Consider the classical compound Poisson model of risk theory, in which dividends are paid to the shareholders according to a barrier strategy. Let b* be the level of the barrier that maximizes the expectation of the discounted dividends until ruin. This paper is inspired by Dickson and Waters (2004). They point out that the shareholders should be liable to cover the deficit at ruin. Thus, they consider b0 , the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. In this paper, b* and b0 are compared, when the claim amount distribution is exponential or a combination of exponentials.

We are proposing a parametric model to rate insurance for vehicles belonging to a fleet. The tables of premiums presented take into account past vehicle accidents, observable characteristics of the vehicles and fleets, and violations of the road-safety code committed by drivers and carriers. The premiums are also adjusted according to accidents accumulated by the fleets over time. The proposed model accounts directly for explicit changes in the various components of the probability of accidents. It represents an extension of bonus malus-type automobile insurance models for individual premiums (Lemaire, 1985; Dionne and Vanasse, 1989 and 1992; Pinquet, 1997 and 1998; Frangos and Vrontos, 2001; Purcaru and Denuit, 2003). The extension adds a fleet effect to the vehicle effect so as to account for the impact that the unobservable characteristics or actions of carriers can have on truck accident rates. This form of rating makes it possible to visualize what impact the behaviors of owners and drivers can have on the predicted rate of accidents and, consequently, on premiums. The results are compared to those of the semiparametric approach.

It is now widely accepted that stochastic mortality – the risk that aggregate mortality might differ from that anticipated – is an important risk factor in both life insurance and pensions. As such it affects how fair values, premium rates, and risk reserves are calculated.

This paper makes use of the similarities between the force of mortality and interest rates to examine how we might model mortality risks and price mortality-related instruments using adaptations of the arbitrage-free pricing frameworks that have been developed for interest-rate derivatives. In so doing, the paper pulls together a range of arbitrage-free (or risk-neutral) frameworks for pricing and hedging mortality risk that allow for both interest and mortality factors to be stochastic. The different frameworks that we describe – short-rate models, forward-mortality models, positive-mortality models and mortality market models – are all based on positive-interest-rate modelling frameworks since the force of mortality can be treated in a similar way to the short-term risk-free rate of interest. While much of this paper is a review of the possible frameworks, the key new development is the introduction of mortality market models equivalent to the LIBOR and swap market models in the interest-rate literature.

These frameworks can be applied to a great variety of mortality-related instruments, from vanilla longevity bonds to exotic mortality derivatives.

Kaas, Dannenburg & Goovaerts (1997) generalized Jewell’s theorem on exact credibility, from the classical Bühlmann model to the (weighted) Bühlmann-Straub model. We extend this result further to the “Bühlmann-Straub model with a priori differences” (Bühlmann & Gisler, 2005). It turns out that exact credibility holds for a class of Tweedie models, including the Poisson, gamma and compound Poisson distribution – the most important distributions for insurance applications of generalized linear models (GLMs). Our results can also be viewed as an alternative to the HGLM approach for combining credibility and GLMs, see Nelder and Verrall (1997).

Estimation risk is known to have a huge impact on mean/variance optimized portfolios, which is one of the primary reasons to make standard Markowitz optimization unfeasible in practice. This issue has attracted new interest in the last years, and several approaches to incorporate estimation risk into portfolio selection have been developed only recently. In this article, we review these approaches as well as some older ones and compare them in an empirical out-of-sample study. The approaches can be classified along two criteria. First, we can differentiate heuristic approaches (restricting portfolio weights and employing simulation techniques) and those based on Bayesian statistics (shrinking the portfolios towards a pre-determined target). Second, the assumptions about the return-generating process differ, either assuming returns to be IID distributed or to be partly predictable. The central result of our empirical study is that all of the IID approaches, whether they account for estimation risk or not, are not superior to simple investment strategies like holding the market portfolio. A risk-adjusted outperformance is possible only if sample means are substituted with conditional expected return estimates. Furthermore, the Bayesian approaches reduce turnover and stabilize portfolio weights.

Consider a portfolio containing heterogeneous risks, where the policyholders’ premiums to the insurance company might not cover the claim payments. This risk has to be taken into consideration in the premium pricing. On the other hand, the premium that the insureds pay has to be fair. This fairness is measured by the distance between the risk and the premium paid. We apply a non-linear programming formulation to find the optimal premium for each class so that the risk is below a given level and the weighted distance between the risk and the premium is minimized. We consider also the dual problem: minimizing the risk level for a given weighted distance between risks and premium.

Standard optimal portfolio choice models assume that investors maximise the expected utility of their future outcomes. However, behaviour which is inconsistent with the expected utility theory has often been observed.

In a discrete time setting, we provide a formal treatment of risk measures based on distortion functions that are consistent with Yaari’s dual (non-expected utility) theory of choice (1987), and set out a general layout for portfolio optimisation in this non-expected utility framework using the risk neutral computational approach.

As an application, we consider two particular risk measures. The first one is based on the PH-transform and treats the upside and downside of the risk differently. The second one, introduced by Wang (2000) uses a probability distortion operator based on the cumulative normal distribution function. Both risk measures rank-order prospects and apply a distortion function to the entire vector of probabilities.

An exchange economy is considered, where agents (insurers/banks) trade risks. Decision making takes place under distorted probabilities, which are used to represent either rank-dependence of preferences or ambiguity with respect to real-world probabilities. Pricing formulas and risk allocations, generalising the results of Bühlmann (1980, 1984) are obtained via the construction of aggregate preferences from heterogeneous agents’ utility and distortion functions. This involves the introduction of a novel ‘collective ambiguity aversion’ coefficient. It is shown that probability distortion changes insurers’ behaviour, who trade not only to share the aggregate market risk, but are also found to bet against each other. Moreover, probability distortion tends to increase the price of insurance (increase asset returns). While the cases of rank-dependence and ambiguity are formally similar, an important distinction emerges as for rank-dependent preferences equilibria are determinate, while for ambiguity they are generally indeterminate.

Quadratic optimization is the classical approach to optimal control of pension funds. Usually the payment stream is approximated by a diffusion process. Here we obtain semiexplicit solutions for quadratic optimization in the case where the payment process is driven by a finite state Markov chain model commonly used in life insurance mathematics. The optimal payments are affine in the surplus with state dependent coefficients. Also constraints on payments and surplus are studied.

This paper proposes a multivariate extension of the equilibrium pricing transforms for pricing general financial and insurance risks. The multivariate Esscher and Wang transforms are derived from Bühlmann’s equilibrium pricing model (1980) under some assumptions on the aggregate risk. It is shown that the Esscher and Wang transforms coincide with each other when the underlying risks are normally distributed.

This paper examines the validity of some stylized statements that can be found in the actuarial literature about random effects models. Specifically, the actual meaning of the estimated parameters and the nature of the residual heterogeneity are discussed. A numerical illustration performed on a Belgian motor third party liability portfolio supports this discussion.