1. RISKGLOBALX.PLC
RiskGlobalX.plc Marine Reinsurance
Claims Report
Modelling and Pricing Strategy Review
M.Zhang
6/17/2015
This report reviews the copula model used for correlation modelling for marine reinsurance claims
between 2 areas and analysed the impact on premiums and risk adjusted returns for RiskGlobalX.plc.
2. 1
Contents Spread Sheet Page in Report
1. Overview Page 1
2. Methods Page 1
3. Fitting Distribution 'Area 1', 'Area 2', 'Comparison Plot' Page 2-3
4. Fitting Copula for Gaussian,
Gumbel and Clayton 'Fitting Copula'
Page 4
5. Method of Moment Check 'MoM check' Page 4
6. Simulation for Gaussian and
Clayton Copula 'Simulation'
Page 5-6
7. Premium Calculation 'Premium', ‘Simulated Claim Sizes’ Page 7
8. Conclusion Page 8
9. Appendix Page 9-10
1. Overview
This report analyses past 20 years’ claims made by the 10 marine insurers in 2 geographical
areas and the correlation between them with copula models.
Currently RiskGlobalX is using Gaussian copula for analysing the correlations between claim
sizes in the 2 areas. In this project, data are fitted to alternative copula models and tested to
see whether there is a better model for analysing the correlation and how the new model
effects the economic capital needed and premium charged by RiskGlobalX.
The main finding of this project is that Clayton is a better model for modelling the claim sizes.
The simulation results from Clayton copula show that we would be able to charge a lower
premium or achieve a higher risk adjusted return with the same level of premium. Therefore
we are able to reduce the economic capital we hold for writing these policies.
2. Methods
Steps taken:
Fitting the marginal distribution for each set of data.
Select best fit marginal distribution based on maximum likelihood estimation (MLE).
Fit three copula models, Clayton, Gaussian, and Gumbel to the data for analysing the
correlation between claims of the 2 areas.
The most suitable copula would be chosen based on MLE.
50,000 simulations performed for calculating required economic capital.
Premiums that gives 15% RAROC based on the new model would be calculated.
3. 2
3. Fitting Distribution
After visual checking on the shapes of the histogram of the claim data and comparing them to
the shapes of various distributions, I choose the following four to fit the data:
Gaussian distribution;
Gamma distribution;
Lognormal distribution;
Weibull distribution.
The fitting is assessed by computing and comparing MLE for each distribution (See
Appendix A for main functions used).
Parameters are found by using Excel solver so that they maximise the sum of log-likelihood
function. This would give us the parameters of the marginal distribution that has the highest
probability to give the values from observations we make.
Results summary for Area 1:
Gaussian Gamma LogNormal Weibull
MLE -450.210 -352.134 -365.431 -352.026
Results summary for Area 2:
Gaussian Gamma LogNormal Weibull
MLE -606.439 -588.808 -597.442 -588.788
The fitted distributions are plotted with the original data for comparison as shown below for
visual check of fit.
Comments:
0
10
20
30
40
50
60
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
16.5
17.5
Frequency
claim size
Distribution fitting for Area 1
Annual claims in area 1
Gaussian
Gamma
LogNormal
Weibull
4. 3
Gaussian fails to capture the tail of the claim beyond 14 therefore is not suitable.
Lognormal does not adhere to the actual claim well and gives a fatter tail than the actual
claims.
Both Gamma and Weibull fit the claim data well.
Comments:
Gaussian doesn’t capture the positive skew and fat tail of claim sizes therefore is not a
good fit.
Overall, Lognormal doesn’t adhere to the data well. Its mean is too small and tail is too
fat compared to the actual data.
Both Gamma and Weibull give a good fit to the data by visual inspection.
Results and decision
According to the Results Summary above, Weibull distribution gives the greatest MLE
values among the four distributions for both areas. Therefore it is suitable to be used as
marginal distribution for copula fitting with the following parameter values.
For Weibull (α, β):
Area 1 Area 2
α 0.947 1.847
β 2.091 9.966
These results could be refined by collecting and analysing more claim data.
0
2
4
6
8
10
12
14
0.5
2.5
4.5
6.5
8.5
10.5
12.5
14.5
16.5
18.5
20.5
22.5
24.5
26.5
28.5
Frequency
claim size
Distribution fitting for Area 2
Annual claims in area 2
Gaussian
Gamma
LogNormal
Weibull
5. 4
4. Fitting Copula
A copula model would be fitted to analyse the correlation between claim sizes in 2 areas. The
suitable model would be chosen based on MLE for each copula (Clayton, Gumbel and
Gaussian).
Firstly, cumulative distribution function (CDF) of each set of data (u and v) is calculated
with the marginal distribution and parameters obtained in previous session.
Then for each copula model, copula parameters are found by using Excel solver function so
that the sum of log-likelihood function ln[c(u,v)] is maximised. (See Appendix A for MLE
function and copula density functions for each model.)
Summary of fitted parameter values and MLE:
Clayton Copula Gumbel Copula Gaussian Copula
α 2.803 α 1.832 ρ 0.722
MLE 115.475 MLE 51.719 MLE 72.441
5. Method of Moment (MOM) Check
Parameters of Clayton and Gumbel are then checked by MOM.
Kendall tau Formula α estimated by MOM α estimated by MLE
Clayton 0.554 τ=α/(α+2) 2.480 2.803
Gumbel 0.554 τ=1-1/α 2.240 1.832
Kendall τ for the claim data is obtained from website calculator1
. By comparing the MOM
estimated α to the MLE estimated α, we can see that the values are close therefore results are
reliable. A bigger sample size could be obtained and used for a more accurate result.
Results and decision
Clayton copula gives greater MLE than the current Gaussian copula. Therefore it would be an
improved model for modelling the claim sizes. Gumbel copula gives the smallest MLE
therefore it is inferior to the other 2 models.
Therefore in the next session, simulations will be performed on Clayton copula and Gaussian
copula to further analyse the required economic capital and the impact on premium pricing.
1
http://www.wessa.net/rwasp_kendall.wasp#output
6. 5
6. Simulation for Gaussian and Clayton copula
Steps taken for simulation
Clayton
1. Simulate a Gamma variate X ~ Gamma(
1
𝛼
, 1)
2. Simulate 50000 independent standard uniform random variables sets V1 and V2
3. Return U=(𝑈1, 𝑈2) = [(1 −
𝑙𝑛𝑉1
𝑥
)
1
𝛼
, (1 −
𝑙𝑛𝑉2
𝑥
)
1/𝛼
]
Gaussian
1. Perform a Cholesky-decomposition ∑ 𝑨′𝑨
2. Generate iid standard Normal pseudo random variables 𝑋1
̃, 𝑋2
̃
3. Compute (𝑋1, 𝑋2)′ = 𝑿 = 𝑨𝑿̃ from 𝑿̃ = (𝑋1
̃, 𝑋2
̃ )′
4. Return U=(𝑈1, 𝑈2)=[Φ(X1),Φ(X2)]
Where Φ(X1) is a standard normal cumulative distribution function.
50,000 simulations are performed for each area with each model. For more accurate results,
more simulations can be done in a similar manner.
The copula values are then inversed from joint CDF back to simulated claim sizes with
inverse Weibull distribution function (see Appendix). Total claim sizes for each simulation is
obtained by combing the values for Area1 and Area 2.
Scatter plots are made with 200 simulations as shown below. By visual inspection, we can
see that simulated Clayton copula is a better representation of the original claim sizes
correlation than the Gaussian copula.
The actual claim sizes shows a higher correlation between area 1 and area 2 when claims are
small and no clear correlation when claims are large.
0
5
10
15
20
25
30
35
0 5 10 15 20
Area 2
Area 1
Actual claims
7. 6
Clayton copula captures the correlation between smaller claims and has as similar shape as
the actual claims.
Gaussian copula does not represent the actual claim’s shape as well as Clayton by visual
check.
Results and decision
Premiums should be recalculated with Clayton copula simulation results and compare to
premiums obtained with current Gaussian copula. Further pricing and capital holding
decisions can be taken after we apply this improved model.
0
5
10
15
20
25
30
35
0 5 10 15
Area 2
Area 1
Simulated Clayton Copula
0
5
10
15
20
25
30
0 5 10 15
Area 2
Area 1
Simulated Gaussian Copula
8. 7
7. Premium Calculation
Economic capital is obtained by formula:
𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 = 𝑉𝑎𝑅99.9% − 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑙𝑖𝑎𝑚 𝑠𝑖𝑧𝑒
VaR99.9% is obtained by finding size of the largest 0.1% total 50,000 simulated claims.
Average claim size is obtained by taking the mean of simulations.
Given that the required RAROC is 15% and required return on capital is 2% p.a., premium
can be calculated with equation:
𝑃𝑟𝑒𝑚𝑖𝑢𝑚2 = 𝑅𝐴𝑅𝑂𝐶 ∗ 𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 − 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 + 𝐶𝑙𝑎𝑖𝑚 𝐶𝑜𝑠𝑡
Illustration result table:
These results are taken from the average of 5 simulated economic capital and average claims.
The detailed tables can be found in Appendix B.
Gaussian Clayton Premium unchanged
Premium 15.63 14.29 15.57
Claim cost 10.96 11.00 11.00
Return on Capital 2% 2% 2%
Economic Capital 31.23 22.11 22.42
RAROC 15% 15% 20.48%
From the illustration table, we can see that modelling with Clayton copula lowers economic
capital required by 8.83, and lowered the premiums by 1.43. While maintaining the current
premium of 15.56, we would be able to obtain a higher risk adjusted return (increased by over
6%). This means that if we remain the 15% RAROC requirement, economic capital held
could be reduced.
Due to size limit, the 50,000 simulations are not in the spreadsheets. But it can be reproduced
with functions provided in it.
The simulated results varies slightly for each simulation and the 5 examples are copied and
pasted figures for illustration. The simulated figures for first column of the example tables are
presented in spread sheet ‘simulated claim size’ for verification.
2
See Appendix A for equation for RAROC.
9. 8
8. Conclusion
MLE values and scatter plot shows that Clayton copula is better than Gaussian copula. With
the Clayton model, a lower premium rate can be charged for the claims and we would be able
to hold less economic capital for writing these policies.
If we charge premiums according to Gaussian copula (as current premium level), we can
obtain a higher RAROC with the improved model, which means we would be able to hold
lower capital level than previously calculated.
Further analysis could be made with more data and larger number of simulations before
further actions being taken regards to premiums or capital reserving.
10. 9
Appendix:
Appendix A: Main formulas used
Likelihood function:
𝐿 = ∏ 𝑓(𝑥𝑡)𝑇
𝑡=1
T is the total number of observations
𝑓(𝑥𝑡) is the probability density function of marginal distribution.
Log-likelihood function:
𝑙𝑛𝐿 = ∑ 𝑙𝑛𝑓(𝑥𝑡)𝑇
𝑡=1
Likelihood function for copula:
𝐿 = ∏ 𝑐(𝐹(𝑥1,𝑡), 𝐹(𝑥2,𝑡), . . , 𝐹(𝑥 𝑁,𝑡))𝑇
𝑡=1
𝑐(𝐹(𝑥1,𝑡), 𝐹(𝑥2,𝑡), . . , 𝐹(𝑥 𝑁,𝑡)) is the copula density function
Clayton copula density function:
c(u, v) = (1 + 𝛼) ∗ (𝑢𝑣)−1−𝛼
∗ (𝑢−𝛼
+ 𝑣−𝛼
− 1)−
1
𝛼
−2
Gaussian copula density function:
𝑐(𝑢) =
1
(2𝜋)
𝑑
2| 𝑅|
1
2
𝑒
−
1
2
𝑥′ 𝑅−1 𝑥
∏
1
(2𝜋)
1
2
𝑑
𝑗=1 𝑒
−
1
2
𝑥 𝑗
2
=
𝑒
−
1
2
𝑥′ 𝑅−1 𝑥
|𝑅|
1
2 𝑒
−
1
2
∑ 𝑥 𝑗
2𝑑
𝑗=1
𝑐(𝑢) =
1
√1−𝜌2
𝑒
−(𝑥2+𝑦2−2𝜌𝑥𝑦)
2(1−𝜌2)
𝑒
−(𝑥2+𝑦2)
2
Gumbel copula density function:
𝑐(𝑢, 𝑣) = 𝐶(𝑢, 𝑣) ∗ (𝑢𝑣)−1−𝛼
∗ [(−𝑙𝑛𝑢) 𝛼
+ (−𝑙𝑛𝑣) 𝛼]−2+
2
𝛼 ∗ (𝑙𝑛𝑢 ∗ 𝑙𝑛𝑣) 𝛼−1
∗ {1 + (𝛼 − 1)[(−𝑙𝑛𝑢) 𝛼
+ (−𝑙𝑛𝑣) 𝛼
]−1/𝛼
}
With u, v being the CDF of each set of the data.
Inverse Weibull function:
𝑥 = 𝛽 (ln (
1
1 − 𝑢
))
1/𝛼
