Transcript Basic Concepts in Credibility CAS Seminar on Ratemaking Salt Lake City, Utah

March 13-15, 2006
Basic Concepts in Credibility
CAS Seminar on Ratemaking
Salt Lake City, Utah
Paul J. Brehm, FCAS, MAAA
Minneapolis
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
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Outline

Background
– Definition
– Rationale
– History

Methods, examples, and considerations
– Limited fluctuation methods
– Greatest accuracy methods

Bibliography
3
Background
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
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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?
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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
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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
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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 (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
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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
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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)
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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:
Z = kE[T]/zpVar[T]1/2
becomes:
Z = kE[T]/zpE[T]1/2 = kE[T]1/2 /zp = kN1/2 /zp

Solving for N (# of claims for full credibility, i.e., Z=1):
N = (zp/k)2
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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%
4,326
1,082
481
291
95%
6,147
1,537
683
584
99%
10,623
2,656
1,180
664
90%
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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]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}
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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.
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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?

Z = (N/ Nfull)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 = E1/ (E0 + E1)
– Relative exposure volume
– Based on the relative contribution of the new exposures to the
whole, but doesn’t use N

Z = N / (N + k)
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Limited Fluctuation Credibility
(continued)
100%
90%
80%
70%
60%
50%
40%
30%
20%
1100
900
700
500
300
(n/1082)^.5
n/n+191
100
Credibility
Partial credibility
Number of Claims
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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 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
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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
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
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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
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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.
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Greatest Accuracy Credibility
Greatest Accuracy Credibility
Illustration
Steve Philbrick’s target shooting example...
B
A
S1
S2
C
E
D
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Greatest Accuracy Credibility
Illustration (continued)
Which data exhibits more credibility?
A
B
S1
S2
E
C
D
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Greatest Accuracy Credibility
Illustration (continued)
Average class variance =
“Expected Value of Process Variance” =
or EVPV; denoted s2/n
Class loss costs per exposure...
0
A
B
E
A
B

Higher credibility:
less variance within,
more variance between
Variance between the means =
“Variance of Hypothetical Means”
or VHM; denoted t2
0
D
C
E
C
D
Lower credibility:
more variance within,
less variance between

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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)
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Greatest Accuracy Credibility
Derivation (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)
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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
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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.
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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
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Introduction to Credibility