A model for general insurance pricing is developed which represents a stochastic generalisation of the discrete model proposed by Taylor (1986). This model determines the insurance premium based both on the breakeven premium and the competing premiums offered by the rest of the insurance market. The optimal premium is determined using stochastic optimal control theory for two objective functions in order to examine how the optimal premium strategy changes with the insurer’s objective. Each of these problems can be formulated in terms of a multi-dimensional Bellman equation.

In the first problem the optimal insurance premium is calculated when the insurer maximises its expected terminal wealth. In the second, the premium is found if the insurer maximises the expected total discounted utility of wealth where the utility function is nonlinear in the wealth. The solution to both these problems is built-up from simpler optimisation problems. For the terminal wealth problem with constant loss-ratio the optimal premium strategy can be found analytically. For the total wealth problem the optimal relative premium is found to increase with the insurer’s risk aversion which leads to reduced market exposure and lower overall wealth generation.

“War is the father of all things”. This sentence of Heraklit (between 540 and 535 BC) has a lot of truth in general. I am quoting it in connection with World War II (1939-1945) and inventions originating in this period.

The best known examples of inventions (or first practical use of such invention) are

• Nuclear Fission leading to the construction of the atomic bomb and nuclear reactors,

• The Programmable Electronic Computer. Best known is the machine of John von Neumann as Los Alamos: Mathematical Analyser Numerical Integrator and Computer (MANIAC),

• Radar to guide the airplanes particularly at night and in bad weather conditions,

• Penicillin to fight bacteriological diseases.

Some of these inventions relied on original discoveries already made before the war, but during the war they were for the first time used on a large scale.

The Capital Asset Pricing Model arises in an economy where agents have exponential utility functions and aggregate consumption is normally distributed, and gives the prices of assets with payoffs which are jointly normal with consumption. Such assets have normal marginal distributions and have dependence with consumption characterised by a normal copula. Wang has derived a transform which extends the CAPM by allowing pricing of assets in such an economy which have non-normal marginal distributions but still are normal-copula with consumption.

Here we set out the stochastic discount factors corresponding to this version of the CAPM and to Wang’s transform, and show how to calculate stochastic discount factors and hence asset prices for assets whose dependence with consumption is non-normal. We show that the impact of dependency structure on asset prices is significant.

For the classical Cramér-Lundberg risk model, a dividend strategy of threshold type has recently been suggested in the literature. This strategy consists of paying out part of the premium income as dividends to shareholders whenever the free surplus is above a given threshold level. In contrast to the well-known horizontal barrier strategy, the threshold strategy can lead to a positive infinite-horizon survival probability, with reduced profit in terms of dividend payments. In this paper we extend several of these results to a Sparre Andersen model with generalized Erlang(n)-distributed interclaim times. Furthermore, we compare the performance of the threshold strategy to a linear dividend barrier model. In particular, (partial) integro-differential equations for the corresponding ruin probabilities and expected discounted dividend payments are provided for both models and explicitly solved for n = 2 and exponentially distributed claim amounts. Finally, the explicit solutions are used to identify parameter sets for which one strategy outperforms the other and vice versa.

The objective of this paper is to present an analysis of a bonus-malus system (BMS) within the framework of the theory of ergodic Markov set-chains. It is shown that this type of Markov chains enables the evaluation of BMS, even in steady-state, under the assumption that transition probabilities change in a definite range. We introduce a model that allows the determination of the consequences of changes in the claim frequency of a policyholder. In a numerical example we examine the BMS employed by one of the Polish insurance companies.

In this paper we propose a discrete-time model with fixed maximum time to maturity of traded bonds. At each trading time, a bond matures and a new bond is introduced in the market, such that the number of traded bonds is constant. The entry price of the newly issued bond depends on the prices of the bonds already traded and a stochastic term independent of the existing bond prices. Hence, we obtain a bond market model for the reinvestment risk, which is present in practice, when hedging long term contracts. In order to determine optimal hedging strategies we consider the criteria of super-replication and risk-minimization.

For the martingale case Föllmer and Sondermann (1986) introduced a unique admissible risk-minimizing hedging strategy for any square-integrable contingent claim H. Schweizer (1991) developed their theory further to the semimartingale case introducing the notion of local risk-minimization. Møller (2001) extended the theory of Föllmer and Sondermann (1986) to hedge general payment processes occurring mainly in insurance. We expand local risk-minimization to the theory of hedging general payment processes and derive such a hedging strategy for general unit-linked life insurance contracts in a general Lévy process financial market.

Operational risk has become an important risk component in the banking and insurance world. The availability of (few) reasonable data sets has given some authors the opportunity to analyze operational risk data and to propose different models for quantification. As proposed in Dutta and Perry [12], the parametric g-and-h distribution has recently emerged as an interesting candidate.

In our paper, we discuss some fundamental properties of the g-and-h distribution and their link to extreme value theory (EVT). We show that for the g-and-h distribution, convergence of the excess distribution to the generalized Pareto distribution (GPD) is extremely slow and therefore quantile estimation using EVT may lead to inaccurate results if data are well modeled by a g-and-h distribution. We further discuss the subadditivity property of Value-at-Risk (VaR) for g-and-h random variables and show that for reasonable g and h parameter values, superadditivity may appear when estimating high quantiles. Finally, we look at the g-and-h distribution in the one-claim-causes-ruin paradigm.

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

In the present paper we give a short proof of a result of Zaks, Frostig and Levikson [2006] on the solution of an optimization problem which is related to the problem of optimal pricing of a heterogeneous portfolio.

Traditional Chain Ladder models are based on a few cells in an upper triangle and often give inaccurate projections of the reserve. Traditional stochastic models are based on the same few summaries and in addition are based on the often unrealistic assumption of independence between the aggregate incremental values. In this paper a set of stochastic models with weaker assumptions based on the individual claims development are described. These models can include information about settlement and can handle seasonal effects, changes in mix of business and claim types as well as changes in mix of claim size. It is demonstrated how the distribution of the process can be specified and especially how the distribution of the reserve can be determined. The method is illustrated with an example.

Cordeiro (2002a) has presented a multiple state model for Income Protection (formerly known as Permanent Health Insurance) which enables us to analyse claims by cause of disability. In that paper average claim durations conditioned on recovery have been calculated.

Since, when an Income Protection claim is reported, the insurance company never knows whether it will end in recovery, in death or in expiry, to analyse only average claim durations conditioned on recovery could be misleading, specially for the people responsible for the claims control process.

The main purpose of this paper is to calculate average claim durations conditioned on death and average claim durations not conditioned on any particular mode of claim termination. We calculate these claim durations for different deferred periods, causes of disability and ages at the beginning of sickness and we analyse the results obtained.

A generalized estimating equations (GEE) approach is developed to estimate structural parameters of a regression credibility model with independent or moving average errors. A comprehensive account is given to illustrate how GEE estimators are worked out within an extended Hachemeister (1975) framework. Evidenced by results of simulation studies, the proposed GEE estimators appear to outperform those given by Hachemeister, and have led to a remarkable improvement in accuracy of the credibility estimators so constructed.

A new method for analysing and projecting mortality is proposed and examined. The method takes observed time series of survival probabilities, finds the corresponding z-scores in the standard normal distribution and forecasts the z-scores. The z-scores appear to follow a common simple linear progression in time and hence forecasting is straightforward. Analysis on the z-score scale offers useful insights into the way mortality evolves over time. The method and extensions are applied to Australian female mortality data to derive projections to the year 2100 in both survival probabilities and expectations of life.

The formulation of generalized linear models in Klugman, Panjer and Willmot (2004) is a bit more general than is often seen, in that the residuals are not restricted to following a member of the exponential family. Some of the distributions this allows have potentially useful applications. The cost is that there is no longer a single form for the likelihood function, so each has to be fit directly. Here the use of loss distributions (frequency, severity and aggregate) in generalized linear models is addressed, along with a few other possibilities.

The aim of the paper is twofold. Firstly, to analyze the historical data of the earthquakes in the boarder area of Greece and then to produce a reliable model for the risk dynamics of the magnitude of the earthquakes, using advanced techniques from the Extreme Value Theory. Secondly, to discuss briefly the relevant theory of incomplete markets and price earthquake catastrophe bonds, combining the model found for the earthquake risk and an appropriate model for the interest rate dynamics in an incomplete market framework. The paper ends by providing some numerical results using Monte Carlo simulation techniques and stochastic iterative equations.

In this paper we explore the bias in the estimation of the Value at Risk and Conditional Tail Expectation risk measures using Monte Carlo simulation. We assess the use of bootstrap techniques to correct the bias for a number of different examples. In the case of the Conditional Tail Expectation, we show that application of the exact bootstrap can improve estimates, and we develop a practical guideline for assessing when to use the exact bootstrap.

The paper is devoted to the local moment matching method and its links with the discrete version of the s-convex extremal distributions. It is well-known that the local moment matching method can produce some negative masses. Connecting the local moment matching method to the discrete s-convex extrema gives an explicit criterion that explains why (and says when) the local moment matching method gives some negative mass.