Math 479 / 568
Casualty Actuarial Mathematics
Fall 2014
University of Illinois at Urbana-Champaign
Professor Rick Gorvett
Session 14: Credibility
October 21, 2014
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Agenda
• Credibility
– Basic concept
– Uses of credibility theory
– Types of credibility calculations
• Classical / limited fluctuation
• Bühlmann
• Bayesian
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The Basic Concept
• “The concept of credibility has been the
casualty actuaries’ most important and
enduring contribution to casualty actuarial
science.”
– Matthew Rodermund, Preface, Foundations of
Casualty Actuarial Science, 1st edition
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The Basic Concept (cont.)
• Credibility:
– Ultimately, we want to know what the “optimal”
combination of data is
• New versus old data
• Individual versus group data
– A function of variability
• Target-shooting example
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Uses of Credibility Theory
Example 1 – New versus old (current)
• Premium rate in effect during 2014, based on
2013 data
– For 2015, we could just “update” the 2014 rate
(e.g., for loss trend)
– But we now also have 2014 data
– How should we “combine” the current rate with
the new data?
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Uses of Credibility Theory (cont.)
Example 2 – Individual versus group
• Insurance applicant has $0 of actual historical
losses
– Applicant is classified in a group with an annual
rate of $1,000
– How shall we “combine” actual ($0) individual
loss experience with the group indication?
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Uses of Credibility Theory (cont.)
• Mathematically, the credibility-weighted
premium is:
where
PC  ZX  (1  Z )M
Z = credibility factor, 0 ≤ Z ≤ 1.
X = new data, or individual experience.
M = old data, or class rate
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Types of Credibility
(1) Classical (limited fluctuation)
– Quantity (number of exposures, claims, or dollars
of loss) needed for data to achieve “full
credibility” (Z = 1)
– Formula for cases where Z < 1.
(2) Bühlmann (least squares)
– Motivation: target-shooting example
(3) Bayesian
– Prior and posterior probabilities
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Classical Credibility
• Two “parameters”: within +/- k of the
expected value at least 100p% of the time
• Volume needed for full credibility can be
expressed in terms of
– Number of exposures
– Number of expected claims
– Aggregate dollars of loss
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Classical Credibility (cont.)
• Formula for full-credibility standard (in
terms of number of claims) for aggregate
losses (or pure premium):
2 

 y   f
S 
nF   2 
 2


k


  f
S 
2
2
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Classical Credibility (cont.)
• Full-credibility formula parameters:
–
F(y) = (1+P) / 2
– P = probability that losses (or pure premium) is
within +/- k of true mean value
– f indicates frequency distribution; S indicates
severity distribution
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Classical Credibility (cont.)
• Partial credibility
– If volume is greater than or equal to the standard
for full credibility, then Z = 1.00
– If volume is less, then
Actual Volume
Z=
Full Credibility Standard
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Bühlmann Credibility
• Z = n / (n + k), where k = v / a
– n = “volume” of data or experience
– v = expected value of the process variance
– a = variance of the hypothetical means
• Relationships:
– n increases  Z increases
– v increases  Z decreases
– a increases  Z increases
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Bayesian Credibility
• Adjust the probabilities associated with
possible distributions or classifications, in
light of emerging experience
– A priori
– A posteriori
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Example # 1
• What is the full credibility standard if the
desired precision is a 90% chance of being
within +/- 5% of the true value, when the
frequency is assumed to be Poisson, and all
claims are assumed to have the same size?
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Example # 2
• What is the full credibility standard if the
desired precision is a 95% chance of being
within +/- 10% of the true value, when the
frequency is assumed to be Poisson, and the
claim size distribution has a coefficient of
variation of 2.0?
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Example # 3
• If the full-credibility standard is 2,000 claims,
what is the credibility assigned to data with
800 claims?
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Example # 4
You are given the following information:
(i) Claim counts follow a Poisson distribution.
(ii) Claim sizes follow a lognormal distribution with a mean of 5,000 and a
variance of 64,000,000.
(iii) Claim sizes and claim counts are independent.
(iv) The number of claims in the first year was 431.
(v) The aggregate loss in the first year was 2.25 million.
(vi) The manual pure premium for the first year was 1.80 million.
(vii) The exposure in the second year is identical to the exposure in the first
year.
(viii) The full credibility standard is to be within 4% of the expected aggregate
loss 90% of the time.
(a) Using classical (or “limited fluctuation”) credibility, determine the full
credibility standard based on the number of claims.
(b) Using your result from (a), determine the credibility associated with the
observed number of claims in the first year.
(c) Use your result from (b) to estimate the classical credibility pure
premium estimate, in millions, for the second year.
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