Karl Borch

Equilibrium in a Reinsurance Market

This paper investigates the possibility of generalizing the classical theory of commodity markets to include uncertainty. It is shown that if uncertainty is considered as a commodity, it is possible to define a meaningful price concept, and to determine a price which makes supply equal to demand. However, if each participant seeks to maximize his utility, taking this price as given, the market will not in general reach a Pareto optimal state. If the market shall reach a Pareto optimal state, there must be negotiations between the participants, and it seems that the problem can best be analysed as an n-person cooperative game.

The safety loading of reinsurance premiums

Abstract 1. Introduction 1.1. In the older forms of reinsurance, often referred to as “proportional” reinsurance, there is no real problem involved in determining the correct safety loading of premiums. It seems natural, in fact almost obvious, that reinsurance should take place on “original terms”, and that any departure from this procedure would need special justification. The only problem which may be troublesome is to determine the three components of the gross premium, i.e. net premium, safety loading and loading for expenses. The last of these components is calculated to cover costs connected with the direct underwriting, such as agents commission, and does not, in principle concern the reinsurer.

Reciprocal reinsurance treaties

In this paper we shall study the situation of two insurance companies which are negotiating with the view of concluding a reciprocal reinsurance treaty. We assume that the two companies are under no compulsion to reach an agreement. This means that if the companies conclude a treaty, the treaty must be such that both companies consider themselves better off than without any treaty. We futher assume that no third company can break into the negotiations. This means that the two companies either have to come to terms, or be without any reinsurance.How the two parties reach an agreement in a situation like this, is one of the classical problems of theoretical economics. It is usually referred to as the “Bargaining Problem†. The problem appears very simple, but this is a deception. It has proved extremely difficult to formulate generally acceptable assumptions which give the problem a determinate solution. The “Theory of Games†, developed by von Neumann and Morgenstern (10), does not give a determinate solution, but it has greatly increased our understanding of such problems, and the present paper will draw heavily on that theory.The situation which we propose to study, is very simple, may be too simple to have any bearing on reinsurance negotiations in real life. If there exists a reinsurance market, which also is a perfect market in the sense given to this term in economic theory, bartering between two companies does not make any sense. They could both do equally well or better by dealing in the market at the market price.

Personal probabilities of probabilities

By definition, the subjective probability distribution of a random event is revealed by the (‘rational’) subjects choice between bets — a view expressed by F. Ramsey, B. De Finetti, L. J. Savage and traceable to E. Borel and, it can be argued, to T. Bayes. Since hypotheses are not observable events, no bet can be made, and paid off, on a hypothesis. The subjective probability distribution of hypotheses (or of a parameter, as in the current ‘Bayesian’ statistical literature) is therefore a figure of speech, an ‘as if’, justifiable in the limit. Given a long sequence of previous observations, the subjective posterior probabilities of events still to be observed are derived by using a mathematical expression that would approximate the subjective probability distribution of hypotheses, if these could be bet on. This position was taken by most, but not all, respondents to a ‘Round Robin’ initiated by J. Marschak after M. H. De-Groots talk on Stopping Rules presented at the UCLA Interdisciplinary Colloquium on Mathematics in Behavioral Sciences. Other participants: K. Borch, H. Chernoif, R. Dorfman, W. Edwards, T. S. Ferguson, G. Graves, K. Miyasawa, P. Randolph, L. J. Savage, R. Schlaifer, R. L. Winkler. Attention is also drawn to K. Borchs article in this issue.

The Utility Concept Applied to the Theory of Insurance

In some recent papers ((1), (2) and (3)) about reinsurance problems I have made extensive use of utility concepts. It has been shown that if a company follows well defined objectives in its reinsurance policy, these objectives can be represented by a utility function which the company seeks to maximise. This formulation of the problem will in general make it possible to determine a unique reinsurance arrangement which is optimal when the companys objectives and external situation are given.More than 50 years ago Guldberg (4) wrote (about the probability of ruin): “Wie hoch diese Wahrscheinlichkeit gegriffen werden soil, muss dent subjektiven Ermessen oder von Aussen kommenden Bedingungen A¼berlassen bleiben†. This is the traditional approach to reinsurance problems. It does obviously not lead to a determinate solution. Most authors taking this approach conclude their studies by giving a mathematical relation between some measure of “stability†, such as the probability of ruin, and some parameter, for instance maximum retention, to which the company can give any value within a certain range. Such studies do usually not state which particular value the company should select for this parameter, i.e. what degree of stability it should settle for. This question is apparently considered as being outside the field of actuarial mathematics.

Reciprocal reinsurance treaties seen as a two-person co-operative game

Abstract 1.1. In 1930 Professor Cramer wrote: “The object of the theory of risk is to give a mathematical analysis of the random fluctuations in an insurance business, and to discuss the various means of protection against their inconvenient effects” ([6], page 7). This definition has obviously been adequate, since Cramer still subscribes to it when 25 years later he makes a comprehensive survey of the subject [7]. However, the mild complaint Cramer made in 1930 that “practical insurance business has hitherto made little or no application of the results offered by the mathematical theory of risk” still has some validity. The very impressive results achieved by the mathematical theory of risk during the last thirty years have certainly found practical applications, but not so many as one would expect, considering that the theory proposes to analyse the very foundations of the insurance business.

General Equilibrium in the Economics of Uncertainty

(1) Most of the papers presented to this Conference deal with various aspects of economic decisions under uncertainty. It is clear that such decisions present a number of very intricate problems. It is also clear that we can solve these problems only if we know the objectives of the decision-maker, and the environment in which the decisions are made.

Application of Game Theory to Some Problems in Automobile Insurance

In this paper we shall study the problem of determining “correct†premium rates for sub-groups of an insurance collective. This problem obviously occurs in all branches of insurance. However, it seems at present to be a really burning issue in automobile insurance. We shall show that the problem can be formulated as a conflict between groups which can gain by co-operating, although their interests are opposed. When formulated in this way, the problem evidently can be analysed and solved by the help ot the “Game Theory†of Von Neumann and Morgenstern (5).We shall first illustrate the problem by a simple example. We consider a group of n1 = 100 persons, each of whom may suffer a loss of 1, with probability p1 = 0.1. We assume that these persons consider forming an insurance company to cover themselves against this risk. We further assume that for some reason, government regulations or prejudices of managers, an insurance company must be organized so that the probability of ruin is less than 0.001If such a company is formed, expected claim payment will beand the standard deviation of the claim payments will beIf the government inspection (or the companys actuary) agrees that the ruin probability can be calculated with sufficient approximation by assuming that the claim payments have a normal distribution, the company must have funds amounting to

Optimal Insurance Arrangements

In a recent paper on the theory of demand for insurance Arrow [1] has proved that the optimal policy for an insurance buyer is one which gives complete coverage, beyond a fixed deductible. The result is proved under very general assumptions, but its content can be illustrated by the following simple example. Assume that a person is exposed to a risk which can cause him a loss x , represented by a stochastic variable with the distribution F(x) . Assume further that he by paying the premium P(y) can obtain an insurance contract which will guarantee him a compensation y(x) , if his loss amounts to x . The problem of our person is to find the optimal insurance contract, i.e. the optimal function y(x) , when the price is given by the functional P(y) . In order to give an operational formulation to the problem we have outlined, we shall assume that the persons attitude to risk can be represented by a Bernoulli utility function u(x) , and we shall write S for his “initial wealth”. His problem will then be to maximize when the functional P(y) is given, and y(x) є Y. The set Y can be interpreted as the set of insurance policies available in the market. It is, natural to assume that o ≤ y(x) ≤ x , but beyond this there is no need for assuming additional restrictions on the set Y .

The rescue of an insurance company after ruin

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.