This paper provides an introduction to asset pricing theory and its applications in non-life insurance. The first part of the paper presents a basic review of asset pricing models, including discrete and continuous time capital asset pricing models (the CAPM and ICAPM), arbitrage pricing theory (APT), and option pricing theory (OPT). The second part discusses applications in non-life insurance. Among the insurance models reviewed are the insurance CAPM, discrete time discounted cash flow models, option pricing models, and more general continuous time models. The paper concludes that the integration of actuarial and financial theory can provide major advances in insurance pricing and financial management.

This paper analyzes the framework of German participating payout life annuities (PLAs), which offer guaranteed minimum benefits as well as participation in insurers' surpluses. We show that the process of sharing surpluses between shareholders and policyholders follows transparent and consistent rules. Subsequently, we develop an asset-liability model for a stylized German life insurer that offers PLAs to evaluate benefit variability and insurer stability given stochastic mortality and capital market developments. Our results suggest that guaranteed benefits can be provided with high credibility via PLAs, while, at the same time, annuitants receive attractive money's worth ratios. Moreover, we show that it might be difficult to offer a fixed benefit annuity providing the same lifetime utility as a PLA for the same premium and a comparably low insolvency risk. Overall, PLA schemes may be an efficient way to deal with risk factors that are highly unpredictable and difficult to hedge over the long run, such as systematic longevity and investment risks.

In a paper entitled “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance” (XVIth Int. Congr. Act. Bruxelles 1960) Karl Borch has shown that, if the reinsurance premium is given, the smallest variance of the cedent's payments is obtained by a stop-loss reinsurance contract. Paul Markham Kahn, in “Some Remarks on a Recent Paper by Borch”, a paper read to the 1961 Astin Colloquium, has given an elegant proof of this theorem which appears to apply also to cases not considered by Borch. In this paper we study the problem from the reinsurer's point of view and it will be seen that, under natural conditions which are also used in the proof of the Borch-Kahn theorem, the minimum variance of the reinsurer's payments is obtained by a quota contract. This focusses attention on a peculiar opposition of interests of the two partners of a reinsurance contract. However, we do not enter any further into the investigation of a possible resolution of this conflict.

We study a problem concerning the division of risk between a cedent and his reinsurer. The risk may refer to a whole portfolio (in which case one might consider a Stop-Loss contract), or to a single contract (when an Excess-Loss contract is a possibility). We shall here use the nomenclature of a portfolio reinsurance.

Let it be assumed that a function F(x) is known which gives the probability of a total claim not exceeding x. We have then in Stieltjes integral notation

The two partners to a reinsurance arrangement agree that the reinsurer reimburses m(x).x out of a claim of x, where m(x) is a continuous and differentiate function of x and o ≤ m(x) ≤ 1.

Consider a classical compound Poisson model. The safety loading can be positive, negative or zero. Explicit expressions for the distributions of the surplus prior and at ruin are given in terms of the ruin probability. Moreover, the asymptotic behaviour of these distributions as the initial capital tends to infinity are obtained. In particular, for positive safety loading the Cramer case, the case of subexponential distributions and some intermediate cases are discussed.

This paper focuses on the estimation and application aspects of the Erlang count logit-weighted reduced mixture of experts model (EC-LRMoE), which is a fully flexible multivariate insurance claim frequency regression model. We first prove the identifiability property of the proposed model to ensure that it is a suitable candidate for statistical inference. An expectation conditional maximization (ECM) algorithm is developed for efficient model calibrations. Three simulation studies are performed to examine the effectiveness of the proposed ECM algorithm and the versatility of the proposed model. The applicability of the EC-LRMoE is shown through fitting an European automobile insurance data set. Since the data set contains several complex features, we find it necessary to adopt such a flexible model. Apart from showing excellent fitting results, we are able to interpret the fitted model in an insurance perspective and to visualize the relationship between policyholders’ information and their risk level. Finally, we demonstrate how the fitted model may be useful for insurance ratemaking.

Finite time ruin methods typically rely on diffusion approximations or discretization. We propose a new method by looking at the surplus process embedded at claim instants and develop a recursive scheme for calculating ruin probabilities. It is assumed that claim sizes follow a phase-type distribution. The proposed method is exact. The application of the method reveals where in the future the relative vulnerability to the company lies.

The aim of this editorial is to increase the acceptance of neural net modeling in the actuarial community. Neural nets may substantially improve classical actuarial models, if appropriately applied. We illustrate this on a toy example but, in fact, this should be understood as a universal concept. Assume we have a classical regression problem where the distribution of a response Y = Y(x) can be described by covariates x. A common actuarial problem is to determine the premium $\mu ({\textit{\textbf{x}}}) = {\mathbb E} [Y({\textit{\textbf{x}}})]$ as a function of the covariates x. Actuaries have developed excellent skills to solve such problems through finding appropriate regression functions x ↦ μ(x). This editorial shows how these skills can further be improved using the toolbox of neural nets.

The problem of distribution-free parameter estimation in recent credibility theory is discussed in the papers [1], [3] and [4] of the bibliography. Here, we consider a multiclass model with regression assumption. In that case, already treated by Ch. Hachemeister, [3], this author obtains an unsymmetrical matrix as an estimator of a covariance matrix. Of course, for practical use, this matrix is symmetrized in the obvious way. We show that this procedure can be avoided and that a lot of symmetrical unbiased estimators can be obtained at once.

By particularisations to the 1-rank model, we find the estimators given by Bühlmann and Straub, [1], [4].

In the multirank case, a generalization of the minimumvariance principle (minimization of the trace of the covariance matrix) leads to an optimal estimator of the mean regression vector. A noteworthy conclusion of our discussion is that there is no difference at all between the various credibility formulae (the inhomogenous formula, the homogeneous formula, the meanfree formula) if the mean regression vector is estimated optimally.

Finally we show that it must not be hoped to find, in a large set of unbiased estimators of the covariance matrix, one estimator furnishing, always, a semidefinite positive estimate.

Conditions under which the natural conjugate prior is not zero on its boundary are given, correcting an argument about conditions for exact credibility given in another paper.

In recent years, market-consistent valuation approaches have gained an increasing importance for insurance companies. This has triggered an increasing interest among practitioners and academics, and a number of specific studies on such valuation approaches have been published.

In this paper, we present a generic model for the valuation of life insurance contracts and embedded options. Furthermore, we describe various numerical valuation approaches within our generic setup. We particularly focus on contracts containing early exercise features since these present (numerically) challenging valuation problems.

Based on an example of participating life insurance contracts, we illustrate the different approaches and compare their efficiency in a simple and a generalized Black-Scholes setup, respectively. Moreover, we study the impact of the considered early exercise feature on our example contract and analyze the influence of model risk by additionally introducing an exponential Lévy model.

Probabilities of ruin are solutions of differential or integrodifferential equations. Solving such equations numerically can be performed by means of approximate quadrature formulae for the convolution part of the equation. In this contribution it is shown how applicable recursion formulae, giving results within a prescribed tolerance level, can be obtained. Some numerical results are displayed.

Personal financial decision making plays an important role in modern finance. Decision problems about consumption and insurance are in this article modelled in a continuous-time multi-state Markovian framework. The optimal solution is derived and studied. The model, the problem, and its solution are exemplified by two special cases: In one model the individual takes optimal positions against the risk of dying; in another model the individual takes optimal positions against the risk of losing income as a consequence of disability or unemployment.

Alzheimer's disease (AD) accounts for a significant proportion of long-term care costs. The recent discovery that the ε4 allele of the ApoE gene indicates a predisposition to earlier onset of AD raises questions about the potential for adverse selection in long-term care insurance, about long-term care costs in general, and about the potential effects on costs of gene therapy, or better targetted treatments for AD. This paper describes a simple Markov model for AD, and the estimation of the transition intensities from the medical and epidemiological literature.

It is essential for insurance regulation to have a clear picture of the risk measures that are used. We compare different mathematical interpretations of the Solvency Capital Requirement (SCR) definition from Solvency II that can be found in the literature. We introduce a mathematical modeling framework that enables us to make a mathematically rigorous comparison. The paper shows similarities, differences, and properties such as convergence of the different SCR interpretations. Moreover, we generalize the SCR definition to future points in time based on a generalization of the value at risk. This allows for a sound definition of the Risk Margin. Our study helps to make the Solvency II insurance regulation more consistent.

The class of distortion riskmetrics is defined through signed Choquet integrals, and it includes many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on general model spaces, extending various results in the existing literature on distortion risk measures and signed Choquet integrals. This paper offers a comprehensive toolkit of theoretical results on distortion riskmetrics which are ready for use in applications.

In this paper we compare ruin functions for two risk processes with respect to stochastic ordering, stop-loss ordering and ordering of adjustment coefficients. The risk processes are as follows: in the Markov-modulated environment and the associated averaged compound Poisson model. In the latter case the arrival rate is obtained by averaging over time the arrival rate in the Markov modulated model and the distribution of the claim size is obtained by averaging the ones over consecutive claim sizes.

In this paper we shall study the problem of determining “correct” premium rates for sub-groups of an insurance collective. This problem obviously occurs in all branches of insurance. However, it seems at present to be a really burning issue in automobile insurance. We shall show that the problem can be formulated as a conflict between groups which can gain by co-operating, although their interests are opposed. When formulated in this way, the problem evidently can be analysed and solved by the help ot the “Game Theory” of Von Neumann and Morgenstern (5).

We shall first illustrate the problem by a simple example. We consider a group of n1 = 100 persons, each of whom may suffer a loss of 1, with probability p1 = 0.1. We assume that these persons consider forming an insurance company to cover themselves against this risk. We further assume that for some reason, government regulations or prejudices of managers, an insurance company must be organized so that the probability of ruin is less than 0.001

If such a company is formed, expected claim payment will be

and the standard deviation of the claim payments will be

If the government inspection (or the company's actuary) agrees that the ruin probability can be calculated with sufficient approximation by assuming that the claim payments have a normal distribution, the company must have funds amounting to

An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. A previous paper developed a continuous-time model of the IBNYR process in which both the Poisson rate at which events occur and the parameters of the delay distribution are unknown random quantities; a full-distributional Bayesian method was then developed to predict the number of unreported events. Using a numerical example, the success of this approach was shown to depend upon whether or not the occurrence dates were available in addition to the reporting dates. This paper considers the more usual practical situation in which only discretized epoch information is available; this leads to a loss of predictive accuracy, which is investigated by considering various levels of quantization for the same numerical example.

We consider particular semi-Markov risk models M/SM and /SM for which the interarrival distributions are exponential with parameters depending of the risk type. We obtain theoretical expressions for the ruin probabilities on an infinite horizon.

In the special case of exponential distributions for the claim amounts, ruin probabilities are in both models solutions of linear differential systems. These systems are explicitly solved when there are only two risk types.

In this paper we give a recursive scheme, involving Panjer's recursion, to compute the distribution of a compound sum of integer claims, when the number of summands follows a Generalized Poisson distribution. Also, an elegant derivation is given for some basic properties of this counting distribution.