3. Introduction to Credibility CAS Seminar on Ratemaking New Orleans March 10-11, 2005

4. 3 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  Methods for increasing credibility

5. 4 Outline  Background m Definition m Rationale m History  Methods, examples, and considerations m Limited fluctuation methods m Greatest accuracy methods  Bibliography

6. Background

7. 6 Background Definition  Common vernacular (Webster): m “Credibility:” the state or quality of being credible m “Credible:” believable m So, “the quality of being believable” m Implies you are either credible or you are not  In actuarial circles: m Credibility is “a measure of the credence that…should be attached to a particular body of experience” -- L.H. Longley-Cook m Refers to the degree of believability; a relative concept

8. 7 Background Rationale 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? Or, alternatively, how much data do we need before it’s sufficiently believable?

9. 8 Background History  The CAS was founded in 1914, in part to help make rates for a new line of insurance -- Work Comp  Early pioneers: m Mowbray -- how many trials/results need to be observed before I can believe my data? m Albert Whitney -- focus was on combining existing estimates and new data to derive new estimates New Rate = Credibility*Observed Data + (1-Credibility)*Old Rate m 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

10. 9 Background Methods “ Frequentist ” Bayesian Greatest Accuracy Limited Fluctuation Limit the effect that random fluctuations in the data can have on an estimate Make estimation errors as small as possible “Least Squares Credibility” “Empirical Bayesian Credibility” Bühlmann Credibility Bühlmann-Straub Credibility “Classical credibility”

11. Limited Fluctuation Credibility

12. 11 Limited Fluctuation Credibility Description  “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?  Sounds like statistical quality control.

13. 12 = (1-Z)*E 1 + ZE[T] + Z*(T - E[T]) Limited Fluctuation Credibility Derivation E 2 = Z*T + (1-Z)*E 1 Add and subtract ZE[T] regroup StabilityTruthRandom Error New Estimate = (Credibility)(Data) + (1- Credibility)(Previous Estimate) = Z*T + ZE[T] - ZE[T] + (1-Z)*E 1

14. 13 Limited Fluctuation Credibility Mathematical 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 = E[T] + z p Var[T] 1/2 (assuming T~Normally) -so- kE[T]/Z = z p Var[T] 1/2  Z = kE[T]/z p Var[T] 1/2

15. 14 N = (z p /k) 2 Limited Fluctuation Credibility Mathematical formula for Z (continued)  If we assume m That we are dealing with an insurance process that has Poisson frequency, and m 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]/z p Var[T] 1/2 becomes: Z = kE[T] 1/2 /z p = kN 1/2 /z p

16. 15 Limited Fluctuation Credibility Standards for full credibility Claim counts required for full credibility based on the previous derivation:

17. 16 N = (z p /k) 2 {Var[N]/E[N]+ Var[S]/E[S] 2 } Limited Fluctuation Credibility Mathematical formula for Z – Part 2  Relaxing the assumption that severity doesn’t matter, m let T = aggregate losses = (frequency)(severity) m then E[T] = E[N]E[S] m and Var[T] = E[N]Var[S] + E[S] 2 Var[N]  Plugging these values into the formula Z = kE[T]/z p Var[T] 1/2 and solving for N (@ Z=1):

18. 17 N = (z p /k) 2 {Var[N]/E[N]+ Var[S]/E[S] 2 } Limited Fluctuation Credibility Mathematical formula for Z – Part 2 (continued) This term is just the full credibility standard derived earlier Think of this as an adjustment factor to the full credibility standard that accounts for relaxing the assumptions about the data. The term on the left is derived from the claim frequency distribution and tends to be close to 1 (it is exactly 1 for Poisson). The term on the right is the square of the c.v. of the severity distribution and can be significant.

19. 18 Limited Fluctuation Credibility Partial credibility  Given a full credibility standard for a number of claims, N full, what is the partial credibility of a number N < N full ?  The square root rule says: Z = (N/ N full ) 1/2  For example, m let N full = 1,082, and m say we have 500 claims. Z = (500/1082) 1/2 = 68%

20. 19 Limited Fluctuation Credibility Partial credibility (continued) Full credibility standards:

21. 20 Limited Fluctuation Credibility Complement of credibility Once partial credibility has been established, the complement of credibility, 1-Z, must be applied to something else. E.g., If the data analyzed is…A good complement is... Pure premium for a classPure premium for all classes Loss ratio for an individual Loss ratio for entire class risk Indicated rate change for a Indicated rate change for territoryentire state Indicated rate change for Trend in loss ratio or the entire stateindication for the country

22. 21 Limited Fluctuation Credibility Example Calculate the expected loss ratios as part of an auto rate review for a given state, given that the expected loss ratio is 75%.  Data: Loss RatioClaims 199567% 535 199677% 616 199779% 634 199877% 615 199986% 686Credibility at: Weighted Indicated 1,0825,410Loss Ratio Rate Change 3 year 81%1,935 100% 60% 78.6%4.8% 5 year 77%3,086 100% 75% 76.5%2.0% E.g., 81%(.60) + 75%(1-.60) E.g., 76.5%/75% -1

23. 22 Limited Fluctuation Credibility Increasing credibility  Per the formula, Z = (N/ N full ) 1/2 = [N/(z p /k) 2 ] 1/2 = k N 1/2 /z p  Credibility, Z, can be increased by: m Increasing N = get more data m increasing k = accept a greater margin of error m decrease z p = concede to a smaller P = be less certain

24. 23 Limited Fluctuation Credibility Weaknesses 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 -- only holds for a normal approximation of the underlying distribution of the data. Insurance data tends to be skewed.  Treats credibility as an intrinsic property of the data.

25. Greatest Accuracy Credibility

26. 25 Greatest Accuracy Credibility Illustration Steve Philbrick’s target shooting example... A D B C E S1 S2

27. 26 Greatest Accuracy Credibility Illustration (continued) Which data exhibits more credibility? A D B C E S1 S2

28. 27 Greatest Accuracy Credibility Illustration (continued) A D BC E ADBC E Class loss costs per exposure... 0 0   Higher credibility: less variance within, more variance between Lower credibility: more variance within, less variance between

29. 28  Suppose you have two independent estimates of a quantity, x and y, with squared errors of u and v respectively  We wish to weight the two estimates together as our estimator of the quantity: a = zx + (1-z)y  The squared error of a is w = z 2 u + (1-z) 2 v  Find Z that minimizes the squared error of a – take the derivative of w with respect to z, set it equal to 0, and solve for z: m dw/dz = 2zu + 2(z-1)v = 0 Z = v/(u+v) Greatest Accuracy Credibility Derivation #1 (with thanks to Gary Venter)

30. 29 The formula Z = v/(u+v) can be reformulated in terms of variances by substituting v = n  v 2 and u = n  u 2 and reducing : Z = (1/  u 2 )/[(1/  u 2 ) + (1/  v 2 )] or Z = 1/(1 +  u 2 /  v 2 ) Greatest Accuracy Credibility Derivation #1 (continued)

31. 30 Greatest Accuracy Credibility Derivation #2 – a typical problem Consider a set of classes (i) of risks with losses per exposure observed over n years (j). Losses in class i for year j are denoted L ij, and can be modeled as L ij = C + M i +  ij Where C is the mean loss over all classes, M i is the mean loss differential for class i, and  ij is the random error. Assume that the M i ’s and the  ij ’ s average 0. Let: t 2 = the variance between the M’s. (variance of hypothetical means - - VHM). s 2 /n = E(s i 2 ) = average of the variances of the within the M’s (expected value of process variance -- EVPV).

32. 31 Greatest Accuracy Credibility Derivation #2 (continued) EVPV VHM Class 1Class 2 Pictorially this looks like:

33. 32  Using the formula that establishes that the least squares value for Z is proportional to the reciprocal of expected squared errors: Z = (n/s 2 ) / (n/s 2 + 1/ t 2 ) = = n/(n+ s 2 /t 2 ) = n/(n+k) where k = EVPV/VHM Greatest Accuracy Credibility Derivation #2 (continued) This is the original Bühlmann credibility formula

34. 33  Per the formula, Z =n n + s 2 t 2  Credibility, Z, can be increased by: m Increasing n = get more data m decreasing s 2 = less variance within classes, e.g., refine data categories m increase t 2 = more variance between classes Greatest Accuracy Credibility Increasing credibility

35. 34 Greatest Accuracy Credibility Strengths and weaknesses  The greatest accuracy or least squares credibility result is more intuitively appealing. m It is a relative concept m It is based on relative variances or volatility of the data m There is no such thing as full credibility  Issues m Greatest accuracy credibility is can be more difficult to apply. Practitioner needs to be able to identify variances. m Credibility, z, in the original Bühlmann construct, is a property of the entire set of data.

36. Bibliography

37. 36 Bibliography  Herzog, Thomas. Introduction to Credibility Theory.  Longley-Cook, L.H. “An Introduction to Credibility Theory,” PCAS, 1962  Mayerson, Jones, and Bowers. “On the Credibility of the Pure Premium,” PCAS, LV  Philbrick, Steve. “An Examination of Credibility Concepts,” PCAS, 1981  Venter, Gary and Charles Hewitt. “Chapter 7: Credibility,” Foundations of Casualty Actuarial Science.  ___________. “Credibility Theory for Dummies,” CAS Forum, Winter 2003, p. 621

38. Introduction to Credibility