1. Introduction to Credibility
CAS Seminar on Ratemaking
San Antonio, Texas
March 27-28, 2003

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

4. Background

5. 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. Why do we need “credibility” anyway?
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. Bayesian views resurrected in the 40’s, 50’s, and 60’s
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. Limited Fluctuation Greatest Accuracy
Background Methods
Limit the effect that random fluctuations in the data can have on an estimate
Limited
Fluctuation
“Frequentist”
Bayesian
“Classical credibility”
Greatest
Accuracy
Make estimation errors as small as possible
“Least Squares Credibility”
“Empirical Bayesian Credibility”
Bühlmann Credibility
Bühlmann-Straub Credibility

9. 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
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?

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
= (1-Z)*E1 + ZE[T] + Z*(T - E[T])
regroup
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 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

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:
Solving for N (# of claims for full credibility, i.e., Z=1):
Z = kE[T]/zpVar[T]1/2 becomes:
Z = kE[T]1/2 /zp = kN1/2 /zp
N = (zp/k)2

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

15. 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]}

16. 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%

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

18. 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

19. 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.

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
% 535
% 616
% 634
% 615
% Credibility at: Weighted Indicated
1,082 5,410 Loss Ratio Rate Change
3 year 81% 1, % 60% % 4.8%
5 year 77% 3, % 75% % 2.0%
E.g.,
81%(.60) + 75%(1-.60)
E.g.,
76.5%/75% -1

21. Greatest Accuracy Credibility

22. Greatest Accuracy Credibility Derivation (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 = u/(u+v)
Each estimate gets a weight equal to the reciprocal of its squared error

23. Greatest Accuracy Credibility Derivation (continued)
Suppose that xi is the mean loss ratio for a class, and y is the overall mean. Let the variance between the class means (xi) be denoted by t2 and the variance about the total be s2.
t2 is called the variance of hypothetical means (VHM)
s2 is called the expected value of process variance (EVPV) or just the process variance
Assume we have enough data about y to know its variance, t2. Also assume we have n observations of a given xi, making our estimate of its variance s2/n. Then,
Z = (n/s2)/(n/s2 + 1/ t2) =
= n/(n+ s2/t2)
= n/(n+k)

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

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

26. Greatest Accuracy Credibility Illustration (continued)
Higher credibility:
less variance within,
more variance between
Class loss costs per exposure... D  A B E C
Lower credibility:
more variance within,
less variance between D A B E C 

27. 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

28. 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 need to be able to identify variances.
Credibility, z, is a property of the entire set of data

29. Bibliography

30. 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

31. Introduction to Credibility