The paper examines a classical risk model where both insurance and reinsurance policies are chosen by the insurer in order to minimize the expected maximal loss. We show that the optimal control problem reduces to a static case. We found that the optimal reinsurance is excess of loss reinsurance and describe the set optimal insurance policies. Such a policy providing the minimal variance of the risk left with insured turns out to be a combination of stop loss and deductible policies. The results are illustrated by two numerical examples.

We propose a capital allocation method for insurance companies. The amount of capital is directly related to the default risk. The expected value of default can be distributed among the liabilities based on the rule of asset payoff at the time of default. We derive a capital allocation scheme from this allocation of the expected default. Assets, liabilities, and other risky items on the balance sheet are treated in a uniform framework. The insurer’s capital is allocated among all these risk contributors. The allocated capitals are given in closed-form formulas, which have straightforward interpretations and are easy to compute. Connections with other allocation methods are also discussed.

After Hurricane Andrew the U.S. Congress entertained proposals to allow insurers to employ tax-deferred loss reserves. Interest was strong at first, but as the events receded interest waned. However, after the most recent severe hurricane seasons the proposals are again being discussed. In this paper we examine the institution of catastrophe loss reserves in a stylized model of insurance provisions. First, we find that the benefits of the tax-deferred loss reserves depend on the actuarial assumptions regarding the expected loss distribution. Second, we make the first attempt at estimating the change in consumer behavior and the social welfare implications for permitting tax deferred loss reserves. In sum, we find under specific circumstances there are large welfare gains for allowing the tax deferral of reserves.

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

We study the continuous-time portfolio optimization problem of an insurer. The wealth of the insurer is given by a classical risk process plus gains from trading in a risky asset, modelled by a geometric Brownian motion. The insurer is not only interested in maximizing the expected utility of wealth but is also concerned about the ruin probability. We thus investigate the problem of optimizing the expected utility for a bounded ruin probability. The corresponding optimal strategy in various special classes of possible investment strategies will be calculated. For means of comparison we also calculate the related mean-variance optimal strategies.

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

The US workers compensation system is different from those in many countries, but it is reinsured in the world-wide market and so has international impact. From its origin in the early 20th century it has been a laboratory for actuarial credibility techniques. In recent years deductibles have been increasing, so that fairly high excess coverage is now commonplace. This puts growing emphasis on estimation of the percentage of loss that is excess of high deductibles. A key element of the excess percentage is the frequency of loss by injury type. Fatalities and permanent disabilities cost more than other injury types, so when they have high relative frequency, more of the claims cost arises from large losses. The vector of claim frequency by injury type can be estimated by class of business using multi-dimensional credibility techniques. Historically the fraction of costs excess of various retentions has been calculated for large groups of classes (hazard groups) and not individual classes. We show, by testing a hold-out sample, that credibility estimation by class does produce additional information in comparison to a widely-used seven-hazard-group system.

The concept of economic equilibrium under uncertainty is applied to a model of insurance market where, in distinction to the classic Borch’s model of a reinsurance market, risk exchanges are allowed between the insurer and each insured only, not among insureds themselves. Conditions characterizing an equilibrium are found. A variant of the conditions, based on the Pareto optimality notion and involving risk aversion functions of the agents, is derived. An existence theorem is proved. Computation of the market premiums and optimal indemnities is illustrated by an example with exponential utility functions.

We provide a unified analytical treatment of first passage problems under an affine state-dependent jump-diffusion model (with drift and volatility depending linearly on the state). Our proposed model, that generalizes several previously studied cases, may be used for example for obtaining probabilities of ruin in the presence of interest rates under the rational investement strategies proposed by Berk & Green (2004).

Together with the Chain Ladder (CL) method, the Bornhuetter/Ferguson (BF) method is one of the most popular claims reserving methods. Whereas a formula for the prediction error of the CL method has been published already in 1993, there is still nothing equivalent available for the BF method. On the basis of the BF reserve formula, this paper develops a stochastic model for the BF method. From this model, a formula for the prediction error of the BF reserve estimate is derived.

Moreover, the model gives important advice on how to estimate the parameters for the BF reserve formula. E.g. it turns out that the appropriate BF development pattern is different from the CL pattern. This is a nice add-on as it makes BF to a standalone reserving method which is fully independent from CL. The other parameter required for the BF reserve is the well-known initial estimate for the ultimate claims amount. Here the stochastic model clearly shows what has to be meant with ‘initial’.

In order to apply the formula for the prediction error, the actuary must assess his uncertainty about both sets of parameters, about the development pattern and about the initial ultimate claims estimates. But for both, much guidance can be drawn from the estimates itself and from the run-off data given. Finally, a numerical example shows how the resulting prediction error compares to the one of the CL method.

We present the first step in a program to develop a comprehensive, unified equilibrium theory of asset and liability pricing. We give a mathematical framework for pricing insurance products in a multiperiod financial market. This framework reflects classical economic principles (like utility maximization) and generates pricing algorithms for non-hedgeable insurance risks.

In this paper, we consider the problem of Pareto optimal allocation in a general framework, involving preference functionals defined on a general real vector space. The optimization problem is equivalent to a modified sup-convolution of the different agents’ preference functionals. The results are then applied to a multi-period setting and some further characterization of Pareto optimality for an allocation is obtained for expected utility for processes.

We investigate a concept of multivariate pricing, which includes claim history for more than one line of business and is a generalization of the Bühlmann-Straub model. The multivariate credibility model is extended to allow for the age of claims to influence the estimation of future claims. The model is applied to data from a portfolio of commercial lines of business.

Evaluating the risk of disorders in long-term insurance often relies on rates of onset estimated from quite small epidemiological studies. These estimates can carry considerable uncertainty, hence so may functions of them, such as a premium rate. In the case of genetic disorders, where it may be required to demonstrate the reliability of genetic information as a risk factor, such uncertainty may be material. Epidemiological studies publish their results in a variety of forms and it is rarely easy to estimate the sampling distribution of a premium rate without access to the original data. We found a large study of breast and ovarian cancer that cited relative risks of breast and ovarian cancer onset, with confidence intervals, in 10-year age groups. We obtained critical illness premium rates and their sampling distributions by parametric bootstrapping, and investigated the effect of possible patterns of sampling correlations. We found that this study provides ample statistical evidence that known BRCA1 or BRCA2 mutations, or a typical family history of breast or ovarian cancer, are reliable risk factors, but the sampling covariances of the relative risks could be important at some ages and terms. Studies that cite only standard errors of parameter estimates erect a small but awkward barrier between the models they describe, and some important actuarial questions.

We consider a dependent portfolio of insurance contracts. Asymptotic tail probabilities of the ECOMOR and LCR reinsurance amounts are obtained under certain assumptions about the dependence structure.

Much attention has been focused recently on the issue of valuing guaranteed minimum death benefits embedded in annuity contracts. These benefits resemble a sequence of put options and their value should obey a differential equation similar to the Black-Scholes equation for simple put options. This paper derives a number of analytic solutions to this equation for a number of simple mortality laws.

We consider the chain ladder reserving method in a Bayesian set up, which allows for combining the information from a specific claims development triangle with the information from a collective. That is, for instance, to consider simultaneously own company specific data and industry-wide data to estimate the own company's claims reserves. We derive Bayesian estimators and credibility estimators within this Bayesian framework. We show that the credibility estimators are exact Bayesian in the case of the exponential dispersion family with its natural conjugate priors. Finally, we make the link to the classical chain ladder method and we show that using non-informative priors we arrive at the classical chain ladder forecasts. However, the estimates for the mean square error of prediction differ in our Bayesian set up from the ones found in the literature. Hence, the paper also throws a new light upon the estimator of the mean square error of prediction of the classical chain ladder forecasts and suggests a new estimator in the chain ladder method.

We apply simple geometrical arguments to show that well-known approaches to determine the premium in insurance contract minimize a weighted squared differences both between the individual premiums and the individual claims and between the total premiums for classes of homogeneous risks and total claims from these blocks of business.

In this paper we explore the problem of economic capital allocations in the context of non-negative multivariate (insurance) risks possessing a dependence structure. We derive a general result and illustrate it with a number of useful examples. One such example, for instance, develops explicit expressions for the discussed economic capital decomposition rule when the underlying portfolio consists of dependent compound Poisson risks.

In an arbitrage-free economy, it is well-known that financial risks can be priced using equivalent martingale measures. We establish in this paper that, for general stochastic processes, the Wang Transform does not lead to a price which is consistent with the arbitrage-free price. Based on these results we must conclude that the Wang Transform cannot be a universal framework for pricing financial and insurance risks.