Mixed Poisson distributions are widely used in various disciplines including actuarial applications. The family of mixed Poisson distributions contains several members according to the choice of the mixing distribution for the parameter of the Poisson distribution. Very few of them have been studied in depth, mainly because of algebraic intractability. In this paper we will describe an EM type algorithm for maximum likelihood estimation for mixed Poisson distributions. The main achievement is that it reduces the problem of estimation to one of estimation of the mixing distribution which is usually easier. Variants of the algorithm work even when the probability function of the mixed distribution is not known explicitly but we have only an approximation of it. Other discrete distributions are treated as well.

We consider d identically and continuously distributed dependent risks X 1,…, Xd . Our main result is a theorem on the asymptotic behaviour of expected shortfall for the aggregate risks: there is a constant cd such that for large u we have . Moreover we study diversification effects in two dimensions, similar to our Value-at-Risk studies in [2].

We consider several one-period reinsurance models and derive a rule which minimizes the ruin probability of the cedent for a fixed reinsurance risk premium. The premium is calculated according to the economic principle, generalized zero-utility principle, Esscher principle or mean-variance principles. It turns out that a truncated stop loss is an optimal treaty in the class of all reinsurance contracts. The result is also valid for models not involving ruin probability. An example is the Arrow model with the Kahneman-Tversky value function.

We derive an expression for the density of the time to ruin in the classical risk model by inverting its Laplace transform. We then apply the result when the individual claim amount distribution is a mixed Erlang distribution, and show how finite time ruin probabilities can be calculated in this case.

In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

In distinction to the Borch’s model of a reinsurance market, this paper treats the problem of optimal risk exchange in an insurance market where treaties are allowed between the insurer and each insured only, not among insureds themselves. A characterization of the Pareto-optimal contract is found. It is shown that the indemnity function in the contract is of a coinsurance kind. We also present a way of finding Pareto-optimal contracts under individual rationality constraints. The obtained results are compared with those of the known model of risk exchange in a reinsurance market.

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

Multivariate stochastic processes with Poisson marginals are of interest in insurance and finance; they can be used to model the joint behaviour of several claim arrival processes, for example. We discuss various methods for the construction of such models, with particular emphasis on the use of copulas. An important class of multivariate counting processes with Poisson marginals arises if the events of a background Poisson process with constant intensity are moved forward in time by a random amount and possibly deleted; here we think of the events of the background process as triggering later claims in different categories. We discuss structural aspects of these models, their dependence properties together with stochastic order aspects, and also some related computational issues. Various actuarial applications are indicated.

In this paper we present an approach to market based valuation of life insurance policies, in the spirit of the NUMAT proposed by Hans Bühlmann (2002) in an editorial in the ASTIN Bulletin. We have experienced the valuation method for more than one decade, both as a pricing procedure applied to policy portfolios of leading insurance companies, and by including the valuation principles into several actuarial teaching activities.

Our interest is mainly focused here on participating policies that in Italy are characterized by contractually binding profit sharing rules. The problem of the fair valuation of the liabilities generated to the insurer by these contracts can be conveniently addressed using the methods of contingent claims pricing. These allow to price correctly the options embedded into the policies and to implement consistent plans of asset-liability management. The approach also provides a market based measurement of the value of business in force for outstanding policy portfolios and consistent assessments of the financial risk based capitals.

In this paper we discuss the application of the proportional hazard premium calculation principle.

In Section 2 we propose a method to calculate the proportional hazard premium of a compound risk when the severity distribution is subexponential.

In Section 3 we use the empirical distribution to calculate the premium when the proportional hazard principle is applied, which leads to a systematic underestimation of the premium. After studying the bias of the premium calculated using this non-parametric approach we take advantage of the bootstrap technique with subsampling to reduce it.

The purpose of this paper is to show that, for the classical risk model, explicit expressions for survival probabilities in a finite time horizon can be obtained through the inversion of the double Laplace transform of the distribution of time to ruin. To do this, we consider Gerber and Shiu (1998) and a particular value for their penalty function. Although other methods to address the problem exist, we find this approach, perhaps, more direct and simple. For the analytic inversion, we have applied twice, after some algebra, the Laplace complex inversion formula.

Insurance premiums are calculated using optimal control theory by maximising the terminal wealth of an insurer under a demand law. If the insurer sets a low premium to generate exposure then profits are reduced, whereas a high premium leads to reduced demand. A continuous stochastic model is developed, which generalises the deterministic discrete model of Taylor (1986). An attractive simplification of this model is that existing policyholders should pay the premium rate currently set by the insurer. It is shown that this assumption leads to a bang-bang optimal premium strategy, which cannot be optimal for the insurer in realistic applications.

The model is then modified by introducing an accrued premium rate representing the accumulated premium rates received from existing and new customers. Policyholders pay the premium rate in force at the start of their contract and pay this rate for the duration of the policy. It is shown that, for two demand functions, an optimal premium strategy is well-defined and smooth for certain parameter choices. It is shown for a linear demand function that these strategies yield the optimal dynamic premium if the market average premium is lognormally distributed.

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

In this paper we model the life-history of LTC-patients using a Markovian multi-state model in order to calculate premiums for a given LTC-plan. Instead of estimating the transition intensities in this model we use the approach suggested by Andersen et al. (2003) for a direct estimation of the transition probabilities. Based on the Aalen-Johansen estimator, an almost unbiased estimator for the transition matrix of a Markovian multi-state model, we calculate so-called pseudo-values, known from Jackknife methods. Further, we assume that the relationship between these pseudo-values and the covariates of our data are given by a GLM with the logit as link-function. Since the GLMs do not allow for correlation between successive observations we use instead the “Generalized Estimating Equations” (GEEs) to estimate the parameters of our regression model. The approach is illustrated using a representative sample from a German LTC portfolio.

Phase-type distributions, defined as the distributions of absorption times of certain Markov jump processes, constitute a class of distributions on the positive real axis which seems to strike a balance between generality and tractability. Indeed, any positive distribution may be approximated arbitrarily closely by phase-type distributions whereas exact solutions to many complex problems in stochastic modeling can be obtained either explicitly or numerically. In this paper we introduce phase-type distributions and retrieve some of their basic properties through appealing probabilistic arguments which, indeed, constitute their main feature of being mathematically tractable. This is illustrated in an example where we calculate the ruin probability for a rather general class of surplus processes where the premium rate is allowed to depend on the current reserve and where claims sizes are assumed to be of phase-type. Finally we discuss issues concerning statistical inference for phase-type distributions and related functionals such as e.g. a ruin probability.

Merton and Perold (1993) offered a framework for determining risk capital in a financial firm based on the cost of the implicit guarantee the firm provides to its subsidiaries to make up any operating shortfall. Merton and Perold assume the price of such guarantees is observable from the market at large. For an insurer, this may not be a realistic assumption. This paper proposes an insurance-specific framework for determining the cost of those parental guarantees, and utilizing that cost in pricing and portfolio mix evaluation. An insurer’s capital is treated as a shared asset, with the insurance contracts in the portfolio having simultaneous rights to access potentially all that shared capital. By granting underwriting capacity, an insurer’s management team is implicitly issuing a set of options to draw upon the common capital pool — similar in structure to letters of credit (LOC), except they are not loans but grants. The paper will (i) discuss the valuation of parental guarantees, beginning with Merton and Perold; (ii) treat insurer capital as a shared asset and explore the dual nature of insurer capital usage; (iii) offer a method for determining insurer capital usage cost; and (iv) demonstrate how this method could be used for product pricing and portfolio mix evaluation using economic value added concepts.

When insurance claims are governed by fat-tailed distributions considerable uncertainty about the value of the tail-index is often inescapable. In this paper, using the theory of risk aversion, a new premium principle (the power principle – analogous to the exponential principle for thin-tailed claims) is established and its properties investigated. Applied to claims arising from generalized Pareto distributions, the resultant premium is shown to be the ratio of the two largest expected claims, for which the ratio of the actual claims is an unbiased as well as a consistent estimator. Whereas thin-tailed claim premiums are determined largely by the first two moments of the claims distribution, fat-tailed claim premiums are determined by the first two extremes. The context of risk-aversion leads to a natural model for incorporating tail-index uncertainty into premiums, which nevertheless leaves the basic ratio structure unaltered. To illustrate the theory, possible ‘premiums’ for US hurricane data are examined, which utilize the consistent pattern of observed extremes.