Distribucije teškog repa i primjena u reosiguranju

master's thesis

• Title (english): Heavy tail distributions and applications in reinsurance
• Author: Dominik Mandić
• Mentor: Zoran Vondraček (mentor)
• Committee member: Vanja Wagner (predsjednik povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-07-20, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada bio je opisati modele za veličinu šteta u osiguranju i odgovarajuće vjerojatnosne distribucije, pri čemu najveću ulogu igraju distribucije teškog repa. U prvom dijelu govori se o konceptu reosiguranja, što je to i koje su situacije u kojima dolazi do potrebe za istim. Također, navode se osnovni oblici proporcionalnog i neproporcionalnog reosiguranja, kao i njihove prednosti i mane. Nastavak rada govori o različitim interpretacijama veličine štete pa u tom kontekstu definiramo neke klase distribucija kao što su regularno varirajuće, subeksponencijalne i distribucije dugog repa te njihova osnovna svojstva i primjere. Iskazan je i osnovni teorem teorije ekstremnih vrijednosti, takozvani Fisher-Tippett-Gnedenko teorem, koji karakterizira oblik granične distribucije za maksimume slučajnih varijabli i definira distribucije ekstremnih vrijednosti. Za sva tri slučaja izvedene su maksimalne domene privlačnosti te je definirana generalizirana Paretova distribucija koja igra veliku ulogu u teoriji ekstremnih vrijednosti. Za kraj, navedeni su primjeri najbitnijih distribucija koje se koriste u modeliranju šteta u reosiguranju s naglaskom na one teškog repa (Pareto, lognormalna, Weibull(α < 1)...) te vizualno prikazane njihove funkcije distribucija i gustoće u usporedbi s ostalima.

Abstract (english)

The aim of this thesis was to describe models for claim sizes in insurance and appropriate probability distributions, with heavy tail distributions playing the largest role. The first chapter talks about the concept of reinsurance, what it is and what are the situations where it is necessary. Also, the basic forms of proportional and disproportionate reinsurance are listed, as well as their advantages and disadvantages. The sequel of the thesis discusses different approaches on what we might righteously call a large claim and how one can perhaps distinguish it from others, so in this context we define distribution classes such as regularly varying, subexponential and long tail distributions with their basic properties and examples. The so-called Fisher-Tippett-Gnedenko theorem, the basic result of the extreme-value theory is introduced, which characterizes the limit distribution for maximum of random variables. For all three cases the maximum domains of attraction are derived and the generalized Pareto distribution that plays a major role in the theory of extreme values is defined. Finally, examples of the most important distributions used in modelling reinsurance claims with an emphasis on one heavy tail are given (Pareto, lognormal, Weibull(α < 1)...), and their distribution and density functions are visually shown compared to other distributions

• Keywords: modeli za veličinu šteta; osiguranje; vjerojatnosne distribucije; proporcionalno reosiguranje; neproporcionalno reosiguranje; Fisher-Tippett-Gnedenko teorem; Paretova distribucija
• Keywords (english): models for claim sizes; insurance; probability distributions; proportional reinsurance; disproportionate reinsurance; Fisher-Tippett-Gnedenko theorem; Pareto distribution
• Language: croatian
• URN:NBN: urn:nbn:hr:217:332518
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-09-14 10:46:17