UBC Theses and Dissertations

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UBC Theses and Dissertations

Topics in stochastic dominance : theory and application Kira, Dennis Seiho 1977

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TOPICS IN STOCHASTIC DOMINANCE: THEORY B . S c , Simon F r a s e r U n i v e r s i t y , 1970 H . S c , Simon F r a s e r U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e : F a c u l t y o f GRADUATE STUDIES (Management S c i e n c e D i v i s i o n ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r equ i red s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u n e , 1977 (c} Dennis Seiho Kira, 1977. AND APPLICATION by DENNIS SEIHO KIRA In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s for an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g or pub l i ca t ion of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 ABSTRACT The d i s s e r t a t i o n i n v e s t i g a t e s some i m p o r t a n t a s p e c t s of m a n a g e r i a l d e c i s i o n mak ing under c o n d i t i o n s o f u n c e r t a i n t y . In the l a s t t h r e e decades two p r o m i n e n t g e n e r a l app roache s have e v o l v e d t o d e a l e x p l i c i t l y w i t h r i s k i n m a n a g e r i a l d e c i s i o n s . They a r e : (1) t h e c e n t r a l t e n d e n c y - d i s p e r s i o n t r a d e o f f a p p r o a c h , a n d (2) e x p e c t e d u t i l i t y a n a l y s i s . The f i r s t t a s k u n d e r t a k e n i n t h i s i n v e s t i g a t i o n i s t o i n t e g r a t e t h e s e two a p p r o a c h e s . T h i s i s a c c o m p l i s h e d by i d e n t i f y i n g t h o s e s i t u a t i o n s i n w h i c h decision r u l e s o b t a i n e d by e i t h e r app r oach a r e e q u i v a l e n t . Once e q u i v a l e n c e between t h e two b a s i c app roache s t o d e c i s i o n mak ing under u n c e r t a i n t y i s e s t a b l i s h e d , t he f o c u s s h i f t s t o t he e x t e n s i o n o f t h e s e d e c i s i o n t h e o r i e s i n t o s i t u a t i o n s i n v o l v i n g m u l t i - a t t r i b u t e o u t -come s p a c e s . In p a r t i c u l a r , s t o c h a s t i c dominance r u l e s f o r m u l t i v a r i a t e outcome d i s t r i b u t i o n s a r e d e v e l o p e d . Two a p p l i c a t i o n s o f s t o c h a s t i c dominance c r i t e r i a a r e then p r e -s e n t e d , i l l u s t r a t i n g t h e r e l e v a n c e o f t he a p p r o a c h t o t h e o r y deve l opment and management o f r e s o u r c e s y s t e m s . The f i r s t i l l u s t r a t i o n d e m o n s t r a t e s t h e a p p l i c a t i o n o f s t o c h a s t i c dominance t o p o r t f o l i o d i v e r s i f i c a t i o n p r o b l e m s . S e v e r a l r e s u l t s a r e o b t a i n e d d e s c r i b i n g t h e s e n s i t i v i t y o f o p t i m a l m ixe s w i t h r e s p e c t t o changes i n o p p o r t u n i t i e s f o r i n v e s t m e n t . i i The second i l l u s t r a t i o n d e m o n s t r a t e s t h e r o l e s t o c h a s t i c dominance c r i t e r i a can p l a y i n e c o s y s t e m p o l i c y a n a l y s i s . A methodo l ogy o f s t o c h a s t i c dominance p o l i c y s c r e e n i n g f o r f o r e s t management s y s tems i s d e v e l o p e d and a p p l i e d . TABLE UF CONTENTS Page ABSTRACT i i L I ST OF TABLES v i L I ST OF FIGURES v i i ACKNOWLEDGMENT v i i i C h a p t e r 1 INTRODUCTION 1 2 EQUIVALENCE AMONG ALTERNATIVE PORTFOLIO SELECTION CRITERIA 9 3 MULTIVARIATE STOCHASTIC DOMINANCE 15 3.1 I n t r o d u c t i o n 15 3.2 E q u i v a l e n c e o f U t i l i t y F u n c t i o n s 17 3.3 I ndependent A t t r i b u t e s 19 3.4 Dependent A t t r i b u t e s 22 4 QUALITATIVE ASPECTS OF OPTIMAL PORTFOLIO BALANCING 28 4.1 I n t r o d u c t i o n 28 4 .2 One S a f e and One R i s k A s s e t s 30 4 . 3 The Two R i s k y A s s e t Case 44 5 METHODOLOGY DEVELOPMENT IN RESOURCE MANAGEMENT 49 5.1 I n t r o d u c t i o n 49 i v C h a p t e r Page 5.2 The F o r e s t Management Sys tem 50 5.3 The F o r e s t Sy s tem P r ob l em 52 5.4 D e r i v a t i o n o f Outcome D i s t r i b u t i o n s 58 5.5 The S c r e e n i n g A l g o r i t h m 59 5.6 S t o c h a s t i c Dominance T e s t R e s u l t s 61 5.7 S e n s i t i v i t y A n a l y s i s and Robu s tne s s o f R e s u l t s 70 5.8 C o n c l u s i o n s 73 SUMMARY 74 BIBLIOGRAPHY 75 APPENDIX 84 v LIST OF TABLES T a b l e Page I Po l i c y R e t u r n s 1 66 I I P o l i c y R e t u r n s 2 67 I I I Dominance Rank ing s o f P o l i c i e s Under Fou r P e r f o r m a n c e C r i t e r i a 68 IV Changes G r e a t e r than 10% i n Means and/o r S t a n d a r d D e v i a t i o n O b t a i n e d by T r u n c a t i n g P o l i c i e s 72 v i LIST OF FIGURES F i g u r e Page 4.1 I l l u s t r a t i o n o f a s s u m p t i o n s C l ' , C 2 1 and C 3 ' 35 4 .2 The o p t i m a l m ix f o r c h a n g i n g p o r t f o l i o c o m p o s i t i o n 36 5.1 The b a s i c s t r u c t u r e o f t h e budworm/fo re s t model 53 5.2 Dominance r e l a t i o n s d e r i v e d f o r budwo rm/ fo re s t p o l i c i e s - p r o f i t / y r a t 0% d i s c o u n t 62 5.3 Dominance r e l a t i o n s d e r i v e d f o r budwo rm/ fo re s t p o l i c i e s - p r o f i t / y r a t 10% d i s c o u n t 63 5.4 Dominance r e l a t i o n s d e r i v e d f o r budworm/ fo re s t p o l i c i e s - f r e q u e n c y w i t h no s p r a y i n g 64 5.5 Dominance r e l a t i o n s d e r i v e d f o r budwo rm/ fo re s t p o l i c i e s - employment s t a b i l i t y i n d e x 65 v i i ACKNOWLEDGMENT I t i s my g r e a t p l e a s u r e t o thank Dr . I I a n V e r t i n s k y , my t h e s i s s u p e r v i s o r , f o r h i s c o n t i n u o u s g u i d a n c e and encou ragement . To D r s . C .C . Huang, W.A. Thompson, G.A. Whi tmore and W.T. Z i emba , who u n d e r t o o k t h e t a s k o f r e a d i n g t h e e n t i r e t h e s i s , I o f f e r my t h a n k s . T h e i r c r i t i c i s m and h e l p f u l s u g g e s t i o n s have enhanced t h e e x p o s i t i o n immense l y . S p e c i a l t h ank s go t o Ms. C a r o l y n e Smart and Ms. Sharon H a l l e r . C a r o l y n e f o r e d i t i n g and Sharon f o r t y p i n g t h i s t h e s i s . I wou ld a l s o l i k e t o thank my f r i e n d s , f o r mak ing my s t a y a t U.B.C. an e n j o y a b l e one . F i n a l l y , I t hank Kaz and Min f o r t h e i r °° s u p p o r t . v i i i C h a p t e r 1 •INTRODUCTION The c o m p l e x i t i e s c h a r a c t e r i z i n g modern m a n a g e r i a l d e c i s i o n e n v i r o n -ments have i n t e n s i f i e d t h e need t o d e v e l o p a p p r o p r i a t e d e c i s i o n s t r u c t u r e s d e a l i n g e x p l i c i t l y w i t h u n c e r t a i n t i e s . A t y p i c a l c h a r a c t e r i z a t i o n o f a d e c i s i o n p r o b l e m under u n c e r t a i n t y i s as f o l l o w s : a d e c i s i o n maker must choose an a c t i o n f r om a s e t o f p o s s i b l e a c t i o n s a s s o c i a t e d w i t h a p r o b -a b i l i t y d i s t r i b u t i o n o v e r p o s s i b l e c o n s e q u e n c e s . T h i s p r o b l e m has a l o n g h i s t o r y d a t i n g back t o t he e a r l y work i n t h e f i e l d o f "games o f c h a n c e " [ 1 8 , 8 5 ] where t h e e x p e c t e d v a l u e was t h o u g h t t o be an a p p r o p r i a t e c r i t e r i o n i n i n d u c i n g t h e p r e f e r e n c e o r d e r among d i s t r i b u t i o n s . The f i r s t a t t e m p t t o d e a l f o r m a l l y w i t h the above p r o b l e m was made by B e r n o u l l i [ 6 ] who i n t r o d u c e d some o f t h e f undamen ta l i d e a s o f e x p e c t e d u t i l i t y t h e o r y . B e r n o u l l i ' s s o l u t i o n t o h i s now famous pa radox o f g a m b l i n g was t o choose a f o rm w i t h d i m i n i s h i n g m a r g i n a l u t i l i t y f o r t h e u t i l i t y f u n c t i o n ( e . g . c h o o s i n g t h e l o g a r i t h m i c u t i l i t y f u n c t i o n i n d i c a t e s t h a t m a r g i n a l u t i l i t y i s i n v e r s e l y p r o p o r t i o n a l t o w e a l t h ) . R i s k t a k i n g a t t i t u d e s were n o t e x p l i c i t l y i n c o r p o r a t e d i n t o h i s d e f i n i t i o n o f t h e f u n c t i o n . Two m a j o r s c h o o l s o f t h o u g h t have s u b s e q u e n t l y e v o l v e d a t t e m p t i n g t o d e a l w i t h r i s k a t t i t u d e s e x p l i c i t l y : 1 2 (1) C e n t r a l t e n d e n c y - d i s p e r s i o n ("mean-risk ana 1ys i s") (2) Expected u t i l i t y a n a l y s i s . C e n t r a l t e n d e n c y - d i s p e r s i o n a n a l y s i s p r e s u p p o s e s a r i s k measure and i n d i f -f e r e n c e c u r v e s between t h i s r i s k measure and e x p e c t e d p a y o f f s . The d e c i s i o n maker w i l l p r e f e r an a c t i o n ( o r a con sequence ) w h i c h i s a s s o c i a t e d w i t h a h i g h e r i n d i f f e r e n c e c u r v e ( " l e s s r i s k and more p a y o f f " ) . The re have been many p r o p o s a l s f o r a l t e r n a t i v e c o m b i n a t i o n s o f c e n t r a l t e n d e n c y and d i s p e r s i o n measurements . M a r k o w i t z [ 62 ] p r o p o s e d t h e v a r i a n c e as a d i s p e r s i o n measure i n t r a d e o f f w i t h e x p e c t e d p a y o f f (mean). Baumol [ 4 ] e x t e n d e d M a r k o w i t z ' s m e a n - v a r i a n c e c r i t e r i o n t o t h e mean - l owe r c o n f i d e n c e r u l e , where t he d i f f e r e n c e between t he mean and a c o n s t a n t m u l t i p l e o f s t a n d a r d d e v i a t i o n i s u sed as a r i s k measure . The p o s s i b l e use o f t h e semi v a r i a n c e , f i r s t s u g g e s t e d by M a r k o w i t z , was e x p l o r e d and d e v e l o p e d by Mao [ 6 1 ] . O t h e r r i s k measures were s u g g e s t e d by Roy [ 8 9 ] , and P h i l i p p a t o s and W i l s o n [ 7 4 ] . A s suming t h a t i n v e s t o r s a r e p r i n c i p a l l y c o n c e r n e d w i t h a v o i d i n g a p o s s i b l e d i s a s t e r , Roy s u g g e s t e d t h a t r i s k s h o u l d be d e f i n e d as t h e p r o b a b i l i t y o f o c c u r r e n c e o f a d i s a s t e r . The mean -en t r opy r u l e was r e c e n t l y p r opo sed by P h i l i p p a t o s and W i l s o n where h i g h e r e n t r o p y i m p l i e s h i g h e r r i s k . No a x i o m a t i c s y s t em u n d e r l i e s t h e s e p r o p o s a l s f o r m e a n - r i s k a n a l y s i s . They a r e p r i m a r i l y based upon o b s e r v a t i o n s o f d e c i s i o n h e u r i s t i c s . In c o n t r a s t t h e d e v e l o p m e n t o f e x p e c t e d u t i l i t y t h e o r y has empha s i z ed t h e deve l opment o f a s a t i s f a c t o r y n o r m a t i v e a x i o m a t i c s y s t em t o d e s c r i b e " r a t i o n a l " c h o i c e . E x p e c t e d u t i l i t y t h e o r y i s based on t h e p r e m i s e t h a t d e c i s i o n mak ing i s p r i m a r i l y a s u b j e c t i v e p r o c e s s and t h e a p p r o p r i a t e c r i t e r i o n i s a " m o r a l e x p e c t a t i o n . " H e r e , t h e t e rm " m o r a l 3 e x p e c t a t i o n " means t h a t v a l u e judgement s on con sequence s and/o r p r o b -a b i l i t i e s have been i n c o r p o r a t e d i n c a l c u l a t i n g t he e x p e c t a t i o n . Under t h i s t h e o r y , t he v a l u e o f con sequence s i s measured by a u t i l i t y f u n c t i o n and t he i n f o r m a t i o n on t he d i s t r i b u t i o n o f con sequence s i s r e p r e s e n t e d by t he s u b j e c t i v e p r o b a b i l i t y . The deve l opment o f t h e weake s t a x i o m a t i c s y s t em i s n o t a t r i v i a l p r o b l e m s i n c e a c o m p l e t e l y p r e o r d e r e d m i x t u r e s e t does n o t g u a r a n t e e t h e e x i s t e n c e o f a u t i l i t y f u n c t i o n [ 1 9 , 2 5 , 3 9 , 1 1 0 ] . The most i m p o r t a n t t h e o r e t i c a l de ve l opmen t i n t h i s r e s p e c t i s t h e work o f Von Neumann and M o r g e n s t e r n who l a i d t he f o u n d a t i o n f o r a c o m p r e h e n s i v e t h e o r y o f " r a t i o n a l d e c i s i o n m a k i n g " f a c i n g u n c e r t a i n t y [ 1 1 0 ] . U n f o r t u n a t e l y , t he a p p l i e d v a l u e o f t h e i r work i s somewhat c o n s t r a i n e d by t h e f a c t t h a t t he f o c u s o f t h e t h e o r y i s upon i n d i v i d u a l d e c i s i o n - m a k e r ' s p r e f e r e n c e p r o f i l e s , hence t h e t h e o r y does n o t a p p l y t o o r g a n i z a t i o n a l d e c i s i o n s and measurement p r o c e d u r e s d e r i v e d on t h e b a s i s o f t h e t h e o r y a r e d i f f i c u l t t o a p p l y and a r e o f t e n u n r e l i a b l e [ 1 0 5 ] . R e c e n t l y , u t i l i z i n g t h e con sen su s on t h e p r o p e r t i e s o f u t i l i t y f u n c t i o n s [ 3 , 8 1 ] , v a r i o u s a u t h o r s have p r opo sed and d e v e l o p e d d e c i s i o n r u l e s w h i c h r e q u i r e l i m i t e d knowledge o f u t i l i t y f u n c t i o n s and a r e c o n s i s t e n t w i t h e x p e c t e d u t i l i t y t h e o r y [ 3 0 , 3 5 , 3 6 , 6 3 , 8 4 , 8 7 , 1 0 8 , 1 1 1 ] . These a u t h o r s have p r o p o s e d t h a t c e r t a i n c h a r a c t e r i s t i c s o f u t i l i t y f u n c t i o n s a r e common t o a l a r g e number o f d e c i s i o n makers and can be v e r i f i e d e a s i l y . These c h a r a c t e r i s t i c s d e f i n e f o u r c l a s s e s o f u t i l i t y f u n c t i o n s . (1) [ii = { u j u i s an i n c r e a s i n g u t i l i t y f u n c t i o n } , [ 3 0 , 3 5 , 3 6 , 6 3 , 8 4 , 8 7 ] . (2 ) U2 = { u j u i s a concave u t i l i t y f u n c t i o n } n U i , [ 3 0 , 3 5 , 3 6 , 6 3 , 8 4 , 8 7 ] . (3) U 3 = (u|u has a n o n - n e g a t i v e t h i r d d e r i v a t i v e } n U2, [ 1 0 0 , 1 1 1 ] . (4) U D = { u | r ' = - u < 0} n U 2 , [ 3 , 8 1 , 1 0 6 , 1 0 7 , 1 0 8 ] , 4 F o r each c l a s s , r u l e s w h i c h p e r m i t some c o m p a r i s o n between p r o b a b i l i t y d i s t r i b u t i o n s o f con sequence s have been d e v i s e d . The r u l e s f o r compa r i n g c u m u l a t i v e d i s t r i b u t i o n s o f con sequence s f o r each o f t h e above c l a s s e s o f u t i l i t y f u n c t i o n s a r e known as (1) F i r s t degree Stochast ic Dominance (FSD) (2) Second degree Stochast ic Dominance (SSD) (3) Th ird degree Stochast ic Dominance (TSD) (k) DARA dominance. The s t o c h a s t i c dominance o r d e r i n g r u l e s a r e c o n s i s t e n t w i t h t h e o r d e r i n g based upon e x p e c t e d u t i l i t y . The random v a r i a b l e X i s s a i d t o domina te t he random v a r i a b l e Y f o r t he u t i l i t y f u n c t i o n u i f and o n l y i f E F u (X) u ( x ) d F ( x ) > u ( x ) d G ( x ) - Eg u ( Y ) . S t o c h a s t i c dominance r u l e s f o r c l a s s e s Ui, U2, U 3 and U Q a r e l i s t e d be l ow . L e t F = {F|F i s a c u m u l a t i v e d i s t r i b u t i o n o f a random v a r i a b l e x) and U = {u| u: R+R, udF < °°, F e F}, t hen t h e s t o c h a s t i c dominance o r d e r i n g c o r r e s p o n d i n g t o Ui, U2, U 3 and Up c o n t a i n e d i n U a r e : (1) L e t F, G e F, t hen F dom ina te s G i n f i r s t deg ree (>i) i f and o n l y i f F ( x ) < G (x ) f o r a l l x e R and a s t r i c t i n e q u a l i t y h o l d s f o r some x e R. (2) L e t F, G e F, t hen F dom ina te s G i n second deg ree (> 2 ) rx i f and o n l y i f rX F ( t ) d t < G ( t ) d t f o r a l l x e R and a s t r i c t i n e q u a l i t y h o l d s f o r some x e R. 5 (3) Let F, G e Ft then F dominates G i n t h i r d degree (> 3) i f and on ly i f rx ry ( i ) ( i i ) F(z)dzdy < G(z)dzdy f o r a l l x e R and a s t r i c t i n e q u a l i t y holds f o r some x e R, and xdF(x) > xdG(x). Let H = F - G e S n i f there e x i s t s (n+1) i n t e r v a l s [ 0 , a i ) , [ a i , a 2 ) , * * * , [ a ,°°) w i th 0 < aj < a 2 , * * * s < a n such t h a t : i ) In each i n t e r v a l , H | 0 and have constant s igns and i i ) The s i gn a l t e r n a t e s between succes s i ve i n t e r v a l s . Then the f o l l o w i n g r u l e ho lds . (4) i ) I f H E S n ( a 1 , " » , a ) and H < 0 on [ C a J , then F »>, G. i i ) I f H E S 2 ( a ! , a 2 ) , then F >Q G i f and on ly i f -EpU > EgU f o r a l l concave exponent ia l u. i i i ) I f H £ S 3 ( a i , a 2 , a 3 ) , l e t b = Sup{z|z < a 3 ; H(x)dx = 0} then F >Q G i f and on ly i f EpU > EgU on [0,b] f o r a l l concave exponent ia l u. i v ) I f H £ S ? n ( a ! , a 2 , • • • , a 2 n ) w i th n > 2, then F > p G H(x) u 1 ( x )dx > 0 f o r a l l rX 3 2n ' i f and on ly i f where wi u ' ( x ) = u ' (a ) exp [ - r ( y ) d y ] Ja r ( y ) = k i , x £ [O .y i ) = k.j, x £ [ y ^ - j . y ^ . i=2, — , n - l = k n , x £ [ y n _ v « 0 , th k i > " ->k n >0 and y . £ ( a 2 - , a 2 i + 1 ) » i= l ,2 ,• • • , n - l 6 (5) If H e S 2 n + ] ( a i , a 2 , " * , a 2 n + 1 ) with n > 2, l e t b = r°° Sup{z|z > a 2ri+T H(x)dx = 0>, then F >D G i f and only J z i f F >p G on [0,b] e S2m with m < n. Each of the classes of u t i l i t y functions U i , U 2 , U 3 and Up represent preference structures considered interest ing from the economic point of view [3], and many of the studies in stochastic dominance have been concerned with U i , U 2 , and Up. The stochastic dominance orderings related to Ui and U 2 have a long history dating back at least to Hardy-Littlewood-Polya [38]. They induced a second degree stochastic dominance on a set of simple measures defined on a one-dimensional space. Other important mathematical develop-ments concerning f i r s t and second degree stochastic dominance are included in Sherman [96], Blackwell [9], Lehmann [48], Cartier-Fel1-Meyer [16] and Strassen [99]. The progress made by these authors was the general ization of consequence space from one-dimensional to a multi-dimensional. The results in the mathematical l i t e r a t u r e c i ted above were l a te r reintroduced to the economic l i t e r a t u r e by Masse* and Mori at [63], Quirk and Saposnik [84], Hadar and Russell [30], Hanoch and Levy [36], Fishburn [26] and Hammond [35]. In a l l these references the consequence space was taken to be a subset of Rn, the n-dimensional real space except for Fishburn 1s work which extended the results to a set of f i n i t e consequences. If one accepts the argument in Arrow-Pratt [3,81] that the r i s k -averse decision maker should have non-increasing absolute r i sk-avers ion ( r ' ( x ) < 0), then the stochastic dominance concept should be extended to the class Up. Whitmore [111] i n i t i a t e d th i s l i ne of invest igat ion 7 by c o n s i d e r i n g t he c l a s s U 3 3 Up. The a n a l y s i s was l a t e r c o m p l e t e d by V i c k s o n [ 1 0 8 ] . S e v e r a l a u t h o r s have a t t e m p t e d t o p r o v i d e a t h e o r e t i c a l b a s i s t o c e n t r a l t e n d e n c y - d i s p e r s i o n a p p r o a c h e s t o r i s k d e c i s i o n a n a l y s i s . These a t t e m p t s c o n s i d e r e d c o n d i t i o n s under w h i c h c r i t e r i a o b t a i n e d by e m p l o y i n g c e n t r a l t e n d e n c y - d i s p e r s i o n a n a l y s i s a r e e q u i v a l e n t t o employment o f m a x i m i z a t i o n o f e x p e c t e d u t i l i t y . P h i l i p p a t o s and G r e s s i s [ 73 ] showed t h a t t h e m e a n - v a r i a n c e , second deg ree dominance and m e a n - e n t r o p y c r i t e r i a a r e e q u i v a l e n t when t h e random v a r i a b l e s have u n i f o r m o r normal d i s t r i b u t i o n s . Hanoch -Levy [ 3 6 ] showed t h a t when random v a r i a b l e s a r e r e s t r i c t e d t o d i s -t r i b u t i o n s h a v i n g two p a r a m e t e r s t h a t a r e i n d e p e n d e n t monotone f u n c t i o n s o f mean and v a r i a n c e , SSD i s e q u i v a l e n t t o t h e m e a n - v a r i a n c e c r i t e r i o n . The e q u i v a l e n c e between t h e mean-semi v a r i a n c e and t h e s t o c h a s t i c dominance c r i t e r i a has been a l s o i n v e s t i g a t e d by P o r t e r [ 76 ] and Jean [ 4 4 ] . P o r t e r showed t h a t second deg ree dominance i m p l i e s semi v a r i a n c e p r e f e r e n c e under r e a s o n a b l y g e n e r a l c o n d i t i o n s . Jean showed t h a t under c e r t a i n e x t r e m e l y r e s t r i c t i v e c o n d i t i o n s , t h i r d deg ree dominance i m p l i e s mean-semi v a r i a n c e p r e f e r e n c e . The f i r s t t a s k w h i c h i s u n d e r t a k e n i n t h i s d i s s e r t a t i o n ( C h a p t e r 2) i s t o b r i d g e t h e r e m a i n i n g gaps between c r i t e r i a d e r i v e d on t h e b a s i s o f c e n t r a l t e n d e n c y - d i s p e r s i o n a p p r o a c h e s and t h o s e d e r i v e d on t he b a s i s o f e x p e c t e d u t i l i t y t h e o r y . A summary theo rem p r e s e n t s e q u i v a l e n c e r e l a t i o n -s h i p s among t h e v a r i o u s c r i t e r i a o f c h o i c e f o r a v a r i e t y o f i m p o r t a n t p r o b a b i l i t y d i s t r i b u t i o n s o f c o n s e q u e n c e s . The a p p e n d i x p r e s e n t s some a d d i t i o n a l r e s u l t s on e q u i v a l e n c e s between s t o c h a s t i c dominance c r i t e r i a and c h o i c e c r i t e r i a based upon p a r t i c u l a r p a r a m e t e r s o f a d i s t r i b u t i o n . 8 The a p p l i c a b i l i t y o f choice r u l e s developed by e i t h e r approach has been l i m i t e d by the focus upon s i n g l e a t t r i b u t e consequences. Yet many important d e c i s i o n s i t u a t i o n s are described by a m u l t i - a t t r i b u t e out-come space. Levy [51] and Levy and Paroush [56,57] pioneered i n the economic l i t e r a t u r e i n developing s t o c h a s t i c dominance r u l e s f o r comparing m u l t i -v a r i a t e d i s t r i b u t i o n s . Huang et al. [43] extended these r e s u l t s to other c l a s s e s of u t i l i t y f u n c t i o n (U 3 and U^). Further extensions, focusing p r i m a r i l y on U 2, were made by Lehmann [48], Sherman [96], Strassen [99], Meyer [65], Brummelle-Vickson [15] and Levhari et al. [50]. Unfortunately these authors provided necessary and s u f f i c i e n t c o n d i t i o n s which are d i f f i c u l t to apply. The prime c o n t r i b u t i o n of Chapter 3 i s to f u r t h e r g e n e r a l i z e the r e s u l t s i n Levy [51], Levy and Paroush [56] and Huang et al. [43] hence extending the a p p l i c a b i l i t y of s t o c h a s t i c dominance. The l a s t two chapters present a p p l i c a t i o n s of s t o c h a s t i c dominance. Chapter 4 presents a t h e o r e t i c a l a p p l i c a t i o n of s t o c h a s t i c dominance to two period p o r t f o l i o s e l e c t i o n problems. In p a r t i c u l a r the chapter inves-t i g a t e s the e f f e c t s upon p o r t f o l i o composition of: (1) i n t r o d u c t i o n o f a s t o c h a s t i c a l l y d o m i n a t i n g s e c u r i t y to the c h o i c e s e t , and (2) a change in l o c a t i o n and s c a l e parameters of a s e c u r i t y in t h a t s e t . Chapter 5 presents an a p p l i c a t i o n of s t o c h a s t i c dominance r u l e s to a system's management problem. This chapter adds to a p p l i c a t i o n s of s t o c h a s t i c dominance an important area c h a r a c t e r i z e d by u n c e r t a i n t i e s , the area of ecosystem management. Chapter 2 EQUIVALENCE AMONG ALTERNATIVE PORTFOLIO SELECTION CRITERIA In t h i s c hap te r we attempt to integrate choice c r i t e r i a for s ituat ions involving uncertainty: those which are derived from the dichotomy of central tendency-dispersion and those which are based on maximization of expected u t i l i t y . We f i r s t provide the formal de f in i t i ons of the various c r i t e r i a and then the summary theorem representing the i r in ter re la t ionsh ips . Let X and Y be random variables with r ight continuous cumulative d i s t r i bu t i on functions F and G having f i n i t e means y , y_ and f i n i t e non-zero variances a 2 p and a 2 g , respect ively. Then the fol lowing de f in i t i ons can be stated. Mean-Variance (E - V). X i s preferred to Y in (E - V) i f f ( i ) y p > y Q , and ( i i ) a 2 p < a 2 G with s t r i c t inequal i ty holding for at least one inequal i ty. As a select ion ru l e , th i s c r i t e r i on has serious drawbacks [3,4,8,11,24,30,36,84,87]. In pract ice, however, i t i s a very popular and useful rule because of i t s s imp l i c i t y . For theoret ical support of E-V r u le , see [8,12,53,94,103,104]. 9 10 Mean-Lower C o n f i d e n c e (E - L ) . Baumol [ 4 ] i n t r o d u c e d t h i s r u l e i n o r d e r t o e l i m i n a t e some s h o r t c o m i n g s i n t he E - a 2 r u l e . L e t Lp = ^ -aap and Lg = - a o g , where a > 0 i s a g i v e n c o n s t a n t . Then X i s s a i d t o be p r e f e r r e d t o Y i n E - L i f f ( i ) Up > U G ( i i ) L F > L Q w i t h s t r i c t i n e q u a l i t y h o l d i n g f o r a t l e a s t one i n e q u a l i t y . Moeseke [ 6 6 ] i n d e p e n d e n t l y p r opo sed t h i s c r i t e r i o n and c a l l e d i t t h e t r u n c a t e d min imax c r i t e r i o n . Fo r a r e c e n t s t a t e m e n t o f t h e o r y o f t h i s method and an a l g o r i t h m f o r c a l c u l a t i n g t he e f f i c i e n t f r o n t i e r , see [ 6 7 ] . A more e f f i c i e n t a l g o r i t h m appea r s i n Hohenba lken [ 4 1 ] . Hanoch and Levy [ 3 7 ] c r i t i c i z e t h i s r u l e s t a t i n g t he f a c t t h a t a v a i l a b l e i n f o r m a t i o n f o r t h e c h o i c e o f an o p t i m a l a i s n o t u t i l i z e d . In a d d i t i o n , E - L i s no t e n t i r e l y an e f f i c i e n t c r i t e r i o n s i n c e i t depends on i n d i v i d u a l t a s t e s t h r o u g h t he c h o i c e o f a . rh M e a n - S e m i v a r i a n c e (E - S V ( h ) ) . L e t SVp(h) = ( x - h ) 2 d F ( x ) and S V G ( h ) E rh -oo ( x - h ) 2 dG(x ) be t he s e m i v a r i a n c e o f F and G. Then X i s p r e -f e r r e d t o Y i n E - SV(h) i f f (i.) Up > U G ( i i ) S V p ( h ) < s v G ( h ) , where one i n e q u a l i t y i s s t r i c t . M a r k o w i t z [ 6 2 ] c o n s i d e r e d t h e s e m i v a r i a n c e as a r i s k measu re , b u t because o f c o m p u t a t i o n a l d i f f i c u l t i e s he r e j e c t e d i t , . Mao [ 6 1 ] a p p l i e d t h i s r u l e t o a c a p i t a l b u d g e t i n g p r o b l e m . An e f f i c i e n t 11 a l g o r i t h m t o compute t he e f f i c i e n t f r o n t i e r i n mean-semi v a r i a n c e space a p p e a r s i n Hogan-Warren [ 4 0 ] . M e a n - A s p i r a t i o n (E - F ( s ) ) . L e t s be t h e i n d i v i d u a l ' s a s p i r a t i o n o r t a r g e t l e v e l . Then X i s s a i d t o be p r e f e r r e d t o Y i n E - F ( s ) i f f ( i ) y p > u G ( i i ) F ( s ) < G ( s ) , where one i n e q u a l i t y i s s t r i c t . T h i s c r i t e r i o n i s s i m i l a r i n s p i r i t t o t he chance c o n s t r a i n e d r u l e s [ 7 2 ] . E - SV(h) and E - F ( s ) r u l e s s u f f e r f r o m a s i m i l a r d e f i c i e n c y t o t he E - L r u l e (Hanoch and Levy [ 3 7 ] ) . l n ( F ' ( x ) ) d F ( x ) and H Q = j — CO Mean -En t r opy (E - H ) . L e t Hp E -l n ( G ' ( x ) ) d G ( x ) be t he e n t r o p i e s o f X and Y, r e s p e c t i v e l y . Then X i s p r e -f e r r e d t o Y i n E - H i f f ( i ) y p > u G ( i i ) H p < H Q w i t h s t r i c t i n e q u a l i t y h o l d i n g f o r a t l e a s t one i n e q u a l i t y . In g e n e r a l , H i s l a r g e whenever t h e d i s t r i b u t i o n s a r e c o m p l e t e l y random, i . e . d i s t r i b u t i o n s a r e u n i f o r m . S t o c h a s t i c Dominance L e t F : ( x ) = F ( t ) d t , G : ( x ) = j — CO fX G ( t ) d t F 2 ( x ) = F i ( y ) d y and G 2 ( x ) = i-x G i ( y ) d y , 12 then X is said to dominate Y in . a) (FSD) i f f F(x) < G(x) for a l l x e R and F ( x 0 ) < G(x 0 ) for some x 0 £ R, b) (SSD) i f f F i ( x ) < Gi(x) for a l l x e R and F i ( .x 0 ) < G i ( x 0 ) for some x 0 e R, c) (TSD) i f f F 2 (x) < G 2 (x) for a l l x e R, with s t r i c t inequal i ty Vip > yg holding for at least one inqua l i t y . Theorems which re late the above rules to expected u t i l i t y theory can be found in [30,36,84,87,111]. We can now state the fol lowing theorem which demonstrates the equivalence of various e f f i c i e n t sets. Theorem 2.1, Let the random variables X and Y have d i s t r ibut ions F and G, respect ively. Suppose (1) F i n t e r s e c t s G o n l y a t yo-(2) F and G are members o f a two parameter f a m i l y whose parameters a r e independent i n c r e a s i n g f u n c t i o n s o f mean and v a r i a n c e . (3) The a s p i r a t i o n and s e m i v a r i a n c e l e v e l s s and h a r e l e s s t h a t yo-Then holds, 13 where t he s o l i d l i n e deno te s g e n e r a l i m p l i c a t i o n s and t h e d o t t e d l i n e deno te s t h e u n i f o r m and normal c a s e s . PROOF: (1) See [ 3 6 ] , (2) L e t F > 2 G, t hen rx [ G i ( t ) - F i ( t ) ] d t > 0 f o r a l l x e R. Hence by i n t e g r a t i o n by p a r t s S V c ( h ) - S V „ ( h ) rh f x F G - 2 [ G ( t ) - F ( t ) ] d t dx < 0. Thus X dom ina te s Y i n E - S V ( h ) . Now suppose X dom ina te s Y i n E - S V ( h ) , t h e n rh rX [ G ( t ) - F ( t ) ] d t dx > 0 and [ G ( t ) - F ( t ) ] d t > 0 . L e t H ( t ) = G ( t ) - F ( t ) . The re a r e two p o s s i b i l i t i e s : e i t h e r H ( t ) > 0 and changes t o H ( t ) < 0 o r H ( t ) < 0 and changes t o H ( t ) > 0. I t i s i m p o s s i b l e f o r H ( t ) ' s i n i t i a l s i g n t o be n e g a t i v e and ^ f* H ( t )dtdx > 0 s i n c e h < y 0 . I » —CO —oo I f H ( t ) s i n i t i a l s i g n i s p o s i t i v e t h e n rx H ( t ) d t > 0 f o r a l l x e R s i n c e H ( t ) d t > 0 and o n l y one s i g n change i s p o s s i b l e f o r x -oo H ( t ) d t . (3) F o l l o w s f r om 1 and 2. (4) Suppose X ( > 2 ) Y , t h e n y p > u G and F ( x ) < G ( x ) f o r a l l x < y" 0 Thus F ( s ) < G ( s ) s i n c e s < y 0 . I f Up > U Q and F ( s ) < G ( s ) t h e n F ( > 2 ) G . T h i s f o l l o w s f r o m [ 3 6 ; Thm. 3 ] . (5) F o l l o w s f r o m 2 and 4. (6) F o l l o w s f r o m 1 and 4. 14 (7) Let F(x) b F - a p and G(x) = T-^—• , then Hc= lnflv-a,-) b G - a G F F F and H G = 1n(b G -a G ) . Therefore F(E-V)G «=» F(E-H)G. S imi lar impl icat ion follows for F(x) 1 (2TOV and G(x) l (2Tr)'af exp exp dt dt Then the equivalence follows from 1 (8) Follows from 4 and 7. (9) Since L p - L g - ( y p - ao^) - ( y p - a<3g) = ( u F - yg ) + a(a.g - a p ) . (10) This follows immediately from the assumption (1) and the def in i t ions of SSD and TSD. Let the sets of e f f i c i e n t por t fo l io s under E - V, E - L, SSD, TSD, E - F(s), E - SV(h) and E - H be e(E - V), e(E - L ) , e(SSD), e(TSD), e(E - F(s ) ) , e (E - SV(h)) and e(E - H) respect ively, then e(TSD) = e(SSD) = e(E - V) = e(E - F(s)) = e(E - SV(h)) c e(E - L) e(SSD) = e(TSD) = e(E - H) = e(E - F(s) ) . C h a p t e r 3 MULTIVARIATE STOCHASTIC DOMINANCE 3.1 I n t r o d u c t i o n In C h a p t e r 2 we have a t t e m p t e d t o i n t e g r a t e t h e p r e v i o u s r e s u l t s i n t h e a r e a o f c h o i c e under u n c e r t a i n t y w i t h a u n i v a r i a t e outcome s p a c e . In t h i s c h a p t e r we g e n e r a l i z e r e s u l t s i n s t o c h a s t i c dominance t h e o r y t o c h o i c e s i n v o l v i n g m u l t i v a r i a t e outcome s p a c e s . The m u l t i v a r i a t e d e c i s i o n p r o b l e m i n v e s t i g a t e d i s as f o l l o w s . L e t Fn - { F n | F n i s t h e j o i n t d i s t r i b u t i o n o f random v e c t o r X = ( X i , " * , X )} and U n = { u n | u n i s a m u l t i - a t t r i b u t e u t i l i t y f u n c t i o n } . Then , what i s a p a r t i a l o r d e r i n g > p on F n , G n e F n such t h a t F n > p G n i f and o n l y i f E u n ( X ) > E u n ( Y ) f o r a l l u n e U p c U n and > f o r some u n , where p d e t e r m i n e s t h e p r o p e r t y o f u t i l i t y f u n c t i o n u n ? . S e v e r a l a t t e m p t s have been made r e c e n t l y t o e x t e n d t h e g e n e r a l f ramework o f s t o c h a s t i c dominance f r om t h e s i n g l e v a r i a b l e t o t h e m u l t i -v a r i a t e c a s e . Levy [ 5 1 ] d e v e l o p e d s u f f i c i e n t r u l e s f o r f i r s t and second deg ree dominance when u t i l i t y f u n c t i o n s a r e d e f i n e d on t e r m i n a l w e a l t h and t h e r e i s i ndependence among outcomes i n d i f f e r e n t p e r i o d s . Levy and Pa r ou sh [ 5 6 ] e x t e n d e d t h e s e r e s u l t s f o r r u l e s o f f i r s t deg ree dominance o m i t t i n g t he i ndependence r e q u i r e m e n t . They a l s o d e v e l o p e d n e c e s s a r y and 15 16 s u f f i c i e n t r u l e s f o r f i r s t deg ree dominance f o r a d d i t i v e u t i l i t y f u n c t i o n s . Huang et al. [43] e x t e n d e d a l l t h e r e s u l t s i n Levy [51] and t h e r e s u l t s o b t a i n e d f o r a d d i t i v e u t i l i t y f u n c t i o n s i n Levy and Pa rou sh [56] t o c l a s s e s U i , U 2 , U 3 and U Q . One can a l s o e a s i l y o b t a i n t h e a d d i t i v e r e s u l t s u t i l i z i n g F i s h b u r n ' s work [25] . He showed t h a t u n s e p a r a t e s i n t o an a d d i t i v e f o rm i f t h e d e s i r a b i l i t y o f any l o t t e r y X depends o n l y on t he m a r g i n a l p r o b -a b i l i t y d i s t r i b u t i o n s o f t h e X ^ ' s and n o t on t h e i r j o i n t d i s t r i b u t i o n s . Of c o u r s e , t h e r e s t r i c t i o n imposed by t h e a d d i t i v i t y o r t h e i ndependence a s s u m p t i o n s on t h e members o f U n , Fn r e s p e c t i v e l y , s e v e r e l y l i m i t s t h e u s e f u l n e s s o f such a r e s u l t . Lehmann [ 48 ] , Sherman [ 96 ] , S t r a s s e n [99 ] , Meyer [65] and B r u m e l l e - V i c k s o n [15] a n a l y z e d s t o c h a s t i c o r d e r i n g s i n g e n e r a l s e t t i n g s . In t h e i r s t u d i e s t h e u t i l i t y f u n c t i o n s were assumed t o be e i t h e r n o n d e c r e a s -i n g o r n o n d e c r e a s i n g and concave w i t h o u t a d d i t i o n o f any o t h e r a s s u m p t i o n s . B r u m e l l e - V i c k s o n base t h e i r a n a l y s i s on a d i f f e r e n t c h a r a c t e r i z a t i o n o f s t o c h a s t i c dominance c o n d i t i o n s due t o R o t h s c h i l d and S t i g l i t z [87 ] . L e v h a r i et at. [50] p r o v i d e s u f f i c i e n t and n e c e s s a r y c o n d i t i o n s f o r t h e g e n e r a l c a se o f monotone i n c r e a s i n g ( f i r s t deg ree dominance) and q u a s i c oncave u t i l i t y f u n c t i o n s . U n f o r t u n a t e l y , t h e c o n d i t i o n s d e v e l o p e d i n t h e above s t u d i e s a r e d i f f i c u l t t o a p p l y . In ou r a n a l y s i s we i n v e s t i g a t e s t o c h a s t i c dominance r u l e s f o r m u l t i - a t t r i b u t e u t i l i t y f u n c t i o n s . A f t e r e s t a b l i s h i n g (by means o f theo rem 3.3.1) t h e e q u i v a l e n c e o f r u l e s f o r m u l t i v a r i a t e u t i l i t y f u n c t i o n s u = u n ( x i , • • • , x n ) and u n i v a r i a t e u t i l i t y f u n c t i o n s d e f i n e d on m u l t i v a r i a t e outcome space u = (cj)(xi,«»«,x ) ) , we f o c u s upon t he l a t t e r . 17 3.2 Equiva lence of U t i l i t y Funct ions Def ine the f o l l o w i n g c l a s s e s of m u l t i - a t t r i b u t e u t i l i t y f u n c t i o n s . i } i } < 0 f o r a l l i } These c l a s s e s correspond to the important c l a s se s of u t i l i t y f unc t i on s i n v e s t i g a t e d e x t e n s i v e l y i n the l i t e r a t u r e f o r the s i n g l e v a r i a b l e case. In t h i s s e c t i on we show tha t they are i d e n t i c a l to the f o l l o w i n g c l a s s e s : V" = { u n ( x i , - - « , x n ) = u . ( ( ( ) ( x i > «« - , x n ) ) |u i e U. and <J> e u"} f o r each i= l ,2 ,3,D. Theorem 3.2. I 1 V? = u" f o r i = l ,2 ,3 ,D. PROOF: F i r s t we show v" c uJ f o r i= l ,2 ,3,D. ^ I t i s assumed t h a t the c h a i n r u l e o f d i f f e r e n t i a t i o n h o l d s . For s u f f i c i e n t c o n d i t i o n s , see e.g. [112]. We a l s o assume t h a t the f u n c -t i o n s u , u. and <j> a r e t h r i c e d i f f e r e n t i a t e u n l e s s s t a t e d o t h e r w i s e . Ui = { u n | ~ - > 0 f o r a l l i } 3 x i " 9 n ,,n r rii n ,,n , d u n £ , U 2 = {u |u e Hi and -^-—^ < 0 f o r a l i : ,,n r n| n ,,n , 3 3 u n n x n U 3 = i u , |u e U 2 and ^ - y > 0 f o r a l i UJ = {u n|u e U3n and g f -. 2 ,n ~ n 9 x . 2 / 8 x . 18 (1) (2) (3) 3LT 3u i U 3x i 3cj) 3x i > 0 since e Ux and <$> e u". 3 2 u n ~3xT 3 2U, 3c(>< 3(j) 9 X i 3u i 3 <ft 9<j) 3x. < 0 since u... e U 2 and <j) e U 2. ^3 n 3 u 3x. a 3 3u, 34>; 3<|) 3x, 3 2u, " ^ i 3 u . 33(j) 3 1 9 <1> 9^_ , _ L „ 9(J)Z 3 x . 2 9x i 3c|> l^TT > 0 since u. e U 3 and cj) e U 3 . (4) 9 X i 3 u ,9u 3x. 2 / 3x. ^3 n „ n 3 u 9u 9x. 3 9x. 9 2 u ^ 2 9 x i 2 3iT 3x. By subst i tut ing (1) and (3) 3 u 3u 3x i 3 9x. 3 3u. 3u. l 1 9<j)3 3(f) 3d) 3x. + 3 3 z u. 3u. „ 2 , ]_ T_ d (p 9<J>2 3<j) 9x..2 9(j) 9x. 3u. [9* J : 9a(j) 9(j) 9x. 3 3x. f 3 u-i i 2 '3(j) _9(j)2 j 3x. + 3 9 V 3 U i ^ 9cj>2 3c|) 9xj 3(j> 9X. 1 9u, [9CJ, j N2A~I 3x-since u. e U n and cj) e IL. i D r D 2,,n 3 zu 9x,« 3 2u. 3u i i 3'cj) 9cJ>2 3cj) 3x. 3(j) 3x. 3 2 u n ^ 2 3X,2 19 Therefore 3x. ^2 n „ n 9 u ,3u 3 x . 2 / 3 x . < 0. Hence v " c u" for i=l,2,3,D. Lett ing u.(<|>) = cb and u. e U. y ie lds v " = u " , for i = l,2,3,D. In what follows the stochastic dominance rules are obtained for the classes V?, i=l,2,3,D. This i s done mainly for consistency with respect to the current l i t e r a t u r e . 3.3 Independent Attr ibutes Consider two pol icy options, with impacts upon at t r ibute levels ( X i , * * * , X ). The impacts of these po l i c ie s are described by the cumula-t i ve d i s t r ibut ions F n = F ( x i , * " , x ) for the f i r s t option and 6 n = G ( x i , « " for the second. In th i s section we consider only the case of independence of x^ ' s 2 n Lernma 3.3.1 Let u^ e U- and <p e IK, then H(x ) = for i=l,2,3,D. u. (*) d F n _ 1 e U. PROOF: By d i f f e r e n t i a t i o n , (1) (2) 9H_ 3x. 3 2H 3 x 2 3 x n since u.. (cb) e Ui since u.{§) e u" We assume t h a t the i n t e r c h a n g e o f i n t e g r a t i o n and d i f f e r e n t i a -t i o n i s 1 e g i t i m a t e , . f o r d e t a i l s see e.g. Loeve [ 6 0 ] . 20 (3) 3 3 H 3 x 3 n 33u.(<j>) . 1 . 9 x n d F n - ] > 0 s i n c e (cj)) e U 3 (4) 3H 33 H 3 x n 3 x n ^ ( c p ) u r 3x H F n - l T~ dF 3x n 3u.((i>) 3 3 u i c c p r i 2 3x„ 3x n 3 (by t h e Cauchy S c h w a r t z i n e q u a l i t y ) ^ V * * d F n - l " 9 x n ( s i n c e u.O) e u[J) 3 2 H T h e r e f o r e , H(x ) e IK f o r i = l , 2 , 3 , D . The f o l l o w i n g theo rem g i v e s n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r F n t o dom ina te G n . F i r s t we need t h e f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 3.3.1 F i s s a i d t o be p r e f e r r e d t o G i n f i r s t , s e c o n d , t h i r d and DARA degree i f and o n l y i f E ^ [ c f> (x i , • • • , x p ) ] > E n u . [<j>(xi, • • • , x n ) ] F G f o r a l l u- e U . , j = l , 2 , 3 and D, r e s p e c t i v e l y (> h o l d s f o r a t l e a s t one u . e U . ) . i J J n n We deno te such r e l a t i o n s h i p o f p r e f e r e n c e by F >. G f o r u £ U . j i = 1,2,3 and D. J Theorem 3.3.1 I f u e U. and cf) £ U1?, t hen F n >. G n i f and o n l y i f F. >. J J J i J G i f o r a l l i , f o r j = l , 2 , 3 , D . 21 PROOF: The p r o o f o f s u f f i c i e n c y i s by i n d u c t i o n on n. L e t n=2, t h e n E p 2 u i (<J>) u . U ) d F ^ x J d F 2 ( x 2 ) i ^ U ) d G ^ X i ) d F 2 ( x 2 ) o ' (by theo rem 3.2.1 and F ^ > . G j 3 o Jo ^(cb) d G x U i ) d G 2 ( x 2 ) (by lemma 3.3.1 and F 2 >. G 2 ) 3 E g 2 u i (<J>). Assume t h a t t h e theo rem h o l d s f o r n=k. L e t n = k+1, t hen Ep k + l u i ^ E p k u i ( ^ d F k + l ( x k + l } 0 r E k u . ( * ) d F k + 1 ( x k + 1 ) o « (by i n d u c t i o n a s s u m p t i o n ) > j E• u.(d,) d G k + 1 ( x k + 1 ) (by lemma 3.3.1 and F, ^ n >. G. , , ) v J k+1 j k+1 EG k + l u i ( < i ) ) For t h e c o n v e r s e , l e t (cb(xj, • • • , x n ) ) = cb(xi, • • • ,x ) = I i =1 w i t h cj). e U. . Then 1 3 E u. (4,) = I E <bAx.) F i = l i G i = 1 l 22 I f F... |>j G i f o r some i , t hen Ep. < Eg cb.. f o r i = Then by r e p l a c i n g ~ 1 n 1 <J>i* by a * . * , E u (cb) - E 11.(4.) = £ [E (j> (x.) - E R <j>(x.)] + (E cb,* -I 1 F 1 G i = l i i h i * 1 E G . * * i * ) * < 0 f o r some a . > 0 But cb and cb have t h e same p r e f e r e n c e o r d e r i n g . T h e r e f o r e |>j G^ f o r some i i m p l i e s F n }>. G n f o r j = l , 2 , 3 , D . Thus F. >. G. f o r a l l i f o r j i j i j = l , 2 , 3 , D i s a n e c e s s a r y c o n d i t i o n f o r F n >. G n f o r j = l , 2 , 3 ,D . J The above theo rem i s a n a t u r a l r e s u l t f o r U1? s i n c e t he c a l c u l a -f f n J t i o n o f ••• u ( x i , " * , x n ) d F ^ X i ) ••• d F n ( x n ) f o r any i n d e x i depends J J e s s e n t i a l l y on t h e g e n e r a l p r o p e r t i e s o f u n w i t h r e s p e c t t o x ^ . Note t h a t any s e p a r a b l e u t i l i t y f u n c t i o n u 1 1 e l ) n b e l o n g s t o Un- i f each component J u t i l i t y f u n c t i o n u. e U. f o r j = l , 2 , 3 , D . ' J 3.4 'Dependent A t t r i b u t e s Levy and Pa rou sh [ 5 6 ] have d e v e l o p e d s u f f i c i e n t c o n d i t i o n s f o r t h e s p e c i a l m u l t i v a r i a t e f u n c t i o n < M x i , x 2 ) = x x • x 2 . In t h i s s e c t i o n we c o n s i d e r t he more g e n e r a l p r ob l em o f s t o c h a s t i c dominance where cb s a t i s f i e s 9cb  3 x i > 0 V i I f t h e i ndependence a s s u m p t i o n o f F n and G n i s weakened, then t h e f o l l o w i n g f i r s t deg ree dominance r e l a t i o n h o l d s . 23 3 2 Theorem 3.4.1 I f cb e U1 and t h e f o l l o w i n g t h r e e c o n d i t i o n s - h o l d ( i ) F i >i G i ( i i ) F2|1 >i G2|1 3G 2 1 1 ( 1 11 ) ° t h e n F >i G' PROOF: W r i t i n g x 2 i m p l i c i t l y , x 2 = x 2 ( c b , x 1 ) . F 2(4») F 2 i i ( x 2 ( cb ,X i ) I x J d F i ( x i ) (cb e u f ) * •'0 .00 G 2 , i(x 2(<J>,x 1) I X i ) d F i ( x i ) ( f r om i i ) G21 i ( x 2 ( ( j ) , x i ) | x i ) dGifXi) = G2(cb) The l a s t i n e q u a l i t y i s due t o t h e f o l l o w i n g f a c t . d G 2 , i ( x 2 ( cb ,X i ) |x-i) 3 G 2 M 3 x 2 3G 2 , 1 d x i 3 x 2 3X i 9X i + a 1 • < 0 ( f r om i i i and cb e U j The r e s u l t e x t e n d s t o t h e m u l t i v a r i a t e c a s e . In t h e f o l l o w i n g H., deno te s 11 * t h e c o n d i t i o n a l c u m u l a t i v e d i s t r i b u t i o n o f random v a r i a b l e z . g i v e n t h e r e a l i z a t i o n Z i , * » » , z . y It i s assumed t h a t the i n v e r s e f u n c t i o n o f cb e x i s t s (a s u f f i c i e n t c o n d i t i o n f o r t h i s i s to have s t r i c t l y i n c r e a s i n g cb) . It i s a l s o assumed t h a t the e x p e c t a t i o n w i t h r e s p e c t to c o n d i t i o n a l d i s t r i b u t i o n i s w e l l def i ned. Theorem 3 .4 .2 p., >i G.• V i = l , 2 , • • • ,n => F n > l G i f ^ ){x1, • • • ,x ) e U1? 3G. . • . 1 * 1 ' n and ^I * < 0 V i=l,2,«««,n and j < i . j PROOF: L e t 9 = (p(xi,-",x n), we have F n ( * ) = .n-1 .M-.n-l F , ( x n ( x " ' , , - j , ) | x , , - , ) d F n _ 1 , ( x ^ ^ x " - 2 ) ••• d F ^ x J 3*(x1,•••,x ) s i n c e — — > 0 (cp e U r ' ) n G , . (x n(x n- 1,cp)| x n- 1)dF n_ 1, ( x ^ ^ x " " 2 ) . . . d F ^ x J s i n c e F i < G , by a s s u m p t i o n VV*"" 1* d Fn-l|.(Vl' x n" 2) d F^ x^ where H,(x ,x n _ 1) E G„ , ( x n (x 1 1" 1 ,<$>) | x n _ 1 ) . H (x ,x n _ 1) has t h e f o l l o w i n g c p n n | * n cpn p r o p e r t y . dx. < 0 , j < n-1 P r o o f o f t h i s s t a t e m e n t i s p a r a l l e l t o t h e p r o o f t h a t d G 2 i (x2(xi,<p) |-xi) 1 j < 0 i n theo rem 3 .4 .1 . Thus dxi 25 n-1 .n-2 V V X " " ' ) d G n - l | . ( x n - l l X " " ^ d F n - 2 | . ( x n - 2 l x n " 3 ) -d V x n ' x n _ 1 ) s i n c e ^ d ^ - ^ 0 a n d F n - l | - >i G n -1 H ^ x ^ x " " 2 ) d F n _ 2 , ( x n _ 2 l x n _ 3 ) . . . d F^Xx) H ^ ( x n , x n - 2 ) = H.(x , x n _ 1 ) dG i I (x , | x n " 2 ) dr n ' n-1 • v n - 1 1 ; dH , ( x ,x ) d x . d x . J J I n t e g r a t i o n by p a r t s y i e l d s n-1 n-2, H, (x , x " ~ ' ) G i i (x n | x M - t ) <j>v n ' n-1 • v n-1 ' d H j x ^ x " " 1 ) r „ o an, ix ,x G (x J x n - 2 ) r n n-1 • n-1 1 ' dx , 1 n-1 dx n-1 d H A ( x , ~ , x n - 2 ) d x . (j>v n J dG , | (x , | . x n _ 2j dH , (x , x n _ 1 ) n-1 • n-1 1 ' cbv n ' dx . dx n-1 9 d 2 h \ ( x , x n _ 1 ) G ( X . | x n - 2 ) . r i n-1 • n - 1 1 ' d x . dx , 1 j n-1 dx n-1 I n t e g r a t i o n by p a r t s o f t h e t h i r d t e rm y i e l d s < 0 V • •< n-2 J -dH , ( x , x n _ 1 ) s i n c e ? d " < 0 M . < n-1 , dG ' < 0 V j < n _ 2 by a s s u m p t i o n , dG and n-1 dx > 0 n-1 by d e f i n i t i o n . i . e . dG , < 0 V . < n-1 and I" 1 d x j " J " d X 0 < 0 V . < n - 2 J -i m p l y dH , ( x , x n ~ 2 ) -< oy: < n-2. d x , - -Hence, by r e p e a t i n g t h e above p r o c e d u r e we o b t a i n F " ( * ) < H ^ x n ' x n " 2 ) d G n - 2 | . ( V 2 l x n " 3 ) d F n - 3 | . ( X n - 3 l x n " 4 ) - d F ^ x .n-3 n-4. H,(x , x " " ° ) d F , i (x J x " " H ) ••• d F i ( x i ) cbx n ' n-3 • v n-31 ' i v i / H ( | ) ( x n , x 1 ) d F 1 ( x 1 ) H ( x n , X i ) d G i ( x i ) G n(cO), 27 Remarks (1) The r e s u l t s i n Levy [ 5 1 ] , Levy and Pa rou sh [ 5 6 ] and Huang et al. [ 4 3 ] ' s a r e t he s p e c i a l c a se o f theorems 3 . 3 . 1 , 3.4.1 and 3 . 4 . 2 . w i t h n <f>(xi,"*,x ) = n x . . I t seems t h a t a s u f f i c i e n t c o n d i t i o n under w h i c h n i = l 1 F dom ina te s G i n > 2 , > 3 and >Q deg ree i s d i f f i c u l t t o o b t a i n . (2) These theorems a l s o a p p l y d i r e c t l y t o u t i l i t y f u n c t i o n s o f t h e f o l l o w i n g s p e c i f i c fo rms (a) l o g a d d i t i v e [ 75 ] (b) q u a s i s e p a r a b l e [ 22 ,46 ] (3) F o r i n d e p e n d e n t x . ' s ( F . . = F . , G . i = G. and 3 G . . . 3G. ^ I * 1 n l * 1 3 x J = = 0) t heo rem 3 .4 .2 y i e l d s t h e same s u f f i c i e n t J J c o n d i t i o n s as s t a t e d i n theo rem 3 . 3 . 1 . C h a p t e r 4 QUALITATIVE ASPECTS OF OPTIMAL PORTFOLIO BALANCING 4.1 I n t r o d u c t i o n S i n c e a c e n t r a l theme i n p o r t f o l i o t h e o r y i s t h e i n v e s t i g a t i o n o f t h e r o l e o f d i v e r s i f i c a t i o n we now f o c u s upon t he i m p l i c a t i o n o f s t o c h a s t i c dominance r u l e s t o t h e p r ob l em o f o p t i m a l p o r t f o l i o b a l a n c i n g . In p a r t i c u l a r , t h e e f f e c t on t h e o p t i m a l m i x unde r t h e f o l l o w i n g two c o n d i t i o n s i s c o n s i d e r e d ; (1) P o r t f o l i o s formed by one s a f e a s s e t and one r i s k y a s s e t , and (2) P o r t f o l i o s formed by two r i s k y a s s e t s . We examine t h e e f f e c t on t h e o p t i m a l m ix due t o changes i n t h e p r o b a b i l i t y d i s t r i b u t i o n o f t h e r e t u r n o f a r i s k y a s s e t i n c a s e s (1) and ( 2 ) . F o r ca se (1) F i s h b u r n and P o r t e r [ 2 8 ] have a n a l y z e d t h e e f f e c t on t he o p t i m a l mix o f t h e two a s s e t s due t o changes i n t he r a t e o f r e t u r n o f t h e r i s k l e s s a s s e t and s h i f t s i n t he p r o b a b i l i t y d i s t r i b u t i o n o f r e t u r n o f t h e r i s k y a s s e t . They o b t a i n e d a g e n e r a l c r i t e r i o n , i n te rms o f t h e i n d e x o f a b s o l u t e r i s k a v e r s i o n w h i c h i n d i c a t e s t h a t any improvement ( i n FSD s en se ) i n t h e p r o b a b i l i t y d i s t r i b u t i o n f o r t he r i s k y a s s e t must l e a d t o an i n c r e a s e , o r a t l e a s t no d e c r e a s e , i n t h e o p t i m a l p r o p o r t i o n i n v e s t e d i n t he r i s k y 28 29 a s s e t . However, t h e i r r e s u l t i s r e s t r i c t e d t o t he ea se where t h e c u m u l a t i v e d i s t r i b u t i o n o f r e t u r n has compact s u p p o r t [ 0 , h ] , h f i n i t e . In t h i s a n a l y s i s we e x t e n d t h e i r r e s u l t i n t he f o l l o w i n g manner. Improvements i n p r o b a b i l i t y d i s t r i b u t i o n s i n SSD and TSD as w e l l as FSD sense and t h e d i s t r i b u t i o n s w h i c h we c o n s i d e r a r e g e n e r a l n o n n e g a t i v e random v a r i a b l e s . The e f f e c t o f l o c a -t i o n and s c a l e p a r a m e t e r changes i n t h e p r o b a b i l i t y d i s t r i b u t i o n o f r e t u r n f o r a r i s k y a s s e t i s e xam ined . Fo r a s p e c i a l c l a s s o f d i s t r i b u t i o n s t h i s t y p e o f change i s e q u i v a l e n t t o FSD and SSD improvements i n c u m u l a t i v e d i s t r i b u t i o n s . We a l s o g e n e r a l i z e F i s h b u r n and P o r t e r ' s work (FSD s en se ) by c o n s i d e r i n g t h e e f f e c t on t h e o p t i m a l mix when changes i n t h e p r o b a b i l i t y d i s t r i b u t i o n o c c u r under c o n d i t i o n ( 2 ) . In a d d i t i o n t o t h e s e a n a l y s e s we examine t h e t i m e e f f e c t on t he o p t i m a l m i x . F o r m a l l y t h e p r o b l e m upon w h i c h we f o c u s i s Max E u Max E u u(W x + Y X 0 . X Q . ) X i e A i H z A 2£A 2 V j = l 2 j 2 j W l = Wo + l=] X l d X ] . (P) u e U 2 where = (X^- j , , X t n ) and t h e n o n n e g a t i v e r e t u r n s i n p e r i o d t , t = l , 2 ; Hj. r e p r e s e n t c u m u l a t i v e j o i n t d i s t r i b u t i o n s o f X^, t = l , 2 ; H 2 | i r e p r e s e n t s t he c o n d i t i o n a l d i s t r i b u t i o n o f X 2 . g i v e n X x ; A = {\^  = (A ^ - j ,• ,A T N) | n £ X^.j = W^ . - j , X^j > 0 } , t = l , 2 ; E^ r e p r e s e n t s t h e e x p e c t a t i o n , o p e r a t o r f o r a c u m u l a t i v e d i s t r i b u t i o n K, and W^, a r e t h e i n i t i a l w e a l t h a t t h e b e g i n n i n g o f p e r i o d t +1 , t = 0 , l , 2 . 30 I t i s c o n v e n i e n t t o assume t h a t t h e random v a r i a b l e s a r e n e t r a t h e r than t h e g r o s s r e t u r n s ( i . e . 1 + r e t u r n ) . We f i r s t c o n s i d e r c a se 1. 4 .2 One S a f e and One R i s k A s s e t s We c o n s i d e r t he s i m p l e p r ob l em where a t the b e g i n n i n g o f each p e r i o d t he c h o i c e o f p o r t f o l i o mix i s l i m i t e d t o one s a f e and one r i s k y a s s e t . L e t p. = t he r e t u r n on t h e r i s k l e s s a s s e t a t t h e b e g i n n i n g o f p e r i o d t , t = l , 2 . X. = t h e random v a r i a b l e r e p r e s e n t i n g t he r e t u r n on t h e r i s k y a s s e t a t t he b e g i n n i n g o f p e r i o d t , t = l , 2 . F. = t h e c u m u l a t i v e d i s t r i b u t i o n o f X . , t = l , 2 . r x 2 Z F 2 / l : dP[X2 < t | X i = X i ] , t he c o n d i t i o n a l d i s t r i b u t i o n o f X 2 g i v e n X i = X i . Then (P) becomes Max E Max E F u.[Wi(l + p 2 ) + X 2 ( X 2 - p 2 ) ] X ieA i 1 X 2 e A 2 ' '2/1 s . t . : . Wi = W 0(1 + Po) + X i ( X i - p i ) , where X 2 = X 2 2 and X 2 i = 1 - X 2 . L e t X^ deno te an o p t i m a l X^ . a t t h e b e g i n n i n g o f p e r i o d t = l , 2 . The i n n e r m a x i m i z a t i o n p r ob l em g i v e s a " d e r i v e d " u t i l i t y f u n c t i o n cf)(Wi) w h i c h r e p r e s e n t s t h e p r e f e r e n c e o f p e r i o d one d i s t r i b u t i o n i n c o r p o r a t i n g t h e u n c e r t a i n n a t u r e o f p e r i o d two p r o s p e c t s [ 6 9 ] . Note t h a t X 2 = 0 i f and o n l y i f t h e s l o p e o f E P u(W 2) i s n o n p o s i t i v e a t X 2 = 0 and X 2 = Wj i f h 2 / l 31 and o n l y i f t h i s s l o p e i s n o n n e g a t i v e a t X 2 = W,. Thus n o n t r i v i a l a l l o c a -t i o n o f Wi o c c u r s i f and o n l y i f x 2 u ' ( W i ( l + x 2 ) ) d F ? / 1 (.xajxi) : ^ •—< p 2 < E (X 2 |Xx = x i ) . u ' (Wi( l + x 2 ) ) d F 2 / - | ( x 2 | x 1 ) * C l e a r l y , p4 > E (X 2 |X i = x j i m p l i e s X1 = 0. In f a c t , Hadar and R u s s e l l p r o ve Theorem 4.2.1 I f » > p > E ( X ) , t hen W + XX + (W - X)p > 2 W + X ' X + (W - X ' ) ; where 0 < X < X' < W. PROOF: See [ 3 1 , Thm. 1 0 ] . I f t h e d i s t r i b u t i o n a s s e s smen t o f changes t o a n c ' G2/1 i s p r e f e r r e d t o i n s t o c h a s t i c dominance s e n s e , t hen do we n e c e s s a r i l y * * have X r > X F ? A s o l u t i o n t o t h i s que r y was o b t a i n e d by F i s h b u r n and b 2 / l " ""2/1 P o r t e r [ 28 ] when t h e p r e f e r e n c e r e l a t i o n i s FSD and t h e d i s t r i b u t i o n s a r e r e s t r i c t e d t o t h o s e d e f i n e d on [ 0 , h ] . They showed t h a t t h e a p p r o p r i a t e s u f f i c i e n t c o n d i t i o n i s r ( w * ) W 0 ^ (x -p) < 1, w* = W 0[X (x -p ) + p ] . We show t h a t t h e i r r e s u l t can be g e n e r a l i z e d t o FSD as w e l l as t h e SSD and TSD c a s e s w i t h a r b i t r a r y n o n n e g a t i v e random v a r i a b l e s . The s e t t i n g f o r ou r a n a l y s i s i s o v e r a two p e r i o d h o r i z o n . The r e s u l t s f o r p e r i o d 2 a r e com-p a r a b l e t o t h o s e o f F i s h b u r n and P o r t e r [ 2 8 ] who u t i l i z e d a s t a t i c m o d e l . * L e t f ( x 2 ) = u ' ( w i ( l + p 2 ) + X ( x 2 - p 2 ) ) ( x 2 - p 2 ) W 2 = W i ( l + p 2 ) + X ( x 2 - p 2 ) . C o n s i d e r t h e f o l l o w i n g c o n d i t i o n s : ( C l ) r ( w 2 ) A (X2-P2) < 1 x 2 e (p 2 ,°°) (C2) - ^ ' , ( ^ 2 ) A * ( x 2 - p 2 ) < 2 x 2 e ' ( p 2 , » ) ie .. mi. / \ ^ (C3) - |]„,|^j A ( x 2 - p 2 ) < 3 x 2 e ( p 2 , » ) Lemma 4.2.1 I f C l , C l and C2 , and C l , C2 and C3 h o l d then f e U . , j = l , 2 , r e s p e c t i v e l y . "k "k ie PROOF: f ' ( x 2 ) = u " ( w 2 ) A ( x 2 - p 2 ) + u ' ( w 2 ) f " ( x 2 ) =.1?" ( w t ) ( A * ) 2 ( x 2 - p 2 ) + 2 X * u " ( w * ) f " ( x 2 ) = u" " ( w t ) ( A * ) 3 ( x 2 - p 2 ) ' + 3 ( A * ) V " (w*) . T h e r e f o r e i f C l h o l d s t h e n f ' ( x 2 ) > 0, i . e . f e U r , i f C l and C2 h o l d t hen f " ( x 2 ) < 0 and f ' ( x 2 ) > 0, i . e . f e U 2 ; i f C l ' , C2 , and C3 h o l d t hen f " > 0, f " < 0 and f > 0, i . e . f E U 3 . We can now s t a t e t h e f o l l o w i n g t heo rem. Theorem 4 . 2 . 2 I f u £ U 3 and > 2 F ^ t hen s u f f i c i e n t c o n d i t i o n s f o r 33 (1) r ( w ^ ) X F ( x 2 - p 2 ) < 1 r 2 / l <2> - ^ K 2 / 1 < * ^ > - 2 x 2 e (P2, 0 0). x 2 e (P2, 0 0) PROOF: d E G u ( W 2 ) dX 2 A 2 =A r d E G u ( W 2 ) d A 2 d E p u ( W 2 ) dX ; "2/1 A 2=A 2/1 u 1 ( w 2 ) ( x 2 - p 2 ) d G 2 / 1 ( x 2 | x 1 ) - F 2 / 1 ( x 2 | x ! ) f ( x 2 ) d G 2 ^ 1 ( x 2 | x 1 ) - F 2 ^ ( x 2 | x i ) > 0 s i n c e f e U 2 and G ^ > 2 F 2 ^ . Theorem 4 . 2 . 3 I f u"" < 0 , u e U 3 and G^ - j > 3 F 2 ^ then s u f f i c i e n t c o n d i -t i o n s f o r Xr > X a r e b 2 / l " h 2 / l •k "k (1) r ( w 2 ) X ( x 2-p 2) < 1 h 2 / l <2> - f ^ % } ^ < - 2 x 2 e (P2 J°°) x 2 e (p2 ' ° ° ) (3) - " I " W X* (X2 -P2) < 3 v ' u (w*) F 2 ^ K z y -x 2 e ip2>m) PROOF: The p r o o f p a r a l l e l s t he p r o o f o f theo rem 4 . 2 . 2 . u " " < 0 assumption is re lated to the peakness measurement, 34 u ' ' 1 u" Remark: Note t h a t i f u e UD t hen > - ^jr, i . e . c o n d i t i o n s C l and C2 a r e n o n c o n t r a d i c t o r y . F i s h b u r n and P o r t e r ' s FSD r e s u l t f o l l o w s i m m e d i a t e l y f r om f e Ui. I f we r e w r i t e t he c o n d i t i o n s C l , C2 and C3 as ( C T ) - j ^ ^ r (w ), V ( x 2 - p 2 ) h 2 / l A ( x 2 - p 2 ) " u ( w > ] F 2 / l <C3') ^ > ^ w 4 y . V ( x 2 - p 2 ) " , u ( W 2 )  h 2 / l t h e n each o f t h e s e r e s t r i c t i o n s g i v e s us a band f o r a c c e p t a b l e u t i l i t y f u n c -t i o n s . F i g u r e 4.T on t h e f o l l o w i n g page i l l u s t r a t e s t h e s e c o n d i t i o n s . C o n s i d e r t h e f o l l o w i n g u t i l i t y f u n c t i o n . u l ( x ) = - e x p ( - a x ) , a > 0, u2 ( x ) = (x + 3) Y, 3 > 0, 0 < y < 1» u3 ( x ) = l o g ( x + 6); 6 > 0, and u 4 ( x ) = - ( x + ( f i ) " ^ <p > 0, > 0. Note t h a t u l and u4 do no t s a t i s f y c o n d i t i o n s C1 -C3 . u2 s a t i s f i e s c o n d i -t i o n s C1-C3 w h i l e u3 s a t i s f i e s C l and C2. A b a s i c r e a s o n why t h e c o n d i t i o n s a r e s a t i s f i e d by u2 and u3 and n o t by u l and u4 i s t h a t t h e r a t e o f . F i g u r e 4 . 1 . I l l u s t r a t i o n o f a s s u m p t i o n s C l 1 , C 2 1 and C 3 ' 36 c o n v e r g e n c e as x 2 app roache s i n f i n i t y o f r ( w ) , ~ u , " ) w ( and ~ u m /K f u n c -U \.YI} U \ W) t i o n s i s f a s t e r t han t he r a t e o f c onve r gence o f - * — . X p ( x 2-p 2) In t h e above a n a l y s i s we r e q u i r e a d d i t i o n a l p r o p e r t i e s f o r u as we expand ou r a n a l y s i s f r om FSD t o SSD and TSD. The r e s t r i c t i o n C l , C2 , and C3 on u p r e v e n t " u n e x p e c t e d movement" o f the o p t i m a l m ix when t h e improvement i n t he p r o b a b i l i t y d i s t r i b u t i o n o f r e t u r n f o r t he r i s k y a s s e t i s i n t r o d u c e d w i t h t h e r a t e o f r e t u r n o f t h e s a f e a s s e t h e l d f i x e d . F i g u r e 4.2 d e p i c t s t he s i t u a t i o n . * * V ^ F b 2 / l r 2 / l ( u n e x p e c t e d ) G 2 / l ( e x p e c t e d ) F i g u r e 4 . 2 . The o p t i m a l mix f o r c h a n g i n g p o r t f o l i o c o m p o s i t i o n . 37 We now c o n s i d e r t he e f f e c t o f changes i n l o c a t i o n and s c a l e p a r -amater s 6 and 3 on X; , r e s p e c t i v e l y . T o b i n [ 101 ] and R o t h s c h i l d and S t i g l i t z [ 88 ] a n a l y z e d changes i n the o p t i m a l p o r t f o l i o mix due t o changes i n mean and v a r i a n c e o f r e t u r n f o r t h e r i s k y a s s e t . T o b i n c l a i m e d ( i n c o r r e c t l y ) t h a t i f u s i s q u a d r a t i c then an i n c r e a s e i n t he e x p e c t e d r a t e o f r e t u r n , E ( X ) , w i l l l e a d to an i n c r e a s e i n X. T o b i n [ 101 ] and R o t h s c h i l d and S t i g l i t z [ 88 ] showed t h a t an i n c r e a s e i n t h e v a r i a n c e , V ( X ) , o f t he r a t e o f r e t u r n o f t h e r i s k y a s s e t w i t h E(X) h e l d f i x e d r e d u c e s X, i f u i s q u a d r a t i c . R o t h s c h i l d and S t i g l i t z [ 88 ] a l s o showed t h a t f o r any u e Up, an i n c r e a s e i n V(X) w i t h E (X ) h e l d f i x e d w i l l r educe X i f t he r e l a t i v e r i s k a v e r s i o n , R ( x ) , i s l e s s t han o r equa l t o one and R 1 ( x ) > 0. Mo s s i n [ 6 8 ] , T o b i n [ 1 0 1 ] , R i c h t e r [ 8 6 ] , S t i g l i t z [ 98 ] and N a s l u n d [ 7 0 ] d e a l w i t h t h e e f f e c t o f t a x a t i o n on s o c i a l r i s k - t a k i n g . R e s u l t s o b t a i n e d by t h e s e a u t h o r s a r e o f t h e f o l l o w i n g f o r m : I f an i n c r e a s e i n income t a x i s i n t r o d u c e d by t h e gove rnment , t hen t h e p u b l i c w i l l t a k e a g r e a t e r r i s k (X i n c r e a s e s ) p r o v i d e d t he p u b l i c u t i l i t y f u n c t i o n i s o f t h e t y p e w h i c h b e l o n g s t o Up and has i n c r e a s i n g r e l a t i v e r i s k a v e r s i o n . F e l d s t e i n [ 2 3 ] showed t h a t t h e t a x has no e f f e c t on X when R(x ) i s c o n s t a n t ( e . g . u ( x ) = l n ( x ) ) . L e l a n d [79 ] d e a l s w i t h t he a l l o c a t i o n p r ob l em between s a f e and r i s k y f o r e i g n exchange . He a l s o o b t a i n e d a s i m i l a r r i s k - t a k i n g b e h a v i o u r by t h e p u b l i c . S i n c e t h e s t o c h a s t i c dominance r e l a t i o n s a r e d e t e r m i n e d by t he above p a r a m e t e r s f o r t h e s p e c i a l c l a s s o f d i s t r i b u t i o n s ( see C h a p t e r 2) we s h a l l c o n s i d e r t h e l o c a t i o n and s c a l e p a r a m e t e r change s . The f i r s t and second o r d e r o p t i m a l i t y c o n d i t i o n s a r e : ,00 u ' ( w 2 ) ( x 2 - p 2 ) d F 2 ^ 1 ( x 2 | x 1 ) = 0, and o ( 4 . 2 . 38 u " ( w 2 ) ( x 2 - p 2 ) d F 2 ^ 1 ( . x 2 | x i ) < 0 ( 4 . 2 . I f u i s s t r i c t l y concave then A 2 i s u n i q u e . L e t t h e r i s k y a s s e t Z 2 ( 6 , 3 ) be 3 X 2 + 6, where 6, 3 > 0 and p 2 > 6. The e f f e c t o f changes i n t he l o c a t i o n p a r a m e t e r 6 on A 2 i s d e t e r m i n e d by d i f f e r e n t i a t i n g ( 4 . 2 . 1 ) w i t h r e s p e c t t o 6. dA d6 2 _ fOO ^ ^ u " ( w 2 ) A 2 ( 3 x 2 + 6 - p 2 ) d F 2 / 1 ( x 2 | x 1 ) + u ' ( w 2 ) d F 2 / 1 ( x 2 | x i ) u " ( w 2 ) ( 3 x 2 + 6 - p 2 ) d F 2 / 1 ( x 2 | x ! ) * * dA where w 2 = w x ( l + p 2 ) + A 2 ( 3 x 2 + 6 - p 2 ) . The s i g n o f depends on t h e I * 2 v a l u e o f t h e n u m e r a t o r , s i n c e u " ( w 2 ) ( 3 x 2 + 6 - p 2 ) d F 2 / 1 ( x 2 | x ! ) < 0. Theorem 4.-2.4 I f u e U 3 and E ( x 2 | X i ) < £ ^ t hen ^ > 0. PROOF: * * u" ( w 2 ) A 2 ( 3 x 2 + 6 - p 2 ) d F 2 / 1 ( x 2 | x i ) P2-5 u " ( w 2 ) A 2 ( 3 x 2 + 6 - P 2 ) d F 2 / 1 ( x 2 | x i ) P2-5 u " ( w 2 ) A 2 ( 3 x 2 + 5 - p 2 ) d F 2 / 1 ( x 2 | x ! ) i i " ( w i ( l + p 2 ) ) A 2 ( 3 x 2 + 6 -P2) d F 2 / 1 ( x 2 | x i ) 39 3 u " ( w i ( l + p 2 ) ) A 2 (3x 2 + 5 - p 2 ) d F 2 ^ ( x 2 | x i ) ( s i n c e u e U 3 ) = u " ( w i ( l + p 2 ) ) A 2 j (3x2 + 6-.p2) d F 2 / 1 ( x 2 | x 1 ) , T h e r e f o r e > 0 i f (3x2 + 6-p 2 ) d F 2 / 1 ( x 2 | x i ) < 0 . S i n c e E ( X 2 | X x ) < P2-5 3 . t h e r e s u l t f o l l o w s . Note t h a t i f we c o n s i d e r t h e change i n l o c a t i o n p a r a m e t e r f r om 6=0 and 3=1, then t h e c o n d i t i o n E ( X 2 | X x ) < P ' 2 ~ 5 r e s t r i c t s Xz f r om becoming W 2, The f o l l o w i n g theo rem g i v e s an e f f e c t o f t h e s c a l e p a r a m e t e r change on A 2 . Theorem 4 . 2 . 5 I f u e U 3 and t h e r a t i o o f c o n d i t i o n a l second moment and t h e f i r s t moment, R = x 2 2 d F 9 / 1 ( x 2 I x x ) 2 " ' ' " < ^ t h e n ^ > 0 . x 2 d F 2 / 1 ( x 2 | x ! ) 3 d3 PROOF: s i n c e * d A 2 "k "k u " ( w 2 ) x 2 X 2 ( 3 x 2 + 6 - p 2 ) d F 2 / 1 ( x 2 | x i ) + ~k ~k u ' ( w 2 ) x 2 X 2 d F 2 / 1 ( x 2 | x i ) d3 _ "k u " (w 2 ) ( 3x 2 + 6 - P 2 ) 2 d F 2 / 1 ( x 2 | x i ) dX d3 > 0 i f u " ' (w 2 ) x 2 ( 3 x 2 + 6-p 2 ) d F 2 ^ 1 ( x 2 | x 1 ) > 0. But 40 * * u"(w 2)x2A 2(3x2 + 6 - p 2 ) d F 2 / 1 ( x 2 | x i ) 0 P 2 - 6 u" (w 2 )A 2X2 (p2-3x2-6) d F 2 / 1 ( x 2 | x x ) P2-6 3 P2-6 u " ( w 2 ) A 2 x 2(3x2+6-p 2) d F 2 / 1 ( x 2 | x i ) 6 * u " ( w 1 ( l + p 2 ) ) A 2 x 2 ( p 2 - 3 x 2 - 6 ) d F 2 / 1 ( x 2 | x i ) ( s i n c e u e U 3 ) u " (w 1 ( 1 + p 2 ) ) A 2 x 2 ( 3 x 2 + 6 - p z ) d F 2 / 1 ( x 2 | x i ) p2«5 3 u " ( w (l+p2)xt u " ( w 1 ( 1 + p 2 ) ) A 2 [ 3 [ x 2 2 3 + x 2 (6-p 2)] d F 2 / - | ( x 2 | x 1 ) x 2 2 d F 2 / 1 ( x 2 | x i ) + (5-P2) x 2 d F 2 / 1 ( x 2 | x i ) ] > 0 ( f r om R < p We now c o n s i d e r t h e e f f e c t s o f changes i n A l 9 F i and pi upon A 2 ( t h i s w i l l i n t r o d u c e t h e t i m e e l e m e n t t o t h e p o r t f o l i o s e l e c t i o n p r o b l e m ) . We f i r s t a n a l y z e t h e e f f e c t o f Wi on A 2 s i n c e Wi summar izes a l l t he r e l e v a n t i n f o r m a t i o n i n c l u d e d i n p a r a m e t e r s o f t h e f i r s t p e r i o d . The e f f e c t o f changes i n Wx on A 2 i s o b t a i n e d by d i f f e r e n t i a t i n g ( 4 . 2 . 1 ) w i t h r e s p e c t t o W i . 41 d A 2 dWi •oo ^ u " ( w 2 ) ( l + p 2 ) ( x 2 - p 2 ) d F 2 / 1 ( x 2 | x i ) u " ( w 2 ) ( x 2 - p 2 ) d F 2 / 1 ( x 2 | x i ) where w 2 = W i ( l + p 2 ) + A 2 ( x 2 - p 2 ) . The s i g n o f 4rp- depends on t h e v a l u e o f t he numera t o r i n t h e above e q u a t i o n s i n c e dWi * u " ( w 2 ) ( x 2 - p 2 ) d F 2 ^ ( x 2 | x i ) < 0. The s i g n o f u " ( w 2 ) ( l + p 2 ) ( x 2 - p 2 ) dF 2^-| ( x 2 | x i ) c a n n o t be d e t e r m i n e d f r om u " a l o n e and i n f o r m a t i o n on t h e r i s k a v e r s i o n i n d e x i s needed. The f o l l o w i n g r e s u l t i s due t o A r row [ 3 ] and Cass and S t i g l i t z [ 1 7 ] . Fo r c o m p l e t e n e s s we s t a t e and p rove t he r e s u l t f o r t h e model under c o n s i d e r a t i o n . Theorem 4 . 2 . 6 I f r 1 = 0 t h e n ^ r r = 0. < dWi > PROOF: •oo u " ( . w 2 ) ( l + p 2 ) ( x 2 - p 2 ) d F 2 / 1 ( x 2 | X i ) P2 * f 0 0 * u " ( w 2 ) ( l + p 2 ) ( p 2 - x 2 ) d F 2 / 1 ( x 2 | . x 1 ) - u " ( w 2 ) ( l + p 2 ) ( x 2 - p 2 ) d F 2 / 1 ( x 2 | X i ) 0 J p2 P2 P2 -u ' (w 2 ) . r ( w 2 ) ( p 2 - x 2 ) ( l + p 2 ) d F 2 / 1 (x 2 IxJ ie ie u'(w2) r (w 2 ) ( l+p 2 ) ( x 2 - p 2 ) d F 2 / 1 ( x 2 | x 1 ) rP2 P2 u'(w2) r ( w i ( l + p 2 ) ) ( p 2 - x 2 ) ( l + p 2 ) d F 2 ^ ( x 2 | x i ) " 0 u ' ( w 2 ) r ( w i ( l + p 2 ) ) ( x 2 - p 2 ) ( l + p 2 ) d F 2 / 1 ( x z l x j ( s i n c e r ' = 0) S i n c e t h e l a s t i n t e g r a l i s equa l t o z e r o r 1 = 0 i m p l i e s = 0. 42 L e t E be t h e e x p e c t e d v a l u e o p e r a t o r w i t h r e s p e c t t o t h e j o i n t d i s t r i b u t i o n o f X i and X2. then we have C o r o l l a r y 4.2.1 I f r ' = 0 t hen = 0. PROOF: S i n c e = ( E ( X 2 ) - P i ) and t h e n o n t r i v i a l s o l u t i o n r e q u i r e s E ( X 2 | X i = X i ) > pi f o r each X i . Then f rom t h e above theo rem ^ ( E ( X 2 ) - p 2 ) I 0 i f r ' I 0 A r a t i o n a l i n d i v i d u a l i s t y p i c a l l y assumed t o have r ' < 0 [ 3 , 8 1 ] , Examples where r ' > 0, r ' = 0 and r ' < 0 can be f ound i n [ 3 4 , 8 1 , 9 9 , 1 0 1 ] . Note t h a t r ' < 0 i m p l i e s u ' " > 0 b u t no t c o n v e r s e l y . Now c o n s i d e r t he e f f e c t o f changes i n W0 on A 2 . | £ - (HP.) • j j f c f X , - P l ) and d ^ M H P l ) + d ^ ( E ( X 1 ) - p 1 ) , where At i s t h e o p t i m a l p a r a m e t e r o f a l l o c a t i o n f o r p e r i o d one d A i Jo i d e n t i f y t h e s i g n s o f ^ and d E ^ ^ . T h e r e f o r e , we need t o d e t e r m i n e t he s i g n o f -n-r- i n o r d e r t o + JL 3 dWr dW0 dW0 F i r s t , c o n s i d e r t h e f o l l o w i n g lemma. 43 Lemma 4.2.1 <j>(wi) i s s t r i c t l y i n c r e a s i n g and concave . PROOF: cj) '(Wi) = E F u ( w i ( l + p 2 ) + X 2 ( x i - p 2 ) ) h 2 / l * = E p u ' ( w i ( l + p 2 ) + x t ( x 2 - p 2 ) ) [ ( l + p 2 ) + ^ ( x 2 - p 2 ) ] = E F u ' ( w t ) ( l + p 2 ) > 0 h 2 / l ^ ' ( w j = E F u " ( w i ( l + p 2 ) + X 2 ( x 2 - p 2 ) ) [ ( l + p 2 ) + ( x 2 - p 2 ) ] + E F u ' U i C l + p i ) + X 2 ( x 2 - p 2 ) ) [ ^ — 2 - ( x 2 - p 2 ) ] < 0 F 2 / l d W l Note t h a t i f u i s q u a d r a t i c , t h e n cj> i s q u a d r a t i c . Then t h e f i r s t p e r i o d s o l u t i o n X i s a t i s f i e s E F i ( p ' l W J l X i - p J = 0 ( 4 . 2 . 3 ) E F i < } ) " (W 1 ) ( X 1 - p 1 ) 2 < 0 ( 4 . 2 . 4 ) Hence, d x * - E F i 4 » " ( w t ) ( x 1 - P l ) ( l . + P l ) dW7 = E F | ( J ) " ( W T ) ( X 1 - P l ) ; i Theorem 4 . 2 . 7 I f i ) r' I 0 , 44 Then i i ) ~ t | r ] > 0 f o r a l l X i , and i i i ) x i < p i . * d A 2 > dW0 < PROOF: dX_2 dWn" d A 2 dwi dwf dW0 d A 2 dw* ( 1 + P i ) + ^ ( x i - P l ) The l a s t i n e q u a l i t i e s f o l l o w s i n c e ^ f - ^ 0 f r om ( i ) and d ~K* ^ -rrp - ( x j - p i ) > 0 f r o m ( i i ) and ( i i i ) . d W o * * dXj . We n o t e t h a t r ' = 0 t hen dW, 0 s i n c e ^ = 0 , i f - f i r ] ' = 0 t hen c^- 2~ = 0 , i f x x = p i , t hen t h e s i g n o f ^ j - 2 - depends on r ' . dW 0 " " x K 1 ' " " " " dW 0 The c o n d i t i o n ( i i i ) r e q u i r e s t h e a c t u a l r e a l i z a t i o n o f a r i s k y v e n t u r e i n p e r i o d one , however , s i n c e t he change i n W0 must t a k e p l a c e b e f o r e t h e r e a l i z a t i o n , t he p r o p e r q u a n t i t y t o be c o n s i d e r e d i s t h e e x p e c t e d change i n A 2 . From t h e c o r o l l a r y 4 . 2 . 1 , t h e e x p e c t e d w e a l t h c hange , ^j^1^ , i s n o n n e g a t i v e i f -[^r] > 0. Hence t h e e x p e c t e d change * cb" 1 o f A 2 mus t -be p o s i t i v e , g i v e n t h e a s s u m p t i o n s r 1 > 0 and - O T T ] > 0 . 4 . 3 , The Two R i s k y A s s e t Case We now c o n s i d e r t he e f f e c t on t h e o p t i m a l mix when bo th a s s e t s a r e random. The model i s 45 Max E X ieA i Ho Max E u(Wi(1 + X 2 2 ) + X 2 ( X 2 1 - X 2 2 ) ) A 2 e A 2 M 2 / l s . t . Wi = W 0 ( l + X 1 2 ) + X ^ X n - x 1 2 ) . The o p t i m a l a l l o c a t i o n X 2 i n p e r i o d two s a t i s f i e s En LI1 ( W 2 ) ( X 2 1 - X 2 2 ) = 0 H 2 / l ( 4 .3 and d X 2 dWi En u " ( W 2 ) ( X 2 i - X u ) < 0 H 2 / l - E „ u"(wt) (1 + X 2 2 ) ( X 2 1 - X 2 2 ) •H 2/J_ H 2 / 1 u " ( W 2 ) ( X 2 1 - X 2 2 ) ' ( 4 . 3 C l e a r l y , t h e s i g n o f -^r1- i s d e t e r m i n e d by t h e numera to r o f * t h e above f r a c t i o n . However, t h e s i g n o f E n u " ( W 2 ) ( l + X 2 2 ) ( X 2 i - X 2 2 ) H 2 / l i s d i f f i c u l t t o d e t e r m i n e s i n c e t h e c a l c u l a t i o n o f t h e above i n t e g r a l i n v o l v e s a j o i n t d i s t r i b u t i o n . I t i s c o n v e n i e n t t o assume t h a t i n v e s t m e n t r e t u r n s a r e s e r i a l l y i n d e p e n d e n t . A s i m i l a r * m o d e l was c o n s i d e r e d by B r u n i e l l e [ 14 ] and P y l e [ 8 3 ] . r * L e t I|J(X22) = - u " ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d H ( x 2 i | x 2 2 ) , where H ( x 2 i | x 2 2 ) d eno te s t h e c o n d i t i o n a l d i s t r i b u t i o n o f X 2 i g i v e n X 2 2 = x 2 2 -S i m i l a r r e s u l t s t o t h o s e i n S e c t i o n 4 .2 can be d e r i v e d bu t a r e n o t i n c l u d e d h e r e . We s t a t e and p rove t h e f o l l o w i n g two r e s u l t s t o . i l l u s t r a t e t he method o f p r o o f u s ed . 46 Theorem 4.3.1 I f r 1 - 0 and u ' ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d H ( x 2 i | x 2 2 ) - 0 V x 2 2 t h e n dWi > u " PROOF: ,00 ,00 0 r°° •oo 0^ J 0 r°° •'o u " ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d H 2 / 1 ( x 2 i , x 2 2 | X i i , x 1 2 ) u " ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d P ( X 2 i < x 2 1 , X 2 2 < x 2 2 ) ty{x22) d P ( X 2 2 :< x 2 2 ) Now (x 2 2 ) x 2 2 * * r ( w 2 ) ( u ( w 2 ) ( l + x 2 2 ) ( x 2 2 - x 2 i ) d H ( x 2 i | x 2 2 ) r°° * * r ( w 2 ) u ' ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d H ( x 2 i | x 2 2 ) x 2 2 - r ( w i ( l + x 2 2 ) ) u ' ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d H ( x 2 i | x 2 2 ) ( f r om r ' '0 T h e r e f o r e dX'7 < u ' ( w 2 ) ( l + x 2 2 ) ( x 2 i - x 2 2 ) d H ( x 2 i | x 2 2 ) - 0 i m p l i e s - 0. The f o l l o w i n g theo rem g e n e r a l i z e s F i s h b u r n and P o r t e r ' s r e s u l t . Theorem 4.3.2 L e t G ( x , y ) = H ( y | x)Gi ( x ) and F ( x , y ) = H ( y | x ) F ! ( x ) be t h e * * j o i n t d i s t r i b u t i o n s o f random v a r i a b l e s X and Y, t hen Ag > Ap whenever t h e f o l l o w i n g c o n d i t i o n s h o l d . ( i ) G i > i F i d 2 u ( w 2 ) dH(,y|x) d u ( w 2 ) d 2 H ( y l x )  1 1 1 ' dxdX dy dX dydx > u t o r eacn x dx Tim H(a .x) d u ( w 2 ) a-*» dX 1 im a-*» d H ( a l x ) d u ( w 2 ) dx dX , and l i m H(a|x) d !M iw 2 i < % c o a n d , i m M a l A l d u ^ J . ^ a^o d x d x a^o d x d x * * where w 2 = W i ( l + y ) + X ( x - y ) and Xg , Xp deno te t h e o p t i m a l mix under G and F , r e s p e c t i v e l y . PROOF: d E G dX u ( w i ( l + Y) + X ( X - Y) x=xr dE„ d E . G [ u ( w i ( l + Y) + X ( X - Y ) ) ] - ^ ± [ u ( w i ( l + Y) + X ( X - Y ) ) ] dX x=x d l»oo dX , 0 u ( w i ( l + y ) + X ( x - y ) dH(y|x ) d ( G i - F i ) x=xr r ° ° r-o 'o ^ u ( w i ( l + y ) + . X ( x - y ) ) dH(y|x ) x=x d ( G i - F x ) > 0 s i n c e d x L ~ * ^ d H ( y | x ) ] •'o _d_ dx H (y|x ) d u ( w 2 ) dX y=oJ _d_ dx H(y|.x) d 2 u ( w 2 ) dydX dy _d_ dx H(y|x ) M s d y=o d 2 u ( w 2 ) dxdA H(y|x ) y=o f°° d u ( w 2 ) d 2 H ( y l x ) , dA dydx a y r H2 d 2 u ( w 2 ) dH(.y|x) . , f d u ( w 2 ) d 2 H ( y [ x ) dxdA dy a y dA dydx ( f r o m a s s u m p t i o n i i i ) > 0 ( f r om a s s u m p t i o n i i ) , ie ie T h e r e f o r e Ag > A p ( f r om u e U 2 ) , C h a p t e r 5 METHODOLOGY DEVELOPMENT IN RESOURCE MANAGEMENT 5.1 I n t r o d u c t i o n W h i l e t h e l a s t c h a p t e r f o c u s e d upon a t h e o r e t i c a l a p p l i c a t i o n o f s t o c h a s t i c dom inance , i n t h i s c h a p t e r we d e v e l o p a methodo l ogy t o w iden t h e scope o f m a n a g e r i a l a p p l i c a t i o n s o f t h e t h e o r y . Most p r e v i o u s m a n a g e r i a l a p p l i c a t i o n s f o c u s e d upon f i n a n c i a l p r ob l ems ( see [ 1 3 , 5 2 , 5 8 , 7 7 , 7 8 , 7 9 , 8 0 , 9 5 , and 1 0 9 ] ) . In t h i s c h a p t e r we f o c u s upon e c o s y s t e m management. Two m a j o r app roache s t o i n c o r p o r a t e r i s k a t t i t u d e s i n t o p o l i c y a n a l y s i s have been commonly u t i l i z e d i n e c o s y s t e m management. (1) Fail-safe and chance constraint formulations: R i s k a v e r s i o n i s i n d i c a t e d by p o s t i n g e i t h e r a b s o l u t e c o n s t r a i n t s on p rob lems o r p r o b -a b i l i t y c o n s t r a i n t s . A b s o l u t e l e v e l c o n s t r a i n t s a r e i n c o r p o r a t e d t o e n s u r e m a r g i n s o f s a f e t y t h r o u g h b u i l t - i n i n e f f i c i e n c i e s ( f a i l - s a f e s y s tem d e s i g n ) w h i l e p r o b a b i l i s t i c c o n s t r a i n t s e n s u r e s a t i s f a c t i o n o f a p a r t i c u l a r o b j e c t i v e ( o r en su r e a g a i n s t a c o n s t r a i n t v i o l a t i o n ) w i t h a g i v e n m in ima l p r o b a b i l i t y . (2) Substitution of individual for social preferences and  risk neutrality assumptions: R i s k a t t i t u d e s o f p a r t i c u l a r i n d i v i d u a l 49 50 d e c i s i o n makers a r e used t o r e p r e s e n t s o c i a l p r e f e r e n c e s , o r i n o t h e r c a s e s e x p e c t a t i o n s o f p a y o f f a r e t a k e n as t h e o b j e c t i v e f u n c t i o n , a s suming i m p l i c i t l y a u t i l i t y l i n e a r i n p a y o f f . S t o c h a s t i c dominance t h e o r y o f f e r s an improvement i n t h e a r e a o f p o l i c y a n a l y s i s . T h i s a pp r oach t o t he p r o b l e m a l l e v i a t e s t h e above d e f i c i e n c i e s by t a k i n g e x p l i c i t a c c o u n t o f common f e a t u r e s o f r i s k a t t i t u d e s i n p o l i c y c o m p a r i s o n s . In t h i s c h a p t e r we o u t l i n e a methodo l ogy f o r e v a l u a t i o n o f f o r e s t s y s tem management p o l i c i e s and p r o v i d e an example o f i t s a p p l i c a t i o n i n t h e a n a l y s i s o f a l t e r n a t i v e management p o l i c i e s f o r t h e New B r u n s w i c k f o r e s t s o f Canada. The methodo l ogy c o n s i s t s o f (1) c o n s t r u c t i o n o f a v a l i d a t e d s i m u l a t i o n m o d e l , f r om w h i c h (2) r eward p r o b a b i l i t y d i s t r i b u t i o n s f o r a l t e r n a t i v e p o l i c i e s a r e i n f e r r e d , w h i c h a r e (3) compared t o i d e n t i f y p o s s i b l e p r e f e r e n c e h i e r a r c h i e s among p o l i c i e s . 5.2 The F o r e s t Management System The b o r e a l f o r e s t s o f N o r t h A m e r i c a have , f o r c e n t u r i e s , e x p e r i e n c e d p e r i o d i c o u t b r e a k s o f a d e f o l i a t i n g i n s e c t , t h e s p r u c e budworm. In any one o u t b r e a k c y c l e a ma jo r p r o p o r t i o n o f t h e mature s o f t w o o d f o r e s t i n a f f e c t e d a r e a s can d i e , w i t h m a j o r con sequence s t o t h e economy and employment o f r e g i o n s l i k e New B r u n s w i c k , w h i c h a r e h i g h l y dependent on t he f o r e s t i n d u s t r y . An e x t e n s i v e i n s e c t i c i d e s p r a y i n g programme i n i t i a t e d i n New B r u n s w i c k i n 1951 has s u c ceeded i n m i n i m i z i n g t r e e m o r t a l i t y , bu t a t t h e p r i c e o f m a i n t a i n i n g i n c i p i e n t o u t b r e a k c o n d i t i o n s o v e r an a r e a c o n -s i d e r a b l y more e x t e n s i v e t han i n t h e p a s t . The p r e s e n t management app r oach 51 i s , t h e r e f o r e , p a r t i c u l a r l y s e n s i t i v e t o u n e x p e c t e d s h i f t s i n e c o n o m i c s , s o c i a l and r e g u l a t o r y c o n s t r a i n t s , and t o u n a n t i c i p a t e d b e h a v i o u r o f t he f o r e s t e c o s y s t e m . A s i m u l a t i o n model o f t he budworm/ fo re s t e c o s y s t e m was d e v e l o p e d by I IASA and IARE s c i e n t i s t s [ 4 2 ] t o be used as a l a b o r a t o r y w o r l d t o a i d i n t he d e s i g n and e v a l u a t i o n o f t h e a l t e r n a t i v e p o l i c i e s . The key r e q u i r e m e n t o f . t h a t l a b o r a t o r y w o r l d was t h a t i t c a p t u r e t h e e s s e n t i a l q u a l i t a t i v e b e h a v i o u r o f t h e budwo rm/ fo re s t e c o s y s t e m i n bo th space and t i m e . E x t e n s i v e d a t a c o n c e r n i n g f o r e s t - p e s t and economic i n t e r r e l a t i o n s has been c o l l e c t e d o v e r t h e p a s t 30 y e a r s by E n v i r o n m e n t Canada as one o f t he e a r l i e s t i n t e r -d i s c i p l i n a r y e f f o r t s i n t h e f i e l d o f r enewab l e r e s o u r c e management. The e s s e n t i a l q u a l i t a t i v e b e h a v i o u r i n t i m e has been i d e n t i f i e d t h r o u g h an a n a l y s i s o f t r e e r i n g s t u d i e s . Four o u t b r e a k s have been d e t e c t e d s i n c e 1770 each l a s t i n g 7 t o 16 y e a r s , w i t h a 34 t o 72 y e a r p e r i o d between t he o u t b r e a k s . D u r i n g t h e i n t e r - o u t b r e a k p e r i o d s t h e budworm i s p r e s e n t i n b a r e l y d e t e c t a b l e d e n s i t i e s w h i c h , when a p p r o p r i a t e c o n d i t i o n s o c c u r , can i n c r e a s e e x p l o s i v e l y o v e r f o u r o r d e r s o f magn i t ude d u r i n g a 3 t o 4 y e a r p e r i o d . The d i s t i n c t i v e p a t t e r n i n t i m e i s p a r a l l e l e d by one i n s p a c e . The h i s t o r i c a l o u t b r e a k s t y p i c a l l y i n i t i a t e d i n one t o t h r e e o r f o u r l o c a l a r e a s o f E a s t e r n Canada and f rom t h o s e c e n t r e s s p r e a d t o c o n t a m i n a t e p r o -g r e s s i v e l y l a r g e r a r e a s . C o l l a p s e o f t h e o u t b r e a k s o c c u r r e d i n t he o r i g i n a l c e n t r e s o f i n f e s t a t i o n i n c o n j u n c t i o n w i t h m o r t a l i t y o f t he t r e e s and s i m i l a r l y s p r e a d t o t he a r e a s i n f e s t e d a t l a t e r t i m e s . The r e s u l t i n g h i g h deg ree o f s p a t i a l h e t e r o g e n e i t y i n t h e f o r e s t age and s p e c i e s com-p o s i t i o n i s c l o s e l y c o u p l e d . t o t h e " c o n t a m i n a t i o n " f e a t u r e c au sed by t h e h i g h d i s p e r s a l p r o p e r t i e s o f t h i s i n s e c t . 52 F i g u r e 5.1 d e p i c t s t he b a s i c model s t r u c t u r e f o r t he budworm s i m u l a t i o n m o d e l . In ou r i n v e s t i g a t i o n we have used t h e model f o r a s i n g l e s i t e ( s i t e - m o d e l ) , t h a t i s Budworm S u r v i v a l M o d e l , F o r e s t Response M o d e l , Budworm C o n t r o l P o l i c y and F o r e s t Management P o l i c y . D i s p e r s a l between s i t e s i s m o d e l l e d as s t o c h a s t i c w i t h o u t d i r e c t r e f e r e n c e t o o t h e r s i t e s . In t h e unmanaged s i t u a t i o n budworm numbers r ema i n low u n t i l b e t t e r w e a t h e r i n d u c e s an o u t b r e a k . Once beyond t h e c o n t r o l o f n a t u r a l e n e m i e s , budworm numbers grow r a p i d l y r e g a r d l e s s o f w e a t h e r , and m o r t a l i t y t o o l d e r t r e e s i s h i g h . However, budworm l a r v a e have poor s u r v i v a l on t r e e s l e s s t h a n t h i r t y y e a r s o l d . Hence, budworm numbers d e c l i n e d r a m a t i c a l l y a f t e r t he few y e a r s t a k e n t o d e s t r o y t h e o l d e r age c l a s s e s o f f i r -F o l l o w i n g t he p o p u l a t i o n c r a s h , budworm rema in s a t a low endemic l e v e l u n t i l t he f o r e s t r e c o v e r s and an ample s u p p l y o f o l d e r t r e e s becomes a v a i l -a b l e t o s u p p o r t a n o t h e r o u t b r e a k . The t i m e between o u t b r e a k s i s t y p i c a l l y t h i r t y t o s e v e n t y y e a r s . In a managed f o r e s t t he above p a t t e r n can be a l t e r e d p r i n c i p a l l y i n two ways. F i r s t , t he budworm's f o o d s u p p l y can be r e d u c e d by l o g g i n g . Second , t he s u r v i v a l o f l a r g e l a r v a e can be r e d u c e d by s p r a y i n g . The p r i n c i p a l p o l i c y p r o b l e m i s t o d e t e r m i n e t h e a p p r o p r i a t e l e v e l s o f l o g g i n g and s p r a y i n g . 5.3 The F o r e s t Sys tem P r o b l e m The most a p p a r e n t o b j e c t i v e i n f o r e s t management i s m a x i m i z a -t i o n o f p r o f i t f r om l o g g i n g . A d d i t i o n a l c o n s i d e r a t i o n s a r e m a i n t e n a n c e o f h i g h r e c r e a t i o n a l v a l u e , low l e v e l o f p e s t i c i d e s p r a y i n g From [42]. BUDW0RW1 CONTROL POLICY B U D W O R M SURVIVAL MODEL 4or e a c h S i - t e FOREST RESPONSE M O D E L tor e a c h s i - t e F O R E S T MANr\&£MENT| P O L I C Y 1 > DISPERSAL B E T W E E N S I T E S t E. N D T I M E LOOP Figure 5.1. The basic structure of the budworm/forest model. 54 and s t a b l e l e v e l s of employment i n the f o r e s t i n d u s t r i e s . In t h i s case high r e c r e a t i o n a l value proved to be commensurate with p r o f i t maximization [42]. Thus, we w i l l confine ourselves to comparing p o l i c i e s i n terms of discounted p r o f i t , spraying frequency and employment s t a b i l i t y . A time horizon of one hundred years was chosen in order to allow time f o r two outbreaks. The three performance measures are defined as f o l l o w s . (1) The t o t a l discounted p r o f i t . i s 100 P = I (1-p) where p i s the discount rate and i = l 1 75 R. = I H.. Q,(V-L.) i s the net p r o f i t i n year i , j _ j i J J J H.. i s harvest i n year i of trees of age j , Q. i s the average value of 1 umber.from one tree of age j , V i s the sal e p r i c e of one u n i t minus the stumpage charge minus the average cost of t r a n s p o r t i n g a log to the mi 11, and L, i s the cost of logging one u n i t of wood from trees J of age j . (2) The spraying index i s S = l-N/100 where N i s the number of years with spraying. 55 (3) The employment s t a b i l i t y index i s 100 E = 12000/ I T. i=2 75 where T. = I max(H. , . - H. ., 0) Q. i l - l , J i j J Two approaches have been taken to d e r i v e management p o l i c i e s f o r the budworm system. In the f i r s t a s i m p l i f i e d v e r s i o n of the budworm s i t e model was used to formulate a dynamic programming problem [42]. This model required a high degree o f aggregation of v a r i a b l e s and the e l i m i n a t i o n of d i s p e r s a l . The f o l l o w i n g p o l i c i e s were derived. In the absence of budworm and f o r a given discount rate p and p r i c e P ($45 per c u n i t here-a f t e r ) , there i s an age of trees i * ( p , P ) such that in each year a l l trees of age i > i * ( p , P ) would be harvested. Trees of age i < i * ( p , P ) should be l e f t . In the presence of budworm, the age of c u t t i n g i * ( p , P ) s t i l l holds. That is, t rees of age i > i * ( p , P ) should be harvested. But f o r trees of age i < i * ( p , P ) the c o r r e c t p o l i c y depends upon not only p and P, but a l s o on the c o n d i t i o n of the f o l i a g e , the budworm de n s i t y l e v e l , and the age of the t r e e * , Examples of these p o l i c i e s are given by H o l l i n g et al. [40]. In general they take on the f o l l o w i n g form: ( i ) At some c r i t i c a l age ( 2 0 - 2 5 , y e a r s ) begin s p r a y i n g when budworm d e n s i t y is high and f o l i a g e i s in moderate to good c o n d i t i o n . ( i i ) With advancing age o f t r e e s , spray a l s o when budworm d e n s i t y i s h i g h and f o i l age is i n f a i r to moderate c o n d i t i o n . ( i i i ) A t another c r i t i c a l age ( 5 0 - 6 0 , y e a r s ) log when budworm d e n s i t y is ex t r e m e l y h i g h . 56 ( i v ) At another s l i g h t l y h i g h e r age, a l s o l og whenever f o l i a g e i s in poor to f a i r cond i t i o n . (v) At another s l i g h t l y h i g h e r age, a l s o l og whenever f o l i a g e i s in f a i r to moderate cond i t i o n . ( v i ) F i n a l l y age i * ( p , P ) i s reached a t which p o i n t t r e e s a r e h a r v e s t e d r e g a r d l e s s o f budworm d e n s i t y and f o r e s t c o n d i t i o n s . Under t h i s p o l i c y there are no budworm outbreaks over a 100 year period f o r p r i c e of lumber greater than $45 per u n i t and discount r a t e p < .05. Under these c o n d i t i o n s i t i s economically sound to prevent budworm outbreaks through i n t e n s i v e spraying. For higher discount rates or lower lumber p r i c e s outbreaks do occur. Two of the p o l i c i e s w i l l be examined here; WI - 0% discount rate - and W2 - 10% discount r a t e . The second approach has been to apply h e u r i s t i c " o p t i m i z a t i o n " techniques to the simu l a t i o n model f o r one s i t e . The r e s u l t i n g p o l i c i e s are s t r u c t u r e d somewhat d i f f e r e n t l y . Each s p e c i f i e s some age i * ( p,P) at which to log and that logging s h a l l not be done on younger t r e e s . Further-more, each computes a hazard index H such that a l l stands of trees f o r which H > H* are sprayed, and stands with the H < H* are l e f t as i s . These p o l i c i e s d i f f e r only i n t h e i r s p e c i f i c a t i o n f o r the f u n c t i o n a l forms of i * (p,P) and H, and the c r i t i c a l l e v e l H*. The f i r s t one has the f o l l o w i n g s p e c i f i c a t i o n : a) H* = 1000, b) H = a + b x T, c) i * ( p,P) = p + qT + rH, where T i s the mean age of trees and x i s the budworm de n s i t y . 57 The parameters p, q, r , and b were chosen to maximize logging p r o f i t . Using an h e u r i s t i c search technique, parameter values a = 206, b = .17, p = 55, q = .08, and r = -.004 were found. One should note that t h i s p o l i c y i s independent of both the p r i c e of lumber and the p r e v a i l i n g discount r a t e . Fourteen examples of t h i s p o l i c y w i l l be examined. The f i r s t f i v e cases — HI,—,H5 — w i l l have H* = 200, 400, 600, 1000 and 2000 r e s p e c t i v e l y . The second f i v e — H6 s... 5H10 — w-jn again have H* = 200, 400, 600, 1000 and 2000 r e s p e c t i v e l y and p = 70 (rather than 55). A l l ten of the preceding s t a r t with current f o r e s t c o n d i t i o n s . The four remaining cases s t a r t with an ' i d e a l ' f o r e s t , i . e . a f o r e s t with equal numbers of trees in each age c l a s s . P o l i c i e s II and 12 have 55 age c l a s s e s , p = 55 and H* = 200, H* > 400 r e s p e c t i v e l y . P o l i c i e s 13 and 14 have 70 age c l a s s e s , p = 70 and H* = 200, H* > 400 r e s p e c t i v e l y . The second set of p o l i c i e s of the above type s p e c i f y : a) H* = 60 b) H = budworm egg d e n s i t y c) i*(p,P) = p + qT + r ( a + bxT) with parameters p, q, r , a and b as above. Again we have a p o l i c y independent of lumber p r i c e and discount r a t e . We examine three cases - J l , J2 and J3 - where H* = 30, 60 and 120 r e s p e c t i v e l y . F i n a l l y , four p o l i c i e s are examined which convert a f o r e s t with the current age s t r u c t u r e i n t o a f o r e s t with an i d e a l age s t r u c t u r e . P o l i c i e s Tl and T2 s p e c i f y c u t t i n g the o l d e s t 1/55th (or 1/70th) of the trees per year f o r the f i r s t 55 years. P o l i c i e s T3 and T4 are s i m i l a r using 1/70th and 70 years. T h e r e a f t e r , Tl through T4 are i d e n t i c a l to II 58 through 14 r e s p e c i t v e l y . Spraying during the t r a n s i t i o n period i s .determined as i n II through 14 r e s p e c t i v e l y . A l l together, twenty-three p o l i c i e s w i l l be examined. In order to compare these p o l i c i e s using s t o c h a s t i c dominance r u l e s , the d i s t r i b u t i o n s a s s o c i a t e d with these p o l i c i e s must be determined. Moreover, a s a t i s f a c t o r y comparison of these random v a r i a b l e s requires development of e f f i c i e n t algorithms. Topics of the next two s e c t i o n s deal with these s u b j e c t s . 5.4 D e r i v a t i o n of Outcome D i s t r i b u t i o n s The complexity of the model makes i t impossible to d i r e c t l y determine the p r o b a b i l i t y d i s t r i b u t i o n of outcomes f o r a given p o l i c y . Hence, we must use Monte Carlo Techniques to estimate the d i s t r i b u t i o n . By sampling from the d i s t r i b u t i o n of d i s p e r s a l p o s s i b i l i t i e s and the d i s -t r i b u t i o n of weather sequences, we can use the s i t e model to obtain a sample point i n the outcome d i s t r i b u t i o n . As the number of independent samples i n c r e a s e s , the sample outcome d i s t r i b u t i o n converges to the prob-a b i l i t y outcome d i s t r i b u t i o n . The sampling schemes f o r d i s p e r s a l and weather were as f o l l o w s . Since we are c o n s i d e r i n g only one s i t e ( 6 x 9 m i l e s ) , the d i s p e r s a l over the whole area can only be t r e a t e d s t o c h a s t i c a l l y . A s a t i s f a c t o r y f i t was obtained by s e t t i n g I = E«e , where I i s immigrants to the s i t e , E i s emigrants from the s i t e and X i s a normally d i s t r i b u t e d random v a r i a b l e with mean 0 and standard d e v i a t i o n .04. Weather sequences f o r New Brunswick have been analysed [42] and l i t t l e year to year c o r r e l a t i o n was found. An adequate model i s t h a t poor weather occurs with p r o b a b i l i t y .25, average 59 weather with p r o b a b i l i t y .50, and good weather with p r o b a b i l i t y .25. Sample weather sequences were randomly generated using the above prob-a b i l i t i e s f o r the three weather types. 5.5 The Screening Algorithm One can screen those p o l i c i e s which are s u p e r i o r to others ( i n the expected u t i l i t y sense) by comparing: 1) cumulative simulated per-formance d i s t r i b u t i o n s at a l l p o i n t s , or 2) areas under the cumulative d i s t r i b u t i o n s at a l l p o i n t s , or 3) areas under the i n t e g r a l s of the cumu-l a t i v e d i s t r i b u t i o n s at a l l p o i n t s . For the purpose of e m p i r i c a l s t u d i e s , a d i s c r e t e v e r s i o n of s t o c h a s t i c dominance r u l e s must be adopted [77]. Let the points X i , x 2 , * * , , x n be the r e a l i z a t i o n s of the random v a r i a t e X with d i s t r i b u t i o n F, and l e t x x < x 2 < ••• < x . Then the cumulative e m p i r i c a l d i s t r i b u t i o n F (x) i s a step f u n c t i o n with steps at x = x.., i = l,2,«««,n. F (x) = P(X < x) = I p., p. = P(X = x.)/q and q = I P(X = x.). x . < x i = l i -If the observations are from completely random experiments, then the p r o b a b i l i t y p. = -^ f o r a l l i can be used as an approximation to the true d i s t r i b u t i o n [29]. We are now prepared to r e s t a t e FSD, SSD and TSD c o n d i t i o n s i n d i s c r e t e form. Let S = {x i,x 2,•••,x } be the set of sample points x 1 n determining F p ( x ) and S^ = {y i , y 2 , , y p } be the set of sample points determine G n ( y ) . Let S z = S xU S y = {zx,z2,• • • ,z^} with Z i < z 2 < < z 9 , that i s , the z's are the x's and y's combined i n t o one ordered set. 60 Rule 1 F n >x G n i f and only i f F ^ ( z . ) < G ^ 1 ^ ) , i = l ,2,---,2n. Rule 2 F n > 2 G n i f and only i f F ^ 2 ^ . ) < G ^ 2 ^ . ) , i=l,2,•••,2n, where F ^ = F n n F ^ ( z i ) = 0 and G ^ , G ^ ( z . ) are defined s i m i l a r l y . n \ n n v i J For both f i r s t and second degree dominance the d i s t r i b u t i o n s are compared only at the observation p o i n t s . Such comparisons are s u f f i c i e n t s i n c e (2) (2) G (x) - F'('x) i s a step f u n c t i o n and G v - F v ' i s piecewise l i n e a r . Let F ( 3 ) ( z . ) = \ I [ F ( 2 ) ( z . ) + F { 2 ) ( z . J ] ( z . - z. , ) , i=2,--.,2n n v v 2 j=2 n J n J " 1 J J - l F ^ ( z , ) = 0, and G ^ ( z . ) i s defined s i m i l a r l y . Then the comparison a t the observations points only i s not s u f f i c i e n t to conclude the dominance of F ^ over G ^ (TSD). This f o l l o w s from the n o n l i n e a r i t y n n v J of G ^ - F ^ . I t has a l o c a l minima at a l l points z* where G^ 2^(z*) -n n ^ n F ^ ( z * ) = 0 and G ( z * ) - F (z * ) > 0. Thus the algorithm of Porter ..et al. n v n n 3 must be modified to include such p o i n t s . Rule 3 F N >3 G N i f and only i f F * 3 ^ ) _< G J 3 ) ( Z . ) , i=l ,2,• • • ,2n, and F ^ 2 ) ( z 2 n ) < G ( 2 ) ( z Q ), and n - n 2n 61 F ( 3 ) ( z * ) < G ( 3 ) ( Z * ) , where n v ' - n F ( 3 ) ( z * ) = F ( 3 ) ( z . ) + 1 [ F ( 2 ) ( z . ) + F ( 2 ) ( z * ) ] ( z * - z. ), n \ / n v k 2 n v k n v / J x k Points z * must sa t i s f y four condit ions: i ) z* £ ( z k , z k + 1 ) , i i ) F^ 2 ) ( z * ) - G^ 2 ) ( z * ) = 0, i i i ) F ^ 2 ) ( z k ) - G j 2 ) ( z k ) > 0 and G ^ ( z * ) i s d e f i n e d s i m i l a r l y , n 5.6 Stohastic Dominance Test Results Pol icy rankings under a l l three stochastic dominance c r i t e r i a for four performance measures are given in Figures 5.2-5.5. Tabulated values of mean, standard deviation and range are in Tables I and II and a summary of dominance re lat ions is given in Table I I I. Examination of these results y ie lds the fol lowing general observation: po l i c ie s which promote a high spraying frequency do poorly under both economic c r i t e r i a (po l i c ie s J2, J3, II, 13, Tl and T3). It i s also noted that those po l i c ie s s p e c i f i c a l l y designed to y i e l d nearly equal harvests each year are successful in minimizing socio-economic d i s l oca t i on , but at the expense of low p ro f i t leve l s . If we f i r s t r e s t r i c t our attention to p r o f i t , we f ind the optimal pol icy is to cut trees at age 70 and never spray with an ' i d e a l ' forest . However, such a pol icy cannot be applied un t i l the ideal forest 62 Figure 5.2. Dominance^relations derived f o r budworm/forest p o l i c i e s -p r o f i t / y r at 0% discount. 64 %3 J l , W I ,W2 J I , W | ( W 2 H5 H4--HI0 H3 H3 H 2 H 2 H9 H 9 H7, H8 H 7 , H 8 J 2 . J 2 J 3 J 3 H I . H 6 HI ,H6 Figure 5.4. Dominance r e l a t i o n s derived f o r budworm/forest p o l i c i e s frequency'with no spraying. Figure 5.5. Dominance r e l a t i o n s derived f o r budworm/forest p o l i c i e s -employment s t a b i l i t y index. 66 Table I P o l i c y Returns 1. Mean, Standard Deviation and Range of Return are Given f o r each P o l i c y . Results were obtained from Twenty Simulation Runs of One Hundred Years Time Horizon using Twenty D i f f e r e n t S t o c h a s t i c Weather Sequences:. $ P r o f i t / y r . @ 0% Discount $ P r o f i t / y r . @ 10% Discount Pol i c y Mean ± S.D. Range Mean ± 1 S.D. Range H 1 - 77 818 + 0 - 8349 + 0 2 3 134 + 2413 - 890, 6382 1 + 58 - 102, 71 3 2 155 + 3144 - 3972, 6382 - 13 + 71 - 174, 71 4 1 738 + 3402 - 3322, 6382 - 27 + 86 - 237, 71 5 6 158 + 689 3898, 6848 71 + 13 27, 91 6 - 51 670 + 0 - 8312 + 0 7 18 929 + 2063 15 093, 21 853 - 50 + 51 - 160, 22 8 16 787 + 1923 13 099, 19 977 - 40 + 51 - 147, 27 9 . 14 531 + 3582 10 588, 18 883 10 + 40 - 52, 84 10 14 853 + 3478 10 714, 20 768 49 + 39 - 42, 91 J 1 2 931 + 3634 - 9 137, 6382 - 23 + 120 - 455, 70 2 - 46 664 + 0 - 2761 + 0 3 - 58 452 + 0 - 4831 + 0 W 1 7 118 + 2684 2 469, 12 115 48 + 13 24, 72 2 9 212 + 2912 4 792, 15 883 74 + 20 36,112 I 1 - 79 927 + 0 - 7963 + 0 2 4 973 + 0 457 + 0 3 - 38 763 + 0 - 2414 + 0 4 45 990 + 161 45 302, 46 075 6016 + 2 6011,6020 T 1 -105 115 + 0 -13 225 ± 0 2 - 20 915 + 0 - 4 806 ± 0 3 - 80 593 + 0 -12 386 ± 0 4 3 607 + 0 - 3 966 i : 0 67 Table II P o l i c y Returns 2. Mean, Standard Deviation and Range of Return are Given f o r each P o l i c y . Results were obtained from Twenty Simulation Runs of One Hundred Years Time Horizon using Twenty D i f f e r e n t S t o c h a s t i c Weather Sequences Non-spray Frequency Employment S t a b i l i t y Index P o l i c y Mean + 1 S.D. Range Mean t 1 S.D. Range H 1 0 + 0 1648 + 0 2 .972 .0218 .94, 1 1384 + 463 815, 2608 3 .975 + .0199 • 94, 1 1270 + 442 427, 2121 4 .986 + .0136 .95, 1 958 + 616 434, 2608 5 .993 + .0047 • 99, 1 1285 + 435. 651, 2608 6 0 + 0 1552 + 0 7 .848 + .0237 .82, .88 1450 + 275 1119, 2121 8 .842 + .0217 .80, .88 920 + 144 720, 1267 9 .896 + .0246 .86, .96 720 + 72 579, 862 .10 .979 + .0095 .96, .99 1817 + 1695 435, 8361 J 1 1 + 0 1079 + 559 441, 2608 2 .370 + 0 1648 + 0 3 .230 + 0 1648 + 0 W 1 1 0 1271 + 670 558, 2517 2 1 + 0 1573 + 787 618, 2857 I 1 0 + 0 29860 + 0 2 1 + 0 29860 + 0 3 0 + 0 4284 + 0 4 1 + 0 4847 + 129 4284, 4877 T 1 0 + 0 24350 + 0 2 1 + 0 24350 + 0 3 0 + 0 27300 + 0 4 1 + 0 27300 + 0 68 Table III Dominance Rankings of P o l i c i e s under Four Performance C r i t e r i a The c r i t e r i a are: 1. p r o f i t at 0% discount rate - 0% 2. p r o f i t a t 10% discount r a t e - 10% 3. frequency of years with no F N S spraying - . . . 4. employment s t a b i l i t y index - E.S.I. P o l i c y rankings are i n d i c a t e d by a s t e r i s k s : * means the p o l i c y was dominated by only one other p o l i c y ; ** means the p o l i c y was not dominated. A l l other p o l i c i e s were dominated by two or more others. No separate t a b l e i s given f o r t h i r d degree dominance s i n c e i t s t a b l e was i d e n t i c a l to that f o r second degree dominance, i n t h i s case. 1st Degree 2nd & 3rd Degre Pol i c y 0% 10% F.N.S. E.S.I. 0% 10% F.N.S. E.S.I. H 1 2 3 4 5 ** * ** ** ** ** ** * ** 6 7 ** * ** ** * 8 9 10 * * * ** * * ** J 1 ** ** 2 ** ** 3 ** ** W 1 * ** * * ** 2 ** ** ** ** ** * 69. i s created. A p l a u s i b l e p o l i c y then i s one which combines a t r a n s i t i o n s t r a t e g y to an ' i d e a l ' f o r e s t and then a switch to a long term steady s t a t e s t r a t e g y . With a time horizon of one hundred years, none of the teste d t r a n s i t i o n s t r a t e g i e s worked w e l l . I n v a r i a b l y the lo s s e s accrued i n the e a r l y years of t r a n s i t i o n were not repaid by the l a t e r years of high p r o f i t . Thus, f o r the remainder of the d i s c u s s i o n we s h a l l r e s t r i c t our a t t e n t i o n to p o l i c i e s f o r the management of the f o r e s t with i t s current age d i s t r i b u t i o n . Low discount rate favours c u t t i n g o l d e r trees and spraying with moderate frequency, while higher discount rate favours e a r l i e r c u t t i n g and lower frequency of spraying. I f we are constrained to cut at age 55, the same p o l i c i e s are favoured regardless of discount r a t e . But, i f we are constrained to cut at age 70, then a s h i f t from low to high discount r a t e i s accompanied by a s h i f t from low to high spray t h r e s h o l d . A d d i t i o n a l points of i n t e r e s t i n c l u d e : the best p o l i c i e s derived v i a formal techniques (Wl and W2, dynamic programming with s i m p l i f i e d model) d i d no b e t t e r than the best h e u r i s t i c a l l y derived p o l i c i e s ; the dynamic programming p o l i c y derived under 10% discount rate was s t o c h a s t i c a l l y s u p e r i o r to the one derived under 0% discount rate regardless of the discount r a t e used i n the simulation model; p o l i c i e s which d i f f e r e d only i n H* tended to produce bimodal d i s t r i b u t i o n s of p r o f i t , suggesting that spraying as soon as budworm numbers pass a small t h r e s h o l d or only when budworm numbers are high are both s u p e r i o r to spraying when budworm numbers are moderate and threatening to outbreak. This observation a r i s e s because budworm numbers are c o n t r o l l e d n a t u r a l l y i n the many years of poor to average weather, provided that the population i s not 'too l a r g e ' . By keeping 70 numbers down in good years through spraying, t h i s natural c o n t r o l i s e f f e c t i v e l y u t i l i z e d . Less frequent spraying allows budworm epidemics anyway. Hence, one might as well drop spraying e n t i r e l y . Examining those p o l i c i e s which r a r e l y spray (say l e s s than 5% of the y e a r s ) , we f i n d the most p r o f i t a b l e are W2, r e g a r d l e s s of discount r a t e , and H5, i f discount rate i s high. F i n a l l y , i f we examine d i s l o c a t i o n , we f i n d that p o l i c i e s which s p e c i f y equal or n e a r l y equal harvests per year dominate. However, such p o l i c i e s are not reasonable economic choices under c u r r e n t f o r e s t condi-t i o n s . S e t t i n g them aside, we f i n d a l a r g e number of p o l i c i e s c l u s t e r e d together. Employing the second degree dominance c r i t e r i o n reduces t h i s set to four non-dominated p o l i c i e s : H10, HI, J2 and J3. The l a t t e r three p o l i c i e s are poor choices since they r e q u i r e spraying in over 50% of the years. 5.7 S e n s i t i v i t y A n a l y s i s and Robustness of Results L e t t i n g F and G be the given t h e o r e t i c a l d i s t r i b u t i o n s , the question of dominance r e l a t i o n s h i p between F and G i s merely an e x e r c i s e i n c a l c u l a t i o n . In e m p i r i c a l s t u d i e s these d i s t r i b u t i o n s are estimated by sample d i s t r i b u t i o n s F^ and Gn- I f F n dominates G n, then we proceed to conclude that F dominates G. However, i t i s c l e a r that F may or may not dominate G. In t h i s s e c t i o n the "confidence" l e v e l attached to the statement F n dominates G^ implies F dominates G i s discussed. The p r o b a b i l i t y of f i n d i n g F >i G or G >i F„ when F and ^ J 3 n - n n - n n G n have the same underlying d i s t r i b u t i o n can be computed d i r e c t l y . Let the sample s i z e be 2n, n points each f o r F and G. Since Fn and G„ are 71 from the same d i s t r i b u t i o n s , each of the 2n sample po in t s are observed f o r F or G w i th equal l i k e l i h o o d . Thus, there are ( 2 n ) sequences of observat ions w i th equal p r o b a b i l i t y . Then F^ >! G^ i f and on l y i f f o r each subsequence {z^)^_y r= l ,2,•••,2n, the number of x ' s belonging to F^ i s no l a r g e r than the number of y ' s belonging to F . Let W. = 1 i f z . = x. -1 i f z . = y . r Then F >i G i f and on ly i f I W. < 0 f o r a l l r = 1,2, — 9 2n. n - n i = l i -r S i m i l a r l y G n >i F^ i f and on ly i f I W. > 0 f o r a l l r = 1,2,•••,2n. i = 1 P r There are (n"-|) sequences f o r which I W. = - 1 . Thus the p r o b a b i l i t y of 2 i = 1 1 observ ing F >x G or G >i F i s - ^ y . This r e s u l t may be used as one c r i t e r i o n f o r s e l e c t i n g sample s i z e . However, no such nonparametric r e s u l t i s a v a i l a b l e f o r second and t h i r d degree dominance. This i s due to the f a c t t ha t the c a l c u l a t i o n of a reas , F 2 ( x ) , G 2 ( x ) , r equ i re s the measurement of z.. - z.. ^, i = l ,2, • • • ,2n. Lack ing any proven s t a t i s t i c a l t e s t s , we chose to compare t runcated sample d i s t r i b u t i o n s . S ince the t a i l s of a d i s t r i b u t i o n are g e n e r a l l y the l e a s t a c c u r a t e l y est imated par t of the d i s t r i b u t i o n , we recompared the dominance r e l a t i o n s a f t e r removing the sma l l e s t 5% and l a r g e s t 5% of the observat ions f o r each emp i r i c a l d i s t r i b u t i o n . Resu l t s of t h i s a n a l y s i s appear i n Table IV. Two types of changes were obta ined i n the dominance r e l a t i o n s . In on ly one case was a dominance r e l a t i o n rever sed ; f o r p r o f i t a t 0% d i scount HI0' >3 H9 was reversed ( reduc t i on in r i s k of H9, see Table IV).. Second and more common 72 Table IV Changes Greater than 10% i n Mean and/or Standard Deviation obtained by Truncating P o l i c i e s . The Maximum and Minimum Value f o r P o l i c y was Deleted, l e a v i n g 18 Outcomes Per P o l i c y of the O r i g i n a l 20 Index Pol i c y Mean St. Dev P r o f i t / y r . J] 2.93 x 10 3 -> 3.41 x 10 3 3.63 x 10 3 + 2 39 x 10 3 @ 0% H9 3.58 x 10 3 -> 1 69 x 10 3 P r o f i t / y r . H5 13 •+ 7 @ 10% J l -23 + -4 120 ->• 69 Employment H5 435 -> 299 S t a b i 1 i t y H10 1817 -> 1530 1695 -> 791 73 were changes of the form F G to F >.. G. Such changes occur p r i m a r i l y from t r u n c a t i o n when cumulative d i s t r i b u t i o n f u n c t i o n s cross a t the f a r l e f t . 5.8 Conclusions Of primary i n t e r e s t i s the question of whether a p p l i c a t i o n o f s t o c h a s t i c dominance r u l e s enhance the a b i l i t y to r a t i o n a l i z e resource management. As more and more of our natural resources become an i n t e g r a l part of the p u b l i c domain ( e i t h e r d i r e c t l y through ownership and management or i n d i r e c t l y through r e g u l a t i o n ) , the need to consider s o c i a l preferences in t h e i r management i n t e n s i f i e s . Rather than making a r b i t r a r y assumptions about the wishes of the amorphous p u b l i c or spending resources e x c e s s i v e l y to obtain information about p r e v a i l i n g preferences among p o s s i b l e system opti o n s , i t i s useful to u t i l i z e some of the more un i v e r s a l and s t a b l e r a t t r i b u t e s of these preferences i n forming p o l i c y . As demonstrated i n t h i s chapter, s t o c h a s t i c dominance o f f e r s a t h e o r e t i c a l l y a t t r a c t i v e and prac-t i c a l l y f e a s i b l e tool f o r p o l i c y option e v a l u a t i o n . Reasonable, rather weak and e a s i l y v e r i f i a b l e assumptions about the commonality among i n d i v i d u a l s ' preferences are used to produce an e f f e c t i v e a l gorithm with good screening power. SUMMARY We have attempted i n t h i s d i s s e r t a t i o n to i n t e g r a t e the two t r a d i t i o n a l approaches to managerial d e c i s i o n making under u n c e r t a i n t y : the c e n t r a l tendency-dispersion approaches and expected u t i l i t y maximization. We then proceeded to advance the f r o n t i e r s of these t h e o r i e s by extending s t o c h a s t i c dominance r u l e s obtained f o r uni-dimensional outcome spaces to m u l t i - a t t r i b u t e d e c i s i o n s i t u a t i o n s . The d i s s e r t a t i o n concluded with demonstration of the use of s t o c h a s t i c dominance c r i t e r i a in development of t h e o r i e s ( p o r t f o l i o s e l e c t i o n problems) and managerial d e c i s i o n making ( f o r e s t management). 74 BIBLIOGRAPHY A i i , M.M., " S t o c h a s t i c Dominance and P o r t f o l i o A n a l y s i s , " Journal of Financial Economics, II (1975), 205-227. 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(eds.), Stochastic Optimization Models in Finance, New York: Academic Press, 1975. APPENDIX ADDITIONAL RESULTS From the development of necessary and s u f f i c i e n t s t o c h a s t i c dominance orderings f o r U i , U 2 , U 3 and U^, i t i s c l e a r that the F > 1 G = * F > 2 G ^ F > 3 G = > F > D G since Ux D U 2 D U 3 D \) Furthermore, one can obtain higher degrees of dominance r e l a t i o n s by c o n s i d e r i n g the set U n where U n i s a convex cone contained in Up. The e x i s t e n c e of a s t o c h a s t i c o r d e r i n g follows from the existence of a dual cone i n / f : , - t h e ' s e t of p r o b a b i l i t y measures [65]. However, the economic s i g n i f i c a n c e of higher s t o c h a s t i c orderings i s not c l e a r . The preceding d i s c u s s i o n i n d i c a t e s t h a t the stronger ordering can be obtained i f the set of u t i l i t y f u n c t i o n s becomes small. In f a c t , in the l i t e r a t u r e d e a l i n g with d e c i s i o n making under u n c e r t a i n t y , u t i l i t y f u n c t i o n s are often assumed to be of a s p e c i f i c form. For example, U log {u|u(x) log(.x + d), d > 0} U exp (u|u(x) d > 0} U (u|u(x) (x+d) 1-c , d > 0, 0 < c < .1}' pow U quad {u|u(x) x - c x 2 , c > 0, x < ~- c} 84 85 The members of U-|og, U g x p and U p Q w e x h i b i t a l l the e s s e n t i a l c h a r a c t e r i s t i c s necessary f o r r a t i o n a l d e c i s i o n making. The members of the l a s t set do not belong to Up. The economic a n a l y s i s which u t i l i z e s the s p e c i a l p r o p e r t i e s of the above u t i l i t y f u n c t i o n s i n d e c i s i o n making under u n c e r t a i n t y can be found i n [7,30,34,35,37,71,82]. S i m i l a r l y , a stronger s t o c h a s t i c o r d e r i n g on F can be achieved by p l a c i n g some a d d i t i o n a l c o n d i t i o n s on the d i s t r i b u t i o n f u n c t i o n s rather than on U, as was done i n theorem 2.1. Let h - {F F has an equal mean} {F F(x) <|>((x - e F ) / s F ) , s F > 0} FI = {F F(x) = <K(t(x) - e F ) / S p ) , t'(x) > 0, Sp > 0} v = (F F i s a X 2 - d i s t r i b u t i o n } {F F i s a T - d i s t r i b u t i o n } FH - {F F i s a normal d i s t r i b t u i o n } ^logN = {F F i s a lognormal d i s t r i b u t i o n } F = s t a b l e .{F F i s a s t a b l e d i s t r i b u t i o n } 7 — uniform {F F i s a uniform d i s t r i b u t i o n } The s t o c h a s t i c dominance a n a l y s i s within these subclasses was i n i t i a t e d by various authors [1,5,30,36]. In p a r t i c u l a r , A l i [1] and Bawa [5] give simple s t o c h a s t i c dominance r u l e s i n terms of s p e c i f i c parameters of the d i s t r i b u t i o n s belonging to F 0 , Fi, F 2 and F . Levy and Hanoch X i [36] and La Cava [47] have a l s o obtained s i m i l a r r e s u l t s on F 0 , F i . 6 e F and denote a locat ion and sca le parameter of F , re spec t i ve l y . 86 In studies of d i v e r s i f i c a t i on analys i s , rates of returns on common stocks are often assumed to be normally d i s t r ibuted. Recent empirical studies, however, have shown that the returns are often better described by lognormal (see Lintner [59] and references there in) , or stable d i s t r ibut ions [20,21]. In view of these empirical re su l t s , the examination of stochastic orderings on the r e s t r i c ted class of F-^0g^ or ^ s t a t ) i e "is useful . Levy provided the f i r s t and second degree dominance c r i t e r i a for lognormal d i s t r ibut ions [55]. Since lognormal d i s t r ibut ions intersect at most at one point, the extension of SSD to TSD i s immediate. If X and Y have symmetric stable d i s t r ibut ions with < °°, 0 < s^ < °°, common character i s t i c exponent a , 1 < a < 2, yp > and F(x) > G(x) for some x e R, then Ziemba [113] shows that EpU. > EgU for a l l convergent con-1/a 1/a cave non-decreasing u i f and only i f < s 2 ' . This resu l t i s used to derive the e f f i c i e n t f r on t i e r in mean-a dispersion space, a general izat ion of Tobin's separation theorem, and an algorithm for computing approximately optimal port-f o l i o a l l ocat ions . Let F ( y , s, 3 , a ) be a cumulative d i s t r i bu t i on of a stable random var iab le, where parameters y , s, 3 and a are locat ion, scale, skewness, and charac te r i s t i c exponent respect ively. Then the fol lowing addit ional results can be obtained. Theorem. Suppose F(x) ~ F (0 , l , 0 ,ap ) and G(x) ~ G(0, l ,0,oig). Then EpU > EgU for a l l concave non-decreasing d i f f e r e n t i a t e u t i l i t y functions i f and only i f tip > a g . PROOF: Au = EpU - EgU fCO f CO fCO u(x) d(F-G) (F-G)u'(x)dx = (G-F)u'(x)dx -co -oo C d rO (G-F)u'(x)dx + (G-F)u'(x)dx. 87 Suppose Au > 0 and ctp < ctg, then F(x) > G(x) on (-°°,0).' Then r 0 from symmetry (3=0), G(x) > F(x) on (0,°°). Thus (G-F)u'(x)dx < 0 and — oo (G-F)u'(x)dx > 0. But then Au < 0 since u'(x) decreasing implies ,0O (G-F)u' (x)dx| > (G-F)u 1 (x)dx, a c o n t r a d i c t i o n . Therefore otp > a G i s a necessary c o n d i t i o n . If.ctp > a^, then we have the opposite s i t u a -t i o n , thus Au > 0. This r e s u l t i s s i m i l a r to that of La Cava [47]. La Cava gives a necessary and s u f f i c i e n t c o n d i t i o n f o r FSD and SSD in terms of the l o c a t i o n and the s c a l e parameters. He al s o shows that in order to r e l a t e the SSD r u l e to the s c a l e parameter, i t i s necessary to assume t h a t the d i s t r i b u t i o n s are symmetric. Jean [44] attempted to show the f o l l o w i n g TSD and E-SV(h) r e l a t i o n . I f F?(b) < G i ( b ) , R = [a,b], F 2 ( x ) = G 2(x) f o r a l l x e R, then Up > y Q and SV(pp) < S V ( y G ) . However, the above r e s u l t cannot hold, since 2F 2(yp) = SV(iip), 2 G 2 ( u G ) = SV (u G ) and, f o r some F and G, F 2 ( y p ) > F 2 ( y Q ) = G 2 ( u G ) - T n u s SV(yp) > S V ( y G ) . The above problem stems from the f a c t that the d i s t r i b u -t i o n s X and Y do not have a common h point. I f X and Y have a s i n g l e i n t e r s e c t i o n property, then the SSD e f f i c i e n t set i s equal to the TSD ' e f f i c i e n t s e t . Hence e(TSD) = e(E - SV(h)) from Theorem 2 .1. ^See page 11. 

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