cs-12 iaa progress on rbc life case study les rehbeli

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

4

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

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

6

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

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

9

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)

11

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

13

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

15

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

16

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)

17

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

19

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)

20

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

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)

23

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)

25

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

27

Lapse Risks

Level risk– Misestimation of the best estimate

Volatility risk– Statistical fluctuations

28

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

32

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

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)

34

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

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

38

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)

39

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