Undertaking Specific Parameters or a Partial Internal Model under Solvency 2?

Solvency 2 directive provides a range of methods to calculate the Solvency Capital Requirement (SCR), which allows undertakings to choose a method that is proportionate to the nature, scale and complexity of the risk that are measured: 1.full internal model (IRM) 2.standard formula and partial internal model (PIRM) 3.standard formula with undertaking-specific parameters (USPs) 4.standard formula (SF) 5.simplification. Regarding the third point, the Technical Specifications (TS) of Quantitative Impact Study5 (QIS5) describes a subset of the SF market parameters that may be replaced by USPs. In particular the TS report three different standardised methods to calculate the standard deviations of premium risk. The aim of this paper is to compare the methodologies proposed in QIS5 with a PIRM for premium risk. Given an insurance portfolio, which are associated claims and any costs incurred by year of occurrence and that constitute the technical basis of a Non-life insurance cover, two pricing models are used to identify premium for different risk profiles: Generalized Linear Model (GLM) and Generalized Additive Model (GAM).These models allow to describe the Aggregate Claim Amount (ACA) as a function of the tariff variables detectable in the insurance contract and consequently a propensity of each insurers to produce a loss for the undertaking. If GLM represents a benchmark within this technical framework, GAM is an interesting alternative for its non-parametric or semi-parametric structure and furthermore when the distribution of the aggregate claim amount for the tariff variable is not linear.First an explorative data analysis will be developed to check the distribution assumption about the number of claims and claim amount, respectively Poisson and Gamma distribution. Than this paper will show a summary of the outputs, statistics and graphical analysis of residuals necessary to validate the “optimum” GLM and GAM, but also to exhibit which model predicts better the expected value of the ACA.Solvency 2 directive requests for the premium risk to define an “error” in term of standard deviation of the Estimated ACA to state a SCR. Using one of the two models outlined above, the “Best PIRM”, will be determined its standard deviation. Moreover a comparison between the SF market parameters, USPs and the standard deviation of the model will shown, using Motor Third Party Liability insurance data of Italian Market. Finally some considerations regards a way to calculate a prediction error of the model will conclude the work.