Exposure Rating Casualty Reinsurance Excess Layers with Closed Form Annuity Models

Casualty excess reinsurance terms are typically stated in fixed attachment and limit amounts. Unless a lump sum settlement or commutation is made ultimate recoveries are settled years later as total payments penetrate the excess layer. This paper demonstrates that annuity models incorporating claim life, late reporting, benefit inflation, and discounting can be formulated with simple functional form components that lead to closed form solutions, or at least efficient numerical integration solutions, for exposure rating quantities. Applications are shown to problems such as commutation of existing claims, prospective reinsurance rating, excess loss development factors, and volatility of excess layers.