1. Testing Reserve Models
Gary G Venter
Guy Carpenter Instrat

2. Testing 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

3. Issues to Be Tested Is development multiplicative?
Is development contingent on previous losses?
Are there diagonal effects in the triangle?

4. 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

5. Development Contingent on 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

6. 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

7. 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)

8. Model Formulas Chain ladder Parameterized B-F Cape Cod Additive
q(w,d+1) = f(d)c(w,d) + e [f(d) means different factor for each lag]
Parameterized B-F
q(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]
Additive
q(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

9. Flavors of the Chain Ladder q(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

10. Hard to say j  0 for this sample data

11. RAA Fac GL 16

12. ISO Zone Rate Trucks $1M Limit or Less
Incremental Paid Loss and LAE

13. RAA: Statistical Significance of Factors
Regressions of incremental on previous cumulative
Constants almost all significant, no factors are
Constants even more significant without factors

14. RAA: 0 to 1 Is Constant + Random

15. ISO: Statistical Significance of Factors
No constants are significant
Large factors are
More factors are significant without constants

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

17. Testing 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) %-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)

18. Fitting BF Parameters – RAA Case19

19. Reducing # 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

20. Goodness of Fit Comparisons - RAA OS Adjusted SSE’s
SSE Model Params Generation Formula
157, 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, CC 9 q(w,d+1) = f(d)h + e
57, CC q(w,d+1) = .78d 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, Sep 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

21. Goodness of Fit Comparisons - ISO OS Adjusted SSE’s
SSE Model Params Generation Formula
59, CL q(w,d+1) = f(d)c(w,d) + e
69,302 BF q(w,d+1) = f(d)h(w) + e
302, CC q(w,d+1) = g(d) + e
54, Sep q(w,d+1) = f(d)c(w,d) diagonal additive dummies g(w+d+1)+e
42,558 Reduced 13 same as separation
7-8, 8-9, 9-10, factors all equal, plus 4 diagonal dummies

22. Regression Array with Diagonal Dummies
3 8 9
7 10 7

23. Which Diagonals? Too many would over determine regression
4,5,10,11 most likely suspects
Reversals in alternative years
No apparent trend

24. Regression Coefficients for Dummies Full Factor Model
Similar but slightly more significant in combined factor model

25. Issues to Be Tested Is development multiplicative?
RAA: no, ISO: yes
Is development contingent on previous losses?
Are there diagonal effects in the triangle?
Some high and low diagonals in both, but no trends

26. finis