Why Pooling Works. CAJPA Spring Mujtaba Datoo Actuarial Practice Leader, Public Entities Aon Global Risk Consulting

Transcription

1 Why Pooling Works CAJPA Spring 2017 Mujtaba Datoo Actuarial Practice Leader, Public Entities Aon Global Risk Consulting

2 Discussion Points Mathematical preliminaries Why insurance works Pooling examples Loss distributions Simulation Extreme events: low frequency, high severity To pool or not to pool 1

3 Mathematical and Statistical Preliminaries

4 Trivia Statistics Defined Latin origin: statisticum collegium meaning council of state Italian connection: statista meaning statesman German context: statistik meant analysis of data about the state English term: political arithmetic After 19th Century, statistics was introduced to refer to any collection and classification of data 3

5 Trivia Mathematics Defined Pythagorean origin That which is learned 4

6 Definitions Risk: Deviation or variability around an expected outcome Uncertainty in timing and amount of payment Liquidity: Ability to pay in cash short term liabilities, usually in one year Solvency: Ability to pay all liabilities in the long run 5

7 Law of Large Numbers Sample mean converges to population mean as sample size increases Variance of means As n gets larger, variance is minimized Coefficient of Variation (CV) Ratio of the standard deviation to the mean Normalizes the scale Compare entities (not distorted by volume or size) 6

8 Law of Large Numbers CV decreases with increasing sample size 7

9 Central Limit Theorem For data from any distribution, if we repeatedly take n independent random samples As n becomes large The distribution of the sample means will approach a normal distribution. 8

10 Common Measures of Risk Measure dispersion: Variance, standard deviation, average absolute deviation, Value at Risk (VaR), Tail Value at Risk (TVaR), Coefficient of Variation (CV) Other measures: Mean (aka average, expected), median, mode 9

11 Key Measure: Coefficient of Variation Coefficient of Variation (CV) = The smaller the CV, the smaller the variability around the average 10

12 Random Number Generation Used for simulation To mimic a distribution where standard functions cannot be accurately fit Can generate random numbers in Excel RAND() generates numbers between 0 and 1 RANDBEWTEEN(low, high) generates whole numbers between selected low and high number 11

13 Generating 1,000 Random Numbers Between 0 and Histogram Bin Freq Cum % Bin Freq Cum % Inc % % % 9.9% % 10.7% % 9.3% % 8.1% % 11.2% % 9.8% % 10.7% % 9.6% % 10.4% % 10.3% Frequency 120.0% 100.0% 80.0% 60.0% 40.0% 20.0% 0.0% 12

14 Law of Large Numbers Coin toss example Coin toss Probability (heads) = 0.5 As more coins tossed, Pr(H) 0.5 (p=1/2) e.g. random 10 tosses produces Pr(H) = 0.7, however as number of tosses increase Pr(H) will converge to 0.5 Trial # Result H = 1 T = o Sum of Heads Pr(H) (4)/(1) (1) (2) (3) (4) (5) 1 H T T H H H H T H H H

15 Simulated Coin Toss H=1, Sum Trial # Rand() Result T=0 of H Pr(H) T 0 0 0% H % T % T % T % H % T % T % T % T % T % H % T % H % H % T % T % T % H % H % 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Pr(H) Trial # 14

16 Why Insurance Works

17 Why Insurance Works Example from Society of Actuary paper Risk and Insurance The example is about the loss amount of a car owner for a certain year Loss Amount Average Loss per Policyholder Average Total Losses (μ) Probability Number of Policyholders ,000 1,000,000 $0 80% $0 $0 $ % ,000 5,000 8% 400 4, ,000 15,000 2% 300 3, ,000 $750 $750 $ , ,000 Variance (σ ) 6 million 60 million 6 billion Standard Deviation(σ) Coefficient of Variation (CV) 2,442 7,722 77,

18 Why Insurance Works People seek security, insurance spreads the risk and can provide the security To measure the potential variability of the loss (Risk), we use the standard deviation. The existence of the insurance industry does not decrease the frequency or severity of loss. Policyholders are willing to pay a gross premium for an insurance contract, which exceeds the expected value of their loss, in order to substitute the fixed low variance premium payment for an unmanageable amount of risk inherent in not insuring 17

19 Pooling Example

20 Scatter Diagram Below $100,000 Single Entity Pool $100,000 $100,000 $75,000 $75,000 $50,000 $50,000 $25,000 $25,000 $0 $0 19 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Jun-14 Jun-15 Jun-16 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Jun-14 Jun-15 Jun-16

21 Scatter Diagram Above $100,000 Single Entity Pool $1,000,000 $900,000 $800,000 $700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100, Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Jun-14 Jun-15 Jun-16 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Jun-14 Jun-15 Jun-16 $1,000,000 $900,000 $800,000 $700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100,000

22 Scatter Diagram Combined Below $100,000 Above $100,000 $100,000 $1,000,000 $900,000 $75,000 $800,000 $700,000 $50,000 $600,000 $500,000 $25,000 $400,000 $300,000 $200,000 $0 $100,000 Jun-13 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Jun-14 Jun-15 Jun-16 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-14 Jun-15 Jun-16 Single Entity Pool Single Entity Pool 21

23 Loss Rates: Pool vs Single Entities $8.00 $7.00 $6.00 $5.00 Pool A B $4.00 $3.00 C D E $2.00 F $1.00 $0.00 G H

24 Risk Measures Pool has lower CV than individual entities Std. Dev. Average CV 1.60 CV 1.51 POOL 0.3 $ A 0.3 $ B 0.5 $ C 0.9 $ D 1.3 $ E 1.6 $ F 2.1 $ G 2.6 $ H 0.8 $ Pool A B C D E F G H 23

25 Funding: Pool vs Single Entity Single Entity is relatively volatile, therefore Need more capital to absorb greater variability Potential overfunding or underfunding in any given year is more likely 24

26 Building a Workers Compensation Probability Distribution

27 WC Empirical Distribution payroll $100,000,000 frqunce per $1M 1 expected number of claims 100 avg annual salary $50,000 number of workers 2,000 Prob of claim per worker 5% prob of NO claims = 95% Med only % of claims 70% TD % of claims 25% PD% of claims 5% severity average range - can fit to a loss ditribution Med Only $1, to 1,500 TD $20,000 10,000 to 30,000 PD $200, ,000 to 300,000 Probabilties random number range NO claim 95.00% to Med only 3.50% to TD 1.25% to PD 0.25% to

28 Loss Distribution Curve Fitting

29 Fit a Curve Empirical histogram, (from scatter diagram) Fit data to standard curves Estimate parameters from data e.g. Normal (bell-shaped symmetrical curve) e.g. skewed distributions: Weibull, lognormal, Burr, Gamma, Pareto Simulation Ensure it mirrors the (empirical) data Allows interpolating or extrapolating beyond observed experience e.g. if no large claims emerged, but can extrapolate from the fitted curve 28

30 Distribution of Incurred Losses $1,000,000 $900,000 $800,000 $700,000 $600,000 $500,000 $400,000 $300,000 $200,000 $100,000 $0 29 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Jun-14 Jun-15 Jun-16

31 Claim Data Adjustments Check for reasonability Detect outliers Develop bring claims to ultimate value Trend bring to a common date, like CPI 30

32 Data Morphing Historical data may not reflect future changes e.g. medical technological improvement such as nano-surgery Impacts future medical costs Tort threshold changes, e.g. North Dakota Judicial impact e.g. Florida Castellanos, et al. Increase in life expectancy, e.g. impact on PD claims Economic disturbances, structural transformation (technology impact) Make judgmental adjustments 31

33 Stratified Losses Histogram Used to fit a curve ,000 4,000 Number of Claims 8,000 20,000 40,000 80, , , ,000 1,500,000 Claim Size

34 Fit the Best Curve: Lognormal, Weibull, Gamma Lognormal Weibull Gamma ,000 4,000 8,000 20,000 40,000 80, , , ,000 1,500,000 Number of Claims Claim Size 33

35 Excess Pool Loss Distribution Example

36 $8 $7 $6 $5 $4 $3 $2 $1 $0 35 Jun-06 Jun-07 Jun-08 Jun-09 Jun-10 Jun-11 Jun-12 Jun-13 Millions Jun-14 Jun-15 Jun-16 Distribution of Losses (Developed, Trended) Losses> $25,000 about 1,300 over 10 years

37 Methodology, Stratify Claims Above $100,000 Project losses above $100,000: Use last 10 years of data Sparse data Volatile, less predictable Use modeling: Pareto distribution 36

38 Methodology: Claims Above $100,000 Frequency x Severity Frequency Number of claims per exposure unit Adjust historical data to a common year, say 2016/17 Select from 10-year history Severity Average cost per claim Adjust historical data to a common year, say 2016/17 Fit a Pareto statistical distribution to the underlying data 37

39 10-year history of claims Developed and trended $8 Millions $7 $6 $5 $4 $3 Over 10 Years, 600 claims over $100K 250 claims over $250K 100 claims over $500K 40 claims over $1M 15 claims over $2M 6 claims over $3M 5 claims over $4M 3 claims over $5M $2 $1 $0 38

40 Pareto Statistical Distribution Used in actuarial literature to estimate large claims Calculate within each layer, say $1,000,000 to $2,000,000: % of claims Average cost per claim Has fat tail allows for very large claims to be considered 39

41 Stratify Claims Over $100,000 Fit to a theoretical curve Claims Above Threshold Number of Claims Ratio to $100,000 Claims Pareto Fit to $100,000 $100 K % 100% $250 K % 38% $500 K % 19% $1 M 40 7% 9% $2 M 15 3% 4% $3 M 6 1% 3% $4 M 5 0.8% 2.1% $5 M 3 0.5% 1.7% 40

42 Frequency Claims Above $100,000 Calculate number of claims per exposure unit Review for reasonability Select frequency, say for 2016/17 = claims for every $1 million of payroll / / / / / / / / / /16 Selected 41

43 Projected 2016/17 Claims Over $100,000 From Pareto Fit Layer Payroll Frequency % of Claims in Layer Projected Number of Claims $1 - $2 M $1.5 B % 5.0 $2 - $3 M $1.5 B % 2.5 $3 - $4 M $1.5 B % 1.6 $4 - $5 M $1.5 B %

44 Projected 2016/17 Severity Over $100,000 From Pareto Fit Layer Straight Average Pareto Fit $1 - $2 M $1,500,000 $1,683,067 $2 - $3 M 2,500,000 2,804,004 $3 - $4 M 3,500,000 3,857,807 $4 - $5 M 4,500,000 4,888,368 Pareto fit reflects skewed nature of large claims Of the 600 claims above $100,000, about 9% will exceed $1 million 43

45 Projected Losses for 2016/17 From Pareto Fit Layer Projected Number of Claims Severity Projected Losses $1 - $2 M 5.0 $683,067 $3.4 M $2 - $3 M , M $3 - $4 M , M $4 - $5 M , M 44

46 Size of Loss Distribution Layer Total Reported Claims Percent of Total Cumulative Percent of Total Total Trended Developed Losses Percent of Total Cumulative Percent of Total $25,000 to $500,000 1, % 89.8% $139,694, % 49.1% $500,000 to $1M % 96.9% 62,628, % 71.1% $1M to 2M % 99.0% 33,370, % 82.9% $2M to 3M 7 0.5% 99.5% 16,252, % 88.6% $3M to 4M 1 0.1% 99.6% 3,639, % 89.9% $4M to 5M 2 0.2% 99.8% 9,029, % 93.0% $5M to 10M 3 0.2% 100.0% 19,795, % 100.0% Over $10M 0 0.0% 100.0% 0 0.0% 100.0% Total 1, % $284,410, % 45

47 Simulation Demonstration

48 Why Simulate? Simulated results generate a distribution of aggregate losses (frequency times severity) From this aggregate distribution, can derive: Average (expected value) Percentiles Rank results in increasing order Divide (or mark off) various percentiles, e.g. 70 th, 90 th Calculate VaR, TVaR Adjust for various limits (SIR), aggregate losses, deductibles, etc. 47

49 Monte Carlo Simulation Use claim stratification information to simulate claims process Number of claims (frequency) Average claim size (severity) $6,000 $4,000 $5,000 $1,000 $2,000 $3,000 Select number of claims For each claim, select size 48

50 Monte Carlo Simulation Number of Claims 20% Uniform Distribution, i.e. equally likely to occur Poisson Distribution, Mean = 100 % of Observations 15% 10% 5% 0% Number of Claims Number of Claims 49

51 Monte Carlo Simulation Claim Size 35% Lego Distribution Lognormal Distribution 30% 25% % of Claims 20% 15% 10% 5% 0% 1,000 2,000 3,000 4,000 5,000 6,000 Claim Size ,500 10,000 50, , ,000 Claim Size 50

52 Monte Carlo Simulation Example Simulated Claims Trial Number Number of Claims Claim Amount for Claim # Aggregate Losses ,999 10,000 Average 51

53 Monte Carlo Simulation Example Simulated Claims Trial Number Number of Claims Claim Amount for Claim # Aggregate Losses 1 2 $2,000 $4,000 $6, ,000 3,000 4,000 3,000 15, ,000 3,000 7, ,000 6,000 4,000 1,000 3,000 15, ,000 4,000 4,000 2,000 16,000 9, ,000 4,000 4,000 12,000 10, ,000 3,000 3,000 4,000 3,000 15,000 Average 3.5 $12,000 52

54 Monte Carlo Simulation Claim Size Comparison 35% Lego Distribution Simulation Distribution 30% 25% % of Claims 20% 15% 10% 5% 0% 1,000 2,000 3,000 4,000 5,000 6,000 Claim Size 1,000 2,000 3,000 4,000 5,000 6,000 Claim Size 53

55 Monte Carlo Simulation Example Ranked Simulated Claims Trial Number Aggregate Losses Rank Aggregate Losses Percentile Confidence Level Factor 1 $5,348, ,884, ,925, ,551, ,166, ,647, ,522,083 9,999 3,825,756 10,000 5,353,735 Average $5,571,671 1 $2,282, ,393, ,522,083 5,000 5,310,869 7,000 6,117,601 9,000 7,615,029 10,000 27,533,442 Average $5,571,671 $6.1M / $5.6M = 70% % 1.37 $7.6M / $5.6M = 54

56 Monte Carlo Simulation Example Confidence Level 99.5% 1 in 200 year 100% 90% 80% 70% 70% confidence level % of Trials 60% 50% 40% 30% 20% 10% 0% Expected level Aggregate Losses ($Million) 55

57 Model Risk One in a 100-year event ,396 games played 215 no-hitters 19 perfect games 2,500 2, , ,

58 Model Risk Parameters shift with time 8 2,500 Frequency first 80 years % 7 Frequency next 20 years % 5 times 2, , ,

59 Risk Margin An amount that recognizes uncertainty VaR (say at 90%) minus E(X) = risk margin Other measures to set risk margin Financial ratios Risk-based capital Solvency II, ORSA Several considerations in setting risk margin Solvency Catastrophic events Ratings Etc. 58

60 Extreme Events Low Frequency, High Severity

61 Extreme Events The black swan event Cannot quantify readily extreme events in tails need for complex statistical approaches (copulas) and simulations Be wary of its limitations Isolated or correlated extreme events is where upper bound of surplus can be targeted Understand well and use judgment to evaluate 60

62 Extreme Event Pictorially Probability of Claim Costs 50 th Percentile Median Expected Claim Costs Black swan lives here! 10% 20% 30% 40% 50% 90% Assets = Reserves + Surplus RBC Expected Policyholder Deficit of 1% to 2% 61

63 VaR and TVaR Value-at-Risk (VaR) Threshold value that losses to a certain confidence level, say 95% of cases would not be exceeded Solvency II calibrates 99.5% over a one-year horizon Tail Value-at-Risk (TVaR) Takes the average of all the values in the tail above VaR threshold for a specific time period Average loss amount of extreme events Actual experience may not exist Mostly estimated by simulation 62

64 VaR and TVaR Simulation Example Trial Number Aggregate Losses Rank Aggregate Losses 1 $5,348, ,884, ,925, ,551, ,166, ,647, ,522,083 9,999 3,825,756 10,000 5,353,735 Average $5,571,671 1 $2,282, ,393, ,522,083 5,000 5,310,869 9,500 8,464,392 9,501 8,473,869 10,000 27,533,442 Average $5,571,671 VaR at 95% = $8,464,392 Average = $9,794,388 = TVaR at 95% 63

65 Asset Liability Matching Duration of investments Investment policy Restrictions, conservative (usually bonds) Payout pattern Large claim WC Liability Liquidate bond (asset) Pricing loss (risk) 64

66 The Formula That Killed Wall Street By Felix Salmom WIRED MAGAZINE

67 To Pool or Not to Pool... That is the Question

68 Minimizing TCOR Expected losses + expenses (overhead, reins, claims admin) + risk margin Most (public entities) seek stability for budgeting Stability is key Smooth out large random, volatile claims If not stable, need source of funding 67

69 Types of Risk Process risk: Associated with projection of future contingencies that are inherently variable Parameter risk: Associated with selection of parameters of the model (e.g., selecting inapplicable LDF) Model risk: Misidentifying a process model (e.g., Poisson for frequency) Surplus provides protection against variation 68

70 Risk Categories Public entities: mostly underwriting risk Investment risk Credit risk Reinsurance recoverable Other credit risks Underwriting risk Premium (pricing) risk Loss reserve risk Operating risk Catastrophe risk Floods Earthquake credit 10% premium written 32% loss reserves 27% investm't 31% 69

71 Balance Sheet $Millions Real Estate Receivables Cash Stocks Bonds Assets Other ULAE IBNR Case Reserves Liabilities Surplus 70

72 Variability Pooling in itself does NOT reduce frequency or severity Reduces variability Yields stability Higher SIRs imply greater volatility more liquidity needed More surplus needed 71

73 Pooling: Long Term Stability Pooling Produces more stable long term averages Individual entity Volatile Not conducive to budgeting 72

74 Pooling Advantages Stability of rates Economies of scale Services Loss control Litigation management Leveraging expertise Purchasing power Equity belongs to members Adds homogeneous risks to pooling, increases volume (credibility) of data Concerns (from individual entity s perspective) Assessable Joint & Several liability Sharing allocation equity Diversification Geographical Homogeneity of risks 73

75 Ken Hearnsberger, thanks Master practitioner Suggested the subject matter 74

76 Questions? Mujtaba Datoo, ACAS, MAAA, FCA Actuarial Practice Leader Aon Global Risk Consulting (949)

77 Thank You! Pyx Chamber, Westminster Abbey 76

78 Thank You! The name Pyx refers to small boxes, containing the official samples of gold and silver coinage which were also kept here. New coins were annually tested against these samples in a public Trial of the Pyx, held in the Palace of Westminster engraving, showing the Pyx Chamber still containing cupboards for state documents. 77