The homogeneous (in time) model of credibility theory is defined by a sequence Θ, X1, X2, … of random variables, where for Θ = θ fixed, the variables X1, X2, … are independent and equidistributed. The structure variable Θ may be interpreted as the parameter of a contract chosen at random in a fixed portfolio, the variable Xk as the total cost (or number) of the claims of the kth year of that contract.

Bühlmann's linear credibility premium of the year t + 1 may be written in the form

where f is a linear function. In optimal semilinear credibility, we look for an optimal f, not necessarily linear, such that (1) is closest to Xt+1 in the least squares sense. In the first section we prove that this optimal f, denoted by f*, is solution of an integral equation of Fredholm type, which reduces to a system of linear equations in the case of a finite portfolio. That is a portfolio in which Θ and Xk can assume only a finite number of values.

In the second section we see that the structure of such a portfolio is closely connected with the decomposition of a quadratic form in a sum of squares of linear forms.

In the last section we calculate numerically the optimal premium for a concrete portfolio in automobile insurance. We limit ourselves to the consideration of the number of claims. The optimal premium is compared with the usual linear premium. The difference is far from negligible.