THEORY OF RISK

Course objectives

The course introduces students to risk theory in the insurance context, with particular focus on probabilistic techniques and models for managing a non-life insurance company, as well as on risk measures and reinsurance contracts. Students will gain an in-depth understanding of the mathematical and probabilistic tools necessary to analyze key issues in insurance markets, such as assessing the solvency of an insurance company, analyzing various reinsurance contracts, and estimating their impact on the risk measures used by insurers. By the end of the course, students will be able to analyze and formalize practical problems in insurance and reinsurance, selecting the most appropriate mathematical and probabilistic models and applying them effectively, including stochastic aspects when relevant. They will be able to critically evaluate results and draw well-founded conclusions, developing independent judgment in identifying the most suitable modeling approach and selecting the most efficient methodologies for each problem. Students will also develop communication skills to present analyses and arguments clearly and rigorously, both orally and in writing, tailoring the language to the type of audience, whether specialist or non-specialist. Simultaneously, they will acquire the ability to organize and independently update their learning, consolidating a study method that enables them to acquire new knowledge and skills, both in professional contexts and in advanced academic pathways, such as second-level master programs or doctoral studies.

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CLAUDIA CECI Lecturers' profile
MARCO NICOLOSI Lecturers' profile

Program - Frequency - Exams

Course program
1. Introduction to risk management in insurance. Principles of premium calculation and their properties. Examples. Structure of claims. The random variable of aggregate claims cost. Calculation of the distribution function of aggregate claims cost. Calculation of moments. The random variable of the number of claims. Some explicit distributions: binomial distribution, Poisson distribution, finite mixture of Poisson. Compound distributions: compound binomial, compound Poisson. Calculation of expected value, variance, distribution function, and moment generating function of the aggregate claim. The random variable of single claim cost. Light-tailed and heavy-tailed probability distributions. Subexponential probability distributions. 2. Ruin theory. Discrete-time model. Probability of ruin in finite and infinite time. Net Profit Condition. The adjustment coefficient. Lundberg’s inequality. The classical model of collective risk theory. Renewal processes and their properties. Poisson and compound Poisson processes. Net Profit Condition. The adjustment coefficient. Lundberg’s inequality. Integro-differential equation for the probability of ruin. The case of exponential claims. 3. Introduction to reinsurance. Linear and non-linear forms of reinsurance. The Cramér–Lundberg model with reinsurance. Unilateral policies of optimal risk retention. 4. Introduction to risk measures. Value-at-Risk and Conditional Tail Expectation. Coherent risk measures. Examples. Theoretical lectures will be complemented by computer lab sessions on the following topics: - Simulation of random numbers and random variables - Light-tailed and heavy-tailed distributions - The random variable of the number of claims. Distribution of the number of claims for a risk portfolio. The random variable of single claim cost. - Calculation of the distribution function of aggregate claims cost - Simulation of the Cramér–Lundberg model, including with reinsurance - Simulation techniques for the calculation of risk measures, with particular focus on Value-at-Risk and Conditional Tail Expectation.
Prerequisites
To successfully attend the course, students are expected to have basic knowledge of mathematics, elements of financial mathematics, introductory probability, and basic statistics.
Books
- Textbook: David C. M. Dickson, Insurance Risk and Ruin, Cambridge University Press

- Reference book: T. Mikosch, Non-Life Insurance Mathematics, Springer

Supplementary teaching material is available on the page: Teoria del rischio

Frequency
Attendance is not mandatory, but it is strongly recommended.
Exam mode
Oral examination
Lesson mode
Face-to-face lectures with examples and exercises. Computer lab sessions. The exercises will be implemented in MATLAB.
  • Lesson code1038108
  • Academic year2025/2026
  • CourseFinance and insurance
  • CurriculumAssicurazioni
  • Year1st year
  • Semester2nd semester
  • SSDSECS-S/06
  • CFU9