Basic Concepts in Credibility

CAS Seminar on RatemakingSalt Lake City, Utah

Paul J. Brehm, FCAS, MAAAMinneapolis

March 13-15, 2006

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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

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 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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BackgroundRationale

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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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 = 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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BackgroundMethods

“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

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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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= (1-Z)*E1 + ZE[T] + Z*(T - E[T])

Limited Fluctuation CredibilityDerivation

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 CredibilityMathematical 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 CredibilityMathematical 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:

Solving for N (# of claims for full credibility, i.e., Z=1):

Z = kE[T]/zpVar[T]1/2 becomes:

Z = kE[T]/zpE[T]1/2 = kE[T]1/2 /zp = kN1/2 /zp

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Limited Fluctuation CredibilityStandards 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 CredibilityMathematical 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):

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N = (zp/k)2{Var[N]/E[N]+ Var[S]/E[S]2}

Limited Fluctuation CredibilityMathematical 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?

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

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

(n/1082) .̂5n/n+191

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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.”

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 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

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 CredibilityIncreasing 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 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.

Greatest Accuracy Credibility

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Greatest Accuracy CredibilityIllustration

Steve Philbrick’s target shooting example...

A

D

B

C

E

S1

S2

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Greatest Accuracy CredibilityIllustration (continued)

Which data exhibits more credibility?

A

D

B

C

E

S1

S2

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Greatest Accuracy CredibilityIllustration (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

Variance between the means =“Variance of Hypothetical Means”

or VHM; denoted t2

Average class variance =“Expected Value of Process Variance” =

or EVPV; denoted s2/n

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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 = u/(u+v)

Greatest Accuracy CredibilityDerivation (with thanks to Gary Venter)

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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)

Greatest Accuracy CredibilityDerivation (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 CredibilityIncreasing credibility

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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.

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

Introduction to Credibility