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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 . , U n i v e r s i t y 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 A g r i c u l t u r a l  Economics)  We accept t h i s t h e s i s as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA July,  1977  Co) Ramon Eugenio Lopez  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  agree  fulfilment  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  this  thesis  f o r s c h o l a r l y purposes may be granted by the Head of my Department  of  this  thesis for  It  financial  The  gain s h a l l not  A g r i c u l t u r a l Fnnnnnfnr'.s  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  Serrtembp.-p  6, 1977.  or  i s understood that copying or p u b l i c a t i o n  written permission.  Department o f  that  reference and study.  t h a t p e r m i s s i o n for e x t e n s i v e copying o f  by h i s r e p r e s e n t a t i v e s .  for  be allowed without my  - 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 t h e e f f i c i e n c y 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 . c o n s i d e r e d v/as t h e QP-VAR method w h i c h m i n i m i z e s  of  The f i r s t method  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 algorithm.  The second method i s t h e MOTAD method w h i c h 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 Semi v a r i a n c e method w h i c h m i n i m i z e s  The t h i r d method i s  t h e 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 e l e m e n t s  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 m a g n i t u d e o f t h e b i a s e s and t h e 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  income-risk  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 prising a theoretical  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  comtheore-  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 o f t h e a s s u m p t i o n s Furthermore, sed.  u n d e r l y i n g t h e use of such m e a s u r e s .  the p l a u s i b i l i t y of these assumptions  was t h o r o u g h l y d i s c u s -  U s i n g t h e c o n c l u s i o n s drawn from t h e t h e o r e t i c a l s t u d y , a s e t o f  experiments  ( 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 h e e f f i c i e n c y  o f t h e 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 . 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 p e r f o r m a n c e  The purpose o f  o f t h e methods when a p p l i e d  u s i n g sample d a 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 distributions of a c t i v i t y returns.  frequency  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 t h e o t h e r w i t h low d e g r e e s 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 ) 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 , t h e 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 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 s were d e t e r m i n e d a p p l y i n g t h e a p p r o p r i a t e method i n each c a s e .  Estimates o f the income-risk 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 t h e 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 bias.  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 as p r o v i d e d by each  method was a l s o compared. the  I f two methods were u n b i a s e d , t h e method w i t h  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 c h o o s e t h e  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 t h e n a t u r e o f t h e 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 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 returns.  However, t h e QP-SEMIV method a p p e a r s 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 t h e QP-VAR and MOTAD methods.  - iv  -  TABLE OF CONTENTS Page LIST OF TABLES  vi  LIST OF FIGURES  ix  CHAPTER I  1  1.1 1.2 1.3 1.4 1.5  INTRODUCTION The Problem Objectives Important h y p o t h e s e s t o be t e s t e d Research Procedure O r g a n i z a t i o n o f the study  CHAPTER I I THEORETICAL REMARKS 2.1 2.2 2.3 2.4 2.5  The Income-Risk E f f i c i e n t f r o n t i e r The E x p e c t e d U t i l i t y F u n c t i o n Risk Aversion 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 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 2.5.1 2.5.2  Q u a d r a t i c Programming: The Variance Approach The S e m i v a r i a n c e A p p r o a c h 2.5.2.1 2.5.2.2  2.5.3 2.6  8 8 12 19 21 25 25 31  The O r d i n a l C l a s s i f i c a t i o n of t h e Expected U t i l i t y Conclusions regarding the  37  S e m i v a r i a n c e Method  39  The MOTAD model  Conclusion  CHAPTER I I I THE EMPIRICAL MODEL 3.1 3.2 3.3 3.4  1 2 4 5 6  General Overview o f the Research Procedure Generation o f Populations The S a m p l i n g P r o c e s s S o l u t i o n s o f t h e Models 3.4.1 D e s c r i p t i o n o f t h e G e n e r a l Model 3.4.2 The " T r u e " Income-Risk F r o n t i e r 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 3.4.4 Analysis of the Solutions  40 43 45 45 49 52 55 55 58 59 62  -  V  -  Page 3.5  3.6  The OP-SEMIV Method as S u b s t i t u t e o f t h e Income Semi v a r i a n c e Method  64  Summary  65  CHAPTER IV THE RESULTS 4.1 T h e 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 Returns  67 67  4.2  T h e Normal Case 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 Returns  71  4.3  Gamma D i s t r i b u t i o n a n d Lew Degree o f C o r r e l a t i o n Among A c t i v i t y Returns  75  4.4  Gamma D i s t r i b u t i o n a n d 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 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 4.6.2 4.6.3 4.6.4  86 87 88 88  4.7 CHAPTER V  Hypothesis Hypothesis Hypothesis Hypothesis  I 2 3 4  Summary  89  A CASE STUDY FARM  CHAPTER VI SUMMARY, CONCLUSIONS AND RECOMMENDATION RESEARCH 6.1  Summary and c o n c l u s i o n s  6.2  Recommendations f o r F u r t h e r S t u d i e s  91 FOR FURTHER  98 99 102  REFERENCES  105  APPENDIX  108  — vi -  L I S T OF TABLES 3.1  Page  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 t h e P o p u l a t i o n s Generated  50  3.2  Variance-Covariance Matrix o f the A c t i v i t y Returns Corresponding t o the D i f f e r e n t Populations Generated  51  3.3  Semivariance-Cosemivariance Matrix o f the A c t i v i t y R e t u r n s 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  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 R e t u r n s C a l c u l a t e d 'from a Sample drawn from a Normal Popul a t i o n ( I I ) : An Example  54  Semivariance-Cosemivariance, M a t r i x and Mean A c t i v i t y R e t u r n s C a l c u l a t e d f r o m a Sample drawn from a Gamma I Population  55  The MOTAD model as A p p l i e d t o a Sample O b t a i n e d a Normal I I P o p u l a t i o n  61  3.4  3.5  3.6  from  4.1  R i s k as E s t i m a t e d by QP-VAR and MOTAD Methods and t h e True Population Values f o r Three Levels o f Expected 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 t h e MOTAD and QP-VAR E s t i m a t e 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  69  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 T r 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  R i s k as E s t i m a t e d by QP-VAR and MOTAD Methods and t h e P o p u l a t i o n V a l u e s 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. 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  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 R i s k a t T h r e e 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  73  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 t h e MOTAD Methods as Compared t o t h e T r 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 , High Degree o f C o r r e l a t i o n  74  4.3  4.4  4.5  4.6  - vii Page  4.7  4.8  4.9  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 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 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 Low Degree of Correlation  76  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 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 Low Degree o f C o r r e l a t i o n  77  Range o f t h e R i s k L e v e l as E s t i m a t e d by QP-SEMIV QPVAR, and MOTAD Methods as Compared t o t h e T r 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 s , Low Degree o f C o r r e l a t i o n  78  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 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 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 of Correlation  79  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 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 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 t h e T r 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 P o s t w i t h t h e 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 84  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 t h e Semiv a r i a n c e ) as P r o v i d e d by t h e QP-SEMIV, QP-VAR and MOTAD Model S o l u t i o n s f o r a Farm i n t h e Peace R i v e r D i s t r i c t of B r i t i s h Columbia  93  L e v e l s o f A c t i v i t i e s as P r o p o s e d by MOTAD, QP-SEMIV and QP-VAR Models f o r a T y p i c a l Small Farm i n t h e Peace R i v e r A r e a o f B r i t i s h C o l u m b i a ( N e t Income $6,0000)  qg  5.2  A.l  A.2  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 t h e 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  109  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 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  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 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]  - viii -  Page A.4  A.5  A.6  A.7  A.8  A.9  A.10  A..11  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 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  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 U s i n g t h e 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  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 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  ^^  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 Low Degree o f Co r r e l a t i o n  115  E s t i m a t e s o f R i s k a s O b t a i n e d U s i n g t h e 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  E s t i m a t e s o f R i s k a s O b t a i n e d U s i n g t h e 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  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  11.8  Variance - Covariance Matrix o f 5 o f the 8 Year A c t i v i t y R e t u r n s Date C o r r e s p o n d i n g t o a Case 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  <  - ix -  L I S T OF FIGURES  2.1  Income I s o - R i s k C u r v e s 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 t h e A c t i v i t y R e t u r n s  2.2  Income-Risk E x p a n s i o n L i n e f o r a Two  Page 9  Activities  Situation  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 Risk Averse 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 C u r v e  15  2.6  U t i l i t y as a F u n c t i o n o f t h e 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 t h e O p t i m a l L e v e l i n an Income-Risk  2.10 3.1 -3.2 3.3 4.1 4.2  5.1 5.2  Frontier  18  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  An O v e r v i e w o f t h e R e s e a r c h P r o c e d u r e  48  B i a s i n t h e Income-Risk F r o n t i e r E s t i m a t e  63  A D i s p e r s i o n Comparison Between t h e E s t i m a t e s o f Two Methods 63 The T r 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 IncomeR 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 The T r 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 IncomeR 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 , H i g h Degree o f C o r r e l a t i o n 80 S t r u c t u r e o f t h e Model The 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 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 of B r i t i s h Columbia  90  95  ACKNOWLEDGEMENTS  I would l i k e t o t h a n k 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 w h i c h l e d t o s u b s t a n t i a l improvements i n t h e final  product.  - 1 -  CHAPTER I INTRODUCTION  1.1  The P r o b l e m 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 e c o n o m i c 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 t h e 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 m i g h t be m e n t i o n e d , a s f a c t o r s w h i c h c o n t r i b u t e t o e x p l a i n t h e 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 t o c o n s i d e r t h e unc 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 w h i c h use o n l y d e t e r m i n i s t i c t o o l s have cons 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 being d i r e c t e d to 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, 1 6 ) . The method w h i c h m i n i m i z e s t o a minimum e x p e c t e d  the variance of a c t i v i t y returns subject  income l e v e l u s i n g a q u a d r a t i c programming  r i t h m ( t h e QP-VAR method) and t h e method w h i c h m i n i m i z e s  algo-  t h e 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 t h e S e m i v a r i a n c e method w h i c h m i n i m i z e s  ( 1 , 9, 1 6 ) .  Although  the negative semivariance 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 n o t been used f r e q u e n t l y , i t h a s , 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 ( 2 3 , 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, 2 1 , 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 t h e  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 c a n t h e 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 t h e 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 t h e 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 An answer t o t h i s p r o b l e m < i s  i s missing.  i m p o r t a n t s i n c e knowledge o f t h e  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 w i s h t o d e c i d e w h i c h ( i f a n y ) o f t h e methods s h o u l d be a p p l i e d under c e r t a i n specific empirical conditions.  In any r e s e a r c h t h e q u a l i t y o f t h e 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 t h e adequacy w i t h w h i c h t h e 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  Objectives A general 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 study t h e e f f i c i e n c y  o f t h e 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 t h e incomer 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 t h e activity returns*.  In s t u d y i n g t h e e f f i c i e n c y o f t h e 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 provided i n Chapter I I .  - 3 -  elements are considered, namely the magnitude o f the bias and the variance o f the sampling estimates o f the income-risk f r o n t i e r . This primary o b j e c t i v e may be s p e c i f i e d i n a number o f more s p e c i f i c sub-objectives as f o l l o w s : 1.  (a)  To review some t h e o r e t i c a l concepts concerning the u t i l i t y  function o f a d e c i s i o n maker when the expected income and r i s k are considered to be v a r i a b l e s . (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 s i t u a t i o n s under which these three methods are  a p p l i c a b l e given c e r t a i n assumptions regarding d e c i s i o n makers' u t i l i t y functions and the p r o b a b i l i t y d i s t r i b u t i o n o f the returns. 2.  (a)  In order t o extend the t h e o r e t i c a l concepts o u t l i n e d i n (1)  above, a small farm planning model c o n s i s t i n g of three a c t i v i t i e s and s i x c o n s t r a i n t s w i l l be employed.  Assuming that the a c t i v i t y returns o f the  three v a r i a b l e s are normally d i s t r i b u t e d and that the degree o f correl a t i o n , 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 o f 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 i n t h i s instance  the degree of c o r r e l a t i o n 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 s i t u a t i o n s the complete set (population) o f  observations i s not a v a i l a b l e , 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 to the population data.  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 t h e randomly drawn sample d a t a w i l l be compared t o d e t e r m i n e t h e 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 f r o m the former. (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 e l e m e n t 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 t h e 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-  n o r m a l , i n a d d i t i o n t o t h e QP-VAR and MOTAD methods, t h e 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 p e r f o r m a n c e  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 n o t 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 highly correlated.. 1•^  I m p o r t a n t Hypotheses  t o be t e s t e d  The f o l l o w i n g f o u r h y p o t h e s e s 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 study.  These  hypotheses  - 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 in the area.  Chapter II provides a t h e o r e t i c a l basis supporting these  hypotheses*. 1.  The QP-VAR a p p r o a c h as a p p l i e d t o s m a l l sample d a t a p r o v i d e s an  unbiased estimate o f t h e actual population income-risk 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 normally 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.  T h e 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 t h e 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 t h e f o l l o w i n g two c o n d i t i o n s a r e satisfied. (a)  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  (b)  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 t h e a c t i v i t y r e t u r n s are c l o s e to zero.  3.  I f a c t i v i t y returns a r e non-normally  d i s t r i b u t e d , t h e 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 t h e a c t u a l p o p u l a t i o n incomerisk frontier. 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 , t h e 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 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 frontier. Other hypotheses  r e l a t e d t o comparisons  o f t h e 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 t h e methods s h o u l d a l s o be i n c l u d e d .  However,  b e c a u s e 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 h e d i s p e r s i o n o f t h e  * The r e a d e r may be seen alternative hypothesis  w i l l n o t i c e t h a t t h e e x c e p t i o n i s h y p o t h e s i s 3. As i n Chapter I I , t h e t h e o r e t i c a l study supports the 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 h e n u l l formulated.  - 6 -  estimates these hypotheses a r e n o t formulated.  Nevertheless, the dis-  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 e l e m e n t t o c o n s i d e r i n t h e e v a l u a t i o n o f t h e methods.. 1.4  Research  Procedure  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 study 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 p e r f o r m a n c e  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  frontiers. 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 t h e a s s u m p t i o n s 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 d e g r e e 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 m o d e l s " phase, where t h e QP-VAR, MOTAD and i n  some i n s t a n c e s t h e 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 a n d f o r a number o f samples drawn f r o m 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 h e 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 h e 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 h e two f r o n t i e r s , i . e . , whether the estimate i s b i a s e d o r not.  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  ( 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  * T h e "mean i n c o m e - r i s k f r o n t i e r " i s t h e 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 f r o m mean v a l u e s o f t h e 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 i m i t e d t r a d e - o f f between bias and variance  o f the estimates w i l l be considered in evaluating methods with d i f f e r e n t degrees of d i s p e r s i o n and b i a s . 1.5  Organization of the Study Chapter 1 includes the statement of the problem, the o b j e c t i v e s  and the basic methodology to be f o l l o w e d . Chapter 2 i s devoted to a t h e o r e t i c a l study of the V a r i a n c e , Total Absolute Deviation and Semivariance as r i s k i n d i c a t o r s which may be used i n farm planning.  The d i s c u s s i o n centers around the accuracy  with which these i n d i c a t o r s may represent r i s k . The next chapter describes the procedures used to generate p o p u l a t i o n s , the sampling process and the models used when each method i s applied. The fourth chapter reports on the a n a l y s i s of the r e s u l t s and the t e s t i n g of the hypotheses.  It examines the performance of the QP-VAR,  MOTAD and Semivariance methods in four p o s s i b l e experimental e m p i r i c a l s i t u a t i o n s and t h e i r r e l a t i v e e f f i c i e n c y as estimators of the incomerisk frontier. Chapter 5 describes an a p p l i c a t i o n 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 conclusions.  - 8 -  CHAPTER II THEORETICAL REMARKS  The purpose of t h i s chapter i s to discuss some t h e o r e t i c a l aspects of the farm planning methods used i n t h i s study.  The a b i l i t y  o f these methods to adequately generate estimates o f i ncome-risk f r o n t i e r s and the c l o s e l y r e l a t e d problem regarding farmers'  utility  functions w i l l be considered i n an evaluation of the methods. 2.1  The Income-Risk E f f i c i e n t  Frontier  As a preliminary step i t i s necessary to define the concept o f an e f f i c i e n t  plan.  Markowitz (23) notes that a plan i s e f f i c i e n t  if  i t i s not p o s s i b l e 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 smaller v a r i a b i l i t y of income f o r the same l e v e l of expected income. Risk may be measured as the degree of v a r i a b i l i t y of  return.  One or a combination of the f o l l o w i n g measures of v a r i a b i l i t y may be used as r i s k i n d i c a t o r s : skewness and k u r t o s i s .  v a r i a n c e , semivariance , absolute d e v i a t i o n , Each of the p o s s i b l e measures of  variability  has d i f f e r e n t c h a r a c t e r i s t i c s and represent c e r t a i n s p e c i f i c aspects of risk differently. 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 s k l e v e l 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 possible  activities  FIGURE 2.1  two  (16).  Income I s o - R i s k C u r v e s Snowing D i f f e r e n t Degrees o f Correlation  Activity  f o r the c a s e o f  between the A c t i v i t y R e t u r n s  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 are 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  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 r e t u r n s o f a c t i v i t y X-j and  family  c o e f f i c i e n t between  ( c l o s e to - 1 . 0 ) .  the  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 one.  As the c o r r e l a t i o n  a c t i v i t i e s r a t h e r than producing j u s t  c o e f f i c i e n t a p p r o a c h e s z e r o o r becomes  the i s o - r i s k c u r v e s a r e l e s s c o n c a v e to the o r i g i n ( f i g u r e 2.1 meaning t h a t d i v e r s i f i c a t i o n does not r e d u c e r i s k g r e a t l y .  positive (b)),  Finally,  -  when the c o r r e l a t i o n  coefficient  10  -  i s equal to.+1.0, the i s o - r i s k curves  are straight l i n e s as shown i n Figure 1(c), In t h i s case there are no benefits from d i v e r s i f i c a t i o n as f a r as the r i s k s i t u a t i o n  i s considered.  By introducing the f a m i l i a r concept of iso-incorne (or i s o revenue) l i n e s , an income-risk expansion l i n e may be defined.  FIGURE 2.2  Income-Risk Expansion Line f o r a Two A c t i v i t y S i t u a t i o n  A c t i v i t y X-,  f  Activity 1  Figure 2.2 shows the income-risk expansion l i n e f o r 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 o f points  which define the minimum r i s k l e v e l at each level of expected income, represented by the points at which the iso-income l i n e s are t a n g e n t i a l 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  combinations  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  t h e maximum income i s t o be a c h i e v e d a t t h e minimum l e v e l o f r i s k . i s i m p o r t a n t t o note t h a t i'thare i s a u n i q u e c o m b i n a t i o n levels  (1)  of a c t i v i t y  (L-J, l_2> L^) w h i c h g e n e r a t e any e f f i c i e n t p l a n *.  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  The income assumptions:  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 ) i s an  function.  2  homothetic  T h i s i m p l i e s t h a t t h e 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  a l l have t h e same shape) t h r o u g h o u t  It  a l l levels of r i s k :  (2)  The  (they relative  p r i c e s o f X-. and X^ r e m a i n 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 not a b l e to a l t e r the s l o p e of the iso-income 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 ) . l i n e s w i l l be s t r a i g h t  l i n e s when i t  Income-risk  expansion  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 are g e n e r a l l y  met.  If points l_i, l_ , 2  o f F i g u r e 2.2 w h i c h r e p r e s 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 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 .  different  i n an income-  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 , w h i c h shows t h e maximum l e v e l o f income o b t a i n a b l e at 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 des 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  plans  g(xx) r  Area o f impossible  2  income-risk frontier Area o f i n e f f i c i e n t plans o  risk  =  f(XiX ) 2  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 t h e e x 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 ( 1 5 , 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 t h e maximum p o s s i b l e income w h i c h c a n 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 t h e i n c o m e - r i s k  frontier  i s e f f i c i e n t and t h e a c t u a l p o i n t o r p l a n chosen w i l l depend upon t h e 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 t h e d e c i s i o n maker, 2.2  The E x 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 d e c a d e s a g o , i t was assumed t h a t t h e 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 expected monetary r e t u r n (23), o b j e c t i v e does n o t r e f l e c t r e a l i t y .  I t was l a t e r found t h a t t h i s  Instead, the expected u t i l i t y r u l e  - 13  -  was p r o p o s e d 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 approach  assumed t h a t t h e i n d i v i d u a l t r i e s t o m a x i m i z e h i s e x p e c t e d  utility  r a t h e r than h i s expected r e t u r n s .  Expected u t i l i t y i s a f u n c t i o n of the  expected 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 . Furthermore, expected 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 the expected r e t u r n s but a l s o a f u n c t i o n o f the degree o f r i s k i n v o l v e d i n t r y i n g to p u r s u e such r e t u r n s . * M a r k o w i t z (23) a c c e p t e d t h e 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 -  ship.  FIGURE 2.4  utility  U t i l i t y Curve f o r a Risk Averse I n d i v i d u a l  *  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 t h e 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 uti l i t y level.  - 14 -  For 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 o n c a v e  utility  f u n c t i o n t h e g a i n i n u t i l i t y f r o m w i n n i n g a d o l l a r w i l l be l e s s t h a n t h e 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 ) . p a r t i c i p a t e i n a " f a i r " game o f c h a n c e .  This individual will  never  F o r example, g i v e n a game i n  which he has an e q u a l 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, t h e e x p e c t e d u t i l i t y o f s u c h a game i s s m a l l e r t h a n t h e u t i l i t y of certainty of  $550,,  i . e . , U-|>  UQ  i n the graph.  The s t r i c t l y c o n c a v e u t i l i t y f u n c t i o n may e x p l a i n why  individuals  are 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 t h e 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 t h e 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 n o t 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 s u c h 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 a v e r a g e r e t u r n s t h a n 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 p r o b l e m 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 with 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 t h e shape o f t h e  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 c o n c a v e 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 t h e 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 l o w e r o r h i g h e r one.  - 15 -  T y p i c a l Shape o f U t i l i t y C u r v e  FIGURE 2.5  utility  *  I  II  I  III  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 c o n v e x (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 c o n c a v e 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 t h e f a r m e r ' s s t a t u s i s n o t changed i n t h e s h o r t run by t h e outcome o f d e c i s i o n s t h a t he makes.  socio-economic production  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 t h e 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 t h e f a r m l e v e l be enough t o move a f a r m e r i n t o lower or higher q u a l i t a t i v e socio-economic average to a r i c h  p o s i t i o n s ( s a y f r o m an  farmer).  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 where  be w r i t t e n a s : (2.1)  U  F(E,R) ,  U  Level of U t i l i t y  E  Expected  R  Level of Risk  Income  - 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 e x 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 ,  _3U_  > o  •  hence:  ^iL<  and  0 .  (2,2) ^  3R  3E  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 :  1UL 32  < 0  and  3U 32  E  2  < 0 >  (2.3)  R  G r a p h i c a l l y , t h e r e l a t i o n between U and R  given a fixed  level  o f E c a n 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 t h e 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 shown i n F i g u r e 2.7.  f u n c t i o n o f both  The n e g a t i v e p a r t o f t h e u t i l i t y f u n c t i o n i s n o t shown  because o f d i f f i c u l t i e s i n drawing to t h e n e g a t i v e a r e a .  income and r i s k l e v e l s i s  i t , b u t o b v i o u s l y i t s h o u l d be e x t e n d e d  - 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  utility  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 c o n c a v e and i t i s p o s s i b l e t o draw c o n v e x 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  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  Iso-Utility  Curves  income  risk  of  -  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 h e p l a n c h o s e n by an i n d i v i d u a l  who m a x i m i z e s h i s p a r t i c u l a r  utility function.  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 . A i n d i c a t e s t h e l e v e l s o f income and r i s k w h i c h maximize the given income-risk  FIGURE 2.9  Point  utility for  frontier.  D e t e r m i n a t i o n o f t h e O p t i m a l 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 t h e 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 t h e d e c i s i o n maker, as d e t e r m i n e d by t h e d e g r e e o f risk aversion.  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 w h i c h  m a x i m i z e s 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  farmers.  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 .  T h u s , 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 u n r e a l i s t i c by the majority  of  farmers and therefore f o r farm planning purposes i t may be a b e t t e r procedure to present farmers a set of e f f i c i e n t plans (the income-risk  frontier),  where they may choose one which maximizes t h e i r expected u t i l i t y . To derive the proper income-risk f r o n 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 c h a r a c t e r i s t i c s of the d e c i s i o n maker's u t i l i t y function and of the d i s t r i b u t i o n of returns have to be considered. 2.3  Risk Aversion P r a t t (26) has defined a measure of th.e absolute r i s k aversion  (r) as f o l l o w s ;  • r  =  ,  (2.4)  U'  where U' i s the f i r s t d e r i v a t i v e 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 a v e r s i o n . efficient  The r i s k aversion c o -  i s the rate of change of the s l o p e , rather than the absolute  change of the s l o p e . assigned to U".  It i s important to understand the negative sign  I f u" i s negative the u t i l i t y function i s  strictly  concave and t h e r e f o r e , the r i s k aversion c o e f f i c i e n t must be p o s i t i v e s i n c e U' i s p o s i t i v e .  But i f the u t i l i t y function i s convex, i . e . ,  the d e c i s i o n maker i s a r i s k t a k e r , 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 e g r e e 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 . an i n d i v i d u a l  The maximum amount o f r i s k premium w h i c h  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 t h e mean o r e x p e c t e d r e t u r n o f a combination o f returns  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 t h e 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  Function  utility  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  risk  a v e r s i o n measure ( r * ) :  r*  =  r,E  ,  (2.5)  Most a u t h o r s a g r e e (4,6,10,30) t h a t an a p p r o p r i a t e  utility  -  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 t h e f o l l o w i n g b a s i c properties: [a)  IT  >  0, i . e . m a r g i n a l  (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  ( )  SLLnl  c  =  r  '  <  u t i l i t y of. income i s p o s i t i v e  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  df, i f a n y t h i n g , d e c r e a s e when t h e income increases. (d) v  J  d ( r * ^ >  ^—  L  -  0,  i . e . , the r e l a t i v e risk aversion  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 t h e income i n c r e a s e s . It i s important t o mention t h a t these p r o p e r t i e s are not 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  function (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 t h e 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 t w o - d i m e n s i o n s p a c e .  A relevant question  to a s k i s w h e t h e r 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 p a r a m e t e r ( s a y t h e s t a n d a r d d e v i a t i o n o r t h e 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 w h i c h i s t h e r e s u l t a n t o f two o r more p a r a m e t e r s .  Furthermore,  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 h e e x p e c t e d income and a s i n g l e p a r a m e t e r 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 h e mean income, E\ i n s u c h a way t o t r a n s f o r m i t to a polynomial, U(E)  =  U(E)  +  U'CC).  (E-E) +  U"(E). 2!  (E-E)  2  +  (2.6) '•  U'"(E)  . (E-E) +  . + U  3  31  where  R  (E)  (E-E)**  +  n+1  nl  =U  p + ]  ( n )  (E) p (n+1)'.  .  ( n + I )  CE-E)  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 [lJ(E)|  = ' U(E)  +  UJiJl  m  2  U" '(E)  +  2 1  +  where m^, m^,  ;  +  ••••  J  J  %  ^  a r e  %  n+1  . c a n be n e g l e c t e d .  3  3 :  +  E [  R  n  +  ]  ,  (2.7)  s e c o n d , t h i r d and s u c c e s s i v e  t n e  h i g h e r moments w i t h r e s p e c t t o t h e mean. R  m  I f the s e r i e s i s convergent  I t i s i m p o r t a n t t o note t h a t t h e f i r s t term o f  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 t h e u t i l i t y c o r r e s p o n d i n g t o t h e mean o r e x p e c t e d income ( E ) , so f o r t h .  i s the variance,  t h e skewness and  I t i s a l s o n o t e d t h a t t h e 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 , expected u t i l i t y i s not merely a f u n c t i o n  o f two parameters, s a y 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 t h e e x p e c t e d income and n - l p a r a m e t e r s  representing risk.  -  23  -  The a c t u a l v a l u e o f n, i . e . , the number o f p a r a m e t e r s w h i c h 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 s e c y t i v e d e r i v a t i v e s w h i c h can be o b t a i n e d function. (E)  =  con-  f r o m the o r i g i n a l u t i l i t y  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"  U'"(E)  =  =  U  (E)  n  =  0.  An i n d i v i d u a l w i t h  s u c h 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 decisions. =  I f 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 l /  n  ^  (E)  =  U" '' {I)  =  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  e x p e c t e d income and the v a r i a n c e m  in his decisions.  2  the  But i f the  utility  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 p a r a m e t e r s i n h i s d e c i s i o n s . equation  2.7 may  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  v a n i s h i f some o f the m e l e m e n t s become 0.  c l e a r l y r e l a t e d to the frequency  d i s t r i b u t i o n o f income.  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^ ffi  n  are i n a f i x e d r e l a t i o n with m  .  2  =  0  This i s  In symmetric and m^,  ...  T h e r e f o r e , the expected  utility (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 ,  even i f U ' "  2  (E)  f  0.  In any skewed d i s t r i b u t i o n riv,  f  0  and m  3  (E), 4  ...U 0 and  h e n c e , 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 p a r a m e t e r s ( E , 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 p a r a m e t e r s w h i c h s h o u l d be c o n s i d e r e d  equal  are  m, 2  t o t h e number o f t i m e s 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 p a r a m e t e r s w h i c h 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 returns, whichever i s smaller; n  =  where d^ r e p r e s e n t s f e r e n t i a b l e and P  s  Min ( d  p  ? ), s  (2.8)  the number o f t i m e s w h i c h t h e u t i l i t y f u n c t i o n i s d i f i s the number o f p a r a m e t e r s w h i c h d e f i n e t h e random  distribution of returns.  -  24  -  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 p a r a m e t e r o n l y when 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 o r when t h e r e t u r n s a r e n o r m a l l y tributed.  dis-  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 t h e income-  r i s k f r o n t i e r , p r o v i d e s t h e s e t o f e f f i c i e n t p l a n s from w h i c h choose t h a t p l a n w h i c h m a x i m i z e s t h e i r e x p e c t e d u t i l i t y . a l t e r n a t i v e s has n o t been d e t e r m i n e d  farmers  If this setof  c o n s i d e r i n g t h e a p p r o p r i a t e number  o f p a r a m e t e r s as d e f i n e d by f o r m u l a 2.8, t h e d e c i s i o n maker may t a k e erroneous  decisions.  For i n s t a n c e , i f returns are not normally  t r i b u t e d ( a s may be e x p e c t e d  dis-  i n many c a s e s ) t h e r e s e a r c h e r s h o u l d  present  the i n c o m e - r i s k f r o n t i e r b a s e d on two p a r a m e t e r s ( 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 . If 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 plan chosen might not 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, t h e 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 i s i m p o r t a n t i n j u d g i n g t h e 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 u n d e r 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 d e t e r m i n e  the s e t o f  e f f i c i e n t plans ( t h e income-risk f r o n t i e r ) l e a v i n g t o the farmer t o pick 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 t h e 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 h e r i s k i n d i c a t o r s i m p l i c i t l y 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 .  con-  The nature 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 t y p e o f f r e q u e n c y  distribution  o f t h e a c t i v i t y r e t u r n s a r e t h e main e l e m e n t s 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 w h i c h method, i f a n y , may be used i n o r d e r t o d e r i v e an incomerisk frontier.  - 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 h e 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 h e d i s c u s s i o n was  c e n t e r e d on t h e 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 t h e 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 plan.  In s e c t i o n 2.4 t h e d i s c u s s i o n  number o f p a r a m e t e r s  focused- on t h e  required to express r i s k i n order to provide the  d e c i s i o n maker w i t h t h e 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 utility.  The r e l a t i o n between t h i s p r o b l e m and t h e 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 t h e QP-VAR, MOTAD and S e m i v a r i a n c e methods u s i n g t h e 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.  A r e t h e 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 a d e q u a t e  to allow f o r maximization o f expected 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 h e 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 e a c h method?  Are the variance, semivariance or  t h 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 a s 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 a t t e m p t e d i n t h i s s e c t i o n g i v e n t h e a n a l y t i c a l framework d e v e l o p e d i n t h e p a s t sections. 2.5.1  Q u a d r a t i c Programming:  The V a r i a n c e A p p r o a c h  M a r k o w i t z (23) f i r s t f o r m u l a t e d t h e r i s k p r o b l e m i n a mathem a t i c a l programming m o d e l .  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 t h e t o t a l r e t u r n s as a r i s k measure (QP-VAR).  Total  - 26 -  r e t u r n s (E) may  be d e f i n e d as  !  where c  folows: C2.9)  °J J , X  i s t h e r e t u r n p e r 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  to that expected  return i s calculated  as f o l l o w s : ton V  =  E E  X  1  i J  a  X  • (2.10).  i j J,  where a., i s t h e 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.i s t h e c o v a r i a n c e between t h e r e t u r n s o f x. and  Defining equation V  =  (2.10) i n m a t r i x  = a-.-  x..  notation, (2,11)  X' QX,  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 , 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 i s t h e t r a n s p o s e  v e c t o r o f X.  1  The p r o b l e m o f d e t e r m i n i n g  the e f f i c i e n t set of plans  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 ramming m o d e l , w h i c h may  be s t a t e d as f o l l o w s  Min  V =  s.t. -  b  cX  =  A  X  -  0  (16): (2,12)  X' QX  AX  a Quadratic  ,  using Prog-  - 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 expected values.  return which i s parameterized  forn different  The s o l u t i o n t o t h i s p r o b l e m 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 ,  w h i c h i s p r e s e n t e d t o t h e f a 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 n o t e some c h a r a c t e r i s t i c s o f t h e o b j e c t i v e function 1.  V  =  X'  QX;  I t i s a p o s i t i v e d e f i n i t e or semidefinite function since the variances  are a l l p o s i t i v e . This i s a d e s i r a b l e f e a t u r e o f the variance 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 h e f e a s i b l e s e t i s convex). 2.  Matrix Q includes not only the individual v a r i a b i l i t y of the a c t i v i t y  returns but a l s o the 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 returns ( c o variances). 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 parameter, the variance.  In o t h e r w o r d s , 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 t h e mean.  Re-  c a l l i n g from s e c t i o n 2,4, i f 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 s q u a d r a t i c , then I T " (E)  =  l) (E)  =,  l v  = l/")(.E) . =  T h i s i m p l i e s t h a t t h e d e c i s i o n maker o n l y c o n s i d e r s t h e e x p e c t e d (E)and t h e v a r i a n c e i n h i s d e c i s i o n s .  income  Thus, i f the e f f i c i e n t s e t o f plans  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 -  formation r e q u i r e d t o maximize h i s expected method i s a p p r o p r i a t e i n t h i s c a s e ,  0.  utility.  Hence, t h e v a r i a n c e  Even i f t h e u t i l i t y f u n c t i o n i s n o t  - 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 . frig rr^  =  0  In t h i s c a s e ,  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  or vanish.  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 or 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 , t h e 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 n o r m a l .  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 to 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 c h o o s e s f r o m 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 t h e 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 m a x i m i z e s h i s e x p e c t e d utility.  B u t , 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 does n o t 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" ' (E) does not v a n i s h . 1  Once t h e 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 s e n s e t h a t t h e e f f e c t o f U'''  (E)fn  3  in  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 t h a n compensate f o r t h e d i f f e r e n c e i n t h e second t e r m , UJ_f£)m , 2! The m a g n i t u d e o f t h e m i s t a k e o f n o t c o n s i d e r i n g IT ' ( E ) m depends 9  L  1  3  on t h e u t i l i t y f u n c t i o n ( t h e v a l u e o f U"(F.)) distribution.  and t h e . s k e w n e s s o f t h e  T h i s i s a q u a n t i t a t i v e p r o b l e m o v e r w h i c h a number o f  a u t h o r s , T s i a n g ( 3 0 ) , B o r c h ( 4 ) , F e l d s t e i n (10) and o t h e r s have a r g u e d . I t seems c l e a r t h a t i t i s n o t p o s s i b l e t o make any g e n e r a l about t h e magnitude o f the higher order terms, searcher  i s c e r t a i n that the value o f ^''(E^m^  r e l a t i v e t o U (I) method.  I t 1s  +  U"(E)fn  seldom  conclusion  Hence, u n l e s s t h e r e i s small  f o r a l l E he s h o u l d n o t u s e t h e QP-VAR  2  possible  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 n o t be a p p l i e d . trictive,  Unfortunately,  these c o n d i t i o n s a r e very  res-  A quadratic 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 accepted  10, 30) b e c a u s e i t does n o t s a t i s f y t h e f o u r c o n d i t i o n s w h i c h  ( 4 , 5,  should  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 m e n t i o n e d i n S e c t i o n 2.3. The absolute risk aversion c o e f f i c i e n t r , i n the quadratic u t i l i t y  function  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, utility  To i l l u s t r a t e , assume any q u a d r a t i c  function: U  =  E  -  bE  (2.13)  2  then ' - ^ r •= dt  U'  =  1  -  2bE  (2,14)  - 30 -  and dU  = . U"  2  =  -2b,  (2,15)  U s i n g t h e 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"  =  U  2b  =  (2.15)  l-2bE  l  and. t h e 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:  dr -— dE As i t c a n be seen r  4b —=— 2 (l-2bE) 2  ••= 1  r  <  =  .  (2.17)  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 t o t a l income i n c r e a s e s ;  a g a i n s t t h e same a b s o l u t e r i s k as h i s  t h i s c o n t r a d i c t s one o f t h e p r o p e r t i e s w h i c h t h e  u t i l i t y f u n c t i o n o f a r i s k averse i n d i v i d u a l should  have.  The a s s u m p t i o n t h a t 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 i s also d i f f i c u l t to sustain.  Hazel 1 (.15) t r i e d t o j u s t i f y t h i s a s s u m p t i o n  f o r a m u l t i a c t i v i t y f a r m b a s e d on t h e C e n t r a l L i m i t t h e o r e m , b u t 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 t h e o r e m was i n correct.  Hazel! t r i e d t o demonstrate that i f a farm produces s e v e r a l  a c t i v i t i e s the d i s t r i b u t i o n o f the t o t a l returns obtained  from a l l these  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 t h e d i s t r i b u t i o n o f t h e i n d i vidual a c t i v i t y returns.  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  are independent situation.  among e a c h o t h e r ;  -  o b v i o u s l y t h i s is. n o t a g e n e r a l  T h u s , 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 t h e 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 t h e i n d i v i d u a l a c t i v i t y returns are a l l approximately  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 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 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  normally  d i s t r i b u t e d t o t a l r e t u r n s and h e n c e , would t h e m a g n i t u d e o f t h e 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  p r o b l e m and a t h e o r e t i c a l a n a l y s i s  may n o t 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 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 w h i c h do n o t c o r r e s p o n d  when i t  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 m e n t i o n e d 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 h e 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 t h e incomer 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 n o t 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 n o t answered by u s i n g only a n a l y t i c a l concepts  i s the following:  What i s t h e m a g n i t u d e 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 t h e c o n d i t i o n s d e s c r i b e d above a r e met? of the s e t o f experiments  One o f t h e p u r p o s e s  reported i n the f o l l o w i n g chapters i s t o pro-  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 p r o b l e m . 2.5.2  The S e m i v a r i a n c e  Approach  One o f t h e l i m i t a t i o n s o f t h e v a r i a n c e a p p r o a c h i s t h a t i t c o n s i d e r s a s 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 a r o u n d t h e e x p e c t e d  income.  I t i s c l e a r t h a t farmers 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 b u t n o t p o s i t i v e deviations.  I t i s reasonable t o s a c r i f i c e part o f the expected  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 t h e income b u t 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 a p p r o a c h 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 with the negative semivariance.  concerned  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  =• J _  E  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 t h e 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 a v e r a g e income o f a c t i v i t y i . As can be s e e n , 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 t h e 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 t h e t o t a l income, t h e incomes 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.  (for a l l g=l, —-,  where S i s t h e t o t a l s e m i v a r i a n c e and t  (1,  K),  (2,19)  , K) a r e t h e y e a r s  i n which t h e p l a n i m p l i e s r e t u r n s below t h e 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,,Ct ) n  The i n c o m e - s e m i v a r i a n c e  =  —  lf °i ' ^ t g ' ^ r ' A g , (  (2.20)  f r o n t i e r c a n be o b t a i n e d i n t h e same way as u s i n g  the v a r i a n c e , b u t i n s t e a d o f m i n i m i z i n g t h e 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 t h e 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 -  A n o t h e r f o r m u l a t i o n o f t h e s e m i v a r i a n c e w h i c h c o u l d be used i s t h e f o i l owing: S  mn EE  (2,21)  x. s . . x.  where T-l  EEE m i n . (c, . - c \ ) , 0} . m i n . { ( c , . - c . ) , 0 } ,  hi j  where. T i s t h e 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  Semivariance  s i n c e i t c o n s i d e r s the semivariance 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 t h e c o s e m i v a r i a n c e s among t h e a c t i v i t i e s ) as a c r i t e r i o n t o c h o o s e t h e 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 import-  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 t h e income semivariance.  B u t i t has t h e a d v a n t a g e t h a t i t c a n 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 t h e income s e m i v a r i a n c e c a n 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 solution. method. 2.12,  approximate  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 t h e QP-SEMIV U s i n g t h e 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  equation  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 t h e v a r i a n c e model.  I t i s important t o note t h a t the a c t i v i t y cosemivariances can never  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 t h e 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 t h e 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 variance.  T h u s , i n t h i s c a s e t h e 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 t h a n p l a n B a c c o r d i n g t o t h e 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 t h e 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 the income.is not normally 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 t h e o t h e r w i t h a g r e a t e r skewness, t h e d i f f e r e n c e s i n t h e s e m i v a r i a n c e o f t h e 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 t h a n 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  to changes i n t h e skewness b u t t h e v a r i a n c e i s n o t .  I f t h e skewness  as-  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 t e n d t o d i m i n i s h even i f t h e v a r i a n c e does n o t d e c r e a s e .  I f t h e skewness becomes more  n e g a t i v e t h e 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 t h e 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 t h a n 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 n o t quadratic.  F o r 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 t h a n 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  E {U (E)}  =  U(E)  +  U  "  ( E ) m  2  be; +  U  2! Assuming a r i s k a v e r s e i n d i v i d u a l , U">0 U  , M  >0.  "  , ( E ) m  3;  3  .  (2.23)  ~"  and a l s o assume t h a t  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 , t h e t h i r d t e r m 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 , w h i c h o b v i o u s l y would a l t e r t h e v a l u e  - 35 -  of E  {U CE)},  T h e i m p o r t a n t p o i n t , however, i s t h a t U " ' (E)m  3  may  3!  a l t e r the r e l a t i v e order o f a set o f plans.  Thus, i f t h e variance i s  s u b s t i t u t e d f o r t h e s e m i v a r i a n c e i n m^, t h e v a r i a t i o n s o f m' a r e a l s o 3  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 If m  3  explicitly.  becomes more p o s i t i v e , m'^ measured by t h e 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 £ w i l l be l e s s n e g a t i v e  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,  which t e n d s t o compensate t h e 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 s k e w n e s s , t h e n t h e 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 t h a n i n t h e f o 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 negative value o f u " ' ' ( E ) m 31  3  . Therefore, despite the fact  • •  t h a t t h e S e m i v a r i a n c e method does n o t f o r m a l l y c o n s i d e r t h e e f f e c t o f moment iri^, 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 h e combined e f f e c t o f m^ a n d m , 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 t h a n t h e v a r i a n c e method o f 3  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 t h e p l a n s . may n o t be s u f f i c i e n t i n c o m p a r i s o n s  The c o m p e n s a t i o n  effect  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 t h e 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 -  Plan I, which w i l l  give; 0 w i t h p r o b a b i l i t y 0.5  or  4 w i t h p r o b a b i l i t y 0.5  Here;  Mean {£)  =2  semivariance  V a r i a n c e (V) = 4 Plan I I , which w i l l  skewness  (SV) (SK)  = 2 = 0  give:  -2 w i t h p r o b a b i l i t y 0.2 or Here:  3 w i t h p r o b a b i l i t y 0.8 E  =  2  SV  =  V  =  4  SK  = -12,0.  Plan I I I , which w i l l  3,2  give:  0.7 w i t h p r o b a b i l i t y 0,68 or Here:  5,0 w i t h p r o b a b i l i t y 0.32 E  = 2  V  =  4  SV  =  1.14  SK  =  7.16.  These t h r e e p l a n s have t h e 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 f r o m t h e p o i n t o f view o f t h e v a r i a n c e analysis.  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 I I 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 ) } t h a n p l a n I I . A l l p l a n s have t h e same mean E , and t h e same v a r i a n c e b u t 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, o f t h e a c t u a l v a l u e o f LTJT)' 2! "  and  U ' (E), 3! 1 1  This holds  regardless  The mean-semi v a r i a n c e  - 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 I I and p l a n I b e c a u s e 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 t h a n p l a n I I  b e c a u s e p l a n I has a s m a l l e r  semivariance,  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 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 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 , t h e v a r i a n c e a n a l y s i s would p r o -  vide correct r e s u l t s  a n d the, S e m i v a r i a n c e  w o u l d n o t . Hence t h e S e m i -  v a r i a n c e method s h o u l d n o t be used i n s i t u a t i o n s where 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 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 E x p e c t e d  Utility  As was p o i n t e d o u t b e f o r e , t h e 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 w h i c h a l l o w s t h e d e c i s i o n maker t o c h o o s e a p l a n w h i c h m a x i m i z e s h i s e x p e c t e d  utility.  In o r d e r w o r d s , a method i s ; s u i t a b l e i f i t c a n p r o v i d e a l l t h e 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 B o r c h ( 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 a r o u n d w h e t h e r t h e terms i n t h e e x p 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  neglected  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 the expected  utility  t o t h e h i g h e r o r d e r moments i n  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 h e v a r i a n c e  t e r m , a n d hence t h e income v a r i a n c e a n a l y s i s w o u l d be v a l i d o n l y when 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 o r when t h e outcomes a r e n o r m a l l y tributed,  T s i a n g (.17) c o n c l u d e d  dis-  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 t h e  - '38-  h i g h e r o r d e r moment t e r m s may be n e g l e c t e d and t h e r e f o r e t h e 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 t h e 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 and t h e outcome d i s t r i b u t i o n i s n o t 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 the f o l l o w i n g remarks: 1.  T h e s e a u t h o r s a g r e e 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 ( a s s u m i n g a n o n - q u a d r a t i c f u n c t i o n ) , f o r e x a m p l e , i f t h e outcome d i s t r i b u t i o n i s v e r y However, t h i s argument i s n o t a l w a y s t r u e ,  utility  skewed.  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^ If  8  (2.24)  does n o t change s u b s t a n t i a l l y among t h e 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  provide the c o r r e c t ordering o f the  s e t o f p l a n s , b e c a u s e t h e r e i s a c o n s t a n t m o n o t o n i c 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 t h e  two measures and t h e r e f o r e , t h e normal d i s t r i b u t i o n r e q u i r e m e n t s t r i c t l y n e c e s s a r y f o r t h e v a l i d i t y o f t h e income v a r i a n c e  i s not  analysis.  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 s k e w n e s s , 6, does n o t a b r u p t l y change among t h e d i f f e r e n t possible plans.  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 p r o b l e m  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 r e m a i n s a p p r o x i m a t e l y stant;. 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  say m^.  con-  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 :  o r d e r moments  I f Y does n o t s u b s t a n t i a l l y c h a n g e , 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 provides the c o r r e c t ordering o f plans regardless 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 t h e 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 n o t e t h a t m^ i s n o t z e r o i n t h e normal d i s t r i b u t i o n , b u t a l l t h e a b o v e - m e n t i o n e d a u t h o r s a g r e e d t h a t when t h e 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 correct.  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 t h e normal d i s t r i b u t i o n i s  a l w a y s 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 t h e v a r i a n c e and m^. 3,  The a d v a n t a g e 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 t h e skewness t h a n t h e v a r i a n c e . c o e f f i c i e n t o f skewness v a r i e s among  Even i f t h e  plans, the 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^ m i g h t n o t c h a n g e .  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, w h i c h i s stable. 2.5.2.2. (a)  Conclusions Regarding the Semivariance  Method.  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 t h e r e t u r n s i s n o t 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 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 non-normal d a t a i f t h e k u r t o s i s c o e f f i c i e n t does n o t change much among t h e 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 t h r o u g h non a n a l y t i c a l methods, 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 p r o p o s e d as an a p p r o x i m a t i o n o f t h e T o t a l Income S e m i v a r i a n c e method. The a d v a n t a g e 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 c a n 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 s u c h as t h e 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 a d e q u a t e  linear  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 t h e q u a d r a t i c ( v a r i a n c e ) and s e m i q u a d r a t i c [semivariance) i n d i c a t o r s reviewed i n t h e former s e c t i o n s . T h i s i s due m a i n l y t o t h e 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 t h e 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 t h a n 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 h e 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 t h e l i n e a r models most commonly used i s t h e 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 t h e MOTAD model i s t h e T o t a l A b s o l u t e D e v i a t i o n o f t h e 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 over  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.A.D,  =  t n E. z h j  ( c, . - c .  )  x  ,  (2.26)  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 t h e j a c t i v i t y and c . i s hj J t h e e x p e c t e d income o f t h e a c t i v i t y X., T h u s , t h e m i n i m i z a t i o n o f t h e T.A.D, c a n be c a s t i n an LP p r o b l e m as f o l l o w s : t n Min z 2 (c c.) x  (2,27)  - At -  s,t,  n  ... x.  <;  b.  CFor a l l i =  1,  •. •  t  m )  X  X j  where a . , i s t h e ith b. A  i s the i  >  0  resource required f o r the j  activity  r e s o u r c e c a p a c i t y and  i s any l e v e l o f e x p e c t e d income which c a n be p a r a m e t e r i z e d . As H a z e l l p r o p o s e d  i t , i t i s p o s s i b l e t o minimize only the  n e g a t i v e p a r t o f t h e T,A.D. i f t h e e x p e c t e d r e t u r n s o f t h e a c t i v i t i e s , c - , a r e t h e sample.mean income, j  T h u s , when c - c o m p l i e s w i t h t h i s 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 t h e 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 m i n i m i z e t h e T.A.D. The MOTAD model bases i t s e s t i m a t i o n o f r i s k on t h e f i r s t 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 n o t c o n s i d e r e d i n t h e d e t e r m i n a t i o n of the expected 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  of 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 h e expected utility.  T h u s , 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 t h e p r o c e s s of maximization o f expected u t i l i t y .  Hence, t h i s p r o c e d u r e would be  appropriate only i f the u t i l i t y function i s linear, i . e . , the decision maker does n o t 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 t h e 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 t h e outcomes a r e n o r m a l l y  - 42 <•  distributed^ m m  2  _ ~  , 7(n-1)" 11  .  CMAD)  (2.28)  2  >  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 t h e sample.  G i v e n 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^ f r o m MAD and t h e d e c i s i o n maker may c h o o s e f r o m 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 h e same a d v a n t a g e s 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 .  Further,  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 t h e mean and MAD  as a c r i t e r i o n w o u l d 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 t h e  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 n o t e 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 f r o m s i n g l e a c t i v i t y mean a b s o l u t e deviations.  I t i s not appropriate f o r c a l c u l a t i o n s o f the variance o f  the t o t a l income g e n e r a t e d  by t h e 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 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 among t h e a c t i v i t i e s i s z e r o .  There-  f o r e , t h e mean a b s o l u t e d e v i a t i o n i s n o t a good e s t i m a t o r o f t h e income v a r i a n c e 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 among t h e 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 zero.  However, MAD has some f u r t h e r d i s a d v a n t a g e s f^.  with respect to  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 t h e  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^. f i c i e n c y i s d e p e n d e n t on t h e sample s i z e n, 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^  This e f -  I f n i s small the r e l a t i v e  w i l l be s m a l l and as n  increases  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 t h e sample v a r i a n c e i n  - 43 '  estimating the population  variance.  T h u s , t h e 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 w h i c h may be used when t h e outcomes a r e  normally  d i s t r i b u t e d , 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 among t h e 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 w o r d s , 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 w h i c h 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 n o t 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  Conclusions 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 t h e  income r i s k f r o n t i e r f o r a n 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  utility  function regardless o f the frequency d i s t r i b u t i o n o f a c t i v i t y returns. 2.  T h e 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 t h e  income r i s k f r o n t i e r i f t h e 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 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 . 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 t h e income  r i s k f r o n t i e r i f t h e 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 t h e income  risk f r o n t i e r only i f the following conditions are simultaneously  satis-  fied, (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 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 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) 5,  The sample s i z e i s l a r g e .  The QP-SEMIV method i s p r o p o s e d as a good i n d i c a t o r o f t h e income -  r i s k f r o n t i e r when 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 s n o t 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 t h a n m^  are not important.  6,  s e c t i o n conclusions regarding the a b i l i t y o f the  In t h i s a n a l y t i c a l  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 t h e i m p l i c i t assumption  t h a t t h e y were a p p l i e d u s i n g t h e c o m p l e t e  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 a s d a t a base. s i t u a t i o n s when t h e d a t a base  frequency  L i t t l e was s a i d  c o n s i s t s o f r e l a t i v e l y small  about  samples  r a t h e r t h a n t h e c o m p l e t e 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 w h e t h e r t h e c o n c l u s i o n s drawn from t h i s d i s cussion are also v a l i d to the estimates o f the income-risk 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 ,  frontier  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 t h e 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 evaluate their relative efficiency. i s made i n C h a p t e r I I I .  An a t t e m p t t o c l a r i f y t h e s e p o i n t s  -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 , t h e main e l e m e n t s t o 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 t h e methods were t h e magnitude o f t h e b i a s and t h e d i s p e r s i o n o f t h e income-risk  frontier.  estimates o f the  The a n a l y t i c a l s t u d y f r o m 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 a s s u m p t i o n t h a t t h e c o m p l e t 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 was used as t h e d a t a base.  However, i n o r d e r t o t e s t t h e h y p o t h e s e s 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 t h e 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 t h a n c o m p l e t e frequency d i s t r i b u t i o n s of a c t i v i t y returns,  The p u r p o s e o f t h i s c h a p t e r  i s to d e s c r i b e the procedure used i n e v a l u a t i n g the 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 f r o m normal and non-normal  distributions.  The f a c t t h a t t h e 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 f o r m a n c e under c o n d i t i o n s w h i c h are s i m i l a r 3.1  to those p r e v a i l i n g i n f i e l d research applications.  General Overview o f the Research Procedure D e c i s i o n makers c h o o s e f r o m a l t e r n a t i v e s i n v o l v i n g  r e t u r n s and d i f f e r e n t d e g r e e s o f r i s k .  different  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 h y p o t h e s i z e d - ,  the  -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 -  garding the frequency 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 , the  c o m p l e t e 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  there-  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 w h i c h 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  f r o n t i e r s , u s i n g t h e sample d a t a p r o v i d e d .  This procedure  income-risk  i s repeated  a number o f t i m e s ( u s i n g d i f f e r e n t s a m p l e s ) 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 verifiable results.  The same p r o b l e m i s s o l v e d u s i n g t h e c o m p l e t e 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 d e p e n d i n g on t h e 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 o r 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  c o n s i d e r e d t o be t h e " t r u e " income r i s k f r o n t i e r s .  thus o b t a i n e d  are  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 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 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 t h e 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 .  F i g u r e 3.1  i n v o l v e s d e f i n i n g t h e p o p u l a t i o n (normal  Step 1 i n  o r non - n o r m a l ) and  the p a r a m e t e r s w h i c h d e f i n e i t . The m a g n i t u d e o f t h e s e p a r a m e t e r s 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 w h i c h i s s i m u l a t e d . 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 t h e  Step 2 i n v o l v e s characteristics  d e f i n e d i n S t e p 1,  S t e p 3 i n v o l v e s t h e d r a w i n g o f samples from t h e popu-  l a t i o n s generated.  S t e p 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  Step 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  Definition of the population parameter* or clvirecteristics  Step 2  Generation of the set of data representing the population distribution of a c t i v i t v returns according to Step 1.  Step 3  Step t  Draw random samples:  Solving a plannijip model applying the appropriate  15 for each method  Solving a planning model applying the tttree -methods using the sample data.  method using the completej population data  Step 5  Step 6  J  -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 t h e 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 t h e  methods a r e a p p l i e d t o t h e samples  data.  In S t e p 5 t h e 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 t h e " 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 w h e t h e r 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 S t e p 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 h e " 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 h e 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, b u t t h e assumptions underlying the d i s t r i b u t i o n of a c t i v i t y returns differed. t h 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.  In  - 49 -  3.2  Generation o f the Populations 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 frequency 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 h e 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 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 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 t h e " t r u e " o n e . F o r t h i s r e a s o n randomly were g e n e r a t e d t h u s r e p r e s e n t i n g returns f o r three a c t i v i t i e s .  d i s t r i b u t e d sets o f data  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  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 a n d l o w d e g r e e 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 gamma t y p e .  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 t h 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 , n a m e l y , 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 n o t 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 t h e d e g r e e 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 t h e 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 t h e U n i t e d S t a t e s . In a l l c a s e s , t h e 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  significant.  •  - 50 -  The u s e o f a d i s t r i b u t i o n whose v a l u e s c a n n o t 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 t h a n n e t 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 ( 1 2 , 15, 16) b e c a u s e 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 o v e 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 , 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 u s , i t appears t h a t t h e  the p o s s i b l e d i s t r i b u t i o n o f gross  a c t i v i t y r e t u r n s w h i c h o b v i o u s l y c a n n o t 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 t h e 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 a n d 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 . 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  Table  generated  and T a b l e 3.2 p r e s e n t 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 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 t h e P o p u l a t i o n s G e n e r a t e d  Type o f Distribution  Mean G r o s s X  l  ; 2 X  Returns ;  X  3  Mean C o r r e l a t i o n Coefficient  Skewness Coefficient  Population 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  Population III  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 t h e 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 w h i c h 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 t h e skewness c o efficients.  In T a b l e 3.2 t h e v a r i a n c e s a r e t h e d i a g o n a l elements  m a t r i x and o f f d i a g o n a l e l e m e n t s  are the covariances.  o f each  T A B L E  3  -  Variance-Covariance Matrices o f the A c t i v i t y  2  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  • I  h  3  x  i  0.5  h .; x  Normal I  ;  x  2  0.18  1.0  : x  3  j  Normal 11 X  0.15 • 0.5  0.30  ;  1.5  :  i  X  2  : X  3  : h  0.46  0.63 •  1.0  0.76  1.5  Gamma I I I  23.1  \ 14.3  222.1  :  Returns  Generated.  :  Gamma IV  3  :  x  7.5  :  49.8  x  22.5  ;  58.2  :  .  x  2  x  ,  96.1  56.1  515.1  179.8  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 t h e 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  Semivariance-Cosemivariance Matrices o f the A c t i v i t y 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  ; : x  l  X  2  X  3  Gamma ( I I I ) X  l  7.9  X  2  Returns  Gamma ( I V ) X  3  12.2  7.1  72.6  16.6 22.4  X  19.2  3 -  90.0  49.8  197.1  154.2 72.6  I t i s i m p o r t a n t t o n o t e t h a t t h e 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 t h e 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 t h e 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 S a m p l i n g  Process  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 t h e 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 e n v i r o n m e n t . 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 t h e QP-VAR method ( s o l v e d 15 t i m e s ) and t h e 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 h e MOTAD method ( t h e same number o f t i m e s ) .  Forty five  samples  were drawn f r o m 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 t h e QP-SEMIV, QP-VAR and MOTAD methods, f i f t e e n t i m e s each.  This allowed s u f f i c i e n t income-risk estimates f o r s t a t i s t i c a l  testing. From e a c h o f t h e samples drawn f r o m t h e normal p o p u l a t i o n s , t h e 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 . was used i n t h e o b j e c t i v e f u n c t i o n o f t h e  This information  QP-VAR method.  The MOTAD  method used t h e c o m p l e t e 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 f r o m t h e mean o r e x p e c t e d income.  F o r t h e samples drawn from  t h e gamma p o p u l a t i o n s t h e same p a r a m e t e r s were c a l c u l a t e d and i n a d d i t i o n t h e 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 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 were used i n t h e QP-VAR method and 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 x i n a d d i t i o n t o t h e mean a c t i v i t y r e t u r n s were used i n t h e QP-SEMIV method. T a b l e 3,4 shows an example o f t h e p a r a m e t e r s c a l c u l a t e d from one sample drawn f r o m t h e 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 h e QP-VAR.  -.. 54  method.  T.  I t i s n o t e d 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  p a r a m e t e r s o f one  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 R e t u r n s 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  X  X  x  l  0.41  l  x  the  o f t h e numerous samples o b t a i n e d ,  Variance -Covariance Matrix  x  represent  2  3  T a b l e 3.5  2  X  3  Mean A c t i v i t y Returns  0.38  0.60  6.3  0.63  0,46  9.7  1.50  9.5  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 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 t h e Gamma I population.  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 t h e QP-SEMIV u s i n g each  o f t h e 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 R e t u r n s 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 Matrix X  l  x  x X  2  5,6  x  l  2  X  Mean A c t i v i t y Returns  3  9.9  7.0  10.3  44.2  20,4  13.4  20.5  20.5  3  As may be e x p e c t e d , t h e p a r a m e t e r s c a l c u l a t e d f o r t h e  samples  a p p r o x i m a t e d t h e p o p u l a t i o n p a r a m e t e r s , and t h e 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 t h e p o p u l a t i o n p a r a m e t e r s i n each c a s e . 3.4  S o l u t i o n o f t h e Models T h i s s e c t i o n d e s c r i b e s the process o f determining the " t r u e "  income r i s k f r o n t i e r and t h e sample 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 f o r the 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  developed.  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 G e n e r a l  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 t h e method used) s u b j e c t t o s e v e n 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)  Subject to  (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  levels, x X  2  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 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  levels,  returns,  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 c h o s e n 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 b o u n d a r y 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 u s e d . tants  The c o n s -  o f t h e model ( m a t r i x A and v e c t o r b) r e m a i n t h e 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 t h e QP-VAR method t h e 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 t h e t o t a l v a r i a n c e o f t h e income: V  =  [ x-j, x , x > ] 2  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 t h e a c t i v i t y returns data. 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 h e QP-SEMIV method, b u t 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 h e 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 h e MOTAD method, t h e 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 absolute 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 with r e s p e c t t o  - 58 -  t h e i r means. 3.4.2  The ' T r u e ' 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 h e 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 tributed data.  In o r d e r t o d e t e r m i n e t h e 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 h e parameters which c h a r a c t e r i z e these normally 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 sJ x  0.50  0,18  0 .15  0.18  1.00  0 .30  0.15  0.30  1 .50  Subject to  where Xy  AX  £  b  cX  -  X  X , x 2  3  (3.2)  are the a c t i v i t y levels.  A i s t h e 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 t h e 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 ( s e e 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 t h e 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 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 o f t h e 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 t h e 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 h e 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 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  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 t h e 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 h e 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 h e 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 .  Instead o f using 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 i n t h e o b j e c t i v e f u n c t i o n , t h e 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 h e gamma p o p u l a t i o n s , was used ( s e e 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 h e gamma p o p u l a t i o n s (see T a b l e 3 . 1 ) . 3.4.3  T h e 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 h e 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  their estimates.  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 h e t h r e e methods were s o l v e d u s i n g t h e 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 than t h e c o m p l e t e  b u t 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  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 p a r a m e t e r s o f  - 60 -  t h e o b j e c t i v e f u n c t i o n and t h e 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 a n d 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 t h e  income r i s k f r o n t i e r f o r each o f t h e f o u r d i s t r i b u t i o n s  generated,  T h e 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 t h e incomer 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 . applied t o the normally distributed  T h i s method was n o t  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 t h e QP-VAR method a s was shown i n C h a p t e r I I . The model used t o e s t i m a t e t h e 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 t h e 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 t h e 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 -  s o 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 t h e 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 t h e e x p e c t e d income c o n s t r a i n t depend on 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 s a m p l e s , 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 t h e d e t e r m i n a t i o n 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 ,  The  a b s o l u t e d e v i a t i o n s were t h e 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 ; t h e 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 t h e 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 . T h u s , 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 t h e t o t a l a b s o l u t e negat 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 t h e r e s o u r c e constraints.  Table  3.6  The MOTAD Model as A p p l i e d t o a Sample  X  Objective  x  l  2  X  3  Function  Y  l  1  Y  2  1  Y  3  Y  1  Obtained From the Normal P o p u l a t i o n I T  4  1  Y  1  5  1  Y  7  1" •  Y  8  1  Y  9  1  Y  10  1  l l  Y  1  Y  RHS  12  1  Minimize  2. 0  0. 4  0. 3  9.0  2. 5  1. 2  1.0  .14.5  1. 0  0 . 7 " 1.2  4 .0  3 .9  3.0  1. 5  2 4  2.5  3 0  0 3  C. 2  5 9  10 1  8.9  TI  0 1  0 4  -1.5  Absolute  T2  -0 2  0 9  0.6  Deviations  T3  0 5  1 8  1.7  Constraints  T4  0 3  0 0  -0.4  T5  -0 5  0 4  -2.9  T6  0 3  0 1  -  T7  -0 9  T8 T9  Resource Constraints  Exoected Income Constraint  -2  9.0 33.5 L  10.5  x 1  0 •1  0 1  0 1  1.0  0 5  -0 5  2.5  -0 4  -0 9  -0.4  T10  0. 3  -0 1  0.8  Til  0. 1  0. 5  -1.1  T12  -o'. 2  -0. 2  0.3  0 1  .5  3  20. 5  0 1•  0 0  1 V  1  0  1 i  0  0  1  0  1 1  0  -  62  -  In o r d e r t o compare t h e QP-VAR a n d MOTAD e s t i m a t e s o f t h e i n c o m e risk f r o n t i e r with the true income-risk f r o n t i e r f o r normally  distributed  d a t a , t h e l e v e l s o f r i s k a r e measured as t h e m a g n i t u d e o f t h e s t a n d a r d , d e v i a t i o n o f t h e t o t a l income ( c a l c u l a t e d  e x - p o s t from t h e MOTAD  solutions).  T h i s p r o c e d u r e was c h o s e n by a s s u m i n g 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 t h e 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 tributed  ( s e e S e c t i o n 2 . 4 ) . In C h a p t e r I I 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  s q u a r e r o o t o f t h e 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 comp 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 t h e gamma d i s t r i butions.  T h e 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 t h e QP-VAR and MOTAD  solutions. 3.4.4  .  Analysis o f the Solutions 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 h e r e s u l t s o b t a i n e d u s i n g t h e  t h r e e methods were t h e 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*,  establishing  i f t h e 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 are 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.  -  FIGURE 3.2  63  -  B i a s i n t h e Income-Risk F r o n t i e r E s t i m a t o r  true income-risk  r  l  r  l"  r  2  r  2  r  3  I f t h e d i f f e r e n c e s shown as AA', BB' and C C  r  frontier  3* risk  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 h e 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  frontier  is considered biased. (2)  Dispersion o f the estimates;  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 as  p r o v i d e d by t h e 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 Income  ^  A D i s p e r s i o n C o m p a r i s o n Between t h e E s t i m a t e s P r o v i d e d by Two Methods 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 t h e 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 t h e 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 t h a n t h e v a r i a n c e o f t h e 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 t h e 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 t h e t r u e income risk frontier.  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 ( s e e C h a p t e r I I , f i g u r e 2 . 2 ) , the r e s u l t s o f t h i s comparison risk levels criterion.  w i l l be t h e same as t h o s e c o n s i d e r i n g t h e  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 t h e 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 t h e 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 t h e 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 t h e o t h e r hand, t h e QP-SEMIV method, which uses t h e A c t i v i t y as a r i s k m e a s u r e , was p r o p o s e d as an a p p r o x i m a t i o n o f t h e Income  Semivariance Semivariance  method, which uses t h e 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.  This  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 t h e 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 t h e s o l u t i o n s p r o v i d e d by t h e t h r e e methods, t h e T o t a l Income  S e m i v a r i a n c e ( a s 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 t h e QP-SEMIV method p r o v i d e d t h e s m a l l e s t income s e m i v a r i a n c e o f t h e 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 w o u l d i m p l y t h a t t h e QP-SEMIV p r o v i d e s t h e b e s t s o l u t i o n o f t h e methods u n d e r s t u d y when t h e d i s t r i b u t i o n o f t h e 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 r e e s o f skewness were  generated.  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. ment was t o d e t e r m i n e  The purpose o f t h i s e x p e r i -  t h e d i f f e r e n c e s among t h e s o l u t i o n s p r e s e n t e d by each  method when t h e d e g r e e o f skewness c h a n g e s .  A p r i o r i one may e x p e c t t h a t as  the d e g r e e o f skewness i n c r e a s e s , t h e d i f f e r e n c e s i n t h e 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 w o r d s , 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 t h e 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 t h e degree o f skew-  ness o f t h e p o p u l a t i o n s . 3.6  Summary A s e t o f experiments  was d e s i g n e d t o t e s t t h e a b i l i t y o f t h e 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  frontiers.  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 determined.  E s t i m a t e s o f t h e 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  r a n d o m l y drawn sample d a t a from t h e p o p u l a t i o n s and t h e mean r i s k  estimates  o b t a i n e d w i t h each method were compared t o t h e t r u e l e v e l s o f r i s k t o establish bias.  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 as p r o v i d e d u s i n g  each method was. a l s o compared. Two a s s u m p t i o n s (1)  were i m p l i c i t l y made t h r o u g h o u t  this analysis:  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  -  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 and h i s e x p e c t e d u t i l i t y i s n o t 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 t h e skewness moment. The f i r s t a s s u m p t i o n i s w i d e l y a c c e p t e d and i t has been used  implicitly  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 a s s u m p t i o n  gamma d i s t r i b u t i o n s .  i s r e l e v a n t f o r t h e a n a l y s i s as a p p l i e d t o  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, t h e QP-  SEMIV method as a p p l i e d t o t h e p o p u l a t i o n d a t a c a n n o t g e n e r a t e a t r u e i n c o m e r i s k f r o n t i e r ( s e e 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 n o t p o s s e s s a quadratic u t i l i t y  function.  - 67  -  CHAPTER IV THE RESULTS  The p u r p o s e 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 obt 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 popul 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 returns varied.  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 t h e 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 c o m p l e t e s e t o f sample e s t i m a t e s o f r i s k may be f o u n d on T a b l e s A . l t o A.10 o f t h e  4.1  Appendix.  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 Returns This s e c t i o n presents the r e s u l t s corresponding to the f i r s t  situation analyzed, i . e . , normally d i s t r i b u t e d a c t i v i t y returns with 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 . QP-VAR and MOTAD mean e s t i m a t e s o f r i s k o f the t o t a l income.  T a b l e 4.1 shows the  measured by the s t a n d a r d d e v i a t i o n  B r a c k e t e d f i g u r e s b e s i d e t h e mean e s t i m a t e s o f  r i s k are the value 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 values.  - 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 t h e T r u e P o p u l a t i o n V a l u e s 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. Normal D i s t r i b u t i o n s w i t h Low Degree of C o r r e l a t i o n  Levels o f Expected High  Income  Medium  True Risk Values  5.7  QP-VAR E s t i m a t e s  5.9 ( 1 . 4 2 )  MOTAD E s t i m a t e s  6.0 (2.00)  ( 2 )  Low  4.0  2.6  4.1 (1.11)  2.7 (1.66)  4.1 (1.00)  2.7 (1.24)  ^ R i s k i s measured by t h e s t a n d a r d d e v i a t i o n o f t h e t o t a l (2) ' . 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 v a l u e s .  income.  v  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 n o t s i g n i f i c a n t l y d i f f e r e n t t o t h e 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 t h e d i f f e r e n c e s between t h e t r u e  and t h e mean e s t i m a t e s a r e no l a r g e r t h a n 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 t h e methods a r e a p p l i e d t o t h e same number o f samples randomy drawn from t h e p o p u l a t i o n ( s e e T a b l e s A . l \8I.;A>2. o f t h e A p p e n d i x . )  - 69 -  W i t h r e s p e c t t o t h e d i s p e r s i o n o f t h e QP-VAR and MOTAD e s t i m a t e s , t h e v a r i a n c e s o f t h e QP-VAR e s t i m a t e s were always s m a l l e r . than t h e v a r i a n c e s o f t h e 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 t h e MOTAD and QP-VAR E s t i m a t e 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  L e v e l s o f e x p e c t e d Income High  Medium  Low  Mean V a r i a b i lity Coefficient  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 1=1  SD. 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 t h e e s t i m a t e s a t an income i and MR i s t h e mean r i s k l e v e l e s t i m a t e d a t income i . T h e 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 t h e c a s e o f t h e MOTAD e s t i m a t e s .  However,  none o f t h e 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 t h e F s t a t i s t i c t e s t was applied.  Most o f t h e F v a l u e s (1,18, 1,38 and 1.43 f o r l o w , 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 d e g r e e o f d i s p e r s i o n o f t h e 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 b o t h methods.  T a b l e 4.3 shows t h e range v a l u e s o f t h e  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 t h e t r u e v a l u e s of risk.  F o r example, a t a h i g h l e v e l o f income, 9.5% o f t h e e s t i m a t e s  made u s i n g t h e 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 t h e T r u e R i i F V a l u e s . 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  Levels o f Expected High  True Risk Values  •  •  5.7  QP-VAR R i s k Range:  Norma"! ~~  Income  lied i urn  Low  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  68% P r o b a b i l i t y  5.3 -  6 5  3,7 -  4.5  2.4 -  3.0  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  - 3.2  MOTAD R i s k Range  i s r e a s o n a b l y low.  None o f t h e 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 t h e 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 o u t  - 71 -  o f 20 e s t i m a t e s o f r i s k t h e m a g n i t u d e o f t h e 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 m a g n i t u d e o f t h e e r r o r when compared t o t h e 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 d e g r e e 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 h e 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 t h e i n c o m e - r i s k frontier.  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 n o t s u f f i c i e n t t o  d e m o n s t r a t e 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 t h e other.  I t s h o u l d be n o t e d 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 small samples.  However, t h e 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 t h e 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.  O n l y 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 t h a n 8 y e a r s d a t a o r w i t h more t h a n 15 y e a r s d a t a i n t h i s t y p e o f a n a l y s i s (1,- 12, 1 6 ) .  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 Returns 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 d e 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 t h e 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 t h e 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 •'  Risk levels Estimated by QP-VAR and M O T A Q e j ^ ^ t j g W u e Risk Values for Three Levels ofTxpected Income NonriaT D i s t r i b u t i o n with "High Degree of C o r r e l a t i o n ( 1 }  Mpan  Levels  6.4  QP-VAR Estimates  6.5  MOTAD Estimates  7.6  (1)  Expected  Income  Medium  High True Risk Values  of  Low  5.1  3.4  (0.38)  4.8 (1.57)  3.2  (4.28**)  5.8  3.9 (2.3*)  (3.68**)  (1.3)  Risk i s measured by the standard deviation o f the t o t a l income. Figures between brackets are the t s t a t i s t i c values.  *  S i g n i f i c a n t at 5% l e v e l of s i g n i f i c a n c e .  **  S i g n i f i c a n t at 1% l e v e l of s i g n i f i c a n c e . The data shown i n Table 4.4 may be graphed i n an income r i s k plane  as i l l u s t r a t e d i n Figure 4 . 1 . Figure 4.1 Income  The True Population Income-risk F r o n t i e r and the Income-Risji F r o n t i e r as Estimated by QP-VAR and MO IAD. Normal D i s t r i b u t i o n . High Degree of C o r r e l a t i o n True income r i s k f r o n t i e r  high  OP-VAR estimate  MOTAD estimate edium.  .ow  S3  ztpzs  T T  Risk (Standard Deviation o f 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 h e QP-VAR e s t i m a t e s and t h e t r u e l e v e l 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% o r 5% LOS ( s e e 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 t h e 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 w h i c h 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 t h e 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, t h e MOTAD e s t i m a t e s a r e p o s i t i v e l y b i a s e d and t h e m a g n i t u d e o f t h e b i a s f l u c t u a t e d f r o m a p p r o x i m a t e l y 1 3 % ( 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 i n c o m e ) . 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 t h e 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 a n d 3.36 f o r t h e 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  t h e 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 a n d 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 a n d QP-VAR E s t i m a t e s o f R s k a t T h r e e L e v e l s o f Income. Normal D i s t r i b u t i o n , High D e g r e e o f C o r r e l a t i o n  Levels o f Expected Hi gn  Med i. urn  Income  Mean V a r i a b i Low" 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  1.06  0.58  0.36  0.17  MOTAD E s t i m a t e s  1,19  1.39  1.21  0.21  - 74 -  As may be seen i n T a b l e 4.5, t h e v a r i a n c e s o f t h e MOTAD e s t i m a t e s more than d o u b l e d t h e QP-VAR e s t i m a t e s e x c e p t a t t h e h i g h l e v e l o f income *.  T h e s e 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 h e 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 t h e QP-VAR and MOTAD estimates 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 T r 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 , 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  True Risk Values  6.4  5.1  QP-VAR R i s k Range: 95%Probability . 4 . 5  - 8.6  3.3 -  5.5  - 7.6  98% P r o b a b i l i t y  5.4  -  68% P r o b a b i l i t y  6.5  68% P r o b a b i l i t y  Low 3.1  6.3  2.0  - 4.4  4.0  -. 5.6  2.6  - 3.8  9.8  3.4  - 8.2  1.7  - 6.1  - 8.7  4,6  - 7.0  3.8  - 5.0  MOTAD R i s k Range:  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 t h e 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 n o t even i n c l u d e t h e t r u e v a l u e a t t h e 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 t h e 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 t h a n t h o s e a t l o w e r 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 not  include  -  o f r i s k made u s i n g t h e MOTAD method w i l l  the true values o f r i s k .  F u r t h e r m o r e , t h e r i s k range o f t h e  MOTAD e s t i m a t e s a r e w i d e r t h a n t h e QP-VAR e s t i m a t e s , w h i c h i m p l i e s the p r o b a b i l i t y o f e r r o r s i n t h e e s t i m a t e s i s s m a l l e r  that  i n t h e QP-VAR method.  I t i s a l s o i m p o r t a n t t o n o t e t h a t both methods a r e l e s s e f f i c i e n t i n t h i s c a s e t h a n 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 case.  This i s r e f l e c t e d i n larger variances  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 estimates.  A reason  and l a r g e r mean 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 , t h e 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 t h e 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 levels.  Thus, small  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 w h i c h o c c u r when t h e  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 s a m p l e s , g e n e r a t e l a r g e r  fluctuations  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 d e g r e e 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 unbiased estimator o f the income-risk f r o n t i e r .  an  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  estimates  o f t h e i n c o m e - r i s k f r o n t i e r and t h e 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 t h a n t h a t o f t h e QP-VAR e s t i m a t e s .  Additionally, the levels o f e f f i c i e n c y  o f b o t h methods a r e l o w e r t h a n i n t h e 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 Returns 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 t h e 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 -  returns.  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 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 ) as compared t o t h e true levels o f risk. T a b l e 4.7  Mean R i s k L e v e l s ^ a s E s t i m a t e d by QP-SEMIV, QP-VAR and MOTAD Methods and t h e T r 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 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 o f Correlation  Levels o f Expected High  Income  Medium  Low 18.1  True Risk Values  44.4  27.2  QP-SEMIV E s t i m a t e s  46.3 (0.78)  29.1  (1.57)  19.6 (1.32)  QP-VAR E s t i m a t e s  64.4 (7.69**)  39.7  (7.96**)  27.1  MOTAD E s t i m a t e s  68.7 (6.53**)  45.7  (6.63**)  29.9 (6.70**)  (7.14**)  0) R i s k i s measured as 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 t h e 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 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 t h e t r u e v a l u e a t 5% and 1% LOS a t t h e d i f f e r e n t l e v e l s o f e x p e c t e d income.  The QP-VAR a n d 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 t h e m a g n i t u d e o f t h e QP-VAR b i a s f l u c t u a t e d between 4 5 % a t t h e h i g h and medium income l e v e l s and 4 9 % a t t h e l o w 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 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 each l e v e l o f e x p e c t e d TABLE 4.8  income,  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 T h r e e L e v e l s o f E x p e c t e d Income. Gamma Dist 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 E x p e c t e d Income High QP-SEMIV QP-VAR MOTAD  Medium  Low  88.6  22.1  19.4  102.0  37.2  24.0  68.9  42.2  207.4  The v a r i a n c e s o f t h e QP-VAR e s t i m a t e s were n o t s i g n i f i c a n t l y  different  from t h e v a r i a n c e s o f t h e 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 t h e low l e v e l o f income. T a b l e 4.9 shows t h e range o f t h e r i s k l e v e l s by each method as compared t o t h e t r u e r i s k v a l u e s .  estimated  The ranges o f t h e  MOTAD e s t i m a t e s a r e w i d e r t h a n t h e QP-SEMIV a n d QP-VAR r a n g e s o f t h e i r estimates.  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 n o t even i n c l u d e t h e 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 t h e QP-  SEMIV e s t i m a t e s i s n a r r o w e r and i t i n c l u d e s t h e 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 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 t h e T r 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 s , Low Degree o f C o r r e l a t i o n .  Levels o f Expected  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%  Probability  High  Medium  44.4  27.2  Income Low 18.1  27.9  -  64.7  19.7  -  38.5  10.8  -  28.4  37.1  -  55.5  24.4  -  33-8  15.2  -  24.0  QP-VAR R i s k Range: 95%  Probability  44.2  -  84.6  27.5  -  51.9  17.3  -  36.9  68%  Probability  54.3  -  75.5  33.6  -  45.8  22.2  -  32.0  MOTAD R i s k Range: 95%  Probability  40.1  -  97.7  29.1  -  62.3  15.7  -  4,1.7  68%  Probability  54.5  -  83.3  37.4  -  54.0  22.3  -  35.2  Thus t h e QP-SEMIV method i s t h e o n l y method w h i c h p r o v i d e s unb i a s e d e s t i m a t e s o f t h e t r u e income r i s k f r o n t i e r .  Furthermore, the  d i s p e r s i o n o f t h e 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 t h a n t h e MOTAD e s t i m a t e s , b u t n o t s m a l l e r t h a n t h e d i s p e r s i o n o f t h e QP-VAR e s t i -  - 79 -.  mates.  T h e r e s u l t s s u g g e s t t h a t u n d e r 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 s m 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 t h e 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 t h e QP-VAR method.  T h e MOTAD method may be c o n s i d e r e d t h e l e a s t e f f i c i e n t  s i n c e t h e m a g n i t u d e o f i t s b i a s and t h e 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 t h e v a r i a n c e i s t h e 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 Returns 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 t h e 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 d e g r e e o f c o r r e l a t i o n among activity returns.  T a b l e 4.10 shows t h e 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 t h e 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 at three l e v e l s o f expected  T a b l e 4.10  income.  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 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 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 Levels o f Expected High  Income  Medium  Low  QP-SEMIV  45.3 (1.12)  29.6 (1,18)  24.2  (0.86)  QP-VAR  76,1  (5.63**)  52.9 (6.72**)  38.7  (7.75**)  MOTAD  89,8  (6,51**)  60.2  43.3 (6.70**)  True  Values  49.8  32,6  (6.63**)  22.5  - 80 -  Legend:^  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 t h e  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  risk  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 T r 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 IncomeR 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 t h e QP-SEMIV mean e s t i m a t e s and t h e 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 t h e t r u e v a l u e s a t 1% LOS ( s e e T a b l e s 9 and 10, A p p e n d i x ) 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 -  biased.  The m a g n i t u d e o f t h e QP-VAR b i a s f l u c t u a t e d from 52% a t t h e h i g h  l e v e l o f e x p e c t e d income t o 7 2 % a t t h e 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 t h e h i g h l e v e l o f income and 92% f o r t h e 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 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 each l e v e l o f e x p e c t e d T a b l e 4.11  income.  V a r i a n c e s o f t h e QP-SEMIV, QP-VAR a n d MOTAD E s t i m a t e 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. 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  Levels o f Expected  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 h e 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 t h e MOTAD e s t i m a t e s have t h e 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  t h e v a r i a n c e s o f t h e 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 t h e 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 t h e QP-SEMIV v a r i a n c e s were compared t o t h e QP-VAR v a r i a n c e s . results  These  s u g g e s t t h a t t h e QP-SEMIV i s t h e o n l y u n b i a s e d method and t h a t t h e  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 t h a n t h e d i s p e r s i o n o f t h e MOTAD  - 82 -  estimates and not l a r g e r than the d i s p e r s i o n 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 s k and a better one than the QP-VAR and MOTAD methods for gamma d i s t r i b u t e d r e turns with high degree of c o r r e l a t i o n among them. It i s i n t e r e s t i n g to observe that as occurs in the normal d i s t r i b u t i o n c a s e , the l e v e l of e f f i c i e n c y of a l l methods i s lower when the degree of c o r r e l a t i o n among a c t i v i t y returns i s high than when i t i s low.  The reason for t h i s may be s i m i l a r to that discussed for the  normally d i s t r i b u t e d returns case,  F i n a l l y , Table 4,12 shows the range of the r i s k l e v e l s 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 b u t i o n s , High Defence of C o r r e l a t i o n  Levels of Expected Income High  True Risk Values  Medium  49,8  Low  32.6  22.5  QP-SEMIV Risk Range: 95% P r o b a b i l i t y  14.3  -  76.3  10.0  -  49.2  9.0  -  39.7  68% P r o b a b i l i t y  29.8  -  60.8  19.8  -  38.8  16.6  -  31.8  95% P r o b a b i l i t y  39.9  - 112.3  29,5  76.3  22.5  -  54.9  68% P r o b a b i l i t y  58.0  -  41.2  64.6  30.6  -  46.8  QP-VAR Risk Range 94.2  -'83  r  .. table, cont'd.  Levels of Expected Income High  Medium  Low  MOTAD Risk Range 95% P r o b a b i l i t y  42.4  -  137.2  27.8  -  92.4  19.3  -  67.3  68% P r o b a b i l i t y  66.1  -  113.5  44.1  -  76.3  31.3  -  55.3  Risk ranges are wider in t h i s case than f o r the case of gamma d i s t r i b u t i o n s with low c o r r e l a t i o n among a c t i v i t i e s .  This f a c t implies that the method  are less p r e c i s e when the degree of c o r r e l a t i o n among a c t i v i t y returns i s high. 4.5  Two V a l i d a t i o n s of the QP-SEMIV Method The QP-SEMIV method has been used to determine the true  r i s k f r o n t i e r for the gamma case.  income-  Doubts may be r a i s e d regarding the  a b i l i t y of t h i s method to approximate the true minimum income semivariance.  Two v a l i d a t i o n s of t h i s method are provided here.  These  v e r i f i c a t i o n s may not be a b s o l u t e l y conclusive but may provide an i n d i c a t i o n o f the aptitudes of the QP-SEMIV method to minimize income semivariance. The income semivariance  was c a l c u l a t e d ex post with the s o l u t i o n s  provided by the three methods as applied to samples from the gamma population.  In a l l cases, the QP-SEMIV method provided s o l u t i o n s with the  smallest income semivariance.  Table 4.13 shows the mean income semi-  variance as c a l c u l a t e d 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 d e g r e e o f c o r r e l a t i o n among t h e 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 P o s t w i t h t h e 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 ~  Levels o f Expected Method Used:  Income  High  Medium  Low  QP-SEMIV  1415,2  650.3  226.2  QP-VAR  1623,0  680.3  289.9  MOTAD  1704.1  763.4  334.1  T h e s e 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 t h a n t h e QP-VAR and MOTAD methods.  Hence, t h e 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 d e g r e e s o f s k e w n e s s . 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 t h e d e g r e e 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 t h e a c t i v i t i e s as p r o p o s e d by t h e  - 85 -  two methods was p e r f 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 t h e  following: TD =  0.04 + 0.73 SK ,  (4.1)  where TD i s a measure o f t h e m a g n i t u d e o f t h e d i v e r g e n c e between t h e QP-VAR and QP-SEMIV s o l u t i o n s and SK i s t h e skewness c o efficient. 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 t h e 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, there  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 t h e r e s u l t s p r o v i d e d by t h e methods and t h e skewness c o e f f i c i e n t . of skewness the g r e a t e r the d i v e r g e n c e s .  The g r e a t e r t h e d e g r e e  When t h e 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 t h e s o l u t i o n s p r o v i d e d by b o t h methods.  E x a c t l y t h e same i s e x p e c t e d t o o c c u r i f t h e 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 t h e 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 t h a n t h e QP-VAR. 4.6  T e s t i n g the  Hypothesees  Four h y p o t h e s e s were s t a t e d i n C h a p t e r I . On t h e 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 .  This section reports I t i s important to r e -  c a l l t h a t t h e 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 t h e e f f i c i e n c y o f t h e 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 c o m p l e t e  frequency 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 w h e t h e r 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 c o m p l e t e population data).  S i n c e i n r e a l w o r l d p r o b l e m s t h e 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  using  c o m p l e t e 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 h y p o t h e s e s were s t a t e d i n terms o f t h e p r o p e r t i e s o f t h e methods when a p p l i e d u s i n g sample d a t a .  Thus, i n order  t o t e s t t h e h y p o t h e s e s 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 as d e s c r i b e d i n C h a p t e r  experiments  I I I where t h e methods were a p p l i e d u s i n g numerous  samples r a n d o m l y 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  experiments  I I have been used t o  and t h e 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  t e s t the hypotheses. 4.6.1  Hypothesis I "The QP-VAR a p p r o a c h 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 t h e 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 t h e a c t i v i t y returns a r e normally 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 h e 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 t h e income r i s k f r o n t i e r when a p p l i e d using normally d i s t r i b u t e d population data.  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 t h e 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 populations o f a c t i v i t y returns.  distributed  In o t h e r w o r d s , t h e QP-VAR n o t 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 t h e income r i s k f r o n t i e r u s i n g normally d i s t r i b u t e d population data but also the estimates obtained 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  using  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 l a r g e r than the variance of- the MOTAD estimates when the c o r r e l a t i o n 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 c o r r e l a t i o n was high.. 4.6.2  Hypothesis 2 "The MOTAD method provides an unbiased estimator of the actual  population income-risk f r o n t i e r only i f the f o l l o w i n g two conditions are satisfied: a)  The a c t i v i t y returns are normally 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 the a c t i v i t y  returns  are c l o s e to z e r o " . It was shown i n Chapter II  that the MOTAD method only provides  close approximations t o . t h e income r i s k f r o n t i e r when applied using normally d i s t r i b u t e d data with low degree of c o r r e l a t i o n among a c t i v i t y  returns.  It was also shown that i f these conditions are not met the MOTAD s o l u t i o n s are not appropriate.  The r e s u l t s of the experiments confirmed these  conclusions for the estimates of the income r i s k f r o n t i e r obtained using sample d a t a .  The MOTAD estimates were s i g n i f i c a n t l y biased i n a l l  except when the samples were drawn from normally d i s t r i b u t e d - l o w •populations. only  situations correlation  Therefore, hypothesis 2 i s accepted. The MOTAD method was not ••  biased but also the method that provided the more dispersed estimates  under a l l s i t u a t i o n s .  Thus the MQTAD method was the l e a s t e f f i c i e n t of the  methods when applied to highly c o r r e l a t e d 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 t h e b i a s o f t h e QP-VAR e s t i m a t e s ) . 4.6.3  Hypothesis  3.  " I f t h e a c t i v i t y returns a r e non-normally  d i s t r i b u t e d , t h e 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 o r s o f t h e a c t u a l population income-risk 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 o u t t h a t t h e QP-  VAR method does n o t 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 t h e 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 4.4  d i s t r i b u t e d data.  S e c t i o n s 4.3 and  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 t h e d e g r e e 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. h y p o t h e s e s 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  Hence  t o t h e QP-VAR  estimates. The c o n c l u s i o n s o f t h e 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  I I 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 t h e second p a r t o f h y p o t h e s i s 3 as well. 4.6.4  Hypothesis  4.  "When t h e 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 t h e 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 t h e a c t u a l t i o n income r i s k f r o n t i e r . "  popula-  - 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 f r o m t h e d i s c u s s i o n i n C h a p t e r I I : (1) T h e 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 t h 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 t o n o n - n o r m a l l y  distri-  buted frequency 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 higher order moments a r e n o t 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 t h e d e c i s i o n maker i s n o t q u a d r a t i c ) ;  (2) T h e QP-SEMIV was p r o p o s e d as an a p p r o x i m a -  t i o n o f t h e 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 t h e 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  risk  f r o n t i e r when a p p l i e d u s i n g samples r a n d o m l y drawn from gamma p o p u l a t i o n s . Thus, hypothesis 4 i s accepted.  Furthermore,  t h e QP-SEMIV may be c o n s i d e r e d  t h e most e f f i c i e n t o f t h e t h r e e methods i n s i t u a t i o n s where t h 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 t h e r e s u l t s o b t a i n e d from t h e  experiments  d e s c r i b e d i n Chapter I I I .  R e s u l t s have been r e p o r t e d f o r t h e  four cases regarding frequency d i s t r i b u t i o n o f a c t i v i t y returns considered, 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 a c t i v i t y returns.  U s i n g t h e s e r e s u l t s and t h e c o n c l u s i o n s drawn from t h e t h e o r e -  t i c a l study, t h e f o u r hypotheses 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  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 .  Furthermore,  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 w h i c h 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 h e a s s e r t i o n t h a t t h e 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 process  o f r e p o r t i n g t h e r e s u l t s i t was ob-  s e r v e d t h a t t h e e f f i c i e n c y o f t h e methods t e n d t o be l o w e r when t h e d e g r e e 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 t h e h i g h e r t h e d e g r e e 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 h e l a r g e r must be t h e sample s i z e . cords  T h u s , a g r e a t e r number o f a c t i v i t y r e -  a r e r e q u i r e d i n o r d e r t o keep t h e e f f i c i e n c y o f t h e methods a t  acceptable  levels.  -  -  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 p e r f o r m a n c e  o f t h e methods i n a more  r e a l i s t i c m o d e l , d a t a f r o m 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 c r o p 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 c r o p 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 m o d e l : (1)  Wheat grown a f t e r summer f a l l o w ;  (2)  (3)  B a r l e y grown a f t e r summer f a l l o w ;  (4)  (5)  O a t s grown a f t e r summer f a l l o w ;  (7)  Rapeseed;  (8)  Fescue seed;  (9)  (6)  Wheat grown a f t e r s t u b b l e ; B a r l e y grown a f t e r s t u b b l e ; O a t s grown a f t e r s t u b b l e ; (10)  A l s i k e seed;  Alfalfa.  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 t h e 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.  v a r i e d w i t h t h e method used.  The o b j e c t i v e f u n c t i o n  I t i s intended to minimize the t o t a l absolute  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 SEMIV).  (QP-  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 t h e 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  anc 2  '  *  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  3)  2  renting land (A  2  g)  or buying other v a r i a b l e inputs 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 m a c h i n e r y (A  + a  2,  and so f o r t h  "  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  i n mind t h e l e g e n d i n F i g u r e  4.7.  keeping  rltSUKt  b. I:  R.H.S.  Variable Inputs  Cropping Activities  Hired Labour  •Family Labour  Rented Land  Own Land  H < M H M tt CO  Borrowed Capital  Own Capital  > o  s t r u c t u r e o f t h e Models  Constraints Objective Capital  Control  A A 2 .3 + a  + a  A  Land  Control  Labour  A2 . 5 A + a  a  s'  * 0  4,7  * 0  A 5,7  * 0  + a  A  A" A" 6 ,1 6 , 2 a  a  A  A  m  Bounds  i  A" 6,3 a  A  m B  2  A" A" 6 ,4 6 ,5 A " 6,6 a  A  3  a  A  m  m B  a  A  B  4  m B  5  A  + a  6,7 A ~ 6,8 a  A  m B  6  ( S u p e r s c r i p t s , when shown, r e p r e s e n t t h e type o f non zero element i n t h e s u b m a t r i x . ) A i s a general B  submatrix with  some n o n z e r o  elements,  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  * 0  l 3,7  A A  3  A  B  Legend:  A  +  D  Cropping A c t i v i t y Resource Use.  Structural  A 2,8  + a  A  Control  E x p e c t e d N e t Income Summary Row  Minimize  M e a s u r e o f r i s k w h i c h d e p e n d s on method IIRPH  Function  submatrix.  i  R  - 93 -  A l l submatrices V e c t o r s Ag ••. and Ag ^ capital. Ag crops  i n t h e n e t income summary row a r e row v e c t o r s .  account f o r t h e costs o f using d i f f e r e n t types o f  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 ) ,  ( f a m i l y l a b o u r c o s t s ) , A g 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 s  from  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 e x p e n s e s f o r t h e d i f f e r e n t a c t i v i t i e s .  Negative 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 t h e 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 n e t income i s not e q u i v a l e n t t o f a r m p r o f i t b e c a u s e t h e 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 a n d 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 a n d 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 m a t e l y 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 c o l u m n s . . 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 t h e p o s 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 c a s h c a p i t a l p l u s a c a s h b o r r o w i n g a t $25,000 p e r y e a r .  A t o t a l o f 1500 hours  s i d e r e d t o be a v a i l a b l e  capacity  estimated  o f f a m i l y l a b o u r was  con-  i n a d d i t i o n t o 1000 h o u r s 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 h e l a s t e i g h t y e a r s o f r e c o r d s regarding y i e l d s p e r acre, p r i c e s and costs o f t h e d i f f e r e n t production a c t i v i t i e s considered.  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 ,  variances o r absolute deviations o f the  activity  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 .  semi-  - 94-  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 u n d e r t h e same constraints  a n  d  a  t t h e same l e v e l s o f n e t income.  The maximum n e t 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 t h e e x p e c t e d r e t u r n o f t h e 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 p e r y e a r .  The model was  solved to minimize  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 p e r y e a r . T a b l e 5.1  shows t h e minimum r i s k l e v e l s as e s t i m a t e d by t h e  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 t h e S q u a r e Root o f t h e S e m i - v a r i a n c e ) as P r o v i d e d by t h e QP-SEMIV, QP-VAR and MOTAD Model S o l u t i o n s f o r 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 LEVELS  M E T H O D U S E D  OF  $6,000  NET  INCOME $5,000  $4,000  QP-SEMIV  2315.9  1380.9  748.3  QP-VAR  4053.0  2666.9  1340.4  MOTAD  4114.1  2708.2  1485.8  The t o t a l s e m i v a r i a n c e was c a l c u l a t e d ex p o s t from t h e s o l u t i o n s p r o v i ded by t h e QP-VAR and MOTAD methods, F i g u r e 5.2 shows t h e e f f i c i e n t i n c o m e - r i s k f r o n t i e r as e s t i m a t e d by t h e t h r e e methods.  As i n T a b l e 5,1, r i s k i s e x p r e s s e d as t h e s q u a r e  root o f the semivariance.  -95 -  The I n c o m e - R i s k F r o n t i e r as E s t i m a t e d by QP-SEMIV. QP-VAR and MOTAD Methods f o r a Farm i n t h e Peace R i v e r D i s t r i c t o f QP-VAR estimate  Income ($)  6,000 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 t h e QP-SEMIV method i m p l i e d t h e s m a l l e s t l e v e l s o f r i s k (measured as t h e s q u a r e r o o t of the semivariance).  The MOTAD method a l w a y s 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 t h a n t h e 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 a s recommended i n t h e s o l u t i o n s p r o v i d e d by t h e 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 t h e Peace R i v e r A r e a o f B r i t i s h Columbia ( N e t Income = $6000) ACTIVITIES  MODE L  Acres Acres Acres Acres of of of of WAS WAF BAF BAS  PROPOSED Acres Acres of of OAS FES  Acres of ALF  U S E D  -  5  -  -  42  82  62  75  20  MOTAD  77  QP-SEMIV QP-VAR  164  -  61  34  -  -  91  -  -  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 = O a t A f t e r S t u b b l e ; FFS = F e s c u e ; A L F = A l f a l f a . As may be seen i n T a b l e 5..2., t h e l e v e l o f a c t i v i t i e s p r o p o s e d  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 . O n l y 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 n o t a p p e a r 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 t h e 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 t h e 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 .  *  T h u s , 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 t h 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 .  O n l y t h e r e t u r n s o f f e s c u e and  a l s i k e a p p e a r t o be w e a k l y 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 ( s e e 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 so d i f f e r e n t .  explain  why t h e QP-VAR and MOTAD s o l u t i o n s a r e  T h e 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 a c -  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 , w h i c h would e x p l a i n t h e 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 t h e QP-SEMIV method.  T h u s , t h i s example i l l u s t r a t e s t h e m a g n i t u d e o f t h e e r r o r w h i c h may o c c u r i f t h e wrong method i s used.  Considering the relatively  h i g h d e g r e e o f c o r r e l a t i o n o f t h e a c t i v i t i e s and t h e f a c t t h a t t h e d a t a appears t o be n o 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 t h e b e s t s o l u t i o n . To u s e t h e MOTAD method i n s t e a d o f t h e 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 t h e a v e r a g e 8 0 % more r i s k y and t h e QP-VAR method p r o v i d e d p l a n s w h i c h were 70% more r i s k y than t h e QP-SEMIV method. importance  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  o f c h o o s i n g t h e 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-  certainty conditions.  T h u s , t h e phase o f d e c i d i n g w h i c h method s h o u l d be  used i s c r u c i a l f o r t h e r e s u l t s ( a s shown by t h e i m p o r t a n t d i f f e r e n c e s i n t h e r e s u l t s p r o v i d e d by t h e methods i n t h i s e x a m p l e ) , worthwhile  and hence, i t i s  t o d e v 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 t o 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 t h e s p e c i f i c o f t h e problem.  nature  - 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 t h e s t u d y and p r i n c i p a l findings.  Given the 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  p r o p o s e d i n o r d e r t o be a b l e t o a s s e s s t h e e m p i r i c a l r e l e v a n c e o f these 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 f a r m 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  experiments  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  concerned  w i t h t h e c h a r a c t e r i s t i c s o f the methods under t h e a s s u m p t i o n  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. used t o e v a l u a t e t h e methods was t h e i r  The main c r i t e r i o n  a b i l i t y to provide the i n f o r -  mation r e q u i r e d by t h e d e c i s i o n maker t o m a x i m i z e h i s e x p e c t e d  utility.  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 t h e 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 p l a n which m a x i m i z e s h i s e x p e c t e d  utility.  the  - 39  -  The e m p i r i c a l work t e s t e d t h e methods' a b i l i t y t o p r o v i d e incomer 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, t h e 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 w h i c h 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 , t h e 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 s i d e r e d the " t r u e " ones.  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-  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 t h e " t r u e " i n c o m e - r i s k 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 t h e e s t i m a t e s .  frontier  G i v e n 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 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  of  frontier.  A g e n e r a l c o n c l u s i o n w h i c h 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 a r e 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 a s s u m p t i o n ) 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 t h e 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 the d e c i s i o n maker.  This  i n c r e a s e s the complexity to these s t u d i e s given the d i f f i c u l t i e s i n v o l v e d i n knowing t h e 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, there  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 e a s y 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 t h e t h r e e methods a r e :  - 100  1.  -  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 q u a d r a t i c t h e QP-  VAR method may be u s e d , i r r e s p e c t i v e o f t h e d i s t r i b u t i o n o f a c t i v i t y returns. 2.  T h e QP-SEMIV method i s p r o p o s e d as t h e most s u i t a b l e method i f  t h e f o l l o w i n g c o n d i t i o n s a r e 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 n o t 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 n o r m a l , b u t moments o f o r d e r h i g h e r t h a n 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 n o t recommended as b e i n g u s e f u l  b e c a u s e i t i s b i a s e d a n d l e s s e f f i c i e n t t h a n t h e QP-SEMIV method. T h e o n l y s i t u a t i o n where t h e MOTAD method may be used i s when 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 e g r e e 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 is low/ 4.  No o n e o f t h e t h r e e methods was found t o be a p p r o p r i a t e i f t h e  following conditions occur  5.  simultaneously:  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 t h e u t i l i t y f u n c t i o n do n o t v a n i s h  c.  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 t h e a c t i v i t y r e t u r n s i s nonnormal and moments h i g h e r t h a n skewness moments a r e n o t negligible.  As i m p o r t a n t as t h e a c t u a l r e s u l t s o b t a i n e d i s t h e 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 distribution  frequency  f o r i n e m p i r i c a l work,the data source i s normally a r e l a -  t i v e l y s m a l l sample drawn from t h e 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 incomer i s k f r o n t i e r u n d e r a p e r f e c t knowledge s i t u a t i o n i s n o t a s u f f i c i e n t n o r 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 v i e w o f t h e problem  an o b j e c t i v e e v a l u a t i o n o f t h e methods c o n s i d e r e d .  required  The c r i t e r i o n used was  t h 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 t h e 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 j u d g e t h e 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  frequency d i s t r i b u t i o n data i s not s u f f i c i e n t nor necessary 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 t h e 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 c o m p l e t e  frequency  distribution.  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 b u t i f i t has a s m 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 w h i c h p r o v i d e s h i g h l y dispersed estimates.  Thus, the study provides a research procedure t o  t e s t d i f f e r e n t methods w h i c h may be used i n f a r m 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 from a n a l y t i c a l  used i s based on t h e 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  d i s c u s s i o n s based on t h e a s s u m p t i o n  frequency d i s t r i b u t i o n o f the a c t i v i t y returns  o f f u l l knowledge o f t h e  are 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 w h i c h 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  in. designing the experiments  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 b a s e d on t h e c o n c l u s i o n s o b t a i n e d from t h e t h e o r e t i c a l study d e v e l o p e d 6.2  i n the p r e c e d i n g  chapter.  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 f a r m p l a n n i n g u n d e r u n c e r t a i n t y depends., on t h e 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 of a c t i v i t y returns.  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 nature of the u t i l i t y f u n c t i o n o f farmers makers.  In o t h e r words, i t i s i m p o r t a n t  as d e c i s i o n  t o know w h e t h e r 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 w h e t h e r 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 w h i c h a r e p e c u l i a r t o them. would a l s o be i m p o r t a n t  It  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 the d i s t r i b u t i o n o f the gross 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 t h e f o l l o w i n g directions: 1.-  Empirical s t u d i e s to e s t a b l i s h whether quadratic u t i l i t y  a r e i n d e e d unusual i s important applications.  among f a r m e r s  functions  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 .  This  b e c a u s e i n s u c h a c a s e t h e QP-SEMIV method c o u l d have w i d e r 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  frequently  among f a r m e r s , b e f o r e c h o o s i n g t h e method t o be u s e d , i t would be  necessary  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 e a c h c a s e (unless there i s c e r t a i n t y that returns are normally d i s t r i b u t e d ) .  -  2. -  103  -  Empirical studies could i n v e s t i g a t e t h e frequency  distribution of  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 ; necessary  In t h e f i r s t p l a c e , i t would be  t o e s t a b l i s h w h e t h e r g r o s s a c t i v i t y r e t u r n s f o r most  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 p r o d u c e d have been a p p r o x i m a t e l y d i s t r i b u t e d during the l a s t decades. t u r n s a p p e a r t o be non - n o r m a l l y  important normally  In t h e s e c o n d p l a c e , i f t h e r e -  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 t h e s i g n o f skewness o f t h e d i s t r i b u t i o n s and the m a g n i t u d e o f h i g h e r o r d e r moments r e l a t i v e t o t h e skewness moment. I f t h e d i s t r i b u t i o n s a p p e a r 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 w o u l d n o t be n e c e s s a r y .  If this i s  n o t t h e c a s e t h e QP-SEMIV method w o u l d be u s e f u l ( p r o v i d e d  non-quadratic  u t i l i t y functions).  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 p e r f o r m a n c e o f  t h e s e methods i s a f f e c t e d by changes i n t h e sample s i z e .  I t c a n 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 t h e sample s i z e i n c r e a s e s , b u t t h e s e improvements a r e n o t 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 t h e r a t e o f i n c r e a s e i n the sample s i z e . information  G i v e n 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  regarding gross returns over time i s necessary).  be p o s s i b l e t o f i n d an o p t i m a l used i n e m p i r i c a l  4. -  sample s i z e ( o r an o p t i m a l  I t would  r a n g e ) t o be  studies,  I t i s a l s o necessary  t o t e s t t h e p e r f o r m a n c e o f t h e s e methods  in p r e d i c i t i n g actual behaviour o f farmers.  I t i s quite clear that the  -•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 f r o m t h e a c t u a l  p l a n s w h i c h f a r m e r s ' make.  I t would-be important t o study t h e plans  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 to farmers' actual plans.  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 w o u l d 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 ' actual production plans.  I f t h e a p p r o x i m a t i o n s , p r o v i d e d by t h i s method  a r e b e t t e r , t h e QP-SEMIV method c o u l d be used as a t o o l i n p r e d i c t i n g p r o d u c t i o n , p r i c e and f a r m income f l u c t u a t i o n s a t t h e m a c r o - e c o n o m i c level.  - 105 -  REFERENCES  A n d e r s o n , J.R. "Programming f o r E f f i c i e n t P l a n n i n g A g a i n s t Non normal R i s k . " A u s t r a l i a n J o u r n a l o f A g r i c u l t u r a l E c o n o m i c s , 19: 107 - 115, 1975. 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"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 " , A m e r i c a n J o u r n a l o f A g r i c u l t u r a l E c o n o m i c s , 53: 53-62, 1971.  16.  H a z e l l , P.B. a n d 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 submitted to t h e C o n t r i b u t e d Papers s e c t i o n of t h e International 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 Economists a t M i n s k , 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 Y o r k , 1971. .  18.  Kmenta, J . , E l e m e n t s 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 Y o r k , 1971.  19.  L e v y , H., Comment",  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 . 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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 A p p r o a c h to D e c i s i o n Models 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.  Non L i n e a r and Dynamic Programming, A d d i s o n Wesley M a s s a c h u s e t t s , 1964.  "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 : A m e r i c a n Economic Review, 64: 343-441, 1974.  ?  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 t o t h e T h e o r y o f S t a t i s t i c s , M c G r a w - H i l l , New Y o r k , 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 t h e 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 , 1968  Addison-Wesley, Reading, Massachusetts,  - 107 -  S c o t t , J . and B a k e r , 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 " , A m e r i c a n 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 t h e 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 . " A m e r i c a n J o u r n a l o f A g r i c u l t u r a l E c o n o m i c s , 54: 503-506, 1972 Tsiang, S.C, "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 , Skewness P r e f e r e n c e and t h e Demand f o r Money", The A m e r i c a n Economic Review, 62: 354-371, 1972. W o l f , F.L. E l e m e n t s 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 c G r a w - H i l l I n c . , New Y o r k , 1974.  - 108 -  APPENDIX  - 109 -  TABLE A.I  Estimates o f Risk*  as O b t a i n e d U s i n g t h e 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 Distributed Population.  Low Degree o f C o r r e l a t i o n Among  A c t i v i t y Returns.  Data s o u r c e used Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6  L e v e l s o f E x p e c t e d Income i  High  ;  Medium  :  Low  6.0 5.8  4.0 . 4,0  2.5 3.1  6.1  4.1 4.2 3.9 4.7 4.6 4,0  2.7  6.0  2.7 3.2 3.1 2.9  Sample 10  5.7 6.6 7.1 5.8 5.8 6.2  Sample Sample Sample Sample Sample  6.1 4.6 5.9 5.9 5.3  4.3 3.3 4.2 4,2 3.7  2.8 2.2 2.6 2.8 2.4  Mean E s t i m a t e s  5.9  4,1  2.7  Variance  0,32  0,13  0.07  Sample 7 Sample 8 Sample 9  *  11 12 13 14 15  Standard D e v i a t i o n o f Total  Income  4.0 4.4  2.6 2.6 2.8  - 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 t h e MOTAD Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn f r o m Normal P o p u l a t i o n w i t h Low Degree o f Correlation  Data s o u r c e used  ' ' :  i i ^ r ^ j r L e v e l s o f E x p e c t e d Income ^ High  ;  Medium  :  Low  Sample 1  6.5  4,3  2.8  Sample 2 Sample 3  6,0 6.1  4,1 4.2  2.6  Sample 4 Sample 5  5.9 5.9  2.6  Sample Sample Sample Sample Sample  6.7 6.7 5.1 6.2 4.8 6.0  ' 4.0 4,1 4.6 4,6 3.2 4.3 3,4 4,1  6.9 5.8  4.7 3.9  2.7 3.2 2.4  6.0 4.9  4.1 3.6  2.8 2.4  0,18  0.10  6 7 8 9 10  Sample 11 Sample 12 Sample 13 Sample 14 Sample 15 Mean E s t i m a t e s  6,0  Variance  0.38  2.8 2.7 3.3 3.0 2.2 2.9 2.2  - fii  T A B L E  A  -  -  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 t h e 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 of Correlation  3  Data S o u r c e Used  ; L e v e l s o f E x p e c t e d Income ;  High  Sample 1  6.5  Sample 2  :  Medium  :  Low 2.6  6,6  5.2 5.0  Sample 3  6.4  5.1  Sample 4  3.9  5.8 6.2 4.5  4,4 4.8 3.5 4,6 5,5 3.9 3,4 6.3  3.0 3.5 2.3 2.9  Sample Sample Sample Sample Sample Sample  5 6 7 8 9 10  Sample Sample Sample Sample Sample  11 12 13 14 15  6,7 7.6 5.4 5.1 10.2 6.9 7.1  3,3  '  3.5 2.4 2.1 4,2 3.9 3.3 3.2  6.6 6.8  5,1 5.3 4.8 5.1  3.3  Mean E s t i m a t e s  6.5  4,8  3.2  Variance  1.06  0,58  0,36  *  Standard D e v i a t i o n o f Total  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 t h e 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 H i g h Degree o f C o r r e l a t i o n  Data S o u r c e Used  L e v e l s o f E x p e c t e d Income .  High  Sample 1 Sample 2 Sample 3  10.5  ?.  Medium  ;  Low  9.3 '  7.4  7.1 8.4  5.0 6.4  Sample 4  3.1 4.1  7.7  5.9  Sample 5  4.0  7.3 6.6 8.7 7.0  6.6 7.7  5,3 5.0 6.5 5,0 5.5 3.9 5.7 6.0 5.8 5.0 6.2  3.6 3.3 4.5 3.3 3.5 2.4 3.9 4.2 3.9 3.2 4.1  7.6  5,8  3.9  Sample Sample Sample Sample Sample Sample  6 7 8 9 10 11  Sample Sample Sample Sample  12 13 14 15  Mean E s t i m a t e s  7.6 5.7 7.5 7.9 7.5  Variance  *  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 t h e QP-SEMIV Method as A p p l i e d t o F i f t e e n Samples Randomly Drawn f r o m 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  Levels o f Expected  Data S o u r c e Used  High  ;  Income  Medium  Low  23.1 23,9  15.0 18.8  Sample Sample Sample Sample  1 2 3 4  44,2  56.7  27,3 32.1  Sample Sample Sample Sample Sample Sample Sample  5 6 7 8 9 10 11  43.9  29.0  . 19.1  29.1 45.5 38.1 47.3 49.7  25.1 24,3 26.8 25.6 26.5  Sample 14 Sample 15  50.3 58.4 57.8 59.2 42.1  30.8 31 .3 30.2 35.6 37.1  18.3 14.6 19.3 17.4 19.1 21.3 29.2 15.4 28.6 22.2  Mean E s t i m a t e s  46.3  29.1  19.6  Variance  88,6  22.1  19.4  Sample 12 Sample 13  *  34.7 37.2  S q u a r e 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.  15.6 19.6  - 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 t h e 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 S o u r c e Used  L e v e l s o f E x p e c t e d Income !  High  <  ;  Medium  Sample 1  54.5  31,9  Sample 2 Sample 3  62.9  41,9 49.8  Sample 4  10.8 56.5  Sample 5 Sample 6 Sample 7  :  Low 25.3 30.2  32,2  29.3 16.2  56.8  35,1  28.6  55,7 58.2 55.4 73.2 78.6 59.6  24.8 24.7 23.7 24.4 35.7 22.4 24.9  8 9 10 11 12  66.7  36.2 33.1 39,7 4.15 48.9 38.7 37.8  Sample 13 Sample 14  73.7 60.9  51.1 39.3  34.7 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  102,0  37,2  24.0  Sample Sample Sample Sample Sample  Variance  *  S q u a r e Root o f S e m i v a r i a n c e o f t h e 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 * a s 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 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 E x p e c t e d Income .  High  ;  Medium  ;  Low  Sample 1  77.5  47,7  33,4  Sample 2 Sample 3  66.4  44.0  28.4  44,0  32.0  18.5  Sample 4  94.0 66.2 94.3 81.2 75.6 68.0 69.4  52,3 42.5  32.9  Sample 5  58.0 54.4 51,6 54.7  28.1 42.1 28.7 29.1 37.2 27.6 19,9 33.6  Sample 13 Sample 14 Sample 15  61.2 69.7 47.1 .56.4 62.8  45,9 35.2 49.1 29.0 40.6 45.8  19.4 24.6 26.5  Mean E s t i m a t e s  68.7  45,7  29.9  207,4  68,9  42,2  Sample Sample Sample Sample Sample  6 7 8 9 10  Sample 11 Sample 12  Variance  * S q u a r e Root o f S e m i v a r i a n c e o f t h e t o t a l income.  - 116 -  TABLE A.9  Estimates of Risk*  as O b t a i n e d U s i n g t h e 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 S o u r c e Used  :  L e v e l s o f E x p e c t e d Income High  ;  Medium  ;  Low  Sample 1  62.1  44,5  35.0  Sample 2 Sample 3  116.2  85,4  66.1  51.9  59.8 40.1  Sample 4  71.0 67.8 69.3 52.6 108.4 89.9 90.6  45,9 49.0 49,5 37.2  79.0 60.1 78.6 61.2  52.7 45.8  67.8  50.3  38.0 32.4 38.8 29.2 38.0  76,1  52,9  38,7  327,6  136.8  65.6  Sample 5 Sample 6 Sample 7 Sample Sample Sample Sample Sample  3 9 10 11 12  Sample 13 Sample 14 Sample 15  Mean E s t i m a t e s Variance  *  S q u a r e Root o f S e m i v a r i a n c e o f T o t a l  75,2 54.8 62.8  49.1 40.7  Income,  33.7 35.3 35.9 29.8 49.8 41.4 47.0  - 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 ,  of Correlation  Data  Source  v" «• •—  Levels o f Expected  Income  Used :  High  Sample 1 Sample 2  120.2  Sample 3  92.7 60.6  Sample 4 Sample 5 Sample 6 Sample Sample Sample Sample Sample Sample Sample Sample Sample  7 8 9 10 11 12 13 14 15  Mean E s t i m a t e s Variance  ;  65.2  102,4 84.1  Medium  :  Low  84,3 39,3  59.0 30.1  63.0  45.2  40.9 73.5  35.7 47.8 44.5  81.7 56.7 121.4 101.6 64.0 69.1  59,3 70.5 66,3 83,5 53.7 36.7 88.6 74.0 45,0 49.6  57.8 42.5 41.0 32.3 32.4 69.3 51 .1 32.6 28.6  89.8  60,2  43,3  561,7  259.2  144.0  123.0 112.1 92.5  S q u a r e r o o t s e m i v a r i a n c e o f t o t a l income,  - 118  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 t h e 8 Year  Activity  R e t u r n Data C o r r e s p o n d i n g t o a Case 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  Activities  WAFC BAFC OAFC RAFC FESC ALFALFA ALSIKE  :  WAFC  981.7  ;  BAFC  : OAFC  Columbia  ;  RAFC  ;  FESC  806  564.2  502,9  1276,6  774.0  833.4  61.2  681.2  517.1  -167.8  219.2  220.1  10.7  351.1  3.8  497.1  -361.3  400  -147.2  629  -443  : ALFALFA : ALSIKE  1836.1  244.4 443  43.2 1 27.4  235  

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