VI Congresso Nazionale di Scienza e Tecnica delle Assicurazioni Bologna, 18-20 Gennaio 2004

1. MANAGEMENT SIMULATION MODELS AND SOLVENCY IN GENERAL INSURANCE___________________Nino SavelliUniversità Cattolica di Milano VI Congresso Nazionale di Scienza e Tecnica delle Assicurazioni Bologna, 18-20 Gennaio 2004

2. Insurance Risk Management and Solvency : • MAIN PILLARS OF THE INSURANCE MANAGEMENT: market share - financial strength - return for stockholders’ capital. • NEED OF NEW CAPITAL: to increase the volume of business is a natural target for the management of an insurance company, but that may cause a need of new capital for solvency requirements and consequently a reduction in profitability is likely to occur. • STRATEGIES: an appropriate risk analysis is then to be carried out on the company, in order to assess appropriate strategies, among these reinsurance covers are extremely relevant. • SOLVENCY vs PROFITABILITY: at that regard risk theoretical models may be very useful to depict a Risk vs Return trade-off.

3. SOLVENCY II: simulation models may be used for defining New Rules for Capital Adequacy; • A NEW APPROACH OF SUPERVISORY AUTHORITIES: assessing the solvency profile of the Insurer according to more or less favourable scenarios (different level of control) and to indicate the appropriate measures in case of an excessive risk of insolvency in the short-term; • INTERNAL RISK MODELS: to be used not only for solvency purposes but also for management’s strategies.

4. Solvency II: A “3-Pillars” Approach to Supervision • FIRST PILLAR: Minimum Financial Requirements Involves the maintenance of a) Appropriate technical provisions b) appropriate assets supporting those obbligations c) a minimum amount of capital for each insurer (developed from a set of available and required capital elements)

5. Solvency II: A “3-Pillars” Approach to Supervision • SECOND PILLAR: Supervisory Review Process Is needed in addition to the first pillar, since not all types of risk can be adequately assessed through solely quantitative measures. This phase will require an indepenedent review (by the the Supervisor or by a designated qualified party), expecially when Internal Models are used. The second pillar is intended to ensure not only that insurers have adequate capital to support all the risks in their business but also to encourage insurers to develop and use better risk management techniques concerning the insurer’s risk profile and in monitoring and managing these risks. Such review will enable supervisory intervention if the insurer’s capital will not sufficiently buffer the risks.

6. Solvency II: A “3-Pillars” Approach to Supervision • THIRD PILLAR: Measures to Foster Market Discipline Serves to strengthen market discipline by introducing disclosure requirements. It is expected that through these requirements, teh industry “best practices” will be fostered. The actuarial profession can assist supervisors within the second pillar by providing independent peer review of an insurer’s liability determination, risk management and/or capital assessment capabilities and within the third pillar in the design of appropriate disclosure practices to serve the public interest.

7. Solvency II: • Within Pillar 1 capital requirements, it is generally believed that next types of insurer risks should be involved: - underwriting risk; - credit risk; - market risk. These risks should be determined by a a factor-driven formula where an appropriatetime horizon and highconfidence level will play a preminent role. • At this regard it is worth to mention the proposal of the IAA Solvency Working Party (draft 2003), where two measures are recommended:

8. Solvency II: • 1) SHORT-TERM: determined for all risks at a very high confidence level (e.g. 99% TVaR) which includes at the end of ONE YEAR the value of future obligations, including a margin (or perhaps at a moderate confidence level such as 75% TVaR); • 2) LONG-TERM: for the complex nature of some insurer risks, a second condition may also be imposed, whereby, if the present value amount of the policy liabilities determined at time zero for all future durations (2-3 years in general insurance and 5 years in life insurance) at a fairly high confidence level (say 90 or 95% TVaR) is greater, then this amount should be held. This second measure picks up all risks for all years including both systematic and non-systematic risks. A lower level of confidence (90/95% instead of 99%) is appropriate given that the company can take some actions after one year to manage its risks.

9. A Management Simulation Model for a General Insurer: • Company: General Insurance • Lines of Business: Casualty or Property (only casualty is here considered) • Catastrophe Losses:may be included (e.g. by Pareto) • Time Horizon: 1<T<5 years • Total Claims Amount: Compound (Mixed) Poisson Proc. • Reinsurance strategy: Traditional (Quota Share, XL, SL) • Investment Return: deterministic or stochastic • Dynamic Portfolio: the total amount of premiums increases year by year according real growth (number of risk unit) and claim inflation (affecting claim size) • Simulations: Monte Carlo Scenario

10. Risk-Reserve Process (Ut): • Ut = Risk Reserve at the end of year t • Bt = Gross Premiums of year t • Xt = Aggregate Claims Amount of year t • Et = Actual General Expenses of year t • BRE= Premiums ceded to Reinsurers • XRE= Amount of Claims recovered by Reinsurers • CRE= Amount of Reinsurance Commissions • j = Investment return (annual rate)

11. Total Claims Amount (Xt):collective approach – one or more lines of business • kt= Number of claims of the year t (Poisson, Mixed Poisson, Negative Binomial, ….) • Zi,t = Claim Size for the i-th claim of the year t. Here a LogNormal distribution is assumedwith values increasing year by year only according to claim inflation • all claim size random variables Zi are assumed to be i.i.d. • random variables Xt are usually independent variables along the time, unless long-term cycles are present and then strong correlation is in force.

12. Number of Claims (k): • POISSON: the unique parameter is nt=n0*(1+g)t depending on the time - risks homogenous - no short-term fluctuations - no long-term cycles • MIXED POISSON: in case a structure random variable q with E(q)=1 is introduced and then parameter nt is a random variable (= nt*q) - only short-term fluctuations have an impact on the underlying claim intensity (e.g. for weather condition – cfr. Beard et al. (1984)) - in case of heterogeneity of the risks in the portfolio (cfr. Buhlmann (1970)) • POLYA: special case of Mixed Poisson when the p.d.f. of the structure variable q is Gamma(h,h) and then p.d.f. of k is Negative Binomial

13. Some simulations of k: • Poisson p.d.f. n = 10.000 results of 10.000 simulations • Negative Binomial p.d.f. n = 10.000 σ(q) = 2,5% results of 10.000 simulations

14. Some simulations of k: • Negative Binomial p.d.f. n = 10.000 σ(q) = 5% results of 10.000 simulations • Negative Binomial p.d.f. n = 10.000 σ(q) = 10% results of 10.000 simulations

15. Some simulations of the Claim Size Z m = € 10.000 cZ = 10 m = € 10.000 cZ = 5

16. Some simulations of the Claim Size Z m = € 10.000 cZ = 1,00 m = € 10.000 cZ = 0,25

17. A Measure for Risk:UES – Unconditional Expected Shortfall • UES = Probability to be in ruin state at time t * MES (MES = Mean Excess Shortfall = Exp.value of ruin deficit); • UES can be regarded as the risk premium of an insurance contract which would cover the shortfall of the company in case it occurs.

18. Other Measures for Risk: • Minimum Risk Capital Required (Ureq)

19. A Theoretical Single-Line General Insurer (Insurer “A”):

20. The simulation results of Insurer “A”:(300.000 simulations)

21. Simulation Moments of capital ratio U/B and loss ratio X/P:

22. Ruin Probabilitieswith two different ruin barriers% values

23. A comparison of U/B Distribution (t =1 and 5):u0=25%, n0=10.000, σq=5%, u0=25%, n0=10.000, σq=5%,E(Z)=3.500, cZ=4 and λ=1.8%E(Z)=10.000, cZ=10 and λ=5% Mean = 24.94 % Std = 4.82 % Skew = - 0.26 Mean = 27.92 % Std = 7.99 % Skew = - 4.79 t=1 Mean = 24.73 % Std = 9.45 % Skew = - 0.11 Mean = 38.78 % Std = 16.25 % Skew = - 1.66 t=5 a

24. Effect of a 20% QS Reinsurance: (with fixed reins. commission = 20% unfavourable for the Insurer because exp=25%): Gross of reins. Net of reins.

25. Effects on Finite Time Ruin Probability Ureq/B0 Conf. level = 99.0%

26. Effects on Ureq/B0 with confidence levels 99.9% and 99.0%

27. A comparison between Ureq and EU MSM ? • Here are disregarded many sources of risk (as e.g. claim reserving run-off, investment risk and underwriting cycles) • Here only a single-line Insurer is regarded (with a reduced variability – cZ) • No dividends and taxation are assumed (but it does not affect so much the downside risk for the usual short time horizon used for solvency analyses).

28. Simulating a trade-off function • Ruin Probability (or UES) vs Expected RoE can be figured out for all the reinsurance strategies available in the market, with a minimum and a maximum constraint • Minimum constraint: Capital Return (e.g. E(RoE)> 5%) Maximum constraint: Risk of Default (e.g. PrRuin < 1%) • Clearly both Risk and Performance measures will decrease as the Insurer reduces its risk retention, but treaty conditions (commissions and loadings mainly) are heavily affecting the most efficient reinsurance strategy, as much as the above mentioned min/max constraints.

29. Risk vs Profitability:(Ruin barrier = 0)UES vs E(RoE) Ruin Prob. vs E(RoE)

30. Effects of other Reinsurance covers: • 5% Quota Share with cRE=22.5% (instead of 20%) • XL with kM=8 and λRE=10.8%

31. The effects on Risk and Profitability of the three reinsurance covers:under management constraints for T=3min(RoE)=25% and max(UES)=0.04 per mille

32. Part II Multi-line Insurer Introducing the Investment Risk (partly) Introducing Taxation and Dividends A larger Risk Loading on Premiums

33. An Insurer with 2 casualty lines:Insurer “B” • Line 1 as the (single-line) Insurer “A” (gross and net of reins.); • Line 2 (independent of line 1) has a very small expected number of claims (n=100 and σ(q)=25%) but with a large expectation and variability of the claim size (m=50.000 Eur and cZ=20) and a significant expected profitability (λ=25%); • Real growth rate: g=5% Claim Inflation: i=5%; • Reinsurance cover for line 2: QS 60% with fixed reins. comm. = 35% (< exp. loading = 40%); • Total Premium Volume: Eur 57,9 mln at year 1 Eur 94,4 mln at year 5; • Premium Mix (constant): 82.0% for line 1 18.0% for line 2 • Retained Premium: 72.8 % • Initial aggregate Capital Ratio (U/B): 25.0 %

34. INSURER BCapital ratio U/B and Loss ratio X/P(gross reins.) (net reins.)

35. An Insurer with 3 casualty lines:Insurer “C” • Lines 1 and 2 as the Insurer “B” (gross and net of reins.); • Line 3 (independent from both lines 1&2) has n=2.500 (with σ(q)=5%) and exp. claim size m=2.000 Eur with size variability cZ=1. Safety loading λ=5% and exp.loading=30%; • Real growth rate: g=5% Claim Inflation: i=5%; • Reinsurance cover for line 3: QS 10% with fixed reins. comm. = exp. loading = 30%; • Total Premium Volume: Eur 65,4 mln at year 1 Eur 106.6 mln at year 5; • Premium Mix (constant): 72.6% for line 1 15.9% for line 2 11.5% for line 3 • Retained Premium: 74.8 % • Initial aggregate Capital Ratio (U/B): 25.0 %.

36. INSURER CCapital ratio U/B and Loss ratio X/P(gross reins.) (net reins.)

37. Comparison among Insurers A/B/C: Min. Capital Ratio Required ureq MAIN ASSUMPTIONS: no claim reserving risk no Investment risk no correlation among lines of bus. no premium cycles no dividends no taxation Insurer A Insurer C Insurer B

38. The impact of Investment Risk, Dividends and Taxation (for a single-line Insurer) Insurer Abis = Insurer A but with the next differences: • Investment return stochastic: j is simulated by an autoregressive model (with an expected value of 4% and st. dev. at year 1 of approx. 1%) ; • Taxation: is assumed a flat rate equal to 35% of the gross year profit (if positive); • Dividends: is assumed a flat rate equal to 20% of the net year profit (if positive);

39. INSURER A INSURER AbisCapital ratio u=U/B Insurer A – gross reins. Insurer Abis – gross reins. Insurer A – net reins. Insurer Abis – net reins.

40. Comparison Insurers A and Abis: Min. Capital Ratio Required ureq MAIN ASSUMPTIONS: no claim reserving risk no Investment risk no correlation among lines of bus. no premium cycles no dividends no taxation no claim reserving risk yes Investment risk no correlation among lines of bus. no premium cycles yes dividends yes taxation Insurer A Insurer B Insurer A bis

41. The presence of the Investment risk (here only partly regarded because no claim reserve is assumed) implies a slightly larger ureq: e.g. for a confidence level 99.0% ureq(1)=11.35% and ureq(2)= 15.37% 11.26% and 15.17% (if j=4%=const.) for the Insurer here regarded Taxation (35%) and Dividends (20%) do not have a significant impact on the downside risk and then on the solvency requirements; e.g. for a confidence level 99.0% ureq(1)=11.38% and ureq(2)= 15.44% 11.26% and 15.17% (j=4% tx=div=0%) Comparison Insurers A and Abis: some comments

42. The impact of Risk Loading on Premiums: Insurer A λ=1.80% Insurer A λ=5.00%

43. Final Comments

44. Final Comments : • The risk of insolvency is heavily affected by, among others, the tail of Total Claims Amount distribution; • Variability and skewness of some variables are extremely relevant: structure variable, claim size variability, investment return, etc.; • A natural choice to reduce risk and to get an efficient capital allocation is to give a portion of the risks to reinsurers, possibly with a favorable pricing. As expected, the results of simulations show how reinsurance is usually reducing not only the insolvency risk but also the expected profitability of the company. In some extreme cases, notwithstanding reinsurance, the insolvency risk may result larger because of an extremely expensive cost of the reinsurance coverage: that happens when the reinsurance price is incoherent with the structure of the transferred risk

45. In many cases the EU “Minimum Solvency Margin” is not reliable and an unsuitable risk profile is reached also for a short time horizon (T≤2) in the results of simulations. It is to emphasize that in our simulations neither (appropriate) investment risk nor claims reserve run-off risk have been considered; • It is possible to define an efficient frontier for the trade-off Insolvency Risk / Shareholders Return according different reinsurance treaties and different retentions according the available pricing in the market;

46. Insurance Solvency II: these simulation models may be used for defining New Rules for Capital Adequacy (also for consolidated requirements); • A new approach of Supervising Authorities: assessing the solvency profile of the Insurer according to more or less favourable scenarios (different level of control) and to indicate the appropriate measures in case of an excessive risk of insolvency in the short-term. • An available measure for the Guarantee Funds: additional premiums related to the effective Insurer’s risk of insolvency

47. Internal Risk Models: to be used not only for solvency purposes but also for management’s strategies and rating; • Appointed Actuary: appropriate simulation models are useful for the role of the Appointed Actuary or similar figures in General Insurance (e.g. for MTPL in Italy), in order to analyse the effect of the strategic triangle Pricing/Reserving/Reinsurance on the Solvency profile of the company. • IAS: to measure in advance the effects of the main forthcoming IAS

48. Further Researches and Improvements of the Model: • Run-Off dynamics of Claims Reserving; • Modelling Investment Risk; • Premium Rating and Premium Cycles; • Dynamic dividends policy and taxation; • Correlation among different insurance lines (Copula analyses); • Comparison of different risk measures

49. Variable reinsurance commissions and profit/losses participation; • Financial Reinsurance and ART; • Asset allocation strategies and non-life ALM; • Modelling Catastrophe Losses; • Modelling a multiplayer market with high policyholders’ sensitivity to either premium measure and insurer’s financial strength, with special reference to TPML (Games Theory). • The impact of forthcoming IAS.

50. Main References : • Beard, Pentikäinen, E.Pesonen (1969, 1977,1984) • Bühlmann (1970) • British Working Party on General Solvency (1987) • Bonsdorff et al. (1989) • Daykin & Hey (1990) • Daykin, Pentikäinen, M.Pesonen (1994) • Taylor (1997) • Klugman, Panjer, Willmot (1998) • Coutts & Thomas (1998) • Cummins et al. (1998) • Savelli (2002 e 2003) • IAA Solvency Working party (2003) • FSA (CP 190, 2003)