Actuarial comparisons for aggregate claims with randomly right-truncated claims

In this note, we consider an extension of the largest claims reinsurance treaty (LCR) with random upper thresholds for the claim sizes, that we call retention levels. The Laplace transform order for insurer’s aggregate claims is obtained assuming dependence among the random retention levels. Different results about the influence of dependence on the insurer total claim amount are also given including the connections with LCR and the case of combination with quota-share. Algebraic bounds for the insurer aggregate claims are obtained when a common fixed threshold is considered.     

Reinsurance of large claims

The large claims reinsurance treaties ECOMOR and LCR are well known not to be very popular. They have been largely neglected by most reinsurers because of their technical complexity. In this paper, we derive new mathematical results connected to asymptotic problems of these reinsurance forms. Perhaps these results can reopen the discussion on the usefulness of including the largest claims in the decision making procedure. Apart from asymptotic estimates for the tail of the distribution of the ECOMOR-quantity, we find its weak laws. We also deal with the weak laws of the LCR-quantity. Finally, we illustrate the outcomes with a number of simulations.     

The price of quality claims

Quality claims and quality in fact of financial and industrial products as well as financial and business services are essential to entice and attract customers. For these reasons, firms are often tempted to assert claims that might or might not be met. These claims have risks which cannot always be prevented when interpreted as a ‘sure thing’ while in fact, quality performance is probabilistic. This paper considers a financial (utility based) approach to pricing a quality claim. To do so, we assume that ex‐ante, a true quality performance is defined by a density function while claims are advertised—setting expectations for a quality performance. On the basis of these assumptions we determine the price associated with such claims. Copyright © 2010 John Wiley & Sons, Ltd.     

Paid-incurred chain claims reserving method

We present a novel stochastic model for claims reserving that allows us to combine claims payments and incurred losses information. The main idea is to combine two claims reserving models (Hertig’s (1985) model and Gogol’s (1993) model ) leading to a log-normal paid-incurred chain (PIC) model. Using a Bayesian point of view for the parameter modelling we derive in this Bayesian PIC model the full predictive distribution of the outstanding loss liabilities. On the one hand, this allows for an analytical calculation of the claims reserves and the corresponding conditional mean square error of prediction. On the other hand, simulation algorithms provide any other statistics and risk measure on these claims reserves.     

Stock options as barrier contingent claims

A comprehensive model is suggested that values securities as options and consequently ordinary stock options as compound options. Extending the basic Black–Scholes model, it can incorporate common contractual features and stylized facts. More specifically, a closed form solution is derived for the price of a call option on a down‐and‐out call. It is then shown how the result obtained can be generalized in order to price options on complex corporate securities, allowing among other things for corporate taxation, costly financial distress and deviations from the absolute priority rule. The characteristics of the model are illustrated with numerical examples.     

Probabilistic models for medical insurance claims

The paper develops two probabilistic models for claim size in health insurance based on the claims of families and individuals covered by the policy. First, general models for the numbers of families and persons covered by a medical insurance are developed. These are then used to construct models for claim size. Applications of these general models are then analysed and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A claims persistence process and insurance

The purpose of this paper is to introduce and construct a state dependent counting and persistent random walk. Persistence is imbedded in a Markov chain for predicting insured claims based on their current and past period claim. We calculate for such a process, the probability generating function of the number of claims over time and as a result are able to calculate their moments. Further, given the claims severity probability distribution, we provide both the claims process generating function as well as the mean and the claim variance that an insurance firm confronts over a given period of time and in such circumstances. A number of results and applictions are then outlined (such as a Compound Claim Persistence Process).     

Recursive evaluation of aggregate claims distributions

In this paper, we give a survey of recursive methods presented in the actuarial literature for exact and approximate evaluation of univariate and multivariate aggregate claims distributions. For the approximations, we present error bounds.     

Moments of claims in a Markovian environment

This paper considers discounted aggregate claims when the claim rates and sizes fluctuate according to the state of the risk business. We provide a system of differential equations for the Laplace–Stieltjes transform of the distribution of discounted aggregate claims under this assumption. Using the differential equations, we present the first two moments of discounted aggregate claims in a Markovian environment. We also derive simple expressions for the moments of discounted aggregate claims when the Markovian environment has two states. Numerical examples are illustrated when the claim sizes are specified.     

Full Bayesian analysis of claims reserving uncertainty

We revisit the gamma–gamma Bayesian chain-ladder (BCL) model for claims reserving in non-life insurance. This claims reserving model is usually used in an empirical Bayesian way using plug-in estimates for the variance parameters. The advantage of this empirical Bayesian framework is that allows us for closed form solutions. The main purpose of this paper is to develop the full Bayesian case also considering prior distributions for the variance parameters and to study the resulting sensitivities.