1. Valuing an Insurance Enterprise and Reserve Estimates Using Bootstrapped Statutory Loss Information William C. Scheel DFA Technologies, LLC scheel@mindspring.com CAS 2001 DFA Seminar

2. 2 Concepts  Target sufficiency level  Minimum sufficiency level  Capital release

3. 3 Techniques  Bootstrapping loss triangles  Simulation of ultimate loss links, payment patterns and asset returns  Non-Linear optimization using proxy assets and target sufficiency levels  Programming with COM in Visual Basic for Applications (Excel VBA)

4. 4 Target Sufficiency Level  Fair value of future cash flows  Present value of deferred annuity valued at riskless rate of return  Net amount necessary to transfer future claims payments

5. 5 Minimum Sufficiency Level  Amount needed to fund future claims within confidence levels that adjust for uncertainty in amount and timing of payments and for uncertainty in investment returns  Chance-constrained valuation of claims  More than a reserve  Basis for the expectation of capital release  Basis of enterprise solidity

6. 6 Capital Release  Initial excess sufficiency plus present value of expected capital release is the value of the enterprise  Amount of capital held in excess of expected sufficiency. It is expected to be released to shareholders rather than to be used to pay obligations.  Capital release occurs when minimum sufficiency levels decrease over time  Like reserves, capital release can only be measured probabilistically

7. 7 Sources of Expected Capital Release  Target sufficiency level is higher than expected present value of claims  Value of assets is higher than expected value needed to achieve this deferred target sufficiency level

8. 8 Sources of Insurance Enterprise Value  Assets exceeding minimum sufficiency level  Expectation of capital release in the future

9. 9 Relationship Among Sufficiency Levels and Capital Release EOP Target sufficiency level BOP Minimum sufficiency level Time Investment return and claims experience expected to produce operating result in this area High probably area and source of capital release expectation

10. 10 Capital Release where: = Capital released at the end of period t, = Minimum sufficiency level at the beginning of period t, = Portfolio return during period t, = Claims payments during period t.

11. 11 Current Excess Value Source of ValueAmount Current market value5,534,719 Less: Current min sufficiency level1,591,549 Current ultimate loss for lines not analyzed2,565 Net Current excess value3,940,605 PV E(capital release) (@.05)297,109

12. 12 Distribution of Capital Release Period 1Period 2Period 3Period 4Period 5Period 6Period 7Period 8Period 9 Mean133,44159,22251,99939,57517,90015,2228,4906,7664,395 Standard Deviation113,28345,98348,34436,44413,69412,4717,2205,7092,895 Median127,13158,01250,62337,48617,59614,7118,3086,5584,360 5 percentile-42,909-15,214-23,636-16,731-4,230-4,442-3,078-1,956-347 10 percentile-6,4341,953-8,050-4,987402-195-905-535678 25 percentile54,01827,82618,29213,8488,2346,4383,5112,6812,431 75 percentile208,57889,46483,84263,17127,06423,71913,38510,6346,361 90 percentile280,925118,269114,96988,25335,39031,34817,71914,4508,200 95 percentile325,144135,291134,078101,40441,02135,71720,37516,5229,103

13. 13 Bootstrapping Loss Triangles  Would rather use individual claims information  Multivariate sampling of lines of business yields covariance matrix  Ultimate link ratios  Paid/Ultimate ratios

14. 14 Feasible Region for Bootstrap Sampling of a Link Ratio

15. 15 Portions of a Bootstrap Sample in Shaded Regions

16. 16 Statistics for Link1

17. Steps in Valuation of Target Sufficiency 1.Perform a bootstrap of link ratios for ultimate loss. 2.Use bootstrapped ultimate link ratios to derive correlation matrix and other statistics. 3.Using the correlation matrices and statistics, simulate ultimate links for each line of business using multinormal methods. 4.Apply the simulated ultimate link ratios to the latest ultimate loss triangle diagonal. 5.Perform a second-stage simulation using the probability distribution of paid-to-ultimate ratios (payment patterns). 6.Use the cash flows to calculate annuity-equivalent values for future loss cash flow. Do this at each forward calendar period.

18. 18 Target Sufficiency Statistics All Lines Period 1Period 2Period 3Period 4Period 5Period 6Period 7Period 8Period 9Period 10 Mean1,798,9211,282,873896,766650,352484,940369,549278,479199,570124,02347,013 Standard Deviation91,55755,18537,40521,52812,3888,8186,4514,5712,833845 Median1,798,3431,281,362895,937649,768484,449369,120278,192199,447123,92046,980 5 percentile1,649,1691,195,019837,174615,604465,394355,541268,231192,349119,50845,660 10 percentile1,682,1851,214,548850,383623,497469,507358,660270,411193,797120,43245,956 25 percentile1,737,0401,244,796870,057634,645476,207363,456274,039196,427122,08346,435 75 percentile1,860,9551,319,754920,838664,331492,981375,434282,777202,531125,78647,557 90 percentile1,913,8781,352,208944,923678,150500,876380,722286,849205,513127,76148,135 95 percentile1,948,8871,374,690960,082686,303506,422384,138289,258207,396128,89948,491

19. 19 Moving from Targets to Minimum Sufficiency Levels  Targets risk-adjust only for uncertainty in amount of payments and timing of payments  What asset levels are required to meet sufficiency targets?  What portfolio allocation?  Technique: use non-linear optimization

20. 20 Optimization Techniques  Objective: Minimize level of assets necessary to meet sufficiency targets  Subject to: Portfolio constraints for proxy assets and minimum sufficiency probability constraint What BOP assets should be held to met EOP sufficiency levels within acceptable levels of risk?

21. 21 Techniques Using Microsoft Excel and Frontline Premium Solver  Workbook A: Non-linear version of Solver posits trial solution (portfolio allocation)  Workbook B (COM object instantiated by A): Has 2,500 simulated asset returns, target sufficiency level and chance-constrained probability (all provided by A). Gets trial portfolio allocation from A.  B determines BOP value of target for each asset simulation using trail solution weights.  Using this BOP distribution, B returns chance-constrained minimum sufficiency level to A as objective value.  Workbook A repeats trial solution tests until it finds the minimum value returned by workbook B

22. 22 Minimum Sufficiency Levels and Optimal Investment Portfolios Period 1Period 2Period 3…Period 8Period 9Period 10 Min Sufficiency Level1,591,5491,064,347680,513 … 78,97737,7110 Required EOP target assets1,683,7851,128,672722,708 … 83,43939,8790 EAFEU.067.073.146 ….063.015 INTLUHD.189.039.126 ….226.036 S&P5.163.023.0 ….082.039 USTB.067.465.063 ….058.559 R_MID.050.0.097 ….119.023 HIYLD.122.148.002 ….098.159 CONV.036.055.131 ….041.003 LBCORP.032.068.023 ….105.0 LBGVT.123.011.325 ….046.148 LBMBS.151.117.088 ….164.017 Expected Return1,770,9391,162,538759,151 … 87,86740,883 Standard Deviation105,18640,05144,333 … 5,2841,176 0.1 Percentile1,635,3571,112,691704,420 … 81,04339,420 0.2 Percentile1,683,8021,128,682722,719 … 83,44739,880 0.9 Percentile1,908,0911,215,841816,402 … 94,72642,410 Sharpe ratio.712.697.759 ….697.580

23. 23 Capital Release Measurement Technique 1.Randomly generate investment scenario. Using portfolio allocation, determine period’s return and apply to minimum sufficiency value for period 1 (MSL 1 ). 2.Generate liabilities and subtract from (1) 3.Compare (2) with MSL 2 to get observation on capital release distribution 4.Repeat steps (1) to (3) many times to obtain distribution for capital release

24. 24 Expected Capital Release

25. 25 Summary  Target sufficiency measured from bootstrapped ultimate links and paid/ultimate ratios. Risk adjustment for amount and timing of losses.  Minimum sufficiency levels derived from (1) using non-linear optimization applied to simulated asset returns and required targets. Risk adjustment for uncertainty in investment returns.  Release of capital measured from simulations of assets and liabilities using minimum sufficiency levels.

26. 26 Questions? William C. Scheel, Ph.D. DFA Technologies, LLC 93 Silkey Road North Granby, CT 06060-1419 (860) 653-7169 scheel@mindspring.com