Pricing for rare large claims

Sahil Mutual Insurance (a fictional insurer) is re-pricing its private motor book for next year. The pricing actuary has one policy year of data on 1,500 policies. Last year a single claim of SAR 480,000 shocked the management team; the sales director now argues that “claims are usually small — about SAR 6,884 — so the premium can come down”, while the CFO wants a large increase “because of the big claim”.

Your job, as the analyst supporting the actuary:

1. check the data and put frequency on a fair basis (claims per policy-year);
2. test whether a simple Poisson model describes how often policyholders claim;
3. describe the size of claims honestly — including the tail;
4. propose a pure premium (expected claims cost per policy-year) and explain how to treat the large claim.

The data. insurance_claims has one row per policy: region, driver age band, vehicle value, exposure (the fraction of the year insured), number of claims and total amount claimed. It is synthetic — generated with a documented frequency–severity model and one planted very large claim — and describes no real insurer or policyholder.

Stakeholders. The pricing actuary (signs off rates), the CFO (solvency and profit), sales (competitiveness), and the regulator (premiums must be fair and justified).