Stochastic Model to Evaluate the Fair Value of the Motor Third-Party Liability Under the Direct Reimbursement Scheme and Quantification of the Capital Requirement in a Solvency II Perspective

As commonly known, to evaluate the claims reserve (otherwise known as the provision for outstanding claims), the loss adjuster uses as a first component the claims reserve given by the sum of the estimated provision for each outstanding claim (known as case reserve). Traditional statistical-actuarial methods are used to control and/or asseverate the evaluation and recent developments have tended to ensure that these enable an independent assessment of the claims reserve. In some European countries, a large subset of motor liability claims are managed within a direct reimbursement (DR) scheme.In Italy, the introduction of the direct compensation CARD system for third-party liability insurance has resulted in greater attention in the use of these traditional methods due to the heterogeneity of the data available for evaluations.This paper presents the results of a study undertaken to define a calculation method that, using the different assumptions that describe the evolution of settlement mechanisms, is able to quantify the claims reserve.The proposed methodology reduces the loss adjuster's discretion in applying statistical methods since all assumptions must be made explicit and can hence be monitored and controlled. Furthermore, a stress test can be performed on all the parameters that influence the settlement and thus the claims reserve. Finally, in a backtesting perspective, the proposed methodology enables an ex-post analysis of actual cash flow deviations compared to expected values, identifying the variables that lead to these differences.In particular, a calculation method is presented that using the different assumptions describing the evolution of the settlement of different claim handling procedures (Non-Card, Card, Handler Forfait and Debtor Forfait) is able to quantify the claims reserve. Via simulation the future payments, the expected value of the claims reserve and some indicators of variability are also estimated. The numerical application allows comparing the results obtained with those deriving from the application of traditional statistical methods. The problem of aggregating the four claim handling procedures is dealt with by using copulas with the goal of building a single distribution of the aggregate claims reserve.Furthermore, in a Solvency II perspective, the capital requirement is determined against the reserve risk based on the internal model proposed and is compared with the standard formula proposed by EIOPA