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

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

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

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

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

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

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

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

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.

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.

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.

Relationship between WC & CMP Correlation coefficient: 0.1510

Fitted Marginal Distribution WC loss ratio CMP loss ratio Distribution AIC K-S stat* 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

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

Different “correlation matrices”

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

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 0.09136 <0.0001 0.6634 .0900 Shape/mu 10.6546 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 1000.52 Exchange structure scores the best fit

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 11.4205 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.

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

Histogram of Total Loss Using Different Assumptions

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 99.5 698.080 732.394 631.948 655.245 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

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

Questions and comments? Thank You!