On joint densities involving the number of claims until ruin, assuming dependent claim sizes and inter-claim times

This thesis generalizes the classical risk model in which the total premium income paid by customers follows a linear function of time with a constant positive premium rate. A more realistic model taking into account the uncertainty of customer arrivals is the one with stochastic premiums. The current work studies various models for the surplus process with the amounts of individual premiums being random, as well as some dependence structures between individual claim size amounts, individual premium size amounts, inter-claim times and inter-premium times. Various risk measures are considered, such as the ruin probability or the ruin time, through the analysis of the expected discounted penalty function, also known as Gerber-Shiu function. For some of the models under consideration, the distribution of dividend payments to shareholders is additionally studied, considering for the respective stochastic surplus process of the portfolio various dividend payment strategies. Chapter 1 is an introductory section which gives basic concepts from ruin theory, a brief description of the classical compound Poisson model, known as the Cramer-Lundberg model, the expected discounted penalty function, and gives the solution of the defective renewal equation for the Gerber-Shiu function. In Chapter 2 two risk models with stochastic premiums without dependencies are examined. In the first model the aggregate premium amount is equal to the number of insured persons which is described by a Poisson process, while in the second model premiums and claims occur in the time according to compound Poisson processes. Chapter 3 considers two risk models with stochastic premiums and dependencies. The case in which the size of a claim controls the distributions of the time until the arrival of the next claim and individual premium sizes is first examined, while in the second case under consideration the time between successive claims determines the distributions of the next claim size and individual premium size. Chapter 4 examines two models with stochastic premiums, dependencies and dividend strategies. Namely, in the first model, under a constant dividend barrier, there is dependence between the claim sizes and their interarrival times, while in the second model, under a threshold dividend strategy, dependencies between premiums and inter-premium times as well as dependencies between claim sizes and their inter-claim times are considered. The Appendix provides the relevant mathematical tools (Laplace transform, Dickson-Hipp operator, etc.) used in this study. Mathematica was used to perform symbolic calculations, arithmetic operations, and graphs.

Dependent frequency–severity modeling of insurance claims

Βασικό αντικείμενο της παρούσας διπλωματικής εργασίας είναι η μελέτη του πλεονάσματος για ανανεωτικές στοχαστικές διαδικασίες ασφαλιστικών κινδύνων. Πραγματοποιείται ενδελεχής ανάλυση σημαντικών μεγεθών όπως η πιθανότητα χρεοκοπίας, ο χρόνος χρεοκοπίας, το πλεόνασμα πριν τη χρεοκοπία, το έλλειμμα που δημιουργείται μετά την επέλευση της χρεοκοπίας κα. Στο Κεφάλαιο 1 γίνεται παρουσίαση των μαθηματικών μεγεθών που θα φανούν χρήσιμα στη συνέχεια. Πιο συγκεκριμένα, παρουσιάζονται το πολυώνυμο Lagrange, ο τελεστής Dickson – Hipp, η ελλειμματική ανανεωτική εξίσωση, καθώς επίσης και δύο κατανομές, πάνω στις οποίες βασίζεται το μεγαλύτερο μέρος της εργασίας, οι κατανομές Erlang και Cox. Στο Κεφάλαιο 2 παρουσιάζεται η συνάρτηση Gerber – Shiu για το κλασσικό μοντέλο κινδύνου Poisson, όπως επίσης και εφαρμογές της συγκεκριμένης συνάρτησης μέσω της ελλειμματικής ανανεωτικής εξίσωσης. Στο Κεφάλαιο 3 μελετάται η συνάρτηση Gerber – Shiu για το μοντέλο Sparre – Andersen (ανανεωτικό μοντέλο της Θεωρίας Κινδύνου) με εξάρτηση μεταξύ των ενδιάμεσων χρόνων εμφάνισης των κινδύνων και των αντίστοιχων μεγεθών των απαιτήσεων ζημιών. Στο Κεφάλαιο 4 δίνονται αναλυτικά αποτελέσματα για τη συνάρτηση των Gerber – Shiu που εξετάστηκε στο Κεφάλαιο 3, θεωρώντας ότι τα ύψη των απαιτήσεων ακολουθούν κατανομές Erlang και Cox. Το Κεφάλαιο 5 εξετάζεται η κατανομή του χρόνου χρεοκοπίας για το κλασσικό μοντέλο κινδύνου Poisson, ενώ δίνεται η από-κοινού συνάρτηση πυκνότητας του χρόνου χρεοκοπίας και του ελλείμματος κατά τη χρεοκοπία. Τέλος, στο Κεφάλαιο 6 παρέχονται χρήσιμα συμπεράσματα για τον υπολογισμό του συνολικού ύψους των απαιτήσεων και το πλήθος αυτών πριν τη χρεοκοπία, μέσω της γενικευμένης συνάρτησης των Gerber - Shiu.

Gerber–Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times

This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.

An insurance claim is a form of request from the policy holder to obtain protection against financial losses due to a risk that occurs. Claims that occur every time there is a risk are called individual claims, while the total of individual claims during one insurance period is called aggregate claims. Claims are an important factor in optimizing insurance company expenses, where one of the calculations that insurance companies need to know based on claims is aggregate loss. Aggregate loss is the total loss in a period experienced by policy holders covered by an insurance company. This study aims to determine the average and variance of claims for the number of claims (frequency) with a Negative Binomial distribution and the amount of claims (severity) with a Discreate Uniform distribution in claim payments according to all types of guarantees and the nature of PT injuries. Jasa Raharja (Persero) Purwakarta Representative during the 2018-2020 period. This research uses a collective risk model and the help of Easyfit software to determine the best distribution for the number and size of claims. The results of the research show that from the recapitulation data of claim payments according to all types of coverage and nature of injury in PT. Jasa Raharja (Persero) Purwakarta Representative during the 2018-2020 period, with the number of claims having a Negative Binomial distribution and the amount of claims having a Discrete Uniform distribution, the average aggregate claim occurrence was IDR with a variance of IDR during the 2018-2020 insurance period.

A discrete-time risk model with a mathematically tractable dependence structure between interclaim times and claim sizes is considered in the presence of an impulsive dividend strategy. Under such a strategy, once the insurer's reserve upcrosses the level b, the excess of the reserve over is paid off as dividends. We derive difference equations for both the expected discounted penalty function and the expected present value of dividend payments. Solution procedures for these difference equations are provided. When the joint distribution of the interclaim time and claim size is a finite mixture of bivariate geometric distributions, closed-form expressions are given. Numerical results for several sets of parameters are also provided to illustrate the applicability of the results obtained.

We consider an extension to the classical compound Poisson risk model for which the increments of the aggregate claim amount process are independent. In Albrecher and Teugels (2006), an arbitrary dependence structure among the interclaim time and the subsequent claim size expressed through a copula is considered and they derived asymptotic results for both the finite and infinite-time ruin probabilities. In this paper, we consider a particular dependence structure among the interclaim time and the subsequent claim size and we derive the defective renewal equation satisfied by the expected discounted penalty function. Based on the compound geometric tail representation of the Laplace transform of the time to ruin, we also obtain an explicit expression for this Laplace transform for a large class of claim size distributions. The ruin probability being a special case of the Laplace transform of the time to ruin, explicit expressions are therefore obtained for this particular ruin related quantity. Finally, we measure the impact of the various dependence structures in the risk model on the ruin probability via the comparison of their Lundberg coefficients.

In this paper, models for claim frequency and average claim size in non-life insurance are considered. Both covariates and spatial random effects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model, we allow for dependencies between claim size and claim frequency. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion and the predictive model choice criterion, we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial effects significantly improves the models for both claim frequency and claim size, and also leads to more accurate predictions of the total claim sizes. Further, we detect significant dependencies between the number of claims and claim size. Both spatial and number of claims effects are interpreted and quantified from an actuarial point of view.

In this paper, we consider a risk process in which the distribution of the inter-claim time is the sum of two independent exponential random variables. We introduce a dependence structure between the claim size and the inter-claim time. The structure is based on FGM copula. An integro-differential equation for the expected discounted penalty function is derived and an explicit expression for the Laplace transform of ruin probability is given for exponential claim size.

In this paper, we study the discounted renewal aggregate claims with a full dependence structure. Based on a mixing exponential model, the dependence among the inter-claim times, the claim sizes, as well as the dependence between the inter-claim times and the claim sizes are included. The main contribution of this paper is the derivation of the closed-form expressions for the higher moments of the discounted aggregate renewal claims. Then, explicit expressions of these moments are provided for specific copulas families and some numerical illustrations are given to analyze the impact of dependency on the moments of the discounted aggregate amount of claims.

On a risk model with stochastic premiums income and dependence between income and loss

On the number of near-maximum insurance claim under dependence

In this paper, we consider a Sparre Andersen risk model where the interclaim time and claim size follow some bivariate distribution. Assuming that the risk model is also perturbed by a jump-diffusion process, we study the Gerber–Shiu functions when ruin is due to a claim or the jump-diffusion process. By using a q-potential measure, we obtain some integral equations for the Gerber–Shiu functions, from which we derive the Laplace transforms and defective renewal equations. When the joint density of the interclaim time and claim size is a finite mixture of bivariate exponentials, we obtain the explicit expressions for the Gerber–Shiu functions.

ABSTRACTIn the present paper, we consider the classical compound Poisson risk model with dependence between claim sizes and claim inter-arrival time. We attempt to analyze the approximation of finite time ruin probability. The finite time ruin probabilities are plotted for fixed threshold value associated to the claim inter-arrival time and also for fixed dependence parameter in Nelsen (2006) copula separately. Additionally, a general form for joint density of the interclaim times and claim sizes is considered. With respect to the classical Gerber-Shiu's (1998) function, first some structural density properties of dependent collective risk model is obtained. Then the ladder height probability density function of claim sizes is computed and the dependency structure investigated for Erlang interclaim time. As the application, some dependent models of the interclaim times and claim sizes are studied.