Non-Linear Hierarchical Bayesian

Transcription

1 A Credble Approach to Reservng Non-Lnear Herarchcal Bayesan Models for Loss Reservng CAS Sprng Meetng Vancouver Jm Guszcza, FCAS, MAAA Delotte Consultng LLP May 20, 2013

2 Anttrust Notce The Casualty Actuaral Socety s commtted to adherng strctly to the letter and sprt of the anttrust laws. Semnars conducted under the auspces of the CAS are desgned solely to provde a forum for the expresson of varous ponts of vew on topcs descrbed n the programs or agendas for such meetngs. Under no crcumstances shall CAS semnars be used as a means for competng companes or frms to reach any understandng expressed or mpled that restrcts competton or n any way mpars the ablty of members to exercse ndependent busness udgment regardng matters affectng competton. It s the responsblty of all semnar partcpants to be aware of anttrust regulatons, to prevent any wrtten or verbal dscussons that appear to volate these laws, and to adhere n every respect to the CAS anttrust complance polcy. 2 Delotte Analytcs Insttute 2011 Delotte LLP

3 Themes Motvatons Herarchcal Models Bayesan Concepts Our Model Case Study

4 Preamble

5 Why Bayes, Why Now From John Kruschke, Indana Unversty: t An open letter to Edtors of ournals, Chars of departments, Drectors of fundng programs, Drectors of graduate tranng, Revewers of grants and manuscrpts, Researchers, Teachers, and Students : Statstcal methods have been evolvng rapdly, and many people thnk t s tme to adopt modern Bayesan data analyss as standard procedure n our scentfc practce and n our educatonal currculum. Three reasons: 1. Scentfc dscplnes from astronomy to zoology are movng to Bayesan data analyss. We should be leaders of the move, not followers. 2. Modern Bayesan methods provde rcher nformaton, wth greater flexblty and broader applcablty than 20th century methods. Bayesan methods are ntellectually coherent and ntutve. Bayesan analyses are readly computed wth modern software and hardware. 3. Null-hypothess sgnfcance testng (NHST), wth ts relance on p values, has many problems. There s lttle reason to persst wth NHST now that Bayesan methods are accessble to everyone. My concluson from those ponts s that we should do whatever we can to encourage the move to Bayesan data analyss. (I couldn t have sad t better myself ) 5 Delotte Analytcs Insttute 2010 Delotte LLP

6 Why Bayes, Why Now From an Intervew wth Sharon Bertsch McGrayne n Chance Magazne: When I started research on [my] book, I could Google the word Bayesan and get 100,000 hts. Recently I got 17 mllon. 6 Delotte Analytcs Insttute 2010 Delotte LLP

7 Our Professon s Bayesan Hertage: Early Late 18th Century: Thomas Bayes and Perre-Smon Laplace formulate the prncples of nverse probablty Probablstc nference from data to model parameters Bayes ntellectual executor, Rchard Prce, became perhaps the world s frst consultng actuary (Equtable Lfe Assurance company, London) Prce s and perhaps Bayes thnkng was nfluenced by the publcaton of Davd Hume s Treatse on Human Nature (1740) 1918: A. W. Whtney The Theory of Experence Ratng. Advocated combnng the clams experence of a sngle rsk wth that of a cohort (class, portfolo, ) of smlar rsks. Z ˆ (1 Z) ˆ, rsk class w w k Et Estmated tdpure premum should be a weghted average of the ndvdual d rsk s clam experence wth that of the cohort k s udgmentally determned. Z 7 Delotte Analytcs Insttute 2010 Delotte LLP

8 Our Professon s Bayesan Hertage: Early-Modern 1950: Arthur Baley publshes Credblty Procedures: Laplace s Generalzaton of Bayes Rule and the Combnaton of Collateral Knowledge wth Observed Data. Antcpates Hans Bühlmann's subsequent work. Quoted Rchard Prce on makng nferences from avalable data. At present, practcally all methods of statstcal estmaton appearng n textbooks are based on an equvalent to the assumpton that any and all collateral nformaton or a pror knowledge s worthless. There have been rare nstances of rebellon aganst ths phlosophy by practcal statstcans who have nssted that they actually had a consderable store of knowledge apart from the specfc observatons beng analyzed However t appears to be only n the actuaral feld that t there has been an organzed revolt aganst dscardng all pror knowledge when an estmate s to be made usng newly acqured data. 8 Delotte Analytcs Insttute 2010 Delotte LLP

9 Our Professon s Bayesan Hertage: Md-Century Modern 1967: Bühlmann s greatest accuracy Bayes credblty model. Let X denote dollars of loss assocated wth rsk at tme. Assume X 1,, X m are d, condtonal on a parameter (vector) θ Let m(θ ) denote rsk premum : m(θ ) E[X θ ] Bühlmann mnmzes mean squared errors: Em( ) X 2 to arrve at an estmator for m(θ ): z X 1 z ) ( where: n E Var X z n k, k Var m The wthn/between varances n k are estmated from the data. 9 Delotte Analytcs Insttute 2010 Delotte LLP

10 Our Professon s Bayesan Hertage: Modern? 10 Delotte Analytcs Insttute 2010 Delotte LLP

11 Loss Reservng and ts Loss Reservng and ts Dscontents

12 Loss Reservng and ts Dscontents Much loss reservng practce s stll pre-theoretcal n nature. Technques lke chan ladder, BF, and Cape Cod aren t performed n a statstcal modelng framework. (Do people agree wth ths statement?) Tradtonal methods aren t necessarly optmal from a statstcal POV. Potental of over-fttng small datasets Dffcult to assess goodness-of-ft, compare nested models, etc Often no concept of out-of-sample valdaton or dagnostc plots Related pont: tradtonal methods produce pont estmates only. Reserve varablty estmates are often ad-hoc 12 Delotte Analytcs Insttute 2010 Delotte LLP

13 Models vs Methods Rather than promulgatng a collecton of loss reservng methods, we buld statstcal models of loss development. Attempt to place loss reservng practce on a sound scentfc footng. Feld s developng rapdly Today: Sketch non-lnear herarchcal Bayesan models Natural, parsmonous models of the loss development process Intally motvated by Dave Clark s [2003] paper as well as herarchcal Bayesan modelng theory. By the way: the debate over models vs methods s msleadng Rather we want to have a flexble and extensble modelng methodology A framework that can always be talored to the specfcs of a gven stuaton Spreadsheets aren t the only way to accomplsh ths 13 Delotte Analytcs Insttute 2010 Delotte LLP

14 Four Essental Features of Loss Reservng Repeated measures Loss reservng s longtudnal data analyss Cumulatve Losses n 1000's AY premum , ,342 1,582 1,736 1,833 1,907 1,967 2,006 2, , ,336 1,580 1,726 1,823 1,903 1,949 1, , ,037 1,401 1,604 1,729 1,821 1,878 1, , ,029 1,195 1,326 1,395 1, , , , , , , A bundle of tme seres A loss trangles s a collecton of tme seres that are related to one another but no guarantee that the same development pattern s approprate to all Non-lnear Each year s development patter s nherently non-lnear Ultmate loss (rato) s an asymptote Incomplete nformaton Few loss trangles contan all of the nformaton needed to make forecasts Most reservng exercses must ncorporate udgment and/or background nformaton Loss reservng s nherently Bayesan! 14 Delotte Analytcs Insttute 2010 Delotte LLP

15 Towards a More Realstc Loss Reservng Framework How many stochastc reservng technques reflect all of these consderatons? 1. Repeated Measures (sn t loss reservng a type of longtudnal l data analyss?) 2. Multple tme seres 3. Non-lnear (are GLMs really approprate?) 4. Incomplete Informaton ( Bayes or Bust!) 1-2 We need herarchcal models 3 They should use growth curves 4 Non-lnear herarchcal models should be Bayesan Cumulatve Losses n 1000's AY premum pe , ,342 1,582 1,736 1,833 1,907 1,967 2,006 2, , ,336 1,580 1,726 1,823 1,903 1,949 1, , ,037 1,401 1,604 1,729 1,821 1,878 1, , ,029 1,195 1,326 1,395 1, , , , , , , Delotte Analytcs Insttute 2010 Delotte LLP

16 Another Bg Motvaton: Predctve Dstrbutons Gven any value (estmate of future payments) and our current state of knowledge, what s the probablty that the fnal payments wll be no larger than the gven value? -- Casualty Actuaral Socety Workng Party on Quantfyng Varablty n Reserve Estmates, 2004 Ths can be read as a request for a Bayesan analyss Bayesans (unlke frequentsts) are wllng to make probablty statements about unknown parameters Ultmate losses are sngle cases dffcult to conceve as random draws from a samplng dstrbuton n the sky. Frequentst probablty nvolved repeated trals of setups nvolvng physcal randomzaton. In contrast t s meanngful to apply Bayesan probabltes to sngle case events The Bayesan analyss yelds an entre posteror probablty dstrbuton not merely moment estmates Bayesan statstcs s the deal framework for loss reservng! 16 Delotte Analytcs Insttute 2010 Delotte LLP

17 The Bayesan Perspectve For Bayesans as much as for any other statstcan, parameters are (typcally) fxed but unknown. It s the knowledge about these unknowns that Bayesans model as random typcally t s the Bayesan who makes the clam for nference n a partcular nstance and the frequentst who restrcts clams to nfnte populatons of replcatons. - Andrew Gelman and Chrstan Robert 17 Delotte Analytcs Insttute 2010 Delotte LLP

18 Orgn of the Approach: Dave s Idea + Random Effects + 18 Delotte Analytcs Insttute 2010 Delotte LLP

19 Current State of uh Development 19 Delotte Analytcs Insttute 2010 Delotte LLP

20 Components of Our Approach Growth curves to model the loss development process (Clark 2003) Parsmony obvates the need for tal factors Loss reservng treated as longtudnal data analyss (Guszcza 2008) A type of herarchcal modelng Parsmony; smlar approach to non-lnear mxed effects models used n bologcal/socal scences Further usng the herarchcal modelng framework to smultaneously model multple loss trangles (Zhang-Dukc-Guszcza 2012) Borrow strength from other loss reservng trangles Smlar n sprt to credblty theory Insuffcent tme to cover ths aspect today Buldng a fully Bayesan model by assgnng pror probablty dstrbutons to all hyperparameters (Zhang-Dukc-Guszcza 2012) Provdes formal mechansm for ncorporatng background knowledge and expert opnon wth datadrven ndcatons. Results n full predctve dstrbuton of all quanttes of nterest Conceptual advantages: Bayesan paradgm treats data as fxed and parameters are randomly varyng 20 Delotte Analytcs Insttute 2010 Delotte LLP

21 The Noton of Herarchcal Structure s Key NB: Bayesan models Herarchcal models! E.g. we could ft a Bayesan chan ladder by puttng prors on the parameters of an overdspersed Posson regresson model but ths wouldn t make t herarchcal. Smlarly non-bayesan herarchcal models are a useful way to quckly ft exploratory models whle gearng up to do a fully Bayesan analyss. 21 Delotte Analytcs Insttute 2010 Delotte LLP

22 Herarchcal Models

23 What s Herarchcal Modelng? Herarchcal modelng s used when one s data s grouped n some mportant way. Clam experence by state or terrtory Workers Comp clam experence by class code Clam severty by nury type Churn rate by agency Multple years of loss experence by polcyholder. Multple observatons of a cohort of clams over tme Often grouped data s modeled ether by: Buldng separate models by group Poolng the data and ntroducng dummy varables to reflect the groups Herarchcal modelng offers a mddle way. Parameters reflectng group membershp enter one s model through approprately specfed probablty sub-models. 23 Delotte Analytcs Insttute 2010 Delotte LLP

24 Common Herarchcal Models Classcal Lnear Model Equvalently: Y ~ N(+X, 2 ) S f h d t t X Y Same, for each data pont Random Intercept Model X Y ] [ Where: Y ~ N( [] +X, 2 ) And: ~ N(, 2 ) Same for each data pont; but vares by group Random Intercept and Slope Model Both and vary by group X Y ] [ ] [ Both and vary by group 2 24 Delotte Analytcs Insttute 2010 Delotte LLP ] [ ] [,, ~, ~ N where X N Y

25 Smple Example: PIF by Regon Smple example: Change n PIF by regon from PIF Growth by Regon regon1 regon2 regon3 regon4 32 data ponts 4 years regons But we could as easly have 80 or 800 regons 2000 Our model would not change 2600 regon5 regon6 regon7 regon8 We vew the dataset as a bundle of very short tme seres Delotte Analytcs Insttute 2010 Delotte LLP

26 Classcal Lnear Model Opton 1: the classcal lnear model Y ~ N ( X X 2, ) PIF Growth by Regon regon1 regon2 regon3 regon4 Complete Poolng Don t reflect regon n the model desgn Just throw all of the data nto one pot and regress 2200 Same and for each regon regon5 regon6 regon7 regon8 Ths obvously doesn t cut t. But fllng 8 separate regresson models or throwng n regonspecfc dummy varables sn t an attractve opton ether. Danger of over-fttng e. credblty ssues Delotte Analytcs Insttute 2010 Delotte LLP

27 Randomly Varyng Intercepts Opton 2: random ntercept model Y ~ N ( X, ) 2 N (, ) 2 [ ] PIF Growth by Regon regon1 regon2 regon3 regon4 ~ Y = [] + X Ths model has 9 parameters: { 1, 2,, 8, } And t contans 4 hyperparameters: {,,, } regon5 regon6 regon7 regon8 A maor mprovement Delotte Analytcs Insttute 2010 Delotte LLP

28 Randomly Varyng Intercepts and Slopes Opton 3: the random slope and ntercept model Y ~ N 2 2 X, where ~ N, [ ] [ ], 2 PIF Growth by Regon regon1 regon2 regon3 regon4 Y = [] + [] X Ths model has 16 parameters: { 1, 2,, 8, 1, 2,, 8 } (note that 8 separate models also contan 16 parameters) regon5 regon6 regon7 regon And t contans 6 hyperparameters: {,,,,, } To repeat: the same number of hyperparameters f we had 80 or 800 regons Delotte Analytcs Insttute 2010 Delotte LLP

29 A Compromse Between Complete Poolng and No Poolng t PIF 8 1,2,.., k k k k t PIF Complete Poolng No Poolng p g Ignore group structure altogether g Estmate a separate model for each group Compromse Herarchcal Model Estmates parameters 2 usng a compromse between complete poolng and no poolng. 29 Delotte Analytcs Insttute 2010 Delotte LLP ] [ ] [,, ~, ~ N where X N Y

30 A Credble Approach For llustraton, recall the random ntercept model: ) ( ) ( 2 2 N X N Y Ths model can contan a large number of parameters { 1,, J,}. ), ( ~ ), ( ~ 2 2 ] [ N X N Y g p { 1,, J,} Regardless of J, t contans 4 hyperparameters {,,, }. Here s how the hyperparameters relate to the parameters: n 2 2 ˆ ) (1 ) ( ˆ n n Z where Z x y Z 30 Delotte Analytcs Insttute 2010 Delotte LLP Bühlmann credblty s a specal case of herarchcal models.

31 A Fully Bayesan Model Wth a Case Study

32 Case Study Data A garden-varety Workers Comp Schedule P loss trangle: Cumulatve Losses n 1000's AY premum CL Ult CL LR CL res , ,342 1,582 1,736 1,833 1,907 1,967 2,006 2,036 2, , ,336 1,580 1,726 1,823 1,903 1,949 1,987 2, , ,037 1,401 1,604 1,729 1,821 1,878 1,919 1, , ,029 1,195 1,326 1,395 1,446 1, , ,007 1, , , , , chan lnk ,067 1,543 chan ldf growth curve 21.2% 50.1% 67.9% 79.0% 86.1% 90.7% 94.2% 96.6% 98.5% 100.0% Let s model ths as a longtudnal dataset. Groupng dmenson: Accdent Year (AY) We can buld a parsmonous non-lnear model that uses random effects to allow the model parameters to vary by accdent year. 32 Delotte Analytcs Insttute 2010 Delotte LLP

33 Growth Curves At the Heart of the Model We want our model to reflect the non-lnear nature of loss development. p GLM shows up a lot n the stochastc loss reservng lterature 1.0 but are GLMs natural models 0.8 for loss trangles? Growth curves (Clark 2003) = ultmate loss rato = scale = shape ( warp ) Heurstc dea We udgmentally select a growth curve form umulatve Perc cent of Ultmate C Let vary by year (herarchcal) 0.0 Add prors to the hyperparameters (Bayesan) Webull and Loglogstc Growth Curves Heurstc: Ft Curves to Chan Ladder Development Pattern G ( x, ) x G ( x, ) 1 exp ( x / ) x Loglogstc Webull Development Age 33 Delotte Analytcs Insttute 2010 Delotte LLP

34 Cumulatve Loss An Exploratory Non-Bayesan Herarchcal Model It s easy to ft non-bayesan herarchcal t y ( t ) * p * ( t t 2 ~ N, models as a data exploraton step. Log-Loglstc Herarchcal Model (non-bayesan) ( t ) ( t 1 ) ( t ) ) Delotte Analytcs Insttute 2010 Delotte LLP Development Tme

35 Addng Bayesan Structure Our herarchcal model s half-way Bayesan On the one hand, we place probablty sub-models on certan parameters But on the other hand, varous (hyper)parameters are estmated drectly from the data. To make ths fully Bayesan, we need to put probablty dstrbutons on all quanttes that are uncertan. We then employ Bayesan updatng: the model ( lkelhood functon ) together wth the pror results n a posteror probablty dstrbuton over all uncertan quanttes. Includng ultmate t loss rato parameters and hyperparameters! We are drectly modelng the ultmate quantty of nterest. Ths s not as hard as t sounds: We do not explctly calculate the hgh-dmensonal posteror probablty dstrbuton. We do use Markov Chan Monte Carlo [MCMC] smulaton to sample from the posteror. Technology: JAGS ( Just Another Gbbs Sampler ), called from wthn R. 35 Delotte Analytcs Insttute 2010 Delotte LLP

36 Results of a Fully Bayesan Herarchcal Model Now we ft a fully Bayesan verson of the model by provdng pror dstrbutons for all of the model hyperparameters, and smulatng the posteror dstrbuton. 36 Delotte Analytcs Insttute 2010 Delotte LLP

37 Results of a Fully Bayesan Herarchcal Model Here we are usng the most recent Calendar Year (red) as a holdout sample. The model fts the holdout well. 37 Delotte Analytcs Insttute 2010 Delotte LLP

38 Bayesan Credble Intervals Now reft the model on all of the data and re-calculate the posteror credble ntervals. 38 Delotte Analytcs Insttute 2010 Delotte LLP

39 Comparson wth the Chan Ladder For comparson, supermpose the at 120 months chan ladder estmates on the posteror credble ntervals. 39 Delotte Analytcs Insttute 2010 Delotte LLP

40 Posteror Dstrbuton of Aggregate Outstandng Losses In the top two mages, we sum up the proected losses for all estmated AY s evaluated at 120 (180) months; then subtract losses to date (LTD). For the 120 month estmate, the posteror medan (1519) comes very close to the chan ladder estmate (1543) Outstandng Loss Estmates at Dfferent Evaluaton Ponts Estmated Ultmate Losses Mnus Losses to Date At 120 Months chan ladder estmate At 180 Months In the bottom mage, we multply the estmated ultmate loss rato parameters by premum and subtract LTD. Decdng whch of these optons s most approprate s akn to selectng a tal factor At Ultmate Delotte Analytcs Insttute 2010 Delotte LLP