Credibility Theory

Citation preview

WILLIAM RUSSELL ｐｕｾ＠

C1

NATIONALE 1\JEDERLANDEN N.V. RESEARCH DEP·\R'nfEL\T P.O. BOX796 3000 AT ROTTERDA.\1 THE :--.'ETHERI.A.'\'DS

M.J . GOOVAERTSANDW.J. HOOGSTAD

SURVEYS OF ACTUARIAL STUDIES NO. 4

HG 80,5 .G"S .1987 PREFACE In the second volume of these Surveys of Actuarial Studies, on "Rate Making" , the subJects of "Credibility Theory" and "Large Claims" were considered , but only to a liml. ted extent. This was mentioned in two reviews, and it was suggested that separate volumes might be devoted to these subjects . With regard to "Large Claims" , in a review by Mr. B. Ajne (Astin Bulletin , vol. 15, nr 1 (1985), p. 67) it is suggested to further elaborate on the topic of large claims : "The problem of large claims is a nuisance in tariff construction work, at least as soon as personal injury cla1ms or fire cla1ms are present . So, as a practitioner , one could have hoped for a fuller treatment, perhaps including the division of claims into more than two groups (e.g. normal cla1ms , excess claims, superexcess claims) and/or some help from the theory of outlying observations. Maybe one could hope for another volume in the series on this subject?" . We have considered this issue, but we have ser1ous doubts whether we will be able to publish a volume on large claims. Up t1ll now the number of articles published on this topic is rather limited, especially as far as the influence of large claims on standard products, such as automobile and home owners insurances , is concerned. In another review , by Mr. E. Straub (Ml.tteilungen der Vere1nigung Schweizerischer Versicherungsmathematiker, 1984, Heft 1, p. 113) the subject of credibility theory was mentioned: " ... es ist kein Kompendium und kann es mit 130 Seiten auch gar nicht se1n (allein iiber Credibility liesse sich mehr schreiben) , . . . " . This part 4 of our ser1es " Surveys oi_ Actuarial Studies" specifically deals with credibility theory , and thus ｾｩｬ＠ treat the SubJect into far more depth than was the case l.n volume 2. This volume was written by Will Hoogstad, who works in our company, and Marc Goovaerts , Professor at the unl.versities of Leuven (Belgium) and Amsterdam. We would especially like to thank Mr . Goovaerts for his valuable contribution . We hope this volume will contribute towards a better understanding of credibility theory and thereby wl.ll prov1de a link to further pract1cal applications .

April 1987

Research Department Nationale- Nederlanden N.V . G. W. de Wit

3

893638

TABLE OF CONTENTS

Preface

3

Table of contents

4

Introduction

7

General Guideline

15

Chapter 1 .

A mathematical model

19

Appendix 1 .1

31

Chapter 2.

Exact credibility

33

Chapter 3 .

The classical model of Blihlmann

37

a. b. c. d. e.

37

Chapter 4 .

The BUhlmann- Straub model a. b. c. d. e. f.

Chapter 5 .

Model and assumptions Comments Computations Remarks Numerical example

The De Vylder non-linear regression model a. b. c. d. e. f.

4

Model and assumptions Comments Computations Remarks Alternative estimators Numerical example

The Hachemeister regression model a. b. c. d. e.

Chapter 6.

Model and assumptions Comments Computations Remarks Numerical example

Model and assumptions Comments Computations Remarks Alternative estimators Numerical example

38 39 39

41 43 43

44 47 48

51 51 53

53 53

54 55 57

61 61 62 63

64 65 66

Chapter 7 .

Chapter B.

Chapter 9 .

The De Vylder semi- linear model

71

a. b. c. d. e.

71 71 73 74 75

Model and assumptions Conunents Computations Remarks Numerical example

The De Vylder optimal semi-linear model

79

a. b. c. d. e.

79 79 80

Model and assumptions Conunents Computations Remarks Numerical example

82 84

The hierarchical model of Jewell

87

a. Model and assumptions b. Computations c. Numerical example

87 89 91

Chapter 10. Special applications of credibility theory

93

10.1 Loss reserving methods by credib1lity

93

a. b. c. d. e.

Model and assumptions Conunents Computations Remarks Numerical example

10 . 2. Large claims and credibility theory

a. b. c. d. e.

Model and assumptions Comments Computations Remarks Numerical example

Chapter 11. Credibility for loaded premiums

93 95 96

98 98 99

99 100 101 102 103 105

11.1 . Credibility for variance loaded premiums

105

11.2. Credibility for Esscher premiums

107

Bibliography

111

5

I NTRODU CT I ON Credibility premiums

theory

for

prov1des

contracts

us

that

with

techn1ques

belong

to

a

to determine

more

or

less

insurance

heterogeneous

in case there is limited or irregular claims exper1ence for

portfolio,

each contract but ample claims experience for the portfolio. It is the art and

sc1ence

of

using both kinds of experience to adjust the insurance

premiums and to improve their accuracy.

The general and by now famous credibility formula

C

=

(1 - Z) .B + Z.A

originated in the United States during the years before World War I and was suggested in the field of workmen's compensation insurance. The industry- wide premium rate charged for a particular occupational class is represented by B. But an employer having a favourable record w1th this class

tries

experience.

to

his

lower

premium

to

A,

the

rate

based

Because observat1ons of one employer are to a

his own

on

large extend

ruled by random fluctuations, Whitney [1918) suggested a balance C between the two extremes A and B. Some 70 years ago he wrote: "The problem of experience rating arises out of the necess1ty , from the

standpoint

of

equity

to

the

individual

risk ,

of

stnk1ng

a

balance between class-experience on the one hand and risk exper1ence on the other" . It was felt that

the mixing-factor Z should reflect the volume of the

employer's experience.

The

larger this volume,

the more credib1lity, by

means of a high value of Z, is attached to the desired premium A. Thus it became common parlance to denote Z as "the credibility factor" or simply "the

credibility".

quest1on

of

experience.

how Of

The

much

course ,

theory weight not

of

credibility

should

only

be

g1ven

downward

but

1s

concerned

to

th1s

also

w1th

actual

upward

the

cla1ms

sh1fts

1n

individual premiums are possible, although the employer's pressure 1n such cases will not be felt strongly.

7

After these early findings credibility theory developed in the direction of what is now called "limited fluctuation credibJ.lity theory". Due to the fact that it was created by North- American actuaries, to some it is also known as " American credibility theory". branch

in the present survey,

Although we will not treat this

we will now briefly outline some of its

features. Without mak1.ng reference to the formula above, this theory or1.ginated with a paper by Mowbray [1914] "How extensive a payroll exposure is necessary to give

a dependable

pure

premium?". Also in a workmen's compensation

context , he poses the quest1.on of how many insureds, covered by the same contract, are necessary to have a fully credible estimate of A that can serve

as

a

(individual)

premium

for

the

next

year.

Or,

reformulated,

how

much

claims experience is needed? We quote his solution to this

problem: "A dependable pure premium is one for which the probability is high, that it does not differ from the true pure premium by more than an arbitrary limit". With a relatJ.vely simple mathematical model it is possible , after setting some tolerance-lJ.mits and using the Central Limit Theorem, to compute the number of insureds required for a

reliable

"true"

interpreted

premium.

credib1.l1.ty. number,

This

number

is

(credible) as

a

estimate of this standard

for

full

In cases where the number of insureds at least equals this

it amounts to putting Z

1. However , this solution left as an

open question how to act when the number of insureds is too small. Of course,

Mowbray

himself did

not

raise that question because his paper

predated that of Whitney. This problem, known as partial credibility, led to numerous articles and a number of popular, heuristic formulas for Z. All these formulas assess partial credibility as a value between 0 and 1 and most of them are dependent on the actual and the required number of insureds·

Despite all of these practical efforts ,

the need for a sound

mathematical model was felt deeply. For a survey of limited fluctuation credibility theory that includes a bibliography and a mathematical addendum we refer to Longley-Cook [1962].

8

The theoretical foundation of credibility theory was not established unt1l the

1965 ASTIN Colloquium,

where Bi.ihlmann presented his "distribution

free" credibility formula (published in [1967)), based on a least squares criterion. This initiated a new branch in the theory , now called "greatest accuracy credibility theory" or simply "European credibility theory". Both Bailey [1950) and Robbins [1955) published results before, but these were not derived in a distribution free context. Bailey [1945) vaguely pointed at this approach. However,

his article is hardly understandable due to

notational difficulties . This rapidly growing branch of credibility theory forms the scope of this publication. But before focussing on th1s and some of the mathematical background, let us cast some light on the place of credibility within the rate making process. Most of the actuaries working in practice probably agree with the top-down approach in tarification as proposed by H. Bi.ihlmann during an Oberwolfach meeting.

He

explained how,

for

an insurance portfolio, the collective

matching of liabilities and premium income is the primary concern (this is the top level) while a fair distribution (the down level) of the premium income among the different contracts has to be realized afterwards. This distribution of the total revenue among the d1fferent contracts could e.g.

be

done

by

means

of premium

principles.

However ,

credibility theory

provides us with a sound statistical tool for a fair distribution of the premium income among the different contracts in a portfolio. The matching of premiums and liabilities must be the insurer's main goal. Apart from situations where the prem1um is prescribed by the government or ruled by considerations of solidarity, payable consists of the future claim amount.

(approx1mated)

the larger part of the premium risk prem1um, i.e. the expected

In the sequel we will ma1nly restrict ourselves to

these risk premiums and shall often denote them by "premiums" only. The treatment of credibility theory for loaded premiums is only briefly dealt with in one of the last chapters . A limited number of specialized papers is ment1oned in the references.

9

For a general actuarial guideline in the rate making process, we refer to the former issue "Rate making" [1983] in this ser1es. Upon read1ng this, it

will

become

clear

that

credibil1ty

is

only

a

part,

and

even

a

dispensable part , of the whole process .

The actuary ' s first task is to determine the characteristics of insureds which, them ,

in his (subjective) opinion, are essential to distinguish between the

so-called tariff variables.

Unfortunately,

not all of these

tariff variables will be observable , only some of them have data available and even fewer appear to be of statistical relevance. A special and very important tariff variable is past claims experience, a representative of both observable and non- observable variables. Preferably simultaneously, but often afterwards, the opt1mal tariff classes (i.e. optimal sub- sets of the

tariff

variables)

are

to

be

calculated.

These

classes

imply

a

structure that consists of so-called cells. Within every cell there are a number

(possibly zero) of insureds with identical risk characteristics.

Now the question of the determinat1on of the insurance premiums for each of the cells arises. There are two solutions. The first possibility is to specify an additive or multiplicative model, in which the (transformed) claims experience variable should be described, as well as possible, by the chosen tariff structure. Belonging to the same methodological approach is the method of maximum likelihood to estimate the parameters of a ment1oned variable.

pre-specified distribution

function

for the above

The other possibihty consists of our credibility approach. The Biihlmann model and 1ts generalizations allow for a distribution free estimation of the 1nsurance premiums as a weighted average of the cell-experience and the

portfol1o

models

exper1ence.

However,

it should be mentioned that

in general are only suited to deal

variable

and

exactly

one

other

these

with the claims experience

tariff variable.

In case more

tariff

variables are involved, the models should be modified . In theory, combinat1ons of both methods are also possible. To be specif1c, consider the follow1ng example in automobile insurance.

10

On the basis of available and relevant tariff factors, such as the we1ght of the car and age of the driver the heterogeneous portfolio 1s (given the tariff classes)

split into groups of insureds which are less heteroge-

neous. With the aid of one of the f1rst techniques, a premium for each group is calculated. This premium reflects the average claim amount within that group. Nevertheless , not all drivers are equally skillful or careful. Ind1vidual claims experience can ristics.

individual group's

tell us more about these hidden

risk characte-

So , within each group a second selection is possible: base the driver's

claims

credibility

experience.

adjusted

premium on

Another part of

h1s

own

and the

the heterogeneity will

be

eliminated and the new premium is closer to the true premium. Of course, insurers

in pract1ce

use

a

this procedure 1s too laborious to handle and

bonus-scale

with

fixed

discounts

and

surcharges

1n

percentages of the group-premium to incorporate the individual (number of) claims experience. Another example can be found in Biihlmann and Straub [ 1970] . We fix our attention to the annual l oss ratios of the different kinds of treaties of a reinsurance portfolio observed during a number of years. treaty,

we

are

observed ratio

interested in the expected indiv1dual (A in our general formula)

loss

For a fixed ratio.

The

is easy to calculate from the

data available. Nevertheless, in most cases the data is too scarce and far too irregular to provide for a reliable estimate. Hence credib1lity theory is

a

useful

collective

tool

loss

to

ratio

apply. (B)

If

the

portfolio

is

large

enough ,

the

can be considered a good estimate for the

expected loss ratio over the portfolio . Now credibility theory g1ves for every ind1v1dual treaty a weighing factor Z that reflects the reliability of the individual and group data. Some credibility techniques are able to handle inflated cla1ms figures. An example

is

private

passenger

Hachemeister ' s bodily

method 1njury

[1975].

His article also deals with

insurance.

Claim

amounts

for

a

few

U.S.A . -states are observed for a number of quarters . They show a tendency to increase in time, due to (inter alia) 1nflat1on. We are interested 1n the state-specific inflation factor. It 1s supposed that 1nflation is not the same in all U. S.A.-states, hence, these states form a heterogeneous collective. 11

The observed state inflation factor factor because of the poor data.

(A)

is a

bad estimate for the true

The countrywide inf l ation

(B) ,

taking

into account the observations of all states is more trustworthy . ｡ｧｾｮＬ＠

Hence

credibility theory applies.

In Chapter 1 , we will first give a rather extensive but simpl e treatment of the model of Biihlmann .

Al though surveys in general do not contain a

mathematical derivation of the model , we think that the reason ing behind this

model

will

provide the reader with a better understanding of the

basic principles of credibility theory. Moreover , the models that emerged later are ,

one way or another, a straightforward generalization of this

model. For practical linear

reasons credibility premiums

functions

experience.

of

the

observed

ｾｮ＠

However ,

some

are

often

(individual

cases

this

and

linear

restricted to be

collective)

approximation

claims of

the

premium turns out to be the optimal premium . This phenomenon is known as exact credibility and is treated in Chapter 2. Chapters 3 through 9 deal with the generalizations mentioned above . For the BUhlmann model it is considered a drawback to have equal weights for all

the

observations.

The

Biihlmann-Straub model

i ncorporates so- called

natural weights to deal with this disadvantage . Hachemeister and De Vylder have developed models that allow for inflation or trends in t h e data . We end up with a review of recent developments in regression and semi-linear models ｾｮ＠ credibility theory. Although every chapter here bears the most familiar name for the model , we

ｷｾｬ＠

not

only

developments

･ｳｴｾｭ｡ｩｯｮ＠ ｣ｯｮｴｲｾ｢ｵｩｳ＠

review

the

afterwards. and we only.

will

original

These

are

mention

article mainly

the

(in

pay attention

but also in our

the

field

opinion)

to

of parameter most

relevant

In Chapter 10 , two applications of credibility theory are presented. The first of these consists of a loss reserving method in which IBNR- claims are forecasted. The second application deals with the determination of the optimal ｴｲｾｭｩｮｧ＠ As ｭ･ｮｴｾｯ､＠ premiums.

12

point in case the data contain outliers . already,

in Chapter 11 ,

we

discuss

a

method

for

loaded

In several papers, De Vylder constructs a uniform credibility theory where extensive use is made of Hilbert space theory and from which all models considered here are derived as special cases. Because the mathematical framework is rather specialized, the interested reader is referred t o the references on this subject given at the end of this monograph. Apart from some exceptions, all models and applications are stated without proofs. Moreover , it is not our intention to give a lengthy and exhaustive review

of

because

every

the

increasing .

article

number

now

that

has appeared .

amounts

Our intention is

to

This would be impossible

approximately

300 and it is ever

to provide the reader with some gene ral

insight and the formulas necessary to ｾｭｰｬ･ｮｴ＠

the methods . By means of

illustration ,

every model is accompanied by a numerical example. These

examples

serve

may

as

a

check

for

those

who

implement

the

model s

themselves on a personal computer or mainframe.

13

GENERAL GUIDELINE Due to different market situations, changing claim patterns or other major shifts,

every

tariff

needs amendment from time to time .

Then,

after a

careful determination of the flaws of the existing structures one should make a set-up for a new tariff that copes with the observed shortcomings .

ｾｮ＠

As mentioned ｣ｯｮｳｾｴ＠

the introduction ,

of a

determination

the process towards the new ratebook

few different stages. of

the

premiums,

is

Perhaps

the

last stage, the actual

the most difficult one.

We

stressed

before that several possibilities are available . This is especially true if the new set- up indicates a credibility approach because the actuary is confronted with a large number of different methods, each benefits.

ｨ｡ｶｾｮｧ＠

their own

To indicate the applicability of these methods for the parti-

cular situation at hand we present the next guideline . This schedule provides the reader with the features of the credibJ.lity methods treated in this issue. Although with this information it is hardly possible to select the method best suited to your problem, we think that it at least may serve as a first indicator . We refer to the corresponding chapter for additional details.

Apart from verbal information on the methods we included some mathematical information

as

well.

At

first

reading,

without

being

ｦ｡ｭｾｬｲ＠

with

credibility theory, we advise you to skip these columns. Nevertheless, you will

find

out

later

that

consultation

of

the mathematics can be very

fruitful.

15

Method

2. Exact

E( E( X]

cred1b1hty

J9 Jix 1

1

, ••• ,Xk)•

l1near in X J

E(XJ 8 J9])• 11(9) J

Cov(x J9 J• o ' (9J).It 1 1 It: (t X t) 1dent1ty Utr.lX

Requ.ued data

Main assuapt lons

R-ru

Structural p•u...aters

depends on selected dutnbution

depends on selected dutnbutton

requ1res knowledge of dutnbution

none

funct1ons

funct1ons

deflated (average) c:la111

data do not

a1110unts, loss ratios

O!tC:. 4. BuhlNnnStraub

E(Xj 5 j9 J• 1J(9j) 1

Cov(X j9j)• o' (9 ) .v )

v.:

J

J

_1_, J

dug (-1-, ... , w]l wJt

see 3, 1n addition: nwnber of cla1ms ,

exh1b1t trend, observations

funct1ona

simple IIOdel, l1a1ted prac:ttc:al value

"o·

useful practical extension of 3

"o• .,

atta1n equal wughts see 3 , d1fferent we1ghts are possible

-·

s •

a

s ' ii

premtums etc: . E(x J9jJ• y.8(9jl 1 y: (t x n) matnx 8(9 ): n vector 1 6. De Vylder

Cov(Xjj9jl• o' (9j) .v v.: (t x t) matux

f: ｒｮ

ｾｒ

ｴ＠

see 4

J

E(x J9 J• f(8!9 )) 1 1 1

(non-ltnear regresuon)

1

see 4 v.: (t x t) aaatnx J

7. De Vylder (se•l-ll.near)

E[fp(X

f

Method

8

)

j9)=

see 3

l'p(9) p

llnear approximation of trend, d1fferO!nt we1ghts are posSible

Required data

extension of 4

non-linear (usually exponentlal) approximation of trend, different wetghts are possible

for practical reasons, extension of 5

see 3,

suited for 1..a.

functions fp(Xsl

truncatlon of data, semi-l i ne ar extens1on of 3

known

: ｒｫｾ＠

iterative

solution,

Mai n assuaptions

iterative solution,

often exponent1al approach used

Structural par...aters

.... thod

B(X j l&j )

Cov (xjle j J

8. De Vylder (optuaal

Required data

see 3

｡ｾＱＭｮ･ｲＩ＠

Rai n assu.ptions

Re.arluo

aee 3,

1terative solution,

approxiaatlnq funct1on f (X ) 5 unkno"'"

prov1des opt1mal result of 7,

Structural parUIIlt ers

computational

probleJilS, extension of 3

9. Jewell

El xP15 1eP, ep J• 1 1'(8p, 8 PJ)

Var(XPJSI&p' ePJJ · l o' (8P, e ) w PJ PJ•

see 4

aee 4 ,

linear

hierarch1cal structure of data

extension of 4

..

HpO' s • ii , 6

Chapter 1. It

is

A MATHEMATICAL MODEL

possible

to

review

the

models

without stating any mathematical

available

background .

in

credibility

theory

However , we feel that some

1ntroductory information on this subject is necessary to have a understanding of the

techniques in use today.

better

In this chapter we try to

explain some of the basic ideas of credibility without going into abstruse mathematics. Although a very general set-up of the theory is on hand using H1lbert

space

theory ,

we

shall

not

adopt

that

line

here .

For

the

interested reader , we refer to the references at the end of th1s booklet , e.g.

De

Vylder

[1976).

Instead ,

the

less

general

assumptions

made

by

BUhlmann [1967] in his celebrated paper form our starting point. As a preamble we first state some basic results and introduce notational conven1ences.

As mentioned in the introduction, similar ,

a

portfolio consists of more or less

but never completely ident1cal ,

contracts.

In modern actuar1al

theory it is customary to think of a risk parameter , say e. , that totally JS describes the risk character1st1cs of contract j (j = 1, 2 , ... , k) in period

s

(s =

1,

2,

t).

Intuitively,

one

assumes

that

all

the

differences between the contracts and periods are caused by the different parameters ell

I

e 12 I

• •• I

ekt. Note that I

all risk parameters are equal,

1n the hypothetical case where

it is useless to apply cred1bil1ty theory.

In nearly all models in credibility theory, it is assumed that the e . are JS time-homogeneous , i.e. the risk parameter for a fixed contract does not change over time. Hence we drop the subscript sand write e , e , ek . 2 1 Evolutionary models , not based on th1s assumpt1on, by, for instance , Kremer

[1982],

bibliography.

Sundt

(1982,1983)

Unfortunately ,

in

and Albrecht most

[1985)

practical

are

listed in

situations

the

the r1sk

parameters are unknown or unobservable and we should try to estimate them as accurately as possible in order to have reliable credibility premiums.

19

From now on these parameters will be lnterpreted as rando we adopt

the

terminology

of

Biihlmann

[19701

distribution funct1on of all these variables, structure funct1on.

and

vanables. Here

denote by

the

U(S)

which 1s also called the

Now the contracts are sim1lar 1n the way the1r nsk

.Parameters 9J. have identical structure functions U (9) but d1ffer because the realizations contract . Just like the

J e.,

6.J

(although

unobservable)

are

d1fferent

for

each

the structure function is usually hard to trace. As thls

is a drawback for most statist1ca1 methods, it is not for most credlbility procedures. These procedures limited information is needed.

are

called

"distr1but1on

free"

and only

In the lntroductlon we remarked that most credibillty procedures, apart from

the

claims expenence

var1able

wh1ch

lies

in

the

nature of all

ｾ＠

methods , are able to deal with one tariff variable only. This corresponds . ｾ＠ a Slngle Valued risk parameter e . . In case one considers more t n JUst this t ff J say one arl variable we have a vector valued risk parameter • 9 9.J = (9 jl' 9 j2 ' · · · • jq), where q denotes the number of tar iff vanables. If these tarl· ff varl· abies are t be true ln dependent, which often happens o practice we ref t model by ' Jewell (1975]

°

er Chapter 9 where we discuss the h1erarch1cal If th . 1" e (ln · e varlables are independent one should genera lZ "Yl the model, in Chaptors 2 through 8, thoj ' deal With one tariff variable only.

Ｂｩｧｨｴｦｯｾｲ､＠

｢ｾ｡ｵＢ＠

Tho 'hi-. "perienoo

· t> l variable is represented by X . with subscnp Ond' ' ' •bove. In the JS' · ble bUt models now With ob,er,.ble . .X.)S is also considered a random varla s tho reall.Zatlons x )S We usually interpret X·)S a "O"ge . .. f (xt l el.u(e ) .dx 1 ... dxt.de 1

1

where u(el deno tes the structure function's

､･ｮｳｾｴｹＮ＠

25

Of

course ,

one

· ｡ｰｲｯｸｾｭｴｮ＠

could

｣ｲｾｴ･ｩｯｮ＠

another

to define

the optimal

but t h a t wou ld not result •n a linear credibility formula . ｾ＠

Bayes ' ａｰｬｹｾｮｧ＠

choose

theorem one ｯ｢ｴ｡ｾｮｳ＠

Also recall that

J ll(9).f(6lx 1 ,

8

x , ... , xt).u(el . dB 2

and hence

j j .. . j

t[g]

e x1

((1!(6)- E[ll(6liX , 1

xt

+ (E[ll(6liX , .. . , Xt]-g(x , . .. , xt)) 2 )*

1

1

We conclude that the optimal function g (the exact credibility formula) using this least squares ｣ｲｾｴ･ｩｯｮ＠

ｾｳ＠

To statisticians this function g is known as the posterior Bayes estimator of ll(8)

ｷｾｴｨ＠

respect to quadratic loss and prior function U(6) .

Unfortunately , ｾｮ＠

order to apply this result to practical situations , the

distribution ｦｵｮ｣ｴｾｯｳ＠ knows

that

this

F (xI 6 l and U ( 6 l have to be known . Every actuary is

mere

fiction.

One

can

show

t hat

even

when

the

distribution functions are known, the mathematics involved to compute the ｰｲ･ｭｾｵ＠

above is, in general, much too complicated . To avoid this kind of trouble for the moment we restrict ourselves to functions g of the linear type, so we are interested in the minimum

over all possible values of c 0 , c 1 , ··· • ct.

26

Note

that

we

st1ll

suppose

the

variables

x 19 , 1

x 19, ... , Xtl9 are 2

independently and identically distributed . Taking first tives

partial deriva -

with respect to the coefficients we have a linear system of t+l

equations t

E[ll (9)

-

Ls=l

co -

c .X l s s

0

t

E[ Xr. (1!(9) - c

-

0

L

s=l

0

c .X ) ] s s

r = 1 , 2,

..., t

Mult1plying the first equation by E[Xr] and subtract1ng 1t from the second success1vely for r = 1 , 2 , . .. , t the latter t equat1ons are equivalent to t

fu

r = 1, 2 , ... , t

c .Cov[X , X ] s r s

The cond1tional variables are identically distributed , so one can use the covariance relations described above.

After some reshuffling our system

reduces to

t

L

Because

of

the

r

= a

c

s=l

= 1,

2 , .. . , t

s

symmetry

of

this

system

of

to c 1 I c 2 I • • • I c t it is •;mmed1ately clear that c 1 we may write our last system as c

0

equations

= c2 = ..·

with

respect

= ct = c and

+ m.t . c = m

s 2 .c + a.t.c

=a

hence a

c = -----

ｾ＠

t

27

where we have put a.t

z = ----s 2 + a. t and consequently

c

0

=

(1 - z) .m

The (linear) credibility adjusted estimator for

ｾＨＹＩ＠

then becomes

t

L

co+

s=l

c .X s s

(1 - z) .m + z.X

where t

Ls=l

X

X

s

t

This

estimator

has

been

used

intuitively appealing formulas that

it

is

possible

by on

､･ｲｾｶ＠

to

American

actuaries ,

however,

with

z, for decades. It should be mentioned this

linear

｣ｲ･､ｾ｢ｩｬｴｹ＠

premium

in a

Bayesian-free way; see Gerber [1982]. Let us next concentrate on the new and elegant result on z and investigate some limiting

In

case

t

ｰｲｯ･ｴｾｳＮ＠

then

+ ..,

experience, it is

z + 1.

ｯ｢ｶｾｵｳ＠

This

is

acceptable because

in case of

full

to give full confidence to the individual risk

premium. In case a

=

0 then z

=

0. When the expected individual claim amounts are

perfectly the same, there is no heterogeneity within the portfolio. Thus m is the best linear estimator f or

ｾＨＹＩＮ＠

In case a .. .., then z t 1. Now the collective is extremely heterogeneous and the

collective

ｩｮ､ｾｶｵ｡ｬ＠

result

d oes

not

contain

ｾｮｦｯｲｭ｡ｴｩ＠

on

the

ｳｰ･｣ｾｦｩ＠

risk.

+ 0. This is also intuitively clear. When the claims experience variable for fixed e shows a ｨｾｧ＠ degree of randomness, the In case s 2

.....

then z

collective information is worthless in

28

･ｳｴｾｭ｡ｩｮｧ＠

the real

ｾＨＸＩ

Ｎ＠

In accordance parameters .

with Norberg

He

practical

used

following

these

in order

chapters

we

call m,

a and s

2

the structural

the term structural to express the fact

purposes,

observations

[1979)

to

that

parameters compute

this

is a

z

be

should

and Ma. ｳ･ｲｾｯｵ＠

that,

for

from

the

estimated

It will problem.

turn

out

in

Many authors

the have

suggested different solutions for different models with every estimator having

its

quality, where

own

features.

We consider unbiasedness

a

highly desirable

so with every model these estimators are represented . In cases

other

important

(unbiased)

estimators

exist ,

they

are

also

mentioned.

Note that we have

independent of the value of z. A faulty estimate for z will result in poor credibility

adjusted

premiums

but

without direct

consequences

for

the

insurer. Of his primal importance is a correct estimate of the collective premium .

This

insurer ' s

is a

good example of the top bottom approach where the

main task consists in calculating a correct collective premium

and subsequently a fair distribution of it among the different contracts. Nevertheless,

in the long run the insurer's portfolio shall reflect the

correctness

of

pushes

overrated

off

his

estimate z.

A bad

contracts.

The

estimate balance

attracts underrated and of

ｰｲ･ｭｾｵｳ＠

and

claims

deteriorates and the tariff needs amendment.

We may also study the problem from a different point of view. One could not be interested in ll (9)

but in the expected ｣ｬ｡ｾｭ＠

amount (loss ratio)

Xt+l in the next period. The solution to this problem is easily obtained in case we suppose the same hypothesis to hold for the new random variable xt+l · For the credibility formula we have

29

which is the same result because ll (9)

E[Xt+lieJ

so

Subsequently minimizing c .X )1 s s

over

all values of

ct leads

]

to our previous system of

equations and hence the same solution for these coefficients because of the covar1ance relations

=a

s

=

l , 2, . . . , t

After this classical result in the literature a number of generalizations appeared. These will be the contents of the following chapters . Although we will not give a derivation of these models, a reasoning similar to this one applies to all of them. For a very readable introduction to statistical theory which involves the techniques used in this chapter see Mood, Graybill and Boes [1974].

30

Appendix 1.1

As

mentioned

in

the

introduction,

all

models

are

accompanied

by

a

numerical illustration. To make a comparison useful throughout the book we will use one set of data. We did not gather a new set but decided to work with

Hachemeister's

well-known and ,

[1975]

in fact,

a

data

set.

It

has

the

advantage

of

being

number of papers and articles have appeared

that have a numerical example on this set. Hachemeister considered five different states and twelve quarters of claims experience

(our "contracts", so k= 5)

(our "periods", so t= 12). This

experience consists of average cla1m amounts for total _private passenger bodily injury insurance from July 1970 until June 1973 . Hence our x. (in US$) read JS

s= 1 2 3 4 5 6 7 8 9 10 11 12

j= 1

j= 2

j= 3

j= 4

j= 5

1738 1642 1794 2051 2079 2234 2032 2035 2115 2262 2267 2517

1364 1408 1597 1444 1342 1675 1470 1448 1464 1831 1612 1471

1759 1685 1479 1763 1674 2103 1502 1622 1828 2155 2233 2059

1223 1146 1010 1257 1426 1532 1953 1123 1343 1243 1762 1306

1456 1499 1609 1741 1482 1572 1606 1735 1607 1573 1613 1690

Up till now we did not discuss (but just mentioned) the fact that several contracts might have a b1gger impact on the overall figures than others. A number of credibility models are apt to incorporate this information. the

sequel

we denote

this

measure

of

volume

by

In

the weights

w. . The JS weights in this set of data reflect the number of claims that correspond

to x . . JS

31

These weights w . are JS

s= 1

2 3 4 5 6 7 8 9 10 11 12

j= 1

j= 2

j= 3

7861 9251 8706 8575 7917 8263 9456 8003 7365 7832 7849 9077

1622 1742 1523 1515 1622 1602 1964 1515 1527 1748 1654 1861

1147 1357 1329 1204 998 1077 1277 1218 896 1003 1108 1121

j= 4

407 396 348 341 315 328 352 331 287 384 321 342

j= 5

2902 3172 3046 3068 2693 2910 3275 2697 2663 3017 3242 3425

At first sight we observe relatively h1gh and 1ncreasing average claim amounts 1n state 1 and a large number of claims in state 1 and extent also in state 5.

32

to some

Ch a p ter 2 . In

the

EXACT CREDIB ILI TY

foregoing

we

derived

the

exact

credibility

formula

which

is

explicitly stated as follows

ＯｾｊＮＡ･Ｉｦｸ

Ｑ

ｬ･ＮｦＡｸ

Ｒ

Ｑ･＾＠

... f(xtle>. dU(e)

] t!x 1 1e>.f!x 2 1e> .. . f(xtle>. dU(e) We also spoke of the necessity to know the functions U(8) and F(xle> and the

difficulty

in evaluating the

remaining

solution is to consider a linear estimate for

integration. 1J.(9)

The classical

in which only first and

second order moments of the (mostly) unknown distributions are involved. However,

one may try to evaluate the formula above for different pairs

(F(xle>, U(e))

and find out that ,

in most cases, l.t is only a waste of

time . Nevertheless , several people have done a great job on this subject and discovered non- linear

pairs of distributions

that did not

turn

into lengthy

expressions but astonishing linear credibility premiums.

See

for example Bailey [1950] and Mayerson [1964]. This phenomenon, where the exact premium equals the linear premium , is called exact credibility. Jewell [1974],

in

a Bayesian formulation , has shown that this equality

occurs for a family of distributions in combination with thel.r so-called natural conjugate priors. He pointed at the sl.ngle-parameter exponential family with natural parametrization , i.e.

f (xi e)

p(x) . e

- e.x

q(e) where p(x) and q(e) are arbitrary functions . The sample sum or mean is the sufficient statistic for this family. The conjugate prior turns out to be

u( e)

where t

and x are defined in the subsequent proof and c ( t , x ) is a 0 0 0 0 normalization constant.

33

For a general treatment of sufficient statistics and Bayes theory we once again refer to Mood, Graybill and Boes [1974). Apart from mathematical refinements the proof is as follows. Because f(xle> must be a density it is necessary to require q(el

=1

p(x) .e

- e.x

. dx

X

and because of the shape of f(xle> one can write E[xleJ

IJ ( e)

d ln q(e) de

Var(xleJ=- d IJ(e) de Taking the derivative of u(e) with respect to e gives du(el de Again 1ntegrating this result over the entire domain of e (it is assumed that u(e) equals zero in both endpoints) leads to m

J

e

IJ(e).u(el. de

］ｾ＠

to

Inserting these results into the exact formula above gives + X

t

(1- z).m + z . X

where we have put t z =--t + t 0

This

linear

expression

proves

that

credibility

is

exact

for

single

exponential families. Goel [1982) even conjectures that this is the only family of functions with this property.

34

Finally , we show that t

0

=

s 2 /a.

Note that d 2 u( e)

d2 6

Integrating with respect to 6 gives

and the result on t

0

The next classical cases

of this

appl1.cable ,

a

follows immediately.

results on exact credibility all follow as special

general

result.

transformation

However, is

before th1.s general

often needed

result

is

to obtain the necessary

natural parametrization. For details we refer to the original article. f (xI 6)

u (6)

Bernoulli

Beta

Geometric

Beta

Poisson

Gamma

Exponential

Gamma

Normal

Normal

By way of illustration, we will evaluate the exact credibility formula for the third of these couples. Because the Poisson d1.stribut1.on

l.S

a discrete

one, we write Ns instead of Xs' where Ns denotes the number of cla1.ms. We ment1.oned before that credibility was not necessarily restricted to claim amounts.

By definition 6n

f (n I 6)

e

-6

n! u (9)

6

a-1 e

-a. 6

.a

a

(B-1)!

35

so, inserting these functions into our general formula, we arrive at

N ! . . . Nt! (8-1)! 1

a

8

(nl+. · .+nt+ 8 )

J e

6

.e

Je e

0 are given for j JS 1 and where ｾＨＹＮＩ＠ and o (9.) are unknown functions. )

1, 2 1

•

• • t

t

)

As usual 6rs denotes the Kronecker symbol r

=s

r ;. s

b. Comments Of course ,

the independence between and within the contracts still

holds. However, only the expected observations are homogeneous in time, because E[X. 19.]= ｾＨＹＮＩＬ＠ independent of s= 1, 2, ... , t. JS ) ) The Blihlmann-Straub model is a very interesting and, to some extent, a straightforward extension of the classical Blihlmann model. To show this, one can rewrite the original assumptions for the Blihlmann

model as (Bl ' l

The contracts j= 1, 2, ... , k are independent. The var1ables 9 1 , 9 2 , ... , 9k are identically distr1buted .

(B2 ' l

k and r,s

For all j .. 1, 2, E[X . 19.] • JS )

ｾ＠

1, 2, ... , tone has

j.1(9 . )

)

Cov(X . , X. 19 l = 6 .o 2 (9.) Jr JS J rs J where 1!(9 . ) and o 1 (9 . ) are not depending on j, r and s. )

)

That means, as has been remarked already, that assumptions only have to be made concerning the two first order moments.

44

To describe

another approach,

we consider a portfolio with , say , Q

contracts and we suppose the first q

to have the same parameter

1

= Q-q , to have the parameter 2 1 Hence we have the following situation the remaining q

e

2

e and 1

.

contract

structure variables s= 1

j= 1

j= 2

el

el

. ..

xll

x21

...

X

X ql +1 , 1

X ql,2

X

X ql,t

X ql+l,t

j= Q

2

x12

x22

...

t

xlt

x2t

...

el

e2

ql , l

ql+l , 2

62 X q +2 11 1 X ql+2,2

... . ..

62 X

Q, l

. ..

xQ , 2

...

xQ,t

'0 0

observable ..... ｾ＠ variables Q)

a. X ql+2 , t

and e to be variables e and 6 and i f the 2 1 1 2 condition of time - homogeneity is also introduced , we have the following hypotheses Here again we consider

(i)

e

The set of variables (6 , x , x , 2 1 1 set (e 2 , xq +l' xq + 2 , ... , XQ). 1

1

X ) is independent of the ql The variables e and 6 are 2 1

identically distributed. Iii) x1 !6 , x 1e , 1 2 1

. .. , x j6 are independent with the same distribuql 1 x 16 are tion function F(x!e) and X +lle2 , X + 2 16 2 , Q 2 ql ql independent with the same distribution function F(xl&l ·

45

THE BUBLMANN-STRAOB MODEL [1970]

Chapter 4.

a. Model and assumptions In the previous model we ment1oned that 1n practical situations it is often considered a drawback not to be able to have a weighing of the contracts . certain

for instance, the impact on the insurer's figures of a

contract,

procedure first

If,

due

its

volume,

is

substantial ,

the

Biihlmann

ignores this 1nformation . Therefore, a few years after this

model

was

introduced

Biihlmann-Straub model (j = 1 , 2,

to

... , k;

a

generalization

was

presented.

The

not

only deals w1th the variables 9 J and X JS s = 1, 2 , .. . , t) but also incorporates (natural)

weights w. (j 1 , 2 , ... , k; s = 1 , 2, ... , t) . This considerably JS enlarges the field of application , as will be evident from the example. Again we present a scheme contract

j= 1 structure variables

j= 2

........

j= k

92

..... .. .

9k

91 s= 1

xll (wll l

x21 (w21 1

. .. .. ...

xkl (wkl)

2

x12(w12 1

x22 (w22 1

....... .

xk2 (wk2 1

'0

observable variables (natural weights)

0

.... k Q)

a.. t

The hypotheses are sl1ghtly different , i.e. (BSl)

The

contracts

independent.

j= The

1,

2,

variables

k

(i.e .

(9 , X . l) are J J are ident1cally

the pairs

distributed.

43

(BS2)

For all j = 1, 2, ... , k and r,s 1, 2, ... , t one has E[X . Ia.] = ll(a.) )S

J

J

Cov [X . ' X . Ia . ] = 6 . 1 Jr JS J rs w.

)S

where w.JS >O are given for j

1, 2, ... , k and s

2

and where ll(a.) and o (a.) are unknown functions. J

1, 2, ... ' t

J

As usual 6 rs denotes the Kronecker Symbol r

=s

r t- s

b. Comments Of course,

the independence between and within

the contracts still

holds. However, only the expected observations are homogeneous in time, because E[X.JS Ia.J ]= ll (a.), independent of s= 1, 2, . .. , t. J The Blihlmann-Straub model is a very interesting and, to some extent, a straightforward extension of the classical Buhlmann model. To show this, one can rewrite the original assumptions for the Blihlmann model as (Bl') ( 82 ' )

The contracts j= 1, 2, ... , k are independent. The ｶ｡ｲｾ｢ｬ･ｳ＠ a 1 , a 2 , ... , ak are identically distributed. For all j = 1 , 2 , k and r,s = 1, 2, ... , tone has ll (9 .) J

Cov[X. , X. Ia . ] = 6 .o 2 ca.) Jr JS J rs J 2

where ll(a.) J and o ca.) J are not depending on j, rand s. That means, as has been remarked already, that assumptions only have to be made concerning the two first order moments.

44

To describe another approach ,

we consider a portfolio with , say , Q

contracts and we suppose the first q

1

to have the same parameter e 1 and

= Q- q , to have the parameter e 2 . 1 2 Hence we have the following situation the remaining q

contract j= Q

j= 1

j= 2

el

el

...

s= 1

xll

x21

.. .

X

2

xl2

x22

.. .

X ql , 2

structure variables

ql , l

e2

e2

el X

X

X

X

ql+l , l

ql+l , 2

ql+2,1 ql+2 , 2

.. .

92

.. .

xQ,l

.. .

xQ,2

. ..

X

'0 0

observable variables

·rl

>-< Q)

c.

. t

xlt

Here again we consider e

x2t

.. .

X ql , t

X

ql+l , t

X

ql+2 , t

Q,t

to be variables 9 1 and 9 2 and if the 2 condition of time - homogeneity is also introduced, we have the following 1

a nd e

hypotheses (i)

The set of va r iables !9 , x 1 , x 2 , 1 set (9 , X l, X , . .. , XQ) · 2 ql + ql + 2

X ) is independent of the ql The vanables e and e 2 are 1

identically distributed. (ii)

X 19 , x 1e , .. . , x 1e are independent with the same distribu2 1 1 1 ql 1 are tion fun ction F!xle> and Xql +lle 2 , Xql +2 192 ,

i ndependen t with the same distrlbution function F!xle> ·

45

Let

th.,

!Xl + x2 +

1

y2

Wo

c.

1 ql

yl

...

+ X ) ql

(X + X + ql+l ql+2

q2

+ X ) Q

Y J ond 1

tho now con t cacts do f inod by (9 , 1

consid"

J.

(9 • Y

2 2 For these two combined contracts we then have the revised hypotheses (B1 " J

Tho

contract


x: Jr ' JS

m. = ｅ｛ｾＨＹＮＩ＠ J estimator only

X is not ww variance estimator within the class of unbiased estimators. Parameter

JS

This

of

contract j. The estimator X is obtained by taking the expectations of zw Xjw over all contracts taking the credibility weights z , z , ... , zk 1 2 as weights. In case the natural weights are used an unbiased estimator is

then

E[x: Ie .1

estimator

compare 0

1

rs

'W':""

three

estimators

so-called Biihlmann-Straub,

a' 19j)

belonging

to

the

same

class,

namely

the

Bichsel-Straub and quadratic weights esti-

mators.

JS

where the transformed natural weights are equal to

f. Numerical example

w' js

In

the

previous

numerical

example

of

Chapter

3

we

only

used

the

information concerning the average claim amounts Xjs and disregarded The

variables

and weights with apostrophes

Blihlmann-Straub model.

then define

the ordinary

the corresponding number of claims. take advantage of this

In

this model

it

is possible to

information via the parameters w.

proper weighing of the observations.

JS

and have a

50

51

Applying the formulas of section c we obtain

Chapter 5. state

fIll

THE HACHEMEISTER REGRESSION MODEL [1975]

a. Model and assumptions

j= 1

j= 2

j= 3

j= 4

j= 5

2061

1511

1806

1353

1600

a regression technique. The variables relevant for the j-th contract are again ej and Xjs (j= 1, 2, ... , k; s = 1, 2,

The Hachemeister model extends the Blihlmann-Straub model by introducing individual estimator credibility adjusted estimator

,, ' :1'

credibility factor

M.= J a M.= J z.= J

2055

1524

1793

1443

1603

0.98

0.93

0.90

0.73

0.96

considers for every contract j a so-called fixed (t x n) design matrix y of full rank n(.

general

Hachemeister

a

.

+ •

model model

reduces by

, a respective sect1ons

the

to

the

BUhlmann,

introduction

of

BUhlmann-Straub

the

hypotheses

and a su;.... table adapted notation.

of

or the

For instance,

for Hachemeister's model, we have the notations

superscript G denotes the fact that this is the general formulation) , hence

so so

y

x1 x2

y

XG

y

-& G

G

so -&

where

as

usual

in

linear

regression E[ E]:::: E[a ]:::: 0 and Cov[e, a]

is

block diagonal with entries E and r.

The generalized

least

squares

so

y

xk and Cov[E, a]

block diagonal with entries 0

2

(9j) .vj (j:::: 1, 2, ... , k)

and a.

estimator

for

this

mixed model ,

cf .

e. Numerical example

Maddala [1977] p. 463, 464, is As explained earlier, G T -1 G ( (y ) .E .y

G T -1 G .y ( (y ) .E

+ r

+ r

-1

-1

-1

)

>•r

. (r -1

-1

.s

G

G T G -1 G .a + (y ) .E .X ) 0

G T

0 + ( (y > • E

-1

•y

G

+r

the data we are using

throughout the book are

from the original article of Hachemeister. These data deal with average claim amounts and his model is an excellfnt tool to incorporate both -1 -l G T -1 G ) . (y l .E •x

weights intercept

and

inflational

and

slope

of

trends. the

His

purpose

trend-line

per

development of the average amount over time.

to

estimate

the

state,

showing

the

was

It is easy to see that,

given this goal, the optimal choice for the design matrix is

I I

II 1!!,

,,,

ｾＶ＠

57

1

12

1

1

These

1

11

1

10

1 1

2 3

filled with diagonal elements one.

y

'ｾ＠

1 1

have

been

obtained

with

initial

diagonal matrices z.

J

Using the stop-criterion with respect to B0

or

•I

results

2 1

1

11

1

12

BOk(t) - BOk(t+1)

< 0.0001

max k = 1, 2

BOk(t)

We prefer the first matrix because the forecast for the next quarter

then equals the intercept.

convergence occurs after 18 iterations.

Following Hacherneister we will also use the diagonal version of the

The

covariance matrix, so

purposes.

intercepts

computed

It was

to ｢ｾ＠

above

are

for

relevant

obviously

rating

expected that both individual and collective

estimators would shift upwards compared to the Bi.ihlmann-Straub model.

v. J

_1_)

diag

All states,

wjt

These

and especially state 1, exhibit strong trends over time.

outcomes

show that one

should be

very

careful in choosing a

and assume uncorrelated claim amounts for fixed j.

credibility model. Only a well-considered methodological choice between

The results of the regression method are

the

models

purposes.

state

j= 1

j= 2

slope

results

For instance,

which are

a

sound

basis

for

rate

making

disregarding the trend in our data leads to

underrating in all states. Note however that this trend is not only due

j= 3

j= 4

j= 5

to

inflation

but

also

to

other

external

factors

like

driving

environment, price and safety measures of the cars and so on.

individual estimators intercept

gives

2470

1621

2096

1538

1676

We conclude this section with a graphical comparison of the Bi.ihlmann-

-62

-17

-43

-28

-12

Straub and the Hachemeister model. In Figure 1 the results of state 2 (and not for the other states, because the figure would then turn out

credibility adjusted

to be too complex)

estimators

intercept slope

and the collective are represented. The horizontal

lines correspond with the model of Bi.ihlmann and Straub. The collective

2437

1650

2073

1507

1759

-57

-21

-41

-15

-26

estimator

is

denoted

by c,

z.

J

I

ｾ＠

I

I'

estimator by

i

and

the

FIGURE = l.35

7,77 -0.06 -0.35

1.30 7.33 1.30 0.06 -0.33 -0.06

7.71 1.17 6.86 0.35 -0.05 -0.31

intercept s slope

s

01 02

expected variance 5 2 variance of mean

a

1885 -32 5.0.10 =(145394 -6628

1.32 7.46 0.06 -0.34

1.800

!

structural parameters: collective estimators

7 -6628) 302

ｾ＠

'-700 1.800

0 •.00

2

58

individual

credibility adjusted estimator by a.

credibility factors

' I

the

3

• • , • QUARTER

go

10

11

12

13

59

'•

ﾷｾ＠,(

1'

,,

II

ii:

Again note that for rate making purposes one should be fully aware of the

features

of the models.

adjusted estimators amounts to $ 126, an increase of 8. 3% for the 13th quarter compared to the Blihlmann-Straub model. As a

final remark,

THE DE VYLDER NON-LINEAR REGRESSION MODEL [1986)

Chapter 6.

The difference between both credibility

This non-linear regression model is a direct extension of Hachemeis-

observe that our results differ from the original

article. This stems from the fact that we used another estimator introduced an iterative solution.

a. Model and assumptions

a and

ter' s

linear

regression model.

contract are

e.

Instead

a

of

J

The variables

relevant

for the

and X. (j= 1, 2 . . . . , k; s = 1, 2, ... , t). JS regression hypothesis, now we have the more

j-th

general

assumption that for every contract there exists a function

a(e . )

where n is the· length of the vector

J

in the previous model and

where t i s , of course, the length of X .. J

For mathematical exist

and

reasons we

assume that all partial derivatives

by 9 fb , b , ••. , bn) 1

i

matrix formed

the pairs

are

we denote the

2

ｾ＠

{t x n)

with these partial derivatives as columns, so

at = [ abl '

ＺｾＲ＠

' ... ' Ｚｾｮ＠ l

The hypotheses of this model then are (Dll The

contracts

independent.

j= 1,

The

2,

... ,

k

(i.e.

variables e , e , ... , ek are 2 1

identically dis-

tributed. 1, 2, ... , k one assumes

(02) For all j

f ＨｾＹＮＩ＠

J

Cov[x.le.)

J

where

f3(9j)

J

is a vector of length n and vj is a (t x t) positive

semi-definite matrix as in the previous model.

60

61

I 1

I, b. Comments

c. Computations

In this model the time stationarity assumption is dropped completely. It is replaced by any other assumption, not necessarily of the linear type.

In

inflation,

cases this

where

we

model

deal

is

with

even

data

more

that

suited

includes than

the

effects

of

method

of

Hachemeister.

g

T

-1

(B . ) • v .

J

J

. IX . J

j=

o,

J

is the vector B. such that J

f (B . )) = 0 J

scalar

The collective estimator for E[a (8 .) ] is J k

L

As a special case consider n

a (8.)

The individual estimator for

2 and write

(n x 1)

z .. B.

j=1

J

J

where the z. are matrices defined as J

1, 2, ... , k

Z,

Define

+ s ' .a. !Sj) -1

(1

J

(n x n)

where blj blj'b2j

.

ｾ＠

J

｢ｬｪＢｾ＠

T

-1

(d .v . • d) J

-1

(n x n)

,,

and

,," d

Cov[f[B(9) ], BT(9)]

(t x n)

a

Cov[B(9.)] J

(n x n)

Obviously this implies the model

•

scalar The credibility adjusted estimator for 0

Ba =

I I

j

(1 -

z.) • B

J

0

+

a (8.) J

is (n x 1)

z .• B.

J

J

b1j

Estimators for s 2

2.b1j"b2j

5'

This model [1986]).

is called the exponential regression model

Although we will

(see De Vylder

adopt this model for our example,

with the

notation blj = cj and b j = uj, we first display the general formulas 2 and return to this special case in section d.

1

k. (t-n)

1

k-1

,

a and d are

Lk

j=1

(X . - f (B , )) J J

T

-1 . v . . (X . - f (BJ.)) J J

scalar

k

c j=1

(n x n)

k

c

j=1

z j. (B j - B ).(f(BJ.)- f(B ))T 0 0

(n x t)

to be used iteratively.

62

63

the matrix a being symmetrical, §. is replaced by (d+dT)/2 in

In fact,

and t

every iteration.

2:

These estimators are unbiased.

s=l t

L: s=l

d. Remarks

Several expressions above are based on the best linear approximation to the general regression model. For this closest linear model there holds

I'

y = d.a

In general, however, recursive estimation of this system of equations is relatively difficult, especially in case t is large.

-1

As a final remark we should stress that this linear approach to the This means that,

for linear f ( B(9 . ) ) , these expressions coincide with J the equi val en ts of Hachemeister. For instance, the reader may easily

verify that

general model is a decision based on considerations on computational ease. For a more general treatment of this model we once again refer to

tor a recent

the original article.

T

a

-1

zj = (1 + s .a.(d .vi .d)

-1 -1

context,

)

follow-up in a multidimensional

see De Vylder [19861. Of particular interest are the general

formulas on zj. 2

a. (a + s .

(y

T

-1

.v . . y)

-1 -1

I

e. Alternative estimators

J

Often,

holds in this case.

the

credibility

adjusted estimators are not

for

the

entire

domain of j in the interval delimited by the individual and collective Now we return to the exponential regression model of section b. The

Bj= (blj' b j) 2 simplified considerably with the additional assumption

general

system

of

equations

to

determine

can

be

estimators. phenomenon,

Although

there

is

a

reasonable

explanation

for

this

because compensation takes place with the initial severi-

ties (we also observed this in the numerical example of Chapter 5), it is sometimes hard to accept. Then one may choose to revise the figures

､ｩ｡ｧＨｾＬ＠

wjl

1

_1_)

wj2'

wjt

j= 1, 2, ... ' k

by

the

introduction

diagonal

credibility

matrices,

where

the

diagonal elements are between 0 and 1. So, with the results obtained with the

we then have

of

formulas

of section c, in a final

stage the next diagonal

matrices are introduced to compute the credibility adjusted estimators with the desired property

t

variance of mean variance

b

After setting a and b equal to 5 2 k

p

-

\

\

- - L._

L._

e p= 2 p-

b

2016

> 2

.

PJ

1

MpO

s' a.

expected variance

k

p

p= 2

the relatively low credibility factors. p

L:

p=1

z

p"

(X

pzw

-

X

zzw

)

2

These unbiased estimators should be used iteratively.

90

91

,.

'r·

••

I

I

I Chapter 10.

SPECIAL APPLICATIONS OF CREDIBILITY THEORY

{ In this

chapter we

deal

with

some

special

theory provides us with alternative tools methods.

Purely

by

coincidence

the

problems

where

credibility

to the (established} existing

applications

here

are

topics

also

treated in two previous publications in this series of surveys, namely the problems of loss reserving and rate making (here in case of large claims) .

J

10 .1. Loss reserving methods by credibility

a. Model and assumptions In

this

paragraph we follow the method of De Vylder [ 19821

estimation of theory.

ｉｂｾｒ＠

(Incurred But Not Reported)

the

A stochastic multiplicative model for the forecasting of IBNR

claims is given. year

for

claims by credibility

The multiplication factor depending on the accident

(or, in the wording of the first issue "Loss reserving methods'',

year of origin) is credibility adjusted. Again we consider the variables X.

JS

(j= 1, 2, ..• ' k; s= 1, 2, ...

r

t).

Now they reflect the total amount paid in development year s on claims that originate from accident year j. Unlike our previous models, here the variables X,

are partitioned in two subsets, namely; the observed

variables

are

JS

(which

in

the

polygon

below)

and

the

non-observed

(outside the polygon) . Based on realizations of the observed variables we shall have to make predictions concerning the remaining variables. To make our results comparable with those in the first issue, we use a transposed representation of the Xjs' viz.

.,

•

I J

93

l

1 :.

structure variables

observable and non-observable variables

j=

1

X 11

91

2 3

accident year

1

92

Ｍｾ＠ｲ : x21

93

1X

s= 2

s= 3 1

x12

s= t

xl3

is independent

x1 / ,t · · · · · · · ..· .... x2t ' .. ,

ﾷｾ［ｲＢ＠

.. ｾＬＮ＠

X

3t

where

....... .

is a

unit matrix,

(t " tl

and pj is the volume) of accident year j.

"

we denote by Ks the set of subscripts j such that XJ.S is observed. For all J·- 1 ' 2 • · • · , k 1 e t T j b e the set of subscripts such that X, is observed. t

JS

Again the distribution of Xj depends on a parameter .

is

an

unknown

scalar,

natural weight

(mostly

(D4) The e , e , ... , ek are identically distributed. 1 2

kt

b. Comments by a multiplicative expression e

JS

• • • 1

r'

(known)

j

of

The idea of approximating X.

For s= 1, 2,

a., that as usual

also the basis of the classical

stochastic model

is

X . ""

JS

e

s"

S

.m. is J

chain-ladder method. A corresponding

m + R j

where

js

the error R

js

has

zero

expectation. The next step is to replace mj by a random variable Mj and to consider the model

J

as an unknown realization of a structure variable e

j"

"·

JS

The hypotheses of this model are (Dl)

It

independent

possible.

ｾｮｴ･ｲｰ､＠

It

pi

J

Of course many other sets of observed and non-observed variables are

ｾｳ＠

and

r'

Cov[Y .]

"" '....... .

k

.

of j

········

x22

31

I

J

ｩＭ［ｾ＠

k) are random variables such that

y

E[Y .] =

development year 5""

... ,

(D3) The Y. (j= 1, 2, J

The pairs

(91' X1)'

are supposed to be independent. This model is close to

where MJ. and R,

(92, X2) ' •.• '

(9 k '

Xk l

are ｾ＠

to exclude negative values in case M is small.

100

101

d. Remarks

For the calculation of the constants

In case the variables Xjs take only a limited number of values, the formulas for optimal semi-linear credibility are reduced to systems of linear equations. This is, of course, also the case for this trimming

[1980].

The model

ae, ae, ... , Ee we refer to Gisler

can be generalized to

take weights

(volumes)

to

account. The interested reader is referred to the former article or to Blihlmann, Gisler and Jewell [1982].

problem. If

the

random variables Xjs are supposed to assume only the values ·· ., at one gets

e. Numerical example The theory of large claims is primarily concerned with individual claim amounts.

R(M)

0

M

R(x ) 2

a 2_ M2_ a

fe(MI

a < M< a

R(xt)

at.::_M

2. al

Hence,

represent

1

these

this

model

amounts.

is

best

for

suited

Our data however deal

case

the

the

with average

claim

amounts, which means that an exceptional claim for a certain contract

2

e

e- - e+l

2, 3,

... ,

t-1

in a year will not necessarily have a substantial i'rnpact on the X. • JS However, we will apply the model to our usual data but beforehand we change

x10

to 5000. Now the optimal trimming point is 2034 and the 2 resulting figures are

where

'

state

where a e,

ae, · · ·, Ee are constants which can be calculated from the

values a 1 , a 2 , ... , at and from the probabilities

i, j""' 1, 2, ... , k; r, s= 1, 2, ... , t

a I m

credibility adjusted estimator ｍｾ＠ credibility factor

J

z

j= 1

j= 2

j= 3

j= 4

j= 5

2015

1613

1856

1457

1679

0.93

0.93

0.93

0.93

0.93

structural parameters: collective estimator Mo collective estimator for f

The optimal trimming point M* is obtained as M*= M*

aft

36599

ｾｦｏ＠

67665

covariance of mean

max

e

102

50862

bfO

50337

point. First, the maximal and minimal values among all observations X, JS are selected. With these two endpoints a number of equidistant points

and

M*

bff

The following method was used in our search for the optimal trimming

2, 3, ... , t-1

ae.oe-2.ae·Ye e ae.oe-2.ae.Ee

1647

expected covariance

where

e=

Mf

expected variance

variance of mean

e•

1724

is determined. Then the two points next to the point with the highest

M=

Me E

ae

Me

t

[ae; ae+1 1 and R(ae) >R(ae+ll

ae+l

Me

I

(a · ?e+l 1 and R(ae) .::_ R(ae+ll e'

[ae; ae+1 1

R(M) are the new endpoints and the search is repeated. If we denote the function value of pointe by Re(M), convergence is defined as Re-1 (M) - Re+1 (M)

< 0.0001

Re(M)

103

Our routine gave converging and satisfactory results in case a small

CREDIBILITY FOR LOADED PREMIUMS

Chapter 11.

number of points was used. For the above-mentioned results we used six points

and

convergence

occurred

after

seven

iterations.

We did not

In

the

previous

chapters,

credibility

methods

have

been

examined

to

program the third advised transformation of section c to obtain results

estimate normal premiums. A question still unanswered is how credibility

which are comparable with the other chapters.

can

be

applied

briefly,

just

obtained

by

for

to

estimating

give

an

loaded

idea of

Biihlmann [1970]

the

using a

premiums.

We

will

technique involved,

recall

here

the results

distribution free approach and the

recent result on credibility for the Esscher principle by Gerber [1980].

11.1. Credibility for variance loaded premiums Biihlmann {19701 extended his own model by the introduction of a variance loading for the individual premiums.

We will briefly state some of his

findings here in a way analogous to Chapter 1. In general terms define the loaded individual premium by

p(Bl

H[X[BI

=

E[X[el + •.var[X[el

where A is an arbitrary number equal to or greater than zero. For the collective premium we have

E[XI + •. Var[XI

H[XI ｅ｛ｾＨｂＩ＠

given

the

claims

I + L (E[o' (9) I + ｖ｡ｲ｛ｾＨｓＩ＠ experience

ll

variables

x , x , ... , Xt one requires an 2 1 approximation to the individual premium H(X[e], because, as usual, e is an unobservable variable. In accordance with the first chapter and introducing the same hypotheses, the new exact credibility premium is defined as

which can be cast into the form

104

105

The first two terms

(the expected value and the variance part) represent

the so-called approximation part

As

far

as the estimation of the credibility factors is concerned, after

some trivial mathematics we have

t.Var[XJ which

is

the

solution

to

our

well-known quadratic

loss

function,

now

Var[X]

(t-1) .Var[X]

applied with respect to p(8). The latter term

(the fluctuation part)

is

(t-1) .Var(E

the price to be paid for the

the

next

linear approximations to each of the three

1 - ｖ｡ｲ｛ｅｾ｝＠

(t 2). Var[E

risk still contained in the approximation part. Blihlmann obtained

2

]

where

parts constituting the loaded credibility premium.

Here z

is nothing but the reshuffled expression of Chapter 3 on z and z 1 2 is more or less the variance analogon of it.

where a.t

a Var[Xl

Of course this is our previous result.

11.2. Credibility for Esscher premiums

Both for

the Bayesian and the distribution free approach to Credibility

Gerber [1980 1 presents where

premium Var[ 0"

2

Var[ I:

(8) 2

1

principle.

principle

of

interesting generalizations of the expected value

The

premium

Bayesian

calculation

approach in

a

can

rather

be

implemented

straightforward

for

any

way.

It

amounts to computing conditional distributions of the risk in question.

]

The

and

distribution free

method

is

more

difficult,

but

before

focussing

thereon we first define the Esscher principle as t

C

s=l

H' [X] t

-

E[X.eh.X] = M' (h) E[eh.XJ

1

where M(h)

M(h)

is the moment generating function of the risk X. Note that ｦｯｾ＠

any X, H[X] is an increasing function of the loading-parameter h. where z

1

is defined above.

Obviously lim h+O

106

H' (X]

E[X]

107

In general, H' [X) is the constant f that minimizes where and H'[XIYl minimizes

t. Var'

z ｴＮｖ｡ｲＧ｛ｾＭＨＹＩ｝＠

｛ｾ＠

J

(9)

+ E'[o 1 (9)]

and where

Y

is

a

random

variable

and

g=

g (Y) .

Now

consider

the

claims

experience variables x , x , xt. Using the quadratic loss function of 1 2 Chapter l, the distribution free method now amounts to the search for the c .X + c 0 1

linear expression

(1 -

• e

J + z.c

h.X

l

t+ l

c

1

.x

+ c

0

= (1 - z) .E' [p(9)] +

-1 ｺＮｰＨｾ＠

(X))

our classical formula! Note that E' [p(S)]

The result is t.Cov' ｛ｾＨＹＩＬ＠

p(9)]

ｴＮｖ｡ｲＧ｛ｾＨｓＩ｝＠

+

log E[e

tX

l9l

= ＹＮｾ

Ｑ＠

E'[a' (9)]

(t) + ｾ

for arbitrary functions 41 E'[p(9)]- ｣

Ｑ

ＮｅＧ｛ｾＨＹＩ｝＠

1

Ｒ

and

H'(X]. One can prove that if

ＨｴＩ＠ ﾢｾ

Ｒ

Ｌ＠

p(9)

is linear in 1J.(9). This is the

case in the next two examples where the Bayesian approach is followed for two couples of distributions (F(xlal, U(6)) that, for the expected value

where

premium principle, ､ｩｳｴｲｾ｢ｵｯｮＬ＠

E[Y.m(S)] E[m(S))]

E' [Y]

0

so

H'[Xt+llx , x , ... , Xt]. Hence, minimize with respect to c 0 and c 1 1 2 - co) '

z) .E' [p(9)

closest to the exact credibility p:r:emhun

result in exact credibility. First,

consider no:nnal

so

m(9)

U(9) "'N(ll, ｡ｾＩ＠ Ccv' [U,V] = E' [U.V] -

Because

X

E' [U].E' [V]

is a consistent estimator of

ll(9)

we have

'I'

It is well-known that limH'[xt+llx , x , ... , xtl c:p(e) 1

ｴＫｾ＠

2

Hence, the linearized premium is asymptotically correct if

We

have H' [X]

for general c

108

1

and c . In this case we find 0

H'

rxle J

ll + Ｈ｡ｾ＠

p(6)

+ ｡ｾＩ＠

a

•h

+ ｡ｾ＠

.h

109

Note

the

resemblance

of A and

h.

After

evaluating

some

conditional

distributions, we arrive at

BIBLIOGRAPHY For

a

complete

list

of

references on credibility related papers

the

interested reader is referred to "Insurance: Abstracts and Reviews" where (1 - z) .H' [X] + z.p(X)

where

a special issue is devoted to the abstracts of credibility papers. The present list of references contains a selection of papers and books considered to be either important in historical context or because they have shown some original paths in credibility theory.

Abbreviations

which coincides with the classical formula in case h • 0.

AB

Astin Bulletin

IME

Insurance: Mathematics and Economics

MVSVM

Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker (Bulletin of the Association of Swiss Actuaries)

PCAS

Proceedings of the Casualty Actuarial Society

SAJ

Scandinavian Actuarial Journal

The second example is concerned with the gamma-Poisson case, hence y (o, B)

0 (9)

FCxle> = P(e)

which results in a negative binomial F(x). With

H' [X]

o.e B+ 1

H'[XJeJ

h

-

e

Albrecht, P. [1985]: An evolutionary credibility model for claim numbers; AB, val. 15, no 1, pp. 1-17

h

p(e)= e.e

Albrecht, P. [1981]: Kredibilitat, Erfahrungstarifierung und sekundare Pramiendifferenzierung; Money, Banking and Insurance, val. 2, pp. 687-701

Ammeter, H. A. pp. 327-342

h

Bailey, A.L. pp. 13-20

we have (1 -

z) .H'

[X] + z.p(X)

[1980]:

Potenzmittel-Credibility; MVSVM, vol. 80,

z

= Ｍｾｴ＠

____-c-

t+S+1-e

h

[1945]: A generalized theory of credibility; PCAS, val. 32,

Bailey, A.L. [1950]: Credibility procedures, Laplace's generalization of Bayes' Rule and the combination of collateral knowledge with observed data; PCAS, vol. 37, pp. 7-23

Surprisingly, this credibility weight now is an increasing function of h. Note that in both examples the credibility adjusted premium again is a linear function of

X.

Bichsel, F. [1964]: Erfahrungstarifierung in der Motorfahrzeughaftpflichtversicherung; MVSVM, vol. 64, Heft 1, pp. 119-130 Blihlmann, H. [1964]: Heft 2, pp. 193-214

Optimale

Pramienstufensysteme;

MVSVM,

val. 64,

Biihlmann, H. pp. 199-207

[1967]: Experience rating and credibility; AB, val. 4, no 3,

Blihlmann, H. pp. 157-165

[1969]: Experience rating and credibility; AB, val. 5, no 2,

Blihlmann, H. [1970]: Mathematical methods in risk theory; Springer Verlag

110

•' '

Heft 3,

Bichsel, F. [1959]: Une methode pour calculer une ristourne adequate pour annees sans sinistres; AB, val. 1, no 3, pp. 106-112

where

••

111

!I

I

..

Bi.ihlmann, H. (1975, 1976]: Minimax credibility; in: Credibility, theory and applications, Proceedings of the Berkeley Actuarial Research Conference on Credibility, Academic Press, New York; also published in: SAJ, 1976, no 2, pp. 65-78

Hachemeister, application to dings of the Academic Press,

Bi.ihlmann, H., A. Gisler and W.S. Jewell {1982]: Excess claims and data trimming in the .context of credibility rating procedures; MVSVM, vol. 82. Heft 1, pp. 117-147

Jewell, w.s. [19741: Credible means are exact Bayesian for exponential families; AB, vol. 8, no 1, pp. 77-90

Bi.ihlmann, H. and E. Straub [1970]: MVSVM, val. 70, Heft 1, pp. 111-133

Glaubwi.irdigkeit

flir

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