The course aims at providing general and issue-specific knowledge of the risk theory and applications in insurance contracts. Firstly, necessary mathematical tools for risk theory are introduced. Then, theory of decisions under uncertainty, individual/collective risk models and premium in insurance contracts are studied. Finally, ruin theory is presented.
Expected learning outcomes
Knowledge and understanding: the student should be able to understand the themes and problems related to the theory of risk theory and insurance contracts.
Applying knowledge and understanding: the student should be able to apply the acquired knowledge to concrete problems in specific models. To this purpose, the teacher will illustrate some different examples and specific cases during the lessons; for non-attending students, assistance time will be provided.
Making judgements: the student should be able to use the acquired knowledge also in an autonomous way, by also applying them to specific issues and problems that are more general or different with respect to those illustrated by the teacher.
Communication: the student should be able to answer in a clear and detailed way to the questions of the written examination and to those of the oral examination.
Lifelong learning skills: the student should be able to show a good learning ability, by widening, for example, his/her knowledge with use of relevant bibliographic references
Some basic knowledge of mathematics and probability, previously acquired by the students in basic courses of the previous years: Linear algebra, calculus, elements of integration.
I) Preliminary notions of probability theory: Probability space, sigma-algebra, probability measures. Random variables and cumulative distributions functions. Indipendence of random variables. Convolutions. Main distributions: binomial, Poisson, Pareto, Normal, Exponential. Stochastic processes, Markov processes, Poisson processes. (16hrs)
I) Choice under Uncertainty: Expected utility criterion, certainty equivalent, risk premium, probability premium. Risk aversion and Jensen Inequality. Arrow-Pratt risk aversion coefficient and characterizations. Stochastic dominance and relation with expected utility. Mean-variance criterion and relation with with expected utility.(24 hrs)
III) Risk measures: Value at Risk, Tail Conditional Expectation, Conditional Value at Risk, Coherent Risk Measures. (8 hrs).
IV Insurance contracts: the premium problem. The Reinsurance problem and the stop loss transform. Individual risk model, mixed risks, approximations. Collective risk models, distributions of the number of claims, compound distributions, Panjer’s recursion. Premium calculations, equivalence principles and their properties. Hints on life-insurance contracts. (16 hrs)
V Ruin Theory: ruin process and ruin probabilities. Ruin probability and capital at ruin. (8 hrs).
During the lessons the issues mentioned in the study program will be discussed and presented together with applications and examples. Additional teaching and support material is made available through the e-learning online platform Moodle, where notes of the course as well as additional exercises solved and explained in detail.
Freddi. Lezioni di Teoria del Rischio 2007 . Aracne
Kaas, Goovaerts, Dhaene, Denuit. Modern Actuarial Risk Theory. 2008 Springer
Lecture notes of the teacher
The assessment is based on a written and an oral examination. The written examination consists in the resolution of 2/3 problems/exercises in 1 hour. The questions are composed in order to evaluate the actual achievement of the objectives on part the students, but, at the same time, the reasoning ability and the capability to apply the theoretical lessons received. The oral examination has the purpose to evaluate the depth in understanding general theoretical knowledge. In their answers the students should be able to clearly show and illustrate the fundamental concepts acquired during their studies.
The oral and written examinations have identical weights in the final mark.