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Best Estimates for Reserves Glen Barnett and Ben Zehnwirth

Best Estimates for Reserves Glen Barnett and Ben Zehnwirth. Overview. Assessing ratios (loss development) ( old paradigm ) the mathematical framework underlying ratio techniques gives a tool for assessing whether ratio techniques ‘work’

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Best Estimates for Reserves Glen Barnett and Ben Zehnwirth

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  1. Best Estimates for ReservesGlen 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: x1,…,xn - Random Sample from fitted distribution: y1,…,yn Introduction What does it mean to say a model gives a good fit? e.g. lognormal fit to claim size distribution Does not mean we think the model generated the data fitted lognormal - Fitted Distribution y’s look like x’s: — Model has probabilistic mechanisms that can reproduce the data

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

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

  6. Cum.(1) vs Cum.(0) 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 0 2,000 4,000 Ratio techniques We can graph the points in a cumulative array: 8,269 5,012

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

  8. Cum.(1) vs Cum.(0) 12,000 10,000 8,000 6,000 4,000 2,000 0 0 2,000 4,000 Ratio techniques Select ratios to be typical (“average”) ratios e.g. arithmetic average, chain ladder, average of last three years,... Average ratios are “average” trends through the origin

  9. Cum.(1) vs Cum.(0) 12,000 10,000 8,000 6,000 4,000 2,000 0 0 2,000 4,000 Ratio techniques Arithmetic Average ratio 1/n(y/x) Chain Ladder Ratio = y/x (Volume weighted average) Regression Line through origin BUT an average trend line through the origin is a regression line through the origin! Average ratio techniques are forms of regression.

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

  11.  y/x · x  xy ·1/x  y ^ b = = =  x  x2 ·1/x  x Ratio techniques To determine an average ratio (slope) b we use weighted least squares. Method aims to estimateb by minimising  w(y-bx)2, where weight w = 1/x  1/Var() We obtain which is the Chain Ladder Ratio (or Weighted Average by Volume, or Volume Weighted Ratio)

  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(e) = s2xd • By changing d, obtain other ‘averages’: • d = 2: Arithmetic Average • d = 1: Chain Ladder (Average Weighted by Volume) • d = 0: Ordinary regression through the origin (Wtd Volume2 Ratio) Any average ratio is a regression estimator through the origin

  13. Assessing ratio techniques How does this formal regression methodology help us to assess whether ratio techniques ‘work’? • 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.

  14. Assessing ratio techniques What are the major assumptions made by the models based on ratio techniques? • E( 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. • No trend in the payment period (diagonal) direction • ratio techniques do not allow for changes in inflation. • Normality • the models assume that observations are values from a normal distribution.

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

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

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

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

  19. Std. Residuals vs. Fitted Values Assessing ratio techniques Is E( y | x ) = bx satisfied by the Mack data? Underfit small values 2 1 0 Overfit large values -1 0 5000 10000 15000 20000 25000 30000

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

  21. Cum.(1) vs Cum.(0) 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 0 2,000 4,000 Assessing ratio techniques Regression line through origin causes underfit/overfit Overfit large values ‘Best’ line ’Best’ line not through origin - has an intercept. Underfit small values Typical with real, exposure-adjusted data.

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

  23. Assessing ratio techniques y - x = a + (b - 1)x +  Is b - 1significantly 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. Cum.(1) vs Cum.(0) Incr.(1) vs Cum.(0) 12,000 8,000 10,000 7,000 8,000 6,000 5,000 6,000 4,000 4,000 3,000 2,000 0 2,000 4,000 0 2,000 4,000 Corr. = -0.117, P-value = 0.764 Assessing ratio techniques Mack data: “Ratio not a predictor of future emergence”

  25. Cum.(1) vs Cum.(0) Incr.(1) vs Cum.(0) 400,000 220,000 350,000 200,000 300,000 180,000 250,000 160,000 200,000 150,000 140,000 100,000 120,000 50,000 100,000 0 100,000 150,000 0 50,000 100,000 150,000 Corr. = 0.985, P-value = 0.000 Assessing ratio techniques • What about Case 2: b> 1, a = 0? Ratios do seem to work for some arrays... ABC

  26. Incr.(1) vs Acc. Yr Incr.(1) vs Cum.(0) 220,000 220,000 200,000 200,000 180,000 180,000 160,000 160,000 140,000 140,000 120,000 120,000 100,000 100,000 77 78 79 80 81 82 83 84 85 86 100,000 150,000 Corr. = 0.985, P-value = 0.000 Corr. = 0.985, P-value = 0.000 Assessing ratio techniques ... most often arrays with trends down the accident years An increasing trend down the accident years ‘induces’ a correlation between (y-x) and x.

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

  28. j j-1 }y x{ 1 2 w  n Assessing ratio techniques Include a trend parameter in the model: y – x = a0 + a1w + (b – 1) x +  Intercept Acc Yr trend ‘Ratio’ This gives theExtended Link Ratio Family of models

  29. Assessing ratio techniques After adjusting for accident trends, the relationship with the previous cumulative often disappears. Case 3:b = 1, a1 0 • abandon ratios – model ‘correlation’ using trend parameter

  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. Wtd Std Res vs Pay. Yr 2 1.5 1 0.5 0 -0.5 -1 78 79 80 81 82 83 84 85 86 87 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? 0 1 Change in inflation between 1992 and 1993 1990 1991 1992 Average ratio incorporates effect of change in inflation 1993 - Relationship between development years differs over accident years - Must account for payment trends, or can’t understand development patterns

  33. Ave ratio: 1.53 1.19 1.08 Forecast Incremental: Actual Incremental: % increase over actual: Implied inflation: 1033 553 288 968 484 242 6.7% 14.3% 18.8% 6.7% 7.1% 4.0% Assessing ratio techniques What are you projecting? Incremental 1600 800 400 200 1600 880 484 242 1760 968 484 1936 968 1936 Cumulative 10% 10% 0%  1600 2480 2964 3206 1760 2728 3212 1936 2904 1936 Final acci. year: Very unlikely you’d want to assume this!

  34. Assessing ratio techniques - 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

  35. Assessing ratio techniques 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 severalmodelswithoutassessingtheirappropriateness (fidelity to data,assumptions,validation) —whymixgoodwithbad? - 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. Assessing ratio techniques 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

  40. Past Future 1986 1987 1998 Probabilistic modelling Trends occur in three directions: 0 1 Development year d Payment year t = w+d w Accident year

  41. x x x x x x x x x x x x x x x x x x x x x x x x x x Probabilistic modelling e.g. trends in the development year direction If we graph the data for an accident year against development year, we can see changing trends. 0123456789101112

  42. x x x x x x x x x x x x x x x x x x x x x x x x x x Probabilistic modelling Could put a line through the points, using a ruler. Or could do something formally, using regression. 0123456789101112

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

  44. Probabilistic modelling 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

  45. 8 7 6 5 4 3 0 1 2 3 4 5 6 7 8 3000 2500 2000 1500 1000 500 0 0 1 2 3 4 5 6 7 8 Probabilistic modelling Real data show the same features as the model Log scale Original scale - spread related to mean

  46. - 0.2d d Probabilistic modelling Properties (axioms) of trends 0 1 2 3 4 5 6 7 8 9 10 11 12 13 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 -0.2 PAID LOSS = EXP(alpha - 0.2d)

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

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

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

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

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