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.

1. Some years ago I discussed optimal reinsurance treaties, without trying to give a precise definition of this term [1]. I suggested that a reinsurance contract could be called “most efficient” if it, for a given net premium, maximized the reduction of the variance in the claim distribution of the ceding company. I proved under fairly restricted conditions that the Stop Loss contract was most efficient in this respect.

I do not consider this a particularly interesting result. I pointed out at the time that there are two parties to a reinsurance contract, and that an arrangement which is very attractive to one party, may be quite unacceptable to the other.

2. In spite of my own reservations, it seems that this result —which I did not think deserved to be called a theorem—has caused some interest. Kahn [4] has proved that the result is valid under far more general conditions, and recently Ohlin [5] has proved that the result holds for a much more general class of measures of dispersion.

In view of these generalizations it might be useful to state once more, why I think the original result has relatively little interest. In doing so, it is by no means my purpose to reduce the value of the mathematical generalizations of Kahn and Ohlin. Such work has a value in itself, whether the results are immediately useful or not. I merely want to point out that there are other lines of research, which appear more promising, if our purpose is to develop a realistic theory of insurance.

1. In a paper presented to the fifth Astin Colloquium (Lucerne, 1965) Bühlmann has given some propositions with regard to Experience Rating understood as a sequence of estimates of the expectation with respect to the distribution function of θ, H(θ) say, of μ(θ) which is the mean for fixed θ with respect to the distribution function G(x; θ) of the variables for each value of ν = 1, 2 … n. In such problems the estimator function for μ(θ) is generally chosen to be the conditional mean of μ(θ) for a given set of observed values of This is generally justified by the principle of least square deviation. According to Bühlmann this justification is not sufficient. Therefore, he bases the choice of this estimator function upon a postulate of equilibrium, described in the following lines.

Let X′ be a subset of X, and C(X′) a cylinder with the base X′ ⊂ X in the product space X × Θ, where X is the set of all possible and Θ the set of all possible θ, then the postulate of equilibrium implies the equality between the expectations of μ(θ) and of a function of on each cylinder C(X′). This is exactly Kolmogoroff's definition of the conditional expectation:

Bühlmann states, further, that the best linear estimate of (1) based on the arithmetic mean of n sample values xν of ν = 1, 2 … n can be written in the form of the Credibility Formula:

where the symbol E[·] denotes the expectation over the product space X × Θ and

If the are independent and identically distributed, this leads to where k = E [σ2 (θ)] / Var [ν(θ)], and σ2(θ) is the variance for a fixed θ of G(x; θ). This case of the credibility formula is, generally, applied in American practice. (The proposition has later been proved for more general conditions).