Average Annual Loss Estimation

They are different summaries of the same loss distribution. The loss exceedance curve gives, for each loss size, the annual rate or probability of exceeding it; a return-period loss reads a specific point off that curve (e.g., the 1-in-250-year loss). The AAL is the area under the exceedance curve, equivalently the frequency-weighted mean of all event losses, so it captures the whole distribution in one expected value. AAL and return-period losses are complementary: AAL answers what you expect to lose on average each year, while return-period and probable-maximum-loss figures answer how bad a rare year can be, which is what capital and solvency depend on.

Because it is the long-run expected claim cost per year for the risk. An insurer who collected exactly the AAL each year and paid all claims would, over a very long horizon, just break even on losses, before adding loadings for expenses, uncertainty, capital cost, and profit. This is why AAL, the frequency-weighted mean loss from the catastrophe model, is the starting point for technical pricing in Grossi and Kunreuther's framework. Actual premiums add risk and expense loadings on top, but the AAL is the fair actuarial core, which is also why getting the frequencies and the full range of event losses right is essential.

AAL is more robust than tail metrics because it is dominated by the well-sampled bulk of the distribution, but it is not immune to rare-event uncertainty: a low-frequency, high-loss event contributes its loss times its small frequency, and if such events are missing or mis-rated the AAL is biased. Silva and colleagues stress the uncertainty in probabilistic risk metrics and the importance of a complete, credible event set. Best practice therefore reports AAL with an uncertainty range, tests sensitivity to assumptions about rare events, and pairs AAL with tail metrics so users see both the average and the extremes the average conceals.