credibility and persistency

8/13/2019 Credibility and Persistency

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C R E D I B I L IT Y A N D P E R S I S T E N C Y

BY VIRGINI R. YOUNG

Umvet~tty of Wtsconsin-Madt ~on

A B S T R A C T

P o l i c y h o l d e r s o ft e n d e c i d e t o b u y, r e n e w, o r ca n c e l i n s u ra n c e b a s e d o n t he p r e m m mc h a r g e d b y t h e i n s u r e r c o m p a r e d w i th w h a t t h e y e x p e c t t h e i r c l m m s w i l l b e I t i si m p o r t a n t f o r a c t u a r ie s t o c o n s n d e r t h e p e r s i s t e n c y o f p o l n c y h o l d e r s b e c a u s e t h ef i n a n c i a l w e l l - b e i n g o f th e i n s u r e r d e p e n d s o n s p r e a d i n g i ts r i sk o v e r a l a r g e h o o ko f b u s m e s s W e u s e s ta tn s tl ca l d e c i s i o n t h e o r y t o d e v e l o p p r e m m m f o r m u l a s t h a ta c c o u n t f o r th e p a s t e x p e ~ ne n c e o f a g i v e n p o l n c y h o l d e r , t h e e x p e r i e n c e o f t h e e n t i rec o l l e c ti o n o f p o l i c y h o l d e r s , a n d t h e h k e h h o o d o f th e p o l i c y h o l d e r re n e w i n g w i t h o rb u y i n g f r o m a g i v e n i n s u r e r, th a t is , p e r s i s t e n c y

W e a s s u m e t ha t th e p e r s i s t e n c y o f p o l i c y h o l d e r s d e p e n d s o n t h e a r i th m e t i cd i f f e r e n c e b e t w e e n t h e p r e m i u m c h a n ge d a n d th e i r a n t i c ip a t e d c la i m s W e e x t e n dt h e w o r k o f TAY L O R ( 1 9 7 5 ) nn w h i c h h e o b t a i n s l i n e a r c r e d ~ b ] h ty f o r m u l a s b ym i n i m i z i n g l o ss f u n c u o n s t h at i n c o r p o r a t e th e p e r s i s t e n c y o f p o l i c y h o l d e r s . We

c o n s i d e r Ta y l o r s l o ss f u n c n o n s a n d o t h e r o b j e c t w e f u n c ti o n s , i n c l u d i n g t h os e t h ata c c o u n t f o r th e a m o u n t o f b u s i n e s s t h e i n s u r e r w r i te s o r r e n e w s

K E Y W O R D S

C r e d t b d l t y, p e r s i s t e n c y, s t a t i s h c a l d e c n s n o n t h e o r y

1. INTRODUCTION

I t is i m p o r t a n t f o r a c t u a r i e s to c o n s i d e r t h e p e r s i s t e n c y o f p o l , c y h o l d e r s b e c a u s e t h ef i n a n c i a l w e l l -b e n n g o f t h e n n su r er d e p e n d s u p o n l o n g - t e r m p r o f i ta b n h t y a n d u p o ns p r e a d i n g i ts t as k o v e r a la r g e b o o k o f b u s i n e s s A n i n s u r e r a l s o w i s h e s t o re t a inb u s i n e s s b e c a u s e w r m n g i n it ia l b u s i n e s s is m o r e e x p e n s i v e t h a n r e n e w i n g e x i s t i n gb u s i n e s s. W e d e v e l o p p r e m m m f o r m u l a s t h a t a c c o u n t f o r t h e p a st e x p e rn e n c e o f ag w e n p o l i c y h o l d e r , t h e e x p e r t e n c e o f t h e e n tn re c o l l e c t i o n o f p o l i c y h o l d e r s , a n d t h el n k e h h o o d o f t he p o h c y h o l d e r r e n e w i n g w nth o r b u y i n g f r o m a gn v en i n s u r e r, t h at i s,p e r s i s t e n c y

T h e f r a m e w o r k u n d e r w h n c h w e d e t e r m i n e t h e e ff e c t o f th e p e rs n s te n c y o f

p o l i c y h o l d e r s o n p r e m n u m s i s s t a n s n c a l d ec ks ~o n t h e o r y a n d i ts a p p l i c a t i o n t oc r e d n b l h t y t h e o r y C r e d L b ] h ty th e o r y s e e k s t o fi n d s y s t e m a t i c m e t h o d s f o r c a l c u l a t -i n g a p o h c y h o l d e r s i n s u r a n c e p r em u u m b a s e d o n t h at p o h c y h o l d e r s p a s t e x p e r i -e n c e a n d t h e e x p e r, e n c e o f th e e n t ir e g r o u p o f p o l i c y h o l d e r s C u r r e n t f o r m u l a s

A STIN BU LL ETIN Vo l 2 6 N o I 1 9 9 6 p p 5 3-69


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5 4 V I RG I N IA R Y O U N G

in c r e d ~ b i h t y t h e o r y o f t e n c a l c u l a t e p r e m i u m a s a w e i g h t e d s u m o f th e a v e r a g ee x p e r i e n c e o f th e p o h c y h o l d e r a n d t h e a v e r a g e e x p e r i e n c e o f th e e n u r e c o l l e c t i o n o fp o h c y h o l d e r s . I n o r d e r to a v o i d a n o f f - b a l a n c e f r o m t h e a p p h c a t i o n o f c re d a b d l ty ,

t h e s e f o r m u l a s u s u a l l y r e q u i r e t h a t a p o l i c y h o l d e r w a ll r e n e w n o a n a tt er w h a tp r e m a u r n i s c h a r g e d

W e a s s u m e t h at th e p e r s i s te n c y o f p o l i c y h o l d e r s d e p e n d s o n t h e a r i th m e u cd i f f e r e n c e b e tw e e n t h e p r e m i u m c h a r g e d a n d th e ar a n u c l p a t e d c l a i m s C o n s ~ d e n n gt h e a ri t h m e t i c d i f fe r e n c e m a k e s s e n s e b e c a u s e w e a s s u m e t h a t th e p o l i c y h o l d e r s a ret h e s a m e s i z e s e e S e c u o n 2 .1 ) . I f o n e w a s h e s t o m o d e l t h e p e r s i s t e n c y o fp o h c y h o l d e r s o f d i ff e r e n t sa ze s, o n e m i g h t a s s u m e t h at t he p e r s i s t e n c y o f p o h c y -h o l d e r s d e p e n d s o n t h e r e l a t i v e d i f f e r e n c e , i n s t e a d o f t h e a r i t h m e t i c d i f f e r e n c e . W ed e v e l o p c r e d l b l h t y f o r m u l a s t h a t o p t i m i z e f u n c ta o n s t h at c o n s a d er th e a m o u n t o f

b u s i n e s s t h a t a n i n s u r e r w r i t e s , a s w e l l a s th e m o n e t a r y g a i n o f t h e i n s u r e r. W e ,t h u s, e x t e n d t h e w o r k o f TAY L O R 1 9 7 5 ) m w h i c h t h e o b t a i n s l i n e a r c r e d i b i l i t yf o r m u l a s b y m m a m a z i n g l o s s f u n c ta o n s t h at i n c o r p o r a t e t h e p e r s i s t e n c y o f p o h c y -h o l d e r s .

S UN D T 1 9 8 3 ) a l s o c o n s i d e r s t h e e ff e c t o f p e r s i s t e n c y m c r e d i b i l i t y r a ti n g H ~sa p p r o a c h d i f f e r s f r o m o u r s m t h a t h e a s s u m e s t h a t t h e h k e h h o o d o f r e n e w i n g i snota f f e c t e d b y t h e p r e m m m c h a r g e d b y t h e i n su r e r . I n st e a d , he a s s u m e s t h a t t h ep e r s i s t e n c y o f th e p o h c y h o l d e r g i v e s t h e i n s u re r i n f o r m a t io n a b o u t t h e c la i md a s t n b u u o n o f th e p o h c y h o l d e r

W e r e v a e w t h e w o r k o f BU H LM A N N 1 9 6 7 , 1 9 7 0) a n d TAY L O R 1 9 7 5 ) mS e c u o n 2 B u h h n a n n d e r i v e s a c r e d ~ bd l ty f o r m u l a b y m i n i m i z i n g th e e x p e c t e dv a l u e o f a s q u a r e d - e r ro r l o s s fu n c t io n . S a m d a r l y , Ta y l o r m m m l l z e s t h e e x p e c t e dv a l u e o f t h e m o n e t a r y l o s s t o t h e I n s u r e r w h i l e d i s c o u n t i n g t h e l o s s b y t h ep e r s i s t e n c y o f p o h c y h o l d e r s I n S e c t i o n 3 , w e i n t ro d u c e o b j e c u v e s t h a t a n i n s u re rm a g h t c o n s a d e r o p t n n l z m g

W e p r o p o s e a n e x p o n e n t m l p e r s i s t e n c y f u n c ti o n i n S e c ti o n 4 an d d e v e l o p ag e n e r a l c r e d i b i l i t y f o r m u l a i n S e c t i o n 5 . In S e c t i o n s 6 a n d 7 , w e c a l c u l a t e c r e d l b d l t yf o r m u l a s an t w o p a r a m e t r i c c a s e s - - n o r m a l - n o r m a l a n d P o l s s o n - g a m m a F i n a l l y , w es u g g e s t f u t u r e r e s e a r c h i n S e c t i o n 8 .

W O R K O F B U H L M A N N A N D T A Y L O R

2 1 N o t a t i o n a n d A s s u m p t i o n s

A s s u m e t h a t t h e to t a l c l a i m s o f a g w e n p o h c y h o l d e r , o r r i sk , m t h e t th p o h c y p e r i o du s u a l l y o n e y e a r ) , is a r a n d o m v a r i a b l eX , I O = 0 ) , o r m o r e s i m p l y , X , 1 0 , i =

I , , n . F o r a g i v e n v a l u e 6 9 = 0 , a s s u m e th a t t h e r a n d o m v a r i a b l e s X , ] 0 , t =I . . . . . n , a r e i n d e p e n d e n t l y a n d ~ d e n t lc a l ly d i s t r i b u t e d a c c o r d i n g t o a c o n d i t i o n a lp r o b a b i l i ty d e n sa ty ) f u n c u o nf x I 0 ) A s s u m e t h a t t h e v a l u e 0 i s f i x e d f o r a g i v e nr i sk , a l t h o u g h i t i s g e n e r a l l y u n k n o w n F o rext.stmg p o h c y h o l d e r s , d e n o t e t h ep r o b a b d l t y d e n s a t y ) f u n c u o n o f 69 b y ~ 0 ) , a l s o c a l l e d t h es t ruc ture f imct ionBUHLMANN,1 9 7 0) N o t e t h a t w e t a c a tl y a s s u m e t h a t th e p o l i c y h o l d e r s a r e th e s a m e

s i z e b e c a u s e t h e d i s tr i b u t i o n o f t h e t o ta l c la m ~ s o f a p o l i c y h o l d e r s e l e c t e d a t r a n d o mas g i v e n b y t h e m a r g i n a l d a s t n b u u o n o f X


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CRED[BII_[TY AND PERSISTENCY

S u p p o s e t h e in s u r e r h a s n ye a r ,, o c l a i m e x p e r i e n c e fo r a p o h c y h o l d e r : x =( t l , , r ,,) ~ X " I n t h is w o r k , w e c o n s i d e r c r e d l b d ~ t y f o r m u l a s th a t a r e n o tn e c e s s a r i l y l i n e a r , d e n o t e d b yd x ) , m w h i c h d i s a r e a l - v a l u e d f u n c t i o n o n X " . W e

a l s o c o n s i d e r l i n e a r c r e d t b l h t y f o r m u l a s o f t h e f o r m a + b Y', tn w h i c h a a n d b a r ec o n s t a n t s t o b e d e t e r m i n e d a n d Y i s t h e a r t t h m e t , c m e a n s o f th e c la m + s x j , ,x,, W erefer to d x ) a s th e r e n e w a l p r e n l t u m , O l s i m p l y p r e m m m , f o r y e a r n + I . a n d w e a r en o t l o a d i n g f o r a d m i n i s t ra t i v e e x p e n s e s

F o , a p r o s p e c t i v e p o h c y h o l d e r ,d x ) ~s m o r e a c c u l a t e l y t e r m e d t h e i n i ti a lp r e m i u m b a s e d o n p a s t e x p e r i e n c e o f t h e r is k , b u t w e b l u r t h is d i s t ra c t i o n b e c a u s et h e i n s u r a n c e p r o d u c t s t h a t t y p i c a l l y u s e c r e d ~ b t h t y p r e m i u m s a r e a n n u a l l yr e n e w a b l e o n e s . F o r t h is r e a s o n . , ,a le s p e r s o n n e l m u s t o ft e n " s e l l " t h e p o h c ya n m t a l l y e v e n ~f t h e p o h c y h o l d e r t s r e n e w i n g a n d n o t i n i ti a l l y b u y i n g

2 2 W o r k o f B i i h lm a n n

To e s t i m a t e t h e f u t u re c l a , m s o f a r i s k, X , , + , l 0 , w ~th u n k n o w nO. BUHLM NN( 1 9 6 7 , 1 9 7 0) m m l r n l z e s t h e e x p e c t e d v a l u e o f t h e s q u a r e d - e r r o r lo s s fu n c t i o n

L ( E I X , , + , 1 0 ]. d ( x ) ) = ( E I X , , + , 101 - d ( x ) )z

U n d e r o u r a s s u m p t i o n s , t h e r e s u l t i n g o p t m l a l p r e m i u md * x ) is

(2 1) E[X, ,+ , Ix ] = f E [ X , , + , 10] 7 r O I x ) d OB y r e s t r i c t i n g th e f o r m o f t h e r e n e w a l p l ' e m l u md x ) t o b e a h n e a r c o m b i n a t i o n

o f t h e c l a , m e x p e r i e n c e , x ~ , . % . a n d b y u s i n g t h e s a m e s q u a r e d - e tTo r l o t sf u n c t i o n , B UH LM A N N ( 1 9 6 7 , 1 9 70 ) o b t a m s t h e l o l l o w m g c r e d l b , l l t y f o r m u l a

d * x ) = (1 - Z ) E I X ]+ Z . ? ,

m w h i c h E [ X ] = E o E [ X ] O ] ts t il e o v e r a l l m e a n , Z = n / ( n + k ) ; a n d k =Eo [Var IX I 0 ]] /Var o [ E I X I 0 I] T h e n u m e r a t o r o f k ~s c a l l e d t h eexpec ted processv a r u m c e , t h e d e n o m i n a t o r , t h evar iance o / the hypothe tmal mean~ .

I n c e r t a i n c a s e s , t h e p r e m , u mE[X,,+ ~ l x ] i t h n e a r a n d , t h u s, e q u a l s t h e h n e a rc r e d l b d n y f o r m u l a JE W t~ LL ( 1 9 7 4 a , 1 9 7 4 b ) v e r i f i e s c o n d i t i o n s u n d e r w h i c h t h~ se.~act credtbdttyo c c u r s A l s o , p l e a s e r e f e r t o W n .L M O T ( 1 9 9 4 ; C h a p t e r 4 ) n l w h i c hh e c l e a r ly e x p l a i n s t he f o r e g o m g t h e o r y a n d d l u s t ra t e s tt b y p r o v i d i n g m a n ye x a m p l e s

2 3 W o r k o f Ta y l o r

O n e o f th e p r o p e r t i e s th a td* x ) = E [X ,,+ ,J x ] s a t i s f i e s ~s t h a t t h e s un 1 o fp r e m i u m s o v e r t h e p o r t f o h o o f r t sk s e q u a l s th e e x p e c t e d c l am a s f r o m t h e p o r t f o h o(B U HL M A NN , 1 9 6 7) I n c o n f i r m i n g t h i s p r o p e r t y, B u h l n l a n n l m p h c , t l y a s s u m e s t h a tt h e s tr u c t u r e o f t h e p o r t f o h o d o e s n o t c h a n g e a s a i e s u l t o f th e r a t i n g f o r m u l aTAY LO R ( 1 9 7 5 ) c h a l l e n g e s t h , s a s s u m p t i o n a n d a s s e r t s th a t ~f a p o h c y h o l d e r t e n d st o c a n c e l w h e n i t i s r e n e w e d w i t h a p r e m i u m t h a t ~s h i g h e r t h a n ~ ts a n t i c i p a t e d


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56 VIRGINIA R YOUNG

claims, then the premmm income will not necessarily equal the expected clmms. Heproposes a loss funcuon that explicitly accounts for the decision of policyholders tobuy, cancel, or renew their insura nce He suggest~ that the Insurer mm nm ze itsexpected financial loss while discounting for the persistency of policyholders.

Taylor defines a persis tency ]unct ton p(0, x, d x ) ) to be the ratio of the exposurein year n + 1 to the exposure m year n for the risk class given by O = 0 with n-yearclmm experience x and premmm d x ) for year n + 1. The exposure in year n Is thenum ber of existing policyholders To be somewhat rigorous fol a conu nuo usstructure parameter O , think of p as an ins ta nt ane ou s ratio or a densityluncuon

His loss funcuon is the financml loss, discounted by persistency,

(2.2) L O , x ~ , a + b x ~ ) = p O , x ] , a + b x ~ ) E l X 2 l O l - a + b ~ ) ) ,

and he finds values for a and b that mm m]lz e the expected loss Note that herestricts n = 1 and d x) = a + bx]

Taylol assumes that p is a linear function of the arithmetic difference betweenwhat the insure r charges, a + bx I, and the a mou nt of cl aims the p ohcy hold erexpects to recur. He then consi ders two cases In the first, Taylor calls thepohcyholder unb ta sed because the policyholder expects to incur EIX2 [Olitshypothetical mean, m the second, the pohcyholder is btased and expects to incur.v~, its recent clau n experie nce (Note that Taylo r does not use the term btased m astatistical sense because the expected value of.~ I is E [ X [ O ] . ) His two persistencyfunctions are, thus,

Unbiased risk p(0,.l. I , a , b ) = 1 - e a + b x i - E [ X 2 [ 0 ] ) ,

Biased risk p( 0, x I , a , b ) = 1 - e a + b x I - .~ l );

in which e is a poslhve constant; Taylor calls the parameter e a ptte-ela.sttctty oJexposure

The linear credibility premiums that Taylor finds are

(2 3a) Unb ias ed risk: (1 - Z ) E I X I + Z x l +I/(2e),

(2.3b) Biased risk. (1 - Z * ) E [ X I + Z * x l +I/(2e);

In these formulas, Z is the Buh lma nn credibi lity weight, and Z* = V2(I + Z ) > ZNote that in each case, the cred~blhty premmm ~s a weighted average of x~ andE [ X ] , plus a flat load, l/(2e), indepen dent of the claim dl stn buu on or thepohcyholder's experience. One can consider this load a risk charge for thepolicyholder selecting against the insurer

3 OBJECTIVES OF THE INSURANCE COMPANY

3 1 M a x i m i z e U n d e r w r i t i n g G a in a n d A m o u n t o f B u s in e s s

One of the goals of an insurance company, as tor any company, is to earn a profit.The profit or und erwr mng gain is the excess of income over outgo Insma nce


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CREDIBILITY ND PERSISTENCY 5 7

p r e m i u m s c o n t a i n p r o v i s i o n s f o r c l a i m s ( i n c l u d i n g r i sk m a r g i n s ) a n d f o r th e

e x p e n s e s o f a d m i n i s t e r i n g t h e in s u r a n c e p o h c y O u t g o c o n s i s ts o f c l a i m s , e x p e n s e s ,a n d e x p e r i e n c e r a t i n g r e f u n d s ( M O R E W O O D , 1 9 9 2 )

I n t h is w o r k , w e c o n s i d e r th e c o m p o n e n t o f t h e u n d e r w r i t i n g g a i n e q u a l to th ee x c e s s o f t h e pr o v B ~ o n f o r c l a i m s a n d r i s k m t h e p r e m i u m o v e r t h e c l a i m st h e m s e l v e s I n o t h e r w o r d s , w e ig n o r e s e m c e f e es , i n v e s t m e n t m c o l n e , l o a d m g s f o re x p e n s e s , a s w e l l a s t h e e x p e n s e s t h e m s e l v e s , a n d e x p e r i e n c e r a t in g l e f u n d s .

A n o t h e r p o s s i b l e g o a l o f an i n s u ra n c e c o m p a n y is t o , n c re a s e , ts b o o k o fb u s i n e s s . M O R I-:W O OD ( 1 9 9 2 ) n o t e s t h a t w r i t in g n e w b u s i n e s s d e p e n d s o n c o m p e t i -t iv e p r e m i u m r a te s a n d t ha t r e n e w i n g e x i s t in g b n s l n e s s d e p e n d s o n h o w t a i r t h ep o h c y h o l d e r p e r c e , v e s th e p r ic e H e p o i n t s o u t th a t u n d e r w r i t in g g a i n a n d g r o w t ha r e , n t e r d e p e n d e n t R a p i d g r o w t h a n d l a r g e p r o fi t m a r g i n s m t h e p r e m i u m s a r eu s u a l l y i n v e r s e l y r e l a te d . We , t h e r e f o r e , p r o p o s e f i n d i n g a f u n c t i o n d * :X --+Rs u c h t h a t t h e c o m b i n a t i o n

3 ~ ) U G+ h B = E [ p 0 , x , d x ) ) d x ) - e l X I O l ) ] + h E [ p 0 , x , d x ) ) ]

3 2 ) = E [ p O , x , d x )) d x ) + h -E [ X I 0 ] ) ]

~s m a x i m u m w h e n d = d * T h e p a r a m e t e r h is a n o n - n e g a t i v e c o n s t a n t , a n d w e ta k e

t h e e x p e c t a t i o n w i t h r e s p e c t to t h e j o i n t d i s t r i b u t i o n o fX~ . . . . X,, a n d O I nS e c t i o n 4 , w e p r o p o s e a s p e c i f i c f o r m u l a f o r t h e p e r s i s t e n c y f u n c t i o n p , b u t h e r e i ti s a n y r e a l - v a l u e d f u n c t i o n d e f i n e d o n O x X x R R e c a l l th a t p a c c o u n t s f o r i n it i alb u s i n e s s a s w e l l a s r e n e w a l

T h e f ir s l t e r m o n t h e r ig h t h a n d s i d e in e q u a t i o n ( 3 .1 ) is th e e x p e c t e d v a l u e o f t h eu n d e r w r m n g g a i n d i s c o u n t e d f o r t h e p e r s B t e n c y o f p o l i c y h o l d e r s W e w r i t eUG t od e n o t e t h is fi rs t t e r m T h e s e c o n d t e r m , s h m u l t i p l i e d b y th e e x p e c t e d r e l a t w ea m o u n t o f b u s i n e s s w r i t t e n , o r B I t i s r e a s o n a b l e t o c o n s t r a i nUG>--O, B > 1 , orb o t h . I f w e l e t h a p p r o a c h 0 , th e n b y m a x n n l z l n gUG + h B, w e m a x m l l z e t h e

e x p e c t e d u n d e r w r i t i n g g a i nUG; i f w e l e t h a p p r o a c h oo, t h e n w e m a x i m i z e t h ee x p e c t e d a m o u n t o f b u s i n e s s w r i tt e n B .T h e p a r a m e t e r h c o n v e r t s t h e re l a t i v e a m o u n t o f b u s i n e s s i n t o m o n e t a r y u n i ts ( s e e

e q u a t i o n ( 5 .1 ) b e l o w ) . To c h o o s e i t s v a l u e s , a n a c t u a r y m a y w i s h t o c o n s i d e r t h ep o t e n ti a l l o ss o r g a i n o f r e v e n u e t o c o v e r f i x e d a d m m ~ s t r a tw e e x p e n s e s A l s o , o n em a y c h o o s e h a c c o r d i n g t o o n e o f th e f o l l o w i n g c n t e r , a

h - --- -0 IS t h e s m a l l e s t v a l u e s u ch t h a tB>--M, f o r s o m e M - - 1, i n w h i c h B i se v a l u a t e d a t t h e o p t i m a l d *

h > 0 i s t h e l a r g e s t v a l u e su c h t h a tUG>--M, fo r s o me M- - - - -0 , m w h i chUG ise v a l u a t e d a t t h e o p t n n a l d *

I n t h i s c o n t e x t , o n e c a n t h i n k o f h o rI/h a s a L a g r a n g e m u l t J p h e r i n a c o n s t r a i n e do p t , m l z a t~ o n p r o b l e m . A t t h e e n d o f S e c t i o n 6 2 , w e a p p l y t h e s e c r i t e r ia m ah y p o t h e t i c a l e x a m p l e .


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58 VIRGINIA R YOUNG

3 2 O p t i m i z e P r o p e r t i e s o f t h e Bo o k o f B u s i n e ss

An insu rance comp any may set goals concerni ng the structure of its book ofbusiness. If the structure of the comp any 's business is given by =( 0) during year I,and the credlblhty premium for year 2 is d(.~), then the expected structure of thebusiness m year 2 is described by

~2 (0) = g J p (0 , x , , d x , ) ) f x , I O) s t O) d . , , ,

in which K IS a normahzmg constantOne goal may be to ensure that the grand mean decreases from year to year. This

occurrence indicates that the company Is writing risks with lower expected claims.The grand mean in year 2, expected at time 0, is

,u,2 = K J',f x2 p(0, x . , d x , ) ) f x I O ) ~ 0 ) d x d O ,

in which x = (x~, x2) Note tha t/t0 : is the expected value of X2, with respect to thedistribution I f x 2 ] O) ~z 0 ) dO

Anot her goal ma y be to ensur e that the total variance decreases over um e Thisoccurrence may enable actuaries to price more accurately by making claims morepredictabl e The variance m year 2, expected au m e 0, is

E I X 2 - ,u02)21 =Eo E I x ~ I O I -E I x

2 1 0 l 2 + EO[EIx

2 1 0 l ~ ] - / , ~ , 2 )

= E x p e c t e d P r oc es s V a r i a n c e + Va r i a n c e o f H y p o t h e t i c a lM e a n s ,

m w h i c h O is d i s tr ib u t e d a c c o r d in g t o ~ 2 0 ) W e e x a m i n e th e r e t w o g o a ls m t w oparamet r i c cases in Sec t ions 6 and 7

4 PERSISTENCY

We assume, as does TAYLOR (1975), that pers istency depend s on the ar~thmeucdifference between premium charged and anticipated claims. Such an assumptionmay be suitable because we assume that risks are the same size. In our work, weexplicitly account for Taylor's belief that policyholders most likely expect clamlssomewhere between E [ X [ O I and 2 We do so by expressing the pohcy hold er'santicipated clauns as a linear combmauon of E [ X ] O ] and 2, namely,(I - ~) E IX [ 0 ] + c 2, 0-< c-< I The dif fere nce bet wee n what the insu rer chargesand what the pohcyholder expects is, therefore,

d = d x ) - [ I - c ) E I X I 0 l + c V ]

Note that when n = 1. the amou nt that the pol icyho lder expects to incur,( 1 - c ) E I X 2 I O ]+aw l, include s as special cases the two that Tayl or exam ines Forunb ias ed risks, c = 0; for biased risks, c = I.

Taylor points out one major weakness of the linear persistency functions that heuses. They may take on negative values, implying that the anaount of business Isnegative We, therefore, propose an exponenn al persistency function

p A ) = c~ ex p( -2 A) ,


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6 0 VIRGIN I R YOUNG

f o r a n a r b i tr a r y s a m p l eX = x , m w h i c h w e t a k e t h e e x p e c t a t i o n w i t h r e s p e c t t o t h ep o s t e r i o r d l s t r t b u n o n o f O I x . Tr e a t i n g e x p r e s s i o n ( 5 . 2 ) a s a f u n c t i o n o fd x) , a n da p p l y i n g s t a n d a r d t e c h m q u e s f r o m c a l c u l u s , w e f i n d t h a t t h e m a x m m m o c c u r sw h e n

2 d * ( x ) E o l , [e x p { 2 1- c ) E I X I 0 ] } ] =

= 2 d * x ) E o l , [ E [ X l O ] e x p { 2 l - c ) E [ X l O ] } ]

+ ( 1 - 2 h ) E o l x [ e x p { 2 l - c ) E [ X I O ] } ],

or, a f t e r so lv ing fo rd * x ) ,

d 4 x) = I / 2 - h + E o l , [ E [ X l O ] e x p { 2 I - c ) E [X ]O ]} ]

+ E o l , [ e x p{ 2 ( I - c ) E I X I0 ] } ]= I / 2 - h + D , { I n E o l , [ e x p l t e [ X l O l } ] }1 , = 2 1 . - , , . [ ]

O ne m ay in te rp re t the t e rms m equ an on (5 1 ) a s fo l lo w s ' T he f i r st , I /2 , is a fl a tl o a d s l n n l a r t o t h e o n e f o u n d b y Ta y l o r m e q u a t i o n s ( 2 3 a , b ), n a m e l y, I / ( 2 e ) ; itp a m a l l y a c c o u n t s f o r t he s e n si ti v it y o f p o h c y h o l d e r s t o th e p r e n u u m c h a rg e d . T h es e c o n d t e r m , - h , o f f s e t s ['o r h o w m u c h t h e in ~ u re r w e i g h t s t h e r e la t iv e a l n o u n t o fbus ines s , B No te tha td * x ) d e c r e a s e s o n e u m t f o r e v e r y u n i t o f in c r e a s e o f h ;t h e r e f o r e , t h e m o r e w e w e i g h t B , t h e m o r e w e d e c r e a s e t h e o p t i m a l p r e l m u m , a s o n em i g h t e x p e c t T h e t h ir d t e rm is a n E s s c h e r p r e m i u m ( G E RB E R , 1 9 8 0 ) , I t e q u a l s a ( x )t h a t l m n m f i z e s t h e e x p e c t e d v a l u e o f t h e f o l l o w i n g l o ss t u n c t l o n

a x ) - E I X l O 1 ) 2e x p { 2 ( 1 - c ) E I X ] O I } .

6 NORMAL-NORMAL

6 1 U n c o n s t r a i n e d M a x i m i z a t i o n o fU G h B

To m a x m a l z e UG + / 1 B m t h e n o r m a l - n o r m a l c a s e , w e a p p l y T h e o r e m 5 1 t o o b t a int h e f o l l o w i n g p r o p o s m o n .

P r o p o s i t i o n 6 1 :Let X, [O - N O , 02 ) , t = 1 , ., n , be indep end en t and iden t ica l lyd i s t r i b u t e d n o r m a l r a n d o m v a r i a b l e s , w i t h u n k n o w n m e a n 0 a n d k n o w n v a r i a n c ea - ' > 0 L e t O ~ N u , T : ) ,w l t hk n o w n m e a n ,u an d v a n a n c e r 2 > 0 . T h enU G + h Bi s m a x i m i z e d w h e n

(6 .1) d * x ) = l / 2 - h + {(1- Z ) / ~ + Z 2 }+ 2 r 2 ( 1 - Z ) (1 - c ) ,

in whi h Z = n r 2 / n r 2 + a 2 ) , t h e B u h l m a n n c r e b l h t y w e i g h t

P r o o f : T h e p o s t e r io r d l s m b u t l o n o f 0 I x is n o r m a l w i t h m e a n

(6 2) / l* = 1 - Z) ,u + Z 2

a n d v a r i a n c e


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C R E D I B I L I T Y A N D P E R S I S T E N C Y 6

(6 .3 ) ( r 2 ) * = O 2 Z ' 2 / ( n ~ 2 + 0 2 ) = r 2 ( I - Z ) .

B e c a u s e E [ x J o I = O , s u b s t i t u t e th e m o m e n t g e n e r a t i n g f u n c t i o n o f O J x i n t o

e q u a t io n (5 I ) T h e m o m e n t g e n e r a t i n g f u n c ti o n o f a n o r m a l r a n d o m v a r i a b l eO ~ N B , r 2) Is

(6 .4 ) M o t) = EO[ e x p {Ot }1 = e x p /a + r 2 t 2/2 }

T h e d e n v a t w e , w i t h r e s p e c t t o t, o f th e n a tu r a l l o g a r i t h m o f th i s m o m e n t g e n e r a t i n gf u n c t i o n i s

(6 5) ,u + r2 t

To c a l c u l a t e t h e c r e c h b l h t y f o r m u l a m( 6 2 ) f o r , u , ( r 2 ) * f r o m e q u a t i o n ( 6 3 ) f o ro b t a i n

d * x ) = I / R - h +

= l / 2 - h +

e q u a t i o n ( 5 . 1 ) , s u b s t i t u t e / ~ * f r o m e q u a t i o nr 2, a n d 3.(1 - c ) f o r t m e q u a u o n ( 6 5 ) W e

u*+ r2) 2 1 -c)){ ( 1 - Z ) / l + Z i : J + A r 2 ( I - Z ) (1 - c ) . [ ]

N o t e t h a t d * x ) i s a h n e a r f u n c t i o n o f t he a v e r a g e c l a n n e x p e r i e n c e 2 , t h e r e f o r e ,w e h a v e a t y p e o f e x a c t c r e d i b i l it y in t hi s c a s e T h i s c r e d i b i l i t y p l e m m m is t h e su mo f fi v e i n t e r e s t in g t e r m s . W e d m c u s s t h e f ix st t w o a t t h e e n d o f S e c t i o n 5 , t h e yo c c u r m d * x ) m g e n e r a l T h e s u m o f th e t h ir d a n d f o u r th t e r m s Is t h e s t a n d a r d

B u h l m a n n c r e d i b i l i t y e s n m a t e m t h e n o r m a l - n o r m a l c a s e . T h e f i ft h e x p r e s s l o n ,d e p e n d s o n h o w m u c h t h e p o l i c y h o l d e r s w e i g h t t h e ir o w n c l a i m e x p e r i e n c e r e l a u v et o t h e i r t r u e m e a n

To s e e h o w t h e o p m n a l p r e m i u m d * ( x ) c h a n g e s w h e n t h e p a r a m e t e r 2 c h a n g e s ,e x a m i n e

D 2 d * x ) = - 1 / 2 2 + r 2 l - Z ) l - c ) .

O b s e r v e t h at i f c i s s u f f i c i e n t l y c l o s e t o I (t h a t is , t h e p o l i c y h o l d e r s w e i g h t t h e i rc l a i m e x p e r i e n c e h e a v i l y ) o r i f n i s s u f f i c i e n t l y l a l g e , t h e nd * x ) d e c r e a s e s a s 2i n c r e a s e s In th i s c a s e , as th e p o h c y h o l d e r s b e c o m e m o r e s e n s i t i v e to th e a r i th m e t i cd i f f e r e n c e z l, th e l o w e r t h e o p t n n a l p r e m m m .

6 . 2 . C o n s t r a i n e d M a x i m i z a t i o n o fUG + h B

U p t o th i s p o in t , w e h a v e n o t c o n s t r a i n e d t h e v a l u e s o fU G a n d B I t i s r e a s o n a b l e t or e q m r e t h a t UG->O, B_>1, o r b o t h , i n o t h e r w o r d s , t h e i n s u r e r d o e s n o t l o s e m o n e y ,b u s i n e s s, o r b ot h . We e x a m i n e t w o s e ts o f c o n s t r a in t s a n d o b t a i n t h e t o l l o w m gp r o p o s i t i o n s

P r o p o s i t i o n 6 . 2 : G i v e n t h e a s s u m p t i o n s In P r o p o s i t i o n 6 1 a n d t h e re s tr lc tN o n sU G - > O a n d d x ) = a + b 2 , w e m a x l n n z e U G + h B w h e n d = d * , w i th d * g i v e nb y

6 6 ) d * x ) = l / A - h + { 1 - Z ) f l + Z .~ } + , ~ r 2 I - Z ) l - c ) ,


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6 2 VIRGIN I R YOUNG

fo r h- < I /)1. , an d

(6 7)

fo r h > I/3.

d * ( x ) = { ( 1- Z ) u + Z . r } + 3 .r 2( I - Z ) ( I - c ) ,

P r o o f : U G + h B = E [ 6 e x p { - 2 ( a + b Y ) - ( l - c ) O - c x } ( a + h + b 2 - O ) ]

(6 8) = 6 5 e x p { - 2 a } ( a + h ) E o [ e x p { 0 3 . ( I - c ) } E x 1 0 [ e x p { - 2 ( b - c ) 2 } ] ]

+ 6 ex p { - 2 a } b E 0 [ e x p { 0 2 (I - c ) }E x l o l X e x p { - 3 . ( b - c ) 2 } ] ]

-6 5 e x p { - 2 a } E o [ 0 e x p {0 3.(1 - c )} E x l o [ e x p { - 3 . ( b - c ) 2 } ] ] .

U s e M x l o ( t ) a n d E x l o [ x e x p {x t} ] t o c a l c u l a t e t h e e x p e c t e d v a l u e s mU G + h BM o r e s p e c i f i c a l l y,

E x l o [ e x p { 2 t ] = { M x [ o ( t /n ) }

= e x p {Ot + o 2 2/(217)},

S u b s t i t u t e t h e s e

U G + h B =

E x l o [ x e x p { Yt }l= { M x l o ( t / n ) } - I E x l o l, t e x p { x t / n I ]

= e x p {Ot+ ~5 2 2 / ( 2 n ) }[0 + ~r zritz]

e x p r e s s i o n s i n t oU G + It B ,e q u a t i o n ( 6 8 ) , to o b t a i n

d e x p { - 2 a } ( a + h) E [ e x p { 3 . ( l - b ) 0 + 2 2 ( b - c ) 2 ~ r2 /( 2n )} ] +

+65 e x p { - 3 . a } b E [ e x p { 2 ( 1 -b )O+3 .Z(b -c ) 2 02 / (2 n)} x

x [ 0 - 3 . ( b - c) o 2 / n J ] - d e x p { - 2 a } x

x E [0 ex p { 2 ( 1 - b) 0 + 2 - (b - c)-~ o z/(2,7) } ]

= d e x p { - 2 a + 2 2 ( b - c ) 2 c r 2 /( 2 n ) } xx { ( a + h - 2 b ( b - c ) o 2 /n )E [ e x p { 0 2( I - b ) } ] - ( 1 - b ) x

x E [ O e x p 1 02 ( I -b)}] }

( 6 9 ) = 6 5 e x p { - 2 a + X 2 ( b - c f a 2 / ( 2 n ) + 2 ( l - b ) , u + X 2 ( l - b f r 2 / 2 }

x { a 1 7 - 2 b (b - c) 02171 -(1 - b),u - 2 ( 1 - b ) 2 r 2 }

B y s e m n g h = 0 m e q u a u o n ( 6 9 ), w e s e e th a t

(6 10) U G > - O f f a n d o n l y I f a > - 2 b ( b - c ) c r z / n + ( I - b ) t + ( l - b ) 2 r 2

T h e v a l u e s o f th e p a r a m e t e r s a a n d b t h a t a p p e a r md * ( x ) m e q u a t i o n ( 6 . 1 ) s a t i s f yt h e m e q u a h t y ( 6 . 1 0 ) f f a n d o n l y i f h - I/3 .. T h e r e f o r e , i f h - 1 /2 , t h e n th e in s u r a n c ec o m p a n y d o e s n o t e x p e c t t o l o s e m o n e y b y u s i n g t h e c r e & b l h t y f o r m u l ad * ( x ) mequa t ion (6 1 ) .

a n d


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CRIzDIBILITY AN D PERSISTENCY 6 3

O n t h e o th e r h a n d , i f h > I / 2 , t h e n i n v o k e th e c o n s t r a i n t a n d s e t

a = a l = 2 b ( b - c ) ~ 2 / n + (1 - b ) /~ + Z ( 1 - b ) 2 r 2 ,

tha t i s , s e t U G = O I n t h . s i n s t a n c e ,

U G + h B = h B

(6 1 I ) = h 6 e x p { - 2 a + ) ~ 2 ( b - c ) 2 o 2 / ( 2 , 7 )+ 2 ( l - b ) / ~ l+ 2 . 2 (1 - b ) 2r~ /2}

T h e v a l u e b~ o f b th a t m a x i m i z e s h B a l s o m a x i m i z e s t h e e x p o n e n t i n th e b r a c e s i ne q u a t i o n ( 6 I I ) . T h e m a x i m u m o t th e e x p o n e n t o c c u r s a t

b j = n T Z / n T z +O 2 = Z

I n t hi s c a se , a ~ = ( I - Z ) t l + 2 r 2 ( l - Z ) ( I - c ) , a n d w e a r e d o n e N o t e t ha t t h ef o r m u l a m e q u a t i o n ( 6 7 ) is t he o n e w e o b t a i n w h e n w e m a x i m i z e B s u b j e c t toU G >--O. []

W e o f f e r th e f o l l o w i n g p r o p o s i t i o n w i t h o u t p l o o f , t h e r e b y s p a r i n g t h e r e a d e r o ft h e m e s s y d e t a i l s th a t a r e s m a f l a r t o t h e o n e s m t h e p l o o f o f P r o p o s i t i o n 6 . 2 .

P r o p o s i t io n 6 3 : G i v e n t h e a s s u l n p t l o n s i n P r o p o s i t i o n 6 1 a n d t h e r e st r ic t i o n sB ~ I a n d d x ) = a + bY, w e m a x i m i z e U G + l iB w h e n d = d * , w i th d * g i v e n b y

d * ( x ) = l / 2 - h + {(1- Z ) u + Z Y } + 2 r 2 ( I - Z ) ( I - c ) ,

fo r h z 1 /2- In d) /2 +2 r ~- (I - Z ) / 2 - 2 c 2 o 2 / ( 2 n ) , a n d

d * ( x ) = {( I- Z ) u + Z f } + In 5)/2 + 2 r 2 ( I - Z ) ( I- 2 c ) / 2 + 2 c ~ - c I 2 / 2 n ) ,

f o r h o t h e r w i s e . [ ]

To i ll u s tr a t e h o w o n e m i g h t c h o o s e t he p a r a m e t e r h , w e o f f e r t h e f o l lo w i n ge x a m p l e

E x a m p l e : In t h e n o rn l a l - n o rn a a l c a s e , l e t ~. = 0 0 1 , c~ 2 = 2 5 0 , ,u = 1 0 0 0 , r 2 = 2 5 0 ,d = 1 .5 , a n d n = 1. T h e r e f o r e , Z = 0 . 5 0 , s o l e t c = 0 7 5 . A s h i n c r e a s e s f r o m 0 t o1 / 2 = 1 0 0 , U G d e c r e a s e s f r o m a b o u t 5 5 to 0. A s h i n c r e a s e s b e y o n d 5 9 3 7 5 , a sd e t e r m i n e d i n P r o p o s i t io n 6 3 , B i n c r e a s e s f r o m 1 u p w a r d .

S u p p o s e w e t a r g e t a n u n d e r w r i t i n g g a i n o f at le a s t 2 5 T h e l a rg e s t v a l u e o f h i s,

t h e r ef o r e , 7 9 .5 7 1 I f w e u s e th i s v a l u e o f h , t h e n w e m a x n n l z e t h e re l a t iv e a m o u n to f b u s i n e s s B , s u b J e c t t o t h e c o n s t r a i n t t h a tUG>--25. A l s o , s u p p o s e w e t a i g e t ar e l a t i v e a m o u n t o f b u s i n e s s o f a t l e a s t 1 1 T h e s m a l l e s t v a l u e o f h ~s, t h e r e f o r e ,6 8 9 3 9 . I f w e u s e t h is v a l u e o f h, th e n w e m a x i m i z e t h e u n d e r w r i t i n g g a i nUG,s u b J e c t t o t h e c o n s t r a i n t th a t B - I I. To a c h i e v e b o t hU G > 2 5 a n d B - - 1 1, u s e a n yv a l u e o f h b e t w e e n 6 8 9 3 9 a n d 7 9 5 71 [ ]


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64 VIRGINIA R YOUNG

6 3 O p t im i z e P r o p e r t ie s o f th e B o o k o f B u s i n e s s

An insurance company may set goals concerning the structure of its book of

busine ss. We obtain the fol lowi ng lemma that we use to rninmll ze the future grandmean and the future total variance.

Lemma 6.4: Assume the condmons in Theorem 6.1. If the structure of thecompany's business is gwen by ~r(0) during year I, and the credibility premium foryear 2 is a + br~, then the expected structure of the business in year 2 is distributednormally with mean u + J . ~ 2 l - b )and variance r 2

P r o o f : zt2(0) ~ J p O , a I , a + b a I ) J x I J O ) s z O ) d x I

exp { - {0 2- 20(/~ +2r2 (1 - b )) ] / (2 r2 )} .

It follows that the structure parameter O is expected to be distributed normally withmean ,u + 2r 2 (i b) and varia nce r 2 []

The density ~z will be independent of the rating parameter a in every case becausethe term e 2 a factors In the nor mal -no rma l case, the distr ibuti on ~s also inde pen-dent of c.

P r o p o s i t i o n 6.5: Given the condition s in Lemma 6 4, the grand mean in year 2,expected at tune 0. is minimized when the insurer gives full weight to thepohcyholder's experience

P r o o f : The tuture grand mean is /q)2 =EO EIX 2] 0] =E [0 l= u+ 2r 2( I- b) , inwhich 0- -


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CREDIBILITY AND PERSISTENCY 6

(7 1) d * ( x ) = I / 2 - h + e t + n ) / f l + n - 2 l - c ) ) ,

fo r f l + n > 2 ( l - c ) .

P r o o f : T h e p o s t e r i o r d l s m b u t l o n o f O I x l~, g a m m a ,G c~+ n , f l + n ) . B e c a u s eE[X210]=Os u b s t i tu t e t h e m o m e n t g e n e r a t i n g f u n c t i o n o f O I x i n to e q u a t i o n(5 1) T h e m o m e n t g e n e r a t i n g f u n c ti o n o f a g a m m a r a n d o m v a r i a b l eO ~ G a , f l)IS

M o t ) = E ol e x p { 0 t} ] = f l ~ / ( f l - t ) ,

f o r f l > t T h e d e n v a u v e , w i t h r e s p e c t to t, o f th e n a t u ra l l o g a r i t h m o f th i s m o m e n tg e n e r a t i n g f u n c t i o n i s

(7 2 ) c ~ l f l - t )

To c a l c u l a t e t h e c l e d l b d l t y f o r m u l a m e q u a t i o n ( 5 .1 ) , s u b s t i t u t e a + n X f o r a , f l + nf o r f l, a n d 2 ( I - c ) f o r t in e q u a t i o n ( 7 . 2 ) W e o b t a i n

d * ( x ) = l / 2 - h + ( a + n ) / ( f l + n - 2 ( I - c ) ) . []

N o t e t h a t d * x ) Is a l i n e a r fu n c t i o n o f t h e a v e r a g e c la m 1 e x p e r i e n c e , t h e r e f o r e ,w e h a v e a t y p e o f e x a c t c r e d l b d ~ t y m t h i s c a s e , a s m th e n o r m a l - n o r m a l c a s e . T h et h i rd te r m in e q u a t i o n ( 7 .1 ) i s s m l d a r to t h e B u h l m a n n c r e d i b d l t y f o r m u l a ,a + n ~ ) / f l+ n ) , e x c e p t f or - 2 ( I - c ) .

To s e e h o w t h e o p u m a l p r e m i u md * x ) c h a n g e s w h e n t h e p a r a m e t e r 2 c h a n g e s ,e x a m i n e

D 2 d ~ x ) = - 1 / 2 2 + I - c ) o L + n ) / f l + n - 2 I - c ) ) 2

A s I n t h e n o r m a l - n o r m a l c a s e , o b s e r v e t h a t i f c ~s s u f f i c i e n t l y c l o s e t o 1 o r ~ f n i ss u f f i c i e n t l y l a r g e , t h e n d * (x ) d e c r e a s e s a s 2 i n c r e a s e s ; t h e r e f o r e , a s th e p o l i c y h o l d -e r s b e c o m e m o r e s e n s m v e t o t h e a n t h m e t t c d i f f e r e n c e A , th e l o w e r th e o p t n n a lp r e m i u m w i ll b e

7 2 C o n s t r a in e d M a x i m i z a t i o n o fU G h B

A s in S e c t i o n 6 2 , w e e x a m i n e t w o s e t s o f c o n s t r a i n t s a n d o b t a i n t h e f o l l o w i n gp r o p o s m o n s

P r o p o s i t i o n 7 2 : G i v e n t h e a s s u m p t i o n s m P r o p o s m o n 7 1 a n d t h e r e s tr i c ti o n sUG-->O a n d d x ) = a + b Y, w e m a x m l l z e U G + h B w h e n d = d * , w i t h d * g i v e nb y

(7 3 ) d * ( x ) = 1 / 2 -h + c~ + n Y ) l f l + n - 2 I - c ) ) ,

f o r h --< I / 2 , an d

(7 .4 )

fo r h > 1 /3 ..

d * x ) = (c~ + ,z Y )/ (f l + ,1 - 2 (I - c) )


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VIRGINI R YOUNG

Proof: The moment generating function of a Poisson random variable P O) is

M x l o ( t ) = E x l o l e x p { r t}1 = e x p { 0 ( e x p { t } - 1 ) } ,

a n d s i m i l a r l y

E x l o [ x e x p {x t} l = e x p { 0 ( e x p { t } - 1 )} [ 0 e x p I t II -

U s e M x l o ( t ) a n d E v l o [ -~ e x p { . ~ t }1 t o c a l c u l a t e so m e o f t h e e x p e c t e d v a l u e s mUG + h B M o r e s p e c i f i c a l l y,

E x l o [ e x p { y t}1 = m x l o ( t / n ) }

= e x p { n O ( e x p{t /n} - 1 } ,

a n d

E x l o [ 2 exp {x t } l = M x l o ( t / n ) } - l E x l o I r e x p{x t /n}]

= e x p { n O ( e x p { t / n } -I ) } 1 0 e x p { t / n } l

A l s o w e h a v e [ o r a g a l n l n a l a n d o m v a r i a b l e,O ~ G ( e ~ , 1 3 ) ,

E [ O exp {0 t}] =o~t3'~ ([3 - t ) ~ + i

A s s um e th at / : 3 > 2 ( I - c ) + n ( e x p { - J . ( b - c ) / n } - l ) , f o r a ll b ~ [0 , 1 ] .S u b s t i t u t e t h e s e e x p r e s s i o n s i n t oUG + h B , e q u a t i o n ( 6 . 8 ) , t o o b t a i n

UG + h B == d e x p { - 2 a } (a + h ) E [ e x p { 0 ( 2 ( 1 - c ) + n ( e x p { - 2 ( b - c ) /n } - l ) ) } ] +

+ d e xp { - 2 a } b E [ e x p { 0( 2. ( I - c ) + n ( e x p{ - 2 ( b - c ) /n }- 1) )} x

x [ 0 e x p { - 2 ( b - c / n } ] ] - c5 e x p { - 2 a } x

x E [ O e x p { 0 ( 2 ( 1 - c ) + n ( e x p { - 2 ( / , - c ) / n} - I ) )} ]

= d e x p { -2 a } { (a + / 7) E [ e x p { 0 ( 2 ( I - c ) + n ( e x p { - 2 ( b - c ) /, ,} - I ) ) ] +

+ ( b ex p { - 2 ( b - c ) / n } - 1) E [ O e xp { 0 ( 2 ( I - c ) +

+ ,1 ( e x p {- 2 ' ( b - c ) / n } - ~ } 1 }

( 7 5 ) = d e x p { - 2 a } / 3 V ( / 3 + n - 2 ( 1 - c ) - n e x p { - 2 ( b - c ) / n } ) ~+

x { ( a + h ) ( ~ + n - 2 ( 1 - c ) - , 1 e x p{ - 2 ( b - c ) /n } )

+ e ~ (b e x p { - 2 ( b - ~ ) / , 1 ) - 1 } .

B y s e t t i n g h = 0 m e q u a t i o n ( 7 5 ) , w e s e e t h a t

( 7 6 ) U G - > O i f a n d o n l y i f

a > - ~ ( I - b e x p{ - 2 ( b -c ) l n } ) / ( [ 3 + n - 2 ( I - c ) - n e x p{ - 2 ( b - c ) / n } )

T h e v a l u e s o f th e p a r a m e t e r s a a n d b t h a t a p p e a r md ~ ( x ) m e q u a t i o n ( 7 I ) s a t i s f yt h e i n e q u a l i t y ( 7 . 6 ) i f a n d o n l y f f h - ~ 1 /2 . T h e r e f o r e , f f 11 - 1 / 2 , t h e n t h e i n su r a n cc o m p a n y d o e s n o t e x p e c t t o l o s e m o n e y b y u s i n g t h e c r e d i b i l i t y f o r m u l ad ' ( x ) me q u a t i o n ( 7 . I )


15/18

C R E D I B I L I T Y N D P E R S I S T E N C Y 6 7

O n t h e o t h e r h a n d , ~f h > l / i t, th e n r e v o k e t h e c o n s t r a i n t a n d s e t

a = a~ = c ~ ( I - b e x p { - i t ( b - c ) / n } ) / ( / 3 + n - 2 ( l - c ) - n e x p { - i t ( b - c ) /n } ) ;tha t 1 se t U G = O . i n t h i s i n s t a n c e ,

U G + h B = h B

= h 6 e x p { - 2 a } / 3'* /( /3 + n - 2 ( 1 - c ) - n e x p { - i t ( b - c ) / n } ) '~

T h e f u n c t i o n h B a t ta i n s ~ ts m a x i m u m a t

b l = n / / 3 + n - i t I-c) ) .I n t h i s c a s e , a l = e ~ / ( f l + n - i t ( l - c ) ) ,a n d w e a r e d o n e [ ]

W e o f f e r t h e f o l l o w i n g p r o p o s i t i o n w i t h o u t p r o v J d , n g a p ro o f .

Proposition 7 . 3 : G i v e n t h e a s s u m p t i o n s m P r o p o s i t i o n 7 .1 a n d t h e r e s t r i c ti o n sB > - 0 a n d d ( x ) = a + b .r,w e m a x m v z e U G + h B w h e n d = d * , w i t h d * g i v e n b y

d * ( x ) = I / i t - h + (e~ + n Y ) / ( f l + n - it ( l - c ) ) ,

f o r 17> -- l / i t - ( In 6 ) 1 2 + o L l ( f l+ n - i t ( I - c) )- ( a / i t ) I n ( / 3 / ( / 3 + n - it ( l - c ) - , 7e x p { - i t ( h i - c ) / n } ) ) ,m w h ic h b I = n / ( f l + n - 2 ( l - c ) ) , a n d

d * (x ) = ( l n 6 ) / i t + ( c t / i t )In ( /3 / ( /3 + n - i t ( I - c ) - n exp { - i t (b , - c ) /n } ) )

+ n V / / 3 + n - i t I - c ) ) .

f o r h o t h e r w i s e [ ]

7 3 Op timize Properties of the Book o f Business

A n i n s u r an c e c o m p a n y m a y s e t g o a l s c o n c e r n i n g t h e s tr u c tt ,r e o f i t s b o o k o fb u s i n e s s W e o b t a i n t h e f o l l o w i n g l e m m a t ha t w e u se to m , m m l z e t h e f u t u re g r a n d

m e a n a n d t h e f u t u r e t o t a l v a r , a n c e

L e m m a 7 . 4 : A s s u m e th e c o n d ~t io n s m T h e o r e m 7 1 I f t h e s t ru c t u re o f t h ec o m p a n y ' s b u s i n e s s i s g , v e n b y 7 r (0 ) d u r i n g y e a r 1, a n d t h e c r e d , b i l l ty p r e m , u m f o ry e a r 2 is a + b X l , t h e n t h e e x p e c t e d s t r u c t u r e o f t h e b u s i n e s s m y e a r 2 ,s g a m m ad is tr ib u te d G ( c q f l + l - 2 ( I - c ) - e x p { - 2 ( b - c ) } )

Proof :0 , 2 0 ) o~ f p O , X l , a + b x l )J Xll0) ~ 0)dXl

oc 0 - ' e x p { - 0 [ / 3 + I - 2 ( I - c ) - e x p { - 2 ( b - c ) } ] } .I t f o l l o w s t h a t th e s t r u c t u r e p a r a m e t e r 6 ) ~s e x p c c t e d t o b e g a m m a d ~ s t r, b u te dG ( o t , / 3 + l - 2 ( l - c ) - e x p { - i t ( b - c ) } ) [ ]

Proposition 7 . 5 : G i v e n t he c o n d m o n s m L e m m a 7 .4 , t h e g r a n d m e a n i n y e a r 2 ,e x p e c t e d a t t m l e O, i s m m u m z e d w h e n t h e I n s u r e r g i v e s f u ll w e i g h t t o th ep o h c y h o l d e r ' s e x p e r i e n c e


16/18

6 8 VIRGINIA R YOUNG

P r o o f : T h e f u t u r e g r a n d m e a n I s Uo2=EoE[X, IOI=E IOI= oH [3+ I- 2 ( I - c ) - e x p { - A ( b - c ) } ) , l n w h m h 0 - - < b - < l In o rd er t o m m m u z e t h l s r n e a n ,s e t b e q u a l t o I , t h a t ~s, g i v e f u ll w m g h t t o t h e p o h c y h o l d e r ' s e x p e r i e n c e [ ]

P r o p o s i t i o n 7 . 6 : G i v e n th e c o n d m o n s m L e m m a 7 4 , th e v ar ia n c e m y e a r 2 ,expec ted a t u rne 0 , i s mlmm~zed when the in su re r g~ves fu l l we tgh t to thep o h c y h o l d e r ' s e x p e ri e nc e .

P r o o f : T h e f u t u r e t o t a l v a r m n c e ~ s

El (X 2 -1102) 2 ]= Eo [ Va r I X 2 1 0 1 ] +Var0 [EIX2]OI ]= E l 0 ] + Va r l O l = a / / 3 + 1 - 2 ( I - c ) - e x p { - 2 ( b - c )} )

+ a / l ? + I - 2 ( 1 - c ) - e x p { - 2 ( b - c )} ) 2.

T h i s v a r i a n c e ~s m m m a t z e d w h e n b e q u a l s 1 [ ]

FUTURE RESEARCH

C r e d i b i l i ty t h e o r y c o n t i n u e s t o b e a n m ~ p o r ta n t a n d d y n a t m c a r e a o f re s e a r c h ma c t u a ri a l s c m n c e a s w i t n e s s e d b y t h e r e c e n t w o r k o f NO R BE R G ( 1 9 9 2 ) a n d PA NJL Ra n d L I ( 1 9 9 4 ) A n a s p e c t o f c r e d ~b f ln y t h e o r y th a t h a s n o t b e e n c o n s i d e r e d v e r ye x t e n s i v e l y i s p e r s i s te n c y, w h i c h w e h a v e a d d r e s s e d h e r e T he ,, w o r k o n l y b e g i n s t od e a l w i t h t h e p r o b l e m o f c r e b d ~ t y a n d p e r s i s te n c y, a n d w e i n t e n d t o e x p l o r e t hi s~ ss ue f u r t h e r S o m e o f t h e o u t s t a n d i n g p r o b l e m s a r e .

Long te rm e ffec t s

W e h a v e o p t u m z e d g a i n f u n c t io n s t h at s p a n o n l y o n e y e a r o r p o h c y p e r i o d B e c a u s e

a c t u a r i e s c o n s t d e r l o n g e r l e n g th s o f t im e , i t m a y b e m o r e a p p r o p r i a t e t o c o n s i d e r t h ef o l l o w i n g o b j e c u v e s . M a x i m i z e th e p r e s e n t v a l u e o f u n d e r w r i ti n g g a i n M a x i m i z e t h e sta b d~ ty o f th e n u m b e r o f m s u r e d s ; f o r e x a m p l e , o n e c o u l d

m t m m l z e t h e c h a n g e i n t h e e x p e c t e d n u m b e r o f m s u r e d s f r o m y e a r t o y e a r O p u m t z e p r o p e m e s o f th e l o n g - t er m s t r u c tu r e o f th e b o o k o f b u s in e s s.A n o t h e r t im e e f f e c t t o m v e s u g a t e ~s t h e c h a n g e , o rtrend, m t h e n u m b e r o r a m o u n to f c la u n s f r o m y e a r t o y e a r. S u c h w o r k c o u l d f o l lo w t he m o d e l s g w e n b y K R EM ER(198 2) o r LEDOLTER, KLUGMAN and LEE (1991)

Different r i sk s izes and empir ica l s tudy

W e h ave n o t con s ide red d i ff e ren t r i sk s izes . Th t s f ac to r ~s an ~m por tan t one toi n c l u d e m f u t u r e m o d e l s b e c a u s e , m r e a li ty, p o l i c y h o l d e r s a r e n o t t h e s a m e s iz e . Inf u t u re w o r k , w e w i ll c o n s i d e r t h e B u h l m a n n - S t r a u b m o d e l ( B U H L M A N N a n dSTRAUB, 1970) and o the r m od e ls tha t a l low fo r va ry ing r i sk s i ze (VENTER, 1990) ,(GOOVAERTS and HOOGSTAD, 1987) .


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CREDIBILITY AN D PERSISTENCY 9

S u c h m o d e l s c o u l d b e u s e d t o fi t e m p t r t c a l d a t a m p r a ct ic a l r e s e a r ch , a s mK L U G M A N 1 9 9 2 ) A n e m p ~ r t ca l s t u d y m a y a l s o te s t t h e v a h d t t y o f t h e m o d e lp r o p o s e d m t h ts p a p e r w i t h t h e o n e m e n t i o n e d a t t h e e n d o f S e c t t o n 4

A C K N O W L E D G E M E N T S

1 t h a n k m y c o l l e a g u e s a n d f r i e n d s fo r g r a c t o u s l y g ~ v m g t h et r t i m e t o h e l p m eo r g a m z e m y t h o u g h t s a n d th i s p a p e r m o r e c l e a r l y. J e d F re e s, J i m H t c k m a n , a n dD a v t d L t . 1 a l s o t h a n k t w o a n o n y m o u s r e f e re e s f o r s u g g e s t m g v a l u a b l e c h a n g e s .

R EFER EN ES

BUHLMANN H (1967) Exper ienc e ra t ing ,md c red]b ih tyA S T IN B u l l e tm4, 199-207BUHL~.I,,XNN, H (1 97 0)M a th e ma tw a l M o d e l s m R t~ k T h e o ryS p rm g e r-Ve l l a g , N e w Yo rk

BUIILMANN. H and SIRAUB E (1970) G lau bwu rd lgke l t fm Schaden , , a tzeMttteHungen der Veremtgun,qSch ~ etzet t s~ her Vet ~wh erun g s-Matl~emattkel70 , I I I - 133

GLRBEr, H U (1980) . Cred ib i l i ty fo r Es , ;cher p re m m m sMlt te thcngen der Vete ln tg lcng Sch we tzer t rc herVel s~che~u n g ~ -M a th e ma n k e t8 0 . 3 0 7 -3 1 2

GOOVAI2RTS M J an d HOOGS3,XD, W J (19 87 )C re d tb th tv T h e o tTSurv eys o l Ac tuar ia l S tud ies . No 4 ,N a u o n a [e -N e d e r l a n d e n , R o t t e rd a m, T h e N e th e r{ a n d s

JEWI-:LL, W S (1974a) Cred[b lh ty i s exac t B , tyesm n for expon enh a l ta m lhe sA S T IN B u l l e t i n8 , 77-90J~-WLLL. W S (1974b) Reg ula r i ty ondmon '~ fo r exac t c red tbdHyA S T IN B u l l e t i n8, 336-341KLUGMAN, S A (19 92)Baye~tan Sta t t~tu s t i t Actt ta t ta l Se tel tce w tth Ent/~h tst l o il Cr edtb thtvK l u w e r

A c a d e m i c P u b h s h e r s , B o s t o nKREMER, E (1982) Cred~b~hty theory fo r som e evo lu t ion ary m ode lsScandtna t , ta t t Ac tuar t t t l Journa l ,

129-142

LEDOLFFR, J , KLUGMAN, S and I .LE , C S (1991) , C le d lbd l ty mo dels w i th m ne -v m ym g t rendc o m p o n e n t s ,ASTIN Bul le t in ,21 73-86

M om :woo rJ , F G (1992) P lanm ng and con t ro l ~ssues InG r o u p h t s u t a n c e(e d W F B lu h m ) , c h a p te r 3 4 ,A C T E X P u b h c a t~ o n s , W re s te d , C o n n e e u c u t

NO,BERG, R (1992) L inea r es tn 'n ,mo u and c red~b~hty m c on t in uou s t~meASTIN Bul le t in22 , 14%165PAN JE R, H H a n d L I, D X (1 9 9 4 ) C re d lb lh ty mo d e N A n e s tm m tm g tu n c t lo n a p p ro a c h W o lk m g

p a p e rSUNDT, B (1983) Cred lbd[ ty m ode ls a l lowing dura uona l e ffec tsM t t te t lu n f fe n d e t Ve te m tg u n g S c h w e t -

ze t t~Lher Ver~ tcherungs-Mathen ta t t~e t 83 ,63-87I',xVLOR, G C (1975) Cred lbd l ty unde r cond i t ions o l mlpe r lec t pe rs is ten cy InCted tb th t~ Theo ry t tnd

A p p h ~ a t to n s(ed P M Kahn ) . pp 391--400, Ac ade m ic Press , New YorkVENiER . G (199 0) Cred~b~hty InFoun dalton3 ofCa.~t~alO, Actu art t l l Scie nce ,C a s u a l ty A c tu a r ia l S o c i e ty,

N e w Yo rkWiLl.MOT, G E (19 94)h t t t o d u c to rv C te d tb th O , I h e o tyIn s h tu l e o f I n s u ra n c e a n d P e n s io n R e s e a rc h ,

Um vers~ ty o f Water loo , W ater loo , Onta r io

V I R G I N I A R . Y O U N G

School of Busme.~s G ramger HallU m v e r s t t v o f W t ~ c o n s t n - M a d t . s o nM a d t s o n , W I , U S A 5 3 7 0 6


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credibility and persistency

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