analytic Hierarchy/Network Processes. Eur. J. Oper. Res., 168: Sarkis, J., Theory and Methodology: Evaluating Environmentally Conscious Business Practices. Eur. J. Oper. Res., 107 (1): Sarkis, J. and R.P. Sundarraj, Hub Location at Digital Equipment Corporation: A Comprehensive Analysis of Qualitative and Quantitative Factors. Eur. J. Oper. Res., 137 (2): Wu, C.R., C.T. Lin and H.C. Chen, Integrated Environmental Assessment of the Location Selection with Fuzzy Analytical Network Process. Qual. Quan., Forthcoming. 14. Published by Taiwan Insurance Institute, From <http://www.tii.org.tw/>. 61

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