Multiplicative background risk models in which the idiosyncratic risk factors are assumed to be distributed exponentially and the systemic risk factor has an arbitrary distribution on the nonnegative half of the real line have seen a great variety of applications in actuarial science. Admittedly, these structures, which are well-known to mathematical statisticians under the name of exponential mixtures, enjoy remarkable level of technical tractability and so are a very

convenient tool for modelling dependency. That said, the assumption of exponentiality is merely a mathematical nicety and does not have to reflect the reality, yet the works that loosen this assumption are rare. In this talk, we pursue a holistic approach by considering a multiplicative background risk model with arbitrarily distributed idiosyncratic and systemic risk factors. We reveal links between the general structure and the one with the exponentially distributed idiosyncratic risk factors, study relevant theoretical properties of the former, and discuss important special cases. Also, we construct realistic numerical examples borrowed from the context of the determination and allocation of economic capital. The examples suggest that a little departure from exponentiality can have substantial impacts on the outcome of risk analysis.