A pricing technique to calculate the Solvency Capital Requirement for non-life Premium Risk

Transcription

1 A pricing technique to calculate the Solvency Capital Requirement for non-life Premium Risk Andrea Tronconi Department of Statistical Science, La Sapienza University of Rome Abstract The birth of Solvency II has pushed insurance Companies to build their own models in order to represent consistently their risk profiles. If the focus is on the non-life Premium Risk model, the perception is a lack of connection with actuarial best practices of pricing. The Premium Risk results from fluctuation in timing of frequency and severity, of insured events, which ensure that the premiums income will be not enough to pay future claims. Since the level of the premiums income is based on actuarial techniques of pricing, which define the risk level of each profile in portfolio, it is clear that a coherent model, used to define the Premium Risk SCR, should be based on these best practices. Further points of attention must be the CARD system and the business strategy of the Company. The actuarial department used to build GLM to define the expected frequency and severity for each claim types defined by the CARD system and for each possible combination of policyholder characteristics. Using these results it s possible to estimate the future expected frequency and severity if their overall levels will be not change. The model developed considers all these aspects and helps the Company Board to take decisions, answering to the following fundamental question: What will happen in terms of expected profitability, loss ratio, SCR, size of portfolio if.? The milestones are the policies in force on an established date. This portfolio is projected, on the time horizon referred to the calculation of the SCR, considering new business, renewal process, ageing of policyholders, bonus-malus class changing and every features that could modify the risk itself. The projected portfolio expected frequencies are estimated for each profile and for each claim types using the GLM-technique results, and are corrected to consider external factors that could change their overall levels. Meanwhile even the severity parameters are estimated analyzing the past experience.

2 Variable ToT Variable ToT COMPANY ALFA CD Claim Size Group Scenario Base CD Claim Size Group Frequency NC 1,10 CD Claim Size Group Frequency CD 4,67 CTT(CD) 8,16 Frequency CG 5,21 CTT Claim Size Mean DN 3,81 CTT Claim Size CV 4,15 CTT 2,70 CG Claim Size Group 1 Mean NC Attritional Claim Size Mean CG Claim Size Group 2 Mean NC Attritional Claim Size CV 4,45 CG Claim Size Group 3 Mean NC Large Claims 0,19 CG Claim Size Group 1 CV 1,12 NC Large Claim Size Mean CG Claim Size Group 2 CV 1,31 NC Threshold CG Claim Size Group 3 CV 1,29 Table 1 : Simulation parameters estimated (frequency parameters estimated with GLM technique) Once estimated these parameters a simulation model is built using the risk theory collective approach in order to calculate possible outcomes of the total claims amount. The main hypothesis concern the distributions used to model the cost of each claim types, both attritional and large, and the non-correlation between the claim types themselves. = 0 ()*! "#$, (.. "-- / 0 4()*! 2 3"#$ 4(.. 3"-- / 5 #$ A.1 0,.. 72 #$ -- 89:;<=>?@A =BCD Once simulated the total claim cost, the Value at Risk at 99.5 and the Tail Value at Risk at 99 have been calculated and compared with both the expected total claims amount and the future expected Pure Premiums Income. In addition even the expected claims amount and the expected Pure Premiums income are compared in order to clarify the expected profitability of the tariffs in force.

3 To calculate the real premiums income of the portfolio and the connected pure premiums the company tariffs are applied to the future portfolio and even the discount policy of the Company is taken into account in order to have a real situation. Variable COMPANY ALFA Scenario Base TVaR VaR Total Amount of Earned Premiums Total Amount of Earned Premiums Net Expenses RBC "Mean"/Premiums Earned Net 35,2 SCR "Mean"/Premiums Earned Net 23,5 RBC "Premiums"/Premiums Earned Net 39,7 SCR "Premiums"/Premiums Earned Net 28,1 Premiums Earned Net/Mean 95,65 Table 2 : Simulation outcomes Being the main aim of that job building an useful instrument for the company board to compare future scenarios depending on their choices, the last part of the paper is focused on comparing different outcomes of the model depending on underwriting and discount policies and market scenarios. Scenario Two is characterized by a big increase of the market frequency, in Scenario Three the New Business subscribed by the Company is five times the Base Scenario and finally in Scenario Four each New Business policies is underwritten in a very risky city. Variable COMPANY ALFA ALFA ALFA ALFA Scenario Base Two Three Four TVaR VaR Total Amount of Earned Premiums Total Amount of Earned Premiums Net Expenses RBC "Mean"/Premiums Earned Net 35,2 37,9 30,4 36,4 SCR "Mean"/Premiums Earned Net 23,5 26,3 21,0 24,8 RBC "Premiums"/Premiums Earned Net 39,7 52,2 41,5 49,3 SCR "Premiums"/Premiums Earned Net 28,1 40,5 32,0 37,7 Premiums Earned Net/Mean 95,65 87,54 90,08 88,60 Table 3 : Simulation outcomes for different Scenarios Finally the Solvency II standard formulas are applied in order to underline their main lacks and to be compared with the internal model results.

4 SCR/Premiums Figure 1 : Comparison between Internal Model and SII standard formulas

5 REFERENCES Swiss Federal Office of Private Insurance, White Paper of the Swiss Solvency Test, ; IAA, A Global Framework for Insurer Solvency Assessment, ; Harry H. Panjer, Measurement of risk, Solvency requirements and allocation of capital within financial conglomerates, ; Arne Sandström, Solvency- a historical review and some pragmatic solutions, ; Teivo Pentikainen, On the Solvency of Insurance Companies, Joint Committee of OSFI,AMF and Assuris, Framework for a New Standard Approach to Setting Capital Requirements, ; Allan Brender, Advanced Methods in Insurance Capital Requirements, ; Min Wang and Lasse Koskinen, Various Faces of Risk Measures: Internal Model s Perspective, ;

6 European Commission, Solvency II: Presentation of the proposed work, ; S. Taylor, Solvency in Europe: Solvency II, ; Kathryn Morgan and Annette Olesen, The Solvency II Actuary, ; European Commission, Risk-based capital systems, ; European Commission, QIS5: Technical Specification, ; EIOPA, Technical Specification on the Long Term Guarantee Assessment, ; A. Gisler, The Insurance Risk in the SST and in Solvency II: Modelling and Parameter Estimation, ; N. Savelli, Dispense di Teoria del Rischio I, Università Cattolica Milano, 2008; N.Savelli, Dispense di Teoria del Rischio II, Università Cattolica Milano, 2010; FOPI, White Paper of the Swiss Solvency Test, ; FOPI, New Swiss Insurance Regulations from a EU Solvency II Perspective, ; FOPI, Technical document on the Swiss Solvency Test, ;

7 T. Luder, Swiss Solvency Test in Non-life Insurance, ; Damir Filipovi c and Nicolas Vogelpoth, A Note on the Swiss Solvency Test Risk Measure, ; P. Keller, Internal Models for the Swiss Solvency Test, ; FOPI, A primer for calculating the SST cost of capital margin, ; FOPI, SST Template 2008, ; Preeti Chandra Shekhar, Nandha Kumar and S. R. Warrier, Journey of Insurer Solvency regulations and beyond, ; ANIA, Appendice statistica 2009, ; Angelo M. Mineo, Modelli Lineari e Non Lineari in R, ; Jae Myung, Tutorial on Maximun Likelihood estimation, ; Ohlsson, Esbjörn, Johansson, Björn, Non-Life Insurance Pricing with Generalized Linear Models, EAA series Springer, 2005; Ulf Olsson, Genaralized Linear Models, An Applied Approach, Studentlitteratur, 2002; Towers Watson, A Practitioners Guide to Generalized Linear Models,