Introduction to Credibility CAS Seminar on Ratemaking New Orleans March 10-11, 2005
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Transcript of Introduction to Credibility CAS Seminar on Ratemaking New Orleans March 10-11, 2005
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Introduction to Credibility
CAS Seminar on RatemakingNew OrleansMarch 10-11, 2005
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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
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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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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
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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? Or, alternatively, how much data do we need before it’s sufficiently believable?
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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
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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”
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Limited Fluctuation Credibility
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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.
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= (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
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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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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
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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:
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N = (zp/k)2{Var[N]/E[N]+ Var[S]/E[S]2}
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):
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N = (zp/k)2{Var[N]/E[N]+ Var[S]/E[S]2}
Limited Fluctuation Credibility
Mathematical formula for Z – Part 2 (continued)
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?
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%
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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:
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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 ratio or theentire state 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 expected loss ratio is 75%.
Data:
Loss Ratio Claims
1995 67% 5351996 77% 6161997 79% 6341998 77% 6151999 86% 686 Credibility at: Weighted Indicated
1,082 5,410 Loss Ratio Rate Change3 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)
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
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Greatest Accuracy Credibility
Illustration
Steve Philbrick’s target shooting example...
A
D
B
C
E
S1
S2
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Greatest Accuracy Credibility
Illustration (continued)
Which data exhibits more credibility?
A
D
B
C
E
S1
S2
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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
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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)
Greatest Accuracy Credibility
Derivation #1 (with thanks to Gary Venter)
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The formula
Z = v/(u+v)
can be reformulated in terms of variances by substituting v = nv
2 and u = nu2 and reducing:
Z = (1/u2)/[(1/u
2) + (1/v2)]
or
Z = 1/(1 + u2 /v
2)
Greatest Accuracy Credibility
Derivation #1 (continued)
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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 + ij
Where C is the mean loss over all classes, Mi is the mean loss differential for class i, and ij is the random error. Assume that the Mi’s and the ij
’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).
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Greatest Accuracy Credibility
Derivation #2 (continued)
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
Class 1 Class 2
Pictorially this looks like:
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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/(n+k)
where k = EVPV/VHM
Greatest Accuracy Credibility
Derivation #2 (continued)
This is the original Bühlmann credibility
formula
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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
Increasing credibility
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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, in the original Bühlmann construct, is a
property of the entire set of data.
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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