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SOLVIBILITA’ E RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI DANNI. N. Savelli Università Cattolica di Milano Seminario Università Cattolica di Milano Milano, 17 Marzo 2004. Insurance Risk Management and Solvency :. MAIN PILLARS OF THE INSURANCE MANAGEMENT:

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SOLVIBILITA’ E RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI DANNI

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Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

SOLVIBILITA’ E

RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI DANNI

N. Savelli

Università Cattolica di Milano

Seminario

Università Cattolica di Milano

Milano, 17 Marzo 2004


Insurance risk management and solvency

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.


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

  • 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.


Framework of the model

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


Conventional target of risk theory models

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.


Risk reserve process u t

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)


Gross premiums b t

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


Total claims amount x t collective approach one or more lines of business

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.


Number of claims k

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


Number of claims k moments

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 = 10.000Std = 100,0 Skew = + 0.01

  • if n = 10.000 and σ(q)=2,5%

    Mean = 10.000Std = 269,3 Skew = + 0.05

  • if n = 10.000 and σ(q)=5%

    Mean = 10.000Std = 509,9 Skew = + 0.10


Some simulations of k

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


Some simulations of k1

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


Claim size z distribution and moments

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


Some simulations of the claim amount z

Some simulations of the Claim Amount Z

m = € 10.000

cZ = 10

m = € 10.000

cZ = 5


Some simulations of the claim amount z1

Some simulations of the Claim Amount Z

m = € 10.000

cZ = 1,00

m = € 10.000

cZ = 0,25


Total claims amount x t moments

Total Claims Amount XtMoments:

If structure variable q is not present


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

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

and Z LogNormal distributed


The capital ratio u u b

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


Expected value of the capital ratio u u b

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)


Mean st dev and skew u b an example in the long run

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


N 1 000 n 100 000

n=1.000n=100.000


Some simulations of u u b n 1 000 vs n 10 000 n 200 simulations

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


Some simulations of u u b n 10 000 vs n 100 000 n 200 simulations

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


Confidence region of u u b for a time horizon t 5 n 10 000 n 5 000 simulations

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)


Simulation moments of u b

Simulation Moments of U/B :


Some comments

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).


Loss ratio x p mean and percentiles

Loss Ratio X/PMEAN AND PERCENTILES


Capital ratio u b the simulation p d f at year t 1 2 3 5

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


The effects of some traditional reinsurance covers

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%withFixed 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


Confidence region u b no reins net of quota share no reins net of xl

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


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 ShareNo Reins. Net of XL


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 ShareNo Reins. Net of XL


A measure for performance expected roe if r 1

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:


The link between expected capital and profitability

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


A measure for risk probability of ruin

A Measure for Risk:Probability of Ruin

  • Probability to be in ruin state at time t:

  • Finite-Time Ruin probability:

  • One-Year Ruin probability:


A measure for risk ues unconditional expected shortfall

A Measure for Risk:UES - Unconditional Expected Shortfall


Other measures for risk

Other Measures for Risk:

  • Capital-at-Risk (CaR)

    (Uε = quantile of U e.g. ε=1%)


Other measures for risk1

Other Measures for Risk:

  • Minimum Risk Capital Required (Ureq)


A theoretical single line general insurer

A Theoretical Single-Line General Insurer:


Some simulations

Some simulations:


Results of 300 000 simulations

Results of 300.000 Simulations:


Percentiles of u b and x p

Percentiles of U/B and X/P:


Ruin probabilities

Ruin Probabilities:


Expected roe

Expected RoE:


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

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


Minimum risk capital required

Minimum Risk Capital Required:


Effect of a 20 qs reinsurance with reinsurance commission 20

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


Effects on ruin probability and u req

Effects on Ruin Probability and Ureq:


Simulating a trade off function

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.


Risk vs profitability u ruin 0 ues vs e roe ruin prob vs e roe

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


Risk vs profitability u ruin 1 3 msm ues vs e roe ruin prob vs e roe

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


Effects of other reinsurance covers

Effects of other Reinsurance covers:

  • 5% Quota Share

    with cRE=22.5%

    (instead of 20%)

  • XL

    with kM=8 and λRE=10.8%


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

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


Conclusions

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


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

  • 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 (T≤2) 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.


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

  • 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.


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

  • 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).


Further researches and improvements of the model

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.


Main references

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)

  • Venter (2001)

  • Savelli (2002)

  • IAA Solvency Working Party (2003)


Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

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Solvibilita e riassicurazione tradizionale nelle assicurazioni danni

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