STUDY OF OPTIMIZATION PROBLEMS IN THE INSURANCE INDUSTRY

In the insurance industry, optimization problems play a pivotal role in various aspects of business operations, such as risk management, pricing, and claim settlement. As competition in the market intensifies, insurance companies are increasingly turning to advanced analytical techniques and mathematical modeling approaches to optimize their strategies and maximize profitability. This thesis investigates the most significant and emerging optimization problems in the insurance industry, with a focus on cutting-edge techniques and methodologies that enhance efficiency and effectiveness in these areas. An extensive review of the literature is presented, the latest trends and innovations in optimization techniques are discussed, and novel solutions to some of the most pressing challenges faced by (re)insurance providers are proposed. Through case studies and empirical analyses, this research demonstrates the value of adopting advanced optimization methods and tools in the insurance industry, providing valuable insights for both academics and practitioners.

The first part deals with the multi-constrained Pareto-optimal reinsurance problems based on general distortion risk measures, which become technically challenging and have only been solved usingad hoc methods for certain special cases. In this research, the method developed by Lo (2017) is extended by proposing a generalized Neyman-Pearson framework to identify the optimal forms of the solutions. Then a dual formulation is developed, which shows that the infinite-dimensional constrained optimization problems can be reduced to finite-dimensional unconstrained ones. With the support of the Nelder-Mead algorithm, the optimal solutions can be obtained efficiently. To illustrate the versatility of our approach, several detailed numerical examples are provided, many of which were only partially resolved in the literature.

In the second part, an evolutionary game model is developed based on a cost-benefit analysis of the insurer, the managing general agency (MGA)/ broker, and the consumer in a digitalization setting. Our findings suggest that an MGA partnership could lead to higher social welfare, considering the underwriting cost and mismatching cost. The MGA partnership is particularly beneficial for small- and medium-sized insurers, and can also expand consumer demand. The evolutionary stable state of the inter-mediated insurance market is determined by the relationship between the consultation fee and its critical value. Finally, the empirical data and simulation results are consistent with the predictions of our model.

[An, H., Yang, R., Ma, X., Zhang, S., and Islam, S. M. (2021). An evolutionary game theory model for the inter-relationships between financial regulation and financial innovation. The North American Journal of Economics and Finance, 55:101341.Aon (2022). Mgas: A market on the move. available at https://mga.aon.com/2022-mgas-on-the-move/marketplace, accessed Dec. 30, 2022.Applied Systems (2022). Applied digital agency report. appliedsystems.com.Arrow, K. J. (1963). Uncertainty and the welfare economics of medical care. The American Economic Review, 53(5):941–973.Arrow, K. J. (2012). Social choice and individual values, volume 12. Yale university press.Asimit, A. V., Badescu, A. M., and Cheung, K. C. (2013a). Optimal reinsurance in the presence of counterparty default risk. Insurance: Mathematics and Economics, 53(3):690–697.Asimit, A. V., Badescu, A. M., and Verdonck, T. (2013b). Optimal risk transfer under quantile-based risk measurers. Insurance: Mathematics and Economics, 53(1):252–265.Asimit, A. V., Cheung, K. C., Chong, W. F., and Hu, J. (2020). Pareto-optimal insurance contracts with premium budget and minimum charge constraints. Insurance: Mathematics and Economics, 95:17–27.Assa, H. (2015). On optimal reinsurance policy with distortion risk measures and premiums. Insurance: Mathematics and Economics, 61:70–75.Audet, C. and Hare, W. (2017). Nelder-mead. In Derivative-Free and Blackbox Optimization, pages 75–91. Springer.Ba, S. and Pavlou, P. A. (2002). Evidence of the effect of trust building technology in electronic markets: Price premiums and buyer behavior. MIS quarterly, pages 243–268.Bakos, Y. (2001). The emerging landscape for retail e-commerce. Journal of economic perspectives, 15(1):69–80.Barton, R. and Ivey, J. (1991). Modifications of the nelder-mead simplex method for stochastic simulation response optimization. In 1991 Winter Simulation Conference Proceedings., pages 945–953.Barton, R. R. and Ivey Jr, J. S. (1996). Nelder-mead simplex modifications for simulation optimization. Management Science, 42(7):954–973.Barvinok, A. (2002). A course in convexity, volume 54. American Mathematical Soc.Baye, M. R. and Morgan, J. (2001). Information gatekeepers on the internet and the competitiveness of homogeneous product markets. American Economic Review, 91(3):454–474.Boonen, T. J. (2016). Optimal reinsurance with heterogeneous reference probabilities. Risks, 4(3):26.Boonen, T. J., De Waegenaere, A., and Norde, H. (2017). Redistribution of longevity risk: The effect of heterogeneous mortality beliefs. Insurance: Mathematics and Economics, 72:175–188.Boonen, T. J. and Ghossoub, M. (2019). On the existence of a representative reinsurer under heterogeneous beliefs. Insurance: Mathematics and Economics, 88:209–225.Boonen, T. J. and Ghossoub, M. (2021). Optimal reinsurance with multiple reinsurers: Distortion risk measures, distortion premium principles, and heterogeneous beliefs. Insurance: Mathematics and Economics, 101:23–37.Boonen, T. J. and Jiang, W. (2022). A marginal indemnity function approach to optimal reinsurance under the vajda condition. European Journal of Operational Research, 303(2):928–944.Borch, K. (1960). An attempt to determine the optimum amount of stop loss reinsurance. In Transactions of the 16th international congress of actuaries, volume 1, pages 597–610.Borch, K. (1969). The optimal reinsurance treaty. ASTIN Bulletin: The Journal of the IAA, 5(2):293–297.Borenstein, S. and Saloner, G. (2001). Economics and electronic commerce. Journal of Economic Perspectives, 15(1):3–12.Braun, A. and Schreiber, F. (2017). The current InsurTech landscape: Business models and disruptive potential. Number 62. I. VW HSG Schriftenreihe.Brynjolfsson, E. and Kahin, B. (2002). Understanding the digital economy: data, tools, and research. MIT press.Cai, J., Lemieux, C., and Liu, F. (2016). Optimal reinsurance from the perspectives of both an insurer and a reinsurer. ASTIN Bulletin: The Journal of the IAA, 46(3):815–849.Cai, J., Liu, H., and Wang, R. (2017). Pareto-optimal reinsurance arrangements under general model settings. Insurance: Mathematics and Economics, 77:24–37.Cai, J. and Tan, K. S. (2007). Optimal retention for a stop-loss reinsurance under the var and cte risk measures. ASTIN Bulletin: The Journal of the IAA, 37(1):93–112.Cai, J., Tan, K. S., Weng, C., and Zhang, Y. (2008). Optimal reinsurance under var and cte risk measures. Insurance: mathematics and Economics, 43(1):185–196.Cai, J. and Weng, C. (2016). Optimal reinsurance with expectile. Scandinavian Actuarial Journal, 2016(7):624–645.Cairncross, F. (2002). The death of distance. RSA Journal, 149(5502):40–42.Cheikbossian, G. (2021). Evolutionarily stable in-group altruism in intergroup conflict over (local) public goods. Games and Economic Behavior, 127:206–226.Chelouah, R. and Siarry, P. (2005). A hybrid method combining continuous tabu search and nelder–mead simplex algorithms for the global optimization of multiminima functions. European Journal of Operational Research, 161(3):636–654.Cheung, K. C. (2010). Optimal reinsurance revisited–a geometric approach. ASTIN Bulletin: The Journal of the IAA, 40(1):221–239.Cheung, K. C., Chong, W. F., and Lo, A. (2019). Budget-constrained optimal reinsurance design under coherent risk measures. Scandinavian Actuarial Journal, 2019(9):729–751.Cheung, K. C., Liu, F., and Yam, S. (2012). Average value-at-risk minimizing reinsurance under wang’s premium principle with constraints. ASTIN Bulletin: The Journal of the IAA, 42(2):575–600.Cheung, K. C. and Lo, A. (2017). Characterizations of optimal reinsurance treaties: a cost-benefit approach. Scandinavian Actuarial Journal, 2017(1):1–28.Cheung, K. C., Sung, K., Yam, S., and Yung, S. (2014). Optimal reinsurance under general law-invariant risk measures. Scandinavian Actuarial Journal, 2014(1):72–91.Chi, Y. and Tan, K. S. (2011). Optimal reinsurance under var and cvar risk measures: a simplified approach. ASTIN Bulletin: The Journal of the IAA, 41(2):487–509.Cressman, R., Ansell, C., and Binmore, K. (2003). Evolutionary dynamics and extensive form games, volume 5. MIT Press.Cui, W., Yang, J., and Wu, L. (2013). Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance Mathematics & Economics, 53:74–85.Cummins, J. D. and Mahul, O. (2004). The demand for insurance with an upper limit on coverage. Journal of Risk and Insurance, 71(2):253–264.Dana, R.-A. and Scarsini, M. (2007). Optimal risk sharing with background risk. Journal of Economic Theory, 133(1):152–176.Deprez, O. and Gerber, H. U. (1985). On convex principles of premium calculation. Insurance: Mathematics and Economics, 4(3):179–189.Dhaene, J., Denuit, M., Goovaerts, M., Kaas, R., and Vyncke, D. (2002). The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics, 31(1):3–33. Special Issue: Papers presented at the 5th IME Conference, Penn State University, University Park, PA, 23-25 July 2001.Dhaene, J., Vanduffel, S., Goovaerts, M. J., Kaas, R., Tang, Q., and Vyncke, D. (2006). Risk measures and comonotonicity: A review. Stochastic Models, 22(4):573–606.Diamond, P. A. (1971). A model of price adjustment. Journal of economic theory, 3(2):156–168.Dub´e, J.-P. and Misra, S. (2017). Scalable price targeting. Number w23775. National Bureau of Economic Research Cambridge, MA.Eling, M. and Lehmann, M. (2018). The impact of digitalization on the insurance value chain and the insurability of risks. The Geneva papers on risk and insurance-issues and practice, 43(3):359–396.Ellison, G. and Fisher Ellison, S. (2005). Lessons about markets from the internet. Journal of Economic perspectives, 19(2):139–158.Fang, Y., Cheng, G., and Qu, Z. (2020). Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles. AIMS Mathematics, 5(4):3231–3255.Fernando, J. (2022). Independent insurance agents and brokers of america (iiaba). available at https://www.investopedia.com/terms/i/independent-insurance-agents-brokers-america.asp, accessed May 26, 2022.Friedman, D. (1991). Evolutionary games in economics. Econometrica: Journal of the Econometric Society, pages 637–666.Friedman, D. (1998). On economic applications of evolutionary game theory. Journal of evolutionary economics, 8(1):15–43.Friedman, T. L. (2006). The world is flat [updated and expanded]: A brief history of the twenty-first century. Macmillan.Fritzsch, S., Scharner, P., and Weiß, G. (2021). Estimating the relation between digitalization and the market value of insurers. The Journal of risk and insurance, 88(3):529–567.Gerber, H. U. (1978). Pareto-optimal risk exchanges and related decision problems. ASTIN Bulletin: The Journal of the IAA, 10(1):25–33.Ghossoub, M. (2019). Budget-constrained optimal insurance with belief heterogeneity. Insurance: Mathematics and Economics, 89:79–91.Goldfarb, A. (2014). What is different about online advertising? Review of Industrial Organization, 44(2):115–129.Goldfarb, A. and Tucker, C. (2019). Digital economics. Journal of Economic Literature, 57(1):3–43.Golubin, A. Y. (2006). Pareto-optimal insurance policies in the models with a premium based on the actuarial value. The Journal of Risk and Insurance, 73(3):469–487.Grzadkowska, A. (2021). What is an mga? available at https://www.insurancebusinessmag.com/us/news/breaking-news/whatis-an-mga-115496.aspx, accessed Sep 21, 2021.Hagiu, A. and Jullien, B. (2011). Why do intermediaries divert search? The RAND Journal of Economics, 42(2):337–362.Hersch, K., Baumann, N., Canaan, M., and Friedman, S. (2022). 2023 insurance outlook: Global insurance industry at a crossroads to shaping long-term success. Deloitte Insights. available at https://www2.deloitte.com/content/dam/Deloitte/cn/Documents/financial-services/deloitecn-fsi-2023-insurance-outlook-en-221021.pdf, accessed Nov. 15, 2022.Hofbauer, J. and Sigmund, K. (1988). The theory of evolution and dynamical systems. mathematical aspects of selection. CAMBRIDGE UNIVERSITY PRESS, NEW YORK, NY(USA). 1988.Hofbauer, J. and Sigmund, K. (2003). Evolutionary game dynamics. Bulletin of the American mathematical society, 40(4):479–519.Houser, D. and Wooders, J. (2006). Reputation in auctions: Theory, and evidence from ebay. Journal of Economics & Management Strategy, 15(2):353–369.Huang, Y. and Yin, C. (2018). A unifying approach to constrained and unconstrained optimal reinsurance. Journal of Computational and Applied Mathematics, 360:1–17.Hui, X., Saeedi, M., Shen, Z., and Sundaresan, N. (2016). Reputation and regulations: Evidence from ebay. Management Science, 62(12):3604–3616.H¨urlimann, W. (2011). Optimal reinsurance revisited – point of view of cedent and reinsurer. ASTIN Bulletin, 41(2):547–574.Jiang, W., Hong, H., and Ren, J. (2018). On pareto-optimal reinsurance with constraints under distortion risk measures. European Actuarial Journal, 8(1):215–243.Johnson, J. P. (2013). Targeted advertising and advertising avoidance. The RAND Journal of Economics, 44(1):128–144.Keener, R. W. (2010). Theoretical statistics: Topics for a core course. Springer.Kelley, C. T. (1999). Iterative methods for optimization. SIAM.Kuhn, H. W. and Tucker, A. W. (1953). Contributions to the Theory of Games. Number 28. Princeton University Press.Lehmann, E. L., Romano, J. P., and Casella, G. (2005). Testing statistical hypotheses, volume 3. Springer.Lendle, A., Olarreaga, M., Schropp, S., and V´ezina, P.-L. (2016). There goes gravity: ebay and the death of distance. The Economic Journal, 126(591):406–441.Lo, A. (2017a). A neyman-pearson perspective on optimal reinsurance with constraints. ASTIN Bulletin: The Journal of the IAA, 47(2):467–499.Lo, A. (2017b). A unifying approach to risk-measure-based optimal reinsurance problems with practical constraints. Scandinavian Actuarial Journal, 2017(7):584–605.Lo, A. and Tang, Z. (2019). Pareto-optimal reinsurance policies in the presence of individual risk constraints. Annals of Operations Research, 274(1):395–423.Lu, Z., Meng, L., Wang, Y., and Shen, Q. (2016). Optimal reinsurance under var and tvar risk measures in the presence of reinsurer’s risk limit. Insurance: Mathematics and Economics, 68:92–100.Lucking-Reiley, D., Bryan, D., Prasad, N., and Reeves, D. (2007). Pennies from ebay: The determinants of price in online auctions. The journal of industrial economics, 55(2):223–233.Nance, E. (2018). What is an mga? available at https://www.ilsainc.com/insurance-industry/what-is-an-mga/, accessed June 11, 2018.Nash Jr, J. F. (1950). Equilibrium points in n-person games. Proceedings of the national academy of sciences, 36(1):48–49.Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. The computer journal, 7(4):308–313.Newton, J. (2018). Evolutionary game theory: A renaissance. Games, 9(2):31.Panjer, H. H. (2006). Operational risk: modeling analytics, volume 620. John Wiley & Sons.Peitz, M. and Waldfogel, J. (2012). The Oxford handbook of the digital economy. Oxford University Press.Porter, M. E. (2011). Competitive advantage of nations: creating and sustaining superior performance. simon and schuster.Raviv, A. (1979). The design of an optimal insurance policy. The American Economic Review, 69(1):84–96.Rochet, J.-C. and Tirole, J. (2003). Platform competition in two-sided markets. Journal of the European economic association, 1(4):990–1029.Rysman, M. (2007). An empirical analysis of payment card usage. The Journal of Industrial Economics, 55(1):1–36.Shanghai Securities News (2022). Beijing banking and insurance regulatory bureau: Batch to establish trial institutions for managed insurance intermediary business. available at https://news.cnstock.com/news,bwkx202208-4936629.html, accessed Aug 10, 2022.Shao, J. (2003). Mathematical statistics. Springer Science & Business Media.Shapiro, C., Varian, H. R., Carl, S., et al. (1999). Information rules: A strategic guide to the network economy. Harvard Business Press.Shubik, M. (1960). Games decisions and industrial organization. Management Science, 6(4):455–474.Sigma (2018). World insurance in 2017: solid, but mature life markets weigh on growth. available at https://www.swissre.com/institute/research/sigma-research/sigma-2018-03.html, No.3/2018.Sigma (2019). World insurance: the great pivot east continues. available at https://www.swissre.com/institute/research/sigmaresearch/sigma-2019-03.html, No.3/2019.Sigma (2020). World insurance: riding out the 2020 pandemic storm. available at https://www.swissre.com/institute/research/ sigma-research/sigma-2020-04.html, No.4/2020.Sigma (2021). World insurance: the recovery gains pace. available at https://www.swissre.com/institute/research/sigmaresearch/sigma-2021-03.html, No.3/2021.Sigma (2022). World insurance: Global insurance premium volumes to reach new high in 2022. available at https://www.swissre.com/institute/ research/sigma-research/sigma-2022-04.html, No.4/2022.Smale, S. (1974). Differential equations, dynamical systems, and linear algebra. Academic press.Stigler, G. J. (1961). The economics of information. Journal of political economy, 69(3):213–225.Stoeckli, E., Dremel, C., and Uebernickel, F. (2018). Exploring characteristics and transformational capabilities of insurtech innovations to understand insurance value creation in a digital world. Electronic markets, 28(3):287–305.Szab´o, G. and Fath, G. (2007). Evolutionary games on graphs. Physics reports, 446(4-6):97–216.Tan, K. S., Weng, C., and Zhang, Y. (2011). Optimality of general reinsurance contracts under cte risk measure. Insurance: Mathematics and Economics, 49(2):175–187.Tanimoto, J. (2015). Fundamentals of evolutionary game theory and its applications. Springer.Taylor, P. D. and Jonker, L. B. (1978). Evolutionary stable strategies and game dynamics. Mathematical biosciences, 40(1-2):145–156.Traulsen, A. and Hauert, C. (2009). Stochastic evolutionary game dynamics. Reviews of nonlinear dynamics and complexity, 2:25–61.Varian, H. R. (1980). A model of sales. The American economic review, 70(4):651–659.Vero (2022). The future of insurance: Bye-bye boomers, hello digital natives. vero.com.au/broker. available at https://www.vero.com.au/content/dam/suncorp/insurance/vero/documents/sme-insurance-index/SMEInsurance-Index-Report-2022.pdf, accessed Nov. 15, 2022.Von Neumann, J. and Morgenstern, O. (1947). Theory of games and economic behavior, 2nd rev. Princeton university press.Weibull, J. W. (1997). Evolutionary game theory. MIT press.Weyl, E. G. (2010). A price theory of multi-sided platforms. American Economic Review, 100(4):1642–72.Wright, S., Nocedal, J., et al. (1999). Numerical optimization. Springer Science, 35(67-68):7.Wu, B., Cheng, J., and Qi, Y. (2020). Tripartite evolutionary game analysis for “deceive acquaintances” behavior of e-commerce platforms in cooperative supervision. Physica A: Statistical Mechanics and its Applications, 550:123892.Xu, X. and Zweifel, P. (2020). A framework for the evaluation of insurtech. Risk Management and Insurance Review, 23(4):305–329.Yan, T. C., Schulte, P., and Chuen, D. L. K. (2018). Insurtech and fintech: banking and insurance enablement. Handbook of Blockchain, Digital Finance, and Inclusion, Volume 1, pages 249–281.Young, V. R. (1999). Optimal insurance under wang’s premium principle. Insurance: Mathematics and Economics, 25(2):109–122.Zheng, Y. and Cui, W. (2014). Optimal reinsurance with premium constraint under distortion risk measures. Insurance: Mathematics and Economics, 59:109–120.Zheng, Y., Cui, W., and Yang, J. (2015). Optimal reinsurance under distortion risk measures and expected value premium principle for reinsurer. Journal of Systems Science and Complexity, 28(1):122–143.Zhu, Q., Zheng, K., and Wei, Y. (2021). Three-party stochastic evolutionary game analysis of reward and punishment mechanism for green credit. Discrete Dynamics in Nature and Society, 2021.Zhuang, S. C., Weng, C., Tan, K. S., and Assa, H. (2016). Marginal indemnification function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 67:65–76.