Greg Taylor. Taylor Fry Consulting Actuaries University of Melbourne University of New South Wales. CAS Spring meeting Phoenix Arizona May 2012

Transcription

1 Chain ladder correlations Greg Taylor Taylor Fry Consulting Actuaries University of Melbourne University of New South Wales CAS Spring meeting Phoenix Arizona 0-3 May 0 Overview The chain ladder produces cell-by-cell forecasts of future claims experience There are two distinct families of model that support the chain ladder algorithm Consideration will be given to the correlations between the forecasts of different cells (conditional on information to date) Separately for the two families Paper to appear shortly as: Chain ladder correlations Variance 5()

2 Framework and notation Claims reserving trapezium Accident period k = 3 J Y kj Observations Often in the literature = J and the trapezium becomes a triangle Framework and notation Claims reserving trapezium Accident period k = 3 J Y kj Observations Calendar time moves in this direction Calendar periods are represented along diagonals 3

3 Framework and notation Claims reserving trapezium Accident period k = 3 J Y kj F U T U R E Observations To be predicted on the basis of past 4 Framework and notation Claims reserving trapezium Accident period k = 3 J X kj F U T U R E Cumulative row sums X kj min J, k i To be predicted on the basis of past 5 Y ki 3

4 Chain ladder algorithm Claims reserving trapezium Accident period k = 3 J X kj F U T U R E Cumulative row sums X kj Calculate min J, k i j j Y ˆ f X / X, j k, j kj k k Forecast Xˆ X fˆ ˆf kj k, k k j ki Last diagonal from the past 6 Correlations of predictions Accident period k = 3 J F U T U R E 7 4

5 Correlations of predictions Accident period k = 3 J F U T U R E Latest information available for row (conditions forecasts) Forecasts of future experience 8 Correlations of predictions Accident period k = 3 J F U T U R E Latest information available for row (conditions forecasts) Forecasts of future experience Consider correlations of within-row forecasts conditioned on most recent information Corr [X k,j+m,x k,j+m+n X kj ] 9 5

6 Correlations of predictions Accident period k = 3 J F U T U R E Latest information available for row (conditions forecasts) Forecasts of future experience Consider correlations of within-row forecasts conditioned on most recent information Corr [X k,j+m,x k,j+m+n X kj ] = ρ k,j+m,j+m+n j 0 ODP Mack model (ODPM) Accident periods are stochastically independent, ie Y kj, Y hi are independent if k h (ODPM) For each k the X kj (j varying) form a Markov chain (ODPM3) For each k=,,, and j=,,,j-, X k,j+ X kj ~ ODP(f j X kj,φ j+ ) Parameters f j are referred to as age-to-age factors Example of a recursive model Recursive because each observation depends on predecessor in same row 6

7 ODP cross-classified model (ODPCC) All random variables Y kj are stochastically independent (ODPCC) For each k =,,, and j =,,, J, a)y kj ~ ODP(α k β j,φ j ) b) J j= β j = Example of a non-recursive model Non-recursive because each observation independent d of predecessors in same row Relevance of ODP Mack and cross-classified models Very different models But in both cases chain ladder algorithm gives MLE forecasts But what about correlations of the forecasts under the respective models? 3 7

8 Conditional correlations of predictors Explicit expressions for ρ k,j+m,j+m+n j = Corr[X k,j+m,x k,j+m+n X kj ] can be obtained (see paper) These are not especially informative in themselves Subsequent discussion concentrates on their properties 4 Properties of conditional correlations ODP Mack model 5 8

9 Properties of conditional correlations ODP Mack model Accident period k = 3 J Correlation decreases as separation increases 6 Properties of conditional correlations ODP Mack model φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n 7 9

10 Properties of conditional correlations ODP Mack model φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n Accident period k = 3 J Correlation increases as overdispersion increases in this region and decreases in this region 8 Properties of conditional correlations ODP Mack model φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n ρ kj+mj+m+n j k,j+m,j+m+n j as any f i, i=j+,,j+m+n provided that φ i f i /(f i -), i=j+,,j+m φ i f i /(f i -), i=j+m+,,j+m+n 9 0

11 Properties of conditional correlations ODP Mack model 3 J Accident period k = φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n ρ kj+mj+m+n j k,j+m,j+m+n j as any f i, i=j+,,j+m+n provided that φ i f i /(f i -), i=j+,,j+m φ i f i /(f i -), i=j+m+,,j+m+n Correlation increases as claims development increases in this region and decreases in this 0 region Properties of conditional correlations ODP Mack model ODP cross-classified model φ i, i=j+,,j+m φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n φ i, i=j+m+,,j+m+n ρ kj+mj+m+n j k,j+m,j+m+n j as any f i, ρ k,j+m,j+m+n j m as any i=j+,,j+m+n provided that β i, i=j+,,j+m φ i f i /(f i -), i=j+,,j+m β i, i=j+m+,,j+m+n φ i f i /(f i -), i=j+m+,,j+m+n Again effect of increasing or decreasing claims development

12 Claims development in recursive and nonrecursive models Claims development effects of correlations expressed in terms of: The f i for recursive models The β i for non-recursive models BUT the two can be shown to be related: f j = j+ i=j β i / j i=j β i Then Properties of conditional correlations (cont d) ODP Mack model ODP cross-classified model φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n f i, i=j+,,j+m+n provided that φ i f i /(f i -), i=j+,,j+m φ i f i /(f i -), i=j+m+,,j+m+n φ i, i=j+,,j+m φ i, i=j+m+,,j+m+n f i, i=j+,,j+m+n provided that Φ i+ f i /(f i -), i=j+,,j+m Φ i+ f i /(f i -), i=j+m+,,j+m+n Now almost identical 3

13 Comparison between correlations of recursive and non-recursive models () Consider the case of recursive and non-recursive models with common values of the f i, φ i The models have been seen to have similar ordering properties Are they actually different? 4 Comparison between correlations of recursive and non-recursive models () Denote ρ k,j+m,j+m+n j by ρ NR k,j+m,j+m+n j for the non-recursive model ρ k,j+m,j+m+n j by ρ R k,j+m,j+m+n j for the recursive model Then ρ NR k,j+m,j+m+n j ρ R k,j+m,j+m+n j ρ NR k,j+m,j+m+n j / ρ R k,j+m,j+m+n j as j 5 3

14 Conclusion The recursive and non-recursive models considered here are quite different but generate identical (chain ladder) forecasts However their prediction errors differ How should one decide which of these chain ladder models to adopt? Correlation properties of forecasts might provide one criterion for the decision eg if one wishes to assume heavy correlations, one might adopt the recursive form 6 Questions? 7 4