Cure frailty models for survival data : Application to recurrences for breast cancer and to hospital readmissions for colorectal cancer

Risk measurement for insurance supervisory purposes requires a first «market consistent» valuation of usually non hedgeable obligations cash-flows. It is now well agreed to use expectations (or «best estimates») with respect to a pricing measure of the traded securities. It is then possible to speak of «risk bearing capital» as the difference of market value of assets (less insurance obligations payments) and some market consistent value of remaining obligations. These values are adapted and respectively retrospective and prospective processes. We present an axiomatic way to give a definition of solvency. One can indeed posit the existence of an operator from risk bearing capital processes to «free capital» processes, positivity of free capital meaning solvency. It is natural to require from the free capital assessment to be «supervisory arbitrage free» to prevent manipulations of solvency standards by too clever market instruments. It is also natural to expect a «time consistency» of the free capital operator. Given an exogeneous obligations portfolio, asset portfolio choice is being used as control variable. The «supervisory provision» is the minimum asset value allowing some hedging of the obligations as well as solvency (or «acceptability») of the hedged obligations. The supervisory provision is a market consistent assessment of obligations. If the current asset value is smaller than the supervisory provision the sine qua non condition is not fulfilled and the «exit value» or «technical provision» is required from the non acceptable company for a transfer of its obligations as well as of the technical provision to a new company. The technical provision, another market consistent assessment of obligations, equals the sum of the best estimate and of the «risk margin» while the new shareholders provide an «absolute capital requirement». Finally solvency capital requirement is defined to deal with the case where shareholders / managers want a definite, non optimal asset portfolio. It adds an «hedgeable» term to the absolute capital requirement and fulfills the supervisory accounting equality. Application of the axiomatic approach to the case of non hedgeable assets will be mentionned, in connection with supervisory treatment of reinsurance. Multiperiod risk measurement : problems, examples, comments