Exponential smoothing and credibility theory

Insurance: Mathematics and Economics

I ( 1982) 2 I3- 2 I7

213

North-Holland Publishing Company

Exponential smoothing and credibility theory Erhard KREMER Unrv~rs;i_v 01Homburg,

(A-3)

X,. i = I.. . . , n independent given 8

Wesr Germon~

Received 20 January 1982 Revised IO March 1982

Some years ago Gerber & Jones(1975a. 1975b) investigated simple recursion formulas for tht linear credibility estimator. In the present paper we reexamine such formulas and show their connection to models of modem times seriesanalysis.

1. lnboduction Consider a portfolio of risks, e.g. a set of insurance policies. Let X,. i = I . . . . . n denote the claims amounts of a risk in past years i = I,. . . , n. Furthermore let 8, be some parameter of the distribution of X,. usually called structure (or risk) parameter. For given risk the structure parameter t?,of period i is unknown and therefore interpreted as realization of a random variablq8, in a measure space (9, &‘). Subject of the credlbdity theory is the prediction of the (pure) risk premium of future yearn+ I: m ##+I =E(X,+,P,+,). X#I+1 resp. t9,+, denoting claims structure parameter of year n + I. claims X,, i = I,..., n. Already in proposed an estimator rit, +, for m, fi ,+,=C,-~“+tl-C,).E(m,+,)

amount resp. from the past 1918 Whitney +, similar to (1.1)

with

It was not before I%5 that this estimator, usually called credibility formula, was justified in a mathematically satisfactory manner by the Swiss actuary BUhlmann (see Biihlmann (1967)). He made the following assumptions: (Al)

tl, = 8, Vi,

(A-2)

X,, i = I,... , n identical distributed given 8,

016746g7/g2/0000-0000/~2.75

Biihlmann’s approach led to a rapid development of credibility theory and nowadays there exists a vast body of literature on this subject (for references see Norberg ( 1979)). One criticism on the classical model refers IO assumption (A.1). meaning the homogenity of the risk characteristics in time. In the last few years some authors investigated models. allowing the risk parameter to change in time (see Gerber & Jones (1975a. 1975b). Jewel1 (1975, 1976). Sundt (1981). Eckhardt (1981). Kremer (1981)). Assuming the risk parameters of different years being dependent, the problem of credibility estimation becomes rather messy. However, assuming some special structure on the covariances of the claimsprocess, simple recursion formulas can be developed (see Gerber & Jones (197Sa. 1975b). Jewel1 ( 1976). Sundt ( 1981). Kremer ( I981 )). Rather simple formulas, called credibility formulas of the updating type, e.g. * n+l =u;X,+(l

-un).tim,

(I .a

were investigated by Gerber & Jones (1975a. 1975b). In case the factors a, = Q E (0. I) are independent of n, the resulting credibility formulas, e.g. rir?I+1=a.X,*(I

--~).a,,

(1.3)

give geometric weights to the past cl:Gms AT,,i = I ,...,n. In the present note we reexamine such credibility formulas. We show that asymptotic formulas can easily be derived from classical results of the theory of forecasting, the corresponding finite sample formulas are deduced from a theorem of Sundt (1981). The models generating formulas of type (1.2). (1.3) are nothing else than special nonstationary AtttMA-processes, which are basic to the Box-Jenkins-approach (see Box & Jenkins (1970)) of forecasting. The present paper has to be understood as supplement to another paper (Kremer (1982)). where the author investigated stationary processes. and has the aim to underline the strong

8 I982 North-Holland

214

connection t~:ulcen credibility theory and modem time series analysis.

2. tM4-~meemes and simple expoueutial sllloothing For fixed probability space (Q. B,P) let L, be the set of measurable, square-integrable functions /:(Q, w”)+ (R, B), B denoting the a-algebra of Bore1 sets on the real line R. Identifying/with the equivalence class of all 1 a-e. equal to j. and introducing the scalar product (/.g>

:=E(f-g).

L, becomes a Hilbert space with norm

is stationary and satisfies with pr = E( Y)

forwnstantsa,.j=O... .p(uO= I),b,,j=O . . . . . q (6, = I) and a white noise process (E,. i E Z). ARIhWp. d, q)_processe s with p = 0 are denoted as IMA( d. q )-processes ( Integrated Mooing Average processes of order (d, q)). and MA( q)-processes ( Mooing Average processes of order q) if in addition d = 0. Notice that ARIMA( p” d, q) models are nonstationary for d >O (the stationary models considered in Kremer (1981) correspond to the case d = 0). Define for stationary process Y = ( U,. i E Z) the wvariance function

Ilfll :=[E(/Z)]? For a linear subspace L C L, the projection PL( f ) of /E L, is defined as unique solution of:

II/-P,(f)II=~~II/-gil.

PL(/)EL

llmwem

In Section 3 we investigate real-valued stochastic processes X=(X,. iEZ) (2 denoting the integers in R) such that ViEZ

1. Let Y=(V,.

iEZ)

be a real-oolued,

stationary process with

( L denoting the closure of L in L, ).

X,E L,.

with cr = E( Y,). We need in Section 3:

(2.1)

(I) Y,EL,,Vi, (2) E(Y,)=O.

Vi.

(3) y,f’ = 0. Vh * 2, (4) Ir:l (!r,‘. fien Y is a hfA( I )-process.

El

and satisfying the recursion formula x,=x,_,

+/3-e,_,

+e,.

WiEZ,

(2.2)

where /I is a real constant and (E,, i E Z) a u&ire noise process, meaning: E(e,)=O,Vi,

Var(e,)=e,2E(O, (e,.e,)=O,

co).

Vi.

Vi#j.

Processes of type (2.2) are special cases of a general class of processes called Autoregressiue Integruted Moving Aueruge ( = ARIMA) pmcesses (see Box & Jenkins ( 1970). Newbold 8c Granger ( 1977)). Defining the backward shift operator B by B(Y,)

This well-known result is an easy consequence of a theorem of Wold ( 1954) (compare Granger & Newbold ( 1977). p. 23). One of the main applications of the ARIMAprocesses is forecasting of future values. For defining what is meant under an optimal forecast (prediction) we introduce the linear subspace A,. spanned by ZE I and all X,. icn. Then the projection )i,+, := P”#(X”.,) is called (optimal linear) prediction of X, +, from X,. i
:= y,_,,

a process X=(X,. i E Z) is called ARIMA( p, d. q)-process (p, d, q nonnegative integers) iff the transformed process Y = (U,. i E Z) with

IIeoran 2 (see Granger & Newbold (1977). p, 172). Let X = ( X,. i E Z) be a real-cahudstochastic process sutis@ng (2. I). (2.2) and/or a : = j3 + I

u,:=(l-B)“(X,)

a E (0.2).

(2.3)

71ren

L!

(c) Condmon I/Ii<

-a)*ri,.

=4x-X,+(1

arrd the prediction 8X”,,

-

equals

at’.

wn.

(2.4)

Ic=[(l

+h?).y+(l

+/I).

+4(1 +P).v.rpj’

D

In case a E (0, I) formula (2.4) represents the basic algorithm for the famous forecasting method known as (simple) exponential smoothmg.

X,. i E 2 be random variables satisfyrng (3.1)

Vi.

i G n are conditionally independent given 8,. i c n, Vn.

E(X,I8,,j
Vk)O.Vi.

E(Var( X,( t9,)) = 0.

Vi

(0 independent of i). Assume (m,. iEL) m, = E( X,)6?) follows an tMA(I. I)_model: m,=m

,__, +b*e,_,

where lb[<

+e,.

ViEL.

(3.2)

(e,.e,)=O.

with (3.2) -(3.6)

=z.c+y.

y: = y,” -

Q

=

y,’ = yh” = 0.

I)-pro-

y

---

Q.

(3.10)

Vhs2.

y(: “(I

i pZ)e,‘.

(3.1 I)

y: =&0,2.

ho&.

Equations

/?=a-1

(3.7)

with

(3.9)-(3. I I ) yield the system

(I ‘pZ).o,*=2.4,+y. h p . a,> = I+b2

(I +b’)’

-&

/3 and a whrte notse

(b) More exu+:

’

(3.9)

Furthermore (3.2)- (3.6) yield for the covariance functions y:. y,” of Y. A4

Since also condition (4) of Theorem I can easily be verified Y builds a MA( 1)-process. Consequently X is an IMA( I. I )-process. e.g. there cx~str a constant p such that (2.2) holds. implying:

X=(X,,iEL)isunlMA(l.

process (E,, i E 2) such that (2.2)

JI=o+y-

and from (3.1). (3.4)

with

(3.4)

there exist a constatu

E):=EM,=O

Var( V,) = 2 . fp + Var( M, )

Under theseconditions one has:

CPU. e.g.

From (3.5). (3.6) we get

(3.4)

ViZj.

Tkorea3.(a)

1.

implying

(3.5)

Vi.

M,=m,-m,

M=(M,,

Var(X,)=p+Var(m,).

Vi.

Var(e,) = a,’ E (0. 43).

I EL).

V=(k;.

(3.3)

I and (e,, i E 2) satisfies

E(e,) =O.

+I?‘).

Remark 4. In time series analysis the assumption lhl< I resp. f/31( I 1s called rnrurrrhrl~r,r conLrron. garanteeing the uniqueness of the representation of the corresponding MA( I )-process.

1: = x, - X, ,.

x,.

.‘],2.(,

Prouf. Define processes iEZ) by

3.AqIeci8Icfa#ilitymo&l

X, E L,.

{(I -h)‘.y’

error

/tm+ ,u2

Let a,.

1 rmpbes

_(b-rWb’)d~38~ c+s

*

.

(3.12) ‘--’

from which one deduces with Q := ~7: -Q after some routine calculations (3.7). (3.8) and part fc) of the statement. Deriving the formula for +Lin part (c) one has to choose between the negative and positive square root. The negative root can be shown to contradict the first equation in (3.52) and assumption J/3/ ( I. 0

Similar as in Kremer (1982) we define the asymptotic credibility esrimator 61”. , of m, _ , as

Defining a, by replacing 4 through formula of Q in Theorem 3. e.g.

rit n-l :=PA,Jm,+l).

b am=(lib2)

for which one derives with the cited results Section 2:

of

(3.2H3.3)

imply for each /E

T,

VfE&

ThUS

Consequently the statement follows by Theorems 2 and 3 (remember in the proof of Theorem 3 we defined + : = 0.’ - cp). U In practice one only has a finite past X ,._... X,. consequently the above result has to be understood as limit solution. One could start recursion formula (1.3) with ti, = E( X, ). but this approach might be inadequate for rather small n. In this case the application of the fmire sumple credtbilrty estimator l

Var( X,)=

V-k W,.

Vna

4 rr, =g+y

(3.17)

I.

W”. Vn*l.h,l.

(3.18)

Consequently condition (5) in Sundt ( 198 I) is satisfied (with p,, = I. Vn). allowing the application of Sundt’s (1981) Theorem 2 (or alternatively Theorem 2a in Gerber & Jones (1975a)). For 4,. defined by recursion formula (3.14) let 7” :=&++

(3.19)

Since b y,=-’

l+b’

’

we get with (3.16)-(3.18) and definition (3.15) Var(X,,)-Cov(X,,,.X,)=(I

, ).

where !*imis the linear subspace of L.: spanned by %E I and X,...., X,. would be preferable. This estimator can be developed from the asymptotic results by a short heuristic argument: Upon substituting $ by J;+, (interpreted as II ’ ) on the left and # by &, on the Ilm #I*, -ifir+, right side of (3.8) one gets the recursion formula

-

(3.16)

one derives with (3-t)-(3.6)

Cov(X,,,.X,)= I =.tr*,.

m n. I : = p., (nr,

V=g,-y,

y,=(m,-m,_,.m,_,-mm,_,>

= Urn,, , -/?I’+fp.

(3.13) . m,,+

Roof. Defining

with

implying with (3.4) -fll*

(3.15)

lkoreat 6. (a) The estimator 61~+ , &fined by (1.2). (3.14). (3.15). sfarfing with 61, :=E(m,). +, : = Var(m, ). is the finite sample credibility estimator iii, * ,. is equal to the estimation error (b) k+, Blmn+i -i%,,,112.

W,=Var(m,)+y,.

(X”.,--m,,.,.mn.,-/)=O.

II%+,

1G

and substituting a by a,, in (1.3). we get an updating formula of type (1.2).

Theorem 5. (a) Under the assumptions and notations of Theorem # b ) the credibili[p estimator th, + , satisfies recursiun /ormula ( I _3)_ (b) #. defined in (3.8). is equal to the estimation error

Proof. Assumptions

.##&+*.

rl;l in the

-a,,)-~,,.

(3.20)

From (3.5)-( 3.6) one derives Wil. I - rv,=y+2.y,. yielding with (3.16)-(3.18) Vaf(X,.,)-Cov(X,.,.X,) =v+y+y,.

Vn>l.

(3.21)

Now (3.16)~(3.18). (3.21) imply Var(X,,,)+Var(X,)-2*Cov(X,.,.X,)

I ( I + b= )’

_(h-r-(t+b%)’ k;+c

(3 14) .

.

=2~?+y.

Vn*l.

(3.22)

Because of (320). (3.22) (and p,, = 1, Vn) formula

6 Kremer / Exponenrral smoothrng and credjbilrg

(6) in Sundt (1981) turns out to be equivalent to (3.14) and formula (7) in Sundt (1981) is identical with (l-2), (3.15). Since (3.13) implies @%+I=P,,(X+,>* the statement (a) is proved. Part (b) follows from (3.13), (3.19) and the equality T”= IIX”., -P&“+,W.

a

Remark 7. As already mentioned in Section 1, Gerber & Jones ((1975a). (1975b)) investigated credibility formulas of type (1.2j. (1.3) without reference to the prediction theory of time series analysis. The model in Gerber & Jones (1975b) uxresponds to our model with the special choice b = 0. Our formulas (l-2), (3.14). (3.15) reduce for b=O to formulas (12), (13) in Gerber & Jones (1975b) (u,‘, i*, v* ag= with our 4,. Y. T), and our asymptotic formulas (1.3). (3.8) correspond in case b = 0 to their result ( 19), (20).

217

References BOX, E.P. and Jenkins. G.M. (1970). Time Series Ana/ysis -Forecasting and Control Holden Day, San Francisco. (21 Buhhnann. H. (1967). Experience raring and credibility. Astin Bulletin 4, 199-207. 131 Eckhardb B.H. (1981). Beatimmung und Vergleich verschiedener Kredibitit&sformelo bei erfahrungstarifiertern Portefeuiffe. Bldner &r Deutschen GeselLwhafi N 20 I - 2 I3. VersicherwtgsmushP 141 Gerber. H.U. and D.A. Jones (197Sa). Credibility formulas of the updating lype. In: Cre&r/iry: Theory and Appbcarions, 3 I-46. AcaJemic Press, New York. [S] Gerber, H.U. and D.A. Jones (1975b). Credibility formulaa with geometric weighti. Transaciions o/ rhe Sociery of Acraanes 27, 229-230. [6] Grangcr, C.WJ. and Newbold. P. (1977). Forecasting Economic Time Series. Academic Preaa, New York. [7) Jewel1 W.S. (1975). Model variations in credibility theory. In: Credibiiity: Tlreory and Applications, 133-244. Acz&mic Press. New York. [8j Jewelf, W.S. (1976). Two classes of covariance matrices giving simple linear forecasts. Scandinavian Acuarial Journol. 15-25. 191 Krcawr, E. (1982). Credibility theory for some evolutionary models. To appear in Scan&avian Acmarial Journal. [IO) Nwbq R (1979). The credibility approach to experience rating. Scaxdinacmn Acwsrial Journal I8 I -22 i [I I) Sundt, B. (1981). Recunive credibility atimation. Scana%avian Actuarial /our& l49- 168. 1121 Wofd, H. (1954). A Sru& in rhe Anafysis 01 Srafionary Time Series. Almquist and Wickaell. Uppaala.

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