Introduction to Credibility CAS Seminar on Ratemaking Las Vegas, Nevada March 12-13, 2001

Purpose Today’s session is designed to encompass: 4Credibility in the context of ratemaking 4Classical and Bühlmann models 4Review of variables affecting credibility 4Formulas 4Practical techniques for applying 4Methods for increasing credibility

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

Background

Background Definition 4Common 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 4In 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

Background Rationale Why do we need “credibility” anyway? 4P&C insurance costs, namely losses, are inherently stochastic 4Observation of a result (data) yields only an estimate of the “truth” 4How much can we believe our data? Consider an example...

Background Simple example

Background History 4The CAS was founded in 1914, in part to help make rates for a new line of insurance -- Work Comp 4Early 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? 4Bayesian views resurrected in the 40’s, 50’s, and 60’s

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

Limited Fluctuation Credibility Description 4“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 4How 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?

= (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

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] + z p Var[T] 1/2 -so- kE[T]/Z = z p Var[T] 1/2  Z = kE[T]/z p Var[T] 1/2

N = (z p /k) 2 Limited Fluctuation Credibility Mathematical formula for Z (continued) 4If 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 4Then E[T] = number of claims (N), and E[T] = Var[T], so: 4Solving 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

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

N = (z p /k) 2 {Var[N]/E[N] + Var[S]/E[S]} Limited Fluctuation Credibility Mathematical formula for Z II 4Relaxing 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] 4Plugging these values into the formula Z = kE[T]/z p Var[T] 1/2 and solving for N Z=1):

Limited Fluctuation Credibility Partial credibility 4Given a full credibility standard, N full, what is the partial credibility of a number N < N full ? 4The square root rule says: Z = (N/ N full ) 1/2 4For example, let N full = 1,082, and say we have 500 claims. Z = (500/1082) 1/2 = 68%

Limited Fluctuation Credibility Partial credibility (continued) Full credibility standards:

Limited Fluctuation Credibility Increasing credibility 4Per the formula, Z = (N/ N full ) 1/2 = [N/(z p /k) 2 ] 1/2 = k N 1/2 /z p 4Credibility, 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

Limited Fluctuation Credibility Weaknesses The strength of limited fluctuation credibility is its simplicity, therefore its general acceptance and use. But it has weaknesses… 4Establishing a full credibility standard requires arbitrary assumptions regarding P and k, 4Typical use of the formula based on the Poisson model is inappropriate for most applications 4Partial 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. 4Treats 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. 4Data: 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

Greatest Accuracy Credibility

4Find the credibility weight, Z, that minimizes the sum of squared errors about the truth 4For illustration, let m L ij = loss ratio for territory i in year j; L.. is the grand mean m L at = loss ratio for territory “a” at some future time “t” 4Find Z that minimizes E{L at - [ZL a. + (1-Z)L.. ]} 2 4Z takes the form Z = n/(n+k) Greatest Accuracy Credibility Derivation

4k takes the form k = s 2 /t 2 4where m s 2 = average variance of the territories over time, called the expected value of process variance (EVPV) m t 2 = variance across the territory means, called the variance of hypothetical means (VHM) 4The 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 Greatest Accuracy Credibility Derivation (continued)

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

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

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

Greatest Accuracy Credibility Increasing credibility 4Per the formula, Z =n n + s 2 t 2 4Credibility, 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 Example Herzog EVPV = (9+1)/2 = 5 VHM k = EVPV/VHM = 5/(19/3) = n = 3 Z = 3/( ) = Next loss estimate: A = 8(.792) + (1-.792)*10 = 8.4 B = 12(.792) + (1-.792)*10 = 11.6

Bibliography

4Herzog, Thomas. Introduction to Credibility Theory. 4Longley-Cook, L.H. “An Introduction to Credibility Theory,” PCAS, Mayerson, Jones, and Bowers. “On the Credibility of the Pure Premium,” PCAS, LV 4Philbrick, Steve. “An Examination of Credibility Concepts,” PCAS, Venter, Gary and Charles Hewitt. “Chapter 7: Credibility,” Foundations of Casualty Actuarial Science.

Introduction to Credibility