Non-life insurance payouts consist of two factors: claimsizes and claim frequency. When calculating e.g. next years premium, it is vital to correctly model these factors and to estimate the unknown parameters. A standard way is to separately estimate in the claimsize and the claim frequency models.

Often there is a deductible with each single claim, and this deductible can be quite large, particularly in inhomogeneous cases such as industrial fire insurance or marine insurance. Not taking the deductibles into account can lead to serious bias in the estimates and consequent implications when applying the model.

When the deductibles are nonidentical, in a full maximum likelihood estimation all unknown parameters have to be estimated simultaneously. An alternative is to use pseudo-maximum likelihood, i.e. first estimate the claimsize model, taking the deductibles into account, and then use the estimated probability that a claim exceeds the deductible as an offset in the claim frequency estimation. This latter method is less efficient, but due to complexity or time considerations, it may be the preferred option.

In this paper we will provide rather general formulas for the relative efficiency of the pseudo maximum likelihood estimators in the i.i.d. case. Two special cases will be studied in detail, and we conclude the paper by comparing the methods on some marine insurance data.

The paper analyses the questions: Should – or should not – an individual buy insurance? And if so, what insurance coverage should he or she prefer? Unlike classical studies of optimal insurance coverage, this paper analyses these questions from a bonus-malus point of view, that is, for insurance contracts with individual bonus-malus (experience rating or no-claim) adjustments. The paper outlines a set of new statements for bonus-malus contracts and compares them with corresponding classical statements for standard insurance contracts. The theoretical framework is an expected utility model, and both optimal coverage for a fixed premium function and Pareto optimal coverage are analyzed. The paper is an extension of another paper by the author, see Holtan (2001), where the necessary insight to – and concepts of – bonus-malus contracts are outlined.

The calculation of mean claim sizes, in the presence of a deductible, is usually achieved through numerical integration. In case of a Lognormal or Gamma distribution, the quantities of interest can easily be expressed as functions of the cumulative distribution function, with modified parameters. This also applies to the F-distribution, where the incomplete Beta function enters the scene; see for instance the appendix in Hogg and Klugman (1984).

The purpose of this paper is to derive an explicit formula for the first two moments of the Inverse Gaussian distribution, in the presence of censoring. For reasons of completeness we also consider truncation of the Inverse Gaussian distribution by an upper limit.

The tractability of the derivation depends in a crucial way on two properties of the Inverse Gaussian distribution. Firstly, the cumulative distribution function of the Inverse Gaussian can be written as a simple function using the Normal probability integral. Secondly, the moment generating function of a censorized Inverse Gaussian distribution boils down to an expression containing the cumulative Inverse Gaussian distribution. This manifests itself most clearly in case of life insurance where the quantity of interest is the expectation of a present value. In case of non-life insurance, where the dimension of the Inverse Gaussian random variable is money instead of time, a further step is required: differentiation of the moment generating function.

So, a natural order of this paper is to address ourselves first to the derivation for the life case and afterwards tackling the more laborious derivation for the non-life case.

Holtan (1994) suggests to replace traditional bonus-malus systems by a high deductible financed by a short-term loan. Practical consequences of this proposal are investigated here. Simulation is used to evaluate the efficiency of the Taiwanese Bonus-malus system and the variability of premiums of an average policyholder. Holtan's high deductible system is analysed under a compound Poisson assumption, with truncated exponential claims. It is shown that the introduction of a high deductible would increase the variability of payments and the efficiency of the rating system for most policyholders.

A class of claim frequency distributions discussed by Sundt and Jewell (1981) is completely enumerated. Computational techniques for the associated compound total claims distribution in the presence of policy modifications are then derived.

In Industrial Fire insurance an aggregate limit for the amount retained by the policyholder under a deductible policy has been agreed upon more frequently in recent times. This agreement is equivalent to a stop-loss cover on the retained loss amount. For the Poisson-lognormal model the corresponding stop-loss net premium is calculated using various methods (normal power, translated gamma, various discretisations) and the methods are compared. Finally, the influence of the model parameters is examined and it is demonstrated how a variety of parameter value combinations can be reduced to only a few rating curves.