1. CS-12 IAA Progress on RBC Life Case Study Les Rehbeli July 29, 2003

2. 1 Notes: Contents 1.Introduction 2.The Insurance Company 3.Mortality Risk 4.Lapse Risk 5.Market Risk 6.Effects of Reinsurance

3. 2 Notes: Introduction Purpose of case study – To demonstrate approaches to determine solvency provisions for various risks – To illustrate concepts for advanced internal modeling – To highlight issues a factor-based approach must address

4. 3 Notes: Internal Modeling Develop models to quantify various risks being considered – Analyze each risk separately  Generate scenarios in which liabilities vary only on the risk being measured – Aggregate into total company solvency requirement Focus on total solvency provisions – Sum of reserves and capital

5. 4 Notes: Internal Modeling Model cash flows over time horizon appropriate to risk being modeled – Systematic (non-diversifiable) risks over entire term of liability – Non-systematic (diversifiable) risks over shorter horizon Liabilities defined as present value of future liability cash flows discounted at risk-free rate Solvency provision defined as difference between average liabilities of worst 1% of scenarios and best estimate liabilities – CTE(99) minus CTE(0)  approximately equivalent to 99.5 th percentile

6. 5 Notes: Risks Analyzed in the Case Study Mortality (systematic risks) – Mortality level risk – Mortality trend risk Lapse (systematic risks) – Lapse level risk Non-systematic insurance risks – Mortality volatility risk – Mortality catastrophe risk – Lapse volatility risk Market risks – Credit risk – Mismatch risk

7. 6 Notes: Contents 1.Introduction 2.The Insurance Company 3.Mortality Risk 4.Lapse Risk 5.Market Risk 6.Effects of Reinsurance

8. 7 Notes: The Insurance Company Medium-sized insurance company – term, whole life and immediate annuity non-participating products Assets managed at the segment level – segments for insurance products, annuity products and surplus – liabilities supported by high grade fixed income securities – surplus also invested in stocks Various reinsurance arrangements in place

9. 8 Notes: The Insurance Company Company Segmentation Product CodeType of Product Number of Lives Sum Assured or Monthly Payment ALC 1001Term to 100 Insurance56,9713.6 billion ALC 1002Non-Par Whole Life5,0000.9 billion ALC 1003Term to 100 Insurance94,5609.0 billion ALC 10041 Year Renewable Term7,4631.4 billion ALC 10055 Year Renewable Term3,4500.5 billion ALC 1006Payout Annuities2501.5 million / month

10. 9 Notes: Total Solvency Provisions Systematic Insurance Risks Non-Systematic Insurance Risks Market Risks Total Product Segment Mortality Level Mortality Trend Lapse Level Mortality Volatility Mortality Catastr. Lapse VolatilityMismatchDefault T100 – 1 43.150.128.93.46.23.5--73.7 Whole Life 43.817.47.13.33.83.2--49.2 T100 – 2 105.7163.6103.39.535.110.9--227.5 1 yr YRT 53.137.639.921.53.512.8--86.3 5 yr YRT 8.65.83.9 4.42.1--14.8 Total Ins. ------335.73.8335.7 Annuities 16.88.7-0.2(0.1)-15.71.424.7 Surplus -------26.7 Total 178.8265.8152.829.753.026.1351.430.5512.4 ($ millions)

11. 10 Notes: Contents 1.Introduction 2.The Insurance Company 3.Mortality Risk 4.Lapse Risk 5.Market Risk 6.Effects of Reinsurance

12. 11 Notes: Mortality Risks Level risk – misestimation of the mean Trend risk – deterioration of the mean Volatility risk – statistical fluctuations Catastrophe risk – spike in mortality experience

13. 12 Notes: Mortality Level Risk Misestimation of the mean Mortality assumptions based on mortality studies and industry data – but mortality studies are based on observations that are volatile In a mortality study, we may presume that historical observations represent the best estimate level of mortality – but it is possible that the observations are in the tail of the true mortality distribution

14. 13 Notes: Mortality Level Risk % of Industry Table Setting of Best Estimate Mortality Assumption

15. 14 Notes: Mortality Level Risk The smaller the portfolio, the larger the range of possible outcomes for future mortality – might also partially rely on industry data To evaluate mortality level risk, assume that observations were actually at, say, 99 th percentile of the true distribution – by using inverse Normal Power approximation – or by simulating claims experience and using 99 th percentile For case study, revalue liabilities with mortality assumption distribution to calculate CTE(99) – or simply revalue liabilities at 99.5 th percentile of assumptions

16. 15 Notes: Mortality Level Risk Mortality Assumption PercentileT100 – 1 Non-Par Whole LifeT100 – 21 year YRT5 year YRT Payout Annuities 5.0124.431.2736.3(267.1)(27.8)271.9 25.0144.246.8787.0(241.6)(24.0)267.9 50.0157.257.7824.2(225.8)(21.4)263.8 75.0170.068.9860.6(211.1)(19.0)255.6 95.0185.284.9900.8(191.5)(15.8)252.5 99.0195.495.7921.4(179.2)(13.7)251.0 99.5198.799.8926.8(174.9)(13.2)248.0 99.9204.2110.5934.8(167.1)(12.1)243.0 CTE(99) – CTE(0) 43.143.8105.753.18.616.8 Liabilities ($ millions)

17. 16 Notes: Mortality Trend Risk Deterioration of the mean – misestimation of the trend We can estimate a “best estimate trend” based on past observations and expert opinions – uncertain due to volatility in past observations – also due to systematic changes in the trend Quantify trend uncertainty by revaluing liabilities under other trend assumptions

18. 17 Notes: Mortality Trend Risk Percentile Annual Mortality Improvement 0.51.77% 1.01.66% 5.01.32% 10.01.14% 30.00.76% 50.00.50% 70.00.24% 90.0-0.14% 95.0-0.32% 99.0-0.66% 99.5-0.76% For case study, assume annual rate of mortality improvement is normally distributed – mean and standard deviation of 0.50% improvement per year – limit improvement to 40 years – limit range to -3.0% and 3.0% Apply to all products simultaneously – determine which direction will increase liabilities on a company basis – consider reinsurance

19. 18 Notes: Mortality Trend Risk Mortality Trend PercentileT100 – 1 Non-Par Whole LifeT100 – 2 1 year YRT 5 year YRT Payout AnnuitiesTotal 5.0123.444.9715.2(249.4)(25.2)257.3867.2 25.0142.852.5779.2(235.6)(23.1)254.1972.9 50.0156.657.4826.1(225.9)(21.6)251.91,046.0 75.0170.362.2870.5(216.5)(20.0)249.61,116.9 95.0189.168.7928.9(202.7)(17.9)246.41,212.9 99.0201.272.7966.3(193.0)(16.5)243.81,274.1 99.5204.774.2982.2(189.9)(16.0)242.91,296.1 99.9214.076.81,014.5(182.2)(15.0)241.41,339.0 CTE(99) – CTE(0) 50.117.4163.637.65.88.7262.5 Liabilities ($ millions)

20. 19 Notes: Mortality Volatility Risk Statistical fluctuations around the expected assumptions – assume that the best estimate assumption is correct Time horizon – level and trend risks were measured over the entire term of the liability – volatility risk can be diversified by management action  project out for a two year time horizon Simulation approach taken for case study – analytic methods are also feasible to quantify volatility risk

21. 20 Notes: Mortality Volatility Risk Mortality Volatility PercentileT100 – 1 Non-Par Whole LifeT100 – 2 1 year YRT 5 year YRT Payout Annuities Total Correlated Total Independent 5.010.54.960.115.93.544.6139.5144.6 25.011.25.562.417.33.944.7144.8147.8 50.011.86.064.218.64.344.7149.6150.4 75.012.56.766.220.44.844.8155.5153.4 95.013.77.969.725.15.944.9166.4159.1 99.014.79.072.532.17.244.9176.7165.5 99.515.19.373.637.07.945.0180.7170.0 99.916.110.175.654.19.945.0190.3182.7 CTE(99) – CTE(0) 3.43.39.521.53.90.231.722.7 Claims over two year horizon ($ millions)

22. 21 Notes: Mortality Volatility Risk Product Capital Based on Two Years Claims Capital Based on All Liability Cash Flows T100 – 13.46.2 Whole Life3.35.4 T100 – 29.516.8 1 Year YRT21.523.9 5 Year YRT3.912.9 Annuities0.27.6 ($ millions)

23. 22 Notes: Mortality Catastrophe Risk One-time spike in mortality experience – for example, Spanish Flu Highly subjective Deterministic approach taken for case study – doubling of mortality for one year Interaction between catastrophe risk and volatility risk – capital for catastrophe risk is difference between CTE(99) at higher mortality and CTE(99) at normal mortality

24. 23 Notes: Mortality Catastrophe Risk Risk Measure Expected Mortality BasisT100 – 1 Non-Par Whole LifeT100 – 21 year YRT 5 year YRT Payout Annuities CTE(99)100%15.39.574.040.88.345.0 CTE(0)100%11.96.264.519.44.444.7 Capital for volatility 3.43.39.521.53.90.2 CTE(99)200%21.513.3109.044.312.844.9 CTE(99)100%15.39.574.040.88.345.0 Capital for catastrophe 6.23.835.13.54.4(0.1) Total9.67.244.624.98.30.1 Claims over two year horizon ($ millions)

25. 24 Notes: Contents 1.Introduction 2.The Insurance Company 3.Mortality Risk 4.Lapse Risk 5.Market Risk 6.Effects of Reinsurance

26. 25 Notes: Lapse Risks Can be analyzed in similar fashion to mortality risks But several other factors to consider: – lapse rates may be correlated with economic assumptions for some portfolios  very difficult to model – lapse assumption highly dependent on product and how it is sold – impact to company can vary for different policy durations and products Case study analyzes inaccuracies due to statistical error

27. 26 Notes: Lapse Risks Level risk – Misestimation of the best estimate Volatility risk – Statistical fluctuations

28. 27 Notes: Lapse Level Risk Misestimation of the best estimate From lapse studies, we can determine best estimate lapse rates and their standard deviations – we can assume a distribution for the lapse rates and solve for lapse rates at alternate percentiles  e.g. assume lapses are normally distributed and grade from 10% to 1% over 12 years –90 th percentile lapse assumption may be 12.4% grading to 1.2% –10 th percentile lapse assumption may be 8.7% grading to 0.8% Need to account for policyholder behavior / economic environment Statistical error may not always be one-sided

29. 28 Notes: Lapse Level Risk Lapse Level Percentile Lapse RatesT100 – 1 Non-Par Whole LifeT100 – 2 1 year YRT 5 year YRT Total Correlated Total Independent 5.0Higher138.149.2742.5(178.4)(17.1)965.3951.0 25.0Higher148.752.3787.6(187.9)(17.7)1,006.1999.7 50.0Exp.155.954.5818.1(196.8)(18.6)1,033.71,032.2 75.0Lower163.256.5847.0(216.2)(20.5)1,061.81,064.6 95.0Lower173.959.1884.7(224.2)(21.3)1,097.51,105.6 99.0Lower181.360.7910.3(228.1)(21.7)1,119.71,133.8 99.5Lower183.861.3917.0(236.1)(22.6)1,126.71,143.1 99.9Lower188.962.4933.4(250.4)(24.2)1,147.41,160.7 CTE(99) – CTE(0)28.97.1103.339.93.997.2115.2 Liabilities ($ millions)

30. 29 Notes: Contents 1.Introduction 2.The Insurance Company 3.Mortality Risk 4.Lapse Risk 5.Market Risk 6.Effects of Reinsurance

31. 30 Notes: Market Risks Mismatch risk – ALM risk Asset default risk – credit risk

32. 31 Notes: Mismatch Risk ALM risk – the risk that best estimate asset cash flows do not match best estimate liability cash flows – reinvestment and disinvestment risk – the risk that the market price of assets changes unfavorably at a time when those assets need to be liquidated Case study projects best estimate asset and liability liabilities under many future reinvestment rate scenarios

33. 32 Notes: Mismatch Risk PercentileInsuranceAnnuities 5.0294.6221.0 25.0406.0226.3 50.0489.2230.4 75.0577.0236.5 95.0807.9243.6 99.0841.9246.1 99.5842.7246.6 99.9843.3247.0 CTE(99) – CTE(0)335.715.7 Assets Required to Back Liabilities ($ millions)

34. 33 Notes: Asset Default Risk Credit risk Case study uses factors derived from existing regulatory regime Since other provisions for risk use the risk-free discount rate, the provision for credit risk on assets backing liabilities is not necessary  included all assets in case study for demonstration purposes

35. 34 Notes: Asset Default Risk Capital for Asset Default Asset Type Book Value of Assets Credit Risk FactorsInsuranceAnnuitySurplusTotal Bank Notes 77.50.25%0.20.0 0.2 Corp. Bonds AAA 134.50.25%0.20.10.00.3 Corp. Bonds AA 263.70.50%0.90.40.01.3 Corp. Bonds A 286.41.00%1.20.51.12.9 Corp. Bonds BBB 99.52.00%0.90.40.62.0 Mortgage Residential 4.02.00%0.10.0 0.1 Mortgage Commercial 8.74.00%0.30.0 0.3 Common Stocks 145.815.00%0.0 21.8 Preferred Stocks 63.52.00%0.0 1.3 Real Estate 15.84.00%0.0 0.6 Other 12.58.00%0.0 1.0 Total1,576.83.81.426.731.9 Capital Requirements ($ millions)

36. 35 Notes: Contents 1.Introduction 2.The Insurance Company 3.Mortality Risk 4.Lapse Risk 5.Market Risk 6.Effects of Reinsurance

37. 36 Notes: Effects of Reinsurance Factor-based systems cannot fully capture the characteristics of the risks a company faces – especially when reinsurance is used Case study analyzes six reinsurance arrangements: – YRT 45% coinsurance at neutral reinsurance rates – YRT excess reinsurance at neutral insurance rates – YRT 90% coinsurance at neutral reinsurance rates – YRT 45% coinsurance at low reinsurance rates – YRT excess reinsurance at low insurance rates – Quota share

38. 37 Notes: Effects of Reinsurance Reinsurance TypeCeded Reinsurance PremiumsLevelTrendVolatility Catastro- phe Gross Basis43.150.13.46.2 Coins.45%70% Table20.920.31.83.4 Excess Retention > $50K70% Table22.321.70.93.5 Coins.90%70% Table2.29.20.30.6 Coins.45%45% Table23.323.41.93.5 Excess Retention > $50K45% Table23.625.20.93.6 Quota Share 45%N/A24.327.21.93.4 Capital for Mortality Risks ($ millions)

39. 38 Notes: Conclusions Advanced models can be developed to better understand the net risks faced by an insurance company These models can be used to develop a standardized approach for risks that are well understood and for which there is ample historical data – difficult to accurately capture the impact of reinsurance Must exercise care for risks not modeled in the case study: – impact of policyholder behavior – complex options in policies – complex interactions between risks