SEMINAIRE SCIENTIFIQUE 29/01/2003 METHOLOGIES NON VIE DANS LA PERSPECTIVE DES NORMES IFRS/IAS DESPEYROUX, GIE AXA

1. SEMINAIRE SCIENTIFIQUE 29/01/2003 METHOLOGIES NON VIE DANS LA PERSPECTIVE DES NORMES IFRS/IAS DESPEYROUX, GIE AXA C. LEVI, GuyCarpenter C. PARTRAT, ISFA Université Lyon 1 Reprennent leur présentation à la GIRO Convention des Actuaires non vie britanniques, octobre 2002

2. Introduction Run off Triangle Cash Flows Stochastic Methods Discount of Cash Flows Extreme Claims Risks Dependence References Conclusion Agenda

3. DSOP : Entity Specific Value Assessment date : 31/12/n Assets Cash Securities (Bonds, equities) in Fair Value Real estate Liabilities Yearly cash flows run off (no new business) gross paid claims (for contracts in force before 31/12/n) Calculations on gross paid claims (no reinsurance taken into account) Introduction

4. Date 31/12/n 1 set of contracts (no new business, no renewals) Claims developing during (n+1) years Run Off Triangle Cash Flows

5. Run Off Triangle Cash Flows

6. Data in the rectangle are incremental values xij = claims amounts paid for underwriting year i during development year j Data : xij i+j n Unknown : Xij i+j >n Run Off Triangle Cash Flows

7. Future cash flows (without discounting) For k=1,…,n and year (n+k) Total To be compared to available assets A at 31/12/n Run Off Triangle Cash Flows

8. For evaluation of CFn+k or CF, we can use the same approaches (deterministic or stochastic) and the same methods as for reserving. Deterministic methods : Chain Ladder Separation (arithmetic) because diagonals effects etc.. Run Off Triangle Cash Flows

9. Modelling more possibilities including uncertainty measures on results but specification error risk Thanks to the City University group (England, Haberman, Renshaw, Verrall) and T. Mack for their work on stochastic reserving Stochastic Methods

10. For each model Assumption 1 : For i,j = 1,..,n, Xij are independent random variables(r.v.) Standard models now : Generalized Linear Models (with the support of Genmod procedure in SAS) Assumption 2 : For i,j = 1,..,n, distribution of Xij belongs to the same exponential family with where V(m) is the “variance function” of the family Stochastic Methods

11. Parameters (factors) mean for year i for delay j (possibly=1) dispersion parameter Stochastic Methods

12. Aims Let FCF distribution function (d.f.) of the r.v. (FCF) selected parameter to be estimated (risk?) Central values : average E(CF), median, fractiles… Dispersion : V(CF) Insufficiency probability : P(CF>A) Tail : VaR with P(CF>VaR)= Expected shortfall E(CF/ CF> VaR ) Stochastic Methods

13. D.f. FCF m.g.f. And inversion (Fast Fourier Transform) Determining a predictor of Xij(i+j>n), CFn+k then CF Stochastic Methods

14. Means Data : in the superior triangle Maximum likehood method , we obtain estimators of For i+j>n estimator of E(Xij) Stochastic Methods

15. for E(CF) with uncertainty measure or more generally, we obtain estimator of Stochastic Methods

16. is a predictor of Xij and for CF, with uncertainty measure or Difficult to obtain analytic expression (even with some approximation) of and Easier by bootstrapping Stochastic Methods

17. Bootstrapping Pearson’s residuals after modelling the superior triangle gives Confidence interval for the parameter Prediction interval Estimation of probability distribution of CF finding again insufficiency probability, VaR.. Cf England, Verrall, 1999 Pinheiro et al., 2001 Stochastic Methods

18. Which risk / discount rate Risk free Risk premium for liabilities risk Risk premium for assets risk others? IASB current proposal Risk free Plus eventually premium independent of assets dependent of liabilities if not reflected in the market value margin. Discounting Cash Flows

19. Market Value Margin There is always some risk or uncertainty about future cash flows, because of occurrence risk severity risk development risk Adjustment for risks and uncertainty must be reflected preferably in the cash flows. Discounting Cash Flows

20. How evaluate the discount rate risk free Market value of discount rate (yield curve) models(like Vasicek/ Cox Ingersoll Ross/ Wilkie…) risk adjusted discount rate CAPM State price deflators Discounting Cash Flows

21. State Price Deflators State price deflators can be thought of as stochastic discount factors allow for investment risk time value of money a cash flow at date t has a value E[DtCt]/D0 Dt are random variable, vary with scenarios Discounting Cash Flows

22. Example yield curve Discounting Cash Flows

23. Example : non discounted cash flows 1000 Discounting Cash Flows

24. Impact of discounting ( long tail development) Discounting Cash Flows

25. Impact of payment pattern Discounting Cash Flows

26. Profit and loss impact Increase of rate => Profit recognition Decrease of rate => Reduction of profit Discounting Cash Flows

27. Measures Given a line (natural events, casualty,…) X r.v. claim amount D.f. F Tail Distribution Speed of convergence of (x) to 0 closely linked with the existence of moments of X Extreme Claims

28. Value-at-Risk (VaR) e (0.05;0.01;0.005;…) VaRe P(X>VaR)= e VaRe x Extreme Claims

29. Tail VaR - Mean excess Tail VaR 0.01=E( X / X  VaR 0.01) More generally Mean excess : e such that e(u)= E( X-u / Xu) Remark : the d.f. of X can be derived from e. Extreme Claims

30. These measures are used too for other problems: Solvency Capital Allocation Coherent measures Etc.. Extreme Claims

31. Classification of theoritical distributions for modelling extreme claims. Extreme Claims

32. Uncertainty F unknown Historical data : x1,…,xn realization of X1,…,Xn (n-sample) Interest measure p(F) Aims : estimation estimator Estimation uncertainty : standard error, confidence interval, analytic or bootstrap. Extreme Claims

33. Return Period Claims frequency excluded r.v. N(u)=min{i  1:Xiu} (rank of the smallest claim exceeding of u) Return period of level u : (in number of claims) u100 such that E[N(u100)]=100 => Extreme Claims

34. Including claims frequency Assumption : Poisson process (l) for the claims frequency Yn(u)=r.v. interoccurence time between two claims u (years) Extreme Claims

35. Extreme claim development ? GLM : existence of the moments supposed Heavy tail distribution : no assumption on the moments Extreme Claims

36. 2 sub-lines of business “Claims correlated” give 2 run-off triangles of increments. Aims : modelling stochastic dependence to obtain the bivariate distribution of (CF;CF’) Risks dependence

37. Modelling If we need to go over correlation Assumption : dependence is just between Xij and X’ij (i,j=0,…,n) we need the bivariate distribution of (Xij ; X’ij) Common shock models Xij = Yij + Sij Yij , Y’ij , Sij independent r.v. X’ij = Y’ij + Sij Sij : common shock. Dist. of dist of (Xij ; X’ij) Risks dependence

38. Copula Nelsen R. B. (1999) : “An introduction to Copulas” Springer Risks dependence

39. Methods developped in an actuarial dissertation : Gillet A., Serra B. (2002) : “Effets de la dépendance entre différentes branches sur le calcul des provisions “ ENSAE Presented to the Institut des Actuaires for AA (next November) Paper submitted to Astin Colloquium (Berlin, August 2003) Risks dependence

40. Blondeau J., Partrat C. (2002) : “La réassurance : approche technique “ Economica (to be published) Embrechts P., Kluppelbegr C., Mikosh T. (1997) “Modelling extremal events for insurance and finance” Springer Daykin C.D. , Hey G.B. (1991) : “ A management model of General Insurance Company using Simulation Techniques in Managing the Insolvency Risk of Insurance Companies” eds : Cummins J.D et al., Kluwer Academic Publ. Daykin C.D., Pentikäinen T., Pesonen M. (1994) : “Practical Risk Theory for Actuaries” Chapman & Hall. Duffie D. (1994) “Modèles dynamiques d’évaluation” PUF References

41. Efron B., Tibshirani P.J. (1993) : “An introduction to the Bootstrap” Chapman & Hall. England P.D., Verrall R.J. (1999) : “Analytic Bootstrap estimates of prediction error in claims reserving” Insurance : Math. and Econ. Vol. 25, 281-293. England P.D., Verrall R.J. (2002) : “Stochastic claims reserving in General Insurance” Institute of Actuaries. IASB (2001) : “Draft Statements of Principles” Jarvis S., Southall F., Varnell E. (2001) “Modern Valuation Techniques” References

42. Kaufman R., Gardmer A., Klett R.(2001) : “Introduction to Dynamic Financial Analysis” Astin Bull. Vol.31,217-253. KPMG (2002) : “Study into the methodologies to assess the overall financial position of an insurance undertaking from the perspective of prudential supervisor” Report for European Commission. Kaas R., Goovaerts M., Dhaene J., Denuit M.(2001) : “Modern Actuarial Risk Theory” Kluwer Academic Publ. Mack T. (1993) : “Distribution free calculation of the standard error of Chain Ladder reserve estimates” Astin Bull. Vol.23, 213-225. McCullagh P.,Nelder J.A. (1985) : “Generalized Linear Models” 2e ed. Chapman & Hall. Quittard-Pinon F. (1993) “Marchés des capitaux et théorie financière” Economica References

43. Shao J., Tu D. (1995) : “The Jackknife and Bootstrap “ Springer. Pinheiro P., Andrade e Silvo J., Centeno M. (2002) “Bootstrap methodology in claims reserving” Astin Colloquium Washington. Taylor G. (2002) : “Loss reserving - An actuarial Perpective” Kluwer Academic Publ. References