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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

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Agenda Today • Multi-year, multi-line reinsurance

• A Framework Using Copulas to model time dependence

• Application using real data

• Concluding remarks

• Q & A

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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

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1,, )(

ttTtCtTtW YPXPlossTotal

),,( ,1, tTiTi xx ),,( ,1, tTiTi yy

DYPXP

ttTtCtTtW ,)(max

3

1,,

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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

Percentage (%)

VaR CTE VaR CTE

99.5 698.080

732.394

631.948

655.245

99 660.840

704.613

610.872

637.249

95 595.420

637.094

568.998

595.911

90 563.536

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?