SOLVIBILITA’ E RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI DANNI

1. SOLVIBILITA’ E RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI DANNI N. Savelli Università Cattolica di Milano Seminario Università Cattolica di Milano Milano, 17 Marzo 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. Framework of the Model • Company: General Insurance • Lines of Business: Casualty or Property (only casualty is here considered) • Catastrophe Losses:may be included (e.g. by Pareto distr.) • Time Horizon: 1<T<5 years • Total Claims Amount: Compound (Mixed) Poisson Process • Reinsurance strategy: Traditional (Quota Share, XL, Stop-Loss) • Investment Return: deterministic or stochastic • Dynamic Portfolio: increase year by year according real growth (number of risks and claims) and inflation (claim size) • Simulations: Monte Carlo Scenario

5. Conventional Target of Risk-Theory Models: • Evaluate for the Time Horizon T the risk of insolvency and the profitability of the company, according the next main strategic management variables : - capitalization of the company - safety loadings - dimension and growth of the portfolio - structure of the insured portfolio - reinsurance strategies - asset allocation - etc.

6. 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)

7. Gross Premiums (Bt): Bt = (1+i)*(1+g)*Bt-1 i = claim inflation rate (constant) g = real growth rate (constant) Bt = Pt + λ*Pt + Ct = (1+λ)*E(Xt) + c*Bt P = Risk Premium = Exp. Value Total Claims Amount λ = safety loading coefficient c = expenses loading coefficient

8. 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.

9. 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

10. Number of Claims (k): Moments • If structure variable q is not present: Mean = E(kt) = nt Variance =σ2(kt) = nt Skewness = γ(kt) = 1/(nt)1/2 • If structure variable q is present (Gamma(h;h) distributed): Mean = E(kt) = nt Variance = σ2(kt) = nt + n2t*σ2(q) Skewness = γ(kt) = ( nt +3n2t*σ2(q)+2n3t*σ4(q) ) / σ3(kt) Some numerical examples: • if n = 10.000 Mean = 10.000 Std = 100,0 Skew = + 0.01 • if n = 10.000 and σ(q)=2,5% Mean = 10.000 Std = 269,3 Skew = + 0.05 • if n = 10.000 and σ(q)=5% Mean = 10.000 Std = 509,9 Skew = + 0.10

11. 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

12. 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

13. Claim Size ZDistribution and Moments: • LogNormal is here assumed, with parametrs changing on the time for inflation only; • cZ = coefficient variability σ(Z)/E(Z) • Moments at time t=0: E(Z0) = m0 σ(Z0) = m0*cZ γ(Z0) = cZ*(3+cZ2) (skewness always > 0 and constant along the time because not dependent on inflation) • if m0 = € 10.000 and cZ = 10 Mean = € 10.000 Std = € 100.000 Skew = + 1.010 • if m0 = € 10.000 and cZ = 5 Mean = € 10.000 Std = € 50.000 Skew = + 140 • if m0 = € 10.000 and cZ = 1 Mean = € 10.000 Std = € 10.000 Skew = + 4

14. Some simulations of the Claim Amount Z m = € 10.000 cZ = 10 m = € 10.000 cZ = 5

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

16. Total Claims Amount XtMoments: If structure variable q is not present

17. If structure variable q is present and Gamma(h;h) distributed and Z LogNormal distributed

18. The Capital Ratio u=U/B • If VP=ΔVX=TX=DV=0 • If Investment Return = constant = j • No reinsurance • r = (1+j) / ((1+i)(1+g)) Joint factor (frequently r<1) • P/B = (1-c)/(1+λ) Risk Premium / Gross Premium • p = (1+j)1/2 P/B

19. Expected Value of the Capital Ratio u=U/B • In usual cases joint factor r < 1 • Consequently the relevance of the initial capital ratio u0 is more significant in the first years, but after that the relevance of the safety loading λp (self-financing of the company) is prevalent to express the expected value of the ratio u • If r<1 for t=infinite the equilibrium level of expected ratio is obtained: u = λp / (1-r)

20. Mean, St.Dev. and Skew. U/BAn example in the long run Initial Capital ratio: 25 % U0=25%*B0 Expenses Loading (c*B):25 % of Gross Premiums B Safety Loading (λ*P): + 5 % of Risk-Premium P Variability Coefficient (cZ): 10 Claim Inflation Rate (i): 2 % Invest. Return Rate (j): 4 % Real Growth Rate (g): 5 % Joint Factor (r): 0,9711 No Structure Variable (q): std(q)=0

21. n=1.000 n=100.000

22. Some Simulations of u=U/B :n=1.000 vs n=10.000(N=200 simulations)

23. Some Simulations of u=U/B :n=10.000 vs n=100.000(N=200 simulations)

24. Confidence Region of u = U/Bfor a Time Horizon T=5n=10.000(N=5.000 simulations) • Number of Claims k: Poisson Distributed with n0=10.000 (no structure variable q) • Claim Size Z: LogNormal Distributed (m0=€ 10.000 and cZ=10)

25. Simulation Moments of U/B :

26. Some comments : • Expected Value of the ratio U/B is increasing from the initial value 25% to 40% at year t=5. It is useful to note that for the Medium Insurer the expected value of the Profit Ratio Y/B is increasing approximately from 4,50% of year 1 to 5% of year 5; • The amplitude of the Confidence Region is rising time to time according the non-convexity behaviour of the standard deviation of the ratio u=U/B; • Because of positive skewness of the Total Claim Amount Xt, both Risk Reserve Ut and Capital ratio u=U/B present a negative skewness, reducing year by year for: - the increasing volume of risks (g=+5%) - the assumption of independent annual technical results (no autocorrelations – no long-term cycles).

27. Loss Ratio X/PMEAN AND PERCENTILES

28. Capital Ratio U/B:the simulation p.d.f. at year t=1-2-3-5

29. The effects of some traditional reinsurance covers: • QUOTA SHARE: Commissions - fixed share of ceded gross premiums (no scalar commissions and no participation to reinsurer losses are considered). - Quota retention = 80% with Fixed Commissions = 25% • EXCESS OF LOSS: Insurer Retention Limit for the Claim Size = M = E(Z) + kM*σ(Z) Insurer Retention 20% of the Claim Size in excess of M: - with kM = 25 and reinsurer safety loading 75% applied on Ceded Risk-Premium Reins. Risk-Premium = 80% * 3.58% * P

30. Confidence Region U/B No Reins. Net of Quota ShareNo Reins. Net of XL

31. Distribution of U/B (t=1)No Reins. Net of Quota ShareNo Reins. Net of XL

32. Distribution of U/B (t=5)No Reins. Net of Quota ShareNo Reins. Net of XL

33. A Measure for Performance:Expected RoE (if r<1) • Expected RoE for the time horizon (0,T): • Forward annual Rate of Expected RoE (year t-1,t): Limit Value:

34. The link between (expected) capital and profitability: • Case u0 > equilibrium level Comparison between expected values of Capital ratio and forward RoE E(U/B) and E(Rfw) time horizon T=20 years • Case u0 < equilibrium level

35. A Measure for Risk:Probability of Ruin • Probability to be in ruin state at time t: • Finite-Time Ruin probability: • One-Year Ruin probability:

36. A Measure for Risk:UES - Unconditional Expected Shortfall

37. Other Measures for Risk: • Capital-at-Risk (CaR) (Uε = quantile of U e.g. ε=1%)

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

39. A Theoretical Single-Line General Insurer:

40. Some simulations:

41. Results of 300.000 Simulations:

42. Percentiles of U/B and X/P:

44. Ruin Probabilities:

45. Expected RoE:

46. A comparison of U/B Distribution (t =1 and 5)u0=25%, n0=10.000, σq=5%,E(Z)=3.500, cZ=4 and λ=1.8% u0=25%, n0=10.000, σq=5%, E(Z)=10.000, cZ=10 and λ=5% t=1 t=5

47. Minimum Risk Capital Required:

48. Effect of a 20% QS Reinsurance: (with reinsurance commission = 20%):

49. Effects on Ruin Probability and Ureq:

50. 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: for the Capital Return (e.g. E(RoE)>5%) Maximum constraint: for the Ruin Probability (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.