Introduction to Credibility CAS Seminar on Ratemaking Las Vegas, Nevada March 12-13, 2001 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 Outline Background Definition Rationale History Methods, examples, and considerations Limited fluctuation methods Greatest accuracy methods Bibliography 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 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? Consider an example... Background Simple example 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 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 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? 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 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 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 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 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 Limited Fluctuation Credibility Mathematical 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]} Limited Fluctuation Credibility Partial 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, and say we have 500 claims. Z = (500/1082)1/2 = 68% Limited Fluctuation Credibility 100% 90% 80% 70% 60% 50% 40% 30% 20% Full credibility standards: Number of Claims 1100 900 700 500 300 683 1,082 100 Credibility Partial credibility (continued) 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 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. Limited Fluctuation Credibility Example Calculate the expected loss ratios as part of an auto rate review for a given state. 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 Greatest Accuracy Credibility Greatest Accuracy Credibility Derivation Find the credibility weight, Z, that minimizes the sum of squared errors about the truth For illustration, let Lij = loss ratio for territory i in year j; L.. is the grand mean Lat = loss ratio for territory “a” at some future time “t” Find Z that minimizes E{Lat - [ZLa. + (1-Z)L..]}2 Z takes the form Z = n/(n+k) Greatest Accuracy Credibility Derivation (continued) 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) 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 Greatest Accuracy Credibility Illustration Steve Philbrick’s target shooting example... B A S C C E D Greatest Accuracy Credibility Illustration (continued) Which data exhibits more credibility? A B S C E C D Greatest Accuracy Credibility Illustration (continued) Higher credibility: less variance within, more variance between Class loss costs per exposure... 0 A B E C D Lower credibility: more variance within, less variance between 0 A B E C D 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 Greatest Accuracy Credibility Example Herzog Year Policy 1 2 3 2 EVPV = A $5 8 11 8 9 (9+1)/2 = B 11 13 12 12 1 10 19/3 k = EVPV/VHM = 5/(19/3) = 0.7895 n=3 Z = 3/(3 + 0.7895) = 0.792 Next loss estimate: A = 8(.792) + (1-.792)*10 = 8.4 B = 12(.792) + (1-.792)*10 = 11.6 5 VHM 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. Introduction to Credibility