Aggregate risk is everywhere in insurance mathematics. A recent example, borrowed from the modern regulatory accords, is the determination of the aggregate economic capital and its consequent allocation to risk drivers. A traditional illustration of the importance of risk aggregation in insurance applications is the celebrated collective risk theory that dates back to the early years of the 20th century.

For a mathematician, aggregating risks boils down to characterizing the stochastic properties of the sums of possibly dependent random variables. The problem is very involved and does not have a general solution unless specific assumptions about the type of the random variables and the underlying dependence are made. In particular, when the aforementioned random variables are jointly normal, then their sum is also normal. Otherwise, as a rule, one has to contend with bounds for the desired sum.

In this presentation, I will demonstrate a method to approximate sums of random variables to any required precision. The proposed approach is fast and can tackle equally well sums with just a few or thousands of summands. Moreover, the new method can be applied to an overwhelming variety of random variables with a long-history of applications in insurance (e.g., Weibull, lognormal, Pareto, to name just a few) and exchangeable and non-exchangeable dependencies.