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1 SOLVIBILITA E RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI DANNI N. Savelli Università Cattolica di Milano Seminario Università Cattolica di Milano Milano, 17 Marzo 2004

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2 Insurance Risk Management and Solvency : MAIN PILLARS OF THE INSURANCE MANAGEMENT: MAIN PILLARS OF THE INSURANCE MANAGEMENT: market share - financial strength - return for stockholders capital. NEED OF NEW 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: 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: SOLVENCY vs PROFITABILITY: at that regard risk theoretical models may be very useful to depict a Risk vs Return trade-off.

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3 SOLVENCY II: simulation models may be used for defining New Rules for Capital Adequacy; A NEW APPROACH OF SUPERVISORY AUTHORITIES: 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: INTERNAL RISK MODELS: to be used not only for solvency purposes but also for managements strategies.

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4 Framework of the Model Company: General Insurance Company: General Insurance Lines of Business: Casualty or Property Lines of Business: Casualty or Property (only casualty is here considered) Catastrophe Losses:may be included (e.g. by Pareto distr.) Catastrophe Losses:may be included (e.g. by Pareto distr.) Time Horizon: 1

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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 : 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.

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6 Risk-Reserve Process (U t ): Risk-Reserve Process (U t ): U t = Risk Reserve at the end of year t U t = Risk Reserve at the end of year t B t = Gross Premiums of year t B t = Gross Premiums of year t X t = Aggregate Claims Amount of year t X t = Aggregate Claims Amount of year t E t = Actual General Expenses of year t E t = Actual General Expenses of year t B RE = Premiums ceded to Reinsurers B RE = Premiums ceded to Reinsurers X RE = Amount of Claims recovered by Reinsurers X RE = Amount of Claims recovered by Reinsurers C RE = Amount of Reinsurance Commissions C RE = Amount of Reinsurance Commissions j = Investment return (annual rate) j = Investment return (annual rate)

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

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

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9 Number of Claims (k): POISSON: the unique parameter is n t =n 0 *(1+g) t depending on the time POISSON: the unique parameter is n t =n 0 *(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 n t is a random variable (= n t *q) MIXED POISSON: in case a structure random variable q with E(q)=1 is introduced and then parameter n t is a random variable (= n t *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 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

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10 Number of Claims (k): Moments If structure variable q is not present: If structure variable q is not present: Mean = E(k t ) = n t Variance = σ 2 (k t ) = n t Skewness = γ(k t ) = 1/(n t ) 1/2 Skewness = γ(k t ) = 1/(n t ) 1/2 If structure variable q is present (Gamma(h;h) distributed): If structure variable q is present (Gamma(h;h) distributed): Mean = E(k t ) = n t Variance = σ 2 (k t ) = n t + n 2 t *σ 2 (q) Variance = σ 2 (k t ) = n t + n 2 t *σ 2 (q) Skewness = γ(k t ) = ( n t +3n 2 t *σ 2 (q)+2n 3 t *σ 4 (q) ) / σ 3 (k t ) Skewness = γ(k t ) = ( n t +3n 2 t *σ 2 (q)+2n 3 t *σ 4 (q) ) / σ 3 (k t ) Some numerical examples: if n = if n = Mean = Std = 100,0 Skew = if n = and σ(q)=2,5% if n = and σ(q)=2,5% Mean = Std = 269,3 Skew = if n = and σ(q)=5% if n = and σ(q)=5% Mean = Std = 509,9 Skew =

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

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

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13 Claim Size Z Distribution and Moments : LogNormal is here assumed, with parametrs changing on the time for inflation only; LogNormal is here assumed, with parametrs changing on the time for inflation only; c Z = coefficient variability σ(Z)/E(Z) c Z = coefficient variability σ(Z)/E(Z) Moments at time t=0: Moments at time t=0: E(Z 0 ) = m 0 σ(Z 0 ) = m 0 *c Z γ(Z 0 ) = c Z *(3+c Z 2 ) (skewness always > 0 and constant along the time because not dependent on inflation) if m 0 = and c Z = 10 Mean = Std = Skew = if m 0 = and c Z = 10 Mean = Std = Skew = if m 0 = and c Z = 5 Mean = Std = Skew = if m 0 = and c Z = 5 Mean = Std = Skew = if m 0 = and c Z = 1 Mean = Std = Skew = + 4 if m 0 = and c Z = 1 Mean = Std = Skew = + 4

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14 Some simulations of the Claim Amount Z m = c Z = 10 m = c Z = 5

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15 Some simulations of the Claim Amount Z m = c Z = 1,00 m = c Z = 0,25

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16 Total Claims Amount X t Moments : If structure variable q is not present

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

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

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19 Expected Value of the Capital Ratio u=U/B In usual cases joint factor r < 1 In usual cases joint factor r < 1 Consequently the relevance of the initial capital ratio u 0 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 Consequently the relevance of the initial capital ratio u 0 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) If r<1 for t=infinite the equilibrium level of expected ratio is obtained: u = λp / (1-r)

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20 Mean, St.Dev. and Skew. U/B An example in the long run Initial Capital ratio: 25 %U 0 =25%*B 0 Expenses Loading (c*B): 25 %of Gross Premiums B Safety Loading (λ*P):+ 5 % of Risk-Premium P Variability Coefficient (c Z ): 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

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21 n=1.000n=

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

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

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24 Confidence Region of u = U/B for a Time Horizon T=5 n= (N=5.000 simulations) Number of Claims k: Poisson Distributed with n 0 = (no structure variable q) Number of Claims k: Poisson Distributed with n 0 = (no structure variable q) Claim Size Z: LogNormal Distributed (m 0 = and c Z =10) Claim Size Z: LogNormal Distributed (m 0 = and c Z =10)

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25 Simulation Moments of U/B :

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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; 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; 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 X t, both Risk Reserve U t and Capital ratio u=U/B present a negative skewness, reducing year by year for: Because of positive skewness of the Total Claim Amount X t, both Risk Reserve U t 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). (no autocorrelations – no long-term cycles).

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27 Loss Ratio X/P MEAN AND PERCENTILES

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28 Capital Ratio U/B: the simulation p.d.f. at year t=

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

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

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

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

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

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34 The link between (expected) capital and profitability: Case u 0 > equilibrium level Case u 0 > equilibrium level Comparison between expected values of Capital ratio and forward RoE E(U/B) and E(Rfw) time horizon T=20 years Case u 0 < equilibrium level Case u 0 < equilibrium level

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

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36 A Measure for Risk: UES - Unconditional Expected Shortfall

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

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38 Other Measures for Risk: Minimum Risk Capital Required (U req ) Minimum Risk Capital Required (U req )

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39 A Theoretical Single-Line General Insurer:

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40 Some simulations:

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41 Results of Simulations:

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42 Percentiles of U/B and X/P:

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43

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44 Ruin Probabilities:

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45 Expected RoE:

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46 A comparison of U/B Distribution (t =1 and 5) u 0 =25%, n 0 =10.000, σ q =5%, E(Z)=3.500, c Z =4 and λ=1.8% u 0 =25%, n 0 =10.000, σ q =5%, E(Z)=10.000, c Z =10 and λ=5% t=1 t=5

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47 Minimum Risk Capital Required:

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

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49 Effects on Ruin Probability and U req :

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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 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%) 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. 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.

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51 Risk vs Profitability: (U RUIN =0) UES vs E(RoE)Ruin Prob. vs E(RoE)

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52 Risk vs Profitability: (U RUIN =1/3 * MSM) UES vs E(RoE)Ruin Prob. Vs E(RoE)

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53 Effects of other Reinsurance covers: 5% Quota Share 5% Quota Share with c RE =22.5% (instead of 20%) XL XL with k M =8 and λ RE =10.8%

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

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55 Conclusions : 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

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56 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; In many cases the EU Minimum Solvency Margin is not reliable and an unsuitable risk profile is reached also for a short time horizon (T2) in the results of simulations. It is to emphasize that in our simulations neither investment risk nor claims reserve run-off risk have been considered, and all the amounts are gross of taxation.

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57 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: 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.

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58 Internal Risk Models: Internal Risk Models: to be used not only for solvency purposes but also for managements strategies and rating; Appointed Actuary: 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).

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59 Further Researches and Improvements of the Model: Modelling a multi-line Insurer (the right-tail of Claim Distribution might have a local maximum point) ; Run-Off dynamics of the Claims Reserve; Premium Rating and Premium Cycles; Dividends barrier and taxation; Modelling Financial Risk; Reinsurance commissions and profit/losses participation; Long-term cycles in claim frequency; Correlation among different insurance lines; Financial Reinsurance and ART; Asset allocation strategies and non-life ALM; Modelling Catastrophe Losses.

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60 Main References : Beard, Pentikäinen, E.Pesonen (1969, 1977,1984) Beard, Pentikäinen, E.Pesonen (1969, 1977,1984) Bühlmann (1970) Bühlmann (1970) British Working Party on General Solvency (1987) British Working Party on General Solvency (1987) Bonsdorff et al. (1989) Bonsdorff et al. (1989) Daykin & Hey (1990) Daykin & Hey (1990) Daykin, Pentikäinen, M.Pesonen (1994) Daykin, Pentikäinen, M.Pesonen (1994) Taylor (1997) Taylor (1997) Klugman, Panjer, Willmot (1998) Klugman, Panjer, Willmot (1998) Coutts, Thomas (1998) Coutts, Thomas (1998) Cummins et al. (1998) Cummins et al. (1998) Venter (2001) Venter (2001) Savelli (2002) Savelli (2002) IAA Solvency Working Party (2003) IAA Solvency Working Party (2003)

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61 Grazie per lattenzione

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62 DOMANDE

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