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A further definition from The (Alternative) Legal Costs Dictionary:
After-the-event (ATE) premium n. a sum of money paid or payable for insurance against the risk of incurring a costs liability in proceedings. The premium is rarely paid at the outset and usually not paid at all if the claim is unsuccessful. The amount of the premium payable is often arbitrarily concocted as the case progresses and subsequently adjusted to whatever figure the court arbitrarily allows. An amount which market forces have never knowingly influenced.