Risk Modeling of Multi-year,
Multi-line Reinsurance Using
Copulas
by Ping Wang
St John’s University, New York
on CICIRM 2011 at Beijing, China
1
Agenda Today
• Multi-year, multi-line reinsurance
• A Framework Using Copulas to model
time dependence
• Application using real data
• Concluding remarks
• Q&A
2
Multi-year, multi-line
reinsurance policies
• Cover losses arising from multiple lines of
business over multiple years (3 or 5 most
common)
• Stop-loss type, commonly. Reinsurer pays
claims only if the accumulated losses from
several business lines over an extended period
exceed a fairly high threshold.
• Reduced volatility compared to separate
coverage
3
Difficulty Facing Actuaries
• Simultaneous modeling dependence
– Across time, and
– Across business lines (e.g., workers
compensation and commercial multiple
perils)
4
Modeling Product Risk With Copula
• Assume independence between
business lines
• Model time-dependence of each
line using copula
• Simulate the distribution of future
accumulated losses
• Estimate the payoff of multi-year,
multi-line reinsurance
5
Marginal Distribution
• Suppose that there are Ti years data for a
business line of the ith primary insurer
Y  Y , Y , Y ,, Y
i
i1
i2
i3
iTi


• Univariate marginal distribution functions
Pit  yit   ProbYit  yit   Pit  P yit ,it 
• Fit with Gamma, normal, lognormal, t-dist’n
6
Modeling Time Dependencies
Using Copulas
• With Copula C, the joint distribution function of
Yi can be expressed as Pi yi1 , , yiT  C Pi1 , , PiT

i
 
i
• The log-likelihood of ith primary insurer is
li 
Ti
 ln p( y
t 1
it , θ it )

 ln c Pi1 , Pi 2 , , PiTi

• where c(.) is the probability density function
corresponding to the copula function
• Predictive distribution is obtained based on the
results of maximum likelihood estimation
7

Estimate Product Risk
• Simulation of joint distribution of each
business line over multiple years
• Calculate the policy payoff
• Analyze the risk using VaR and CTE
8
Real Data
• Loss ratios of workers compensation (WC)
and commercial multiple perils (CMP)
• 32 primary insurers
• Task: based on the loss history of 5 years,
fit the multivariate distribution, simulate the
future losses, then model the risk of the
reinsurance policy that covers
accumulated losses of both lines over next
three years.
9
Correlations across Time: WC
• Loss ratios among years are not independent.
WC04
WC04
WC03
WC02
WC03
WC02
WC01
WC00
.6483
(<.0001)
.6640
(<.0001)
.4611
(.0079)
.6128
(.0002)
.6586
(<.0001)
.3132
(.0809)
.3398
(.0571)
.6144
(.0002)
.3796
(.0321)
WC01
.5617
(.0008)
Reported are the value of Pearson correlations and corresponding p-values.
10
Correlations across Time: CMP
CMP04
CMP04
CMP03
CMP02
CMP03
CMP02
CMP01
CMP00
.4771
(.0058)
.3327
(.0628)
.3200
(.0742)
.3661
(.0394)
.4999
(.0036)
.1510
(.4093)
.1225
(.5041)
.4212
(.0164)
.2571
(.1554)
CMP01
.3589
(.0437)
Reported are the value of Pearson correlations and corresponding p-values.
11
Relationship between WC & CMP
• Correlation coefficient: 0.1510
Scatter plot of WC vs CMP loss ratio
140
120
WC loss ratio
100
80
60
40
20
0
0
20
40
60
CMP loss ratio
80
100
120
12
Fitted Marginal Distribution
WC loss ratio
CMP loss ratio
Distribution
AIC
K-S stat*
AIC
K-S stat
Lognormal
2176.4276
0.0383
2087.3092 0.0538
Gamma
2176.0656
0.0399
2087.6407 0.0709
t-dist’n
2588.6599
0.2707
2411.356
0.2561
*: kolmogorov-Smirnov test statistic
13
t-copula
• t-copula:
c(u1 ,, u m ) 

pT G -r1 (u1 ),..., G -r1 (u m )

m
i 1
1
g r (G -r1 (ui ))
• where Gr is CDF of t-distribution function and
rm
( r  m )
)
2
1

1 
2
p T (t; r ,  ) 
1  t  Σ t 
r
r

(r ) m / 2 ( ) | Σ |1 / 2 
2
(
14
Different “correlation matrices”
1

0
I  
...

0

 AR
 1


 2

3
 4

0 ... 0 

1 ... 0 
... ... ...

0 ... 1  5 X 5

1

2
3
2

1

2
3
2

1

 EX
1


 





1





1





1







1  5 X 5
4 

3 
 2 


1 5 X 5
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Maximum Likelihood Estimation
• Parameters to be estimated:
– of copula:  in correlation matrix Σ and
degrees of freedom r
– of marginal distribution, e.g. shape and scale
parameters for Gamma
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MLE Results: WC
t-copula + Gamma margin
parameter
estimate
StdError

0.6443
Shape/mu
p-value
t-copula + lognormal margin
estimate
StdError
p-value
0.09136 <0.0001
0.6634
.0900
<0.0001
10.6546
1.9740
<0.0001
4.1954
0.0455
<0.0001
Scale/sigma
6.6438
1.2528
<0.0001
0.3235
0.0310
<0.0001
DF r
4.2362
0.2704
<0.0001
4.2519
0.2704
<0.0001
AIC
999.77
1000.52
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MLE Results: CMP
t-copula + Gamma margin
t-copula + lognormal margin
parameter
estimate
StdError
p-value
estimate
StdError p-value

0.4339
0.0925
<0.0001
0.4493
.0947
<0.0001
Shape/mu
11.4205
1.6132
<0.0001
3.9882
0.0296
<0.0001
Scale/sigma
4.9811
0.7206
<0.0001
0.3083
0.0222
<0.0001
DF r
4.2524
0.2703
<0.0001
4.2641
0.2703
<0.0001
AIC
979.07
981.18
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Simulation and Analysis
• Based on the multivariate distribution of the loss ratio
for business lines (WC, CMP separately) for the
primary insurer
• Simulate the multivariate variables( xi,T 1,, xi,T t ) and
( yi ,T 1 ,, yi ,T t )
• The overall loss across two lines over three years is
3
Total loss   ( PW ,t X T t  PC ,t YT t )
t 1
• Where P denotes the annual premium
• Payment on the reinsurance policy after deductible D
 3

max   ( PW ,t X T t  PC ,tYT t )  D,0 
 t 1

19
Histogram of Total Loss Using Different
Assumptions
20
VaR and CTE of Total Loss (in millions)
Using Different Assumptions
• Of 10,000 simulations of Total
Loss
• Based on temporal
independent loss ratios 196
are greater than the threshold;
the reinsurer expects claims at
a frequency of one in about
fifty years, with average claims
of $24.50 million.
• Based on copula dependence
the frequency of claims is
about 5% (495 of 10,000), or
one in twenty years, and the
average claims $41.71 million.
VaR and CTE of Total Loss (in millions)
Using Different Assumptions
Copula dependence
Percentag
e (%)
99.5
99
95
90
Independence
VaR
CTE
VaR
CTE
698.080
660.840
595.420
563.536
732.394
704.613
637.094
607.559
631.948
610.872
568.998
545.428
655.245
637.249
595.911
576.016
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Remarks
• Copulas
– can use information developed over time to
better fit the multi-year claims experience
– Can use information from similar risk classes
22
Questions and
comments?
Thank You!
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