Credibility Session SPE-42 CAS Seminar on Ratemaking Tampa, Florida March 2002

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


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])


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


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


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


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


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}


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):


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%


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:


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


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


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.


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)


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


Derivation (continued)

Class 1 Class 2


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


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


Illustration

Steve Philbrick’s target shooting example...

A

D

B

C

E

S

Next


Illustration (continued)

Which data exhibits more credibility?

A

D

B

C

E

S

Next


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

Credibility Session SPE-42 CAS Seminar on Ratemaking Tampa, Florida March 2002

Documents

Transcript of Credibility Session SPE-42 CAS Seminar on Ratemaking Tampa, Florida March 2002