Ragnar Norberg

Power tailed ruin probabilities in the presence of risky investments

The present paper addresses the situation where the reserve of an insurance business is currently invested in an asset that may yield negative interest. Upper and lower bounds for the probability of ruin are obtained in the case where the cash flow of premiums less claims and the logarithm of the asset price are both Levy processes. These bounds are in general power functions of the initial reserve.

Ruin problems with assets and liabilities of diffusion type

Ruin and related problems are studied for a risk business with compounding assets when the cash flow and the cumulative interest rate are diffusion processes with coefficients depending on the time and on the current cash balance. Differential equations are obtained for the probabilities of ruin at a given date, in finite time, and in infinite time. Some previously known explicit formulas related to Brownian motion come out as special cases. Relationships between crossing probabilities and transition probabilities are investigated and, in particular, existing results on the probability distribution of the running maximum of a Brownian motion and on the relationship between the probability of ruin and on the probability distribution of the discounted total payments are generalized. Proofs rest on a martingale technique.

Prediction of Outstanding Liabilities in Non-Life Insurance

A fully time-continuous approach is taken to the problem of predicting the total liability of a non-life insurance company. Claims are assumed to be generated by a non-homogeneous marked Poisson process, the marks representing the developments of the individual claims. A first basic result is that the total claim amount follows a generalized Poisson distribution. Fixing the time of consideration, the claims are categorized into settled, reported but not settled, incurred but not reported, and covered but not incurred. It is proved that these four categories of claims can be viewed as arising from independent marked Poisson processes. By use of this decomposition result predictors are constructed for all categories of outstanding claims. The claims process may depend on observable as well as unobservable risk characteristics, which may change in the course of time, possibly in a random manner. Special attention is given to the case where the claim intensity per risk unit is a stationary stochastic process. A theory of continuous linear prediction is instrumental.

Differential equations for moments of present values in life insurance

Abstract Ordinary differential equations are obtained for first and higher order conditional moments of present values of payments in respect of a life insurance policy described as a time-continuous Markov chain. Those for the first moments are the well-known Thieles differential equations for the reserve. It is shown how the differential equations can be used to construct untraditional insurance products. Numerical computations of moments are performed for some forms of insurance common in practice. Applications to problems in pure probability theory are demonstrated by examples.

A credibility theory for automobile bonus systems

Summary This paper deals with the problem of designing experience rating systems of the bonus type, commonly used in automobile insurance. On the basis of a simple model the mean squared deviation between a policys expected claim amount and its premium in the nth insurance period as n→∞, is taken as a measure of the efficiency of a bonus system. It is shown that to any set of bonus rules (which determines the bonus class transitions of the policies), there is an optimal premium scale, which coincides with the one proposed by Pesonen in 1963. Thus the problem of choosing an efficient bonus system reduces to choosing efficient bonus rules. Examples are given of comparison between different bonus rules. In one example the present Norwegian bonus system is compared to alternative systems. Comments are made on earlier papers on bonus systems. The credibility theoretic foundation is laid in a separate section.

Reserves in Life and Pension Insurance

Prospective and retrospective reserves are defined as conditional expected values, given some information available at the time of consideration. Each specification of the information invoked gives rise to a corresponding pair of reserves. Relationships between reserves are established in the general set-up. For the prospective reserve the present definition conforms with, and generalizes, the traditional one. For the retrospective reserve it appears to be novel. Special attention is given to the continuous time Markov chain model frequently used in the context of life and pension insurance. Thiele’s differential equation for the prospective reserve is shown to have a retrospective counterpart. It is pointed out that the prospective and retrospective differential equations have, respectively, the Kolmogorov backward and forward differential equations as special cases. Practical uses of the differential equations are demonstrated by examples.

Prediction of Outstanding Liabilities II. Model Variations and Extensions

Thas as a follow-up of a previous paper by the author, where claims reserving m non-hfe insurance as treated in the framework of a marked Polsson claims process A key result on decomposmon of the process as generahzed, and a number of related results are added. Their usefulness is demonstrated by examples and, in pamcular, the connection to the analogous discrete tame model is clarified. The problem of predmtmg the outstanding part of reported but not settled clmms is revisated and, by way of example, solved in a model where the partml payments are governed by a Dmchlet process The process of reported clmms is examined, and ats dual relationship to the process of occurred claims is pointed out.

Empirical Bayes credibility

Abstract A credibility estimator is Bayes in the restricted class of linear estimators and may be viewed as a linear approximation to the (unrestricted) Bayes estimator. When the structural parameters occurring in a credibility formula are replaced by consistent estimators based on data from a collective of similar risks,we obtain an empirical credibility estimator, which is a credibility counterpart of empirical Bayes estimators. Empirical credibility estimators are proposed under various model assumptions, and sufficient conditions for asymptotic optimality are established.

The credibility approach to experience rating

Summary Chapter 1 presents the problem of experience rating and the objective of credibility theory. In Chapter 2 the so-called “limited fluctuation credibility theory” is presented briefly. The far more important and developed “greatest accuracy credibility theory” is introduced in Chapter 3. The discussion in Chapters 1–3 is carried through with a minimum of mathematics, the aim being to explain the basic ideas and techniques. A fuller account of the greatest accuracy theory is given in Chapter 4. The division of the matter into paragraphs reflects the distinction between what can be called the classical or empirical Bayes point of view and the (purely) Bayesian point of view. In Chapter 5 are mentioned some selected problems related to practical applications of the theory.

Hierarchical credibility: analysis of a random effect linear model with nested classification

Abstract A random coefficient regression model with multi-stage nested classification is considered. Linear Bayes estimators are obtained for random effects at all stages, and estimators of the structural parameters are proposed.