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An evaluation of quadratic programming and the MOTAD model as applied to farm planning under uncertainty Lopez, Ramon Eugenio 1977

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AN EVALUATION OF QUADRATIC PROGRAMMING AND THE MOTAD MODEL AS APPLIED TO FARM PLANNING UNDER UNCERTAINTY by RAMON EUGENIO LOPEZ B . S c . , Un ivers i ty of C h i l e , 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Agr icu l tu ra l Economics) We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1977 Co) Ramon Eugenio Lopez In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of A g r i c u l t u r a l Fnnnnnfnr'.s The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Serrtembp.-p 6, 1977. - i -ABSTRACT AN EVALUATION OF QUADRATIC PROGRAMMING AND THE MOTAD MODEL AS APPLIED TO FARM PLANNING UNDER UNCERTAINTY. by Ramon E. Lopez The o b j e c t i v e o f t h i s t h e s i s was to s t u d y the e f f i c i e n c y o f t h r e e methods used i n farm p l a n n i n g under u n c e r t a i n t y . The f i r s t method c o n s i d e r e d v/as the QP-VAR method which m i n i m i z e s the v a r i a n c e o f a c t i v i t y r e t u r n s s u b j e c t t o a minimum income l e v e l u s i n g a q u a d r a t i c programming a l g o r i t h m . The second method i s t h e MOTAD method which m i n i m i z e s the mean a b s o l u t e d e v i a t i o n o f a c t i v i t y r e t u r n s s u b j e c t to a minimum i n -come l e v e l u s i n g a l i n e a r programming a l g o r i t h m . The t h i r d method i s the Semi v a r i a n c e method which m i n i m i z e s the n e g a t i v e 'semi v a r i a n c e o f a c t i v i t y r e t u r n s s u b j e c t t o a minimum income l e v e l . The main elements used t o e v a l u a t e the e f f i c i e n c y o f t h e s e methods were the magnitude o f the b i a s e s and the d i s p e r s i o n o f the e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r o b t a i n e d u s i n g each method. In o r d e r to a c h i e v e t h i s o b j e c t i v e , a r e s e a r c h p r o c e d u r e com-p r i s i n g a t h e o r e t i c a l and an e m p i r i c a l s t u d y was d e v e l o p e d . The t h e o r e -t i c a l s t u d y i n c l u d e d an a n a l y s i s o f the measures o f r i s k used by each - i i -method and of t h e assumptions u n d e r l y i n g the use of such measures. F u r t h e r m o r e , the p l a u s i b i l i t y of t h e s e assumptions was t h o r o u g h l y d i s c u s -sed. U s i n g the c o n c l u s i o n s drawn from the t h e o r e t i c a l s t u d y , a s e t of e x p e r i m e n t s ( t h e e m p i r i c a l s t u d y ) was d e s i g n e d t o t e s t t he e f f i c i e n c y o f t he methods as e s t i m a t o r s o f i n c o m e - r i s k f r o n t i e r s . The purpose o f t h e s e e x p e r i m e n t s was t o t e s t t h e performance of the methods when a p p l i e d u s i n g sample da t a o f r e l a t i v e l y s m a l l s i z e r a t h e r than complete f r e q u e n c y d i s t r i b u t i o n s o f a c t i v i t y r e t u r n s . Two t r i v a r i a t e n o r m a l l y d i s t r i b u t e d p o p u l a t i o n s (one w i t h h i g h and the o t h e r w i t h low degrees of c o r r e l a t i o n among a c t i v i t y r e t u r n s ) and two t r i v a r i a t e gamma d i s t r i b u t e d p o p u l a t i o n s (one w i t h h i g h , the o t h e r w i t h low degrees o f c o r r e l a t i o n among a c t i v i t y r e t u r n s ) r e p r e s e n t i n g a c t i v i t y r e t u r n s d a t a were g e n e r a t e d u s i n g a random number g e n e r a t o r . U s i n g t h e s e p o p u l a t i o n s as d a t a b a s e s , t h r e e p o i n t s on the " t r u e " i n c o m e - r i s k f r o n t i e r s were d e t e r m i n e d a p p l y i n g the a p p r o p r i a t e method i n each c a s e . E s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r s were o b t a i n e d u s i n g randomly drawn samples from t h e p o p u l a t i o n s and the mean r i s k e s t i m a t e s o b t a i n e d u s i n g each method were compared t o e s t a b l i s h b i a s . The degree o f d i s p e r s i o n o f the e s t i m a t e s as p r o v i d e d by each method was a l s o compared. I f two methods were u n b i a s e d , the method w i t h the s m a l l e s t d i s p e r s i o n o f i t s e s t i m a t e s was c o n s i d e r e d more e f f i c i e n t . A g e n e r a l c o n c l u s i o n drawn from t h i s t h e s i s was t h a t t h e r e i s not an o p t i m a l method t o be used i n a l l c a s e s . In o r d e r t o choose the b e s t method, i t i s n e c e s s a r y t o c o n s i d e r the n a t u r e of the f a r m d e c i s i o n -maker's u t i l i t y f u n c t i o n and the f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s . However, the QP-SEMIV method appears t o be a p p r o p r i a t e under a w i d e r range o f e m p i r i c a l s i t u a t i o n s than the QP-VAR and MOTAD methods. - iv -TABLE OF CONTENTS Page LIST OF TABLES v i LIST OF FIGURES i x CHAPTER I INTRODUCTION 1 1.1 The Problem 1 1.2 O b j e c t i v e s 2 1.3 Important hypotheses t o be t e s t e d 4 1.4 Res e a r c h P r o c e d u r e 5 1.5 O r g a n i z a t i o n o f the s t u d y 6 CHAPTER II THEORETICAL REMARKS 8 2.1 The Income-Risk E f f i c i e n t f r o n t i e r 8 2.2 The Expe c t e d U t i l i t y F u n c t i o n 12 2.3 R i s k A v e r s i o n 19 2.4 The U t i l i t y F u n c t i o n and the Income-Risk F r o n t i e r 21 2.5 Some Methods used i n Farm P l a n n i n g under U n c e r t a i n t y 25 2.5.1 Q u a d r a t i c Programming: The V a r i a n c e Approach 25 2.5.2 The S e m i v a r i a n c e Approach 31 2.5.2.1 The O r d i n a l C l a s s i f i c a t i o n o f t h e Expected U t i l i t y 37 2.5.2.2 C o n c l u s i o n s r e g a r d i n g the S e m i v a r i a n c e Method 39 2.5.3 The MOTAD model 40 2.6 C o n c l u s i o n 43 CHAPTER I I I THE EMPIRICAL MODEL 45 3.1 G e n e r a l Overview o f the Research P r o c e d u r e 45 3.2 G e n e r a t i o n o f P o p u l a t i o n s 49 3.3 The Sampling P r o c e s s 52 3.4 S o l u t i o n s o f the Models 55 3.4.1 D e s c r i p t i o n o f the General Model 55 3.4.2 The "True" Income-Risk F r o n t i e r 58 3.4.3 The Income R i s k - F r o n t i e r E s t i m a t e s 59 3.4.4 A n a l y s i s o f t h e S o l u t i o n s 62 - V -Page 3.5 The OP-SEMIV Method as S u b s t i t u t e o f the Income Semi v a r i a n c e Method 64 3.6 Summary 65 CHAPTER IV THE RESULTS 67 4.1 The Normal c a s e w i t h Low Degree o f C o r r e l a t i o n Among A c t i v i t y R e t u rns 67 4.2 The Normal Case w i t h High Degree o f C o r r e l a t i o n Among A c t i v i t y Returns 71 4.3 Gamma D i s t r i b u t i o n and Lew Degree o f C o r r e l a t i o n Among A c t i v i t y R e t u r n s 75 4.4 Gamma D i s t r i b u t i o n and High Degree o f C o r r e l a t i o n Among A c t i v i t y Returns 79 4.5 V a l i d a t i o n s o f t h e QP-SEMIV Method 83 4.6 T e s t i n g t h e Hypotheses 85 4.6.1 H y p o t h e s i s I 86 4.6.2 H y p o t h e s i s 2 87 4.6.3 H y p o t h e s i s 3 88 4.6.4 H y p o t h e s i s 4 88 4.7 Summary 89 CHAPTER V A CASE STUDY FARM 91 CHAPTER VI SUMMARY, CONCLUSIONS AND RECOMMENDATION FOR FURTHER RESEARCH 98 6.1 Summary and c o n c l u s i o n s 99 6.2 Recommendations f o r F u r t h e r S t u d i e s 102 REFERENCES 105 APPENDIX 108 — v i -LIST OF TABLES Page 3.1 Mean A c t i v i t y R e t u r n s , Mean C o r r e l a t i o n C o e f f i c i e n t s and Skewness C o e f f i c i e n t o f the P o p u l a t i o n s Generated 50 3.2 V a r i a n c e - C o v a r i a n c e M a t r i x o f the A c t i v i t y Returns C o r r e s p o n d i n g t o the D i f f e r e n t P o p u l a t i o n s Generated 51 3.3 S e m i v a r i a n c e - C o s e m i v a r i a n c e M a t r i x o f the A c t i v i t y R e t u rns C o r r e s p o n d i n g t o t h e Gamma P o p u l a t i o n s 52 3.4 V a r i a n c e - C o v a r i a n c e M a t r i x and Mean A c t i v i t y Returns C a l c u l a t e d 'from a Sample drawn from a Normal Popu-l a t i o n ( I I ) : An Example 54 3.5 S e m i v a r i a n c e - C o s e m i v a r i a n c e , M a t r i x and Mean A c t i v i t y R e t u rns C a l c u l a t e d from a Sample drawn from a Gamma I P o p u l a t i o n 55 3.6 The MOTAD model as A p p l i e d t o a Sample O b t a i n e d from a Normal II P o p u l a t i o n 61 4.1 R i s k as E s t i m a t e d by QP-VAR and MOTAD Methods and the Tru e P o p u l a t i o n V a l u e s f o r Three L e v e l s o f Ex p e c t e d Income. Normal D i s t r i b u t i o n s w i t h Low Degree o f C o r r e l a t i o n 68 4.2 V a r i a n c e and Mean V a r i a b i l i t y C o e f f i c i e n t o f the MOTAD and QP-VAR E s t i m a t e s o f R i s k a t Thre e L e v e l s o f Ex p e c t e d Income 69 4.3 Range L e v e l s o f the R i s k E s t i m a t e s P r o v i d e d by QP-VAR and MOTAD Methods as Compared t o the Tr u e R i s k V a l u e s . Normal D i s t r i b u t i o n , Low Degree o f C o r r e l a t i o n 70 4.4 R i s k as E s t i m a t e d by QP-VAR and MOTAD Methods and the P o p u l a t i o n V a l u e s f o r Thr e e L e v e l s o f Expe c t e d Income. Normal D i s t r i b u t i o n w i t h High Degree o f C o r r e l a t i o n 72 4.5 V a r i a n c e s and Mean V a r i a b i l i t y C o e f f i c i e n t o f the MOTAD and QP-VAR E s t i m a t e s o f R i s k a t Three L e v e l s o f Income. Normal D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n 7 3 4.6 Range L e v e l s o f the R i s k E s t i m a t e s P r o v i d e d by QP-VAR and the MOTAD Methods as Compared t o the True R i s k V a l u e s . Normal D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n 74 - v i i -Page 4.7 R i s k as E s t i m a t e d by QP-SEMIV, QP-VAR and MOTAD Methods and t h e True P o p u l a t i o n V a l u e s o f R i s k f o r Three L e v e l s o f E x p e c t e d Income, Gamma D i s t r i b u t i o n w i t h Low Degree 76 o f C o r r e l a t i o n 4.8 V a r i a n c e s o f the QP-SEMIV, QP-VAR and MOTAD E s t i m a t e s o f R i s k a t Three L e v e l s o f Expected Income. Gamma D i s t r i b u t i o n Low Degree o f C o r r e l a t i o n 77 4.9 Range o f the R i s k L e v e l as E s t i m a t e d by QP-SEMIV QP-VAR, and MOTAD Methods as Compared t o the True R i s k V a l u e s . Gamma D i s t r i b u t i o n s , Low Degree o f C o r r e l a t i o n 78 4.10 R i s k as E s t i m a t e d by QP-SEMIV, QP-VAR and MOTAD Methods and t h e Tr u e P o p u l a t i o n V a l u e s o f R i s k f o r T h r e e L e v e l s o f E x p e c t e d Income. Gamma D i s t r i b u t i o n w i t h High Degree o f C o r r e l a t i o n 79 4.11 V a r i a n c e s o f the QP-SEMIV, QP-VAR and MOTAD E s t i m a t e s o f R i s k a t Thre e L e v e l s o f Expe c t e d Income. Gamma D i s t r i -b u t i o n s , High Degree o f C o r r e l a t i o n 4.12 Range o f t h e R i s k L e v e l s as E s t i m a t e d by QP-SEMIV QP-VAR and MOTAD Methods as Compared t o the Tr u e R i s k V a l u e s . Gamma D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n 32 4.13 The Income S e m i v a r i a n c e as C a l c u l a t e d Ex Post w i t h t he QP-SEMIV, QP-VAR and MOTAD S o l u t i o n s , Gamma D i s t r i b u t i o n s , Low Degree o f C o r r e l a t i o n 8 4 5.1 R i s k L e v e l s ( E x p r e s s e d as t h e Square Root o f the Semi-v a r i a n c e ) as P r o v i d e d by the QP-SEMIV, QP-VAR and MOTAD Model S o l u t i o n s f o r a Farm i n the Peace R i v e r D i s t r i c t o f B r i t i s h Columbia 93 5.2 L e v e l s o f A c t i v i t i e s as Proposed by MOTAD, QP-SEMIV and QP-VAR Models f o r a T y p i c a l Small Farm i n the Peace R i v e r A r e a o f B r i t i s h Columbia (Net Income $6,0000) qg A . l E s t i m a t e s o f R i s k as O b t a i n e d U s i n g t he QP-VAR Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn from a No r m a l l y D i s t r i b u t e d P o p u l a t i o n , Low Degree o f C o r r e l a t i o n 109 among A c t i v i t y Returns A.2 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the MOTAD Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn from Normal P o p u l a t i o n w i t h Low Degree o f C o r r e l a t i o n II 0 A.3 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the QP-VAR Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Normal P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n H] - v i i i -Page A.4 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the MOTAD Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Normal P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n 112 A.5 E s t i m a t e s o f R i s k as O b t a i n e d as Usi n g the QP-SEMIV Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h Low Degree o f C o r r e l a t i o n 113 A.6 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the QP-VAR Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h Low Degree o f C o r r e l a t i o n ^ ^ A.7 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the MOTAD Method A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h Low Degree o f Co r r e l a t i o n 115 A.8 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the QP-SEMIV Method A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h Degree o f C o r r e l a t i o n 116 A.9 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the QP-VAR Method A p p l i e d t o F i f t e e n Samples Randomly Drawn from a. Gamma P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n 11:7 A.10 E s t i m a t e s o f R i s k as O b t a i n e d U s i n g the MOTAD Method A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n 11.8 A..11 V a r i a n c e - C o v a r i a n c e M a t r i x o f 5 o f the 8 Year A c t i v i t y R e t u rns Date C o r r e s p o n d i n g t o a Case Farm i n the Peace < R i v e r D i s t r i c t o f B r i t i s h Columbia - ix -LIST OF FIGURES Page 2.1 Income I s o - R i s k Curves Showing D i f f e r e n t Degree o f C o r r e l a t i o n Between the A c t i v i t y Returns 9 2.2 Income-Risk E x p a n s i o n L i n e f o r a Two A c t i v i t i e s S i t u a t i o n 10 2.3 The Income R i s k F r o n t i e r 12 2.4 U t i l i t y Curve f o r a R i s k A v e r s e I n d i v i d u a l 13 2.5 T y p i c a l Shape o f a U t i l i t y Curve 15 2.6 U t i l i t y as a F u n c t i o n o f the R i s k L e v e l s 16 2.7 U t i l i t y as a F u n c t i o n o f Income and R i s k 17 2.8 I s o - U t i l i t y Curves 17 2.9 D e t e r m i n a t i o n o f the Optimal L e v e l i n an Income-Risk F r o n t i e r 18 2.10 R i s k Premium i n a Concave U t i l i t y F u n c t i o n 20 3.1 An Overview o f the Research P r o c e d u r e 48 -3.2 B i a s i n the Income-Risk F r o n t i e r E s t i m a t e 63 3.3 A D i s p e r s i o n Comparison Between the E s t i m a t e s o f Two Methods 63 4.1 The True P o p u l a t i o n Income-Risk F r o n t i e r and the Income-R i s k F r o n t i e r as E s t i m a t e d by QP-VAR and MOTAD. Normal d i s t r i b u -t i o n , H i g h Degree o f C o r r e l a t i o n 72 4.2 The True P o p u l a t i o n Income-Risk F r o n t i e r and the Income-R i s k F r o n t i e r as E s t i m a t e d by QP-VAR, MOTAD and QP-SEMIV Methods. Gamma D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n 80 5.1 S t r u c t u r e o f the Model 90 5.2 The Income-Risk F r o n t i e r as E s t i m a t e d by QP-SEMIV, QP-VAR . and MOTAD Methods i n a Farm i n t h e Peace R i v e r D i s t r i c t o f B r i t i s h Columbia 95 ACKNOWLEDGEMENTS I would l i k e t o thank my a d v i s o r , P r o f e s s o r John Graham and t h e members o f my T h e s i s Committee, P r o f e s s o r R i c k B a r i c h e l l o and P r o f e s s o r George E a t o n , f o r t h e i r numerous s u g g e s t i o n s which l e d t o s u b s t a n t i a l improvements i n the f i n a l p r o d u c t . - 1 -CHAPTER I INTRODUCTION 1.1 The Problem A g r i c u l t u r a l p r o d u c t i o n may be c o n s i d e r e d a r i s k y a c t i v i t y . Wide v a r i a t i o n s i n y i e l d s and p r i c e s o v e r time a r e common among a g r i -c u l t u r a l commodities. C e r t a i n economic c h a r a c t e r i s t i c s such as r e l a -t i v e l y p r i c e i n e l a s t i c demand f u n c t i o n s and the dependence o f a g r i -c u l t u r a l p r o d u c t i o n on c l i m a t e might be mentioned,as f a c t o r s which c o n t -r i b u t e t o e x p l a i n the g r e a t e r v a r i a b i l i t y o f r e t u r n s from a g r i c u l t u r e as compared t o o t h e r s e c t o r s . G i v e n t h i s s i t u a t i o n , i t i s n e c e s s a r y to c o n s i d e r t he un-c e r t a i n t y o f r e t u r n s as an i m p o r t a n t f a c t o r i n f l u e n c i n g farm d e c i s i o n s . Farm p l a n n i n g t e c h n i q u e s which use o n l y d e t e r m i n i s t i c t o o l s have con-s i d e r a b l e l i m i t a t i o n s . A t t e n t i o n i s i n c r e a s i n g l y b e i n g d i r e c t e d t o s t u d i e s t h a t a l l o w f o r u n c e r t a i n t y , both i n a t h e o r e t i c a l and e m p i r i c a l framework (1, 9, 16). The method which m i n i m i z e s t h e v a r i a n c e o f a c t i v i t y r e t u r n s s u b j e c t t o a minimum e x p e c t e d income l e v e l u s i n g a q u a d r a t i c programming a l g o -r i t h m ( t h e QP-VAR method) and t h e method which m i n i m i z e s the mean t o t a l a b s o l u t e d e v i a t i o n s o f a c t i v i t y r e t u r n s s u b j e c t t o a minimum e x p e c t e d i n -come l e v e l u s i n g a l i n e a r programming a l g o r i t h m (The MOTAD - 2 -method) have been two w i d e l y a p p l i e d p r o c e d u r e s (1, 9, 16). A l t h o u g h the S e m i v a r i a n c e method which m i n i m i z e s t h e n e g a t i v e s e m i v a r i a n c e o f a c t i v i t y r e t u r n s s u b j e c t t o a minimum e x p e c t e d income l e v e l has not been used f r e q u e n t l y , i t has, n e v e r t h e l e s s , been c o n s i d e r e d an i m p o r t a n t method (23, 2 5 ) . D e s p i t e some t h e o r e t i c a l d i s c u s s i o n s about t h e r e l i a -b i l i t y o f t h e s e methods (6, 15, 21, 29) i t i s n o t c l e a r how t o e v a l u a t e t h e i r performance under d i f f e r e n t e m p i r i c a l s i t u a t i o n s . How do the r e s u l t s d i f f e r when QP-VAR, MOTAD and S e m i v a r i a n c e methods a r e a p p l i e d ? Under what c i r c u m s t a n c e s can the r e l a t i v e e f f i c i e n c y o f t h e s e methods be c o n s i d e r e d s i m i l a r ? A s y s t e m a t i c e v a l u a t i o n o f the a b s o l u t e and r e l a t i v e e f f i c i e n c y o f the methods under d i f f e r e n t e m p i r i c a l c i r c u m s t a n c e s i s m i s s i n g . An answer t o t h i s problem<is i m p o r t a n t s i n c e knowledge o f the l i m i t a t i o n s o f t h e s e methods may be i m p o r t a n t f o r r e s e a r c h e r s who wish t o d e c i d e which ( i f any) o f the methods s h o u l d be a p p l i e d under c e r t a i n s p e c i f i c e m p i r i c a l c o n d i t i o n s . In any r e s e a r c h t h e q u a l i t y o f the r e -s u l t s i s g o i n g t o depend, among o t h e r f a c t o r s , on the adequacy w i t h which the method used i s a p p r o p r i a t e f o r t h e e m p i r i c a l s i t u a t i o n . 1.2 O b j e c t i v e s A g e n e r a l o b j e c t i v e o f t h i s t h e s i s i s t o stu d y t h e e f f i c i e n c y o f the QP-VAR, MOTAD and S e m i v a r i a n c e methods as e s t i m a t o r s o f the income-r i s k f r o n t i e r when a p p l i e d t o d i f f e r e n t f r e q u e n c y d i s t r i b u t i o n s o f the a c t i v i t y r e t u r n s * . In s t u d y i n g t h e e f f i c i e n c y o f the methods two main * A d e t a i l e d d e s c r i p t i o n o f t h e s e methods and o f t h e i n c o m e - r i s k f r o n t i e r i s p r o v i d e d i n C h a p t e r I I . - 3 -elements are considered, namely the magnitude of the bias and the variance of the sampling estimates of the income-risk f r o n t i e r . This primary objective may be specified in a number of more sp e c i f i c sub-objectives as follows: 1. (a) To review some theoretical concepts concerning the u t i l i t y function of a decision maker when the expected income and r i s k are considered to be variables. (b) To review the concept of an e f f i c i e n t income-risk f r o n t i e r as defined by the QP-VAR, MOTAD and Semi variance methods. (c) To analyze situations under which these three methods are applicable given certain assumptions regarding decision makers' u t i l i t y functions and the probability d i s t r i b u t i o n of the returns. 2. (a) In order to extend the theoretical concepts outlined in (1) above, a small farm planning model consisting of three a c t i v i t i e s and s i x constraints w i l l be employed. Assuming that the a c t i v i t y returns of the three variables are normally dis t r i b u t e d and that the degree of corre-lation, among them i s low, an income-risk f r o n t i e r w i l l be derived using the complete population of a c t i v i t y returns as the data base. (b) Step (a) as defined above w i l l be repeated but in this instance the degree of correlation among a c t i v i t y returns w i l l be assumed to be r e l a -t i v e l y high. (c) In most empirical situations the complete set (population) of observations i s not available, but a selected sample thereof i s . Therefore, income-risk f r o n t i e r s w i l l be estimated using the QP-VAR and MOTAD methods - 4 -a p p l i e d t o randomly drawn sample d a t a o f a c t i v i t y r e t u r n s , r a t h e r than t o t h e p o p u l a t i o n d a t a . The i n c o m e - r i s k f r o n t i e r s w i l l be e s t i m a t e d f o r samples drawn from the same p o p u l a t i o n s as d e f i n e d i n (a) and ( b ) . (d) The i n c o m e - r i s k f r o n t i e r as d e t e r m i n e d f o r the p o p u l a t i o n d a t a and t h a t e s t i m a t e d by u s i n g the randomly drawn sample da t a w i l l be compared t o d e t e r m i n e the degree o f d e p a r t u r e ( e x t e n t o f b i a s ) o f t h e l a t t e r from the f o r m e r . (e) A measure o f the degree o f d i s p e r s i o n o f t h e i n c o m e - r i s k sample e s t i m a t e s w i l l p r o v i d e an a d d i t i o n a l element w i t h which t o e v a l u a t e t h e e f f i c i e n c y o f the methods. 3. Under c o n d i t i o n s where the d i s t r i b u t i o n o f a c t i v i t y r e t u r n s i s non-normal, i n a d d i t i o n t o the QP-VAR and MOTAD methods, the S e m i v a r i a n c e method w i l l be used as an e s t i m a t o r o f t h e e f f i c i e n t i n c o m e - r i s k f r o n t i e r . Steps ( a ) , ( b ) , ( c ) , (d) and (e) as d e f i n e d i n 2. above, w i l l be r e p e a t e d . 4. In o r d e r t o i l l u s t r a t e t h e performance o f t h e s e methods, d a t a from an a c t u a l f a r m o f t h e Peace R i v e r D i s t r i c t o f B r i t i s h Columbia w i l l be used i n o r d e r t o e s t i m a t e an i n c o m e - r i s k f r o n t i e r u s i n g each o f the methods. Sample o b s e r v a t i o n s f o r t h i s farm ( t h e l a s t e i g h t y e a r s ' r e c o r d s o f y i e l d s , p r i c e s and c o s t s ) do not a l l o w one t o a f f i r m t h a t t h e a c t i v i t y r e t u r n s a r e n o r m a l l y d i s t r i b u t e d and t h e d a t a i n d i c a t e t h a t t h e a c t i v i t y r e t u r n s a r e f a i r l y h i g h l y c o r r e l a t e d . . 1•^ Important Hypotheses t o be t e s t e d The f o l l o w i n g f o u r hypotheses a r e s t a t e d in o r d e r t o a l l o w f o r a c l o s e r s p e c i f i c a t i o n o f the o b j e c t i v e s o f t h i s s t u d y . These h y p o t h e s e s - 5 -a r e based on o b s e r v a t i o n s made by t h e a u t h o r on v a r i o u s t h e o r e t i c a l s t u d i e s i n the a r e a . C h a p t e r II p r o v i d e s a t h e o r e t i c a l b a s i s s u p p o r t i n g t h e s e h y p o t h e s e s * . 1. The QP-VAR approach as a p p l i e d t o sm a l l sample data p r o v i d e s an un b i a s e d e s t i m a t e o f t h e a c t u a l p o p u l a t i o n i n c o m e - r i s k f r o n t i e r i f the a c t i v i t y r e t u r n s a r e n o r m a l l y d i s t r i b u t e d , r e g a r d l e s s o f t h e degree o f c o r r e l a t i o n among t h e a c t i v i t y r e t u r n s . 2. The MOTAD method p r o v i d e s an u n b i a s e d e s t i m a t e o f the a c t u a l popu-l a t i o n i n c o m e - r i s k f r o n t i e r o n l y i f the f o l l o w i n g two c o n d i t i o n s a r e s a t i s f i e d . (a) The a c t i v i t y r e t u r n s a r e n o r m a l l y d i s t r i b u t e d , and (b) The c o r r e l a t i o n c o e f f i c i e n t s among t h e a c t i v i t y r e t u r n s a r e c l o s e to z e r o . 3. I f a c t i v i t y r e t u r n s a r e n o n - n o r m a l l y d i s t r i b u t e d , the QP-VAR method and t h e MOTAD method y i e l d u n b i a s e d e s t i m a t e s o f the a c t u a l p o p u l a t i o n income-r i s k f r o n t i e r . 4. When a c t i v i t y r e t u r n s a r e n o n - n o r m a l l y d i s t r i b u t e d , the S e m i v a r i a n c e method w i l l p r o v i d e u n b i a s e d e s t i m a t e s o f the a c t u a l p o p u l a t i o n i n c o m e - r i s k f r o n t i e r . O t h e r hypotheses r e l a t e d t o comparisons o f the degree o f d i s p e r s i o n o f t h e e s t i m a t e s p r o v i d e d by the methods s h o u l d a l s o be i n c l u d e d . However, because t h e r e i s no a p r i o r i knowledge r e g a r d i n g t he d i s p e r s i o n o f the * The r e a d e r w i l l n o t i c e t h a t t he e x c e p t i o n i s h y p o t h e s i s 3. As may be seen i n Ch a p t e r I I , t he t h e o r e t i c a l study s u p p o r t s the a l t e r n a t i v e h y p o t h e s i s i m p l i c i t t o h y p o t h e s i s 3 r a t h e r than t he n u l l h y p o t h e s i s f o r m u l a t e d . - 6 -e s t i m a t e s t h e s e hypotheses a r e n o t f o r m u l a t e d . N e v e r t h e l e s s , t he d i s -p e r s i o n o f t h e e s t i m a t e s p r o v i d e d by t h e m e t h o d s . w i l l be an i m p o r t a n t element t o c o n s i d e r i n the e v a l u a t i o n o f t h e methods.. 1.4 Research P r o c e d u r e The r e s e a r c h p r o c e d u r e i n c l u d e s f o u r s t e p s : 1. A t h e o r e t i c a l s t u d y o f t h e b a s i c assumptions and c h a r a c t e r i s t i c s o f t h e QP-VAR, MOTAD and Semi v a r i a n c e methods, thus p r o v i d i n g q u a l i t i t a t i v e knowledge o f t h e performance o f t h e s e methods i n e s t i m a t i n g i n c o m e - r i s k f r o n t i e r s . 2. An e x p e r i m e n t a t i o n phase where a random number g e n e r a t o r i s used to g e n e r a t e d i f f e r e n t s e t s o f p o p u l a t i o n s as r e q u i r e d by the assumptions r e g a r d i n g 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 r e t u r n s and t h e degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s . 3. A " s o l v i n g o f t h e models" phase, where t h e QP-VAR, MOTAD and i n some i n s t a n c e s t he S e m i v a r i a n c e method w i l l be used t o s o l v e models f o r d i f f e r e n t s e t s o f p o p u l a t i o n d a t a and f o r a number o f samples drawn from each t y p e o f p o p u l a t i o n g e n e r a t e d . 4. An a n a l y s i s o f t h e r e s u l t s where t he mean i n c o m e - r i s k f r o n t i e r * as e s t i m a t e d u s i n g each method w i l l be compared t o t h e a c t u a l p o p u l a t i o n i n c o m e - r i s k f r o n t i e r ( d e t e r m i n e d w i t h t he a p p r o p r i a t e method) a t t e m p t i n g to e s t a b l i s h i f t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s between t he two f r o n t i e r s , i . e . , whether the e s t i m a t e i s b i a s e d o r not. The d i s p e r s i o n o f the e s t i m a t e s ( v a r i a n c e ) p r o v i d e d by t h e methods w i l l a l s o be compared. I f two methods * The "mean i n c o m e - r i s k f r o n t i e r " i s the i n c o m e - r i s k f r o n t i e r o b t a i n e d from mean v a l u e s o f the numerous sample s o l u t i o n s . - 7 -are unbiased, the method with smaller variance of i t s estimates w i l l be considered more e f f i c i e n t . A l im i ted t rade-of f between bias and variance of the estimates w i l l be considered in evaluat ing methods with d i f f e ren t degrees of d ispers ion and b ias . 1.5 Organization of the Study Chapter 1 includes the statement of the problem, the object ives and the basic methodology to be fo l lowed. Chapter 2 i s devoted to a theoret ica l study of the Var iance, Total Absolute Deviation and Semivariance as r i sk ind icators which may be used in farm planning. The discussion centers around the accuracy with which these indicators may represent r i s k . The next chapter describes the procedures used to generate populat ions, the sampling process and the models used when each method i s app l ied . The fourth chapter reports on the analys is of the resu l ts and the tes t ing of the hypotheses. I t examines the performance of the QP-VAR, MOTAD and Semivariance methods in four poss ib le experimental empir ica l s i tua t ions and t he i r r e l a t i ve e f f i c i ency as estimators of the income-r i s k f r o n t i e r . Chapter 5 describes an app l ica t ion of the QP-VAR, Semivariance and MOTAD methods to data obtained from a case farm of the Peace River D i s t r i c t of B r i t i s h Columbia. F i n a l l y , Chapter 6 summarizes the study and provides basic conc-lus ions . - 8 -CHAPTER II THEORETICAL REMARKS The purpose of t h i s chapter i s to discuss some theore t ica l aspects of the farm planning methods used in th i s study. The a b i l i t y of these methods to adequately generate estimates of i ncome-risk f ron t ie rs and the c lose ly re la ted problem regarding farmers' u t i l i t y funct ions w i l l be considered in an evaluat ion of the methods. 2.1 The Income-Risk E f f i c i e n t Front ie r As a prel iminary step i t i s necessary to define the concept of an e f f i c i e n t p l a n . Markowitz (23) notes that a plan i s e f f i c i e n t i f i t i s not poss ib le to obtain a higher expected income with the same v a r i a b i l i t y of income ( r i s k ) , or i f there i s no other plan with a smal ler v a r i a b i l i t y of income for the same leve l of expected income. Risk may be measured as the degree of v a r i a b i l i t y of re tu rn . One or a combination of the fo l lowing measures of v a r i a b i l i t y may be used as r i s k i nd i ca to rs : var iance, semivariance , absolute dev ia t i on , skewness and ku r tos i s . Each of the possib le measures of v a r i a b i l i t y has d i f fe ren t cha rac te r i s t i cs and represent cer ta in s p e c i f i c aspects of r i s k d i f f e ren t l y . An t s o - r i s k curve represents a l l possible combinations of a c t i v i t i e s which y i e l d the same r i sk level as measured by any of these measures. It has commonly been assumed that the i s o - r i s k curves - 9 -a r e e l l i p t i c a l when r e p r e s e n t e d d i a g r a m a t i c a l l y f o r the c ase o f two p o s s i b l e a c t i v i t i e s ( 1 6 ) . FIGURE 2.1 Income I s o - R i s k Curves Snowing D i f f e r e n t Degrees o f  C o r r e l a t i o n between the A c t i v i t y Returns A c t i v i t y X, (a) F i g u r e 2.1 shows t h r e e f a m i l i e s o f income i s o - r i s k c u r v e s , where r i s k l e v e l s a r e i n c r e a s i n g from the o r i g i n . F i g u r e 2.1(a) shows a f a m i l y where t h e r e i s a s t r o n g n e g a t i v e c o r r e l a t i o n c o e f f i c i e n t between the r e t u r n s o f a c t i v i t y X-j and ( c l o s e t o - 1 . 0 ) . In t h i s c a s e f o r a g i v e n l e v e l o f income t h e r e i s a d e c r e a s e i n "the l e v e l o f r i s k t h r o u g h p r o d u c i n g a c o m b i n a t i o n o f the two a c t i v i t i e s r a t h e r than p r o d u c i n g j u s t one. As the c o r r e l a t i o n c o e f f i c i e n t approaches z e r o o r becomes p o s i t i v e the i s o - r i s k c u r v e s a r e l e s s concave to the o r i g i n ( f i g u r e 2.1 ( b ) ) , meaning t h a t d i v e r s i f i c a t i o n does not reduce r i s k g r e a t l y . F i n a l l y , - 10 -when the correlation c o e f f i c i e n t i s equal to.+1.0, the i s o - r i s k curves are straight lines as shown in Figure 1(c), In this case there are no benefits from d i v e r s i f i c a t i o n as far as the r i s k situation i s considered. By introducing the f a m i l i a r concept of iso-incorne (or iso -revenue) l i n e s , an income-risk expansion l i n e may be defined. FIGURE 2.2 Income-Risk Expansion Line for a Two A c t i v i t y Situation A c t i v i t y X-, f A c t i v i t y 1 Figure 2.2 shows the income-risk expansion l i n e for the two a c t i v i t y case. The income-risk expansion l i n e i s defined as the locus of points which define the minimum r i s k level at each level of expected income, represented by the points at which the iso-income l i n e s are tangential to the i s o - r i s k l i n e s . The income-risk expansion l i n e shows the e f f i c i e n t - 11 -c o m b i n a t i o n s o f the a c t i v i t i e s and which a f i r m w i l l produce i f the maximum income i s t o be a c h i e v e d a t the minimum l e v e l o f r i s k . I t i s i m p o r t a n t t o note t h a t i'thare i s a unique c o m b i n a t i o n o f a c t i v i t y l e v e l s (L-J, l_2> L^) which g e n e r a t e any e f f i c i e n t p l a n *. The income -r i s k l i n e w i l l be a s t r a i g h t l i n e g i v e n the f o l l o w i n g a s s u m p t i o n s : (1) The r i s k f u n c t i o n ( r i s k as a f u n c t i o n o f X-, and X 2 ) i s an h o m o t h e t i c f u n c t i o n . T h i s i m p l i e s t h a t the i s o - r i s k c u r v e s w i l l be p a r a l l e l ( t h e y a l l have the same shape) t h r o u g h o u t a l l l e v e l s o f r i s k : (2) The r e l a t i v e p r i c e s o f X-. and X^ remain c o n s t a n t a t a l l l e v e l s o f p r o d u c t i o n , i . e . , the f i r m i s n o t a b l e t o a l t e r the s l o p e o f t h e i s o - i n c o m e l i n e s when i t expands i t s p r o d u c t i o n ( p e r f e c t c o m p e t i t i o n ) . I n c o m e - r i s k e x p a n s i o n l i n e s w i l l be s t r a i g h t l i n e s i n most farm p l a n n i n g s i t u a t i o n s s i n c e t h e s e assumptions a r e g e n e r a l l y met. I f p o i n t s l _ i , l _ 2 , o f F i g u r e 2.2 which r e p r e s e n t d i f f e r e n t l e v e l s o f income a t t h e s m a l l e s t r i s k l e v e l s a r e o g r a p h e d i n an income-r i s k p l a n e , F i g u r e 2.3 i s o b t a i n e d . T h i s i s d e f i n e d as an e f f i c i e n t i n c o m e - r i s k f r o n t i e r , which shows the maximum l e v e l o f income o b t a i n a b l e a t each l e v e l o f r i s k . * T h i s i m p l i e s t h a t i n e v a l u a t i n g a p a r t i c u l a r method i t i s s u f f i c i e n t t o c o n s i d e r e i t h e r the r i s k l e v e l e s t i m a t e d a t a c e r t a i n l e v e l o f income o r the a c t i v i t y l e v e l s e s t i m a t e d . T h i s c o n c l u s i o n i s i m p o r t a n t i n the de-s i g n o f t h e e x p e r i m e n t s d e s c r i b e d i n the f o l l o w i n g c h a p t e r . - 12 -FIGURE 2.3 The Income-Risk F r o n t i e r Income = A g(xrx2) o In many s t u d i e s i t has been assumed t h a t r i s k l e v e l s i n c r e a s e a t an i n c r e a s i n g m a r g i n a l r a t e when the ex p e c t e d income i s expanded f u r t h e r t h a n a c e r t a i n l e v e l (15, 1 6 ) . T h i s f a c t i s c o r r o b o r a t e d by a number o f e m p i r i c a l e s t i m a t i o n s o f t h e i n c o m e - r i s k f r o n t i e r ( 1 , 1 5 ) . F i g u r e 2.3 shows an i n c o m e - r i s k f r o n t i e r o f t h i s t y p e and p o i n t L n r e p r e s e n t s the maximum p o s s i b l e income which can be o b t a i n e d g i v e n a l i m i t e d r e s o u r c e base. Any p l a n chosen a l o n g the i n c o m e - r i s k f r o n t i e r i s e f f i c i e n t and the a c t u a l p o i n t o r p l a n chosen w i l l depend upon the s p e c i f i c u t i l i t y f u n c t i o n o f the d e c i s i o n maker, 2.2 The Ex p e c t e d U t i l i t y F u n c t i o n U n t i l a number o f decades ago, i t was assumed t h a t the p r o p e r o b j e c t i v e o f an i n d i v i d u a l when f a c e d w i t h u n c e r t a i n s i t u a t i o n s was t o maximize e x p e c t e d monetary r e t u r n ( 2 3 ) , I t was l a t e r found t h a t t h i s o b j e c t i v e does not r e f l e c t r e a l i t y . I n s t e a d , t he e x p e c t e d u t i l i t y r u l e A r e a o f i m p o s s i b l e p l a n s i n c o m e - r i s k f r o n t i e r A r e a o f i n e f f i c i e n t p l a n s r i s k = f ( X i X 2 ) - 13 -was proposed as a s u b s t i t u t e t o t h e e x p e c t e d r e t u r n r u l e . T h i s a p proach assumed t h a t the i n d i v i d u a l t r i e s t o maximize h i s e x p e c t e d u t i l i t y r a t h e r t h a n h i s e x p e c t e d r e t u r n s . E x p e c t e d u t i l i t y i s a f u n c t i o n o f t h e e x p e c t e d r e t u r n s but t h i s f u n c t i o n a l r e l a t i o n i s not n e c e s s a r i l y l i n e a r . F u r t h e r m o r e , e x p e c t e d u t i l i t y i s not o n l y a f u n c t i o n o f t h e e x p e c t e d r e t u r n s b u t a l s o a f u n c t i o n o f t h e degree o f r i s k i n v o l v e d i n t r y i n g t o pursue such r e t u r n s . * Markowitz (23) a c c e p t e d the h y p o t h e s i s o f d e c r e a s i n g m a r g i n a l u t i l i t y as the l e v e l o f income i n c r e a s e s . F i g u r e 2.4 shows t h i s r e l a t i o n -s h i p . FIGURE 2.4 U t i l i t y Curve f o r a R i s k A v e r s e I n d i v i d u a l u t i l i t y * The e x p e c t e d u t i l i t y can a l s o be a f u n c t i o n o f o t h e r f a c t o r s such as p r e s t i g e o f the d e c i s i o n - m a k e r and o t h e r s . T h i s s t u d y w i l l be con c e r n e d o n l y w i t h e x p e c t e d r e t u r n s and r i s k as f a c t o r s a f f e c t i n g the u t i l i t y l e v e l . - 14 -F o r an i n d i v i d u a l who p o s s e s s e s a s t r i c t l y c oncave u t i l i t y f u n c t i o n the g a i n i n u t i l i t y from w i n n i n g a d o l l a r w i l l be l e s s than the l o s s i n u t i l i t y from l o s i n g a d o l l a r ( T T ) . T h i s i n d i v i d u a l w i l l n e v e r p a r t i c i p a t e i n a " f a i r " game o f chance. For example, g i v e n a game i n which he has an equal chance o f w i n n i n g and l o s i n g $450 as shown i n F i g u r e 2.4, the e x p e c t e d u t i l i t y o f such a game i s s m a l l e r than the u t i l i t y o f c e r t a i n t y o f $ 5 5 0 , , i . e . , U-|> U Q i n the g r a p h . The s t r i c t l y concave u t i l i t y f u n c t i o n may e x p l a i n why i n d i v i d u a l s a r e w i l l i n g t o t a k e i n s u r a n c e a g a i n s t b i g l o s s e s even i f the i n s u r a n c e company makes a p r o f i t . Friedman and Savage (11) have p o i n t e d o u t t h a t g i v e n the 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 a s s u m p t i o n , i n d i v i d u a l s would always have t o be p a i d t o i n d u c e them t o b e a r r i s k . However, t h i s s t a t e m e n t i s c l e a r l y c o n t r a d i c t e d by a c t u a l b e h a v i o u r . P e o p l e not o n l y engage i n f a i r games o f c h a n c e , t h e y engage f r e e l y i n such u n f a i r games as l o t t e r i e s . P e o p l e e n t e r r i s k y o c c u p a t i o n s and make r i s k y i n v e s t m e n t s t h a t y i e l d even s m a l l e r average r e t u r n s than r e l a t i v e l y s a f e i n v e s t m e n t s . T h i s problem i s s t i l l more s e r i o u s c o n s i d e r i n g t h a t many i n d i v i d u a l s i n s u r e a g a i n s t damage and s i m u l t a n e o u s l y t h e y buy l o t t e r y t i c k e t s o r i n v e s t i n h i g h l y r i s k y a c t i v i t i e s w i t h average r e t u r n s . Friedman and Savage (11) h y p o t h e s i z e d t h a t the shape o f the u t i l i t y f u n c t i o n f o r most i n d i v i d u a l s i s s i m i l a r t o t h e one shown i n F i g u r e 2.5, w i t h two concave s t a g e s and an i n t e r m e d i a t e convex s t a g e . The convex s t a g e i s a t r a n s i t i o n a l phase which i s r e l e v a n t when the o u t -come may i m p l y l o s s e s o r g a i n s s u f f i c i e n t l y l a r g e t o t r a n s f e r t h e i n d i -v i d u a l from one s o c i o - e c o n o m i c p o s i t i o n t o a q u a l i t a t i v e l y lower o r h i g h e r one. - 15 -FIGURE 2.5 T y p i c a l Shape o f U t i l i t y Curve u t i l i t y * I II I I I I o The u t i l i t y f u n c t i o n as shown i n F i g u r e 2,5 has t h r e e s t a g e s ; two con-cave: ( s t a g e s 1 and I I I ) and one convex (stage 1 1 ) . The shape o f t h e f u n c t i o n assumed i n t h i s s t u d y w i l l be s t r i c t l y concave as shown i n F i g u r e 2.4 and i t i s t h e r e b y assumed t h a t the farmer's s o c i o - e c o n o m i c s t a t u s i s n o t changed i n t h e s h o r t run by the outcome o f p r o d u c t i o n d e c i s i o n s t h a t he makes. Only i n v e r y r a r e o c c a s i o n s w i l l t he outcome o f t h e p r o d u c t i o n d e c i s i o n s a t the farm l e v e l be enough t o move a f a r m e r i n t o l o w e r o r h i g h e r q u a l i t a t i v e s o c i o - e c o n o m i c p o s i t i o n s (say from an average t o a r i c h f a r m e r ) . C o n s i d e r i n g u t i l i t y t o be a f u n c t i o n o f income and r i s k , a u t i l i t y f u n c t i o n may be w r i t t e n a s : U F(E,R) , (2.1) where U L e v e l o f U t i l i t y E E x p e c t e d Income R L e v e l o f R i s k - 16 -The u t i l i t y l e v e l w i l l i n c r e a s e / d e c r e a s e when ex p e c t e d income i n c r e a s e s / d e c r e a s e s and w i l l d e c r e a s e / i n c r e a s e f o r i n c r e a s i n g / d e c r e a s i n g l e v e l s o f r i s k f o r a l l l e v e l s o f income and r i s k , hence: _3U_ > o • and ^ i L < 0 . (2,2) ^  3 E 3R S i n c e u t i l i t y i s a s t r i c t l y c o n c a v e f u n c t i o n o f t h e r e t u r n s and u t i l i t y d e c r e a s e s a t an i n c r e a s i n g r a t e w i t h h i g h e r l e v e l s o f r i s k : 1 U L < 0 and 3U 2 < 0 > (2.3) 3 2 E 3 2 R G r a p h i c a l l y , t he r e l a t i o n between U and R g i v e n a f i x e d l e v e l o f E can be r e p r e s e n t e d as i n F i g u r e 2.6 FIGURE 2.6 U t i l i t y as a F u n c t i o n o f the R i s k L e v e l s U t i l i t y as a s i m u l t a n e o u s f u n c t i o n o f both income and r i s k l e v e l s i s shown i n F i g u r e 2.7. The n e g a t i v e p a r t o f the u t i l i t y f u n c t i o n i s not shown because o f d i f f i c u l t i e s i n drawing i t , but o b v i o u s l y i t s h o u l d be ext e n d e d t o t he n e g a t i v e a r e a . - 17 -FIGURE 2.7 U t i l i t y as a F u n c t i o n o f Income and R i s k f u t i l i t y As d e f i n e d , by e q u a t i o n s 2.3, t h i s f u n c t i o n i s s t r i c t l y concave and i t i s p o s s i b l e t o draw convex i s o - u t i l i t y c u r v e s as shown i n F i g u r e 2.8 where U^> U^ >.....> U^. The i s o - u t i l i t y c u r v e s r e p r e s e n t c o m b i n a t i o n s o f income and r i s k t h a t p r o v i d e a c o n s t a n t l e v e l o f u t i l i t y , FIGURE 2.8 I s o - U t i l i t y Curves income r i s k - 18 -. G i v e n an e f f i c i e n t i n c o m e - r i s k f r o n t i e r , i . e . , knowing the optimum i n c o m e - r i s k c o m b i n a t i o n s , i t i s p o s s i b l e t o f i n d t he p l a n chosen by an i n d i v i d u a l who maximizes h i s p a r t i c u l a r u t i l i t y f u n c t i o n . F i g u r e 2.9 shows a f a m i l y o f i s o - u t i l i t y c u r v e s c o r r e s p o n d i n g t o a s p e c i f i c u t i l i t y f u n c t i o n and an e f f i c i e n t i n c o m e - r i s k f r o n t i e r . P o i n t A i n d i c a t e s the l e v e l s o f income and r i s k which maximize u t i l i t y f o r t h e g i v e n i n c o m e - r i s k f r o n t i e r . FIGURE 2.9 D e t e r m i n a t i o n o f t h e Optimal L e v e l i n an Income-Risk F r o n t i e r The optimum p o i n t depends on the income r i s k f r o n t i e r and on t h e u t i l i t y . f u n c t i o n o f the d e c i s i o n maker, as d e t e r m i n e d by the degree o f r i s k a v e r s i o n . P o i n t M i n F i g u r e 2.9 c o r r e s p o n d s t o t h a t p l a n t which maximizes p r o f i t f o r g i v e n r e s o u r c e s and o n l y i n v e r y r a r e o c c a s i o n s w i l l a p l a n r e p r e s e n t e d by M be c h o s e n . L i n , Dean and Moore (20) t e s t e d the e x p e c t e d u t i l i t y v e r s u s the p r o f i t m a x i m i z a t i o n c r i t e r i o n i n p r e d i c t i n g a c t u a l d e c i s i o n s o f a number o f C a l i f o r n i a f a r m e r s . I t was shown t h a t the u t i l i t y f o r m u l a t i o n s p r o v i d e d g r e a t e r a c c u r a c y i n p r e d i c t i n g a c t u a l and p l a n n e d c r o p p a t t e r n s . Thus, p l a n s o b t a i n e d by u s i n g s i m p l e L i n e a r - 19 -Programming may be considered un rea l i s t i c by the major i ty of farmers and therefore for farm planning purposes i t may be a bet ter pro-cedure to present farmers a set of e f f i c i e n t plans (the income-risk f r o n t i e r ) , where they may choose one which maximizes the i r expected u t i l i t y . To derive the proper income-risk f r on t i e r i t i s necessary to use an appropriate method, and, as w i l l be shown l a t e r , in order to choose the appropriate method some charac te r i s t i cs of the decis ion maker's u t i l i t y funct ion and of the d i s t r i bu t i on of returns have to be considered. 2.3 Risk Aversion Prat t (26) has defined a measure of th.e absolute r i s k aversion (r) as fo l lows ; • r = , (2.4) U' where U' i s the f i r s t der iva t ive of u t i l i t y , U, with respect to income and U" i s the second d e r i v a t i v e . Neither the slope of U (IT) nor the change of the slope (IT) are appropriate measures of r i s k avers ion. The r i s k aversion co-e f f i c i e n t i s the rate of change of the s lope , rather than the absolute change of the s lope. It i s important to understand the negative sign assigned to U". I f u" i s negative the u t i l i t y funct ion i s s t r i c t l y concave and therefore, the r i s k aversion coe f f i c i en t must be pos i t i ve s ince U' i s pos i t i ve . But i f the u t i l i t y funct ion i s convex, i . e . , the decis ion maker i s a r i sk taker , r w i l l be negative. - 20 -I f a d e c i s i o n maker has a h i g h d egree o f r i s k a v e r s i o n , he wi be w i l l i n g t o pay a h i g h premium t o a v o i d r i s k and as r i n c r e a s e s t h i s premium w i l l a l s o i n c r e a s e . The maximum amount o f r i s k premium which an i n d i v i d u a l would be w i l l i n g t o pay i s i l l u s t r a t e d i n F i g u r e 2.10. I F i g u r e 2,10, P o i n t F c o r r e s p o n d s t o the mean o r e x p e c t e d r e t u r n o f a c o m b i n a t i o n o f r e t u r n s A and B w i t h g i v e n p r o b a b i l i t i e s . D i s t a n c e DF c o r r e s p o n d s t o the maximum r i s k premium ( 5 ) , FIGURE 2.10 R i s k Premium i n a Concave U t i l i t y F u n c t i o n u t i l i t y The r i s k a v e r s i o n f u n c t i o n as d e f i n e d i n e q u a t i o n 2.4 i s a measure o f the a b s o l u t e r i s k a v e r s i o n , P r a t t has a l s o d e f i n e d a r e l a t i v e r i s k a v e r s i o n measure ( r * ) : r * = r,E , (2.5) Most a u t h o r s agree (4,6,10,30) t h a t an a p p r o p r i a t e u t i l i t y - 21 -f u n c t i o n f o r r i s k a v e r s e i n d i v i d u a l s s h o u l d have the f o l l o w i n g b a s i c p r o p e r t i e s : [a) IT > 0, i . e . m a r g i n a l u t i l i t y of. income i s p o s i t i v e (b.) U" < 0, i . e . d e c r e a s i n g m a r g i n a l u t i l i t y o f income ( c ) SLLnl = r' < 0, i . e . , a b s o l u t e r i s k a v e r s i o n s h o u l d d f , i f a n y t h i n g , d e c r e a s e when t h e income i n c r e a s e s . (d ) d ( r * ^ > v J ^ — L - 0, i . e . , t he r e l a t i v e r i s k a v e r s i o n dE. s h o u l d , i f a n y t h i n g i n c r e a s e when the income i n c r e a s e s . I t i s i m p o r t a n t t o mention t h a t t h e s e p r o p e r t i e s a r e n o t t o t a l l y met by q u a d r a t i c u t i l i t y f u n c t i o n s ; n e i t h e r by any o t h e r p o l y n o m i a l f u n c t i o n (3, 30). 2.4 . The U t i l i t y F u n c t i o n and t h e Income-Risk F r o n t i e r In S e c t i o n 2.1 the i s o - u t i l i t y c u r v e s and t h e i n c o m e - r i s k f r o n t i e r were r e p r e s e n t e d i n a two-dimension space. A r e l e v a n t q u e s t i o n t o ask i s whether r i s k can be r e p r e s e n t e d by a s i n g l e parameter ( s a y the s t a n d a r d d e v i a t i o n o r the a b s o l u t e d e v i a t i o n ) o r by a more complex c o n c e p t which i s t h e r e s u l t a n t o f two o r more parameters. F u r t h e r m o r e , i s t h e d e c i s i o n maker a b l e t o make c o n s i s t e n t d e c i s i o n s when he i s f a c e d w i t h an i n c o m e - r i s k f r o n t i e r hased on o n l y two p a r a m e t e r s , t he e x p e c t e d income and a s i n g l e parameter r e p r e s e n t i n g r i s k ? - 22 -F i r s t , c o n s i d e r a u t i l i t y f u n c t i o n UC.E) and expand i t i n t o a T a y l o r ' s s e r i e s around t he mean income, E\ i n such a way t o t r a n s f o r m i t t o a p o l y n o m i a l , U(E) = U(E) + U'CC). (E-E) + U " ( E ) . ( E - E ) 2 + 2! '• U ' " ( E ) . ( E - E ) 3 + . + U ( n ) ( E ) (E-E)** + 31 nl where R p + ] = U ( n + I ) (E) p . C E - E ) n + 1 (n+1)'. and p i s some p o i n t between E and E\ Then, t h e e x p e c t e d u t i l i t y w i l l be: E [ l J ( E ) | = ' U(E) + U J i J l m 2 + U" '(E) m 3 2 1 3 : (2.6) n+1 + ; + J J ^ % + E [ R n + ] , (2.7) where m^, m^, •••• % a r e t n e s e c o n d , t h i r d and s u c c e s s i v e h i g h e r moments w i t h r e s p e c t t o the mean. I f the s e r i e s i s c o n v e r g e n t R . can be n e g l e c t e d . I t i s i m p o r t a n t t o note t h a t the f i r s t term o f n+1 the r i g h t hand s i d e o f e q u a t i o n 2.7, U ( E ) , i s the u t i l i t y c o r r e s p o n d i n g t o th e mean o r e x p e c t e d income ( E ) , i s t h e v a r i a n c e , t h e skewness and so f o r t h . I t i s a l s o noted t h a t t he f i r s t moment w i t h r e s p e c t t o t h e mean v a n i s h e d i n 2,7, T h e r e f o r e , e x p e c t e d u t i l i t y i s not me r e l y a f u n c t i o n o f two parameters, say expected income and v a r i a n c e but, i t i s a f u n c t i o n o f the e x p e c t e d income and n - l parameters r e p r e s e n t i n g r i s k . - 23 -The a c t u a l v a l u e o f n, i . e . , the number o f parameters which the d e c i s i o n maker w i l l c o n s i d e r , w i l l depend on the number o f con-s e c y t i v e d e r i v a t i v e s which can be o b t a i n e d from the o r i g i n a l u t i l i t y f u n c t i o n . I f the d e c i s i o n maker's u t i l i t y f u n c t i o n i s l i n e a r t h e n U" (E) = U'"(E) = = U n (E) = 0. An i n d i v i d u a l w i t h such u t i l i t y f u n c t i o n w i l l o n l y c o n s i d e r e x p e c t e d income U(E) i n h i s d e c i s i o n s . I f t h e u t i l i t y f u n c t i o n i s q u a d r a t i c U" '' {I) = = l / n ^ (E) = 0, t h e d e c i s i o n maker w i l l c o n s i d e r t h e e x p e c t e d income and the v a r i a n c e m 2 i n h i s d e c i s i o n s . But i f the u t i l i t y f u n c t i o n i s o f a h i g h e r o r d e r t h e d e c i s i o n maker w i l l have t o a c c o u n t f o r more parameters i n h i s d e c i s i o n s . A l t e r n a t i v e l y , h i g h e r o r d e r terms i n e q u a t i o n 2.7 may v a n i s h i f some o f the m elements become 0. T h i s i s c l e a r l y r e l a t e d t o t h e f r e q u e n c y d i s t r i b u t i o n o f income. In symmetric d i s t r i b u t i o n s , say the normal d i s t r i b u t i o n , m^ = 0 and m^, ... ffi n a r e i n a f i x e d r e l a t i o n w i t h m 2 . T h e r e f o r e , t h e e x p e c t e d u t i l i t y (n) w i l l depend o n l y on t h e e x p e c t e d income and m 2, even i f U ' " ( E ) , ...U (E) f 0. In any skewed d i s t r i b u t i o n riv, f 0 and m 3 4 0 and hence, t h e e x p e c t e d u t i l i t y w i l l depend a t l e a s t on 3 parameters ( E , m 2, m^) when t h e u t i l i t y f u n c t i o n i s n e i t h e r l i n e a r nor q u a d r a t i c . Thus, the a c t u a l number o f parameters which s h o u l d be c o n s i d e r e d a r e equal t o t h e number o f times the u t i l i t y f u n c t i o n i s d i f f e r e n t i a b l e o r t o t h e number o f i n d e p e n d e n t parameters which c h a r a c t e r i z e s the d i s t r i b u t i o n o f r e t u r n s , w h i c h e v e r i s s m a l l e r ; n = Min ( d p ? s ) , (2.8) where d^ r e p r e s e n t s the number o f t i m e s which t h e u t i l i t y f u n c t i o n i s d i f -f e r e n t i a b l e and P s i s the number o f parameters which d e f i n e t h e random d i s t r i b u t i o n o f r e t u r n s . - 2 4 -R e f e r r i n g back t o t h e q u e s t i o n posed e a r l i e r , i t may be s a i d t h a t r i s k s h o u l d be r e p r e s e n t e d i n terms o f a s i n g l e parameter o n l y when the u t i l i t y f u n c t i o n i s q u a d r a t i c o r when t h e r e t u r n s a r e n o r m a l l y d i s -t r i b u t e d . A t t h i s point., i t i s i m p o r t a n t t o remember t h a t the income-r i s k f r o n t i e r , p r o v i d e s t he s e t o f e f f i c i e n t p l a n s from which f a r m e r s choose t h a t p l a n which maximizes t h e i r e x p e c t e d u t i l i t y . I f t h i s s e t o f a l t e r n a t i v e s has not been d e t e r m i n e d c o n s i d e r i n g the a p p r o p r i a t e number o f parameters as d e f i n e d by f o r m u l a 2.8, the d e c i s i o n maker may t a k e e r r o n e o u s d e c i s i o n s . For i n s t a n c e , i f r e t u r n s a r e not n o r m a l l y d i s -t r i b u t e d (as may be e x p e c t e d i n many c a s e s ) t he r e s e a r c h e r s h o u l d p r e s e n t the i n c o m e - r i s k f r o n t i e r based on two parameters ( e x p e c t e d income and v a r i a n c e ) o n l y when he i s c e r t a i n t h a t t h e u t i l i t y f u n c t i o n i s q u a d r a t i c . I f t h e u t i l i t y f u n c t i o n i s not q u a d r a t i c , t h e p l a n chosen might n o t maximize t h e d e c i s i o n maker's u t i l i t y f u n c t i o n . To sum up, the n a t u r e o f the u t i l i t y f u n c t i o n i s i m p o r t a n t i n j u d g i n g t he s u i t a b i l i t y o f d i f f e r e n t methods used i n farm p l a n n i n g under u n c e r t a i n t y , even i f t h e s e methods a r e used o n l y t o dete r m i n e the s e t o f e f f i c i e n t p l a n s ( t h e i n c o m e - r i s k f r o n t i e r ) l e a v i n g t o the farmer t o p i c k one o f them. The i n d i c a t o r s o f r i s k used i n e s t i m a t i n g the i n c o m e - r i s k f r o n t i e r must be c o n s i s t e n t w i t h t he r i s k i n d i c a t o r s i m p l i c i t l y con-s i d e r e d i n t h e d e c i s i o n maker's u t i l i t y f u n c t i o n . The n a t u r e o f t h e d e c i s i o n maker's u t i l i t y f u n c t i o n and t h e type o f f r e q u e n c y d i s t r i b u t i o n o f the a c t i v i t y r e t u r n s a r e t h e main elements t o be c o n s i d e r e d i n d e t e r -m i n i n g which method, i f any, may be used i n o r d e r to d e r i v e an income-r i s k f r o n t i e r . - 25 -2.5 Some Methods Used i n Farm P l a n n i n g Under U n c e r t a i n t y In t he e a r l y s e c t i o n s o f t h i s c h a p t e r t he d i s c u s s i o n was c e n t e r e d on the c o n c e p t s o f an e f f i c i e n t i n c o m e - r i s k f r o n t i e r , t h e u t i l i t y f u n c t i o n , r i s k a v e r s i o n and on the d e t e r m i n a t i o n o f an optimum e f f i c i e n t p l a n . In s e c t i o n 2.4 the d i s c u s s i o n focused- on the number o f parameters r e q u i r e d t o e x p r e s s r i s k i n o r d e r to p r o v i d e the d e c i s i o n maker w i t h the n e c e s s a r y i n f o r m a t i o n t o maximize h i s e x p e c t e d u t i l i t y . The r e l a t i o n between t h i s problem and the n a t u r e o f t h e u t i l i t y f u n c t i o n assumed was a l s o s t r e s s e d . T h i s s e c t i o n i s d e v o t e d t o an e v a l u a t i o n o f the QP-VAR, MOTAD and S e m i v a r i a n c e methods u s i n g t he framework p r o v i d e d i n s e c t i o n s 2.1 t o 2.4. Are the r i s k i n d i c a t o r s used by t h e s e t h r e e methods adequate t o a l l o w f o r 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 ? What u t i l i t y f u n c t i o n s s h o u l d be assumed i n o r d e r t o a p p l y t h e s e methods? Are the u t i l i t y f u n c t i o n s assumed c o n s i s t e n t w i t h t he b a s i c c o n d i t i o n s enumerated i n s e c t i o n 2.3 ? What f r e q u e n c y d i s t r i b u t i o n o f t h e r e t u r n s s h o u l d be assumed i n o r d e r t o a p p l y each method? A r e t h e v a r i a n c e , s e m i v a r i a n c e o r th e t o t a l a b s o l u t e d e v i a t i o n , when used as a s i n g l e measure o f r i s k , s u f f i c i e n t t o e x p r e s s i t ? An answer t o t h e s e q u e s t i o n s w i l l be attempted i n t h i s s e c t i o n g i v e n the a n a l y t i c a l framework d e v e l o p e d i n the p a s t s e c t i o n s . 2.5.1 Q u a d r a t i c Programming: The V a r i a n c e Approach Markowitz (23) f i r s t f o r m u l a t e d t h e r i s k p roblem i n a mathe-m a t i c a l programming model. He used a Q u a d r a t i c Programming method w i t h t h e v a r i a n c e o f the t o t a l r e t u r n s as a r i s k measure (QP-VAR). T o t a l - 26 -r e t u r n s (E) may be d e f i n e d as f o l o w s : ! ° J X J , C2.9) where c i s the r e t u r n per u n i t a c t i v i t y and x. i s the a c t i v i t y l e v e l . The t o t a l v a r i a n c e , V, c o r r e s p o n d i n g t o t h a t e x p e c t e d r e t u r n i s c a l c u l a t e d as f o l l o w s : ton V = E E X a X i J 1 i j J , • (2.10). where a., i s the v a r i a n c e o f r e t u r n c^ f o r a c t i v i t y x^ and a.- = a-.-i s t h e c o v a r i a n c e between the r e t u r n s o f x. and x.. D e f i n i n g e q u a t i o n (2.10) i n m a t r i x n o t a t i o n , V = X' QX, (2,11) where Q i s the v a r i a n c e - c o v a r i a n c e m a t r i x , X i s a column v e c t o r o f a c t i v i t i e s and X 1 i s t h e t r a n s p o s e v e c t o r o f X. The problem o f d e t e r m i n i n g the e f f i c i e n t s e t o f p l a n s u s i n g v a r i a n c e as a r i s k i n d i c a t o r i s o f t e n t a c k l e d t h r o u g h a Q u a d r a t i c Prog-ramming model, which may be s t a t e d as f o l l o w s ( 1 6 ) : Min V = X' QX (2,12) s.t. AX - b cX = A X - 0 , - 27 -where A i s a m a t r i x o f t e c h n i c a l c o e f f i c i e n t s , b i s a v e c t o r o f r e s o u r c e r e s t r a i n t s , c i s a v e c t o r o f a c t i v i t y r e t u r n s and X i s a parameter o f t o t a l e x p e c t e d r e t u r n w h i c h i s p a r a m e t e r i z e d f o r n d i f f e r e n t v a l u e s . The s o l u t i o n t o t h i s problem y i e l d s an i n c o m e - r i s k f r o n t i e r , which i s p r e s e n t e d t o t h e fa r m e r s i n o r d e r t o a l l o w them t o choose a p l a n a c c o r d i n g t o t h e i r u t i l i t y f u n c t i o n s (.14). I t i s i m p o r t a n t t o note some c h a r a c t e r i s t i c s o f the o b j e c t i v e f u n c t i o n V = X' QX; 1. I t i s a p o s i t i v e d e f i n i t e o r s e m i d e f i n i t e f u n c t i o n s i n c e the v a r i a n c e s a r e a l l p o s i t i v e . T h i s i s a d e s i r a b l e f e a t u r e o f the v a r i a n c e approach s i n c e i t means t h a t any l o c a l minimum w i l l be a g l o b a l minimum ( s i n c e t he f e a s i b l e s e t i s c o n v e x ) . 2. M a t r i x Q i n c l u d e s n o t o n l y t he i n d i v i d u a l v a r i a b i l i t y o f the a c t i v i t y r e t u r n s but a l s o t h e c o r r e l a t e d v a r i a t i o n s o f the a c t i v i t y r e t u r n s ( c o -v a r i a n c e s ) . The QP-VAR method assumes t h a t r i s k may be r e p r e s e n t e d as a s i n g l e p a r a m e t e r , the v a r i a n c e . In o t h e r words, t h i s method n e g l e c t s t h e i n f l u e n c e o f o t h e r h i g h e r o r d e r moments w i t h r e s p e c t t o the mean. Re-c a l l i n g from s e c t i o n 2,4, i f the u t i l i t y f u n c t i o n o f the d e c i s i o n maker i s q u a d r a t i c , t h e n I T " (E) = l ) l v ( E ) =, = l / " ) ( . E ) . = 0. T h i s i m p l i e s t h a t t he d e c i s i o n maker o n l y c o n s i d e r s t he e x p e c t e d income (E)and the v a r i a n c e i n h i s d e c i s i o n s . Thus, i f the e f f i c i e n t s e t o f p l a n s i s p r e s e n t e d t o him i n an i n c o m e - v a r i a n c e p l a n e he w i l l have a l l t h e i n -f o r m a t i o n r e q u i r e d t o maximize h i s e x p e c t e d u t i l i t y . Hence, the v a r i a n c e method i s a p p r o p r i a t e i n t h i s c a s e , Even i f t h e u t i l i t y f u n c t i o n i s not - 28 -q u a d r a t i c t h e v a r i a n c e may s t i l l be a good i n d i c a t o r o f r i s k i f the outcomes (t h e t o t a l income) a r e n o r m a l l y d i s t r i b u t e d . In t h i s c a s e , frig = 0 and h i g h e r o r d e r moments a r e e i t h e r i n a f i x e d r e l a t i o n w i t h rr^ o r v a n i s h . T h i s i m p l i e s t h a t a l t h o u g h the d e c i s i o n maker would l i k e t o c o n s i d e r o t h e r h i g h e r moments, he may make h i s d e c i s i o n s based o n l y on the mean and v a r i a n c e because a l l o t h e r moments a r e a t a z e r o l e v e l o r i n a f i x e d r e l a t i o n w i t h m^. T h u s , the QP-VAR method may be used o n l y when a t l e a s t one o f the two f o l l o w i n g c o n d i t i o n s o c c u r : a. The u t i l i t y f u n c t i o n o f t h e d e c i s i o n maker i s q u a d r a t i c , o r b. The d i s t r i b u t i o n o f r e t u r n s i s normal. I f t h e s e c o n d i t i o n s do not h o l d the QP-VAR method c o u l d l e a d t o m i s l e a d i n g r e s u l t s . Assume f o r i n s t a n c e , t h a t a d e c i s i o n maker w i t h a n o n - q u a d r a t i c u t i l i t y f u n c t i o n chooses from an i n c o m e - v a r i a n c e s e t o f p o s s i b l e p l a n s and t h a t the d i s t r i b u t i o n o f the outcomes i s skewed w i t h rr^ f 0, f 0 and assume t h a t a l l o t h e r terms i n the r i g h t - h a n d s i d e o f e q u a t i o n 2.7 can be n e g l e c t e d . The d e c i s i o n maker must choose a p l a n from t h e i n c o m e - v a r i a n c e f r o n t i e r , which maximizes h i s e x p e c t e d u t i l i t y . But, the i n c o m e - v a r i a n c e f r o n t i e r does not c o n s i d e r m^ , a l t h o u g h i t i s d i f f e r e n t from z e r o and u" 1' (E) does not v a n i s h . Once the d e c i s i o n maker chooses a p l a n from the i n c o m e - v a r i a n c e f r o n t i e r w i t h o u t c o n s i d e r i n g m^ he may make a m i s t a k e , i n t h e sense t h a t t h e e f f e c t o f U''' ( E ) f n 3 i n e q u a t i o n 2.7 has not been c o n s i d e r e d and hence, i t i s p o s s i b l e t h a t t h e r e a r e o t h e r f e a s i b l e p l a n s which p r o v i d e a g r e a t e r e x p e c t e d u t i l i t y , t h a t i s , p l a n s which have a s m a l l e r n e g a t i v e m3. T h i s d i f f e r e n c e might - 29 -more than compensate f o r the d i f f e r e n c e i n the second term, UJ_f£)m 9, 2! L The magnitude o f t h e m i s t a k e o f not c o n s i d e r i n g IT '1 ( E ) m 3 depends on t h e u t i l i t y f u n c t i o n (the v a l u e o f U"(F.)) and the.skewness o f the d i s t r i b u t i o n . T h i s i s a q u a n t i t a t i v e problem o v e r which a number o f a u t h o r s , T s i a n g ( 3 0 ) , Borch ( 4 ) , F e l d s t e i n (10) and o t h e r s have argued. I t seems c l e a r t h a t i t i s not p o s s i b l e t o make any g e n e r a l c o n c l u s i o n about t h e magnitude o f the h i g h e r o r d e r terms, Hence, u n l e s s t he r e -s e a r c h e r i s c e r t a i n t h a t t he v a l u e o f ^ ' ' ( E ^ m ^ i s small r e l a t i v e t o U (I) + U"( E ) f n 2 f o r a l l E he s h o u l d not use the QP-VAR method. I t 1s seldom p o s s i b l e t o make t h i s e s t i m a t i o n and t h e r e f o r e , i f c o n d i t i o n s (a) o r (b) a r e n o t s a t i s f i e d t h e mean-variance a n a l y s i s s h o u l d not be a p p l i e d . U n f o r t u n a t e l y , t h e s e c o n d i t i o n s a r e v e r y r e s -t r i c t i v e , A q u a d r a t i c u t i l i t y f u n c t i o n i s not g e n e r a l l y a c c e p t e d ( 4, 5, 10, 30) because i t does not s a t i s f y t h e f o u r c o n d i t i o n s which s h o u l d c h a r a c t e r i z e a r i s k a v e r s e i n d i v i d u a l mentioned i n S e c t i o n 2.3. The a b s o l u t e r i s k a v e r s i o n c o e f f i c i e n t r , i n the q u a d r a t i c u t i l i t y f u n c t i o n i s i n c r e a s i n g t h r o u g h o u t a l l l e v e l s o f income c o n t r a d i c t i n g c o n d i t i o n (c) as enumerated i n s e c t i o n 2.3, To i l l u s t r a t e , assume any q u a d r a t i c u t i l i t y f u n c t i o n : U = E - b E 2 (2.13) t h e n ' - ^ r •= U' = 1 - 2bE (2,14) dt - 30 -and d U 2 = . U" = -2b, (2,15) U s i n g t he r i s k a v e r s i o n measure as s t a t e d i n e q u a t i o n 2.4, s e c t i o n 2.3, and s u b s t i t u t i n g 2.14 and 2.15 i n 2.4, r = ~U" = 2b (2.15) U l l - 2 b E and. t he d e r i v a t i v e o f r w i t h r e s p e c t t o E w i l l be: d r 4 b 2 . (2.17) -— ••= r< = — = — 2 dE ( l - 2 b E ) As i t can be seen r 1 i s p o s i t i v e a t a l l l e v e l s o f income, E. T h i s i s q u i t e a b s u r d s i n c e i t would mean f o r i n s t a n c e , t h a t an i n d i v i d u a l would be w i l l i n g t o pay more i n s u r a n c e a g a i n s t t he same a b s o l u t e r i s k as h i s t o t a l income i n c r e a s e s ; t h i s c o n t r a d i c t s one o f the p r o p e r t i e s which the u t i l i t y f u n c t i o n o f a r i s k a v e r s e i n d i v i d u a l s h o u l d have. The assumption t h a t t he r e t u r n s a r e n o r m a l l y d i s t r i b u t e d i s a l s o d i f f i c u l t t o s u s t a i n . Hazel 1 (.15) t r i e d t o j u s t i f y t h i s assumption f o r a m u l t i a c t i v i t y farm based on t h e C e n t r a l L i m i t theorem, but Chen (8) showed t h a t t h i s a p p l i c a t i o n o f t h e C e n t r a l L i m i t theorem was i n -c o r r e c t . H a z e l ! t r i e d t o d e m o n s t r a t e t h a t i f a farm produces s e v e r a l a c t i v i t i e s t h e d i s t r i b u t i o n o f the t o t a l r e t u r n s o b t a i n e d from a l l t h e s e a c t i v i t i e s s h o u l d be normal r e g a r d l e s s o f the d i s t r i b u t i o n o f t h e i n d i -v i d u a l a c t i v i t y r e t u r n s . T h i s i s t r u e o n l y when t h e a c t i v i t y r e t u r n s - 31 -a r e i n d e p e n d e n t among each o t h e r ; o b v i o u s l y t h i s is. not a g e n e r a l s i t u a t i o n . Thus, when t h e a c t i v i t y r e t u r n s a r e c o r r e l a t e d the t o t a l r e t u r n w i l l n o t be n o r m a l l y d i s t r i b u t e d u n l e s s the i n d i v i d u a l a c t i v i t y r e t u r n s a r e a l l a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d . However, what r e s u l t s may one e x p e c t i f the c o r r e l a t i o n c o e f f i c i e n t s among a c t i v i t y r e t u r n s a r e r e l a t i v e l y low? Would t h i s l e a d t o a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d t o t a l r e t u r n s and hence, would the magnitude o f the e r r o r be n e g l i g i b l e ? T h i s i s a q u a n t i t a t i v e problem and a t h e o r e t i c a l a n a l y s i s may not p r o v i d e c l e a r c u t c o n c l u s i o n s . In summary, t h e QP-VAR method p r e s e n t s i n c o n v e n i e n c e s when i t i s a p p l i e d . t o e m p i r i c a l s i t u a t i o n s which do n o t c o r r e s p o n d t o e i t h e r one o f t h e two c o n d i t i o n s mentioned above ( q u a d r a t i c u t i l i t y f u n c t i o n o r normal d i s t r i b u t i o n o f r e t u r n s ) . I f a t l e a s t one o f t h e s e c o n d i t i o n s o c c u r s , t he QP-VAR p r o v i d e s an a p p r o p r i a t e r e p r e s e n t a t i o n o f the income-r i s k f r o n t i e r . U n f o r t u n a t e l y , t h e s e c o n d i t i o n s do not seem t o be p r e s e n t f r e q u e n t l y i n e m p i r i c a l s i t u a t i o n s . A q u e s t i o n not answered by u s i n g o n l y a n a l y t i c a l c o n c e p t s i s the f o l l o w i n g : What i s the magnitude o f t h e e r r o r i n e s t i m a t i n g i n i n c o m e - r i s k f r o n t i e r u s i n g t h e QP-VAR method when n e i t h e r o f the c o n d i t i o n s d e s c r i b e d above a r e met? One o f t h e purposes o f t h e s e t o f e x p e r i m e n t s r e p o r t e d i n the f o l l o w i n g c h a p t e r s i s t o p r o -v i d e some i n d i c a t i o n s r e g a r d i n g t h i s problem. 2.5.2 The S e m i v a r i a n c e Approach One o f the l i m i t a t i o n s o f t h e v a r i a n c e approach i s t h a t i t c o n s i d e r s as e q u a l l y u n d e s i r a b l e n e g a t i v e and p o s i t i v e f l u c t u a t i o n s o f income around t he e x p e c t e d income. I t i s c l e a r t h a t f a r m e r s a r e i n t e r -e s t e d i n m i n i m i z i n g n e g a t i v e v a r i a t i o n s o f t h e i r income but not p o s i t i v e d e v i a t i o n s . I t i s r e a s o n a b l e t o s a c r i f i c e p a r t o f t h e e x p e c t e d income - 32 -i n o r d e r t o d i m i n i s h n e g a t i v e v a r i a t i o n s o f the income but i t would be f o o l i s h t o do so i n o r d e r t o d i m i n i s h , p o s i t i v e v a r i a t i o n s . The S e m i v a r i a n c e approach c o n s i d e r s t h i s problem, Markowitz (23) s t u d i e d s e m i v a r i a n c e as an i n d i c a t o r o f r i s k , b e i n g m a i n l y c o n c e r n e d w i t h t he n e g a t i v e s e m i v a r i a n c e . The n e g a t i v e s e m i v a r i a n c e f o r one a c t i v i t y may be e x p r e s s e d as f o l l o w s : S E =• J _ t (min. { ( c h i - c . ) , 0 } ) 2 , (2.18) where T i s t h e t o t a l number o f y e a r s , c ^ i s the a c t u a l income d u r i n g y e a r h o f t h e a c t i v i t y i and c^ i s t h e e x p e c t e d o r average income o f a c t i v i t y i . As can be seen, t h e S e m i v a r i a n c e method i s o n l y c o n c e r n e d w i t h v a r i a t i o n s o f the income below i t s e x p e c t e d v a l u e . The s e m i v a r i a n c e c o r r e s p o n d i n g t o the t o t a l income, t h e income-s e m i v a r i a n c e , may be c a l c u l a t e d i n a s i m i l a r way t o t h e v a r i a n c e . mn s- = ZL x. s.. ( t ) x. ( f o r a l l g = l , — - , K ) , (2,19) where S i s the t o t a l s e m i v a r i a n c e and t ( 1 , , K) a r e the y e a r s i n which t he p l a n i m p l i e s r e t u r n s below the e x p e c t e d income and s.. i s d e f i n e d as f o l l o w s : s , , C t n ) = — l f ( ° i ' ^ t g ' ^ r ' A g , (2.20) The i n c o m e - s e m i v a r i a n c e f r o n t i e r can be o b t a i n e d i n the same way as u s i n g the v a r i a n c e , but i n s t e a d o f m i n i m i z i n g t he t o t a l v a r i a n c e , e q u a t i o n 2.19 s h o u l d be m i n i m i z e d , s u b j e c t t o t h e same r e s t r a i n t s as the v a r i a n c e m i n i m i -s a t i o n problem as shown i n e q u a t i o n 2.12. - 33 -An o t h e r f o r m u l a t i o n o f the s e m i v a r i a n c e which c o u l d be used i s t h e f o i l owing: S mn EE x. s. . x. (2,21) where T - l EEE min. (c, . - c \ ) , 0} . min. { ( c , . - c . ) , 0 } , h i j where. T i s the t o t a l number o f y e a r s c^- i s t h e a c t u a l r e t u r n d u r i n g y e a r h o f a c t i v i t y i . c. . i s t h e a c t u a l r e t u r n d u r i n g y e a r h o f a c t i v i t y j . T h i s f o r m u l a t i o n c o u l d be c a l l e d t h e A c t i v i t y S e m i v a r i a n c e s i n c e i t c o n s i d e r s t he s e m i v a r i a n c e o f t h e i n d i v i d u a l a c t i v i t i e s (and the c o s e m i v a r i a n c e s among the a c t i v i t i e s ) as a c r i t e r i o n t o choose the o p t i m a l p l a n , namely t h e o p t i m a l c o m b i n a t i o n o f a c t i v i t i e s . I t i s i m p o r t -a n t t o s t r e s s t h a t t h i s f o r m u l a t i o n i s o n l y an a p p r o x i m a t i o n o f the income s e m i v a r i a n c e . But i t has the advantage t h a t i t can be m i n i m i z e d u s i n g a q u a d r a t i c programming a l g o r i t h m , whereas the income s e m i v a r i a n c e can o n l y be s o l v e d t h r o u g h s i m u l a t i o n t e c h n i q u e s , thus a l s o p r o v i d i n g an ap p r o x i m a t e s o l u t i o n . The A c t i v i t y S e m i v a r i a n c e w i l l be r e f e r r e d t o as the QP-SEMIV method. U s i n g the QP a l g o r i t h m , i t i s j u s t n e c e s s a r y t o m i n i m i z e e q u a t i o n 2.12, s u b s t i t u t i n g s'.^ i n m a t r i x Q f o r ° ^ as c a l c u l a t e d i n e q u a t i o n 2.21. T h i s m a t r i x Q w i l l keep i t s d e s i r a b l e c h a r a c t e r i s t i c s as i n the v a r i a n c e model. I t i s i m p o r t a n t t o note t h a t the a c t i v i t y c o s e m i v a r i a n c e s can n e v e r have a n e g a t i v e s i g n , t h e i r s m a l l e s t p o s s i b l e v a l u e i s z e r o . - 34 -I f t o t a l income i s n o r m a l l y d i s t r i b u t e d the s e m i v a r i a n c e i s e x a c t l y o n e - h a l f o f the v a r i a n c e a t a l l l e v e l s o f e x p e c t e d income and v a r i a n c e . Thus, i n t h i s c a s e the s e m i v a r i a n c e f o l l o w s a l l changes o f t h e v a r i a n c e and i f p l a n A i s c o n s i d e r e d b e t t e r , e q u i v a l e n t o r worse than p l a n B a c c o r d i n g t o the v a r i a n c e c r i t e r i o n i t w i l l a l s o be c o n s i d e r e d b e t t e r , e q u i v a l e n t o r worse r e s p e c t i v e l y , i f the s e m i v a r i a n c e i s used as a c r i t e r i o n . I f t h e i n c o m e . i s not n o r m a l l y d i s t r i b u t e d , the semi-v a r i a n c e w i l l not n e c e s s a r i l y f o l l o w the v a r i a t i o n s c f the v a r i a n c e . I t s h o u l d be noted t h a t i f two d i s t r i b u t i o n s a r e compared, one w i t h a low skewness and the o t h e r w i t h a g r e a t e r skewness, the d i f f e r e n c e s i n t he s e m i v a r i a n c e o f the two d i s t r i b u t i o n s w i l l be g r e a t e r than d i f f e r e n -ces i n the v a r i a n c e v a l u e . In o t h e r words, t h e s e m i v a r i a n c e i s s e n s i t i v e t o changes i n t h e skewness but t h e v a r i a n c e i s n o t . I f the skewness a s -sumes a h i g h e r p o s i t i v e v a l u e t h e s e m i v a r i a n c e w i l l tend t o d i m i n i s h even i f the v a r i a n c e does not d e c r e a s e . I f the skewness becomes more n e g a t i v e the s e m i v a r i a n c e w i l l t e n d t o i n c r e a s e even i f the v a r i a n c e does not i n c r e a s e . To i l l u s t r a t e , assume a skewed d i s t r i b u t i o n where m^, m^ a r e g r e a t e r than z e r o and assume t h a t t h e u t i l i t y f u n c t i o n i s not quad-r a t i c . For s i m p l i c i t y a l s o assume t h a t a l l terms h i g h e r than m^ can be n e g l e c t e d . Then t h e e x p e c t e d u t i l i t y w i l l be; E {U (E)} = U(E) + U " ( E ) m 2 + U " , ( E ) m 3 . (2.23) 2! 3; ~" Assuming a r i s k a v e r s e i n d i v i d u a l , U">0 and a l s o assume t h a t U , M >0. A p p l y i n g t h e income v a r i a n c e c r i t e r i o n , the t h i r d term o f t h e r i g h t - h a n d s i d e would be n e g l e c t e d , which o b v i o u s l y would a l t e r t he v a l u e - 35 -o f E {U CE)}, The i m p o r t a n t p o i n t , however, i s t h a t U " ' (E)m3 may 3! a l t e r the r e l a t i v e o r d e r o f a s e t o f p l a n s . Thus, i f t h e v a r i a n c e i s s u b s t i t u t e d f o r t he s e m i v a r i a n c e i n m^, the v a r i a t i o n s o f m'3 a r e a l s o t a k e n i n t o c o n s i d e r a t i o n , d e s p i t e t h a t rfi^ does n o t appear e x p l i c i t l y . I f m 3 becomes more p o s i t i v e , m'^ measured by the s e m i v a r i a n c e w i l l be s m a l l e r and t h e r term U " ( E)m£ which i s n e g a t i v e r e c a l l i n g t h a t U " (E) < 0, w i l l be l e s s n e g a t i v e which t e n d s t o compensate the n e g l e c t e d e f f e c t o f t h e g r e a t e r p o s i t i v e v a l u e o f U ' " ' ^ ^ . I f a n o t h e r p l a n has a " 31 n e g a t i v e skewness, t h e n t he s e m i v a r i a n c e w i l l be l a r g e r and hence U".(E)m'2 w i l l become more n e g a t i v e than i n t h e fo r m e r example, which a c c o u n t s f o r the g r e a t e r n e g a t i v e v a l u e o f u " ' ' ( E ) m 3 . T h e r e f o r e , d e s p i t e t h e f a c t 31 • • t h a t t h e S e m i v a r i a n c e method does not f o r m a l l y c o n s i d e r the e f f e c t o f moment i r i ^ , i t i s i m p l i c i t l y c o n s i d e r i n g t he combined e f f e c t o f m^ and m3, which l e a d s t o a b e t t e r a p p r o x i m a t i o n than t he v a r i a n c e method o f the t r u e o r d i n a l c l a s s i f i c a t i o n o f the p l a n s . The compensation e f f e c t may not be s u f f i c i e n t i n comparisons among p l a n s where t h e d i s t r i b u t i o n o f outcomes i s e x t r e m e l y skewed o r when t h e v a l u e o f I T 1 1 (E) i s e x t r e m e l y l a r g e as compared t o t h e a b s o l u t e v a l u e o f U" (E ) . An example may h e l p t o u n d e r s t a n d the i d e a s j u s t mentioned. Assume t h r e e d i f f e r e n t p l a n s ( I , I I , I I I ) w i t h t h e f o l l o w i n g c h a r a c t e r i s t i c s : - 36 -P l a n I , which w i l l g i v e ; 0 w i t h p r o b a b i l i t y 0.5 o r 4 w i t h p r o b a b i l i t y 0.5 Here; Mean {£) = 2 s e m i v a r i a n c e (SV) = 2 V a r i a n c e (V) = 4 skewness (SK) = 0 P l a n I I , which w i l l g i v e : -2 w i t h p r o b a b i l i t y 0.2 o r 3 w i t h p r o b a b i l i t y 0.8 Here: E = 2 SV = 3,2 V = 4 SK = -12,0. P l a n I I I , which w i l l g i v e : 0.7 w i t h p r o b a b i l i t y 0,68 o r 5,0 w i t h p r o b a b i l i t y 0.32 Here: E = 2 SV = 1.14 V = 4 SK = 7.16. These t h r e e p l a n s have the same mean and v a r i a n c e and hence t h e y would be c o n s i d e r e d e q u a l l y e f f i c i e n t from the p o i n t o f view o f the v a r i a n c e a n a l y s i s . However, a c c o r d i n g t o e q u a t i o n 2,23 c o n s i d e r i n g V and SK, p l a n I I I s h o u l d g i v e a h i g h e r E{u(E) } t h a n p l a n s I and II and p l a n I s h o u l d g i v e a g r e a t e r E{U(E ) } than p l a n I I . A l l p l a n s have the same mean E, and the same v a r i a n c e b ut p l a n I I I i s p o s i t i v e l y skewed, skewness o f p l a n I i s z e r o and p l a n II i s n e g a t i v e l y skewed, T h i s h o l d s r e g a r d l e s s o f the a c t u a l v a l u e o f LTJT)' and U 1 1 ' ( E ) , The mean-semi v a r i a n c e 2! " 3! - 37 -a n a l y s i s p r o v i d e s t h e same o r d e r i n g w i t h o u t e x p l i c i t l y c o n s i d e r i n g skewness. P l a n I I I i s b e t t e r t h a n p l a n II and p l a n I because p l a n I I I has a s m a l l e r s e m i v a r i a n c e and p l a n I i s ranked h i g h e r than p l a n II because p l a n I has a s m a l l e r s e m i v a r i a n c e , I t i s i m p o r t a n t t o n o t e , however, t h a t i f the u t i l i t y f u n c t i o n o f t he d e c i s i o n maker were q u a d r a t i c t h e s e t h r e e p l a n s would be e q u a l l y d e s i r a b l e t o him. I f t h i s i s t h e c a s e , the v a r i a n c e a n a l y s i s would p r o -v i d e c o r r e c t r e s u l t s and the, S e m i v a r i a n c e would n o t . Hence t h e Semi-v a r i a n c e method s h o u l d not be used i n s i t u a t i o n s where the d e c i s i o n maker's u t i l i t y f u n c t i o n i s q u a d r a t i c . 2,5.2,1 The O r d i n a l C l a s s i f i c a t i o n o f t h e Ex p e c t e d U t i l i t y As was p o i n t e d o u t b e f o r e , t he aim o f u s i n g i n c o m e - r i s k methods i s t o o b t a i n an o r d i n a l c l a s s i f i c a t i o n o f a l t e r n a t i v e p l a n s which a l l o w s t h e d e c i s i o n maker t o choose a p l a n which maximizes h i s e x p e c t e d u t i l i t y . In o r d e r words, a method i s ; s u i t a b l e i f i t can p r o v i d e a l l the i n f o r m a t i o n n e c e s s a r y t o o r d e r a l t e r n a t i v e p l a n s on a r e l a t i v e s c a l e . Most o f t h e d i s c u s s i o n among Borch ( 5 ) , F e l d s t e i n ( 1 0 ) , and T s i a n g ( 3 0 ) , w i t h r e g a r d t o t h e v a l i d i t y o f income v a r i a n c e a n a l y s i s , has c e n t e r e d around whether the terms i n t h e e x p e c t e d u t i l i t y n e g l e c t e d by t h e income v a r i a n c e a n a l y s i s a r e l a r g e o r n o t . The f i r s t two a u t h o r s c o n c l u d e d t h a t t h e terms c o r r e s p o n d i n g t o t h e h i g h e r o r d e r moments i n the e x p e c t e d u t i l i t y f u n c t i o n may be l a r g e , even l a r g e r than t he v a r i a n c e t e rm, and hence t h e income v a r i a n c e a n a l y s i s would be v a l i d o n l y when th e u t i l i t y f u n c t i o n i s q u a d r a t i c o r when the outcomes a r e n o r m a l l y d i s -t r i b u t e d , T s i a n g (.17) c o n c l u d e d t h a t under f r e q u e n t c i r c u m s t a n c e s the - '38-h i g h e r o r d e r moment terms may be n e g l e c t e d and t h e r e f o r e t he income v a r i a n c e a n a l y s i s i s v a l i d even when the u t i l i t y f u n c t i o n i s not q u a d r a t i c and t h e outcome d i s t r i b u t i o n i s not n o r m a l . In c o n n e c t i o n w i t h t h i s d i s c u s s i o n i t i s i n t e r e s t i n g t o note t h e f o l l o w i n g remarks: 1. These a u t h o r s agree t h a t i f m^, m^, ,,. m n a r e l a r g e enough, t h e income v a r i a n c e a n a l y s i s i s n o t v a l i d (assuming a n o n - q u a d r a t i c u t i l i t y f u n c t i o n ) , f o r example, i f t h e outcome d i s t r i b u t i o n i s v e r y skewed. However, t h i s argument i s n o t always t r u e , The c o e f f i c i e n t o f skewness, 3, i s d e f i n e d : _!_3 m^ (2.24) I f 8 does n o t change s u b s t a n t i a l l y among the d i f f e r e n t a l t e r n a t i v e p l a n s , the income v a r i a n c e a n a l y s i s w i l l p r o v i d e the c o r r e c t o r d e r i n g o f t h e s e t o f p l a n s , because t h e r e i s a c o n s t a n t monotonic r e l a t i o n between m^ and m^. Hence, i t i s s u f f i c i e n t t o c l a s s i f y t h e p l a n s a c c o r d i n g t o any o f the two measures and t h e r e f o r e , t he normal d i s t r i b u t i o n r e q u i r e m e n t i s not s t r i c t l y n e c e s s a r y f o r the v a l i d i t y o f the income v a r i a n c e a n a l y s i s . T h i s method i s a l s o v a l i d f o r skewed (even v e r y skewed) d i s t r i b u t i o n s whose c o e f f i c i e n t o f skewness, 6, does not a b r u p t l y change among the d i f f e r e n t p o s s i b l e p l a n s . Even i f m^ changes s u b s t a n t i a l l y , t h e r e i s no problem i n a p p l y i n g t h e income v a r i a n c e a n a l y s i s i f 6 remains a p p r o x i m a t e l y c o n -s t a n t ; . -2. A s i m i l a r argument may be made i n c o n n e c t i o n w i t h h i g h e r o r d e r moments say m^. A c o e f f i c i e n t o f K u r t o s i s , ^, i s d e f i n e d as f o l l o w s : I f Y does not s u b s t a n t i a l l y change, a g a i n t h e income v a r i a n c e a n a l y s i s p r o v i d e s t h e c o r r e c t o r d e r i n g o f p l a n s r e g a r d l e s s o f which might be v e r y l a r g e and f l u c t u a t e s g r e a t l y among the a l t e r n a t i v e . p l a n s . I t i s i n t e r e s t i n g t o not e t h a t m^ i s not z e r o i n the normal d i s t r i b u t i o n , but a l l t he above-mentioned a u t h o r s agreed t h a t when the outcomes a r e n o r m a l l y d i s t r i b u t e d , t h e income v a r i a n c e a n a l y s i s i s c o r r e c t . The c o e f f i c i e n t o f K u r t o s i s f o r the normal d i s t r i b u t i o n i s always e q u a l t o 3 and t h u s , t h e r e i s a f i x e d r e l a t i o n s h i p between the v a r i a n c e and m^. 3, The advantage o f t h e income-semi v a r i a n c e a n a l y s i s i s t h a t i t i s more c l o s e l y r e l a t e d t o the skewness than t he v a r i a n c e . Even i f the c o e f f i c i e n t o f skewness v a r i e s among p l a n s , t he r e l a t i o n c o e f f i c i e n t between t h e s e m i v a r i a n c e and m^ might n o t change. Indeed, t h e r e i s a f u n c t i o n a l r e l a t i o n between t h e s e m i v a r i a n c e and t h e skewness, which i s s t a b l e . 2.5.2.2. C o n c l u s i o n s R e g a r d i n g t he S e m i v a r i a n c e Method. (a) The s e m i v a r i a n c e may be c o n s i d e r e d a b e t t e r i n d i c a t o r o f r i s k t h a n t h e v a r i a n c e when t h e f r e q u e n c y d i s t r i b u t i o n o f the r e t u r n s i s not normal and i f t h e d e c i s i o n maker's u t i l i t y f u n c t i o n i s n o n - q u a d r a t i c , (b) The s e m i v a r i a n c e may p r o v i d e c l o s e r r e p r e s e n t a t i o n s o f the income-r i s k f r o n t i e r when a p p l i e d t o non-normal d a t a i f the k u r t o s i s c o e f f i c i e n t does not change much among the a l t h e r n a t i v e p l a n s , - 40 -Cc) S i n c e t h e T o t a l Income S e m i v a r i a n c e method needs t o be a p p r o x i -mated thr o u g h non a n a l y t i c a l methods, t he A c t i v i t y S e m i v a r i a n c e method i s proposed as an a p p r o x i m a t i o n o f the T o t a l Income S e m i v a r i a n c e method. The advantage o f t h e A c t i v i t y S e m i v a r i a n c e method i s t h a t i t can be e s t i m a t e d u s i n g an a n a l y t i c a l method such as the Q.P a l g o r i t h m . 2.5.3 The MOTAD Model Many e f f o r t s have been d e v o t e d t o d e v e l o p adequate l i n e a r i n d i c a t o r s o f r i s k as a l t e r n a t i v e s t o the q u a d r a t i c ( v a r i a n c e ) and semi-q u a d r a t i c [ s e m i v a r i a n c e ) i n d i c a t o r s r e v i e w e d i n the former s e c t i o n s . T h i s i s due m a i n l y t o the f a c t t h a t l i n e a r i n d i c a t o r s o f r i s k a l l o w the use o f L i n e a r Programming a l g o r i t h m s which a r e b e t t e r known than QP a l g o r i t h m s and a l s o c h e a p e r from t he p o i n t o f view o f c o m p u t a t i o n a l c o s t s . One o f the l i n e a r models most commonly used i s the MOTAD model d e v i s e d by Hazel 1 ( 1 5 ) . The r i s k c r i t e r i o n used i n the MOTAD model i s the T o t a l A b s o l u t e D e v i a t i o n o f the income (T.A.D.) f o r a s e t o f t o b s e r v a t i o n s o v e r n a c t i v i t i e s which may be c a l c u l a t e d as f o l l o w s : t n T.A.D, = E. z ( c, . - c . ) x , (2.26) h j w h e r e c . i s t h e h o b s e r v a t i o n o f income o f the j a c t i v i t y and c . i s hj J th e e x p e c t e d income o f t h e a c t i v i t y X., Thus, t h e m i n i m i z a t i o n o f the T.A.D, can be c a s t i n an LP problem as f o l l o w s : t n Min z 2 (c - c.) x (2,27) - At -s,t, n ... x. <; b. C F o r a l l i = 1, •. • t m ) X X j > 0 th where a., i s the i r e s o u r c e r e q u i r e d f o r the j a c t i v i t y b. i s the i r e s o u r c e c a p a c i t y and A i s any l e v e l o f e x p e c t e d income which can be p a r a m e t e r i z e d . As H a z e l l proposed i t , i t i s p o s s i b l e t o m i n i m i z e o n l y t he n e g a t i v e p a r t o f t h e T,A.D. i f the e x p e c t e d r e t u r n s o f the a c t i v i t i e s , c - , a r e the sample.mean income, Thus, when c- c o m p l i e s w i t h t h i s j j r e q u i r e m e n t t h e m i n i m i z a t i o n o f the n e g a t i v e T.A.D. i s e q u i v a l e n t t o mi n i m i z e t he T.A.D. moment w i t h r e s p e c t t o t h e mean (Mean a b s o l u t e d e v i a t i o n , MAD). As can be seen i n e q u a t i o n 2.7, MAD i s not c o n s i d e r e d i n the d e t e r m i n a t i o n o f t h e e x p e c t e d u t i l i t y ; d e s p i t e t h a t MAD i s one o f t h e d e t e r m i n a n t s o f t h e l e v e l s o f u t i l i t y , i t i s n o t an e l e n e n t d e t e r m i n i n g t he e x p e c t e d u t i l i t y . T hus, t h e MOTAD model c l a s s i f i e s p l a n s u s i n g an i n d i c a t o r o f r i s k which i s n o t even c o n s i d e r e d by t h e d e c i s i o n maker i n the p r o c e s s 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 . Hence, t h i s p r o c e d u r e would be a p p r o p r i a t e o n l y i f t h e u t i l i t y f u n c t i o n i s l i n e a r , i . e . , t he d e c i s i o n maker does not c o n s i d e r iii,? ^3 ••<• However, Thomson and H a z e l l (29) have p o i n t e d o u t t h a t t h e r e i s a c o n s t a n t r e l a t i o n between the v a r i a n c e and t h e mean a b s o l u t e d e v i a t i o n when the outcomes a r e n o r m a l l y The MOTAD model bases i t s e s t i m a t i o n o f r i s k on the f i r s t - 42 <• d i s t r i b u t e d ^ m _ , 1 1 . CMAD) 2 (2.28) m 2 ~ 7 ( n - 1 ) " > Where n i s t h e t o t a l number o f o b s e r v a t i o n s i n the sample. Given n, i t i s p o s s i b l e t o c a l c u l a t e m^ from MAD and t h e d e c i s i o n maker may choose from a s e t o f p l a n s based on t h e mean v a r i a n c e c r i t e r i o n , h a v i n g t he same advantages and l i m i t a t i o n s as u s i n g t h e income v a r i a n c e a n a l y s i s . F u r t h e r , any o r d i n a l c l a s s i f i c a t i o n o f t h e a l t e r n a t i v e p l a n s u s i n g the mean and MAD as a c r i t e r i o n would be e q u i v a l e n t t o a c l a s s i f i c a t i o n u s i n g the mean and m^ as a c r i t e r i o n . I t i s i m p o r t a n t t o note t h a t f o r m u l a 2,28 a p p l i e s o n l y t o e s t i m a t i o n s o f v a r i a n c e c a l c u l a t e d from s i n g l e a c t i v i t y mean a b s o l u t e d e v i a t i o n s . I t i s n o t a p p r o p r i a t e f o r c a l c u l a t i o n s o f the v a r i a n c e o f the t o t a l income g e n e r a t e d by the combined e f f e c t o f a number o f a c t i v i t i e s , u n l e s s the c o r r e l a t i o n c o e f f i c i e n t among the a c t i v i t i e s i s z e r o . T h e r e -f o r e , the mean a b s o l u t e d e v i a t i o n i s not a good e s t i m a t o r o f the income v a r i a n c e when the c o r r e l a t i o n c o e f f i c i e n t s among the a c t i v i t i e s i s s i g n i f i c a n t l y d i f f e r e n t from z e r o . However, MAD has some f u r t h e r d i s a d v a n t a g e s w i t h r e s p e c t t o f ^ . As Thomson and H a z e l l (25) r e c o g n i z e d , t h e r e a r e d i f f e r e n c e s i n the r e l a t i v e s t a t i s t i c a l e f f i c i e n c y o f MAD w i t h r e s p e c t t o m^. T h i s e f -f i c i e n c y i s dependent on the sample s i z e n, I f n i s sm a l l the r e l a t i v e e f f i c i e n c y o f MAD w i t h r e s p e c t t o m^ w i l l be sm a l l and as n i n c r e a s e s the e f f i c i e n c y i n c r e a s e s a s y m p t o t i c a l l y t o 88% o f the sample v a r i a n c e i n - 43 ' e s t i m a t i n g t h e p o p u l a t i o n v a r i a n c e . Thus, the a b s o l u t e d e v i a t i o n may be seen as an i n d i r e c t e s t i m a t o r o f r i s k which may be used when the outcomes a r e n o r m a l l y d i s t r i b u t e d , when the c o r r e l a t i o n c o e f f i c i e n t s among the a c t i v i t i e s a r e c l o s e t o z e r o and when samples a r e l a r g e . In o t h e r words, two f u r t h e r r e s t r i c t i o n s a r e added t o t h o s e which a f f e c t t h e a p p l i c a b i l i t y o f t h e mean v a r i a n c e a n a l y s i s . I f any o f t h e s e r e s t r i c t i o n s i s not met t h e MOTAD model w i l l p r o v i d e u n r e l i a b l e r e s u l t s . 2,6 C o n c l u s i o n s The f o l l o w i n g c o n c l u s i o n s may be drawn from t h i s d i s c u s s i o n : 1. The QP-VAR method p r o v i d e s an a p p r o p r i a t e r e p r e s e n t a t i o n o f the income r i s k f r o n t i e r f o r an i n d i v i d u a l who p o s s e s s e s a q u a d r a t i c u t i l i t y f u n c t i o n r e g a r d l e s s o f the f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s . 2. The QP-VAR method p r o v i d e s an a p p r o p r i a t e r e p r e s e n t a t i o n o f the income r i s k f r o n t i e r i f the d i s t r i b u t i o n o f a c t i v i t y r e t u r n s i s n o r m a l , r e g a r d l e s s o f the d e c i s i o n maker's u t i l i t y f u n c t i o n . 3. The QP-VAR method y i e l d s an u n r e l i a b l e r e p r e s e n t a t i o n o f the income r i s k f r o n t i e r i f the a c t i v i t y r e t u r n s a r e n o n - n o r m a l l y d i s t r i b u t e d and i f t h e d e c i s i o n maker's u t i l i t y f u n c t i o n i s n o t q u a d r a t i c , 4. The MOTAD method p r o v i d e s a p p r o p r i a t e r e p r e s e n t a t i o n o f the income r i s k f r o n t i e r o n l y i f the f o l l o w i n g c o n d i t i o n s a r e s i m u l t a n e o u s l y s a t i s -f i e d , (a) The a c t i v i t y r e t u r n s a r e n o r m a l l y d i s t r i b u t e d o r the d e c i s i o n maker's u t i l i t y f u n c t i o n i s q u a d r a t i c , - 44 " (b) The c o r r e l a t i o n c o e f f i c i e n t s among a c t i v i t y r e t u r n s a r e c l o s e t o z e r o and Cc) The sample s i z e i s l a r g e . 5, The QP-SEMIV method i s proposed as a good i n d i c a t o r o f the income -r i s k f r o n t i e r when the u t i l i t y f u n c t i o n o f the d e c i s i o n maker i s not q u a d r a t i c o r l i n e a r , f o r any d i s t r i b u t i o n o f a c t i v i t y r e t u r n s where moments h i g h e r than m^ a r e n o t i m p o r t a n t . 6, In t h i s a n a l y t i c a l s e c t i o n c o n c l u s i o n s r e g a r d i n g the a b i l i t y o f the methods t o d e t e r m i n e an income r i s k f r o n t i e r were o b t a i n e d under the i m p l i c i t a ssumption t h a t t h e y were a p p l i e d u s i n g t he complete f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s as d a t a base. L i t t l e was s a i d about s i t u a t i o n s when the d a t a base c o n s i s t s o f r e l a t i v e l y s m a l l samples r a t h e r than t he complete f r e q u e n c y d i s t r i b u t i o n and t h e r e f o r e , i t i s n e c e s s a r y t o d e t e r m i n e whether the c o n c l u s i o n s drawn from t h i s d i s -c u s s i o n a r e a l s o v a l i d t o the e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r u s i n g samples as d a t a b a s e s . F u r t h e r m o r e , i t i s a l s o i m p o r t a n t t o compare the d i s p e r s i o n o f the e s t i m a t e s p r o v i d e d by t h e methods i n o r d e r t o e v a l u a t e t h e i r r e l a t i v e e f f i c i e n c y . An attempt t o c l a r i f y t h e s e p o i n t s i s made i n C h a p t e r I I I . -45 -CHAPTER I I I THE EMPIRICAL MODEL As i t was p o i n t e d o u t i n the i n t r o d u c t o r y c h a p t e r , the main elements to be c o n s i d e r e d i n t h e e v a l u a t i o n o f the methods were the magnitude o f the b i a s and the d i s p e r s i o n o f t h e e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r . The a n a l y t i c a l s t u d y from C h a p t e r II p r o v i d e d a q u a l i t a t i v e e v a l u a t i o n o f t h e methods under t h e assumption t h a t t h e complete f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s was used as the d a t a base. However, i n o r d e r t o t e s t t he hypotheses more p r e c i s e l y , q u a n t i t a t i v e i n f o r m a t i o n r e g a r d i n g b i a s and d i s p e r s i o n i s r e q u i r e d f o r the c a s e where t h e d a t a base c o n s i s t s o f samples r a t h e r than complete f r e q u e n c y d i s t r i b u t i o n s o f a c t i v i t y r e t u r n s , The purpose o f t h i s c h a p t e r i s t o d e s c r i b e t h e p r o c e d u r e u s e d i n e v a l u a t i n g t h e performance o f t h e methods when a p p l i e d i n a p l a n n i n g model u s i n g randomly drawn samples from normal and non-normal d i s t r i b u t i o n s . The f a c t t h a t the methods a r e a p p l i e d i n a p l a n n i n g model u s i n g d a t a c o n s i s t i n g o f r e l a t i v e l y s m a l l samples a l l o w s one t o e v a l u a t e t h e i r p e r formance under c o n d i t i o n s which a r e s i m i l a r t o t h o s e p r e v a i l i n g i n f i e l d r e s e a r c h a p p l i c a t i o n s . 3.1 G e n e r a l Overview o f t h e R e s e a r c h P r o c e d u r e D e c i s i o n makers choose from a l t e r n a t i v e s i n v o l v i n g d i f f e r e n t r e t u r n s and d i f f e r e n t d e g rees o f r i s k . In o r d e r t o examine t h i s s i t u a t i o n a model o f a f i r m w i t h t h r e e p r o d u c t i o n a c t i v i t i e s i s hypothesized-, t h e - v 46 ^ upper l e v e l s o f the a c t i v i t i e s a r e l i m i t e d by a number o f c o n s t r a i n t s . Under most c i r c u m s t a n c e s a d e c i s i o n maker has l i m i t e d i n f o r m a t i o n r e -g a r d i n g the f r e q u e n c y d i s t r i b u t i o n s o f a c t i v i t y r e t u r n s , That i s , t h e complete f r e q u e n c y d i s t r i b u t i o n ( p o p u l a t i o n ) , i s seldom known and t h e r e -f o r e i t i s assumed t h a t the i n f o r m a t i o n a v a i l a b l e t o the d e c i s i o n maker may be r e p r e s e n t e d by a sample o f l i m i t e d s i z e which i s randomly drawn from the p o p u l a t i o n s . The t h r e e methods a r e used t o e s t i m a t e i n c o m e - r i s k f r o n t i e r s , u s i n g the sample d a t a p r o v i d e d . T h i s p r o c e d u r e i s r e p e a t e d a number o f t i m e s ( u s i n g d i f f e r e n t samples) i n o r d e r t o o b t a i n s t a t i s t i c a l l y v e r i f i a b l e r e s u l t s . The same problem i s s o l v e d u s i n g the complete popu-l a t i o n d i s t r i b u t i o n o f a c t i v i t y r e t u r n s as a d a t a s o u r c e by a p p l y i n g the a p p r o p r i a t e method depending on the c h a r a c t e r o f the d i s t r i b u t i o n (.normal or non - n o r m a l ) . The income r i s k c o m b i n a t i o n s thus o b t a i n e d a r e c o n s i d e r e d t o be the " t r u e " income r i s k f r o n t i e r s . The s o l u t i o n s o b t a i n e d from t h e sample d a t a a r e compared t o the t r u e i n c o m e - r i s k f r o n t i e r i n o r d e r t o e s t a b l i s h b i a s and r e l a t i v e e f f i c i e n c y measures o f the methods. F i g u r e 3.1 p r e s e n t s a g e n e r a l view o f t h e r e s e a r c h p r o c e d u r e . Step 1 i n F i g u r e 3.1 i n v o l v e s d e f i n i n g the p o p u l a t i o n (normal o r non - normal) and the parameters which d e f i n e i t . The magnitude o f t h e s e parameters w i l l depend on the e m p i r i c a l s i t u a t i o n which i s s i m u l a t e d . Step 2 i n v o l v e s the p r o c e s s o f g e n e r a t i n g p o p u l a t i o n s a c c o r d i n g t o the c h a r a c t e r i s t i c s d e f i n e d i n Step 1, Step 3 i n v o l v e s t h e drawing o f samples from t h e popu-l a t i o n s g e n e r a t e d . Step 4 c o n s i d e r s the models t o be s o l v e d i n o r d e r t o o b t a i n two t y p e s o f r e s u l t s ; -47-Fiamr: 3 . 1 A I I O V E R V I E W or T H E R E S T : A R C H P R O C E D U R E Step 1 Step 2 Step 3 Step t Definition of the population parameter* or clvirecteristics Generation of the set of data representing the population distribution of activitv returns according to Step 1. Draw random samples: 15 for each method Solving a plannijip model applying the appropriate method using the completej population data Solving a planning model applying the tttree -methods using the sample data. J Step 5 Step 6 -48-( a ) The " t r u e " income r i s k f r o n t i e r , which i s o b t a i n e d when a method j u d g e d a p p r o p r i a t e i s a p p l i e d d i r e c t l y t o the complete p o p u l a t i o n d a t a : (b) The e s t i m a t e d income r i s k f r o n t i e r s , which a r e o b t a i n e d when the methods a r e a p p l i e d t o t h e samples d a t a . In Step 5 the mean income r i s k f r o n t i e r as e s t i m a t e d by each method i s compared t o the " t r u e " income r i s k f r o n t i e r i n o r d e r t o e s t a b l i s h whether t h e d i f f e r e n t methods p r o v i d e b i a s e d e s t i m a t e s . F i n a l l y , i n Step 6, t h e v a r i a n c e s o f t h e e s t i m a t e s o b t a i n e d w i t h each method a r e compared. The p r o c e e d u r e o u t l i n e d i n F i g u r e 3.1 i s r e p e a t e d f o u r t i m e s . In each i n s t a n c e t he " t r u e " i n c o m e - r i s k f r o n t i e r i s compared w i t h t he mean i n c o m e - r i s k f r o n t i e r as e s t i m a t e d by each method, but the assump-t i o n s u n d e r l y i n g t he d i s t r i b u t i o n o f a c t i v i t y r e t u r n s d i f f e r e d . In th e f o l l o w i n g s e c t i o n t h e s e s t e p s w i l l be e l a b o r a t e d upon. - 49 -3.2 G e n e r a t i o n o f the P o p u l a t i o n s In o r d e r t o e v a l u a t e t h e methods under s t u d y , i t i s assumed t h a t a complete s e t o f f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s i s known. I t i s a l s o assumed t h a t i f t he a p p r o p r i a t e method i s used t o d e r i v e an i n c o m e - r i s k f r o n t i e r u s i n g the complete f r e q u e n c y d i s t r i b u t i o n as i t d a t a s o u r c e , then t h i s i n c o m e - r i s k f r o n t i e r may be c o n s i d e r e d as the " t r u e " one. For t h i s r e a s o n randomly d i s t r i b u t e d s e t s o f d a t a were g e n e r a t e d thus r e p r e s e n t i n g complete f r e q u e n c y d i s t r i b u t i o n s o f r e t u r n s f o r t h r e e a c t i v i t i e s . Four t r i v a r i a t e d i s t r i b u t i o n s o f r e -t u r n s were r e q u i r e d , two normal d i s t r i b u t i o n s w i t h h i g h and low c o r -r e l a t i o n among a c t i v i t y r e t u r n s and two non-normal d i s t r i b u t i o n s w i t h h i g h and low degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s . The non - normal p o p u l a t i o n d i s t r i b u t i o n s chosen were o f the gamma t y p e . The gamma d i s t r i b u t i o n has two c h a r a c t e r i s t i c s t h a t a r e o f i m p o r t a n c e t o t h i s s t u d y , namely, i t i s p o s i t i v e l y skewed and i t s v a l u e s c a n n o t - be n e g a t i v e . The use o f a p o s i t i v e l y skewed d i s t r i b u t i o n may be j u s t i f i e d because i t i s e x p e c t e d t h a t h i s t o r i c a l s e r i e s o f a c t i v i t y r e t u r n s w i l l be p o s i t i v e l y skewed because o f c o n t i n u o u s t e c h n o l o g i c a l improvements i n a g r i c u l t u r e , p r o v i d e d t h a t r e a l p r i c e changes do not o f f -s e t such t r e n d s . L u t t r e l l and G i l b e r t (22) measured the degree o f skewness which c h a r a c t e r i z e d y i e l d d i s t r i b u t i o n s d u r i n g the l a s t 41 y e a r s f o r a number o f c r o p s i n numerous s t a t e s o f the U n i t e d S t a t e s . In a l l c a s e s , t he skewness c o e f f i c i e n t s c a l c u l a t e d were p o s i t i v e a l t h o u g h not a l l were s t a t i s t i c a l l y s i g n i f i c a n t . • - 50 -The use o f a d i s t r i b u t i o n whose v a l u e s cannot be n e g a t i v e i s j u s t i f i e d when r i s k i s measured u s i n g g r o s s r e t u r n r a t h e r than net r e t u r n s . Many e m p i r i c a l s t u d i e s have used g r o s s r e t u r n s (12, 15, 16) because i t i s d i f f i c u l t t o c a l c u l a t e n e t r e t u r n s f o r each, a c t i v i t y f o r a l l y e a r s o f a h i s t o r i c a l s e r i e s , s i n c e t h i s n e c e s s i t a t e s knowing v a r i a b l e and ove r h e a d c o s t s f o r each a c t i v i t y o v e r t h a t p e r i o d , Thus, i t appears t h a t the gamma d i s t r i b u t i o n c l o s e l y a p p r o x i m a t e s t h e p o s s i b l e d i s t r i b u t i o n o f g r o s s a c t i v i t y r e t u r n s which o b v i o u s l y cannot be n e g a t i v e . In o r d e r t o g e n e r a t e the d i f f e r e n t p o p u l a t i o n s t h e i r means and v a r i a n c e - c o v a r i a n c e m a t r i c e s o f a c t i v i t y r e t u r n s were p r e d e f i n e d . T a b l e 3.1 shows t h e mean a c t i v i t y r e t u r n s f o r t h e f o u r p o p u l a t i o n s g e n e r a t e d and T a b l e 3.2 p r e s e n t s the v a r i a n c e - c o v a r i a n c e m a t r i c e s . TABLE 3.1 Mean A c t i v i t y R e t u r n s , Mean C o r r e l a t i o n C o e f f i c i e n t s and  Skewness C o e f f i c i e n t o f the P o p u l a t i o n s Generated Type o f D i s t r i b u - Mean Gross Returns Mean C o r r e - Skewness t i o n l a t i o n C o ef- C o e f f i -X l ; X 2 ; X 3 f i c i e n t c i e n t P o p u l a t i o n 1 Normal 6.0 10.0 9.0 0.2 0 P o p u l a t i o n II Normal 6.0 10,0 9,0 0.6 0 P o p u l a t i o n I I I Gamma 8.9 17,0 13.5 0,2 1-1 P o p u l a t i o n IV Gamma 8.9 17.0 13,5 0,6 1.4 T a b l e 3,1 a l s o shows the mean c o r r e l a t i o n c o e f f i c i e n t among a c t i v i t y r e t u r n s which i s c a l c u l a t e d from t h e v a r i a n c e - c o v a r i a n c e v a l u e s and the skewness c o -e f f i c i e n t s . In T a b l e 3.2 the v a r i a n c e s a r e the d i a g o n a l elements o f each m a t r i x and o f f d i a g o n a l elements a r e the c o v a r i a n c e s . T A B L E 3- 2 V a r i a n c e - C o v a r i a n c e M a t r i c e s o f the A c t i v i t y Returns  C o r r e s p o n d i n g to the. D i f f e r e n t P o p u l a t i o n s Generated. • Normal I : Normal 11 : Gamma I I I : Gamma IV I x i x 2 x 3 j X i X 2 X 3 : h \ x 3 : x . x 2 x , h 0.5 0.18 0.15 • 0.5 0.46 0.63 • 23.1 14.3 7.5 : 49.8 96.1 56.1 h .; 1.0 0.30 ; 1.0 0.76 222.1 22.5 ; 515.1 179.8 x 3 ; 1.5 : 1.5 : 58.2 : 175.3 - 52 -The s e m i v a r i a n c e - c o s e m i v a r i a n c e m a t r i c e s were c a l c u l a t e d f o r the gamma d i s t r i b u t e d p o p u l a t i o n s a c c o r d i n g t o e q u a t i o n 2.22 from C h a p t e r I I . T a b l e 3.3 shows t h e s e m i v a r i a n c e - c o s e m i v a r i a n c e m a t r i c e s o f the gamma p o p u l a t i o n s TABLE 3.3 S e m i v a r i a n c e - C o s e m i v a r i a n c e M a t r i c e s o f the A c t i v i t y Returns  C o r r e s p o n d i n g t o the Gamma P o p u l a t i o n s ; Gamma ( I I I ) Gamma (IV) : X l X 2 X 3 X 3 -x l 7.9 12.2 7.1 19.2 90.0 49.8 X 2 72.6 16.6 197.1 154.2 X 3 22.4 72.6 I t i s i m p o r t a n t t o note t h a t the d a t a r e p r e s e n t i n g each p o p u l a t i o n c o n s i s t e d o f a d i s c r e t e s e t o f 500 o b s e r v a t i o n s . T h i s made i t p o s s i b l e t o c a l c u l a t e the a c t i v i t y semi v a r i a n c e s and c o s e m i v a r i a n c e s from the gamma p o p u l a t i o n s . As w i l l be seen i n s e c t i o n 3.4, mean a c t i v i t y r e t u r n s , ^ e m i v a r i a n c e - c o s e m i v a r i a n c e m a t r i c e s and v a r i a n c e - c o v a r i a n c e m a t r i c e s w i l l be used i n s o l v i n g t h e QP-SEMIV and QP-VAR methods. 3.3 The Sampling P r o c e s s A number o f samples ( s i z e 12) a r e randomly drawn from each - 53 -p o p u l a t i o n d e f i n e d above. Each sample r e p r e s e n t s i n f o r m a t i o n r e g a r d i n g a c t i v i t y r e t u r n s p r o v i d e d by h i s t o r i c a l r e c o r d s o v e r a number o f y e a r s i n an e m p i r i c a l s e t t i n g . The s i z e o f 12 was j u d g e d t o be a r e a s o n a b l e a p p r o x i m a t i o n o f the number o f y e a r s d a t a t h a t would n o r m a l l y be a v a i l a b l e t o a d e c i s i o n maker i n a r e a l w o r l d environment. T h i r t y samples were randomly drawn from each normal p o p u l a t i o n . F i f t e e n o f t h e s e were used t o o b t a i n s o l u t i o n s u s i n g the QP-VAR method ( s o l v e d 15 t i m e s ) and the r e m a i n i n g f i f t e e n samples were used i n s o l v i n g f o r t he MOTAD method ( t h e same number o f t i m e s ) . F o r t y f i v e samples were drawn from each gamma p o p u l a t i o n d i s t r i b u t i o n and t h e s e were used i n s o l v i n g f o r the QP-SEMIV, QP-VAR and MOTAD methods, f i f t e e n t i mes each. T h i s a l l o w e d s u f f i c i e n t i n c o m e - r i s k e s t i m a t e s f o r s t a t i s t i c a l t e s t i n g . From each o f t h e samples drawn from the normal p o p u l a t i o n s , the means, v a r i a n c e s and c o v a r i a n c e s were c a l c u l a t e d . T h i s i n f o r m a t i o n was used i n the o b j e c t i v e f u n c t i o n o f the QP-VAR method. The MOTAD method used the complete sample d i s t r i b u t i o n t o m i n i m i z e t h e a b s o l u t e d e v i a t i o n from the mean o r e x p e c t e d income. For the samples drawn from the gamma p o p u l a t i o n s t h e same parameters were c a l c u l a t e d and i n a d d i t i o n the s e m i v a r i a n c e and c o s e m i v a r i a n c e m a t r i c e s were o b t a i n e d . The mean a c t i v i t y r e t u r n s and the v a r i a n c e - c o v a r i a n c e m a t r i x were used i n t he QP-VAR method and the s e m i v a r i a n c e - c o s e m i v a r i a n c e m a t r i x i n a d d i t i o n to the mean a c t i v i t y r e t u r n s were used i n the QP-SEMIV method. T a b l e 3,4 shows an example o f t h e parameters c a l c u l a t e d from one sample drawn from the normal p o p u l a t i o n ( I I ) and used i n s o l v i n g t he QP-VAR. -.. 54 T . method. I t i s noted t h a t t h e d a t a p r e s e n t e d i n T a b l e 3,4 r e p r e s e n t the parameters o f one o f the numerous samples o b t a i n e d , TABLE 3.4 V a r i a n c e - C o v a r i a n c e M a t r i x and Mean A c t i v i t y Returns C a l c u l a t e d from a Sample Drawn from a Normal P o p u l a t i o n  " ( I I ) : An Example V a r i a n c e - C o v a r i a n c e M a t r i x Mean A c t i v i t y R eturns X l x2 X3 x l 0.41 0.38 0.60 6.3 x2 0.63 0,46 9.7 X3 1.50 9.5 T a b l e 3.5 shows the s e m i v a r i a n c e - c o s e m i v a r i a n c e m a t r i x and mean a c t i v i t y r e t u r n s as c a l c u l a t e d from a sample drawn from the Gamma I p o p u l a t i o n . T h i s i n f o r m a t i o n i s used i n s o l v i n g the QP-SEMIV u s i n g each o f the samples drawn. - 55 -TABLE 3 .5 S e m i v a r i a n c e - C o s e m i v a r i a n c e M a t r i x and Mean A c t i v i t y Returns  C a l c u l a t e d from a Sample Drawn from a Gamma~(l) P o p u l a t i o n h _ Semi. v a r i a n c e - C o s e m i v a r i a n c e M a t r i x Mean A c t i v i t y Returns X l x 2 X 3 x l 5,6 9.9 7.0 10.3 x 2 44.2 20,4 13.4 X 3 20.5 20.5 As may be e x p e c t e d , t h e parameters c a l c u l a t e d f o r the samples approximated the p o p u l a t i o n p a r a m e t e r s , and the mean v a l u e s o b t a i n e d from a l l samples were s i m i l a r t o the p o p u l a t i o n parameters i n each c a s e . 3.4 S o l u t i o n o f the Models T h i s s e c t i o n d e s c r i b e s the p r o c e s s o f d e t e r m i n i n g the " t r u e " income r i s k f r o n t i e r and the sample e s t i m a t e s o f the income r i s k f r o n t i e r f o r t he h y p o t h e t i c a l f i r m s i t u a t i o n d i s c u s s e d e a r l i e r . A s m a l l model c h a r a c t e r i z i n g t h e s e t o f c o n s t r a i n t s f o r t h i s f i r m g i v e n an o b j e c t i v e o f m i n i m i z i n g r i s k f o r c e r t a i n l e v e l s o f e x p e c t e d income was d e v e l o p e d . T h i s model was s o l v e d u s i n g each o f t h e methods under s t u d y . 3 . 4 . 1 . D e s c r i p t i o n o f the General Model The g e n e r a l model m i n i m i z e s r i s k (measured d i f f e r e n t l y a c c o r d i n g t o the method used) s u b j e c t t o seven l i n e a r c o n s t r a i n t s , s i x s i m u l a t i n g - 56 -r e s o u r c e c o n s t r a i n t s and one a minimum e x p e c t e d r e t u r n . The g e n e r a l model used was t h e f o l l o w i n g : Min. R = 0 (X) S u b j e c t t o (3.1) AX - b cX - X X - 0, where X i s t h e column v e c t o r o f a c t i v i t y l e v e l s , x 2  X 3 R i s the r i s k l e v e l as a f u n c t i o n o f t h e a c t i v i t y l e v e l s , c i s a row v e c t o r o f e x p e c t e d a c t i v i t y r e t u r n s , X i s the t o t a l e x p e c t e d r e t u r n , A i s a 6 x 3 m a t r i x o f t e c h n i c a l c o e f f i c i e n t d e f i n e d as f o l l o w s *: A 4 2.0 0.4 0.3 2.5 1.2 1.0 1.0 0.7 1.2 4.0 3.9 3.0 1.5 2.4 2.5 3.0 0.3 0.2 The a c t u a l m a t r i x A and v e c t o r b were chosen so as t o g e n e r a t e a smooth p r o d u c t i o n p o s s i b i l i t y f r o n t i e r a l l o w i n g f o r a l a r g e number o f boundary s o l u t i o n s . The b a s i c c o n s i d e r a t i o n i n d e t e r m i n i n g m a t r i x A and v e c t o r b was t o a v o i d c o r n e r s o l u t i o n s which would have d i m i n i s h e d the s e n s i t i v i t y o f t h e r e s u l t s t o the d i f f e r e n t methods used. and b i s a column v e c t o r o f r e s o u r c e c o n s t r a i n t s , , g i v e n as f o l l o w s : 9.0 14.5 9,0 33.5 20.5 10.5 The r i s k f u n c t i o n , 0 , depends on t h e method used. The cons-t a n t s o f t h e model ( m a t r i x A and v e c t o r b) remain the same f o r a l l methods. The v a l u e X i s p a r a m e t e r i z e d f o r t h r e e l e v e l s o f e x p e c t e d i n -come, o b t a i n i n g t h r e e s o l u t i o n s f o r each method. In s o l v i n g f o r the QP-VAR method the o b j e c t i v e f u n c t i o n r e -p r e s e n t e d the t o t a l v a r i a n c e o f the income: V = [ x-j, x 2 , x 3 > ] • [Q] • •> (2 (3 where Q i s t h e v a r i a n c e - c o v a r i a n c e m a t r i x C a l c u l a t e d from the a c t i v i t y r e t u r n s d a t a . The same q u a d r a t i c o b j e c t i v e f u n c t i o n was used i n s o l v i n g t he QP-SEMIV method, but m a t r i x Q i s s u b s t i t u t e d by a. s e m i v a r i a n c e - c o s e m i -v a r i a n c e m a t r i x as c a l c u l a t e d from t he a c t i v i t y r e t u r n s d a t a . In s o l v i n g t he MOTAD method, the o b j e c t i v e f u n c t i o n r e p r e s e n t e d the t o t a l a b s o l u t e d e v i a t i o n o f the a c t i v i t y r e t u r n s w i t h r e s p e c t t o - 58 -t h e i r means. 3.4.2 The 'True' Income R i s k F r o n t i e r s , As shown i n C h a p t e r I I , t he QP-VAR method p r o v i d e s a ' t r u e ' a p p r o x i m a t i o n o f t h e income r i s k f r o n t i e r when a p p l i e d t o n o r m a l l y d i s -t r i b u t e d d a t a . In o r d e r t o d e t e r m i n e t he t r u e income r i s k f r o n t i e r f o r n o r m a l l y d i s t r i b u t e d p o p u l a t i o n , t h e QP-VAR method was a p p l i e d u s i n g t he parameters which c h a r a c t e r i z e t h e s e n o r m a l l y d i s t r i b u t e d p o p u l a t i o n s . The model used i n s o l v i n g t h e QP-VAR method a p p l i e d t o t h e normal p o p u l a t i o n (1) d a t a i s t h e f o l l o w i n g : Min, [ V V x s J 0.50 0,18 0 .15 0.18 1.00 0 .30 0.15 0.30 1 .50 S u b j e c t t o AX £ b cX - X (3.2) where Xy X 2 , x 3 a r e t h e a c t i v i t y l e v e l s . A i s the m a t r i x o f t e c h n i c a l c o e f f i c i e n t s shown i n e q u a t i o n 3.1, b i s a column v e c t o r o f r e s o u r c e c o n s t r a i n t s shown i n e q u a t i o n 3.1, X i s the e x p e c t e d t o t a l income t o be p a r a m e t e r i z e d a t t h r e e l e v e l s , and c i s a row v e c t o r o f mean a c t i v i t y r e t u r n s which f o r t h e normal p o p u l a t i o n (1) i s d e f i n e d as f o l l o w s (see T a b l e 3,1): c = [ 6 , 0 10.0 9 ,o] . - 59 -The m a t r i x shown i n the o b j e c t i v e f u n c t i o n o f e q u a t i o n 3.2 i s the v a r i a n c e - c o v a r i a n c e m a t r i x o f the normal p o p u l a t i o n (1) as p r e s e n t e d i n T a b l e 3.2. • J The model i s s o l v e d f o r t h r e e l e v e l s o f e x p e c t e d income, thus r e p r e s e n t i n g t h r e e p o i n t s o f the t r u e income r i s k f r o n t i e r . In d e t e r -m i n i n g t he t r u e income r i s k f r o n t i e r f o r t h e normal p o p u l a t i o n ( 1 1 ) , the same model i s used e x c e p t t h a t the v a r i a n c e - c o v a r i a n c e m a t r i x used i n t h e o b j e c t i v e f u n c t i o n i s c a l c u l a t e d from the normal p o p u l a t i o n ( 1 1 ) . The QP-SEMIV method was used i n d e t e r m i n i n g t he t r u e p o p u l a t i o n i n c o m e - r i s k f r o n t i e r s f o r t h e gamma p o p u l a t i o n s , b e c a u s e , as may be r e c a l l e d from C h a p t e r I I , t he s e m i v a r i a n c e p r o v i d e s a p p r o p r i a t e r e -p r s e n t a t i o n s o f r i s k f o r skewed d i s t r i b u t i o n s . I n s t e a d o f u s i n g t he v a r i a n c e - c o v a r i a n c e m a t r i x i n the o b j e c t i v e f u n c t i o n , the semi v a r i a n c e -c o s e m i v a r i a n c e m a t r i x , as d e f i n e d f o r t he gamma p o p u l a t i o n s , was used (see T a b l e 3.5). A d d i t i o n a l l y , t h e mean a c t i v i t y r e t u r n s were t h o s e d e f i n e d f o r t he gamma p o p u l a t i o n s (see T a b l e 3.1). 3.4.3 The Income R i s k F r o n t i e r E s t i m a t e s . C o n s i d e r i n g t h a t t h e samples r e p r e s e n t l i m i t e d i n f o r m a t i o n a v a i l a b l e t o t h e d e c i s i o n maker i n r e a l s i t u a t i o n s , t h e e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r u s i n g t h i s sample d a t a were used t o e v a l u a t e the t h r e e methods. In o t h e r words, t he methods were e v a l u a t e d con-s i d e r i n g t h e d e p a r t u r e from t h e t r u e i n c o m e - r i s k f r o n t i e r ( c a l c u l a t e d a c -c o r d i n g t o S e c t i o n 3.4.2) o f t h e i r e s t i m a t e s . In o r d e r t o o b t a i n t h e s e e s t i m a t e s , t he t h r e e methods were s o l v e d u s i n g t he same g e n e r a l model d e s c r i b e d i n S e c t i o n 3.3.1 but i n t h i s i n s t a n c e sample d a t a r a t h e r than t h e complete p o p u l a t i o n d a t a was used t o c a l c u l a t e t h e parameters o f - 60 -the o b j e c t i v e f u n c t i o n and the e x p e c t e d a c t i v i t y r e t u r n s . The income r i s k f r o n t i e r was e s t i m a t e d a t t h r e e l e v e l s o f e x p e c t e d income u s i n g each method. The QP-VAR and MOTAD methods p r o v i d e d f i f t e e n e s t i m a t e s o f the income r i s k f r o n t i e r f o r each o f the f o u r d i s t r i b u t i o n s g e n e r a t e d , The QP-SEMIV method a l s o p r o v i d e d f i f t e e n sample e s t i m a t e s o f the income-r i s k f r o n t i e r f o r each o f t h e gamma d i s t r i b u t i o n s . T h i s method was n o t a p p l i e d t o the n o r m a l l y d i s t r i b u t e d d a t a because i t s e s t i m a t e s a r e e q u i v a l e n t t o t h o s e o f the QP-VAR method as was shown i n Ch a p t e r I I . The model used t o e s t i m a t e t he income r i s k f r o n t i e r , u s i n g t h e QP-VAR and QP-SEMIV methods, was s i m i l a r t o t h a t used i n d e t e r m i n i n g the t r u e p o p u l a t i o n income r i s k f r o n t i e r as shown i n e q u a t i o n s 3.1 and 3.2. The model used f o r the MOTAD method as a p p l i e d t o one sample drawn from a normal p o p u l a t i o n ( I ) i s p r e s e n t e d i n T a b l e 3.6. The o b j e c t i v e f u n c t i o n o f t h e MOTAD model c o n s i s t s o f 12 v a r i a b l e s which a c c o u n t f o r the t o t a l n e g a t i v e a b s o l u t e d e v i a t i o n o f income f o r each o b s e r v a t i o n . The r e -so u r c e c o n s t r a i n t s and t h e i r maximum l e v e l s a r e d e f i n e d by the v a l u e s shown i n m a t r i x A and v e c t o r b i n t h e g e n e r a l model as s t a t e d i n e q u a t i o n 3.1. The v a l u e s o f the e x p e c t e d income c o n s t r a i n t depend on the mean a c t i v i t y r e t u r n s f o r t h e samples, p a r a m e t e r i z e d f o r t h e same l e v e l s o f income as i n the d e t e r m i n a t i o n o f the t r u e i n c o m e - r i s k f r o n t i e r , The a b s o l u t e d e v i a t i o n s were t he f u n c t i o n a l p a r t o f t h e o b j e c t i v e f u n c t i o n ; the n e g a t i v e a b s o l u t e d e v i a t i o n s w i t h r e s p e c t t o the mean a c t i v i t y r e t u r n s a r e summed i n t h e o b j e c t i v e f u n c t i o n . Thus, t h e model i s d e s i g n e d t o choose t h a t l e v e l o f a c t i v i t i e s which m i n i m i z e s the t o t a l a b s o l u t e nega-t i v e d e v i a t i o n g i v e n an e x p e c t e d l e v e l o f income s u b j e c t t o the r e s o u r c e c o n s t r a i n t s . Table 3.6 The MOTAD Model as Applied to a Sample Obtained From the Normal P o p u l a t i o n I T X l x 2 X 3 Y l Y2 Y 3 Y4 Y5 Y7 Y8 Y9 Y10 Y l l Y12 RHS Objective Function 1 1 1 1 1 1 1" • 1 1 1 1 1 Minimize 2 . 0 0 . 4 0. 3 9.0 Resource 2 . 5 1 . 2 1.0 .14.5 Constraints 1 . 0 0 . 7 " 1.2 9.0 4 .0 3 .9 3.0 33.5 1 . 5 2 4 2.5 L 20. 5 3 0 0 3 C. 2 10.5 Exoected Income 5 9 10 1 8.9 x Constraint TI 0 1 0 4 -1.5 1 0 Absolute T2 -0 2 0 9 0.6 •1 0 Deviations T3 0 5 1 8 1.7 1 0 Constraints T4 0 3 0 0 -0.4 1 0 T5 -0 5 0 4 -2.9 1 0 T6 0 3 0 1 - .5 1 • 0 T7 -0 9 -2 3 1.0 1 0 T8 0 5 -0 5 2.5 1 V 0 T9 -0 4 -0 9 -0.4 1 0 T10 0. 3 -0 1 0.8 i 1 0 T i l 0. 1 0. 5 -1.1 1 0 T12 -o'. 2 -0. 2 0.3 1 0 - 62 -In o r d e r t o compare t he QP-VAR and MOTAD e s t i m a t e s o f the income-r i s k f r o n t i e r w i t h t he t r u e i n c o m e - r i s k f r o n t i e r f o r n o r m a l l y d i s t r i b u t e d d a t a , t h e l e v e l s o f r i s k a r e measured as the magnitude o f the s t a n d a r d , d e v i a t i o n o f the t o t a l income ( c a l c u l a t e d e x - p o s t from t he MOTAD s o l u t i o n s ) . T h i s p r o c e d u r e was chosen by assuming t h a t r i s k may be a d e q u a t e l y r e -p r e s e n t e d by the v a r i a n c e when t h e a c t i v i t y r e t u r n s a r e n o r m a l l y d i s -t r i b u t e d (see S e c t i o n 2.4). In C h a p t e r II t h e s e m i v a r i a n c e was shown to be an a p p r o p r i a t e measure o f r i s k i n skewed d i s t r i b u t i o n s . Thus, the square r o o t o f the t o t a l s e m i v a r i a n c e was used t o measure r i s k when com-p a r i n g t h e QP-SEMIV, QP-VAR and MOTAD s o l u t i o n s f o r the gamma d i s t r i -b u t i o n s . The s e m i v a r i a n c e was c a l c u l a t e d ex p o s t f o r the QP-VAR and MOTAD s o l u t i o n s . . 3.4.4 A n a l y s i s o f the S o l u t i o n s T h i s s u b s e c t i o n i s c o n c e r n e d w i t h s t e p s 5 and 6 shown i n F i g u r e 3.1. The main c r i t e r i o n used t o e v a l u a t e t he r e s u l t s o b t a i n e d u s i n g the t h r e e methods were the f o l l o w i n g : (1) B i a s o f an i n c o m e - r i s k f r o n t i e r mean estimate*, e s t a b l i s h i n g i f the l e v e l o f r i s k e s t i m a t e d by t h e methods a t each l e v e l o f income a r e s i g n i f i c a n t l y d i f f e r e n t from t h e t r u e r i s k l e v e l s . In F i g u r e 3.2 the t r u e i n c o m e - r i s k f r o n t i e r and an e s t i m a t e d i n c o m e - r i s k f r o n t i e r a r e shown. - 63 -FIGURE 3.2 B i a s i n the Income-Risk F r o n t i e r E s t i m a t o r t r u e i n c o m e - r i s k f r o n t i e r r l r l " r 2 r2 r 3 r 3 * r i s k I f t h e d i f f e r e n c e s shown as AA', BB' and C C a r e s i g n i f i c a n t ( a t 5% l e v e l o f s i g n i f i c a n c e ) , t he method used t o e s t i m a t e t h a t i n c o m e - r i s k f r o n t i e r i s c o n s i d e r e d b i a s e d . (2) D i s p e r s i o n o f the e s t i m a t e s ; the v a r i a n c e s o f the e s t i m a t e s as p r o v i d e d by the methods w i l l be compared, as i l l u s t r a t e d i n F i g u r e 4.2. FIGURE 4.2 A D i s p e r s i o n Comparison Between the E s t i m a t e s P r o v i d e d by Two Methods Income ^ Method I Income Method II - 64 -In F i g u r e 4.2 two u n b i a s e d e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r as p r o v i d e d by any two methods a r e shown. Method I i s more e f f i c i e n t than method II i f t h e v a r i a n c e o f the e s t i m a t e s p r o v i d e d by method I i s s m a l l e r than the v a r i a n c e o f the e s t i m a t e s o f method I I . I t i s a l s o p o s s i b l e t o compare a c t i v i t y l e v e l s as e s t i m a t e d by the methods t o the a c t u a l l e v e l o f a c t i v i t i e s u n d e r l y i n g the t r u e income r i s k f r o n t i e r . G i v e n t h a t t h e r e i s o n l y one a c t i v i t y c o m b i n a t i o n which m i n i m i z e s r i s k a t each l e v e l o f e x p e c t e d income (see C h a p t e r I I , f i g u r e 2 . 2 ) , t h e r e s u l t s o f t h i s comparison w i l l be the same as t h o s e c o n s i d e r i n g t h e r i s k l e v e l s c r i t e r i o n . Thus comparing t h e income r i s k e s t i m a t e s i t i s a s u f f i c i e n t c r i t e r i o n t o judge the r e l a t i v e e f f i c i e n c y o f the methods. 3.5 The QP-SEMIV Method as a S u b s t i t u t e o f the Income S e m i v a r i a n c e Method In c h a p t e r II i t was shown t h a t the T o t a l Income S e m i v a r i a n c e i s an adequate measure o f r i s k when the d a t a i s non - n o r m a l l y d i s t r i b u t e d . On the o t h e r hand, t h e QP-SEMIV method, which uses the A c t i v i t y S e m i v a r i a n c e as a r i s k measure, was proposed as an a p p r o x i m a t i o n o f the Income S e m i v a r i a n c e method, which uses the T o t a l Income S e m i v a r i a n c e as a r i s k measure. T h i s s e c t i o n p r o v i d e s a d e s c r i p t i o n o f two t e s t s d e s i g n e d t o e v a l u a t e the QP-SEMIV method as a s u b s t i t u t e o f t h e Income S e m i v a r i a n c e Method: 1. From the s o l u t i o n s p r o v i d e d by the t h r e e methods, the T o t a l Income S e m i v a r i a n c e (as d e f i n e d i n E q u a t i o n 2.15) was c a l c u l a t e d ex p o s t , i n o r d e r to see i f the QP-SEMIV method p r o v i d e d the s m a l l e s t income s e m i v a r i a n c e o f the methods as e x p e c t e d a p r i o r i a c c o r d i n g t o t h e d i s c u s s i o n i n C h a p t e r I I . - 65 -I f s o , t h i s would i m p l y t h a t t h e QP-SEMIV p r o v i d e s t he b e s t s o l u t i o n o f the methods under s t u d y when the d i s t r i b u t i o n o f the a c t i v i t y r e t u r n s i s skewed (gamma). 2. A number o f gamma p o p u l a t i o n s o f d i f f e r e n t d e g rees o f skewness were g e n e r a t e d . The QP-VAR and QP-SEMIV methods were a p p l i e d t o each o f t h e s e p o p u l a t i o n s and t h e i r s o l u t i o n s were compared. The purpose o f t h i s e x p e r i -ment was t o d e t e r m i n e the d i f f e r e n c e s among the s o l u t i o n s p r e s e n t e d by each method when the degree o f skewness changes. A p r i o r i one may e x p e c t t h a t as the degree o f skewness i n c r e a s e s , t he d i f f e r e n c e s i n the s o l u t i o n s p r o v i d e d w i l l become more a p p a r e n t . In o t h e r words, t h e r e s h o u l d be a p o s i t i v e c o r r e l a t i o n between the d i v e r g e n c e o f t h e s o l u t i o n s and the degree o f skew-ness o f the p o p u l a t i o n s . 3.6 Summary A s e t o f e x p e r i m e n t s was d e s i g n e d to t e s t the a b i l i t y o f the methods t o g e n e r a t e u n b i a s e d and e f f i c i e n t e s t i m a t e s o f t r u e i n c o m e - r i s k f r o n t i e r s . Four p o p u l a t i o n s r e p r e s e n t i n g a c t i v i t y r e t u r n s d a t a were g e n e r a t e d and u s i n g t h e s e as d a t a bases t h r e e p o i n t s on an i n c o m e - r i s k f r o n t i e r were det e r m i n e d . E s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r were o b t a i n e d u s i n g randomly drawn sample dat a from t he p o p u l a t i o n s and the mean r i s k e s t i m a t e s o b t a i n e d w i t h each method were compared t o the t r u e l e v e l s o f r i s k t o e s t a b l i s h b i a s . The degree o f d i s p e r s i o n o f the e s t i m a t e s as p r o v i d e d u s i n g each method was. a l s o compared. Two assumptions were i m p l i c i t l y made t h r o u g h o u t t h i s a n a l y s i s : (1) a c t i v i t y r e t u r n s may be c o n s i d e r e d randomly d i s t r i b u t e d and (2) - 66 -the d e c i s i o n maker's u t i l i t y f u n c t i o n i s n o n - q u a d r a t i c and h i s e x p e c t e d u t i l i t y i s not s t r o n g l y a f f e c t e d by moments h i g h e r than the skewness moment. The f i r s t a ssumption i s w i d e l y a c c e p t e d and i t has been used i m p l i c i t l y o r e x p l i c i t l y i n most t h e o r e t i c a l and a p p l i e d s t u d i e s o f r i s k ( 1 , 21, 22,23, 28). The second assumption i s r e l e v a n t f o r the a n a l y s i s as a p p l i e d t o gamma d i s t r i b u t i o n s . I f a q u a d r a t i c u t i l i t y f u n c t i o n i s assumed, the QP-SEMIV method as a p p l i e d t o the p o p u l a t i o n d a t a cannot g e n e r a t e a t r u e income-r i s k f r o n t i e r (see C h a p t e r I I , S e c t i o n 2.5). However, as may be seen i n s e c t i o n s 2.3 and 2,4 a l a r g e number o f d e c i s i o n makers may not p o s s e s s a q u a d r a t i c u t i l i t y f u n c t i o n . - 67 -CHAPTER IV THE RESULTS The purpose o f t h i s c h a p t e r i s t o r e p o r t on the r e s u l t s ob-t a i n e d from the e x p e r i m e n t s d e s c r i b e d i n C h a p t e r I I I . As may be r e -c a l l e d from C h a p t e r I I I , the methods were t e s t e d f o r f o u r d i f f e r e n t popu-l a t i o n s where t h e f r e q u e n c y d i s t r i b u t i o n s and degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s v a r i e d . Hence, f o u r s e t s o f r e s u l t s c o r r e s p o n d i n g t o t h e s e f o u r s i t u a t i o n s w i l l be r e p o r t e d i n the f o l l o w i n g s e c t i o n s . In p r e s e n t i n g the r e s u l t s o b t a i n e d f o r each s i t u a t i o n , t h r e e s e t s o f i n f o r -mation w i l l be shown, namely the mean l e v e l s , v a r i a n c e s and ranges o f the r i s k e s t i m a t e s p r o v i d e d by each method a t t h r e e income l e v e l s . The complete s e t o f sample e s t i m a t e s o f r i s k may be found on T a b l e s A . l t o A.10 o f the Appendix. 4.1 The Normal Case w i t h Low Degree o f C o r r e l a t i o n Among A c t i v i t y  R e t u rns T h i s s e c t i o n p r e s e n t s t h e r e s u l t s c o r r e s p o n d i n g t o the f i r s t s i t u a t i o n a n a l y z e d , i . e . , n o r m a l l y d i s t r i b u t e d a c t i v i t y r e t u r n s w i t h a low degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s . T a b l e 4.1 shows the QP-VAR and MOTAD mean e s t i m a t e s o f r i s k measured by the s t a n d a r d d e v i a t i o n o f the t o t a l income. B r a c k e t e d f i g u r e s b e s i d e the mean e s t i m a t e s o f r i s k a r e t h e v a l u e o f the t s t a t i s t i c s ^ c a l c u l a t e d i n o r d e r to e s t a b l i s h whether t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s between the e s t i m a t e d and t r u e v a l u e s . - 68 -T a b l e 4.1 Mean R i s k ^ L e v e l s as E s t i m a t e d by QP-VAR and MOTAD  Methods, and the Tr u e P o p u l a t i o n V a l u e s f o r Three L e v e l s  o f E x p e c t e d Income. Normal D i s t r i b u t i o n s w i t h Low Degree o f C o r r e l a t i o n L e v e l s o f Expe c t e d Income High Medium Low True R i s k V a l u e s 5.7 4.0 2.6 QP-VAR E s t i m a t e s 5.9 ( 1 . 4 2 ) ( 2 ) 4.1 (1.11) 2.7 (1.66) MOTAD E s t i m a t e s 6.0 (2.00) 4.1 (1.00) 2.7 (1.24) ^ R i s k i s measured by the s t a n d a r d d e v i a t i o n o f the t o t a l income. (2) v ' . F i g u r e s between b r a c k e t s a r e the t s t a t i s t i c v a l u e s . As may be seen i n T a b l e 4.1, t h e QP-VAR and MOTAD mean e s t i m a t e s a r e not s i g n i f i c a n t l y d i f f e r e n t t o the t r u e r i s k v a l u e a t 1% o r even 5% l e v e l o f s i g n i c a n c e (LOS). Thus the d i f f e r e n c e s between t he t r u e and the mean e s t i m a t e s a r e no l a r g e r than t h o s e t h a t would a r i s e from s a m p l i n g e r r o r . The mean e s t i m a t e s o f r i s k were c a l c u l a t e d from f i f t e e n e s t i m a t e s o f r i s k o b t a i n e d when the methods a r e a p p l i e d t o the same number o f samples randomy drawn from t he p o p u l a t i o n (see T a b l e s A . l \8I.;A>2. o f the Appendix.) - 69 -With r e s p e c t t o the d i s p e r s i o n o f the QP-VAR and MOTAD e s t i m a t e s , t he v a r i a n c e s o f the QP-VAR e s t i m a t e s were always s m a l l e r . than the v a r i a n c e s o f the MOTAD e s t i m a t e s as i t i s shown i n T a b l e 4.2. T a b l e 4.2 V a r i a n c e s and Mean V a r i a b i l i t y C o e f f i c i e n t o f the MOTAD  and QP-VAR E s t i m a t e s o f R i s k a t Thre e L e v e l s o f Expected Income L e v e l s o f e x p e c t e d Income Mean V a r i a b i -l i t y C o e f f i -High Medium Low c i e n t QP-VAR e s t i m a t e s 0.32 0.13 0.07 0.10 MOTAD e s t i m a t e s 0.38 0.18 0.10 0.12 The Mean V a r i a b i l i t y C o e f f i c i e n t (MVC) i s c a l c u l a t e d as f o l l o w s : 3 SD. 1=1 i where SD^ . i s t h e s t a n d a r d d e v i a t i o n o f the e s t i m a t e s a t an income i and MR i s the mean r i s k l e v e l e s t i m a t e d a t income i . The mean v a r i a b i l i t y c o e f f i c i e n t i s a l s o l a r g e r i n the ca s e o f t h e MOTAD e s t i m a t e s . However, none o f the d i f f e r e n c e s between QP-VAR and MOTAD v a r i a n c e o f t h e i r e s t i m a t e s was s i g n i f i c a n t a t 5% LOS when the F s t a t i s t i c t e s t was a p p l i e d . Most o f the F v a l u e s (1,18, 1,38 and 1.43 f o r low, medium and high, l e v e l s o f e x p e c t e d income, r e s p e c t i v e l y ) were s i g n i f i c a n t o n l y a t 25% LOS. - 70' -The degree o f d i s p e r s i o n o f the r e s u l t s i s i n g e n e r a l s a t i s -f a c t o r y f o r both methods. T a b l e 4.3 shows the range v a l u e s o f the e s t i m a t e s w i t h 95% and 68% p r o b a b i l i t y as compared t o the t r u e v a l u e s o f r i s k . F o r example, a t a h i g h l e v e l o f income, 9.5% o f the e s t i m a t e s made u s i n g the QP-VAR method f a l l between 4.8 t o 7.0 w i t h a t r u e v a l u e o f 5.7 . MOTAD Methods as Compared t o the True R i i F V a l u e s . Norma"! u i s t n b u t i o n . Low Degree o f C o r r e l a t i o n ~~ L e v e l s o f Expected Income High • • l i e d i urn Low True R i s k V a l u e s QP-VAR R i s k Range: 5.7 4.0 2.6 95% P r o b a b i l i t y 4.8 - 7 .0 3.4 - 4.8 2.2 - 3.2 68% P r o b a b i l i t y 5.3 - 6 5 3,7 - 4.5 2.4 - 3.0 MOTAD R i s k Range 95% P r o b a b i l i t y 4,8 - 7. 2 3,3 - 4.9 2.1 - 3.3 68% P r o b a b i l i t y 5.4 - 6. 7 3.5 - 4.7 2,3 - 3.1 i s r e a s o n a b l y low. None o f the e s t i m a t e s a r e more than 26% d i f f e r e n t from the t r u e r i s k v a l u e w i t h 95% p r o b a b i l i t y . T h i s means t h a t i n 19 out - 71 -o f 20 e s t i m a t e s o f r i s k t h e magnitude o f the e r r o r was l e s s than 26%, In a p p r o x i m a t e l y 14 o u t o f 20 e s t i m a t e s t h e magnitude o f t h e e r r o r when compared t o the t r u e r i s k l e v e l was l e s s than 16%. In summary, when t h e r e t u r n s a r e n o r m a l l y d i s t r i b u t e d w i t h a low degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s the QP-VAR and MOTAD methods may be c o n s i d e r e d u n b i a s e d e s t i m a t o r s o f the i n c o m e - r i s k f r o n t i e r . F u r t h e r m o r e , s t a t i s t i c a l e v i d e n c e was not s u f f i c i e n t t o demonstrate c a t e g o r i c a l l y t h a t one method i s more e f f i c i e n t than the o t h e r . I t s h o u l d be noted t h a t t h e s e r e s u l t s may change f o r samples o f s m a l l e r s i z e , s i n c e as may be r e c a l l e d from C h a p t e r I I , E q u a t i o n 2.28 i s not v a l i d f o r s m a l l samples. However, t he sample s i z e used i n t h e s t u d y a p p r o x i m a t e s the amount o f o b s e r v a t i o n s a v a i l a b l e i n a p p l i e d problems. Only on r a r e o c c a s i o n s w i l l a r e s e a r c h e r work w i t h l e s s than 8 y e a r s d a t a o r w i t h more than 15 y e a r s d a t a i n t h i s type o f a n a l y s i s (1,- 12, 16). 4.2 The Normal Case w i t h a High Degree o f C o r r e l a t i o n Among A c t i v i t y  R e t u rns T h i s s e c t i o n r e p o r t s on the r e s u l t s o b t a i n e d when t h e QP-VAR and MOTAD methods were a p p l i e d t o n o r m a l l y d i s t r i b u t e d d a t a w i t h h i g h de-gree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s . T a b l e 4.4 shows the mean l e v e l s o f r i s k e s t i m a t e d a t t h r e e l e v e l s o f e x p e c t e d income by the QP-VAR and MOTAD methods as compared t o t h e t r u e p o p u l a t i o n v a l u e s o f r i s k . - 72 -TABLE 4 4 Mpan Risk ( 1 } l e v e l s Estimated by QP-VAR and M O T A Q e j ^ ^ • ' t j gWue Risk Values for Three Levels ofTxpected Income NonriaT D is t r i bu t ion with "High Degree of Corre la t ion True Risk Values QP-VAR Estimates MOTAD Estimates Levels of Expected Income High Medium Low 6.4 6.5 (0.38) 7.6 (4.28**) 5.1 4.8 (1.57) 5.8 (3.68**) 3.4 3.2 (1.3) 3.9 (2.3*) Risk i s measured by the standard deviat ion of the to ta l income. Figures between brackets are the t s t a t i s t i c values. S ign i f i can t at 5% leve l of s ign i f i cance . S ign i f i can t at 1% leve l of s ign i f i cance . The data shown in Table 4.4 may be graphed in an income r i sk plane as i l l u s t r a t e d in Figure 4 .1 . (1) * ** Figure 4.1 Income high The True Population Income-risk Front ier and the Income-Risji Front ier as Estimated by QP-VAR and MO IAD. Normal D is t r i bu t ion . High Degree of Corre lat ion True income r i s k f r o n t i e r OP-VAR estimate MOTAD estimate edium. .ow S3 ztpzs T T Risk (Standard Deviation : of income) - 73 -The t t e s t s a p p l i e d showed t h a t t h e d i f f e r e n c e s between t he QP-VAR e s t i m a t e s and the t r u e l e v e l s o f r i s k were not s i g n i f i c a n t a t 1% o r 5% LOS (see T a b l e 4.4). The MOTAD e s t i m a t e s were a l l s i g n i f i c a n t l y d i f f e r e n t t o the t r u e r i s k l e v e l s a t 1% LOS e x c e p t f o r t h e low l e v e l o f income which was s i g n i c a n t a t 5%. Hence, t h e MOTAD e s t i m a t e s o f r i s k may be c o n s i d e r e d b i a s e d e s t i m a t e s o f the t r u e r i s k l e v e l s . As may be seen i n T a b l e 4.4, the MOTAD e s t i m a t e s a re p o s i t i v e l y b i a s e d and t h e magnitude o f t h e b i a s f l u c t u a t e d from a p p r o x i m a t e l y 13% ( a t medium income l e v e l ) t o 19% ( a t t h e h i g h l e v e l o f income). The MOTAD method e s t i m a t e s had a l a r g e r v a r i a n c e t h a n t h e QP-VAR e s t i m a t e s a t a l l l e v e l s o f income as may be seen i n T a b l e 4.5. The d i f f e r e n c e s i n t h e v a r i a n c e s as p r o v i d e d by the methods were s i g n i -f i c a n t a t 5% (LOS) f o r t h e medium and low l e v e l incomes when t h e F s t a t i s t i c was a p p l i e d ( t h e F v a l u e s were 1.12, 2.42 and 3.36 f o r the h i g h , medium and low income l e v e l s r e s p e c t i v e l y ) . T a b l e 4.5 a l s o i n c l u d e s the mean v a r i a b i l i t y c o e f f i c i e n t c a l c u l a t e d as i n d i c a t e d i n T a b l e 4.4. T a b l e 4.5 V a r i a n c e s and Mean V a r i a b i l i t y C o e f f i c i e n t o f t h e MOTAD and  QP-VAR E s t i m a t e s o f Rsk a t Three L e v e l s o f Income. Normal  D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n L e v e l s o f Expe c t e d Income Hi gn Med i. urn Low" Mean V a r i a b i -l i t y C o e f f i c i e n t QP-VAR E s t i m a t e s MOTAD E s t i m a t e s 1.06 0.58 1,19 1.39 0.36 1.21 0.17 0.21 - 74 -As may be seen i n T a b l e 4.5, the v a r i a n c e s o f the MOTAD e s t i m a t e s more than doubled the QP-VAR e s t i m a t e s e x c e p t a t the hi g h l e v e l o f income *. These f i g u r e s a r e a l s o more d i s p e r s e d than t he QP-VAR e s t i m a t e s . T a b l e 4.6 shows t h e r i s k ranges o f the QP-VAR and MOTAD e s t i m a t e s a t 95% l e v e l o f p r o b a b i l i t y . T a b l e 4.6 Range L e v e l s o f t h e R i s k E s t i m a t e s P r o v i d e d by QP-VAR  and MOTAD Methods as Compared t o t h e True R i s k V a l u e s . Normal D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n L e v e l s of, E x p e c t e d Income High Medium Low True R i s k V a l u e s 6.4 5.1 3.1 QP-VAR R i s k Range: 9 5 % P r o b a b i l i t y . 4 . 5 - 8.6 3.3 - 6.3 2.0 - 4.4 68% P r o b a b i l i t y 5.5 - 7.6 4.0 -. 5.6 2.6 - 3.8 MOTAD R i s k Range: 98% P r o b a b i l i t y 5.4 - 9.8 3.4 - 8.2 1.7 - 6.1 68% P r o b a b i l i t y 6.5 - 8.7 4,6 - 7.0 3.8 - 5.0 I t i s i n t e r e s t i n g t o note i n T a b l e 4.6 t h a t a t the 68% p r o b a b i l i t y l e v e l the r i s k range o f MOTAD e s t i m a t e s does not even i n c l u d e t he t r u e v a l u e a t the h i g h and low l e v e l s o f e x p e c t e d income. T h i s means t h a t a p p r o x i -* A r e a s o n f o r t h i s may be t h a t the c o m b i n a t i o n s o f a c t i v i t i e s which y i e l d the h i g h income l e v e l a r e fewer than t h o s e a t lower l e v e l s o f income. - 75 -m a t e l y 14 o u t o f 20 e s t i m a t i o n s o f r i s k made u s i n g t h e MOTAD method w i l l not i n c l u d e the t r u e v a l u e s o f r i s k . F u r t h e r m o r e , t he r i s k range o f the MOTAD e s t i m a t e s a r e w i d e r than t he QP-VAR e s t i m a t e s , which i m p l i e s t h a t the p r o b a b i l i t y o f e r r o r s i n the e s t i m a t e s i s s m a l l e r i n the QP-VAR method. I t i s a l s o i m p o r t a n t t o note t h a t both methods are l e s s e f -f i c i e n t i n t h i s c a s e than i n t h e normal d i s t r i b u t i o n w i t h low c o r r e l a t i o n c o e f f i c i e n t c a s e . T h i s i s r e f l e c t e d i n l a r g e r v a r i a n c e s and l a r g e r mean v a r i a b i l i t y c o e f f i c i e n t s o f the e s t i m a t e s . A r e a s o n f o r t h i s may be t h a t t h e l a r g e r t h e c o r r e l a t i o n c o e f f i c i e n t s among a c t i v i t y r e t u r n s , the l a r g e r a r e t h e f l u c t u a t i o n s on the r i s k l e v e l s due t o a g i v e n change i n a c t i v i t y l e v e l s . Thus, s m a l l v a r i a t i o n s on a c t i v i t y l e v e l s which o c c u r when the methods a r e a p p l i e d u s i n g d i f f e r e n t samples, g e n e r a t e l a r g e r f l u c t u a t i o n s on r i s k l e v e l s when t h e c o r r e l a t i o n c o e f f i c i e n t s a r e h i g h . In summary, when r e t u r n s a r e n o r m a l l y d i s t r i b u t e d w i t h a h i g h degree o f c o r r e l a t i o n among them, t h e QP-VAR method may be c o n s i d e r e d an un b i a s e d e s t i m a t o r o f the i n c o m e - r i s k f r o n t i e r . The MOTAD method has shown t o be i n a d e q u a t e i n t h i s s i t u a t i o n s i n c e i t p r o v i d e s b i a s e d e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r and the d i s p e r s i o n o f i t s e s t i m a t e s i s l a r g e r than t h a t o f the QP-VAR e s t i m a t e s . A d d i t i o n a l l y , t h e l e v e l s o f e f f i c i e n c y o f both methods a r e lower than i n the normal - low c o r r e l a t i o n c a s e . 4.3 Gamma D i s t r i b u t i o n s and Low Degree o f C o r r e l a t i o n Among A c t i v i t y R e t u rns T h i s s e c t i o n p r e s e n t s t h e r e s u l t s o b t a i n e d when the methods were a p p l i e d t o gamma d i s t r i b u t e d d a t a w i t h low c o r r e l a t i o n among a c t i v i t y - 76 -r e t u r n s . T a b l e 4. 6 shows t h e MOTAD, QP-VAR and QP-SEMIV mean e s t i m a t e s o f r i s k (measured as t h e square r o o t o f the s e m i v a r i a n c e ) as compared t o the t r u e l e v e l s o f r i s k . T a b l e 4.7 Mean R i s k L e v e l s ^ as E s t i m a t e d by QP-SEMIV, QP-VAR and MOTAD  Methods and t h e Tr u e V a l u e s o f R i s k f o r T h r e e L e v e l s o f  Exp e c t e d Income. Gamma D i s t r i b u t i o n w i t h Low Degree o f C o r r e l a t i o n L e v e l s o f Expe c t e d Income High Medium Low True R i s k V a l u e s 44.4 QP-SEMIV E s t i m a t e s 46.3 (0.78) QP-VAR E s t i m a t e s 64.4 (7.69**) MOTAD E s t i m a t e s 68.7 (6.53**) 0) 27.2 29.1 (1.57) 39.7 (7.96**) 45.7 (6.63**) 18.1 19.6 (1.32) 27.1 (7.14**) 29.9 (6.70**) R i s k i s measured as the square r o o t o f the s e m i v a r i a n c e o f the t o t a l income. F i g u r e s between b r a c k e t s a r e the t s t a t i s t i c v a l u e s . * S i g n i f i c a n t a t 5% LOs ** S i g n i f i c a n t a t 1% LOS. The d i f f e r e n c e s between t h e QP-SEMIV mean e s t i m a t e s o f r i s k and t h e t r u e l e v e l s were n o t s i g n i f i c a n t a t 1% o r 5% as may be seen i n T a b l e 4.7 The QP-VAR e s t i m a t e s and MOTAD method e s t i m a t e s were s i g n i f i c a n t l y d i f f e r e n t t o the t r u e v a l u e a t 5% and 1% LOS a t the d i f f e r e n t l e v e l s o f ex p e c t e d income. The QP-VAR and MOTAD e s t i m a t e s were p o s i t i v e l y b i a s e d - 77 -and the magnitude o f the QP-VAR b i a s f l u c t u a t e d between 45% a t the high and medium income l e v e l s and 49% a t the low l e v e l o f income. The b i a s o f t h e MOTAD e s t i m a t e s were l a r g e r and f l u c t u a t e d between 54% a t t h e h i g h l e v e l o f income and 68% a t t h e medium income. T a b l e 4,8 shows t h e v a r i a n c e s o f the QP-SEMIV, QP-VAR and MOTAD e s t i m a t e s o f r i s k a t each l e v e l o f e x p e c t e d income, TABLE 4.8 V a r i a n c e s o f the QP-SEMIV, QP-VAR and MOTAD E s t i m a t e s  o f R i s k a t Three L e v e l s o f Expe c t e d Income. Gamma Dis-t r i b u t i o n , Low Degree o f C o r r e l a t i o n L e v e l s o f Ex p e c t e d Income High Medium Low QP-SEMIV QP-VAR MOTAD 88.6 102.0 207.4 22.1 37.2 68.9 19.4 24.0 42.2 The v a r i a n c e s o f the QP-VAR e s t i m a t e s were not s i g n i f i c a n t l y d i f f e r e n t from the v a r i a n c e s o f the QP-SEMIV e s t i m a t e s a t 5% LOS o r even a t 10% when the F t e s t was a p p l i e d . The MOTAD and QP-SEMIV v a r i a n c e s were a l l s i g n i -f i c a n t l y d i f f e r e n t a t 10% e x c e p t a t the low l e v e l o f income. T a b l e 4.9 shows the range o f the r i s k l e v e l s e s t i m a t e d by each method as compared t o t h e t r u e r i s k v a l u e s . The ranges o f the MOTAD e s t i m a t e s a r e w i d e r t h a n t he QP-SEMIV and QP-VAR ranges o f t h e i r e s t i m a t e s . A d d i t i o n a l l y , t h e r i s k ranges o f t h e MOTAD and QP-VAR e s t i m a t e s do not even i n c l u d e the t r u e v a l u e s w i t h 68% p r o b a b i l i t y . The range o f the QP-SEMIV e s t i m a t e s i s narrower and i t i n c l u d e s the t r u e v a l u e s o f r i s k a t a l l l e v e l s as e x p e c t e d income. T a b l e 4.9 Range o f the R i s k L e v e l s as E s t i m a t e d by QP-SEMIV, QP-VAR and  MOTAD Methods as Compared t o t h e True R i s k V a l u e s . Gamma  D i s t r i b u t i o n s , Low Degree o f C o r r e l a t i o n . L e v e l s o f Expected Income True r i s k v a l u e s : QP-SEMIV R i s k Range: 95% P r o b a b i l i t y 68% P r o b a b i l i t y QP-VAR R i s k Range: 95% P r o b a b i l i t y 68% P r o b a b i l i t y MOTAD R i s k Range: 95% P r o b a b i l i t y 68% P r o b a b i l i t y High 27.9 - 64.7 37.1 - 55.5 44.2 - 84.6 54.3 - 75.5 40.1 - 97.7 54.5 - 83.3 Medium 19.7 - 38.5 24.4 - 33-8 27.5 - 51.9 33.6 - 45.8 29.1 - 62.3 37.4 - 54.0 Low 10.8 - 28.4 15.2 - 24.0 17.3 - 36.9 22.2 - 32.0 15.7 - 4,1.7 22.3 - 35.2 44.4 27.2 18.1 Thus the QP-SEMIV method i s the o n l y method which p r o v i d e s un-b i a s e d e s t i m a t e s o f the t r u e income r i s k f r o n t i e r . F u r t h e r m o r e , the d i s p e r s i o n o f the QP-SEMIV e s t i m a t e s i s s i g n i f i c a n t l y s m a l l e r than the MOTAD e s t i m a t e s , but not s m a l l e r than the d i s p e r s i o n o f the QP-VAR e s t i -- 79 -. mates. The r e s u l t s s u g g e s t t h a t under c o n d i t i o n s o f gamma d i s t r i b u t i o n o f r e t u r n s and sm a l l c o r r e l a t i o n c o e f f i c i e n t among a c t i v i t y r e t u r n s the QP-SEMIV method p r o v i d e s good e s t i m a t e s o f t h e income r i s k f r o n t i e r . The QP-SEMIV method i s c l e a r l y t h e most e f f i c i e n t method f o l l o w e d by the QP-VAR method. The MOTAD method may be c o n s i d e r e d the l e a s t e f f i c i e n t s i n c e t h e magnitude o f i t s b i a s and the d i s p e r s i o n o f i t s e s t i m a t e s as measured by the v a r i a n c e i s the l a r g e s t o f a l l methods. 4.4 Gamma D i s t r i b u t i o n s and High Degree o f C o r r e l a t i o n Among A c t i v i t y  R e t u rns T h i s s e c t i o n p r e s e n t s t he r e s u l t s o b t a i n e d when the methods were a p p l i e d u s i n g gamma d i s t r i b u t e d d a t a w i t h h i g h degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s . T a b l e 4.10 shows the QP-SEMIV, QP-VAR and MOTAD mean e s t i m a t e s o f r i s k as compared t o the t r u e p o p u l a t i o n l e v e l s o f r i s k a t t h r e e l e v e l s o f e x p e c t e d income. T a b l e 4.10 Mean R i s k L e v e l s (1) as E s t i m a t e d by QP-SEMIV, QP-VAR and MOTAD Methods and the True P o p u l a t i o n V a l u e s o f R i s k f o r Three  L e v e l s o f Ex p e c t e d Income. Gamma D i s t r i b u t i o n w i t h High  Degree o f C o r r e l a t i o n QP-SEMIV QP-VAR MOTAD True V a l u e s High 45.3 (1.12) 76,1 (5.63**) 89,8 (6,51**) 49.8 L e v e l s o f Expe c t e d Income Medium 29.6 (1,18) 52.9 (6.72**) 60.2 (6.63**) 32,6 Low 24.2 (0.86) 38.7 (7.75**) 43.3 (6.70**) 22.5 - 80 -L e g e n d : ^ R i s k i s measured by t h e s q u a r e r o o t o f t h e s e m i v a r i a n c e o f the t o t a l income, F i g u r e s between b r a c k e t s a r e t h e t s t a t i s t i c c a l c u a l t e d . * s i g n i f i c a n t a t 5% LOS ** s i g n i f i c a n t a t 1% LOS The d a t a shown i n T a b l e 4.10 may be graphed i n an income r i s k p l a n e as i l l u s t r a t e d i n F i g u r e 4.2. F i g u r e 4.2 The Tr u e P o p u l a t i o n Income-Risk F r o n t i e r and t h e Income-R i s k F r o n t i e r as E s t i m a t e d by QP-VAR, MOTAD and QP-SEMIV Methods. Gamma D i s t r i b u t i o n , High Degree o f C o r r e l a t i o n of income ) The t t e s t a p p l i e d showed t h a t t h e d i f f e r e n c e s between the QP-SEMIV mean e s t i m a t e s and the t r u e v a l u e s o f r i s k were n o t s i g n i f i c a n t a t 1% LOS (see T a b l e 8, A p p e n d i x ) , The MOTAD and QP-VAR mean e s t i m a t e s were s i g n i f i c a n t l y d i f f e r e n t t o the t r u e v a l u e s a t 1% LOS (see T a b l e s 9 and 10, Appendix) and hence, MOTAD and QP-VAR e s t i m a t e s may be c o n s i d e r e d - 81 -b i a s e d . The magnitude o f the QP-VAR b i a s f l u c t u a t e d from 52% a t the hi g h l e v e l o f e x p e c t e d income to 72% a t the low l e v e l o f e x p e c t e d income. The MOTAD b i a s f l u c t u a t e d between 80% f o r the h i g h l e v e l o f income and 92% f o r t he low l e v e l o f income. T a b l e 4,11 shows t h e v a r i a n c e s o f the QP-SEMIV, QP-VAR and MOTAD e s t i m a t e s o f r i s k a t each l e v e l o f e x p e c t e d income. T a b l e 4.11 V a r i a n c e s o f t h e QP-SEMIV, QP-VAR and MOTAD E s t i m a t e s o f  R i s k a t Thre e L e v e l s o f Expected Income. Gamma D i s t r i - b u t i o n , High Degree o f C o r r e l a t i o n L e v e l s o f Expe c t e d Income High Medium Low QP-SEMIV . 240.2 96.1 57.8 QP-VAR 327.6 136.8 65.6 MOTAD 561.7 259.2 144.0 As may be seen i n T a b l e 4.11, t h e QP-SEMIV e s t i m a t e s have t he s m a l l e s t v a r i a n c e o f a l l t h r e e methods, ahd the MOTAD e s t i m a t e s have the l a r g e s t v a r i a n c e a t t h e t h r e e l e v e l s o f e x p e c t e d income. The d i f f e r e n c e s between the v a r i a n c e s o f the QP-SEMIV and MOTAD e s t i m a t e s were a l l s i g n i f i c a n t a t 5% LOS when the F t e s t was a p p l i e d . T h e r e were no s i g n i f i c a n t d i f f e r e n c e s when the QP-SEMIV v a r i a n c e s were compared t o the QP-VAR v a r i a n c e s . These r e s u l t s s u g g e s t t h a t t he QP-SEMIV i s the o n l y u n b i a s e d method and t h a t t he d i s p e r s i o n o f i t s e s t i m a t e s i s s m a l l e r than the d i s p e r s i o n o f the MOTAD - 82 -estimates and not larger than the d ispers ion of the QP-VAR estimates. Thus, the QP-SEMIV may be considered an e f f i c i e n t estimator of r i sk and a better one than the QP-VAR and MOTAD methods for gamma d is t r ibu ted re-turns with high degree of co r re la t ion among them. It i s in te res t ing to observe that as occurs in the normal d i s -t r i bu t i on case, the leve l of e f f i c i ency of a l l methods i s lower when the degree of co r re la t ion among a c t i v i t y returns i s high than when i t i s low. The reason for th is may be s im i la r to that discussed for the normally d is t r ibu ted returns case, F i n a l l y , Table 4,12 shows the range of the r i sk leve ls estimated by the QP-SEMIV, QP-VAR and MOTAD methods. Table 4.12 Range of the Risk Levels as Estimated by QP-SEMIV, QP-VAR and MOTAD Methods as Compared to the True Risk Values, Gamma  D i s t r i bu t i ons , High Defence of Corre la t ion Levels of Expected Income High Medium Low True Risk Values QP-SEMIV Risk Range: 95% Probab i l i t y 68% Probab i l i t y QP-VAR Risk Range 95% Probab i l i t y 68% Probab i l i t y 49,8 14.3 - 76.3 29.8 - 60.8 39.9 - 112.3 58.0 - 94.2 32.6 10.0 - 49.2 19.8 - 38.8 29,5 41.2 76.3 64.6 22.5 9.0 - 39.7 16.6 - 31.8 22.5 - 54.9 30.6 - 46.8 - '83 r . . t ab le , con t ' d . Levels of Expected Income High Medium Low MOTAD Risk Range 95% Probab i l i t y 42.4 - 137.2 27.8 - 92.4 19.3 - 67.3 68% Probab i l i t y 66.1 - 113.5 44.1 - 76.3 31.3 - 55.3 Risk ranges are wider in th i s case than for the case of gamma d is t r i bu t ions with low co r re la t i on among a c t i v i t i e s . This fac t impl ies that the method are less precise when the degree of co r re la t ion among a c t i v i t y returns i s high. 4.5 Two Val idat ions of the QP-SEMIV Method The QP-SEMIV method has been used to determine the true income-r i sk f r on t i e r for the gamma case. Doubts may be raised regarding the a b i l i t y of th is method to approximate the true minimum income semi-variance. Two va l idat ions of th is method are provided here. These ve r i f i ca t i ons may not be absolutely conclusive but may provide an ind ica t ion o f the aptitudes of the QP-SEMIV method to minimize income semivariance. The income semivariance was calcu lated ex post with the solut ions provided by the three methods as appl ied to samples from the gamma popu-l a t i o n . In a l l cases, the QP-SEMIV method provided solut ions with the smallest income semivariance. Table 4.13 shows the mean income semi-variance as ca lcu lated ex post for the three methods in the case of - 84 -gamma p o p u l a t i o n s w i t h low degree o f c o r r e l a t i o n among the a c t i v i t i e s . TABLE 3.13 The Mean Income S e m i v a r i a n c e as C a l c u l a t e d Ex Post w i t h t he  QP-SEMIV, QP-VAR and MOTAD S o l u t i o n s . Gamma D i s t r i b u t i o n s , Low Degree o f C o r r e l a t i o n ~ Method Used: QP-SEMIV QP-VAR MOTAD High 1415,2 1623,0 1704.1 L e v e l s o f Expected Income Medium 650.3 680.3 763.4 Low 226.2 289.9 334.1 These o b s e r v a t i o n s i n d i c a t e t h a t t h e QP-SEMIV method p r o v i d e s p l a n s w i t h s m a l l e r l e v e l s o f income s e m i v a r i a n c e than the QP-VAR and MOTAD methods. Hence, the QP-SEMIV method i s more e f f i c i e n t s i n c e i t p r o v i d e s p l a n s w i t h lower r i s k l e v e l s . A second v e r i f i c a t i o n was o b t a i n e d by g e n e r a t i n g a s e t o f p o p u l a t i o n s o f d i f f e r e n t degrees o f skewness. The QP-SEMIV and t h e QP-VAR methods were a p p l i e d t o t h e s e p o p u l a t i o n s i n o r d e r t o m i n i m i z e r i s k a t t h r e e l e v e l s o f income. A s i m p l e r e g r e s s i o n between the degree o f skewness and t h e t o t a l a b s o l u t e d i f f e r e n c e s i n the a c t i v i t i e s as proposed by the - 85 -two methods was perf o r m e d . The f i t t e d r e g r e s s i o n l i n e o b t a i n e d was the f o l l o w i n g : TD = 0.04 + 0.73 SK , (4.1) where TD i s a measure o f the magnitude o f the d i v e r g e n c e between the QP-VAR and QP-SEMIV s o l u t i o n s and SK i s the skewness c o -e f f i c i e n t . The c o r r e l a t i o n c o e f f i c i e n t o b t a i n e d was 0.78 and t h i s c o -e f f i c i e n t and the s l o p e c o e f f i c i e n t were s i g n i f i c a n t a t 99%. Thus, t h e r e i s a r a t h e r s t r o n g c o r r e l a t i o n between t h e d i v e r g e n c e s i n the r e s u l t s p r o -v i d e d by the methods and the skewness c o e f f i c i e n t . The g r e a t e r the degree o f skewness t h e g r e a t e r the d i v e r g e n c e s . When the skewness i s z e r o , t h e r e a r e p r a c t i c a l l y no d i f f e r e n c e s between the s o l u t i o n s p r o v i d e d by both methods. E x a c t l y the same i s e x p e c t e d t o o c c u r i f the Income-Semi v a r i a n c e method were used. Hence t h i s i s a n o t h e r i n d i c a t i o n t h a t the QP-SEMIV method p r o v i d e s good a p p r o x i m a t i o n s t o t h e minimum income s e m i v a r i a n c e , a t l e a s t c l o s e r r e s u l t s than t h e QP-VAR. 4.6 T e s t i n g the Hypothesees Four hypotheses were s t a t e d i n C h a p t e r I. T h i s s e c t i o n r e p o r t s On the t e s t s and i m p l i c a t i o n s o f t h e s e h y p o t h e s e s . I t i s i m p o r t a n t t o r e -c a l l t h a t the c o n c l u s i o n s from C h a p t e r II r e g a r d i n g the e f f i c i e n c y o f the methods i n d e t e r m i n i n g i n c o m e - r i s k f r o n t i e r s a r e v a l i d assuming t h a t t h e methods were a p p l i e d u s i n g complete f r e q u e n c y d i s t r i b u t i o n s o f a c t i v i t y r e t u r n s as d a t a b a s e s . I t was n o t c l e a r whether t h e s e c o n c l u s i o n s s t i l l - 86 -h o l d i f t h e methods were used t o e s t i m a t e ( r a t h e r than t o d e t e r m i n e ) an income r i s k f r o n t i e r u s i n g sample i n f o r m a t i o n ( r a t h e r than t h e complete p o p u l a t i o n d a t a ) . S i n c e i n r e a l w o r l d problems the methods a r e a p p l i e d u s i n g r e l a t i v e l y s m a l l sample d a t a and o n l y i n v e r y r a r e o c a s s i o n s u s i n g complete f r e q u e n c y d i s t r i b u t i o n s , t h e hypotheses were s t a t e d i n terms o f the p r o p e r t i e s o f the methods when a p p l i e d u s i n g sample d a t a . Thus, i n o r d e r t o t e s t t h e hypotheses i t was n e c e s s a r y t o d e v e l o p a s e t o f ex p e r i m e n t s as d e s c r i b e d i n C h a p t e r I I I where the methods were a p p l i e d u s i n g numerous samples randomly drawn from h y p o t h e t i c a l p o p u l a t i o n . The r e s u l t s o f t h e s e e x p e r i m e n t s and the c o n c l u s i o n s o b t a i n e d from C h a p t e r II have been used t o t e s t t h e h y p o t h e s e s . 4.6.1 H y p o t h e s i s I "The QP-VAR approach as a p p l i e d t o sample d a t a p r o v i d e s an u n b i a s e d e s t i m a t o r o f the a c t u a l p o p u l a t i o n income r i s k f r o n t i e r , i f the a c t i v i t y r e t u r n s a r e n o r m a l l y d i s t r i b u t e d , r e g a r d l e s s o f the degree o f c o r -r e l a t i o n among t h e a c t i v i t y r e t u r n s . " A p r i o r i i t was shown ( C h a p t e r I I ) t h a t t he QP-VAR method p r o -v i d e s an a p p r o p r i a t e r e p r e s e n t a t i o n o f the income r i s k f r o n t i e r when a p p l i e d u s i n g n o r m a l l y d i s t r i b u t e d p o p u l a t i o n d a t a . As i t was shown i n S e c t i o n s 4.1 and 4.2 t h e QP-VAR method a l s o p r o v i d e s u n b i a s e d e s t i m a t e s o f the income r i s k f r o n t i e r when a p p l i e d u s i n g samples drawn from n o r m a l l y d i s t r i b u t e d p o p u l a t i o n s o f a c t i v i t y r e t u r n s . In o t h e r words, the QP-VAR not o n l y p r o v i d e s a p p r o p r i a t e r e p r e s e n t a t i o n s o f the income r i s k f r o n t i e r u s i n g n o r m a l l y d i s t r i b u t e d p o p u l a t i o n d a t a but a l s o t he e s t i m a t e s o b t a i n e d u s i n g sample d a t a were u n b i a s e d and t h e r e f o r e h y p o t h e s i s I may be a c c e p t e d i n f u l l . - 87 -Furthermore, the QP-VAR method was shown to be not only unbiased but also an e f f i c i e n t method since the variance of i t s estimates was not larger than the variance of- the MOTAD estimates when the cor re la t ion among a c t i v i t y returns was low and i t was s i g n i f i c a n t l y smaller when the degree of cor -re la t ion was high.. 4.6.2 Hypothesis 2 "The MOTAD method provides an unbiased estimator of the actual population income-risk f ron t i e r only i f the fo l lowing two condit ions are s a t i s f i e d : a) The a c t i v i t y returns are normally d is t r ibu ted and b) The cor re la t ion coe f f i c ien ts among the a c t i v i t y returns are c lose to ze ro" . I t was shown in Chapter II that the MOTAD method only provides close approximations to. the income r i sk f ron t ie r when appl ied using normally d is t r ibu ted data with low degree of co r re la t ion among a c t i v i t y returns. It was also shown that i f these condit ions are not met the MOTAD solut ions are not appropriate. The resu l ts of the experiments confirmed these conclusions for the estimates of the income r isk f r on t i e r obtained using sample data. The MOTAD estimates were s i g n i f i c a n t l y biased in a l l s i tuat ions except when the samples were drawn from normally d is t r ibuted- low cor re la t ion •populations. Therefore, hypothesis 2 i s accepted. The MOTAD method was not •• only biased but also the method that provided the more dispersed estimates under a l l s i t ua t i ons . Thus the MQTAD method was the least e f f i c i e n t of the methods when appl ied to highly corre lated a c t i v i t y returns and/or gamma - 88 -d i s t r i b u t e d data, ( c o n s i d e r i n g t h a t i n t h i s l a t e r c a s e i t s b i a s was l a r g e r than the b i a s o f t h e QP-VAR e s t i m a t e s ) . 4.6.3 H y p o t h e s i s 3. " I f t h e a c t i v i t y r e t u r n s a r e no n - n o r m a l l y d i s t r i b u t e d , the QP-VAR method and the MOTAD method y i e l d u n b i a s e d e s t i m a t o r s o f the a c t u a l p o p u l a t i o n i n c o m e - r i s k f r o n t i e r . " C o n c l u s i o n s o b t a i n e d from C h a p t e r II p o i n t e d out t h a t the QP-VAR method does not p r o v i d e a p p r o p r i a t e r e p r e s e n t a t i o n s o f the income r i s k f r o n t i e r when a p p l i e d . u s i n g n o n - n o r m a l l y d i s t r i b u t e d d a t a . S e c t i o n s 4.3 and 4.4 r e p o r t e d on t h e QP-VAR s o l u t i o n s o b t a i n e d u s i n g sample d a t a drawn from gamma p o p u l a t i o n s . The QP-VAR e s t i m a t e s were b i a s e d when the degree o f c o r r e l a t i o n o f a c t i v i t y r e t u r n s was h i g h and a l s o when i t was low. Hence hypotheses 3 s h o u l d be r e j e c t e d i n i t s p a r t c o r r e s p o n d i n g t o the QP-VAR e s t i m a t e s . The c o n c l u s i o n s o f the a n a l y t i c a l d i s c u s s i o n from C h a p t e r II and the f a c t t h a t t h e MOTAD e s t i m a t e s were a l s o b i a s e d when a p p l i e d t o gamma d i s t r i b u t e d d a t a a l l o w one t o r e j e c t the second p a r t o f h y p o t h e s i s 3 as w e l l . 4.6.4 H y p o t h e s i s 4. "When the a c t i v i t y r e t u r n s a r e non - n o r m a l l y d i s t r i b u t e d the s e m i v a r i a n c e method w i l l p r o v i d e u n b i a s e d e s t i m a t e s o f the a c t u a l p o p u l a -t i o n income r i s k f r o n t i e r . " - 89 -Two b a s i c c o n c l u s i o n s were o b t a i n e d from t h e d i s c u s s i o n i n Ch a p t e r I I : (1) The income s e m i v a r i a n c e method may p r o v i d e a p p r o p r i a t e r e p r e s e n -t a t i o n s o f the i n c o m e - r i s k f r o n t i e r when a p p l i e d t o no n - n o r m a l l y d i s t r i -b uted f r e q u e n c y d i s t r i b u t i o n s ( p r o v i d e d t h a t k u r t o s i s and h i g h e r o r d e r moments a r e not i m p o r t a n t and t h a t t h e u t i l i t y f u n c t i o n o f the d e c i s i o n maker i s not q u a d r a t i c ) ; (2) The QP-SEMIV was proposed as an approxima-t i o n o f the Income-Semi v a r i a n c e method. As r e p o r t e d i n S e c t i o n s 4.3 and 4.4 the QP-SEMIV method p r o v i d e d unbiased e s t i m a t e s t n e i n c o m e r i s k f r o n t i e r when a p p l i e d u s i n g samples randomly drawn from gamma p o p u l a t i o n s . Thus, h y p o t h e s i s 4 i s a c c e p t e d . F u r t h e r m o r e , t h e QP-SEMIV may be c o n s i d e r e d the most e f f i c i e n t o f the t h r e e methods i n s i t u a t i o n s where th e y a r e a p p l i e d u s i n g gamma d i s t r i b u t e d d a t a . 4.7 Summary T h i s c h a p t e r has r e p o r t e d on the r e s u l t s o b t a i n e d from the experiments d e s c r i b e d i n C h a p t e r I I I . R e s u l t s have been r e p o r t e d f o r the f o u r c a s e s r e g a r d i n g f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s c o n s i d e r e d , i . e . , normal d i s t r i b u t i o n w i t h low and h i g h degree o f c o r r e l a t i o n among ac-t i v i t y r e t u r n s . U s i n g t h e s e r e s u l t s and the c o n c l u s i o n s drawn from the t h e o r e -t i c a l s t u d y , the f o u r hypotheses as s t a t e d i n C h a p t e r I were t e s t e d . A l l hypotheses e x c e p t h y p o t h e s i s 3 were a c c e p t e d . F u r t h e r m o r e , two t e s t s d e s i g n e d t o v a l i d a t e t h e QP-SEMIV method y i e l d r e s u l t s which p r o v i d e - 90 "-an a d d i t i o n a l e v i d e n c e t o s u s t a i n t he a s s e r t i o n t h a t t he QP-SEMIV method i s a c l o s e s u b s t i t u t e t o t h e Income-Semi v a r i a n c e Method, Throughout the p r o c e s s o f r e p o r t i n g t he r e s u l t s i t was ob-s e r v e d t h a t the e f f i c i e n c y o f t h e methods tend t o be lower when the degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s i s h i g h as compared t o when i t i s low. T h i s f a c t may have a d i r e c t p r a c t i c a l i m p l i c a t i o n s i n c e i t would mean t h a t the h i g h e r t h e degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s , t he l a r g e r must be the sample s i z e . Thus, a g r e a t e r number o f a c t i v i t y r e -c o r d s a r e r e q u i r e d i n o r d e r t o keep the e f f i c i e n c y o f the methods a t a c c e p t a b l e l e v e l s . - 91' -CHAPTER V A CASE STUDY FARM In o r d e r t o i l l u s t r a t e t h e performance o f t h e methods i n a more r e a l i s t i c model, d a t a from t h e Peace R i v e r D i s t r i c t o f B.C. was used t o e s t i m a t e i n c o m e - r i s k f r o n t i e r s . T h e farmer chosen was m a i n l y a crop p r o -d u c e r w i t h no l i f e s t o c k a c t i v i t i e s . The f o l l o w i n g crop p r o d u c t i o n a c t i v i t i e s were c o n s i d e r e d i n t h e model: (1) Wheat grown a f t e r summer f a l l o w ; (2) Wheat grown a f t e r s t u b b l e ; (3) B a r l e y grown a f t e r summer f a l l o w ; (4) B a r l e y grown a f t e r s t u b b l e ; (5) Oats grown a f t e r summer f a l l o w ; (6) Oats grown a f t e r s t u b b l e ; (7) Rapeseed; (8) F e s c u e s e e d ; (9) A l s i k e s e e d ; (10) A l f a l f a . The main c o n s t r a i n t s c o n s i d e r e d were a r a b l e l a n d owned by the o p e r a t o r , a r a b l e l a n d a v a i l a b l e f o r r e n t i n g , c a s h c a p i t a l owned by the farm o p e r a t o r , c a s h b o r r o w i n g c a p a c i t y and f a m i l y l a b o u r a v a i l a b l e . F i g u r e 5.1 o u t l i n e s a g e n e r a l o v e r v i e w o f t h e model used. The o b j e c t i v e f u n c t i o n v a r i e d w i t h t h e method used. I t i s i n t e n d e d t o m i n i m i z e the t o t a l a b s o l u t e d e v i a t i o n (MOTAD), t o t a l v a r i a n c e (QP-VAR) o r t o t a l s e m i v a r i a n c e (QP-SEMIV). The c a p i t a l c o n t r o l rows c o n s i d e r the f l o w o f o p e r a t i n g and o v e r -head c a p i t a l i n t o the model t h r o u g h s u b m a t r i c e s D 2~^ -j and D 2~^ 2 a n c' * n i s c a p i t a l , may be used f o r v a r i o u s a c t i v i t i e s such as c o s t s o f owning l a n d ( A 2 3) r e n t i n g l a n d ( A 2 g) o r b u y i n g o t h e r v a r i a b l e i n p u t s f o r p r o d u c i n g c r o p s such as f e r t i l i z e r s , s e e d , r e n t i n g machinery and so f o r t h (A + a " 2, 8 ) . A l l o t h e r c o n t r o l rows may be i n t e r p r e t e d i n t h e same way k e e p i n g i n mind t h e l e g e n d i n F i g u r e 4 . 7 . rltSUKt b. I: s t r u c t u r e o f the Models > o H < M H M tt CO C o n s t r a i n t s Own Capital Borrowed Capital Own Land Rented Land •Family Labour Hired Labour Cropping Activities Variable Inputs R.H.S. O b j e c t i v e F u n c t i o n Measure o f r i s k which depends on method IIRPH M i n i -mize C a p i t a l C o n t r o l A + a  A 2 .3 A+ a A + a  2 . 5 A + a A a  A2,8 * 0 Land C o n t r o l D s ' 3 A+ l A3,7 * 0 Labour C o n t r o l A4,7 * 0 Croppi n g A c t i v i t y Resource Use. A + a  A5,7 * 0 Exp e c t e d Net Income Summary Row A"a A 6 ,1 A" a  A6,2 A"a  A6,3 A"a  A 6 ,4 A"a  A 6 ,5 A"a A6,6 A + a 6,7 A~ a  A6,8 i R S t r u c t u r a l Bounds m B i m B2 m B 3 m B 4 m B 5 m B6 Legend: ( S u p e r s c r i p t s , when shown, r e p r e s e n t the typ e o f non z e r o element i n t h e s u b m a t r i x . ) A i s a g e n e r a l s u b m a t r i x w i t h some non z e r o e l e m e n t s , B i s a s t r u c t u r a l upperbound, D i s a d i a g o n a l s u b m a t r i x . - 93 -A l l s u b m a t r i c e s i n the net income summary row a r e row v e c t o r s . V e c t o r s Ag •. and Ag ^ ac c o u n t f o r t h e c o s t s o f u s i n g d i f f e r e n t t y p e s o f c a p i t a l . The v e c t o r s Ag ^ (own l a n d c o s t s ) Ag ^ ( l a n d r e n t a l c o s t s ) , Ag ( f a m i l y l a b o u r c o s t s ) , A g s g ( h i r e d l a b o u r c o s t s ) , Ag ^ ( g r o s s r e t u r n s from c r o p s and Ag g ( v a r i a b l e c o s t s f o r c r o p p r o d u c t i o n ) a l l a c c o u n t f o r income and expenses f o r the d i f f e r e n t a c t i v i t i e s . N e g a t i v e c o e f f i c i e n t s i n d i c a t e c o s t s and the p o s i t i v e c o e f f i c i e n t s , r e t u r n s . The e x p e c t e d net income i s not e q u i v a l e n t t o farm p r o f i t because t he d e p r e c i a t i o n c o s t s o f f i x e d c a p i t a l and i n t e r e s t c h a r g e d on c a p i t a l o t h e r t h a n t h o s e o f l a n d and l i v e -s t o c k have n o t been i n c l u d e d . The model as s o l v e d f o r t h e MOTAD method has 38 rows, a p p r o x i -mately 28 columns and 10 s t r u c t u r a l bounds. The QP-VAR and QP-SEMIV models have 34" rows and 28 columns. . The c a s e farm c o n s i d e r e d had 370 a c r e s o f a r a b l e l a n d and the pos-s i b i l i t y e x i s t e d f o r r e n t i n g a n o t h e r 80 a c r e s . The farm o p e r a t i o n had $5,ooo a v a i l a b l e i n cash c a p i t a l p l u s a cash b o r r o w i n g c a p a c i t y e s t i m a t e d a t $25,000 per y e a r . A t o t a l o f 1500 hours o f f a m i l y l a b o u r was con-s i d e r e d t o be a v a i l a b l e i n a d d i t i o n t o 1000 hours o f h i r e d l a b o u r . The b a s i c i n p u t s used were t he l a s t e i g h t y e a r s o f r e c o r d s r e g a r d i n g y i e l d s p e r a c r e , p r i c e s and c o s t s o f t h e d i f f e r e n t p r o d u c t i o n a c t i v i t i e s c o n s i d e r e d . T h i s d a t a was used t o c a l c u l a t e t h e v a r i a n c e s , semi-v a r i a n c e s o r a b s o l u t e d e v i a t i o n s o f the a c t i v i t y r e t u r n s t o be used on the QP-VAR, QP-SEMIV and MOTAD methods r e s p e c t i v e l y . - 9 4 -A l l methods were s o l v e d f o r m i n i m i z i n g r i s k under t h e same c o n s t r a i n t s a n d a t the same l e v e l s o f net income. The maximum net income a t t a i n a b l e from t h i s farm c o n s i d e r i n g the e x p e c t e d r e t u r n o f the a c t i v i t i e s was a p p r o x i m a t e l y $6,000 per y e a r . The model was s o l v e d t o m i n i m i z e r i s k a t $6,000, $5,000 and $4,000 e x p e c t e d n e t income per y e a r . T a b l e 5.1 shows the minimum r i s k l e v e l s as e s t i m a t e d by the methods a t t h r e e l e v e l s o f e x p e c t e d n e t income. TABLE 5.1: R i s k L e v e l s ( E x p r e s s e d as the Square Root o f the S e m i - v a r i a n c e ) as P r o v i d e d by the QP-SEMIV, QP-VAR and MOTAD Model S o l u t i o n s f o r a Farm i n the Peace River D i s t r i c t o f B r i t i s h Columbia LEVELS OF NET INCOME M E T H O D U S E D $6,000 $5,000 $4,000 QP-SEMIV 2315.9 1380.9 748.3 QP-VAR 4053.0 2666.9 1340.4 MOTAD 4114.1 2708.2 1485.8 The t o t a l s e m i v a r i a n c e was c a l c u l a t e d ex p o s t from the s o l u t i o n s p r o v i -ded by t h e QP-VAR and MOTAD methods, F i g u r e 5.2 shows t h e e f f i c i e n t i n c o m e - r i s k f r o n t i e r as e s t i m a t e d by the t h r e e methods. As i n T a b l e 5,1, r i s k i s e x p r e s s e d as the square r o o t o f the s e m i v a r i a n c e . - 9 5 -Income ($) 6,000 The Income-Risk F r o n t i e r as E s t i m a t e d by QP-SEMIV. QP-VAR and MOTAD Methods f o r a Farm i n the Peace R i v e r D i s t r i c t o f QP-VAR estimate MOTAD estimate 5,000 4,000 I RISK (V semivar) A t a l l l e v e l s o f income t h e p l a n s recommended by the QP-SEMIV method i m p l i e d t he s m a l l e s t l e v e l s o f r i s k (measured as the square r o o t o f t h e s e m i v a r i a n c e ) . The MOTAD method always p r o v i d e d s o l u t i o n s s l i g h t l y worse than t he QP-VAR method. T a b l e 5.2 shows t h e l e v e l o f p r o d u c t i o n a c t i v i t i e s as recommended i n t h e s o l u t i o n s p r o v i d e d by the methods a t s i x thousand d o l l a r s e x p e c t e d n e t income l e v e l . -'96 -TABLE 5,2 L e v e l s o f A c t i v i t i e s as Proposed by MOTAD, QP-SEMIV and QP-VAR  Models f o r a T y p i c a l Small Farm i n the Peace R i v e r Area o f  B r i t i s h Columbia (Net Income = $6000) ACTIVITIES PROPOSED M O D E L U S E D A c r e s o f WAF A c r e s o f WAS A c r e s o f BAF A c r e s o f BAS A c r e s o f OAS A c r e s o f FES A c r e s o f ALF MOTAD 77 - 5 - 164 - 61 QP-SEMIV - 42 82 34 - - 91 QP-VAR 62 75 20 - - 57 58 WAF = Wheat A f t e r F a l l o w ; WAS = Wheat A f t e r S t u b l e ; BAF = B a r l e y A f t e r F a l l o w ; BAS - B a r l e y A f t e r S t u b b l e ; OAS = Oat A f t e r S t u b b l e ; FFS = Fe s c u e ; ALF = A l f a l f a . As may be seen i n T a b l e 5..2., the l e v e l o f a c t i v i t i e s proposed by each o f the t h r e e methods d i f f e r s s u b s t a n t i a l l y . Only t h r e e p r o d u c t i o n a c t i v i t i e s do not appear i n any s o l u t i o n , o a t s a f t e r f a l l o w , r a p e s e e d and a l s i k e . A l l o t h e r a c t i v i t i e s a r e a t l e a s t i n one o f the s o l u t i o n s and o n l y two a c t i v i t i e s a r e i n the t h r e e s o l u t i o n s , b a r l e y a f t e r f a l l o w and a l f a l f a . * Thus, t h e t h r e e methods c o n s i d e r e d i n t h i s s t u d y have p r o v i d e d v e r y d i f f e r e n t s o l u t i o n s when th e y have been a p p l i e d t o a Peace R i v e r d i s t r i c t farm. I t i s i n t e r e s t i n g t o note t h a t most c o r r e l a t i o n c o e f f i c i e n t s among the a c t i v i t y r e t u r n s a r e r e l a t i v e l y l a r g e . Only the r e t u r n s o f f e s c u e and a l s i k e appear t o be weakly c o r r e l a t e d w i t h any o t h e r a c t i v i t y r e t u r n and i n - 97 -some c a s e s h a v i n g n e g a t i v e s i g n (see T a b l e A.11 A p p e n d i x ) . The mean c o r r e l a t i o n c o e f f i c i e n t among a l l o t h e r a c t i v i t i e s i s a p p r o x i m a t e l y 0.66. T h i s f a c t would e x p l a i n why t h e QP-VAR and MOTAD s o l u t i o n s a r e so d i f f e r e n t . The e i g h t y e a r g r o s s r e t u r n r e c o r d s s u g g e s t t h a t most ac-t i v i t y r e t u r n s a r e skewed d i s t r i b u t e d , which would e x p l a i n the a p p a r e n t l y b e t t e r p l a n s p r o v i d e d by the QP-SEMIV method. Thus, t h i s example i l l u s t r a t e s the magnitude o f the e r r o r which may o c c u r i f t h e wrong method i s used. C o n s i d e r i n g t h e r e l a t i v e l y h i g h degree o f c o r r e l a t i o n o f the a c t i v i t i e s and the f a c t t h a t t h e data appears t o be no n - n o r m a l l y d i s t r i b u t e d t h e QP-SEMIV method may be c o n s i d e r e d as p r o v i d i n g the b e s t s o l u t i o n . To use t h e MOTAD method i n s t e a d o f the QP-SEMIV method i m p l i e d t h a t p l a n s g e n e r a t e d were on the average 80% more r i s k y and the QP-VAR method p r o v i d e d p l a n s which were 70% more r i s k y than t h e QP-SEMIV method. T h i s example i s u s e f u l i n i l l u s t r a t i n g t h e importance o f c h o o s i n g the a p p r o p r i a t e method i n farm p l a n n i n g under un-c e r t a i n t y c o n d i t i o n s . Thus, the phase o f d e c i d i n g which method s h o u l d be used i s c r u c i a l f o r the r e s u l t s (as shown by the i m p o r t a n t d i f f e r e n c e s i n the r e s u l t s p r o v i d e d by the methods i n t h i s example), and hence, i t i s wo r t h w h i l e t o dev o t e a g r e a t d e a l o f r e s o u r c e s and time i n o r d e r to be a a b l e t o s e l e c t t h e most a p p r o p r i a t e method a c c o r d i n g t o the s p e c i f i c n a t u r e o f the problem. - 98 -CHAPTER VI SUMMARY, CONCLUSIONS AND RECOMENDATION FOR FURTHER STUDIES T h i s c h a p t e r p r e s e n t s a r e v i e w o f the s t u d y and p r i n c i p a l f i n d i n g s . G iven t h e l i m i t a t i o n s o f the approach f u r t h e r r e s e a r c h i s proposed i n o r d e r t o be a b l e t o a s s e s s the e m p i r i c a l r e l e v a n c e o f t h e s e f i n d i n g s . 6.1 Summary and C o n c l u s i o n s The b a s i c o b j e c t i v e o f t h i s s t u d y was t o e v a l u a t e the p e r f o r -mance o f t h r e e methods used i n farm p l a n n i n g under u n c e r t a i n t y , namely, the QP-VAR, MOTAD and S e m i v a r i a n c e methods. The work was d e v e l o p e d i n two p a r t s , a t h e o r e t i c a l o r c o n c e p t u a l s e c t i o n and a s e t o f e x p e r i m e n t s c o m p r i s i n g an e m p i r i c a l s e c t i o n . The t h e o r e t i c a l s t u d y was c o n c e r n e d w i t h the c h a r a c t e r i s t i c s o f the methods under t h e assumption t h a t the p o p u l a t i o n d i s t r i b u t i o n o f a c t i v i t y r e t u r n s was known. The main c r i t e r i o n used t o e v a l u a t e t h e methods was t h e i r a b i l i t y t o p r o v i d e the i n f o r -mation r e q u i r e d by the d e c i s i o n maker t o maximize h i s e x p e c t e d u t i l i t y . T h a t i s , t o p r o v i d e an o r d i n a l c l a s s i f i c a t i o n o f a l t e r n a t i v e p l a n s c o n s i s t e n t w i t h the l e v e l o f e x p e c t e d u t i l i t y which each p l a n i m p l i e s to a d e c i s i o n maker. To meet t h i s r e q u i r e m e n t , a n e c e s s a r y c o n d i t i o n was t h a t t h e method s h o u l d p r o v i d e a s e t o f e f f i c i e n t p l a n s , i , e , , an i n c o m e - r i s k f r o n t i e r which would e n a b l e a d e c i s i o n maker t o choose the p l a n which maximizes h i s e x p e c t e d u t i l i t y . - 39 -The e m p i r i c a l work t e s t e d the methods' a b i l i t y t o p r o v i d e income-r i s k f r o n t i e r s as a p p l i e d u s i n g sample d a t a o f l i m i t e d s i z e r a t h e r than complete f r e q u e n c y d i s t r i b u t i o n p.f a c t i v i t y r e t u r n s . Under a s i t u a t i o n o f p e r f e c t knowledge, the t h e o r e t i c a l a n a l y s i s i n d i c a t e d which methods p r o v i d e a p p r o p r i a t e r e p r e s e n t a t i o n s o f the i n c o m e - r i s k f r o n t i e r . In the e m p i r i c a l work s e c t i o n , the i n c o m e - r i s k f r o n t i e r s p r o v i d e d by such methods as a p p l i e d t o complete f r e q u e n c y d i s t r i b u t i o n s d a t a were con-s i d e r e d t h e " t r u e " ones. The e s t i m a t e s o f the i n c o m e - r i s k f r o n t i e r ob-t a i n e d u s i n g sample d a t a were compared t o the " t r u e " i n c o m e - r i s k f r o n t i e r i n o r d e r t o measure b i a s and d i s p e r s i o n o f the e s t i m a t e s . Given the r e s u l t s , c o n c l u s i o n s were drawn r e g a r d i n g t h e r e l a t i v e e f f i c i e n c y o f the methods as e s t i m a t o r s o f t h e t r u e i n c o m e - r i s k f r o n t i e r . A g e n e r a l c o n c l u s i o n which may be drawn from t h i s s t u d y i s . t h a t u n l e s s a c t i v i t y r e t u r n s are assumed t o be n o r m a l l y d i s t r i b u t e d (which may be an u n r e a l i s t i c assumption) p l a n n i n g under u n c e r t a i n t y needs t o c o n s i d e r the n a t u r e o f the u t i l i t y f u n c t i o n o f the d e c i s i o n maker. T h i s i n c r e a s e s t h e c o m p l e x i t y t o t h e s e s t u d i e s g i v e n the d i f f i c u l t i e s i n -v o l v e d i n knowing the u t i l i t y f u n c t i o n s o f d e c i s i o n makers. Thus, t h e r e i s not an o p t i m a l method t o be used i n a l l c a s e s and hence t h e r e i s n o t an easy r u l e t o be a p p l i e d i n farm p l a n n i n g under u n c e r t a i n t y . More s p e c i f i c c o n c l u s i o n s r e g a r d i n g the t h r e e methods a r e : - 100 -1. I f the d e c i s i o n maker's u t i l i t y f u n c t i o n i s q u a d r a t i c the QP-VAR method may be used, i r r e s p e c t i v e o f the d i s t r i b u t i o n o f a c t i v i t y r e t u r n s . 2. The QP-SEMIV method i s proposed as the most s u i t a b l e method i f the f o l l o w i n g c o n d i t i o n s a re met: a. The d e c i s i o n maker's u t i l i t y f u n c t i o n i s not q u a d r a t i c nor l i n e a r b. The f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s i s non -nor m a l , b u t moments o f o r d e r h i g h e r than t h e skewness moment a r e n e g l i g i b l e . 3. The MOTAD method i s i n g e n e r a l not recommended as b e i n g u s e f u l because i t i s b i a s e d and l e s s e f f i c i e n t t h a n t h e QP-SEMIV method. The o n l y s i t u a t i o n where t he MOTAD method may be used i s when a c t i v i t y r e t u r n s are n o r m a l l y d i s t r i b u t e d and the degree o f c o r r e l a t i o n among a c t i v i t y r e t u r n s i s l o w / 4. No o n e o f the t h r e e methods was found t o be a p p r o p r i a t e i f the f o l l o w i n g c o n d i t i o n s o c c u r s i m u l t a n e o u s l y : a. A d e c i s i o n maker's u t i l i t y f u n c t i o n i s n o n - q u a d r a t i c b. F o u r t h and h i g h e r o r d e r d e r i v a t i v e s o f the u t i l i t y f u n c t i o n do not v a n i s h c. The f r e q u e n c y d i s t r i b u t i o n o f the a c t i v i t y r e t u r n s i s non-normal and moments h i g h e r than skewness moments a r e not n e g l i g i b l e . 5. As i m p o r t a n t as the a c t u a l r e s u l t s o b t a i n e d i s the g e n e r a l p r o c e d u r e used t o e v a l u a t e t h e d i f f e r e n t methods. To e x p l a i n , t h e r e i s 101 -a r e c o g n i t i o n t h a t t h e most i m p o r t a n t f e a t u r e o f a method i s i t s p e r f o r -mance when a p p l i e d u s i n g sample d a t a r a t h e r than complete f r e q u e n c y d i s t r i b u t i o n f o r i n e m p i r i c a l work,the d a t a s o u r c e i s n o r m a l l y a r e l a -t i v e l y s m a l l sample drawn from the p o p u l a t i o n o f a c t i v i t y r e t u r n s . Indeed, t h e f a c t t h a t a method p r o v i d e s an a p p r o p r i a t e r e p r e s e n t a t i o n o f t h e income-r i s k f r o n t i e r under a p e r f e c t knowledge s i t u a t i o n i s not a s u f f i c i e n t nor a n e c e s s a r y c o n d i t i o n f o r such a method t o g e n e r a t e e q u a l l y a p p r o p r i a t e e s t i m a t e s when u s i n g s m a l l sample d a t a , T h i s view o f the problem r e q u i r e d an o b j e c t i v e e v a l u a t i o n o f the methods c o n s i d e r e d . The c r i t e r i o n used was th e c o n c e p t o f e f f i c i e n c y o f e s t i m a t o r s as d e f i n e d i n the s t a t i s t i c s s e n s e . B i a s and d i s p e r s i o n o f t h e e s t i m a t e s were used t o judge the s o l u t i o n s p r o v i d e d by t h e d i f f e r e n t methods. U s i n g t h e s e c o n c e p t s i t becomes c l e a r why t h e a p p r o p r i a t e n e s s o f t h e i n c o m e - r i s k f r o n t i e r d e r i v e d when u s i n g complete f r e q u e n c y d i s t r i b u t i o n d a t a i s n o t s u f f i c i e n t nor n e c e s s a r y f o r e f f i c i e n c y as e s t i m a t o r s o f t h e i n c o m e - r i s k f r o n t i e r . J u s t as the s t a n d a r d e r r o r measure i s a b i a s e d e s t i m a t o r o f t h e s t a n d a r d d e v i a t i o n (E (<r) £ 0 " ) , a method may p r o v i d e b i a s e d e s t i m a t e s o f t h e income r i s k f r o n t i e r d e s p i t e p r o v i d i n g a " t r u e " o n e when a p p l i e d t o t h e complete f r e q u e n c y d i s t r i b u t i o n . S i m i l a r l y , a method may be s l i g h t l y b i a s e d but i f i t has a sm a l l d i s p e r s i o n i t may be more e f f i c i e n t than an u n b i a s e d method which p r o v i d e s h i g h l y d i s p e r s e d e s t i m a t e s . Thus, the s t u d y p r o v i d e s a r e s e a r c h p r o c e d u r e t o t e s t d i f f e r e n t methods which may be used i n farm p l a n n i n g under u n c e r t a i n t y . The r e s e a r c h p r o c e d u r e used i s based on the r e c o g n i t i o n t h a t , c o n c l u s i o n s from a n a l y t i c a l d i s c u s s i o n s based on the assumption o f f u l l knowledge o f the f r e q u e n c y d i s t r i b u t i o n o f the a c t i v i t y r e t u r n s a r e not d i r e c t l y a p p l i c a b l e i n t h e p r a c t i c a l p r o c e s s o f d e c i d i n g which method s h o u l d be used. However, - 102 -the a n a l y t i c a l r e s u l t s were h i g h l y i m p o r t a n t i n . d e s i g n i n g t h e e x p e r i m e n t s t o t e s t the d i f f e r e n t methods and t h e m a j o r i t y o f t h e s t e p s f o l l o w e d i n c h a p t e r t h r e e were based on t h e c o n c l u s i o n s o b t a i n e d from the t h e o r e t i c a l study d e v e l o p e d i n the p r e c e d i n g c h a p t e r . 6.2 Recommendations f o r f u r t h e r s t u d i e s I t has been shown t h a t t h e e f f i c i e n c y o f the d i f f e r e n t methods used i n farm p l a n n i n g under u n c e r t a i n t y depends., on the n a t u r e o f the u t i l i t y f u n c t i o n and on t h e f r e q u e n c y d i s t r i b u t i o n o f a c t i v i t y r e t u r n s . Hence, i t would be a d v i s a b l e t o i n v e s t i g a t e t h e p r e s e n c e o f any s o r t o f r e g u l a r i t y i n the n a t u r e o f t h e u t i l i t y f u n c t i o n o f f a r m e r s as d e c i s i o n makers. In o t h e r words, i t i s i m p o r t a n t t o know whether c e r t a i n u t i l i t y f u n c t i o n s can be r u l e d o u t f o r t h e m a j o r i t y o f f a r m e r s and whether t h e r e a r e some f a m i l i e s o f u t i l i t y f u n c t i o n s which a r e p e c u l i a r t o them. I t would a l s o be i m p o r t a n t t o d e t e r m i n e i f t h e r e a r e some f r e q u e n c y d i s t r i b u t i o n s which c h a r a c t e r i z e b e t t e r t h e d i s t r i b u t i o n o f t h e g r o s s a c t i v i t y r e t u r n s . A c c o r d i n g l y , f u r t h e r s t u d i e s s h o u l d be o r i e n t a t e d m a i n l y i n the f o l l o w i n g d i r e c t i o n s : 1.- E m p i r i c a l s t u d i e s t o e s t a b l i s h whether q u a d r a t i c u t i l i t y f u n c t i o n s a r e i n d e e d unusual among f a r m e r s as t h e o r e t i c a l s t u d i e s s u g g e s t . T h i s i s i m p o r t a n t because i n such a c a s e t h e QP-SEMIV method c o u l d have w i d e r a p p l i c a t i o n s . But i f t h e q u a d r a t i c u t i l i t y f u n c t i o n o c c u r s f r e q u e n t l y among f a r m e r s , b e f o r e c h o o s i n g the method t o be used, i t would be n e c e s s a r y t o d e t e r m i n e t h e u t i l i t y f u n c t i o n o f t h e d e c i s i o n maker i n each c a s e ( u n l e s s t h e r e i s c e r t a i n t y t h a t r e t u r n s a r e n o r m a l l y d i s t r i b u t e d ) . - 103 -2. - E m p i r i c a l s t u d i e s c o u l d i n v e s t i g a t e t h e f r e q u e n c y d i s t r i b u t i o n o f g r o s s a c t i v i t y r e t u r n s f o r d i f f e r e n t c r o p s and l i v e s t o c k e n t e r p r i s e s . Two b a s i c p o i n t s need i n v e s t i g a t i o n ; In the f i r s t p l a c e , i t would be n e c e s s a r y t o e s t a b l i s h whether g r o s s a c t i v i t y r e t u r n s f o r most i m p o r t a n t c r o p s and l i v e s t o c k e n t e r p r i s e s produced have been a p p r o x i m a t e l y n o r m a l l y d i s t r i b u t e d d u r i n g the l a s t d e c a d e s . In the second p l a c e , i f the r e -t u r n s appear t o be non - n o r m a l l y d i s t r i b u t e d i t would be n e c e s s a r y t o o b t a i n some i d e a r e g a r d i n g the s i g n o f skewness o f the d i s t r i b u t i o n s and the magnitude o f h i g h e r o r d e r moments r e l a t i v e t o the skewness moment. I f t h e d i s t r i b u t i o n s appear t o be a p p r o x i m a t e l y normal t h e QP-VAR method may be used and t h e QP-SEMIV method would n o t be n e c e s s a r y . I f t h i s i s n o t t he c a s e t h e QP-SEMIV method would be u s e f u l ( p r o v i d e d n o n - q u a d r a t i c u t i l i t y f u n c t i o n s ) . 3. - A d d i t i o n a l l y , i t would be i m p o r t a n t t o s t u d y how t h e performance o f t h e s e methods i s a f f e c t e d by changes i n the sample s i z e . I t can be e x p e c t e d t h a t r e s u l t s become more a c c u r a t e as the sample s i z e i n c r e a s e s , but t h e s e improvements a r e not n e c e s s a r i l y p r o p o r t i o n a l t o the r a t e o f i n c r e a s e i n the sample s i z e . G iven t h a t t o i n c r e a s e sample s i z e has a c o s t (more i n f o r m a t i o n r e g a r d i n g g r o s s r e t u r n s o v e r time i s n e c e s s a r y ) . I t would be p o s s i b l e t o f i n d an o p t i m a l sample s i z e ( o r an o p t i m a l range) t o be used i n e m p i r i c a l s t u d i e s , 4. - I t i s a l s o n e c e s s a r y t o t e s t t he performance o f t h e s e methods i n p r e d i c i t i n g a c t u a l b e h a v i o u r o f f a r m e r s . I t i s q u i t e c l e a r t h a t t he -•104'-l i n e a r programming r e s u l t s may > be v e r y d i f f e r e n t from t he a c t u a l p l a n s which f a r m e r s ' make. I t would-be i m p o r t a n t t o s t u d y t h e p l a n s o b t a i n e d u s i n g QP-VAR and s p e c i a l l y QP-SEMIV methods, and compare t h e s e t o f a r m e r s ' a c t u a l p l a n s . A p r i o r i , i t c o u l d be e x p e c t e d t h a t t h e QP-SEMIV method would p r o v i d e t h e c l o s e s t a p p r o x i m a t i o n t o t h e f a r m e r s ' a c t u a l p r o d u c t i o n p l a n s . 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Graham, J.D. and Lopez, R., Farm P l a n n i n g Under U n c e r t a i n t y - An  a p p l i c a t i o n o f MOTAD Programming t o the Peace R i v e r B l o c k o f B.C., Department o f A g r i c u l t u r a l "Economics, UBC, Vancouver, B.C. 1976 - 106 -13. H a d l e y , G., I n t r o d u c t i o n t o P r o b a b i l i t y and S t a t i s t i c a l D e c i s i o n  T h e o r y , Holden-Day, I n c . , San F r a n c i s c o , C a l i f o r n i a , 1967 14. Hadley, G., Non L i n e a r and Dynamic Programming, A d d i s o n Wesley P u b l i s h i n g , M a s s a c h u s e t t s , 1964. 15. H a z e l ! , P.B. "A L i n e a r A l t e r n a t i v e t o Q u a d r a t i c and S e m i v a r i a n c e Programming f o r Farm P l a n n i n g Under U n c e r t a i n t y " , American  J o u r n a l o f A g r i c u l t u r a l Economics, 53: 53-62, 1971. 16. H a z e l l , P.B. and How, R.B., " O b t a i n i n g A c c e p t a b l e Farm P l a n s Under U n c e r t a i n t y " , paper s u b m i t t e d to the C o n t r i b u t e d Papers s e c t i o n o f t h e I n t e r n a t i o n a l A s s o c i a t i o n o f A g r i c u l t u r a l E c onomists a t Minsk, U.S.S.R., 1970. 17. I n t r i l i g a t o r , M., M a t h e m a t i c a l O p t i m i z a t i o n and Economic T h e o r y , P r e n t i c e - H a l l , New York, 1971. . 18. Kmenta, J . , Elements o f E c o n o m e t r i c s , M a c M i l l a n P u b l i s h i n g Co., New York, 1971. 19. Levy, H., "The R a t i o n a l e o f t h e Mean-Standard D e v i a t i o n A n a l y s i s : Comment", American Economic Review, 64: 343-441, 1974. 20. L i n , W. Dean, G. and Moore, C , "An E m p i r i c a l T e s t o f U t i l i t y v s . P r o f i t M a x i m i z a t i o n i n A g r i c u l t u r a l P r o d u c t i o n " , American J o u r n a l o f A g r i c u l t u r a l Economics, 56: 497-508, 1974. 21. Lopez, R.E., " L i n e a r and Non L i n e a r I n d i c a t o r s o f R i s k as A p p l i e d t o Farm P l a n n i n g under U n c e r t a i n t y " , U n p u b l i s h e d , Department o f A g r i c u l t u r a l Economics, U.B.C, 1976. 22. L u t t r e l l , C. and G i l b e r t , R., "Crop Y i e l d s : Random, C y c l i c a l , o r Bunchy?" American J o u r n a l o f A g r i c u l t u r a l Economics, 58: 521-531, 1976. 23. M a r k o w i t z , H., P o r t f o l i o S e l e c t i o n : E f f i c i e n t D i v e r s i f i c a t i o n o f  Investments, John W i l e y and Sons, I n c . , New York, 1959. 24. M c M i l l a n , C. and G o n z a l e s , R., Systems A n a l y s i s a Computer Approach  t o D e c i s i o n M o d e l s ? R i c h a r d I r w i n I n c . , Homewood, I l l i n o i s , 1968. 25. Mood, A. and G r a y b i l l , F. I n t r o d u c t i o n to the Theory o f S t a t i s t i c s , M c G r a w - H i l l , New York, 1963 26. P r a t t , J.W., " R i s k A v e r s i o n i n t h e Small and the L a r g e " , E c o n o m e t r i c a , 32: 122-136, 1964. 27. R a i f f a , H., D e c i s i o n A n a l y s i s , A d d i s o n - W e s l e y , Reading, M a s s a c h u s e t t s , 1968 - 107 -S c o t t , J . and Baker, C , "A P r a c t i c a l Way t o S e l e c t an Optimum Farm P l a n Under R i s k " , American J o u r n a l o f A g r i c u l t u r a l Economic?; 54: 657-660, 1972. Thomson, K., and H a z e l l , P., " R e l i a b i l i t y o f U s i n g the Mean A b s o l u t e D e v i a t i o n t o D e r i v e E f f i c i e n t E.V. Farm P l a n s . " American J o u r n a l o f A g r i c u l t u r a l Economics, 54: 503-506, 1972 T s i a n g , S . C , "The R a t i o n a l e o f the Mean-Standard D e v i a t i o n A n a l y s i s , Skewness P r e f e r e n c e and t h e Demand f o r Money", The American Economic Review, 62: 354-371, 1972. Wolf, F.L. Elements o f P r o b a b i l i t y and S t a t i s t i c s , M cGraw-Hill I n c . , New York, 1974. - 108 -APPENDIX - 109 -TABLE A.I E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the QP-VAR Method as  A p p l i e d t o F i f t e e n Samples Randomly Drawn from a N o r m a l l y  D i s t r i b u t e d P o p u l a t i o n . Low Degree o f C o r r e l a t i o n Among  A c t i v i t y R e t u r n s . Data s o u r c e used L e v e l s o f Expe c t e d Income i High ; Medium : Low Sample 1 6.0 4.0 . 2.5 Sample 2 5.8 4,0 3.1 Sample 3 6.1 4.1 2.7 Sample 4 6.0 4.2 2.7 Sample 5 5.7 3.9 3.2 Sample 6 6.6 4.7 3.1 Sample 7 7.1 4.6 2.9 Sample 8 5.8 4,0 2.6 Sample 9 5.8 4.0 2.6 Sample 10 6.2 4.4 2.8 Sample 11 6.1 4.3 2.8 Sample 12 4.6 3.3 2.2 Sample 13 5.9 4.2 2.6 Sample 14 5.9 4,2 2.8 Sample 15 5.3 3.7 2.4 Mean E s t i m a t e s 5.9 4,1 2.7 V a r i a n c e 0,32 0,13 0.07 * S t a n d a r d D e v i a t i o n o f T o t a l Income - 110 -TABLE A.2 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the MOTAD  Method as A p p l i e d t o F i f t e e n Samples Randomly  Drawn from Normal P o p u l a t i o n w i t h Low Degree o f  C o r r e l a t i o n Data s o u r c e ' i i ^ r ^ j r L e v e l s o f E x p e c t e d Income used ' ^ : High ; Medium : Low Sample 1 6.5 4,3 2.8 Sample 2 6,0 4,1 2.6 Sample 3 6.1 4.2 2.8 Sample 4 5.9 ' 4.0 2.6 Sample 5 5.9 4,1 2.7 Sample 6 6.7 4.6 3.3 Sample 7 6.7 4,6 3.0 Sample 8 5.1 3.2 2.2 Sample 9 6.2 4.3 2.9 Sample 10 4.8 3,4 2.2 Sample 11 6.0 4,1 2.7 Sample 12 6.9 4.7 3.2 Sample 13 5.8 3.9 2.4 Sample 14 6.0 4.1 2.8 Sample 15 4.9 3.6 2.4 Mean E s t i m a t e s 6,0 V a r i a n c e 0.38 0,18 0.10 - fii -T A B L E A- 3 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n q the QP-VAR  Method as A p p l i e d t o F i f t e e n Samples Randomly  Drawn from a Normal P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n Data Source Used ; L e v e l s o f Ex p e c t e d Income ; High : Medium : Low Sample 1 6.5 5.2 2.6 Sample 2 6,6 5.0 3,3 Sample 3 6.4 5.1 3.9 Sample 4 5.8 4,4 3.0 Sample 5 6.2 4.8 3.5 Sample 6 4.5 3.5 2.3 Sample 7 6,7 4,6 2.9 Sample 8 7.6 5,5 ' 3.5 Sample 9 5.4 3.9 2.4 Sample 10 5.1 3,4 2.1 Sample 11 10.2 6.3 4,2 Sample 12 6.9 5,1 3.9 Sample 13 7.1 5.3 3.3 Sample 14 6.6 4.8 3.2 Sample 15 6.8 5.1 3.3 Mean E s t i m a t e s 6.5 4,8 3.2 V a r i a n c e 1.06 0,58 0,36 * S t a n d a r d D e v i a t i o n o f T o t a l Income, 112 -E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the MOTAD Method as A p p l i e d  t o F i f t e e n Samples Randomly Drawn from a Normal P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n Data Source Used L e v e l s o f Ex p e c t e d Income . High ?. Medium ; Low Sample 1 10.5 9.3 ' 7.4 Sample 2 7.1 5.0 3.1 Sample 3 8.4 6.4 4.1 Sample 4 7.7 5.9 4.0 Sample 5 7.3 5,3 3.6 Sample 6 6.6 5.0 3.3 Sample 7 8.7 6.5 4.5 Sample 8 7.0 5,0 3.3 Sample 9 7.6 5.5 3.5 Sample 10 5.7 3.9 2.4 Sample 11 7.5 5.7 3.9 Sample 12 7.9 6.0 4.2 Sample 13 7.5 5.8 3.9 Sample 14 6.6 5.0 3.2 Sample 15 7.7 6.2 4.1 Mean E s t i m a t e s 7.6 5,8 3.9 V a r i a n c e * S t a n d a r d D e v i a t i o n o f T o t a l Income, - 1T3 :'-TABLE A.5 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the QP-SEMIV Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn  from a Gamma P o p u l a t i o n w i t h Low Degree o f C o r r e l a t i o n Data Source L e v e l s o f E x p e c t e d Income Used H i g h ; Medium Low Sample 1 44,2 23.1 15.0 Sample 2 34.7 23,9 15.6 Sample 3 37.2 27,3 18.8 Sample 4 56.7 32.1 19.6 Sample 5 43.9 29.0 . 19.1 Sample 6 29.1 25.1 18.3 Sample 7 45.5 24,3 14.6 Sample 8 38.1 26.8 19.3 Sample 9 47.3 25.6 17.4 Sample 10 49.7 26.5 19.1 Sample 11 50.3 30.8 21.3 Sample 12 58.4 31 .3 29.2 Sample 13 57.8 30.2 15.4 Sample 14 59.2 35.6 28.6 Sample 15 42.1 37.1 22.2 Mean E s t i m a t e s 46.3 29.1 19.6 V a r i a n c e 88,6 22.1 19.4 * Square r o o t o f s e m i v a r i a n c e o f t o t a l income. - 114 -TABLE A.6 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the QP-VAR  Method A p p l i e d t o F i f t e e n Samples Randomly Drawn  From a Gamma P o p u l a t i o n w i t h Low Degree o f C o r r e l a t i o n Data Source Used L e v e l s o f Expected Income ! H i g h < ; Medium : Low Sample 1 54.5 31,9 25.3 Sample 2 62.9 41,9 30.2 Sample 3 10.8 49.8 29.3 Sample 4 56.5 32,2 16.2 Sample 5 56.8 35,1 28.6 Sample 6 55,7 36.2 24.8 Sample 7 58.2 33.1 24.7 Sample 8 55.4 39,7 23.7 Sample 9 73.2 4.15 24.4 Sample 10 78.6 48.9 35.7 Sample 11 59.6 38.7 22.4 Sample 12 66.7 37.8 24.9 Sample 13 73.7 51.1 34.7 Sample 14 60.9 39.3 29.8 Sample 15 62.8 38.6 28.7 Mean E s t i m a t e s 64.4 39.7 27.1 V a r i a n c e 102,0 37,2 24.0 * Square Root o f S e m i v a r i a n c e o f the T o t a l Income, - 115 -TABLE A.7 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the MOTAD Method A p p l i e d  to F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h  Low Degree o f C o r r e l a t i o n Data o u r c e Used L e v e l s o f Expected Income . High ; Medium ; Low Sample 1 77.5 47,7 33,4 Sample 2 66.4 44.0 28.4 Sample 3 44,0 32.0 18.5 Sample 4 94.0 52,3 32.9 Sample 5 66.2 42.5 28.1 Sample 6 94.3 58.0 42.1 Sample 7 81.2 54.4 28.7 Sample 8 75.6 51,6 29.1 Sample 9 68.0 54.7 37.2 Sample 10 69.4 45,9 27.6 Sample 11 61.2 35.2 19,9 Sample 12 69.7 49.1 33.6 Sample 13 47.1 29.0 19.4 Sample 14 .56.4 40.6 24.6 Sample 15 62.8 45.8 26.5 Mean E s t i m a t e s 68.7 45,7 29.9 V a r i a n c e 207,4 68,9 42,2 * Square Root o f S e m i v a r i a n c e o f the t o t a l income. - 116 -TABLE A.9 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g the QP-VAR Method  A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma  P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n Data Source Used : L e v e l s o f Exp e c t e d Income Hig h ; Medium ; Low Sample 1 62.1 44,5 35.0 Sample 2 116.2 85,4 59.8 Sample 3 66.1 51.9 40.1 Sample 4 71.0 45,9 33.7 Sample 5 67.8 49.0 35.3 Sample 6 69.3 49,5 35.9 Sample 7 52.6 37.2 29.8 Sample 3 108.4 75,2 49.8 Sample 9 89.9 54.8 41.4 Sample 10 90.6 62.8 47.0 Sample 11 79.0 52.7 38.0 Sample 12 60.1 45.8 32.4 Sample 13 78.6 49.1 38.8 Sample 14 61.2 40.7 29.2 Sample 15 67.8 50.3 38.0 Mean E s t i m a t e s 76,1 52,9 38,7 V a r i a n c e 327,6 136.8 65.6 * Square Root o f S e m i v a r i a n c e o f T o t a l Income, - U7 -TABLE A.10 E s t i m a t e s o f R i s k * as O b t a i n e d U s i n g t h e MOTAD Method A p p l i e d t o F i f t e e n Samples Randomly Drawn from a Gamma P o p u l a t i o n w i t h High Degree o f C o r r e l a t i o n , v" «• •— Data Source L e v e l s o f Expected Income Used : High ; Medium : Low Sample 1 120.2 84,3 59.0 Sample 2 65.2 39,3 30.1 Sample 3 92.7 63.0 45.2 Sample 4 60.6 40.9 35.7 Sample 5 102,4 73.5 47.8 Sample 6 84.1 59,3 44.5 Sample 7 123.0 70.5 57.8 Sample 8 112.1 66,3 42.5 Sample 9 92.5 83,5 41.0 Sample 10 81.7 53.7 32.3 Sample 11 56.7 36.7 32.4 Sample 12 121.4 88.6 69.3 Sample 13 101.6 74.0 51 .1 Sample 14 64.0 45,0 32.6 Sample 15 69.1 49.6 28.6 Mean E s t i m a t e s 89.8 60,2 43,3 V a r i a n c e 561,7 259.2 144.0 Square r o o t s e m i v a r i a n c e o f t o t a l income, TABLE A.11 V a r i a n c e - C o v a r i a n c e M a t r i x o f the 8 Year A c t i v i t y  R e turn Data C o r r e s p o n d i n g t o a Case Farm i n the Peace R i v e r D i s t r i c t o f B r i t i s h Columbia - 118 -A c t i v i t i e s : WAFC ; BAFC : OAFC ; RAFC ; FESC : ALFALFA : ALSIKE WAFC 981.7 806 564.2 502,9 -443 244.4 43.2 BAFC 1276,6 774.0 833.4 61.2 443 1 27.4 OAFC 681.2 517.1 -167.8 219.2 220.1 RAFC 629 10.7 351.1 3.8 FESC 1836.1 497.1 -361.3 ALFALFA 400 -147.2 ALSIKE 235 

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