On Additive Premium Calculation Principles

A premium calculation principle is a general rule that assigns a premium P to any given risk S. Intuitively, P is what the insurance carrier charges (apart from an expense allowance) for taking over the risk S (see [3], p. 85-87). Mathematically, S is a random variable, and P depends on S through its distribution function. The value of P may be finite or infinite; in the latter case we speak of an uninsurable risk.A premium calculation principle is called additive, if the premium assigned to the sum of two independent risks is the sum of the premiums that are assigned to the two risks individually. For example, the variance principle, P = E[S] + β Var [S] (β > o), is additive, because the variance of the sum of independent random variables equals the sum of the variances. Additivity is a very desirable property, from a theoretical as well as from a practical point of view (as pointed out by Borch [2], p. 429).The variance principle is not entirely satisfactory for various reasons. For one thing it does not take account of the skewness of S (a risk whose distribution is skewed to the right seems to be more dangerous than one with a symmetrical distribution). Furthermore, it produces in some cases a premium P that exceeds S with probability one (example: β = .3, S = o or 10, each with probability 1/2).