Randomized observation periods for the compound Poisson risk model: the discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities.     

Optimal dividend policies for piecewise-deterministic compound Poisson risk models

This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.     

Analysis of ruin measures for the classical compound Poisson risk model with dependence

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.     

Optimal dividend problems for Sparre Andersen risk model with bounded dividend rates

ABSTRACT

This paper concerns the optimal dividend problem with bounded dividend rate for Sparre Andersen risk model. The analytic characterizations of admissible strategies and Markov strategies are given. We use the measure-valued generator theory to derive a measure-valued dynamic programming equation. The value function is proved to be of locally finite variation along the path, which belongs to the domain of the measure-valued generator. The verification theorem is proved without additional assumptions on the regularity of the value function. Actually, the value function may have jumps. Under certain conditions, the optimal strategy is presented as a Markov strategy with space-time band structure. We present an iterative algorithm to approximate the optimal value function and the optimal dividend strategy. As applications, some numerical examples are given.     

On a multi-threshold compound Poisson surplus process with interest

We consider a multi-threshold compound Poisson surplus process. When the initial surplus is between any two consecutive thresholds, the insurer has the option to choose the respective premium rate and interest rate. Also, the model allows for borrowing the current amount of deficit whenever the surplus falls below zero. Starting from the integro-differential equations satisfied by the Gerber–Shiu function that appear in Yang et al. (2008), we consider exponentially and phase-type(2) distributed claim sizes, in which cases we are able to transform the integro-differential equations into ordinary differential equations. As a result, we obtain explicit expressions for the Gerber–Shiu function.     

Some results on the compound Markov binomial model

Abstract

Some years ago, in the course of an analysis of upper and lower limits for incomplete moments of statistical distributions I established an elementary summation formula¹ which proved rather useful for the purpose I had in view. Subsequently the formula was generalized by professor Steffensen, who showed² that the formula in question could be looked upon as giving the first term of an expansion in a certain type of series. Professor Steffensen established recurrence formulae for the coefficients of the series and computed the second, third and fourth term and the corresponding remainders¹, but did not arrive at a general, explicite expression for the coefficient of the n-th term and the corresponding remainder. A year later I found these expressions accidentally while I was working on some other problem. I also discovered the real nature of the procedure in question which proved to be a certain kind of least square fitted polynomial approximation. I did not, however, at the time publish the result. Taking the question up again later I found that the whole problem could be considerably generalized. The type of generalization in question is analogous to the generalization from polynomials to arbitrary functions.     

A Markov-switching generalized additive model for compound Poisson processes,with applications to operational loss models

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.     

Optimal dividend strategy with transaction costs for an upward jump model

In this paper, we consider the optimal dividend problem with transaction costs when the incomes of a company can be described by an upward jump model. Both fixed and proportional costs are considered in the problem. The value function is defined as the expected total discounted dividends up to the time of ruin. Although the same problem has already been studied in the pure diffusion model and the spectrally negative Lévy process, the optimal dividend problem in an upward jump model has two different aspects in determining the optimal dividends barrier and in the property of the value function. First, the value function is twice continuous differentiable in the diffusion case, but it is not in the jump model. Second, under the spectrally negative Lévy process, downward jumps will not cause any payment actions; however, it might trigger dividend payments when there are upward jumps. In deriving the optimal barriers, we show that the value function is bounded by a linear function. Using this property, we establish the verification theorem for the value function. By solving the quasi-variational inequalities associated with this problem, we obtain the closed-form solution to the value function and hence the optimal dividend strategy when the income sizes follow a common exponential distribution. In the presence of a fixed transaction cost, it is shown that the optimal strategy is a two-barrier policy, and the optimal barriers are only dependent on the fixed cost and not the proportional cost. A numerical example is used to illustrate how the fixed cost plays a significant role in the optimal dividend strategy and also the value function. Moreover, an increased fixed cost results in larger but less frequent dividend payments.     

Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1) process

In this paper, we study the retention levels for combinations of quota-share and excess of loss reinsurance by maximizing the insurer’s adjustment coefficient, which in turn minimizes the asymptotic result of ruin probability. Assuming that the premiums are determined by the expected value principle, we consider a discrete risk model, in which a dependence structure is introduced based on Poisson MA(1) process between the claim numbers for each period. The impact of dependence parameter on the adjustment coefficient is discussed and numerical examples are provided to illustrate the results obtained in this paper.     

Semiparametric estimation in the optimal dividend barrier for the classical risk model

In the context of an insurance portfolio which provides dividend income for the insurance company’s shareholders, an important problem in risk theory is how the premium income will be paid to the shareholders as dividends according to a barrier strategy until the next claim occurs whenever the surplus attains the level of ‘barrier’. In this paper, we are concerned with the estimation of optimal dividend barrier, defined as the level of the barrier that maximizes the expected discounted dividends until ruin, under the widely used compound Poisson model as the aggregate claims process. We propose a semi-parametric statistical procedure for estimation of the optimal dividend barrier, which is critically needed in applications. We first construct a consistent estimator of the objective function that is complexly related to the expected discounted dividends and then the estimated optimal dividend barrier as the minimizer of the estimated objective function. In theory, we show that the constructed estimator of the optimal dividend barrier is statistically consistent. Numerical experiments by both simulated and real data analyses demonstrate that the proposed estimators work reasonably well with an appropriate size of samples.     

On a risk model with dependence between interclaim arrivals and claim sizes

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 Albrecher, H. and Teugels, J. 2006. Exponential behavior in the presence of dependence in risk theory. Journal of Applied Probability, 43(1): 257–273. [Crossref], [Web of Science ®] , [Google Scholar]), 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.     

Analysis of a threshold dividend strategy for a MAP risk model

We consider a class of Markovian risk models in which the insurer collects premiums at rate c₁(c₂) whenever the surplus level is below (above) a constant threshold level b. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin. By interpreting that the insurer pays dividends continuously at rate c₁?c₂ whenever the surplus level is above b, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained by making use of an existing connection which links an insurer's surplus process to an embedded fluid flow process.     

Extremes on the discounted aggregate claims in a time dependent risk model

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rv's). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rv's. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.     

Payout Policy Tax Clienteles,Ex‐dividend Day Stock Prices and Trading Behavior in Germany: The Case of the 2001 Tax Reform

This paper examines the impact of the German 2001 tax reform, where Germany switched from a full imputation system to a classical system. Theory suggests that both price drop ratios and trading volume decrease following the reform. We document a significant reduction in the valuation of net dividends–in particular for high dividend yield stocks–and weakening payout policy tax clienteles. Ex‐dividend day returns are likely to be driven by short‐term traders. Though the reform removed incentives for cross‐border dividend stripping and reduced tax heterogeneity among investors, we show that the high trading volume around ex‐dividend days persists.     

Risk model based on the first-order integer-valued moving average process with compound Poisson distributed innovations

This paper considers a discrete-time risk model by introducing a temporal dependence structure between the number of claims for each period. The risk model is based on the first-order integer-valued moving average (INMA(1)) process with compound Poisson distributed innovations. We derive the explicit expression for the moment generating function of the aggregate claim amount, which can be used for the calculation of some related quantities. We examine the properties of the adjustment coefficient for measuring the dangerousness of an insurance portfolio. Some special cases are included and numerical examples are provided to illustrate the results obtained in the paper.     

Prediction of outstanding payments in a Poisson cluster model

We consider a simple Poisson cluster model for the payment numbers and the corresponding total payments for insurance claims arriving in a given year. Due to the Poisson structure one can give reasonably explicit expressions for the prediction of the payment numbers and total payments in future periods given the past observations of the payment numbers. One can also derive reasonably explicit expressions for the corresponding prediction errors. In the (a, b) class of Panjer's claim size distributions, these expressions can be evaluated by simple recursive algorithms. We study the conditions under which the predictions are asymptotically linear as the number of past payments becomes large. We also demonstrate that, in other regimes, the prediction may be far from linear. For example, a staircase-like pattern may arise as well. We illustrate how the theory works on real-life data, also in comparison with the chain ladder method.     

On semiparametric estimation of ruin probabilities in the classical risk model

The ruin probability of an insurance company is a central topic in risk theory. We consider the classical Poisson risk model when the claim size distribution and the Poisson arrival rate are unknown. Given a sample of inter-arrival times and corresponding claims, we propose a semiparametric estimator of the ruin probability. We establish properties of strong consistency and asymptotic normality of the estimator and study bootstrap confidence bands. Further, we present a simulation example in order to investigate the finite sample properties of the proposed estimator.     

Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes

In this paper, we study optimal dividend problem in the classical risk model. Transaction costs and taxes are required when dividends occur. The problem is formulated as a stochastic impulse control problem. By solving the corresponding quasi-variational inequality, we obtain the analytical solutions of the optimal return function and the optimal dividend strategy when claims are exponentially distributed. We also find a formula for the expected time between dividends. The results show that, as the dividend tax rate decreases, it is optimal for the shareholders to receive smaller but more frequent dividend payments.     

On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier

In the framework of classical risk theory we investigate a model that allows for dividend payments according to a time-dependent linear barrier strategy. Partial integro-differential equations for Gerber and Shiu's discounted penalty function and for the moment generating function of the discounted sum of dividend payments are derived, which generalizes several recent results. Explicit expressions for the nth moment of the discounted sum of dividend payments and for the joint Laplace transform of the time to ruin and the surplus prior to ruin are derived for exponentially distributed claim amounts.