Basic Concepts in Credibility CAS Seminar on Ratemaking Salt Lake City, Utah

1. March 13-15, 2006 Basic Concepts in CredibilityCAS Seminar on RatemakingSalt Lake City, Utah Paul J. Brehm, FCAS, MAAA Minneapolis

2. Topics Today’s session will cover: • Credibility in the context of ratemaking • Classical and Bühlmann models • Review of variables affecting credibility • Formulas • Complements of credibility • Practical techniques for applying credibility • Methods for increasing 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, credibility is “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?

7. BackgroundHistory • The CAS was founded in 1914, in part to help make rates for a new line of insurance -- Workers Compensation – and credibility was born out the problem of how to blend new experience with initial pricing • Early pioneers: • Mowbray (1914) -- how many trials/results need to be observed before I can believe my data? • Albert Whitney (1918) -- 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

8. “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

9. Limited Fluctuation Credibility

10. 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 (1916) • Alternatively, the credibility, Z, of an estimate, T, is defined by the probability, P, that it within a tolerance, k%, of the true value

11. 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

12. 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 = E[T] + zpVar[T]1/2 (assuming T~Normally) -so- kE[T]/Z = zpVar[T]1/2  Z = kE[T]/(zpVar[T]1/2)

13. N = (zp/k)2 Limited Fluctuation CredibilityMathematical formula for Z (continued) • If we assume • we are measuring 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]/zpE[T]1/2 = kE[T]1/2 /zp = kN1/2 /zp

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

15. Limited Fluctuation CredibilityMathematical formula for Z – Part 2 • Relaxing the assumption that severity doesn’t matter, • Let “data” =T = aggregate losses = frequency x severity = N x S • 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}

16. Limited Fluctuation CredibilityMathematical formula for Z – Part 2 (continued) N = (zp/k)2{Var[N]/E[N]+ Var[S]/E[S]2} Think of this as an adjustment factor to the full credibility standard that accounts for relaxing the assumptions about the data. This term is just the full credibility standard derived earlier 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.

17. Limited Fluctuation CredibilityPartial credibility Given a full credibility standard for a number of claims, Nfull, what is the partial credibility of a number N < Nfull? • Z = (N/ Nfull)1/2 • “The square root rule” • Based on the belief that the correct weights between competing estimators is the ratios of the reciprocals of their standard deviations • Z = E1/ (E0 + E1) • Relative exposure volume • Based on the relative contribution of the new exposures to the whole, but doesn’t use N • Z = N / (N + k)

18. Limited Fluctuation CredibilityPartial credibility (continued)

19. 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 or the entire state indication for the country Limited Fluctuation CredibilityComplement of credibility Once partial credibility, Z, has been established, the mathematical complement, 1-Z, must be applied to something else – the “complement of credibility.”

20. E.g., 81%(.60) + 75%(1-.60) 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, given that the target loss ratio is 75%. Loss Ratio Claims 1995 67% 535 1996 77% 616 1997 79% 634 1998 77% 615 1999 86% 686 Credibility at: Weighted Indicated 1,0825,410 Loss RatioRate Change 3 year 81% 1,935 100% 60%78.6%4.8% 5 year 77% 3,086 100% 75% 76.5% 2.0%

21. 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

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 -- 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.

23. Greatest Accuracy Credibility

24. Greatest Accuracy CredibilityIllustration Steve Philbrick’s target shooting example... B A S1 S2 E D C

25. Greatest Accuracy CredibilityIllustration (continued) Which data exhibits more credibility? A B S1 E S2 C D

26. Average class variance = “Expected Value of Process Variance” = or EVPV; denoted s2/n Variance between the means = “Variance of Hypothetical Means” or VHM; denoted t2 Greatest Accuracy CredibilityIllustration (continued) Class loss costs per exposure...  0 D A B E C Higher credibility: less variance within, more variance between D A B E C 0  Lower credibility: more variance within, less variance between

27. Greatest Accuracy CredibilityDerivation (with thanks to Gary Venter) • 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 ais w = z2 u + (1-z)2v • 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: • dw/dz = 2zu + 2(z-1)v = 0 Z = u/(u+v)

28. Greatest Accuracy CredibilityDerivation (continued) Using the formula that establishes that the least squares value for Z is proportional to the reciprocal of expected squared errors: Z = (n/s2)/(n/s2 + 1/ t2) = = n/(n+ s2/t2) = n/(n+k) This is the original Bühlmann credibility formula

29. 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

30. Greatest Accuracy CredibilityStrengths and weaknesses • The greatest accuracy or least squares credibility result is more intuitively appealing. • It is a relative concept • It is based on relative variances or volatility of the data • There is no such thing as full credibility • Issues • Greatest accuracy credibility is can be more difficult to apply. Practitioner needs to be able to identify variances. • Credibility, z, is a property of the entire set of data. So, for example, if a data set has a small, volatile class and a large, stable class, the credibility of the two classes would be the same.

31. Bibliography

32. 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

33. Introduction to Credibility