Teorija povjerenja u aktuarstvu

master's thesis

• Author: Antonio Roić
• Mentor: Hrvoje Šikić (mentor)
• Committee member: Hrvoje Šikić (predsjednik povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Damir Bakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu bavimo se Teorijom povjerenja, granom matematike osiguranja koja se bavi procjenjivanjem budućih potraživanja od ugovaratelja osiguranja prema osiguravajućem društvu, a samim time i budućih premija. U prvome dijelu upoznajemo se sa pojmovima iz Bayesovske statistike kao što su apriori i a posteriori distribucije, konjugirane familije i funkcije gubitka. Zatim krećemo s proučavanjem različitih pristupa Teoriji povjerenja koji se koriste za procjenu potraživanja. Klasični pristup se bavi situacijama u kojima podacima možemo dati puno povjerenje i tu se upoznajemo sa samim pojmom punog povjerenja. Proučavamo i situacije u kojima ne možemo dati puno povjerenje podacima. Bayesovski pristup koristi Bayesovsku statistiku za procjenu budućih potraživanja i premija. Tu se upoznajemo i s različitim modelima poput Poisson | Gamma modela, koji su nam zanimljivi jer kod njih a posteriori očekivanje možemo prikazati u obliku koji nam je pogodan za naše procjene. Empirijski Bayesovski pristup nam je jako zanimljiv jer tu promatramo situacije u kojima imamo određeni broj potraživanja prethodnih godina i zanima nas broj potraživanja sljedeće godine, a to je situacija koju jako cesto srećemo u praksi. Tu se susrećemo sa dva modela, Model 1 i Model 2, kod kojih imamo znanje i podatke o određenom broju rizika tijekom određenog broja godina i cilj nam je za te rizike procijeniti premiju za sljedeću godinu. Model 2 je bolji od ta dva jer on u obzir uzima i obujam poslovanja, dok ga Model 1 zanemaruje. Na kraju donosimo zaključak o naučenom o Teoriji povjerenja.

Abstract (english)

In this work we focused on Credibility theory, line of insurance mathematics, which deals with estimating future claims and premiums. In the first chapter are introduced with some terms from Bayesian statistics such as, prior and posterior distribution, conjugate families and loss functions. Then we study different approaches to Credibility theory which are used in estimating claims. Classical approach deals with situations where we can give full credibility to our data and here we are introduced with term full credibility. We also study situations where we cannot give full credibility to our data. Bayesian approach uses Bayesian statistics to estimate future claims and premiums. Here we are introduced with different models such as Poisson | Gamma model, which are of interest to us because their posterior mean can be expressed in our desired form. Empirical Bayesian approach is of interest to us because here we observe situations in which we have possess knowledge about number of claims over the past few years and we want to estimate number of claims for the next year. This is situation we often see in a real world. Here we have two models, Model 1 and Model 2, where we possess knowledge and data on a certain number of risks over a certain number of years. Are main goal is to estimate premiums for those risks, for the next year. Model 2 is better than Model 1 because he takes in account volume of business for those risks. In the end, we draw conclusion about what we have learned about Credibility theory.

• Keywords: teorija povjerenja; Bayesovska statistika; Poisson | Gamma model
• Keywords (english): credibility theory; Bayesian statistics; Poisson | Gamma model
• Language: croatian
• URN:NBN: urn:nbn:hr:217:305030
• Study programme: Title: Finance and Business Mathematics; 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: 2019-06-17 11:31:29