1. Best Estimates for Reserves Glen Barnett and Ben Zehnwirth

2. Overview Assessing ratios (loss development) (old paradigm) –the mathematical framework underlying ratio techniques –gives a tool for assessing whether ratio techniques ‘work’ –Find: often powerless to provide meaningful forecasts The solution? (new paradigm) –a probabilistic modelling framework –powerful and flexible enough to capture the information in the data –benefits unimaginable under the old paradigm – Segmentation, Pricing Excess Layers, Value-at-risk

3. - Real Sample: x 1,…, x n - Fitted Distribution fitted lognormal - Random Sample from fitted distribution: y 1,…, y n What does it mean to say a model gives a good fit? e.g. lognormal fit to claim size distribution Model has probabilistic mechanisms that can reproduce the data Does not mean we think the model generated the data Introduction y’s look like x’s: —

4. PROBABILISTIC MODEL Real Data S1 S2 Simulated triangles cannot be distinguished from real data – similar trends, trend changes in same periods, same amount of random variation about trends Introduction

5. 8,269 / 5,012= 1.65 8,992 / 3,410= 2.64 Ratio (loss development) techniques start with a (cumulative) loss development array: Ratio techniques - analysed by Mack (1993) Incurred Losses, Historical Loss Development Study, 1991 Ed. (RAA)

6. We can graph the points in a cumulative array: Ratio techniques 8,269 5,012

7. Any ratio is the slope of a line through the origin. Ratio techniques Slope= rise/run (the ratio) = 8,269/5,012 = 1.65 What is the SLOPE of this line? Cum.(1) vs Cum.(0) 4,0002,0000 12,000 11,000 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0 2.64 1.65 8,269 5,012

8. Ratio techniques Average ratios are “average” trends through the origin Select ratios to be typical (“average”) ratios e.g. arithmetic average, chain ladder, average of last three years,...

9. Arithmetic Average ratio 1 / n  (y/x) Chain Ladder Ratio =  y/  x (Volume weighted average) Regression Line through origin Ratio techniques BUT an average trend line through the origin is a regression line through the origin! Average ratio techniques are forms of regression.

10. j-1 j x{x{ }y}y (Chain Ladder - Mack 1993) Regression equation: y = bx + , where E(  ) = 0 and Var(  ) =  2 x Ratio techniques If ratios are regressions, we can describe ratio techniques using formal statistical language. Equivalently E( y | x ) = bx and Var( y | x ) =  2 x

11. which is the Chain Ladder Ratio (or Weighted Average by Volume, or Volume Weighted Ratio) Ratio techniques To determine an average ratio (slope) b we use weighted least squares. Method aims to estimate b by minimising  w(y-bx) 2, where weight w = 1 / x  1 / Var(  ) We obtain b = = = ^  x y · 1 / x  x 2 · 1 / x  y / x · x  x x  y y  x x

12. Ratio techniques We have now seen that the Chain Ladder Ratio is a regression estimator through the origin Mack (‘93) ctd: Can generalise by Var (  )  2 x  By changing  obtain other ‘averages’: –  = 2: Arithmetic Average –  = 1: Chain Ladder (Average Weighted by Volume) –  = 0: Ordinary regression through the origin (Wtd Volume 2 Ratio) Any average ratio is a regression estimator through the origin

13.  We know that when we apply a ratio technique, we are actually fitting a regression model through the origin.  Any regression model is based on a set of assumptions.  If the assumptions of a model are not supported by the data then any subsequent calculations made using the model (eg forecasts) are meaningless – they are based on the model, not the data.  Regression methodology allows us to test the assumptions made by a model. Assessing ratio techniques How does this formal regression methodology help us to assess whether ratio techniques ‘work’?

14. 1E( y | x ) = bx –i.e. to obtain the mean cumulative at development period j, take the cumulative at the previous period and multiply by the ratio. 2No trend in the payment period (diagonal) direction –ratio techniques do not allow for changes in inflation. 3Normality –the models assume that observations are values from a normal distribution. Assessing ratio techniques What are the major assumptions made by the models based on ratio techniques?

15. Fitted values - value given by the model (the value on the line) called the fitted (or predicted) value, y Assessing ratio techniques ^ Fitted line x y xixi yiyi yiyi ^

16. Assessing ratio techniques Residual = Observed value - Fitted value Fitted line x y xixi residual yiyi yiyi yiyi ^

17. Residual Analysis Residual = Data – Fit Raw residuals have different standard deviations - need to adjust to make them comparable Many model checks use standardized residuals Standardized residual = Residual Std. dev.(residual) ______________ Assessing ratio techniques

18. What can we do with the residuals? What features of the data does this model not capture? 1990 6 1996 x e.g.: Plot vs dev. yr Plot vs pmt. yr Plot vs acci. yr Plot vs fitted Residual plots should appear random about 0, without pattern. Assessing ratio techniques

19. Is E( y | x ) = bx satisfied by the Mack data? Std. Residualsvs. Fitted Values 050001000015000200002500030000 0 1 2 Underfit small values Overfit large values

20. 1982 is underfitted 1982: low incurred development 1984: high incurred development 1984 is overfitted Why is E(y|x) = bx not satisfied by the data? Assessing ratio techniques

21. Regression line through origin causes underfit/overfit ‘Best’ line Overfit large values Underfit small values Typical with real, exposure-adjusted data. ’Best’ line not through origin - has an intercept.

22. Assessing ratio techniques Including an intercept term : y = a + bx + , or equivalently y–x = a + (b – 1) x + , IncrementalCumulative at j at j -1 j-1 j }y-x Cumulative Incremental (Murphy 1995)

23. Assessing ratio techniques y - x = a + (b - 1)x +  Is b - 1 significantly different from zero? (Venter 1998) Case 1: b = 1 a ≠ 0 - ratio has no power to predict next incremental - abandon ratios and predict next incremental by: a = Average(incrementals) (  =0) Case 2: b > 1 - could use link-ratio techniques for projection with possibly an intercept ^

24. “Ratio not a predictor of future emergence” Assessing ratio techniques Mack data:

25. Ratios do seem to work for some arrays... Assessing ratio techniques What about Case 2: b > 1, a = 0? ABC

26. ... most often arrays with trends down the accident years An increasing trend down the accident years ‘induces’ a correlation between (y-x) and x. Assessing ratio techniques

27. Condition 1: Condition 2:            y-x w w j-1 j }y-x Incremental x{x{ j-1 j Cumulative }y}y x{x{ y-x w  Constant Trend            y - x = a + (b - 1) x +  usually ratios not helpful ratios work: - see acc.yr. trend

28. Assessing ratio techniques y – x = a 0 + a 1 w + (b – 1) x +  InterceptAcc Yr trend ‘Ratio’ This gives the Extended Link Ratio Family of models Include a trend parameter in the model: j-1 j x{x{ }y}y 1 2 n  w

29. Assessing ratio techniques Case 3: b = 1, a 1  0 abandon ratios – model ‘correlation’ using trend parameter After adjusting for accident trends, the relationship with the previous cumulative often disappears.

30. Assessing ratio techniques Summary of Assumption 1: E( y | x ) = bx Very often not satisfied by the data –residuals suggest intercept is needed Include intercept term: –Case 1: ratios abandoned in favour of intercepts –Case 2: projection using link ratios A relationship between y and x is often better modeled using a trend parameter –Case 3: ratios abandoned in favour of trend terms ‘Optimal’ model in the ELRF is not likely to include ratios!

31. Assessing ratio techniques Assumption 2: Are there trends in the payment period direction? Commonly have changing payment period trends Indicated here by Chain Ladder Ratio model residuals Assumption 2 is not supported by this data

32. Assessing ratio techniques Why are changing payment period trends important? 01 1990 1991 1992 1993 Change in inflation between 1992 and 1993 - Relationship between development years differs over accident years - Must account for payment trends, or can’t understand development patterns Average ratio incorporates effect of change in inflation

33. 10%10%0%  Incremental 1600248029643206 176027283212 19362904 1936 Cumulative Ave ratio:1.531.191.08 968484242 1033553288Forecast Incremental: Actual Incremental: % increase over actual: Implied inflation: 6.7%14.3%18.8% 6.7% 7.1% 4.0% Final acci. year: Assessing ratio techniques What are you projecting? 1600 800 400 200 1600 880 484 242 1760 968 484 1936 968 1936 Very unlikely you’d want to assume this!

34. - Cumulatives “smear” over the breaks - With randomness and cumulation means no hope of seeing the trend changes - Lack control over future inflation assumption - Even ‘best’ model in ELRF can’t capture such trend changes Assessing ratio techniques

35. Pan6 Assumption 3: Often have non-normality: Considerable right-skewness - even if model for mean was correct, estimates may be very poor

36. Assessing ratio techniques Combining answers from several techniques - It is a misguided exercise to combine information from several models without assessing their appropriateness (fidelity to data, assumptions, validation) — why mix good with bad? - Answers should not be selected merely on the basis of their similarity to each other — not borrowing strength - Projections from the best models unlikely to come from the centre of the range of answers.

37. Assessing ratio techniques - Misguided to try to fit a continuous probability distribution to the results from a collection of methods. Different methods do NOT simply yield random values from some underlying process centered on the correct answer - many methods will share similar biases. e.g. may miss the same features. - More methods do NOT mean more information about the process. - the range of answers from a variety of methods does NOT reflect the uncertainty in the process generating the losses.

38. Assessing ratio techniques Summary of ratio techniques Average ratios are regression estimators through the origin. Regression methods allow us to test the assumptions of a model. Often find that major assumptions of ratios model are not satisfied by the data –lack predictive power –cannot capture payment period trend changes –non-normality. Need to work from a different paradigm.

39. Where to from here? - very important to check if the technique is appropriate! - lack of predictive power of cumulative to predict next incremental suggests abandoning ratios - changing trends (e.g. against payment years) suggests modeling trend changes - non-normality and nonlinear trends suggests transformation - particularly log transform Assessing ratio techniques

40. 10 d t = w+d Development year Payment year Accident year w Trends occur in three directions: 1986 1987 1998 Future Past Probabilistic modelling

41. 0 1 2 3 4 5 6 7 8 9 10 11 12 If we graph the data for an accident year against development year, we can see changing trends. e.g. trends in the development year direction xxxxxxxxxxxxx x x x x x x x x xx x x x Probabilistic modelling

42. 0 1 2 3 4 5 6 7 8 9 10 11 12 x x x x x x x x xx x x x Could put a line through the points, using a ruler. Or could do something formally, using regression. xxxxxxxxxxxxx Probabilistic modelling

43. The model is not just the trends in the mean, but the distribution about the mean 0 1 2 3 4 5 6 7 8 9 10 11 12 x x x x x x x x xx x x x ( Data = Trends + Random Fluctuations ) Probabilistic modelling

44. The modeling framework (containing many possible models) consists of four components: Model the mean in each cell by: - the level of its own accident year - plus the development trends to that development - plus the payment trends to that payment year Model the distribution about the mean Probabilistic modelling

45. Real data show the same features as the model Log scale Original scale - spread related to mean Probabilistic modelling

46. 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 7427 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 100000 81873 67032 54881 44933 36788 30119 24660 20190 100000 81873 67032 54881 44933 36788 30119 24660 100000 81873 67032 54881 44933 36788 30119 100000 81873 67032 54881 44933 36788 100000 81873 67032 54881 44933 100000 81873 67032 54881 100000 81873 67032 100000 81873 100000 - 0.2d d 0 1 2 3 4 5 6 7 8 9 10 11 12 13 PAID LOSS = EXP(alpha - 0.2d) -0.2 Probabilistic modelling Properties (axioms) of trends

47. Underlying Trends in the Data Probabilistic modelling 0.1 0.3 0.15

48. Accident year 1983 Accident year 1978 Accident year 1979 Development year trends Projection of trends onto other directions Probabilistic modelling

49. Changing trends hard to pick without removing main development and payment year trend. Trends + randomness Development Year Probabilistic modelling

50. Wtd. Std. Residualsvs. Payment Year 7879808182838485868788899091 -2 0 1 2 Fitting a single trend to changing trends - estimates as an average trend - changes show up in the residuals Probabilistic modelling log data Payment Period Trends Payment Periods 78 79 80 81 82 83 84 85 86 87 88 89 90 91 15.63% trend 10% trend 78-82, 30% trend 82-83, and 15% trend 83+ 13.25 12.75 12.25 11.75 11.25 13.75

51. Summary of trend properties of loss development arrays Probabilistic modelling Trends in payment year direction project onto the other two directions and vice versa You cannot determine payment period trend changes without including parameters in that direction Changing trends can be hard to pick up in the presence of noise, unless main trends are removed first (regression as a form of adjustment) Modeling a changing trend as a single trend will result in pattern in the residual plots

52. Checking the modeling framework Probabilistic modelling if the modeling framework “works”, it should be hard to differentiate between real data and data simulated from an identified model for that data –trends are the ‘same’ and change in the same periods –amount of randomness is the same if you create (simulate) data, you should be able to identify the (known) changing trends in the data; mean forecasts should usually be within about 2 standard errors of the true mean

53. Smooth data can conceal changing trends Very smooth data (ABC) Very smooth ratios Probabilistic modelling

54. Smooth data can conceal changing trends Residuals after removing all accident year and development year trends. Probabilistic modelling

55. Volatile (noisy) data can be predictable (within model uncertainty) if trends are stable Noisy data is not necessarily hard to predict Probabilistic modelling Very noisy data 2 orders of magnitude

56. residual plots and other diagnostics for a simple model are good forecast yields an outstanding mean forecast of $20.6 million and a standard deviation of $9.3 million, so the standard deviation is high (volatile data). compare forecasts as most recent years removed (validation): Probabilistic modelling

57. We can simulate from the distribution of the sums of log- normals (accident year totals, payment year totals, total reserve). Probabilistic modelling can obtain means (and s.d.’s) for reinsurance layers. can calculate margins at given reserve/premium

58. distributions can be computed for every cell and sums of cells include both process risk and parameter risk. Ignoring parameter risk leads to underreserving and underpricing (especially in reinsurance). forecast distributions are accurate if assumptions about the future remain true. Prediction intervals and uncertainty Prediction and Reserving

59. Prediction intervals and uncertainty Distribution of sum of payment year totals important for dynamic financial analysis. Distributions for future underwriting years important for pricing. For a fixed security level on all the lines combined, the risk margin per line decreases as the number of lines increases. Prediction and Reserving

60. future uncertainty in loss reserves should be based on a probabilistic model, which might not be related to reserves carried by the in the past. uncertainty for each line should be based on a probabilistic model that describes the particular line experience may be unrelated to the industry as a whole. Security margins should be selected formally. Implicit risk margins may be much less or much more than required. Risk Based Capital Prediction and Reserving

61. Booking the Reserve identify trends, stability of trends and distributions about trends. Validation analysis. formulate assumptions about future. If recent trends unstable, try to identify the cause, and use any relevant business knowledge. select percentile (use distribution of reserves, combined security margin, and available risk capital). Increased uncertainty about future trends may require a higher security margin. Prediction and Reserving

62. Credibility - if a trend parameter estimate for an individual company is not credible, it can be formally shrunk towards an industry estimate. Segmentation and layers - often the statistical model (parameter structure) identified for a combined array applies to some of its segments. These ideas can also be applied to territories etc. and to layers. Other Benefits of the Statistical Paradigm

63. Value at Risk - PALD results useful for Value-at-Risk calculations - for any given quantile, the Value at Risk is simply that quantile minus the provision. - maximum dollar loss at the 100(1–  )% confidence level is the V-a-R for the (1  ) quantile. Other Benefits of the Statistical Paradigm

64. Value at Risk Selected Statistics (Kernel Density Estimate) Mean Median Std Dev. Sel.Value % #Std.Devs V-a-R 479.942474.40673.610 550.000 83.0840.958 70.490 / Other Benefits of the Statistical Paradigm

65. Fitted vs Predictive Distributions - predictive distributions incorporate parameter uncertainty - fitted distributions underestimate the uncertainty (variance) - since mean and variance are related for losses, fitted distributions lead to under-reserving - predictive/fitted distinction a critical consideration when performing DFA Other Benefits of the Statistical Paradigm

66. Predictive Aggregate Loss Distributions (PALD) - can’t calculate distribution of totals directly (acci year, pmt year, total outstanding) - generally can’t use CLT:skewness, dependence, different distributions  larger sample size - can simulate from the predictive distribution of aggregates e.g. predictive distribution of payment or calendar year totals important for liability-side of DFA Other Benefits of the Statistical Paradigm

67. Predictive Aggregate Loss Distributions (PALD) Simulated distribution of total outstanding + various approximations to predictive distribution Histogram Kernel Lognormal Gamma Density Comparisons 1 Unit = $1,000 800600400 0.006 0.005 0.004 0.003 0.002 0.001 0 Other Benefits of the Statistical Paradigm

68. Pricing - Can simulate future year for pricing purposes (assumptions about level, etc) - Obtain entire distribution of predicted losses, not just mean and variance Other Benefits of the Statistical Paradigm

69. Several Reinsurance Layers Layer 1 Ground-Up Data Trends change together - “same” model works for each triangle Layer 2 Other Benefits of the Statistical Paradigm

70. Example Same parameter structure - payment year trends change together Same thing in other directions XS10 (used here like GU data)40XS10 (a layer of XS10) Other Benefits of the Statistical Paradigm

71. Correlation between layers Layer1 Ground-Up Data Layer2 Correlation: matters when simulating or forecasting layer sums/differences Corresponding wtd residuals -2.5 0 2.5 -3-2012 40XS10 XS10 DY 0-4 DY 5+ (wt = 0.219) Other Benefits of the Statistical Paradigm

72. Equality of parameters across layers - Often development and payment year trend parameters nearly identical (same percentage changes). - Can set equal in a combined regression model, taking the correlations into account. - More parsimonious model. Other Benefits of the Statistical Paradigm

73. Equality of parameters across layers XS10 40XS10 No significant differences between layers Other Benefits of the Statistical Paradigm

74. Equality of parameters across layers reduced model As we eliminate parameters, the estimates of the remaining parameters generally become more precise Other Benefits of the Statistical Paradigm

75. Segments often related too ALL2M (combined data)Segment of ALL2M (Hospital S) Payment year trends Trends change in same years Wtd Std Res vs Pay. Yr 89909192939495969798 2 1.5 1 0.5 0 -0.5 -1.5 -2 -2.5 -3 -3.5 -4 Wtd Std Res vs Pay. Yr 89909192939495969798 3 2.5 2 1.5 1 0.5 0 -0.5 -1.5 -2 -2.5 -3 Other Benefits of the Statistical Paradigm

76. Excess layers as differences of limited layers Limited to $2M - Still same parameter structure as the limited layers - Excess layer has larger relative uncertainty - Larger uncertainty on parameter estimates Limited to $1M $1M XS $1M — = Other Benefits of the Statistical Paradigm

77. Iota (inflation) Parameter Estimates Payment 2M 1M 1Mxs1M Years IotaS.E.IotaS.E.IotaS.E. 1989-980.110.030.080.030.060.05 - Superimposed inflation estimate not credible on 1Mxs1M alone Other Benefits of the Statistical Paradigm

78. - Fit a combined model for the two limited layers, or the base layer and the excess layer - If some parameters are equal (particularly the inflation parameter in this case), may yield a better inflation estimate for the excess layer Other Benefits of the Statistical Paradigm

79. I. Ratio techniques and extensions Ratios are regressions Regressions have assumptions (know what you assume when using ratios) Assumptions need to be checked (When do ratios work?) Assumptions often don't hold What does this suggest? Summary

80. II. Statistical modeling framework (Probabilistic Trend Family of models) Model the logarithms of the incrementals Parameters to pick up trends in the three directions Probability Distribution to every cell Assumptions generally met Assessing stability of trends (confidence about the future) III. Reserve Figure Summary

81. RATIO TECHNIQUES EXTENDED LINK RATIO FAMILY PREDICTIVE AGGREGATE LOSS DISTRIBUTION OPTIMAL ASSET ALLOCATION REINSURANCE RISK BASED CAPITAL SIMULATION PROBABILISTIC TREND MODELS Summary

82. Contact: BenZehnwirth@insureware.com or see some of the papers and examples on our website: www.insureware.com Further Information