2.1 Random variable, density function, enumerative density function and distribution function

Transcription

1 Risk Theory I Prof. Dr. Christian Hipp Chair for Science of Insurance, University of Karlsruhe (TH Karlsruhe) Contents 1 Introduction 1.1 Overview on the insurance industry Insurance in Benin Internationalisation of the market 1.2 Operating mode of insurance 1.3 Relevant rates and factors 1.4 Learning targets 1.5 The concept of risk 2 Bases of statistics 2.1 Random variable, density function, enumerative density function and distribution function 2.2 Moments and expectation values 2.3 Stochastic independence and convolutions Stochastic independence Fubini s theorem Correlation Convolution 2.4 Examples for distribution functions Uniform distribution Pareto distribution Gamma distribution IBAN: DE , BIC (SWIFT)-Code: WELA DE DN Page 1 of 5

2 2.4.4 Weibull distribution Logarithmic distribution Negative binomial distribution 2.5 Quantile transformation and transformation of distributions Maximum limit of liability (insurance of first risk) Self-insurance (insurance of second risk) 2.6 Characteristic function, moment generating function, generating function 2.7 Convolutions (II) 2.8 Chebyshev inequality 2.9 Laws of large numbers 2.10 Limit theorems 3 Models of risk theory 3.1 Individual model 3.2 Collective model Sums of distributions Convolutions of sums of distributions Straightforward calculation of sums of distributions Recursive calculation of sums of distributions 3.3 Compound Poisson Distribution Proof of the theorem on PSV-Approximations 3.4 Standard Approximation for sum of Poisson distributions 3.5 Sums of distributions for major damages 3.6 Approximation of tail probabilities IBAN: DE , BIC (SWIFT)-Code: WELA DE DN Page 2 of 5

3 4 Theory of ruin 4.1 Discrete probability of ruin 4.2 Adjustment coefficient and inequality of Cramer-Lundberg 4.3 Convolution formula for probability of ruin 4.4 Classic process of reserving for risk 4.5 Problems of major damages 5 Markov processes 5.1 Finite state space 5.2 Countable state space, countable processes 5.3 Sums of processes 5.4 Conditional expected value 6 Principles of premium calculation 6.1 Overview 6.2 Economic and mathematical characteristics Characteristics of principles of premium calculation Exponential principle - adjustment coefficient Exponential principle - variance principle 7 Statistical methods 7.1 Non-parametric methods 7.2 Parametric methods Maximum likelihood method Maximum likelihood method with transformed observations Moment estimator Models for non-homogeneous portfolios, linear modes and alternatives Co-variate variable IBAN: DE , BIC (SWIFT)-Code: WELA DE DN Page 3 of 5

4 7.2.4 Bayes method 7.3 Theory of credibility and tariffing by experience Credibility estimator 8 Exercices and solutions 8.1 Exercices 8.2 Solutions 9 Concise overview on distributions 9.1 Discrete distribution Mixed Poisson distribution Poisson distribution Binomial distribution Negative binomial distribution Logarithmic distribution Generalised Poisson distribution 9.2 Continuous distribution Uniform distribution Exponential distribution Gamma distribution Normal distribution Weibull distribution Log normal distribution Log gamma distribution Beta distribution Inverse Gaussian distribution Pareto distribution Shifted Pareto distribution IBAN: DE , BIC (SWIFT)-Code: WELA DE DN Page 4 of 5

5 not included are: 1. Capital allocation 2. The premium principle of Wang 3. Differential equation for the probability of ruin 4. Probability of ruin with interest on the reserves 5. Non-homogeneous Poisson process IBAN: DE , BIC (SWIFT)-Code: WELA DE DN Page 5 of 5