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Planning Problems Of A Gambling-House with Application to Insurance Business

In the classical risk theory the interdependence between the security loading and the initial risk reserve is studied. It could be said that the purpose of these studies are to state how large the security loading must be in order to avoid ruin of the insurance business. It has often been said that the classical approach with an infinite planning horizon is unrealistic. The main reason for this attitude is that if the security loading is equal to zero then ruin is certain. Since in practice it is often difficult to estimate the true size of the security loading the whole problem of ruin or non-ruin seems to rest on a rather shaky foundation. This attitude to the problem is reflected in studies in risk theory performed in recent years. The infinite planning horizon is then often replaced by a finite time period. Since the probability of ruin during a short period of time depends to a minor extent on the size of the security loading these studies are concentrated mainly on the shape of the claim distribution, while the security loading is of minor interest. Let us think of a gambling-house, where coin-tossing is practised. Let us assume that the gambling-house for reason of fairness decides to pay two dollars to each winner who has staked one dollar. Probability theory tells us that however rich the gambling-house may be, it will be ruined in the long run. This simple example reminds us of the trivial fact that insurance business without a sufficient security loading in the premium is commercially impossible.