Bootstrapping

1. Bootstrapping • Identify some of the forces behind the move to quantify reserve variability. • Review current regulatory requirements regarding reserves and how they have been traditionally addressed. • Walk through an example of the traditional chain-ladder reserving approach. • Contrast the differences between the chain-ladder and bootstrap approaches (or deterministic and stochastic models more generally). • Walk through an example of a Bootstrap iteration.

2. All Booked Reserves are “Estimates” of the Ultimate Liability.

3. Partial List of Sources • CAS Working Party on Quantifying Variability in Reserve Estimates. The Analysis and Estimation of Loss & ALAE Variability: A Summary Report. CAS Forum (Fall 2005): 29-146. • England, P. D. and R. J. Verrall. 2002. Stochastic Claims Reserving in General Insurance. British Actuarial Journal 8:443-544. • Kirschner, Gerald S., Colin Kerley, and Belinda Isaacs. Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business. CAS Forum (Fall 2002): 211-46.

4. Reserve Estimation Variability • Actuaries dissatisfied with “point estimates”. • Companies Developing ERM Practices. • Technology Allows for Company Simulations. • Rating Bureaus (like AM Best) and Regulators have an interest in Reserve Variability.

5. RED ALERT!! Australia’s Prudential Regulatory Authority: “Technical Reserves To Be Determined as the Present value of a Central Estimate, with a Risk Margin to approximate the 75% Confidence Level.”

6. Statements of Statutory Accounting Principles • “Management’s best estimate” of its liabilities is to be recorded. • Accrue the midpoint of range when no single estimate is better than any other. • Accrue best estimate by line of business. Redundancies in one line cannot offset inadequacies in another.

7. Statement of Actuarial Opinion • Governed by Actuarial Standard of Practice (that is ASOP) 36. • When reserve is within “range of reasonable estimates”, it is assumed the reserve is reasonable. • Range of Reasonable Estimates determined by appropriate methods or sets of assumptions judged to be reasonable.

8. Historically, the Range of Reasonable Estimates have been developed by varying methods and/or assumptions, NOT by using statistics to evaluate the loss distribution.

9. Cumulative Paid Losses

14. Traditional Approaches: Deterministic – No Randomness In Outcomes. Bootstrapping: Stochastic – Randomness is allowed to influence the outcomes. Allows for the estimation of the Probability Distribution. Traditional Reserving vs. Bootstrapping

15. Stochastic models complement Deterministic methods by providing more information on the possible outcomes.

16. Bootstrapping • Resampling with Replacement Method • Incorporates Parameter Variance • Incorporates Process Variance • Cannot Incorporate Model Uncertainty (but no model can)

17. Bootstrapping • Resamples Pearson Residuals • Relies on the “Over-Dispersed Poisson Distribution” Which Can Model the Traditional Link Ratio Method • Thus, a Generalized Liner Model Underlies the Traditional Link Ratio Method

18. The Gamma Distribution • Used in place of Over-Dispersed Poisson Distribution in Bootstrapping • Models Process Variance in Bootstrapping • Sum of n exponentially distributed random variables • Described by a shape parameter a and a scale parameter B • Mean = aB , Variance = aB2 • Always > 0 • Moderately Skewed

20. Create New Triangle through Backward Recursion

21. New Triangle Preserves Parameter Variance

24. Scale Factor & Bias Adjustment

25. Triangle From Which Random Draws Will Be Made (excluding top right and bottom left zeros)

26. Iteration Begins: First Cell in “Pseudo” Triangle

27. "Pseudo" Incremental Paid Loss Triangle

28. Completed “Pseudo Square

29. Process Variance (Random Paid Loss)