Knowledge and understanding: At the end of the course, students must have acquired the ability of building models, proposing solutions concerning problems and applying the introduced techniques in order to evaluate and handle risks in non-life insurance. Applying knowledge and understanding: At the end of the course, students must be able: to apply the mathematical methods developed in the course to the solution of problems concerning non-life insurance theory, by using the appropriate mathematical tools. Making judgments: At the end of the course, students must be able to create links among notions acquired during the course and to tackle problems concerning non-life insurance through the use of logical and formal mathematical tools. Communication skills: At the end of the course, students must acquire an ability to communicate clearly and effectively the acquired knowledge. Learning skills: At the end of the course, students must have developed good learning skills, which allow them to independently apply the knowledge acquired during their studies.

In order to understand the topics dealt with in the course, the concepts provided in the following courses are necessary: calculus, financial mathematics, probability, statistics and actuarial mathematics.

Models for the claim number process: Poisson process. Inhomogeneous Poisson process. Mixed Poisson processes. Mixed Poisson processes with gamma mixing distribution. Collective risk theory: The risk process. Distribution of the cumulative claim process. Asymptotic ruin probability. Lundberg adjustment coefficient and Lundberg inequality. Application of the model to determine the safety loading, the risk capital and retention in quota share and excess-of-loss reinsurance. Bayesian approach in experience rating: A-priori premium, individual premium, Bayesian premium. Poisson-gamma model for the claim number process: Bayesian premium and parameter estimates. The Bayesian credibility approach to experience rating: linear credibility premium. Bühlmann model: credibility premium and parameter estimates. Bühlmann-Straub model: credibility premium and parameter estimates. Semi parametric mixture Poisson model with regression components. Basics on exact credibility. Experience rating in motor insurance: Bonus-Malus (BM) systems. Evolution of the distribution of the insureds among the merit classes: Markov model, mixture of Markov model, Poisson-gamma model. Evaluation of BM systems. Setting BM scales. Stochastic simulation for actuarial problems. Risk measures: Value-at-Risk, Conditional tail expectation, Tail VAR, Wang risk measure. Some connections with premium principles. Risk capital. Technical reserves: generalities on premium and claims reserves. Legislation in force and IFRS 17 principle. Premium reserve evaluation. Traditional claims reserving methods: chain-ladder, modified chain-ladder, Taylor separation, loss-ratio, Bornhuetter-Ferguson, Fisher-Lange. Deterministic methods for the evaluation of IBNR reserve. Solvency capital requirement: Solvency II Directive.

References M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas (2005), Actuarial Theory for Dependent Risks. Measures, Orders and Models, Wiley M. Denuit, A. Charpentier (2005), Mathématiques de l'assurance non-vie, Economica R. Kaas, M. Goovaerts, J. Dhaene, M. Denuit (2008), Modern Actuarial Risk Theory.Using R, Second Edition, Springer S. A. Klugman, H. H. Panjer, G. E. Willmot (2008), Loss Models: From Data to Decisions, 3th Edition, Wiley T. Mikosch (2004), Non-Life Insurance Mathematics. An Introduction with Stochastic Processes, Springer G.C. Taylor (1986), Claim reserving in non-life insurance, North Holland More references will be suggested during the lessons.

Models for the claim number process: Poisson process. Inhomogeneous Poisson process. Mixed Poisson processes. Mixed Poisson processes with gamma mixing distribution. Collective risk theory: The risk process. Distribution of the cumulative claim process. Asymptotic ruin probability. Lundberg adjustment coefficient and Lundberg inequality. Application of the model to determine the safety loading, the risk capital and retention in quota share and excess-of-loss reinsurance. Bayesian approach in experience rating: A-priori premium, individual premium, Bayesian premium. Poisson-gamma model for the claim number process: Bayesian premium and parameter estimates. The Bayesian credibility approach to experience rating: linear credibility premium. Bühlmann model: credibility premium and parameter estimates. Bühlmann-Straub model: credibility premium and parameter estimates. Semi parametric mixture Poisson model with regression components. Basics on exact credibility. Experience rating in motor insurance: Bonus-Malus (BM) systems. Evolution of the distribution of the insureds among the merit classes: Markov model, mixture of Markov model, Poisson-gamma model. Evaluation of BM systems. Setting BM scales. Stochastic simulation for actuarial problems. Risk measures: Value-at-Risk, Conditional tail expectation, Tail VAR, Wang risk measure. Some connections with premium principles. Risk capital. Technical reserves: generalities on premium and claims reserves. Legislation in force and IFRS 17 principle. Premium reserve evaluation. Traditional claims reserving methods: chain-ladder, modified chain-ladder, Taylor separation, loss-ratio, Bornhuetter-Ferguson, Fisher-Lange. Deterministic methods for the evaluation of IBNR reserve. Solvency capital requirement: Solvency II Directive.

Lectures supported by slide presentations that are made available on the MOODLE platform.

The arguments developed in the course are directly connected with the other actuarial courses, in particular with Matematica attuariale delle assicurazioni danni, for which it is the natural continuation, and with Statistica assicurativa. Non-attending students are invited to contact the teacher for further suggestions on the course program and bibliographic references

The examination consists of an oral exam with open questions, that aims to test a sound knowledge and comprehension of the arguments presented during the lessons. The students have to prove their comprehension of the fundamental concepts explained in the course and their ability of connecting the different topics; moreover, they have to be able to present the acquired knowledges clearly.

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