Actuarial and statistical aspectsof reinsurance in R

Tom Reynkens Jan Beirlant Roel Verbelen

useR!2017, Brussels

Reinsurance

Insurance companies want to cover themselves against large losses.

Reinsurance: reinsurer covers part of the insurance risk of the client(which is typically an insurance company).

⇒ Part of the risk of the insurance company is transferred to the reinsurer.

Premium paid by the client in return.

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 2/

Excess-loss reinsurance

Reinsurer covers the client’s losses above a certain retention level R.

Moral hazard ⇒ limits.

R

Insurance XL reinsurance

InsurerReinsurer

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 3/

Excess-loss reinsurance

Reinsurer covers the client’s losses above a certain retention level R.

Moral hazard ⇒ limits.

R

R+L

Insurance XL reinsurance XL reinsurance with limit

InsurerReinsurer

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 3/

Extreme Value Theory

Large insurance claims can be a threat to the solvency of the company.

Loss models to

set suitable premiumscalculate risk measuresdetermine capital requirements for solvency regulations.

Company needs to remain solvent, even in the case of catastrophes.

Reinsurance!

Extreme Value Theory (EVT) models rare events with a large impact.

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 4/

EVT in R

(Main) R packages related to EVT:

actuar (Dutang et al., 2008)evir (Pfaff and McNeil, 2012)fExtremes (Wurtz and Rmetrics Association, 2013)QRM (Pfaff and McNeil, 2016)

CRAN task view “Extreme Value Analysis”.

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 5/

ReIns package

ReIns package (Reynkens and Verbelen, 2017)

Basic extreme value theory (EVT) estimatorsand graphical methods (Beirlant et al.,2004).

EVT estimators and graphical methodsadapted for censored and/or truncated data.

Splicing models.

Risk measures: Value-at-Risk (VaR),Conditional Tail Expectation (CTE) andexcess-loss premium estimates.

+ Unified framework for all estimators andplots.

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 6/

Splicing

Global fits are needed for a risk analysis that does not only focus onextreme events.

Different behaviour of small and large claims⇒ combine two distributions in a splicing model.

Body Tail

Exponential Pareto Beirlant et al. (2004); Klug-man et al. (2012)

Log-normal Pareto Cooray and Ananda (2005);Scollnik (2007); Pigeon andDenuit (2011)

Weibull Pareto Ciumara (2006); Scollnikand Sun (2012)

......

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 7/

Splicing of ME and Pareto

Mixture of Erlangs (ME) with common scale parameter.

+ Class of ME distributions with common scale parameter is dense in thespace of positive continuous distributions (Tijms, 1994).

− ME distribution has an asymptotically exponential tail (Neuts, 1981).

Splicing of ME and Pareto distributions (Reynkens et al., 2017).

⇒ Flexibility of ME distribution to model the body.

⇒ Pareto distribution provides a suitable tail fit.

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MTPL insurance

Motor third party liability (MTPL) insurance in Europe between 1995and 2005, see Albrecher et al. (2017).

596 claims, 45% not closed in 2011.

0

200,000

400,000

600,000

800,000

0

200,000

400,000

600,000

800,000

1995 2000 2005 2010 1995 2000 2005 2010Year

Figure: Cumulative indexed payments (full) and indexed incurreds (dashed).

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Fitting a general splicing model to censored data

tl t T

i.xi

ii.xi

iii.li uixi

iv.li uixi

v.li uixi

li uixi

Observed data point

Unobserved data point

Direct likelihood optimisation is not ideal.

Fit spliced distribution using expectation maximisation (EM)algorithm.

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MTPL: fitted model

S3 class containing details on fitted object: SpliceFit.

Several plots to assess goodness-of-fit.

SpliceTB(x, L, U, censored, splicefit).

0 500000 1000000 1500000 2000000 2500000

0.0

0.2

0.4

0.6

0.8

1.0

x

1−F

(x)

Fitted survival functionTurnbull estimator95% confidence intervals

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 11/

MTPL: fitted model

S3 class containing details on fitted object: SpliceFit.

Several plots to assess goodness-of-fit.

SplicePP TB(L, U, censored, splicefit).

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Splicing PP−plot

Turnbull survival probability

Fitt

ed s

urvi

val p

roba

bilit

y

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 11/

MTPL: fitted model

S3 class containing details on fitted object: SpliceFit.

Several plots to assess goodness-of-fit.

SpliceQQ TB(L, U, censored, splicefit).

0 500000 1000000 1500000 2000000 2500000 3000000

050

0000

1000

000

1500

000

2000

000

Splicing QQ−plot

Quantiles of splicing fit

Turn

bull

quan

tiles

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 11/

MTPL: risk measures

Excess-loss premium: E((X −R)+)

ExcessSplice(R, L, splicefit)

0 500000 1000000 1500000 2000000 2500000 3000000

050

000

1500

0025

0000

R

Exc

ess−

loss

pre

miu

m

No limitL=2R

Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 12/

MTPL: risk measures

Value-at-Risk: VaR1−p = F−1(1− p)

VaR(p, splicefit)

0.0 0.2 0.4 0.6 0.8 1.0

050

0000

1000

000

1500

000

p

VaR

1−p

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Conclusions

Extreme value theory in R.

Focus on EVT for censored and/or truncated data.

Applications to (re)insurance: global models, risk measures, etc.

ReIns package available at CRAN and GitHub:https://github.com/TReynkens/ReIns.

Introduction to the package in vignette on CRAN.

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References I

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and StatisticalAspects. John Wiley & Sons, Ltd, Chichester, UK. To appear.

Beirlant, J., Fraga Alves, I. and Reynkens, T. (2017). Fitting Tails Affected by Truncation.Electron. J. Stat., 11(1), 2026–2065.

Beirlant, J., Fraga Alves, M. I. and Gomes, M. I. (2016). Tail Fitting for Truncated andNon-Truncated Pareto-Type Distributions. Extremes, 19(3), 429–462.

Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes:Theory and Applications. Wiley, Chichester.

Ciumara, R. (2006). An Actuarial Model Based on the Composite Weibull-ParetoDistribution. Math. Rep. (Bucur.), 8(4), 401–414.

Cooray, K. and Ananda, M. M. (2005). Modeling Actuarial Data With a CompositeLognormal-Pareto Model. Scand. Actuar. J., 2005(5), 321–334.

Dutang, C., Goulet, V. and Pigeon, M. (2008). actuar: An R package for Actuarial Science.J. Stat. Softw., 25(7), 1–37.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012). Loss Models: From Data toDecisions. John Wiley & Sons, Inc., Hoboken, NJ. 4th edition.

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References II

Neuts, M. F. (1981). Matrix-geometric Solutions in Stochastic Models: An AlgorithmicApproach. John Hopkins University Press, Baltimore, MD.

Pfaff, B. and McNeil, A. (2012). evir: Extreme Values in R. url:https://CRAN.R-project.org/package=evir, R package version 1.7-3.

Pfaff, B. and McNeil, A. (2016). QRM: Provides R-Language Code to ExamineQuantitative Risk Management Concepts. url:https://CRAN.R-project.org/package=QRM, R package version 0.4-13.

Pigeon, M. and Denuit, M. (2011). Composite Lognormal-Pareto Model With RandomThreshold. Scand. Actuar. J., 2011(3), 177–192.

Reynkens, T. and Verbelen, R. (2017). ReIns: Functions from “Reinsurance: Actuarial andStatistical Aspects”. url: https://CRAN.R-project.org/package=ReIns, R packageversion 1.0.4.

Reynkens, T., Verbelen, R., Beirlant, J. and Antonio, K. (2017). Modelling CensoredLosses Using Splicing: a Global Fit Strategy With Mixed Erlang and Extreme ValueDistributions, available on arXiv:1608.01566.

Scollnik, D. P. M. (2007). On Composite Lognormal-Pareto Models. Scand. Actuar. J.,2007(1), 20–33.

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References III

Scollnik, D. P. M. and Sun, C. (2012). Modeling With Weibull-Pareto Models. N. Am.Actuar. J., 16(2), 260–272.

Tijms, H. C. (1994). Stochastic Models: an Algorithmic Approach. John Wiley & Sons,Ltd, Chichester, UK.

Wurtz, D. and Rmetrics Association (2013). fExtremes: Rmetrics - Extreme FinancialMarket Data. url: https://CRAN.R-project.org/package=fExtremes, R packageversion 3010.81.

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Truncation

tl T

Uncensoredxi

Lower truncatedxi

Upper truncatedxi

Observed data point

Unobserved data point

Realisations of X are observed with

X =d Y | tl < Y < T.

Lower truncation: e.g. deductible in insurance.

Upper truncation: e.g. earthquake magnitudes.

Upper endpoint T often unknown ⇒ needs to be estimated.Actuarial and statistical aspects of reinsurance in R – Tom Reynkens 17/

Groningen

Gas extraction induces earthquakes in Groningen (the Netherlands).

Estimate maximum possible earthquake magnitude.

⇒ Worst-case damage estimates.

EVT methods of Beirlant et al. (2016); Beirlant et al. (2017).

0 50 100 150 200 250

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

k

End

poin

t

Truncated Pareto Truncated GPD

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