1st Edition

# Risk Measures and Insurance Solvency Benchmarks Fixed-Probability Levels in Renewal Risk Models

**Also available as eBook on:**

** Risk Measures and Insurance Solvency Benchmarks: Fixed-Probability Levels in Renewal Risk Models** is written for academics and practitioners who are concerned about potential weaknesses of the Solvency II regulatory system. It is also intended for readers who are interested in pure and applied probability, have a taste for classical and asymptotic analysis, and are motivated to delve into rather intensive calculations.

The formal prerequisite for this book is a good background in analysis. The desired prerequisite is some degree of probability training, but someone with knowledge of the classical real-variable theory, including asymptotic methods, will also find this book interesting. For those who find the proofs too complicated, it may be reassuring that most results in this book are formulated in rather elementary terms. This book can also be used as reading material for basic courses in risk measures, insurance mathematics, and applied probability. The material of this book was partly used by the author for his courses in several universities in Moscow, Copenhagen University, and in the University of Montreal.

**Features**

- Requires only minimal mathematical prerequisites in analysis and probability
- Suitable for researchers and postgraduate students in related fields
- Could be used as a supplement to courses in risk measures, insurance mathematics and applied probability.

**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.

### Biography

**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.