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.

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

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): England, P. D. and R. J. Verrall Stochastic Claims Reserving in General Insurance. British Actuarial Journal 8: Kirschner, Gerald S., Colin Kerley, and Belinda Isaacs. Two Approaches to Calculating Correlated Reserve Indications Across Multiple Lines of Business. CAS Forum (Fall 2002):

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.

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.”

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.

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.

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

Cumulative Paid Losses Age inAccident Year Mos ,0001,5001,6001, ,2501,7852, ,5002, ,725

Age to Age Factor (Link Ratio) Age inAccident YearAverage Mos to to to TailSelected………………………………1.000

Age to Age Factor (Link Ratio) Age inAccident Year Mos to to to Tail(48-Ult)1.000

Cumulative Paid Losses Age inAccident Year Mos ,0001,5001,6001, ,2501,7852,0322, ,5002,1062,4182, ,7252,4222,7813,046 Ultimate1,7252,4222,7813,046

Cumulative Paid Losses Age inAccident Year Mos ,0001,5001,6001, ,2501,7852,0322, ,5002,1062,4182, ,7252,4222,7813,046 Ultimate1,7252,4222,7813,046 Reserve ,246 TotalReserve=2,311

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

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

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

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

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  and a scale parameter  Mean = , Variance =  2 Always > 0 Moderately Skewed

“Original” Incremental Paid Loss Triangle Age inAccident Year Mos ,0001,5001,6001, Cumulative Paid Loss Triangle Age inAccident Year Mos ,0001,5001,6001, ,2501,7852, ,5002, ,725 Age to Age Factor (Link Ratio) Age inAccident YearAverage Mos to to to TailSelected………………… 1.000

Create New Triangle through Backward Recursion Original Cumulative Paid Loss Diagonal Age inAccident Year Mos Selected 121, , , ,

New Triangle Preserves Parameter Variance “New” Cumulative Paid Loss Triangle Age inAccident Year Mos Selected 121,0191,4311,6431, ,2611,7702, ,5002, , ,500=1,725 / 1.150

“New” Incremental Paid Loss Triangle Age inAccident Year Mos ,0191,4311,6431,

"Original" Incremental Paid Loss Triangle Age inAccident Year Mos ,0001,5001,6001, "New" Incremental Paid Loss Triangle Age inAccident Year Mos ,0191,4311,6431, Unscaled Pearson Residuals [Original - New/Sqrt(New) ] Age inAccident Year Mos

Scale Factor & Bias Adjustment Square of Unscaled Pearson Residuals Age inAccident Year Mos Sum of Squares = N = # of Data Points In Triangle10 P=# Parameters Estimated = 2x(# Accident Years)-1=7 Scale Factor = Sum of Squares / (N - P) =6.517 Bias Adjustment = Sqrt(N) / (N - P) =1.054

Triangle From Which Random Draws Will Be Made (excluding top right and bottom left zeros) Bias Adjustment x Unscaled Pearson Residuals Age inAccident Year Mos Exclude These

Iteration Begins: First Cell in “Pseudo” Triangle Bias Adjusted Unscaled Pearson Residuals (select a random draw) Age inAccident Year Mos Random Draw "New" Incremental Paid Loss Triangle Age inAccident Year Mos ,0191,4311,6431,800 "Pseudo" Incremental Paid Loss Triangle Age inAccident Year Mos , ,019 + [ x Sqrt(1,019) ]

"Pseudo" Incremental Paid Loss Triangle “New” Increment + Random Pearson Residual x Sqrt(“New” Increment) Age inAccident Year Mos ,0931,4541,7371,

Completed “Pseudo Square "Pseudo" Incremental Paid Loss Square Age inAccident Year Mos ,0931,4541,7371,

Process Variance (Random Paid Loss) "Pseudo" Incremental Paid Loss Square Age inAccident Year Mos Random Number for Each Future "Pseudo Cell" "Pseudo" Incremental Paid Loss Square Scale=  =6.517  = 327 /  = Gamma Inverse (0.154, ,  ) = 280

"Pseudo" Cumulative Paid Losses with Process Variance Age inAccident Year Mos ,0931,4541,7371, ,3211,7812,1632, ,5132,0972,5652, ,7482,3762,9442,818 Ultimate1,7482,3762,9442,818 Reserve ,149 TotalReserve=2,210 End of First Iteration

AFTER 5,000 BOOTSTRAP ITERATIONS: Comparison of Bootstrapped Percentiles Vs. The Point Estimate From Average Link Ratios BootstrapPoint PercentileReserve 01,4102, ,0802, ,2712,311 Mean2,2792, ,4732, ,3802,311 Std Dev = +/-288