Distribution Of Claims Research Articles

The Time to Ruin in Some Additive Risk Models with Random Premium Rates

The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

The Time to Ruin in Some Additive Risk Models with Random Premium Rates

The risk processes considered in this paper are generated by an underlying Markov process with a regenerative structure and an independent sequence of independent and identically distributed claims. Between the arrivals of claims the process increases at a rate which is a nonnegative function of the present value of the Markov process. The intensity for a claim to occur is another nonnegative function of the value of the Markov process. The claim arrival times are the regeneration times for the Markov process. Two-sided claims are allowed, but the distribution of the positive claims is assumed to have a Laplace transform that is a rational function. The main results describe the joint Laplace transform of the time at ruin and the deficit at ruin. The method used consists in finding partial eigenfunctions for the generator of the joint process consisting of the Markov process and the accumulated claims process, a joint process which is also Markov. These partial eigenfunctions are then used to find a martingale that directly leads to an expression for the desired Laplace transform. In the final section, three examples are given involving different types of the underlying Markov process.

Delegation of Specification: An Agency Theory of Organizations

To date, much of the literature on institutional economics has relied on abstract metaphors based in exchange. Thus, Williamson introduced the fundamental insights surrounding his “transaction costs” model and discussed the governance of contracts in exchange relationships. Yet organizations are horizontal exchange systems only in metaphor; we also need to understand how vertical agency is directed -- how it is that people arrive with skills and learn what they need to know to make the enterprise successful. The distribution of higher-level decision patterns or residual claims do not fully explain the processes of specification of what organizational members actually do. This paper rejects the view common in the agency literature that an organization should be seen as a nexus of contracts. It develops an explicit theory of delegation of specification in which economizing principals prefer to hire imperfect agents as organizational members, allowing (and manipulating) third-party mechanisms in the organization to perfect or correct for agent shortcomings. In this way, collective or emergent features of organization settings are joined to dyadic concerns. A number of propositions follow. The logic allows us to link organization literatures now treated as disjunct. Organizations are created -- exist -- as efficient means of directing (and correcting) the behavior of agents. Organizations exist to tell people what to do.

Allocation Rules and the Stability of Mass Tort Class Actions

This paper studies the effects of allocation rules on the stability of mass tort class actions. I analyze a two-stage model in which a defendant faces multiple plaintiffs with heterogeneous damage claims. In stage 1, the plaintiffs play a noncooperative coalition formation game. In stage 2, the class action and any individual actions by opt-out plaintiffs are litigated or settled. I examine how the method for allocating the class recovery interacts with other factors -- the shape of the damage claim distribution, the scale benefits of the class action, and the plaintiffs' probability of prevailing at trial and bargaining power in settlement negotiations -- to determine the asymptotic stability of the global class. My results suggest criteria to attorney and courts for structuring and approving efficient allocations plans in mass tort class actions and for evaluating the predominance and superiority requirements for class certification in mass tort cases.

Modeling insurance claims via a mixture exponential model combined with peaks-over-threshold approach

We consider a model which allows data-driven threshold selection in extreme value analysis. A mixture exponential distribution is employed as the thin-tailed distribution in view of the special structure of insurance claims, where individuals are often grouped into categories. An EM algorithm-based procedure is described in model fitting. We then demonstrate how a multi-level fitting procedure will substantially reduce computation time when the data set is large. The fitted model is applied to derive statistics such as return level and expected tail loss of the claim distribution, and ruin probability bounds are obtained. Finally we propose a statistical test to justify the choice of mixture exponential distribution over the homogeneous exponential distribution in terms of improved fit.

Modelling component reliability using warranty data

Accelerated testing is often used to gauge the reliability of a product before it is released to the market. Warranty data provide valuable additional information about the reliability of a product and its constituent components via field data. In this study, warranty data provided by an Australian automobile manufacturer is used. This article considers the need to clean the data before commencing any analysis. It discusses the distribution of warranty claims and the use of warranty data to model the reliability of elemental components of an automobile. Various model options are considered before suitable modelling is adopted. The reality of modelling the reliability of a vehicle containing a large number of components is considered and a suitable method is adopted. Verification of the reliability modelling is made by comparing an estimated warranty cost with the actual cost. References Blischke, W. R., Karim, M. R. and Murthy, D. N. P. Warranty Data Collection and Analysis. Springer, London, 2011. Blischke, W. R. and Murthy, D. N. P. Warranty Cost Analysis. Marcel Dekker, Inc, New York, 1994. Blischke, W. R. and Murthy, D. N. P. Reliability: Modelling, Prediction, and Optimisation. John Wiley and Sons, New York, 2000. Campean, I. F. and Brunson, D. Reliability modelling using warranty data. International Conference on Statistics and Analytical Methods in Automotive Engineering. IMechE Conference Transactions 2002, 225--236. Cox, D. R. Renewal Theory. Chapman and Hall, New York, 1982. Cox, D. R. and Oakes, D. Analysis of Survival Data. Chapman and Hall, London, 1984. Hill, V. L. and Blischke, W. R. An assessment of alternative models in warranty analysis. Journal of Information and Optimization Sciences, 8, 1987, 33--55. Iskandar, B. P. and Blischke, W. R. Reliability and Warranty Analysis of a Motorcycle Based on Claims Data. In Case Studies in Reliability and Maintenance (eds W. R. Blischke and D. N. P. Murthy.) John Wiley and Sons, Hoboken, NJ, USA, 2003, 623--656. doi:10.1002/0471393002.ch27 Lawless, J. F. Statistical Models and Methods for Lifetime Data. John Wiley and Sons, New York, 1982. Lawless, J. F., Hu, J. and Cao, J. Methods for the estimation of failure distributions and rates from automobile warranty data. Lifetime Data Analysis, 1, 1995, 227--240. Lu, M. Automotive reliability prediction based on early field failure warranty data. Quality And Reliability Engineering International, 14, 1998, 103--108. doi:10.1002/(SICI)1099-1638(199803/04)14:2<103::AID-QRE147>3.0.CO;2-5 Majeske, K. D. Evaluating product and process design changes with warranty data. International Journal of Production Economics, 50, 1997, 79--89. doi:10.1016/S0925-5273(97)00034-0 Majeske, K. D. and Herrin, G. D. Determining the warranty benefits for automobile design changes. Proceedings Annual Reliability and Maintainability Symposium, 1998, 94--99. http://ieeexplore.ieee.org/xpls/abs\protect \global \let \OT1\textunderscore \unhbox \voidb@x \kern .06em\vbox {\hrule width.3em}\OT1\textunderscore all.jsp?arnumber=653636andtag=1 Nelson, W. Applied Life Data Analysis. John Wiley and Sons, New York, 1982. Rai, B. and Singh, N. Hazard rate estimation from incomplete and unclean warranty data. Reliability Engineering and System Safety, 81, 2003, 79--92. doi:10.1016/S0951-8320(03)00083-8 Singh, N. and Rai, B. K. Reliability Analysis and Prediction with Warranty Data: Issues, Strategies, and Methods. CRC Press, 2009. doi:10.1201/9781439803264 Summit, R. Estimating the reliability model parameters through a simulation of warranty claims: How much data is needed? ANZIAM Journal, 53, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/5114 Tijms, H. C. Stochastic Models: An Algorithmic Approach. John Wiley and Sons, New York, 1994. Xie, M. On the Solution of Renewal-Type Integral Equations. Communications in Statistics, 18, 1989, 281--293. doi:10.1080/03610918908812760

Modeling dependent yearly claim totals including zero claims in private health insurance

In insurance applications yearly claim totals of different coverage fields are often dependent. In many cases there are numerous claim totals which are zero. A marginal claim distribution will have an additional point mass at zero, hence this probability function (pf) will not be continuous at zero and the cumulative distribution functions will not be uniform. Therefore using a copula approach to model dependency is not straightforward. We will illustrate how to express the joint pf by copulas with discrete and continuous margins. A pair-copula construction will be used for the fit of the continuous copula allowing to choose appropriate copulas for each pair of margins.

On Fairness and Claims

Perhaps the best-known theory of fairness is John Broome's: that fairness is the proportional satisfaction of claims. In this article, I question whether claims are the appropriate focus for a theory of fairness, at least as Broome understands them in his current theory. If fairness is the proportionate satisfaction of claims, I argue, then the following would be true: fairness could not help determine the correct distribution of claims; fairness could not be used to evaluate the distribution of claims; fairness could not guide us in distributing claims (or unowed goods); we could not have a claim to be treated fairly; and we would not be wronged when treated unfairly. These entailments mean that it is questionable that fairness is concerned with claims in the way Broome suggests. At the very least, the relationship between fairness and claims appears to be more complex than the picture painted by Broome.

Distribution of litigation claims across a generalised psychiatric patient journey

Aims and methodWe analysed all 1213 negligence claims made against contributing psychiatric services since the inception of the National Health Service Litigation Authority (NHSLA) in 1995 and until 1 June 2009. More than half (55%) were settled, at a cost of £47.2 million, 26% were closed without penalty and 19% were still in progress at the time of review. Five individual claims exceeded £1 million.ResultsBy allocating 43 NHSLA-assigned causes for a claim to the 11 stages of a generalised patient journey, we noted that assessment of patient risks was the single largest cause of claims (32%) and the single largest cost of settlements (£16.2 million, 34%).Clinical implicationsAt the individual level it is difficult to see patterns of errors, whereas increased volumes reveal systemic trends. This analysis presents a new perspective from which to improve patient safety.

Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

In this paper we examine the large deviations principle (LDP) for sequences of classic Cramer-Lundberg risk processes under suitable time and scale modifications, and also for a wide class of claim distributions including (the non-super- exponential) exponential claims. We prove two large deviations principles: first, we obtain the LDP for risk processes on D∈[0,1] with the Skorohod topology. In this case, we provide an explicit form for the rate function, in which the safety loading condition appears naturally. The second theorem allows us to obtain the LDP for Aggregate Claims processes on D∈[0,∞) with a different time-scale modification. As an application of the first result we estimate the ruin probability, and for the second result we work explicit calculations for the case of exponential claims.

Summarizing Insurance Scores Using a Gini Index

Individuals, corporations and government entities regularly exchange financial risks y at prices Π. Comparing distributions of risks and prices can be difficult, particularly when the financial risk distribution is complex. For example, with insurance, it is not uncommon for a risk distribution to be a mixture of 0’s (corresponding to no claims) and a right-skewed distribution with thick tails (the claims distribution). However, analysts do not work in a vacuum, and in the case of insurance they use insurance scores relative to prices, called “relativities,” that point to areas of potential discrepancies between risk and price distributions.Ordering both risks and prices based on relativities, in this article we introduce what we call an “ordered” Lorenz curve for comparing distributions. This curve extends the classical Lorenz curve in two ways, through the ordering of risks and prices and by allowing prices to vary by observation. We summarize the ordered Lorenz curve in the same way as the classic Lorenz curve using a Gini index, defined as twice the area between the curve and the 45-degree line. For a given ordering, a large Gini index signals a large difference between price and risk distributions.We show that the ordered Lorenz curve has desirable properties. It can be expressed in terms of weighted distributions functions. In special cases, curves can be ranked through a partial ordering. We show how to estimate the Gini index and give pointwise consistency and asymptotic normality results. A simulation study and an example using homeowners insurance underscore the potential applications of these methods.

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by De Vylder and Goovaerts (1999, 2000). First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.

Extremal Subexponentiality in Ruin Probabilities

In this article, we consider risk models with a heavy-tailed parametric claim distribution from the subexponential class 𝒮 with at least two parameters. We choose the proper convergence of a parameter, such that the tail of the claims distribution becomes heavier, and then we study the limit behavior of the ruin probability. Further, we employ an appropriate functional normalization in order to keep intact the distributional properties.

Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error’s asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.

Matrix-form Recursive Evaluation of the Aggregate Claims Distribution Revisited

AbstractThis paper aims to evaluate the aggregate claims distribution under the collective risk model when the number of claims follows a so-called generalised (a, b, 1) family distribution. The definition of the generalised (a, b, 1) family of distributions is given first, then a simple matrix-form recursion for the compound generalised (a, b, 1) distributions is derived to calculate the aggregate claims distribution with discrete non-negative individual claims. Continuous individual claims are discussed as well and an integral equation of the aggregate claims distribution is developed. Moreover, a recursive formula for calculating the moments of aggregate claims is also obtained in this paper. With the recursive calculation framework being established, members that belong to the generalised (a, b, 1) family are discussed. As an illustration of potential applications of the proposed generalised (a, b, 1) distribution family on modelling insurance claim numbers, two numerical examples are given. The first example illustrates the calculation of the aggregate claims distribution using a matrix-form Poisson for claim frequency with logarithmic claim sizes. The second example is based on real data and illustrates maximum likelihood estimation for a set of distributions in the generalised (a, b, 1) family.

On optimality of the barrier strategy for the classical risk model with interest

In this paper, we consider the optimal dividend problem for a classical risk model with a constant force of interest. For such a risk model, a sufficient condition under which a barrier strategy is the optimal strategy is presented for general claim distributions. When claim sizes are exponentially distributed, it is shown that the optimal dividend policy is a barrier strategy and the maximal dividend-value function is a concave function. Finally, some known results relating to the distribution of aggregate dividends before ruin are extended.

Matrix-Form Recursions for a Family of Compound Distributions

AbstractIn this paper, we aim to evaluate the distribution of the aggregate claims in the collective risk model. The claim count distribution is firstly assumed to belong to a generalised (a, b, 0) family. A matrix form recursive formula is then derived to evaluate the related compound distribution when individual claim amounts follow a discrete distribution on non-negative integers. The corresponding formula is also given for continuous individual claim amounts. Secondly, we pay particular attention to the recursive formula for compound phase-type distributions, since only certain types of discrete phase-type distributions belong to the generalised (a, b, 0) family. Similar recursive formulae are obtained for discrete and continuous individual claim amount distributions. Finally, numerical examples are presented for three counting distributions.

On the Lagrangian Katz family of distributions as a claim frequency model

The Panjer (Katz) family of distributions is defined by a particular first-order recursion which is built on the basis of two parameters. It is known to characterize the Poisson, negative binomial and binomial distributions. In insurance, its main usefulness is to yield a simple recursive algorithm for the aggregate claims distribution. The present paper is concerned with the more general Lagrangian Katz family of distributions. That family satisfies an extended recursion which now depends on three parameters. To begin with, this recursion is derived through a certain first-crossing problem and two applications in risk theory are described. The distributions covered by the recursion are then identified as the generalized Poisson, generalized negative binomial and binomial distributions. A few other properties of the family are pointed out, including the index of dispersion, an extended Panjer algorithm for compound sums and the asymptotic tail behaviour. Finally, the relevance of the family is illustrated with several data sets on the frequency of car accidents.

INVESTIGATION OF THE GERBER-SHIU DISCOUNTED PENALTY FUNCTION ON FINITE TIME HORIZON

In this paper, the classical risk model with exponential claim sizes is considered. The explicit expression of the Gerber-Shiu discounted penalty function (x; ; t) and discounted moments m(x; ; t) on finite time horizon is obtained, where > 0 is the force of interest, x - the initial reserve. Also the expression of the Gerber-Shiu discounted penalty function on time interval [t1; t2] is derived. The dependance of discounted penalty function on the main parameters x; ; ; ; is presented in diagrams, where > 0 is the parameter of Poisson process, > 0 is the safety loading coefficient and is the parameter of claim distribution.

Completeness, Interconnectedness and Distribution of Interbank Exposures: A Parameterized Analysis of the Stability of Financial Networks

This paper assesses the impact of a certain structure of interbank exposures on the stability of a stylized financial system. Given a certain balance sheet structure of financial institutions, a large number of valid matrices of interbank exposures is created by a random generator. Assuming a certain loss given default, domino effects are simulated. The main results are, first, that financial stability depends not only on the completeness and interconnectedness of the network but also on the distribution of interbank exposures within the system (measured by entropy). Second, looking at random graphs, the sign of the correlation between the degree of equality of the distribution of claims and financial stability depends on the connectivity of the financial system as well as on additional parameters that affect the vulnerability of the system to interbank contagion. Third, the more concentrated assets are within a money center model, the less stable it is. Fourth, a money center model with asset concentration among core banks is less stable than a random graph with banks of homogeneous size. Results obtained in this paper extend existing theoretical literature that exclusively focuses on completeness and interconnectedness of the network as well as empirical literature that exclusively focuses on one particular financial network.