1. 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

2. Agenda Today Multi-year, multi-line reinsurance
A Framework Using Copulas to model time dependence
Application using real data
Concluding remarks
Q & A

3. 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

4. Difficulty Facing Actuaries
Simultaneous modeling dependence
Across time, and
Across business lines (e.g., workers compensation and commercial multiple perils)

5. 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

6. Marginal Distribution
Suppose that there are Ti years data for a business line of the ith primary insurer
Univariate marginal distribution functions
Fit with Gamma, normal, lognormal, t-dist’n
Regression framework

7. Modeling Time Dependencies Using Copulas
With Copula C, the joint distribution function of Yi can be expressed as
The log-likelihood of ith primary insurer is
where c(.) is the probability density function corresponding to the copula function
Predictive distribution is obtained based on the results of maximum likelihood estimation

8. 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

9. 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.

10. Correlations across Time: WC
Loss ratios among years are not independent.
WC04
WC03
WC02
WC01
WC00
.6483
(<.0001)
.6640
.4611
(.0079)
.6128
(.0002)
.6586
.3132
(.0809)
.3398
(.0571)
.6144
.3796
(.0321)
.5617
(.0008)
Reported are the value of Pearson correlations and corresponding p-values.

11. Correlations across Time: CMP
.4771
(.0058)
.3327
(.0628)
.3200
(.0742)
.3661
(.0394)
.4999
(.0036)
.1510
(.4093)
.1225
(.5041)
.4212
(.0164)
.2571
(.1554)
.3589
(.0437)
Reported are the value of Pearson correlations and corresponding p-values.

12. Relationship between WC & CMP
Correlation coefficient:

13. Fitted Marginal Distribution
WC loss ratio
CMP loss ratio
Distribution
AIC
K-S stat*
K-S stat
Lognormal
0.0383
0.0538
Gamma
0.0399
0.0709
t-dist’n
0.2707
0.2561
*: kolmogorov-Smirnov test statistic

14. t-copula
t-copula:
where Gr is CDF of t-distribution function and

15. Different “correlation matrices”

16. 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

17. t-copula + Gamma margin t-copula + lognormal margin
MLE Results: WC
t-copula + Gamma margin
t-copula + lognormal margin
parameter
estimate
StdError
p-value 
0.6443
<0.0001
0.6634
.0900
Shape/mu
1.9740
4.1954
0.0455
Scale/sigma
6.6438
1.2528
0.3235
0.0310
DF r
4.2362
0.2704
4.2519
AIC
999.77
Exchange structure scores the best fit

18. t-copula + Gamma margin t-copula + lognormal margin
MLE Results: CMP
t-copula + Gamma margin
t-copula + lognormal margin
parameter
estimate
StdError
p-value 
0.4339
0.0925
<0.0001
0.4493
.0947
Shape/mu
1.6132
3.9882
0.0296
Scale/sigma
4.9811
0.7206
0.3083
0.0222
DF r
4.2524
0.2703
4.2641
AIC
979.07
981.18
AR(1) structure fits best.

19. 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 and
The overall loss across two lines over three years is
Where P denotes the annual premium
Payment on the reinsurance policy after deductible D

20. Histogram of Total Loss Using Different Assumptions

21. 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
Independence
Percentage (%)
VaR
CTE
99.5 99 95 90

22. Remarks
Copulas
can use information developed over time to better fit the multi-year claims experience
Can use information from similar risk classes
Potentially useful in developing experience rating methods for long-tailed distributions

23. Questions and comments?
Thank You!