11

SOLVIBILITA’ E RIASSICURAZIONE TRADIZIONALE

NELLE ASSICURAZIONI DANNI

N. SavelliUniversità Cattolica di Milano

Seminario Università Cattolica di MilanoMilano, 17 Marzo 2004

 

 

22

Insurance Risk Insurance Risk Management and Management and

Solvency :Solvency : MAIN PILLARS OF THE INSURANCE MANAGEMENT:MAIN PILLARS OF THE INSURANCE MANAGEMENT:market share - financial strength - return for stockholders’ market share - financial strength - return for stockholders’ capital. capital.

NEED OF NEW CAPITAL: NEED OF NEW CAPITAL: to increase the volume of business is a natural target for the to increase the volume of business is a natural target for the management of an insurance company, but that may cause a management of an insurance company, but that may cause a need of new capital for solvency requirements and need of new capital for solvency requirements and consequently a reduction in profitability is likely to occur.consequently a reduction in profitability is likely to occur.

STRATEGIES: STRATEGIES:

an appropriate risk analysis is then to be carried out on the an appropriate risk analysis is then to be carried out on the company, in order to assess appropriate strategies, among company, in order to assess appropriate strategies, among these reinsurance covers are extremely relevant.these reinsurance covers are extremely relevant.

SOLVENCY vs PROFITABILITY: SOLVENCY vs PROFITABILITY:

at that regard risk theoretical models may be very useful to at that regard risk theoretical models may be very useful to depict a Risk vs Return trade-off.depict a Risk vs Return trade-off.

33

SOLVENCY II: SOLVENCY II:

simulation models may be used for defining New Rules for Capital simulation models may be used for defining New Rules for Capital Adequacy;Adequacy;

A NEW APPROACH OF SUPERVISORY AUTHORITIES:A NEW APPROACH OF SUPERVISORY AUTHORITIES:assessing the solvency profile of the Insurer according to more or assessing the solvency profile of the Insurer according to more or less favourable scenarios (different level of control) and to indicate less favourable scenarios (different level of control) and to indicate the appropriate measures in case of an excessive risk of the appropriate measures in case of an excessive risk of insolvency in the short-term;insolvency in the short-term;

INTERNAL RISK MODELS:INTERNAL RISK MODELS: to be used not only for solvency purposes but also for to be used not only for solvency purposes but also for management’s strategies.management’s strategies.

44

Framework of the ModelFramework of the Model

Company: Company: General Insurance General Insurance Lines of Business:Lines of Business: Casualty or Property Casualty or Property

(only casualty is here considered) (only casualty is here considered) Catastrophe Losses:Catastrophe Losses: may be included (e.g. by Pareto may be included (e.g. by Pareto

distr.) distr.) Time Horizon:Time Horizon: 1<T<5 years1<T<5 years Total Claims Amount: Total Claims Amount: Compound (Mixed) Poisson Compound (Mixed) Poisson

ProcessProcess Reinsurance strategy: Reinsurance strategy: Traditional Traditional

(Quota Share, XL, Stop-Loss)(Quota Share, XL, Stop-Loss) Investment Return: Investment Return: deterministic or stochastic deterministic or stochastic Dynamic Portfolio:Dynamic Portfolio: increase year by year according increase year by year according

real growth (number of risks and real growth (number of risks and claims) and inflation (claim claims) and inflation (claim

size)size) Simulations: Simulations: Monte Carlo Scenario Monte Carlo Scenario

55

Conventional Target of Conventional Target of Risk-Theory Models:Risk-Theory Models:

Evaluate for the Time Horizon T the Evaluate for the Time Horizon T the risk of risk of insolvency insolvency and theand the profitability profitability of the company, of the company, according the next main strategic management according the next main strategic management variables :variables :- - capitalizationcapitalization of the company of the company- - safety loadingssafety loadings- - dimension and growthdimension and growth of the portfolio of the portfolio- structure of the insured - structure of the insured portfolioportfolio - - reinsurancereinsurance strategies strategies- - assetasset allocation allocation- etc.- etc.

66

Risk-Reserve Process Risk-Reserve Process (U(Utt):):

UUtt = = Risk Reserve at the end of year t Risk Reserve at the end of year t

BBtt = = Gross Premiums of year tGross Premiums of year t

XXtt = = Aggregate Claims Amount of year tAggregate Claims Amount of year t

EEtt = = Actual General Expenses of year tActual General Expenses of year t

BBRERE = = Premiums ceded to ReinsurersPremiums ceded to Reinsurers

XXRERE = = Amount of Claims recovered by ReinsurersAmount of Claims recovered by Reinsurers

CCRERE = = Amount of Reinsurance CommissionsAmount of Reinsurance Commissions j j = = Investment return (annual rate)Investment return (annual rate)

2/11 )1()

~()

~(

~)1(

~jCXBEXBUjU RE

tREt

REtttttt

77

Gross Premiums (BGross Premiums (Btt):):

BBtt = (1+i)*(1+g)*B = (1+i)*(1+g)*Bt-1t-1

i = claim inflation rate i = claim inflation rate (constant)(constant)

g = real growth rate (g = real growth rate (constant)constant)

BBtt = P = Ptt + + λ*λ*PPtt + C + Ctt = (1+ = (1+λλ)*E(X)*E(Xtt) + c*B) + c*Btt

P = Risk Premium = Exp. Value Total Claims AmountP = Risk Premium = Exp. Value Total Claims Amountλλ = = safety loadingsafety loading coefficient coefficientc = expenses loading coefficientc = expenses loading coefficient

88

Total Claims Amount (XTotal Claims Amount (Xtt):):collective approach – one or more lines of collective approach – one or more lines of

businessbusiness

kktt = = Number of claimsNumber of claims of the year t of the year t ((PoissonPoisson, , Mixed PoissonMixed Poisson, , Negative BinomialNegative Binomial, ….), ….)

ZZi,ti,t = = Claim SizeClaim Size for the i-th claim of the year t. for the i-th claim of the year t. Here a Here a LLogNormal ogNormal distribution is assumeddistribution is assumed with with values increasing year by year only according to values increasing year by year only according to claim inflation claim inflation

all claim size random variables all claim size random variables ZZii are assumed to be i.i.d. are assumed to be i.i.d. random variables random variables XXt t are usually independent variablesare usually independent variables

along the timealong the time, , unless long-term cycles are presentunless long-term cycles are present and then and then strong correlation is in force.strong correlation is in force.

tk

itit ZX

~

1,

~~

99

Number of Claims (k):Number of Claims (k):

POISSONPOISSON: the unique parameter is : the unique parameter is nntt=n=n00*(1+g)*(1+g)tt depending on the time depending on the time - risks homogenous- risks homogenous- no short-term fluctuations- no short-term fluctuations- no long-term cycles- no long-term cycles

MIXED POISSONMIXED POISSON: in case a structure random variable q with : in case a structure random variable q with E(q)=1E(q)=1 is is

introduced and then parameter n introduced and then parameter ntt is a random variable is a random variable (= n (= ntt*q)*q)

- only - only short-term fluctuationsshort-term fluctuations have an impact on the underlying claim have an impact on the underlying claim intensity (e.g. for weather condition – cfr. Beard et al. (1984))intensity (e.g. for weather condition – cfr. Beard et al. (1984))- in case of - in case of heterogeneity of the risksheterogeneity of the risks in the portfolio (cfr. Buhlmann in the portfolio (cfr. Buhlmann (1970)) (1970))

POLYAPOLYA: special case of Mixed Poisson : special case of Mixed Poisson when the p.d.f. of the structure when the p.d.f. of the structure variable q is Gamma(h,h) variable q is Gamma(h,h) and then p.d.f. of k is and then p.d.f. of k is Negative Negative Binomial Binomial

1010

Number of Claims (k): Number of Claims (k): MomentsMoments

If structure variable If structure variable q is not presentq is not present::Mean = E(kMean = E(ktt) = n) = ntt

Variance =Variance =σσ22(k(ktt) = n) = ntt

Skewness = Skewness = γγ(k(ktt) = 1/(n) = 1/(ntt))1/21/2

If structure variable If structure variable q is present (Gamma(h;h) q is present (Gamma(h;h) distributed)distributed)::Mean = E(kMean = E(ktt) = n) = ntt

Variance = Variance = σσ22(k(ktt) = n) = ntt + n + n22tt**σσ22(q)(q)

Skewness = Skewness = γγ(k(ktt) = ( n) = ( ntt +3n +3n22tt**σσ22(q)+2n(q)+2n33

tt**σσ44(q) ) / (q) ) / σσ33(k(ktt))

Some numerical examplesSome numerical examples:: if n = 10.000if n = 10.000

Mean = 10.000Mean = 10.000 Std = 100,0 Skew = + 0.01Std = 100,0 Skew = + 0.01 if n = 10.000 and if n = 10.000 and σσ(q)=2,5%(q)=2,5%

Mean = 10.000Mean = 10.000 Std = 269,3 Skew = + 0.05Std = 269,3 Skew = + 0.05 if n = 10.000 and if n = 10.000 and σσ(q)=5%(q)=5%

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

1111

Some simulations of k:Some simulations of k:

Poisson p.d.f.Poisson p.d.f.n = 10.000n = 10.000results of 10.000 results of 10.000 simulationssimulations

Negative Binomial p.d.f.Negative Binomial p.d.f.n = 10.000n = 10.000σσ(q) = (q) = 2,5%2,5%results of 10.000 results of 10.000 simulationssimulations

1212

Some simulations of k:Some simulations of k:

Negative Binomial p.d.f.Negative Binomial p.d.f.n = 10.000n = 10.000σσ(q) = (q) = 5%5%results of 10.000 results of 10.000 simulationssimulations

Negative Binomial p.d.f.Negative Binomial p.d.f.n = 10.000n = 10.000σσ(q) = (q) = 10%10%results of 10.000 results of 10.000 simulationssimulations

1313

Claim Size ZClaim Size ZDistribution and MomentsDistribution and Moments::

LogNormal LogNormal is here assumed, with parametrs changing on the time for inflation only;is here assumed, with parametrs changing on the time for inflation only; ccZZ = coefficient variability = coefficient variability σσ(Z)/E(Z)(Z)/E(Z) Moments at time t=0:Moments at time t=0:

E(ZE(Z00) = m) = m00

σσ(Z(Z00) = m) = m00*c*cZZ

γγ(Z(Z00) = c) = cZZ*(3+c*(3+cZZ22) ) (skewness always > 0 and constant along the time (skewness always > 0 and constant along the time

because not dependent on inflation) because not dependent on inflation)

if mif m00 = € 10.000 and c = € 10.000 and cZZ = 10 = 10 Mean = € 10.000 Std = € 100.000 Skew = + 1.010Mean = € 10.000 Std = € 100.000 Skew = + 1.010 if mif m00 = € 10.000 and c = € 10.000 and cZZ = 5 = 5 Mean = € 10.000 Std = € 50.000 Skew = + 140Mean = € 10.000 Std = € 50.000 Skew = + 140 if mif m00 = € 10.000 and c = € 10.000 and cZZ = 1 = 1 Mean = € 10.000 Std = € 10.000 Skew = + 4Mean = € 10.000 Std = € 10.000 Skew = + 4

0,0,, )1()~

()1()~

( jZtjj

itjj

ti aiZEiZE

1414

Some simulations of Some simulations of the Claim Amount Zthe Claim Amount Z

m = € 10.000m = € 10.000

ccZZ = 10 = 10

m = € 10.000m = € 10.000

ccZZ = 5 = 5

1515

Some simulations of Some simulations of the Claim Amount Zthe Claim Amount Z

m = € 10.000m = € 10.000

ccZZ = 1,00 = 1,00

m = € 10.000m = € 10.000

ccZZ = 0,25 = 0,25

1616

Total Claims Amount XTotal Claims Amount Xtt MomentsMoments::

If structure variable If structure variable q is not presentq is not present

)~

()1(

1

)(

1)

~(

)~

()1()1()~

(

)~

()1()1()~

(

02/3,2

,3

022

,22

0,1

Xga

a

nX

XiganX

XEiganXE

ttZ

tZ

t

t

tttZtt

tttZtt

1717

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

and and Z LogNormal distributed Z LogNormal distributed

)~

(

)~(2)~(3)

~(

)~()~

(

)~

(

3

4332,2

2,3

222,2

2

,1

t

tttZtttZtt

tttZtt

tZtt

X

qmnqamnanX

qmnanX

anXE

1818

The Capital Ratio u=U/BThe Capital Ratio u=U/B If VP=If VP=ΔΔVX=TX=DV=0VX=TX=DV=0 If Investment Return = constant = jIf Investment Return = constant = j No reinsuranceNo reinsurance

r = (1+j) / ((1+i)(1+g)) r = (1+j) / ((1+i)(1+g)) Joint factor (frequently Joint factor (frequently r<1)r<1)

P/B = (1-c)/(1+P/B = (1-c)/(1+λλ) ) Risk Premium / Gross Risk Premium / Gross Premium Premium

p = (1+j)p = (1+j)1/21/2 P/B P/B

t

k

kt

k

kt

k

ktt r

P

Xrpuru

1

1

00

~)1(~

1919

Expected Value of Expected Value of the Capital Ratio u=U/Bthe Capital Ratio u=U/B

In usual cases joint factor r < 1In usual cases joint factor r < 1 Consequently the relevance of the initial capital ratio uConsequently the relevance of the initial capital ratio u00 is is

more significant in the first years, but after that the relevance more significant in the first years, but after that the relevance of the safety loading of the safety loading λλp (self-financing of the company) is p (self-financing of the company) is prevalent to express the expected value of the ratio uprevalent to express the expected value of the ratio u

If r<1 for t=infinite the equilibrium level of expected ratio is If r<1 for t=infinite the equilibrium level of expected ratio is obtained: obtained: u u = = λλp / (1-r) p / (1-r)

11

1

)~(

1

0

0

rifr

rpur

uE

riftpu

tt

t

2020

Mean, St.Dev. and Skew. Mean, St.Dev. and Skew. U/BU/B

An example in the long runAn example in the long run

Initial Capital ratio:Initial Capital ratio: 25 %25 % UU00=25%*B=25%*B00

Expenses Loading (c*B):Expenses Loading (c*B): 25 %25 % of Gross Premiums of Gross Premiums BB

Safety Loading (Safety Loading (λλ*P):*P):+ 5 % of Risk-Premium P+ 5 % of Risk-Premium P

Variability Coefficient (cVariability Coefficient (cZZ):): 10 10

Claim Inflation Rate (i):Claim Inflation Rate (i): 2 % 2 %

Invest. Return Rate (j):Invest. Return Rate (j): 4 %4 %

Real Growth Rate (g):Real Growth Rate (g): 5 %5 %

Joint Factor (r): Joint Factor (r): 0,97110,9711

No Structure Variable (q):No Structure Variable (q): std(q)=0 std(q)=0

2121

n=1.000n=1.000 n=100.000n=100.000

2222

Some Simulations of Some Simulations of u=U/B :u=U/B :

n=1.000 vs n=1.000 vs n=10.000n=10.000 (N=200 simulations)(N=200 simulations)

2323

Some Simulations of Some Simulations of u=U/B :u=U/B :

n=10.000 vs n=10.000 vs n=100.000n=100.000 (N=200 simulations)(N=200 simulations)

2424

Confidence Region of u = Confidence Region of u = U/BU/B

for a Time Horizon T=5for a Time Horizon T=5n=10.000n=10.000 (N=5.000 simulations)(N=5.000 simulations)

Number of Claims k: Number of Claims k: Poisson Distributed with nPoisson Distributed with n00=10.000 (no structure variable q)=10.000 (no structure variable q) Claim Size Z: Claim Size Z: LogNormal Distributed (mLogNormal Distributed (m00=€ 10.000 and c=€ 10.000 and cZZ=10)=10)

2525

Simulation Moments of Simulation Moments of U/B :U/B :

EXACT AND SIMULATION MOMENTS OF THE CAPITAL RATIO U/B Time EXACT MOMENTS SIMULATION MOMENTS FREQUENCY OF RUIN

AT YEAR t t MEAN ST.DEV. SKEWN. MEAN ST.DEV. SKEWN. % VALUES

0 0.2500 0.2500 1 0.2792 0.0714 - 9.91 0.2793 0.0730 - 7.34 0.54 2 0.3075 0.0984 - 6.92 0.3077 0.1006 - 6.42 0.89 3 0.3350 0.1173 - 5.58 0.3353 0.1184 - 4.21 1.10 4 0.3618 0.1318 - 4.77 0.3617 0.1335 - 3.46 1.19 5 0.3877 0.1435 - 4.22 0.3876 0.1441 - 2.77 1.22

2626

Some comments :Some comments :

Expected ValueExpected Value of the ratio U/B is increasing from the initial 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 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 Insurer the expected value of the Profit Ratio Y/B is increasing approximately from 4,50% of year 1 to 5% of year 5;approximately from 4,50% of year 1 to 5% of year 5;

The The amplitude of the Confidence Regionamplitude of the Confidence Region is rising time to time is rising time to time according the non-convexity behaviour of the standard deviation of according the non-convexity behaviour of the standard deviation of the ratio u=U/B; the ratio u=U/B;

Because of positive skewness of the Total Claim Amount XBecause of positive skewness of the Total Claim Amount X tt, both Risk , both Risk Reserve UReserve Ut t and Capital ratio u=U/B present a and Capital ratio u=U/B present a negative skewness, negative skewness, reducing year by yearreducing year by year for: for:- the increasing volume of risks (g=+5%)- the increasing volume of risks (g=+5%)- the assumption of independent annual technical results - the assumption of independent annual technical results (no autocorrelations – no long-term cycles).(no autocorrelations – no long-term cycles).

2727

Loss Ratio X/PLoss Ratio X/P MEAN AND PERCENTILESMEAN AND PERCENTILES

2828

Capital Ratio U/B:Capital Ratio U/B:the simulation p.d.f. the simulation p.d.f.

at year t=1-2-3-5at year t=1-2-3-5

2929

The effects of some traditional The effects of some traditional reinsurance covers:reinsurance covers:

QUOTA SHAREQUOTA SHARE: : Commissions - fixed share of ceded gross premiums Commissions - fixed share of ceded gross premiums (no scalar commissions and no participation to reinsurer losses are (no scalar commissions and no participation to reinsurer losses are considered).considered).- Quota retention = - Quota retention = 80%80% withwith Fixed Commissions = 25%Fixed Commissions = 25%

EXCESS OF LOSS: EXCESS OF LOSS: Insurer Retention Limit for the Claim Size = M = E(Z) + kInsurer Retention Limit for the Claim Size = M = E(Z) + kMM**σσ(Z) (Z) Insurer Retention 20% of the Claim Size in excess of M:Insurer Retention 20% of the Claim Size in excess of M:- with - with kkMM = 25 and = 25 and reinsurer safety loading 75%reinsurer safety loading 75%

applied on Ceded Risk-Premiumapplied on Ceded Risk-Premium Reins. Risk-Premium = 80% * 3.58% * Reins. Risk-Premium = 80% * 3.58% * PP

3030

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

3131

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

No Reins. No Reins. Net of XLNet of XL

3232

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

No Reins. No Reins. Net of XLNet of XL

3333

A Measure for A Measure for Performance:Performance:

Expected RoE (if Expected RoE (if r<1)r<1) Expected RoE for the Expected RoE for the

time horizon (0,T):time horizon (0,T):

Forward annual Rate of Forward annual Rate of

Expected RoE (year t-Expected RoE (year t-1,t):1,t):

Limit Value:Limit Value:

1)~(

)1()1(~

),0(00

0

u

uEig

U

UUETR TTTT

1)1()1()1(),0(0

u

urrigTR TTTT

)1(1

)1()1)(1(

)~(

)1)(1(),1(

011

uu

r

jigj

uE

igpjttfwR

tt

1)1)(1(),1(lim

igttfwRt

3434

The link between The link between (expected) capital and (expected) capital and

profitability:profitability: Case uCase u0 0 > equilibrium level> equilibrium level

Comparison between expected Comparison between expected values of Capital ratio and values of Capital ratio and

forward RoEforward RoE E(U/B) and E(Rfw)E(U/B) and E(Rfw)

time horizon T=20 yearstime horizon T=20 years

Case uCase u0 0 < equilibrium level< equilibrium level

3535

A Measure for Risk:A Measure for Risk:Probability of RuinProbability 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:

UUUtU t 0/0~

Pr);(

UUTtoneleastatforUTU t 0/,...2,10

~Pr);(

1,...,2,100

~Pr),1;( thforUandUttU ht

3636

A Measure for Risk:A Measure for Risk:UES - Unconditional UES - Unconditional Expected ShortfallExpected Shortfall

)(),,(

)(~/~)()(~Pr

)(~

/~

)(/)(~

Pr

)~

,0max();,(

0

tMEStUU

BtuuutuEtuu

tUUUtUEUUtUU

UUEtUUUES

RUIN

tRUINttRUINRUINt

RUINttRUINRUINt

tRUINRUIN

3737

Other Measures for Risk:Other Measures for Risk:

Capital-at-Risk Capital-at-Risk (CaR)(CaR)

(U(Uεε = quantile of U = quantile of U e.g. e.g. εε=1%)=1%)

)(),0( 0 tUUtCaR

)(),0( 0 tUUtCaR

t

t

CaR r

j

u

tu

U

tCaRtu

)1()(1

),0(),0(

00

3838

Other Measures for Risk:Other Measures for Risk:

Minimum Risk Minimum Risk Capital Required Capital Required (U(Ureqreq))

)(),0( 0 tUUtCaR

)(),0()1( Re tUtUj qt

t

qq r

tuu

B

tUtu

)(),0(),0( 0

0

ReRe

3939

A Theoretical Single-Line A Theoretical Single-Line General Insurer:General Insurer:

Parameters :

Initial risk reserve ratio u0 0,250

Initial Expected number of claims n0

10.000 St.Dev. structure variable q q 0.05 Skewness structure variable q + 0.10

Initial Expected claim amount (EUR) m0 3.500 Variability coeffic. of Z cZ 4

Safety loading coeffic. + 1.80 % Expense loadings coefficient c 25.00 % Real growth rate g +5.00 % Claim inflation rate i 5.00 % Investment return rate j 4.00 %

Initial Risk Premium (mill EUR) P 35,00 Initial Gross Premiums (mill EUR) B 47,51

Joint factor (1+j)/(1+g)(1+i) r 0,9433

4040

Some simulations:Some simulations:

4141

Results of 300.000 Results of 300.000 Simulations:Simulations:

SIMULATION MOMENTS OF THE CAPITAL RATIO U/B AND PURE LOSS RATIO X/P (% VALUES)

SIMULATION MOMENTS OF U/B SIMULATION MOMENTS OF X/P

t MEAN ST.DEV. SKEW. KURT. MEAN ST.DEV. SKEW. KURT.

0 25.00 100.000 - 1 24.94 4.82 - 0.26 3.43 99.998 6.41 + 0.25 3.43 2 24.88 6.60 - 0.18 3.25 99.993 6.37 + 0.28 3.70 3 24.82 7.82 - 0.15 3.15 100.001 6.30 + 0.26 3.68 4 24.78 8.73 - 0.13 3.12 99.983 6.24 + 0.24 3.32 5 24.73 9.45 - 0.11 3.10 99.999 6.17 + 0.23 3.31

4242

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

Time SIMULATION PERCENTILES OF X/P t MEAN 0.1% 1% 5% MEDIAN 99.9%

0 100.000 1 99.998 81.98 86.05 89.92 99.77 123.23 2 99.993 82.19 86.07 89.94 99.77 122.83 3 100.001 82.23 86.24 90.04 99.78 122.30 4 99.983 82.44 86.36 90.13 99.78 122.03 5 99.999 82.43 86.51 90.21 99.81 121.67

Time SIMULATION PERCENTILES OF U/B t MEAN 0.1% 1% 5% MEDIAN 99.9%

0 25.00 1 24.94 7.48 12.97 16.77 25.11 38.47 2 24.88 2.40 8.75 13.77 25.04 44.17 3 24.82 1.45 5.92 11.75 24.97 47.51 4 24.78 -4.17 3.62 10.19 24.95 50.44 5 24.73 -6.44 1.94 8.98 24.89 52.69

4343

4444

Ruin Probabilities:Ruin Probabilities:

WITH RUIN BARRIER URUIN =0 WITH RUIN BARRIER URUIN=1/3*MSM

Time t

ANNUAL

RUIN PROB.

ONE-YEAR

RUIN PROB. FINITE TIME RUIN

PROB.

ANNUAL

RUIN

PROB.

ONE-YEAR

RUIN

PROB.

FINITE

TIME

RUIN

PROB.

1 0.01 0.01 0.01 0.05 0.05 0.05 2 0.05 0.04 0.05 0.31 0.28 0.33 3 0.16 0.13 0.18 0.87 0.68 1.01 4 0.36 0.25 0.44 1.60 1.03 2.05 5 0.61 0.38 0.81 2.33 1.19 3.27

4545

Expected RoE:Expected RoE:

Tim

e t

FORWARD

RATE FINITE-TIME

RATE

1 9.96 9.96 2 9.98 20.54 3 9.99 32.58 4 10.01 45.85 5 10.02 60.47

4646

A comparison of U/B Distribution (t =1 A comparison of U/B Distribution (t =1 and 5)and 5)uu00=25%, n=25%, n00=10.000, =10.000, σσqq=5%,=5%,E(Z)=3.500, cE(Z)=3.500, cZZ==44 and and λλ==1.8%1.8%

u u00=25%, n=25%, n00=10.000, =10.000, σσqq=5%,=5%, E(Z)=10.000, c E(Z)=10.000, cZZ==1010 and and

λλ==5%5%

t=1

t=5

4747

Minimum Risk Capital Minimum Risk Capital Required:Required:

MINIMUM RISK CAPITAL REQUIRED FOR A DIFFERENT TIME HORIZON AS A PERCENT

VALUE OF THE INITIAL GROSS PREMIUMS (UREQ(0,T)/B0) ACCORDING TWO DIFFERENT

CONFIDENCE LEVELS GROSS OF REINS.

Time Horizon

T

CONFID. 99.9%

CONFID. 99.0%

UREQ(T) / UREQ(1)

T = 1 17.07 % 11.26 % 1.00 T = 2 22.31 % 15.17 % 1.35 T = 3 26.73 % 17.94 % 1.59 T = 4 30.27 % 20.43 % 1.81 T = 5 33.62 % 22.40 % 1.99

4848

Effect of a 20% QS Effect of a 20% QS Reinsurance: Reinsurance:

(with reinsurance commission = 20%):(with reinsurance commission = 20%):

4949

Effects on Effects on Ruin Probability Ruin Probability andand

UUreqreq:: MINIMUM RISK CAPITAL REQUIRED FOR A DIFFERENT TIME HORIZON AS A PERCENT

VALUE OF THE INITIAL GROSS PREMIUMS (UREQ(0,T)/B0) ACCORDING TWO DIFFERENT

CONFIDENCE LEVELS GROSS OF REINS. NET OF REINS.

Time

Horizon T

CONFID. 99.9%

CONFID. 99.0%

UREQ(T) / UREQ(1)

CONFID. 99.9%

CONFID. 99.0%

UREQ(T) / UREQ(1)

T = 1 17.07 % 11.26 % 1.00 14.73 % 10.09 % 1.00 T = 2 22.31 % 15.17 % 1.35 20.07 % 14.36 % 1.42 T = 3 26.73 % 17.94 % 1.59 24.83 % 17.80 % 1.76 T = 4 30.27 % 20.43 % 1.81 28.94 % 21.07 % 2.09 T = 5 33.62 % 22.40 % 1.99 32.99 % 24.01 % 2.38

FINITE-TIME EXPECTED ROE AND FINITE-TIME RUIN PROBABILITY GROSS AND NET OF A QUOTA

SHARE REINSURANCE (URUIN=0). (% VALUES)

Time Horizon GROSS. OF REINS. NET OF 20% QS REINS.

T FINITE TIME

EXP. ROE R(0,T)

FINITE-TIME

RUIN PROB. FINITE TIME

EXP. ROE R(0,T)

FINITE-TIME

RUIN PROB.

0 - - - - 1 9.96 0.01 4.28 0.00 2 20.54 0.05 8.77 0.02 3 32.58 0.18 13.45 0.10 4 45.85 0.44 18.42 0.36 5 60.47 0.81 23.56 0.91

5050

Simulating a trade-off Simulating a trade-off functionfunction

Ruin Probability (or UES) vs Expected RoE can be figured out Ruin Probability (or UES) vs Expected RoE can be figured out for all the reinsurance strategies available in the market, with a for all the reinsurance strategies available in the market, with a minimum and a maximum constraint minimum and a maximum constraint

Minimum constraintMinimum constraint: for the Capital Return (e.g. E(RoE)>5%): for the Capital Return (e.g. E(RoE)>5%)Maximum constraintMaximum constraint: for the Ruin Probability (e.g. PrRuin<1%): for the Ruin Probability (e.g. PrRuin<1%)

Clearly both Risk and Performance measures will decrease as the Clearly both Risk and Performance measures will decrease as the Insurer reduces its risk retention, but Insurer reduces its risk retention, but treaty conditions treaty conditions (commissions and loadings mainly) are heavily affecting the (commissions and loadings mainly) are heavily affecting the most efficient reinsurance strategy, as much as the above most efficient reinsurance strategy, as much as the above mentioned min/max constraintsmentioned min/max constraints..

5151

Risk vs Profitability:Risk vs Profitability:(U(URUINRUIN=0)=0)

UES vs E(RoE)UES vs E(RoE) Ruin Prob. vs E(RoE)Ruin Prob. vs E(RoE)

5252

Risk vs Profitability:Risk vs Profitability:(U(URUINRUIN=1/3 * MSM)=1/3 * MSM)

UES vs E(RoE)UES vs E(RoE) Ruin Prob. Vs E(RoE)Ruin Prob. Vs E(RoE)

5353

Effects of other Reinsurance Effects of other Reinsurance covers:covers:

5% Quota Share 5% Quota Share

with cwith cRERE=22.5%=22.5%(instead of 20%)(instead of 20%)

XL XL

with kwith kMM=8 and =8 and λλRERE=10.8%=10.8%

5454

The effects on Risk and The effects on Risk and Profitability of the three Profitability of the three

reinsurance covers:reinsurance covers:under management constraints for T=3 under management constraints for T=3

min(RoE)=25% and max(UES)=0.04 per millemin(RoE)=25% and max(UES)=0.04 per mille (% VALUES)

TIME

HORIZON NO REINSURANCE TREATY A:

20% QS AND CRE=20% TREATY B:

5% QS AND CRE=22.5% TREATY C:

XL AND RE=10.8%

T EXP. RETURN

UES(T) EXP. RETURN

UES(T) EXP. RETURN

UES(T) EXP. RETURN

UES(T)

(0,T) (0,T) (0,T) (0,T) % per mille % per mille % per mille % per mille

0 1 9.97 0.0064 4.28 0.0026 9.16 0.0087 8.18 0.0000 2 20.97 0.0228 8.77 0.0112 19.11 0.0123 17.07 0.0005 3 33.05 0.0584 13.45 0.0304 30.11 0.0476 26.84 0.0068 4 46.44 0.1259 18.42 0.0934 42.28 0.1093 37.42 0.0288 5 61.12 0.2253 23.56 0.2375 55.37 0.2093 49.03 0.0761

5555

ConclusionsConclusions : :The risk of insolvency is heavily affected by, among others, The risk of insolvency is heavily affected by, among others, the tail of Total Claims Amount distributionthe tail of Total Claims Amount distribution;;

Variability and skewness of some variables are extremely Variability and skewness of some variables are extremely relevant: relevant: structure variablestructure variable, , claim size variabilityclaim size variability, , investment returninvestment return, etc.;, etc.;

A natural choice to reduce risk and to get an efficient A natural choice to reduce risk and to get an efficient capital allocation is to give a portion of the risks to capital allocation is to give a portion of the risks to reinsurers, possibly with a favorable pricing. As expected, reinsurers, possibly with a favorable pricing. As expected, the results of simulations show how the results of simulations show how reinsurance is reinsurance is usually reducing not only the insolvency risk but also usually reducing not only the insolvency risk but also the expected profitability of the companythe expected profitability of the company. . In some In some extreme cases, notwithstanding reinsurance, the extreme cases, notwithstanding reinsurance, the insolvency risk may result larger because of an extremely insolvency risk may result larger because of an extremely expensive cost of the reinsurance coverage: that happens expensive cost of the reinsurance coverage: that happens when the reinsurance price is incoherent with the structure when the reinsurance price is incoherent with the structure of the transferred riskof the transferred risk

5656

It is possible It is possible to define an efficient frontier for the to define an efficient frontier for the trade-off Insolvency Risk / Shareholders Returntrade-off Insolvency Risk / Shareholders Return according different reinsurance treaties and different according different reinsurance treaties and different retentions according the available pricing in the market;retentions according the available pricing in the market;

In many cases the EU “Minimum Solvency Margin” is In many cases the EU “Minimum Solvency Margin” is not reliablenot reliable and an unsuitable risk profile is reached also and an unsuitable risk profile is reached also for a short time horizon (T≤2) in the results of simulations. for a short time horizon (T≤2) in the results of simulations. It is to emphasize that It is to emphasize that in our simulations neither in our simulations neither investment risk nor claims reserve run-off risk have been investment risk nor claims reserve run-off risk have been consideredconsidered, and all the amounts are gross of taxation., and all the amounts are gross of taxation.

5757

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 assessing the solvency profile of the Insurer according to more or less favourable scenarios (different level of control) more or less favourable scenarios (different level of control) and to indicate the appropriate measures in case of an and to indicate the appropriate measures in case of an excessive risk of insolvency in the short-term. excessive risk of insolvency in the short-term.

5858

Internal Risk Models: Internal Risk Models:

to be used not only for solvency purposes but also for to be used not only for solvency purposes but also for management’s strategies and rating;management’s strategies and rating;

Appointed Actuary: Appointed Actuary:

appropriate simulation models are useful for the role of the appropriate simulation models are useful for the role of the Appointed Actuary or similar figures in General Insurance Appointed Actuary or similar figures in General Insurance (e.g. for MTPL in Italy). (e.g. for MTPL in Italy).

5959

Further Researches and Further Researches and Improvements of the Improvements of the

Model: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.

6060

Main References :Main References :

Beard, PentikBeard, Pentikääinen, E.Pesonen (1969, 1977,1984)inen, E.Pesonen (1969, 1977,1984) BBüühlmann (1970)hlmann (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, PentikDaykin, Pentikääinen, M.Pesonen (1994)inen, 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)

6161

Grazie per l’attenzioneGrazie per l’attenzione

6262

DOMANDEDOMANDE