Introduction to Credibility
CAS Seminar on Ratemaking
New Orleans
March 10-11, 2005
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
Outline
 Background



Definition
Rationale
History
 Methods, examples, and considerations


Limited fluctuation methods
Greatest accuracy methods
 Bibliography
4
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
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? Or,
alternatively, how much data do we need before it’s
sufficiently believable?
7
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
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
9
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?
 Sounds like statistical quality control.
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
regroup
= (1-Z)*E1 + ZE[T] + Z*(T - E[T])
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 = E[T] + zpVar[T]1/2
(assuming T~Normally)
-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:
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
14
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
15
Limited Fluctuation Credibility
Mathematical formula for Z – Part 2
 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]2}
16
Limited Fluctuation Credibility
Mathematical formula for Z – Part 2 (continued)
N = (zp/k)2{Var[N]/E[N]+ Var[S]/E[S]2}
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, 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%
18
Limited Fluctuation Credibility
100%
90%
80%
70%
60%
50%
40%
30%
20%
Full credibility
standards:
1100
900
700
500
300
683
1,082
100
Credibility
Partial credibility (continued)
Number of Claims
19
Limited Fluctuation Credibility
Complement of credibility
Once partial credibility has been established, the complement
of credibility, 1-Z, must be applied to something else. E.g.,
If the data analyzed is…
A good complement is...
Pure premium for a class
Pure premium for all
classes
Loss ratio for an individual
risk
Loss ratio for entire class
Indicated rate change for a
territory
Indicated rate change for
entire state
Indicated rate change for
entire state
Trend in loss ratio or the
indication 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 expected loss ratio is
75%.
 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
21
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
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.
23
Greatest
Accuracy
Credibility
Greatest Accuracy Credibility
Illustration
Steve Philbrick’s target shooting example...
B
A
S1
S2
C
E
D
25
Greatest Accuracy Credibility
Illustration (continued)
Which data exhibits more credibility?
A
B
S1
S2
E
C
D
26
Greatest Accuracy Credibility
Illustration (continued)
Higher credibility:
less variance within,
more variance between
Class loss costs per exposure...
0
A
B
E

D
C
Lower credibility:
more variance within,
less variance between
0
A
B
E
C
D

27
Greatest Accuracy Credibility
Derivation #1 (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 = v/(u+v)
28
Greatest Accuracy Credibility
Derivation #1 (continued)
The formula
Z = v/(u+v)
can be reformulated in terms of variances by
substituting v = nsv2 and u = nsu2 and reducing:
Z = (1/su2)/[(1/su2) + (1/sv2)]
or
Z = 1/(1 + su2 /sv2)
29
Greatest Accuracy Credibility
Derivation #2 – a typical problem
Consider a set of classes (i) of risks with losses per exposure
observed over n years (j). Losses in class i for year j are denoted Lij,
and can be modeled as
Lij = C + Mi + eij
Where C is the mean loss over all classes, Mi is the mean loss
differential for class i, and eij is the random error. Assume that the
Mi’s and the eij’s average 0.
Let:
t2 = the variance between the M’s. (variance of hypothetical means - VHM).
s2/n = E(si2 ) = average of the variances of the within the M’s
(expected value of process variance -- EVPV).
30
Greatest Accuracy Credibility
Derivation #2 (continued)
Pictorially this looks like:
Distribution of U.S. Reserves
VHM
0.3%
0.3%
0.2%
0.2%
0.1%
0.1%
EVPV
EVPV
0.0%
5,500
6,000
6,500
Class 1
7,000
7,500
8,000
8,500
9,000
Class 2
31
Greatest Accuracy Credibility
Derivation #2 (continued)
 Using the formula that establishes that the least squares value
for Z is proportional to the reciprocal of expected squared
errors:
Z = (n/s2)/(n/s2 + 1/ t2) =
= n/(n+
s2/t2)
This is the original
Bühlmann credibility
formula
= n/(n+k)
where k = EVPV/VHM
32
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
33
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, in the original Bühlmann construct, is a
property of the entire set of data.
34
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.
 ___________. “Credibility Theory for Dummies,” CAS Forum, Winter
2003, p. 621
36
Introduction to Credibility