Introduction to Credibility

CAS Seminar on RatemakingSan Antonio, TexasMarch 27-28, 2003

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

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?

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

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]}

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, and say 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

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.

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

Greatest Accuracy Credibility

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 Credibility

Derivation (with thanks to Gary Venter)

Suppose that xi is the mean loss ratio for a class, and y is the overall mean. Let the variance between the class means (xi) be denoted by t2 and the variance about the total be s2. t2 is called the variance of hypothetical means (VHM) s2 is called the expected value of process variance (EVPV) or just

the process variance

Assume we have enough data about y to know its variance, t2. Also assume we have n observations of a given xi, making our estimate of its variance s2/n. Then,

Z = (n/s2)/(n/s2 + 1/ t2) =

= n/(n+ s2/t2)

= n/(n+k)

Greatest Accuracy Credibility

Derivation (continued)

Greatest Accuracy Credibility

Illustration

Steve Philbrick’s target shooting example...

A

D

B

C

E

S1

S2

Greatest Accuracy Credibility

Illustration (continued)

Which data exhibits more credibility?

A

D

B

C

E

S1

S2

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

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

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 need to be able to identify variances. Credibility, z, is a property of the entire set of data

Bibliography

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