This paper considers the portfolio selection and capital injection problem for a diffusion risk model within the classical Black–Scholes financial market. It is assumed that the original surplus process of an insurance portfolio is described by a drifted Brownian motion, and that the surplus can be invested in a risky asset and a risk-free asset. When the surplus hits zero, the company can inject capital to keep the surplus positive. In addition, it is assumed that both fixed and proportional costs are incurred upon each capital injection. Our objective is to minimize the expected value of the discounted capital injection costs by controlling the investment policy and the capital injection policy. We first prove the continuity of the value function and a verification theorem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation. We then show that the optimal investment policy is a solution to a terminal value problem of an ordinary differential equation. In particular, explicit solutions are derived in some special cases and a series solution is obtained for the general case. Also, we propose a numerical method to solve the optimal investment and capital injection policies. Finally, a numerical study is carried out to illustrate the effect of the model parameters on the optimal policies.

A conventional practice in standard risk theory considerations has been to assume that claims are paid immediately as they have incurred (see BPP, item 3.1c, BPP is used as an abbreviation for the book “Risk Theory”, 1984 edition, by Beard, Pentikäinen, Pesonen). The delay of the claims settlement has been, of course, a central aspect in reserve calculation theory and practices, and numerous valuable works have been published on this topic in recent years. However, its regard in general model building and in risk theory considerations has gained little attention until recent years. The purpose of this paper is to contribute to this research work by discussing how the “run-off” risk, i.e., the variability due to the delay of the claims payment, could be incorporated into the standard risk theory models as a separate entry (see BPP, item 10.2e) and to find some evaluation of the order of magnitude of the “extra” (if any) fluctuation so rendered. We expect that the proposed technique can also be utilized in testing different reserve calculation methods and in comparing their effectiveness. The main ideas follow very much along the lines given by Rantala in his doctoral thesis (1984).

One should appreciate the fact that any risk theory model can never be more than an idealization of real-life processes. An intricate problem for practitioners is to evaluate the uncertainties ensuing from the fact that the model, more or less, ignores or only approximates the factors affecting the real events, and that the practical applications are often based on and their necessary parameters estimated from observed data that are subject to random fluctuations and to many other kinds of uncertainties. The problem complex of the run-off risk, when understood in a broad sense, is so wide that it requires a series of studies, and this paper should be regarded as a first step only. The posing of the problem follows the conventional risk theory approach by using the mixed compound Poisson process further allowing for long-term variations of risk exposure (“cycles”), and now extending the model to cope with the delayed settlement of the claims. At this stage of the on-going researchwork the impact of the parameter estimation is excluded from consideration. Therefore, our results and the numerical examples, as given in what follows, do not describe the total uncertainty of the claims or the reserves.

We investigate the usual method of discretizing loss reserving data by calendar year and show how this procedure may introduce fluctuations in the delay probabilities. These fluctuations, when treated as random fluctuations, possess a special correlation structure and we present a simple credibility method accounting for these fluctuations. The results are illustrated by a numerical example.

In this paper we discuss the application of the proportional hazard premium calculation principle.

In Section 2 we propose a method to calculate the proportional hazard premium of a compound risk when the severity distribution is subexponential.

In Section 3 we use the empirical distribution to calculate the premium when the proportional hazard principle is applied, which leads to a systematic underestimation of the premium. After studying the bias of the premium calculated using this non-parametric approach we take advantage of the bootstrap technique with subsampling to reduce it.

Albrecher et al. (Albrecher, H., Mayer Ph., Schoutens, W. (2008) General lower bounds for arithmetic Asian option prices. Applied Mathematical Finance, 15, 123–149) have proposed model-independent lower bounds for arithmetic Asian options. In this paper we provide an alternative and more elementary derivation of their results. We use the bounds as control variates to develop a simple Monte Carlo method for pricing contracts with Asian-style features. The conditioning idea that is inherent in our approach also inspires us to propose a new semi-analytic pricing approach. We compare both approaches and conclude that these both have their merits and are useful in practice. In particular, we point out that our newly proposed Monte Carlo method allows to deal with Asian-style products that appear in insurance (e.g., unit-linked contracts) in a very efficient way, and outperforms other known Monte Carlo methods that are based on control variates.

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.

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.

This paper deals with experience rating of claims processes of ARIMA structures. By experience rating we mean that future premiums should be only a function of past values of the claims process. The main emphasis is on demonstrating the usefulness of the control-theoretical approach in the search for optimal rating rules. Optimality is here defined to mean as smooth a flow of premiums as possible when the variation in the accumulated profit is restricted to a certain amount. First it is shown how the underlying model in its simplest form can be transformed into the state-space form. Then the Kalman filter technique is used to find the optimal rules. Also a time delay in information is taken into account. The optimal rules are illustrated by examples.

In this paper, we are interested in a model related to a number of periods of Company's activity.

The Company calculates an amount s which then could be given as a supplementary interest to the shareholders. This calculated amount is taken from the risk reserve. Let us assume the risk reserve = S at the beginning of an operating period. If the Company gives an amount s to the shareholders, then the risk reserve is S — s. (Borch 1972; Seal 1969).

We must determine the best policy of dividends, that is a rule which determines the payments to be made each year to the shareholders of the Company, maximising a definite criterion.

The problem of dividends must be approached in the “dynamic programming manner”. Indeed, the payments of dividends have an effect upon further gains of the Company and its capacity to pay dividends in the future.

The objective of the Company is, for example, maximising the average utility of the dividend payments, which is calculated according to the distribution of claims. (Borch 1964a; Wolff 1966).

We will in this paper consider the risk process from the point of view of random walk in one dimension. The particle starts out at the origin. Each claim is equivalent to a step in the random walk. The length of the step is equal to the amount of the claim minus the amount of the premium which has been obtained since the preceding claim. If the difference is positive the particle advances to the right and if the difference is negative to the left. At distance U to the right from the origin there is a barrier. The problem is to find the distribution function of X, the time it takes the particle to cross the barrier for the first time.

In most practical applications of risk theory U is large in comparison to the individual steps of the particle. We will in this paper assume that U is large in comparison to the individual steps and draw certain conclusions about the risk processes from this assumption.

The individual steps of the particle have a certain distribution. The corresponding characteristic function is ϕ. For reasons which will be seen later we will consider ϕ to be a function of it = θ instead of t. This means that

The mean value and the standard deviation of each step is equal to m and σ respectively. We now write

We now define two random variables X and Y.

X = time to cross the barrier for the first time

Y = X σ2/U2.

The purpose of this paper is to show that, for the classical risk model, explicit expressions for survival probabilities in a finite time horizon can be obtained through the inversion of the double Laplace transform of the distribution of time to ruin. To do this, we consider Gerber and Shiu (1998) and a particular value for their penalty function. Although other methods to address the problem exist, we find this approach, perhaps, more direct and simple. For the analytic inversion, we have applied twice, after some algebra, the Laplace complex inversion formula.

This article summarizes some main results in modern portfolio theory. First, the Markowitz approach is presented. Then the capital asset pricing model is derived and its empirical testability is discussed. Afterwards Neumann–Morgenstern utility theory is applied to the portfolio problem. Finally, it is shown how optimal risk allocation in an economy may lead to portfolio insurance.

In this paper, based on the additive measure integral representation of a non-additive measure integral, it is shown that any comonotonically additive premium principle can be represented as an integral of the distorted decumulative distribution function of the insurance risk. Furthermore, a sufficient and necessary condition that a premium principle is a distortion premium principle is given.

This paper deals with bonus systems used in Denmark, Finland, Norway, Sweden, Switzerland and West Germany. These systems are studied by methods given by Mr. Loimaranta). The bonus rules of Denmark and Sweden have been modified because they contradict to one of the assumptions of the theory.

It is assumed that the number of claims in a year follows the Poisson distribution with a mean λ. Further it is assumed that the value of λ is independent of time.

In each case bonus rules are given in form of transformations Tk defined in ref. 1., i.e. Tk(i) = j when a policy moves from class i to class j after k claims. The class where a new policy starts from is called the initial class. Bonus scales, the vectors B, are normed so that the premium of the initial class is 100.

For each bonus system the efficiency η of the system and the discrimination power d of the bonus rules as a function of the mean claim frequency λ have been calculated. The graphs of these functions are presented in the figures 2 to 4 respectively. In the following pages, different bonus rules are described in detail and some simple analysis based on the curves on pp. 211-214 has been made. The calculations were performed quite recently and any deeper analysis of the results has not been possible because of lack of time.

In order to be able to apply the theory of Markov chains to a bonus system we must require among others that the transition to a certain class depends only on the number of claims occurred during last period ignoring possible former claims. The Danish system has originally four bonus classes labeled from 0 to 3.

This article discusses risk classification and develops and discusses a framework for estimating the effects of restrictions on risk classification. It is shown that expected losses due to adverse selection depend only on means, variances and covariances of insurance factors and rates of uptake of insurance. Percentage loadings required to avoid losses are displayed. Correlated information, such as family history, is also incorporated and it is seen how such information limits losses and decreases required loadings. Although the evidence suggests that adverse selection is not, at present, a severe problem for insurers, this might change if the authorities impose restrictions on risk classification and/or customers gain an informational advantage (such as better knowledge of their own risk levels). Application is made to unisex annuity pricing in the UK insurance market.

More than ten years ago I wrote a paper with the title “The Economic Theory of Insurance” [6]. I was not particularly happy about this paper, and I do not think it contributed much to the development of a sastifactory theory. The paper did however make me—and I hope some readers—acutely aware of the difficulties and problems which must be overcome before a proper theory can be constructed. These problems are still unsolved, so I have on the present occassion chosen a more modest title for a paper on substantially the same subject.

Insurance is an economic activity of some importance, and there is an obvious need for a theory to explain and analyse the activity in the insurance sector of the economy. During the last decade many economists seem to have felt the need, and to have taken it as a challenge. The results have been a fair amount of research, and a number of publications, which I shall not try to review here. It may however be useful to refer to three very recent survey articles by Farny [10], Ferry [11] and Rosa [14], which give extensive bibliographies. The three articles seem to indicate that the economics of insurance is becoming a fashionable subject of research.

1.1. In the different versions of the “Theory of Risk” it is almost universally assumed that ruin or bankruptcy marks the end of the game. The earlier versions of the theory tried to estimate the probability of this event, and studied the steps which an insurance company could take to bring probability of ruin down to an acceptable level. The more modern versions of the theory of risk tend to formulate the problem in economic terms, and study the cost of postponing or avoiding ruin.

In a recent discussion of a paper [4] surveying the development of the theory of risk, Professor Bather suggested that ruin may not necessarily be the end. If an otherwise sound insurance company gets into difficulties, so that ruin looms large, it is very likely that steps will be taken to rescue the company, for instance by refinancing, or in more extreme cases, by a merger.

1.2. To practical insurance men the simple suggestion of Professor Bather may seem next to trivial. Insurance companies get into difficulties fairly regularly, and rescue operations are considered in the insurance world, if not daily, at least annualy. The suggestion has, however, far-reaching implications for the theory of risk, and these do not seem to have been fully realised. If ruin does not mean the end of the game, but only the necessity of raising additional money, the current theories of risk may have to be radically revised. In this paper we shall discuss some of these implications.

Let us consider the following problem.

A motorist has decided to effect an accident insurance under the following conditions. The insurance runs for one year. The premium for the first year amounts E0. If no damages have been claimed during i successive years, i = 1, 2 or 3 the premium is reduced to Ei. After four years of damagefree driving no further premium reduction is granted, so the premium remains E3. The premium is due on the first day of the year. The own risk amounts a0.

The number of accidents of our motorist during a time period T is assumed to be Poisson distributed with parameter λT. The extent of the damage has distribution function F(s) with finite mean and variance.

The problem of our motorist will be to decide whether to claim a damage or not. He will have to develop a strategy that specifies his decisions in every possible situation. His strategy will be called optimal if it minimizes the expected costs in the long run.

We may expect that in view of the premium reduction, it will be unprofitable to claim damages which are not much larger than a0. Once a damage is claimed it will be profitable to claim all following damages that exceed a0 during the remaining part of the year.

Hence his decisions will also depend on the time of the year and the premium paid at the beginning of that year. So we distinguish between four types of years, for each premium one.

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.