Basic Concepts in Credibility CAS Seminar on Ratemaking Salt Lake City, Utah Paul J. Brehm, FCAS, MAAA Minneapolis March 13-15, 2006

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

Background

5 Background Definition 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 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?

7 Background History 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 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”

Limited Fluctuation Credibility

10 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 (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 = (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

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] + 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 )

13 N = (z p /k) 2 Limited Fluctuation Credibility Mathematical 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]/z p Var[T] 1/2 becomes: Z = kE[T]/z p E[T] 1/2 = kE[T] 1/2 /z p = kN 1/2 /z p

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

15 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, – 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] 2 Var[N] Plugging these values into the formula Z = kE[T]/z p Var[T] 1/2 and solving for N Z=1):

16 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.

17 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 ? Z = (N/ N full ) 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 = E 1 / (E 0 + E 1 ) – 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 Credibility Partial credibility (continued)

19 Limited Fluctuation Credibility Complement of credibility Once partial credibility, Z, has been established, the mathematical complement, 1-Z, must be applied to something else – the “complement of credibility.” 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 territory entire state Indicated rate change for Trend in loss ratio or the entire stateindication 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 target loss ratio is 75%. Loss RatioClaims % % % % % 686Credibility at: Weighted Indicated 1,0825,410Loss Ratio Rate Change 3 year 81%1, % 60% 78.6%4.8% 5 year 77%3, % 75% 76.5%2.0% E.g., 81%(.60) + 75%(1-.60) E.g., 76.5%/75% -1

21 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: – Increasing N = get more data – increasing k = accept a greater margin of error – decrease z p = 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.

Greatest Accuracy Credibility

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

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

26 Greatest Accuracy Credibility Illustration (continued) A D BC E A D BC E Class loss costs per exposure   Higher credibility: less variance within, more variance between Lower credibility: more variance within, less variance between Variance between the means = “Variance of Hypothetical Means” or VHM; denoted t 2 Average class variance = “Expected Value of Process Variance” = or EVPV; denoted s 2 /n

27 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: – dw/dz = 2zu + 2(z-1)v = 0 Z = u/(u+v) Greatest Accuracy Credibility Derivation (with thanks to Gary Venter)

28 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) Greatest Accuracy Credibility Derivation (continued) This is the original Bühlmann credibility formula

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

30 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, 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.

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

Introduction to Credibility