Bootstrapping the triangles

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Bootstrapping the triangles

Bootstrapping the trianglesPrečo, kedy a ako (NE)bootstrapovat’ v trojuholníkoch?Michal PeštaCharles University in PragueFaculty of Mathematics and PhysicsActuarial Seminar, Prague19 October 2012


Overview- Idea and goal of bootstrapping- Bootstrap in reserving triangles- Residuals- Diagnostics- Real data example• Support:Czech Science Foundation project “DYME DynamicModels in Economics” No. P402/12/G097


Prologue forBootstrap- computationally intensive method popularized in1980s due to the introduction of computers instatistical practice- a strong mathematical background bootstrapdoes not replace or add to the original data- unfortunately, the name “bootstrap” conveys theimpression of “something for nothing” idlyresampling from their samples


Reserving Issue- consider traditional actuarial approach to reservingrisk . . . the uncertainty in the outcomes over thelifetime of the liabilities- bootstrap can be also applied under Solvency II. . . outstanding liabilities after 1 year- distribution-free methods (e.g., chain ladder) onlyprovide a standard deviation of theultimates/reserves (or claims developmentresult/run-off result)? another risk measure (e.g., V aR @ 99.5%) moreover, distributions of ultimate cost of claimsand the associated cash flows (not just a standarddeviation)?! claims reserving techniqueapplied mechanically and without judgement


Bootstrap- simple (distribution-independent) resamplingmethod- estimate properties (distribution) of an estimatorby sampling from an approximating (e.g., empirical)distribution- useful when the theoretical distribution ofa statistic of interest is complicated or unknown


Resampling withreplacement- random sampling with replacement from theoriginal dataset for b = 1, . . . , B resample fromX 1 , . . . , X n with replacement and obtain X ∗ 1,b , . . . , X∗ n,b


Bootstrap example- input data (# of catastrophic claims per year in 10yhistory): 35, 34, 13, 33, 27, 30, 19, 31, 10, 33 mean = 26.5, sd = 9.168182Case sampling◮.◮bootstrap sample 1 (1st draw with replacement):30, 27, 35, 35, 13, 35, 33, 34, 35, 33 mean ∗ 1 = 31.0, sd ∗ 1 = 6.847546bootstrap sample 1000 (1000th draw withreplacement): 19, 19, 31, 19, 33, 34, 31, 34, 34, 10 mean ∗ 1000 = 26.4, sd ∗ 1000 = 8.771165- mean ∗ 1 , . . . , mean∗ 1000 provide bootstrap empiricaldistribution for mean and sd ∗ 1 , . . . , sd∗ 1000 providebootstrap empirical distribution for sd (REALLY!?)


Terminology- X i,j . . . claim amounts in development year j withaccident year i- X i,j stands for the incremental claims in accidentyear i made in accounting year i + j- n . . . current year – corresponds to the mostrecent accident year and development period- Our data history consists ofright-angled isosceles triangles X i,j , wherei = 1, . . . , n and j = 1, . . . , n + 1 − i


Run-off (incremental)triangleAccidentDevelopment year jyear i 1 2 · · · n − 1 n1 X 1,1 X 1,2 · · · X 1,n−1 X 1,n2 X 2,1 X 2,2 · · · X 2,n−1. ..... X i,n+1−in − 1 X n−1,1 X n−1,2n X n,1


Notation- C i,j . . . cumulative payments in origin year i after jdevelopment periodsC i,j =j∑k=1X i,k- C i,j . . . a random variable of which we have anobservation if i + j ≤ n + 1- Aim is to estimate the ultimate claims amount C i,nand the outstanding claims reserveR i = C i,n − C i,n+1−i ,i = 2, . . . , nby completing the triangle into a square


Run-off (cumulative)triangleAccidentDevelopment year jyear i 1 2 · · · n − 1 n1 C 1,1 C 1,2 · · · C 1,n−1 C 1,n2 C 2,1 C 2,2 · · · C 2,n−1. ..... C i,n+1−in − 1 C n−1,1 C n−1,2n C n,1


Theory behind thebootstrap- validity of bootstrap procedure- asymptotically distributionally coincidêR ∗ i − ̂R i∣ ∣∣{Xi,j: i + j ≤ n + 1} and ̂Ri − R i- approaching (each other) in distributionin probability along D (n)n = {X i,j : i + j ≤ n + 1}̂R ∗ i − ̂R i∣ ∣∣D (n)nD←→ ̂R i − R i in probability, n → ∞◮∀ real-valued bounded continuous function f[ (E f ̂R∗ i − ̂R) ∣ ] [ ( )]∣∣D (n)Pi n − E f ̂Ri − R i −−−→ 0n→∞


Raw residuals(R)r i,j = X i,j − ̂X i,j or (R) r i,j = C i,j − Ĉi,j- in homoscedastic linear regression, CL with α = 0,GLM with normal distribution and identity link


Pearson residuals(P )r i,j = X i,j − ̂X i,j√V ( ̂X i,j )- in GLM or GEE- idea is to standardize raw residuals- ODP:(P )r i,j = X i,j − ̂X i,j√̂Xi,j- Gamma GLM:(P )r i,j = X i,j − ̂X i,ĵX i,j


Anscombe residuals I- in GLM or GEE(A)r i,j = A(X i,j) − A( ̂X i,j )A ′ ( ̂X√i,j ) V ( ̂X i,j )- disadvantage of Pearson residuals: often markedlyskewed- idea is to obtain residuals, which are not skewed(to "normalize" residuals)


Anscombe residuals II- ODP:- Gamma GLM:- inverse Gaussian:∫A(·) =(A)r i,j = 3 2dµV 1/3 (µ)X 2/3i,j−(A)r i,j = 3 X1/3̂X 1/6i,ĵX2/3i,j1/3i,j− ̂Xi,ĵX 1/3i,j(A)r i,j = log X i,j − log ̂X i,ĵX 1/2i,j


Deviance residuals(D)r i,j = sign(X i,j − ̂X i,j ) √ d i ,where d i is the deviance for one unit, i.e., ∑ d i = D- in GLM or GEE- similar advantageous properties like Anscomberesiduals (see Taylor expansion)- ODP:√(D)r i,j = sign(X i,j − ̂X i,j ) 2(X i,j log(X i,j / ̂X i,j ) − X i,j + ̂X)i,j


Scaled residuals- scaling parameter φ(sc) r i,j = r i,j√̂φ- to obtain the bootstrap prediction error, it isnecessary to add an estimate of the processvariance . . . bias correction in the bootstrapestimation variance- Pearson scaled residuals in ODP for thebootstrap prediction error:(sc)(P ) r′ i,j = (P ) r i,j ×( ∑i,j≤n+1 (P )ri,j2 ) −1/2n(n + 1)/2 − (2n + 1)


Adjusted residuals- alternative to scaled residuals . . . bias correction inthe bootstrap estimation variance(adj) r i,j = r i,j ×√n − pn


Diagnostics forresiduals- residuals and squared residuals- type of plots:◮ accident years◮ accounting years◮ development years- no pattern visible- iid with zero mean, symmetrically distributed,common variance


Bootstrap procedure I- Ex: bootstrap in ODP(1) fit a model estimates ̂α i , ̂β j , ̂γ, ̂φ(2) fitted (expected) values for trianglêX i,j = exp{̂γ + ̂α i + ̂β j }(3) scaled Pearson residuals(sc)(P ) r i,j = X i,j − ̂X i,j√̂φ ̂Xi,j


Bootstrap procedure II{ } (sc)(4) resample residuals(P ) r i,j B-times withreplacement B}triangles of bootstrappedresiduals{ (sc)(P,b) r∗ i,j, 1 ≤ b ≤ B(5) construct B bootstrap triangles(b)X ∗ i,j = (sc)(P,b) r∗ i,j√̂φ ̂X i,j + ̂X i,j(6) perform ODP on each bootstrap triangle bootstrapped estimates (b)̂α i , (b) ̂βj , (b)̂γ, (b) ̂φ


Bootstrap procedure III(7) calculate bootstrap reserves(b) ̂R ∗ i =n∑j=n+2−i(b) ̂X ∗ i,j = exp{ (b)̂γ + (b)̂α i }n∑j=n+2−i◮ (4)–(7) is a bootstrap loop (repeated B-times)exp{ (b) ̂βj }(8) empirical distribution of size B for the reserves empirical (estimated) mean, s.e., quantiles, . . .


Taylor and Ashe (1983)data- incremental triangle357 848 766 940 610 542 482 940 527 326 574 398 146 342 139 950352 118 884 021 933 894 1 183 289 445 745 320 996 527 804 266 172290 507 1 001 799 926 219 1 016 654 750 816 146 923 495 992 280 405310 608 1 108 250 776 189 1 562 400 272 482 352 053 206 286443 160 693 190 991 983 769 488 504 851 470 639396 132 937 085 847 498 805 037 705 960440 832 847 631 1 131 398 1 063 269359 480 1 061 648 1 443 370376 686 986 608344 014227 229 67 948425 046- R software, ChainLadder package


Development of claimsClaims1e+06 3e+06 5e+06123456789012346789587234561427356134265142351234123121 12 4 6 8 10Development period


CL with Mack’s s.e.8e+066e+061Chain ladder developments by origin periodChain ladder dev.Mack's S.E.2 4 6 8 102 4 6 8 102 34 54e+062e+06Amount0e+006 78 9108e+066e+064e+062e+060e+002 4 6 8 102 4 6 8 102 4 6 8 10Development period


Chain ladder diagnosticsValue0e+00 2e+06 4e+06 6e+06Mack Chain Ladder ResultsForecast●Latest●●● ● ●●●●●1 2 3 4 5 6 7 8 9 10AmountChain ladder developments by origin period1e+06 3e+06 5e+06 7e+06223234434234 2 51 1611 127358 561723456 11123465789Standardised residuals−1 0 1 2●●●●●●● ●●● ●● ●● ●●●●●●●● ●●●●● ●●● ● ●●● ●●● ●●●●●1234567890●2 4 6 8 101e+06 2e+06 3e+06 4e+06 5e+06Origin periodDevelopment periodFittedStandardised residuals−1 0 1 2●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●Standardised residuals−1 0 1 2●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●Standardised residuals−1 0 1 2●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2 4 6 8Origin period2 4 6 8Calendar period1 2 3 4 5 6 7 8Development period


Bootstrap resultsHistogram of Total.IBNRecdf(Total.IBNR)Frequency0 50 150 250Fn(x)0.0 0.4 0.81.0e+07 2.0e+07 3.0e+071.0e+07 2.0e+07 3.0e+07Total IBNRTotal IBNRultimate claims costs5.0e+06 1.5e+07●Simulated ultimate claims costMean ultimate claim●● ●●● ●● ●●●●●●●●●●●●●●●●●latest incremental claims0 1000000 2000000●Latest actual incremental claimsagainst simulated values● Latest actual●●●●●●●●●●●●●●●●●1 2 3 4 5 6 7 8 9 10origin period1 2 3 4 5 6 7 8 9 10origin period


Mack CL vs bootstrapGLM (ODP)Accident Chain Ladder Bootstrapyear Ultimate IBNR S.E. Ultimate IBNR S.E.1 3 901 463 0 0 3 901 463 0 02 5 433 719 94 634 75 535 5 434 680 95 595 106 3133 5 378 826 469 511 121 699 5 396 815 487 500 222 0014 5 297 906 709 638 133 549 5 315 089 726 821 265 6965 4 858 200 984 889 261 406 4 875 837 1 002 526 313 0156 5 111 171 1 419 459 411 010 5 113 745 1 422 033 377 7037 5 660 771 2 177 641 558 317 5 686 423 2 203 293 487 8918 6 784 799 3 920 301 875 328 6 790 462 3 925 964 789 3299 5 642 266 4 278 972 971 258 5 675 167 4 311 873 1 034 46510 4 969 825 4 625 811 1 363 155 5 148 456 4 804 442 2 091 629Total 53 038 946 18 680 856 2 447 095 53 338 139 18 980 049 3 096 767


Comparison ofdistributional properties- why to bootstrap ?- moment characteristics (mean, s.e., . . . ) does notprovide full information about the reserves’distribution- additional assumption required in the classicalapproach- 99.5% quantile necessary for VaR◮ assuming normally distributed reserves. . . 24 984 154◮ assuming log-normally distributed reserves. . . 25 919 050◮ bootstrap . . . 28 201 572


Conclusions- mean and variance do not contain full informationabout the distribution cannot providequantities like VaR- assumption of log-normally distributed claims log-normally distributed reserves (far morerestrictive)- various types of residuals- bootstrap (simulated) distributionmimics the unknown distribution of reserves(a mathematical proof necessary)- R software provides a free sufficient actuarialenvironment for reserving


ReferencesEngland, P. D. and Verrall, R. J. (1999)Analytic and bootstrap estimates of predictionerrors in claims reserving.Insurance: Mathematics and Economics, 25.England, P. D. and Verrall, R. J. (2006)Predictive distributions of outstanding claims ingeneral insurance.Annals of Actuarial Science, 1 (2).Pesta, M. (2012)Total least squares and bootstrapping withapplication in calibration.Statistics: A Journal of Theoretical and AppliedStatistics.


Thank you !pesta@karlin.mff.cuni.cz

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