CS-12 IAA Progress on RBCLife Case Study
Les Rehbeli
July 29, 2003
2
Contents
1. Introduction2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
3
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
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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
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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 99.5th percentile
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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
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Contents
1. Introduction
2. The Insurance Company3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
8
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
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The Insurance CompanyCompany Segmentation
Product Code Type of ProductNumber of
LivesSum Assured or Monthly Payment
ALC 1001 Term to 100 Insurance 56,971 3.6 billion
ALC 1002 Non-Par Whole Life 5,000 0.9 billion
ALC 1003 Term to 100 Insurance 94,560 9.0 billion
ALC 1004 1 Year Renewable Term 7,463 1.4 billion
ALC 1005 5 Year Renewable Term 3,450 0.5 billion
ALC 1006 Payout Annuities 250 1.5 million / month
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Total Solvency Provisions
Systematic Insurance Risks Non-Systematic Insurance Risks Market Risks
TotalProduct Segment
Mortality Level
Mortality Trend
Lapse Level
Mortality Volatility
Mortality Catastr.
Lapse Volatility Mismatch Default
T100 – 1 43.1 50.1 28.9 3.4 6.2 3.5 - - 73.7
Whole Life 43.8 17.4 7.1 3.3 3.8 3.2 - - 49.2
T100 – 2 105.7 163.6 103.3 9.5 35.1 10.9 - - 227.5
1 yr YRT 53.1 37.6 39.9 21.5 3.5 12.8 - - 86.3
5 yr YRT 8.6 5.8 3.9 3.9 4.4 2.1 - - 14.8
Total Ins. - - - - - - 335.7 3.8 335.7
Annuities 16.8 8.7 - 0.2 (0.1) - 15.7 1.4 24.7
Surplus - - - - - - - 26.7 26.7
Total 178.8 265.8 152.8 29.7 53.0 26.1 351.4 30.5 512.4
($ millions)
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Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
12
Mortality Risks
Level risk– misestimation of the mean
Trend risk– deterioration of the mean
Volatility risk– statistical fluctuations
Catastrophe risk– spike in mortality experience
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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
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20% 30% 40% 50% 60% 70% 80% 90% 100% 110% 120%
Mortality Level Risk
20% 30% 40% 50% 60% 70% 80% 90% 100% 110% 120%
% of Industry Table
Setting of Best Estimate Mortality Assumption
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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, 99th percentile of the true distribution– by using inverse Normal Power approximation– or by simulating claims experience and using 99th percentile
For case study, revalue liabilities with mortality assumption distribution to calculate CTE(99) – or simply revalue liabilities at 99.5th percentile of assumptions
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Mortality Level Risk
Mortality Assumption Percentile T100 – 1
Non-Par Whole Life T100 – 2 1 year YRT 5 year YRT
Payout Annuities
5.0 124.4 31.2 736.3 (267.1) (27.8) 271.9
25.0 144.2 46.8 787.0 (241.6) (24.0) 267.9
50.0 157.2 57.7 824.2 (225.8) (21.4) 263.8
75.0 170.0 68.9 860.6 (211.1) (19.0) 255.6
95.0 185.2 84.9 900.8 (191.5) (15.8) 252.5
99.0 195.4 95.7 921.4 (179.2) (13.7) 251.0
99.5 198.7 99.8 926.8 (174.9) (13.2) 248.0
99.9 204.2 110.5 934.8 (167.1) (12.1) 243.0
CTE(99) – CTE(0) 43.1 43.8 105.7 53.1 8.6 16.8
Liabilities ($ millions)
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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
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Mortality Trend Risk
Percentile
Annual Mortality
Improvement
0.5 1.77%
1.0 1.66%
5.0 1.32%
10.0 1.14%
30.0 0.76%
50.0 0.50%
70.0 0.24%
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
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Mortality Trend Risk
Mortality Trend
Percentile T100 – 1Non-Par
Whole Life T100 – 21 year YRT
5 year YRT
Payout Annuities Total
5.0 123.4 44.9 715.2 (249.4) (25.2) 257.3 867.2
25.0 142.8 52.5 779.2 (235.6) (23.1) 254.1 972.9
50.0 156.6 57.4 826.1 (225.9) (21.6) 251.9 1,046.0
75.0 170.3 62.2 870.5 (216.5) (20.0) 249.6 1,116.9
95.0 189.1 68.7 928.9 (202.7) (17.9) 246.4 1,212.9
99.0 201.2 72.7 966.3 (193.0) (16.5) 243.8 1,274.1
99.5 204.7 74.2 982.2 (189.9) (16.0) 242.9 1,296.1
99.9 214.0 76.8 1,014.5 (182.2) (15.0) 241.4 1,339.0
CTE(99) – CTE(0) 50.1 17.4 163.6 37.6 5.8 8.7 262.5
Liabilities ($ millions)
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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
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Mortality Volatility Risk
Mortality Volatility
Percentile T100 – 1
Non-Par Whole
Life T100 – 21 year YRT
5 year YRT
Payout Annuities
Total Correlated
Total Independent
5.0 10.5 4.9 60.1 15.9 3.5 44.6 139.5 144.6
25.0 11.2 5.5 62.4 17.3 3.9 44.7 144.8 147.8
50.0 11.8 6.0 64.2 18.6 4.3 44.7 149.6 150.4
75.0 12.5 6.7 66.2 20.4 4.8 44.8 155.5 153.4
95.0 13.7 7.9 69.7 25.1 5.9 44.9 166.4 159.1
99.0 14.7 9.0 72.5 32.1 7.2 44.9 176.7 165.5
99.5 15.1 9.3 73.6 37.0 7.9 45.0 180.7 170.0
99.9 16.1 10.1 75.6 54.1 9.9 45.0 190.3 182.7
CTE(99) – CTE(0) 3.4 3.3 9.5 21.5 3.9 0.2 31.7 22.7
Claims over two year horizon ($ millions)
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Mortality Volatility Risk
ProductCapital Based on Two
Years ClaimsCapital Based on All Liability Cash Flows
T100 – 1 3.4 6.2
Whole Life 3.3 5.4
T100 – 2 9.5 16.8
1 Year YRT 21.5 23.9
5 Year YRT 3.9 12.9
Annuities 0.2 7.6
($ millions)
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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
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Mortality Catastrophe Risk
Risk Measure
Expected Mortality
Basis T100 – 1Non-Par
Whole Life T100 – 21 year YRT
5 year YRT
Payout Annuities
CTE(99) 100% 15.3 9.5 74.0 40.8 8.3 45.0
CTE(0) 100% 11.9 6.2 64.5 19.4 4.4 44.7
Capital for volatility 3.4 3.3 9.5 21.5 3.9 0.2
CTE(99) 200% 21.5 13.3 109.0 44.3 12.8 44.9
CTE(99) 100% 15.3 9.5 74.0 40.8 8.3 45.0
Capital for catastrophe 6.2 3.8 35.1 3.5 4.4 (0.1)
Total 9.6 7.2 44.6 24.9 8.3 0.1
Claims over two year horizon ($ millions)
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Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk5. Market Risk
6. Effects of Reinsurance
26
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
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Lapse Risks
Level risk– Misestimation of the best estimate
Volatility risk– Statistical fluctuations
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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– 90th percentile lapse assumption may be 12.4% grading to 1.2%– 10th 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
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Lapse Level Risk
Lapse Level
PercentileLapse Rates T100 – 1
Non-Par Whole
Life T100 – 21 year YRT
5 year YRT
Total Correlated
Total Independent
5.0 Higher 138.1 49.2 742.5 (178.4) (17.1) 965.3 951.0
25.0 Higher 148.7 52.3 787.6 (187.9) (17.7) 1,006.1 999.7
50.0 Exp. 155.9 54.5 818.1 (196.8) (18.6) 1,033.7 1,032.2
75.0 Lower 163.2 56.5 847.0 (216.2) (20.5) 1,061.8 1,064.6
95.0 Lower 173.9 59.1 884.7 (224.2) (21.3) 1,097.5 1,105.6
99.0 Lower 181.3 60.7 910.3 (228.1) (21.7) 1,119.7 1,133.8
99.5 Lower 183.8 61.3 917.0 (236.1) (22.6) 1,126.7 1,143.1
99.9 Lower 188.9 62.4 933.4 (250.4) (24.2) 1,147.4 1,160.7
CTE(99) – CTE(0) 28.9 7.1 103.3 39.9 3.9 97.2 115.2
Liabilities ($ millions)
30
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk6. Effects of Reinsurance
31
Market Risks
Mismatch risk– ALM risk
Asset default risk– credit risk
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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
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Mismatch Risk
Percentile Insurance Annuities
5.0 294.6 221.0
25.0 406.0 226.3
50.0 489.2 230.4
75.0 577.0 236.5
95.0 807.9 243.6
99.0 841.9 246.1
99.5 842.7 246.6
99.9 843.3 247.0
CTE(99) – CTE(0) 335.7 15.7
Assets Required to Back Liabilities ($ millions)
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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
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Asset Default Risk
Capital for Asset Default
Asset TypeBook Value of Assets
Credit Risk Factors Insurance Annuity Surplus Total
Bank Notes 77.5 0.25% 0.2 0.0 0.0 0.2Corp. Bonds AAA 134.5 0.25% 0.2 0.1 0.0 0.3Corp. Bonds AA 263.7 0.50% 0.9 0.4 0.0 1.3Corp. Bonds A 286.4 1.00% 1.2 0.5 1.1 2.9
Corp. Bonds BBB 99.5 2.00% 0.9 0.4 0.6 2.0Mortgage Residential 4.0 2.00% 0.1 0.0 0.0 0.1Mortgage Commercial 8.7 4.00% 0.3 0.0 0.0 0.3
Common Stocks 145.8 15.00% 0.0 0.0 21.8 21.8Preferred Stocks 63.5 2.00% 0.0 0.0 1.3 1.3
Real Estate 15.8 4.00% 0.0 0.0 0.6 0.6Other 12.5 8.00% 0.0 0.0 1.0 1.0
Total 1,576.8 3.8 1.4 26.7 31.9
Capital Requirements ($ millions)
36
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
37
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
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Effects of Reinsurance
Reinsurance Type Ceded
Reinsurance Premiums Level Trend Volatility
Catastro-phe
Gross Basis 43.1 50.1 3.4 6.2
Coins. 45% 70% Table 20.9 20.3 1.8 3.4
Excess Retention > $50K 70% Table 22.3 21.7 0.9 3.5
Coins. 90% 70% Table 2.2 9.2 0.3 0.6
Coins. 45% 45% Table 23.3 23.4 1.9 3.5
Excess Retention > $50K 45% Table 23.6 25.2 0.9 3.6
Quota Share 45% N/A 24.3 27.2 1.9 3.4
Capital for Mortality Risks ($ millions)
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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