Bootstrapping the triangles

Transcription

1 Bootstrapping the triangles Prečo, kedy a ako (NE)bootstrapovat v trojuholníkoch? Michal Pešta Charles University in Prague Faculty of Mathematics and Physics Actuarial Seminar, Prague 19 October 01

2 Overview - Idea and goal of bootstrapping - Bootstrap in reserving triangles - Residuals - Diagnostics - Real data example Support: Czech Science Foundation project DYME Dynamic Models in Economics No. P40/1/G097

3 Prologue for Bootstrap - computationally intensive method popularized in 1980s due to the introduction of computers in statistical practice - a strong mathematical background bootstrap does not replace or add to the original data - unfortunately, the name bootstrap conveys the impression of something for nothing idly resampling from their samples

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

5 Bootstrap - simple (distribution-independent) resampling method - estimate properties (distribution) of an estimator by sampling from an approximating (e.g., empirical) distribution - useful when the theoretical distribution of a statistic of interest is complicated or unknown

6 Resampling with replacement - random sampling with replacement from the original dataset for b = 1,..., B resample from X 1,..., X n with replacement and obtain X 1,b,..., X n,b

7 Bootstrap example - input data (# of catastrophic claims per year in 10y history): 35, 34, 13, 33, 7, 30, 19, 31, 10, 33 mean = 6.5, sd = Case sampling. bootstrap sample 1 (1st draw with replacement): 30, 7, 35, 35, 13, 35, 33, 34, 35, 33 mean 1 = 31.0, sd 1 = bootstrap sample 1000 (1000th draw with replacement): 19, 19, 31, 19, 33, 34, 31, 34, 34, 10 mean 1000 = 6.4, sd 1000 = mean 1,..., mean 1000 provide bootstrap empirical distribution for mean and sd 1,..., sd 1000 provide bootstrap empirical distribution for sd (REALLY!?)

8 Terminology - X i,j... claim amounts in development year j with accident year i - X i,j stands for the incremental claims in accident year i made in accounting year i + j - n... current year corresponds to the most recent accident year and development period - Our data history consists of right-angled isosceles triangles X i,j, where i = 1,..., n and j = 1,..., n + 1 i

9 Run-off (incremental) triangle Accident Development year j year i 1 n 1 n 1 X 1,1 X 1, X 1,n 1 X 1,n X,1 X, X,n X i,n+1 i n 1 X n 1,1 X n 1, n X n,1

10 Notation - C i,j... cumulative payments in origin year i after j development periods C i,j = j k=1 X i,k - C i,j... a random variable of which we have an observation if i + j n Aim is to estimate the ultimate claims amount C i,n and the outstanding claims reserve R i = C i,n C i,n+1 i, i =,..., n by completing the triangle into a square

11 Run-off (cumulative) triangle Accident Development year j year i 1 n 1 n 1 C 1,1 C 1, C 1,n 1 C 1,n C,1 C, C,n C i,n+1 i n 1 C n 1,1 C n 1, n C n,1

12 Theory behind the bootstrap - validity of bootstrap procedure - asymptotically distributionally coincide R i R i {Xi,j : i + j n + 1} and Ri R i - approaching (each other) in distribution in probability along D (n) n = {X i,j : i + j n + 1} R i R i D (n) n D R i R i in probability, n real-valued bounded continuous function f [ ( E f R i R ) ] [ ( )] D (n) P i n E f Ri R i 0 n

13 Residuals - measure the discrepancy of fit in a model - can be used to explore the adequacy of fit of a model - may also indicate the presence of anomalous values requiring further investigation - in regression type problems, it is common to bootstrap the residuals, rather than bootstrap the data themselves - should mimic iid r.v. having zero mean, common variance, symmetric distribution

14 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

15 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,j X i,j

16 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 markedly skewed - idea is to obtain residuals, which are not skewed (to "normalize" residuals)

17 Anscombe residuals II - ODP: - Gamma GLM: - inverse Gaussian: A( ) = (A)r i,j = 3 dµ V 1/3 (µ) X /3 i,j (A)r i,j = 3 X1/3 X 1/6 i,j X /3 i,j 1/3 i,j X i,j X 1/3 i,j (A)r i,j = log X i,j log X i,j X 1/ i,j

18 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 Anscombe residuals (see Taylor expansion) - ODP: (D)r i,j = sign(x i,j X i,j ) (X i,j log(x i,j / X i,j ) X i,j + X ) i,j

19 Scaled residuals - scaling parameter φ (sc) r i,j = r i,j φ - to obtain the bootstrap prediction error, it is necessary to add an estimate of the process variance... bias correction in the bootstrap estimation variance - Pearson scaled residuals in ODP for the bootstrap prediction error: (sc) (P ) r i,j = (P ) r i,j ( i,j n+1 (P )ri,j ) 1/ n(n + 1)/ (n + 1)

20 Adjusted residuals - alternative to scaled residuals... bias correction in the bootstrap estimation variance (adj) r i,j = r i,j n p n

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

22 Bootstrap procedure I - Ex: bootstrap in ODP (1) fit a model estimates α i, β j, γ, φ () fitted (expected) values for triangle X 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

23 Bootstrap procedure II { } (sc) (4) resample residuals (P ) r i,j B-times with replacement B} triangles of bootstrapped residuals { (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

24 Bootstrap procedure III (7) calculate bootstrap reserves (b) R i = n j=n+ i (b) X i,j = exp{ (b) γ + (b) α i } n j=n+ 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,...

25 Taylor and Ashe (1983) data - incremental triangle R software, ChainLadder package

26 Development of claims e+06 3e+06 5e+06 Development period Claims

27 CL with Mack s s.e. 8e+06 6e+06 1 Chain ladder developments by origin period Chain ladder dev. Mack's S.E e+06 e+06 Amount 0e e+06 6e+06 4e+06 e+06 0e Development period

28 Chain ladder diagnostics Forecast Latest Mack Chain Ladder Results Origin period Value 0e+00 e+06 4e+06 6e e+06 3e+06 5e+06 7e+06 Chain ladder developments by origin perio Development period Amount e+06 e+06 3e+06 4e+06 5e Fitted Standardised residuals Origin period Standardised residuals Calendar period Standardised residuals Development period Standardised residuals

29 Bootstrap results Histogram of Total.IBNR ecdf(total.ibnr) Frequency Fn(x) e+07.0e e e+07.0e e+07 Total IBNR Total IBNR ultimate claims costs 5.0e e+07 Simulated ultimate claims cost Mean ultimate claim latest incremental claims Latest actual incremental claims against simulated values Latest actual origin period origin period

30 Mack CL vs bootstrap GLM (ODP) Accident Chain Ladder Bootstrap year Ultimate IBNR S.E. Ultimate IBNR S.E Total

31 Comparison of distributional properties - why to bootstrap? - moment characteristics (mean, s.e.,... ) does not provide full information about the reserves distribution - additional assumption required in the classical approach % quantile necessary for VaR assuming normally distributed reserves assuming log-normally distributed reserves bootstrap

32 Conclusions - mean and variance do not contain full information about the distribution cannot provide quantities like VaR - assumption of log-normally distributed claims log-normally distributed reserves (far more restrictive) - various types of residuals - bootstrap (simulated) distribution mimics the unknown distribution of reserves (a mathematical proof necessary) - R software provides a free sufficient actuarial environment for reserving

33 References England, P. D. and Verrall, R. J. (1999) Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, 5. England, P. D. and Verrall, R. J. (006) Predictive distributions of outstanding claims in general insurance. Annals of Actuarial Science, 1 (). Pesta, M. (01) Total least squares and bootstrapping with application in calibration. Statistics: A Journal of Theoretical and Applied Statistics.

34 Thank you!