SOLVIBILITA’ E
RIASSICURAZIONE TRADIZIONALE NELLE ASSICURAZIONI
DANNI
N. Savelli
Università Cattolica di Milano
Seminario
Università Cattolica di Milano
Milano, 17 Marzo 2004
1
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.
2
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.
3
Framework of the Model






Company:
Lines of Business:
Catastrophe Losses:
Time Horizon:
Total Claims Amount:
Reinsurance strategy:

Investment Return:
Dynamic Portfolio:

Simulations:

General Insurance
Casualty or Property
(only casualty is here considered)
may be included (e.g. by Pareto distr.)
1<T<5 years
Compound (Mixed) Poisson Process
Traditional
(Quota Share, XL, Stop-Loss)
deterministic or stochastic
increase year by year according
real growth (number of risks and
claims) and inflation (claim size)
Monte Carlo Scenario
4
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.
5
Risk-Reserve Process (Ut):








Ut =
Bt =
Xt =
Et =
BRE =
XRE =
CRE =
j
=
Risk Reserve at the end of year t
Gross Premiums of year t
Aggregate Claims Amount of year t
Actual General Expenses of year t
Premiums ceded to Reinsurers
Amount of Claims recovered by Reinsurers
Amount of Reinsurance Commissions
Investment return (annual rate)


~
~
~
~ RE RE
RE
U t  (1  j)  U t 1  (Bt  X t  Et )  (Bt  X t  Ct )  (1  j)1/ 2
6
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
7
Total Claims Amount (Xt):
collective approach – one or more lines of
business
~
Xt 




~
kt
~
Z
 i ,t
i 1
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 assumed with
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.
8
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
9
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

if n = 10.000 and σ(q)=2,5%
Mean = 10.000

if n = 10.000 and σ(q)=5%
Mean = 10.000
Std = 100,0
Skew = + 0.01
Std = 269,3
Skew = + 0.05
Std = 509,9
Skew = + 0.10
10
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
11
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
12
Claim Size Z
Distribution 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
if m0 = € 10.000 and cZ = 5
if m0 = € 10.000 and cZ = 1
Mean = € 10.000 Std = € 100.000 Skew = + 1.010
Mean = € 10.000 Std = € 50.000 Skew = + 140
Mean = € 10.000 Std = € 10.000 Skew = +
4
~j
~j
jt
E ( Z i ,t )  (1  i)  E ( Z i ,0 )  (1  i) jt  a jZ ,0
13
Some simulations of
the Claim Amount Z
m = € 10.000
cZ = 10
m = € 10.000
cZ = 5
14
Some simulations of
the Claim Amount Z
m = € 10.000
cZ = 1,00
m = € 10.000
cZ = 0,25
15
Total Claims Amount Xt
Moments:
If structure variable q is not present
~
~
t
t
E ( X t )  nt  a1Z ,t  (1  g )  (1  i )  E ( X 0 )
2 ~
t
2t
2 ~
 ( X t )  nt  a2 Z ,t  (1  g )  (1  i )   ( X 0 )
a3 Z ,t
1
~
 (Xt ) 


3/ 2
nt (a2 Z ,t )
1
(1  g ) t
~
 (X0)
16
If structure variable q is present and Gamma(h;h)
distributed
and
Z LogNormal distributed
~
E ( X t )  nt  a1Z ,t
2 ~
 ( X t )  nt  a2 Z ,t  nt2  mt2   2 (q~ )
2
2 ~
3 3
4 ~
nt a3 Z ,t  3nt mt a2 Z ,t   (q )  2nt mt   (q )
~
 (Xt ) 
3 ~
 (Xt )
17
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))
P/B = (1-c)/(1+λ)
p = (1+j)1/2 P/B
Joint factor (frequently r<1)
Risk Premium / Gross Premium
~

X k t k 
t
k
~
ut  r  u0  p  (1   )   r  
r 
k 0
k 1 Pk


t 1
t
18
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)
u0    p  t
if r  1
E (u~t ) 
t
1

r
r t  u0    p 
1 r
if r  1
19
Mean, St.Dev. and Skew. U/B
An 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
20
n=1.000
n=100.000
21
Some Simulations of
u=U/B :
n=1.000
vs
n=10.000
(N=200 simulations)
22
Some Simulations of
u=U/B :
n=10.000
vs
n=100.000
(N=200 simulations)
23
Confidence Region of u = U/B
for a Time Horizon T=5
n=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)
24
Simulation Moments of U/B :
EXACT AND SIMULATION MOMENTS OF THE CAPITAL RATIO U/B
Time
EXACT MOMENTS
t
MEAN
0
1
2
3
4
5
0.2500
0.2792
0.3075
0.3350
0.3618
0.3877
ST.DEV.
0.0714
0.0984
0.1173
0.1318
0.1435
SIMULATION MOMENTS
SKEWN.
MEAN
ST.DEV.
SKEWN.
FREQUENCY OF RUIN
AT YEAR t
% VALUES
- 9.91
- 6.92
- 5.58
- 4.77
- 4.22
0.2500
0.2793
0.3077
0.3353
0.3617
0.3876
0.0730
0.1006
0.1184
0.1335
0.1441
- 7.34
- 6.42
- 4.21
- 3.46
- 2.77
0.54
0.89
1.10
1.19
1.22
25
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).
26
Loss Ratio X/P
MEAN AND PERCENTILES
27
Capital Ratio U/B:
the simulation p.d.f.
at year t=1-2-3-5
28
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
29
Confidence Region U/B
No Reins.
No Reins.
Net of Quota Share
Net of XL
30
Distribution of U/B (t=1)
No Reins.
Net of Quota Share
No Reins.
Net of XL
31
Distribution of U/B (t=5)
No Reins.
Net of Quota Share
No Reins.
Net of XL
32
A Measure for Performance:
Expected RoE

~
Expected RoE for the time
~
 UT  U 0 
T
T E (uT )
  (1  g ) (1  i) 
R (0, T )  E
1
horizon (0,T):

U
u


(if r<1)
0

0

u
R (0, T )  (1  g ) T (1  i) T  r T  (1  r T )    1
u0 

Forward annual Rate of
Expected RoE (year t-1,t):
R fw(t  1, t )  j  p 
Limit Value:
(1  g )(1  i )
(1  g )(1  i )  (1  j )

j

~ )
u
E (u
t 1
1  r t 1  ( 0  1)
u
lim R fw(t  1, t )  (1  g )(1  i)  1
t 
33
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
34
A Measure for Risk:
Probability of Ruin

Probability to be in ruin state at time t:
~

 (U ; t )  PrU t  0 / U 0  U 



Finite-Time Ruin probability:
~


 (U ;T )  Pr U t  0 for at least one t  1,2,...T / U 0  U 





One-Year Ruin probability:
~

 (U ; t  1, t )  Pr U t  0 and U h  0 for h  1,2,..., t  1


35
A Measure for Risk:
UES - Unconditional
Expected Shortfall


~
UES (U ,U RUIN ; t )  E max( 0,U RUIN  U t ) 




~
~
~
 Pr U t  U RUIN (t ) / U 0  U  E U RUIN (t )  U t / U t  U RUIN (t )  




 Pr u~t  u RUIN (t )   E  u RUIN (t )  u~t / u~t  u RUIN (t )   Bt 


  (U ,U RUIN , t )  MES (t )
36
CaR(0, t )  U 0  U (t )
Other Measures for Risk:

Capital-at-Risk (CaR)
(Uε = quantile of U
e.g. ε=1%)
CaR(0, t )  U 0  U (t )
u (t ) (1  j )t
CaR(0, t )
uCaR (0, t ) 
 1

U0
u0
rt
37
CaR(0, t )  U 0  U (t )
Other Measures for Risk:

Minimum Risk Capital
Required (Ureq)
u Re q (0, t ) 
(1  j) t  U Re q (0, t )  U  (t )
U Re q (0, t )
B0
u (t )
 u0 
rt
38
A Theoretical Single-Line
General Insurer:
Parameters :
Initial risk reserve ratio
u0
0,250
Initial Expected number of claims n0
St.Dev. structure variable q
q
Skewness structure variable q
10.000
0.05
+ 0.10
Initial Expected claim amount (EUR) m0
Variability coeffic. of Z
cZ
3.500
4
Safety loading coeffic.
Expense loadings coefficient
Real growth rate
Claim inflation rate
Investment return rate
Initial Risk Premium (mill EUR)
Initial Gross Premiums (mill EUR)
Joint factor (1+j)/(1+g)(1+i)
r

c
g
i
j
+ 1.80 %
25.00 %
+5.00 %
5.00 %
4.00 %
P
B
35,00
47,51
0,9433
39
Some simulations:
40
Results of 300.000
Simulations:
SIMULATION MOMENTS OF THE CAPITAL RATIO U/B AND PURE LOSS RATIO X/P
(% VALUES)
SIMULATION MOMENTS OF U/B
t
MEAN
0
1
2
3
4
5
25.00
24.94
24.88
24.82
24.78
24.73
ST.DEV.
4.82
6.60
7.82
8.73
9.45
SKEW.
- 0.26
- 0.18
- 0.15
- 0.13
- 0.11
SIMULATION MOMENTS OF X/P
KURT.
MEAN
3.43
3.25
3.15
3.12
3.10
100.000
99.998
99.993
100.001
99.983
99.999
ST.DEV.
6.41
6.37
6.30
6.24
6.17
SKEW.
KURT.
+ 0.25
+ 0.28
+ 0.26
+ 0.24
+ 0.23
3.43
3.70
3.68
3.32
3.31
41
Percentiles of U/B and X/P:
SIMULATION PERCENTILES OF U/B
Time
t
MEAN
0
1
2
3
4
5
25.00
24.94
24.88
24.82
24.78
24.73
0.1%
7.48
2.40
1.45
-4.17
-6.44
1%
12.97
8.75
5.92
3.62
1.94
5%
16.77
13.77
11.75
10.19
8.98
SIMULATION PERCENTILES OF X/P
Time
MEDIAN 99.9%
t
MEAN
0.1%
1%
5%
MEDIAN 99.9%
38.47
44.17
47.51
50.44
52.69
0
1
2
3
4
5
100.000
99.998
99.993
100.001
99.983
99.999
81.98
82.19
82.23
82.44
82.43
86.05
86.07
86.24
86.36
86.51
89.92
89.94
90.04
90.13
90.21
99.77
99.77
99.78
99.78
99.81
25.11
25.04
24.97
24.95
24.89
123.23
122.83
122.30
122.03
121.67
42
43
Ruin Probabilities:
WITH RUIN BARRIER URUIN =0
WITH RUIN BARRIER URUIN=1/3*MSM
Tim
e
t
ANNUAL
RUIN
PROB.
ONE-YEAR
RUIN PROB.
FINITE
TIME
RUIN
PROB.
ANNUAL
RUIN
PROB.
1
2
3
4
5
0.01
0.05
0.16
0.36
0.61
0.01
0.04
0.13
0.25
0.38
0.01
0.05
0.18
0.44
0.81
0.05
0.31
0.87
1.60
2.33
ONEYEAR
RUIN
PROB.
FINITE
TIME
RUIN
PROB.
0.05
0.28
0.68
1.03
1.19
0.05
0.33
1.01
2.05
3.27
44
Expected RoE:
Tim
e
t
FORWARD
RATE
FINITE-TIME
RATE
1
2
3
4
5
9.96
9.98
9.99
10.01
10.02
9.96
20.54
32.58
45.85
60.47
45
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
46
Minimum Risk Capital 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
Time
Horizon
T
T=1
T=2
T=3
T=4
T=5
GROSS OF REINS.
CONFID. CONFID.
99.9%
99.0%
17.07 %
22.31 %
26.73 %
30.27 %
33.62 %
11.26 %
15.17 %
17.94 %
20.43 %
22.40 %
UREQ(T) /
UREQ(1)
1.00
1.35
1.59
1.81
1.99
47
Effect of a 20% QS Reinsurance:
(with reinsurance commission = 20%):
48
Effects on
Ruin Probability
Ureq:
and
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
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.
CONFIDENCE LEVELS
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
2
3
4
5
9.96
20.54
32.58
45.85
60.47
0.01
0.05
0.18
0.44
0.81
4.28
8.77
13.45
18.42
23.56
0.00
0.02
0.10
0.36
0.91
NET OF REINS.
Time
Horizon
T
CONFID.
99.9%
CONFID.
99.0%
UREQ(T) /
UREQ(1)
T=1
T=2
T=3
T=4
T=5
17.07 %
22.31 %
26.73 %
30.27 %
33.62 %
11.26 %
15.17 %
17.94 %
20.43 %
22.40 %
1.00
1.35
1.59
1.81
1.99
CONFID. CONFID.
99.9% 99.0%
14.73 %
20.07 %
24.83 %
28.94 %
32.99 %
10.09 %
14.36 %
17.80 %
21.07 %
24.01 %
UREQ(T) /
UREQ(1)
1.00
1.42
1.76
2.09
2.38
49
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:
Maximum constraint:

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.
for the Capital Return (e.g. E(RoE)>5%)
for the Ruin Probability (e.g. PrRuin<1%)
50
Risk vs Profitability:
UES vs E(RoE)
(URUIN=0)
Ruin Prob. vs E(RoE)
51
Risk vs Profitability:
UES vs E(RoE)
(URUIN=1/3 * MSM)
Ruin Prob. Vs E(RoE)
52
Effects of other Reinsurance
covers:

5% Quota Share
with cRE=22.5%
(instead of 20%)

XL
with kM=8 and
λRE=10.8%
53
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
(% VALUES)
TIME
NO REINSURANCE
HORIZON
T
0
1
2
3
4
5
UES(T)
TREATY A:
20% QS AND CRE=20%
TREATY B:
5% QS AND CRE=22.5%
TREATY C:
XL AND RE=10.8%
per mille
EXP.
RETURN
(0,T)
%
per mille
EXP.
RETURN
(0,T)
%
per mille
0.0026
0.0112
0.0304
0.0934
0.2375
9.16
19.11
30.11
42.28
55.37
0.0087
0.0123
0.0476
0.1093
0.2093
8.18
17.07
26.84
37.42
49.03
0.0000
0.0005
0.0068
0.0288
0.0761
EXP.
RETURN
(0,T)
%
per mille
EXP.
RETURN
(0,T)
%
9.97
20.97
33.05
46.44
61.12
0.0064
0.0228
0.0584
0.1259
0.2253
4.28
8.77
13.45
18.42
23.56
UES(T)
UES(T)
UES(T)
54
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
55
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.
56
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.
57


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).
58
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.
59
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)
60
Grazie per l’attenzione
61
DOMANDE
62