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