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C L A I M S R E S E R V I N G I N C O N T I N U O U S T I M E ; A N O N P A R A M E T R I CB A Y E S I A N A P P R O A C HS V EN D H A A S T R U P A N D E L J A A R J A S

Laboratory o f Actuarial Mathem atics, Universitetsparken 5,2100 Kobenhavn O, DenmarkDepartment o f Ap plied Mathematics and Statistics, University o f Oulu,Linnanmaa 90570, Oulu , FinlandA B S T R A C T

O c c ur r e nc e s a nd de ve lop m e n t s o f c l a ims a r e mode l l e d a s a ma r ke d po in t p r oc e s s . T heind iv idua l c l a im c ons i s t s o f a n oc c u r r e nc e t ime , tw o c o va r i a t e s , a r e po r t i ng de l a y , a nda p r oc e s s de sc r ib ing pa r t ia l pa ym e n t s a nd se t t le m e n t o f t he c la im . U n de r c e r t a inl i ke l ihood a s sum pt ions t he d i s t ri bu t ion o f t he p r oc e s s i s de sc r ibe d by 14 one -d ime ns iona l c ompone n t s . T he mode l l i ng i s nonpa r a me t r i c Ba ye s i a n . T he pos t e r io rd i s t ri bu t ion o f t he c omp one n t s a nd the pos t e r io r d is t ri bu t ion o f t he ou t s t a nd ing I B N Ra nd R BN S l ia b i li ti e s a re f ound s imu l t a ne ous ly . T he m e thod i s a pp l i e d to a po r t f o l i o o facc ident insurances .

K E Y W O R D SI B N R a n d R B N S l i a b i l i t i e s , m a r k e d p o i n t p r o c e s s , m i s s i n g d a t a , M a r k o v c h a i nM o n t e C a r l o .

I . I N T R O D U C T I O NA m a jo r i s sue i n non - l i f e in su r a nc e is p r e d i c t i on o f ou t s t a n d ing l i a b il i ti e s . O u t -s t a nd ing l i a b i l i t i e s a r e t r a d i t i ona l ly d iv ide d in to oc c u r r e d bu t no t r e po r t e d ( I BN R)c l a im s a n d r e p o r t e d b u t n o t s e t tl e d ( R B N S ) c l a im s . A t e a c h t im e t h e i n s u r a n c e c o m p a n yha s t o p r e d i c t t he ou t s t a nd ing l i a b il i ti e s a nd p r ov ide a r e se r ve c o r r e sp ond ing ly .A v a s t n u m b e r o f a rt i cl e s h a v e b e e n w r i tt e n o n t h e s u b j e ct . I n m o s t m o d e l s t h eda t a a r e a s sume d to be d i s c r e t i z e d . A r j a s ( 1989 ) , J e w e l l ( 1989 ) , N or be r g ( 1993a , b , c )a n d H e s s e l a g e r (1 9 9 4 ) m o d e l i n c o n t i n u o u s t i m e . N o r b e r g ( 1 9 9 3 a ) d e s c r i b e s o c c u r -r e n c e an d d e v e l o p m e n t o f t h e c l a i m s b y a m a r k e d P o i s s o n p r o c e ss . I n N o r b e r g( 1 9 9 3 c ) a n d K i r k e g a a r d ( 1 9 9 4 ) d i f f e r e n t p a r a m e t r i c s p e c i f ic a t i o n s o f th e m o d e l a r ec o n s i d e r e d , a n d r e a l i n s u r a n c e d a t a a r e a n a l y z e d . F u r t h e r m o r e , N o r b e r g ( 1 9 9 3 a )c o n s i d e r s a n e x t e n d e d m o d e l w h e r e t h e o c c u r r e n c e i n t e n s i ty i s a s s u m e d t o b e a s t o -c ha s t i c p r oc e s s , a nd he f i nds t he be s t l i ne a r p r e d i c to r o f the ou t s t a nd ing l i a b il i ti e s .T h e p r e s e n t p a p e r d e a ls , b y w a y o f a c a s e - s tu d y , w i th a p o r t f o l i o o f a c c i d e n ti n s u ra n c e s. T h e m o d e l u s e d i s c l o s e t o th a t o f N o r b e r g ( 1 9 9 3 a ). T h e c l a i m s p r o c e s sg e n e r a t i n g o c c u r r e n c e s , c o v a r i a te s a n d d e v e l o p m e n t s o f t h e c la i m s i s m o d e l l e d a s am a r k e d P o i s s o n p r o c e s s . O u r a p p r o a c h t o e s t i m a t i o n a n d p r e d i c t i o n i s n o n p a r a -ASTIN BULLETIN,Vol. 26, No. 2, 1996, pp. 139-164


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140 SVEND HAASTRUP AND ELJA ARJASm e t r i c B a y e s i a n . A d o p t i n g t h e m e t h o d s o f A r j a s a n d G a s b a r r a ( 1 9 9 4 ) , t h e d is t ri b u -t io n g o v e r n i n g t h e p r o c e s s o f o c c u r r e n c e s , c o v a r i a t es a n d d e v e l o p m e n t s i s m o d e l l e dby p i e c e w ise c ons t a n t c ond i t i ona l i n t e ns i t i e s . T he in t e r va l s on w h ic h the i n t e ns i t i e sa r e c ons t a n t , a nd the va lue s ( l e ve l s ) o f the i n t e ns i t i e s on the d i f f e r e n t i n t e r va l s , a rethe n v i e w e d a s mode l pa r a me te r s . I n p r inc ip l e , suc h a pa r a me te r spa c e i s o f i n f in i t ed im e ns ion . A p r io r d i s t ri bu t ion ( a d i s tr i bu t ion on the pa r a m e te r spa c e ) is a t ta c he d tothe i n t e ns i t i e s o f the c l a im s p r oc e s s .B o t h t h e m o d e l p a r a m e t e r s a n d t h e o u t s t a n d i n g l i a b i l i t i e s ( R B N S a n d I B N Rc l a i m s ) a r e u n k n o w n . T h e p r o b l e m is t o f in d t h e c o n d i t i o n a l d i s tr i b u ti o n o f su c hunknow ns g ive n the obse r va t ions . T h i s d i s t r i bu t ion w i l l be c a l l e d t he pos t e r io r ; i tc o v e r s t h e c o n d i t io n a l d i s t ri b u t io n o f t h e u n k n o w n p a r a m e t e r s w h i c h b y s t a n d a r dusa ge is c a l l e d pos t e r io r , a nd the c ond i t i ona l d i s t r i bu t ion o f f u tu r e obse r va b le sw h ic h by s t a nda r d u sa ge i s c a l l e d p r e d i c t i ve d i s t r i bu t ion . I n c omple x mode l s , i t i so f l : e n d i f f i c u l t t o i de n t i f y t he pos t e r io r . T he pos t e r io r c a n a lw a ys be de t e r mine d upto p r opo r t i on a l i t y , bu t i t c a n be d i f f i c u l t t o no r m a l i z e , w h ic h is ne c e s s a r y e .g . f o r t hec a l c u l a t io n o f m e a n s . R e c e n t l y , a t e c h n i q u e c a l le d M a r k o v c h a i n M o n t e C a r l o( M C M C ) i n t e g ra t io n h a s b e e n u s ed t o s o l v e th i s p r o b l e m n u m e r i c a l l y in c o n n e c t i o nw i th l a r ge s t a t is t i c a l mode l s . A g e ne r a l r e v i e w o f t he t op i c c a n be f ound in S mi tha n d R o b e r t s ( 1 9 9 3 ) . T h e i d e a is to g e n e r a t e a M a r k o v c h a i n w h i c h h a s t h e p o s t e r i o rd i s t r i bu t ion a s i t s e qu i l i b r ium d i s t r i bu t ion . U s ing suc h a c ha in , a l l qua n t i t i e s o fi n te r e st c a n b e e s t i m a t e d / p r e d i c te d . F o r e x a m p l e , a t e a c h s t e p o f th e c h a i n a n e wv a l u e o f t h e R B N S c l a i m s i s s a m p l e d ; t h e e m p i r ic a l d i s tr i b u ti o n o f t h e s e s a m p l e dR B N S c l a im s t h e n c o n v e r g e s t o w a r d s t h e p r e d i c t i v e ( p o s t e ri o r ) d i s tr i b u ti o n o f t h eR B N S c l a i m s .S e c t i o n 2 b e l o w d e s c r i b e s t h e d a ta . A c l a i m i s d e s c r i b e d b y a n o c c u r r e n c e t i m e ,t w o c o v a r i a t e s , a n d a d e v e l o p m e n t . T h e d e v e l o p m e n t c o n t ai n s a r e p o r t in g d e l a y , as e t t l e m e n t d e l a y , a n d a p a r t i a l p a y m e n t p r o c e s s c o n t a i n i n g t h e p a r t i a l p a y m e n t sm a d e f r o m r e p o r t i n g t o t h e s e t tl e m e n t o f th e c l a im .I n S e c t i o n 3 d i s t ri b u t io n a l a s s u m p t i o n s a r e m a d e . C l a i m s a r e a s s u m e d t o o c c u r ina c c o r d a n c e w i t h a P o i s s o n p r o c e s s , a n d c o v a r i a t e s a n d d e v e l o p m e n t s a r e m o d e l l e das m a r k s a s s o c i a te d w i t h t h e o c c u r r e n c e s . T h e d i s t r ib u t i o n o f b o t h o c c u r r e n c e s a n dma r ks i s spe c i f i e d by p i e c e w ise c ons t a n t c ond i t i ona l i n t e ns i t i e s , a nd a p r io r d i s -t r i bu t ion o f t he se i n t e ns i t ie s i s c hose n .

S e c t i o n 4 d e s c r i b e s t h e M C M C a l g o r i t h m ( s a m p l i n g a l g o r i t h m ) . A M a r k o v c h a i nw i th t he de s i r e d p r ope r t i e s i s ge ne r a t e d . T he a lgo r i t hm i s c lo se t o t he one de sc r ibe db y A r j a s a n d G a s b a r r a ( 1 9 9 4 ) .S e c t i o n 5 d e s c r i b e s t h e e s t i m a t i o n s a n d p r e d i c t i o n s . U s i n g t h e M a r k o v c h a i ng e n e r a t e d i n t h e s a m p l i n g a l g o r i t h m w e a p p r o x i m a t e b o t h t h e d is t ri b u t io n o f t h ec l a im s p r oc e s s a nd the d i s t r i bu t ion o f t he ou t s t a nd in g l i ab i l it i es .

2. THE DATAT h e da t a a r e a po r t f o l i o o f a c c ide n t i n su r a nc e s . I n t he f o l l ow ing w e f i r s t de sc r ibe t hege ne r a l s t r uc tu r e o f t he da t a , a nd the n g o on w i th a de t a i l e d de sc r ip t i on o f t hep r e se n t da t a s e t .


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CLAIMS RESERVING IN CONTINUOUS TIME 1412 . 1. S t r u c t u r e o f t h e d a t aW e u s e th e s e t -u p o f N o r b e r g ( 1 9 9 3 a ). B y a c l a i m w e u n d e r s t a n d a c o m b i n a t i o n o fa t i m e o f o c c u r r e n c e , a s e t o f c o v a r i a t e s , a n d a d e v e l o p m e n t . L e t t i m e 0 b e t hei n i ti a l t i m e , a n d l e t T b e t h e t im e a t w h i c h I B N R a n d R B N S l i a b i l it i e s a r e t o b ep r e d i c t ed . E a c h i n d i v id u a l p o l i c y i s d e s c r i b e d b y t h e c o v a r i a t e s ( s, a ) d e n o t i n g s e xa n d a g e o f t h e p o l i c y h o l d e r . F o r e a c h c o m b i n a t i o n o f c a l e n d a r t i m e t in (0 , 7 -] , s e x sin { m a l e , f e m a l e } , a n d a g e a in { 1 , 2 , . . . } , t h e n u m b e r o f p o l i c i e s c o v e r e d i sd e n o t e d

w ( t , s , a ) .T h e f u n c t i o n w i s c a l l e d t h e e x p o s u r e r a t e .U s i n g t h e n o t a t io n o f N o r b e r g ( 1 9 9 3 a ) t h e d e v e l o p m e n t o f a c l a i m c a n b ed e s c r i b e d b y

( V , ( Y ( v ) ) v ~ [ 0 ,~ ) ,w h e r e U is t h e w a i t i n g t i m e f r o m o c c u r r e n c e u n ti l n o t i f i c at i o n ( t h e r e p o r t i n g d e l a y ) ,V i s t h e w a i t i n g t i m e f r o m n o t i f i c a t i o n u n t il f i n a l s e t t l e m e n t , a n d Y(v) i s t h e a m o u n tp a i d v t i m e u n i t s a f t e r n o t i f i c a t i o n . T h e f i n a l c l a i m a m o u n t i s Y(V ), w h i c h i s c a l l e d Yf o r s h o r t. M o r e o v e r , w h e n a c l a i m i s r e p o r t e d , t h e o c c u r r e n c e t i m e T a n d t h ec o v a r i a t e s s e x a n d a g e ( S , A ) a r e k n o w n , a g e b e i n g t h e a g e a t t h e t i m e o f o c c u r r e n c e .S o t he c o m p l e t e d e s c r i p t i o n o f a c l a i m i s

( T , S , A , U , X ) w h e r e , f o r s h o r t , X = ( Y ( v ) ) v [ 0 , v ].T h e p a r t i a l p a y m e n t s p r o c e s s X = ( Y ( v ) ) ~ . [0 v] c o n s i s t s o f a s e r i e s o f l u m pp a y m e n t s . A n i l l u s t r a t i o n c a n b e f o u n d i n N o r b e ' r g ( 1 9 9 3 a ) , F i g u r e 2 .

N o t a l l c l a i m s w h i c h o c c u r r e d b e f o r e t i m e T a r e a c t u a ll y o b s e r v e d . A t t i m e 7- w eh a v e o n l y o b s e r v e d t h e r e p o r t e d c l a i m s , i .e . c l a i m s w i t h T + U _< r , a n d f o r e a c h o ft h e s e w e o n l y k n o w t h e d e v e l o p m e n t u p t o t i m e r . T h i s m e a n s t h a t f o r a r e p o r t e dc l a i m w e a l w a y s k n o w ( T , S , A , U ). F u r t h e r m o r e w e o b s e r v e ,

f o r a s e t t l e d c l a i m ,f o r a r e p o r t e d b u t n o t s e t t l ed c l a i m ,

( V , ( Y ( v ) ) , , e [ o , v ] ) , a n d ( 2 . 1 )( l ( v < T - T - U) = 0 , ( Y ( v ) ) v [0 ,r - T - UI) '

( 2 . 2 )w h e r e l ( v < r - T - U ) = 0 i n d ic a t e s t h a t t h e c l a i m i s n o t s e t tl e d . N o t e t h a t t h e a b o v ei s j u s t t h e p a rt i a l p a y m e n t p r o c e s s , ( Y ( V )) v e[ 0 , E , c e n s o r e d a t c a l e n d a r t i m e 7 -.2 .2 . T h e p r e s e n t d a t a s e tT h e d a t a a r e a p o r t f o l i o o f a c c i d e n t i n s u r a n c e s, s u p p l i e d b y a D a n i s h i n s u r a n c ec o m p a n y . T h e r e a r e f o u r d i f f e r e n t k i n d s o f c l a i m s ; d e n t a l c l a i m s , s p e c t a c l e s c l a i m s ,d i s a b i l i t y c l a i m s , a n d d e a t h c l a i m s . W e h a v e c h o s e n t o m o d e l t h e d e n t a l c l a i m s , a n dl o o k at l e is u r e t i m e c o v e r o n l y . C l a i m s w h i c h o c c u r r e d b e t w e e n J a n u a r y 1, 19 8 2 a n dD e c e m b e r 3 1 , 1 9 9 0 a n d w h i c h w e r e r e p o r t e d b e f o r e M a r c h 3 , 1 9 9 2 a r e o b s e r v e d a n dc o n t a i n e d i n t h e d a t a s e t . H e r e w e h a v e c h o s e n t o c o n s i d e r r = 6 y e a r s . T h i s m e a n s


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142 SVEND HAASTRUP AND ELJA ARJASt h a t o n l y d a t a r e c o r d e d b y D e c e m b e r 3 1 , 1 98 7 a r e c o n s i d e re d . T h e r e s t o f t h e d a tac a n b e u s e d t o c h e c k t h e v a l id i t y o f o u r m o d e l .S o m e m o d i f i c a t i o n s a r e n e e d e d t o g i v e t h e d a t a t h e d e s i r e d s t r u c t u r e . T h ee x p o s u r e i s o n l y k n o w n b y y e a r s , n o t d a y s . A n e s t i m a t e o f t h e e x p o s u r e r a t e w iso b t a i n e d b y i n t e r p o l a t i o n . I n f o r m a t i o n a b o u t c l a i m d e v e l o p m e n t s i s a l m o s t a s w en e e d t h e m . S o m e o f t h e c la i m s r e c o r d s d o n o t c o n t a i n i n f o r m a t i o n a b o u t t hec o v a r i a t e s , a n d f o r s i m p l i c i t y s u c h c l a i m s r e c o r d s ( a s w e l l a s e x p o s u r e ) w e r er e m o v e d . F u r t h e r m o r e , t i m e s o f o c c u r r e n c e , n o t i f ic a t io n , p a r t ia l p a y m e n t s a n ds e t t l e m e n t s a r e o n l y k n o w n b y d a y s . H o w e v e r , w e s h a l l m o d e l o c c u r r e n c e s a n dd e v e l o p m e n t s i n c o n t i n u o u s t i m e , a n d v i e w t h e r e p o r t e d t im e s o f o c c u r r e n c e ,n o t i f i c a t i o n , e t c . a s a p p r o x i m a t i o n s t o t h e t r u e v a l u e s . F i n a l l y , w e h a v e c h o s e n t om o d i f y t h e o b s e r v e d p a r t i a l p a y m e n t p r o c e s s e s . L o o k i n g a t a p a r t i a l p a y m e n tp r o c e s s , t w o p a y m e n t s m a d e d u r i n g t h e s a m e d a y o n t h e s a m e c l a i m a r e l u m p e dt o g e t h e r a n d v i e w e d a s a si n g l e p a y m e n t . F u r t h e r m o r e , a b o u t 2 % o f th e p a r ti a lp a y m e n t s w e r e n e g a ti v e . S o m e o f th e s e h a d a c o rr e s p o n d i n g p o s i ti v e p a y m e n t o nt h e s a m e d a y , a n d i n t h a t c a s e b o t h t h e s e p a y m e n t s w e r e d i s r e g a r d e d . N e g a t i v ep a y m e n t s t h a t d i d n o t h a v e a c o r r e s p o n d i n g p o s i t i v e p a y m e n t n e v e r e x c e e d e d t h ea c c u m u l a t e d c l a i m a m o u n t p a id , a n d t h e y w e r e s e t o f f a g a i n s t th e p r e v i o u s p o s i t i v ep a y m e n t s . F o r e x a m p l e , a r e c o r d e d p a r t i a l p a y m e n t p r o c e s s c o n t a i n i n g t w o p a y -m e n t s , o n e o f 1 2 0 0 D K K a t J a n u a r y 5 , 1 98 5 a n d o n e o f - 5 0 0 D K K a t J a n u a r y 16 ,1 9 8 5 , i s t r a n s f o r m e d i n t o a p a r t i a l p a y m e n t p r o c e s s c o n t a i n i n g o n l y o n e p a r t i a lp a y m e n t o f 7 0 0 D K K m a d e a t J a n u a r y 5 , 1 98 5. L u m p i n g p a y m e n t s th a t a r e m a d ed u r i n g t h e s a m e d a y is e a s y t o j u s t i f y , b u t t o s e t o f f n e g a t i v e p a y m e n t s a g a i n s tp r e v i o u s p o s i ti v e p a y m e n t s re q u ir e s s o m e c o m m e n t s . W e h a v e c h o s e n D e c e m b e r3 1 , 1 98 7 a s t h e ti m e o f p r e d i c t io n . H o w e v e r , i n o u r m o d i f i c a t i o n o f t h e o b s e r v e dp a r t i a l p a y m e n t s p r o c e s s e s w e i n c l u d e d a l l p a r t i a l p a y m e n t s m a d e b e f o r e M a r c h 3 ,1 99 2. T h i s m e a n s t h a t w e u s e n e g a t i v e p a y m e n t s i n th e f u t u re t o m o d i f y p a y m e n t s i nt h e p a s t. A n i n s u r a n c e c o m p a n y c a n n o t d o th a t as it d o e s n o t k n o w a b o u t p a y m e n t si n t h e f u t u r e . T h e r e f o r e , i n s u r a n c e c o m p a n i e s w i t h r e c o r d e d n e g a t i v e p a y m e n t s w i l lt e n d t o s l ig h t l y o v e r e s t i m a t e t h e o u t s t a n d i n g l i a b il i ti e s i f t h e y u s e o u r m o d e l .A f t e r t h e s e m o d i f i c a t i o n s , o u r d a t a c o n t a i n 4 3 4 . 0 0 0 e x p o s u r e y e a r s . T h e r e a r e2 8 0 6 r e p o r t e d c l a i m s ; o f t h e se 2 1 9 1 a r e se t tl e d a n d 6 1 7 u n s e t tl e d . T h e r e a r e 3 7 1 8o b s e r v e d p a r t i a l p a y m e n t s , a n d t h e y a d d u p t o 1 0 . 0 4 0 . 0 0 0 D K K .

3. DISTRIBUTIONAL ASSUMPTIONSW e m o d e l t h e d is t ri b u t io n o f a ll c l a i m s w h i c h o c c u r i n o u r p o r t fo l i o a n d t h e i r fu l ld e v e l o p m e n t . L e t

T i , S i , A i , U i , . ~ i ) i > _ ] (3 .1)d e n o t e t h e s e c l a im s . I t i s im p o r t a n t to n o t e t h a t m a n y o f th e c l a i m s ( 3 .1 ) a r e n o to b s e r v e d c o m p l e t e l y , a n d s o m e o f t h e m n o t a t a ll. A s m e n t i o n e d in th e p r e v i o u ss e c t i o n , w e o n l y o b s e r v e t h e r e p o r t e d c l a i m s a n d t h e i r d e v e l o p m e n t i s c e n s o r e d a tc a l e n d a r t i m e T .


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CLAIMS RESERVING IN CONTINUOUS TIME 143A Ba ye s i a n mode l i s u se d . T he mode l l i ng i s done in tw o s t e ps . F i r s t , t hed i s t r i bu t ion o f t he c l a ims p r o c e s s ( 3 . 1 ) is de sc r ibe d by in t e ns i t i e s , he nc e f o r th

r e f e r r e d t o a s components . O u r m o d e l w i ll h a v e 1 4 s u c h c o m p o n e n t s . S o m e d e s c r ib et h e d i s tr i b u ti o n o f t he o c c u r r e n c e s a n d c o v a r i a te s , s o m e d e s c r ib e t h e d i s t r ib u t i o n o fthe r e po r t i ng de l a ys , a nd som e the d i s t ri bu t ion o f t he pa r ti a l pa y m e n t s p r oc e s se s .T he n , a p r io r d i s t r i bu t ion i s c hose n . T he in t e ns i t i e s ( c ompone n t s ) a r e a s sume d to bep ie c e w ise c ons t a n t . T he in t e r va l s on w h ic h the i n t e ns i t i e s a r e c ons t a n t , a nd theva lue s ( l e ve l s ) o f t he i n t e ns i t i es on the d i f f e r e n t i n t e r va l s ar e t he pa r a m e te r s . A p r io rd i s t r i bu t ion ( a d i s t r i bu t ion on the pa r a me te r spa c e ) i s a t t a c he d to t he i n t e ns i t i e s .T h e t w o s t e p s a r e d e s c r i b e d b e l o w . T h e f i r s t s t e p c o n t a i n s ' l i k e l i h o o d a s s u m p -t i o n s ' , a n d h e r e t h e m a j o r r e s t r i c t i o n s a r e m a d e . T h e s e c o n d s t e p c o n t a i n s ' p r i o ra s s u m p t i o n s ' . S i n c e w e h a v e o n l y l i tt le k n o w l e d g e a t h a n d , w e t r y t o a dd o n l y l i tt les t r uc tu r e i n ou r c ho ic e o f p r io r .3 . 1 . L i k e l i h o o d a s s u m p t i o n sA t c a l e nd a r t ime t t he r e a r e w ( t , s , a ) po l i c i e s w i th s e x s a nd a ge a i n the po r t f o l i o .W e a s s u m e t h a t f o r a n i n d i v i d u a l p o l i c y , c l a i m s o c c u r a c c o r d i n g t o a P o i s s o np r oc e s s w i th i n t e ns i t y

[ ( t , s ,a ) , t E (0 ,7" ]( a g e c h a n g i n g o n c e a y e a r) . A s a c o n s e q u e n c e w e g e t th a t, a m a l g a m a t i n g a ll p o l i c ie si n o u r p o r t f o l i o , t h e o c c u r r e n c e t i m e s a n d c o v a r i a t e s f o l l o w a m a r k e d P o i s s o np r oc e s s w i th i n t e ns i t y

w ( t , s , a ) f ( t , s , a ) , ( t , s ,a ) E ( 0 , T] { m a l e , f e m a l e } x { 1 , 2 , . . . } .I t is a s sum e d tha t t he i n t e ns i t y f c a n be w r i t t e n a s

f ( t , s , a ) = f l ( t ) f z ( s , a ) . 3 . 2 )I n t hi s w a y t h e d i s tr i b u ti o n o f o c c u r r e n c e s a n d c o v a r i a t e s is d e s c r i b e d b y t h r e ec o m p o n e n t s ; a c a l e n d a r ti m e e f f e c t f l , a n a g e e f f e c t f o r m a l e s f 2 ( m a l e , .), a n d a n a g ee f f e c t f o r f e m a l e s f 2 ( f e m a l e , -).T h e d e v e l o p m e n t o f a c la i m c o n t a i n s a r e p o r t in g d e l a y a n d a p a rt ia l p a y m e n tp r o c e s s. F o l l o w i n g N o r b e r g ( 1 9 9 3 a ) , w e a s s u m e t h at t h e d i s t ri b u t io n o f t h e d e v e l -o p m e n t o f a c l a im d e p e n d s o n t h e p a s t h i s t o r y o f t h e p r o c e s s o n l y t h r o u g h t h ea s s o c ia t e d o c c u r r e n c e t i m e a n d c o v a r i a te s . T h i s k i n d o f d e v e l o p m e n t d i s tr i b u t io n i sc a l l e d posit ion dependent, se e K a r r (1991 ) . I n t he f o l l ow ing the d i s t ri bu t ion o f t hed e v e l o p m e n t o f a c la i m , w h i c h o c c u r r e d a t t im e t a n d h a s c o v a r i a t e s (s ,a) , isd e s c r i b e d b y t h e d i s tr i b u ti o n o f th e r e p o r t in g d e l a y U, P u I t , s , a , a nd the d i s t r i bu t ionPxl,, ,t ,s ,. o f the pa r t ia l p ay m en t proce ss X g ive n U = u .T he d i s t r i bu t ion P u I t , s , a i s a s sume d to be a bso lu t e ly c on t inuous ( w i th r e spe c t t ot h e L e b e s g u e m e a s u r e ) . I t i s m o d e l l e d b y t h e c o r r e s p o n d i n g h a z a r d r a t e

gflt,s,~(u) = g(u), (3 .3)w h i c h is a ss u m e d i n d e p e n d e n t o f b o t h t h e t i m e o f o c c u r r e n c e a n d t h e c o v a r i at e s ;


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144 SVEND HAASTRUP AND ELJA ARIASt ha t a p r obab i l i t y P on 'R .+ ha s haza r d r a te g m eans tha t g u ) d u = P d u ) / P u , c x z ) ) .T her e a r e many poss ib l e ex t ens ions . For example , we cou ld a s sume tha t P u l t , s , a hashaza r d r a t e

g u I t, s ,a u ) = gl (s , u ) g 2 s , a ) g 3 t ) ,con ta in ing 5 componen t s . Bo th sexes have then the i r i nd iv idua l r epor t i ng de l ayc o m p o n e n t a n d a g e c o m p o n e n t , a n d t h e n t h e re is a c a le n d a r ti m e c o m p o n e n t . S u c han ex t ens ion would cause no ma thema t i ca l p r ob lems , bu t t he computa t iona l e f f o r twould inc r ea se .T he p a r ti a l pay m ent p r ocess X = (Yv)v~I0 ,v ] a r i s ing f r om a s ing le c l a im i s aj u m p p r o c e s s . L e t

7 - t , , =

be the h i s to r y o f a r epor t ed bu t no t se t t led c l a im; h e r e v i s the t ime s ince the c l a imwas r eport ed . L e t dy deno te a sm a l l i n te r va l o f l eng th dy a round y . T he d i s t ri bu t iono f X i s descr ibe d us in g in tens i t ies . L e t hse(V [ 7" (v-) be the in tens i ty of se t t l ing a tt ime v w i thou t a pa r t ia l pa ym ent a t t im e v , le t h s e p v , d y [ ~ - ) b e t h e i n t en s i ty o fse t t li ng a t t ime v w i th pa r t ia l p aym ent o f s iz e dy , and f ina l ly le t h p v , d y ] ~ _ ) bethe in t ens i ty a t t im e v o f hav ing a pa r t i al pay m ent o f s i z e dy w i thou t se t t lemen t . W ehave to dec ide ho w the in t ens i ti e s hse , hsep and hp depend o f t he h i s to r y ~ _ jus tbefore t ime v .

T h e f o l l o w i n g i n fo r m a t i o n c an b e d e r iv e d f r o m H ~ _ :N v n u m b e r o f p a r t i a l p a y m e n t s i n [0 , v ),T v t im e s ince the l a t e s t pa r t i a l pa y m en t i f an y , e l se Tv = v .

We a ssume tha t t he i n t ens i ty o f se t t l i ng on ly i shse v) i f Nv = 0h s~ V [ 7-/,,_) = h ~ e T v i f Nv > 0 , (3 , 4 )

t he i n t ens i ty o f se t t li ng w i th a pa r ti a l pay m ent o f si z e dy i s{ o ohsep(V, d y I 7-t~_) = h s e p V ) P s e P y ) d Y l i f Nv = 0 (3.5)h s e p T v ) P s e p y ) d y if N~ > O,and the i n t ens i ty o f hav ing a pa r t ia l pay m ent o f s i z e dy w i thou t se t t l ement i s

f h ~ v ) p ~ y ) d y i f N ~ = 0h p v , d y [ ~ _ ) = ~ h l p T v ) p ~ , y ) d y i fN~ > 0 . (3.6)As a consequence , pa r ti a l pay m ent s ar e d i s t ri bu t ed accor d ing to the dens i t ie s 0Psep,P ~ e p , p O and p l . T hese w i ll be de sc r ibed by the i r co r r e spond ing haza r d r a te s deno tedq s e p , q ~ e p , q0p Pand q~ .

T he a s sum pt ions ( 3 .4 ) -( 3 .6 ) need som e com m ent s . T he d i s t r i bu t ion o f the pa rt ia lp a y m e n t p r o c es s P O u , t , s , a iS a s sum ed to be i ndepend en t o f bo th t he t ime o f occur -


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CLAIMS RESERVING IN CONTINUO US TIME 145r e n c e t , t h e r e po r t i ng d e l a y u a nd the c ova r i a t e s ( s , a ) . T he in t e ns i t i e s o f t he pa r t i a lp a y m e n t p r o c e s s d e p e n d o n t h e p a s t h i s to r y o f th e p r o c e s s o n l y t h r o u g h t h e n u m b e ro f p a r ti a l p a y m e n t s ( N v ) a n d t h e t i m e s i n c e l a t e s t p ar ti a l p a y m e n t i f a n y , e l s e t h et ime s inc e no t i f i c a t i on . T he se a s sumpt ions a r e no t c r uc i a l , a nd a t t h i s s t a ge t hem o d e l h a s m a n y p o s s i b l e e x t en s i o n s . F o r e x a m p l e , i t c o u l d b e r e a s o n a b l e t o a s s u m et h a t t h e s i z e o f a p a rt ia l p a y m e n t d e p e n d s o n t h e s i z e s o f t h e p r e v i o u s p a r t ia lp a y m e n t s . A n d o f t e n i t i s r e a s o n a b l e t o a s s u m e t h a t t h e s i z e o f a p a y m e n t a l s od e p e n d s o n t h e t i m e s i n c e t h e l a t es t p ar t ia l p a y m e n t ( i f a n y ) . S u c h e x t e n s i o n s c a n b ed o n e w i t h o u t m a t h e m a t i c a l d i f f ic u l t ie s , b u t t h e c o m p u t a t i o n a l e f f o r t w o u l d i n c re a s e .F o r e x a m p l e , m a k i n g t h e s i z e o f a p a r ti a l p a y m e n t d e p e n d e n t o n t h e c u m u l a t e da m o u n t o f th e p r e v i o u s p a y m e n t s , Y v - ) , c a n be done a s suming tha t t he i n t e ns i t i e s

l a nd I ha ve the s t r uc tu r eqsep qpq ~ e p ( Y l ~ _ ) _ _ la Ibs e p Y ) q s e p Y v - ) )

a n dq ~ Y [ ~ v - ) = q ~ a y ) q ~ b y v _ ) ) .

E v e n t h e p r o d u c t a s s u m p t i o n s m a d e h e r e a r e n o t t h a t c r u c i a l .

3 . 2 . P r i o r a s s u m p t i o n sI n t h e p r e v i o u s s u b s e c t i o n t h e d i s t ri b u t io n o f t h e c l a im s p r o c e s s is d e s c r i b e d b y 1 4c o m p o n e n t s ( i n t e n s i t i e s ) . T h e s e a r ef l , f 2 ( m a le , . ), f 2 ( f e m a le , . ) , g ,

0 0 0 0 0 I 1 I 1 Ihsep, qsep, h p, qp, hse , hsep, qsep, h p, qp, hse .N o w a p r i o r d i s t r i b u t i o n i s t o b e c h o s e n . T h e i n t e n s i t i e s a r e a s s u m e d p i e c e w i s ec ons t a n t . T he in t e r va l s on w h ic h the i n t e ns i t i e s a r e c ons t a n t , a nd the va lue s ( l e ve l s )o f t he i n t e ns i t i e s on the d i f f e r e n t i n t e r va l s a r e t he pa r a me te r s . A p r io r d i s t ri bu t ion ( ad i s t r i bu t ion on the pa r a me te r spa c e ) i s a t t a c he d to t he i n t e ns i t i e s . T he ma in i de a i st a k e n f r o m A r j a s a n d G a s b a r r a ( 1 9 9 4 ) . F o l l o w i n g t h e i r n o t a t i o n , w e s h a l l d e n o t eunknow n pa r a me te r s by L a t in l e t t e r s a nd pa r a me te r s i n t he p r io r d i s t r i bu t ions , t hes o - c a l le d h y p e r p a r a m e t e r s , b y G r e e k l e tt e rs .T o b e g i n w i t h w e l o o k a t t h e c a l e n d a r t im e e f f e c t f l . I t is a s s u m e d t o h a v e ap i e c e w i s e c o n s t a n t s t r u c t u r e

f l t ) = ~ _ j I s j< t < s /+ , ) b j, ( 3 . 7 )j > 0

w h e r e b j , g i v e n ( b 0 , . . . , b j - i ) , f o l lo w s a I o g n o rm a l d is t ri b u ti o n w i t h p a r a m e t e rs( I og ( b . l ) , o -~ .) de n o t ing the m e a n a nd the va r i a nc e i n t he a s soc i a t e d no r m a l d i s -J - ft . . . .t r i bu t ion , a nd 0 = S o < S~ < $2 < . . . f o l l ow s a P o l s so n p r oc e s s w i th i n t e ns i t y AA .I n sho r t , w e w r i t e

b j l ( b 0 , . . . , b j - , ) ~ - L o g N ( l o g ( b j _ i ) , a} , ) , S i ) i _ > o ~ P o i s s o n ( A f , ) . ( 3 . 8 )


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146 S V E N D H A A S T R U P A N D E L J A A R J A SFor un iqueness i t i s a ssumed tha t the in i t ia l l eve l b0 i s 1 .

T he p r io r s t r uc tu r e ( 3 . 7 ) - ( 3 . 8 ) i s e s se n t i a l l y u se d to mode l a l l 14 c ompone n t s . Ap r i o r s h o u l d r e f l e c t t h e k n o w l e d g e o n e h a s a b o u t t h e p r o b l e m a t h a n d , a n d i t ist h e r e f o r e r e a s o n a b l e t o d i s c u ss w h a t k i n d o f p r io r i n f o r m a t i o n t h e s t ru c t u r e ( 3 .7 ) -( 3 . 8 ) w i l l r e p r e se n t . T o sa ve no t a t i on w e d r op the subsc r ip t f l on t he hyp e r pa r a -me te r s .

T h e i n t e n s i t y f j is b y (3 . 7 ) a s s u m e d t o b e a p o s i ti v e s im p l e f u n c t i o n (a p i e c e w i s ec o n s t a n t f u n c t i o n ) . T h e p r i o r d i s tr i b u ti o n o f f l s h o u l d t h e r e f o r e b e a d i s tr i b u ti o n o nthe spa c e o f pos i t i ve s im p le f unc t ions . T he p r io r ( 3 .8 ) i s a pos s ib l e c ho ic e . T h e p r io re xp e c t e d nu m be r o f c ha n ge s i n l e ve l pe r ye a r is A , a nd the l e ve l s ha ve a l og -ma r t inga l e s t r uc tu r e ,

l o g ( h i ) = l o g ( b j _ 1 ) --] -Ej,w h e r e t he e ' s a r e i id no r m a l ly d i s t r i bu t e d w i th z e r o m e a n a n d va r i a n c e o 2. A sm a l l( l a rg e ) v a l u e o f e 2 c o r r e s p o n d s t o a h i g h ( l o w ) c o r r e l a t i o n b e t w e e n t h e l e v e ls a td i f f e r e n t t im e s . W e c o n s i d e r t h i s c h o i c e o f p r i o r v e r y v a g u e . I t is o u r e x p e r i e n c et h at , w i t h a re a s o n a b l e a m o u n t o f d a ta , t h e d at a w i ll ' s p e a k f o r t h e m s e l v e s ' . B ya d j u s t in g t h e v a l u e s o f t h e h y p e r p a r a m e t e r s w e c a n , h o w e v e r , c o n t ro l t h e s m o o t h -n e s s o f t h e e s ti m a t e , t h e e s t im a t e b e i n g t h e p o s t e r i o r m e a n , s a y. N o w , o u r p r i o r a l soh a s s o m e w e a k n e s s e s . T h e p r i o r d i s t ri b u t io n o f f l is n o t s t a ti o n a ry . T h e m e d i a n i nthe p r io r d i s t r i bu t ion o f b j i s b0 , bu t t he me a n i s boe xp ( jo ' 2 /2 ) . T h i s c ou ld be ap r o b l e m i f w e w i s h t o p r e d i c t t h e o c c u r r e n c e i n t e n s i t y in t h e y e a r s to c o m e . T h e r ew e h a v e n o d a ta , a n d i f w e c h o o s e t o p r e d i c t u s in g t h e p o s t e r i o r m e a n , w e w i l lt h e r e f o r e g e t a n i n c r e a s i n g e s t i m a t o r . T h e r e a d e r m i g h t a s k : w h y n o t c o n t r o l t h em e a n b y m u l t i p l y in g b j w i t h e x p ( - j c r 2 / 2 ) ? I f t h at is d o n e , t h e n i t c a n b e s h o w n t h atb j c onve r ge s a lmos t su r e ly t o 0 , a nd ne w d i f f i c u l t i e s a r i s e . A s a n a l t e r na t ive p r io ro n e c o u l d k e e p t h e d i s tr i b u ti o n a l a s s u m p t i o n s a b o u t t h e j u m p t im e s , b u t a s s u m e t h atthe l e ve l s a r e d i s t r i bu t e d a s

b j = b (a., w h e r e 6 ~ G a m m a ( i , , 6 ) , a n d ~'j a r e i id G a m m a ( x , l / x ) . ( 3 .9 )F r o m a c o m p u t a t i o n a l p o i n t o f v i e w , t h i s p r i o r g i v e s n o d i f fi c u l ti e s . F u r t h e r m o r e , i ti s s t a t i ona r y . T he a s sum pt ion is t ha t the i n t e ns i t y va r i e s a r oun d a l e ve l b . I t doe s n o ta l l o w f o r p e r m a n e n t c h a n g e s i n t h e i n te n s i ty . A s a c o n s e q u e n c e , t h e p r i o r (3 . 9 ) a d d smor e s t r uc tu r e t o t he i n t e ns i t y ( sma l l e r va r i a nc e ) , a nd tha t c a n be u se f u l w he npr e d ic t i ng the f u tu r e oc c u r r e nc e i n t e ns i t y . We a r e go ing to u se p r io r ( 3 . 7 ) - ( 3 . 8 ) . T hep r io r i s c hose n in a n a t t e mp t t o a dd on ly l i t t l e p r io r i n f o r ma t ion .

T h e a g e e f f e c t s J ; ( m a l e , -) a n d f 2 ( f e m a l e , .) ar e m o d e l l e d a s d i s c r e t e v e r s io n s o fthe c ons t r uc t ion ( 3 . 7 ) - ( 3 . 8 ) . We a s sume tha tf 2 ( m a l e , a ) = Z l (a e{ r j+ a ,. . . ,~ + ~ } ) c )' , a = 1 , 2 , 3 . . . .j>0

> a b o v e , b u t w i t h n e w h y p e r p a r a m e t e r s e 2he r e ( c j ) j> j a r e m ode l l e d a s ( b j )j 1a nd A f z,, a nd w he r e ( K j + I - K j ) j > t a r e a s sum e d to be l i d ge om e t r i c a l l y d i st r ibu ( e ~w i th pa r a me te r t ~ f ~ . S ome r e a de r s migh t a ga in f i nd i t r e a sona b le t o a dd the p r io ri n f o r m a t i o n t h a t y o u n g m a l e s h a v e a h i g h i n t e n s i ty o f m a k i n g c l a i m s . S u c h


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CLAIMS RESERVING IN CONTINUOUS TIME 147in f or ma t ion i s no t added he r e , bu t our e s t ima te s w i l l show tha t young ma le s a r e ah igh r isk g r oup . T he age e f f ec t fo r f em a le s f 2 ( f em a le , .) is a s sume d to have the sam es t r uc tu r e a s t he age e f f ec t f o r ma le s , bu t w i th new hype r pa r ame te r s . Fur the r mor e ,we need to spec i f y the p r io r o f t he i n i ti a l l eve l s f 2 ( m a le , 0 ) and f 2( f e m a le , 0 ). T hesea r e a s sum ed to be l o gno r ma l ly d i s t ri bu t ed w i th pa r am e te r s ( t z0 f ~ , c r~ f ~ ) f o r ma le sand (#0f2f, ~0 f : ) fo r fema les ; the ~ ' s con t rol l ing the leve ls and the t r ' s co nt rol l in gthe var iabi l i ty .Th e re m ain ing 11 com po ne nts (g, hs.ep,qsep, e tc . ) are a l l a ssu m ed to h av e the sam es t r uc tu r e ( 3 .7 ) - (3 . 8 ) a s t he ca l enda r t ime e f f ec t f l ; e ach c om pon en t h av ing i tsind iv idua l hype r pa r ame te r s . T he in it ia l l eve ls a re a s su m ed to be I ogno r ma l ly d is -t r i bu t ed , aga in w i th i nd iv idua l hype r pa r ame te r s f o r e ach componen t .F i n a l ly , u n k n o w n p a r a m e t e r s a s s o c i at e d w i th d i f f e re n t c o m p o n e n t s a r e a s s u m e dto be independen t w i th r e spec t t o t he p r io r . T h i s i ndependence a s sumpt ion i s no ttha t c ruc ia l , and in some branches i t i s not reasonable . I t might be tha t a la rgenum ber o f occur r ences i nd uces r e l a ti ve ly sma l l c la ims . I n t ha t c a se f l shou ld bepos i t i ve ly cor r e l a t ed w i th t he q ' s . A l so , i t migh t no t be r ea sonab le t o a s sume0 0 0independence be tween the compet ing r i sks , hse hsep and hp, say. I t could be tha t ,i f the in tens i ty of se t t l ing o nly , hse, is h igh, the n the in ten s i ty of se t t l ing wi th a0par t ia l payment , hsep , t ends t o be h igh too . T he r e a r e ways o f mode l l i ng suchdependenc ie s , bu t f o r s impl i c i t y we have chosen no t t o cons ide r t hem he r e .

3 3 Additional remarksT he d i s t ri bu t ion o f the c l a ims p r ocess ( 3 .1 ) i s de sc ribed by 14 com pone n t s . T hesec o m p o n e n t s a r e m o d e l le d n o n p a r a m e t ri c a ll y . S o m e r e a d e rs m i g h t w a n t t o a d d m o r es t r uc tu r e by us ing pa r ame t r i c mode l s . For example , t he d i s t r i bu t ion o f t he pa r t i a lpay m ent s cou ld be de sc r ibed by m ix tu r e s o f logno r ma l d i s t ri bu t ions . Such a cho icecan be mot iva t ed ea s i l y . We wor k w i th den ta l c l a ims ; t he r e a r e examina t ions andthe r e a r e ope r a t ions . Ope r a t ions a r e mor e expens ive than examina t ions , and the r e -f o r e i t c an be a r gued tha t t he d i s t r i bu t ion o f a pa r t i a l payment shou ld have twopeaks . T he e s t ima te s w i ll show tha t the d i s t r ibu t ions o f the pa r ti a l paym ent s a c tua l lyhave two peaks . A l so , a s d i scussed p r ev ious ly , i t cou ld be use f u l t o add mor es t ruc tu r e i n to t he ca l enda r t ime e f f ec t f l wh en p r ed ic t i ng the occur r ence in t ens i ty i nthe yea r s t o come . As t im e goes (7- i nc r ea se s ) mor e and mo r e c l a ims a r e obse r ved ,and the e s t ima te s o f a l l componen t s t ha t a r e i ndependen t o f c a l enda r t ime w i l l becons i s t en t ( t he pos t e r io r d i s t r i bu t ions w i l l conve r ge towar ds un i t mass d i s t r i bu-t ions ) . Ho wev e r , t he unce r t a in ty abou t t he occur r ence in t ens i ty i n t he yea r s to com ewi l l r ema in . A ssum ing tha t the occur r ence in t ens i ty does n o t change w i th ca l enda rt ime ( f l cons tant ) i s a poss ibi l i ty . This i s , however , typica l ly not rea l i s t ic , ands t ruc tures l ike (3 .9) could be chosen.

A long the way we have po in t ed ou t poss ib l e ex t ens ions . Di f f e r en t b r anches o finsur ance ( m otor i nsur ance , f ir e i nsur ance , e t c .) c a ll f o r d i f f e r en t spec i f ic a t ions o fthe mode l . T he insur ance companie s w i l l have an idea abou t which dependenc ie sthe mo de l sh ou ld a l l ow f o r, e. g . tha t a l ong se t t l emen t pe r iod typ i ca l ly i nduces l a r gepa r ti a l paym ent s .


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148 SVEND HAASTRUP AND ELJA ARJAS4. MARKOV CHAIN MONTE CARLO INTEGRATION

W e w i sh t o a p p r o x i m a t e t h e p o s t e r i o r d i s tr i b u ti o n o f t h e u n k n o w n s : i .e . p a r a m e t e r sa n d o u t s t a n d i n g l i a b i l i t i e s . T o d o t h a t , M a r k o v c h a i n M o n t e C a r l o ( M C M C ) i n t e -g r a t i on i s u se d . A M a r ko v c ha in , w h ic h ha s t he pos t e r io r d i s tr i bu t ion a s i ts e qu i l i-b r ium d i s t r i bu t ion , i s ge ne r a t e d . T he a lgo r i t hm c on ta in s a n a r b i t r a r i l y c hose nn u m b e r o f s t e p s w h e r e , a t e a c h s t e p , n e w v a l u e s o f th e u n k n o w n p a r a m e t e r s , t h er e m a i n i n g d e v e l o p m e n t s o f th e r e p o r t ed b u t n o t s e tt le d c l a im s ( R B N S ) a n d t h eo c c u r r e n c e t i m e s , c o v a r i a t e s an d f u l l d e v e l o p m e n t s o f t he o c c u r r e d b u t n o t re p o r t e dc l a i m s ( I B N R ) , a r e s a m p l e d . A s t h e n u m b e r o f s te p s i n c r e a se s , t h e em p i r ic a ld i s t ri b u t i o n o f t h e s e s a m p l e d q u a n t i ti e s c o n v e r g e s t o w a r d s t h e p o s t e r i o r d i s tr ib u t i o n .T h e r e b y t h e p o s t e r i o r c a n b e a p p r o x i m a t e d , a n d b a s e d o n t h i s a p p r o x i m a t i o n t h ee s t i m a t i o n s a n d p r e d i c t i o n s a r e m a d e . B y i n c r e a si n g t h e n u m b e r o f s te p s w e c a nm a k e t h e a p p r o x i m a t i o n a s e x a c t a s w e w i s h .F i g u r e 1 s h o w s a m o d e l g r a p h . Q u a n t i t i e s s u r r o u n d e d b y s q u a r e s a r e k n o w n ; t h e s ea r e e i th e r h y p e r p a r a m e t e r s c h o s e n b y u s , o r th e y a r e d at a. Q u a n t i ti e s s u r r o u n d e d b ye l li p s e s ar e u n k n o w n s ; t h e s e a r e e i t h e r u n k n o w n p a r a m e t e r s , I B N R c l a i m s o r R B N Sc l ai m s . A r r o w s i n d i c a te d e p e n d e n c i e s . I n e a c h s t ep o f t h e a lg o r i t h m n e w v a l u e s o fa l l qua n t i t i e s su r r ounde d by c i r c l e s a r e s a mp le d .

B e f o r e w e g o o n w i t h a d e s c r i p t io n o f t h e a l g o r it h m w e s h al l a g a in r e f e r t o S m i t ha n d R o b e r t s (1 9 9 3 ) f o r an i n t r o d u c t i o n to M C M C i n te g r a ti o n .4 . 1 . L i k e l i h o o dI n o r d e r t o s o l v e th e B a y e s i a n i n fe r e n ti a l p r o b l e m w e n e e d t h e l i k e l i h o o d o f t heda t a , i.e . t he d i s t r i bu t ion o f t he ( obse r v e d ) da t a g ive n the unk now n p a r a me te r s .

T h e o b s e r v a t i o n s a r e t h e c l a i m s r e p o r t e d , i n c l u d i n g t h e i r d e v e l o p m e n t u p t oc a l e n d a r t i m e r . W e d e n o t e t h e s e o b s e r v a t i o n s(~ t ' , S~?, A~ , U~, ~ / ) i > 1. (4 .1 )

N o te t ha t X~ i s t he pa r t i a l pa yme n t p r oc e s s c e nso r e d a t c a l e nda r t ime r , i . e .r - ~ - U~' t im e un i t s a f t e r no t i f i c a t i on ; c f . ( 2 , 1 ) - (2 , 2 ) . In t he p r e v io us s e c t ionthe d i s t r ibu t ion o f a ll c l a ims w h ic h ha d o c c u r r e d , i nc lud ing t h e i r f u l l d e v e l o p m e n tw e r e m o d e l l e d . T h e r e s u lt s o f N o r b e r g ( 1 9 9 3 a ) i m p l y th a t, g i v e n t h e u n k n o w np a r a m e t e r s , t h e p r o c e s s g e n e r a t i n g t h e r e p o r t e d c l a ims inc lud ing the i r f u l l de ve l -o p m e n t , e v e n i f t h e s e g o b e y o n d c a l e n d a r t i m e r , h a s o c c u r r e n c e i n t e n s it y

w ( t , s , a ) f ( t , s, a ) e u ( r - t ). (4 .2)T h e c o n d i t i o n a l d i s t ri b u t io n o f t h e r e p o r t in g d e l a y o f a c l a i m w h i c h o c c u r r e d a t t im et is

P u (d u ) l ( ,, _< ~ _ ,)e u ( r - t ) ' ( 4 .3 )

a n d t h e d i s t r ib u t i o n o f th e c o r r e s p o n d i n g p a r t ia l p a y m e n t p r o c e s s isP x ( d x ) .


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C L A I M S R E S E R V I N G I N C O N T I N U O U S T I M E 149RBNS .claims:r e m a a m n g }a r t i a l p a y m e n t s /

R e p o r t e d c l a i m s :o b s e r v e dp a r t i a l p a y m e n t s

T~ p I B N R cla im s: " ~a x t i a . l p a y m e n t ~

T \e I B N R c l a i m s : ~ ' ~ x f o o o ~ Icu r rence t i m es , I { h . O . . h p . q O . h . . p . q . o , le o v a r i a t e s a n d / ~ h_X h I , x h 1 , , 1 1 Ie p o r ti n g d e l a y s l ~ e , p , u p, , ,p , ~ . e ~ / I\ i--f ZE xp sure : . . . . ..~ . . . . r / \\ l

H y p e r p a x a m e t e r scon t ro l l i ngf l , f 2 , g

R e p o r t e d c l a i m s :o c c u r r e n c e t i m e s ,c o v a r i a t e s a n dr e p o r t i n g d e l a y s

H y p e r p a r a m e t e r scon t ro l l i ngt h e h ' s a n d q ' s

FIGURE I Mo de l graph

The c la ims (4 .1) occu r wi th in tens ity (4 .2) . The d i s t r ibu t ion of a repor t ing d elay o f ac la im w hich occurred a t t ime t is g iven b y (4.3). The d i s t r ibu t ion of the obse rvedpar t o f a par ti a l pa ym ent p rocess o f a c la im, which occu rred a t t ime t and w asrepor ted a f ter u time uni t s , is eas i ly found us ing the ' in tens i ty cons t ruct ion ' o f thedis t ribu t ion Px of the par ti a l paym ent process ; see Subsec t ion 3 .1 . We den ote thedis t r ibut ion

wh ere r - t - u refers to the censor ing o f the proce ss r - t - u t ime uni ts af ternot i f i ca t ion . A l ike l ihood i s a dens i ty . In the fo l lowing we use a somewhat s loppynota t ion w hich hop efu l ly does not cause m isunders tandings . The l ike l ihood o f theobservat ions (4 .1) i s


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150 SVEND HAASTRUP AND ELJA ARJASA ( o b s ) e x H f ( 7 ~ / , S ~# ,A ) e u ( r - 7 ~ ) P v ( d U ) ~ ' - T - u 7i > l e - - ~ = - ~ ) PX ( d ~ i )

x e x p - Z L,.o o o ~ - r ] - u , '= Hf (T~ ,S i ,A i )Pu(dUi )Px ( d X ~ )

i>_l

x e x p ( - Z L,ao < s i>_l i>_l e x p ( - - ~ S o r w ( t ,s ,a ) f l( t ) f2 ( s ,a ) ( 1 - - e - f ; - g ( u ) d U

\ Si

I I g (UT) e x p - g(u)dui>_l7 - t~ t ox I I P 7 ' - ~ ' ( d X ~ ) .

i_>l

( 4 . 4 )

(4 .5 )

H e r e t h e l a s t p a r t ( 4 . 5 ) , d e a l i n g w i t h t h e p a r t i a l p a y m e n t p r o c e s s e s , h a s t o b e w r i t t e ni n m o r e d e t a il . R e c a l l t h a t X i i s t h e o b s e r v e d p a r t o f t h e p a rt i al p a y m e n t p r o c e s s o ft h e i th c l a i m . T h e s u p e r s c r i p t ~ - - ~ - U ~ r e f e r s to t h e c e n s o r i n g o f t h e p r o c e s s a tc a l e n d a r t i m e 7 -. F r o m t h e o b s e r v e d p a r t i a l p a y m e n t p r o c e s s e s ( X i )i _> ~ s o m e u s e f u lq u a n t i t i e s a r e d e r i v e d . F i r s t w e c o n s i d e r e x p o s u r e :W ( v ) : T h e n u m b e r o f c l a i m s i n w h i c h t h e w a i t i n g t i m e f r o m n o t i f i c a t i o n t o fi rs t

p a r t i al p a y m e n t o r s e t t l e m e n t i s a t l e as t v . N o t e t h a t W i s m a j o r i s e d b y t h en u m b e r o f r e p o r t e d c l a i m s a n d d e c r e a s i n g i n v . W e h a v e W ( 0 ) = 2 8 0 6 , t h en u m b e r o f r e p o r t ed c l a im s .

W I ( v ): T h e n u m b e r o f t im e s a n y p a r t i al p a y m e n t p r o c e s s i s o b s e r v e d , w i t h w a i t i n gt i m e a t l ea s t v s i n c e t h e l a t e s t p a r t i a l p a y m e n t . A l s o W ~ i s d e c r e a s i n g i n v. W eh a v e W I ( 0 ) = 1 7 8 0 . H e r e W l i s l e s s t h a n t h e n u m b e r o f r e p o r t e d c l a i m s , b u t inp r i n c i p l e W I c a n e x c e e d t h e n u m b e r o f r e p o r t e d c l a i m s s i n c e a c l a i m c a n h a v es e v e r a l p a r t i a l p a y m e n t s .

T h e n ' e v e n t s ' :( V ~d)j > I: W a i t i n g t i m e s f r o m n o t i f i c a ti o n t o s e t t l e m e n t w i t h o u t p a y m e n t . T h e r e a r eo n l y 9 o f t h e s e i n t h e d a ta .( ~ p ) j > i a n d (Y ~ eJp )j > i : W a i t i n g t i m e s f r o m n o t i f i c a t i o n t o s e t t l e m e n t w i t h o n e

p @ m e n t , a n d t h e ~ o r r e s p o n d i n g a m o u n t s p a i d (i n t h is c a s e t h e p a y m e n t e q u a l st h e t o t a l c l a i m a m o u n t ) . T h e r e a r e 1 2 9 5 o f t h e s e .


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CLAIMS RESERVING IN CONTINUOU S TIME 151~pJ ) j ?_ I and ( Y~J)j >_ I : Wa i t ing t imes f rom not i f i c a t ion to the f i r s t pa r t ia l pa ym en t

w i t h o u t s e t t l e m e n t , a n d t h e c o r r e s p o n d i n g a m o u n t s p a i d . T h e r e a r e 1 3 7 6 o fthese .Vs lJ)j ~ l : Wa i t i n g t i m e s f r o m a p a r t i a l p a y m e n t t o s e t t l e m e n t o n l y ( n o p a r t i a lp a y m e n t a t t h e t i m e o f s e t t l e m e n t ) . T h e r e a r e 24 2 o f ' t h e s e .

V]~pj_l).> and (Y~ep)j_>ld Wa i t i n g t i m e s f r o m a p a r t i a l p a y m e n t t o s e t t l e m e n t w i t hp a y m e n t , a n d c o r r e s p o n d i n g a m o u n t p a i d a t s e t tl e m e n t . T h e r e a r e 6 4 3 o f th e s e.

Vi~ l J )j> l and (Y~J) j _> i" W ai t ing t imes f rom a pa r t i a l p ay m en t to the nex t , wh en thel a t t e r d o e s n o t s e t t l e t h e c l a i m , a n d c o r r e s p o n d i n g a m o u n t s . T h e r e a r e 4 0 4 o fthese .

A s a n i l lu s t r a ti o n w e h a v e i n F i g u r e 2 s h o w n t w o e x a m p l e s o f p a r t ia l p a y m e n tp r o c e s s e s . T h e f i rs t p r o c e s s s e t t l e s w i t h i n t i m e 7-. T h e r e a r e t w o p a y m e n t s m a d e , a n dn o p a y m e n t is m a d e a t th e t im e o f s e t tl e m e n t . T h e s e c o n d p r o c e s s a r i se s f r o m a nR B N S c l a im . T h e r e i s o n e p a y m e n t m a d e a n d t h e p a r ti a l p a y m e n t p r o c e s s isc e n s o r e d ( c a l e n d a r t i m e 7 - i s r e a c h e d ) . E v e n t s a n d a m o u n t s a r e s h o w n a t th e f i g u r e ;w a i t i n g t i m e b e t w e e n j u m p s a r e e v en t s , a nd s i z e s o f j u m p s a r e a m o u n t s . N o t e t h a tV ' i s t h e w a i t i n g t i m e f r o m l a t e s t p a y m e n t u n t i l c e n s o r i n g , a n d i t o n l y a f f e c t s t h ee x p o s u r e . T h e e x p o s u r e f i a n c t i o n s a r e g i v e n b y

ccumulated ~ s l e , 1payments

l=: y l , xVpl,1 ,,

i

aT i m e s in ceA c c u m u l a t e dp a y m e n t s

Vp0,2

n ot i f i ca t ion

V I7i

y o ,2: p,tJ

i_T i m e s in cenot i f icat ion

FIGURE2 Exam ples of observed partial paym ent processes; x denotes a settlement


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152 SVEND HAASTRUPAND ELJA ARJAS2 v ~ ( o , G , : ]

~ ' li O ( I ,) = 1 P E ( l. ~ , 2 , j / ~ , ' ]o v ~ ~ , l , o o )v ~ (o,v~;']

Wsen d W l( v ) = v E ( i . I Vi, l ]E ( v ~ , ' , v ' ]0 v ~ ( v ' , c )

As a f unc t ion o f exposur e and even t s t he l a s t pa r t ( 4 .5 ) o f t he l i ke l ihood can bewr i t ten as

- ~ - ~ - u 7 . . . . . o j h 0 ( ~ jH t'p'r COXr) O( H h s e ( ~ s e ) H -sep, s e p , ~ h g ( ~ p j )i _ > l j > l j > l j ~ l1 I j l d I i d 1 ] h s < ( ~ ; o ) f i ,s e p V ; e p H h p ( V p )j>_l

x e x pj > i j _ > I

- / 0

(4.6)(4.7)

(4.8)(4.9)

(4 .10)(4.11)

x I I Psep ~ /p ) H P~ (~pd)j_>l y_>lx H i t j i t j

psop(r;e.) H p p (r; ,j ~ _ l j > _ lwher e the V , W and Y depe nd on the o bse r ved pa r t i al pay m ent p r ocesse s i n ano b v i o u s w a y . T h e d e n s i t y 0 0sep, say, i s g iven by i t s cor responding hazard ra te qsep"W e h a v e

~2Jp0 1 - 2 ..., [ qsep (y )d y } . (4 .12)] , o p ( ~ : p ) ~ I I o ( . - . , , ` 0 osep (Y~sTp) x e x p

j _> l j Z l \ j _> l J O ]T he l a s t pa r t ( 4 . 5 ) o f t he l i ke l ihood f imc t ion can now be subs t i t u t ed w i th ( 4 . 6 ) -(4.12) .W e have ch osen p r io rs a lon g the l i ne s o f Ar j a s and Gasba r r a ( 1994). T o cop ythe i r s am pl ing a lgor i t hm i t i s nece ssa r y tha t , i n e ach o f t he 14 com pone n t s , t hel ike l ihood i s p r opor t iona l t o an expr e ss ion o f t he f o r m

1 ] , , ( ) e x ( - I (wher e h i s a comp onen t , and Z does no t depen d ( f unc t iona l ly ) on h . T h i s i s the ca sewi th a l l comp onen t s bu t g , which u nf or tun a t e ly occur s i n t he pa r t ( 4 .4 ) o f t hel ike l ihood . A way to dea l w i th t h i s p r ob lem i s t o use data augmentation4 2 Missing dataT he r ea son tha t t he l i ke lihood o f t he ( obse r ved) da t a does no t have a t r a ct ab l e shapei s t ha t t he occur r ed bu t no t r epor t ed c l a ims ( I BNR) a r e no t obse r ved . He r e da t aa u g m e n t a t i o n c a n b e u s e d . B y s a m p l i n g a n d a d d i n g t o o u r d a t a th e o c c u r r en c e t i m e s


15/26

C L A I M S R E S E R V IN G I N C O N T I N U O U S T I M E 153and cova r i a t e s o f the I BN R c l a ims , t he l i ke l ihood ob ta ins t he de s i red shape . T heoccur r ence t imes and cov a r i a te s o f t he I BNR c l a ims a r e ca l led missing data. T h emiss ing da t a a r e deno ted

(~j, S~, AT)j_ 1' (4.13)No te t ha t r epor t ing de l ay s and pa r ti a l paym ent p r ocesses o f t he I BN R c l a ims , andt h e re m a i n i n g d e v e l o p m e n t o f t he R B N S c l a im s , a r e n o t i n c l u d e d a s m i s s i n g d a ta .Ho wev e r , we a r e go ing to samp le t hem a t e ach s t ep o f t he a lgor i t hm, bu t t hey a ren o t u s e d t o s a m p l e u n k n o w n p a r a m e t er s . T h e y a r e o n l y u s e d t o p r e d ic t o u t s t a n d i n gl iabi li t ies , c f . F igu re 1 . The resul t s of N orbe rg (1993a) im ply tha t , g iven theunknown pa r ame te r s , t he mi ss ing da t a ( 4 . 13) a r e i ndependen t o f t he obse r va t ions( 4 . 1 ) and tha t t he mi ss ing da t a a r e d i s t r i bu t ed a s a ma r ked Po i s son p r ocess w i thin t ens i tyw(t,s,a)f(t,s,a)(1 - P u ( r - t)), (t,s,a) E ( O , r) x { m a l e, f e m a le } x { 1 , 2 , . . . } .

(4 .14)T he l i ke l ihood o f t he mi ss ing da t a on ly i s

A ( m i s ) cx H . f ( ~ j , ~ j , A ~ ")(I - P v ( r - 7 ~ n))j > l

x exp - w(t,s,a)f(t,s,a)(1 - P u ( r - t ) ) d t( : : )x H fl(7~n)f2(~j ,A~n)exp - g (u)duj > 1 d O

x exp - w(t,s,a)fl(t)f2(s,a)e- g(")dUdt .Because o f cond i t i ona l i ndependence o f t he obse r ved and miss ing da t a , t he l i ke -l i hood o f t he se com bined d a t a i s t he p r oduc t o f t he cor r e spond ing two l i ke l ihoodexpressions, i .e . ,A ( t o t a l ) = A ( o b s ) A ( m i s ) ( 4 .1 5 )

f i l A Ti _> l j > l i > l j > lxexp(- -s~foTW(t ,s ,a) f~( t) f2(s,a)dt) (4.16)x r I g ( U ~ ) x e x p - g (u)du - Z : g(u)du (4 .17)

i_> 1 " j_> 1 dOp T - - U : d X T ) , I ~ L Xi > l


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154 S V E N D H A A S T R U P A N D E L J A A R J A Swhere the las t par t i s wr i t ten in de ta i l in (4 .6) - (4 .12) . Now the l ike l ihood has thedes i red shape .4 . 3 . S a m p l i n g a l g o r i t h mT he sam pl ing a lgor i t hm goes a s f o l lows : T o beg in w i th , i n i ti a l va lue s o f t heunknown pa r ame te r s a r e de t e r mined . T hese can e i t he r be sampled f r om the p r io rd i s t r ibu t ion o r cho sen a r b i t r a ri l y ; a sym pto t i c a l ly it does no t ma t t e r. A gene r a l s t epo f the a lgo r i t hm con ta ins t h r ee subs teps :Subs t ep I : Sample occur r ence t im es and cova r ia t e s o f t he I BNR c l a ims ( the mi ss ingda ta ) g iven the unkn ow n pa r ame te r s sam pled in the p r ev ious s t ep o f t hea lgor i t hm;Subs t ep 2 : Sam ple t he r ema in ing dev e lopm ent o f the r epor t ed bu t no t se tt l ed c l a ims( R B N S ) a n d t h e f ul l d e v e l o p m e n t s o f t he I B N R c l a i m s , g i v e n th e u n k n o w npa r ame te r s samp led in t he p r ev ious s t ep o f the a lgor i t hm, and g iven bo thobse r ved and miss ing da t a whe r e t he l a t t e r we r e sampled in Subs t ep 1 ;Subs t ep 3 : Sample t he unknown pa r ame te r s g iven bo th the da t a ( obse r va t ions ) andthe mi ss ing da t a sampled in Subs t ep 1 .T hese th r ee subs t eps a r e d i scussed in t he f o l lowing .

4 . 3 .1 . Sampl ing the mis s ing da taT he miss ing da t a a r e t he occur r ence t imes and cova r i a t e s o f t he 1BNR c l a ims , see( 4 .13) . G iven the p r e sen t va lue o f the un kno wn p a r ame te r s , s amp led in the p r ev iouss t ep o f t he a lgor i t hm, t he m i ss ing da t a a re d i s t ri bu t ed a s a ma r k ed Po i s son p r ocesswi th in tens i ty (4 .14) . I t fo l low s f rom N orberg (199 3a) , Th eore m 1, tha t the to ta ln u m b e r o f I B N R c l a im s is P o i ss o n d i s t ri b u te d w i t h m e a nW IBNR = w( t , s ,a ) f l ( t ) f2 ( s ,a ) (1 - Pu(7- - t ) )d t ,

and g iven th i s t o ta l num ber , t he occur r ence t imes and co va r i a te s o f the I BN R c l a imsare i id wi th dens i tyw(t, s, a)fl (t)f2(s, a) ( l - Pu(7- - t ) )/ I V mNR

o n (0 ,7 -] { m a l e , f e m a l e} { 1 , 2 , . . . } . T h e m i s s in g d a t a a re sa m p l e d f ro m th i sdis t r ibut ion.

4 . 3 . 2 . S a m p l e I B N R a n d R B N S c l a i m sI n Subs t ep 2 t he unk now n pa r am e te r s samp led ea r l ie r by the a lgor i t hm, t he da t a, andthe occur r ence t imes and cova r i a t e s o f t he I BN R c l a im s samp led in Subs t ep 1 ( themiss ing da ta ) , a re a l l he ld f ixed.

F i r s t we sam ple t he r epor ti ng de l ays o f the I BN R c l a ims . W e a l r eady kno w the i rcor r e spond ing occur r ence t imes (7~" )j_> l. So , t he j t h r epor t ing d e l ay mu s t exceed7- - ~ j ' , an d i t is d i s tr i bu t ed accor d ing to t he haza r d r a te g . No w th e j th r epor ti ng


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C L A I M S R E S E R V I N G I N C O N T I N U O U S T I M E i 5 5de lay i s sam pled f ro m a dis t r ibut ion on (~- - T j ' , cx~) wi th p iecew ise con stant hazardrate g.

T h e p a r t ia l p a y m e n t p r o ce s s o f a n I B N R c l a im c a n b e s a m p l e d a s f o l lo w s . F i r stt he wa i t i ng t ime to t he f i r s t ' even t ' i s s ampled . T h i s wa i t i ng t ime i s d i s t r i bu t edaccording to the piecewise constant in tens i ty hse + hs0eD+ h ~ . T h e n t h e t y p e o f th eeven t i s s amp led . W i th p r obab i l it y hse/(hse + h~ep+ hp) i t i s a se t t lement only , wi thpr obab i l i ty 0 0 0 0s e p / ( h s e "Jr-h sep "q- h p ) i t is a s e t t l emen t w i th pa r ti a l p aym ent , and w i th

O U O Opr obab i l i t y hp/(hse + hsep + hp) i t is a par t ia l p ay m en t only . I f the f i r s t eventinc ludes a pa r ti a l pay m ent t hen the s i z e o f t h is pay m ent i s s ampled . I f t he f i rs t0 e lse i tven t a lso i nc ludes se t t l emen t , t hen the pa r t ia l pay m ent ha s h aza r d r a t e q s e p ,has haza r d r a t e qO . I f t he f i rs t even t was a pa r t ia l pay m ent on ly then the wa i t i ng t im eto the nex t even t and the t ype o f t h i s even t a r e sampled . H e r e t he com pon en t s1 and I W a i t i ng t imes and even t s a r e sam pled un t i lee de d are hie, h]ep , h~, qsep qp"se t t l ement i s r e ached .T h e r e m a i n i n g p a rt ia l p a y m e n t p r o c es s o f a n R B N S c l a im i s s a m p l e d u s i n g t h es a m e m e t h o d a s a b o v e . F o r e a c h i n d i v id u a l R B N S c l a im w e n e e d t o k n o w w h e t h e rany p r ev ious pa r t i a l paym ent s have been m ade , and the t ime s ince the l a t e s t pa r ti a lpayment i f any , e l se t he t ime f r om r epor t i ng .4.3.3. Sampling the unknown parametersI n t h i s subs t ep we cond i t i on on the obse r ved da t a and the mi ss ing da t a sampled inSubs t ep 1 , and sam ple t he unk now n pa r ame te r s ( j ump t imes and l eve ls o f t he 14com pon en t s ) . T he con d i t i ona l d is t r ibu t ion o f the se un kno wn s ha s a dens i ty tha t isp r opor t iona l t o the l i ke l ihood ( 4 .15) o f t he com bine d obse r ved and m iss ing da t a,mul t i p l i ed w i th t he p r io r dens i ty o f t he unknown pa r ame te r s . Di r ec t sampl ing f r omth i s d i s tr i bu t ion i s no t poss ib le . Ho wev e r , we can iden t i f y the con d i t i ona l d i s t ri bu-t io n s o f e ac h o f t h e u n k n o w n p a r a m e t e r s g i v e n t h e r e m a i n i n g p a r t o f t h e u n k n o w np a r a m e t e r s a n d g i v e n t h e o b s e r v e d a n d m i s s i n g d a t a . F r o m t h e s e o n e - d i m e n s i o n a ld i s t r i bu t ions t he pa r ame te r s c an be sampled one by one . A comple t e de sc r ip t ion o ft h e s a m p l i n g o f th e u n k n o w n p a r a m e t e r s w o u l d t a k e q u it e a f e w p a g e s. W e s h a l lhe r e g ive a shor t ove r v i ew .T h e unk now n pa r ame te r s a r e t he j um p t imes and l eve l s o f a l l t he 14 com pon en t s .T o s a v e n o t a t io n w e s h a l l d e n o t e t h e u n k n o w n p a r a m e t e rs b y X ~ , X 2 , . . . ( h o w t h e yare ordered wi l l be discussed la te r ) . Assume tha t the present s tep i s the kth , andd e n o t e b y X ~ - I , X ~ - I . . . . t h e v a l u e s o f t h e u n k n o w n p a r a m e t e rs s a m p l e d in t h ep r e v io u s s t ep o f t h e a lg o r it h m . T h e y a re n o w s a m p l e d o n e b y o n e . A s m e n t i o n e d ,the cond i t i ona l d i s t r i bu t ion o f e ach ind iv idua l unknown pa r ame te r , g iven ther e m a i n i n g p a rt s o f th e u n k n o w n s a n d g i v e n o b s e r v ed a n d m i s s in g d a t a , is k n o w n( a t l e a s t up to p r opor t iona l i t y ) . F i r s t a new va lue X~ o f X1 i s sampled f r om thed i s t r i bu t ion o f

X ] g iv e n X ~ - t , X ~ - I , . . . , (4 .1 8)then a new va lue X~ o f X2 i s sampled f r om the d i s t r i bu t ion o f

X 2 g iv en )fl~l, ,V3~ -t, .. ., (4.1 9)


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156 S V E N D H A A S T R U P A N D E L J A A R J A Sand so the a lgor i t hm con t inues . T he a lgor i t hm i s c a l l ed G ibbs sampl ing , c f . Smi thand Robe r t s (1993) . Reca l l t ha t the unk now n pa r am e te r s a re t he l eve l s and ju m pt imes o f t he 14 com pon en t s . W hen the unk now n pa r am e te r ( an X) i s a leve l , i t i s no tt r ac t ab l e t o sample f r om the above one - d imens iona l d i s t r i bu t ions and then M e t r o -po l i s - Has t ings i s used . W i th t h i s mo di f i c a t ion the a lgor i t hm is c a l led a ' va r i ab l e - a t-a - ti m e M e t r o p o l is - H a s t i n g s ' , cf . C h a n a n d G e y e r ' s d i s c u ss i o n o f th e p a p e r b yYierney (1994) .T he me thod used f o r do ing the sampl ing ( 4 . 18) - ( 4 . 19) i s t aken f r om Ar j a s andGasba r r a ( 1994). T he unk now n pa r ame te r s c an be g r ouped in to 14 g r oups accor d ingt o t h e c o m p o n e n t t h e y d e te r m i n e ( f l , f 2 ( m a l e , " ), f2 ( fe m a l e , . ), g , . . . ) . I n A rj as a n dGasba r r a ( 1994) i t i s shown how the unknown pa r ame te r s a s soc i a t ed w i th onec o m p o n e n t c a n b e s a m p l e d . D e n o t e b y b j ) a n d S j ) t h e l e v e ls a n d ju m p t im e s o f t h ec o m p o n e n t f ~ , s a y . A r j a s a n d G a s b a r r a o r d e r t h e m a s

b 0, S i , b l , $ 2 , . . . , ( 4.2 0 )and sam ple ne w va lue s a s i nd i ca t ed in ( 4 .18) - (4 . 19) ab ove . T he i r a lgor i t hm ca r r ie sove r w i th min or m odi f i c a t ions ; the i r leve l s a re cor r e l a ted g am m a d i s t ri bu t ed , whi l eour l eve l s a re cor r e l a t ed logn or m a l ly d i s t ri bu t ed , and whe n sam pl ing the l eve l s theyuse r e j ec t ion sampl ing which cor r e sponds to r epea t ing M e t r opo l i s - Has t ings un t i la ccep tance i s r e ached . When sampl ing the unknown pa r ame te r s a s soc i a t ed w i th ani n d i v id u a l c o m p o n e n t , w e a d o p t t h e m e t h o d s o f A r ja s a n d G a s b a rr a , a n d t h e r e b y t h ecom pon en t s a r e sam pled one by one , f o l low ing the o r de r ( 4 .20 ) . B u t in wh ich o r de rshou ld the d i f f e r en t componen t s be sampled? Accor d ing to t he p r io r t he 14com pon en t s a r e i ndepen den t , and b y inspec t ion o f t he l i ke l ihood ( 4 .15) o f t hecombined obse r ved and miss ing da t a , we ge t t ha t a ccor d ing to t he cond i t i ona ld i s t r ibu t ion o f the unk now n pa r am e te r s g iven obse r ved and m iss ing da t a , a l lcom po nen t s bu t f l , f 2 ( m a le , .) and f 2 ( f em a le , -) a r e i ndepen den t . T he r e f o r e , excep tf o r t he m utua l o r de r o f f i , f 2 ( m a le , -) and f f f f em a le , - ), t he o r de r in which thed i f f e re n t c o m p o n e n t s a re s a m p l e d d o e s n o t p la y a r o le . W e h a v e c h o s e n t o s am p l ef i r s t f i , then f2 (m ale , . ), then J~ ( fem ale , . ), and the n the rem ainin g 11 com pon ents .Asympto t i c a l ly , t he o r de r does no t p l ay any r o l e .

5 . R E S U L T S O F T H E A N A L Y SI S

Suppo se w e s top the sampl ing a lgor i t hm a f t e r n s t eps. W ha t we then have i s the f ir s tn s t eps o f a M ar kov cha in . T ha t t he cha in i s M ar kov f o l low s fr om the cons t r uc t ion .E a c h s t ep o f th e c h a i n c o n t a i n s s a m p l e d v a l u e s o f t he u n k n o w npa r ame te r s ,o c c u r r e n c e t i m e s , c o v a ri a te s a n d f u ll d e v e l o p m e n t o f t he I B N Rc la ims , andr e m a i n i n g d e v e l o p m e n t o f t h e R B N S c l a i m s .

By const ruc t ion, the pos te r ior d is t r ibut ion, i . e . the condi t iona l d is t r ibut ion of theabove unk now ns g iven the obse r va t ions , i s inva r i an t fo r t ha t cha in . I t i s obv iou s t ha t


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C L A I M S R E S E R V I N G I N C O N T I N U O U S T I M E 157t h e c h a i n i s b o t h i r r e d u c i b le a n d a p e r i o d i c . F r o m T i e r n e y ( 1 9 9 4 ) i t n o w f o l l o w s t h a tthe c ha in i s ( pos i t i ve ) r e c u r r e n t . We ne e d the c ha in t o be e r god ic , a nd f o r t ha t as t r ong e r t ype o f r e c u r r e nc e , c a l l e d H a r r i s r e c u r r e n c e , is ne e de d . I n the i r d i s c us s iono f t he p a p e r b y T i e r n e y (1 9 9 4 ), C h a n a n d G e y e r s h o w t ha t a c e rt a in t y p e o fa l g o r i t h m , w h i c h t h e y c a l l a ' v a r i a b l e - a t - a - t i m e M e t r o p o l i s - H a s t i n g s ' , i s o f t e nH a r r is r e c u r r e n t . T h e i r p r o o f ca r r ie s o v e r t o o u r s i tu a t io n w i t h o n l y m i n o r m o d i f i c a -t i ons . It t he n f o l l ow s tha t ou r c ha in i s e r god ic . A su r v e y o f t he t he o r y c a n be f ou ndin T ie r ne y ( 1994 ) .

L e t (M j ) j= t , 2 ... de n o te t he c ha in , l e t h be a f unc t ion on the spa c e o f pos s ib l ev a l u e s o f t h e u n k n o w n s a b o v e , a n d a s s u m e t h a t h h a s f i n it e m e a n w i th r e s p e c t t o t h epos t e r io r d i s t r i bu t ion . T he n1 a sh g j l E p o s t or o r h . ( 5 . l )

A s a n e x a m p l e , s u p p o s e w e w i s h t o f in d ( a p p r o x i m a t e ) th e p o s t e r i o r d i s t ri b u t io n o fthe s i z e o f t he ou t s t a nd ing l i ab i l it i es . T h e ou t s t a n d ing l i a b il i ti e s a r e a f unc t ion o f t hec l a ims p r o c e s s ( 3 . 1 ) . T h e l i a b i li t ie s a r e d iv ide d in to li a b i li t ie s a r i s ing f r om the I BN Rc la ims ,X I B N R = ~ I T , _ < ~ , < T i + u , ) Y i ,

i > la nd l i a b i l i t i e s a r i s ing f r om the RBN S c l a ims ,

X RBNs = ~ l(r , + u, _< T < T, + U, + v , ) (Y i - Y i (~ - - T i - Ui ) ) .i > lL e t ~ jB N R . a n d ~ j B N S d e n o t e t h e v a l u e s s a m p l e d i n th e j t h s t ep o f t h e a l g o r i t h m( b o t h a r e f u n c t i o n s o f M j ) . B y ( 5 . 1 ) w e h a v e

I ~ - ~ a . s . o . / v I BN R S R B N S1 x , .~ R ~ . ~ s < ~ - -- ~ + < x ) , ( 5 . 2 )j + j - - ) * p o s t e n o r k " x

a nd th e r e by the pos t e r io r ( p r e d i c t i ve ) d i s t r ibu t ion o f t he ou t s t a nd ing l i a b il i ti e s c a nb e a p p r o x i m a t e d b y t h e l e f t h a n d s id e o f (5 . 2 ). T h e t h e o r y d o e s n o t g i v e a n e x a c ta n s w e r a s t o h o w l a r g e n s h o u l d b e c h o s e n . H o w e v e r , w h a t o n e u s u a l l y d o e s i s t or un se ve r a l i nde p e nd e n t s imu la t i ons a nd u se t he se to e va lua t e t he s t a b i l i t y o f t hea lgo r i t hm.T o b e g i n w i t h w e h a v e t o c h o o s e v a l u e s o f t h e h y p e r p a r a m e t e r s , an d t h e i n i ti a lv a l u e s o f th e u n k n o w n p a r a m e t e r s . R e c a l l t h a t e a c h o f t h e 1 4 c o m p o n e n t s ( i n te n -s i t ie s ) can be wr i t ten a s

~ i ( s j < t < s , ~ . , ) b j . (5 .3)j _ > o

T h e h y p e r p a r a m e t e r s ~ 0 a nd /2 ,0 d e t e r m i n e t h e p r i o r v a r i a n c e a n d m e a n o f t h e i n i ti a ll e ve l ( b0 ) w h i l e A ( o r t o) a nd o 2 de t e r m ine the p r io r va r i a b i l i t y o f t he i n t e ns i t y ; c f .t h e d is c u s s i o n i n S u b s e c t i o n 3 .2 . T a b l e l s h o w s t h e c h o s e n v a l u e s o f t h e h y p e r -


20/26

158 S V E N D H A A S T R U P A N D E L JA A R J A ST A B L E I

H Y P E R P A R A M E T E R S

ao #o A - i 0.2f l 70 0 .01g 0 .2 - 9 6 0 .010 0 .2 - 3 10 0 .02h h a p 0 .2 - 6 I 0 0 .02DhsOe 0.2 - 6 10 0.0 2l 0.2 --3 I 0 0.02h h ~ p 0 .2 - 6 10 0 .02Dhie 0 .2 - 6 10 0 .02qOp 0.02 - I0 250 0.04

0.02 - I 0 250 0.04q~q]ep 0.02 - 10 250 0.04qp 0.02 - 10 250 0.04o ~ # o - I o g ( l - n ) a 2

f2(m ale, .) 0.2 - 1 I 5 O.OIrE(fem ale, .) 0,2 - I I 5 O.Ol

p a r a m e t e r s . W e s h a l l n o t t r y t o m o t i v a t e t h e c h o i c e o f a ll th e v a l u e s. H o w e v e r , a f e wc o m m e n t s a r e n e e d e d . T h e i n d iv i d u a l v a l u e s o f A a n d o 2 w e r e c h o s e n a s f o ll o w s .W e s t a r te d o u t w i t h s o m e a r b i t r a ry v a lu e s . I f t h e e s ti m a t e ( p o s t e r io r m e a n o f t h ei n t e n s i t y ) t u r n e d o u t e i t h e r t o o r a g g e d o r t o o s m o o t h , t h e n n e w v a l u e s o f A a n d o 2w e r e c h o s e n . T h is w a y o f c h o o s i n g h y p e r p a r a m e t e r s c o r r e s p o n d s t o h o w w i n d o ws i z e ( b a n d w i d t h ) is c h o s e n i n k e r n e l d e n s i t y e s t i m a t i o n , a n d i t i s n o t o r t h o d o xB a y e s i a n i n s p ir it . O n t h e o t h e r h a nd , o u r c h o i c e o f t he h y p e r p a r a m e t e r s ~ a n d # 0r e p r es e n t s p r i o r k n o w l e d g e . F o r e x a m p l e , w e b e l ie v e t h a t s m a l l p a y m e n t s ( f r o m 0 t o2 5 D K K ) a r e v e r y u n l i k e l y , a n d t h e r e f o r e t h e v a l u e s o f t h e q ' s s h o u l d s t a rt o u t s m a l l .B y c h o o s i n g # 0 s m a l l w e e x p r e s s t h e p r io r k n o w l e d g e t h at t h e in t e n s it y ( h a z a r d r a te )s t ar t s o u t s m a l l , a n d b y c h o o s i n g ~ s m a l l w e e x p r e s s a h i g h d e g r e e o f b e l i e f i n t h isk n o w l e d g e . O t h e r v a l u e s o f t h e h y p e r p a r a m e t e r s w e r e t r ie d o u t. B u t , w i t h i n w h a t w eb e l i e v e w e r e r e a s o n a b l e v a l u e s , th e h y p e r p a r a m e t e r s h a d o n l y m i n o r i n f l u e n c e o no u r e s t i m a t i o n s / p r e d i c t i o n s .

T h e u n k n o w n p a r a m e t e r s a r e t h e j u m p t i m e s a n d l e v el s (t h e S ' s a n d b ' s , c f . ( 5 .3 ) )o f e a c h o f th e 1 4 c o m p o n e n t s . T h e i r i n it ia l v a l u e s c a n , f o r e x a m p l e , b e s a m p l e df r o m t h e p r i o r d is t r i b u t io n o r b e c h o s e n a r b i tr a r il y . T h e t h e o r y s a y s th a t a s y m p t o -t ic a l ly i t d o e s n o t m a t t e r w h a t y o u d o , w h i c h w a s a l s o o u r e x p e r ie n c e . H o w e v e r , t hea l g o r i t h m t u r n e d o u t t o b e t i m e c o n s u m i n g , a n d t o s a v e i t e r a t io n s w e , th e r e f o r e ,w a n t e d t o s t a r t t h e a l g o r i t h m a t a p l a c e w i t h h i g h p o s t e r i o r p r o b a b i l i t y . F o r e a c hc o m p o n e n t , t h e in it ia l v a l u e s o f t h e j u m p t im e s w e r e c h o s e n s u c h t h a t a r e a s w i t hm a n y o b s e r v a t i o n s ( o c c u r r e n ce s , d e l ay s o r p a y m e n t s ) h a d m a n y j u m p s , G i v e n t hej u m p t im e s , t h e m a x i m u m l i k el i h o o d e s ti m a t o r s o f t h e le v e ls w e r e u s e d a s t h ei rin i t i a l va lue s .


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C L A I M S R E S E R V IN G I N C O N T I N U O U S T IM E 159T h e a l g o r i t h m w a s w r i t t e n i n S - P l u s a n d r u n o n a D E C w o r k s t a t i o n . A s m e n -

t io n e d , t h e a l g o r i t h m t u r n e d o u t t o b e r a t h e r t i m e c o n s u m i n g ; a b o u t 1 h o u r f o r e a c h1 00 s t e p s , b u t m u c h c o u l d h a v e b e e n g a i n e d b y w r i t in g p a r t s o f th e a l g o r i t h m in C ,s a y . W e d i d t w o r u n s ; o n e o f le n g t h 1 0 0 0 a n d o n e o f l e n g th 5 0 0 . A f t e r a f e wi te r a ti o n s ( 4 0 - 6 0 ) t h e a l g o r i t h m s e e m e d s t a b l e ( w e p l o t t e d t h e a p p r o x i m a t i o n s t o th ep o s t e r i o r m e a n o f t h e in d i v i d u a l 1 4 c o m p o n e n t s , a n d f o u n d t h a t t h e s e d id n o t c h a n g em u c h ) . A s a p p r o x i m a t i o n t o th e p o s t e r i o r d i s tr i b u t io n o f al l u n k n o w n s w e u s e t h ee m p i r i c a l d i s t r i b u t io n o f a l l 1 5 0 0 r e a l i z a t i o n s .

E a c h o f t he 1 4 c o m p o n e n t s w a s e s t i m a t e d b y its ( a p p r o x i m a t e d ) p o s t e r io r m e a n( a n a v e r a g e o f t h e 1 5 0 0 i t e ra t io n s ) . I n F i g u r e 3 th e e s t i m a t e s o f th e c o m p o n e n t s f ~ ,

f 2 a n d g a r e s h o w n . F r o m a m a t h e m a t i c a l p o i n t o f v i e w , i t is e a s y to c a l c u l a te( p o s t e r i o r ) p o i n t w i s e 9 5 % c r e d i b l e i n t e r v a l s f o r e a c h c o m p o n e n t , b u t t h e c o m p u t a -t io n a l e f f o r t w o u l d i n c re a s e . E v e n t h o u g h w e h a v e n o c r e d i b l e i n t e rv a l s , w e a l l o wo u r s e l v e s t o c o m m e n t o n t h e e s t i m a t e s . T h e o c c u r r e n c e i n t e n s i t y i s t h e p r o d u c t o ft he c a l e n d a r t im e e f f e c t f l a n d t h e a g e e ff e c t f 2 . L o o k i n g a t f l , t h er e s e e m s t o b e n oo b v i o u s s e a s o n a l e f f ec t . In o b s e r v a t i o n y e a r n o . 3 ( 1 9 8 4 ) t h e o c c u r r e n c e i n t e n s it y i sh i g h ( w e h a v e n o e x p l a n a t i o n w h y ) , o t h e r w i s e t h e r e a r e o n l y m i n o r v a r i a t i o n s .L o o k i n g a t f 2 i t i s o b v i o u s t h a t th e r e i s a n a g e e f f ec t Y o u n g m a l e s a r e a h i g h r i s kg r o u p , w h i l e y o u n g f e m a l e s a re a l o w r i sk g r o u p . F r o m a g e 3 0 a n d o n t h e r e a r e o n l ym i n o r d i f f e r e n c e s b e t w e e n t h e s e x e s . T h e l a s t g r a p h s h o w s t h e r e p o r t i n g d e l a yh a z a r d r a t e g . B a s e d o n t h e e s t i m a t e f r o m F i g u r e 3 , th e 9 5 % q u a n t i l e o f t h e

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23/26

C L A I M S R E S E R V IN G I N C O N T I N U O U S T I M E 161p a y m e n t s n o t m a d e a t t h e t im e o f s e t tl e m e n t w e r e o f th e s i z e o f 1 7 8 , 1 85 , 1 8 6 o r 1 89D K K . T h e s e a m o u n t s a r e p r o b a b l y f e e s f o r s o m e s t a n d a r d d e n t a l e x a m i n a t i o n . O u re s t ima te o f p p the r e f o r e ha s a h igh pe a k in t h i s a r e a ( t he pe a k ha s be e n t r unc a t e d onthe g r a ph ) , a nd i t i s que s t i ona b le w he the r i t i s r e a sona b le t o mode l t he d i s t r i bu t iono f t h es e p a y m e n t s a s a c o n t i n u o u s d i s tr ib u t io n . N o n e o f t h e o t h e r ty p e s o f p a y m e n t( f ir s t -t i m e p a y m e n t a t t im e o f s e t t le m e n t , o r s u b s e q u e n t p a y m e n t s ) h a d s u c h v e r yf r e q u e n t l y o c c u r r i n g s i z e s . T h e a v e r a g e f i r s t - t i m e p a y m e n t n o t m a d e a t t i m e o fs e t t le m e n t is a p p r o x i m a t e l y 1 0 0 0 D K K . T h e d i s t ri b u t io n o f t h e s u b s e q u e n t p a y -i has the sam e shap e as pOp, ex ce pt for thee n t s n o t m a d e a t t i m e o f s et t le m e n t , p p ,h i g h p e a k . B a s e d o n t h e e s t im a t e f r o m F i g u r e 4 , t h e m e a n o f t h e se p a y m e n t s i sa p p r o x i m a t e l y 2 1 0 0 D K K . T h e d i st ri b u ti o n s o f t h e p a y m e n t s t h a t a re m a d e a t t h et ime o f s e t t l e m e n t , Psep a nd P ~ e p, s e e m to be i nde p e nd e n t o f w h e the r a ny p r e v iousp a y m e n t s h a v e b e e n m a d e . T h e d i s t r i b u t i o n s h a v e 2 p e a k s ; o n e a t a p p r o x i m a t e l y2 0 0 D K K a n d o n e a t a p p r o x i m a t e l y 3 5 0 0 D K K . T h e a v e r a g e a m o u n t s a r e a p p r o x i -m a t e ly 42 0 0 a n d 4 6 0 0 D K K .

F i g u r e 5 c o n t a i n s o u r m a i n r e su l ts . H e r e t h e ( a p p r o x i m a t e d ) p r e d i c t i v e d i s t ri b u -t i on o f t he ou t s t a nd ing l i a b il i ti e s a r e show n . Re c a l l t ha t t he o r ig ina l da t a s e tc o n t a i n e d i n f o r m a t i o n a b o u t o c c u r r e n c e t i m e s , c o v a r i a t e s a n d r e p o r t i n g d e l a y s , o na ll c l a i m s w h i c h o c c u r r e d b e t w e e n J a n u a r y 1, 19 82 a n d D e c e m b e r 3 1 , 1 9 9 0 a n dw h i c h w e r e r e p o r t e d b e f o r e M a r c h 3 , 1 9 9 2 . F u r t h e r m o r e , a l l p a y m e n t s m a d e u p o n

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P r e d i c t i v e d i s t r i b u t i o n o f o u t s t a n d i n g l i a b i l i t i e s ; o b s e r v e d a c c u m u l a t e d r u n - o f f M a r c h 3 , 1 9 9 2( 4 .2 y e a r s a f t e r t i m e 7") i s s h o w n w i t h d o t t e d l i n e s


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162 SVEND HAASTRUP AND ELJA ARJASt h e s e c l a im s b e f o r e M a r c h 3 , 1 9 9 2, w e r e o b s e r v e d . I n o u r e s t i m a t i o n / p r e d i c t i o n w eo n l y u s e d a ta t h a t w e r e a v a i l a b l e a t D e c e m b e r 3 1, 1 9 8 7 0 - = 6 y e a r s a f t e r J a n u a r y 1,1 98 2 ). T h e p r e d i c t i v e d i s tr i b u t i o n s o f t h e o u t s t a n d i n g l i a b i li t ie s c o u l d t h e r e f o r e h a v eb e e n c a l c u l a t e d a t t h a t t i m e . T h e d i s t r i b u t i o n s a r e c a l c u l a t e d u s i n g t h e m e t h o di n d i c a te d in ( 5 .2 ) . W e h a v e t h e o b s e r v e d r u n - o f f f o r a p e r i o d o f 4 .2 y e a r s . O n t h eg r a p h s t h e a c c u m u l a t e d r u n - o f f s a r e s h o w n w i t h d o t t e d v e r t i c a l l i n e s . T h e r e a r e 6 8o b s e r v e d I B N R c l a i m s . T h i s c o r r e s p o n d s t o th e 3 9 % q u a n t i le o f t h e p r e d i c ti v ed i s tr i b u t io n o f t h e I B N R c l a i m s . T h e p o s t e r i o r d i s tr i b u ti o n o f th e r e p o r t i n g d e l a y sh a s a l m o s t n o m a s s f o r d e l a y s e x c e e d i n g 3 y e a r s , a n d w e t h e r e f o r e e x p e c t t h a t a llI B N R c l a i m s a r e r e p o r t e d w i t h i n t h e 4 . 2 y e a r s . T a b l e 2 s h o w s m e a n , c o e f f i c i e n t o fv a r i a t io n ( c .v . ) a n d q u a n t i le s , f r o m t h e p r e d i c t i v e d i s t ri b u t io n s o f t h e o u t s t a n d i n gl i a b il i ti e s , c f . F i g u r e 5 . T h e c o e f f i c i e n t o f v a r i a t i o n i s 0 . 22 in t h e p r e d i c t i v ed i s tr i b u t io n o f th e o u t s t a n d i n g I B N R l ia b i li ti e s, w h i l e it is o n l y 0 . 0 9 f o r t h e R B N Sl i a b il i ti e s . I n t u i t i v e l y , t h is i s o b v i o u s ; o n l y t h e s i z e s o f t h e RB N S c l a i m s a r eu n k n o w n , w h i l e f o r t he I B N R c l a i m s a l s o th e n u m b e r o f c l a im s is u n k n o w n . A f t e r4 . 2 y e a r s th e o b s e r v e d r u n - o f f f o r b o t h t h e I B N R a n d t h e R B N S l ia b i li ti e s a r e b e l o wt h e 0 .2 q u a n t i l e o f t h e p r e d i c t i v e d i s t r ib u t i o n o f t h e o u t s t a n d i n g l i ab i l it ie s . T h a t w a st o b e e x p e c t e d : 1 0 o f t h e 6 8 o b s e r v e d I B N R c l a i m s w e r e n o t y e t s e t tl e d a f t e r 4 .2y e a r s , a n d t h e r e f o r e t h e r e st il l r e m a i n e d s o m e p a y m e n t s o n t h e I B N R c l a i m s . R e c a l lt h a t R B N S c l a i m s w i t h p r e v i o u s p a y m e n t s t e n d t o h a v e a l o n g s e t t l e m e n t d e l a y . O ft h e o b s e r v e d 6 1 7 R B N S c l a i m s , 1 82 w e r e n o t s e t tl e d a f t e r 4 .2 y e a r s . S o m e o f th e s ew i l l s e t t l e w i t h o u t f u r t h e r p a y m e n t s , b u t s t i l l , t h e o u t s t a n d i n g l i a b i l i t i e s w i l l e x c e e dt he o b s e r v e d 1 .9 1 0 .0 0 0 D K K . B y c o m b i n i n g t he s a m p l e d v a l u e s o f t he I B N R a n dt h e R B N S l i a b i l i t i e s a n d u s i n g ( 5 . 2 ) , w e g e t a n a p p r o x i m a t i o n t o t h e p r e d i c t i v ed i s tr i b u t io n o f th e o u t s t a n d i n g I B N R a n d R B N S l ia b i li ti e s, s h o w n a t t h e l as t g r a p hi n F i g u r e 5 . T h e m e a n a n d q u a n t i l e s a r e f o u n d i n T a b l e 2 .

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