BONUS-MALUS SYSTEMS IN AUTOMOBILE INSURANCE

Diplomsko delo seznani bralca z bonus-malus sistemom in uporabo v avtomobilskih zavarovanjih. V prvem delu bonus-malus sistem identificiramo z markovsko verigo. Opišemo stohastične procese, na podlagi katerih ustvarimo modela za analizo porazdelitve škod posameznika ter celotnega portfelja. Navedemo osnove Bayesove teorije in statistike, ki je temelj za izpeljavo optimalnega bonus-malus sistema. Osrednji del je namenjen primerjavi bonus-malus sistemov. Uporabljamo različne prijeme analize bonus-malus sistema, izračunanih je več koeficientov, na podlagi katerih uvrščamo bonus-malus sisteme med strožje in milejše. Primerjani so bonus-malus sistemi iz evropskih držav (Švica in Finska) s sistemi s tujih celin (Kenija in Tajvan), vsi pa so primerjani z bonus-malus sistemom, ki se trenutno uporablja v Zavarovalnici Maribor. V zadnjem delu konstruiramo optimalni bonus-malus sistem s pomočjo modela statistične igre med naravo in aktuarstvom. Izpeljava temelji na Bayesovi statistiki s pomočjo linearne kombinacije apriorne premije in škodnega vedenja. Tak bonus-malus sistem je pošten do vseh zavarovancev, kar pomeni, da vsak posameznik plača premijo, sorazmerno njegovi škodni pogostosti. Hkrati pa z zelo strogimi premijskimi razredi malusa izravna premijske razrede bonusa do te mere, da je povprečna premijska stopnja enaka 100.The thesis focuses on the bonus-malus system and its use in the motor vehicle insurance segment. In the first part of the thesis, the bonus-malus system is identified with a Markov chain. Described are stochastic processes on the basis of which we have created two models for analysing the distribution of claims of an individual, and of the total portfolio. The principal features of Bayes\u27 Theorem and statistics, which form a basis for the implementation of an optimal bonus-malus system, are set out in this part of the thesis. The central part of the thesis deals with a comparison of different bonus-malus systems. Various models of the bonus-malus system analysis are presented, and several coefficients are calculated, on the basis of which we classify bonus-malus systems into more rigorous or more lenient ones. A comparison is made of bonus-malus systems used in some European countries (Switzerland and Finland) with those used in countries on other continents (Kenya and Taiwan), all of which are compared with the bonus-malus system currently used in Zavarovalnica Maribor. In the last part of the thesis, we have created an optimal bonus-malus system by means of a statistical match between nature and the actuarial science. The derivation is based on Bayesian statistics by use of a linear combination of an a priori premium and indemnity behaviour. Such bonus-malus system is fair to all insurance holders, meaning that each individual pays a premium in proportion to his/her claim frequency. At the same time, by applying very rigorous malus classes, this system levels off premium classes to the extent that the average premium rate equals 100