Introduction to Credibility CAS Seminar on Ratemaking New Orleans March 10-11, 2005 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 3 Outline Background Definition Rationale History Methods, examples, and considerations Limited fluctuation methods Greatest accuracy methods Bibliography 4 Background Background Definition 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 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? 7 Background History 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 8 Background Methods Limited Fluctuation “Frequentist” “Classical credibility” Greatest Accuracy Bayesian 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 9 Limited Fluctuation Credibility 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. 11 Limited Fluctuation Credibility Derivation New Estimate = (Credibility)(Data) + (1- Credibility)(Previous Estimate) E2 = Z*T + (1-Z)*E1 Add and subtract ZE[T] = Z*T + ZE[T] - ZE[T] + (1-Z)*E1 regroup = (1-Z)*E1 + ZE[T] + Z*(T - E[T]) Stability Truth Random Error 12 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] + zpVar[T]1/2 (assuming T~Normally) -so- kE[T]/Z = zpVar[T]1/2 Z = kE[T]/zpVar[T]1/2 13 Limited Fluctuation Credibility Mathematical 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: Z = kE[T]/zpVar[T]1/2 becomes: Z = kE[T]1/2 /zp = kN1/2 /zp Solving for N (# of claims for full credibility, i.e., Z=1): N = (zp/k)2 14 Limited Fluctuation Credibility Standards for full credibility Claim counts required for full credibility based on the previous derivation: k P 2.5% 5% 7.5% 10% 90% 4,326 1,082 481 291 95% 6,147 1,537 683 584 99% 10,623 2,656 1,180 664 15 Limited Fluctuation Credibility Mathematical formula for Z – Part 2 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} 16 Limited Fluctuation Credibility Mathematical formula for Z – Part 2 (continued) N = (zp/k)2{Var[N]/E[N]+ Var[S]/E[S]2} 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. 17 Limited Fluctuation Credibility Partial credibility Given a full credibility standard for a number of claims, 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, and say we have 500 claims. Z = (500/1082)1/2 = 68% 18 Limited Fluctuation Credibility 100% 90% 80% 70% 60% 50% 40% 30% 20% Full credibility standards: 1100 900 700 500 300 683 1,082 100 Credibility Partial credibility (continued) Number of Claims 19 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 class Pure premium for all classes Loss ratio for an individual risk Loss ratio for entire class Indicated rate change for a territory Indicated rate change for entire state Indicated rate change for entire state Trend in loss ratio or the indication for the country 20 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 Ratio Claims 1995 1996 1997 1998 1999 67% 77% 79% 77% 86% 535 616 634 615 686 3 year 5 year 81% 77% 1,935 3,086 E.g., 81%(.60) + 75%(1-.60) Credibility at: 1,082 5,410 100% 60% 100% 75% Weighted Indicated Loss Ratio Rate Change 78.6% 4.8% 76.5% 2.0% E.g., 76.5%/75% -1 21 Limited Fluctuation Credibility Increasing 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 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. 23 Greatest Accuracy Credibility Greatest Accuracy Credibility Illustration Steve Philbrick’s target shooting example... B A S1 S2 C E D 25 Greatest Accuracy Credibility Illustration (continued) Which data exhibits more credibility? A B S1 S2 E C D 26 Greatest Accuracy Credibility Illustration (continued) Higher credibility: less variance within, more variance between Class loss costs per exposure... 0 A B E D C Lower credibility: more variance within, less variance between 0 A B E C D 27 Greatest Accuracy Credibility Derivation #1 (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 a is 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 = v/(u+v) 28 Greatest Accuracy Credibility Derivation #1 (continued) The formula Z = v/(u+v) can be reformulated in terms of variances by substituting v = nsv2 and u = nsu2 and reducing: Z = (1/su2)/[(1/su2) + (1/sv2)] or Z = 1/(1 + su2 /sv2) 29 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 Lij, and can be modeled as Lij = C + Mi + eij Where C is the mean loss over all classes, Mi is the mean loss differential for class i, and eij is the random error. Assume that the Mi’s and the eij’s average 0. Let: t2 = the variance between the M’s. (variance of hypothetical means - VHM). s2/n = E(si2 ) = average of the variances of the within the M’s (expected value of process variance -- EVPV). 30 Greatest Accuracy Credibility Derivation #2 (continued) Pictorially this looks like: Distribution of U.S. Reserves VHM 0.3% 0.3% 0.2% 0.2% 0.1% 0.1% EVPV EVPV 0.0% 5,500 6,000 6,500 Class 1 7,000 7,500 8,000 8,500 9,000 Class 2 31 Greatest Accuracy Credibility Derivation #2 (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) This is the original Bühlmann credibility formula = n/(n+k) where k = EVPV/VHM 32 Greatest Accuracy Credibility Increasing 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 33 Greatest Accuracy Credibility Strengths 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, in the original Bühlmann construct, is a property of the entire set of data. 34 Bibliography 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 36 Introduction to Credibility