Size-biased risk measures of compound sums

The size-biased, or length-biased transform is known to be particularly useful in insurance risk measurement. The case of continuous losses has been extensively considered in the actuarial literature. Given their importance in insurance studies, this paper concentrates on compound sums. The zero-augmented distributions which naturally appear in the individual model of risk theory are obtained as particular cases when the claim frequency distribution is concentrated on {0,1}. The general results derived in this paper help actuaries to understand how risk measurement proceeds since the formulas make explicit the loadings corresponding to each source of randomness. Some simple and explicit expressions are obtained when losses are modeled by compound Poisson sums and compound mixed Poisson sums, including the compound Negative Binomial sums. Extensions to conditionally independent risks are discussed. As an illustration, the paper considers the case of compound mixed Poisson sums where the Poisson means are proportional to a random common risk factor