CredibilitySession SPE-42 CAS Seminar on RatemakingTampa, FloridaMarch 2002
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 credibility Methods for increasing credibility Complements of 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? What else can we believe?
Consider a simple 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
“Frequentist”
Bayesian
Greatest Accuracy
LimitedFluctuation
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 CredibilityBühlmann-Straub Credibility
“Classical 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?
= (1-Z)*E1 + ZE[T] + Z*(T - E[T])
Limited Fluctuation Credibility
Derivation
E2 = Z*T + (1-Z)*E1
Add and subtract
ZE[T]
regroup
Stability Truth Random Error
New Estimate = (Credibility)(Data) + (1- Credibility)(Previous Estimate)
= Z*T + ZE[T] - ZE[T] + (1-Z)*E1
Limited Fluctuation Credibility
Derivation (continued)
We wish to find the level of credibility, Z, ...such that the error, T-E[T], …is less than or equal to k% of the truth, E[T], …at least P% of the time
Mathematically, this is expressed as
Probability{Z(T-E[T]) < kE[T]} = P
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
N = (zp/k)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:
Solving for N (# of claims for full credibility, i.e., Z=1):
Z = kE[T]/zpVar[T]1/2 becomes:
Z = kE[T]1/2 /zp = kN1/2 /zp
Limited Fluctuation Credibility
Standards for full credibility
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
Claim counts required for full credibility based on the previous derivation:
N = (zp/k)2{Var[N]/E[N] + Var[S]/E[S]2}
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):
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. If we have 500 claims:
Z = (500/1082)1/2 = 68%
Limited Fluctuation Credibility
Partial credibility (continued)
20%30%40%50%60%70%80%90%
100%
100
300
500
700
900
1100
Number of Claims
Cre
dib
ilit
y
683
1,082
Full credibility standards:
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
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 Loss ratio for entire classrisk
Indicated rate change for a Indicated rate change for territory entire state
Indicated rate change for Trend in loss ratioentire state
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 -- is only approximate.
Treats credibility as an intrinsic property of the data. Complement of credibility is highly judgmental.
Limited Fluctuation Credibility
Example
Calculate the expected loss ratios as part of an auto rate review for a given state.
Data:
3 year 81% 1,935 5 year 77% 3,086
Loss Ratio Claims
1996 67% 5351997 77% 6161997 79% 6341999 77% 6152000 86% 686
E.g., 81%(.60) + 75%(1-.60)
E.g., 76.5%/75% -1
Credibility at: Weighted Indicated1,082 5,410 Loss Ratio Rate Change
100% 60% 78.6% 4.8% 100% 75% 76.5% 2.0%
Greatest Accuracy Credibility
Find the credibility weight, Z, that minimizes the sum of squared errors about the truth
Z takes the form
Z = n/(n+k) 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)
Greatest Accuracy Credibility
Derivation
Distribution of U.S. Reserves
0.0%
0.1%
0.1%
0.2%
0.2%
0.3%
0.3%
5,500 6,000 6,500 7,000 7,500 8,000 8,500 9,000
EVPV EVPV
VHM
Greatest Accuracy Credibility
Derivation (continued)
Class 1 Class 2
Greatest Accuracy Credibility
Partial credibility
0%10%20%30%40%50%60%70%80%90%
100%
0 600 1200 1800 2400 3000 3600 4200 4800
n/n+k
1082
K=500
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
Illustration
Steve Philbrick’s target shooting example...
A
D
B
C
E
S
Next
Greatest Accuracy Credibility
Illustration (continued)
Which data exhibits more credibility?
A
D
B
C
E
S
Next
Greatest Accuracy Credibility
Illustration (continued)
A DB CE
A DB CE
Class loss costs per exposure...
0
0
Higher credibility: less variance within, more variance between
Lower credibility: more variance within, less variance between
Conclusion
Credibility
Conclusion
Actuarial credibility is a relative measure of the believability of the data used in analysis
The underlying math can be complex, but concepts are intuitive
Credibility of the data increases with volume and uniqueness and decreases with the data’s volatility
Credibility weighting of data increases the stability in estimates and improves accuracy
Bibliography
Bibliography
Dean, C. Gary. An Introduction to Credibility. PCAS Herzog, Thomas. Introduction to Credibility Theory. Longley-Cook, L.H. “An Introduction to Credibility
Theory,” PCAS, 1962 Mahler, Howard and C Gary Dean. “Chapter 8:
Credibility,” Foundations of Casualty Actuarial Science.
Mayerson, Jones, and Bowers. “On the Credibility of the Pure Premium,” PCAS, LV
Philbrick, Steve. “An Examination of Credibility Concepts,” PCAS, 1981
Introduction to Credibility