Credibility Session SPE-42 CAS Seminar on Ratemaking Tampa, Florida March 2002

1. Credibility Session SPE-42 CAS Seminar on Ratemaking Tampa, Florida March 2002

2. Purpose Today’s session is designed to encompass: • Credibility in the context of ratemaking • Classical and Bühlmann models • Review of variables affecting credibility • Formulas • Practical techniques for applying credibility • Methods for increasing credibility • Complements of credibility

3. Outline • Background • Definition • Rationale • History • Methods, examples, and considerations • Limited fluctuation methods • Greatest accuracy methods • Bibliography

4. Background

5. BackgroundDefinition • Common vernacular (Webster): • “Credibility:” the state or quality of being credible • “Credible:” believable • So, “the quality of being believable” • Implies you are either credible or you are not • In actuarial circles: • Credibility is “a measure of the credence that…should be attached to a particular body of experience”-- L.H. Longley-Cook • Refers to the degree of believability; a relative concept

6. BackgroundRationale Why do we need “credibility” anyway? • P&C insurance costs, namely losses, are inherently stochastic • Observation of a result (data) yields only an estimate of the “truth” • How much can we believe our data? • What else can we believe? Consider a simple example...

7. BackgroundSimple example

8. BackgroundHistory • The CAS was founded in 1914, in part to help make rates for a new line of insurance -- Work Comp • Early pioneers: • Mowbray -- how many trials/results need to be observed before I can believe my data? • Albert Whitney -- focus was on combining existing estimates and new data to derive new estimates New Rate = Credibility*Observed Data + (1-Credibility)*Old Rate • Perryman (1932) -- how credible is my data if I have less than required for full credibility? • Bayesian views resurrected in the 40’s, 50’s, and 60’s

9. “Frequentist” Bayesian BackgroundMethods Limit the effect that random fluctuations in the data can have on an estimate Limited Fluctuation “Classical credibility” Make estimation errors as small as possible Greatest Accuracy “Least Squares Credibility” “Empirical Bayesian Credibility” Bühlmann Credibility Bühlmann-Straub Credibility

10. Limited Fluctuation Credibility

11. Limited Fluctuation CredibilityDescription • “A dependable [estimate] is one for which the probability is high, that it does not differ from the [truth] by more than an arbitrary limit.” -- Mowbray • How much data is needed for an estimate so that the credibility, Z, reflects a probability, P, of being within a tolerance, k%, of the true value?

12. Add and subtract ZE[T] regroup Limited Fluctuation CredibilityDerivation New Estimate = (Credibility)(Data) + (1- Credibility)(Previous Estimate) E2 = Z*T + (1-Z)*E1 = Z*T + ZE[T] - ZE[T] + (1-Z)*E1 = (1-Z)*E1 + ZE[T] +Z*(T - E[T]) Stability Truth Random Error

13. Limited Fluctuation CredibilityDerivation (continued) • We wish to find the level of credibility, Z, • ...such that the error, T-E[T], • …is less than or equal to k% of the truth, E[T], • …at least P% of the time • Mathematically, this is expressed as Probability{Z(T-E[T]) < kE[T]} = P

14. Limited Fluctuation CredibilityMathematical formula for Z Pr{Z(T-E[T]) < kE[T]} = P -or- Pr{T < E[T] + kE[T]/Z} = P E[T] + kE[T]/Z looks like a formula for a percentile: E[T] + zpVar[T]1/2 -so- kE[T]/Z = zpVar[T]1/2  Z = kE[T]/zpVar[T]1/2

15. N = (zp/k)2 Limited Fluctuation CredibilityMathematical formula for Z (continued) • If we assume • That we are dealing with an insurance process that has Poisson frequency, and • Severity is constant or severity doesn’t matter • Then E[T] = number of claims (N), and E[T] = Var[T], so: • Solving for N (# of claims for full credibility, i.e., Z=1): Z = kE[T]/zpVar[T]1/2 becomes: Z = kE[T]1/2 /zp = kN1/2 /zp

16. Limited Fluctuation CredibilityStandards for full credibility Claim counts required for full credibility based on the previous derivation:

17. Limited Fluctuation CredibilityMathematical formula for Z II • Relaxing the assumption that severity doesn’t matter, • let T = aggregate losses = (frequency)(severity) • then E[T] = E[N]E[S] • and Var[T] = E[N]Var[S] + E[S]2Var[N] • Plugging these values into the formula Z = kE[T]/zpVar[T]1/2 and solving for N (@ Z=1): N = (zp/k)2{Var[N]/E[N] + Var[S]/E[S]2}

18. Limited Fluctuation CredibilityPartial credibility • Given a full credibility standard, Nfull, what is the partial credibility of a number N < Nfull? • The square root rule says: Z = (N/ Nfull)1/2 • For example, let Nfull = 1,082. If we have 500 claims: Z = (500/1082)1/2 = 68%

19. Limited Fluctuation CredibilityPartial credibility (continued) Full credibility standards:

20. Limited Fluctuation CredibilityIncreasing credibility • Per the formula, Z = (N/ Nfull)1/2 = [N/(zp/k)2]1/2 = kN1/2/zp • Credibility, Z, can be increased by: • Increasing N = get more data • Increasing k = accept a greater margin of error • Decrease zp = concede to a smaller P = be less certain

21. If the data analyzed is… A good complement is... Pure premium for a class Pure Premium for all classes Loss ratio for an individual Loss ratio for entire class risk Indicated rate change for a Indicated rate change for territory entire state Indicated rate change for Trend in loss ratio entire state Limited Fluctuation CredibilityComplement of credibility Once partial credibility has been established, the complement of credibility, 1-Z, must be applied to something else. E.g.,

22. Limited Fluctuation CredibilityWeaknesses The strength of limited fluctuation credibility is its simplicity, therefore its general acceptance and use. But it has weaknesses… • Establishing a full credibility standard requires arbitrary assumptions regarding P and k. • Typical use of the formula based on the Poisson model is inappropriate for most applications. • Partial credibility formula -- the square root rule -- is only approximate. • Treats credibility as an intrinsic property of the data. • Complement of credibility is highly judgmental.

23. Loss Ratio Claims 1996 67% 535 1997 77% 616 1997 79% 634 1999 77% 615 2000 86% 686 E.g., 81%(.60) + 75%(1-.60) Credibility at: Weighted Indicated 1,0825,410 Loss RatioRate Change 100% 60%78.6%4.8% 100% 75% 76.5% 2.0% E.g., 76.5%/75% -1 Limited Fluctuation CredibilityExample Calculate the expected loss ratios as part of an auto rate review for a given state. • Data: 3 year 81% 1,935 5 year 77% 3,086

24. Greatest Accuracy Credibility

25. Greatest Accuracy CredibilityDerivation • Find the credibility weight, Z, that minimizes the sum of squared errors about the truth • Z takes the form Z = n/(n+k) • k takes the form k = s2/t2 where • s2 = average variance of the territories over time, called the expected value of process variance (EVPV) • t2 = variance across the territory means, called the variance of hypothetical means (VHM)

26. VHM EVPV EVPV Greatest Accuracy CredibilityDerivation (continued) Class 1 Class 2

27. Greatest Accuracy CredibilityPartial credibility K=500

28. Greatest Accuracy CredibilityIncreasing credibility • Per the formula, Z = n n + s2 t2 • Credibility, Z, can be increased by: • Increasing n = get more data • decreasing s2 = less variance within classes, e.g., refine data categories • increase t2 = more variance between classes

29. Greatest Accuracy CredibilityIllustration Steve Philbrick’s target shooting example... B A S Next E D C

30. Greatest Accuracy CredibilityIllustration (continued) Which data exhibits more credibility? A B S E Next C D

31. Greatest Accuracy CredibilityIllustration (continued) Higher credibility: less variance within, more variance between Class loss costs per exposure...  0 D A B E C Lower credibility: more variance within, less variance between D A B E C 0 

32. Conclusion

33. CredibilityConclusion • Actuarial credibility is a relative measure of the believability of the data used in analysis • The underlying math can be complex, but concepts are intuitive • Credibility of the data increases with volume and uniqueness and decreases with the data’s volatility • Credibility weighting of data increases the stability in estimates and improves accuracy

34. Bibliography

35. Bibliography • Dean, C. Gary. An Introduction to Credibility. PCAS • Herzog, Thomas. Introduction to Credibility Theory. • Longley-Cook, L.H. “An Introduction to Credibility Theory,” PCAS, 1962 • Mahler, Howard and C Gary Dean. “Chapter 8: Credibility,” Foundations of Casualty Actuarial Science. • Mayerson, Jones, and Bowers. “On the Credibility of the Pure Premium,” PCAS, LV • Philbrick, Steve. “An Examination of Credibility Concepts,” PCAS, 1981

36. Introduction to Credibility