Risk Measures and Insurance Solvency Benchmarks

Vsevolod K. Malinovskii graduated from the Moscow State University, earned his Ph.D. in Mathematics from the Steklov Mathematical Institute in 1983, and his D.Sc. in Mathematics from the Central conomics and Mathematics Institute (CEMI) of the Russian Academy of Science in 2000. He joined Probability Theory's Department of Steklov Mathematical Institute in 1982 and worked there until 2006. Since 2009, he has been a Chief research fellow at the CEMI.He was Visiting Professor at the University of Copenhagen in 1993 and in 1998, and at the University of Montreal in 2001. He has authored Insurance Planning Models: Price Competition and Regulation of Financial Stability and Level-Crossing Problems and Inverse Gaussian Distributions: Closed-Form Results and Approximations. Professor Malinovskii's main research interests are in Applied Probability and in Mathematical Statistics.

• 1. Risk measures in finance and insurance. 1.1. Risk measures in finance and portfolio management. 1.2. Risk measures in Solvency II system. 1.3. Risk measures in risk theory. 1.4. Aim and structure of the book. 1.5. Readers, to whom this book is addressed. Problems. 2. Fixed-probability level in a diffusion model. 2.1. Diffusion model: an auxiliary tool. 2.2. Direct level-crossing problem. 2.3. Inverse level-crossing problem. 2.4. Asymptotic behaviour of fixed-probability level. 2.5. Primary upper bounds on fixed-probability level. 2.6. Elaborated upper bounds on fixed-probability level. 2.7. Conclusions and perspectives. Problems. 3. Fixed-probability level in an exceptional renewal model. 3.1. Exponential renewal model: an exceptional case. 3.2. Direct level-crossing problem. 3.3. Inverse level-crossing problem. 3.4. Asymptotic behaviour of fixed-probability level. 3.5. Primary upper bounds on fixed-probability level. 3.6. Elaborated upper bounds on fixed-probability level. 3.7. Conclusions. Problems. 4. Implicit function defined by M-equation. 4.1. Analytical properties of core integral expression. 4.2. Proximity between Mu;c(t) and Mu;c(t j v). 4.3. Analytical properties of M-level. Problem. 5. Fixed-probability level in general renewal model. 5.1. General renewal model: main framework. 5.2. Direct level-crossing problem. 5.3. Inverse level-crossing problem. 5.4. Primary upper bounds on fixed-probability level. 5.5. Proximity to M-level. 5.6. Conclusion. Problem. 6. Case study: numerical evaluation of fixed-probability Level. 6.1. Distributions of T and Y selected for numerical calculations. 6.2. Simulation in level-crossing problems. 6.3. Numerically calculated bounds on the fixed-probability level. 6.4. Conclusion. Problems. 7. Probability mechanism of insurance with migration and ERS-analysis. 7.1. Structural model of insurance business: origin and purpose of ERS-analysis. 7.2. Price competition, migration, and market price. 7.3. Compound Poisson risk model with migration. 7.4. ERS-analysis, when Y is exponentially distributed. 7.5. ERS-analysis, when Y is generally distributed. 7.6. Conclusions. Problems. A. Auxiliary results from analysis. B. Auxiliary results from probability. List of Notations. Notes and Comments. Bibliography. Index.