Joseph O. Marker Marker Actuarial Services, LLC a e ctua a Se v - PowerPoint PPT Presentation

Joseph O. Marker Marker Actuarial Services, LLC a e ctua a Se v ces, C and University of Michigan CLRS 2010 Meeting J. Marker, LSMWP, CLRS 1

Expected vs Actual Distribution  Test distributions of:  Number of claims (frequency)  Size of ultimate loss (severity)  Size of ultimate loss (severity)  Sources of significant difference between actual and expected amounts:  Programming or communication errors  Not understanding how statistical language  Not understanding how statistical language (e.g. “R”) works.  Errors or misleading results in “R”. J. Marker, LSMWP, CLRS 2

Display Raw Simulator Output  Cl i  Claims file fil Simulation Occurrence Claim Accident No No No Date Report Date Line Type 1 1 1 1 1 1 20000104 20000104 20000227 20000227 1 1 1 1 1 2 1 20000105 20000818 1 1 ……….  Transactions file  Transactions file Simulation Occurrence Claim Trans ‐ Case No No No Date action Reserve Payment 1 1 1 1 1 1 20000227 REP 20000227 REP 2000 2000 0 0 1 1 1 20000413 RES 89412 0 1 1 1 20000417 CLS ‐ 91412 141531 …….. ………. …….. ……… J. Marker, LSMWP, CLRS 3

Another use for Testing information  Create Ultimate Loss File for Analysis – Layout  Create Ultimate Loss File for Analysis Layout Simula Occur ‐ Claim Accident. Report. Case. Pay ‐ ‐ tion. rence Line Type No Date Date Reserve ment No No  Idea: Another use for this section of paper  If an insurer can summarize its own claim data to this format, then it can use the tests we will discuss to parameterize the Simulator using its data.  We have included in this paper all the “R” code used in testing. J. Marker, LSMWP, CLRS 4

Emphasis in the Paper  Document the “R” code used in performing various tests.  Provide references for those who want to explore the modeling more deeply. ode g o e deep y  Provide visual as well as formal tests  QQPlots, histograms, densities, etc. J. Marker, LSMWP, CLRS 5

Test 1 – Frequency, Zero ‐ Modification, Trend  Model parameters: M d l  # Occurrences ~ Poisson (mean = 120 per year)  1,000 simulations  One claim per occurrence O l i  Frequency Trend 2% per year, three accident years  Pr[Claim is Type 1] = 75%; Pr[Type 2] = 25%  Pr[CNP(“Closed No payment”)] = 40%  Pr[CNP( Closed No payment )] = 40%  “Type” and “Status” independent.  Status is a category variable for whether a claim is closed with payment.  Test output to see if its distribution is consistent with assumptions. p J. Marker, LSMWP, CLRS 6

Test 1 – Classical Chi ‐ square C Contingency Table ti T bl Actual Counts Expected Counts Type 1 T 1 Type 2 T 2 M Margin i T Type 1 1 Type 2 T 2 M Margin i 111,066 37,007 0.398906 111,029.0 37,044.0 0.398906 CNP CNP 167,268 55,857 0.601094 167,305.0 55,820.0 0.601094 CWP CWP 0.749826 0.250174 371,198 0.749826 0.250174 371,198 Margin Margin  2 ( Actual Expected )  Χ 2 = = 0.0819 ij ij Expected i j ij Pr [ Χ 2 > 0.0819 ] = 0.775. The independence of Type and Status is supported. J. Marker, LSMWP, CLRS 7

Test 1 – Regression approach  Previous result can be obtained using xt abs xt abs command in “R”  Result can also be obtained using Poisson GLM  Full model: ll d l m m odel 6x<- odel 6x<- gl m l m ( count ~ Type ( count ~ Type + St at us + Type* St at us, + St at us + Type* St at us, dat a = t em p. dat acc. st ack, f am i l y = poi sson, x=T)  Reduced model: d d d l m m odel 5x<- odel 5x<- gl m l m ( count ~ Type + St at us , ( count ~ Type + St at us , dat a = t em p. dat acc. st ack, f am i l y = poi sson, x=T)  Independence obtains if the interactive variable Type*Status is not significant . J. Marker, LSMWP, CLRS 8

Test 1 – Analysis of variance  anova( anova ( ( m ( m o odel 5 d l 5 d l 5x, x, m m o odel 6 d l 6 d l 6x, x, t t es t est =" Chi " ) t " Chi " ) " Chi " ) " Chi " ) Anal ysi s of Devi ance Tabl e Response: count Ter m s Resi d. Df Resi d. Dev Test Df 1 + Type + St at us 143997 160969. 366 2 Type + St at us + Type * St at us 143996 160969. 284 +Type: St at us 1 Devi ance Pr ( Chi ) 1 2 0. 0819088429 0. 774727081  Result matches the previous Χ 2 Test  Result matches the previous Χ 2 Test.  We did not show here the model coefficients, which will produce the expected frequency for each combination of Type and Status. d f f h bi i f T d S J. Marker, LSMWP, CLRS 9

Test 2 – Univariate size of loss  Model parameters: M d l  Three lines – no correlation in frequency by line  # Claims for each line ~ Poisson (mean = 600 per year)  Two accident years, 100 simulations  Size of loss distributions  Line 1 – lognormal  Line 2 – Pareto  Line 3 ‐‐ Weibull  Zero trend in frequency and size of loss.  Expected count = 600 (freq) x 100 (# sims) x 3 (lines) x 2 (years) = 360,000.  Actual # claims: 359 819  Actual # claims: 359,819. J. Marker, LSMWP, CLRS 10

Size of loss – testing strategy   Person doing testing Person running simulation.  Test all three distributions on each line’s output. T t ll th di t ib ti h li ’ t t  Produce plots to “get a feel” for distributions.  Fit using maximum likelihood estimation.  Produce QQ (quantile ‐ quantile) plots  Run formal goodness ‐ of ‐ fit tests. J. Marker, LSMWP, CLRS 11

Si Size of loss – Histograms and p.d.f. f l Hi t d d f J. Marker, LSMWP, CLRS 12

Size of loss – Histograms and p.d.f. J. Marker, LSMWP, CLRS 13

Size of loss  The plots above compare:  Histogram of empirical distribution  Density of the theoretical distribution with m l e  Density of the theoretical distribution with m.l.e. parameters  The plots show that both Weibull and Pareto fit Lines 2 and 3 well.  QQ plots offer another perspective. J. Marker, LSMWP, CLRS 14

Size of loss – QQ Plots  Example of “R” code to produce a QQ Plot t hqua. w2 <- q r wei bul l ( n2, shape=f i t . w2$est i m at e[ 1] , scal e=f i t . w2$est i m at e[ 2] ) generate a random sample same size n2 as empirical data qqpl ot ( ul t l oss2, t hqua. w2, xl ab=" Sam pl e Q uant i l es" , yl ab= Theor et i cal Q yl ab=" Theor et i cal Q uant i l es" uant i l es , m m ai n=" Li ne 2 W ai n= Li ne 2, W ei bul l " ) ei bul l ) ultloss2 is empirical data, thqua.w2 is the generated sample abl i ne( 0, 1, col =" r ed“ )  One can also replace the sample with the quantiles of the theoretical Weibull c.d.f. J. Marker, LSMWP, CLRS 15

Size of Loss – QQ Plot, Line 1 J. Marker, LSMWP, CLRS 16

Size of Loss – QQ Plot, Line 2 J. Marker, LSMWP, CLRS 17

Size of Loss – QQ Plot, Line 3 . J. Marker, LSMWP, CLRS 18

Size of Loss – Fitted distributions  From QQ Plots, it appears that lognormal fits Line 1, Pareto fits Line 2, and Weibull fits Line 3.  Chi ‐ square is a formal goodness ‐ of ‐ fit test. Section 6 discusses setting up the test for Pareto on Line 2. Appendix B contains g p pp “R” code for all the chi ‐ square tests.  Komogorov ‐ Smirnov test was applied also, but too late to include results in this presentation. J. Marker, LSMWP, CLRS 19

Size of Loss – Chi ‐ square g.o.f. test Setting up bins and the expected and actual # claims by bin is not easy in R. Define break points and bins: s = sqr t ( var ( ul t l oss2) ) s = sqr t ( var ( ul t l oss2) ) ul t 2. cut <- ul t 2. cut <- cut ( ul t l oss2. cut ( ul t l oss2. 0, 0, ##bi nni ng dat a ##bi nni ng dat a br eaks = c( 0, m br eaks = c( 0, m - s/ 2 - s/ 2, m , m , m , m +s/ 4, m +s/ 4, m +s/ 2, m +s/ 2, m +s, m +s, m +2* s, 2* m + 2* s, 2* m ax( ul t l oss2) ) ) ax( ul t l oss2) ) ) Not e: ul t l oss2. 0 i s vect or of l oss si zes, m = m ean The t abl e of expect ed and obser ved val ues by bi n: # E. 2 O . 2 x. sq. 2 #[ 1, ] 43993. 890 44087 0. 19705959 Not es: #[ 2, ] 35651. 989 35680 0. 02200752 E. 2 expect ed num #[ 2, ] 35651. 989 35680 0. 02200752 E. 2 expect ed num ber ber #[ 3, ] 10493. 758 10323 2. 77864169 O . 2 act ual num ber #[ 4, ] 7240. 583 7269 0. 11152721 x. sq. 2 Chi - sq st at i st i c #[ 5, ] 9277. 383 9164 1. 38570182 #[ 6 ] 8063 576 8176 1 56743997 #[ 6, ] 8063. 576 8176 1. 56743997 #[ 7, ] 5289. 820 5312 0. 09299630 J. Marker, LSMWP, CLRS 20