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Actuarial modelling of claim counts and losses in motor third party liability insurance
(07-2013) Tzougas, George J.; Τζουγάς, Γεώργιος Ι.; Athens University of Economics and Business, Department of Statistics; Frangos, Nikolaos
Actuarial science is the discipline that deals with uncertain events where clearly theconcepts of probability and statistics provide for an indispensable instrument in themeasurement and management of risks in insurance and finance. An important aspectof the business of insurance is the determination of the price, typically calledpremium, to pay in exchange for the transfer of risks. It is the duty of the actuary toevaluate a fair price given the nature of the risk. Actuarial literature research covers awide range of actuarial subjects among which is risk classification and experiencerating in motor third-party liability insurance, which are the driving forces of theresearch presented in this thesis. This is an area of applied statistics that has beenborrowing tools from various kits of theoretical statistics, notably empirical Bayes,regression, and generalized linear models, GLM, (Nelder and Wedderburn, 1972).However, the complexity of the typical application, featuring unobservable riskheterogeneity, imbalanced design, and nonparametric distributions, inspiredindependent fundamental research under the label `credibility theory', now acornerstone in contemporary insurance mathematics. Our purpose in this thesis is tomake a contribution to the connection between risk classification and experiencerating with generalized additive models for location scale and shape, GAMLSS,(Rigby and Stasinopoulos, 2005) and finite mixture models (Mclachlan and Peel,2000). In Chapter 1, we present a literature review of statistical techniques that can bepractically implemented for pricing risks through ratemaking based on a priori riskclassification and experience rated or Bonus-Malus Systems. The idea behind a prioririsk classification is to divide an insurance portfolio into different classes that consistof risks with a similar profile and to design a fair tariff for each of them. Recentactuarial literature research assumes that the risks can be rated a priori usinggeneralized linear models GLM, (see, for example, Denuit et al., 2007 & Boucher etal., 2007, 2008). Typical response variables involved in this process are the number ofclaims (or the claim frequency) and its corresponding severity (i.e. the amount theinsurer paid out, given that a claim occurred). In Chapter 2, we extend this setupfollowing the GAMLSS approach of Rigby and Stasinopoulos (2005). The GAMLSSmodels extend GLM framework allowing joint modeling of location and shapeparameters. Therefore both mean and variance may be assessed by choosing a marginal distribution and building a predictive model using ratemaking factors asindependent variables. In the setup we consider, risk heterogeneity is modeled as thedistribution of frequency and cost of claims changes between clusters by a function ofthe level of ratemaking factors underlying the analyzed clusters. GAMLSS modelingis performed on all frequency and severity models. Specifically, we model the claimfrequency using the Poisson, Negative Binomial Type II, Delaporte, Sichel and Zero-Inflated Poisson GAMLSS and the claim severity using the Gamma, Weibull, WeibullType III, Generalized Gamma and Generalized Pareto GAMLSS as these models havenot been studied in risk classification literature. The difference between these modelsis analyzed through the mean and the variance of the annual number of claims and thecosts of claims of the insureds, who belong to different risk classes. The resulting apriori premiums rates are calculated via the expected value and standard deviationprinciples with independence between the claim frequency and severity componentsassumed. However, in risk classification many important factors cannot be taken intoaccount a priori. Thus, despite the a priori rating system, tariff cells will not becompletely homogeneous and may generate a ratemaking structure that is unfair to thepolicyholders. In order to reduce the gap between the individual's premium and riskand to increase incentives for road safety, the individual's past record must taken intoconsideration under an a posteriori model. Bonus-Malus Systems (BMSs) are aposteriori rating systems that penalize insureds responsible for one or more accidentsby premium surcharges or maluses and reward claim-free policyholders by awardingthem discounts or bonuses. A basic interest of the actuarial literature is theconstruction of an optimal or `ideal' BMS defined as a system obtained throughBayesian analysis. A BMS is called optimal if it is financially balanced for theinsurer: the total amount of bonuses must be equal to the total amount of maluses andif it is fair for the policyholder: the premium paid by each policyholder is proportionalto the risk that they impose on the pool. The study of such systems based on differentstatistical models will be the main objective of this thesis. In Chapter 3, we extend thecurrent BMS literature using the Sichel distribution to model the claim frequencydistribution. This system is proposed as an alternative to the optimal BMS obtained bythe Negative Binomial model (see, Lemaire, 1995). We also consider the optimalBMS provided by the Poisson-Inverse Gaussian distribution, which is a special caseof the Sichel distribution. Furthermore, we introduce a generalized BMS that takesinto account both the a priori and a posteriori characteristics of each policyholder, extending the framework developed by Dionne and Vanasse (1989, 1992). This isachieved by employing GAMLSS modeling on all the frequency models consideredin this chapter, i.e. the Negative Binomial, Sichel and Poisson-Inverse Gaussianmodels. In the above setup optimality is achieved by minimizing the insurer's risk.The majority of optimal BMSs in force assign to each policyholder a premium basedon their number of claims disregarding their aggregate amount. In this way, apolicyholder who underwent an accident with a small size of loss will be unfairlypenalized in comparison to a policyholder who had an accident with a big size of loss.Motivated by this, the first objective of Chapter 4 is the integration of claim severityinto the optimal BMSs based on the a posteriori criteria of Chapter 3. For this purposewe consider that the losses are distributed according to a Pareto distribution,following the setup used by Frangos and Vrontos (2001). The second objective ofChapter 4 is the development of a generalized BMS with a frequency and a severitycomponent when both the a priori and the a posteriori rating variables are used. Forthe frequency component we assume that the number of claims is distributedaccording to the Negative Binomial Type I, Poisson Inverse Gaussian and SichelGAMLSS. For the severity component we consider that the losses are distributedaccording to a Pareto GAMLSS. This system is derived as a function of the years thatthe policyholder is in the portfolio, their number of accidents, the size of loss of eachof these accidents and of the statistically significant a priori rating variables for thenumber of accidents and for the size of loss that each of these claims incurred.Furthermore, we present a generalized form of the one obtained in Frangos andVrontos (2001). Finally, in Chapter 5 we give emphasis on both the analysis of theclaim frequency and severity components of an optimal BMS using finite mixtures ofdistributions and regression models (see Mclachlan and Peel, 2000 & Rigby andStasinopoulos, 2009) as these methods, with the exception of Lemaire(1995), have notbeen studied in the BMS literature. Specifically, for the frequency component weemploy a finite Poisson, Delaporte and Negative Binomial mixture, while for theseverity component we employ a finite Exponential, Gamma, Weibull andGeneralized Beta Type II (GB2) mixture, updating the posterior probability. We alsoconsider the case of a finite Negative Binomial mixture and a finite Pareto mixtureupdating the posterior mean. The generalized BMSs we propose adequately integraterisk classification and experience rating by taking into account both the a priori and aposteriori characteristics of each policyholder.
Cluster analysis techniques applications in automobile insurance
(Athens University of Economics and Business, 05-2003) Tsompanaki, Evgenia; Athens University of Economics and Business, Department of Statistics; Frangos, Nikolaos
Thesis - Athens University of Economics and Business. Postgraduate, Department of Statistics
Natural catastrophe models for weather related events and insurance applications
(12-12-2023) Κρουστάλλης, Ευστάθιος; Kroustallis, Efstathios; Athens University of Economics and Business, Department of Statistics; Karlis, Dimitrios; Zazanis, Michael; Ntzoufras, Ioannis; Chadjikonstantinidis, Efstathios; Vrontos, Spyridon; Tzougas, George; Frangos, Nikolaos
Η συχνότητα και η σφοδρότητα των φυσικών καταστροφών σχετικά με τον μετεωρολογικό κίνδυνο αναμένεται να αυξηθούν λόγω της κλιματικής αλλαγής. Οι νέες κλιματικές προβλέψεις αναδεικνύουν ότι στο μέλλον οι ακραίες συνθήκες που σχετίζονται με το κλίμα θα αυξηθούν σε πολλές ευρωπαϊκές περιοχές με σημαντικό κοινωνικό και οικονομικό αντίκτυπο, επηρεάζοντας επίσης τον ασφαλιστικό κλάδο. Επιπλέον, τα συμβάντα που σχετίζονται με τις καιρικές συνθήκες συχνά δεν καταγράφονται πλήρως, επομένως είναι δύσκολο να μοντελοποιήσουμε τη σχέση μεταξύ κλιματικών γεγονότων και της συχνότητας των ασφαλιστικών απαιτήσεων. Παράλληλα, πολλές φορές, υπάρχει δυσκολία στην αναγνώριση της πρωτογενούς αιτίας των σχετικών ασφαλιστικών ζημιών. Με κίνητρο τα παραπάνω ζητήματα, εφαρμόσαμε μια από τις πιο αντιπροσωπευτικές προσεγγίσεις μοντέλου «εποπτευόμενης εκμάθησης», μέθοδο βασισμένη σε δενδροδιαγράμματα αποφάσεων, την «Gradient Boosting», για την ταξινόμηση του πλήθους των ασφαλιστικών απαιτήσεων που προκαλούνται από καταιγίδες στην Ελλάδα και μια νέα κατηγορία μοντέλων σύνθετων κατανομών συχνότητας για την από κοινού μοντελοποίηση της συχνότητας των ασφαλιστικών αποζημιώσεων και του πλήθους των ακραίων μετεωρολογικών φαινομένων. Τα προτεινόμενα μοντέλα καταδεικνύουν την από κοινού κατανομή του πλήθους των μετεωρολογικών συμβάντων και των ασφαλιστικών αποζημιώσεων περιλαμβάνοντας γεωχωρικές μεταβλητές για την αξιολόγηση των επιπτώσεών τους στη συχνότητα εμφάνισης των συμβάντων και των σχετικών ασφαλιστικών απαιτήσεων. Τέλος, προτείνεται μια μεθοδολογική προσέγγιση στον υπολογισμό Κεφαλαιακών Απαιτήσεων Φερεγγυότητας στα πλαίσια των ιδιαιτεροτήτων σχετικά με τα χαρτοφυλάκια ασφαλιστικών εταιρειών, των κινδύνων και των μετεωρολογικών χαρακτηριστικών στην Ελλάδα.
Negative binomial regression with application in autorating
(Athens University of Economics and Business, 1998) Koutsoumbos, Fotios V.; Athens University of Economics and Business, Department of Statistics; Frangos, Nikolaos
Thesis - Athens University of Economics and Business. Postgraduate, Department of Statistics
Pricing options with Black - Scholes model and applications in Athens derivatives exchange
(Athens University of Economics and Business, 10-2011) Toumpakaris, Nikolaos; Frangos, Nikolaos
Thesis - Athens University of Economics and Business. Postgraduate, Department of Statistics
Probabilistic models in financial mathematics
(31-07-2006) Καλπινέλλη, Ευαγγελία; Kalpinelli, Evangelia; Athens University of Economics and Business, Department of Statistics; Zazanis, Michael; Panas, Epaminondas; Frangos, Nikolaos
Ο στόχος ενός στοχαστικού προτύπου είναι να συνδυαστεί η ικανότητα της θεωρίας Πιθανοτήτων να αντιμετωπίσει την αβεβαιότητα με την ικανότητα της παραγωγικής λογικής να προσδιορίσει τη δομή. Το αποτέλεσμα είναι ένας πλουσιότερος και πιο εκφραστικός φορμαλισμός με ένα ευρύ φάσμα τομέων εφαρμογής. Η δυσκολία με τα στοχαστικά πρότυπα είναι ότι τείνουν να πολλαπλασιάσουν την υπολογιστική πολυπλοκότητα των στοχαστικών και ντετερμινιστικών παραγόντων τους. Αυτή η διατριβή είναι μια συνοπτική ανάπτυξη της Μωρίας Πιθανοτήτων, που δίνει κύρια έμφαση στη μαθηματική αυστηρότητα και στις λεπτομερείς ιδιότητες ορισμένων προτύπων παρά στις γενικές έννοιες. Οι τρεις κρίσιμες έννοιες τη θεωρίας Πιθανοτήτων, εκείνη της τυχαίας μεταβλητής και αυτές της κατανομής πιθανότητας και της χαρακτηριστικής συνάρτησης μιας τυχαίας μεταβλητής, αναπτύσσονται συστηματικά σε χώρους πιθανότητας, στα πρώτα τρία κεφάλαια της παρούσας διατριβής. Εξίσου αναπτύσσονται και το Κεντρικό Οριακό Θεώρημα και η θεωρία σχετικά με τις Απείρως Διαιρετές Κατανομές. Στη συνέχεια, έχοντας ήδη εισάγει τις θεμελιώδεις αρχές της Θεωρίας Πιθανοτήτων, παρουσιάζουμε ορισμένα άλλα πεδία αυτής της θεωρίας, με σκοπό και να καλύψουμε τα σημαντικότερα πεδία αλλά και για να κρατήσουμε μια ισχυρή σύνδεση με τις εφαρμογές στα χρηματοοικονομικά. Για να είμαστε αρκετά συγκεκριμένοι, όλες αυτές οι μάλλον περίπλοκες μαθηματικές έννοιες τις οποίες παρουσιάζουμε, όπως τα Martingales, η Κίνηση Brown και το Στοχαστικό Ολοκλήρωμα, είναι εκείνες που οι οικονομικοί αναλυτές χρησιμοποιούν για να περιγράφουν τη συμπεριφορά των αγορών ή για να δημιουργήσουν νέες υπολογιστικές μεθόδους. Τελικά, αξίζει να αναφέρει ότι η Θεωρία Πιθανοτήτων μπορεί να εφαρμοστεί σε διάφορες άλλες επιστήμες, όπως η Βιοπληροφορική, η Επίσημη Επιστημολογία, η Θεωρία Παιγνίων, η Ψυχολογία κλπ., αλλά η παρουσίαση αυτών των τομέων εφαρμογής ξεφεύγει από τους σκοπούς αυτής της εργασίας.
A study of optimal bonus-malus systems in automobile insurance using different underlying approaches
Tzougas, George J.; Τζουγάς, Γεώργιος Ι.; Athens University of Economics and Business, Department of Statistics; Frangos, Nikolaos
The parallel growth of accidents and casualties to the increasing number of motor vehicles during the twentieth century and up to our days, has led the actuarial scientists around the world to develop Bonus-Malus Systems (BMS) that penalize insureds responsible for one or more accidents by premium surcharges or maluses and reward claim-free policyholders by awarding them discounts or bonuses. The Bonus-Malus Systems have played a fundamental role in the automobile insurance since it holds a significant part of the non-life business of many companies. Furthermore, due to the enormous and still growing competitiveness of the market, the Bonus-Malus Systems should be efficient, penalizing the bad drivers and simultaneously competitive. A basic interest of the actuarial literature is the construction of an optimal or ‘ideal’ BMS defined as a system obtained through Bayesian analysis. The main objective of the current thesis will be the study of optimal Bonus-Malus Systems using different underlying approaches. The majority of optimal BMS assign to each policyholder a premium based on his number of claims (claim frequency) disregarding his/hers aggregate claim amount (claim severity). In this way, a policyholder who underwent an accident with a small size of loss will be unfairly penalized in the same way with a policyholder who had an accident with a big size of loss. Motivated by this, in chapter 2 we will present an optimal BMS based on the a posteriori frequency and the a posteriori severity component under the assumption that the number of claims is distributed according to the Negative Binomial distribution and that the losses of the claims are distributed according to the Pareto distribution. Also, we will present a generalized optimal BMS that is based both on the a priori and a posteriori classification criteria by incorporating a priori information for each policyholder in the above design. In chapter 3, we will present a classical optimal BMS that takes into account the claim frequency and one that takes into account both the claim frequency and the claim severity. This time the claim frequency is distributed according to the Geometric distribution and the claim severity is distributed according to the Pareto distribution again. In chapter 4 we will present an optimal BMS that uses a three parameters distribution the Hofmann’s distribution for modeling claim frequency. Furthermore, a non-parametric method, that permits a simple formulation of the stationary and transition probabilities in a portfolio, is presented for the construction of an optimal BMS. In chapter 5, our analysis is based on the fact that for the construction of optimal BMS the distribution of the number of claims is frequently chosen within the “mixed-Poisson” family. We will show the general properties of “mixed Poisson” family distributions and we will give a unifying approach of several particular cases including the geometrical, the P-Erlang, the Negative Binomial and the Poisson inverse gaussian distributions. Also, in order to avoid the problem of adjustment that is the thickness of the tails of the underlying distributions we will present a new family of “mixed-Poisson”, built upon “fatty-tailed” underlying distributions, the “P-rational’’ distributions. In chapter 6, we will present an alternative approach to BMS the Stochastic Vortices Model developed under the assumption that we have an open portfolio, i.e., we consider that a policy can be transferred from one insurance company to another and that the new policies that constantly arrive into a portfolio can be placed not only in the “starting class” but into any of the bonus classes. The Stochastic Vortices Model applies to populations divided into sub-populations which correspond to the transient states of homogeneous Markov chains. Also, by using the limit state probabilities of the Model we can estimate the Long Run Distribution for a BMS and calculate optimal bonus-malus scales. Furthermore, since the Stochastic Vortices Model allows the subscription and the annulment of policies in the portfolio it is an alternative approach to the usual BMS model and the fact that the population is taken as open renders it quite representative of the reality. Finally, in chapter 7 for the first time in actuarial literature, we will propose a combination of a Poisson- Inverse Gaussian distribution for modeling claim frequency and of a Pareto distribution for modeling claim severity for the construction of an optimal BMS.
Utility indifference pricing : application on weather derivatives
(Athens University of Economics and Business, 05-2011) Papanikolaou, Zoi G.; Frangos, Nikolaos
Thesis - Athens University of Economics and Business. Postgraduate, Department of Statistics