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A mathematical procedure for selecting among alternative utility functions Schoot, Gerrit Paul van der 1981

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A MATHEMATICAL PROCEDURE FOR SELECTING AMONG ALTERNATIVE UTILITY FUNCTIONS  by  GERRIT PAUL VAN DER SCHOOT B.Sc,  D e l f t U n i v e r s i t y o f Technology, 1974  M.Sc., D e l f t U n i v e r s i t y o f Technology, 1976  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN BUSINESS ADMINISTRATION in THE FACULTY OF GRADUATE STUDIES (The F a c u l t y o f Commerce and B u s i n e s s A d m i n i s t r a t i o n )  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d s t a n d a r d  THE UNIVERSITY OF BRITISH COLUMBIA March 1981 ©  G e r r i t P a u l v a n der Schoot, 1981  In p r e s e n t i n g  this thesis i n p a r t i a l  f u l f i l m e n t o f the  requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t it  the L i b r a r y s h a l l make  f r e e l y a v a i l a b l e f o r r e f e r e n c e and study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . understood t h a t  copying or p u b l i c a t i o n  f o r f i n a n c i a l gain  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  n r _  C  I O  /no \  Hou^eM ^T,  of t h i s thesis  s h a l l n o t be a l l o w e d without my  permission.  Date  It i s  V<^%\.  Columbia  written  - i -  Abstract  This thesis presents a mathematical procedure, c a l l e d the R^-method, for  selecting among alternative u t i l i t y functions to represent a decision  maker's r i s k preference.  A general class of u t i l i t y functions i s introduced  and for f i v e a l t e r n a t i v e members of this class, the absolute r i s k aversion at the i n i t i a l wealth w , i . e . R.(w ), i s expressed as a function of: o A o (i)  the parameters of a nondegenerate gamble z; and  ( i i ) the decision maker's response to that gamble ( i n terms of r i s k premium, or c e r t a i n t y equivalent, or p r o b a b i l i t y equivalent, or gain equivalent). Mathematical r e s u l t s are obtained for two d i f f e r e n t gambles. calculates the values of R^ for several responses to d i f f e r e n t  The R^-method reference  gambles, and then selects the u t i l i t y function with the least r e l a t i v e standard deviation over the R^ values.  The procedure i s based on the fact,  that f o r the decision maker's actual u t i l i t y function, R^ must t h e o r e t i c a l l y a t t a i n the same value at W, q  assess R^.  namely R^( )> W  n  o  D  Suggestions are made for extending  matter what gamble i s used to the R -method to incorporate A  r i s k proneness as well as attitudes which are r i s k averse over one part of the domain and r i s k seeking over another part. matical extensions  F i n a l l y , a chapter on mathe-  i s provided i n order to improve the R^-method by including  a larger set of a l t e r n a t i v e u t i l i t y functions.  Contents  Abstract Contents 1. Introduction 2. Notation and Preliminaries 3. Mathematical Results f o r R.(w ) A  o  3.1. The idea behind the mathematical r e s u l t s 3.2. A generalized class of u t i l i t y functions 3.3. Risk aversion R^ as a function of certainty equivalent or r i s k premium for the gamble z = <h,p;-h> 3.A. Risk aversion R^ as a function of p r o b a b i l i t y equivalent for the gamble z = <h,p;-h> 3.5. Risk aversion R^ as a function of gain equivalent for the gamble z = <h,p;-h> 3.6. Risk aversion R^ as a function of certainty equivalent or r i s k premium for the gamble z = <h,p;0> 3.7. Risk aversion R  as a function of p r o b a b i l i t y equivalent  . f o r the gamble z = <h,p;0> 3.8. Risk aversion R  as a function of gain equivalent  for the gamble 2 = <h,p;0> 4. The R -method A 4.1. The R -method for r i s k averse decision makers A  4.2. The R^-method including r i s k proneness and r i s k neutrality  - iii -  5. Mathematical Extensions  : 85  5.1. Extended results for u(w;£,a,T,g,n)  :  87  5.2. Extended results f o r the r e l a t i v e r i s k aversion ^(w^^)  : 93  5.3. Extended results for other gambles  : 96  5.4. The results using Taylor series expansions  104  5.5. The sum-of-exponentials u t i l i t y  107  function  5.6. Summary References  •' H I :  113  - iv -  Acknowledgement  The. author Is grateful to his; supervisor, Professor D, Wehrung, for many valuable comments and stimulating discussions throughout the preparation of t h i s thesis-.  He would also l i k e to thank the other  committee members-, Professors B., Schwab and L.G. Mitten, for their assistance. Business  F i n a l l y , suggestions- of Professor J.W. Pratt of Harvard  School have been h e l p f u l i n determining  knowledge i n the f i e l d .  the current state of  - 1 -  1  Introduction  Decision normative and  theory i s o f t e n c l a s s i f i e d under two descriptive.  Normative d e c i s i o n theory i s concerned with  p r e s c r i p t i o n of courses of a c t i o n that maker's p r e f e r e n c e s . describe  the  The  i n t e r r e l a t e d headings,  conform most c l o s e l y to the  the  decision  purpose of d e s c r i p t i v e d e c i s i o n t h e o r y i s to  d e c i s i o n maker's p r e f e r e n c e s and  preferences into his decisions.  he  incorporates  these  In other words, the normative theory  guides the d e c i s i o n maker i n what he t h e o r y f o c u s s e s on what the  the way  should do,  whereas the  d e c i s i o n maker does do.  descriptive  Even though both  l  theories logical  are  concerned w i t h b e h a v i o r a l  a s p e c t s , i t appears t h a t  concepts are m a i n l y used when d e s c r i b i n g  d e c i s i o n making. approach and  The  normative theory u s u a l l y  i s i n the  (and  follows  psycho-  predicting) an  human  axiomatic  l i t e r a t u r e a l s o known under the heading  decision  2  analysis.  As  the main r e s u l t of t h i s r e s e a r c h  d e c i s i o n making, our psychological  focus w i l l be  a s p e c t s w i l l be  i s a normative t o o l f o r 3  on d e c i s i o n a n a l y s i s , but  covered as we  go  behavioral  along.  1. Reviews o f b e h a v i o r a l d e c i s i o n t h e o r y i n c l u d e Edwardst1954,1961), Luce and SuppesC1965), Becker and M c C l i n t o c k ( 1 9 6 7 ) , S l o v i c and L i c h t e n s t e i n ( 1 9 7 1 ) , Rapoport and W a l l s t e n ( 1 9 7 2 ) , F i s c h e r and Edw a r d s ( 1 9 7 3 ) , and S l o v i c , F i s c h h o f f and L i c h t e n s t e i n ( 1 9 7 7 ) . Most o f t h e s e r e f e r e n c e s are m a i n l y o r i e n t e d towards p s y c h o l o g i c a l f a c e t s . 2. See, f o r example, Keeney(1978): " D e c i s i o n A n a l y s i s p r o v i d e s a n o r m a t i v e t h e o r y which p r e s c r i b e s how a d e c i s i o n maker s h o u l d behave i n o r d e r t o be c o n s i s t e n t w i t h h i s judgments and p r e f e r e n c e s . I t does not p r o v i d e a method f o r d e s c r i b i n g how, i n f a c t , i n d i v i d u a l s do behave." 3. I n t r o d u c t o r y r e a d i n g s i n d e c i s i o n a n a l y s i s i n c l u d e P r a t t , R a i f f a and S c h l a i f e r ( 1 9 6 4 ) , SwaltnC1965), Hammond( 1967), How.ird(1968,1980) and Keeney( 1978). S p e c i a l i s s u e s on d e c i s i o n a n a l y s i s appeared i n IEEE T r a n s a c t i o n s on SSC (see lloward(1968)) and O p e r a t i o n s Research (see K i r k w o o d ( 1 9 8 0 ) ) . Many t e x t books i n d e c i s i o n a n a l y s i s — a t d i f f e r e n t l e v e l s — are a v a i l a b l e , such as R a i f f a ( 1 9 6 8 ) , S c h l a i f e r ( 1 9 6 9 ) , Brown, Kahr and P e t e r s o n t 1 9 7 4 ) , Keeney and R a i f f a ( 1 9 7 6 ) , J o n e s ( 1 9 7 7 ) , L a V a l l e ( 1 9 7 8 ) , and Holloway (1979).  and  - 2 -  Decision analysis i s a d i s c i p l i n e providing a normative, l o g i c a l framework for decision making under uncertainty.  Within this framework, the  decision maker's preferences for the possible consequences of any action are a key element i n the problem description. analysis i s provided by a set of axioms. we may  The logic of decision  For the sources of these axioms  actually have to go back to Bernoulli(1738) who  of maximizing u t i l i t y .  set forth the idea  However, the axiomization of u t i l i t y was  rigorously developed with the work of Von Neumann und In an economic, game-theoretic  first  Morgenstern(1947).  context they postulated the set of axioms  which we w i l l adopt i n this research.  Essential i s the idea that i f an  appropriate u t i l i t y (value) i s assigned to each consequence and the expected u t i l i t y of each alternative i s calculated, then the best course of action i s the alternative with the highest expected u t i l i t y .  This so-called expected  u t i l i t y hypothesis has been the basis for most of the research i n u t i l i t y or if  value theory.  The Von Neumann and Morgenstern(1947) u t i l i t y notion was  the  foundation for the work of Marschak(1950), Savage(1954), Luce and Raiffa (1957), Pratt, R a i f f a and Schlaifer(1965), and Fishburn(1970), who  came up  with d i f f e r e n t sets of axioms, which a l l prescribe expected u t i l i t y as an appropriate guide for decision making.  The axioms of u t i l i t y theory assume  that either numerical, objective p r o b a b i l i t i e s e x i s t , or that both (subj e c t i v e ) p r o b a b i l i t i e s and u t i l i t i e s can be j o i n t l y derived.  At this point  we w i l l not dwell on the problems associated with subjective p r o b a b i l i t i e s , but instead assume that the decision maker knows the likelihoods of the possible consequences.  Hence, we w i l l b a s i c a l l y follow Luce and Raiffa(1957)  4. Fishburn(1968) provides a review a r t i c l e on u t i l i t y theory with an extensive bibliography up to 1968. A more recent review i s given by FishburnC1978).  - 3-  in this respect.  The next important step then i s to assess the decision  maker's preferences  for each of the possible consequences.  We have noted that the decision maker's preferences  for the various  consequences i s an essential ingredient i n the analysis of decisions under uncertainty.  These preferences  can be represented by u t i l i t y functions i n  the sense of Von Neumann and Morgenstern(1947). for  Many different  evaluating u t i l i t y functions have been proposed.  techniques  See, for example, t  Mosteller and Nogee(1951), Davidson, Suppes and Siegel(1957), Becker, DeGroot and Marschak(1964), Meyer and Pratt(1968), Schlaifer(1969), and Hammond(1974).  Basic to most assessment procedures are the well known  reference gambles and a c l a s s i f i c a t i o n of attitudes towards r i s k .  Schlaifer  (1971) published a book of computer programs for decision analysis including various assessment techniques.  The preceding discussion together with  the  5  many applications of decision analysis  indicate the need to have e f f e c t i v e  methods for assessing u t i l i t y functions. The major contribution of this research is a suggested procedure for selecting a u t i l i t y function from among a l t e r n a t i v e s . procedure i s the absolute r i s k aversion R.(w)  Basic to the new  over wealth,  introduced by Pratt(1964) and Arrow(1965,1971).  independently  For each of a number of  u t i l i t y functions we w i l l express the absolute r i s k aversion at the wealth l e v e l w , i . e . R,(w ), as a function of: o A o.  initial  (i)  the parameters of a nondegenerate gamble z, and  (ii)  the certainty equivalent, or the r i s k premium, or the p r o b a b i l i t y equivalent, or the gain equivalent of that gamble.  5 . We m e n t i o n o n l y a f e u c o n t r i b u t i o n s t o t h e p u b l i s h e d l i t e r a t u r e : G r a y s o n f 1 9 6 0 ) , Swalm(1966), S p e t z l e r ( 1 9 6 8 ) , H o w a r d , M a t h e s o n a n d N o r t h ( 1 9 7 2 ) , d e N e u f v i l l e a n d K e e n e y ( 1 9 7 2 ) , a n d Hauser and Urban(1979).  - 4 -  The u t i l i t y functions include the l i n e a r function and f i v e functions within the well known Hyperbolic Absolute Risk Ayersipn  (HARA) c l a s s .  After  obtaining the decision maker's responses for two or more reference gambles based on ( i ) and  ( i i ) , the new  procedure (called R.-method) i s then able to  select the u t i l i t y function from among the alternatives that conforms most closely to the decision maker's preferences. The R - method seems quite appealing  i n several ways.  It enables us to  use d i f f e r e n t response modes, thus avoiding problems inherent i n using only one response mode . 6  Also, the R^-method has the advantage of requiring  knowledge about the decision maker's i n i t i a l endowment only a f t e r responses for the reference gambles have been obtained.  In spite of the sometimes  i n t r i c a t e mathematical derivations, the r e s u l t s are not complicated hence the R^-method can be applied i n a straightforward manner.  and  The R -  method i s described for a limited class of u t i l i t y functions, but section and Chapter 5 w i l l give various extensions to how  one may  4.2,  and provide several directions as  proceed to enlarge the usefulness of the R^-method. F i n a l l y ,  the basic idea to express R. = R.(w ) as a function of (i) and A A o  ( i i ) seems to  be a clear basis for extended research. The outline of the thesis i s as follows.  After the necessary notation,  d e f i n i t i o n s , lemmas and other preliminaries are given i n Chapter 2, we present  i n Chapter 3 our r e s u l t s for R (w ) expressed as function of ( i ) and A  (ii).  will  O  The f i r s t section of Chapter 4 w i l l introduce and discuss the R A  method, assuming that the decision maker conforms to one averse u t i l i t y function.  (unknown) r i s k  Section 4.2. w i l l discuss the case where this  6 E x a m p l e s o f s u c h p r o b l e m s c a n be f o u n d i n L i c h t e n s t e i n a n d S l o v i c ( 1 9 7 1 ) , S l o v i c ( 1 9 7 2 ) , s t e i n and S l o v i c < 1 9 7 3 ) , C r e t h e r and P l o t t ( 1 9 7 9 ) , and Kahneman and T v e r s k y ( 1 9 7 9 ) .  Lichten-  assumption  i s violated.  Hence, i n t h i s section we w i l l c e r t a i n l y consider  r i s k proneness and r i s k n e u t r a l i t y . mathematical  extensions and should serve as a d i r e c t i o n for extending the  use of the R.-method. A included.  Chapter 5 w i l l be a c o l l e c t i o n of  At the end of the thesis our reference l i s t i s  -  2  6  -  Notation and  Preliminaries  A gamble z, which r e s u l t s  i n a net change of h^ w i t h p r o b a b i l i t y  (0 < p < 1) and i s denoted by  a net change of h z =  <h^,p;h2> .  e x p l i c i t l y w i t h i n the b r a c k e t s , values  p  w i t h the complementary p r o b a b a i l i t y 1-p, Note t h a t the p r o b a b i l i t y  i s the l i k e l i h o o d  that appear w i t h i n the b r a c k e t s .  degenerate i f  1  The  p, which appears  of the f i r s t  gamble z i s s a i d  of the to be  two  non-  0 < p < 1 .  U s u a l l y gambles are presented  either  by branch diagrams ( F i g u r e  2.1.)  or by c h a r t s ( F i g u r e 2.2.):  Figure z  The  2.1.  gamble  presented  Figure The z  z =  <h p;h > l5  2  as a branch diagram.  2.2.  gamble  presented  z =  <h^,p;h2  >  as a c h a r t , where  the shaded area i s a p of the t o t a l  fraction  area.  1. Wc a s s u m e t h e r e a d e r t o b e q u i t e f a m i l i a r w i t h t h e i d e a s a n d d e f i n i t i o n s g i v e n b y P r a t t ( 1 9 6 4 ) a n d Schlaifer(1969). T h i s c h a p t e r draws h e a v i l y on these r e f e r e n c e s . A v e r y good e x p o s i t i o n i n t h i s c o n t e x t i s p r o v i d e d by Keeney and R a i f f a ( 1 9 7 6 ) .  The s p e c i a l case when p = \ i s c a l l e d a f i f t y - f i f t y gamble and w i l l be denoted if  by  z =  <h j ^ h ^  =  <  ' 1'^2 1  >  '  T  h  e  8 ble a m  i t s expected v a l u e i s z e r o , i . e . E ( z ) = 0 .  gamble can be r e p r e s e n t e d by  <h;-h>  2  *-  s s  a  ^  t  t>  o  e  fair  Hence, a f a i r f i f t y - f i f t y  , which has a s t a n d a r d  d e v i a t i o n equal t o h . Let  now u(w) be a c o n t i n u o u s , u n i d i m e n s i o n a l , m o n o t o n i c a l l y  increasing 2  utility  f u n c t i o n over w £ R, where the a t t r i b u t e i s chosen to be wealth.  We w i l l not c o n s i d e r d e c r e a s i n g f u n c t i o n s , however the d e r i v e d r e s u l t s be q u i t e s i m i l a r f o r m o n o t o n i c a l l y d e c r e a s i n g u t i l i t y f u n c t i o n s . the d e c i s i o n maker's i n i t i a l  will  Assume  endowment to be w o  D e f i n i t i o n 2.1.  A c e r t a i n t y e q u i v a l e n t o f a gamble z i s an amount z = z(w , z ) , such t h a t the d e c i s i o n maker i s i n d i f f e r e n t o between z and the amount z f o r s u r e .  Hence, z = 2(w  o  ,z) i s d e f i n e d by  u(w +z) = E[u(w + Z ) ] . o o  (2.1.)  As we have chosen the a t t r i b u t e w to be wealth, we may  substitute  cash  3  equivalent of  for certainty equivalent.  Note t h a t , of course, z i s a f u n c t i o n  the gamble z i t s e l f , but a l s o o f the i n i t i a l  endowment w  .  In other  words, i t i s assumed t h a t the d e c i s i o n maker may behave d i f f e r e n t l y at  2. T h e c o n c e p t s  are equally  3. C a s h e q u i v a l e n t s  valid  f o r nonmonetary a t t r i b u t e s .  are extensively  discussed  i n L a V a l l e t 1968).  -  various wealth l e v e l s .  8 -  Except for the case of a linear u t i l i t y  function, a  decision maker i s assumed to state d i f f e r e n t cash equivalents at different i n i t i a l wealth positions for the same nondegenerate gamble.  Let us  i l l u s t r a t e this point with the following example.  Example 2.1.  A decision maker i s assumed to behave according to the u t i l i t y 2  function the  fair  u^(w) = w-(.25)w , where f i f t y - f i f t y gamble  z =  0 ^ w ^ 1.5 . <0.5;-0.5>  calculate that at an i n i t i a l wealth of i s z(w',z) « -0.081, as i t holds that o  Suppose he i s faced with  . Using (2.1.) i t i s easy to  w^ = 0.5 h i s certainty equivalent u(0.5-0.081) « J>u(0) + ^u(l) .  Thus, the decision maker i s i n d i f f e r e n t between the gamble and losing a sure amount of ,.0.081 . At an i n i t i a l wealth of is  2(w",z) w -0.118, as o  w^ = 1  his certainty equivalent  u(l-0.118) « %u(0.5) > %u(l.5) .  Hence, z(w',£) o  f 2(w",z) . o  D e f i n i t i o n 2.2.  The r i s k premium TT = TT(W , z) of a gamble z i s i t s expected  t  Q  value minus i t s certainty equivalent.  Hence, TT = TT(W . z) o  (2.2.)  i s defined by  TT = E(z) - 2 ,  From (2.2.) we see that  z = E(z) - TT , which after substitution i n (2.1.)  results i n the following fundamental relationship:  (2.3'.)  u[w  + E(z~) o  where z = 2(w ,z) . o  - TT] = E[u(w +z) ] o  - 9 -  or, more formally  (2.3".)  uLw o  + E(a) - TT(W ,Z)] = E[u(w + 2 ) ] o o  which reduces for a f a i r gamble to  (2.4.)  utw o  Example 2.2.  - TT(W ,z)] = Etu(w +z)] . o o  Referring to Example 2.1. i t can be calculated, that, using  (2.3.), the decision maker's r i s k premium at the i n i t i a l wealth level of w  1  o  ly,  = 0.5  i s TT( ',z) « 0.081, as o w  ^(w^z) ^ 0.118 .  u(0.5+0-0.081) « ^u(0) + ^ u ( l ) . Similar-  Of course, these results could have been obtained  more d i r e c t l y i n this case.  Clearly, for a f a i r gamble (2.2.) becomes  '"' = -£, which proves the following lemma.  Lemma 2.1.  The r i s k premium of a f a i r gamble z i s the negative of  the certainty equivalent of that gamble.  By D e f i n i t i o n 2.2. we have f o r any constant c  (2.5.)  TT(W ,z) = "T(w +c,z-c) o  .  o  It i s often feasible and i l l u s t r a t i v e to present a gambling situation with the corresponding r i s k premium i n a graph of the u t i l i t y function. This point w i l l be explained through the following example.  - 10 -  Example 2.3.  Figure 2.3. provides a picture of the u t i l i t y  function  2 u^(w) = w-(.25)w  (0 £ w ^ 1.5)  as introduced i n Example 2.1.  We w i l l  leave i t for the reader to v e r i f y the resulting graph, as presented  for w" = 1.  u. (w)  *• w  Figure 2.3. 2 A decision maker with u (w) = w-(.25)w , facing a f a i r f i f t y - f i f t y gamble z = <0.5;-0.5> states for h i s r i s k premium IT = TT (w ,z) = 0.118 at h i s i n i t i a l wealth level w" = 1 .  D e f i n i t i o n 2.3.  The insurance premium  u =y ( >z) of a gamble z i s the w  Q  negative of the certainty equivalent of that gamble.  Hence,  y (w ,"z) i s defined by  - 11 -  (2.6.)  u(w ,"2;) = - z(w ,z) . o  o  From Lemma 2.1. we derive, that for a f a i r gamble the r i s k and insurance premium coincide.  D e f i n i t i o n 2.4.  An amount h such that the decision maker i s indifferent between the status quo (his i n i t i a l wealth w ) and the o gamble  z =  <h,p;-h> i s called the gamble's gain  equivalent and i s denoted by  Hence,  h = h(w ,p,h) .  h = h(w ,p,h) i s defined by Q  u(w ) = E[u(w +z)] = p.u(w +h) + (l-p).u(w -h) . o o o o  (2.7.)  In order to have the expression gain equivalent make sense, we w i l l obviousl y require h to be positive.  D e f i n i t i o n 2.5.  A probability p .such that the decision maker i s i n d i f f e rent between the status quo w  and the gamble  is called the gamble's probability  equivalent and i s  denoted by p = p(w ,h) . o Hence,  (2.8.)  z = <h,p;-h>  p = p(w ,h) i s defined by  u(w ) = E[u(w +z)] = p.u(w +h) + (l-p).u(w -h) o o o o  - 12 -  U t i l i t y functions that have i d e n t i c a l implications for action are called s t r a t e g i c a l l y equivalent.  This notion can be formalized through the  following d e f i n i t i o n .  D e f i n i t i o n 2.6.  Two u t i l i t y functions, u^(w) and u^Cw), are s t r a t e g i c a l l y equivalent, written —*  u,(w) ^ u„(w), i f there exist con1 2  stants a and $>0 such that  u^(w) = a + B.u^Cw) .  It i s now straightforward to show that s t r a t e g i c a l l y equivalent  utility  functions imply the same preference ranking for any two gambles.  Also, i f  u^(w) ^ u^(w), then both u^(w) and u^(w) give the same cash equivalents for a gamble.  For example, l e t  u (w) = a + B.u (w)  (2.9.)  1  and consider  z =  (2.10.)  where  2  <h^,p;h2> .  8>0  Then, we have by (2.1.)  u,(w +z,) = p.u. (w +h_) + (1-p).u. (w +h ) l o l 1 o 1 1 o 2 r  r  and (2.11.)  u.(w +z„) = p.u.(w +h.) + (l-p).u (w +h ) 0  l  o  l  I  o  l  l  so that substitution of (2.9.) i n (2.10.) yields p.[  a  + 3.u„(w +h,)] + (1-p).[a + B.u„(w +h„)]  result:  o  \  l  o  l  o  l  a + r3.u (w +2^) = 2  which gives the following  - 13 -  u„(w + z ) = p.u„(w + h ) + ( l - p ) . i i (w +h ) .  (2.12.)  2 o l  Hence, equating  2  o  l  2  o  2  (2.11.) and (2.12.), we find that z, - 2„ .  If u^(w) and u^(w) are s t r a t e g i c a l l y equivalent, we can also say that u^(w)  determines u^(w) up to p o s i t i v e linear  transformations.  It w i l l appear to be quite important to categorize the attitude of a decision maker with respect to r i s k .  Or, the question arises whether a  decision maker i s a r i s k lover or a r i s k avoider.  This i s not to say that  a decision maker w i l l always exhibit the same r i s k attitude over the entire region.  In fact, many researchers  believe that often decision makers are  r i s k averse for gains while being r i s k prone for negative outcomes. For theoretical purposes, however, D e f i n i t i o n 2.7. assumes ho change of r i s k attitude over the entire domain.  D e f i n i t i o n 2.7.  A decision maker i s said to be r i s k averse  i f u[E(w +z)] > E[u(w +£)], o o  r i s k neutral i f u[E(w +z)] = E[u(w +z)], and o o r i s k prone  i f u[E(w +z)] < E[u(w +z)] o o  for a l l nondegenerate gambles z .  This research w i l l focus on the r i s k averse and r i s k neutral decision maker. A r i s k averse decision maker w i l l prefer the expected consequence of any nondegenerate gamble to the gamble i t s e l f . the most common s i t u a t i o n i n practice.  And i t i s believed that this i s  The r i s k neutral case w i l l be i n -  U. S e e , f o r e x a m p l e , S w a l r a C 1 9 6 6 ) a n d K a h n e m a n a n d T v e r s k y ( 1 9 7 9 ) .  - 14 \  eluded f o r completeness sake.  The r i s k prone decision maker i s not con-  sidered here, even though our results can quite e a s i l y be extended i n that direction.  Lemma 2.2.  A decision maker i s r i s k averse i f and only i f his u t i l i t y function i s concave.  The proof of Lemma 2.2. can be found i n Keeney and Raiffa(1976).  Note that  the u t i l i t y function i n Example 2.1. i s concave over the defined range. Therefore,  this decision maker i s r i s k averse.  And indeed, consistent with  D e f i n i t i o n 2.7., i t holds that choosing  w = 0.5 o 1  u[E(w'+z)] > E[u(w'+z)]. For example, o o and z = <0.5.-0.5> , we have u[E(w'+z)] = u[E(w')] = o o  7/16 > 6/16 = JJU(W'-0.5) + %u(w*+0.5) = E[u(w'+z)] . o o o Lemma 2.3.  A decision maker with an increasing u t i l i t y function i s r i s k averse i f and only i f for any nondegenerate gamble 1 i t holds that:  (2.13.)  E(z) > 2 .  For the proof, we f i r s t assume r i s k aversion, so that by D e f i n i t i o n 2.7. we have  u[E(w +z)] > E[u(w +z)] o o  . Using (2.1.) we see that  ULE(W  o  +Z)]  u(w +2), which for an increasing function y i e l d s o  (2.14.)  Noting that  E(w +z) > w +2 . o o E(w +z) = w  + E(z) , we see that (2.13.) follows d i r e c t l y from  - 15 -  (2.14.).  Second, assuming (2.13.) gives d i r e c t l y (2.14.), so that U[E(W +Z)] Q  > u(w +z) = E[u(w +z)], as the u t i l i t y function i s increasing. o o / decision maker i s r i s k averse.  Hence, the  This completes the proof of Lemma 2.3.  •  Using Lemma 2.3. and D e f i n i t i o n 2.2. we e a s i l y arrive at the following  Lemma 2.4.  A decision maker with an increasing u t i l i t y function i s r i s k averse i f and only i f h i s r i s k premium i s positive for  a l l nondegenerate gambles.  For the r i s k neutral decision maker similar lemmas can be derived.  We w i l l  state these lemmas without proofs as these proofs are similar to those of the preceding lemmas.  Lemma 2.5.  A decision maker with an increasing u t i l i t y function i s r i s k neutral i f and only i f for any nondegenerate gamble z i t holds that  (2.15.)  E(z) = z .  Lemma 2.6.  A decision maker with an increasing u t i l i t y function i s r i s k neutral i f and only i f h i s r i s k premium i s zero for a l l nondegenerate gambles.  Now assume u(w) to be l i n e a r , i . e . u(w) = a + Bw ^ w , where 6 > 0 . We see that  E[u(w +z)] = Eta + B ( w + z)] = a + 8[w + E ( z ) ] and o o o  u[E(w +z)] = u[w + E(z)] = a + B t w + E(z)] , so that o o o  u[E(w +£)] = E[u(w + Z ) L o o  tl  - 16 -  Hence, according  to D e f i n i t i o n 2.7.,  the decision maker i s r i s k neutral,  which proves the following lemma.  Lemma 2.7.  A decision maker with a linear u t i l i t y u(w)  We or not.  are now  (8  = a + $w  > 0)  function  i s r i s k neutral.  able to investigate whether a decision maker i s risk averse  What we would l i k e to do next i s to state a measure of r i s k  aversion  so that an indication i s provided when one decision maker is more r i s k averse than another.  The d e f i n i t i o n we are about to give here stems from Pratt  (1964) and Arrow(1965,1971), and i t assumes u(w)  i s continuous and twice  differentiable.  D e f i n i t i o n 2.8.  The  ^ — ^ — — — — i ^ — — — — —  absolute  r i s k aversion at w, R (w), of a u t i l i t y • •  function u(w), (2.16.)  R  A  (  W  A  i s defined  by:  ) . _ « ^ L .  u'(w)  The absolute r i s k aversion R (w) i s a function of w, A to be a measure with many desirable properties.  and i t c e r t a i n l y seems  Note that u'(w)  positive for monotonically increasing functions and that u"(w)  i s always i s negative  for concave functions, so that R (w) i s positive for r i s k averse decision makers.  S i m i l a r l y , i t can be shown that R.(w)  = 0 for r i s k neutral decision  A  makers and R^(w) Two  <  0 for r i s k prone decision makers.  related lemmas follow:  - 17 -  Lemma 2 . 8 .  Two s t r a t e g i c a l l y  e q u i v a l e n t u t i l i t y f u n c t i o n s have the  same a b s o l u t e r i s k a v e r s i o n R,(w) A  Proof:  For  u^(w) = a + Bu^Cw) , where B > 0 , we have  ui'Cw) (2.17.)  R  A 1  Lemma 2 . 9 .  If  =  Bu'(w)  then they are  u"(w) d — R.(w) = , , . = —[log u'(w)] A u (w) dw  after  -JR  u2(w)  A  ( w ). '  Z  strategically  J — R^( w )  dw =  l o g u'(w)  +  risk  equivalent,  so that by i n t e g r a t i o n  e  of  (w)dw  (with C. 1 , which  (2.18.) gives  J e  C =  A  A  1  (with C  (w)dw  -JR  e  dw  =  u'(w)  .  an i n t e g r a t i o n  e  C 1  Now, R.(w) determines u(w) up to p o s i t i v e  u'(w)  u(w) = a + Btje  constant)  + C0  l i n e a r transformations,  that  (2.20.)  = R  exponentiating  (2.18)  Integration  u''(w) = - JL  two u t i l i t y f u n c t i o n s have the same a b s o l u t e  b e i n g an i n t e g r a t i o n c o n s t a n t )  (2.19.)  S  uj(w)  a v e r s i o n R^(w),  yields  Bu"(w)  (w)=--^  A > 1  Proof:  .  —JR (w)dw A dw]  by n o t i n g  - 18 -c where a = —  -c and 8 = e  e  > 0  are two constants.  Finally, i f  R ,(w) = R „(w), then i t follows from (2.20.) and Definition A,l A,2 the corresponding u^(w) and u^(w) are s t r a t e g i c a l l y  2.6., that  equivalent, i . e . u^(w)  ^ u (w) .  •  2  Related to the absolute r i s k aversion R,(w) are two other measures of A r i s k aversion. One i s the r e l a t i v e r i s k aversion R (w) as defined by Pratt R  (1964) and Arrow(1965,1971); the other i s the p a r t i a l r e l a t i v e r i s k aversion -  5  Rp(w;w ), as introduced by Menezes and Hanson(1970) .  We w i l l now provide  o  the definitions  Definition  of R„(w) and R (w;w ) . R P o  2.9.  The r e l a t i v e r i s k aversion at w, R (w), of a u t i l i t y R  function u(w), i s defined by n  (2.21.)  /  U"(W)  N  R (w) = - w u'(w)  R  Definition  2.10.  The p a r t i a l r e l a t i v e r i s k aversion R (w;w ) of a u t i l i t y p  Q  function u(w), i s defined by  (2.22.)  R (w;w ) = — w y  °  where w  o  u"(w + w) o u'(w  + w)  i s some fixed wealth level,  Zeckhauser and Keeler(1970) established a direct  relationship between R (w)  R (w), and R (w;w ), which we w i l l present through the following lemma. R P o 5. T h e s e a r e n o t t h e o n l y m e a s u r e s o f r i s k a v e r s i o n . Especially i n the finance l i t e r a t u r e different m e a s u r e s o f r i s k a v e r s i o n h a v e b e e n i n t r o d u c e d , d i s c u s s e d , a n d a p p l i e d ,• e. g. Y a a r i ( 1 9 6 9 ) , M a y s l i a r (1975), M i 1 1 f r ( 1975), K u b i n s t e i n ( 1 9 7 6 ) , and K a l l b e r g and Ziemba(1978).  A  - 19 -  Lemma 2.10.  R^Cwjw ) = R_(w P o R o  + w) — w .R.(w + w) . 0 A 0  The proof of Lemma 2.10. i s obvious. Let  us now i l l u s t r a t e the idea of a r i s k aversion measure with the  following example. Example 2.4. Consider two decision makers with two different functions:  u^w) = w - (l/4)w  utility  (see Examples 2.1. to 2.3.) and u (w) = 2  2 w - (l/3)w  , both defined over the range  0 <^ w £ 3/2 . Using D e f i n i t i o n  2.8. we calculate R, . (w) = ~ — and R „(w) = -r-nr , so that R (w) A,l 2 - w A,2 3/2 - w A,l < R „(w) when 0 < w < 3/2 . Thus, over the defined wealth range, the A,2 f i r s t decision maker i s less r i s k averse than the second decision maker. But what does this actually mean?  The answer i s that one decision maker i s  less r i s k averse than another implies that for a l l nondegenerate  gambles  within the defined wealth range his r i s k premium ois smaller than that of the other decision maker.  For example, assuming an i n i t i a l wealth level of  w" = 1 , we have R, .(w") = 1 < 2 = R. _(w") . Accordingly, for the gamble o A,l o A,2 o z = <h:-h>  we have TT,(W",Z) « 0.118 , which i s less than 1 o  TT (w",z) « 0.207 . I o  . In fact, we introduced i n Example 2.4. some kind of comparative r i s k aversion.  The following lemma has actually been i l l u s t r a t e d i n the example.  The proof of Lemma 2.11. can be found i n Pratt(1964).  Lemma 2.11. v  I t holds that TT^W.Z) < TT^CW.Z)  for a l l w and z, i f and  only i f R (w) < R _(w) for a l l w . A,l >^ A  - 20 -  In order to make e f f e c t i v e  use of the absolute r i s k aversion R (w) A  we state another important lemma:  Lemma 2.12.  R (w) i s positive (negative) for a l l w i f and only i f the A decision maker i s r i s k averse (risk prone).  Proof:  We show the v a l i d i t y of the r i s k averse case.  The proof for the  r i s k prone case uses similar arguments and i s therefore omitted. now ^ ( ) R  u"(w)  w  >  Assume  0 . Since u'(w) > 0 , because u(w) i s increasing, we have  = - R (w).u'(w) < 0 , so that u(w) i s concave.  follows that the decision maker i s r i s k averse.  Using Lemma 2.2. i t  The converse i s straight-  forward and has been stated e a r l i e r i n this chapter.  In addition, i t i s clear, using Lemma 2.7.,  Lemma 2.13.  I f R.(w) = 0  |  that the following lemma holds:  , the decision maker i s r i s k neutral.  The converses of Lemma 2.7. and Lemma 2.13.  can be proved, though the proofs  w i l l be lengthy and not quite relevant. Hence, this i s not discussed here.  We have seen, that TT(W ,z) i s a function of both w and the gamble z . o o Now we would l i k e to discuss what happens to TT(W ,Z) as w varies. I t i s o o believed that many decision makers w i l l pay a smaller r i s k premium as their ( i n i t i a l ) wealth increases. The reason being that one can better afford to take a certain r i s k as one becomes richer.  Of course, this i s not a general  - 21 -  rule, i t only provides motivation decreasingly,  to investigate whether a decision maker i s  constantly, or increasingly r i s k averse.  Pratt(1964) provides  several d e f i n i t i o n s and theorems i n that d i r e c t i o n , the major d e f i n i t i o n being:  ,  D e f i n i t i o n 2.11.  A r i s k averse decision maker i s decreasingly  (constantly;  increasingly) r i s k averse i f w(w^,z) for any gamble z decreases ( i s constant; decreases) as w  increases. o  Menezes and Hanson(1970) extended Pratt's theorems, while stating and proving  the following lemma.  Lemma 2.14.  Let the r i s k premium for a gamble z be TT(W , Z ) and  A be  l t i p l i c a t i v e factor such that P(w +Az<0) = 0 , then: a multip  (2.24.)  (2.25.)  (2.26.)  | — [TT(W , 2 ) ] dw o  8_  TT(W J A Z ) o  8A  9_  =0  (i)  <  ^ 0 <  (ii)  TT(AW , A Z )  o  8A  ^ 0 <  (iii)  i f the corresponding function  ( i ) R.(w ) , ( i i ) A  Rp(w ;w ) , o  Q  (iii) R^^)  constant, or decreasing  O  *- respectively increasing, s  in w .  - 22 -  A very useful application of the absolute r i s k aversion R^(w) and the r e l a t i v e r i s k aversion R (w) i s provided by.the following two lemmas. R  Lemma 2.15.  I f the absolute r i s k aversion i s constant, say R^( ) w  >  =  c  then: (2.27.)  u(w) ^ w  i f and only i f R (w) = 0 , A  (2.28.)  u(w) ^ - e ~  A  C W  i f and only i f R,(w) = c > 0 . A  u" (w) Assume f i r s t u(w) ^ w , so that R (w) = - ,, ( = 0 . Conversely, A u (w)  Proof:  A  1  assuming R.(w) = 0, (2.20.) gives u(w) = a + Bw , where 6 > 0, so that u(w) ^ w . This proves (2.27.).  Now assume u(w) ^ - e  C W  , so that R^( ) w  =  u"(w) - ^ , = a - 8e  C  W  c . Conversely, assuming ^ ( ) = c > 0, (2.20.) gives u(w) = R  w  , where 8 > 0, so that u(w) 'v* - e  C  W  . This proves (2.28.) and  completes the proof of Lemma 2.15. Lemma 2.16.  I f the r e l a t i v e r i s k aversion i s constant, say R (w) = c , t)  then: (2.29.)  u(w) % w  1 _ C  i f and only i f 0 < R (w) = c < 1 , D  R  (2.30.)  u(w) ^ log w  i f and only i f R (w) = c = 1 , D  R  (2.31.)  u(w) a- - w  1-c  i f and only i f R (w) = c > 1 .  The proof of Lemma 2.16. i s similar  D  R  to the proof of Lemma 2.15. and can be  found i n Pratt(1964). We w i l l f i n i s h this chapter with the following example:  - 23 -  Example 2.5. Following Lemma 2.14. we see that with (2.24.) i f R.(w ) i s constant i n w , so that A o o  (2.32.)  TT(W',2) = TT(W",2)  0  where  0  o  w  1  o  i f R.(w ) i s constant as w varies. A o o that (2.32.) holds i f u(w) ^ - e °  3 ~ |_[Tr(w ,2) ]= 0 o  W  f w" o  Then, according to Lemma 2.15. we have . The l a t t e r i s graphically shown i n  Figure 2.4. below.  u(w)  o Figure 2.4. A decision maker with u(w) a, - e w i l l state the same r i s k premium for a nondegenerate gamble z at any ( i n i t i a l ) wealth l e v e l . C  W  - 24 -  3  Mathematical  T h i s chapter  Results  w i l l derive ^ (  w 0  ^>  for  R (w ) A  Q  b r i e f l y denoted by R^,  as a  function of: (i)  the parameters o f a nondegenerate gamble z;  (ii)  the c e r t a i n t y e q u i v a l e n t , or the r i s k premium, or the p r o b a b i l i t y e q u i v a l e n t , or the g a i n e q u i v a l e n t  Some o f the d e r i v e d R^(WQ)  f u n c t i o n s are not  i n terms of ( i ) and  of p and  ( i i ) ; e.g.  the parameters of  We  will  introduce  f i r s t discuss  gives  the  gamble. implicit  functions  of  i s i m p l i c i t l y expressed i n terms  z.  the i d e a behind the mathematical r e s u l t s ,  a g e n e r a l i z e d c l a s s of u t i l i t y  t h i s chapter  of that  e x p l i c i t but R^  and  function R  functions, while  w i t h i n a subclass  then  the remainder of  of the  generalized  A  c l a s s of u t i l i t y r e s u l t s by  functions.  i n t r o d u c i n g a new  Chapter 4 w i l l g i v e an a p p l i c a t i o n of method f o r s e l e c t i n g a u t i l i t y  mathematical r e s u l t s are extended i n Chapter 5, results  f o r other u t i l i t y  functions.  \  the  function.  f o r i n s t a n c e by g i v i n g  The  - 25 -  3.1.  The idea behind the mathematical results.  Let u(w) be a continuous,  twice d i f f e r e n t i a b l e , and increasing  u t i l i t y function over wealth w. r i s k aversion R.(w)  Pratt(1964) then interprets  the  absolute  by considering a decision maker's r i s k premium for a  A  small, f a i r gamble z. variance a  2  Thus, l e t z be a gamble with E(z) = 0 and  2 . We w i l l now  = O .  z (2.4.), which gives  expand u(w  o  ) around w  o  small  on both sides of  2 (3.1.)  u(w  -TT)  = u(w  o  ) -  TTU'(W ) + J -  o  o  u"(w)  -  2  and ~2 (3.2.)  E[u(w  + 5 ) ] = E[u(w o  ) + zu'(w o  ) + o  u"(w 2  ) +  ]  o  2 = u(w  Equating  ) + ~ o 2  u"(w  ) + o  (3.1.) and (3.2.) and neglecting higher-order  terms (as we  are  actually considering i n f i n i t e s i m a l gambles) gives us  (3.3.)  u(w  ) - TTU'(W  o  o  ) o  (=>  u(w  ) + %a u"(w ) o 2  from which i t follows that  (w)«4  H  (3.4.)  R sR A  i.e. R  A  °  a  V-  2TT(W  2  G  ~  ,Z)  2  i s twice the r i s k premium per unit of variance for i n f i n i t e s i m a l  - 26 -  gambles.  S i m i l a r l y , Pratt(1964) shows, that for z being a nonfair gamble,  i t holds that  (3.5.)  2TT(W ,Z) o  R.[w +E(z")] «  a- 2 z  and for z = <h;-h> we have  (3.6.)  R s R ( ) « |(2p - 1) A A o h  where p = p(w ,h), o  W  and i n both cases z i s assumed to be i n f i n i t e s i m a l . Pratt's interpretation of the absolute r i s k aversion i s certainly useful, which can be i l l u s t r a t e d through the following example.  Example 3.1.  The decision makers i n Example 2.3. were assumed to behave  according to the u t i l i t y functions u^ = w - (l/4)w (0 <^ w <^ 3/2) respectively. that R ^w") A,lo  2  and u^  w - (l/3)w  =  At an i n i t i a l wealth of w^ = 1 i t was shown  = 1, R (w") = 2, TT (w",z) = 0.118 and TT (W",5) = 0.207, A,2o l o 2 o 0  0  where z = <0.5;-0.5>. Using (3.4.) we can estimate R  by R  where  2TT (w"s2) (3.7.)  R, .(w") = A, 1 o  1  ° 2 z  =  0.944  (R. . = 1) A, 1  1.656  (R  and  A ft ^ ( 3  -  8 , ;  6 A,2(w") =  R  2  2TT (w" 5 ) —^—2 2 = a~ z  A  A  „ = 2).  '2  E s p e c i a l l y i n the l a t t e r case the estimation error seems quite large.  - 27 -  We can formalize the estimation error by the following  D e f i n i t i o n 3.1.  d e f i n i t i on.  The R -percentage error i s defined by A .  R (w ) - R (w ) A o A o A  (3.9.)  R % ESTIMATION ERROR = A  A  .100  R.(w ) A o  An i l l u s t r a t i o n i s given by the following example.  Example 3.2. Referring to Example 3.1. we find that by (3.9.) the R  ,% ESTIMATION ERROR = 5.6 and the R. J„ ESTIMATION ERROR = 17.2 A,l A,2  Altogether,  using Pratt's interpretation of R  as a formula i s limited  A  in the sense that the formula i s an approximation and i t requires ranged gambles. function.  small-  On the other hand, the formula applies to any u t i l i t y  The question now arises whether we are able to come up with exact  results and not only approximations or estimations.  The answer has c l e a r l y  proven to be 'yes', however at the price of specifying the shape of the u t i l i t y function (not i t s essential parameter).  This has led to the f i r s t  R^ as a function of the r i s k premium and the parameter of a f a i r  THEOREM 3.1.  For a decision maker with a quadratic u t i l i t y  gamble:  function  u(w) ^ w - cw , where c > 0 and w < — , i t holds that 2c 2TT  (3.10.)  R  A  2  2  where TT = Tr(w ,z) and z a f a i r gamble with standard o  deviation a  = 0~.  - 28 -  Proof:  Noting that u  u(w -TT) = u(w ) o o  (m)  TTU'(W  = 0 for m = 3,4, , (3.1.) becomes , m dw 2 ~ %TT u"(w ) and (3.2.) becomes E[u(w +z)] = o o  (w) = ) +  o  d  u  (  w  )  2 u(w^) + ho~ u"(w ) . Equating the last two equations according to (2.4.) o  2 2 - im'Cw ) + %TT U"(W ) = %C~ u"(w ), from which the desired result o o z o  gives  (3.10.) follows d i r e c t l y .  Notice that i f one uses (3.4.) to estimate  for the u t i l i t y function  u(w) ^ w - cw , the R % ESTIMATION ERROR becomes A 2  2TT  R.% ESTIMATION ERROR = A  a  2  2TT  2  - TT  a  a2  2TT 2 2  .100 = ^ - (100), 2  where a  a  2  = a~ . z 2  - TT  Example 3.3. Using the above expression, we are now able to check the results i n Example 3.2.: The R^ ^% ESTIMATION ERROR = the R % ESTIMATION ERROR = A,2 0  ^'^VX = 17.14. (.1/4;  (  11 ft)  2  =5.57 and  The differences are due to  rounding-off errors.  Theorem 3.1. has provided the stimulus f o r a search for similar results with other u t i l i t y functions. considered  i s addressed  The question of which functions can be  i n the following section.  - 29 -  3.2.  A generalized class of u t i l i t y  In this section we w i l l  functions.  introduce a generalized class of u t i l i t y  functions. We w i l l  THEOREM 3.2.  f i r s t state:  I f U(w) and u(w) have absolute r i s k aversion R (w) A,U and R n  for  f  p  K  £I22E-  some d i f f e r e n t i a b l e function f(w) ^ 0, i t holds  R „(w) = R (w) - ^ T T • A,U A,u f(w)  (3.11.)  T.  (w) respectively, then i f U'(w) = f(w).u'(w) j U  A  y  ( ^ = - " U  w ;  ( w )  = _ f(w).u"(w) + f'(w).u'(w) _ _ u"(w) _ f'(w f(w).u'(w) u'(w) f(w  u'(w)  - » ( ^ = R. (w) A,u  f'(w) —s- .  f(w)  Now l e t us define a generalized class of u t i l i t y functions u(w) by  (3.12.)  u'(w) ^ f(w),  r  Bwa +  —- n 6w  a  with w > 0, B > 0, a > 0, T + 0,  + n > 0, and T) = 1 i f x -»• °°.  Using  Theorem 3.2. i t i s easy to v e r i f y that the absolute r i s k aversion R.(w) corresponding to (3.12.) i s : (3.13.)  p ( , _ u"(w) _ aBw 3 R A (w) - - ~  T  f'(w) ^  +n  - 30 -  we define u(w) = u ( w ; £ , a , T , B , T | ) b y  Choosing f(w) =  u*(w) = u ' C w j ^ . a . T . B . n ) ~ w  (3.14.)  ?  According  :  (3.15.)  R(w) A  5  (3.16.)  +n  !BwC  R (w)  E, = 0, we o b t a i n  a  to (3.14.) i s by D e f i n i t i o n 2.9,  - e.  R  Selecting  +n class  of  u t i l i t y  functions  u(w;0,a,T,8>n)  as  by  r  (3.17.)  u' ( w ; 0 , a , T , 8 , r i )  ^  Bwa  —  ^ +  n  Multiplying the right hand side of (3.17.) by  (3.18.)  corresponding  w  Bwc  5W  _ TIT -  +n  a-l _ aBw'  The r e l a t i v e r i s k aversion corresponding  c  -T  to (3.13.) we have for the absolute r i s k aversion  to u ( w ; £ a , T , B 5 r i )  defined  'Bwa  T  and  substituting  gives  u ' ( w ) 'V/ ( w 3  +  c)  T  so that both (3.14.) and (3.17.) generalize the class of u t i l i t y  functions  - 31 -  (3.18.), which class was proposed by Pratt(1964).  Integration of (3.17.)  for a = 1 gives  l-T  (3.19.)  u(w;0,l,T,B,n) ^  (T  B(1-T)| T  t  1)  which i s the well known Hyperbolic Absolute Risk Aversion (HARA-) class of u t i l i t y functions as defined by Merton(1971). Hence, (3.14.) and (3.17.) are also generalizations of the HARA-class.  Not for a l l functions within  the HARA-class we need to use a l l parameters e x p l i c i t l y ,  for some i t w i l l  TIT  suffice to substitute c = —^ , so that by (3.15.) we have  (3.20.)  R  A  ( ) = _ L _ w +c w  which d i r e c t l y explains the name of the HARA-class. We w i l l now continue to give c r i t e r i a for (3.14.) to be s t r i c t l y decreasingly r i s k  averse.  THEOREM 3.3.  The class of u t i l i t y functions u(w;£,a,T,6,n) as defined by (3.14.) i s s t r i c t l y decreasingly r i s k averse i f :  (3.21.)  w > d, T > 0 and f i n i t e , and 0 <. £ <_ xa, where  (3.22.)  d = < max 0  where  -  ^  8 '  (3(xa-0  - 32 -  2?  i(a-l) + (3.23.)  K  =  26  According  n  2  2  h  T ;  - a  T  Proof:  ^\ ^ M\ a ((a-1) +.a {—M 2  -  to Lemma 2.12. u(w; E,, a, T , 3 >n) i s r i s k averse i f ^ ( ) i s R  w  p o s i t i v e , so that with (3.15.) we have aBw  a-1  (3.24.)  Bw  +  ^ w  > 0  n  which gives  (3.25.)  V. w>  K ±  when  B(Ta-c-)  Ta.  Furthermore, by D e f i n i t i o n 2.11. and Lemma 2.14. i t follows that the u t i l i t y function u(w;E,,a,T,B,r|) i s s t r i c t l y  decreasingly r i s k averse when i n addition  i t holds that ^-[R,(w)] < 0, or: dw A fBw a  1  a(a-l)(Bwa  (3.26.)  We w i l l  2  )  -  Bw  r  o apw • T  a  -  1  i (aBw a  V w  solve (3.26.) for w  by finding the roots of — [R.(w)] = 0 , i . e . the • > ° dw A l e f t hand side of (3.26.) set equal to zero. We now multiply the LHS of 2 3  (3.26.) by -|— and set this equal to zero. find a quadratic equation  i n w , namely  After reordering the terms, we  - 33 -  (3.27.)  (f  - a)|-(w ) a  +  2  [ 1 | + a(a-l)]$nw  + fy\  a  2  =  0  which gives the following two roots for w  c  a(a-l) +  (3.28.)  w1  ± n a (a-l)  T 2  23. i T  1%  + a .-^  =  K.  a  T  Note that 'the LHS of (3.27.) has a negative l i m i t i n g value for w ->- , as r ft (— - ) — < 0 for 6,T > 0 and 0 < £ < T a . Assume now n > 0, so that 00  2  a  T  K  -  T  > K  =  for 8,T > 0  , as — ( - - a) < 0  +  T T  (3.27.) i s negative for w  3  > K  and w  and  < K  3  ,Va ,  w > [K_]  i s one solution of (3.26.).  2, .»2 2 4^1 Va (a-l) + a • —  [a(a-l) + %  T  2  & < a.  i n this case, so that w  3  • ^f(ax-C)! a  k  > a(a-l) +  =  > K_ w i l l  > 0 which i s included i n (3.21.) and (3.22.).  w > [K ] ^  3  and the i n i t i a l conditions •^ — + n > 0 W  T >  where again c =  .  2E T  or  <_ 0, as for  always be overruled  (or: w > [- ^ ]  ^  a  ) and  p  (3.21.) to (3.23.) have been obtained.  For u(w;0,a,T,6>i~|) i t I  (3.29.)  > K  Combining now (3.25.),  T  w > 0, the conditions  3  Assuming n < 0 gives K_ £  0 <. £ < T a , so that for n <_ 0 the condition w by w  Hence, the LHS of  T  0  and  s  •  easy to see that (3.21.) to (3.23.) become  w >  max[0, -c, c ( a - l ) ]  The conditions  V*  (3.29.) were e a r l i e r derived by Pratt(1964)  P  for his class of u t i l i t y functions averse.  (3.18.) to be s t r i c t l y  decreasingly  risk  - 34 -  For u(w;0,l,T,B,n) we find that (3.21.) to (3.23.) reduce to:  (3.30.)  T > 0  and  w > max 0  - 21 6  The remainder of this chapter w i l l be r e s t r i c t e d to the function R A I  for u(w;0,l , T , 8 , T ) ) , where T = -1, \, 1, 2 and u(w;0,0,T,8,ri). twofold.  00  respectively, and for  The reason behind this selection of u t i l i t y  functions i s  F i r s t , for these u t i l i t y functions a l l desired functions R  be obtained, so that a very complete presentation results.  could  Second, for the  purpose of the new assessment procedure, as discussed i n Chapter 4, these functions s u f f i c e i n explaining and i l l u s t r a t i n g  the method.  However, i n  Chapter 5 we w i l l give extended results for other functions within the class defined by (3.17.).  Table 3.1.* l i s t s the above six u t i l i t y functions.  derivation of the contents of Table 3.1. can be found i n Pratt(1964) and Merton(1971).  The  Table 3.1.  Utility  Function:  Name:  E x p l i c i t Expression with FIT c = -5— ( E x p o n e n t i a l : c = 6 ) P  Absolute r i s k  as a f u n c t i o n o f w :  u(w;0,0,T,8,n)  linear  w  constant  u(w;0,1,-1,8,n)  quadratic  - (w + c)  increasing  u(w;0,l,%,B,n)  square r o o t  V  decreasing  u(w;0,l,l,8,n)  logarithmic  log  u(w;0,l,2,8,n)  hyperbolic  u(w;0,l,°°,c,l)  exponential  w + c  (w + c)  1 w + c  - e  -cw  aversion  decreasing  decreasing  constant  - 36 -  3.3.  Risk aversion  as a f u n c t i o n o f c e r t a i n t y e q u i v a l e n t or  r i s k premium f o r the gamble z = <h,p;-h>.  Consider  the gamble z = <h,p;-h>, where E ( z ) = h(2p-l) :  + h F i g u r e 3.1.  (3.31.) z = <h,p;-h> (1-p)^^- h  so that by (2.1.)  (3.32.)  u(w +z) = p.u(w +h) + (l-p).u(w -h) o o o  Hence, f o r u(w;0,1,T,6,n) w i t h T f 1, (3.32.) r e s u l t s i n :  rS(w +z)  l-T  +n  3(1-T) (3.33.)  6(w +h) o B(l-x)  Using  (3.20.) and f u r t h e r r e d u c i n g  hR.l-T  hR^l-T  A l  1+  l-T  +n  +(l-p)  (3.33.) p r o v i d e s f o r T t 1  zR l-T  (3.34.)  6(w -h)  l-T  +n  = P 1+  + (1-p) 1 -  S u b s t i t u t i o n o f z = h(2p-l) - TT p r o v i d e s e a s i l y a r e l a t i o n s h i p c o n t a i n i n  - 37 -  TT = TT(W ,Z). O  R  as a function of z and TT for u(w;0,l , x , B , n ) with x ^ 1  A  can be derived from (3.34.).  The function R  for u(w;0,1,1,8,n) and  A  u(w;0,0,x,6,Tl) w i l l be derived d i f f e r e n t l y .  We continue by specifying the  functions R^ for the u t i l i t y functions of Table 3.1. Again, we w i l l denote R (w ) by R . A o A A  1 .  J  For u(w;0,1,-1,8,11) ^ - (w+c) , with c = -  the relationship (3.34.) p  becomes:  (3.35.)  (1-2RJ  = p(l-hR)  2  A  2  +  (l-p)(l+hR )  2  A  A  A  which gives  (3.36.)  - 2[h(2p-l) - 2j _  R  h  A  - 2  2  2TJ_ 4p(l-p)h  2  2  + 2h(2p-l)TT - TT  2  /  and for the f a i r f i f t y - f i f t y gamble z = <h;-h> the result becomes quite simple:  (3.37.)  R A  = _22 i  h  v  '  2°_.  For u ( w ; 0 , l , % , 8 , n )  (3.38.)  =  ~  2  2  - z  _2TT v  h  - TT  s/w+T , with  ^  y l + 22R  = A  f o r z = <h;-h>. 2  2  c = -~, (3.34.) gives <£p  p V l + 2hR  + ( l - p ) V l - 2hR A •  A  - 38 -  so that  (3.39.)  2p(l-p)[h(2p-l) - z] ,„ 2 „ , 2. 2 .2 (2p -2p+l) h - 2h(2p-l)z + z  =  A  x  2p(l-p)ir . 2,, ,22 2 4p (1-p) h + TT  and for the f a i r f i f t y - f i f t y gamble z = <h;-h> this gives:  (3.40.)  R  A  =  -22  ,2 ,.2 h + 4z x  =  2TT  for z = <h:-h>.  2 2 h + 4TT  3°.  For u(w;0,l,l,B,n) ^ log (w + c) , with c = §, (3.34.) cannot be P applied. However, applying (3.32.) gives  (3.41.)  log(w +c+z) = p.log(w +c+h) + (1-p).log(w +c-h) o  o  o  which gives, after noting that according to (3.20.)  W  q  + c = [R ]  following r e s u l t :  *r  2  5- + h(2p-l) - TT = R  A  (3.42.)  1-p +h  ^- - h  so that for the f a i r f i f t y - f i f t y gamble z = <h;-h> we have  , the  - 39 -  (3.43.)  4^.  v  ~  =  A  2  G  2 =  ,2 „2 h + z  For u(w;0,1,2,3,11) ^ -  +  w  (3.44.)  1 1 + %zR A  TT  c >  with c =  p 1 + %hR A  =  for  .2 2 h + TT  +  z = <h;-h>,  (3.34.) gives  1-p 1 - %hR. A  so that  ^  (3.45.)  2[h(2p-l) - z]  =  h  A  2TT  - h(2p-l)z  2  4p(l-p)h  + h(2p-l)TT  and f o r the f a i r f i f t y - f i f t y gamble z = <h;-h> this gives  (3.46.)  5^.  R  -2z 2 ti  For u ( w ; 0 , l , c o  jC  2TT  =  A =  h  ,1) ^ - e  W  z = <h;-h>.  (3.34.) provides  1-T  zR. lim  C  for  2  1 +  hR.  1-T  p. lim 1 + X-Ko  (3.47.)  hR. + (1-p).lim 1 -  or,  using lim 1 + X-x»  (3.48.)  1-T  -zR„ A i  _  e  =  k  .  , this becomes -hR, A p.e  , +  hR, x A (l-p).e  1-T  - 40 -  Hence,  -zR e  A  _ = e  -[h(2p-l) - Tr]R A 4  (3.49.) hR =  where the hyperbolic For the f a i r  2p.cosh(hR) A  +  (l-2p).e  cosine of x i s defined by  cosh(x) = % ( e + e ) X  X  f i f t y - f i f t y gamble z = <h;-h> the following result then  e a s i l y obtained:  -zR (3.50.)  e  TTR  =  e  A  =  cosh(hR) A  for I = <h;-h>.  we w i l l of course have R = 0, as the A decision maker i s r i s k neutral with a l i n e a r u t i l i t y function. 6 .  For u(w;0,0,T,$,ri) ^ w  A summary of the results i n this section i s provided through Tables 3.2. and 3.3., which appear after section 3.5.  - 41 -  3.4.  Risk aversion  as a f u n c t i o n of p r o b a b i l i t y e q u i v a l e n t f o r  the gamble z = <h,p;-h>.  Consider  the gamble z = <h,p;-h>, so t h a t by D e f i n i t i o n 2.5. and  (2.8.) we have f o r u(w;0,1,x, 8,n) with  6w  l-T  ^  o  x ? 1 :  B(I-T) (3.51.) rB(w  o  B(1-T)  Using  + n  (3.20.) and f u r t h e r r e d u c i n g  1  =  + (1-P)  (3.51.) p r o v i d e s  p 1 +  + n  for x ^ 1  hR. +  l-T  B(w -h)  l-T  hR. (3.52.)  l-T  +h)  l-T  (1-p) 1 -  Hence, f o r p = p(w ,p,h) i t h o l d s o  hR. 1 (3.53.)  P  =  hR.  -  l-T ~  1 -  l-T  1 +  1 -  The d e s i r e d f u n c t i o n s R^ f o r u(w;0,1 , x , B > n ) vith (3.52.) o r (3.53.) by s p e c i f y i n g x more d i r e c t way.  l-T  hR.  x  1 can be d e r i v e d  F o r x = 1 we w i l l d e r i v e a r e s u l t  from in a  - 42 -  1°.  For u(w;0,l,-l,8,ri) * ~ (w + c) , with c = - ^, (3.52.) gives  (3.54.)  1  =  p(l - hR )  +  2  A  (l-p)(l + hR )  2  A  which gives  (3.55.)  R = -(2p - 1) A h 2  so that  (3.56.)  p = khR  A  + k  which can be v e r i f i e d through (3.53.) as well.  2° .  For u(w;0,l,%,8,n) ^ v*w + c , with c= jg,  (3.57.)  1  =  pVl + 2hR  (3.52.) gives  + ( l - p ) v l " 2hR  A  A  which gives, after taking both sides to the power two and after some rearranging:  (3.58.)  (1 - 2p)hR + p(l-p) A  so that  (3.59.)  R, A  -2^2P-D(P-1) h(l-2p+2p ) 2  2  =  p(l-p)vl - 4h R 2  2 A  - 43 -  whereas (3.53.) w i l l give an e x p l i c i t expression for p.  3^.  For u(w;0,l,l,B,n) ^ log (w + c) , with c = §, (3.52.) and (3.53.) p  cannot be applied.  (3.60.)  However, using (2.8.), we have  log(w +c) = p.log(w +h+c) + (l-p).log(w -h+c) o o o  which gives, after noting that according to (3.20.) w  + c = [R ] , o A 1  log(l-hR ) A  ( 3  '  £.  6 1  -  )  P  log(l-hR ) - log(l+hR ) '  =  A  For u(w;0,l,2,6,n) ^ -  (3.62.)  1  =  w  A  ]_  c  , with c =  p [ l + %hR ]  1  +  ( l - p ) [ l - %hR ]  i\  so that  (3.63.)  R = A  2 ( 2  P  ~ h  l )  yielding  (3.64.)  p = %jhR + % A  which can be v e r i f i e d  (3.52.) gives  through (3.53.) as well.  A  1  - 44 -  31.  For u(w;0,l,°°,c,l) ^ - e  (3.65.)  C  w  (3.52.) provides  hR. 1  =  p. 1 im 1 +  lim 1  *  +  = e  T  (3.66.)  6°.  R  A  =  l.  l  o  g  hR. +  l-T  which gives, using  l-T  l-T  (l-p).lim 1 -  -k  _£_  For u(w;0,0,T,6,ri) ^ w  we w i l l simply have R  = 0, so that p  A summary of the results i n this section i s provided through Table 3.2., which appears after section 3.5.  -  3.5.  Risk aversion  45 -  as a f u n c t i o n of g a i n e q u i v a l e n t f o r the  gamble z = < K , p ; - h > .  Consider the gamble z = <fi,p;-h>  , so that by D e f i n i t i o n 2 . 4 .  and v by  ( 2 . 7 . ) we have f o r u ( w ; 0 , 1 , T , B , n ) , w i t h T ? 1:  1-T  Bw B(l-T) (3.67.)  B(w +h)  __£_  B(1-T)  T  + n  +  _  ( 1  U s i n g ( 3 . 2 0 . ) and f u r t h e r r e d u c i n g ( 3 . 6 7 . ) p r o v i d e s  (3.68.)  1 = p 1 •+  hR.  r  p )  B(w -h)  |__£_  +  ^  T  n  f o r T / 1:  1-T +  (1-p) 1 -  h R  A)  1 _ T  /  which we w i l l use i n most of the f o l l o w i n g d e r i v a t i o n s o f R. as a f u n c t i o n A of h f o r the u t i l i t y f u n c t i o n s l i s t e d i n T a b l e 3 . 1 .  1°•  For u ( w ; 0 ) l , - l , g , n )  (3.69.)  - (w + c)  1 = p(l - hRA)2 + (l-p)(l  so that  (3.70.)  , with c = - J  =  R  A  2ph -  2(l-p)h  ph2 + (l-p)h2  P  + hRA)2  , (3.68.)  gives  - 46 -  which becomes for p = \ :  (3-71.)  R  =  A  -P^n h +h  for z = <n>h>.  For u(w;0,l,%,B,n) ^ W + c , w i t h c = -§ , (3.68.) g i v e s 2p  2J\  (3.72.)  1 = [ i + 2hR ]^ p  A  +  (l-p)[l - 2hR ]  %  A  which gives after several calculations:  7  o  x  „  _ 2p(l-p)[ph+(p-l)h]  A  For p = \  (3.74.)  (3.73.) becomes  R. = A  3^.  tp h + (l-p)^h]  For u(w;0,l,l,B,ri)  - H) (h + h )  4 ( H  for z = <h;-h>, 2  ^ log(w + c) , with c = ^ , (3.68.) does not apply.  However, using (2.7.), we have  (3.75.)  log(w +c) = p-log(w +c+h) + (l-p)-log(w +c-h) o o o  which gives, after noting that according to (3.20.)  (3.76.) .  1 = [1 + h R j [ l - h R j A A P  1P  .  w + c = [R 1 o A  1  - 47 -  Formula  (3.77.)  (3.76.) becomes for p ^ h  R. A  4°. =  hh  ^  For u(w;0,l,2,B,n) ^  (3.78.)  for z = <h;-h>.  , with c = ^  — w + c  1 = p[l + %hR ] A  , (3.68.) gives p  _ 1  +  ( l - p ) [ l - ^hR ]  1  A  so that  (3.79.)  =  2[ph - (l-p)h] hh  which for p = % becomes  (3.80.)  5°.  R. =  h -h  for z = <h;-h>  hh  For u(w;0,l,°°,c,l) % - e °  W  (3.68.) provides l-T  (3.81.)  hR.  1 = p•1im 1 +  +  (l-p)-lim 1 -r-Ko  l-T  which gives, using lim 1 + -  ~ A 1 = p•e hR  (3.82.)  For p = h  (3.82.) becomes  -k  h R  +  (1-p)-e  A  l-T  - 48 -  —  83.)  1  =  2  hR  e  For u(w;0,0,T,8,n) ^ w  A  ^ +  hR • e  A  we simply have n" =  1-P h -and R. = 0 A . P .  A summary o f the r e s u l t s i n t h i s s e c t i o n i s p r o v i d e d through es 3.2. and 3.3., which appear on the f o l l o w i n g  pages.  T a b l e 3.2.  Summary t a b l e o f the r e s u l t s f o r the gamble z = <h,p;-h>.  Absolute Utility Function:  Risk  Aversion  Certainty Equivalent 2  R, A  as  a  function  of :  Probability Equivalent p  R i s k Premium IT  Gain Equivalent h  Linear: R  u(w;0,0,T,|3,ri)  A  = 0  R  R  1  Quadratic: u(w;0,l,-l,B,n)  „  R  = 0  h  2  -,2  2TT  4p(l-p)h +2h(2p-l)-Tr  2  A  = 0  R  R  A  - i ( 2 p - 1)  = 0  A  2  „  _ 2ph - 2(l-p)h A - 2 pli +  2  2 (l-p)h  A  Z  A = 2 p ( l - p ) t h ( 2 p - l ) - 2]  u(w;0,l,%,6,ri)  2  _  R  (2p -2p+l) h -2h(2p-l)2+2 2  2  2  2p(l-p)TT 4p (l-p) h  A  2  2  JJ + IT  2  2  h ( 2 p - l ) - ir = ' 1 fl l fl 1 2 = - ~ + £ +h i - h A '•A J '•A J P  1 _ P  1  h[l-2p+2p ] 2  2  2p(l-p)[ph+(p-l)h] [p h+  A  (l-p) h]  2  2  2  P =  l fl  fl  JJ _  _ -2p(2j5-l)(f5-l) A  Logarithmic: u(w;0,l,l,8,n)  R  A =  _ 2 [ h ( 2 p - l ) - 2] "  Square Root:  A  l  p  1  _  log(l-hR )  p  A  1=  log(l-hR )-log(l+hR ) A  (l+hR ) (l-hR ) P  A  1 _ P  A  A  Hyperbolic: u(w;0,l,2,6,n)  _ 2 [ h ( 2 p - l ) - 2]  R  h  A  -2R Exponential: u(w;0,l,°°,c,l)  e  2  r -  - h(2p-l)2  2 7 r  R  4p(l-p)h -[h(2p-D  A  =  +h(2p-l)TT  - TT]R  e A  K  _ 2[ph - ( l - p ) h ]  _  •  *  •  hh  = hR.  A  „  = 1 ( 2 * - 1)  A  hR 2p.cosh(hR ) + ( l - 2 p ) e  A  2p.cosh(hR )+(l-2p)e A  A  -hR 1 = pe  hR  A  + (l-p)e  A  Table 3.3.  Summary table of the results for the gamble z = <h;-h>. Absolute  Utility Function:  Name:  Certainty Equivalent 2  R. as a f u n c t i o n o f : A Gain Risk Premium TT Equivalent h  u(w;0,0,T,6,n)  Linear  R =0  R = 0  u(w;0,l,-l,B,n)  Quadratic  u(w;0,l,%,B,n)  u(w;0,l,l,B,n)  u(w;0,l,2,B,n)  5  '  Square Root  r A  R  r  Logarithmic  "  A  ~  r  " .2 . .2 h + 4z 2 2  -  ~  r  2 z  .2 .2 h +z +  "  2 2  R  2  h  -2R  -  A  u  -  A  -  A  =  A A  = cosh(hR.) A  e  A  21T  A  2  2  h  - TT  27T  „  , 2 . 2 h + 4TT  27T  . 2 2 h + TT  H  2 h  uR  A  e  r  2 2  .2 2 h -2  +  A  R  A  a  R = A  Hyperbolic  Exponential  Aversion  A  A  u(w;0,l,°° c,l)  Risk  A  = 0  _ 2(h-h) 2 .2 h +h c  _  A  ~  h  r  h -h ~  hh  \ -V A  2  (h +  A  R  4(h-h) "  h  hh  , r -hR, hR , 1 A A 1 = - e +e  l1  A  = cosh(hR.) A  - 51 -  3.6.  Risk aversion  as a f u n c t i o n o f c e r t a i n t y e q u i v a l e n t o r  r i s k premium f o r the gamble z = <h,p;0>. -  Consider  the gamble z = <h,p;0>, where E ( z ) = ph :  F i g u r e 3.2.  (3.84.)  z = <h,p;0>  so t h a t by  (3.85.)  (2.1.)  u(w +z) = p.u(w +h) + (l-p).u(w ) o o o r  Hence, f o r u(w;0,1,T,8,n) w i t h T ± 1, (3.85.) r e s u l t s i n :  rB(w +z)  x  l-T  +n  .6(1-T) (3.86.)  B(W +h) o  8(1-T)  Using  — = —  (3.20.) and f u r t h e r r e d u c i n g  ^  + n  l-T  Bw + (l-p)  + n  (3.86.) g i v e s f o r T ^ 1  l-T (3.87.)  l-T  = P 1+  hR , A  l-T + (1 - p)  S u b s t i t u t i o n o f z = ph - TT i n (3.87.) p r o v i d e s e a s i l y a r e l a t i o n s h i p  - 52 -  containing TT = TT(W ,Z). R o  A  as a function of 2 and TT for u ( w ; 0 , l , T , $ , r i )  with T f 1 can be derived from (3.87.).  The case T = 1 w i l l be derived  differently.  ll.  For u(w;0,l,-l,B,n) ~ - (w + c ) , with c = - §, (3.87.) gives 2  P  (3.88.)  (1-2R )  =  2  p(l-hR )  A  + (1-p)  2  A  which gives  (3.89.)  {V -*\ = ,2 „2  R. A  2  I..2 ..  h  ph  - z  2  ; r  p(l-p)h  2  + 2phTT - TT  and for the f i f t y - f i f t y gamble z = <h;0>  (3.90.)  R =  =  A  h  2^.  - 22  For u ( w ; 0 l % g r ) ^  (3.91.)  J  )  J  )  -  for S = <h;0>.  2  h  + 4hfT - 4TT  + c , with c =  1  V l + 22R  A  =  pVl + 2hR  A  (3.87.) gives  + (1-p)  so that  (3.92.)  R A  =  ~2p(l-p)(z-ph) [2-hp ] 2  2  =  2p(l-p)TT [p(l-p)h - TT]  2  and  ex  for the f i f t y - f i f t y gamble z = <h;0> we obtain :  no  \  (3.93.)  rt  4h-82  R  8TT  =  ,  =  (42-hr  A  N  for z = <h;0>. (h-4TT)  Z  For U(W;0,1,1,B,TI) ^ log (w + c) , with c = §, (3.87.) cannot be P applied. However, using (3.85.) gives us:  (3.94.)  log(w +c+z) = p.log(w +c+h) + (l-p).log(w +c) o o o  which gives, after noting that according to (3.20.) the  w + c = [R ] o A  following r e s u l t :  (3.95.)  1 + 2R  A  =  1 + (ph - TT)R  A  A  =  [1 + hR ]  A  P  A  or log(l+zR ) A  (3.96.)  p=  =  log(l+hR ) A A  l o g [ l + (ph—TT)R ] _ l o g ( l + hR ) A A  Hence, for the f i f t y - f i f t y gamble z = <h;0>, we have:  (3.97.)  R  =  ^ - ^ i 2  =  ^ (h - 2TT)  For u(w;0,l,2,B,n) ^ " — J — » w + c  w  i  t  h  c  f  o  r  2  =  <  h  = ^5 , (3.87.) gives p  ;  0  >  ,  - 54 -  (3.98.)  1 + %zR  1 + %hR  A  + (1-p) A  so that  (3.99.)  R  2(ph-z) (l-p)h2  =  A  2jn_ (l-p)h(ph-Tr)  =  and for the f i f t y - f i f t y gamble z = <h;0> we have  (3.100.)  R  A  2h - 4z hz  =  For u(w;0,l,°°,c,l) i> - e  (3.101.)  lim  1 +  zR.  C  8TT  1-T  1-T  -zR  (3.102.)  = e  1-T  + (1-p)  -k  -(ph-Tr)R  A  e - 1 -hR. e - 1  P =  hR.  p. lim 1 +  T  z = <h;0>.  (3.87.) provides  w  which gives, using lim 1 + X"KO  for  h(h-2Tr)  e -hR. e -1  and for the f i f t y - f i f t y gamble z = <h;0>  (3.103.)  6 .  —zR  A  For u(w;0,0,T,6,n) ^ w  -(%h-Tr)R,  -  ;  hR  e  we w i l l again have  A  +1  R  for  z = <h  = 0.  A summary of the results i n this section i s provided through Tables 3.4. and 3.5., which appear after section 3.8.  - 55 -  3.7.  Risk aversion R  as a function of p r o b a b i l i t y equivalent for  the gamble z = <h,p;0>.  Consider the gamble z = <h,p;0>, and note that D e f i n i t i o n 2.5. does not apply for this gamble.  D e f i n i t i o n 2.5. i s the d e f i n i t i o n as known i n  the l i t e r a t u r e , hence i t i s included under the preliminaries. For the purpose of this section we w i l l need:  D e f i n i t i o n 3.2.  A p r o b a b i l i t y p such t h a t the d e c i s i o n maker  is  i n d i f f e r e n t between  z =  W  +  Q  %h  a n  d the gamble  <h,p;0> i s c a l l e d the gamble's p r o b a b i l i t y and i s denoted by  equivalent  p = pCw^jh).  Hence, p = p(w^,h) i s defined by  (3.104.)  u(w +^h) = p.u(w +h) + (l-p).u(w ) o o . o r  so that f o r u(w;0,1,T,3,n)  T B(1-T)  r  with T f 1  B(w +hh) ^ ° . —  -  —  +  we have l-T  n  (3.105.)  B(I-T)  B(W +h) o  1-T +  n  1-T  Bw + (l-p)  Using (3.20.) and further reducing (3.105.) gives for T ^ 1  + Tl  - 56 -  l-T (3.106.)  1 +  l-T  hR. 1 +  2T  +  (1-p)  Hence l-TL X  1 (3.107.)  P =  l-T 1 -  T  We w i l l now specify our results for the gamble z = <h,p;0> for the u t i l i t y functions i n Table 3.1.  lj\  For u(w;0,l,-l,B,n) * - (w + c) , with c =  (3.108.)  (1.- % h R )  =  2  A  p(l - hR )  2  A  +  - j,  (3.106.) gives  (1-p)  wh i ch g ive s  (3.109.)  R  A  =  *<1 - P> h ( l - 4p) 2  so that 4 - hR. (3.110.) 4(2 - hR ) A  which can be v e r i f i e d through (3.107.) as well,  V  - 57 -  2^.  For u(w;0,l,%,B,n) ^ Vw + c , with c = jg,  V I + hR A  (3.111.)  p V l + 2hR A  =  (3.106.) gives  +( 1 - p ) r  which gives . -4p(2p-l)(p-l)  (3.112.)  R. = [2p -l] h  A  2  2  Formula (3.112.) does not permit a way of expressing p as a function of h R , however applying (3.107.) gives  (3.113.)  p  =  1 - V I + hR A  •  1 - V I + 2hR A  3^.  For u(w;0,l,l,6,n) ^ log (w + c) , with c =  (3.106.) and (3.107.) P However, applying (3.104.) gives us  cannot be applied.  (3.114.)  p.log(w +h+c) + o  (1-p).log(w +c) o  which gives, after noting that according to (3.20.)  w + £ = [R ]-1 o A  (3.115.)  log(w +%h+c) o  p  =  =  l o g ( l + %hR ) 2- . l o g ( l + hR ) A  4^.  For u(w;0,l,2,B,n) ^ - £-^r , with c = ^ ,  (3.106.) gives  - 58 -  (3.li6.)  (1 + %jhR ) A  =  1  p ( l + %hR ) A  1  A  +  (1-p)  which gives 2(2p - 1) (3.117.)  R  A  =  (1 - p)h  so that 2 + hR. (3.118.) 4 + hR.  which can v e r i f i e d through (3.107.) as,well,  5jL  For u(w;0,l,°°,c,l) % - e  (3.119.)  hR *\ A lim 1 + 2T  C W  ,  (3.106.) provides  1-T =  p.lim 1 + X"K»  1-T which gives, using  lim 1 + X-K»  (3.120.)  R  A  =  e  -k  h ' S 11 " P.  =  lo  so that —%hR 1 - e (3.121.)  P  =  -hR. 1 - e  which can be v e r i f i e d through (3.107.) as well.  hR.  1-T +  (1-p)  - 59 -  6°.  For u(w;0,0,T,B,r|) ^ w  we simply have  A summary of the results i n this section Table 3.4., which appears after section  3.8.  R  = 0  and  i s provided  p s h.  through  - 60 -  3.8.  Risk aversion R  as a function of gain equivalent for the  gamble z = <h,p;0>.  Consider the gamble z = <h,p;0> , and note that D e f i n i t i o n 2.4. does not apply for this gamble.  D e f i n i t i o n 2.4. concerning gain equivalents i s  best known, hence included under the preliminaries. For the purpose of th section we w i l l now state:  D e f i n i t i o n 3.3.  An amount h such that the decision maker i s indifferent between the status quo W  plus p-h, and the gamble z =  q  <fi,p;0> i s called the gamble's gain equivalent and i s denoted by h = h(w p,h). o>  Hence, h = h(w ,p,h) i s defined by o  (3.122.)  u(w  o  + ph) = p-u(w  o  + h)  +  (l-p)-u(w ) o  so that for u(w;0,1,T,B,n) with T f 1 we have  T BU-T)  B(w +ph) o _ —  ^  1-T  + r,  (3.123.) r  B(1-T)  B(w +h)  i  1-T  rBw + (1-p)  V.  J  Using (3.20.) and furhter reducing (3.123.) gives for T ± 1:  1-T  - 61 -  l-T  phR (3.124.)  l-T  1 +  =  P 1 +  +  (1 " p) •  We w i l l now specify R^ as a function of h for z = <h,p;0> for the u t i l functions l i s t e d i n Table 3.1.  For u ( w ; 0 , l , - l , B , T i ) ^ - (w + c) , with  (3.125.)  (1 - p h R )  =  2  A  p(l - hR ) A  c  2  ~ g » (3.124.) gives  =  +  (1- ) P  which gives  -  2(h - h)  (3.126.)  v  so that for p= %  (3.126.) becomes  (3.127.)  R  =  A  2^.  4(h - h)  " 2h - h 2  for z = <h;0>.  2  For u(w;0 l,%,6,n) ^ Vw + c , with c =  (3.128.)  5  t l + 2phR  =  p [ l + 2hR  which gives after several calculations  (3.129.)  R A  = 2(l-p)(h-h) " (h - p h )  2  ,  +  (3.124.) gives  (1-p)  - 62 -  Hence, for the f i f t y - f i f t y gamble z=<h;0>, i . e . p = % ,  (3-130.)  R  =  M  3^.  for I  ' ^  h  (2h  A  we have  = <h;0>.  h r  -  For u(w;0,l,l,g,n) ^ log(w + c) , with c = ^  (3.124.) does not  P  apply.  However, using (3.122.), we have  (3.131.)  log(w +c+ph) = p-log(w +c+h) + (1-p)•log(w +c) Q  Q  Q  which gives, after noting that according to (3.20.)  (3.132.)  [1 + hR.] A  =  P  1 + phR  A  w  + c = [R ]-1 o A  .  Hence, for p = h this gives  (3.133.)  R  =  4 ( f i  A  4i.  I h  For u(w;0,l,2,B,n) ^  "  (3-134.)  ]— W  [1 + % hR ]  1  P  so that  (3.135.)  R A  =  for z = <h;0>.  h )  2(h-h) (l-p)hh  +  , with c =  = p[l + %h  ,  (3.124.) gives  K  C  R  ]  1  °  + (i- ) p  - 63 -  Hence, f o r p = %  (3.136.)  IV  this  gives  4(h - h) R  A  =  F o r u(w;0,l,°°,c,l) ^ - e  phR (3.137.)  f o r z = <h:0>.  hh  C  (3.124.) p r o v i d e  W  l-T  hR.  lim 1 +  p • lim 1 +  -r-X»  + (1-p)  •r-X»  l-T  which g i v e s , u s i n g l i m 1 +  -hR (3.138.)  l-T  =  A  p-e  =  -phR e  e  -k  A  - (1-p)  so t h a t f o r p = %  (3.139.)  6° .  e  -hR. A  F o r u(w;0,0 , T , 8 5 r))  —%hR „ A 2e  =  ^  w  we w i l l  , -1  again  f o r z = <h;0>.  simply  state  h  A summary o f the r e s u l t s i n t h i s s e c t i o n i s p r o v i d e d T a b l e s 3.4. and 3.5., which appear on the f o l l o w i n g pages.  = h  and R  through  =  0.  Table 3.4.  Summary table of the results for the gamble z = <h,p;0>.  Absolute Utility Function:  Risk  Aversion  R, A  as  a  function Probability Equivalent p  R i s k Premium ir  Certainty Equivalent z  of : Gain Equivalent h  Linear: u(w;0,0,T,6,Tl)  R  = 0  A  R  = 0  A  R  A  = 0  R  A  * °  v  Quadratic: u(w;0,1,-1,6,n)  „  _ A  2(ph-2)  u  R  7  2  ph  - g  -  '  2 1 1  p(l-p)h +2phTT-ir  A  2  A  2  _ 4(1-20) h(l-4§)  „ A  _ 2(h - h) -2 ,2 h - ph  Square Root: _  uU.O.l.if.B.n)  2  Logarithmic: u(w;0,l,l,i3,n)  -2p(l-p)(z-ph) r - K 1J Iz-hp  A  log(l  A n  log(l  P  + hR )  2p(l- )T7 P  -  log[l  + (ph-TT)R  A  2  A  ]  l o g ( l + hR.) A  A  _ -4p(2p-l)(p-l)  R  [p(l-p)h - TT]  A  + zR )  p r  _  R  2  h[2f> -l] 2  2  A  log(l P  jy _ A  + lshR ) A " log(l + hR )  2(l-p)(h"h) - 2 [h-ph] 2  A  1 + phR = [1 + hR J A A  P  A  Hyperbolic: u(w;0,l,2,B,n)  A  _ 2(ph-z) (l-p)hz  R  A  -  2  T  (l-p)h(ph-TT)  „ A K  - 2(2p-l) h(l-p)  2(h-h) * (l-p)hh  A A  Exponential: u(w;0,l, ,c,l)  -(ph-TT)R  O T  e  = 1 + p e  A  -  l  -1+p  , -hR, i e -1  -hR pe  "PhR. = e -(1-p) A  ( Table 3.5.  Summary ruble of the results for the gamble 1 • <h;0>. Absolute  Utility Function:  u(w;0,l,-l,B,n)  u(w;0,l,%,B,ri)  U(W;0,1,1,B,TI)  Name:  Quadratic  Square Root  Logarithmic  u(w;0,l,2,6,n)  Hyperbolic  u(w;0,l,°°,c,l)  Exponential  Risk  Certainty ^ Equivalent z  R  2h - 42 . 2 h~ - 2 z  A  R. = A  |  A  2  4h - 82  A  =  k  p mium TT  ," A . i f / ™ *  re  8TT  2 . ± h +4hTr 4TT  A  R. = A  +  x  Gain Equivalent h  R  - 4(h - h) A " -2 ,2 2h - h A  4(h - h)  8TT  (2h - h )  (h - 4TTV  R. =  2R  of :  —  A  h2  r 2  g  R. =  (42 - hV  2h - 42 R  i  A  T2  =  R  R. = A  h - 22 R  R^ as a f u n c t i o n  Aversion  4(h - h)  8TT R  (h - 2irV  8rr h(h - 2TT)  -(^h-TT)RA  2  A  =  4(h - h) R  A  -hR.  hh  —%hR = 2e " - 1  - 66 -  4 The R -method A  In t h i s c h a p t e r we w i l l function  e s t a b l i s h a new procedure to s e l e c t a u t i l i t y  from among a l t e r n a t i v e s .  The procedure i s based on the mathematical  r e s u l t s i n Chapter 3 and i s c a l l e d the R^-method. alternative u t i l i t y functions l i s t e d  4.1. the  f u n c t i o n s a r e assumed to be the f i v e r i s k a v e r s e u t i l i t y  i n T a b l e 3.1.  the R^-method t o i n c l u d e over t h e i r e n t i r e  In s e c t i o n  region.  utility  The remainder o f t h i s c h a p t e r w i l l f u n c t i o n s t h a t a r e not s t r i c t l y  extend  r i s k averse  - 67  4.1.  -  The R - method for r i s k averse decision makers. A— • • '• •— 1  :  In t h i s section we w i l l assume that the decision maker behaves according to a s i n g l e r i s k averse u t i l i t y function over the entire wealth region.,  In the next section we w i l l eliminate this assumption.  mathematical extensions  Also, the  i n Chapter 5 w i l l give d i r e c t i o n as to how  one  may  expand the set of u t i l i t y functions from which one i s chosen. From a notational point of view we w i l l c a l l A the set of r i s k averse u t i l i t y functions l i s t e d i n Table 3.1.  D e f i n i t i o n 4.1.  The class A of r i s k averse u t i l i t y functions contains utility  (4.1.)  Hence:  functions  u(w;0,l,T,B,n) where  and  x e T,  T = (-l,^,l,2,oo) with c =  (4.2.)  i f  T  <  00  a n d  6  c =  3  and  n = 1  if  T ->• °° .  The R -method w i l l select a u t i l i t y function within class A to which A the decision maker conforms most c l o s e l y .  The s e l e c t i o n procedure starts  off by obtaining the decision maker's response to two or more reference f  gambles.  We  suggest taking at least three d i f f e r e n t gambles, and the result i  of the R^-method w i l l only benefit by taking more gambles. responses may  The required  be chosen from the four indicated i n Chapter 3, i . e . the r i s k  premium, the c e r t a i n t y equivalent, the p r o b a b i l i t y equivalent, or the gain equivalent.  The question as to how we may  be able to a r r i v e at the decision  maker's exact response w i l l be deferred u n t i l l a t e r ; for the moment we  will  - 68 -  assume that the responses are assessed without error, and that the decision maker responds consistently according to one (unknown) r i s k averse u t i l i t y function. Now, l e t us suppose, f o r example, that the decision maker provides us with TT^, ^ 2 a n d TT^ as h i s r i s k premiums f o r the gambles z^ = <h^,p;-h^>, 2  2  =  <  ^ 2' ' " 2 ' 1  P  _  1  >  a m  * 3 2  =  < h  3»P' ^3 _  >  respectively, where  ?  £  • We  are then able to calculate the value of R. f o r the u t i l i t y functions A class A, according to the results l i s t e d special case when p = %) .  within  i n Table 3.2. (or Table 3.3. f o r the  This way we obtained for each u t i l i t y  function  within A three values f o r the function R , which may or may not be equal. A  If a l l three responses lead to the same value for R^ f o r one u t i l i t y function within A, then these responses are consistent with this u t i l i t y function.  Hence, this w i l l be the function selected as the one to which the  decision maker conforms  because R. i s the value of the absolute r i s k aversion A at the i n i t i a l wealth l e v e l w . That i s , R. = R.(w ) has a s p e c i f i c value no o A A o r  matter what gamble the decision maker i s faced with.  Let us c l a r i f y  this  point with the following example, since i t i s an e s s e n t i a l facet of the R method. Example 4.1.  (4.3.)  A decision maker i s assumed to behave according to  u(w;0,l,2,B,n)  ^  -  — r — -  w  T  where c = --£ 2  c  P  He states 25 and 16 as the r i s k premiums f o r 2^ = <50;-50> and 2^  =  respectively.  Is this consistent with the function R  <40;-40>  taking on the same  - 69 -  value at w^?  Yes, according to (3.46.) we find that indeed  2TT  (4.4.)  A, 1  o  1  =  2(25)  =  2(16)  (50)  2  (40)  :TT  1  2 h  2  More often none of the u t i l i t y functions within A w i l l a t t a i n the same values for R^ at W  from several responses.  q  In this case we cannot d i r e c t l y  determine which u t i l i t y function should be selected to represent the decision maker's preferences.  We can s t i l l select a u t i l i t y function within A to •  which the decision maker conforms most closely.  In other words, we w i l l  select the u t i l i t y function within A that represents "best" the decision maker's behavior.  Certainly, the answer to the question as to what i s "best"  i s highly subjective and dependent on the choice of an appropriate c r i t e r i o n . However, i t does seem to make sense to select the u t i l i t y function for which the three R. values have the least r e l a t i v e standard deviation. What we mean A by ' r e l a t i v e ' w i l l be explained i n the formal description of the R^-method. We w i l l now present t h i s stepwise description of the R -method for r i s k averse u t i l i t y functions.  For the time being we w i l l l i m i t ourselves to the  use of r i s k premiums and the gamble z = <h,p;-h> to aid understanding.  Step 1:  Present the decision maker with n.^ 2 d i f f e r e n t gambles 2  <h^,p^;-tu> ( i = l,2,...,n) where h_^ > 0. terms of r i s k premiums i s W  q  =  The decision maker's response i n  = TT^(W ,Z\) ( i = l,2,...,n) respectively, where O  i s the decision maker's (yet unknown) i n i t i a l endowment.  It i s assumed  that the assessment of the r i s k premiums i s completed without errors, and that  the decision maker responds consistently according utility  Step 2:  to a single (unknown)  function.  Check the r i s k premiums to determine whether the decision maker i s  r i s k averse, i . e . TT^ > 0 for a l l i .  (If not, then refer to section 4.2.  where r i s k proneness i s considered.)  Step 3:  Calculate the values f o r the functions  according  to the r e s u l t s  i n Table 3.2. for a l l u t i l i t y functions within A and for a l l i . We w i l l use the following notation: / 2  1  2  u(w;0,l,-l,B,n)  R (w ,-l,l)  u(w;0.1,3s,g,n)  R (w ,^,l)  u(w;0,l,°°,c,l)  A  A  o  R (w ,-l,2) A  R (w ,-l,n) ,  o  A  A  •  R (w ,~,l)  R (w ,-,2)  o  A  o  o  S  •  •  i . e . R.(w ,x,i) i s the value of the function R, according A o A Table 3.2. f o r the gamble z  o  R (w ,J ,n)  o  •  A  z n  2  R (w ,»,n) A  o  to the r e s u l t s i n  and f o r the u t i l i t y function within A for which  x G T.  Step 4:  Let y(R ,x) and a(R ,T) be respectively the mean and standard  deviation of the n R -values R,(w ,x,i) where i = l,2,...,n and x £ T. A A o Identify x* £ T such that i t s r e l a t i v e standard deviation i s  - 71 -  ,x*)  a(R  (4.5.)  —  = Min  |U(R ,T*)|  xeT  A  a(R ,x) -  .  |p(R ,x)| A  Select u(w;0,l,x*,8,n) as the u t i l i t y function within A that conforms most c l o s e l y to the decision maker's behavior.  Step 5: *  Choose as an estimation of R.(w ) : A o  (4.6.)  R (W ) = u(R ,x*) a  Q  A  Step 6:  Obtain the decision maker's status quo W.  Step 7:  I f x*  Q  R. (w ) = c . A o  (4.7.)  °°, then we select the parameter c =  R A  (  W  ) >  a  Q  s  i t :  holds that  I f x* < , we e a s i l y derive from (3.20.) 00  c =  |- - w A ° R  T  so that we w i l l select the parameter  c =  -  w .  We w i l l i l l u s t r a t e the R -method with the following example.  Example 4.2. 2  1  A decision maker i s presented with the following three gambles:  = <1,.75;-1>, z  2  = <2;-2> and  = <3;-3>, for which he indicates  - 72 -  respectively TT^ = .2, TT^ = 1 and TT^ = 2 as the r i s k premiums. Lemma 2.4. i = 1,2,3.  According to  his. responses are consistent with r i s k aversion, as TT_^ > 0 for The calculations i n Steps 3 and 4 can now  results l i s t e d i n Tables 3.2. shown i n Table 4.1.  and 3.3.  on the next page.  be performed using  the  The r e s u l t s of the calculations are We  see from Table 4.1.,  that T* = 2,  so that we w i l l select u(w;0,l,2,g,n) as the u t i l i t y function within A that conforms most c l o s e l y to the decision maker's r i s k attitude. with Step 5, we choose R. (w ) = U(R,,T*) = .472 A o A  In accordance  as an estimation of R.(w  ). o  A  2 Suppose further, that W  q  = 9, so that we have c = —2JT72" ~ 9 = - 4.763 by  (4.7.)  Hence, the selected u t i l i t y function can be written as: ( 4  - ' 8  )  Table 4.1.  U ( W )  - - w - 4~.763  i s shown on page 73.  In Step 1 we assumed that the assessment of the decision maker's responses w i l l be performed without error.  This thesis w i l l not address the  question as to which method should be used for this purpose. the l i t e r a t u r e contains  We  s u f f i c i e n t directions for this problem and we refer  to the suggested method by Becker, DeGroot, and Marschak(1964). discussions i n Grether and Plott(1979), Lichtenstein and Slovic(1972), who  f e e l that  Also, the  Slovic(1973),  and  actually a l l employed this method, provide a good reference.  The c r i t e r i o n we use i n the R -method to determine which u t i l i t y A  function  i s "best", i s the r e l a t i v e standard deviation, or the absolute value of the c o e f f i c i e n t of v a r i a t i o n .  As mentioned before, we  f e e l that the v a r i a t i o n  Table 4.1.  The results of calculations r e f e r r i n g to Example 4.2.  U t i l i t y function  u(w;0,l,-l,f3,n)  R as a function of A the r i s k premium TT  2TT  4p(l-p)h +2h(2p-l)-Tr 2  u(w;O,l,*s,0,n)  2P(1-P)TT 2 2 2 4p (1-p) h +  TT  h(2p-l) - TT =  -  (l l fl ) + i +h ^ -h |R J l J p  1  _  R.Cw ,T,2) A O  R (w ,t,3) A  o  V  a  a l l u  .234  .667  .800  .567  .242  .43  .415  .250  .160  .275  .106  .39  .462  .400  .308  .390  .063  .16  .471  .500  .444  .472  .023  .05  .469  .609  .676  .585  .086  .15  2  2  R  u(w;0,l,l,B.n)  R (w ,T,1) A O  A  | A p  R  A  u(w;0,l,2,8,ri)  A  2TT  4p(l-p)h +h(2p-l)Tr 2  e  -[h(2p-l)-Tr]R _ A  u(w;0,l,oo, ,l)  hR  c  2p.cosh(hR )+(l-2p)e A  - 74 -  of the n R -values should be as small as, possible for the selected function, however allowing greater v a r i a t i o n for a greater mean u of the R^-yalues. This i s exactly what i s implied by using the r e l a t i v e standard deviation as our c r i t e r i o n .  Note further that i n the case where for one u t i l i t y  function  within A a l l R.(w , T , i ) a t t a i n the same value, the r e l a t i v e standard deviation A o v.  i n Step 4 reaches i t s minimum at zero.  Example 4.3.  A decision maker with (yet unknown) i n i t i a l wealth w^ indicates  .25 and 1 as h i s r i s k premiums for the gambles z^ = <1;-1> and z^ = <2;-2> respectively.  The r e s u l t s of the calculations i n the R^-method are shown i n  Table 4.2. below.  Table 4.2.  U t i l i t y function  R (w ,x,l) A  o  R (W ,T,2) a  O  u(w;0,l,-l,B,n)  .533  .667  u(w;0,l,Js,B,n)  .400  .250  u(w;0,l,l,B,n)  .471  .400  u(w;0,l,2,3,n)  .500  .500  u(w;0,l,°°,c,l)  .522  .609  C l e a r l y , the u t i l i t y function u(w;0,l,2,B',n) i s selected.  Note that i n the above example not more than one u t i l i t y function within  - 75 -  class A can have R. (w ,x,l) = R. (w, ,x,2). A o A o forms for R  This i s because the functional  as a function of the r i s k premium i n Table 3.3. are d i f f e r e n t  A  and cannot be equal i f the r i s k premiums are not equal. example two gambles 2^ = <h^;-h^> and z^ = " ^ j l ^ ,  Consider for  where h^ ^ h.^  for  which the decision maker states TT^ and TT^ as h i s respective r i s k .premiums. Suppose R (W ,2,1) = R (W ,2,2), as was a  have by  q  a  q  the case i n Example 4.3.  We  then  (3.46.): 2TT  2TT  1  (4.9.)  0  -fh  =  l  h  2  from which i t follows that -n j TT^ as h^ ^ h^. for instance, R (w ,1,1) / A  the following.  R A  (  W  We are now able to show that,  .1,2), since by (3.43. and (4.9.) we have  A O  O  I f TT^ f 0 ^ n ^ , then  1 R  h  fw ,1,1) A _ O  2 l  2 *1  +  2TT  2 h  1  1  2IT,  1  ±  h  2 1  2 ^ 1  2  2TT„ I  2  1  (4.10.) h  A  2 2  2TT  2  as I T . i  4.  1  2"2  I T „ . Hence, R. (w ,1,1) I A o  R (W ,1,2) a  q  If TT. = 0 and ir. 1 z  R. (w ,1,2). A o  7  0 or vice  versa, then the R -values would be d i f f e r e n t , i . e . R (w ,2,1) ^ R (w ,2,2). A O  A  Other i n e q u a l i t i e s can be shown s i m i l a r l y .  O  The functional forms for R^ as  a function of the r i s k premium i n Table 3.2. do not d i r e c t l y similar i n e q u a l i t i e s .  A  Hence, i n theory we may  guarantee  encounter the same R^-values  for more than one u t i l i t y function within A, however we f e e l that i n practice t h i s problem w i l l not a r i s e .  In circumstances where t h i s problem does occur,  we should increase n (= number of reference gambles) i n the R^-method.  - 76 -  The functional forms f o r  as a function of the probability equivalent  in the gamble z <= <h,p;-h> are equal for the quadratic and the hyperbolic functions.  A similar case can be seen i n Table 3.3. for R. as a function of  A the gain equivalent i n the gamble z = <h;-h> for the logarithmic and the hyperbolic functions.  These equalities do not impose any d i f f i c u l t i e s on the  R -method based upon p r o b a b i l i t y or gain equivalents, as long as multiple A  response modes are employed, as w i l l be discussed i n the next paragraph. The R^-method, as introduced i n this section, employs the r i s k premium for the gamble 2 = <h,p;-h> as the only response mode.  I t should be clear,  however, that with our results i n Chapter 3 the R -method can employ multiple response modes for both z = <h,p;-h> and z = <h,p;0>. must t h e o r e t i c a l l y a t t a i n a single value at W, q  which type of equivalent or gamble are used.  This i s because  namely R^Cw^), regardless of  For example, one reference  gamble may ask for the certainty equivalent for the gamble z^ = <h^,p^;-h^>, another for the p r o b a b i l i t y equivalent for might ask for the gain equivalent f o r  z^ = ^^yp^^»  = <h^;0>.  and  a  t h i r d one  The following example  i l l u s t r a t e s this point for two reference gambles.  Example 4.4. A decision maker indicates that he i s indifferent between the gamble  = <h;-100> and the status quo, when the gain equivalent h = h(w ,z^) Q  = 180. He indicates that he i s also indifferent between the gamble  =  <100,fi;0> and winning 50 for sure, when the p r o b a b i l i t y equivalent £ = p(w ,100) Q  = .55. See D e f i n i t i o n 3.2.  The r e s u l t s of our calculations are summarized  in Table 4.3., where we can e a s i l y select u(w;0,l,2,B,n) without even having to use the c r i t e r i o n of r e l a t i v e standard deviation i n Step 4 of the R ~method. A  - 77 -  Table  4.3. R. aa a f u n c t i o n of A  R, as a f u n c t i o n o f A  h f o r the gamble  f) f o r the gamble  u(w;0,l,-l,g,n)  .00377  .00333  u(w;0,l, 5.S»T))  .00408  .00635  u(w;0,l,l,g,n)  .00444  .00496  u(w;0,l,2,g,n)  .00444  .00444  u(w;0,l,°°,c,l)  .00432  .00401  Utility  function  3  T h i s exmaple t h e r e f o r e the  column  z^  demonstrates t h a t the e q u a l i t y of the two R^-values i n  f o r R^ as a f u n c t i o n of h does not impose any  The f o l l o w i n g examples  difficulties.  p r o v i d e i l l u s t r a t i o n s of s e v e r a l mathematical  c a l c u l a t i o n s , m a n i p u l a t i o n s , and r e l a t i o n s h i p s , which a r e a l l based on our r e s u l t s i n Chapter 3. consistency  Example 4.5.  (4.11.)  I t may  w e l l be used i n the R^-method, f o r i n s t a n c e  checks f o r the assessment of the responses to r e f e r e n c e  A d e c i s i o n maker i s assumed to behave a c c o r d i n g  u(w;0,l,2,B,n) * -  ^  where  as  gambles.  to  c = —\}  - 78 -  He indicates, - 25 as. his. certainty equivalent for the gamble z^ = <50;-50>. What w i l l be his. certainty equivalent for  = <20;-20>?  Using (3.46.) gives.  - 22. (w ,h.) (4.12.)  R, A  Therefore, f o r i = 1 -2[z^/(20) ]  or  2  ^-r —— 5  =  h l  for i = 1,2.  2  R^ = -2(-25)/(50) = - 4.  2  = .02, and for i = 2 we have .02 =  I t i s important  to note, that we do not have to  know the e x p l i c i t values of the i n i t i a l wealth W  q  or the parameter c to obtain  this result.  Example 4.6.  Referring to Example 4.5., what w i l l be the decision maker's  r i s k premium for the gamble 2^ = <50,. 75;-50>?  Expressing  T C \  IT^(W ,2^)  =  o  e x p l i c i t l y by (3.45.) results i n 4p(l-p)h R 2  TT . =  (4.13.)  3  f o r z. = <h.,p;-h.>. 2-h.(2p-l)R  1  For p = .75, h  A  = 50 and R  A  1  1  1  A  = .02, (4.13.) gives for i = 3 :  = 25. Note,  that the certainty equivalent 2^ = h^(2p-l) - TT^ = 50(.5) - 25 = 0.  Example 4.7. A decision maker i s again assumed to behave according to (4.11.) Using  (3.46.), we e a s i l y e s t a b l i s h  z\ = ^ ^ » ^ <  1  instance  (4.14.)  _  >  f°  r  1»2.  =  TT^  2 = [h^/h^J TT^» where  3  with  By the figures of Example 4.5. we then obtain for  2 = [20/50] (25) = 4, so that  TT. =  n\ = ^ ( ^ i ^ )  rM2 T T  1  Ih.J  TT. i  = - 4.  In general:  - 79 -  where TT^ and 2^ are as. defined above.  Similar relationships can be derived  for other u t i l i t y functions within class A.  Example 4.8.  A decision maker i s assumed to behave according to the u t i l i t y  function u(w;0,l,-l,g,n)  2 - (w + c) , with c = - -j}.  ^  p  For the gamble z  1  =  <h,p;-50>, for what value of p w i l l the decision maker state h = 60, given that h i s r i s k premium for we have  = <40;0> i s equal to 5.  = 8(5)/[(40) +4(40)(5)-4(5) ] = ^fj 2  2  •  According to (3.90.)  Then, by (3.70.), we  obtain hR  + 2h — -^—z2(h+h) - R (h -h ) 2  (4.15.)  p =  A  2 so that with R  =  YTK>  h = 60, and h = 50, we have  p ~ .71.  - 80 -  4.2.  The R -method including r i s k proneness and r i s k n e u t r a l i t y . A  \  The R - method as discussed  i n section 4.1. assumed, that the decision  A  maker conforms to a single r i s k averse u t i l i t y function over the entire wealth region.  Actually, t h i s assumption confounds two assumptions  (1)  that the decision maker i s r i s k averse over the entire domain, and (2) that he conforms to j u s t one u t i l i t y function over the whole region. Let us f i r s t consider according  the case where the decision maker s t i l l behaves  to one u t i l i t y function, however he i s not r i s k averse over the  whole domain.  With our r e s u l t s and remarks i n Chapter 3 i t w i l l be s t r a i g h t -  forward to discover r i s k n e u t r a l i t y , i n which case the l i n e a r u t i l i t y u(w;0,0,T,B,ri) w i l l be selected. decision maker i s r i s k prone.  function  I t i s also easy to f i n d out whether the  Namely, i f he indicates negative r i s k  premiums for reference gambles, then the decision maker i s r i s k prone according  to a lemma similar to Lemma 2.4.  However, Chapter 3 does not  provide a t o o l f o r s e l e c t i n g the "best" function from among a l t e r n a t i v e r i s k prone u t i l i t y functions.  Only i f we derive similar mathematical r e s u l t s , can  we employ the R^-method f o r r i s k prone u t i l i t y functions.  Below we use two  examples to give d i r e c t i o n as to how such r e s u l t s may be obtained.  Example 4.9.  (4.16)  Consider the r i s k prone quadratic u t i l i t y function given by  2 u(w) ^ (w + c)  over the region w > - c.  where c > 0  Note that f o r this u t i l i t y function and the gamble  - 81 -  z = <h;-h> (2.4.) becomes  (4.17.)  (w +C-TT) o  2  = ^(w +c+h) + 5g(w +c-h) . p 0 2  2  %  u"(w ) o  Noting that R. = R (w ) = A A o  »/\ u (w ) o  =  , we can substitute w  ;  w + c o  o  +c  - - i — i n (4.17.), so that A R  [  (4.19.)  ~ T~  = -JT  A  H[  +  A  h  ]  2 +  ~R~-  h[  h  ]  2  A  from which we e a s i l y derive the following r e s u l t :  (4.19.)  R  A  = - ~ — j h - TT  Note that t h i s r e s u l t i s equivalent  for z = <h;-h>  to the function R, for the r i s k averse A  quadratic u t i l i t y function u(w;0,l,-l,3 ,n) ^ - (w + c ) , where c = - -JJ-. 2  P  However, according  to Lemma 2.4., we know that TT w i l l be p o s i t i v e , whereas  i t can be proved that f o r a r i s k prone u t i l i t y function IT w i l l be negative. In other words, (3.37.) and (4.19.) d i f f e r i n that (3.37.) w i l l contain a p o s i t i v e and (4.19.) a negative r i s k premium.  Other results for the u t i l i t y  function (4.16.) w i l l follow s i m i l a r l y .  Example 4.9.  (4.20.)  Consider the r i s k prone exponential  u(w) ^ e  u t i l i t y function given by  where c > 0.  - 82 -  For this, u t i l i t y function (2.4.) becomes for the gamble z = <h;-h>:  c(w (4.21.)  e  -TT)  = he  0  c(w -h) °  <  gives  \  TTR  e  he  -c, and some reduction,  which after noting that R^  (4.22.)  c(w +h) ° +  cosh(hR )  A  A  This r e s u l t i s equivalent  to (3.50.) for the r i s k averse exponential  utility  —cw ^ function u(w;0,l,°°,c,l) ^ - e , however d i f f e r e n t i n the sense that TT w i l l be p o s i t i v e f o r the r i s k averse and negative for the r i s k prone utility  exponential  function.  After deriving complete results for the functions R  for a class P of  A  u t i l i t y functions consisting of (4.16.), (4.20.), and other r i s k prone u t i l i t y functions, we would have the foundation for an R -method based upon the A  assumption that the decision maker conforms to a single r i s k prone u t i l i t y function. Let us now consider  the case where the decision maker i s r i s k averse  (or r i s k prone) over the whole region, however he does not conform to a single r i s k averse (or r i s k prone) u t i l i t y function. conforms to u,(w) for w < w  1  constant.  o  Suppose the decision maker  + b, and to u«(w) for w > w + b, where b i s a z — o  We can then employ the R -method for a r i s k averse (or r i s k prone)  decision maker i n order to obtain the u t i l i t y function within class A (or  - 83 -  class P) to which he conforms most closely over the region w < w  + b.  o S i m i l a r l y , we w i l l use the R -method over the region w >_ w A  to assess  (w).  + b i n order O  It should be clear that this process can e a s i l y be extended  to the case where the decision maker conforms to three or more r i s k averse (or r i s k prone) u t i l i t y functions.  In practice, i t w i l l be d i f f i c u l t  to  a r r i v e at the value of b, however several reference gambles may be s u f f i c i e n t to provide a good estimation f o r b.  Also, we should use a smoothing  function around b i n order to connect u^(w) and  utility  ^(w).  An alternative method for assessing the u t i l i t y function of a decision maker that conforms to a single r i s k averse u t i l i t y function over the whole region, or to more than one r i s k averse u t i l i t y function over d i f f e r e n t regions, i s provided by section 5.5., where we discuss the sum-of-exponentials u t i l i t y function.  This well known and widely used u t i l i t y function offers  great f l e x i b i l i t y and can be assessed by a method based upon our mathematical results i n Chapter 3.  Let  us f i n a l l y  consider the case where the decision maker i s assumed  to be r i s k averse over part of his wealth domain and r i s k prone over another part of t h i s domain.  We w i l l simplify this case as follows.  Suppose the  decision maker conforms to the r i s k prone u t i l i t y function u (w) for w < w  o  + b, and to the r i s k averse u t i l i t y function u (w) for w > w + b. a = o J  We w i l l follow the l i t e r a t u r e , e.g. Kahneman and Tversky(1979), by assuming that b w i l l be around zero.  That i s , as compared to h i s i n i t i a l wealth  W, Q  the decision maker i s assumed to be r i s k averse for gains and r i s k prone for losses.  For assessing the r i s k averse u t i l i t y function for w >. w^ we can  - 84 -  employ the R -method using the gamble z J\  = <h  ,p ;0>. A.  J.  However, our r e s u l t s  x  i n Chapter 3 do not provide us with a s i m i l a r gamble to be used for losses. In other words, we would prefer to have mathematical r e s u l t s for the gamble Iry = <0,p^;-h2> so that the r e s u l t s for this gamble can be used i n an method for assessing the r i s k prone u t i l i t y function u^(w) Fortunately, i t i s not hard to obtain r e s u l t s for z^. equivalent to z^ by using the following  (4.23.)  h  so that by applying  2  = -  and  u (w) and u (w) we a p  2  = 1 - p^  and 3.5. we  will  In order to connect the functions  should use a smoothing process.  finding the best u t i l i t y function u(w)  exponentials  Q  Notice that z^ i s  (4.23.) to our r e s u l t s i n Tables 3.4.  h i s taken to be r e l a t i v e l y small.  W.  transformation:  p  d i r e c t l y obtain the desired r e s u l t s .  for w <  R^-  This could be done by  using the gamble 2 = <h,p;-h>, where  We can also think of using the sum-of-  u t i l i t y function (see section 5.5.)  for this purpose.  With the remarks and r e s u l t s i n this section we actually gave d i r e c t i o n to a General R -method.  This method enables us to assess the u t i l i t y function  to which a decision maker conforms. r i s k averse, r i s k prone or even both.  The underlying u t i l i t y function may  be  Also, the General R^-method covers the  case where the decision maker conforms to more than one u t i l i t y function over different  regions.  - 85 -  5  Mathematical  Extensions  i  This chapter w i l l discuss several mathematical extensions r e s u l t s i n Chapter 3, where we expressed R^ =  R A  (  W Q  )  a s  a  of our  function of a  response variable and the parameters of a nondegenerate gamble for a class A of u t i l i t y functions defined by D e f i n i t i o n 4.1.  The purpose of  our r e s u l t s i s to improve the R^-method as discussed i n Chapter 4.  extending The  following sections cover a wide variety of possible extensions. In section 5.1. we w i l l give some d i r e c t i o n as to how  one may be able  to obtain the function R^ for u t i l i t y functions beyond the class defined by A.  The R^-method w i l l c e r t a i n l y be improved i f the u t i l i t y function can be  selected from among a greater number of a l t e r n a t i v e s . In section 5.2. we w i l l note that we could also establish r e s u l t s using R^ = R^Cw^  instead of R^.  This may  p a r t i c u l a r l y be useful when R^  can only be expressed as a function e x p l i c i t l y  including the i n i t i a l wealth.  Another extension can be found i n the choice of the reference gamble. In section 5.3. we w i l l show, that i t i s f e a s i b l e to derive similar r e s u l t s for gambles other than 2 = <h,p;-h> and z = <h,p;0>.  To be s p e c i f i c ,  w i l l b r i e f l y discuss the gamble z = <h^,p;ti2> where h^ ^ h^, s o l e l y given by i t s mean and variance, the normally  we  the gamble  distributed gamble, and  f i n a l l y the uniformly d i s t r i b u t e d gamble. Section 5.4.  focusses on how  to derive our r e s u l t s by means of Taylor  - 86 -  series expansions on both sides of the equation (2.4.) • We f e e l that results f o r other u t i l i t y functions may well be obtained by using this technique. In section 5.5. we w i l l present results similar to those i n Chapter 3 f o r the well known and widely used sum-of-exponentials  u t i l i t y function.  F i n a l l y , section 5.6. w i l l conclude this chapter with a b r i e f summary of the thesis.  - 8,7 -  5.1.  Extended r e s u l t s for u(w;g,a,T,g,ri) .  In section 3.2. we introduced a class of u t i l i t y functions u(w) = u(w;£,a,T,g,ri) defined by (3.14.).  The results of Chapter 3 are confined  to s i x member u t i l i t y functions within this class.  However, i t should be  clear that many other u t i l i t y functions belong to (3.14.) and i t may be possible to a r r i v e at s i m i l a r r e s u l t s for those functions. A subclass of (3.14.) i s formed by (3.18.), of which Pratt(1964) gave several s p e c i a l cases, e.g.  (5.1.)  and  (5.2.)  and  (5.3.)  By s p e c i a l i z i n g (5.1.) one step further we obtain  (5.4.)  - 88 -  where s i n h ^w i s the i n v e r s e h y p e r b o l i c e a s i l y derive  (5.5.)  the f o l l o w i n g  sine  f u n c t i o n o f w.  From (5.2.) we  s p e c i a l case:  u(w;0,2,l,8,g) ^ arctan(w)  A c t u a l l y , Pratt(1964) introduced  h i s s p e c i a l cases (5.1.) to (5.3.) i n -  cluding a transformation,  the s u b s t i t u t i o n w by w + c.  namely  This  way,  (5.4.) and (5.5.) become r e s p e c t i v e l y :  (5.6.)  u(w) ^  s i n h ^(w + c)  and  (5.7.)  u(w) % arctan(w + c)  Other s p e c i a l cases o f u ( w ; £ , a , T , 0 , r i ) i n c l u d e  (5.8.)  u(w;a-l,a,T,8,ri)  ^  f  a-lf8s  T  . 1~  rr, a  (l-T)aBI  Bw  + n  ds  =  T  + c 1-T  a w  a  + c  where c =  jrr B  and  (5.9.)  u(w;0,2,-l,B,n) ^ -w(w  2  + c)  , where c =  3n  ^  - 89 -  To arrive at general and complete results for many of the above u t i l i t y functions, within the class defined by (3.14.), would certainly enhance the application p o s s i b i l i t i e s of the R -method. feel that such results are very hard to obtain.  At this moment we  However, the following  example i l l u s t r a t e s that i t may s t i l l be possible to derive extended results.  Consider the u t i l i t y function (5.7.), for which we have by (2.3.)  with z = <h;-h>:  (5.10.)  arctan(w +C-TT) = %arctan(w +c+h) + %arctan(w +c-h) o o o  which, after taking the tangent of twice both sides, gives:  (5.11.)  tan[2arctan(w +C-TT)] = o =  tan[arctan(w +c+h) + arctan(w +c-h)] o o  Further we have 2(w +C-TT) (5.12.)  tan[2arctan(w +C-TT)] = 1 - (w +C-TT) o  and  (5.13.)  tan[arctan(w +c+h) + arctan(w +c-h)] =  2(w +c) 1 - [(w +c) -h ] o 2  Equating (5.12.) and (5.13.) according to (5.11.) gives after cross-  2  - 90 -  m u l t i p l i c a t i o n and some reduction:  (5.14.)  7T(W + C ) 2  -  (TT2+h2)(w  o  +C)  +  TT(l+h2)  =  0  o  which yields 2 (5.15.)  TT  w + c = o  ,2 AI r + h + y [TT  .2,2 - h J -  2  2 4TT  2TT  though only  TT  (5.16.)  2  + h  2  ]j[-n  -  w + c = o  2  - h ] 2  2  -  4TT  2  27T  is applicable because of the following:  For h > 0 and TT  0, R (w ) should A o A  reach zero, so that after noting that  (5.17.)  u"(w ) o _ R,(w ) = ° u'(w ) o A  2(w +c) o 1 + (w + c ) o  i t follows that only (5.16.) should be considered.  2  F i n a l l y , inserting  (5.16.) into (5.17.) gives  r 2 .2  o  (5.18.)  A  ,  o  \  2TT[TT +h  44 +h  TT  u  / 2 .2,2 . 2, - VLTT -h  J -4TT J  , 2.2. /' 2 ^2,2 . 2 (TT +h )V[TT -h 3 -4TT r  —  so that we derived the absolute r i s k aversion at the i n i t i a l wealth w as a o function of h and the r i s k premium TT for the gamble z = <h;-h> for a decision maker with the u t i l i t y function u(w) y arctan(w + c)  - 91 -  At the end of this section we would l i k e to have a closer look at the absolute r i s k aversion R  as a function of the r i s k premium ir for the A  gamble z = <h;-h> and the u t i l i t y functions u(w;0,1 ,T,B,r)), x = -1, h, 1, 2. From Table 3.3. we see that the corresponding functions R^ can be written as 2TT  (5.19.) h  2  + a(x) -TT  x = - l , h, l , 2,  2  where a ( - l ) = -1, a(%) = 4, a ( l ) = 1, and a(2) = 0.  The function a(x) i s  given i n Figure 5.1. for the specified values T = -1. %, 1, and 2.  O(T)  Figure 5.1. The function a(x) i n (5.19.) for T = -1, k, I, and 2. -;i *-  -1  From (5.19.) we e a s i l y derive the following:  (5.20.)  a(T)  =  2  T = -1, %, 1, 2,  TTR  A  but, of course, i t would be of much more interest a function of the parameter T .  to express a d ) solely as  In that way we would have achieved a  s i g n i f i c a n t generalization of our previous  results.  - 92 -  Note that we can v i s u a l i z e , f o r instance, a hyperbolic shaped l i n e through the three points i n the f i r s t quadrant of Figure 5.1.  Also note  that the fourth point ( r , a ) = (-1,-1) i s the mirror image of (r,a) = (1,1) with respect to the l i n e a = - T.  I f indeed i t were possible to derive a(T)  s o l e l y as a function of T, we would have at once mathematical results f o r a l l functions included by u(w;0,l,T,g,n)•  For example, suppose a(-b) to be  known, then we d i r e c t l y e s t a b l i s h the function  f o r the u t i l i t y function  3/  u(w;0,l,-%,g,n) ^ - (w + c ) through (5.19.) by plugging i n the value of 2  a(-b)results.  At t h i s point, however, we have not established such extended  - 93 -  5.2.  Extended results for the r e l a t i v e r i s k aversion  In Chapter 3 we derived R.(w A  O  R^^)'  ) as a function of the parameters of a  gamble z, and a response variable of that gamble ( i . e . certainty equivalent, or r i s k premium, or p r o b a b i l i t y equivalent, or gain equivalent).  We  are  also able to derive similar results for the r e l a t i v e r i s k aversion R (w ) R o at the status quo w .. By D e f i n i t i o n 2.9. we have R s R (w ) = w R.(w ) = o R R o o A o w^R^. To be s p e c i f i c now, R^ can be expressed as a function of: (i)  the parameters of a nondegenerate gamble z, as a f r a c t i o n of the i n i t i a l wealth w^  (ii)  i n case of gains and losses; and  the certainty equivalent, or the r i s k premium, or the gain equivalent of that gamble, as a f r a c t i o n of  An i l l u s t r a t i o n of the preceding  w^.  i s a decision maker having a quadratic  u t i l i t y function u(w; 0,1 ,--1, B ,Tl) 'v* - (w + c )  where c = - 5.  2  TT  he states TT* = — as his r e l a t i v e r i s k o w ° h f a i r , nondegenerate r e l a t i v e gamble z = <h*;-h*> = <—;w o we have: i n i t i a l wealth w  (5.21.)  R  = w R  2TT  2 o , h -  1. Note that h* be  like  premium for the h i — > . By (3.37.) w o  2TT* TT  2  (h*)  2  -  Hence we expressed for this decision maker R^  •ay  At his  P  can be s t a t e d l i k e "one f i f t h " t e n p e r c e n t of my status quo".  of the  (TT*)  =  2  ^R^ ^  d e c i s i o n Baker's  &  S  Q  status quo"  A  ^  and  U  N  C  his  T  ^  O  N  °^  response  - 94 -  h* s —— , which i s the standard deviation h of a f a i r f i f t y - f i f t y gamble w TT  °  as a f r a c t i o n of w , and TT* * — , which i s the gamble's r i s k premium as a o w o f r a c t i o n of w . The results for the other u t i l i t y functions i n Table 1 o follow s i m i l a r l y . Using  enables us also to obtain results for u t i l i t y functions for  which results regarding R  are d i f f i c u l t — i f not impossible— to obtain.  An  example i s provided^ by the special exponential u t i l i t y function defined by  (5.22.)  Consider for  u(w) % - e  the function  n  (5.22.):  \  (5.23.)  As R  where c > max (0, 2w)  the f a i r f i f t y - f i f t y gamble z = <h;-h>, so that by (2.3.) we have  w / c  c/  e  l  =  = R (w ) = - w R R o o  (5.24.)  -  -TT °  w +h  .  O  ,  --e  u"(w ) — , v u (w ; o  + 2w —  =  + c w  (  w -h  1  --e  O  , i t follows that  o  c = w (R - 2) o R  Substitution of (5.24.) into (5.23.) yields  y (5.25.)  which gives R  e -* 1  R  v  2  -  i-e  1 + l 1  v  2  *  •  2  i.e ""* 1  as an i m p l i c i t function of h* and TT*.  /  - 95 -  So far we have shown, that for some u t i l i t y functions we might be able to express the r e l a t i v e r i s k aversion (ii).  = R^C ) W  0  a  s  a  function of ( i ) and  However, we have not e x p l i c i t l y discussed the purpose of doing  this.  To understand t h i s point, we should note that our results i n Chapter 3 derived the function  to be independent from the i n i t i a l wealth W . q  This  i s an important observation, since therefore we do not require the knowledge of the decision maker's status quo W i n order to obtain his response (e.g. q  r i s k premium) to a nondegenerate gamble, nor do we need to know W i n order q  to calculate the value of the function R^ for this gamble.  So, i f i t i s not  feasible to obtain the function R. without w , we may s t i l l be able to arrive A o at a r e s u l t where R^ i s expressed as a function of ( i ) and ( i i ) .  That i s ,  i n t h i s case the knowledge of W would not be required i n order to evaluate q  the function R,,.  - 96 -  5.3.  Extended results for other gambles.  In this section we w i l l point out that the results of Chapter 3 can be extended to gambles other than  = <h,p;-h> and z^  =  results may be obtained for a generalization of z gamble z = <h^,p;h >. 2  <h,p;0>.  First,  and z^, namely the  Case 1 below w i l l center on this gamble.  Second,  instead of looking at discrete gambles, we may think of considering continuous gambles.  Several continuous gambles are i l l u s t r a t e d i n the Cases  2, 3 and 4 below.  Case ph  1  1:  Consider the gamble z = <h^,p;h > where h^ f h^, and E(z) =  + (l-p)h  2  2  = p(h - h ) + h x  2  2  :  Figure 5.2. z :  The gamble z = <h ,p;h >  so that by (2.1.) we have:  (5.26.)  u(w +z) = p-u(w +h,) + (l-p)-u(w +h„) o o 1 o 2  Hence, for the u t i l i t y function u(w;0,l,T,8,n) with 1 t 1 the following relationship:  (5.26.) yields  - 97 -  r  1-T  B(w +z) o  +n  BU-T) (5.27.)  rB(w +h. ) 2—L-  B(1-T)  1-T r  +  1-T  B(w +h ) o  2  + (1-P)  n  Using (3.20.) and further reducing (5.27.) gives:  (5.28.)  1 +  2R.  1-T  =  p 1+  h  1-T  lV  (1-p)  E x p l i c i t r e s u l t s are given for two cases by specifying T.  h  1 +  1-T  2V  Setting T -  1  gives by (5.28.) f o r u(w;0,l,-l,8,r,) ^ - (w + c ) , with c = - j 2  (5.29.)  (1 - 2R )  =  2  p(l - h R ) x  +  A  (1-P)(1 " h R ) ' 2  A  which gives 2[2 - p h - ( l - p ) h l x  (5.30)  R  ~  a  2  2  - ph - (l-p)h  2  2  2  which i s a generalization of respectively (3.36.) and (3.89.), as can d i r e c t l y be shown by selecting By setting c =  = h and respectively h  1-P 1 + bh R  1 +  1 + b2R,  2  which gives  (5.32.)  = - h and h  T = 2 i n (5.28.) we obtain for u(w;o,l,2,p\n) ^ -  2n  (5.31.)  2  2[2 - p h - ( l - p ) h ] x  A  h h  2  2  - ph 2 - (1-P)h 2 2  1  A  w  +  c  >  w  = 0.  2  i  t  h  - 98 -  which i s a generalization of respectively (3.45.) and (3.99.), as can d i r e c t l y be shown by selecting h^ = h and respectively  Case  2:  = -h and h^ = 0.  Consider a gamble z solely given by i t s mean and variance.  We  are thinking of I being continuous, although our statements are equally v a l i d for z having a discrete d i s t r i b u t i o n .  As no s p e c i f i c a t i o n of the  d i s t r i b u t i o n of z has been given, we do not expect to arrive at complete results for many d i f f e r e n t u t i l i t y functions^  However, the example given  i n this section should serve as a f i r s t step towards other results possibly to be obtained by good approximations. A gamble s o l e l y given by i t s mean and variance i s e s p e c i a l l y i n t e r e s t i n g from a finance theoretic point of view.  For example, the  o r i g i n a l Markowitz(1952) theory of p o r t f o l i o selection assumes that the decision maker's (investor's) preferences  can be ranked completely i n terms  of mean and variance of the p o r t f o l i o return.  Also, the discussion on the  mean-standard deviation analysis i n Tsiang(1972), Bierwag(1974), Borch(1974), Levy(1974), and Tsiang(1974) i l l u s t r a t e s the interest i n gambles solely given by mean and variance  (or standard  deviation).  We w i l l not further digress  2  on the finance theoretic l i t e r a t u r e  of this topic, but instead emphasize  that most of the l i t e r a t u r e on gambles solely given by their mean and variance mention the special role of the quadratic u t i l i t y function i n this respect. The reason behind this i s , that for a quadratic u t i l i t y function u(w) = u(w;  ,1,-1,B,ri) y - (w + c)  , where c = - — , the expected u t i l i t y of a P  2. Closely r e l a t e d references include Biervag(1973), Borch(1969,1973), and Nigro(1972), Hanoch and Levy(1970), and Tobin(1958,1969).  Feldstein(1969), G l u s t o f f  - 99 -  gamble I depends only on the mean u and the variance a hence not on the d i s t r i b u t i o n of the gamble.  E[u(z)]  =  of the gamble, and  Namely:  E[-(z+c) ] Z  - E(2 ) - 2cE(z) - c 2  2  2 2 2 - (y + o ) - 2cu - c  (5.33.)  2 2 - (y + c) - a  =  u(y) - a .  For a decision maker with a quadratic u t i l i t y function u(w;0,1,-1, 6,TI) we have by (2.3.):  (5.34.)  Developing  (5.35.)  2 - (w + y - TT + c) o  2 E [ - (w + 1 + c) ] o  =  the right hand side of (5.34.) gives  E[-(w +z+c) ]= -(w + c ) - 2(w +c)y - (y +o ) o o o 2  2  2  2  Substitution of (5.35.) into (5.34.) yields after some rearranging:  (5.36.)  According  2 2 - 2(w +C)TT = a + 2yrr - TT o  to (3.20.) we have  = - — ^ — - , which gives with (5.36.) o  - 100 -  (5.37.)  R A  -  2 l T  a  2  2 + 2yrr - Tr  Note that (5.37.) generalizes our i n i t i a l Theorem 3.1. for non-fair gambles. That i s , for a f a i r gamble we have i n (3.10.).  y = 0 and hence (5.37.) results d i r e c t l y  Also, considering the gamble z = <h;-h>, which has y = 0 and  2  2 . = h , we obtain (3.37.) from (5.37.). S i m i l a r l y , for the gamble z = 2 2 <h;0> with y = ^h and c = fch we easily derive (3.90.) from (5.37.)  a  Case  3:  Consider the normally distributed gamble z with mean y and  2 „ variance o , i . e . P(z<z) = / f^(z)dz , where 2  —00  _ (g-y) i  (5.38.)  —i=»e  f„(z) = N  "  2  2  °2 where Tt = 3.14. .  OV2TI  For this gamble we have by (2.1.):  +oo  (5.39.)  u(w +z) = / u(w +z)f„(z)dz o o N -co  Selecting for example the u t i l i t y function u(w;0,1,°°,c,1) ^ - e  C  W  we  derive from (5.39.): -c(w +2) -c(w +•£.) - e = - / e f„(z;dz N  J  —CO  (5 40 ) v:>.tu.;  i)  ^ ~ = _f  c w  2 2 o-cy+^c 0 +°° / e  1-;[z-y+co _ 2,2 ] 2o^ d r  5  - 101  -  1.2 2  (5.40.)  -cw -cu+^c o - e °  =  which easily gives  / c / i -v (5.41.)  e  "  c 2  =  e  -cy+%c a 2  2  so that, using 2 = u - TT , where ir i s the r i s k premium for the gamble z, and c = R , we have:  (5.42.)  R  &LZ».  =  a  =  ^ o  2  2  If we select the u t i l i t y function to be the quadratic u(w;0,1 ,-1, (3,ri) 2 x\ - (w + c) , with c = - -pr , we can find i n a similar P  (5 43 ) ' U  J  J  K  R A  =  ~ 2 + u  2 ( y  2 a  g )  ^2 - z  =  2 a  way:  ?! „ 2 + 2yrr - TT  Of course, this r e s u l t also follows d i r e c t l y from (5.37.) i n Case 2,  Case 4 :  Consider the gamble z uniformly distributed h  l  +  n  2  gamble, where E(z) =  (5.44.)  u(w +2) o  where f , i s defined by TTl  UN  ,  =  (2.1.) gives  / u(w +z)f (2)d2 o UN  on [h ,h ] . For this  - 102 -  h  2 "  h  hj < z <  l  h  2  (5.45.) elsewhere  Hence, for u(w;0,1,T,8,n) with x ^ 1  r  B(w  +z)  B(I-T)  (5.44.) results i n :  ^ + n  1-T  +00  r -oo °  h — r h—  -  f u(w + z)dz  2  (5.46.) B(w + h „ ) o "2'  >2 + n  T  r  B(w +h,) o  (h2-h1)B<£(l-T)(2-T)  which gives, using (3.20.), for T  (5.47.)  ,2-T  -- + z A  2  M  1-T  T  i + n  r  2-T  —  (h2-h )(2-T) 1  R  R  h  +  A  1  or  zR.^  (5.48.)  1 +  2-T  2-T  1-T  A  (h2-h )(2-T)  1+  h  2 A  h  R  1+  l A R  1  By selecting d i f f e r e n t values for T we w i l l be able to derive the results  T  - 103 -  Eor example, l e t us select T =  for this uniformly d i s t r i b u t e d gamble. i . e . the quadratic u t i l i t y function.  (5.49.)  R  For this function (5.48.) gives:  A - V " 3lh7h7(1-h.R.) 2 A (1  2  -1,  2  3  - (l-h.R ) 1 A  3  A  from which i t follows that  (5.50.)  p  22 - (h +h ) A  =  Jl  2  l ,2 - 3[h  ^  r  2 +  h ^  +  hj]  Of course, this r e s u l t could have been obtained d i r e c t l y from (5.37.) i n Case 2, by noting that y = E(z) = uniformly d i s t r i b u t e d gamble z.  \  1 2^ 2 l^ h  +h  2 a  n  d  0  =  1 12"^ 2~ l-' h  h  2 f  o  r  t  h  e  -  5.4.  104  -  The results using Taylor series expansions.  The idea behind 3 . 1 . , was  the mathematical r e s u l t s , as pointed out i n section  to expand both sides of the  (2.4.)  u(w  -  TT)  =  E[u(w  +  o using Taylor series.  equation  z)]  o O r i g i n a l l y we derived our results by expanding (2.4.)  e x p l i c i t l y for the functions i n Table 3 . 1 . , i . e . we e x p l i c i t l y calculated (3.1.)  and  For example, consider the hyperbolic u t i l i t y function  (3.2.).  u(w;0,1,1,B,n) y - — - — , where c = w + c p simply denoted by u(w), we have  tc c i •» (5.51.)  so that  u  (3.1.)  (m), \ (w)  _  f -1 * w + c  =  For this function, from now  on  m.  becomes  u(w  o  - TT)  =  u(w  =  u(w  o  TT111/  , m+1  ) -  E. — ( - 1 ) m=l m.  ) -  ? m=l  N  -u  (m),  (w ) o  (5.52.)  and  (3.2.)  o  m , , m+1 (w +c) o N  can also be e x p l i c i t l y calculated for the f a i r gamble z = <h;-h>,  where again the u t i l i t y function i s taken to be the hyperbolic u t i l i t y function u(w) <v u(w;0,1,1,B,n.) • We  derived:  - 105 -  E[u(w +z)] = ^u(w +h) + %u(w -h) o o o , . 1 » h (m), , 1 92 h ,n+l (m) . = u(w )+•-;. I, — r - u (w ) - -r- h — r (( - l ) -u (w ) o 2 m=l m. o 2 m=l m. o (5.53.) .m u(w ) ~ —• L -• , o 2 m=l , .... \m+l (w +c) o  1. ,  ?  si  h m  2 m-1 (w +c)m+1 o m  2m u(w ) o Equating  2m+l m=l (w +c) o  (5.52.) and (5.53.) according to (2.4.) gives: 2m  (5.54.)  m= 1 w  + cj v  w+c o  u"(w ) , , °v = —— u(w) w+c o o 2  so that with R, s R, (w ) = A A o  (5.55.)  m=l  The formula  (5.56.)  as without  (5.57.)  =  A  we obtain  S t(%hR ) ] . m=l A 2  1  m  A  (5.55.) e a s i l y yields:  - 1  1 - %TTR,  - 1  = 1 - (*>hR V A  (%hR V < 1 and A w + c w+c o o loss of generality. Then, (5.56.) gives  h  2  < 1  can be assumed  - 106 -  which was d i f f e r e n t l y d e r i v e d i n (3.46.). an R %  Note t h a t t h i s r e s u l t d i s p l a y s  ESTIMATION ERROR equal to z e r o .  A  Other examples are s i m i l a r l y d e r i v e d , e s p e c i a l l y f o r the q u a d r a t i c utility  function.  The l a t t e r was  shown i n the p r o o f o f Theorem 3.1., where  i t h o l d s , t h a t u ^ ( w ; 0 , 1 ,-1,B,n) = 0 f o r m = 3, 4, r e s u l t s i n Chapter expansions.  C e r t a i n l y , the  3 w i l l not be e a s i e r o b t a i n e d by u s i n g T a y l o r s e r i e s  However, we b e l i e v e t h a t the p r e c e d i n g example may  basis for deriving results  for u t i l i t y  f u n c t i o n s other than those l i s t e d i n  T a b l e 3.1. or f o r gambles other than those d e a l t w i t h i n Chapter  i  serve as a  3.  - 107 -  5.5.  The sum-of-exponentials u t i l i t y  function.  One of the most used u t i l i t y functions in decision analysis i s the sum-of-exponentials u t i l i t y function, defined by  (5.63.)  _ -*w  u ( w )  e  _ . b  where a,b,c>0 , a*c  _ c w e  For this u t i l i t y function we w i l l present i n this section some mathematical r e s u l t s similar to those given i n Chapter 3. Let TT. = TT.(W ,h.), i = 1, 2, 3, be the r i s k premiums for the gambles 1 l o l = <h^;-h.>, i = 1, 2, 3 respectively, for a decision maker who behaves according  to the sum-of-exponentials u t i l i t y function (5.63.).  assumed that  (5.64.)  f  f h^.  According to (2.4.) we e a s i l y obtain  u(w - T T . ) O  It i s  =  E[u(w +2.)] o i  1  i = 1, 2, 3,  which y i e l d s for (5.63.): -a(w e  +  1  (5.65.) 1 2  so  -c(w -TT  -TT.)  °  b-e  0  )  1  f-a(w+h.) -c(w +h ) -a(w -h ) ° + b-e ° + e ° b-e 1  1  e  that after rearranging we obtain the following:  1  +  -c(w  - 108 -  aTT  -aw  i  1  ah. e  -ah. + e  1  1  (5.66.) +  b-e  -cw  CTT.  ch.  1  e  -eh. l + e  I  =  0  Thus aTT.  (5.67.)  -(c-a)w  e  b-e  cosh(ah.) I  CTT.  i = 1, 2, 3.  cosh(ch.) I  I f we define aTT.  (5.68.)  e  - cosh(ah.)  H. s 1  e  CTT. 1  i = 1, 2, 3, - cosh(ch.) I  then (5.67.) i s  (5.69.)  -(c-a)w b-e  i = 1, 2, 3,  so that the following results: -aw -cw I o ,2 o a e + be e 9  R. s R (w ) = A A o' (5.70.) a  _ 2  u"(w ) 2_ u'(w ) o  - -(c-a)w 2, o + c be -(c-a)w + cbe  -aw ae  2  a  2  o ,, + bee  -cw  o  U  - c H. a - cH. I  i = 1, 2, 3.  - 109 -  Hence, we found  a  (5.71.)  2  2 - c H. 1 a - cH^  =  2 2 a - c H „ ?. = a - cH^  2 2 a - c H, i a - cH^ U  which after cross-multiplications reduces to:  (5.72.)  H  l  =  H  2  =  H  3  One of the above three equations i s redundant, while the other two equations (e.g.  H^ = H^ and H^ = H^) provide through existing numerical  procedures  solutions for a and c. I f the value w of the decision maker's i n i t i a l o wealth i s known, the third parameter b i s easily obtained from (5.69.). In other words, with the preceding we established a procedure to determine  the three parameters of (5.63.) i f the decision maker i s known to  behave according to this sum-of-exponentials Example:  utility  function.  A decision maker with a sum-of-exponentials  u t i l i t y function u(w)  —aw — cw . . _ - e - b-e i s known to state the following r i s k premiums for the gambles z^ = <1;-1>, z" = <2;-2>, and z^ = <3;-3>: 2  (5.73.)  TTj = . 50  and  TT = 1.50 2  and  TT = 2.57 3  The values i n (5.73.) are a r t i f i c i a l l y selected inorder to arrive at integer parameter values l a t e r on. The equations (5.72.) with become for these values:  defined by (5.68.)  - 110 -  ' air  aiT  e (5.74.)  - cosh(a)  e  - cosh(c)  e  CTT1  air^  2  - cosh(2a)  e  - cosh(2c)  e  CTT_  e  - cosh(3a) CTT-  A numerically derived solution of (5.74.) y i e l d s :  - cosh(3c)  a « 1 and c =s 2.  One  can check:  1.65 - 1.54 _ 4.48 - 3.76 _ 13.09 - 10.07 2.71 - 3.76 20.09 - 27.31 ~ 171.37 - 201.72 %  (5.75.) 0.1 = H  ( i = 1,2,3)  Hence 1 - 4H. (5.76.)  R  =  « 1.167. 1 - 2H. x  If i n addition the decision maker's i n i t i a l wealth W  q  then according to (5.69.) we have (c-a)w (5.77.)  b =  - e  °H  ±  * 1,  so that we f i n a l l y obtain:  (5.78.)  u(w)  - e~  W  - e"  2 w  i s known to be w  c  - Ill -  5.6.  Summary.  The basis for the R^-method i s provided by the mathematical results i n Chapter 3.  E s s e n t i a l i s the idea that for a number of u t i l i t y  functions,  the absolute r i s k aversion at the i n i t i a l wealth w , i . e . R.(w ), can be O A O expressed as a function of (i)  the parameters of a nondegenerate gamble; and  (ii)  the decision maker's response to that gamble, i n p a r t i c u l a r : the r i s k premium, or the certainty equivalent, or the p r o b a b i l i t y equivalent, or the gain equivalent of that gamble.  In Chapter 3 we derived the results for f i v e r i s k averse u t i l i t y  functions  within the general class u(w;£,a,x,B,n)» as introduced and defined by  (3.14.).  These f i v e alternative u t i l i t y functions also belong to the well known HARAclass.  We obtained  r e s u l t s r e f e r r i n g to both the gamble 2 = <h,p;-h> and  the  gamble z = <h,p;0>. The R -method as discussed i n section 4.1.  employs these mathematical  A  r e s u l t s to obtain a procedure for selecting among a l t e r n a t i v e u t i l i t y functions.  The basic idea of the R -method i s , that R.(w ) has a s p e c i f i c A A o  value, no matter what gamble the decision maker i s faced with. determines  The R^-method  which u t i l i t y function, chosen from the f i v e r i s k averse u t i l i t y  functions, the decision maker conforms to most c l o s e l y . selected should have the least r e l a t i v e standard  The function to be  deviation for two or more  values of the function R, = R.(w ). The values of R. are obtained by the A A o A decision maker's responses for two or more reference gambles, and c a l c u l a t i n g  - 112  the corresponding R  -  values as given i n Chapter 3.  Section 4.2.  extends  A the R^-method to incorporate the r i s k prone and the r i s k neutral cases. Also, the case where the decision maker i s both r i s k averse and r i s k prone over d i f f e r e n t regions was b r i e f l y discussed. 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