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R isk Modeling of Multi-year , Multi-line Reinsurance Using Copulas. by Ping Wang St John’s University, New York on CICIRM 2011 at Beijing, China. Agenda Today. Multi-year, multi-line reinsurance A Framework Using Copulas to model time dependence Application using real data - PowerPoint PPT Presentation
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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-linereinsurance 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
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Difficulty Facing Actuaries
• Simultaneous modeling dependence – Across time, and– Across business lines (e.g., workers
compensation and commercial multiple perils)
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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
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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
iiTiiii
YYYY ,,,,321Y
ititititititit yPyYy ,PProbP
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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
ii iTiiTii PPyy ,,C,,P 11
i
i
iTii
T
tititi PPPθyl ,,,cln),p(ln 21
1
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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
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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.
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Correlations across Time: WC• Loss ratios among years are not independent.
WC04 WC03 WC02 WC01 WC00
WC04 .6483(<.0001)
.6640(<.0001)
.4611(.0079)
.6128(.0002)
WC03 .6586(<.0001)
.3132(.0809)
.3398(.0571)
WC02 .6144(.0002)
.3796(.0321)
WC01 .5617(.0008)
Reported are the value of Pearson correlations and corresponding p-values.
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Correlations across Time: CMP
CMP04 CMP03 CMP02 CMP01 CMP00
CMP04 .4771(.0058)
.3327(.0628)
.3200(.0742)
.3661(.0394)
CMP03 .4999(.0036)
.1510(.4093)
.1225(.5041)
CMP02 .4212(.0164)
.2571(.1554)
CMP01 .3589(.0437)
Reported are the value of Pearson correlations and corresponding p-values.
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Relationship between WC & CMP
• Correlation coefficient: 0.1510
Scatter plot of WC vs CMP loss ratio
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120CMP loss ratio
WC
loss
ratio
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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
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t-copula • t-copula:
• where Gr is CDF of t-distribution function and
m
i irrmrrm ug
uupuu1
1-1-
11-
1 ))(G(1)(G),...,(G),,(c T
2)(
1
2/12/
11||)
2()(
)2
(),;(
mr
m rrr
mr
rp
tΣtΣ
tT
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Different “correlation matrices”
55234
23
22
32
432
11
11
1
X
AR
5511
11
1
X
EX
551...00............0...100...01
X
I
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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: WCt-copula + Gamma margin t-copula + lognormal margin
parameter estimate StdError p-value estimate StdError p-value
0.6443 0.09136 <0.0001 0.6634 .0900 <0.0001
Shape/mu 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 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
3
1,, )(
ttTtCtTtW YPXPlossTotal
),,( ,1, tTiTi xx ),,( ,1, tTiTi yy
0,)(max
3
1,, DYPXP
ttTtCtTtW
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Histogram of Total Loss Using Different Assumptions
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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 VaR CTE
99.5 698.080 732.394 631.948 655.24599 660.840 704.613 610.872 637.24995 595.420 637.094 568.998 595.91190 563.536 607.559 545.428 576.016
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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
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Thank You!
Questions and comments?
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