# From the Bootstrap to the bootstrap and beyond

• Slides: 13

From the “Bootstrap” to the bootstrap and beyond March 2021

What is the bootstrap (1)? • UK actuarial: usually understood to be a specific application of resampling to a chain ladder model. – E. g. , as described in England PD and Verrall RJ (1999). Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics 31, 461 -466 – Black box for many practioners – Frequently the underlying model fits poorly and/or is quite different to the model actually used to estimate the reserve. • Statistically: a sampling methodology that may be used to estimate uncertainty for many types of stochastic model. – The original paper is Efron B (1979). Bootstrap methods: another look at the jackknife. Annals of Statistics, 7, 1– 26. Of particular relevance is Section 7, which discusses regression models. – This point was made in England PD and Verrall RJ (2006). Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science, 1, 221 -270. – Some discussion also in Carrato A, Mc. Guire G and Scarth R. A Practitioner’s Introduction to Stochastic Reserving. GIRO 2015 and Taylor G and Mc. Guire G (2015). Stochastic Loss Reserving using Generalized Linear Models. CAS monograph series, in press March 2021 2

What is the bootstrap (2)? • So stochastic models (both aggregate and individual) beyond the chain ladder may be used in conjunction with the bootstrap to estimate uncertainty – Can estimate independent error – parameter and process error. – Other sources of error not estimated by the bootstrap: • Internal model error • Systemic/future environmental uncertainty • ENIDs/binary events • External and systemic sources of error still need to be considered and allowed for (usually assessed qualitatively)– no matter how fancy the model! March 2021 3

Extensions to aggregate models (1) • GLMs of typical chain ladder triangles – Model effects by accident, development, calendar periods – Smoothing of development factors -> better estimate for tail development – E. g. Taylor G and Mc. Guire G (2015). Stochastic Loss Reserving using Generalized Linear Models. CAS monograph series, in press. • Frequency and severity modelling – Total number of claims – Average size of claims – E. g. Taylor, G. C. (2000), "Loss Reserving: An Actuarial Perspective", Kluwer Academic Publishers, Boston, – Frees E. W. and Derrig, R. A. (eds. ) (2014), "Predictive Modeling Applications in Actuarial Science: Volume 1, Predictive Modeling". Cambridge University Press, New York, NY, USA March 2021 4

Extensions to aggregate models (2) • Exposure based models – Variable of interest depends on another quantity • Models of total claim numbers based on exposure measure (e. g. number of vehicles for motor vehicle) • Number of closed claims based on number of open claims at start of period. • Payments per claims incurred (total number of claims, average payment per claim) • Payments per claim finalised (total number of claims, number of finalised claims, average payment per finalised claim) – E. g. Taylor, G. C. (2000), "Loss Reserving: An Actuarial Perspective", Kluwer Academic Publishers, Boston, – Frees E. W. and Derrig, R. A. (eds. ) (2014), "Predictive Modeling Applications in Actuarial Science: Volume 1, Predictive Modeling". Cambridge University Press, New York, NY, USA • Consider the appropriate granularity level – Month/quarter/half-year/annual – If you are using stochastic models then you can more easily work with more finely aggregated data March 2021 5

Individual models • Individual data permits use of additional claimant and claim specific data • Two broad classes of variables: – Static variables: Constant over life of claim • e. g. gender, pre-injury earnings – Dynamic variables: changeable. Further categorised as • Predictable variables: usually relate to passage of time so future values known with certainty (e. g. development, calendar periods) • Unpredictable variables: not predictable with certainty (e. g. time until a claim closes, spells off work) – Variable types discussed in Taylor G, Mc. Guire G & Sullivan J (2006). Individual claim loss reserving conditioned by case estimates, http: //www. actuaries. org. uk/research-andresources/documents/individual-claim-loss-reserving-conditioned-case-estimates March 2021 6

Implications for uncertainty measurement • Individual models lie on a spectrum from those models with time variables only to those with all types of predictors. • Simpler models may have similar time variables to aggregate models. But increased granularity of data means that data trends can be examined in better detail. – May reduce internal model error – so better forecasts assuming environment remains constant. • E. g. facilitates identification of super-imposed inflation trends, drivers, impacts which can lead to better forecasts • Data set in Taylor G, Mc. Guire G & Sullivan J (2006). Individual claim loss reserving conditioned by case estimates shows considerably lower uncertainty measures for individual models vs Mack chain ladder. – External systemic error still requires consideration March 2021 7

Complex individual models • Often referred to as stochastic case estimate models – Alternative estimates of ultimate claim costs to manual case estimates – High discriminatory power needed->Large numbers of predictors – Taylor G & Campbell M (2002). Statistical case estimation. Research paper No 104 of the Centre for Actuarial Studies, University of Melbourne. Available at http: //fbe. unimelb. edu. au/__data/assets/pdf_file/0005/806396/104. pdf • Use in reserving may be limited – Unpredictable variables need prediction so prediction errors compound. • Taylor G, Mc. Guire G & Sullivan J (2006). Individual claim loss reserving conditioned by case estimates shows considerably lower uncertainty measures for individual models vs Mack chain ladder: Some of the more complicated individual models have greater prediction errors than the simpler models – Characteristics of IBNR claims? – Time consuming to build March 2021 8

Some practical suggestions – not exhaustive! • Fit a good and appropriate stochastic model and bootstrap (or simulate/MCMC) this. – Venturing beyond chain ladder may lead to a better match between the valuation and uncertainty models (ideally these are the same) – Granularity may often lead to a better model – e. g. a quarterly or monthly triangle, or individual data • Consider carefully what variables should be included – trade-off between increased discrimination from more variables vs additional prediction error – What output do you actually need? • Model validation is crucial • Consider and allow for systemic risks/ENIDS (usually qualitatively) March 2021 9

Other references (1) • Triangle free reserving: Parodi 2012, 2013: – Sets out a framework for individual claim reserving and some possible models to use within that framework for reserving – http: //www. actuaries. org. uk/research-and-resources/documents/triangle-free-reserving-nontraditional-framework-estimating-reserv • Individual modelling of claim size using GLMs including comparison with chain ladder model results – Taylor G & Mc. Guire G (2004). Loss reserving with GLMs: a case study. Casualty Actuarial Society 2004 Discussion Paper program, pp 327 -392 – Simple individual model – time variables only [accident, calendar and development/operational time] – Similarities to Parodi framework, though different types of models used – Illustrates how individual GLM model is preferable to chain ladder for one particular motor bodily injury data set. Does not consider uncertainty March 2021 10

Other references (2) • Individual modelling of motor bodily injury data using claim severity – Same Lo. B as previous CAS paper, but ~4 years later with claim severity introduced to improve reserve estimation. Does not consider uncertainty – Mc. Guire G (2007). Individual claim modelling of CTP data. http: //actuaries. asn. au/Library/6. a_ACS 07_paper_Mc. Guire_Individual claim modellingof CTP data. pdf • Individual loss reserving conditioned by case estimates – Taylor G, Mc. Guire G & Sullivan J (2006). Individual claim loss reserving conditioned by case estimates, http: //www. actuaries. org. uk/research-and-resources/documents/individual-claimloss-reserving-conditioned-case-estimates – Considers a number of models, firstly Mack and then a number of individual models. Does examine measures of uncertainty. In particular, the individual models are found to have much lower mean square errors of prediction than the Mack model. March 2021 11

Other references (3) • Model validation – Many statistics text books – Carrato A, Mc. Guire G and Scarth R. A Practitioner’s Introduction to Stochastic Reserving. GIRO 2015 – Taylor G and Mc. Guire G (2015). Stochastic Loss Reserving using Generalized Linear Models. CAS monograph series, in press. • Holistic approach to reserve uncertainty including quantitative and qualitative errors - a view from Australia – O'Dowd C. , Smith A. and Hardy P. (2005). "A framework for estimating uncertainty in insurance claims cost". XVth General Insurance Seminar, 16 -19 October 2005. Institute of Actuaries of Australia. Available at http: //www. actuaries. asn. au/Library/gipaper_odowd-smithhardy 0510. pdf – Risk Margins Taskforce (2008), "A framework for assessing risk margins", XVIth General Insurance Seminar, 9 -12 November 2008. Institute of Actuaries of Australia. Available at http: //www. actuaries. asn. au/Library/Framework%20 for%20 assessing%20 risk%20 margins. pdf March 2021 12

Other references (4) • MCMC models – an alternative to bootstrapping – Scollnik D P M (2001). Actuarial modeling with MCMC and BUGS. North American Actuarial Journal, 5(2), 96 -125. – Scollnik D P M (2002). Implementation of four models for outstanding liabilities in Win. BUGS: A discussion of a paper by Ntzoufras and Dellaportas. North American Actuarial Journal, 6, 128 -136. – Ntzoufras I & Dellaportas P (2002). Bayesian Modelling of Outstanding Liabilities Incorporating Claim Count Uncertainty (with discussion). North American Actuarial Journal, 6, 113 -128. – Ntzoufras I, Katsis A & Karlis D (2005). Bayesian assessment of the distribution of insurance claim counts using reversible jump MCMC. North Americal Actuarial Journal, 9, 90 -108. – Meyers G. CAS monograph no. 1: Stochastic loss reserving using Bayesian MCMC models. http: //www. casact. org/pubs/monographs/index. cfm? fa=meyers-monograph 01 March 2021 13