Quantitative Retention Management for Life Insurers

Transcription

1 Quantitative Retention Management for Life Insurers Kai Kaufhold, Ad Res Advanced Reinsurance Services GmbH 2016 SOA Life & Annuity Symposium, Nashville, TN May 16, 2016

2 Introducing the speaker Kai Kaufhold, Aktuar DAV Zurich Re, Cologne Manulife Re, Toronto & Cologne 2011 present Ad Res, Cologne Page 2

3 Agenda 1. Research objective 2. Statistical experience analysis 3. Case study Part 1 4. Simulation techniques 5. Case study Part 2 6. Implications Page 3

4 Research Objective Impact of Reinsurance on Reserves and Capital Principles-Based Reserves Economic Capital Page 4

5 Principles Based Reserves Probabilistic Valuation required: 1. Experience analysis Relevant and statistically credible 2. Measure uncertainty Adverse Deviation Estimation Error 3. Use Test: Company s overall risk assessment process Page 5

6 Principles Based Reserves NAIC: Valuation Manual 05, Section 12. A. (1) Quantify benefits ( ) and their risks at a level of conservatism that reflects conditions that include unfavorable events that have a reasonable probability of occurring during the lifetime of the contracts. ( ) (2) Incorporate assumptions, risk analysis methods ( ) that are consistent with ( ) those utilized by the company s overall risk assessment process ( ). (3) ( ) b. i) Be established using the company s available experience, to the extent that it is relevant and statistically credible; ( ) (4) Provide margins for uncertainty, including adverse deviation and estimation error, such that the greater the uncertainty the larger the margin and the resulting reserve. Page 6

7 Principles Based Reserves NAIC: Valuation Manual 05, Section 12. A. (1) Quantify benefits ( ) and their risks at a level of conservatism that reflects conditions that include unfavorable events that have a reasonable probabilityof occurring during the lifetime of the contracts. ( ) (2) Incorporate assumptions, risk analysis methods ( ) that are consistent with ( ) those utilized by the company s overall risk assessment process( ). (3) ( ) b. i) Be established using the company s available experience, to the extent that it is relevant and statistically credible; ( ) (4) Provide margins for uncertainty, including adverse deviation and estimationerror, such that the greater the uncertainty the larger the margin and the resulting reserve. Page 7

8 Economic Capital Protection against Insolvency: Risk-Based Capital USA Minimum Continuing Capital & Surplus Requirements Canada Individual Capital Assessments United Kingdom Swiss Solvency Test Switzerland Solvency II European Union Quantify All Risks. One-Year View. Page 8

9 Reserve & Capital Margin Quantify All Risks. 1. Model risk 2. Volatility 3. Misestimation 4. Trend risk Page 9

10 Reserve & Capital Margin Quantify All Risks. 1. Model risk Estimate impact 2. Volatility Random mortality events 3. Misestimation Experience data finite 4. Trend risk Important... another day! Page 10

11 Impact of Reinsurance Margins for Uncertainty. 1. Different forms of reinsurance. Proportional / non-prop reinsurance 2. Reduction in reserve margins? 3. Reduction in solvency capital requirements? Page 11

12 Project Outline Experience Data Statistical Model Monte Carlo Simulation Reserve Margins Impact of Reinsurance 1. Seriatim experience data. 2. Survival Model fitted to the data. 3. Use model output to run simulations. 4. Calculate reserve and capital margins. 5. Change after reinsurance? Page 12

13 Questions for Audience What do you expect: 1. Which factors have an impact on reserves & capital? 2. Does reinsurance make a difference? Page 13

14 Agenda Statistical experience analysis 3. Case study Part 1 4. Simulation techniques 5. Case study Part 2 6. Implications Page 14

15 Survival Models Why Statistical Experience Analysis? 1. Impact of different risk factors (Multivariate analysis) 2. Limit model risk 3. Measure misestimation risk Page 15

16 Survival Models Age at entry, observed time, status variable Death within study period left-truncated data + =1 Survival past end of study period left-truncated and right-censored data + =0 Page 16

17 Experience Data 0-1 Crude Mortality Hazard Rates ln ( actual deaths / exposure ) ln =+ Male Female Attained Age Page 17

18 Experience Data 0 Crude Lapse Hazard Rates ln ( actual lapses / exposure ) -0,5-1 -1,5-2 -2,5-3 -3,5-4 ln = Attained Age Male Female Page 18

19 Survival Models Survival within Portfolio =exp ( ) =exp ( ) Deaths Lapses Total Page 19

20 Model Risk Bootstrap experiment: Random sample of 10,000 lives. Compare actual to modeled deaths. "#$% &$' / &"& &$' %)*& 100%, $./#0 86% Include more risk factors, until $./#0 100% Page 20

21 Model Risk 200% Bootstrap Experiment 180% Actual deaths / Expected deaths 160% 140% 120% 100% 80% 60% 40% Amounts A/E Lives A/E Page 21

22 Model Risk 200% Bootstrap Experiment 180% Actual deaths / Expected deaths 160% 140% 120% 100% 80% 60% 40% Amounts A/E Lives A/E Page 22

23 Agenda Case study Part 1 4. Simulation techniques 5. Case study Part 2 6. Implications Page 23

24 Case Study: US Term Life Overview of Data Business mix 1.6 million lives in total. Product: 10 Product: 15 Product: 20 Product: 30 Product: Other term Page 24

25 Concentration Risk 100% 80% 60% 40% 20% 0% Lives Amounts distribution Amounts Amount band 8 Amount band 7 Amount band 6 Amount band 5 Amount band 4 Amount band 3 Amount band 2 Amount band 1 Amount band Up to $100k $125k $250k $300k $500k $750k $2m unlimited Page 25

26 Risk Factors: Death Category Risk Group Estimate Std error Significance Gender Male 1.00 baseline Female *** Underwriting Preferred *** Residual Std 1.00 baseline Smoker *** Substandard *** Page 26

27 Risk Factors: Death Category Risk Group Estimate Std error Significance Policy Size Band *** Bands 2 & *** Bands Baseline Product T10, T15, T baseline T *** Other term *** Page 27

28 Survival Model Results 120% Select Mortality Pattern Percentage of Ultimate Mortality 100% 80% 60% 40% 20% 0% Policy Year Page 28

29 Risk Factors: Lapse Category Risk Group Estimate Std error Significance Gender Male baseline Female *** Underwriting Preferred *** Residual Std baseline Smoker *** Substandard *** Page 29

30 Risk Factors: Lapse Category Risk Group Estimate Std error Significance Policy Size Bands 1 5 baseline Band *** Band *** Band *** Product T *** T10 baseline T *** T *** Other term *** Page 30

31 Survival Model Results 35% Crude Lapse Hazard Rate Actual lapses / Exposure 30% 25% 20% 15% 10% 5% 0% Policy Year Page 31

32 Agenda Simulation techniques 5. Case study Part 2 6. Implications Page 32

33 Simulation Two step Monte-Carlo simulation: 1. Perturb survival model parameters 2. Simulate future lifetimes Page 33

34 Monte Carlo Simulation 1. Perturb model parameters: Random, but consistent with data Replace:,, 5, 5,, =, :9 78 9: 78 :: < = < > Page 34

35 Monte Carlo Simulation 2. Simulate future lifetimes: a. Draw random number # b. Apply inverse cumulative distribution Probability of exit F(t) # Simulation of one 52-year-old, male, preferred NS Duration t # Page 35

36 Monte Carlo Simulation Page 36

37 Monte Carlo Simulation Random Scatter: Future Lifetimes Remaining lifetime [years] Simulation Illustration for 52-year-old, male, preferred NS Page 37

38 Monte Carlo Simulation Age at death Frequency * Age Illustration for 52-year-old, male, preferred NS Page 38

39 Reinsurance Impact 14,0% 12,0% 10,0% 8,0% 6,0% 4,0% 2,0% 0,0% Illustration for T10 Distribution of liabilities Fully retained Retention $750k Total liabilities [$ millions] Page 39

40 Agenda Case study Part 2 6. Implications Page 40

41 Simulation Results Claims Distributions for different retentions $ $ $ $ $ $ $ $ % 12% 10% Frequency 8% 6% 4% 2% 0% $1.720 $2.000 $2.270 $2.540 $2.820 $3.090 $3.370 $3.640 $3.910 $4.190 $4.460 $4.740 $5.010 $5.280 $5.560 $5.830 $6.100 Total PV of Claims [in $ millions] Page 41

42 Simulation Results $7.000 $6.000 Liabilities against Time PV of Future Claims in US$'000'000 $5.000 $4.000 $3.000 $2.000 $1.000 $ Calendar Year $ $ $ $ $ $ $ $ Page 42

43 Simulation Results 2,80% Reserves (CTE70 ) 2,75% 2,70% (X - BEL) / BEL 2,65% 2,60% 2,55% 2,50% 2,45% 2,40% 2,35% $- $1.000 $2.000 $3.000 $4.000 $5.000 Retention [$k] Page 43

44 Simulation Results Retention Best-Estimate Liabilities Reserve CTE70 at t=0 Reserve Margin $100k $1.87bn $1.91bn 2.40% $500k $4.63bn $4.75bn 2.62% $1m $5.48bn $5.63bn 2.70% $5m $5.96bn $6.12bn 2.79% $150m $5.99bn $6.16bn 2.83% Page 44

45 Simulation Results $70 Capital against time Total Capital Required in US$'000'000 $60 $50 $40 $30 $20 $10 $ Calendar Year $ $ $ $ $ $ $ $ Page 45

46 Simulation Results 1,0% Economic Capital (CTE99) 0,9% (X - BEL) / BEL 0,8% 0,7% 0,6% 0,5% $- $1.000 $2.000 $3.000 $4.000 $5.000 Retention [$k] Page 46

47 Simulation Results Retention Best-Estimate Liabilities Capital (CTE99) Capital Margin $100k $1.87bn $12.1m 0.65% $500k $4.63bn $34.9m 0.75% $1m $5.48bn $45.7m 0.83% $5m $5.96bn $58.7m 0.98% $150m $5.99bn $65.0m 1.09% Page 47

48 Simulation Results Impact of Reinsurance Percetnage of Fully Retained 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% $- $1.000 $2.000 $3.000 $4.000 $5.000 Retention [$k] Capital Impact Reserve Impact Page 48

49 Run-off Capital 8,00% Capital Margin against time Economic Capital / Liabilities 7,00% 6,00% 5,00% 4,00% 3,00% 2,00% 1,00% 0,00% Calendar Year $ $ $ $ $ $ $ $ Page 49

50 Different Blocks Random Sample Number of Lives Best- Estimate Liabilities Reserve Margin Capital Margin 10% 94,140 $556m 9.1% 4.1% 25% 235,980 $1.48bn 5.3% 2.3% 50% 472,350 $2.83bn 3.9% 1.5% 100% 943,250 $5.99bn 2.8% 1.1% Page 50

51 Findings for Reserves 1. Reinsurance has limited influence on margins, 2. Reserve margins are greater for smaller blocks of business. Page 51

52 Findings for Capital 1. Surplus reinsurance reduces Economic Capital significantly. 2. Impact depends on business mix. 3. Importance of reinsurance increases over time. Page 52

53 Optimal Retention Retention Best-Estimate Liabilities Capital Capital Margin Return on EC $100k $1.87bn $12.1m 0.65% 11.2% $500k $4.63bn $34.9m 0.75% 16.3% $1m $5.48bn $45.7m 0.83% 15.4% $5m $5.96bn $58.7m 0.98% 13.2% $150m $5.99bn $65.0m 1.09% 12.0% Assumption: Base Return on Economic Capital = 12%; Cost of R/I = 20% of expected profit Page 53

54 Simulation Results 17% Optimal Retention (Profit - cost of RI/) / EC 16% 15% 14% 13% 12% 11% 10% $- $250 $500 $750 $1.000 $1.250 $1.500 $1.750 $2.000 Retention [$k] Assumption: Base RoEC = 12%; Cost of R/I = 20% of expected profits Page 54

55 Implications 1. Reinsurance levels playing field for small companies. ( 1 st $ quota share) 2. Larger companies can optimize return on Economic Capital by means of reinsurance. ( surplus reinsurance) 3. It pays to model your mortality and lapse experience. Page 55

56 Summary 1. Small companies to hold greater reserve margins irrespective of reinsurance. 2. Reinsurance has greater impact on economic capital than on reserves. 3. Impact of reinsurance depends on business mix. 4. Life insurers can optimize their retention level based on economic capital. Page 56

57 References Gompertz, B. (1825) The nature of the function expressive of the law of human mortality, Philosophical Transactions of the Royal Society 115: Richards, S.J. (2016) Mis-estimation risk: measurement and impact. Presented at IFoA Sessional Meeting 29 th February Kaufhold, K. and W. Lennartz (2016) Quantitative Retention Management for Life Insurers. SOA Research Report (to be published). Page 57