Testing Reserve ModelsTesting Reserve ModelsTesting Reserve ModelsTesting Reserve Models

Gary G VenterGary G Venter

Guy Carpenter InstratGuy Carpenter Instrat

Testing MethodologyTesting MethodologyTesting MethodologyTesting Methodology

Tests based on goodness of fit of models

Test by SSE adjusted for number of parameters

Can also test significance of parameters and validity of model assumptions

Issues to Be TestedIssues to Be TestedIssues to Be TestedIssues to Be Tested

Is development multiplicative? Is development contingent on previous

losses? Are there diagonal effects in the

triangle?

Is Development Multiplicative?Is Development Multiplicative?Is Development Multiplicative?Is Development Multiplicative?

Convenient assumption Allows for differences in levels among

accident years Sometimes too restrictive for actual data Additive models particularly useful when

triangle is built from on-level loss ratios– On leveling puts in known adjustments so

parameters of model do not have to fit them– Parameters can then fit development patterns

only Can test significance of additive and

multiplicative parameters, as well as goodness of fit

Development Contingent on Development Contingent on Previous Losses?Previous Losses?Development Contingent on Development Contingent on Previous Losses?Previous Losses?

Age-to-age factors typically applied to previous losses

But in B-F models they are applied to expected

Can test which is better Contingent vs. non-contingent a

fundamental categorization of models

Diagonal Effects in Triangle?Diagonal Effects in Triangle?Diagonal Effects in Triangle?Diagonal Effects in Triangle?

Calendar year inflation can affect all accident years

Changes in claims department practices can affect selected years

Can test significance of diagonal parameters

Notation Notation Notation Notation

c(w,d): cumulative loss from accident year w as of age d

q(w,d): incremental loss for accident year w at age d– q(w,d) = c(w,d) – c(w,d-1)

f(d): factor applied to c(w,d) to estimate q(w,d+1)

Model FormulasModel FormulasModel FormulasModel Formulas Chain ladder

q(w,d+1) = f(d)c(w,d) + e [f(d) means different factor for each lag]

Parameterized B-Fq(w,d+1) = f(d)h(w) + e [h(w) means different ultimate for each

year] Cape Cod

q(w,d+1) = f(d)h + e [h with no w means all years have the same h]

Additiveq(w,d+1) = g(d) + e [project constant loss for each year varying

by lag]

(Additive = Cape Cod) Separation

q(w,d+1) = f(d)h(w)g(w+d+1) + e OR:q(w,d+1) = f(d)c(w,d)g(w+d+1) + e OR:q(w,d+1) = f(d)h(w)+g(w+d+1) + e OR:q(w,d+1) = f(d)c(w,d)+g(w+d+1) + e

Flavors of the Chain LadderFlavors of the Chain Ladderq(w,d+1) = f(d)c(w,d) + eq(w,d+1) = f(d)c(w,d) + eFlavors of the Chain LadderFlavors of the Chain Ladderq(w,d+1) = f(d)c(w,d) + eq(w,d+1) = f(d)c(w,d) + e

E(e) = 0, but what about variance of e? Jth flavor has standard deviation(e) =

c(w,d)j

Divide both sides by c(w,d)j to get sd = , which is regression assumption

Can do similarly for other models Murphy PCAS 1994 shows that this

adjusted regression based on various j’s reproduces common estimation methods for development factors

But it may be hard to tell what j is

Hard to say j Hard to say j 0 for this 0 for this sample datasample dataHard to say j Hard to say j 0 for this 0 for this sample datasample data

Lag 1 Absolute Error vs. Lag 0 Losses

0

200

400

600

800

1000

1200

1400

1600

1800

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Incremental Incurred Losses

0 1 2 3 4 5 6 7 8 95012 3257 2638 898 1734 2642 1828 599 54 172106 4179 1111 5270 3116 1817 -103 673 535

3410 5582 4881 2268 2594 3479 649 6035655 5900 4211 5500 2159 2658 9841092 8473 6271 6333 3786 2251513 4932 5257 1233 2917557 3463 6926 1368

1351 5596 61653133 22622063

RAA Fac GLRAA Fac GLRAA Fac GLRAA Fac GL

ISO Zone Rate Trucks $1M Limit or ISO Zone Rate Trucks $1M Limit or LessLessISO Zone Rate Trucks $1M Limit or ISO Zone Rate Trucks $1M Limit or LessLess

0 1 2 3 4 5 6 7 8 9 10 11 12 11,305 18,904 17,474 10,221 3,331 2,671 693 1,145 744 112 40 13 0 8,828 13,953 11,505 7,668 2,943 1,084 690 179 1,014 226 16 616 8,271 15,324 9,373 11,716 5,634 2,623 850 381 16 28 558 7,888 11,942 11,799 6,815 4,843 2,745 1,379 266 809 12 8,529 15,306 11,943 9,460 6,097 2,238 493 136 11

10,459 16,873 12,668 9,199 3,524 1,027 924 1,190 8,178 12,027 12,150 6,238 4,631 919 435

10,364 17,515 13,065 12,451 6,165 1,381 11,855 20,650 23,253 9,175 10,312 17,133 28,759 20,184 12,874 19,373 31,091 25,120 18,433 29,131 20,640

Incremental Paid Loss and LAE

RAA: Statistical Significance of RAA: Statistical Significance of FactorsFactorsRAA: Statistical Significance of RAA: Statistical Significance of FactorsFactors

Statistical Significance of Link Ratios and Constants

0 to 1 1 to 2 2 to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8

`a’ 5113 4311 1687 2061 4064 620 777 3724

Std a 1066 2440 3543 1165 2242 2301 145 0

`b’ -0.109 0.049 0.131 0.041 -0.100 0.011 -0.008 -0.197

std b 0.349 0.309 0.283 0.071 0.114 0.112 0.008 0

•Regressions of incremental on previous cumulative

•Constants almost all significant, no factors are

•Constants even more significant without factors

RAA: 0 to 1 Is Constant + RAA: 0 to 1 Is Constant + RandomRandomRAA: 0 to 1 Is Constant + RAA: 0 to 1 Is Constant + RandomRandom

Lag 1 vs. Lag 0 Losses

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 1000 2000 3000 4000 5000 6000

ISO: Statistical Significance of ISO: Statistical Significance of FactorsFactorsISO: Statistical Significance of ISO: Statistical Significance of FactorsFactors

0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10a 8 18 45 -7 6 9 -20 12 1 3Std a 20 32 27 33 22 10 12 17 4 16b 1.57 0.42 0.01 0.14 0.01 -0.02 0.10 -0.04 0.00 -0.01Std b 0.18 0.19 0.13 0.15 0.10 0.04 0.05 0.07 0.02 0.07

•No constants are significant

•Large factors are

•More factors are significant without constants

ISO: Factor Times Lag 0 a ISO: Factor Times Lag 0 a Good Predictor of Lag 1Good Predictor of Lag 1ISO: Factor Times Lag 0 a ISO: Factor Times Lag 0 a Good Predictor of Lag 1Good Predictor of Lag 1

Lag 1 Incremental as Function of Lag 0 with Factor Fit

0

5000

10000

15000

20000

25000

30000

35000

7000 9000 11000 13000 15000 17000 19000

Testing Fit of ModelsTesting Fit of ModelsTesting Fit of ModelsTesting Fit of Models

Multiply SSE by a factor that increases with more parameters, like:

AIC: e2p/n 4% BIC or SIC: sqrt(n)2p/n 8% HQIC: ln(n)2p/n 6% OS: (n-p)-2 4%-8% for p=1…24 (percent shown is needed improvement in

SSE for an extra parameter for a sample of 50 observations)

IC formulas are for nExp(IC/n) with normal residuals

(A=Akaike, B=Bayesian, HQ=Hannan-Quinn, OS=Old Standby)

Fitting BF Parameters – RAA Fitting BF Parameters – RAA CaseCaseFitting BF Parameters – RAA Fitting BF Parameters – RAA CaseCase

h(w) = df(d)q(w,d) / df(d)2

f(d) = wh(w)q(w,d) / wh(w)2

BF Parameters

Age d 0 1 2 3 4 5 6 7 8 9 f(d) 1st 0.106 0.231 0.209 0.155 0.117 0.083 0.038 0.032 0.018 0.011 f(d) ult 0.162 0.197 0.204 0.147 0.115 0.082 0.037 0.030 0.015 0.009 Year w 0 1 2 3 4 5 6 7 8 9 h(w) 1st 17401 15729 23942 26365 30390 19813 18592 24154 14639 12733 h(w) ult 15982 16501 23562 27269 31587 20081 19032 25155 13219 19413

Reducing # of ParametersReducing # of ParametersReducing # of ParametersReducing # of Parameters

CC makes all accident year factors the same– Useful for stable data or on-level loss

ratios Some factors may be the same

– If difference between factors is not significant, can make them the same

Can fit lines or curves to some of the parameters:– f(d) = (1+i)-d

Goodness of Fit Comparisons - Goodness of Fit Comparisons - RAARAA OS Adjusted SSE’s OS Adjusted SSE’s

Goodness of Fit Comparisons - Goodness of Fit Comparisons - RAARAA OS Adjusted SSE’s OS Adjusted SSE’s

SSE Model Params Generation Formula 157,902 CL 9 q(w,d+1) = f(d)c(w,d) + e 81,167 BF 18 q(w,d+1) = f(d)h(w) + e 75,409 CC 9 q(w,d+1) = f(d)h + e 57,527 CC- 2 q(w,d+1) = .78d6756 + e 52,360 BF-CC 9 q(w,d+1) = f(d)h(w) + e

– h5 - h9 same, h3=(h2+h4)/2, f1=f2, f6=f7, f8=f9 44,701 Sep 7 q(w,d+1) =

f(d)h(w)g(w+d+1)+e– 4 unique age factors, 2 diagonal, 1 acc. yr.– Diagonal factors fit like B-F, with 3 step iteration

Goodness of Fit Comparisons - ISOGoodness of Fit Comparisons - ISO OS Adjusted SSE’s OS Adjusted SSE’sGoodness of Fit Comparisons - ISOGoodness of Fit Comparisons - ISO OS Adjusted SSE’s OS Adjusted SSE’s

SSE Model Params Generation Formula

59,185 CL 12 q(w,d+1) = f(d)c(w,d) + e 69,302 BF 24 q(w,d+1) = f(d)h(w) + e 302,168 CC 12 q(w,d+1) = g(d) + e 54,852 Sep 16 q(w,d+1) =

f(d)c(w,d) + 4 diagonal additive dummies g(w+d+1)+e

42,558 Reduced 13 same as separation7-8, 8-9, 9-10, 10-11 factors all equal, plus 4

diagonal dummies

Regression Array with Diagonal Regression Array with Diagonal DummiesDummiesRegression Array with Diagonal Regression Array with Diagonal DummiesDummies

1 2 5 4

3 8 9

7 10

7

2 1 0 0 0 0

8 3 0 0 1 0

10 7 0 0 0

1

5 0 3 0 1 0

9 0 11 0 0

1

4 0 0 8 0

1

Which Diagonals?Which Diagonals?Which Diagonals?Which Diagonals?

Diagonal Mean Residual1 147 2 365 3 107 4 (1,502) 5 693 6 (7) 7 120 8 219 9 (368)

10 765 11 (432) 12 8

•Too many would over determine regression

•4,5,10,11 most likely suspects

•Reversals in alternative years

•No apparent trend

Regression Coefficients for Regression Coefficients for DummiesDummiesFull Factor ModelFull Factor Model

Regression Coefficients for Regression Coefficients for DummiesDummiesFull Factor ModelFull Factor Model

Diagonal 11 Diagonal 10 Diagonal 5 Diagonal 4add -774 926 1239 -1744std dev 522 530 700 780

•Similar but slightly more significant in combined factor model

Issues to Be TestedIssues to Be TestedIssues to Be TestedIssues to Be Tested

Is development multiplicative?– RAA: no, ISO: yes

Is development contingent on previous losses?– RAA: no, ISO: yes

Are there diagonal effects in the triangle?– Some high and low diagonals in both, but

no trends

finis