Insurance Planning Models: Price Competition and Regulation of Financial Stability

Description

"Insurance Planning Models: Price Competition and Regulation of Financial Stability is an exciting new book that takes readers inside the secrets of internal organization of the modern general insurance business. Many people know that it is subject to intensive state regulation, whereby the purpose is to maintain long-term efficiency, honesty, security and stability in the interest and for the protection of policyholders. However, except for knowing that the insurance system is regulated by intensive calculations, that the insurance companies have different positions on the market, that they pursue different goals and even compete with each other, and that one of the tools of this competition is the policy price, not so many people know how to achieve these deserving goals. In developing quantitative recommendations and directives to competing insurers, regulators rely on certain models. In the 1900s, such models were proposed. They were useful for an insight into the probabilistic nature of the insurance process, but not for direct application to practically meaningful problems of insurance regulation. This book is your guide to the rigorously constructed long-term dynamic models with the aim to improve regulatory methods and develop quantitative recommendations using both analytical calculations and computer simulation. It is addressed to a wide range of readers, including interested policyholders, economists whose interest lies in insurance management and regulation, and mathematicians wishing to expand the scope of application for their knowledge. This book is devoted to certain issues that are either not sufficiently presented, or even absent in the literature. It is an attempt to penetrate from the standpoint of mathematical modeling into the goals which face insurance regulators and contending company managers for preventing insolvencies, or even crises pertinent to badly regulated complex reflexive systems.It offers rigorous probabilistic models of long-term insurance business based on the laws of mass phenomena. They mitigate deficiencies of oversimplified risk models. The book presents advances in probabilistic techniques designed to seek quantitative, rather than qualitative, directives and recommendations regarding safe control aiming to achieve different business goals." Contents Preface About the author Economic background 1. Introduction: regulated competitive insurance market 1.1 Competitive insurance system and its regulation 1.1.1 Why insurance protection is possible 1.1.2 Fundamental principles of solvency and equity 1.1.3 Competitive and monopolistic insurance market 1.1.4 Parties of insurance system: insurers, policyholders and regulators 1.2 Participants of insurance company’s business: shareholders, managers and insureds 1.3 Strategic interests of insurers 1.4 Some aspects of modern insurance regulation 1.4.1 Some facts from the practice of insurance regulation 1.4.1.1 Integrity of regulator’s requirements 1.4.1.2 Some aspects of regulation in the UK 1.4.1.3 Some aspects of regulation in the US 1.4.2 Measures for harmonization of solvency regulation 1.4.2.1 Algorithms that are hidden in the directives 1.4.2.2 Critical analysis of the algorithm 1.4.3 Uncertainties external to insurance business 1.4.4 Uncertainties internal to insurance business, and its cyclicity 1.4.4.1 Origin of underwriting cycle, price competition and migration of insureds 1.4.4.2 Competition-originated underwriting cycles and reflexivity 1.4.4.3 Reflexivity and regulation 1.5 Dynamic models in regulation and management 1.5.1 The need for modeling 1.5.2 Lundberg’s collective risk model 1.5.2.1 Central aspects of Lundberg’s model: collective risk 1.5.2.2 Change of time and operational time 1.5.2.3 Random change of time 1.5.3 Advantages and disadvantages of collective risk model 1.5.3.1 Purpose and origin of Lundberg’s model 1.5.3.2 Whether Lundberg’s model is untenable for practice? 1.5.3.3 Main drawbacks of Lundberg’s model 1.5.3.4 Ways to overcome the identified problems 1.5.4 Diffusion risk model: a useful auxiliary tool 1.5.5 Program for building a model of long-term controlled insurance process 1.5.5.1 Model of Solvency Working Parties 1.5.5.2 Model of multi-year controlled insurance process of an insurance company 1.5.6 Simulation and analytical methods 1.5.7 A compromise between simulation and analytical methods 2. Competitive insurance market and complex reflexive systems 2.1 Price competition and underwriting cycles 2.1.1 The insurance system 2.1.2 Regulated competitive market and cycles 2.1.3 Short-term and long-term regulations 2.1.4 Main aspects of integral model 2.1.5 Factors used in quantitative analysis 2.2 Competition-originated cycles: visible manifestations 2.2.1 Quarters of a cycle and periods of stability 2.2.2 Driving forces behind the cycles and two main causal connections 2.2.3 Disequilibrium, strategies and concerted industry’s response 2.2.4 Does insurance provide an example of a complex reflexive system? 2.2.4.1 Major elements of reflexive system 2.2.4.2 Internal model, or multi-year integral model of business of an insurance company 2.2.4.3 Additional conditions for reflexivity and specialized partial models 2.3 Course of competition-originated cycle and main factors that generate cyclicity 2.3.1 A brief description of the natural course of historical changes 2.3.2 How to read the mind of the insurance market 2.3.3 Expansion, revenue and solvency analysis and minimum solvency requirements Mathematical models 3. Integral model of multi-year insurance process 3.1 Some general criteria for selection of a mathematical model 3.1.1 General criteria for modeling 3.1.2 How to define the model rigorously 3.2 Market prices and fundamental risks 3.2.1 Market prices in multi-year periods 3.2.2 What is the market price? 3.2.3 Complete and incomplete information about market prices 3.2.3.1 The autoregression scenario for market prices 3.2.3.2 Scenario with randomness, but without trend 3.3 Multi-year control model of insurance process of a company 3.3.1 The annual probability mechanisms of insurance 3.3.1.1 Selection of the state space 3.3.1.2 Selection of the annual probability mechanisms of insurance 3.3.2 The annual controls and multi-year control strategies 3.3.2.1 Selection of the control space 3.3.2.2 Selection of the annual controls 3.3.3 The rigorous definition of integral multi-year model 3.4 Long-memory and Markov models 3.4.1 Markov and non-Markov models 3.4.2 Markov models with complete information 3.4.3 Markov models with incomplete information 3.5 The growth of complexity and its translation from one form to another 4. Annual probability mechanisms of insurance 4.1 Migration of insureds and variable portfolio’s volume 4.1.1 The annual migration rate function 4.1.1.1 Examples of ultimate migration rate function 4.1.1.2 A simple migration rate function: M-function 4.1.1.3 Cumulative migration rate functions 4.1.2 The process of portfolio volume’s change 4.1.3 Stochastic migration rate functions 4.2 The annual risk reserve 4.3 The annual probability of ruin 4.4 The annual non-ruin capital 4.5 The company’s annual intrinsic value 5. Expansion, revenue, and solvency: a company’s financial position 5.1 Analytical method for assessing solvency and attractiveness for investors in Lundberg’s model with exponential claim amounts 5.1.1 The asymptotic bounds for non-ruin capital 5.1.2 The asymptotic bounds for annual intrinsic value 5.1.2.1 Explicit expression for conservative intrinsic value 5.1.2.2 Conservative intrinsic values for maximal and minimal portfolio’s volumes 5.1.2.3 Conservative intrinsic value and the shape of migration rate function 5.2 Deficiencies of a simplified measure of intrinsic value 5.3 Multi-year partial models configured for different business objectives and market conditions 5.4 Extension of results in Lundberg’s model with exponential claim amounts to more general models 6. Profit-seeking insurers on profitable market with low competition 6.1 What does a profit-seeking insurer seek in years of peace and profit? 6.2 Outline of the set of admissible prices 6.2.1 Impact of the market profitability 6.2.2 Impact of the shape of migration rate function 6.3 Finding optimal prices among admissible prices 6.3.1 Highly profitable market 6.3.2 Moderately profitable market 6.4 Some concluding remarks 6.4.1 What is the magnitude of error when using a bound? 6.4.2 Is it always sensible to choose the price which maximizes the intrinsic value? 6.4.3 Influence of the time-speed of migration 7. Profit-seeking insurers on profitable market with high competition 7.1 What worries a profit-seeking insurer on a profitable market with high competition? 7.2 Annual control and analysis of expansion, revenue and solvency 7.2.1 Multi-year model and the annual control 7.2.1.1 Purpose of the control 7.2.1.2 Selection of the initial portfolio’s volume 7.2.1.3 Admissible prices and target interval for portfolio’s volume 7.2.1.4 Non-ruin capital providing solvency 7.2.2 Optimistic and pessimistic forecasts for intrinsic value 7.3 Some conclusions 8. Pricing when market is at near-equilibrium 8.1 Near-equilibrium: a system consisting of companies seeking stability 8.2 Sources of uncertainty and its consequences 8.2.1 Why in near-equilibrium market prices can sometimes fall below marginal cost of insurance? 8.2.2 Why in near-equilibrium market prices fluctuate around marginal cost of insurance? 8.2.3 Losses associated with failed underwriting optimism and failed underwriting pessimism 8.2.4 Impact of the portfolio’s volume 8.2.5 Additional impact of migration 8.3 Annual control for a company seeking stability 8.3.1 Constraints on the frozen capital 8.3.2 Constraints on the undervalued capital 8.3.3 Maximization of intrinsic value in the interval of admissible prices 8.3.4 The impact of migration on selection of prices 9. How an aggressively expanding company becomes insolvent 9.1 How can an aggressor destroy itself? 9.1.1 The danger of a rapid growth 9.1.2 Default, and unexpectedness of its occurrence 9.1.3 How a rational strategy can lead to default 9.2 Multi-year model of an aggressively growing company 9.2.1 Two rational strategies of aggressive growth 9.2.2 Fund for strategic growth and annual net worth values 9.3 Analysis of rational strategy γ 9.3.1 Annual net worth value for strategy γ 9.3.2 Two illustrative examples 10. Active defense by an incumbent insurer against aggressive newcomers 10.1 How to “delve one yard below their mines” 10.2 Comparative analysis of expansion, revenue and solvency for two competing companies 10.2.1 Model of two competing companies 10.2.2 Price cut up to or below the marginal cost, if risk is of the same quality 10.2.3 Price cut up to or below the marginal cost, if risk is not of the same quality 11. Simulation and numerical results 11.1 Numerical and analytical results for exponentially distributed claim amounts 11.1.1 Numerical calculations based on analytical expressions 11.1.1.1 Convex-concave migration rate functions 11.1.1.2 Linear migration rate functions 11.1.1.3 Concave-convex migration rate functions 11.1.2 Numerical calculations based on the use of simulation 11.1.2.1 Algorithm of simulation 11.2 Simulation analysis for Chapters 8 and 10 11.2.1 Simulation when market is at near-equilibrium 11.2.2 Simulation of active defense 11.3 Simulation for non-exponentially distributed claim amounts Appendices Appendix A Diffusion processes, Poisson processes, and random walks A.1 Diffusion processes A.1.1 Wiener and diffusion processes A.1.2 Samuelson’s exponent of the Brownian motion A.1.3 Mean-reverting diffusion processes A.2 Poisson processes A.2.1 Homogeneous Poisson processes A.2.2 Inhomogeneous Poisson processes A.2.3 Doubly stochastic Poisson processes A.3 Random walks A.3.1 Standard random walks A.3.2 Random walks with random displacements Appendix B Results from calculus and probability B.1 Mill’s ratio B.2 Bessel functions B.3 Differentiation of implicit functions B.4 Straight line tangent to hyperbola B.5 An auxiliary function B.6 Exponential, gamma, beta, and Pareto distributions Appendix C Risk models without migration C.1 Diffusion risk model: one notable simplification C.1.1 Equalities for non-ruin capital and non-ruin price C.1.2 Inequalities for non-ruin capital and non-ruin price C.2 Lundberg’s model with exponentially distributed T and Y C.2.1 Lundberg’s risk models C.2.2 Distribution of the total claim amount C.2.3 Probabilities of ruin C.2.3.1 Probabilities of ruin and random walks C.2.3.2 The probability of ultimate ruin C.2.3.3 The probability of ruin within finite time C.2.3.4 Probability of ruin near the critical value c = λ/μ C.2.4 Non-ruin capital C.2.4.1 Asymptotic equalities for non-ruin capital C.2.4.2 Monotony of non-ruin capital C.2.4.3 Convexity of non-ruin capital C.2.4.5 Elementary bounds on non-ruin capital C.2.4.6 Broken-line bounds on non-ruin capital C.2.4.7 Upper bounds on non-ruin capital that use tangency C.3 Some innovative results in Lundberg’s model with T and Y generally distributed List of notations Notes and comments Bibliography Index