Mathematical Methods in Risk Theory

Transcription

1 Hans Bühlmann Mathematical Methods in Risk Theory Springer-Verlag Berlin Heidelberg New York 1970

2 Table of Contents Part I. The Theoretical Model Chapter 1: Probability Aspects of Risk Random variables explained by the example of claim amount Definition Classification and examples of distribution functions Expected values Characteristics of a probability distribution and auxiliary functions Chebyshev's Inequality Sequences of random variables explained by the example of claim amount reproductions Multi-dimensional distributions and auxiliary functions Conditional distribution functions and conditional expectation Independence Covariance and correlation The law of large numbers 32 Chapter 2: The Risk Process Fundamentals Definitions and intuitive description of risk Stochastic processes with independent increments Markov processes The claim number process Mathematical description The claim interoccurrence time The homogeneous claim number process operational time The case of time-independent intensities of claim frequency: contagion modeis The accumulated claim process Definition as random sum and basic representation Proof of the basic representation of the accumulated claim distribution The reduced basic representation: time-independent claim amounts The reduced basic representation: time-dependent claim amounts An example 60 Chapter 3: The Risk in the Collective Risk-theoretical definitions Risk and collective The structure function The weighted risk process as description of the risk in the collective Weighted laws of probability The risk pattern in the collective 67

3 X Table of Contents The number of Claims process in the couective The weighted Poisson and negative binomial distributions The accumulated claim process in the couective Portfolios in the couective Some definitions Stabilizing in time (Theorem of Ove Lundberg) Stabilizing in size.. 80 Part II. Consequences of the Theoretical Model Chapter 4: Premium Calculation Principles of premium calculation General Some principles of premium calculation Discussion of the principles of premium calculation The risk premium and the couective premium The risk premium The couective premium Statistics and couective premium The dilemma and the connection between risk and couective premium The credibility premium The credibility premium as sequential approximation to the risk premium A new interpretation of the variance principle for calculation of premiums Construction of the credibility premium Assumptions for our further investigations Properties of the credibility premium The credibility formulae for the three components of the credibility premium Determining the weights in the credibüity formulae A practical example: risk, couective and credibility premium in automobile liability insurance 106 Chapter 5: Retentions and Reserves The retention problem General The retention under proportional and non-proportional reinsurance The relative retention problem Proportional reinsurance Non-proportional reinsurance The risk with given risk parameter and the risk in the couective under non-proportional reinsurance Credibility approximation for the relative retention The absolute retention problem Exact Statement of the problem The random walk of the risk carrier's free reserves generated by the risk mass Reserves 129

4 Table of Contents XI Chapter 6: The Insurance Carrier's Stability Criteria The stability problem Decision variables Stability problem and stability criteria The probability of ruin as stability criterion Planning horizon and ruin probability Admissible stability policies Hypotheses about the model variables in calculating the probability of ruin Calculating the probability of ruin in the discrete case with finite planning horizon Calculating the probability of ruin with an infinite planning horizon using the Wiener-Hopf method Calculating the probability of ruin in the continuous case with infinite planning horizon using renewal theory methods The absolute retention when the probability of ruin is chosen as the stability criterion Restatement of the problem and assumptions The optimal gradation of retentions The stability condition Determining the absolute retention when the risk parameter is known Determining the absolute retention when the risk Parameters are drawn from one or more collectives Practical remark on the probability of ruin as stability criterion Dividend policy as criterion of stability General description of the criterion Hypotheses about the model variables when the dividend policy is used as stability criterion Dividend policy in the discrete case Results in the discrete case Barrier strategies in the discrete case Dividend policy in the continuous case The integro-differential equation of the barrier strategy in the continuous case Solving the integro-differential equation for V(Q, ä) Asymptotic formula for a Optimum dividend policy for Q>a 0 and other evaluations Utility as criterion of stability Evaluating the random walk of free reserves Equivalent evaluations; definition of Utility Axioms about Utility Existence theorem for an equivalent Utility Integral evaluation The problem of risk exchange The theorem of Borch A consequence of Borch's theorem Price structures with quadratic Utility kerneis 197

5 xn Table of Contents Appendix: The Generalized Riemann-Stieltjes Integral 201 A.l. Prelirninary 201 A.2. Definition of the generalized Riemann-Stieltjes integral in two special cases 201 A.3. Definition in the general case 203 A.4. Integrable functions 203 A.5. Properties of the generalized Riemann-Stieltjes integral 204 Bibliography 206 Index 209