Basic Concepts in Credibility CAS Seminar on Ratemaking Atlanta, Georgia

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Basic Concepts in Credibility CAS Seminar on
RatemakingAtlanta, Georgia

• March 8-9, 2007

Keith Sunvold, FCAS, MAAA Minneapolis
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Topics

• Todays 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

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Background
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BackgroundDefinition

• 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
  thatshould 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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BackgroundRationale

• Why do we need credibility anyway?
• PC 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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BackgroundHistory

• 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 CredibilityObserved 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 40s, 50s, and
  60s

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BackgroundMethods
Limit the effect that random fluctuations in the
data can have on an estimate
Limited Fluctuation
Classical credibility
Make estimation errors as small as possible
Greatest Accuracy
Least Squares Credibility Empirical Bayesian
Credibility Bühlmann Credibility Bühlmann-Straub
Credibility
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Limited Fluctuation Credibility
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Limited Fluctuation CredibilityDescription

• 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 CredibilityDerivation
New Estimate (Credibility)(Data) (1-
Credibility)(Previous Estimate)
E2 ZT (1-Z)E1
ZT ZET - ZET (1-Z)E1
(1-Z)E1 ZET Z(T - ET)
Stability
Truth
Random Error
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Limited Fluctuation CredibilityMathematical
formula for Z
PrZ(T-ET) lt kET P
-or- PrT lt ET kET/Z P
ET kET/Z ET
zpVarT1/2 (assuming TNormally)
-so- kET/Z zpVarT1/2
? Z kET/(zpVarT1/2)
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Limited Fluctuation CredibilityMathematical
formula for Z (continued)

• If we assume
• we are measuring an insurance process that has
  Poisson frequency, and
• Severity is constant or severity doesnt matter
• Then ET number of claims (N), and ET
  VarT, so
• Solving for N ( of claims for full credibility,
  i.e., Z1)

Z kET/zpVarT1/2 becomes Z
kET/zpET1/2 kET1/2 /zp kN1/2 /zp
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Limited Fluctuation CredibilityStandards for
full credibility
Claim counts required for full credibility based
on the previous derivation
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Limited Fluctuation CredibilityMathematical
formula for Z Part 2

• Relaxing the assumption that severity doesnt
  matter,
• Let data T aggregate losses frequency x
  severity N x S
• then ET ENES
• and VarT ENVarS ES2VarN
• Plugging these values into the formula
• Z kET/zpVarT1/2
• and solving for N (_at_ Z1)

N (zp/k)2VarN/EN VarS/ES2
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Limited Fluctuation CredibilityMathematical
formula for Z Part 2 (continued)
N (zp/k)2VarN/EN VarS/ES2
Think of this as an adjustment factor to the full
credibility standard that accounts for relaxing
the assumptions about the data.
This term is just the full credibility standard
derived earlier
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 CredibilityPartial
credibility

• Given a full credibility standard for a number
  of claims, Nfull, what is the partial credibility
  of a number N lt 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 doesnt use N
• Z N / (N k)

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Limited Fluctuation CredibilityPartial
credibility (continued)
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Limited Fluctuation CredibilityComplement of
credibility

• Once partial credibility, Z, has been
  established, the mathematical complement, 1-Z,
  must be applied to something else the
  complement of credibility.

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Limited Fluctuation CredibilityExample

• 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 2002 67 535 2003 77
616 2004 79 634 2005 77 615 2006 86
686 Credibility at Weighted
Indicated 1,082 5,410 Loss Ratio Rate
Change 3 year 81 1,935 100 60
78.6 4.8 5 year 77 3,086 100 75
76.5 2.0
E.g., 81(.60) 75(1-.60)
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Limited Fluctuation CredibilityIncreasing
credibility

• Per the formula,
• Z (N/ Nfull)1/2 N/(zp/k)21/2
• 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 CredibilityWeaknesses

• 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
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Greatest Accuracy CredibilityIllustration
Steve Philbricks target shooting example...
B
A
S1
S2
E
D
C
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Greatest Accuracy CredibilityIllustration
(continued)
Which data exhibits more credibility?
A
B
S1
E
S2
C
D
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Greatest Accuracy CredibilityIllustration
(continued)
Class loss costs per exposure...
?
D
0
A
B
C
E
Higher credibility less variance within, more
variance between
D
A
B
C
E
0
?
Lower credibility more variance within, less
variance between
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Greatest Accuracy CredibilityDerivation (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/(uv)

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Greatest Accuracy CredibilityDerivation
(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)
• n/(nk)

This is the original Bühlmann credibility formula
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Greatest Accuracy CredibilityIncreasing
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 CredibilityStrengths 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
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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